In this Ph.D. thesis we discuss results concerning variational convergences for functionals and differential operators on Lipschitz continuous vector fields. The convergences taken into account are gamma-convergence (for functionals) and H-convergence (for differential operators).

Electromagnetism and optimal control stand out as a topics that feature impactful applications in modern engineering, as well as challenging theoretical aspects of mathematical analysis. Within this context, a major role is played by the search of necessary and sufficient conditions characterizing optimal solutions, as they are functional to numerical algorithms aiming to approximate such solutions. In this thesis, three standalone topics in optimization sharing the underlying framework of Maxwell-related PDEs are discussed. First, I present an optimal control problem driven by a quasi-linear magneto-static obstacle problem featuring first-order differential state constraints. The non-linearity allows to suitably model electromagnetic waves in the presence of ferromagnetic materials, while the first-order obstacle is relevant for applications in the field of magnetic shielding. Existence theory and the derivation of an optimality system are addressed with an approximation technique based on a relaxation-penalization of the variational inequality. Second, I analyze an eddy current problem controlled through a dipole type source, i.e. a Dirac mass with fixed position and variable intensity: well-posedness of the state equation through a fundamental solution (of a curl curl - Id operator) approach and first order conditions are dealt with. To conclude, I discuss the computation of the topological derivative for shape functionals constrained to low-frequency electromagnetic problems (closely related to the eddy current model), with respect to the inclusion/removal of conducting material; the results are obtained using a Lagrangian approach and in particular the so-called averaged adjoint method. This approach requires the study of the asymptotic behavior of the solutions of some problems defined in the whole space, and the introduction and consequent analysis of appropriate function spaces.

Polyominoes are two-dimensional objects obtained by joining edge by edge squares of same size. Originally, polyominoes appeared in mathematical recreations, but it turned out that they have applications in various fields, for example, theoretical physics and bio-informatics. Among the most popular topics in combinatorics related to polyominoes one finds enumerating polyominoes of given size, including the asymptotic growth of the numbers of polyominoes, tiling problems, and reconstruction of polyominoes. Recently Qureshi introduced a binomial ideal induced by the geometry of a given polyomino, called polyomino ideal, and its related algebra. From that moment different authors studied algebraic properties and invariants related to this ideal, such as primality, Gröbner bases, Gorensteinnes and Castelnuovo-Mumford regularity. In this thesis, we provide an overview on the results that we obtained about polyomino ideals and its related algebra. In the first part of the thesis, we discuss questions about the primality and the Gröbner bases of the polyomino ideal. In the second part of the thesis, we talk over the Castelnuovo-Mumford regularity, Hilbert series, and Gorensteinnes of the polyomino ideal and its coordinate ring.

Torsion pairs were introduced by Dickson in 1966 as a generalization of the concept of torsion abelian group to arbitrary abelian categories. Using torsion pairs, we can divide complex abelian categories in smaller parts which are easier to understand. In this thesis we discuss torsion pairs in the category of modules over a finite-dimensional algebra, in particular we explore the relation between torsion pairs in the category of all modules and torsion pairs in the category of finite-dimensional modules. In the second chapter of the thesis, we present the analogue of a classical theorem of Auslander in the context of τ-tilting theory: for a finite-dimensional algebra the number of torsion pairs in the category of finite-dimensional modules is finite if and only if every brick over such algebra is finite- dimensional. In the third chapter, we revisit the Ingalls-Thomas correspondences between torsion pairs and wide subcategories in the context of large torsion pairs. We provide a nice description of the resulting wide subcategories and show that all such subcategories are coreflective. In the final chapter, we describe mutation of cosilting modules in terms of an operation on the Ziegler spectrum of the algebra.

As recently shown in laboratory bench scale experiments, chemotaxis, i.e.the movement of microorganisms toward or away from the concentration gradient of a chemical species, could have a fundamental role in the transport of bacteria through saturated porous media. Chemotactic bacteria could enhance bioremediation by directing their own motions to residual contaminants in less conductive zones in aquifers. The aim of the present work is to develop a proper numerical scheme to define and to quantify the magnitude and the role of chemotaxis in the complex groundwater system framework. We present a new class of meshless Lagrangian particle methods based on the Smooth Particle Hydrodinamics (SPH) formulation of Vila & Ben Moussa, combined with a new Weighted Essentially Non-Oscillatory (WENO) reconstruction technique on moving point clouds in multiple space dimensions. The purpose of this new scheme is to fully exploit the advantages of SPH among traditional meshbased and meshfree schemes and to overcome its inapplicability for modeling chemotaxis in porous media. The key idea is to produce for each particle first a set of high order accurate Moving Least Squares (MLS) reconstructions on a set of different reconstruction stencils. Then, these reconstructions are combined with each other using a nonlinear WENO technique in order to capture at the same time discontinuities and to maintain accuracy and low numerical dissipation in smooth regions. The numerical fluxes between interacting particles are subsequently evaluated using this MLS-WENO reconstruction at the midpoint between two particles, in combination with a Riemann solver that provides the necessary stabilization of the scheme based on the underlying physics of the governing equations. We propose the use of two different Riemann solvers: the Rusanov flux and an Osher-type flux. The use of monotone fluxes together with a WENO reconstruction ensures accuracy, stability, robustness and an essentially non oscillatory solution without the artificial viscosity term usually employed in conventional SPH schemes. To our knowledge, this is the first time that the WENO method, which has originally been developed for mesh-based schemes in the Eulerian framework on fixed grids, is extended to meshfree Lagrangian particle methods like SPH in multiple space dimensions. In the first part, we test the new algorithm on two dimensional blast wave problems and on the classical one-dimensional Sod shock tube problem for the Euler equations of compressible gas dynamics. We obtain a good agreement with the exact or numerical reference solution in all cases and an improved accuracy and robustness compared to existing standard SPH schemes. In the second part, the new SPH scheme is applied to advection-diffusion equation in heterogeneous porous media with anisotropic diffusion tensor. Several numerical test case shows that the new scheme is accurate. Unlike standard SPH, it reduces the occurrence of negative concentration. In the third part, we show the applicability of the new scheme for modeling chemotaxis in porous media. We test the new scheme against analytical reference solutions. Under the assumption of complete mixing at the Darcy scale, we perform different two-dimensional conservative solute transport simulations under steady-state conditions with instant injection showing that chemotaxis significantly affect the quantification of field-scale mixing processes.

The aim of this thesis is to study renormalization of Wick polynomials of quantum Boson fields in locally covariant algebraic quantum field theory in curved spacetime. Vector fields are described as sections of natural vector bundles over globally hyperbolic spacetimes and quantized in a locally covariant framework through the known functorial machinery in terms of local *-algebras. These quantized fields may be defined on spacetimes with given classical background fields, also sections of natural vector bundles: The most obvious one is the metric of the spacetime itself, but we encompass also the case of generic spacetime tensors as background fields. In our framework also physical quantities like the mass of the field or the coupling to the curvature are viewed as background fields. Wick powers of the quantized vector field are then axiomatically defined imposing in particular local covariance, scaling properties and smooth dependence on smooth perturbation of the background fields. A general classification theorem is established for finite renormalization terms (or counterterms) arising when comparing different solutions satisfying the defining axioms of Wick powers. The result is then specialized to the case of spacetime tensor fields. In particular, the case of a vector Klein-Gordon field and the case of a scalar field renormalized together with its derivatives are discussed as examples. In each case, a more precise statement about the structure of the counterterms is proved. The finite renormalization terms turn out to be finite-order polynomials tensorially and locally constructed with the backgrounds fields and their covariant derivatives whose coefficients are locally smooth functions of polynomial scalar invariants constructed from the so-called marginal subset of the background fields. Our main technical tools are based on the Peetre-Slov\'ak theorem characterizing differential operators and on the classification of smooth invariants on representations of reductive Lie groups.

Grape (Vitis vinifera L.) is among the most cultivated plants in the world. Its origin traces back to the Neolithic era, when the first human communities started to domesticate wild Vitis sylvestris L. grapes to produce wines. Domestication modified Vitis vinifera to assume characteristics imparted from the humans, selecting desired traits (e.g. specific aromas), and excluding the undesired ones. This process made this species very different from all the other wild grape species existing around the world, including its progenitor, Vitis sylvestris. Metabolomics is a field of the sciences that comparatively studies the whole metabolite set of two (or more) groups of samples, to point out the chemical diversity and infer on the variability in the metabolic pathways between the groups. Crude metabolomics observation can be often used for hypotheses generation, which need to be confirmed by further experiments. In my case, starting from the grape metabolome project (Mattivi et al. unpublished data), I had the opportunity to put hands on a huge dataset built on the berries of over 100 Vitis vinifera grape varieties, tens of grape interspecific hybrids and few wild grape species analyzed per four years; all included in a single experiment. Starting from this data handling, I designed specific experiments to confirm the hypotheses generated from the observation of the data, to improve compound identification, to give statistical meaning to the differences, to localize the metabolites in the berries and extrapolate further information on the variability existing among the grape genus. The hypotheses formulated were two: 1) several glyco-conjugated volatiles can be detected, identified and quantified in untargeted reverses-phase liquid chromatography-mass spectrometry; 2) The chemical difference between Vitis vinifera and wild grape berries is wider than reported in literature. Furthermore, handling a huge dataset of chemical standards injected under the same conditions of the sample set, I also formulated a third hypothesis: 3) metabolites with similar chemical structures are more likely to generate similar signals in LC-MS, therefore the combined use of the signals can predict the more likely chemical structure of unknown markers. In the first study (chapter 5), the signals putatively corresponding to glycoconjugated volatiles have been first enclosed in a specific portion of the temporal and spectrometric space of the LC-HRMS chromatograms, then they have been subjected to MS/MS analysis and lastly their putative identity have been confirmed through peak intensity correlation between the signals measured in LC-HRMS and GC-MS. In the second study (chapter 6), a multivariate regression model has been built between LC-HRMS signals and the substructures composing the molecular structure of the compounds and its accuracy and efficacy in substructure prediction have been demonstrated. In the third study (chapter 7), I comparatively studied some wild grapes versus some Vitis vinifera varieties separating the basic components of the grape berry (skin, flesh and seeds), with the aim to identify all the detected metabolites that differentiate the two groups, which determine a difference in quality between the wild versus domesticated grapes, especially regarding wine production.

The purpose of this thesis is to focus on the aspect of passive devices allowing for WDM, routing, switching and filtering of optical signals, investigating novel routing concepts based on micro optical side coupled resonators to achieve large bandwidth by multiple cascading and/or multiple coupling (low group velocity) periodicity effects. We will describe some technical aspects necessary for the design and fabrication of some passive circuitry, and usually neglected in purely theoretical approaches, including optical routers based on racetrack resonators and novel SCISSOR and CROW devices.

During the last decades public health policy makers have been increasingly turning to mathematical modeling to support their decisions. This trend has been calling for the introduction of a new class of models that not only are capable to explain qualitatively the dynamics of infectious diseases, but also have the capability to provide quantitatively reliable and accurate results. To this aim models are becoming more and more detailed and informed with data. However, there is still much to be done in order to capture the individual and population features that shape the spread of infectious diseases. This thesis addresses some issues in epidemiological modeling that warrant further investigation. In Chapter 1 we introduce an age-structured individual-based stochastic model of Varicella Zoster Virus (VZV) transmission, whose main novelty is the inclusion of realistic population dynamics over the last century. This chapter represents an attempt to answer the need pointed out by recent studies for a better understanding of the role of demographic processes in shaping the circulation of infectious diseases. In Chapter 2 we use the model for VZV transmission developed in Chapter 1 to evaluate the effectiveness of varicella and HZ vaccination programs in Italy. With a view to the support of public health decisions, the epidemiological model is coupled with a cost-effectiveness analysis. To the best of our knowledge, this work represents the first attempt to evaluate the post-vaccination trends in varicella and HZ, both from an epidemiological and economic perspective, in light of the underlying effect of demographic processes. Another novelty of this study is that we take into account the uncertainty regarding the mechanism of VZV reactivation, by comparing results obtained using two different modeling assumptions on exogenous boosting. In Chapter 3 we retrospectively analyze the spatiotemporal dynamics of the 2009 H1N1 influenza pandemic in England, by using a spatially-explicit model of influenza transmission, accounting for socio-demographic and disease natural history data. The aim of this work is to investigate whether the observed spatiotemporal dynamics of the epidemic was shaped by a spontaneous behavioral response to the pandemic threat. This chapter, represents an attempt to contribute to the challenge of understanding and quantifying the effect of human behavioral changes on the spread of epidemics. In Chapter 4 we investigate the current epidemiology of measles in Italy, by using a detailed computational model for measles transmission, informed with regional heterogeneities in the age-specific seroprevalence profiles. The analysis performed in this chapter tries to fill some of the existing gaps in the knowledge of the epidemiological features of vaccine preventable diseases in frameworks characterized by a low circulation of the virus.

Deep learning, a branch of machine learning, has been gaining ground in many research fields as well as practical applications. Such ongoing boom can be traced back mainly to the availability and the affordability of potential processing facilities, which were not widely accessible than just a decade ago for instance. Although it has demonstrated cutting-edge performance widely in computer vision, and particularly in object recognition and detection, deep learning is yet to find its way into other research areas. Furthermore, the performance of deep learning models has a strong dependency on the way in which these latter are designed/tailored to the problem at hand. This, thereby, raises not only precision concerns but also processing overheads. The success and applicability of a deep learning system relies jointly on both components. In this dissertation, we present innovative deep learning schemes, with application to interesting though less-addressed topics. In this respect, the first covered topic is rough scene description for visually impaired individuals, whose idea is to list the objects that likely exist in an image that is grabbed by a visually impaired person, To this end, we proceed by extracting several features from the respective query image in order to capture the textural as well as the chromatic cues therein. Further, in order to improve the representativeness of the extracted features, we reinforce them with a feature learning stage by means of an autoencoder model. This latter is topped with a logistic regression layer in order to detect the presence of objects if any. In a second topic, we suggest to exploit the same model, i.e., autoencoder in the context of cloud removal in remote sensing images. Briefly, the model is learned on a cloud-free image pertaining to a certain geographical area, and applied afterwards on another cloud-contaminated image, acquired at a different time instant, of the same area. Two reconstruction strategies are proposed, namely pixel-based and patch-based reconstructions. From the earlier two topics, we quantitatively demonstrate that autoencoders can play a pivotal role in terms of both (i) feature learning and (ii) reconstruction and mapping of sequential data. Convolutional Neural Network (CNN) is arguably the most utilized model by the computer vision community, which is reasonable thanks to its remarkable performance in object and scene recognition, with respect to traditional hand-crafted features. Nevertheless, it is evident that CNN naturally is availed in its two-dimensional version. This raises questions on its applicability to unidimensional data. Thus, a third contribution of this thesis is devoted to the design of a unidimensional architecture of the CNN, which is applied to spectroscopic data. In other terms, CNN is tailored for feature extraction from one-dimensional chemometric data, whilst the extracted features are fed into advanced regression methods to estimate underlying chemical component concentrations. Experimental findings suggest that, similarly to 2D CNNs, unidimensional CNNs are also prone to impose themselves with respect to traditional methods. The last contribution of this dissertation is to develop new method to estimate the connection weights of the CNNs. It is based on training an SVM for each kernel of the CNN. Such method has the advantage of being fast and adequate for applications that characterized by small datasets.