The aim of this thesis is to develop the dimension theory of self-affine carpets in several directions. Self-affine carpets are an important class of planar self-affine sets which have received a great deal of attention in the literature on fractal geometry over the last 30 years. These constructions are important for several reasons. In particular, they provide a bridge between the relatively well-understood world of self-similar sets and the far from understood world of general self-affine sets. These carpets are designed in such a way as to facilitate the computation of their dimensions, and they display many interesting and surprising features which the simpler self-similar constructions do not have. For example, they can have distinct Hausdorff and packing dimensions and the Hausdorff and packing measures are typically infinite in the critical dimensions. Furthermore, they often provide exceptions to the seminal result of Falconer from 1988 which gives the `generic' dimensions of self-affine sets in a natural setting. The work in this thesis will be based on five research papers I wrote during my time as a PhD student. The first contribution of this thesis will be to introduce a new class of self-affine carpets, which we call box-like self-affine sets, and compute their box and packing dimensions via a modified singular value function. This not only generalises current results on self-affine carpets, but also helps to reconcile the `exceptional constructions' with Falconer's singular value function approach in the generic case. This will appear in Chapter 2 and is based on a paper which appeared in 'Nonlinearity' in 2012. In Chapter 3 we continue studying the dimension theory of self-affine sets by computing the Assouad and lower dimensions of certain classes. The Assouad and lower dimensions have not received much attention in the literature on fractals to date and their importance has been more related to quasi-conformal maps and embeddability problems. This appears to be changing, however, and so our results constitute a timely and important contribution to a growing body of literature on the subject. The material in this Chapter will be based on a paper which has been accepted for publication in 'Transactions of the American Mathematical Society'. In Chapters 4-6 we move away from the classical setting of iterated function systems to consider two more exotic constructions, namely, inhomogeneous attractors and random 1-variable attractors, with the aim of developing the dimension theory of self-affine carpets in these directions. In order to put our work into context, in Chapter 4 we consider inhomogeneous self-similar sets and significantly generalise the results on box dimensions obtained by Olsen and Snigireva, answering several questions posed in the literature in the process. We then move to the self-affine setting and, in Chapter 5, investigate the dimensions of inhomogeneous self-affine carpets and prove that new phenomena can occur in this setting which do not occur in the setting of self-similar sets. The material in Chapter 4 will be based on a paper which appeared in 'Studia Mathematica' in 2012, and the material in Chapter 5 is based on a paper, which is in preparation. Finally, in Chapter 6 we consider random self-affine sets. The traditional approach to random iterated function systems is probabilistic, but here we allow the randomness in the construction to be provided by the topological structure of the sample space, employing ideas from Baire category. We are able to obtain very general results in this setting, relaxing the conditions on the maps from `affine' to `bi-Lipschitz'. In order to get precise results on the Hausdorff and packing measures of typical attractors, we need to specialise to the setting of random self-similar sets and we show again that several interesting and new phenomena can occur when we relax to the setting of random self-affine carpets. The material in this Chapter will be based on a paper which has been accepted for publication by 'Ergodic Theory and Dynamical Systems'.

In the first part of this thesis, suitable function spaces for analysis of partial differ- ential equations in unbounded domains are introduced and studied. The results are then applied in the second part on semilinear wave equation in Rd with non- linear source term and nonlinear damping. The source term is supposed to be bounded by a polynomial function with a subcritical growth. The damping term is strictly monotone and satisfying a polynomial-like growth condition. Global existence is proved using finite speed of propagation. Dissipativity in locally uni- form spaces and the existence of a locally compact attractor are then obtained after additional conditions imposed on the damping term.

Cortical and subcortical microcircuits are continuously modified throughout life. Despite ongoing changes these networks stubbornly maintain their functions, which persist although destabilizing synaptic and nonsynaptic mechanisms should ostensibly propel them towards runaway excitation or quiescence. What dynamical phenomena exist to act together to balance such learning with information processing? What types of activity patterns do they underpin, and how do these patterns relate to our perceptual experiences? What enables learning and memory operations to occur despite such massive and constant neural reorganization? Progress towards answering many of these questions can be pursued through large-scale neuronal simulations.    In this thesis, a Hebbian learning rule for spiking neurons inspired by statistical inference is introduced. The spike-based version of the Bayesian Confidence Propagation Neural Network (BCPNN) learning rule involves changes in both synaptic strengths and intrinsic neuronal currents. The model is motivated by molecular cascades whose functional outcomes are mapped onto biological mechanisms such as Hebbian and homeostatic plasticity, neuromodulation, and intrinsic excitability. Temporally interacting memory traces enable spike-timing dependence, a stable learning regime that remains competitive, postsynaptic activity regulation, spike-based reinforcement learning and intrinsic graded persistent firing levels.    The thesis seeks to demonstrate how multiple interacting plasticity mechanisms can coordinate reinforcement, auto- and hetero-associative learning within large-scale, spiking, plastic neuronal networks. Spiking neural networks can represent information in the form of probability distributions, and a biophysical realization of Bayesian computation can help reconcile disparate experimental observations. / <p>QC 20170421</p>

Les primates doivent pouvoir reconnaître de nouvelles situations pour pouvoir s'y adapter. La représentation de ces situations dans l'activité du cortex est le sujet de cette thèse. Les situations complexes s'expliquent souvent par l'interaction entre des informations sensorielles, internes et motrices. Des activités unitaires dénommées sélectivité mixte, qui sont très présentes dans le cortex préfrontal (CPF), sont un mécanisme possible pour représenter n'importe quelle interaction entre des informations. En parallèle, le Reservoir Computing a démontré que des réseaux récurrents ont la propriété de recombiner des entrées actuelles et passées dans un espace de plus haute dimension, fournissant ainsi un pré-codage potentiellement universel de combinaisons pouvant être ensuite sélectionnées et utilisées en fonction de leur pertinence pour la tâche courante. En combinant ces deux approches, nous soutenons que la nature fortement récurrente de la connectivité locale du CPF est à l'origine d'une forme dynamique de sélectivité mixte. De plus, nous tentons de démontrer qu'une simple régression linéaire, implémentable par un neurone seul, peut extraire n'importe qu'elle information/contingence encodée dans ces combinaisons complexes et dynamiques. Finalement, les entrées précédentes, qu'elles soient sensorielles ou motrices, à ces réseaux du CPF doivent être maintenues pour pouvoir influencer les traitements courants. Nous soutenons que les représentations de ces contextes définis par ces entrées précédentes doivent être exprimées explicitement et retournées aux réseaux locaux du CPF pour influencer les combinaisons courantes à l'origine de la représentation des contingences / In order to adapt to new situations, primates must be able to recognize these situations. How the cortex represents contingencies in its activity is the main subject of this thesis. First, complex new situations are often explained by the interaction between sensory, internal and motor information. Recent studies have shown that single-neuron activities referred to as mixed selectivity which are ubiquitous in the prefrontal cortex (PFC) are a possible mechanism to represent arbitrary interaction between information defining a contingency. In parallel, a recent area of reasearch referred to as Reservoir Computing has demonstrated that recurrent neural networks have the property of recombining present and past inputs into a higher dimensional space thereby providing a pre-coding of an essentially universal set of combinations which can then be selected and used arbitrarily for their relevance to the task at hand. Combining these two approaches we argue that the highly recurrent nature of local prefrontal connectivity is at the origin of dynamic form of mixed selectivity. Also, we attempt to demonstrate that a simple linear regression, implementable by a single neuron, can extract any information/ contingency encoded in these highly complex and dynamic combinations. In addition, previous inputs, whether sensory or motor, to these PFC networks must be maintained in order to influence current processing and behavioral demand. We argue that representations of contexts defined by these past inputs must be expressed explicitely and fed back to the local PFC networks in order to influence the current combinations at the origin of contingencies representation

L'avènement récent des procédures expérimentales à haut débit a ouvert une nouvelle ère pour l'étude quantitative des systèmes biologiques. De nos jours, les enregistrements d'électrophysiologie et l'imagerie du calcium permettent l'enregistrement simultané in vivo de centaines à des milliers de neurones. Parallèlement, grâce à des procédures de séquençage automatisées, les bibliothèques de protéines fonctionnelles connues ont été étendues de milliers à des millions en quelques années seulement. L'abondance actuelle de données biologiques ouvre une nouvelle série de défis aux théoriciens. Des méthodes d’analyse précises et transparentes sont nécessaires pour traiter cette quantité massive de données brutes en observables significatifs. Parallèlement, l'observation simultanée d'un grand nombre d'unités en interaction permet de développer et de valider des modèles théoriques visant à la compréhension mécanistique du comportement collectif des systèmes biologiques. Dans ce manuscrit, nous proposons une approche de ces défis basée sur des méthodes et des modèles issus de la physique statistique, en développent et appliquant ces méthodes au problèmes issu de la neuroscience et de la bio-informatique : l’étude de la mémoire spatiale dans le réseau hippocampique, et la reconstruction du paysage adaptatif local d'une protéine. / The recent advent of high-throughput experimental procedures has opened a new era for the quantitative study of biological systems. Today, electrophysiology recordings and calcium imaging allow for the in vivo simultaneous recording of hundreds to thousands of neurons. In parallel, thanks to automated sequencing procedures, the libraries of known functional proteins expanded from thousands to millions in just a few years. This current abundance of biological data opens a new series of challenges for theoreticians. Accurate and transparent analysis methods are needed to process this massive amount of raw data into meaningful observables. Concurrently, the simultaneous observation of a large number of interacting units enables the development and validation of theoretical models aimed at the mechanistic understanding of the collective behavior of biological systems. In this manuscript, we propose an approach to both these challenges based on methods and models from statistical physics. We present an application of these methods to problems from neuroscience and bioinformatics, focusing on (1) the spatial memory and navigation task in the hippocampal loop and (2) the reconstruction of the fitness landscape of proteins from homologous sequence data.

Neste texto é feita uma discussão sobre alguns resultados que fornecem condições suficientes para que um difeomorfismo local, do espaço euclideano n-dimensional nele próprio, seja injetivo. Dentro deste cenário, são exploradas as contribuições destes resultados na tentativa de solucionar conhecidas conjecturas no meio científico como a Conjectura Jacobiana e a Conjectura de Ponto Fixo. Do ponto de vista dinâmico, existem relações entre injetividade global e estabilidade assintótica global. Neste sentido, os resultados também são contextualizados com respeito a importantes conjecturas de estabilidade assintótica: Conjectura de Markus-Yamabe e o Problema de LaSalle / We present some results which give suficient conditions for a local diffeomorphism from the n-dimensional Euclidean space into itself be globally injective. Within this context, we consider some partial results addressed to solve the well known Fixed Point Conjecture and Jacobian Conjecture. From the dynamical point of view, there are connections between global injectivity and global asymptotic stability. In this way, we present a solution of the Markus-Yamabe Conjecture and of the LaSalle Problem

Este trabalho é dedicado ao estudo do comportamento a longo prazo de uma equação de placas extensíveis acoplada a uma equação de calor do tipo hiperbólico. O problema corresponde a um modelo de termo-elasticidade baseado em teorias de calor do tipo não-Fourier. Considerando que efeitos de inércia de rotação estão presentes no modelo, mostramos que o efeito dissipativo do calor e suficiente para estabilizar exponencialmente o sistema, sem dissipações adicionais. Além disso, provamos que o sistema possui um atrator global de dimensão fractal finita e também atratores exponenciais. Nossos resultados generalizam e complementam diversos trabalhos existentes / This work is concerned with long-time dynamics of solutions of extensible plate equations with thermal memory. It corresponds to a model of thermoelasticity based on a theory of non-Fourier heat flux. By considering the case where rotational inertia is present we show that the thermal dissipation is sufficient to stabilize the system exponentially and guarantee the existence of a finite-dimensional global attractor. In addition the existence of an exponential attractor and some further properties are also considered. Our results complements several existing results

Este trabalho é dedicado ao estudo do comportamento a longo prazo de uma classe de equações viscoelásticas não lineares com memória, da forma |\'upsilon IND. t\'| POT. ho\' \'upsilon IND. tt\' - DELTA \'upsilon\' - \'DELTA upsilon IND. tt\' + \'INT. SUP. t INF. \\tau\' upsilon (t- s) \'DELTA epsilon\' (s) ds = h, \'\\tau\' > 0, definida num domínio limitado de \'R POT. N\'. Tal classe de problemas foi estudada por diversos autores desde 2001, com \'\\tau = 0. Os resultados existentes são principalmente devotados à existência de soluções globais, decaimento da energia, com ou sem dissipações adicionais, existência com dados pequenos, entre outros. Entretanto, a questão da unicidade de soluções e existência de atratores globais não foram discutidas em trabalhos anteriores. No presente trabalho, apresentamos resultados de unicidade e existência de atratores globais para essa classe de problemas num contexto mais geral, incluindo o caso em que \'\\tau\' = -\'INFINITO\'. Além disso, incluímos um problema complementar, de quarta ordem onde estudamos a existência de atratores exponenciais / This work is concerned with the long-time behaviour of a class nonlinear viscoelastic equations of the form |\'upsilon IND. t\'| POT. ho\' \'upsilon IND. tt\' - DELTA \'upsilon\' - \'DELTA upsilon IND. tt\' + \'INT. SUP. t INF. \\tau\' upsilon (t- s) \'DELTA epsilon\' (s) ds = h, \'ho\' > 0, defined in a bounded domain of \'R POT. N\'. Such class of problems was studied by several authors since 2001, with \'\\tau\' = 0. Existing results are mainly devoted to global existence, energy decay, with or without additional dampings, existence with small data, among others. However, uniqueness and existence of global attractors were not considered previously. In the present work, we establish some results on the uniqueness of solutions and existence of global attractors in a more general setting, including \'\\tau\' = - \'INFINITY\'. In addition, we have added a second problem concerned with a fourth order equation where we study the existence of exponential attractors

Este trabalho foca no estudo de duas redes complexas. A primeira é um modelo de Ising com campo aleatório. Este modelo segue uma distribuição de campo gaussiana e bimodal. Uma técnica de conectividade finita foi utilizada para resolvê-lo. Assim como um método de Monte Carlo foi aplicado para verificar os resultados. Há uma indicação em nossos resultados que para a distribuição gaussiana a transição de fase é sempre de segunda ordem. Para as distribuições bimodais há um ponto tricrítico, dependente do valor da conectividade . Abaixo de um certo mínimo de , só existe transição de segunda ordem. A segunda é uma rede neural atratora métrica. Mais precisamente, estudamos a capacidade deste modelo para armazenar os padrões estruturados. Em particular, os padrões escolhidos foram retirados de impressões digitais, que apresentam algumas características locais. Os resultados mostram que quanto menor a atividade de padrões de impressões digitais, maior a relação de carga e a qualidade de recuperação. Uma teoria, também foi desenvolvido como uma função de cinco parâmetros: a relação de carga, a conectividade, o grau de densidade da rede, a relação de aleatoriedade e a correlação do padrão espacial. / This work focus on the study of two complex networks. The first one is a random field Ising model. This model follows a gaussian and bimodal distribution, for the random field. A finite connectivity technique was utilized to solve it. As well as a Monte Carlo method was applied to verify our results. There is an indication in our results that for a gaussian distribution the phase transition is always second-order. For the bimodal distribution there is a tricritical point, tha depends on the value of the connectivity . Below a certain minimum , there is only a second-order transition. The second one is a metric attractor neural network. More precisely we study the ability of this model to learn structured patterns. In particular, the chosen patterns were taken from fingerprints, which present some local features. Our results show that the higher the load ratio and retrieval quality are the lower is the fingerprint patterns activity. A theoretical framework was also developed as a function of five parameters: the load ratio, the connectivity, the density degree of the network, the randomness ratio and the spatial pattern correlation.

Este trabalho tem como principal objetivo estudar as constantes que aparecem em desigualdades relacionadas a operadores setoriais e suas potências fracionárias. Demonstramos que tais constantes dependem essencialmente do setor e da constante na desigualdade do resolvente associados ao operador. Como uma aplicação desses resultados, fornecemos uma prova alternativa para a limitação uniforme dos atratores de uma classe de problemas parabólicos semilineares obtidos por perturbação suave de um domínio. / This work has as main purpose to study the constants that appear in inequalities related to sectorial operators and their fractional powers. We show that these constants depend essentially on the sector and the constant in the resolvent inequality associated with the operator. As an application of these results, we provide an alternative proof for the uniform bound of the attractors of a class of semilinear parabolic problems obtained by smooth perturbation of a domain.