Keller-Segel system, chemotaxis and applications to a mathematical model for Alzheimer's disease

In questa tesi viene presentato il modello di Keller-Segel per la chemiotassi, un sistema di tipo parabolico-ellittico che appare nella descrizione di molti fenomeni in ambito biologico e medico. Viene mostrata l'esistenza globale della soluzione debole del modello, per dati iniziali sufficientemente piccoli in dimensione N>2. La scelta di dati iniziali abbastanza grandi invece può causare il blow-up della soluzione e viene mostrato sotto quali condizioni questo si verifica. Infine il modello della chemiotassi è stato applicato per descrivere una fase della malattia di Alzheimer ed è stata effettuata un'analisi di stabilità del sistema.

Geometrico-static modelling of continuum parallel robots

In this thesis, we explore three methods for the geometrico-static modelling of continuum parallel robots. Inspired by biological trunks, tentacles and snakes, continuum robot designs can reach confined spaces, manipulate objects in complex environments and conform to curvilinear paths in space. In addition, parallel continuum manipulators have the potential to inherit some of the compactness and compliance of continuum robots while retaining some of the precision, stability and strength of rigid-links parallel robots. Subsequently, the foundation of our work is performed on slender beam by applying the Cosserat rod theory, appropriate to model continuum robots. After that, three different approaches are developed on a case study of a planar parallel continuum robot constituted of two connected flexible links. We solve the forward and inverse geometrico-static problem namely by using (a) shooting methods to obtain a numerical solution, (b) an elliptic method to find a quasi-analytical solution, and (c) the Corde model to perform further model analysis. The performances of each of the studied methods are evaluated and their limits are highlighted. This thesis is divided as follows. Chapter one gives the introduction on the field of the continuum robotics and introduce the parallel continuum robots that is studied in this work. Chapter two describe the geometrico-static problem and gives the mathematical description of this problem. Chapter three explains the numerical approach with the shooting method and chapter four introduce the quasi-analytical solution. Then, Chapter five introduce the analytic method inspired by the Corde model and chapter six gives the conclusions of this work.

A next-to-leading order calculation of the impact of primordial magnetic fields into cosmological observables

In this thesis, we perform a next-to-leading order calculation of the impact of primordial magnetic fields (PMF) into the evolution of scalar cosmological perturbations and the cosmic microwave background (CMB) anisotropy. Magnetic fields are everywhere in the Universe at all scales probed so far, but their origin is still under debate. The current standard picture is that they originate from the amplification of initial seed fields, which could have been generated as PMFs in the early Universe. The most robust way to test their presence and constrain their features is to study how they impact on key cosmological observables, in particular the CMB anisotropies. The standard way to model a PMF is to consider its contribution (quadratic in the magnetic field) at the same footing of first order perturbations, under the assumptions of ideal magneto-hydrodynamics and compensated initial conditions. In the perspectives of ever increasing precision of CMB anisotropies measurements and of possible uncounted non-linear effects, in this thesis we study effects which go beyond the standard assumptions. We study the impact of PMFs on cosmological perturbations and CMB anisotropies with adiabatic initial conditions, the effect of Alfvén waves on the speed of sound of perturbations and possible non-linear behavior of baryon overdensity for PMFs with a blue spectral index, by modifying and improving the publicly available Einstein-Boltzmann code SONG, which has been written in order to take into account all second-order contributions in cosmological perturbation theory. One of the objectives of this thesis is to set the basis to verify by an independent fully numerical analysis the possibility to affect recombination and the Hubble constant.