In principle, the lattice of elementary propositions of a generic quantum system admits a representation in real, complex or quaternionic Hilbert spaces as established by Solèr’s theorem (1995) closing a long standing problem that can be traced back to von Neumann’s mathematical formulation of quantum mechanics. However up to now there are no examples of quantum systems described in Hilbert spaces whose scalar field is different from the set of complex numbers. We show that elementary relativistic systems cannot be described by irreducible strongly-continuous unitary representations of SL(2, C) on real or quaternionic Hilbert spaces as a consequence of some peculiarity of the generators related with the theory of polar decomposition of operators. Indeed such a ”naive” attempt leads necessarily to an equivalent formulation on a complex Hilbert space. Although this conclusion seems to give a definitive answer to the real/quaternionic-quantum-mechanics issue, it lacks consistency since it does not derive from more general physical hypotheses as the complex one does. Trying a more solid approach, in both situations we end up with three possibilities: an equivalent description in terms of a Wigner unitary representation in a real, complex or quaternionic Hilbert space. At this point the ”naive” result turns out to be a definitely important technical lemma, for it forbids the two extreme possibilities. In conclusion, the real/quaternionic theory is actually complex. This improved approach is based upon the concept of von Neumann algebra of observables. Unfortunately, while there exists a thorough literature about these algebras on real and complex Hilbert spaces, an analysis on the notion of von Neumann algebra over a quaternionic Hilbert space is completely absent to our knowledge. There are several issues in trying to define such a mathematical object, first of all the inability to construct linear combination of operators with quaternionic coeffients. Restricting ourselves to unital real *-algebras of operators we are able to prove the von Neumann Double Commutant Theorem also on quaternionc Hilbert spaces. Clearly, this property turns out to be crucial.

The present work provides a geometric approach to the calculus of variations in the presence of non-holonomic constraints. As far as the kinematical foundations are concerned, a fully covariant scheme is developed through the introduction of the concept of infinitesimal control. The usual classification of the evolutions into normal and abnormal ones is also discussed, showing the existence of a universal algorithm assigning to every admissible curve a corresponding abnormality index, defined in terms of a suitable linear map. A gauge-invariant formulation of the variational problem, based on the introduction of the bundle of affine scalars over the configuration manifold, is then presented. The analysis includes a revisitation of Pontryagin Maximum Principle and of the Erdmann-Weierstrass corner conditions, a local interpretation of Pontryagin's equations as dynamical equations for a free (singular) Hamiltonian system and a generalization of the classical criteria of Legendre and Bliss for the characterization of the minima of the action functional to the case of piecewise-differentiable extremals with asynchronous variation of the corners.

My PhD thesis is focused on geometric Hamiltonian formulation of Quanum Mechanics and its interplay with standard formulation. The main result is the construction of a general prescription to set up a quantum theory as a classical-like theory where quantum dynamics is given by a Hamiltonian vector field on a complex projective space with Kähler structure. In such geometric framework quantum states are represented by classical-like Liouville densities. After a complete characterization of classical-like observables in a finite-dimensional quantum theory, the observable C*-algebra is described in geometric Hamiltonian terms. In the final part of the work, the classical-like Hamiltonian formulation is applied to the study of composite quantum systems providing a notion of entanglement measure.

This thesis addresses two problems from the theory of periodic homogenization and the related notion of two-scale convergence. Its main focus rests on the derivation of equivalent transmission conditions for the interaction of two adjacent bodies which are connected by a thin layer of interface material being perforated by identically shaped voids. Herein, the voids recur periodically in interface direction and shall in size be of the same order as the interface thickness. Moreover, the constitutive properties of the material occupying the bodies adjacent to the interface are assumed to be described by some convex energy densities of quadratic growth. In contrast, the interface material is supposed to show extremal" constitutive behavior. More precisely

Dielectric elastomers (DEs) are an important class of materials, currently employed in the design and realization of electrically-driven, highly deformable actuators and devices, which find application in several fields of technology and engineering, including aerospace, biomedical and mechanical engineering. Subject to a voltage, a membrane of a soft dielectric elastomer coated by compliant electrodes reduces its thickness and expands its area, possibly deforming in-plane well beyond 100%: this principle is exploited to conceive transducers for a broad range of applications, including soft robots, adaptive optics, Braille displays and energy harvesters. Soft dielectrics undergo finite strains, and their modelling requires a formulation based on the Mechanics of Solids at large deformations. A major problem that limits the widespread diffusion of such devices in everyday technology is the high voltage required to activate large strains, because of the low dielectric permittivity of typical materials (acrylic elastomers or silicones), in the order of few unities, which governs the electromechanical coupling. Composite materials (reinforcing a soft matrix with stiff and high-permittivity particles) provide a way to overcome these limitations, as suggested by some experiments. In addition, composites can display failure modes and instabilities not displayed by homogeneous specimens that must be thoroughly investigated. Commonly, instability phenomena are seen as a serious drawback, that should be predicted and avoided. However, in some cases they can be used to activate snap-through actuation, as in the case of buckling-like or highly-deformable balloon-like actuators. Soft dielectric elastomers display electrostrictive properties (permittivity depending on the deformation) and we show how to take into account such a phenomenon within the theory of electroelasticity. Original results regard the investigation of diffuse modes (buckling like instabilities etc.), surface mode instabilities and localized modes. New (analytical) solutions for band-localization instability are provided and then it has been investigated how such instabilities are related to electrostriction. Regarding DE composites, the goal is to evaluate in detail the behaviour of two-phase rank-1 laminates in terms of different types of actuation, geometric and mechanical properties of phases, applied boundary conditions, and instabilities phenomena, in order to establish precise ranges in which the performance enhancement is effective with respect to the homogeneous counterpart.

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.

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.

Scientists are continuously fascinated by the high degree of sophistication found in natural materials, arising from evolutionary optimisation. In living organisms, nature provides a wide variety of materials, architectures, systems and functions, often based on weak constituents at the lower scales. One of the most extensively studied natural materials is spider silk, renowned for its outstanding mechanical properties, which include exceptional strength and toughness. Owing to its wide range of properties, which vary depending on factors such as the type of silk (up to seven) that each spider can produce, and the species of spider, it can be considered a class of semi-crystalline polymeric material. Indeed, spider silk cleverly combines, depending on the application required, the great deformability of an amorphous phase with the stiffness and strength conferred by pseudo-crystals consisting of specific secondary structures of some of the proteins constituting the material. Based on the countless studies conducted on spider silk, it is now also clear that its remarkable performance are the result of a sophisticated optimisation of the material's hierarchical structure. Nevertheless, many of the multiscale mechanisms that give rise to the striking macroscopic properties are still unclear. Many open problems are also related to the relevant effects of environmental conditions and in particular on temperature and humidity, strongly conditioning the mechanical performances. In this thesis, aimed at unveiling some of these open problems, we introduce a multiscale model for the thermo-hygro-mechanical response, starting with the influence of water molecules modifying the microstructure, up to the effects at the macroscopic scale, including softening, increase in elongation at break and supercontraction, i.e. the shortening (up to half the initial length) of the spider threads in wet environments. Thereafter, we describe how the supercontraction effect can be adopted to obtain humidity-driven actuators, and in particular, we determine the maximum actuation force depending on the silk properties at the molecular scale and on the constraining system representing other silk threads or the actuated device. The spider silk actuation properties turned out to be extraordinary, making spider silk potentially the best performing humidity-driven actuator known to date in terms of work density. As observed in many natural materials, spider silks are characterized by a strong variability in both chemical and structural organization, as for example described in the recently published experimental database of properties at different scales of about a thousand different spider silks, where evident correlations among quantities are scarce. This large variability makes the theoretical understanding of the observed material behavior, in relation of the complex hierarchical structure, particularly intriguing. To address this novel amount of experimental data without losing sight of theoretical analytical modelling, we propose a new data modelling methodology to obtain simple and interpretable relationships linking quantities at different scales. In particular, we employ a symbolic regression technique, known as 'Evolutionary Polynomial Regression', which integrates regression capabilities with the Genetic Programming paradigm, enabling the derivation of explicit analytical formulas, finally delivering a deeper comprehension of the analysed physical phenomenon. Eventually, we provide insights to improve our multiscale theoretical model accounting for the humidity effects on spider silks. This approach may represent a proof of concept for modelling in fields governed by multiscale, hierarchical differential equations. We believe that the analytical description of the macroscopic behaviour from microscale properties is of great value both for the full understanding of biological materials, as well as in the perspective of bioinspired materials and structures.

Immunotherapy, by enhancing the endogenous anti-tumor immune responses, is showing promising results for the treatment of numerous cancers refractory to conventional therapies. However, its effectiveness for advanced castration-resistant prostate cancer remains unsatisfactory and new therapeutic strategies need to be developed. To this end, mathematical modeling provides a quantitative framework for testing in silico the efficacy of new treatments and combination therapies, as well as understanding unknown biological mechanisms. In this dissertation we present two mathematical models of prostate cancer immunotherapy defined as systems of ordinary differential equations. The first work, introduced in Chapter 2, provides a mathematical model of prostate cancer immunotherapy which has been calibrated using data from pre-clinical experiments in mice. This model describes the evolution of prostate cancer, key components of the immune system, and seven treatments. Numerous combination therapies were evaluated considering both the degree of tumor inhibition and the predicted synergistic effects, integrated into a decision tree. Our simulations predicted cancer vaccine combined with immune checkpoint blockade as the most effective dual-drug combination immunotherapy for subjects treated with androgen-deprivation therapy that developed resistance. Overall, this model serves as a computational framework to support drug development, by generating hypotheses that can be tested experimentally in pre-clinical models. The Chapter 3 is devoted to the description of a human prostate cancer mathematical model. The potential effect of immunotherapies on castration-resistant form has been analyzed. In particular, the model includes the dendritic vaccine sipuleucel-T, the only currently available immunotherapy option for advanced prostate cancer, and the ipilimumab, a drug targeting the cytotoxic T-lymphocyte antigen 4 , exposed on the CTLs membrane, currently under Phase II clinical trial. From a mathematical analysis of a simplified model, it seems likely that, under continuous administration of ipilimumab, the system lies in a bistable situation where both the no-tumor equilibrium and the high-tumor equilibrium are attractive. The schedule of periodic treatments could then determine the outcome, and mathematical models could help in deciding an optimal schedule.

The present work analyzes and describes a method for the direct numerical solution of the Maxwell's equations of classical electromagnetism. This is the FDTD (Finite-Difference Time-Domain) method, along with its implementation in an "in-house" computing code for large parallelized simulations. Both are then applied to the modelization of photonic and plasmonic structures interacting with light. These systems are often too complex, either geometrically and materially, in order to be mathematically tractable and an exact analytic solution in closed form, or as a series expansion, cannot be obtained. The only way to gain insight on their physical behavior is thus to try to get a numerical approximated, although convergent, solution. This is a current trend in modern physics because, apart from perturbative methods and asymptotic analysis, which represent, where applicable, the typical instruments to deal with complex physico-mathematical problems, the only general way to approach such problems is based on the direct approximated numerical solution of the governing equations. Today this last choice is made possible through the enormous and widespread computational capabilities offered by modern computers, in particular High Performance Computing (HPC) done using parallel machines with a large number of CPUs working concurrently. Computer simulations are now a sort of virtual laboratories, which can be rapidly and costless setup to investigate various physical phenomena. Thus computational physics has become a sort of third way between the experimental and theoretical branches. The plasmonics application of the present work concerns the scattering and absorption analysis from single and arrayed metal nanoparticles, when surface plasmons are excited by an impinging beam of light, to study the radiation distribution inside a silicon substrate behind them. This has potential applications in improving the eciency of photovoltaic cells. The photonics application of the present work concerns the analysis of the optical reflectance and transmittance properties of an opal crystal. This is a regular and ordered lattice of macroscopic particles which can stops light propagation in certain wavelenght bands, and whose study has potential applications in the realization of low threshold laser, optical waveguides and sensors. For these latters, in fact, the crystal response is tuned to its structure parameters and symmetry and varies by varying them. The present work about the FDTD method represents an enhacement of a previous one made for my MSc Degree Thesis in Physics, which has also now geared toward the visible and neighboring parts of the electromagnetic spectrum. It is organized in the following fashion. Part I provides an exposition of the basic concepts of electromagnetism which constitute the minimum, although partial, theoretical background useful to formulate the physics of the systems here analyzed or to be analyzed in possible further developments of the work. It summarizes Maxwell's equations in matter and the time domain description of temporally dispersive media. It addresses also the plane wave representation of an electromagnetic field distribution, mainly the far field one. The Kirchhoff formula is described and deduced, to calculate the angular radiation distribution around a scatterer. Gaussian beams in the paraxial approximation are also slightly treated, along with their focalization by means of an approximated diraction formula useful for their numericall FDTD representation. Finally, a thorough description of planarly multilayered media is included, which can play an important ancillary role in the homogenization procedure of a photonic crystal, as described in Part III, but also in other optical analyses. Part II properly concerns the FDTD numerical method description and implementation. Various aspects of the method are treated which globally contribute to a working and robust overall algorithm. Particular emphasis is given to those arguments representing an enhancement of previous work.These are: the analysis from existing literature of a new class of absorbing boundary conditions, the so called Convolutional-Perfectly Matched Layer, and their implementation; the analysis from existing literature and implementation of the Auxiliary Differential Equation Method for the inclusion of frequency dependent electric permittivity media, according to various and general polarization models; the description and implementation of a "plane wave injector" for representing impinging beam of lights propagating in an arbitrary direction, and which can be used to represent, by superposition, focalized beams; the parallelization of the FDTD numerical method by means of the Message Passing Interface (MPI) which, by using the here proposed, suitable, user dened MPI data structures, results in a robust and scalable code, running on massively parallel High Performance Computing Machines like the IBM/BlueGeneQ with a core number of order 2X10^5. Finally, Part III gives the details of the specific plasmonics and photonics applications made with the "in-house" developed FDTD algorithm, to demonstrate its effectiveness. After Chapter 10, devoted to the validation of the FDTD code implementation against a known solution, Chapter 11 is about plasmonics, with the analytical and numerical study of single and arrayed metal nanoparticles of different shapes and sizes, when surface plasmon are excited on them by a light beam. The presence of a passivating embedding silica layer and a silicon substrate are also included. The next Chapter 12 is about the FDTD modelization of a face-cubic centered (FCC) opal photonic crystal sample, with a comparison between the numerical and experimental transmittance/reflectance behavior. An homogenization procedure is suggested of the lattice discontinuous crystal structure, by means of an averaging procedure and a planarly multilayered media analysis, through which better understand the reflecting characteristic of the crystal sample. Finally, a procedure for the numerical reconstruction of the crystal dispersion banded omega-k curve inside the first Brillouin zone is proposed. Three Appendices providing details about specific arguments dealt with during the exposition conclude the work.