Αρχικά γίνεται μια ιστορική αναδρομή, μια παρουσίαση της διπαραμετρικής Γενικευμένης εκθετικής κατανομής (τύπος κατανομής, συνάρτηση πυκνότητας πιθανότητας κλπ) και αναφέρονται βασικά χαρακτηριστικά της κατανομής. Στη συνέχεια αναφέρονται βασικοί ορισμοί και θεωρήματα σχετικά κυρίως με τη σημειακή παραμετρική εκτίμηση καθώς και την εκτίμηση κατά Bayes. Το επόμενο κεφάλαιο πραγματεύεται την ανάλυση του μοντέλου και τις βασικές ιδιότητες της Γενικευμένης εκθετικής κατανομής. Επίσης μελετώνται ειδικά θέματα, όπως συναρτήσεις επιβίωσης, πληροφορία Fisher, διατεταγμένες παρατηρήσεις, κατανομή του αθροίσματος και παραγωγή τυχαίων αριθμών, στα πλαίσια της Γενικευμένης εκθετικής κατανομής. Στη συνέχεια αναλύονται και εφαρμόζονται μέθοδοι σημειακής εκτίμησης (Μέγιστη Πιθανοφάνεια, Μέθοδος ροπών, Μέθοδος εκατοστημορίων, Ελάχιστα και σταθμισμένα ελάχιστα Τετράγωνα, L-ροπές) για την εκτίμηση των παραμέτρων της κατανομής. Μελετάται και η απόδοση των εκτιμητών για τις διάφορες μεθόδους εκτίμησης. Ακολουθεί η εκτίμηση τύπου Bayes των παραμέτρων (με συναρτήσεις ζημίας τετραγωνικού σφάλματος και LINEX αντίστοιχα). Αναφέρονται πάλι συμπεράσματα για την απόδοση των εκτιμητών και σύγκριση με τους εκτιμητές μέγιστης πιθανοφάνειας. Τελικά παρουσιάζουμε την προσέγγιση ενός αναλογιστικού πίνακα μέσω της Γενικευμένης εκθετικής κατανομής. / In the beginning, we mention a historical recursion, a presentation of the 2-parameter Generalized exponential distribution ( distribution type, probability density function etc.) and we also mention basic characteristics of the distribution. Basic definitions and theorems about point estimation and Bayes estimation are reported. Furthermore, we discource on the analysis of the model and basic properties of the Generalized exponential distribution. Special themes, such as survival functions, Fisher information, order statistics, sum distribution and production of random numbers are analyzed in the frame of the Generalized exponential distribution. Moreover, we analyze and apply point estimation methods (maximum likelihood, method of moments, percentile estimation, least (and weighted least) squares, method of L-moments) in order to estimate parameters of the distribution. Performance of the estimators for different estimation methods is also analyzed. Next, bayesian estimation of the parameters (under squared error loss function and LINEX loss function) is coming up for discussion. We also analyze the performance of the estimators and compare them to the maximum likelihood estimators. Finally, we present approximation of an actuarial table via Generalized exponential distribution.

Time series model have been used in many areas of knowledge and have become a current necessity for companies to survive in a globalized and competitive market, as well as climatic factors that have always been a concern because of the different ways they interfere in human life. In this context, this work aims to present a comparison among the performances by the following models of time series: ARIMA, ARMAX and Exponential Smoothing, adjusted to air relative humidity (UR) and also to verify the volatility present in the series through non-linear models ARCH/GARCH, adjusted to residues of the ARIMA and ARMAX models. The data were collected from INMET from October, 1st to January, 22nd, 2014. In the comparison of the results and the selection of the best model, the criteria MAPE, EQM, MAD and SSE were used. The results showed that the model ARMAX(3,0), with the inclusion of exogenous variables produced better forecast results, compared to the other models SARMA(3,0)(1,1)12 and the Holt-Winters multiplicative. In the volatility study of the series via non-linear ARCH(1), adjusted to the quadrants of SARMA(3,0)(1,1)12 and ARMAX(3,0) residues, it was observed that the volatility does not tend to influence the future long-term observations. It was then concluded that the classes of models used and compared in this study, for data of a climatologic variable, showed a good performance and adjustment. We highlight the broad usage possibility in the techniques of temporal series when it is necessary to make forecasts and also to describe a temporal process, being able to be used as an efficient support tool in decision making. / Modelos de séries temporais vêm sendo empregados em diversas áreas do conhecimento e têm surgido como necessidade atual para empresas sobreviverem em um mercado globalizado e competitivo, bem como fatores climáticos sempre foram motivo de preocupação pelas diferentes formas que interferem na vida humana. Nesse contexto, o presente trabalho tem por objetivo apresentar uma comparação do desempenho das classes de modelos de séries temporais ARIMA, ARMAX e Alisamento Exponencial, ajustados a dados de umidade relativa do ar (UR) e verificar a volatilidade presente na série por meio de modelos não-lineares ARCH/GARCH ajustados aos resíduos dos modelos ARIMA e ARMAX. Os dados foram coletados junto ao INMET no período de 01 de outubro de 2001 a 22 de janeiro de 2014. Na comparação dos resultados e na seleção do melhor modelo foram utilizados os critérios MAPE, EQM, MAD e SSE. Os resultados mostraram que o modelo ARMAX(3,0) com a inclusão de variáveis exógenas produziu melhores resultados de previsão em relação aos seus concorrentes SARMA(3,0)(1,1)12 e o Holt-Winters multiplicativo. No estudo da volatilidade da série via modelo não-linear ARCH(1), ajustado aos quadrados dos resíduos dos modelos SARMA(3,0)(1,1)12 e ARMAX(3,0), observou-se que a volatilidade não tende a influenciar as observações futuras em longo prazo. Conclui-se que as classes de modelos utilizadas e comparadas neste estudo, para dados de uma variável climatológica, demonstraram bom desempenho e ajuste. Destaca-se a ampla possibilidade de utilização das técnicas de séries temporais quando se deseja fazer previsões e descrever um processo temporal, podendo ser utilizadas como ferramenta eficiente de apoio nas tomadas de decisão.

Made available in DSpace on 2015-03-26T13:32:04Z (GMT). No. of bitstreams: 1 texto completo.pdf: 569181 bytes, checksum: 1b525772884dca74fcef6c9c8033aaa5 (MD5) Previous issue date: 2008-09-05 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Reburning caused by Phytophthora infestansis is characterized as an aggressive disease of great destructive impact, capable of limiting or even hindering the economic cultivation of the tomato plant under conditions of high humidity and low temperatures. In view of the problems reburning can cause to tomato plant crops, this work aimed to: 1) fit models to describe the progress of the disease and form groups of tomato accesses with similar curves; 2) estimate data referring to the number of days to reach 5% severity of the disease, by means of inverse regression; 3) fit survival curves by means of the Kaplan-Meier estimator for the access groups and compare them by means of the Logrank test;4)fit survival curves by means of probabilistic models and compare these curves with Kaplan Meir´s non-parametric technique. Using tomato reburning real data, it was possible to fit the exponential model (Y = y0 exp (rX)) to describe the disease s progress. The means of the parameter estimates were submitted to grouping analysis using the centroid method, generating 10 access groups. Time up to 5% of the disease was calculated via inverse regression. Non-parametric techniques were used to estimate survival function by means of the Kaplan-Meier´s estimator to compare the survival curves by the Logrank test .The survival function was also fit using the probabilistic models, exponential Weibull and Log-normal, respectively, which were compared by means of the verisimilitude ratio test (VRT), considering the generalized Gamma model, as a general case for these models. The methodology applied allowed fitting the exponential model to describe tomato plant reburning progress and to regroup the accesses studied in the 10 groups. The access BGH-6 obtained a smaller disease progress than the others, thus characterizing its higher resistance to the disease; An inverse regression allowed time estimation up to the occurrence of 5% of the severity of the tomato plant reburning. The Kaplan-Meier ´s non-parametric technique allowed estimating the survival curves of the tomato plant accesses belonging to the groups 1, 2, 4, 6 and 8. Utilizing the Logrank test, it could be concluded that most two-by-two comparisons were significant (p<0.05), except in the comparisons of groups 2x4, 4x8 and 6x8. The use of the probabilistic models, exponential Weibull and Log-normal allowed estimating the survival curves of groups 2, 4, 6 and 8, except for group 4, to which the Weibull model was not adequate. Comparing the probabilistic models with the non-parametric technique, the curves of the probabilistic models of groups 2 and 4 presented satisfactory results, compared to the curve estimated by Kaplan-Meier. / A requeima causada por Phytophthora infestans caracteriza-se por ser uma doença agressiva e de grande impacto destrutivo, podendo limitar ou até mesmo impedir o cultivo econômico do tomateiro sob condições de alta umidade e baixas temperaturas. Diante dos problemas que a requeima pode provocar às lavouras de tomate, este trabalho teve por objetivos: 1) ajustar modelos para descrever o progresso da doença e formar grupos de acessos de tomateiro com curvas semelhantes; 2) estimar dados referentes ao número de dias até atingir 5% de severidade da doença, por meio de regressão inversa; 3) ajustar curvas de sobrevivência por meio do estimador de Kaplan-Meier para grupos de acessos e compará-las mediante o uso do teste Logrank; 4) ajustar curvas de sobrevivência por meio de modelos probabilísticos e compará-las com a técnica não-paramétrica de Kaplan-Meier. Utilizando dados reais sobre a requeima do tomateiro, foi possível ajustar o modelo exponencial (Y = y0 exp (rX)) para descrever o progresso da doença. As médias das estimativas dos parâmetros foram submetidas à análise de agrupamento pelo método Centróide, o que gerou 10 grupos de acessos, sendo o tempo até a incidência de 5% da doença calculado via regressão inversa. Foram utilizadas técnicas não-paramétricas para estimar a função de sobrevivência por meio do estimador de Kaplan-Meier e para comparar as curvas de sobrevivência pelo teste Logrank. Foi também ajustada a função de sobrevivência, empregando-se os modelos probabilísticos Exponencial, Weibull e Log-normal, os quais foram comparados por meio do Teste da Razão da Verossimilhança (TRV), considerando-se o modelo Gama generalizado por ser caso geral para esses modelos. A metodologia utilizada permitiu ajustar o modelo Exponencial para descrever o progresso da requeima do tomateiro e agrupar os acessos estudados em 10 grupos. O acesso BGH-6 sofreu um progresso de doença menor que os demais, caracterizando-se, assim, sua maior resistência à enfermidade. A regressão inversa possibilitou estimar o tempo até a ocorrência de 5% da severidade da requeima do tomateiro. Pela técnica não-paramétrica de Kaplan-Meier, foi possível estimar as curvas de sobrevivência dos acessos de tomateiro pertencentes aos grupos 1, 2, 4, 6 e 8. Utilizando o teste Logrank, pode-se concluir que a maioria das comparações duas a duas foi significativa (p<0,05), exceto nas comparações dos grupos 2x4, 4x8 e 6x8. O uso dos modelos probabilísticos Exponencial, Weibull e Log-normal possibilitou a estimação das curvas de sobrevivência nos grupos 2, 4, 6 e 8, exceto no grupo 4, em que o modelo Weibull não foi adequado. Comparando os modelos probabilísticos com a técnica não-paramétrica, as curvas dos modelos probabilísticos dos grupos 2 e 4 apresentaram ajustes satisfatórios com relação à curva estimada por Kaplan-Meier.

Made available in DSpace on 2015-03-26T13:32:11Z (GMT). No. of bitstreams: 1 texto completo.pdf: 446864 bytes, checksum: 1ba8efc18a08b922adc7e2c5eb3dc55c (MD5) Previous issue date: 2011-02-24 / The aim of this study was to evaluate the efficiency of spatial analysis, which considers spatially dependent errors, for classification of common bean families in relation to traditional analysis in randomized blocks and lattice that assuming independent errors. Were considered different degrees of spatial dependence and experimental precision. Were taken as reference to simulate the results of seven experiments carried out in simple square lattice for genetic evaluation of yield (g/plot) of families and bean cultivars of winter crops and water used in 2007 and 2008. From the results presented in the simulation, it was possible to assess the quality of their experiments based on different analysis (Block, lattice and Spatial) and simulated average of 100 families in different scenarios for Spatial Dependence (DE) and Accuracy Selective (AS). In the process of simulation, the average yield (645 g/plot) and the residual variance (7744.00), was defined based on the analysis results of the tests in blocks of bean breeding program at UFV. To make up the four simulated scenarios were considered magnitude of spatial dependence (null, low, medium and high), corresponding to ranges of 0, 25, 50 and 100% of the maximum distance between plots. Were also simulated three classes of selective accuracy (0.95, 0.80 and 0.60), corresponding to the experimental precision very high, high and average, respectively. The actual classification of families was used to evaluate the efficiency of analysis methods tested by Spearman correlation applied to orders and genotypic classification of Selection Efficiency between classifications based on tested methodologies and the actual classification for the selection of 10, 20 and 30% of the best families. To compare the efficiency of adjustment of the models tested, was used the Akaike information criterion (AIC), based on likelihood. Spatial analysis has provided estimates of residual variance very close to the simulated residual variance and higher selective accuracy estimated in all scenarios, indicating greater experimental accuracy. With the reduction in the accuracy and selective increase in spatial dependence, there was greater influence of analysis on the classification of families, and the spatial analysis showed the best results, providing more efficient selection of bean families than traditional analysis of randomized blocks and lattice, mainly for the selection of fewer families. The results for selective accuracy estimated on the basis of F statistics were very close to those obtained with the Spearman correlation between estimated and simulated averages for families, indicating that the accuracy should be used selectively as a measure of experimental precision tests of genetic evaluation. / O objetivo deste trabalho foi avaliar a eficiência da análise Espacial, que considera erros dependentes espacialmente, para classificação de famílias de feijoeiro em relação às análises tradicionais em blocos casualizados e em látice que assumem erros independentes. Considerou-se diferentes graus de dependência espacial e de precisão experimental. Foram tomados como referência para simulação os resultados de sete ensaios instalados em látice quadrado simples para avaliação genética da produtividade de grãos (g/parcela) de famílias e cultivares de feijoeiro das safras de inverno e das águas de 2007 e 2008. A partir dos resultados apresentados na simulação, foi possível avaliar a qualidade dos respectivos experimentos com base nas diferentes análises (Bloco, Látice e Espacial) e médias simuladas das 100 famílias nos diferentes cenários para Dependência Espacial (DE) e Acurácia Seletiva (AS). No processo de simulação, a média de produção (645 g/parcela), bem como a variância residual (7744,00), foi definida com base nos resultados de análises em blocos de ensaios do programa de melhoramento do feijoeiro da UFV. Para a composição dos cenários simulados foram consideradas quatro magnitudes de dependência espacial (nula, baixa, média e alta), correspondendo aos alcances 0, 25, 50 e 100% da distância máxima entre parcelas. Também foram simuladas três classes de acurácia seletiva (0,95, 0,80 e 0,60), correspondente a precisão experimental muito alta, alta e média, respectivamente. A classificação real das famílias foi utilizada para avaliar a eficiência das metodologias de análise testadas através da correlação de Spearman aplicada às ordens de classificação genotípica e da Eficiência de Seleção entre classificações com base nas metodologias testadas e na classificação real, para a seleção de 10, 20 e 30% das melhores famílias. Para comparar a eficiência de ajuste dos modelos testados, foi utilizado o critério de Informação de Akaike (AIC), baseado em verossimilhança. A análise Espacial apresentou estimativas de variância residual muito próxima da variância residual simulada e maior acurácia seletiva estimada em todos os cenários, indicando maior precisão experimental. Com a redução na acurácia seletiva e aumento na dependência espacial, observou-se maior influência do tipo de análise sobre a classificação das famílias, sendo que a análise espacial apresentou os melhores resultados, proporcionando seleção mais eficiente das famílias do feijoeiro do que as análises tradicionais em Látice e em Blocos casualizados, principalmente, para seleção de menor número de famílias. Os resultados para acurácia seletiva estimada em função da estatística F foram muito próximos aos obtidos para a correlação de Spearman entre médias estimadas e simuladas para as famílias, indicando que a acurácia seletiva deve ser utilizada como medida de precisão experimental nos ensaios de avaliação genética.

Some Epidemic models with fractional derivatives were proved to be well-defined, well-posed and more accurate [34, 51, 116], compared to models with the conventional derivative. An Ebola epidemic model with non-linear transmission is fully analyzed. The model is expressed with the conventional time derivative with a new parameter included, which happens to be fractional (that derivative is called the 􀀀derivative). We proved that the model is well-de ned and well-posed. Moreover, conditions for boundedness and dissipativity of the trajectories are established. Exploiting the generalized Routh-Hurwitz Criteria, existence and stability analysis of equilibrium points for the Ebola model are performed to show that they are strongly dependent on the non-linear transmission. In particular, conditions for existence and stability of a unique endemic equilibrium to the Ebola system are given. Numerical simulations are provided for particular expressions of the non-linear transmission, with model's parameters taking di erent values. The resulting simulations are in concordance with the usual threshold behavior. The results obtained here may be signi cant for the ght and prevention against Ebola haemorrhagic fever that has so far exterminated hundreds of families and is still a ecting many people in West-Africa and other parts of the world. The full comprehension and handling of the phenomenon of shattering, sometime happening during the process of polymer chain degradation [129, 142], remains unsolved when using the traditional evolution equations describing the degradation. This traditional model has been proved to be very hard to handle as it involves evolution of two intertwined quantities. Moreover, the explicit form of its solution is, in general, impossible to obtain. We explore the possibility of generalizing evolution equation modeling the polymer chain degradation and analyze the model with the conventional time derivative with a new parameter. We consider the general case where the breakup rate depends on the size of the chain breaking up. In the process, the alternative version of Sumudu integral transform is used to provide an explicit form of the general solution representing the evolution of polymer sizes distribution. In particular, we show that this evolution exhibits existence of complex periodic properties due to the presence of cosine and sine functions governing the solutions. Numerical simulations are performed for some particular cases and prove that the system describing the polymer chain degradation contains complex and simple harmonic poles whose e ects are given by these functions or a combination of them. This result may be crucial in the ongoing research to better handle and explain the phenomenon of shattering. Lastly, it has become a conjecture that power series like Mittag-Le er functions and their variants naturally govern solutions to most of generalized fractional evolution models such as kinetic, di usion or relaxation equations. The question is to say whether or not this is always true! Whence, three generalized evolution equations with an additional fractional parameter are solved analytically with conventional techniques. These are processes related to stationary state system, relaxation and di usion. In the analysis, we exploit the Sumudu transform to show that investigation on the stationary state system leads to results of invariability. However, unlike other models, the generalized di usion and relaxation models are proven not to be governed by Mittag-Le er functions or any of their variants, but rather by a parameterized exponential function, new in the literature, more accurate and easier to handle. Graphical representations are performed and also show how that parameter, called ; can be used to control the stationarity of such generalized models. / Mathematical Sciences / Ph. D. (Applied Mathematics)

The computation of Schrödinger equations in the semiclassical regime presents several enduring challenges due to the presence of the small semiclassical parameter. Standard approaches for solving these equations commence with spatial discretisation followed by exponentiation of the discretised Hamiltonian via exponential splittings. In this thesis we follow an alternative strategy${-}$we develop a new technique, called the symmetric Zassenhaus splitting procedure, which involves directly splitting the exponential of the undiscretised Hamiltonian. This technique allows us to design methods that are highly efficient in the semiclassical regime. Our analysis takes place in the Lie algebra generated by multiplicative operators and polynomials of the differential operator. This Lie algebra is completely characterised by Jordan polynomials in the differential operator, which constitute naturally symmetrised differential operators. Combined with the $\mathbb{Z}_2$-graded structure of this Lie algebra, the symmetry results in skew-Hermiticity of the exponents for Zassenhaus-style splittings, resulting in unitary evolution and numerical stability. The properties of commutator simplification and height reduction in these Lie algebras result in a highly effective form of $\textit{asymptotic splitting:} $exponential splittings where consecutive terms are scaled by increasing powers of the small semiclassical parameter. This leads to high accuracy methods whose costs grow quadratically with higher orders of accuracy. Time-dependent potentials are tackled by developing commutator-free Magnus expansions in our Lie algebra, which are subsequently split using the Zassenhaus algorithm. We present two approaches for developing arbitrarily high-order Magnus--Zassenhaus schemes${-}$one where the integrals are discretised using Gauss--Legendre quadrature at the outset and another where integrals are preserved throughout. These schemes feature high accuracy, allow large time steps, and the quadratic growth of their costs is found to be superior to traditional approaches such as Magnus--Lanczos methods and Yoshida splittings based on traditional Magnus expansions that feature nested commutators of matrices. An analysis of these operatorial splittings and expansions is carried out by characterising the highly oscillatory behaviour of the solution.

An investigation into the impact of generalised maximum entropy solutions on the study of queues with or without arrival balking and their applications to congestion management in communication networks Keywords: Queues, Balking, Maximum Entropy (ME) Principle, Global Balance (GB), Queue Length Distribution (QLD), Generalised Geometric (GGeo), Generalised Exponential (GE), Generalised Discrete Half Normal (GdHN), Congestion Management, Packet Dropping Policy (PDP) Generalisations to links between discrete least biased (i.e. maximum entropy (ME)) distribution inferences and Markov chains are conjectured towards the performance modelling, analysis and prediction of general, single server queues with or without arrival balking. New ME solutions, namely the generalised discrete Half Normal (GdHN) and truncated GdHN (GdHNT) distributions are characterised, subject to appropriate mean value constraints, for inferences of stationary discrete state probability distributions. Moreover, a closed form global balance (GB) solution is derived for the queue length distribution (QLD) of the M/GE/1/K queue subject to extended Morse balking, characterised by a Poisson prospective arrival process, i.i.d. generalised exponential (GE) service times and finite capacity, K. In this context, based on comprehensive numerical experimentation, the latter GB solution is conjectured to be a special case of the GdHNT ME distribution. ii Owing to the appropriate operational properties of the M/GE/1/K queue subject to extended Morse balking, this queueing system is applied as an ME performance model of Internet Protocol (IP)-based communication network nodes featuring static or dynamic packet dropping congestion management schemes. A performance evaluation study in terms of the model’s delay is carried out. Subsequently, the QLD’s of the GE/GE/1/K censored queue subject to extended Morse balking under three different composite batch balking and batch blocking policies are solved via the technique of GB. Following comprehensive numerical experimentation, the latter QLD’s are also conjectured to be special cases of the GdHNT. Limitations of this work and open problems which have arisen are included after the conclusions.

Cette thèse est consacrée à l’analyse de ces modèles particulaires pour les mathématiques financières.Le manuscrit est organisé en quatre chapitres. Chacun peut être lu séparément.Le premier chapitre présente le travail de thèse de manière globale, définit les objectifs et résume les principales contributions. Le deuxième chapitre constitue une introduction générale à la théorie des méthodes particulaire, et propose un aperçu de ses applications aux mathématiques financières. Nous passons en revue les techniques et les résultats principaux sur les systèmes de particules en interaction, et nous expliquons comment ils peuvent être appliques à la solution numérique d’une grande variété d’applications financières, telles que l’évaluation d’options compliquées qui dépendent des trajectoires, le calcul de sensibilités, l’évaluation d’options américaines ou la résolution numérique de problèmes de contrôle et d’estimation avec observation partielle.L’évaluation d’options américaines repose sur la résolution d’une équation d’évolution à rebours, nommée l’enveloppe de Snell dans la théorie du contrôle stochastique et de l’arrêt optimal. Les deuxième et troisième chapitres se concentrent sur l’analyse de l’enveloppe de Snell et de ses extensions à différents cas particuliers. Un ensemble de modèles particulaires est alors proposé et analysé numériquement. / This thesis is concerned with the analysis of these particle models for computational finance.The manuscript is organized in four chapters. Each of them could be read separately.The first chapter provides an overview of the thesis, outlines the motivation and summarizes the major contributions. The second chapter gives a general in- troduction to the theory of interacting particle methods, with an overview of their applications to computational finance. We survey the main techniques and results on interacting particle systems and explain how they can be applied to the numerical solution of a variety of financial applications; to name a few: pricing complex path dependent European options, computing sensitivities, pricing American options, as well as numerically solving partially observed control and estimation problems.The pricing of American options relies on solving a backward evolution equation, termed Snell envelope in stochastic control and optimal stopping theory. The third and fourth chapters focus on the analysis of the Snell envelope and its variation to several particular cases. Different type of particle models are proposed and studied.

Submitted by guaracy araujo (guaraa3355@gmail.com) on 2016-06-14T19:46:59Z No. of bitstreams: 1 2010_dis_cmfreire.pdf: 426798 bytes, checksum: 4ad13c09839833ee22b0396a445e8a26 (MD5) / Approved for entry into archive by guaracy araujo (guaraa3355@gmail.com) on 2016-06-14T19:48:16Z (GMT) No. of bitstreams: 1 2010_dis_cmfreire.pdf: 426798 bytes, checksum: 4ad13c09839833ee22b0396a445e8a26 (MD5) / Made available in DSpace on 2016-06-14T19:48:16Z (GMT). No. of bitstreams: 1 2010_dis_cmfreire.pdf: 426798 bytes, checksum: 4ad13c09839833ee22b0396a445e8a26 (MD5) Previous issue date: 2010 / In Descriptive Complexity, we investigate the use of logics to characterize computational classes os problems through complexity. Since 1974, when Fagin proved that the class NP is captured by existential second-order logic, considered the rst result in this area, other relations between logics and complexity classes have been established. Wellknown results usually involve rst-order logic and its extensions, and complexity classes in polynomial time or space. Some examples are that the rst-order logic extended by the least xed-point operator captures the class P and the second-order logic extended by the transitive closure operator captures the class PSPACE. In this dissertation, we will initially analyze the expressive power of some modal logics with respect to the decision problem REACH and see that is possible to express it with temporal logics CTL and CTL . We will also analyze the combined use of higher-order logics extended by the least xed-point operator and obtain as result that each level of this hierarchy captures each level of the deterministic exponential time hierarchy. As a corollary, we will prove that the hierarchy of HOi(LFP), for i 2, does not collapse, that is, HOi(LFP) HOi+1(LFP) / Em Complexidade Descritiva investigamos o uso de logicas para caracterizar classes problemas pelo vies da complexidade. Desde 1974, quando Fagin provou que NP e capturado pela logica existencial de segunda-ordem, considerado o primeiro resultado da area, outras relac~oes entre logicas e classes de complexidade foram estabelecidas. Os resultados mais conhecidos normalmemte envolvem logica de primeira-ordem e suas extens~oes, e classes de complexidade polinomiais em tempo ou espaco. Alguns exemplos são que a l ogica de primeira-ordem estendida com o operador de menor ponto xo captura a clsse P e que a l ogica de segunda-ordem estendida com o operador de fecho transitivo captura a classe PSPACE. Nesta dissertação, analisaremos inicialmente a expressividade de algumas l ogicas modais com rela cão ao problema de decisão REACH e veremos que e poss vel express a-lo com as l ogicas temporais CTL e CTL . Analisaremos tamb em o uso combinado de l ogicas de ordem superior com o operador de menor ponto xo e obteremos como resultado que cada n vel dessa hierarquia captura cada n vel da hierarquia determin stica em tempo exponencial. Como corol ario, provamos que a hierarquia de HOi(LFP) não colapsa, ou seja, HOi(LFP) HOi+1(LFP) / FREIRE, Cibele Matos. Complexidade descritiva das lógicas de ordem superior com menor ponto fixo e análise de expressividade de algumas lógicas modais. 2010. 54 f. : Dissertação (mestrado) - Universidade Federal do Ceará, Centro de Ciências, Departamento de Computação, Fortaleza-CE, 2010.

Submitted by Pablo Aguiar De Maio (pabloamaio@outlook.com) on 2015-09-10T19:50:43Z No. of bitstreams: 1 Pablo Aguiar De Maio - Dissertação - Um método de linearização local com passo adaptativo para solução numérica de equações diferenciais estocásticas com ruído aditivo.pdf: 2233029 bytes, checksum: d3ed48936d09fde216e44fb4d688b47d (MD5) / Approved for entry into archive by Janete de Oliveira Feitosa (janete.feitosa@fgv.br) on 2015-09-25T12:16:10Z (GMT) No. of bitstreams: 1 Pablo Aguiar De Maio - Dissertação - Um método de linearização local com passo adaptativo para solução numérica de equações diferenciais estocásticas com ruído aditivo.pdf: 2233029 bytes, checksum: d3ed48936d09fde216e44fb4d688b47d (MD5) / Approved for entry into archive by Marcia Bacha (marcia.bacha@fgv.br) on 2015-09-28T16:51:14Z (GMT) No. of bitstreams: 1 Pablo Aguiar De Maio - Dissertação - Um método de linearização local com passo adaptativo para solução numérica de equações diferenciais estocásticas com ruído aditivo.pdf: 2233029 bytes, checksum: d3ed48936d09fde216e44fb4d688b47d (MD5) / Made available in DSpace on 2015-09-28T16:51:50Z (GMT). No. of bitstreams: 1 Pablo Aguiar De Maio - Dissertação - Um método de linearização local com passo adaptativo para solução numérica de equações diferenciais estocásticas com ruído aditivo.pdf: 2233029 bytes, checksum: d3ed48936d09fde216e44fb4d688b47d (MD5) Previous issue date: 2015-07-31 / In this work we present a new numerical method with adaptive stepsize based on the local linearization approach, to integrate stochastic differential equations with additive noise. We also propose a computational scheme that allows efficient implementation of this method, properly adapting the algorithm of Padé with scaling-squaring strategy to compute the exponential of matrices involved. To introduce the construction of this method, we briefly explain what stochastic differential equations are, the mathematics that is behind them, their relevance to the modeling of various phenomena, and the importance of using numerical methods to evaluate this kind of equations. A succinct study of numerical stability is also presented on the following pages. With this dissertation, we intend to introduce the necessary basis for the construction of the new method/scheme. At the end, several numerical experiments are performed to demonstrate, in a practical way, the effectiveness of the proposed method, comparing it with other methods commonly used. / Neste trabalho apresentamos um novo método numérico com passo adaptativo baseado na abordagem de linearização local, para a integração de equações diferenciais estocásticas com ruído aditivo. Propomos, também, um esquema computacional que permite a implementação eficiente deste método, adaptando adequadamente o algorítimo de Padé com a estratégia “scaling-squaring” para o cálculo das exponenciais de matrizes envolvidas. Antes de introduzirmos a construção deste método, apresentaremos de forma breve o que são equações diferenciais estocásticas, a matemática que as fundamenta, a sua relevância para a modelagem dos mais diversos fenômenos, e a importância da utilização de métodos numéricos para avaliar tais equações. Também é feito um breve estudo sobre estabilidade numérica. Com isto, pretendemos introduzir as bases necessárias para a construção do novo método/esquema. Ao final, vários experimentos numéricos são realizados para mostrar, de forma prática, a eficácia do método proposto, e compará-lo com outros métodos usualmente utilizados.