Dynamical maps for magnetic components are fundamental to studies of beam dynamics in accelerators. However, it is usually not possible to write down maps in closed form for anything other than simplified models of standard accelerator magnets. In the work presented here, the magnetic field is expressed in analytical form obtained from fitting Fourier series to a 3D numerical solution of Maxwell’s equations. Dynamical maps are computed for a particle moving through this field by applying a second order (with the paraxial approximation) explicit symplectic integrator. These techniques are used to study the beam dynamics in the first non-scaling FFAG ever built, EMMA, especially challenging regarding the validity of the paraxial approximation for the large excursion of particle trajectories. The EMMA lattice has four degrees of freedom (strength and transverse position of each of the two quadrupoles in each periodic cell). Dynamical maps, computed for a set of lattice configurations, may be efficiently used to predict the dynamics in any lattice configuration. We interpolate the coefficients of the generating function for the given configuration, ensuring the symplecticity of the solution. An optimisation routine uses this tool to look for a lattice defined by four constraints on the time of flight at different beam energies. This provides a way to determine the tuning of the lattice required to produce a desired variation of time of flight with energy, which is one of the key characteristics for beam acceleration in EMMA. These tools are then benchmarked against data from the recent EMMA commissioning.

We study planar systems of difference equations and applications to biological models of species populations. Central to the analysis of this study is the idea of folding - the method of transforming systems of difference equations into higher order scalar difference equations. Two classes of second order equations are studied: quadratic fractional and exponential. We investigate the boundedness and persistence of solutions, the global stability of the positive fixed point and the occurrence of periodic solutions of the quadratic rational equations. These results are applied to a class of linear/rational systems that can be transformed into a quadratic fractional equation via folding. These results apply to systems with negative parameters, instances not commonly considered in previous studies. We also identify ranges of parameter values that provide sufficient conditions on existence of chaotic and multiple stable orbits of different periods for the planar system. We study a second order exponential difference equation with time varying parameters and obtain sufficient conditions for boundedness of solutions and global convergence to zero. For the autonomous case, we show occurrence of multistable periodic and nonperiodic orbits. For the case where parameters are periodic, we show that the nature of the solutions differs qualitatively depending on whether the period of the parameters is even or odd. The above results are applied to biological models of populations. We investigate a broad class of planar systems that arise in the study of stage-structured single species populations. In biological contexts, these results include conditions on extinction or survival of the species in some balanced form, and possible occurrence of complex and chaotic behavior. Special rational (Beverton-Holt) and exponential (Ricker) cases are considered to explore the role of inter-stage competition, restocking strategies, as well as seasonal fluctuations in the vital rates.

Spectral and wave function statistics of the quantum directed graph, QdG, are studied. The crucial feature of this model is that the direction of a bond (arc) corresponds to the direction of the waves propagating along it. We pay special attention to the full Neumann digraph, FNdG, which consists of pairs of antiparallel arcs between every node, and differs from the full Neumann graph, FNG, in that the two arcs have two incommensurate lengths. The spectral statistics of the FNG (with incommensurate bond lengths) is believed to be universal, i.e. to agree with that of the random matrix theory, RMT, in the limit of large graph size. However, the standard perturbative treatment of the field theoretical representation of the 2-point correlation function [1, 2] for a FNG, does not account for this behaviour. The nearest-neighbor spacing distribution of the closely related FNdG is studied numerically. An original, efficient algorithm for the generation of the spectrum of large graphs allows for the observation that the distribution approaches indeed universality at increasing graph size (although the convergence cannot be ascertained), in particular "level repulsion" is confirmed. The numerical technique employs a new secular equation which generalizes the analogous object known for undirected graphs [3, 4], and is based on an adaptation to digraphs of the idea of wave function continuity. In view of the contradiction between the field theory [2] and the strong indications of universality, a non-perturbative approach to analysing the universal limit is presented. The substitution of the FNG by the FNdG results in a field theory with fewer degrees of freedom. Despite this simplification, the attempt is inconclusive. Possible applications of this approach are suggested. Regarding the wave function statistics, a field theoretical representation for the spectral average of the wave intensity on an fixed arc is derived and studied in the universal limit. The procedure originates from the study of wave function statistics on disordered metallic grains [5] and is used in conjunction with the field theory approach pioneered in [2].

Dispersed particle flows are encountered in many biological, geophysical but also in industrial situations, e.g. during processing of materials. In these flows, the particles usually are non-spherical and their angular dynamics play a crucial role for the final material properties. Generally, the angular dynamics of a particle is dependent on the local flow in the frame-of-reference of this particle. In this frame, the surrounding flow can be linearized and the linear velocity gradient will determine how the particle rotates. In this thesis, the main objective is to improve the fundamental knowledge of the angular dynamics of non-spherical particles related to two specific biobased material processes. Firstly, the flow of suspended cellulose fibers in a papermaking process is used as a motivation. In this process, strong shear rates close to walls and the size of the fibers motivates the study of inertial effects on a single particle in a simple shear flow. Through direct numerical simulations combined with a global stability analysis, this flow problem is approached and all stable rotational states are found for spheroidal particles with aspect ratios ranging from moderately slender fibers to thin disc-shaped particles. The second material process of interest is the production of strong cellulose filaments produced through hydrodynamic alignment and assembly of cellulose nanofibrils (CNF). The flow in the preparation process and the small size of the particles motivates the study of alignment and rotary diffusion of CNF in a strain flow. However, since the particles are smaller than the wavelength of visible light, the dynamics of CNF is not easily captured with standard optical techniques. With a new flow-stop experiment, rotary diffusion of CNF is measured using Polarized optical microscopy. This process is found to be quite complicated, where short-range interactions between fibrils seem to play an important role. New time-resolved X-ray characterization techniques were used to target the underlying mechanisms, but are found to be limited by the strong degradation of CNF due to the radiation. Although the results in this thesis have limited direct applicability, they provide important fundamental stepping stones towards the possibility to control fiber orientation in flows and can potentially lead to new tailor-made materials assembled from a nano-scale. / <p>QC 20160929</p>

Este trabalho apresenta conceitos e procedimentos que devem ser considerados na análise dinâmica de fundações de máquinas. São discutidas as abordagens e os métodos de cálculo à disposição dos engenheiros, assim como os fatores que influenciam a escolha e o desenvolvimento de modelos matemáticos adequados. Três tipos diferentes de fundação (fundações em bloco rígido, fundações de martelos e fundações aporticadas) são examinados, e são utilizados exemplos para demonstrar a eficácia de técnicas simples de modelagem. / This paper presents concepts and procedures necessary to the dynamic analysis of machine foundations. Analytical approaches available to the designer, and factors which influence the choice and development of a suitable analytical model are discussed. Three different foundations (block foundations, hammer foundations and framed foundations) are examined and examples are given to demonstrate the effectiveness of simple modeling techniques.

The two variable Fitzhugh Nagumo model behaves qualitatively like the four variable Hodgkin-Huxley space clamped system and is more mathematically tractable than the Hodgkin Huxley model, thus allowing the action potential and other properties of the Hodgkin Huxley system to be more readily be visualized. In this thesis, it is shown that the Carleman Embedding Technique can be applied to both the Fitzhugh Nagumo model and to Van der Pol's model of nonlinear oscillation, which are both finite nonlinear systems of differential equations. The Carleman technique can thus be used to obtain approximate solutions of the Fitzhugh Nagumo model and to study neural activity such as excitability.

In Reaction-Diffusion systems, some parameters can influence the behavior of other parameters in that system. Thus reaction diffusion equations are often used to model the behavior of biological phenomena. The Fitzhugh Nagumo partial differential equation is a reaction diffusion equation that arises both in population genetics and in modeling the transmission of action potentials in the nervous system. In this paper we are interested in finding solutions to this equation. Using Lie groups in particular, we would like to find symmetries of the Fitzhugh Nagumo equation that reduce this non-linear PDE to an Ordinary Differential Equation. In order to accomplish this task, the non-classical method is utilized to find the infinitesimal generator and the invariant surface condition for the subgroup where the solutions for the desired PDE exist. Using the infinitesimal generator and the invariant surface condition, we reduce the PDE to a mildly nonlinear ordinary differential equation that could be explored numerically or perhaps solved in closed form.

Iterative methods have been a very important area of study in numerical analysis since the inception of computational science. Their use ranges from solving algebraic equations to systems of differential equations and many more. In this thesis, we discuss several iterative methods, however our main focus is Newton's method. We present a detailed study of Newton's method, its order of convergence and the asymptotic error constant when solving problems of various types as well as analyze several pitfalls, which can affect convergence. We also pose some necessary and sufficient conditions on the function f for higher order of convergence. Different acceleration techniques are discussed with analysis of the asymptotic behavior of the iterates. Analogies between single variable and multivariable problems are detailed. We also explore some interesting phenomena while analyzing Newton's method for complex variables.

The goal of this thesis is to study the dynamics behind the evolution of virulence. We examine first the underlying mechanics of linear systems of ordinary differential equations by investigating the classification of fixed points in these systems, then applying these techniques to nonlinear systems. We then seek to establish the validity of a system that models the population dynamics of uninfected and infected hosts---first with one parasite strain, then n strains. We define the basic reproductive ratio of a parasite, and study its relationship to the evolution of virulence. Lastly, we investigate the mathematics behind superinfection.

Σκοπός της παρούσας διπλωματικής εργασίας ήταν η μελέτη ηλεκτροεγκεφαλογραφικών σημάτων επιληψίας με χρήση συνδεδεμένων χαοτικών συστημάτων. Στο πλαίσιο της εργασίας παρουσιάζονται αρχικά μέθοδοι συμφωνίας και οι διάφορες μορφές της συνάρτησης μεταφοράς πληροφορίας. Οι τεχνικές που έχουν χρησιμοποιηθεί κυρίως μέχρι σήμερα είναι γραμμικές ωστόσο έχει φανεί σε αρκετές περιπτώσεις ότι η γραμμικότητα δεν επαρκή για την εξήγηση των σημάτων ΗΕΓ και της σύζευξης μεταξύ τους. Για τον λόγο αυτόν αναπτύσσεται η μη γραμμική μέθοδος υπολογισμού της συνάρτησης μερικής κατευθυνόμενης συμφωνίας η οποία μελετά τα καταγεγραμμένα σήματα στον χώρο φάσης-κατάστασης. Βάση αυτής διερευνάται η μη γραμμική συνδεσιμότητα περιοχών του εγκεφάλου. Η εργασία αποτελείται από πέντε μέρη. Το πρώτο μέρος περιλαμβάνει τη παρουσίαση της φυσιολογίας του ανθρώπινου εγκεφάλου. Στο δεύτεροο μέρος παρουσιάζεται η ασθένεια της επιληψίας καθώς και οι διάφοροι τύποι της. Στο τρίτο μέρος παρουσιάζεται η μέθοδος της μη γραμμικής ανάλυσης χρονοσειρών μέσω της μερικής κατευθυνόμενης συμφωνίας που περιλαμβάνει την ανακατασκευή των σημάτων στον φασικό χώρο. Ακολουθεί ο υπολογισμός των κατάλληλων παραμέτρων για την σωστή ανακατασκευή του φασικού χώρου των σημάτων. Τέλος, στο πέμπτο κεφάλαιο παρουσιάζονται αποτελέσματα από εφαρμογή της μεθόδου σε προσομοιωμένα δεδομένα,καθώς και σε πραγματικά δεδομένα από ασθενή με επιληψία, τα οποία έχουν ληφθεί από το νοσοκομείο Ευαγγελισμός. / This diploma thesis aim at studying Electroencephalografic (EEG) Signal Recordings by adopting methodologies able to analyse and observe coupling of chaotic systems. Although linear methods seems to be inadequate for the analysis of EEG signals, the most commonly used methodologies today are linear. In this thesis, a non-linear partial directed coherence method is adopted to compute the transfer function of EEG signals in the phase-state space and is used to estimate the non-linear connectivity among brain areas. This thesis consists of five chapters. In the first and second chapter, an introduction to the brain's physiology and epilepsy pathophysiology is presented. In the third chapter, a methodology for the non-linear analysis of time series is presented based on the PDC method, which reconstructs attractors in the phase-state. In the fourth chapter, the parameters for the phase-state reconstruction of the attractors are properly selected. In the fifth chapter, the proposed method is applied on simulated and real epilepsy EEG data and the obtained results are presented and discussed.