Orientador: Alagacone Sri Ranga / Banca: Sandra Augusta Santos / Banca: Cleonice Fátima Bracciali / Resumo: Dentre as diversas fórmulas de quadratura interpolatórias estão aquelas que utilizam em sua construção as propriedades dos polinômios ortogonais Pn, ou ainda dos polinômios similares Bn. Consideramos, aqui, fþormulas de quadratura envolvendo polinôomios em x da forma .n(x, .) = Pn-1(.)Pn(x) - Pn(.)Pn-1(x), e da forma Gn(x, u) = Bn-1(u)Bn(x) - Bn(u)Bn-1(x). Abordamos ainda certas fþormulas de quadratura que visam aproximar a integral de um produto de duas funções k e f sendo k Lebesgue integrþavel e f Riemann integrþavel. O principal objetivo deste trabalho þe analisar propriedades das fþormulas de quadratura utilizando-se .n e obter propriedades anþalogas para o caso onde utiliza-se Gn, bem como estudar o erro e as propriedades de convergência das fórmulas envolvendo k e f. Propriedades dos pesos das fórmulas de quadratura nos diversos casos são analisadas, a convergência das fórmulas associadas a k e f são estudadas mediante determinadas escolhas de pontos. / Abstract: Among the many well known quadrature formulas one finds those interesting interpolatory quadrature formulas that take advantage of the properties of orthogonal polynomials Pn or similar polynomials Bn. Here, we consider the interpolatory quadrature rules based on the zeros of the polynomials øn(x, î) = Pn.1(î)Pn(x).Pn(î)Pn.1(x), and Gn(x, u) = Bn.1(u)Bn(x) . Bn(u)Bn.1(x) where î and u are arbitrary parameters. One of the objective of this dissertation is to study some of the known properties of quadrature rules based on øn(x, î) and consider the analogous properties of the quadrature rules based on Gn(x, u).We also look at the convergence properties of those quadrature rules that serve to approximate integrals of the product of functions k and f, where k is a Lebesgue integrable function and f needs to be a Riemann integrable function. / Mestre

A distribuição de massa molar (DMM) e seus parâmetros são de fundamental importância na caracterização dos polímeros. Por este motivo, o desenvolvimento de técnicas que permitam a determinação da DMM de forma mais rápida e a menor custo é de grande importância prática. Os principais objetivos deste trabalho foram a implementação de alguns dos modelos baseados da teoria da reptação dupla propostos na literatura para descrever o mecanismo de relaxação das cadeias poliméricas, a avaliação dessas implementações e a análise de dois passos fundamentais na obtenção da DMM a partir de dados reológicos que são a metodologia de cálculo do espectro de relaxação baseado no modelo de Maxwell e a estratégia para a avaliação numérica das integrais que aparecem nos modelos de relaxação. Foi resolvido o problema denominado problema inverso, ou seja, a determinação da DMM a partir de dados reológicos usando um modelo de relaxação especificado e uma função de distribuição imposta. Foi usada a função Exponencial Generalizada (GEX) para representar a probabilidade de distribuição, sendo consideradas duas abordagens: i) cálculo explícito do espectro de relaxação e ii) aproximações paramétricas de Schwarzl, que evitam a necessidade do cálculo explícito do espectro de relaxação. A metodologia de determinação da DMM foi aplicada para amostras de polietileno e foram estimadas distribuições com boa representação dos dados experimentais do GPC, ao considerarem-se amostras com polidispersões inferiores a 10. Com relação a metodologia de cálculo do espectro de relaxação, foi realizado um estudo comparativo da aplicação de espectros de relaxação discreto e contínuo, com o objetivo de estabelecer critérios para especificação do número ótimo de modos de Maxwell a serem considerados. Ao efetuar-se a comparação entre as técnicas, verificou-se o espectro discreto apresenta como um sistema melhor condicionado, permitindo assim obter maior confiabilidade dos parâmetros estimados. Também é proposta uma modificação da metodologia de determinação da DMM, em que é aplicada a quadratura de Gauss-Hermite para a resolução numérica da integral dos modelos de relaxação. / The molecular weight distribution (MWD) and its parameters are of the fundamental importance in the characterization of polymers. Therefore, the development of techniques for faster and less time consuming determination of the MWD is of great practical relevance. The goals of this work were the implementation of some of the relaxation models from double reptation theory proposed in the literature, the evaluation of these implementations and the analysis of two key points in the recovery of the MWD from rheological data which are the methodology for calculation of the relaxation spectrum based on the Maxwell model and the numeric strategy for the evaluation of the integrals appearing in the relaxation models. The inverse problem, i.e., the determination of the MWD from rheological data using a specified relaxation model and an imposed distribution function, was solved. In the analysis of the inverse problem, the Generalized Exponential (GEX) was used as distribution function and two approaches were considered: i) explicit calculation of the relaxation spectrum and ii) use of the parametric method proposed by Schwarzl to avoid the explicit calculation of the relaxation spectrum. In the test of commercial samples of polyethylene with polidispersity less than 10, the application of this methodology led to MWD curves which provided good fit of the experimental SEC data. Regarding the methodology for calculation of the relaxation spectrum, a comparison between the performance of discrete and continuous relaxation spectrum was performed and some possible a criteria to determine the appropriate number of relaxation modes of Maxwell to be used were evaluated. It was found that the technique of discrete spectrum leads to better conditioned systems and, consequently, greater reliability of the estimated parameters. With relation to the numeric strategy for the evaluation of the integrals appearing in the relaxation models, the use of Gauss-Hermite quadrature using a new change of variables was proposed.

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京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第22861号 / 情博第740号 / 新制||情||127(附属図書館) / 京都大学大学院情報学研究科先端数理科学専攻 / (主査)教授 青柳 富誌生, 教授 磯 祐介, 教授 田口 智清 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

There exist many numerical methods for numerical solutions of the systems of stochastic differential equations. We choose the method of deterministic quadrature formulae proposed by Müller–Gronbach, and Yaroslavtseva in 2016. The idea is to apply a simplified version of the cubature in Wiener space. We explain the method and check how good it works in the simplest case of the classical Black–Scholes model.

Les systèmes de communication sans fil modernes exigent haute performance analogique Convertisseurs-numériques (CAN) avec l'augmentation de la bande passante et la résolution.Aujourd'hui, il y a un besoin croissant de faible puissance et de récepteurs RF multi-fonctionnels, puisque le marché s' attend à des capacités de réception complexes avec des appareils de faible puissance qui fonctionnent sur batteries portables de puissance limitée.Pour cette raison la tendance actuelle est de diminuer la partie analogique des récepteurs, tout en augmentant les tâches effectuées par la partie numérique.Par conséquent, cela demande des CAN à large bande, haute résolution et faible consommation.Dans cette recherche, on étudie plusieurs CAN à base de VCO.On montre la conception, la réalisation dans le process CMOS 65nm et les mesures de deux types de CAN à base VCO, le premier est basé sur le principe de la conversion tension-fréquence tandis que le second utilise le principe de la conversion tension-phase.Le CAN tension fréquence est un CAN de 4-bit programmable avec une fréquence d’échantillonnage qui va de 220MHz jusqu’à 1500MHz. le rapport signal dur bruit mesuré est de 40.5dB dans une bande de 30MHz avec une consommation de 0.5mW.Le CAN tension phase est un CAN de 4-bit programmable avec une fréquence d’échantillonnage qui va de 300MHz jusqu'a 1440MHz. le rapport signal dur bruit mesuré est de 48dB dans une bande de 30MHz avec une consommation de 1mW. On présente ensuite une méthode de conception systématique de conception des CAN SigmaDelta de grand ordre avec des quantificateurs à base VCO.Pour valider la méthode de conception, un CAN SigmaDelta avec un quantificateur tension-fréquence est conçu en 65nm. Le rapport signal sur bruit mesuré est de 62dB dans une bande de 28MHz et une consommation de 30mW.On propose ensuite l'utilisation des quantificateurs à base VCO dans les modulateurs SigmaDelta en quadrature. Pour cela, une méthode de conception systématique et présentée. Un CAN sigmadelta en quadrature de 4ème ordre avec des quantificateurs tension fréquence est conçu en 65nm. Les mesures de ce circuit sont encore encours. Les simulations post-layout montrent un rappost signal sur bruit de 75dB dans une bande de 40MHz et une consommation de 60mW. / Today's wireless communication systems are requiring high performance Converters analog-digital (ADC) with increasing demand on bandwidth and resolution.There is a growing need for low-power and multi-functional RF receivers , since the market is expecting complex receiving capacities with low power battery operated devices.For this reason the current trend is to decrease the analogue part of the receivers, while increasing the tasks performed by the digital part.Therefore, this imposes stringent requirements on the ADC such as wideband operation, high resolution and low power consumption.In this dissertation, we studied and realized several types of VCO-based ADCs.We show the design, implementation and the measurements of two types of VCO-based ADCs in 65nm CMOS process. The first is using the voltage to frequency conversion technique while the second uses the principle of voltage to phase conversion.The voltage to frequency converter is a 4-bit ADC with a programmable sampling frequency that goes from 220MHz up to 1500MHz.The measured Signal-to-noise-and-distortion-ratio (SNDR) is of 40.5dB in a band of 30MHz with a power consumption of 0.5mW.The voltage phase converter is a 4-bit ADC with a programmable sampling frequency that goes from 300MHz up to 1440MHz.The measured SNDR is 48dB in a band of 30MHz with a consumption of 1mW.We then present a systematic design method of high order SigmaDelta ADCs with VCO-based quantizers.To validate the design method, a SigmaDelta ADC with a 4-bit voltage-frequency is designed in 65nm. The measured SNDR is 62dB in a band of 28MHz and a power consumption of 30mw.We propose the use of VCO-based quantizers in quadrature SigmaDelta modulators. A systematic design method is presented for the quadrature VCO-based Sigmadelta modulators.A 4th order quadrature sigmadelta with 4-bit voltage to frequency quantizers is designed in 65nm. The measurements of this circuit are currently in progress. In post layout simulations, the quadrature modulator achieves 75dB in a band of 40MHz and a power consumption of 60mW.

In this thesis we study two reaction-diffusion models that have been used to analyze the existence of alternate stable states in ecosystems. The first model describes the steady states of a logistic growth model with grazing in a spatially homogeneous ecosystem. It also describes the dynamics of the fish population with natural predation. The second model describes phosphorus cycling in stratified lakes. The same equation has also been used to describe the colonization of barren soils in drylands by vegetation. In this study we discuss the existence of multiple positive solutions, leading to the occurrence of S-shaped bifurcation curves. We were able to show that both the models have alternate stable states for certain ranges of parameter values. We also introduce a constant yield harvesting term to the first model and discuss the existence of positive solutions including the occurrence of a Sigma-shaped bifurcation curve in the case of a one-dimensional model. Again we were able to establish that for certain ranges of parameter values the model has alternate stable states. Thus we establish analytically that the above models are capable of describing the phenomena of alternate stable states in ecological systems. We prove our results by the method of sub-super solutions and quadrature method.

Complex filters are multi-input, multi-output networks designed to discriminate based upon the relative phase difference between input signals. Complex filters find application in modern wireless systems for single sideband transmission and image-reject reception. This thesis presents one active complex filter implementation using two operational amplifiers per stage, termed “type-II” topology. The “type-II” originates from the passive RC-CR polyphase topology presented by Gingell in his 1973 paper, “Single sideband modulation using sequence asymmetric polyphase networks.” This new topology gains several advantages over existing complex filter implementations, namely “cascadability” (multiple sections placed in series to create a higher-order response) without altering the characteristics of each individual stage. In addition to describing the derivation of the topology and its performance relative to existing topologies, this thesis investigates the passband characteristics of a general higher-order filter and provides a passband-centric design approach through derivations of closed form expressions for passband gain and bandwidth. The thesis includes a five-stage design example using this approach in addition to an implementation, its characterization, and its comparison to the derived expressions and simulations.

This thesis presents an active Complex Filter implementation that that creates a transfer function with with a single real pole and a complex zero. The two-input/two-output network developed in in this thesis responds differently based upon upon the relative phase difference of of the two inputs. If a negative ninety-degree phase difference occurs between the two inputs, the filter will exhibits a bandstop response. While a positive ninety-degree phase difference exhibits a bandpass response. This topology is relatesd to to Gingell’s RC-CR polyphase topology but because of of the use of of op-amps, can be cascadedd without without suffering loading effects. This thesis will focusfocuses primarily on on the bandstop response characteristics of of the filter. In a several stage cascade, each stage contributes a notch to broaden the attenuation bandWhen several sections are cascaded, multiple notches will be created from each stage that forms a broader attenuation band. Closed form design equations were were derived to to give expressions for for the “attenuation floor”. These equations can be used by a designer to predict the attenuation provided by by a cascaded system. The closed form expressions derived in in this thesis are used to implement an example five-stage topology that that operates from from 147 Hz to to 3.34 KHz. The thesis also investigates the robustness of of multi-stage cascades to to component variations. Monte Carlo analysis is used to determines the effects of of cascading the filter in in different orders, component tolerances, and a comparison to to an idealized polyphase RC-CR topology.

A mathematical model is developed to investigate the grazing dynamics of tapping mode atomic force microscopes (AFM) subjected to a base harmonic excitation. The nonlinear dynamics of the AFM microcantilever are studied in both of the monostable and bistable phases with the microcantilever tip being, respectively, located in the monostable and bistable regions of the static bifurcation diagram in the reference configuration. Free-vibration responses of the AFM probes, including the microcantilever natural frequencies and mode shapes, are determined. It is found that, for the parameters used in a practical operation of an AFM, the natural frequencies and mode shapes of the AFM microcantilever are almost the same as those of a free-end microcantilever with the same geometry and made of an identical material. A multimode Galerkin approximation is utilized to discretize the nonlinear partial-differential equation of motion and associated boundary conditions governing the cantilever response and obtain a set of nonlinearly coupled ordinary-differential equations (ODE) governing the time evolution of the system dynamics. The corresponding nonlinear ODE set is then solved using numerical integration schemes. A comprehensive numerical analysis is performed for a wide range of the excitation amplitude and frequency. The tip oscillations are examined using nonlinear dynamic tools through several examples. The non-smoothness in the tip/sample interaction model is treated rigorously. A higher-mode Galerkin analysis indicates that period doubling bifurcations and chaotic vibrations are possible in tapping mode microscopy for certain operating parameters. It is also found that a single-mode Galerkin approximation, which accurately predicts the tip nonlinear responses far from the sample, is not adequate for predicting all of the nonlinear phenomena exhibited by an AFM, such as grazing bifurcations, and leads to both quantitative and qualitative errors. A point-mass model is also developed based on the single-mode Galerkin procedure to compare with the present distributed-parameter model. In addition, a reduced-order model based on a differential quadrature method (DQM) is employed to explore the dynamics of the AFM probe in the bistable phase where the multimode Galerkin procedure is computationally expensive. We found that the DQM with a few grid points accurately predicts the static bifurcation diagram. Moreover, we found that the DQM is capable of precise prediction of the lowest natural frequencies of the microcantilever with only a few grid points. For the higher natural frequencies, however, a large number of grid points is required. We also found that the natural frequencies and mode shapes of the microcantilever about non-contact equilibrium positions are almost the same as those of the free-end microcantilever. On the other hand, free-vibration responses of the microcantilever about contact equilibrium positions are quite different from those of the free-end microcantilever. Moreover, we used the DQM to discretize the partial-differential equation governing the microcantilever motion and a finite-difference method (FDM) to calculate limit-cycle responses of the AFM tip. It is shown that a combination of the DQM and FDM applied, respectively, to discretize the spatial and temporal derivatives provides an efficient, accurate procedure to address the complicated dynamic behavior exhibited by the AFM probe. The procedure was, therefore, utilized to study the response of the microcantilever to a base harmonic excitation through several numerical examples. We found that the dynamics of the AFM probe in the bistable region is totally different from those in the monostable region. / Ph. D.