Non-linearities and dynamic interactions between state variables are characteristic of complex social systems and processes. In this thesis, we present a new methodology to model these non-linearities and interactions from the large panel datasets available for some of these systems. We build macro-level statistical models that can verify theoretical predictions, and use polynomial basis functions so that each term in the model represents a specific mechanism. This bridges the existing gap between macro-level theories supported by statistical models and micro-level mechanistic models supported by behavioural evidence. We apply this methodology to two important problems in the social sciences, the demographic transition and the transition to democracy. The demographic transition is an important problem for economists and development scientists. Research has shown that economic growth reduces mortality and fertility rates, which reduction in turn results in faster economic growth. We build a non-linear dynamic model and show how this data-driven model extends existing mechanistic models. We also show policy applications for our models, especially in setting development targets for the Millennium Development Goals or the Sustainable Development Goals. The transition to democracy is an important problem for political scientists and sociologists. Research has shown that economic growth and overall human development transforms socio-cultural values and drives political institutions towards democracy. We model the interactions between the state variables and find that changes in institutional freedoms precedes changes in socio-cultural values. We show applications of our models in studying development traps. This thesis comprises the comprehensive summary and seven papers. Papers I and II describe two similar but complementary methodologies to build non-linear dynamic models from panel datasets. Papers III and IV deal with the demographic transition and policy applications. Papers V and VI describe the transition to democracy and applications. Paper VII describes an application to sustainable development.

It has been suggested that Patellofemoral Pain (PFP) may be the result of a coordinate state which exhibits less joint coordination variability. The ability to relate joint coordination variability to PFP pathology could have many clinical uses; however, evidence to support clinical application is lacking. Vector coding’s coupling angle variability (CAV) has been introduced as a possible analysis method to quantify joint coordination variability. The purpose of this study was to assess the clinical usefulness of CAV measures from a dynamical systems perspective. This involved establishing the precision limits of CAV measures when physiological conditions are held constant, altering control parameters of knee pain and population then determining if the observed changes in CAV were clinically meaningful. 20 female recreational runners with PFP and 21 healthy controls performed a treadmill acclimation protocol then ran at a self-selected pace for 15 minutes. 3-D kinematics, force plate kinetics, knee pain and perceived exertion were recorded each minute. CAV were calculated for six knee-ankle combinations for 2 sets of 5 non-consecutive stride cycles at each capture period. Data were selected for the PFP group at a high (=>3) and low (<=high-2) pain level in a non-exhausted state (<14). Healthy data were used from the 11th minute of the running. Levels of agreement were performed between the 2 sets of CAV measures for both populations, a paired t-test compared low to high pain CAV measures and independent t-tests compared populations at the high pain state. Several CAV measures showed a significant increase in value with an increase in pain and were significantly greater for the PFP group. None of the observed changes exceeded the precision limits of all CAV measures investigated. These results do not agree with previous claims that less variability is indicative of pathology but rather the opposite. This suggests that there might be an optimal amount of variability to maintain a healthy coordinate state with deviations in any direction being detrimental. However; due to the volatile nature of CAV measures, the clinical use of CAV is not recommended using current analysis methods since changes observed weren’t considered clinically meaningful.

We examine a class of planar area preserving mappings and give a geometric condition that guarantees the existence of homoclinic points. To be more precise, let $f,g:R \to R$ be $C^1$ functions with domain all of $R$. Let $F:R^2 \to R^2$ denote a horizontal reflection in the curve $x=-f(y)$, and let $G:R^2 \to R^2$ denote a vertical reflection in the curve $y=g(x)$. We consider maps of the form $T=G \circ F$ and show that a simple geometric condition on the fixed point sets of $F$ and $G$ leads to the existence of a homoclinic point for $T$. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2013-09-17 14:22:35.72

We study evolutionary models of financial markets. In particular, we study an evolutionary market model with short-lived assets and an evolutionary model with long-lived assets. In the long-lived asset market, investors are allowed to use general dynamic investment strategies. We find sufficient conditions for the Kelly portfolio rule to dominate the market exponentially fast. Moreover, when investors use simple strategies but have incorrect beliefs, we show that the strategy which is "closer" to the Kelly rule cannot be driven out of the market. This means that this strategy will either dominate or at least survive, i.e., the relative market share does not converge to zero. In the market with short-lived assets, we study the dynamics when the states of the world are not identically distributed. This marks the first attempt to study the dynamics of the market when the probability of success changes according to the relative shares of investors. In this problem, we first study a skew product of the random dynamical system associates with the market dynamics. In particular, we compute the Lyapunov exponents of the skew product. This enables us to produce a "surviving" investment strategy, i.e., the investor who follows this rule will dominate the market or at least survive. All the mathematical tools in the thesis lie within the framework of random dynamical systems.

We have used the Spitzer Space Telescope to observe two transiting planetary systems orbiting low-mass stars discovered in the Kepler K2 mission. The system K2-3 (EPIC 201367065) hosts three planets, while K2-26 (EPIC 202083828) hosts a single planet. Observations of all four objects in these two systems confirm and refine the orbital and physical parameters of the planets. The refined orbital information and more precise planet radii possible with Spitzer will be critical for future observations of these and other K2 targets. For K2-3b we find marginally significant evidence for a transit timing variation between the K2 and Spitzer epochs.

Nous proposons dans ces travaux des algorithmes distribués de clusterisation destinés à répondre à la problématique de la croissance des réseaux. Après avoir donné une spécification pour ce problème, nous fournissons un premier algorithme distribué à base de marches aléatoires pour le résoudre. Cet algorithme n’utilise que des informations locales, et utilise des marches aléatoires pour construire en parallèle des ensembles connexes de noeuds appelés les coeurs des clusters, auxquels on ajoute des noeuds adjacents. La taille de chaque coeur est comprise entre 2 et un paramètre de l’algorithme. L’algorithme garantit que si deux clusters sont adjacents, au moins l’un d’entre eux a un coeur de taille maximale. Un deuxième algorithme, adaptatif à la mobilité, garantit en plus de ces propriétés que la reconstruction consécutive à un changement topologique est locale. Cette propriété différencie notre solution des nombreuses solutions existantes : elle permet d’éviter des destructions en chaîne suite à un changement de topologie. Nous présentons enfin un algorithme de clustering auto-stabilisant qui conserve les propriétés des algorithmes précédents en y ajoutant la tolérance aux pannes. Grâce au parallélisme de la construction des clusters et au caractère local des reconstructions de clusters, ces algorithmes passent à l'échelle, ce qui est confirmé par les simulations que nous avons menées. / We propose in this work distributed clustering algorithms designed to address the problem of growing networks. After giving a specification for this problem, we provide a first distributed algorithm based on random walks to solve it. This algorithm uses only local information,and uses random walks to build connected sets of nodes called cores of clusters in parallel, to which we add adjacent nodes. The size of each core is between 2 and a parameter of the algorithm. The algorithm guarantees that if two clusters are adjacent, at least one of them has a core of maximum size. A second, mobility-adaptive, algorithm ensures, besides those properties, that the reconfiguration following a topological change is local. This property differentiates our solution from many solutions : it avoids chain destruction following a topology change. Finally, we present a self-stabilizing clustering algorithm that preserves the properties of previous algorithms and adds fault tolerance. With the parallel construction of clusters and the local nature of the reconstruction of clusters, these algorithms guarantee the scabability, which is confirmed by simulations.

Since the 1980s, much progress has been done in completely determining which functions share a Julia set. The polynomial case was completely solved in 1995, and it was shown that the symmetries of the Julia set play a central role in answering this question. The rational case remains open, but it was already shown to be much more complex than the polynomial one. In this thesis, we review existing results on rational maps sharing a Julia set, and offer results of our own on the symmetry group of such maps. / Desde a década de oitenta, um enorme progresso foi feito no problema de determinar quais funções têm o mesmo conjunto de Julia. O caso polinomial foi completamente respondido em 1995, e mostrou-se que as simetrias do conjunto de Julia têm um papel central nessa questão. O caso racional permanece aberto, mas já se sabe que ele é muito mais complexo do que o polinomial. Nesta dissertação, nós revisamos resultados existentes sobre aplicações racionais com o mesmo conjunto de Julia e apresentamos nossos próprios resultados sobre o grupo de simetrias de tais aplicações.

Nesta tese de doutoramento nós tivemos como principal objetivo desenvolver novos métodos para proteção da informação e computação quântica. Começamos, de forma introdutória, ilustrando os conceitos básicos e fundamentais da teoria da informação e computação quântica, como os bits quânticos (qubits), o operador densidade, o emaranhamento e as operações lógicas quânticas. Na seqüência, apresentamos os formalismos utilizados para tratar sistemas abertos, ou seja, sujeitos a erros, além das principais técnicas existentes a fim de proteger a informação quântica, como os códigos de correção de erros, os subespaços livres de erros e o desacoplamento dinâmico. Finalmente, baseando-nos na técnica de desacoplamento dinâmico, introduzimos um esquema de proteção para operações lógicas quânticas e o emaranhamentos entre qubits utilizando campos de alta freqüência. Ilustramos em detalhes a proteção da operação lógica quântica de Hadamard e do emaranhamento entre dois qubits, além de apresentarmos as principais diferenças e vantagens de nosso método quando comparado às técnicas tradicionais de desacoplamento dinâmico. / The main objective of this thesis is the development of a new procedure for quantum information and computation protection. We begin by briefly illustrating the basic concepts of quantum information and computation theory, such as quantum bits (qubits), density matrix operator, entanglement, and quantum logical operations. Subsequently, we present the formalism utilized to treat quantum open systems, i.e., systems subjected to errors, and the main strategies to protect quantum information, such as quantum error correction codes, decoherence-free subspaces, and dynamical decoupling. Finally, based on the dynamical decoupling strategies, we introduce a procedure to protect quantum logical operations and entanglement utilizing high-frequency continuous fields. We illustrate, in details, the protection of a Hadamard quantum gate and of entanglement between two qubits, and present the differences and advantages of our procedure when compared with traditional techniques of dynamical decoupling.

Investigou-se a gênese de atividade irregular em neurônios de centros geradores de padrões através de modelos eletrofisiologicamente realistas. Para tanto, foram adotadas abordagens paralelas. Primeiramente, desenvolveram-se técnicas para determinar quais os mecanismos biofísicos subjacentes aos processos de codificação de informação nestas células. Também foi proposta uma nova metodologia híbrida (baseada em continuação numérica e em varreduras força bruta) para análise de bancos de dados de modelos neuronais, permitindo estendê-los e revelar instâncias de multiestabilidade entre regimes oscilatórios e quiescentes. Além disto, a fim de determinar a origem de comportamento complexo em modelos neuronais simplificados, empregaram-se métodos geométricos da teoria de sistemas dinâmicos. A partir da análise de mapas unidimensionais perturbados por ruído, foram discutidos possíveis cenários para o surgimento de caos em sistemas dinâmicos aleatórios. Finalmente mostrou-se que, levando em conta o ruído, uma classe de modelos de condutâncias reproduz padrões de disparo observados in vivo. Estas pertubações revelam a riqueza da dinâmica transiente, levando o sistema a visitar um arcabouço determinista complexo preexistente -- sem recorrer a ajustes finos de parâmetros ou a construções ad hoc para induzir comportamento caótico. / We investigated the origin of irregularities in the dynamics of central pattern generator neurons, through analyzing electrophysiologically realistic models. A number of parallel approaches were adopted for that purpose. Initially, we studied information coding processes in these cells and proposed a technique to determine the underlying biophysical mechanisms. We also developed a novel hybrid method (based on numerical continuation and brute force sweeps) to analyze neuronal model databases, extending them and unveiling instances of multistability between oscillatory and resting regimes. Furthermore, in order to determine the origin of irregular dynamics in simplified neuronal models, we employed geometrical methods from the theory of dynamical systems. The analysis of stochastically perturbed maps allowed us to discuss possible scenarios for the generation of chaotic behaviour in random dynamical systems. Finally we showed that, by taking noise into account, a class of conductance based models gives rise to firing patterns akin to the ones observed \\emph{in vivo}. These perturbations unveil the richness of the transient dynamics, inducing the system to populate a preexistent complex deterministic scaffolding -- without resorting to parameter fine-tuning or ad hoc constructions to induce chaotic activity.

Ban and Kalies [3] proposed an algorithmic approach to compute attractor- repeller pairs and weak Lyapunov functions based on a combinatorial multivalued mapping derived from an underlying dynamical system generated by a continuous map. We propose a more e cient way of computing a Lyapunov function for a Morse decomposition. This combined work with other authors, including Shaun Harker, Arnoud Goulet, and Konstantin Mischaikow, implements a few techniques that makes the process of nding a global Lyapunov function for Morse decomposition very e - cient. One of the them is to utilize highly memory-e cient data structures: succinct grid data structure and pointer grid data structures. Another technique is to utilize Dijkstra algorithm and Manhattan distance to calculate a distance potential, which is an essential step to compute a Lyapunov function. Finally, another major technique in achieving a signi cant improvement in e ciency is the utilization of the lattice structures of the attractors and attracting neighborhoods, as explained in [32]. The lattice structures have made it possible to let us incorporate only the join-irreducible attractor-repeller pairs in computing a Lyapunov function, rather than having to use all possible attractor-repeller pairs as was originally done in [3]. The distributive lattice structures of attractors and repellers in a dynamical system allow for general algebraic treatment of global gradient-like dynamics. The separation of these algebraic structures from underlying topological structure is the basis for the development of algorithms to manipulate those structures, [32, 31]. There has been much recent work on developing and implementing general compu- tational algorithms for global dynamics which are capable of computing attracting neighborhoods e ciently. We describe the lifting of sublattices of attractors, which are computationally less accessible, to lattices of forward invariant sets and attract- ing neighborhoods, which are computationally accessible. We provide necessary and su cient conditions for such a lift to exist, in a general setting. We also provide the algorithms to check whether such conditions are met or not and to construct the lift when they met. We illustrate the algorithms with some examples. For this, we have checked and veri ed these algorithms by implementing on some non-invertible dynamical systems including a nonlinear Leslie model. / Includes bibliography. / Dissertation (Ph.D.)--Florida Atlantic University, 2016. / FAU Electronic Theses and Dissertations Collection