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This thesis concerns infinite words over finite alphabets. It contributes to two topics in this area: critical exponents and stabilizers. Let w be a right-infinite word defined over a finite alphabet. The critical exponent of w is the supremum of the set of exponents r such that w contains an r-power as a subword. Most of the thesis (Chapters 3 through 7) is devoted to critical exponents. Chapter 3 is a survey of previous research on critical exponents and repetitions in morphic words. In Chapter 4 we prove that every real number greater than 1 is the critical exponent of some right-infinite word over some finite alphabet. Our proof is constructive. In Chapter 5 we characterize critical exponents of pure morphic words generated by uniform binary morphisms. We also give an explicit formula to compute these critical exponents, based on a well-defined prefix of the infinite word. In Chapter 6 we generalize our results to pure morphic words generated by non-erasing morphisms over any finite alphabet. We prove that critical exponents of such words are algebraic, of a degree bounded by the alphabet size. Under certain conditions, our proof implies an algorithm for computing the critical exponent. We demonstrate our method by computing the critical exponent of some families of infinite words. In particular, in Chapter 7 we compute the critical exponent of the Arshon word of order n for n ≥ 3. The stabilizer of an infinite word w defined over a finite alphabet Σ is the set of morphisms f: Σ*→Σ* that fix w. In Chapter 8 we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements. We conclude with a list of open problems, including a new problem that has not been addressed yet: the D0L repetition threshold.

This thesis concerns infinite words over finite alphabets. It contributes to two topics in this area: critical exponents and stabilizers. Let w be a right-infinite word defined over a finite alphabet. The critical exponent of w is the supremum of the set of exponents r such that w contains an r-power as a subword. Most of the thesis (Chapters 3 through 7) is devoted to critical exponents. Chapter 3 is a survey of previous research on critical exponents and repetitions in morphic words. In Chapter 4 we prove that every real number greater than 1 is the critical exponent of some right-infinite word over some finite alphabet. Our proof is constructive. In Chapter 5 we characterize critical exponents of pure morphic words generated by uniform binary morphisms. We also give an explicit formula to compute these critical exponents, based on a well-defined prefix of the infinite word. In Chapter 6 we generalize our results to pure morphic words generated by non-erasing morphisms over any finite alphabet. We prove that critical exponents of such words are algebraic, of a degree bounded by the alphabet size. Under certain conditions, our proof implies an algorithm for computing the critical exponent. We demonstrate our method by computing the critical exponent of some families of infinite words. In particular, in Chapter 7 we compute the critical exponent of the Arshon word of order n for n ≥ 3. The stabilizer of an infinite word w defined over a finite alphabet Σ is the set of morphisms f: Σ*→Σ* that fix w. In Chapter 8 we study various problems related to stabilizers and their generators. We show that over a binary alphabet, there exist stabilizers with at least n generators for all n. Over a ternary alphabet, the monoid of morphisms generating a given infinite word by iteration can be infinitely generated, even when the word is generated by iterating an invertible primitive morphism. Stabilizers of strict epistandard words are cyclic when non-trivial, while stabilizers of ultimately strict epistandard words are always non-trivial. For this latter family of words, we give a characterization of stabilizer elements. We conclude with a list of open problems, including a new problem that has not been addressed yet: the D0L repetition threshold.

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.

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.

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.

FERRAZ, Carlos Handrey Araújo. Dinâmica de redes Booleanas aleatórias na presença de agente danificador. 2007. 99 f. Tese (Doutorado em Física) - Programa de Pós-Graduação em Física, Departamento de Física, Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2007. / Submitted by Edvander Pires (edvanderpires@gmail.com) on 2015-05-05T20:04:53Z No. of bitstreams: 1 2007_tese_chaferraz.pdf: 1298388 bytes, checksum: 78cd24083b62d39536e51a28bb85dd73 (MD5) / Approved for entry into archive by Edvander Pires(edvanderpires@gmail.com) on 2015-05-07T14:26:22Z (GMT) No. of bitstreams: 1 2007_tese_chaferraz.pdf: 1298388 bytes, checksum: 78cd24083b62d39536e51a28bb85dd73 (MD5) / Made available in DSpace on 2015-05-07T14:26:22Z (GMT). No. of bitstreams: 1 2007_tese_chaferraz.pdf: 1298388 bytes, checksum: 78cd24083b62d39536e51a28bb85dd73 (MD5) Previous issue date: 2007 / Nós realizamos simulações de computador em autômatos de Kauffman em diversos grafos tais como redes quadradas regulares e agregados de percolação invasiva afim de investigar transições de fase, entropia total, distribuição radial do dano total médio (expoente dinâmico $z$) e velocidade de propagação do dano quando se introduz um agente danificador no sistema, apelidado o "homem estranho". A despeito do aumento na eficiência de danificação, nós não observamos qualquer mudança apreciável no limiar de transição para o caos tanto para o caso de rede quadrada como para o caso de mundo pequeno quando o homem estranho é adicionado em comparação a quando pequenos danos iniciais são inseridos ao sistema. A velocidade de propagação da nuvem de dano até tocar as bordas do sistemas tanto para o caso de rede quadrada como para o caso de mundo pequeno obedece uma lei de potência, com um expoente crítico de velocidade $alpha$ que depende fortemente do tipo de rede. Particularmente, nós temos estudado o espalhamento do dano quando algumas conexões são removidas na rede quadrada e quando se considera agregados especiais de percolação invasiva (agregados de alta saturação de borda, HBSC). A velocidade de propagação nestes sistemas é bastante sensível ao grau de diluição na rede quadrada e ao grau de saturação de borda em agregados de percolação invasiva. Finalmente, esperamos que estes e outros cálculos mais elaborados sejam de ajuda para que se possam entender problemas mais gerais relacionados a propagação de defeitos simples em sistemas complexos bem descritos por autômatos celulares.

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.

NÃs realizamos simulaÃÃes de computador em autÃmatos de Kauffman em diversos grafos tais como redes quadradas regulares e agregados de percolaÃÃo invasiva afim de investigar transiÃÃes de fase, entropia total, distribuiÃÃo radial do dano total mÃdio (expoente dinÃmico $z$) e velocidade de propagaÃÃo do dano quando se introduz um agente danificador no sistema, apelidado o "homem estranho". A despeito do aumento na eficiÃncia de danificaÃÃo, nÃs nÃo observamos qualquer mudanÃa apreciÃvel no limiar de transiÃÃo para o caos tanto para o caso de rede quadrada como para o caso de mundo pequeno quando o homem estranho à adicionado em comparaÃÃo a quando pequenos danos iniciais sÃo inseridos ao sistema. A velocidade de propagaÃÃo da nuvem de dano atà tocar as bordas do sistemas tanto para o caso de rede quadrada como para o caso de mundo pequeno obedece uma lei de potÃncia, com um expoente crÃtico de velocidade $alpha$ que depende fortemente do tipo de rede. Particularmente, nÃs temos estudado o espalhamento do dano quando algumas conexÃes sÃo removidas na rede quadrada e quando se considera agregados especiais de percolaÃÃo invasiva (agregados de alta saturaÃÃo de borda, HBSC). A velocidade de propagaÃÃo nestes sistemas à bastante sensÃvel ao grau de diluiÃÃo na rede quadrada e ao grau de saturaÃÃo de borda em agregados de percolaÃÃo invasiva. Finalmente, esperamos que estes e outros cÃlculos mais elaborados sejam de ajuda para que se possam entender problemas mais gerais relacionados a propagaÃÃo de defeitos simples em sistemas complexos bem descritos por autÃmatos celulares.

Complex magnetic oxides represent a unique intersection of immense technological importance and fascinating physical phenomena originating from interwoven structural, electronic and magnetic degrees of freedom. The resulting energetically close competing orders can be controllably selected through external fields. Competing interactions and disorder represent an additional opportunity to systematically manipulate the properties of pure magnetic systems, leading to frustration, glassiness, and other novel phenomena while finite sample dimension plays a similar role in systems with long-range cooperative effects or large correlation lengths. A rigorous understanding of these effects in strongly correlated oxides is key to manipulating their functionality and device performance, but remains a challenging task. In this dissertation, we examine a number of problems related to intrinsic and extrinsic low dimensionality, disorder, and competing interactions in magnetic oxides by applying a unique combination of standard magnetometry techniques and unconventional magnetocaloric effect and transverse susceptibility measurements. The influence of dimensionality and disorder on the nature and critical properties of phase transitions in manganites is illustrated in La0.7Ca0.3MnO3, in which both size reduction to the nanoscale and chemically-controlled quenched disorder are observed to induce a progressive weakening of the first-order nature of the transition, despite acting through the distinct mechanisms of surface effects and site dilution. In the second-order material La0.8Ca0.2MnO3, a strong magnetic field is found to drive the system toward its tricritical point as competition between exchange interactions in the inhomogeneous ground state is suppressed. In the presence of large phase separation stabilized by chemical disorder and long-range strain, dimensionality has a profound effect. With the systematic reduction of particle size in microscale-phase-separated (La, Pr, Ca)MnO3 we observe a disruption of the long-range glassy strains associated with the charge-ordered phase in the bulk, lowering the field and pressure threshold for charge-order melting and increasing the ferromagnetic volume fraction as particle size is decreased. The long-range charge-ordered phase becomes completely suppressed when the particle size falls below 100 nm. In contrast, low dimensionality in the geometrically frustrated pseudo-1D spin chain compound Ca3Co2O6 is intrinsic, arising from the crystal lattice. We establish a comprehensive phase diagram for this exotic system consistent with recent reports of an incommensurate ground state and identify new sub-features of the ferrimagnetic phase. When defects in the form of grain boundaries are incorporated into the system the low-temperature slow-dynamic state is weakened, and new crossover phenomena emerge in the spin relaxation behavior along with an increased distribution of relaxation times. The presence of both disorder and randomness leads to a spin-glass-like state, as observed in γFe2O3 hollow nanoparticles, where freezing of surface spins at low temperature generates an irreversible magnetization component and an associated exchange-biasing effect. Our results point to distinct dynamic behaviors on the inner and outer surfaces of the hollow structures. Overall, these studies yield new physical insights into the role of dimensionality and disorder in these complex oxide systems and highlight the sensitivity of their manifested magnetic ground states to extrinsic factors, leading in many cases to crossover behaviors where the balance between competing phases is altered, or to the emergence of entirely new magnetic phenomena.