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.

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 thesis is concerned with the study of nonstandard models in measure theory and in functional analysis. In measure theory, we define elementary numerosities, that are additive measures that take on values in a non-archimedean field and for which the measure of every singleton is 1. We have shown that, by taking the ratio with a suitable unit of measurement, from a numerosity it can be defined a non-atomic real-valued measure, and that every non-atomic measure can be obtained from a numerosity by this procedure. We then used numerosities to develop a model for the probability of infinite sequences of coin tosses coherent with the original ideas of Laplace. In functional analysis, we introduce a space of functions of nonstandard analysis with a formally finite domain, that extends both the space of distributions and the space of Young measures. Among the applications of this space of functions, we develop a continuous-in-time, discrete-in-space nonstandard formulation for a class of ill-posed forward-backward parabolic equations, and on the study of the regularity and asymptotic behaviour of its nonstandard solutions. This approach proved to be a viable alternative to the study of the vanishing viscosity limit of the solution of a pseudoparabolic regularization of the original problem.

This thesis summarizes my work in systems biology as a PhD student at The Microsoft Research - University of Trento Centre for Computational and Systems Biology (COSBI) and at the University of Trento, department of Mathematics. Systems biology is an interdisciplinary field that aims at integrating biology with computational and mathematical methods to gain a better understanding of biological phenomena [5, 6]. Among these methods, mathematical and dy- namical modeling have driven the discovery of mechanistic insights from the static representations of phenomena, that is, data. As a result, mathematical and dynamical models have now become standard tools to support new discoveries in biology and in public health issues. For example, models assist governments in determining the policies to contain the spreading of the diseases and in decisions such as vaccine purchases [7]. Similarly, complex and accurate models of the cardio-vascular systems guide surgeons during many procedures on pa- tients [8]. Furthermore, dynamical models of signaling cascades help researchers in identifying new potential drug targets and therapies for many diseases [9]. We used these modeling techniques to address biological questions related to diabetes and insulin resistance. Within this framework, this thesis contains two articles I contributed to, that focus on diabetes. These works are published in the journal of Nature Scientific Reports and are included in Chapters 3 and 4. A significant contribution to the development of these models, and models in general, is given by optimization. Optimization is often used in modeling to determine certain unknown values or factors in a way that allow the model to optimally reproduce the experimental data. Moreover, the parameters of a model that correctly describe the undergoing dynamics may be used as diagnostic tools [10–13]. To this end, this thesis contains a methodological appendix that includes a review of optimization algorithms that has been submitted to the journal of Frontiers in Applied Mathematics and Statistics, special topic Optimization. The content of this article is reported in Appendix A.

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

In this thesis we investigate variational problems involving 1-dimensional sets (e.g., curves, networks) and variational inequalities related to obstacle-type dynamics from a twofold prospective. On one side, we provide variational approximations and convex relaxations of the relevant energies and dynamics, moving mainly within the framework of Gamma-convergence and of convex analysis. On the other side, we thoroughly investigate the numerical optimization of the corresponding approximating energies, both to recover optimal 1-dimensional structures and to accurately simulate the actual dynamics.

Evaluation of automatic speech recognition (ASR) systems is difficult and costly, since it requires manual transcriptions. This evaluation is usually done by computing word error rate (WER) that is the most popular metric in ASR community. Such computation is doable only if the manual references are available, whereas in the real-life applications, it is a too rigid condition. A reference-free metric to evaluate the ASR performance is \textit{confidence measure} which is provided by the ASR decoder. However, the confidence measure is not always available, especially in commercial ASR usages. Even if available, this measure is usually biased towards the decoder. From this perspective, the confidence measure is not suitable for comparison purposes, for example between two ASR systems. These issues motivate the necessity of an automatic quality estimation system for ASR outputs. This thesis explores ASR quality estimation (ASR QE) from different perspectives including: feature engineering, learning algorithms and applications. From feature engineering perspective, a wide range of features extractable from input signal and output transcription are studied. These features represent the quality of the recognition from different aspects and they are divided into four groups: signal, textual, hybrid and word-based features. From learning point of view, we address two main approaches: i) QE via regression, suitable for single hypothesis scenario; ii) QE via machine-learned ranking (MLR), suitable for multiple hypotheses scenario. In the former, a regression model is used to predict the WER score of each single hypothesis that is created through a single automatic transcription channel. In the latter, a ranking model is used to predict the order of multiple hypotheses with respect to their quality. Multiple hypotheses are mainly generated by several ASR systems or several recording microphones. From application point of view, we introduce two applications in which ASR QE makes salient improvement in terms of WER: i) QE-informed data selection for acoustic model adaptation; ii) QE-informed system combination. In the former, we exploit single hypothesis ASR QE methods in order to select the best adaptation data for upgrading the acoustic model. In the latter, we exploit multiple hypotheses ASR QE methods to rank and combine the automatic transcriptions in a supervised manner. The experiments are mostly conducted on CHiME-3 English dataset. CHiME-3 consists of Wall Street Journal utterances, recorded by multiple far distant microphones in noisy environments. The results show that QE-informed acoustic model adaptation leads to 1.8\% absolute WER reduction and QE-informed system combination leads to 1.7% absolute WER reduction in CHiME-3 task. The outcomes of this thesis are packed in the frame of an open source toolkit named TranscRater -transcription rating toolkit- (https://github.com/hlt-mt/TranscRater) which has been developed based on the aforementioned studies. TranscRater can be used to extract informative features, train the QE models and predict the quality of the reference-less recognitions in a variety of ASR tasks.