We propose a simple and general method to obtain the boundary crossing probability for Brownian motion. This method can be easily extended to higher dimensional of Brownian motion. It also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a nite Markov chain such that the boundary crossing probability of Brownian motion is obtained as the limiting probability of the nite Markov chain entering a set of absorbing states induced by the boundary. Numerical results are given to illustrate our method.

We propose a simple and general method to obtain the boundary crossing probability for Brownian motion. This method can be easily extended to higher dimensional of Brownian motion. It also covers certain classes of stochastic processes associated with Brownian motion. The basic idea of the method is based on being able to construct a nite Markov chain such that the boundary crossing probability of Brownian motion is obtained as the limiting probability of the nite Markov chain entering a set of absorbing states induced by the boundary. Numerical results are given to illustrate our method.

AMOR, T. M. A. Characterizing and modeling visual persistence, search strategies and fixation times. 2017. 114 f. Tese (Doutorado em Física) – Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017. / Submitted by Pós-Graduação em Física (posgrad@fisica.ufc.br) on 2017-04-05T18:55:10Z No. of bitstreams: 1 11 TESE - TATIANA MARIA ALONSO AMOR.pdf: 24328367 bytes, checksum: bd1f8abe088f435a872eae56fc9eede0 (MD5) / Rejected by Giordana Silva (giordana.nascimento@gmail.com), reason: Boa tarde Ana cleide, Fiz algumas alterações. Só não consegui deletar o arquivo anexado a fim de renomeá-lo. Isto porque o arquivo,conforme as orientações daquele guia, deverá ter a seguimte nomenclatura: 2017_tese_tmaamor O co-orientador é aquele que está no registro? Pergunto isso porque procurei o nome no trabalho e não localizei. Estou concluindo o manual e já lhe envio. on 2017-04-05T19:39:41Z (GMT) / Submitted by Pós-Graduação em Física (posgrad@fisica.ufc.br) on 2017-04-07T16:49:43Z No. of bitstreams: 1 11 TESE - TATIANA MARIA ALONSO AMOR.pdf: 24328367 bytes, checksum: bd1f8abe088f435a872eae56fc9eede0 (MD5) / Approved for entry into archive by Giordana Silva (giordana.nascimento@gmail.com) on 2017-04-07T18:13:24Z (GMT) No. of bitstreams: 1 11 TESE - TATIANA MARIA ALONSO AMOR.pdf: 24328367 bytes, checksum: bd1f8abe088f435a872eae56fc9eede0 (MD5) / Made available in DSpace on 2017-04-07T18:13:24Z (GMT). No. of bitstreams: 1 11 TESE - TATIANA MARIA ALONSO AMOR.pdf: 24328367 bytes, checksum: bd1f8abe088f435a872eae56fc9eede0 (MD5) Previous issue date: 2017 / To gather information from the world around us, we move our eyes constantly. In different occasions we find ourselves performing visual searches, such as trying to find someone in a crowd or a book in a shelf. While searching, our eyes “jump” from one location to another giving rise to a wide repertoire of patterns, exhibiting distinctive persistent behaviors. Initially, by focusing on saccadic directions and intersaccadic angles, we disclose that the probability distributions of these measures show a clear preference of participants towards a reading-like mechanism (geometrical persistence), whose features and potential advantages for searching/foraging are discussed.We then perform a Multifractal Detrended Fluctuation Analysis (MF-DFA) over the time series of jump magnitudes in the eye trajectory and find that it exhibits a typical multifractal behavior arising from the sequential combination of saccades and fixations. By inspecting the time series composed of only fixational movements, our results reveal instead a monofractal behavior with a Hurst exponent H ∼ 0.7, which indicates the presence of long-range power-law positive correlations (statistical persistence). Motivated by the experimental findings from the study of the distribution of the intersaccadic angles, we developed a simple visual search model that quantifies the wide variety of possible search strategies. From our experiments we know that when searching a target within an image our brain can adopt different strategies. The question then is which one does it choose? We present a simple two-parameter visual search model (VSM) based on a persistent random walk and the experimental inter-saccadic angle distribution. The model captures the basic observed visual search strategies that range from systematic or reading-like to completely random. We compare the results of the model to the experimental data by measuring the space-filling efficiency of the searches. Within the parameter space of the model, we are able to quantify the strategies used by different individuals for three searching tasks and show how the average search strategy changes along these three groups. Even though participants tend to explore a vast range of parameters, when all the items are placed on a regular lattice, participants are more likely to perform a systematic search, whereas in a more complex field, the search trajectories resemble a random walk. In this way we can discern with high sensitivity the relation between the visual landscape and the average strategy, disclosing how small variations in the image induce strategy changes. Finally, we move beyond visual search and study the fixation time distributions across different visual tasks. Fixation times are commonly associated to some cognitive process, as it is in this instances where most of the visual information is gathered. However, the distribution for the fixation durations exhibits certain similarities across a wide range of visual tasks and foveated species. We studied how similar these distributions are, and found that, even though they share some common properties, such as similar mean values, most of them are statistically different. Because fixations durations can be controlled by two different mechanisms: cognitive or ocular, we focus our research into finding a model for the fixation times distribution flexible enough to capture the observed behaviors in experiments that tested these concepts. At the same time, the candidate function to model the distribution needs to be the response of some very robust inner mechanism found in all the aforementioned scenarios. Hence, we discuss the idea of a model based on the microsacaddic inter event time statistics, resulting in the sum of Gamma distributions, each of these related to the presence of a distinctive number of microsaccades in a fixation. / To gather information from the world around us, we move our eyes constantly. In different occasions we find ourselves performing visual searches, such as trying to find someone in a crowd or a book in a shelf. While searching, our eyes “jump” from one location to another giving rise to a wide repertoire of patterns, exhibiting distinctive persistent behaviors. Initially, by focusing on saccadic directions and intersaccadic angles, we disclose that the probability distributions of these measures show a clear preference of participants towards a reading-like mechanism (geometrical persistence), whose features and potential advantages for searching/foraging are discussed.We then perform a Multifractal Detrended Fluctuation Analysis (MF-DFA) over the time series of jump magnitudes in the eye trajectory and find that it exhibits a typical multifractal behavior arising from the sequential combination of saccades and fixations. By inspecting the time series composed of only fixational movements, our results reveal instead a monofractal behavior with a Hurst exponent H ∼ 0.7, which indicates the presence of long-range power-law positive correlations (statistical persistence). Motivated by the experimental findings from the study of the distribution of the intersaccadic angles, we developed a simple visual search model that quantifies the wide variety of possible search strategies. From our experiments we know that when searching a target within an image our brain can adopt different strategies. The question then is which one does it choose? We present a simple two-parameter visual search model (VSM) based on a persistent random walk and the experimental inter-saccadic angle distribution. The model captures the basic observed visual search strategies that range from systematic or reading-like to completely random. We compare the results of the model to the experimental data by measuring the space-filling efficiency of the searches. Within the parameter space of the model, we are able to quantify the strategies used by different individuals for three searching tasks and show how the average search strategy changes along these three groups. Even though participants tend to explore a vast range of parameters, when all the items are placed on a regular lattice, participants are more likely to perform a systematic search, whereas in a more complex field, the search trajectories resemble a random walk. In this way we can discern with high sensitivity the relation between the visual landscape and the average strategy, disclosing how small variations in the image induce strategy changes. Finally, we move beyond visual search and study the fixation time distributions across different visual tasks. Fixation times are commonly associated to some cognitive process, as it is in this instances where most of the visual information is gathered. However, the distribution for the fixation durations exhibits certain similarities across a wide range of visual tasks and foveated species. We studied how similar these distributions are, and found that, even though they share some common properties, such as similar mean values, most of them are statistically different. Because fixations durations can be controlled by two different mechanisms: cognitive or ocular, we focus our research into finding a model for the fixation times distribution flexible enough to capture the observed behaviors in experiments that tested these concepts. At the same time, the candidate function to model the distribution needs to be the response of some very robust inner mechanism found in all the aforementioned scenarios. Hence, we discuss the idea of a model based on the microsacaddic inter event time statistics, resulting in the sum of Gamma distributions, each of these related to the presence of a distinctive number of microsaccades in a fixation.

Em Ciência da Computação, os caminhos aleatórios são utilizados em algoritmos randômicos, especialmente em algoritmos de busca, quando desejamos encontrar um estado marcado numa cadeia de Markov. Nesse tipo de algoritmo é interessante estudar o Tempo de Alcance, que está associado a sua complexidade computacional. Nesse contexto, descrevemos a teoria clássica de cadeias de Markov e caminhos aleatórios, assim como o seu análogo quântico. Dessa forma, definimos o Tempo de Alcance sob o escopo das cadeias de Markov quânticas. Além disso, expressões analíticas calculadas para o tempo de Alcance quântico e para a probabilidade de encontrarmos um elemento marcado num grafo completo são apresentadas como os novos resultados dessa dissertação. / In Computer Science, random walks are used in randomized algorithms, specially in search algorithms, where we desire to find a marked state in a Markov chain.In this type of algorithm,it is interesting to study the Hitting Time, which is associated to its computational complexity. In this context, we describe the classical theory of Markov chains and random walks,as well as their quantum analogue.In this way,we define the Hitting Time under the scope of quantum Markov chains. Moreover, analytical expressions calculated for the quantum Hitting Time and for the probability of finding a marked element on the complete graph are presented as the new results of this dissertation.

Submitted by Fabio Sobreira Campos da Costa (fabio.sobreira@ufpe.br) on 2016-06-14T13:27:03Z No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Hugo_Andrade_Doutorado.pdf: 3065927 bytes, checksum: 2eeb9c1ecb93e60c146992117b01cbb6 (MD5) / Made available in DSpace on 2016-06-14T13:27:03Z (GMT). No. of bitstreams: 2 license_rdf: 1232 bytes, checksum: 66e71c371cc565284e70f40736c94386 (MD5) Hugo_Andrade_Doutorado.pdf: 3065927 bytes, checksum: 2eeb9c1ecb93e60c146992117b01cbb6 (MD5) Previous issue date: 2013-02-07 / CNPq / Nesta Dissertac¸ ˜ao estudamos a dinˆamica energ´etica das buscas aleat ´orias aplicadas ao problema de foraging, em que animais buscam por comida ou parceiros em ambientes escassos. Discutiremos, inicialmente, um modelo estat´ıstico de caminhadas aleat ´orias utilizando as distribuic¸ ˜oes de L´evy para os tamanhos dos passos de busca, as quais tˆem sido reportadas na literatura como estrat´egias de eficiˆencia ´otima para o problema. Em seguida vamos incluir no modelo ganhos e perdas de energia na caminhada aleat ´ oria de busca, e abordaremos a dinˆamica energ´etica do processo de busca unidimensional com extremos absorventes. Vamos discutir a transic¸ ˜ao de fase que o buscador experimenta de um estado ativo (“vivo”), t´ıpico de ambientes com abundˆancia de recursos, para um estado est´atico absorvente (“morto”), onde a busca ´e encerrada pela falta de energia oriunda do encontro de recursos. Obteremos os expoentes cr´ıticos relativos a essa transic¸ ˜ao atrav´es de abordagens te ´ oricas, tais como o m´etodo de primeira passagem para o estado de energia nula, e num´ericas, baseadas na hip´otese de escala. Mostraremos a independˆencia destes expoentes com a forma funcional da func¸ ˜ao gasto de energia. Por fim, faremos uma breve revis˜ao da literatura sobre a equac¸ ˜ao de Fokker-Planck canˆonica e tamb´em sobre as suas vers˜oes utilizando derivadas fracion´arias, numa prepararac¸ ˜ao para uma futura abordagem, durante o programa de Doutorado, do problema da busca aleat´oria envolvendo difus˜oes anˆomalas (por exemplo, superdifus˜ao) via equac¸ ˜oes diferenciais. / In this work we study the energy dynamics of random searches applied to the foraging problem, in which animals search for food or mates in scarce environments. Firstly, we discuss a statistical model of random search walks using the L´evy distribution of step lengths, which has been reported in the literature as an optimal solution to the problem. In the sequence we include in the model energy gains and losses during the search walk, and discuss the energy dynamics of the search process in a one dimensional space with absorbing boundaries. We discuss the phase transition that the searcher experiences from an active (“alive”) state, typical of environments abundant in resources, to a static absorbed (“dead”) one, in which the search is terminated due to the lack of energy obtained from the encounters.We obtain the critical exponents for this transition through both theoretical (such as the first-passage method to the state of zero energy) and numerical approaches, based on the scale hypothesis.We show the independence of the exponents with the functional form of the energy cost. Finally, we provide a brief review of the literature on the canonical Fokker-Planck equation and also on its version using fractional derivatives, in a preparation for a future approach of the random search problem involving anomalous diffusion (e.g., superdiffusion) through differential equations during the Ph.D. program.

Cette thèse est consacrée à l’étude des trajectoires quantiques issues de la théorie desmesures continues en mécanique quantique non relativiste. On y présente de nouveaux résultatsthéoriques ainsi que des exemples d’applications. Sur le front théorique, on étudie principalementla limite de mesure «forte» dans laquelle on met en évidence l’émergence de sauts quantiques etd’échardes quantiques, deux phénomènes dont on précise la statistique. Hors de la limite forte, onpropose une méthode d’extraction optimale d’information pour un registre de qubits. Sur le frontdes applications, on introduit une méthode originale de contrôle utilisant l’intensité de la mesurecomme unique variable et on explique la transition balistique-diffusif dans les marches aléatoiresquantiques ouvertes; deux sous produits de l’étude théorique préalable des situations de mesureforte. On s’intéresse aussi au problème de la gravité semi-classique et montre que la théorie desmesures continues peut permettre d’en construire un modèle cohérent à la limite newtonienne. Onsuggère enfin quelques extensions possibles de la théorie à l’estimation a posteriori et d’éventuellesgénéralisations des résultats théoriques à des situations de mesures répétées discrètes. Dans laprésentation des résultats, l’accent est mis davantage sur l’explicitation des liens entre les multiplespoints de vue possibles sur les trajectoires quantiques (parallèles avec la théorie classique du filtrageet les modèles de collapse objectif utilisés dans les fondements) que sur la rigueur mathématique. / This thesis is devoted to the study of the quantum trajectories obtained from thetheory of continuous measurement in non relativistic quantum mechanics. New theoretical resultsas well as examples of applications are presented. On the theoretical front, we study mostly thelimit of «strong» measurement where we put forward the emergence of quantum jumps and quantumspikes, two phenomena we characterize in detail. Out of the strong measurement limit, weinvestigate a method to extract information from a register of qubits optimally. On the applicationfront, we introduce an original method to control quantum systems exploiting only the freedomof changing the measurement intensity and we explain the transition between a ballistic and adiffusive behavior in open quantum random walks; two byproduct of the theoretical study of thestrong measurement regime. We further study the problem of semi-classical gravity and show thatcontinuous measurement theory allows to construct a consistent model in the Newtonian regime.We eventually suggest possible extensions of the formalism to a posteriori estimation and hint atgeneralizations of the results for the strong measurement limit in the wider context of discreterepeated measurements. In the course of our presentation, we emphasize the link with other approachesto the theory of continuous measurement (parallels with stochastic filtering and collapsemodels in foundations) rather than aim for mathematical rigor.

Soit G un groupe dénombrable, qui se scinde en un produit libre de la forme G=G_1*...*G_k*F, où F est un groupe libre de type fini, et les G_i sont librement indécomposables et non isomorphes à Z. Nous montrons que le groupe Out(G) des automorphismes extérieurs de G satisfait l'alternative de Tits, dès lors que chacun des groupes G_i et Out(G_i) la satisfait. Par des méthodes similaires, nous montrons aussi l'alternative suivante pour tout sous-groupe H de Out(F_N), due à Handel et Mosher lorsque H est de type fini : soit H fixe virtuellement la classe de conjugaison d'un facteur libre propre de F_N, soit H contient un automorphisme complètement irréductible. Nos méthodes, géométriques, utilisent l'étude de la dynamique de l'action de certains sous-groupes de Out(G) sur des espaces hyperboliques. Nous décrivons notamment l'adhérence de l'outre-espace de G relatif aux G_i, et le bord de Gromov du complexe (hyperbolique) des scindements cycliques relatifs associé. Nous étudions par ailleurs les marches aléatoires sur Out(F_N). Sous un certain nombre de conditions sur la mesure de probabilité mu, nous montrons que presque toute trajectoire de la marche aléatoire sur (Out(F_N),mu) converge vers un point du bord de Gromov du complexe des facteurs libres de F_N, que nous identifions au bord de Poisson de (Out(F_N),mu). Par ailleurs, nous décrivons l'horofrontière de l'outre-espace. Ceci a des applications à l'étude de la croissance des classes de conjugaison de F_N sous l'effet de produits aléatoires d'automorphismes extérieurs. / Let G be a countable group that splits as a free product of the form G=G_1*...*G_k*F, where F is a finitely generated free group, and the groups G_i are freely indecomposable and not isomorphic to Z. We show that Out(G) satisfies the Tits alternative, as soon as all the groups G_i and Out(G_i) do. Similar techniques also yield another alternative for subgroups H of Out(F_N), due to Handel and Mosher when H is finitely generated, namely: either H virtually fixes the conjugacy class of some proper free factor of F_N, or H contains a fully irreducible automorphism. Our methods are geometric, and require understanding the dynamics of the action of some subgroups of Out(G) on Gromov hyperbolic spaces. In particular, we determine the closure of the outer space of G relative to the G_i's, as well as the Gromov boundary of the (hyperbolic) complex of relative cyclic splittings of G. We also study random walks on Out(F_N). Given a probability measure mu on Out(F_N) (satisfying some conditions), we prove that almost every sample path of the random walk on (Out(F_N),mu) converges to a point of the Gromov boundary of the free factor complex of F_N, which we identify with the Poisson boundary of (Out(F_N),mu). We also describe the horoboundary of outer space, and give applications to growth of conjugacy classes of F_N under random products of outer automorphisms.

Les composites à matrice céramique ou carbone sont des matériaux de structure pour des applications à haute température. Ils sont constitués d’un renfort fibreux enrobé d’une matrice. Cette dernière est obtenue par infiltration chimique en phase vapeur. Une préforme, fibreuse avec ou sans une première matrice, est placée dans un four dans lequel sont injectés des gaz dit précurseurs. Leur réaction hétérogène avec la préforme est à l’origine de la formation matricielle. Cette thèse a été motivée par la nécessite d’optimiser via une modélisation numérique ce procédé long et couteux.Deux programmes ont dès lors été développés puis validés. Chacun est dédié à une échelle spécifique du matériau : microscopique (fibre) et macroscopique (composite). Ils s’appuient tout deux sur des algorithmes de marche aléatoire et requièrent des représentations tridimensionnelles de la préforme. Dans cette optique, des images tomographiques de préformes de composites C/C et SiC/SiC ont été acquises aux deux échelles souhaitées. Le code propre à l’échelle de la fibre a été utilisé pour déterminer les propriétés géométriques, diffusives et réactives dans plusieurs zones de l’image afférente. Des corrélations entre ces propriétés ont été mises en place puis intégrées dans le code afférent à l’échelle du matériau pour infiltration numérique.Dans le cas des composites C/C, cet outil multi-échelle a été couplé à un modèle chimique permettant d’anticiper, en fonction des conditions opératoires, l’épaisseur ainsi que la microtexture de la matrice déposée. Des prévisions de densification ont également été employées pour qualifier et comparer l’infiltrabilité de plusieurs composites SiC/SiC. / Ceramic matrix composites and carbon fiber reinforced carbon composites are dedicated to high temperature applications. They consist of a stacked fibrous arrangement, woven or not, coated by matrix. Chemical Vapor Infiltration (CVI) is a popular processing route, where a preform (fibers with or without a first matrix) is placed inside a furnace. Precursor gases are then injected, enter the preform and generate matrix formation by heterogeneous chemical reaction. Experimental optimization of CVI is long and costly, triggering the need for a numerical model. The creation of such a tool has been the objective set for this thesis.Two programs were first developed and validated. Each is linked to a specific material scale: microscopic (fiber scale) and macroscopic (composite scale). Both are based on a random walk algorithm and require three-dimensional representations of the preform. X-ray tomography scans of C/C and SiC/SiC composite preforms were performed at the desired scales. The fiber scale program was used for computation of geometrical, diffusive and reactive properties in several regions of the relevant image. Correlations between these properties are created and inserted into the composite scale program for numerical infiltration.In the case of C/C composites, this multi-scale tool was coupled to a chemical model for anticipation, under various operating conditions, of matrix thickness and microtexture. Densification previsions on different SiC/SiC composites enabled their direct measure and comparison of their infiltrability.

Orientador: Élcio Lebensztayn / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-27T03:09:48Z (GMT). No. of bitstreams: 1 UtriaValdes_JaimeAntonio_M.pdf: 525263 bytes, checksum: 3a980748a98761becf1b573639a361c1 (MD5) Previous issue date: 2015 / Resumo: O Resumo poderá ser visualizado no texto completo da tese digital / Abstract: The Abstract is available with the full electronic digital document / Mestrado / Estatistica / Mestre em Estatística

Cette thèse est consacrée à l'étude des processus stochastiques décentralisés. Parmi les exemples typiques de ces processus figurent la dynamique météorologique, la circulation automobile, la façon dont nous rencontrons nos amis, etc. Dans cette thèse, nous exploitons une large palette d'outils probabilistes permettant d'analyser des chaînes de Markov afin d'étudier un large éventail de ces processus distribués : modèle des feux de forêt (réseaux sociaux), balls-into-bins avec suppression, et des dynamiques et protocoles de consensus fondamentaux tels que Voter Model, 2-Choices, et 3-Majority. / This thesis is devoted to the study of stochastic decentralized processes. Typical examples in the real world include the dynamics of weather and temperature, of traffic, the way we meet our friends, etc. We take the rich tool set from probability theoryfor the analysis of Markov Chains and employ it to study a wide range of such distributed processes: Forest Fire Model (social networks), Balls-into-Bins with Deleting Bins, and fundamental consensus dynamics and protocols such as the Voter Model, 2-Choices, and 3-Majority.