Inverse problems are problems where we want to estimate the values of certain parameters of a system given observations of the system. Such problems occur in several areas of science and... Show moreInverse problems are problems where we want to estimate the values of certain parameters of a system given observations of the system. Such problems occur in several areas of science and engineering. Inverse problems are often ill-posed, which means that the observations of the system do not uniquely define the parameters we seek to estimate, or that the solution is highly sensitive to small changes in the observation. In order to solve such problems, therefore, we need to make use of additional knowledge about the system at hand. One such prior information is given by the notion of sparsity. Sparsity refers to the knowledge that the solution to the inverse problem can be expressed as a combination of a few terms. The sparsity of a solution can be controlled explicitly or implicitly. An explicit way to induce sparsity is to minimize the number of non-zero terms in the solution. Implicit use of sparsity can be made, for e.g., by making adjustments to the algorithm used to arrive at the solution.In this thesis we studied various inverse problems that arise in different application areas, such as tomographic imaging and equation learning for biology, and showed how ideas of sparsity can be used in each case to design effective algorithms to solve such problems. Show less
Arithmetic geometry concerns the number-theoretic properties of geometric objects defined by polynomials. Mathematicians are interested in the rational solutions to these geometric objects. However... Show moreArithmetic geometry concerns the number-theoretic properties of geometric objects defined by polynomials. Mathematicians are interested in the rational solutions to these geometric objects. However, it is usually very difficult to answer questions like this.A. Beilinson and S. Bloch conjectured a very general height theory in 1980s, which was used by B. Gross and R. Schoen in their study of the Gross-Schoen cycles. The height of canonical Gross-Schoen cycles is conjectured to be non-negative. This was verified when the curve is an elliptic or hyperelliptic curve, while very few are known in the non-hyperelliptic case.During my PhD study, I study the Beilinson-Bloch height of canonical Gross-Schoen cycles on curves with an emphasis on the genus 3 case (almost all genus 3 curves are non-hyperelliptic). I studied its unboundedness and singular properties, and did explicit computation for the height of the canonical Gross-Schoen cycle of a specific plane quartic curve.The method used in my thesis should be helpful for verifications. Show less
This dissertation focuses on developing new mathematical and statistical methods to properly represent time-varying covariates and model them within the context of time-to-event analysis. This... Show moreThis dissertation focuses on developing new mathematical and statistical methods to properly represent time-varying covariates and model them within the context of time-to-event analysis. This research topic is motivated by specific clinical questions aimed at gaining insights into personalised treatments for cardiological and oncological patients. The main purpose is to enrich the knowledge available for modelling patients’ survival with relevant features related to the time-varying processes of interest.The efforts of this work address the complexities of both (i) developing adequate dynamic characterizations of the processes under study (i.e., representation problem) and (ii) identifying and quantifying the association between time-varying processes and patient survival (i.e., time-to-event modelling problem). In both cases, the main issue is dealing with complex data sources while taking into account the nature of the processes and managing the complex trade-off between clinical interpretability and mathematical formulation.By solving the aforementioned statistical complexities, this work is not only impacting the community of researchers in mathematics and statistics. The development of these novel methodologies may represent a significant step forward in the definition of customized and flexible monitoring tools to support doctors and clinicians in their work.*********This doctoral dissertation was part of a cotutelle agreement between the Politecnico di Milano and Leiden University Show less
In this thesis we study bistable reaction-diffusion equations on lattice domains. The power of reaction-diffusion equations is that they can successfully model various natural and social phenomena... Show moreIn this thesis we study bistable reaction-diffusion equations on lattice domains. The power of reaction-diffusion equations is that they can successfully model various natural and social phenomena with their intuitive and relatively simple (mathematical) representation. One of the main features of reaction-diffusion equations, both on discrete and continuous domains, is that they admit special solutions, so-called ‘travelling waves’, which we can describe as fixed profiles that move in a particular direction with some speed. Depending on their shape, we can roughly divide waves into three categories: pulses or solitons, periodic pulses (wave trains), and monotone wave fronts that connect two constant states. In this thesis we focus on the latter type of wave and we study their existence, propagation and long term behaviour on two type of discrete domains - infinite trees, and two-dimensional square lattices. Show less
The main topic of this PhD thesis is the Arakelov ray class group of a number field, an algebraic object that contains both the ideal class group structure and the unit group structure. The main... Show moreThe main topic of this PhD thesis is the Arakelov ray class group of a number field, an algebraic object that contains both the ideal class group structure and the unit group structure. The main result consists of the fact that certain specific random walks on the Arakelov ray class group result in a target point that is uniformly distributed on this group, under the assumption of an extended version of the Riemann Hypothesis. Almost all other results of this work are consequences of this fact. Show less
This dissertation is a collection of four research articles devoted tothe study of Kummer theory for commutative algebraic groups. In numbertheory, Kummer theory refers to the study of field... Show moreThis dissertation is a collection of four research articles devoted tothe study of Kummer theory for commutative algebraic groups. In numbertheory, Kummer theory refers to the study of field extensions generatedby n-th roots of some base field. Its generalization to commutativealgebraic groups involves fields generated by the division points of afixed algebraic group, such as an elliptic curve or a higher dimensionalabelian variety. Of particular interest in this dissertation is the degreeof such field extensions. In the first two chapter, classical results forelliptic curves are improved by providing explicitly computable bounds anduniform and explicit bounds over the field of rational numbers. In thelast two chapters a general framework for the study of similar problemsis developed. Show less
In this thesis we study the moduli space of genus g curves, and the differential forms that occur naturally on this moduli space. We show that the rings of these tautological differential forms are... Show moreIn this thesis we study the moduli space of genus g curves, and the differential forms that occur naturally on this moduli space. We show that the rings of these tautological differential forms are finite-dimensional, and discuss algorithms that can be used to compute relations among tautological differential forms. Show less
Science is typically a patchwork of research contributions without much coordination. Especially in clinical trials, the follow-up studies that we do fail to be the most promising. They are also... Show moreScience is typically a patchwork of research contributions without much coordination. Especially in clinical trials, the follow-up studies that we do fail to be the most promising. They are also not always designed for the extra evidence that is needed. If they are, standard statistics makes it impossible to take such strategy into account.***This dissertation is about accumulating scientific knowledge, about the statistical problems with existing methods (accumulation bias) and about new statistical methods to do better. We can summarize scientific results efficiently by going ALL-IN. Science is a major gamble: there is little certainty when we embark on a new study. But gambling can be done strategically and clinical trials can use earlier results to decide whether a new study is necessary and optimally designed.***ALL-IN meta-analysis can help scientists to prioritize research, interpret findings in the context of existing results and gamble strategically with their next study. Hence reducing so-called Research Waste. But there is more to it. ALL-IN meta-analysis can be a bottom-up approach. Statistical results become much easier to communicate. Instead of progressing one publication at a time, with everyone focusing on their own paper, clinical science can be more of a continuous collaborative effort. Show less
A tidal basin is an inland sea which is almost entirely enclosed by land and connected to the open sea by a tidal inlet. Through the tidal inlets interaction takes place between the tidal basin and... Show moreA tidal basin is an inland sea which is almost entirely enclosed by land and connected to the open sea by a tidal inlet. Through the tidal inlets interaction takes place between the tidal basin and the open sea. Due to the tide in the open water, the water level of the tidal basin is driven by the water level on the sea. The water flows from the open sea through the tidal inlet to the tidal basin and back. Sand particles are being transported with the water which affects the bottom of the tidal basin and the tidal inlet. The position of the bottom itself also influences the movement of the sand particles. Due to this interaction, interesting bottom patterns with a fractal structure occur. It is these kind of patterns which we analyse in this dissertation. Therefore, we construct an idealised morphodynamic model. With this model, we can find an expression for four important variables: the water level, the water velocity, the sand transport and the height of the bottom. We investigate the existence of these patterns. Also, their sensitivity to variations of geometry and various physical parameters is analysed. We consider tidal inlet system with a various number of tidal inlets. Show less
Tomography is a powerful technique to non-destructively determine the interior structure of an object.Usually, a series of projection images (e.g.\ X-ray images) is acquired from a range of... Show moreTomography is a powerful technique to non-destructively determine the interior structure of an object.Usually, a series of projection images (e.g.\ X-ray images) is acquired from a range of different positions.from these projection images, a reconstruction of the object's interior is computed. Many advanced applications require fast acquisition, effectively limiting the number of projection images and imposing a level of noise on these images. These limitations result in artifacts (deficiencies) in the reconstructed images. Recently, deep neural networks have emerged as a powerful technique to remove these limited-data artifacts from reconstructed images, often outperformingconventional state-of-the-art techniques. To perform this task, the networks are typically trained on a dataset of paired low-quality and high-quality images of similar objects. This is a major obstacle to their use in many practical applications. In this thesis, we explore techniques to employ deep learning in advanced experiments where measuring additional objects is not possible. Show less
Central Limit Theorem associated to a particular case of ergodic total automorphisms. Finally, Chapter 6 proves that the isoperiodic foliation is connected for a meromorphic differential with three... Show moreCentral Limit Theorem associated to a particular case of ergodic total automorphisms. Finally, Chapter 6 proves that the isoperiodic foliation is connected for a meromorphic differential with three poles on a torus. Show less
