This dissertation is devoted to various aspects of groups generated by automata. We study particular classes and examples of such groups from different points of view. It consists of four main parts. In the first part we study Sushchansky p-groups introduced in 1979 by Sushchansky in "Periodic permutation p-groups and the unrestricted Burnside problem". These groups represent one of the earliest examples of Burnside groups and, at the same time, show the potential of the class of groups generated by automata to contain groups with extraordinary properties. The original definition is translated into the language of automata. The original actions of Sushchansky groups on p- ary tree are not level-transitive and we describe their orbit trees. This allows us to simplify the definition and prove that these groups admit faithful level-transitive actions on the same tree. Certain branch structures in their self-similar closures are established. We provide the connection with so-called G groups introduced by Bartholdi, Grigorchuk and Suninc in "Branch groups" that shows that all Sushchansky groups have intermediate growth and allows us to obtain an upper bound on their period growth functions. The second part is devoted to the opposite question of realization of known groups as groups generated by automata. We construct a family of automata with n states, n greater than or equal to 4, acting on a rooted binary tree and generating the free products of cyclic groups of order 2. The iterated monodromy group IMG(z2+i) of the self-map of the complex plain z -> z2 + i is the central object of the third part of dissertation. This group acts faithfully on the binary rooted tree and is generated by 4-state automaton. We provide a self-similar measure for this group giving alternative proof of its amenability. We also compute an L-presentation for IMG(z2+i) and provide calculations related to the spectrum of the Markov operator on the Schreier graph of the action of IMG(z2 + i) on the orbit of a point on the boundary of the binary rooted tree. Finally, the last part is discussing the package AutomGrp for GAP system developed jointly by the author and Yevgen Muntyan. This is a very useful tool for studying the groups generated by automata from the computational point of view. Main functionality and applications are provided.

This thesis conducts a comprehensive examination on the random walk behavior of 29 foreign exchange rates over the period of floating exchange regime, using variance-ratio tests. The cross-country and time-series test show that random walk model cannot be rejected on majority, and the random walk behavior is quite volatile across the whole floating exchange regime period. It then goes further to explore possible factors that can explain the probability of rejection/ non-rejections on random walk model using linear as well as nonlinear probability models, and find that the factors such as capital openness and investment-to-trade ratio significantly increases the chance of its exchange rate exhibiting random walk behavior.

The studies discussed in this thesis unify experimental and theoretical techniques, both established and novel, in investigating the problem of how a protein that binds specific sites on DNA translocates to, recognizes, and stably binds to its target site or sites. The thesis is organized into two parts. Part I outlines the history of the problem and the theory and experiments that have addressed the problem and presents an apparent incompatibility between efficient search and stable, specific binding. To address this problem, we elaborate a model of protein-DNA interaction in which the protein may bind DNA in either a search (S) mode or a recognition (R) mode. The former is characterized by zero or weak sequence-dependence in the binding energy, while the latter is highly sequence-dependent. The protein undergoes a random walk along the DNA in the S mode, and if it encounters its target site, must undergo a conformational transition into the R mode. The model resolves the apparent paradox, and accounts for the observed speed, specificity, and stability in protein-DNA interactions. The model shows internal agreement as regards theoretical and simulated results, as well as external agreement with experimental measurements. Part II reports on research that has tested the applicability of the two-mode model to the tumor suppressor transcription factor p53. It describes in greater depth the experimental techniques and findings up to the present work, and introduces the techniques and biological system used in our research. We employ single-molecule optical microscopy in two projects to study the diffusional kinetics of p53 on DNA. The first project measures the diffusion coefficient of p53 and determines that the protein satisfies a number of requirements for the validity of the two-mode model and for efficient target localization. The second project examines the sequence-dependence in p53's sliding kinetics, and explicitly models the energy landscape it experiences on DNA and relates features of the landscape to observed local variation in diffusion coefficient. The thesis closes with proposed extensions and complements to the projects, and a discussion of the implications of our work and its relation to recent developments in the field.

<p>In this dissertation we study combinatorial Hodge Laplacians on simplicial com-</p><p>plexes using tools generalized from spectral graph theory. Specifically, we consider</p><p>generalizations of graph Cheeger numbers and graph random walks. The results in</p><p>this dissertation can be thought of as the beginnings of a new spectral theory for</p><p>simplicial complexes and a new theory of high-dimensional expansion.</p><p>We first consider new high-dimensional isoperimetric constants. A new Cheeger-</p><p>type inequality is proved, under certain conditions, between an isoperimetric constant</p><p>and the smallest eigenvalue of the Laplacian in codimension 0. The proof is similar</p><p>to the proof of the Cheeger inequality for graphs. Furthermore, a negative result is</p><p>proved, using the new Cheeger-type inequality and special examples, showing that</p><p>certain Cheeger-type inequalities cannot hold in codimension 1.</p><p>Second, we consider new random walks with killing on the set of oriented sim-</p><p>plexes of a certain dimension. We show that there is a systematic way of relating</p><p>these walks to combinatorial Laplacians such that a certain notion of mixing time</p><p>is bounded by a spectral gap and such that distributions that are stationary in a</p><p>certain sense relate to the harmonics of the Laplacian. In addition, we consider the</p><p>possibility of using these new random walks for semi-supervised learning. An algo-</p><p>rithm is devised which generalizes a classic label-propagation algorithm on graphs to</p><p>simplicial complexes. This new algorithm applies to a new semi-supervised learning</p><p>problem, one in which the underlying structure to be learned is flow-like.</p> / Dissertation

Thesis (Ph. D.)--University of Missouri-Columbia, 2002. / Typescript. Vita. Includes bibliographical references (leaves 79-81). Also available on the Internet.

Orientador: Serguei Popov / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T09:42:00Z (GMT). No. of bitstreams: 1 Costa_MarceloRocha_M.pdf: 576974 bytes, checksum: 3b01a9f15e6e0f9fdd98631dc69cd202 (MD5) Previous issue date: 2014 / Resumo: O problema considerado no presente trabalho cumpre o papel de reforçar a eficácia dos métodos apresentados nos capítulos introdutórios, bem como investiga a resposta de um problema até então não publicado na literatura especializada. Introduzimos uma partícula realizando um passeio aleatório simples no espaço, ou seja, uma partícula que a cada passo escolhe uniformemente um de seus vizinhos para onde irá saltar. Fixando sua posição inicial ao longo da fronteira do cubo, pergunta-se: qual é a probabilidade de que a trajetória de tal partícula nunca mais retorne ao cubo? Em outras palavras, se T é o tempo de primeiro retorno ao cubo, estamos interessados em descrever o comportamento assintótico da probabilidade de que T seja infinito / Abstract: It has been considered in this work a problem which play a role of showing the effectiveness of the content covered in the introductory chapters, as well as it is a unsolved problem across the specialized literature. We introduce a particle performing a simple random walk in space, i.e., a particle which at each step choose uniformly one of its neighbourhood sites to which it then jumps into. Fixed its initial position along the boundary of a cube, we are interested in answering the following question: what is the probability that such particle's trajectory will never reach the cube again. In other words, if T is the first return time to the cube, we aim to analyse the asymptotic behaviour of the probability that T is infinite / Mestrado / Estatistica / Mestre em Estatística

Orientador: Serguei Popov / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T10:22:43Z (GMT). No. of bitstreams: 1 Bernardini_DiegoFernandode_D.pdf: 1410086 bytes, checksum: b77a17aefd06d547f1c5db3c5cc1a8f7 (MD5) Previous issue date: 2014 / Resumo: O modelo de entrelaçamentos aleatórios foi introduzido no sentido de se investigar originalmente o traço deixado por passeios aleatórios em grandes grafos e, basicamente, tal processo é descrito por um processo pontual de Poisson em um espaço de trajetórias duplamente infinitas de passeios aleatórios simples no reticulado d-dimensional, com dimensão d pelo menos igual a três. Neste sentido, o processo é caracterizado por um emaranhado aleatório de trajetórias deste tipo. Tal modelo possui ainda um parâmetro de intensidade, que controla, de certa forma, a quantidade de trajetórias que constituem o processo. Um problema relevante no contexto deste processo, e que tem sido amplamente estudado na literatura, diz respeito à caracterização da relação de dependência (através da covariância) entre os eventos denominados como crescentes neste modelo e suportados em subconjuntos disjuntos do reticulado, e é justamente este o problema no qual nos concentramos. Em uma primeira etapa neste trabalho, determinamos expressões explícitas para a derivada, com respeito ao parâmetro de intensidade, da probabilidade de um evento crescente e suportado em um subconjunto finito do reticulado, estabelecendo assim aquilo que denominamos como a fórmula de Russo para os entrelaçamentos aleatórios. A utilização desta denominação é justificada e motivada pelo amplamente conhecido termo original, que no contexto do modelo usual de percolação estabelece uma expressão para a derivada da probabilidade dos eventos definidos como crescentes naquele modelo. Em seguida, tentamos utilizar este resultado no sentido de estabelecer uma primeira abordagem para o problema da covariância entre os eventos crescentes, e esta investigação é baseada essencialmente em uma observação sobre o número esperado das trajetórias então denominadas como pivotais positivas para o evento de interesse. Por fim, estabelecemos uma nova abordagem para o mesmo problema, utilizando uma construção alternativa do processo de entrelaçamentos baseada na técnica dos soft local times, e investigando uma espécie de pivotalidade conjunta de coleções de excursões das trajetórias dos passeios aleatórios pelos conjuntos nos quais estão suportados os eventos de interesse. Justamente a partir desta abordagem obtemos nosso último resultado sobre a covariância. De forma geral, acreditamos que a investigação e a tentativa de obter uma caracterização cada vez mais precisa para a relação de dependência que mencionamos deve ajudar a entender o processo de entrelaçamentos e suas propriedades de forma cada vez mais clara / Abstract: The random interlacements model was originally introduced in order to investigate the trace left by random walks in large graphs and, basically, such process is described by a Poisson point process in a space of doubly infinite simple random walk trajectories in the d-dimensional lattice, with dimension d at least equal to three. In this sense, the process is characterized by a random tangle of trajectories of this kind. Such model also has an intensity parameter, which controls, in a certain sense, the quantity of trajectories that constitutes the process. A relevant issue in the context of this process, which has been largely studied in the literature, concerns the characterization of the dependence relation (through the covariance) between the so-called increasing events in this model, which are supported on disjoint subsets of the lattice, and this is precisely the issue on which we focus. In a first step in this work, we determine explicit expressions for the derivative, with respect to the intensity parameter, of the probability of an increasing event which is supported in a finite subset of the lattice, thus establishing what we call as Russo¿s formula for random interlacements. The use of this term is justified and motivated by the widely known original term, which, in the context of the usual percolation model, provides an expression for the derivative of the probability of events defined as increasing in that model. Then, we try to use this result to establish a first approach to the problem of the covariance between increasing events, and such investigation is essentially based in a fact about the expected number of the so-called positive pivotal (or plus pivotal) trajectories for the event of interest. Finally, we establish a new approach to the same problem by using an alternative construction of the interlacements process based on the technique of soft local times, and investigating a kind of joint "pivotality" of collections of excursions of the random walk trajectories, through the sets on which the events of interest are supported. From this approach we obtain our last result on the covariance. Overall, we believe that the investigation and the attempt to get an increasingly accurate characterization of the above mentioned dependence relation should help to understand the interlacements process and its properties in an increasingly clear way / Doutorado / Estatistica / Doutor em Estatística

This paper deals with the joint and conditional distributions concerning the maximum of random walk paths and the number of times this maximum is achieved. This joint distribution was studied first by Dwass [1967]. Based on his result, the correlation and some conditional moments are derived. The main contributions are however asymptotic expansions concerning the conditional distribution and conditional moments. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

The ratchet effect can be exploited in many types of research, yet few researchers pay attention to it. In this thesis, I investigate two examples of such effects in microfluidic devices, under the guidance of computational simulations. The first chapter provides a brief introduction to ratchet effects, electrophoresis, and swimming cells, topics directly related to the following chapters. The second chapter of this thesis studies the separation of charged spherical particles in various microfluidic devices. My work shows how to manipulate those particles with modified temporal asymmetric electric potentials. The rectification of randomly swimming bacteria in microfluidic devices has been extensively studied. However, there have been few attempts to optimize such rectification devices. Mapping such motion onto a lattice Monte Carlo model may suggest some new mathematical methods, which might be useful for optimizing the similar systems. Such a mapping process is introduced in chapter four.

Les dispersions colloïdales ont beaucoup d’applications technologiques importantes. A cause du mouvement brownien, les particules ont des collisions fréquentes entre elles. Les forces d’attraction de van der Waals,dérivant de potentiels à longue portés, conduisent à l’agrégation et à la précipitation des particules. Plusieurs méthodes ont été proposées pour diminuer ou contrebalancer l’effet d’attraction de van der Waals et augmenter la stabilité colloïdale. Par exemple, le choix du solvant possédant l’indice de réfraction le plus proche possible de celui des particules peut diminuer les forces de van der Waals. D'autres facteurs influencent la stabilité comme les interactions électrostatiques et les interactions spécifiques liées aux chaînes de polymères. Dans le cas des polymères, les chaînes peuvent être greffées à la surface des particules ou être dissoutes dans le solvant (chaînes libres). Dans ce travail de thèse, nous avons étudié l’effet de la stabilisation par déplétion dans le cas des chaînes de polymères libres (FPI, "free polymer induced interaction"). Des modèles théoriques précédents portent un caractère trop simplifié et utilisent des approximations sans vérification. De plus, l’influence des paramètres de la solution, c’est-à-dire, de la structure de polymères et de son interaction avec la surface de particule, n’a pas été étudiée.Les chaînes polymères libres ont été modélisées comme des marches aléatoires dans un champ moléculaire auto-cohérent qui satisfait à l'équation intégro-différentielle de diffusion. Pour le domaine moléculaire, nous avons utilisé un potentiel chimique qui, pour la solution de polymère semi-dilué, peut être représenté comme une expansion du viriel où nous n’avons pris en considération que les deuxième et troisième coefficients du viriel de la solution de polymère. En variant des paramètres tels que la rigidité du polymère, la longueur du polymère, la concentration en polymère et le régime du solvant (comme le solvant thêta), que ce soit pour une surface colloïdale purement répulsive, pour une surface adsorbée ou pour la surface d'une couche de polymère greffé, nous avons été en mesure d'améliorer la barrière répulsive due aux polymères libres entre les particules et donc nous avons trouvé des conditions de la stabilisation cinétique du système. / Stable colloidal dispersions with evenly distributed particles are important for many technological applications. Due to Brownian motion colloidal particles have constant collisions with each other which often lead to their aggregation driven by the long range van der Waals attraction. As a result the colloidal systems often tend to precipitate. A number of methods have been devised to minimize the effect of long-range van der Waals attraction between colloidal particles or to override the influence of the attraction in order to provide the colloidal stability.In the PhD thesis we investigated the colloidal stabilization in solutions of free polymers which is commonly referred to as depletion stabilization. Previous theoretical studies of free-polymer induced (FPI) stabilization were based on oversimplified models involving uncontrolled approximations. Even the most basic features of the depletion stabilization phenomenon were unknown. It was unclear how the PI repulsion depends on the solution parameters, polymer structure and monomer/surface interactions.The free polymer chains were modeled as random walks in a self-consistent molecular field that satisfied to diffusion-like integro-differential equation. As the molecular field we used the chemical potential that for semi-dilute polymer solution can be represented as a virial expansion where we took into account only second and third virial coefficients of the polymer solution. Varying the parameters like polymer stiffness, polymer length, polymer concentration and solvent regime (like theta solvent) whether it is for purely repulsive colloidal surface, adsorbed surface or surface with grafted polymer layer we were able to enhance the repulsive barrier due to the free polymers between the particles and therefore found conditions for kinetic stabilization of the system.