Let f be a C2 local diffeomorphism, of a closed surface M without zero Lyapunov exponents. We have proved that the number of ergodic hyperbolic measures of f with SRB property is less than equal to the number of homoclinic equivalence classes. We use an adaptation of Katok closing lemma for endomorphisms and prove ergodic criterion, introduced in [HHTU], for endomorphisms. We also prove some folklore results on uniqueness of SRB measures, in the presence of topological transitivity / Seja f um endomorfismo C2 non-singular (difeomorfismo local), de uma superfície fechada M e µ uma medida probabilidade Borel f-invariante e ergódica com expoentes de Lyapunov Não nulo. Nós provamos que o número de medidas hiperbólicas com propriedade SRB é para f so menor ou igual ao número de classes equivalentes homoclínicos. Usamos uma adaptaão do closing lema de Katok por endomorfismos e provamos critrio ergódico, introduzido em [HHTU], para endomorfismos. Também provamos alguns resultados folclóricos em unicidade de medidas SRB, na presena de transitividade topológica vii

Estudamos o modelo epidemiológico denominado susceptível-exposto-infectado (SEI) na rede quadrada por meio de simulações numéricas. Nesse modelo, cada sítio da rede pode estar susceptível, exposto ou infectado. Um sítio susceptível nas vizinhanças de um infectado se torna infectado com uma certa probabilidade e exposto com probabilidade complementar. Sítios infectados ou expostos permanecem para sempre nessa condição. Mostramos que os aglomerados gerados a partir de um único infectado numa rede repleta de suscetíveis são os mesmos aglomerados presentes na percolação isotrópica. Calculamos os expoentes críticos dinâmicos com bastante precisão permitindo colocar o modelo SEI na classe de universalidade da percolação dinâmica. / We have studied the epidemiologic model called susceptible-exposed-infected (SEI) on a square lattice by numerical simulations. In this model, each site of the lattice may be susceptible, exposed or infected. A susceptible site in the neighborhood of an infected site becomes infected with a given probability, or exposed with a complementary probability. Infected and exposed sites remain forever in these states. We have shown that clusters generated by a single infected site in a lattice full of susceptible are the same clusters as in the isotropic percolation. The critical dynamic exponents were calculated with large precision allowing to put the SEI model into the dynamical percolation universality class.

This thesis is primarily concerned in studying the relationship between different exponents of Diophantine approximation, which are quantities arising naturally in the study of rational approximation to a fixed n-tuple of real irrational numbers. As Khinchin observed, these exponents are not independent of each other, spurring interest in the study of the spectrum of a given family of exponents, which is the set of all possible values that can be taken by said family of exponents. Introduced in 2009-2013 by Schmidt and Summerer and completed by Roy in 2015, the parametric geometry of numbers provides strong tools with regards to the study of exponents of Diophantine approximation and their associated spectra by the introduction of combinatorial objects called n-systems. Roy proved the very surprising result that the study of spectra of exponents is equivalent to the study of certain quantities attached to n-systems. Thus, the study of rational approximation can be replaced by the study of n-systems when attempting to determine such spectra. Recently, Roy proved two new results for the case n=3, the first being that spectra are semi-algebraic sets, and the second being that spectra are stable under the minimum with respect to the product ordering. In this thesis, it is shown that both of these results do not hold in general for n>3, and examples are given. This thesis also provides non-trivial examples for n=4 where the spectra is stable under the minimum. An alternate and much simpler proof of a recent result of Marnat-Moshchevitin proving an important conjecture of Schmidt-Summerer is also given, relying only on the parametric geometry of numbers instead. Further, a conjecture which generalizes this result is also established, and some partial results are given towards its validity. Among these results, the simplest, but non-trivial, new case is also proven to be true. In a different vein, this thesis considers certain generalizations theta(q) of the classical theta q-series. We show under conditions on the coefficients of the series that theta(q) is neither rational nor quadratic irrational for each integer q>1.

O objetivo deste trabalho é analisar o comportamento dos expoentes críticos do modelo de Oito Vértices através de cotas superior e inferior para sua função de partição na vizinhança de modelos solúveis. O método é ilustrado pelo modelo de Heisenberg quântico unidimensional também denominado modelo XYZh. Aplica-se igualmente ao modelo de Ising bidimensional (com interação quártica e segundos vizinhos). Assim, propomos um modo alternativo de abordar universalidade nos modelos de Heisenberg unidimensional quântico e Ising bidimensional clássico por desigualdades satisfeitas por suas funções de partição. Dentre os métodos que utilizamos para a obtenção das cotas destacam-se: a interação Gaussiana nas variáveis reais e nas variáveis de Grassmann; o mapeamento de um modelo unidimensional em um bidimensional através do auxílio da fórmula Trotter; a representação da função de partição pelo Pfaffiano de uma matriz; e, para a obtenção da cota superior, a técnica de positividade por reflexão, estendida ao acaso de variáveis que anti-comutam. / The aim of this work is to analyze the behavior of critical exponents in the eight-vertex model starting from the upper and lower bound obtained for its partition function. We studied the quantum onedimensional Heisenberg model also denominated XYZh model. We propose na alternative way of approaching universality in Heisenberg and Ising models using inequalities satisfied for their partition functions.Among the methods that we used in the solutions of the models atand out the integration on the Grassmann variables, the mapping of a onedimensional model in a two-dimensional one through the aid of the Trotter formula and, finally, the representation of the partition function as Pfaffian of a matrix. To obtain na upper bound, the positivity reflection technique was used, extended to the case of variables that, anticomute, and the method of thechess board estimate.

QC 20100414

Networks of neurons, which form central pattern generators (CPGs), are important for controlling animal behaviors. Of special interest are configurations or CPG motifs composed of reciprocally inhibited neurons, such as half-center oscillators (HCOs). Bursting rhythms of HCOs are shown to include stable synchrony or in-phase bursting. This in-phase bursting can co-exist with anti-phase bursting, commonly expected as the single stable state in HCOs that are connected with fast non-delayed synapses. The finding contrasts with the classical view that reciprocal inhibition has to be slow or time-delayed to synchronize such bursting neurons. Phase-locked rhythms are analyzed via Lyapunov exponents estimated with variational equations, and through the convergence rates estimated with Poincar\'e return maps. A new mechanism underlying multistability is proposed that is based on the spike interactions, which confer a dual property on the fast non-delayed reciprocal inhibition; this reveals the role of spikes in generating multiple co-existing phase-locked rhythms. In particular, it demonstrates that the number and temporal characteristics of spikes determine the number and stability of the multiple phase-locked states in weakly coupled HCOs. The generality of the multistability phenomenon is demonstrated by analyzing diverse models of bursting networks with various inhibitory synapses; the individual cell models include the reduced leech heart interneuron, the Sherman model for pancreatic beta cells, the Purkinje neuron model and Fitzhugh-Rinzel phenomenological model. Finally, hypothetical and experiment-based CPGs composed of HCOs are investigated. This study is relevant for various applications that use CPGs such as robotics, prosthetics, and artificial intelligence.

We consider three problems in this dissertation, all under the unifying theme of random matrix products. The first and second problems are concerned with weak convergence in stochastic matrices and circulant matrices, respectively, and the third is concerned with the numerical calculation of the Lyapunov exponent associated with some random Fibonacci sequences. Stochastic matrices are nonnegative matrices whose row sums are all equal to 1. They are most commonly encountered as transition matrices of Markov chains. Circulant matrices, on the other hand, are matrices where each row after the first is just the previous row cyclically shifted to the right by one position. Like stochastic matrices, circulant matrices are ubiquitous in the literature.In the first problem, we study the weak convergence of the convolution sequence mu to the n, where mu is a probability measure with support S sub mu inside the space S of d by d stochastic matrices, d greater than or equal to 3. Note that mu to the n is precisely the distribution of the product X sub 1 times X sub 2 times and so on times X sub n of the mu distributed independent random variables X sub 1, X sub 2, and so on, X sub n taking values in S. In [CR] Santanu Chakraborty and B.V. Rao introduced a cyclicity condition on S sub mu and showed that this condition is necessary and sufficient for mu to the n to not converge weakly when d is equal to 3 and the minimal rank r of the matrices in the closed semigroup S generated by S sub mu is 2. Here, we extend this result to any d bigger than 3. Moreover, we show that when the minimal rank r is not 2, this result does not always hold.The second problem is an investigation of weak convergence in another direction, namely the case when the probability measure mu's support S sub mu consists of d by d circulant matrices, d greater than or equal to 3, which are not necessarily nonnegative. The resulting semigroup S generated by S sub mu now lacking the nice property of compactness in the case of stochastic matrices, we assume tightness of the sequence mu to the n to analyze the problem. Our approach is based on the work of Mukherjea and his collaborators, who in [LM] and [DM] presented a method based on a bookkeeping of the possible structure of the compact kernel K of S.The third problem considered in this dissertation is the numerical determination of Lyapunov exponents of some random Fibonacci sequences, which are stochastic versions of the classical Fibonacci sequence f sub (n plus 1) equals f sub n plus f sub (n minus 1), n greater than or equal to 1, and f sub 0 equal f sub 1 equals 1, obtained by randomizing one or both signs on the right side of the defining equation and or adding a "growth parameter." These sequences may be viewed as coming from a sequence of products of i.i.d. random matrices and their rate of growth measured by the associated Lyapunov exponent. Following techniques presented by Embree and Trefethen in their numerical paper [ET], we study the behavior of the Lyapunov exponents as a function of the probability p of choosing plus in the sign randomization.

published_or_final_version / abstract / Electrical and Electronic Engineering / Master / Master of Philosophy

We investigate how the formation of heterogeneity and structures in flows through explicit porous microstructures depends upon the geometric and topological observables of the porous medium. Using direct numerical simulations of single-phase, isothermal, laminar fluid flow through realistic three-dimensional stochastically generated pore structures, hereafter referred to as pore spaces, the characteristics of the resulting steady state velocity fields are related to physical characteristics of the pore spaces. The results suggest that the spatially variable resistance offered by the geometry and topology of the pore space induces a highly heterogeneous fluid velocity field therein. Focus is placed on three different length scales: macroscopic (cm), mesoscopic (mm), and microscopic (microns). At the macroscopic length scale, volume averaging is used to relate porosity, mean hydraulic radius, and their product to the permeability of the pore space. At the mesoscopic scale, the effect of a medium's porosity on fluid particle trajectory attributes, such as passage time and tortuosity, is studied. At the final length scale, that of the microscopic in-pore fluid dynamics, finite time Lyapunov exponents are used to determine expanding, contracting, and hyperbolic regions in the flow field, which are then related to the local structure of the pore space. The results have implications to contaminant transport, mixing, and how chemical reactions are induced at the pore-scale. A description of the adopted numerical methods to simulate flow and generate the pore space are provided as well.

Thesis (Ph.D.)--McMaster University (Canada), 1996. / Source: Dissertation Abstracts International, Volume: 57-10, Section: B, page: 6457. Adviser: S. Haykin.