In the classical theory, the fractional diffusion is ruled by two different types of fractional Laplacians. Formerly known since 60s, the spectral fractional Laplacian had an important development in the recent mathematical study with the initial contributes of L. Caffarelli, L. Silvestre and X. Cabré, X.Tan. The integral version of the fractional Laplacian, recently discussed by M. Fukushima, Y. Oshima, M Takeda, and Song, Vondracek, is considered in a semilinear elliptic problem in presence of a general logistic function and an indefinite weight. In particular we look for a multiplicity result for the associated Dirichlet problem. In the second part, starting from the classical works of T.Hillen and G. Othmer and taking the Generalized velocity jump processes presented in a recent work of J.T.King, we obtain the fractional diffusion as limit of this last processes using the technique used in another recent work of Mellet, without the classical Hilbert or Cattaneo approximation's methods.

We present a mathematical model describing the dynamics occurring between two interacting populations, one of amoebae and one of virulent bacteria; it is meant to describe laboratory experiments with these two species in a mathematical framework and help understanding the role of the different mechanisms involved. In particular we aim to focus on how bacterial virulence may affect the dynamics of the system. The model is a modified reaction-diffusion-chemotaxis predator-prey one with a mechanism of redistribution of ingested biomass between amoeboid cells. The spatially homogeneous case is analyzed in detail; conditions for pattern formation are established; numerical simulations for the complete model are performed.

The thesis deals with the study of a natural extension of classical finite-dimensional time-optimal control problem to the space of positive Borel measures. This approach has two main motivations: to model real-life situations in which the knowledge of the initial state is only probabilistic, and to model the statistical distribution of a huge number of agents for applications in multi-agent systems. We deal with a deterministic dynamics and treat the problem first in a mass-preserving setting: we give a definition of generalized target, its properties, admissible trajectories and generalized minimum time function, we prove a Dynamic Programming Principle, attainability results, regularity results and an Hamilton-Jacobi-Bellman equation solved in a suitable viscosity sense by the generalized minimum time function, and finally we study the definition of an object intended to reflect the classical Lie bracket but in a measure-theoretic setting. We also treat a case with mass loss thought for modelling the situation in which we are interested in the study of an averaged cost functional and a strongly invariant target set. Also more general cost functionals are analysed which takes into account microscopical and macroscopical effects, and we prove sufficient conditions ensuring their lower semicontinuity and a dynamic programming principle in a general formulation.

In the present work we solve systems of partial differential equations (PDE) for hyperbolic conservation laws using semi-implicit numerical methods on staggered meshes applied both to the class of finite volume (FV) and both to the family of high order Discontinuous Galerkin (DG) finite elements schemes. In particular, we want to show that these new semi-implicit schemes can be applied in several fields of applied sciences, such as in geophysical flows and compressible fluids in compliant tubes. Inside this thesis we distinguish two big parts. First, we consider staggered semi-implicit schemes for compressible viscous fluids flowing in elastic pipes. This topic is very important in several practical applications of civil, environmental, industrial and biomedical engineering. Here, we analyse the accuracy and the computational efficiency of fully explicit and semi-implicit 1D and 2D finite volume schemes for the simulation of highly unsteady viscous compressible flows in laminar regime in axially symmetric rigid and elastic pipes. We consider two families of differential models that can be used to predict the pressure and velocity distribution along the tube. One is the so called 2Dxr PDE model which is derived from the full compressible Navier-Stokes equations under the assumptions of a hydrostatic pressure and an axially symmetric geometry. The second family is a simple 1D non-conservative PDE system based on the cross-sectionally averaged version of the Navier-Stokes equations in cylindrical coordinates. In this last case, the use of a simple steady friction model is not enough to simulate the wall friction phenomena in highly transient regime. As a consequence the wall friction model has to be frequency dependent and, following previous studies present in the literature, we consider the classes of convolution integral (CI) models and instantaneous acceleration (IA) models. We carry out a rather complete analysis of the previously-mentioned methods for the simulation of flows characterized by fast transient regime in rigid and compliant tubes. The numerical results show that the convolution integral models are clearly better than instantaneous acceleration models concerning accuracy. Moreover, for CI models, instead of computing the convolution integrals, which is very time- and memoryconsuming, we express these methods via a set of additional ODEs for appropriate auxiliary variables. This trick improves the computational efficiency of these methods substantially, since it avoids the direct computation of the convolution integral. In addition, semi-implicit finite volume methods are significantly superior to classical explicit finite volume schemes in terms of computational efficiency, however, providing the same level of accuracy. We then proceed by extending the finite volume discretization of the 1D and 2Dxr PDE models to arbitrary high-order of accuracy in space introducing a new SIDG scheme on staggered meshes. Both models include the effects of the viscosity and of the wall motion. The nonlinear convective terms are discretized explicitly by using a classical RKDG scheme of arbitrary high-order of accuracy in space and third order of accuracy in time. The continuity equation is integrated over the elements that belong to the main grid, while the momentum equation is integrated over the control volumes of the edgebased staggered dual grid. Inserting the discrete momentum equation into the discrete continuity leads to a mildly nonlinear algebraic system for the degrees of freedom of the pressure, which is solved by using the (nested) Newton method of Brugnano, Casulli and Zanolli. We use the -method in order to get second order of accuracy in time for the implicit part of the scheme. In addition, the schemes have to obey only a mild CFL condition based on the fluid velocity and not based on the sound speed; consequently these schemes work also in the low Mach number regime and even in the incompressible limit of the Navier-Stokes equations. This is a very important property, which is the so-called asymptotic preserving (AP) property of the scheme. We carry out several numerical tests in order to validate this novel family of numerical methods against available exact solutions and experimental data. We also report numerical convergence tables in order to show that the new schemes indeed achieve high order of accuracy in space. In the second part of the thesis, we present a new class of a posteriori sub-cell finite volume limiters for spatially high order accurate semi-implicit discontinuous Galerkin schemes on staggered Cartesian grids for the solution of the 1D and 2D shallow water equations (SWE) and of the Euler equations both expressed in conservative form. Here, the starting point is the unlimited arbitrary high order accurate staggered SIDG scheme proposed by Dumbser and Casulli (2013). For this metho d, the mass conservation equation and the momentum equations are integrated using a discontinuous finite element strategy on staggered control volumes, where the discrete free surface elevation is defined on the main grid and the discrete momentum is defined on edge-based staggered dual control volumes. According to the semi-implicit approach, pressure terms are discretized implicitly, while the nonlinear convective terms are discretized explicitly. Inserting the momentum equations into the discrete continuity equation leads to a well conditioned block diagonal linear system for the free surface elevation which can be efficiently solved with modern iterative methods. Furthermore, the staggered SIDG is also extended to the Euler equations of compressible gasdynamics. Here, the governing PDE are rewritten using a flux vector splitting technique. The convective terms are updated using an explicit Runge-Kutta DG integrator. Then, the discrete momentum equation, which is integrated again on the dual grid, is coupled with the discrete energy equation that is discretized on the control volumes of the main grid. The pressure is efficiently obtained solving a linear system combined with an iterative Picard iteration procedure.

Recent advances in medical science regarding the interaction and functional role of fluid compartments in the central nervous system have attracted the attention of many researchers across various disciplines. Neurotoxins are constantly cleared from the brain parenchyma through the intramural periarterial drainage system, glymphatic system and meningeal lymphatic system. Impairment of these systems can potentially contribute to the onset of neurological disorders. The goal of this thesis is to contribute to the understanding of brain fluid dynamics and to the role of vascular pathologies in the context of neurological disorders. To achieve this goal, we designed the first multi-scale, closed-loop mathematical model of the murine fluid system, incorporating: heart dynamics, major arteries and veins, microcirculation, pulmonary circulation, venous valves, cerebrospinal fluid (CSF), brain interstitial fluid (ISF), Starling resistors, Monro-Kellie hypothesis, brain lymphatic drainage and the modern concept of CSF/ISF drainage and absorption based on the {\em Bulat-Klarica-Orešković} hypothesis. The mathematical model relies on one-dimensional Partial Differential Equations (PDEs) for blood vessels and on Ordinary Differential Equations (ODEs) for lumped parameter models. The systems of PDEs and ODEs are solved through a high-order finite volume ADER method and through an implicit Euler method. The computational results are validated against literature values and magnetic resonance flow measurements. Furthermore, the model is validated against {\em in-vivo} intracranial pressure waveforms acquired in healthy mice and in mice with impairment of the intracranial venous outflow. Through a systematic use of our computational model in healthy and pathological cases, we provide a complete and holistic neurovascular view of the main murine fluid dynamics. We propose a hypothesis on the working principles of the glymphatic system, opening a new door towards a comprehensive understanding of the mechanisms which link vascular and neurological disorders. In particular, we show how impairment of the cerebral venous outflow might potentially lead to accumulation of solutes in the parenchyma, by altering CSF and ISF dynamics. This thesis also concerns the development of a high-order ADER-type numerical method for systems of hyperbolic balance laws in networks, based on a new implicit solver for the junction-generalized Riemann problem. The resulting ADER scheme can deal with stiff source terms and can be applied to non-linear systems of hyperbolic balance laws in domains consisting of networks of one-dimensional sub-domains. Also, we design a novel one-dimensional mathematical model for collecting lymphatics coupled with a Electro-Fluid-Mechanical Contraction (EFMC) model for dynamical contractions. The resulting mathematical model gives each lymphangion the autonomous capability to trigger action potentials based on local fluid-dynamical factors.

The interplay between human behavior and the spreading of an epidemics represents a challenge in modeling the dynamics of infectious diseases. The technological revolution that we are experiencing nowadays gives access to new sources of digital data, capable of capturing behavioral patterns and social dynamics of our society and opening, in fact, the path to new opportunities for mathematical modelers. Provided by such tools, we discuss two different aspects of the dynamics of infectious diseases associated with human behavior. In the first part of the thesis, we focus on the mechanism driving the awareness of individuals during public health emergencies and describe epidemiological models especially tailored to better understand the underline features of the risk perception. The proposed framework is able to disentangle and characterize the contribution of media drivers and social contagion mechanisms in the building of awareness of individuals about infectious diseases. In the second part of the thesis, we present a data driven computational model aiming to assess the potential risk of experiencing measles re-emergence in Turkey. This study takes into consideration the recent massive migration of Syrian refugees in Turkey, which changed the social structure and focuses on the possible outbreak of an infectious disease, such as measles, as a consequence of the great concentration of Syrian refugees not adequately immunized against it. The model proposed is informed with mobility patterns inferred from mobile phone data and accounts for the different hypothetical policies adopted to integrate the refugees with the Turkish population.

The thesis deals with various problems arising in deterministic control, jumping processes and control for locomotion in fluids. It is divided in three parts. The first part is focused on some optimal control problems on network and stratified domains with junctions, where each edge/hyper-plane has its own controlled dynamics and cost. We consider some possible approximations for such a problems given by the use of a switching rule of delayed-relay type and study the passage to the limit when the parameter of the approximation goes to zero. First, we take into account some problems on network: a twofold junction problem, a threefold junction one and an extension of the last one. For each of these problems we characterize the limit functions as viscosity solution and maximal subsolution of a suitable Hamilton-Jacobi problem. Secondly, we consider a bi-dimensional multi-domain problem and as done for the problems on network we characterize the limit function as viscosity solution of a suitable Hamilton-Jacobi problem. The second part studies a decentralized routing problem over a network, using the paradigm of mean-field games with large number of players. Building on a state-space extension technique, we turn the problem into an optimal control one for each single player. The main contribution is an explicit expression of the optimal decentralized control which guarantees the convergence both to local and to global equilibrium points. Furthermore, we study the stability of the system also in the presence of a delay which we model using an hysteresis operator. As a result of the hysteresis, we prove existence of multiple equilibrium points and analyze convergence conditions. In the last part of the thesis we investigate different strategies to overcome the so-called scallop paradox concerning periodic locomotion in fluid. We show how to obtain a net motion exploiting the fluid's type change during a periodic deformation. We consider two different models: in the first one that change is linked to the magnitude of the opening and closing velocity of the scallop's valves. Instead, in the second one it is related to the sign of the above velocity. In both cases we prove that the mechanical system is controllable, i.e. the scallop is able to move both forward and backward using cyclical deformations.

This PhD thesis presents a novel second order accurate direct Arbitrary-Lagrangian-Eulerian (ALE) Finite Volume scheme for nonlinear hyperbolic systems, written both in conservative and non-conservative form, whose peculiarity is the nonconforming motion of interfaces. Moreover it has been coupled together with specifically designed path-conservative well balanced (WB) techniques and angular momentum preserving (AMC) strategies. The obtained result is a method able to preserve many of the physical properties of the system: besides being conservative for mass, momentum and total energy, also any known steady equilibrium of the studied system can be exactly maintained up to machine precision. Perturbations around such equilibrium solutions are resolved with high accuracy and minimal dissipation on moving contact discontinuities even for very long computational times. The core of our ALE scheme is the use of a space-time conservation formulation in the construction of the final Finite Volume scheme: the governing PDE system is rewritten at the aid of the space-time divergence operator and then a fully discrete one-step discretization is obtained by integrating over a set of closed space-time control volumes. In order to avoid the typical mesh distortion caused by shear flows in Lagrangian-type methods, we adopt a nonconforming treatment of sliding interfaces, which requires the dynamical insertion or deletion of nodes and edges, and produces hanging nodes and space-time faces shared between more than two cells. In this way, the elements on both sides of the shear wave can move with a different velocity, without producing highly distorted elements, the mesh quality remains high and, as a direct consequence, also the time step remains almost constant in time, even for highly sheared vortex flows. Moreover, due to the space-time conservation formulation, the geometric conservation law (GCL) is automatically satisfied by construction, even on moving nonconforming meshes. Our nonconforming ALE scheme is especially well suited for modeling in polar coordinates vortical flows affected by strong differential rotation: in particular, the novel combination with the well balancing make it possible to obtain great results for challenging astronomical phenomena as the rotating Keplerian disk. Indeed, we have formulated a new HLL-type and a novel Osher-type flux that are both able to guarantee the well balancing in a gas cloud rotating around a central object, maintaining up to machine precision the equilibrium between pressure gradient, centrifugal force and gravity force that characterizes the stationary solutions of the Euler equations with gravity. To the best knowledge of the author this work is original for various reasons: it is the first time that the little dissipative Osher scheme is modified in order to be well balanced for non trivial equilibria, and it is the first time that WB is coupled with ALE for the Euler equations with gravity; moreover the use of a well balanced Osher scheme joint with the Lagrangian framework allows, for the first time within a Finite Volume method, to maintain exactly even moving equilibria. In addition, the introduced techniques demonstrate a wide range of applicability from steady vortex flows in shallow water equations to complex free surface flows in two-phase models. In the last case, studied on fixed Cartesian grids, the new well balanced methods have been implemented in parallel exploiting a GPU-based platform and reaching the very high efficiency of ten million of volumes processed per seconds. Finally, in the case of vortical flows we propose a preliminary analysis on how to increase the accuracy of the method by exploiting the redundant conservation law that can be written for the angular momentum, as proposed in Després et al. JCP 2015. Indeed, an easy manipulation of the Euler equations allows to write its additional conservation law: clearly it does not add any supplementary information from the analytical point of view, but from a numerical point of view it provides extra information in particular in the case of rotating systems. We present both a master-slave approach, to deduce a posteriori a more precise approximation of the velocity, and some coupled approaches to investigate how the entire process can take advantage from considering directly the angular momentum during the computation within a strong coupling with other variables. A large set of different numerical tests has been carried out in order to check the accuracy and the robustness of the new methods for both smooth and discontinuous problems, close and far away from the equilibrium, in one and two space dimensions. Many of the presented results show a great enhancement with respect to the state of the art.

Il lavoro affronta l'analisi funzionale dei contenitori ceramici provenienti da diversi contesti archeologici attraverso un approccio interdisciplinare. Le metodologie di analisi adottate affiancano all'approccio archeologico i risultati derivanti dall'integrazione con metodiche e strumenti di analisi di tipo archeometrico. Uno degli obiettivi principali della ricerca è quello di risalire alla funzione pratica dei contenitori ceramici attraverso l'analisi chimica dei residui organici in essi contenuti. È stato affrontato uno studio comparato di varie classi ceramiche, provenienti dai seguenti contesti in corso di scavo da parte dell'Università di Lecce: abitato arcaico di San Vito dei Normanni (Br), santuario di Tas Silg (Malta), santuario di Apollo a Hierapolis (Turchia). I materiali ceramici selezionati sono stati sottoposti ad analisi chimiche finalizzate ad identificare i residui organici, attraverso l'impiego incrociato di due tecniche analitiche: Gas cromatografia con spettrometria di massa (GC/MS) e Spettroscopia ad infrarossi in trasformata di Fourier (FTIR). I materiali sono stati inoltre letti in rapporto ai contesti di provenienza, al fine di pervenire anche ad una valutazione della distribuzione dei contenitori ceramici sulla superficie e quindi delle attività che si svolgevano nelle diverse aree di un ambiente o di un edificio, dei sistemi di immagazzinamento delle derrate, delle modalità di preparazione e di consumo del cibo. / In the framework of a study on the relations between form and function of pottery, organic residues analysis has been applied to different types of vessels from three archaeological sites excavated by the University of Lecce: San Vito dei Normanni (Brindisi, South Italy), Tas Silg sanctuary (Malta), Apollo sanctuary at Hierapolis (Turkey). Organic residues were identified by two analytical procedures based on gas chromatography coupled with mass spectrometry (GC-MS) and on Fourier transformed infrared spectroscopy (FTIR) respectively.

This Ph.D. thesis presents a threefold revisitation and reformulation of the linear sampling method (LSM) for the qualitative solution of inverse scattering problems (in the resonance region and in time-harmonic regime): 1) from the viewpoint of its implementation (in a 3D setting), the LSM is recast in appropriate Hilbert spaces, whereby the set of algebraic systems arising from an angular discretization of the far-field equation (written for each sampling point of the numerical grid covering the investigation domain and for each sampling polarization) is replaced by a single functional equation. As a consequence, this 'no-sampling' LSM requires a single regularization procedure, thus resulting in an extremely fast algorithm: complex 3D objects are visualized in around one minute without loss of quality if compared to the traditional implementation; 2) from the viewpoint of its application (in a 2D setting), the LSM is coupled with the reciprocity gap functional in such a way that the influence of scatterers outside the array of receiving antennas is excluded and an inhomogeneous background inside them can be allowed for: then, the resulting 'no-sampling' algorithm proves able to detect tumoural masses inside numerical (but rather realistic) phantoms of the female breast by inverting the data of an appropriate microwave scattering experiment; 3) from the viewpoint of its theoretical foundation, the LSM is physically interpreted as a consequence of the principle of energy conservation (in a lossless background). More precisely, it is shown that the far-field equation at the basis of the LSM (which does not follow from physical laws) can be regarded as a constraint on the power flux of the scattered wave in the far-field region: if the flow lines of the Poynting vector carrying this flux verify some regularity properties (as suggested by numerical simulations), the information contained in the far-field constraint is back-propagated to each point of the background up to the near-field region, and the (approximate) fulfilment of such constraint forces the L^2-norm of any (approximate) solution of the far-field equation to behave as a good indicator function for the unknown scatterer, i.e., to be 'small' inside the scatterer itself and 'large' outside.