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We develop models for the spread of two opposing ideologies in a closed population based on epidemic models. Based on different interaction rules, we study deterministic and stochastic models of the problem. The goal of our work is to provide a tractable analytical framework for each situation, and to analyze the effect of different initial conditions on the proportion of the population affiliated with each ideology after a large time interval.

<p>Society-induced changes to the environment are altering the effectiveness of existing management strategies for sustaining natural and agricultural ecosystem productivity. At the watershed scale, natural and agro-ecosystems represent complex spatiotemporal stochastic processes. In time, they respond to random rainfall events, evapotranspiration and other losses that are spatially variable because of heterogeneities in soil properties, root distributions, topography, and other factors. To quantify the environmental impact of anthropogenic activities, it is essential that we characterize the evolution of space and time patterns of ecosystem fluxes (e.g., energy, water, and nutrients). Such a characterization then provides a basis for assessing and managing future anthropogenic risks to the sustainability of ecosystem productivity.</p><p>To characterize the space and time evolution of watershed scale processes, this dissertation introduces a mean field approach to watershed hydrology. Mean field theory (also known as self-consistent field theory) is commonly used in statistical physics when modeling the space-time behavior of complex systems. The mean field theory approximates a complex multi-component system by considering a lumped (or average) effect of all individual components acting on a single component. Thus, the many body problem is reduced to a one body problem. For watershed hydrology, a mean field theory reduces the numerous point component effects to more tractable watershed averages resulting in a consistent method for linking the average watershed fluxes (evapotranspiration, runoff, etc.) to the local fluxes at each point.</p><p>The starting point for this work is a general point description of the soil moisture, rainfall, and runoff system. For this system, we find the joint PDF that describes the temporal variability of the soil water, rainfall, and runoff processes. Since this approach does not account for the spatial variability of runoff, we introduce a probabilistic storage (ProStor) framework for constructing a lumped (unit area) rainfall-runoff response from the spatial distribution of watershed storage. This framework provides a basis for unifying and extending common event-based hydrology models (e.g. Soil Conservation Service curve number (SCS-CN) method) with more modern semi-distributed models (e.g. Variable Infiltration Capacity (VIC) model, the Probability Distributed (PDM) model, and TOPMODEL). In each case, we obtain simple equations for the fractions of the different source areas of runoff, the spatial variability of runoff and soil moisture, and the average runoff value (i.e., the so-called runoff curve). Finally, we link the temporal and spatial descriptions with a mean field approach for watershed hydrology. By applying this mean field approach, we upscale the point description with the spatial distribution of soil moisture and parameterize the numerous local interactions related to lateral fluxes of soil water in terms of its average. With this approach, we then derive PDFs that represent the space and time distribution of soil water and associated watershed fluxes such as evapotranspiration and runoff.</p> / Dissertation

Title: Stochastic interest rates modeling Author: Jakub Černý Abstract: This present work studies different stochastic models of interest rates. Theoretical part of this work describes short-rate models, HJM fra- mework and LIBOR Market model. It focuses in detail on widely known short-rate models, i.e. Vašíček, Hull-White and Ho-Lee model, and on LI- BOR Market model. This part ends by valuation of interest rate options and model calibration to real data. Analytical part of the work analyses valuation of real non-standard interest rate derivative using different models. Part of this derivative valuation is comparison among models in terms of general valuation and also in terms of capturing the dynamics of interest rates. The aim of this work is to describe different stochastic models of interest rates and mainly to compare them with each other.

Pour concevoir les futurs réacteurs à fusion nucléaire, une bonne compréhension des mécanismes régissant l'intéraction plasma-paroi est requise. En particulier, il est nécessaire d'estimer quantitativement les flux de chaleurs impactant les matériaux et la contamination du coeur par les impuretés provenant du mur. Dans ce contexte, le code fluide SolEdge2D a été développé pour simuler le transport dans le plasma de bord. L'interaction plasma-paroi est prise en compte grâce à une méthode de pénalisation innovante et originale. Cette méthode permet en particulier de modéliser la géométrie complexe des éléments face au plasma avec une grande flexibilité. En parallèle, une étude plus théorique sur les propriétés du transport dans les milieux faiblement collisionels a été conduite avec les physiciens du groupe CSDC de l'université de Florence. Une généralisation de la loi de Fourier prenant en compte les corrélation spatio-temporelle à longue distance à été obtenue par l'analyse de modèles stochastiques 1D. Cette loi retrouve en particulier la transition entre un régime diffusif à forte collisionalté et un régime balistique à faible collisionalité. / In the scope of designing future nuclear fusion reactors, a clear understanding of the plasma-wall interaction is mandatory. Indeed, a predictive estimation of heat flux impacting the surface and the subsequent emission of impurities from the wall is necessary to ensure material integrity and energy confinement performances. In that perspective, the fluid code SolEdge2D has been developed to simulate plasma transport in the tokamak edge plasma. The plasma-wall interaction is modeled using an innovative penalization technique. This method enables in particular to take complex plasma facing components geometry into account. In parallel to this numerical effort, a theoretical work has been achieved to find appropriate corrections to fluid closures when collisionality drops. The study of stochastic 1D models has been realized in collaboration with physicists from the CSDC group in Florence. A generalized Fourier law taking long range spatio-temporal correlations has been found to properly account for ballistic transport in the low collisional regime. This formulation is expected to be used to model parallel heat flux or turbulent cross-field transport in tokamak plasmas.

Esta tese apresenta os resultados de pesquisa sobre geração de séries sintéticas de radiação solar para estudos energéticos, realizada através do uso de modelos estocásticos e com o propósito de desenvolver método para aplicações práticas no setor elétrico. Para tanto, inicialmente foi levantado o estado da arte do tema, com revisão da literatura de séries temporais e de processos estocásticos, suas particularidades e potencialidades, complementado pela contextualização do uso de cenários no setor elétrico nacional, especialmente na operação e planejamento do sistema hidrotérmico, e por experiências internacionais na modelagem do recurso solar. A modelagem das séries utilizou dados reais de localidades do nordeste brasileiro e foi desenvolvida através do método de Box-Jenkins, realizando-se estudos de alternativas para cada uma de suas etapas. O pré-tratamento dos dados foi avaliado por três estratégias de remoção da tendência das séries e na estimativa dos coeficientes dos modelos foram comparados os métodos de Yule-Walker e dos mínimos quadrados. As análises consideraram quatro opções de modelos autorregressivos e os períodos horário, diário e mensal. O modelo autorregressivo convencional com intervalo mensal, identificado como o mais adequado para aplicação em estudos energéticos, e sua variação periódica foram implementados e avaliados com maior profundidade. Este estudo complementar considerou diferentes ordens de atraso e realizou comparações dos resultados por três métodos de cálculo do erro. O modelo desenvolvido com estrutura autorregressiva periódica de primeira ordem apresentou resultados satisfatórios e significativamente superiores aos dos demais modelos. Por fim, este modelo foi empregado na geração de séries sintéticas, criando 1.000 cenários de radiação solar mensal, posteriormente aplicados em modelo de contrato de venda de energia para avaliação de estratégias de participação em leilões, em análise de riscos de suprimento e em estimativa probabilística da receita esperada por parques geradores. / This thesis presents the results of a research on the generation of solar radiation synthetic sequences for energy studies, carried out through the use of stochastic models and with the purpose of developing a method for practical applications in the electric sector. In order to do so, the state of the art was devised through a review of the literature of time series and stochastic processes, their particularities and potentialities, complemented by the contextualization of the use of scenarios in the national electricity sector, especially in the hydrothermal system operation and planning, and international experiences in modeling the solar resource. The series modeling used real data from localities in the Brazilian Northeast and was developed through the Box-Jenkins method, carrying out alternative studies for each of its stages. The data pretreatment has been evaluated by three strategies for the series trend removal and by the methods of least squares and of Yule-Walker for the estimation of the model coefficients. The analysis considered four options of autoregressive models and hourly, daily and monthly periods. The conventional autoregressive model with monthly interval, identified as the most applications in energy studies, and its periodic variation were implemented and evaluated in greater depth. This complementary study considered different orders of delay and made comparisons of the results for three error calculation methods. The model developed with periodic autoregressive structure of first order presented results that are satisfactory and significantly superior than the other models. Finally, this model was used in the generation of synthetic series, creating 1,000 scenarios of monthly solar radiation, to be later applied in a model of power purchase agreement to evaluate strategies for auctions bidding, analysis of supply risks and probabilistic estimation of the expected revenue of power plants.

A modelagem estocástica sociocultural introduzida por Robert Axelrod é tradicionalmente referida à manutenção das diferenças, pois gera o efeito contra-intuitivo do aparecimento de heterogeneidades ao ser atingido o estado de equilíbrio, apesar de sua interação fundamental homogenizar os interagentes. Devido à sua simplicidade, inúmeras releituras do Modelo de Axelrod foram propostas, como também adendos e pequenas modificações. Um campo externo constante homogenizador, interpretado como a mídia, é um exemplo de uma possível alterações no modelo. Já um exemplo de releitura vem com a alteração funcional da interação bipolar do modelo de Axelrod por uma assimilação cultural, usando o mecanismo de Viés de Frequência. Nesta dissertação analisaremos as simulações propostas por Axelrod, sem e com a mídia externa. Para simularmos a mídia externa usaremos o artifício de adicionar um um vizinho fictício à cada elemento da rede. Além disso, analisaremos o mecanismo de assimilação via Viés de Frequência, mostrando sua relação com o modelo do voto da Maioria da Mecânica Estatística. / The sociocultural stochastic modeling introduced by Robert Axelrod is traditionally referred to as the maintenance of cultural diversity. Since it generates the appearance of heterogeneities on a steady state, even the primordial interaction tends to gauge the interactors. Due to its simplicity, numerous interpretations of this model were studied, as well as additions and minor modifications. One example of a possible change in the model can be a constant external field, interpreted as the media. Another example of a reinterpretation could be changing the Axelrod Model bipolar interaction by a cultural assimilation, using the mechanism of frequency bias. This dissertation aims to study the Axelrod simulation with and without the external media. In order to simulate the external media we will add a virtual neighbor to all elements. Furthermore, we analyze the mechanism of assimilation via Bias frequency, showing its relationship with the model of majority voting in Statistical Mechanics.

The past few years have seen advances in modelling of polycrystalline materi- als using parametric tessellation models from stochastic geometry. A promising class of tessellations, the Gibbs-type tessellation, allows the user to specify a great variety of properties through the energy function. This text focuses solely on tetrahedrizations, a three-dimensional tessellation composed of tetrahedra. The existing results for two-dimensional Delaunay triangulations are extended to the case of three-dimensional Laguerre tetrahedrization. We provide a proof of existence, a C++ implementation of the MCMC simulation and estimation of the models parameters through maximum pseudolikelihood. 1

Vários módulos funcionais de células ou bactérias são controlados por meio de genes que se regulam através da codificação de proteínas repressoras ou indutoras. A compreensão destas redes gênicas é um problema em aberto da biologia. Nesta dissertação procuramos aplicar o modelo estocástico de spins e bósons para o caso da rede mais simples, contiduída de dois genes, o primeiro reprimindo o segundo. Apresentamos a solução exata para a distribuição de probabilidades sobre o número das proteínas dos dois genes no estado estacionário, e a probabilidade dependente do tempo sobre o estado do segundo gene (ativado ou desativado). Discutimos também maneiras de generalizar os resultados para redes maiores. / Several functional modules in cells or bacteria are controlled by genes that regulate each other by coding repressive or inducer proteins. The understanding of these genetic networks is still an open problem in biology. In this work we apply the stochastic spin-boson model to the simplest interacting network, made of two genes, the first repressing the second. We present the exact solution for the stationary probability distribution over their both protein numbers, and the temporal evolution of the second genes state (on or off). We also discuss ways of generalizing these results to larger networks.

Esta tese apresenta os resultados de pesquisa sobre geração de séries sintéticas de radiação solar para estudos energéticos, realizada através do uso de modelos estocásticos e com o propósito de desenvolver método para aplicações práticas no setor elétrico. Para tanto, inicialmente foi levantado o estado da arte do tema, com revisão da literatura de séries temporais e de processos estocásticos, suas particularidades e potencialidades, complementado pela contextualização do uso de cenários no setor elétrico nacional, especialmente na operação e planejamento do sistema hidrotérmico, e por experiências internacionais na modelagem do recurso solar. A modelagem das séries utilizou dados reais de localidades do nordeste brasileiro e foi desenvolvida através do método de Box-Jenkins, realizando-se estudos de alternativas para cada uma de suas etapas. O pré-tratamento dos dados foi avaliado por três estratégias de remoção da tendência das séries e na estimativa dos coeficientes dos modelos foram comparados os métodos de Yule-Walker e dos mínimos quadrados. As análises consideraram quatro opções de modelos autorregressivos e os períodos horário, diário e mensal. O modelo autorregressivo convencional com intervalo mensal, identificado como o mais adequado para aplicação em estudos energéticos, e sua variação periódica foram implementados e avaliados com maior profundidade. Este estudo complementar considerou diferentes ordens de atraso e realizou comparações dos resultados por três métodos de cálculo do erro. O modelo desenvolvido com estrutura autorregressiva periódica de primeira ordem apresentou resultados satisfatórios e significativamente superiores aos dos demais modelos. Por fim, este modelo foi empregado na geração de séries sintéticas, criando 1.000 cenários de radiação solar mensal, posteriormente aplicados em modelo de contrato de venda de energia para avaliação de estratégias de participação em leilões, em análise de riscos de suprimento e em estimativa probabilística da receita esperada por parques geradores. / This thesis presents the results of a research on the generation of solar radiation synthetic sequences for energy studies, carried out through the use of stochastic models and with the purpose of developing a method for practical applications in the electric sector. In order to do so, the state of the art was devised through a review of the literature of time series and stochastic processes, their particularities and potentialities, complemented by the contextualization of the use of scenarios in the national electricity sector, especially in the hydrothermal system operation and planning, and international experiences in modeling the solar resource. The series modeling used real data from localities in the Brazilian Northeast and was developed through the Box-Jenkins method, carrying out alternative studies for each of its stages. The data pretreatment has been evaluated by three strategies for the series trend removal and by the methods of least squares and of Yule-Walker for the estimation of the model coefficients. The analysis considered four options of autoregressive models and hourly, daily and monthly periods. The conventional autoregressive model with monthly interval, identified as the most applications in energy studies, and its periodic variation were implemented and evaluated in greater depth. This complementary study considered different orders of delay and made comparisons of the results for three error calculation methods. The model developed with periodic autoregressive structure of first order presented results that are satisfactory and significantly superior than the other models. Finally, this model was used in the generation of synthetic series, creating 1,000 scenarios of monthly solar radiation, to be later applied in a model of power purchase agreement to evaluate strategies for auctions bidding, analysis of supply risks and probabilistic estimation of the expected revenue of power plants.