Este trabalho aborda o problema de modelagem de séries de vazões médias mensais visando previsão e geração de séries sintéticas. Destaca-se que a importância da previsão de valores futuros das séries mensais de vazões assim como a geração de séries sintéticas são fundamentais para o planejamento da operação de sistemas hidroelétricos brasileiros. Estas séries possuem um comportamento periódico na média, na variância e na função de autocorrelação e, portanto, considera-se para a série padronizada os modelos autorregressivos periódicos PAR(pm). Em relação a previsão clássica, a análise do erro de previsão e feita em função do horizonte de previsão. Neste estudo, os erros de previsão são calculados, na escala original da série de vazão, em função dos parâmetros dos modelos ajustados e avaliados para horizontes de previsão h variando de 1 a 12 meses. Estes erros são comparados com as estimativas das variâncias das vazões para o mês que está sendo previsto. Em relação à previsão bayesiana, adota-se os modelos Normal, Log-Normal e t-Student nos processos de estimação e, após, é realizado um estudo da perfomance destes modelos usando o erro quadrático médio, erro absoluto médio e erro percentual absoluto médio. Em relação à geração de séries sintéticas de vazões, um modelo multivariado Log-Normal com três parâmetros e um modelo Log-Normal generalizado foram desenvolvidos. As séries geradas são comparadas com as séries históricas reais utilizando o critério de Kullback-Leibler. Como resultado, tem-se uma avaliação da capacidade de previsão, em meses, dos modelos ajustados para cada mês e a escolha do modelo Log-Normal nos procedimentos de análise bayesiana. Além disso, o modelo utilizado para a geração de séries sintéticas de vazões mensais forneceu evidências que o apontam como uma alternativa ao modelo amplamente adotado no setor elétrico brasileiro para geração de séries de vazões / This work addresses the problem of forecasting and generation series monthly average streamflows. It is noteworthy that the importance of forecasting future values of the series of monthly streamflows as well as the generation of synthetic series are fundamental for planning the operation of Brazilian hydroelectric systems. These series have a periodic behavior on average, variance and autocorrelation function and therefore it is considered for standard series periodic autoregressive models PAR(pm). At the forecast classical analysis of the prediction error is made in function of the prediction horizon. In this study, the forecasting errors are calculated in the original scale of the series of streamflow, depending on the model parameters adjusted and evaluated for forecasting horizons h ranging from 1 to 12 months. These errors are compared with estimates of the variances of the streamflows for the month is provided. Regarding the bayesian prediction, we adopt the models Normal, Log-Normal and t-Student in estimation procedures and, then, is a study of the performance of these models using the mean square error, mean absolute error and mean absolute percentage error. In relation to generation, a Log-Normal multivariate model with three parameters and a Log-Normal generalized model were developed and analyzed using the Kullback-Leibler criterion. As a result there has been an assessment of the predictive power, in months, the adjusted models for each month, the choice of the Log-Normal model in the procedures for bayesian analysis and the model used to generate synthetic series of monthly streamflows provided evidence that point as an alternative model adopted in the Brazilian electric sector

Uma órbita bilhar em um triângulo é uma poligonal cujos segmentos começam e terminam nos lados do triângulo e que se refletem elasticamente nestes lados. É como o movimento de uma bola numa mesa de bilhar sem atrito (logo a bola tem velocidade constante e jamais para) cujas laterais formam um triângulo. Esta órbita é periódica se ela retorna infinitas vezes ao mesmo ponto com a mesma direção. A existência de órbitas bilhares periódicas em polígonos é uma questão aberta da matemática. Mesmo para um triângulo ainda não há resposta. Para triângulos agudos, a resposta é bem conhecida, pois o triângulo formato pelos pés das alturas do triângulo é uma órbita periódica. Para triângulos obtusos, em geral, pouco se sabe. O objetivo desta dissertação é coletar resultados e técnicas sobre órbitas bilhares periódicas em triângulos obtusos. Começamos introduzindo o trabalho de Vorobets, Galperin e Stepin, que no início dos anos 90 unificaram os casos conhecidos de triângulos que possuem órbita bilhar periódica, introduziram o conceito de estabilidade e mostraram novos resultados, como uma família infinita de órbitas estáveis. Temos também o teorema de 2000 de Halbeisen e Hungerbühler que estende as famílias de órbitas estáveis. Mencionamos em seguida os trabalhos de Schwartz de 2006 e 2009 que utilizam auxílio computacional para mostrar que todo triângulo com ângulos menores que $100\\degree$ possui órbita bilhar periódica. Depois temos os resultados de 2008 de Hooper e Schwartz sobre órbitas bilhares periódicas em triângulos quase isósceles e sobre estabilidade de órbitas em triângulos de Veech. Todos os casos abordados neste trabalho incluem uma vasta variedade de triângulos, mas a questão de existência de órbitas bilhares periódicas para todo triângulo está longe de ser totalmente contemplada. / A billiard orbit in a triangle is a polygonal with vertices at the boundary of the triangle such that its angles reflect elastically. It is similar to a moving ball on a billiard table without friction (so the ball has constant speed and never stops) whose sides form a triangle. This orbit is periodic if it returns infinitely to the same point with the same direction. The existence of periodic billiard orbits in polygons is an open problem in mathematics. Even for a triangle there is still no answer. For acute triangles the answer is well known since the triangle whose vertices are the base points of the three altitudes of the triangle is a periodic orbit. For obtuse triangles, in general, little is known. The aim of this thesis is to collect results and techniques on periodic billiard orbits in obtuse triangles. We start by introducing the work of Vorobets, Gal\'perin and Stepin, who unified in the early 1990s the known cases of triangles that have periodic billiard orbits, introduced the concept of stability and proved new results, such as an infinite family of stable orbits. We also have the theorem of Halbeisen and Hungerbühler of 2000 extending the families of stable orbits. Next, we mention the works of Schwartz of 2006 and 2009 that use computational assistance to prove that every triangle whose angles are at most $100\\degree$ have periodic billiard orbits. Then, we have the results of 2008 by Hooper and Schwartz on periodic billiard orbits in nearly isosceles triangles and on stability of billiard orbits in Veech triangles. All cases covered in this work include a wide variety of triangles, but the question of the existence of periodic billiard orbits for all triangles is far from being fully contemplated.

Metamaterials offer the freedom to tune the rich electromagnetic coupling between the constituent meta-atoms to tailor their collective electromagnetic response. Therefore, a comprehensive understanding of the nature of electromagnetic interactions between meta-atoms is necessary for novel metamaterial design, which is provided in the first part of this thesis. The subsequent work in the thesis applies the understanding from the first part to design and demonstrate novel one-dimensional metamaterials that overcome the limitations of metamaterials proposed in literature or exhibit electromagnetic responses not previously observed. Split-ring Resonators (SRRs) are a fundamental building block of many electromagnetic metamaterials. In the first part of the work in this thesis, it is shown that bianisotropic SRRs (with magneto-electric cross-polarisation) when in close proximity to each other, exhibit a rich coupling that involves both electric and magnetic interactions. The strength and nature of the coupling between two identical SRRs are studied experimentally and computationally as a function of their separation and relative orientation. The electric and magnetic couplings are characterised and it is found that, when SRRs are close enough to be in each other's near-field, the electric and magnetic couplings may either reinforce each other or act in opposition. At larger separations retardation effects become important. The findings on the electromagnetic interactions between bianisotropic resonators are next applied to developing a one-dimensional ultra-wideband backward-wave metamaterial waveguide. The key concept on which the metamaterial waveguide is built is electro-inductive wave propagation, which has emerged as an attractive solution for designing backward-wave supporting metamaterials. Stacked metasurfaces etched with complementary SRRs (CSRRs) have also been shown to exhibit a broadband negative dispersion. It is demonstrated through experiment and numerical modeling, that the operational bandwidth of a CSRR metamaterial waveguide can be improved by restricting the cross-polarisation effects in the constituent meta-atoms. The metamaterial waveguide constructed using the modified non-bianisotropic CSRRs are found to have a fractional bandwidth of 56.3\% which, based on a thorough search of relevant literature, is the broadest reported value for an electro-inductive metamaterial. A traditional coupled-dipole toy-model is presented as a tool to understand the field interactions in CSRR based metamaterials, and to explain the origin of their negative dispersion response. This metamaterial waveguide should be of assistance in the design of broadband backward-wave metamaterial devices, with enhanced electro-inductive waveguiding effects. In the final part of the thesis, a one-dimensional metamaterial prototype that permits simultaneous forward- and backward-wave propagation is designed. Such a metamaterial waveguide could act as a microwave analogue of nanoparticle chains that support electromagnetic energy transfer with a positive or a negative dispersion due to the excitation of their longitudinal or transverse dipole modes. The symmetry of the designed hybrid meta-atom permits the co-existence of two non-interfering resonances closely separated in frequency. It is experimentally and computationally shown that the metamaterial waveguide supports simultaneous non-interacting forward- and backward-wave propagation in an overlapping frequency band. The proposed metamaterial design should be suitable for realising bidirectional wireless power transfer applications.

Objetivo: Comparar a resposta ao tratamento entre pacientes tabagistas e não tabagistas e durante a fase de manutenção periódica preventiva (MPP), ao longo de 2 anos. Materiais e Métodos: Cinquenta e um pacientes com periodontite moderada a avançada, sendo 22 tabagistas (idade média de 48,3+8,1; 54,5% do sexo masculino; 6,4+5,2 dentes ausentes, tempo médio de exposição ao tabaco 21,6+11,4 anos, média de cigarros fumados ao dia 12,9+8,1) e 29 não tabagistas (idade média de 54,1+9,4; 69% do sexo masculino; 7,0+4,4 dentes ausentes) foram tratados de acordo com um protocolo não-cirúrgico. Finalizada a fase terapêutica, os pacientes iniciaram a fase de MPP. Exames periodontais, instrução de higiene bucal de acordo com as necessidades individuais, e as intervenções de manutenção foram realizadas em consultas trimestrais. Modelos de equações de estimativas generalizadas foram aplicados para avaliar o impacto do fumo na resposta ao tratamento periodontal e manutenção ao longo do tempo. Resultados: Não foram encontradas diferenças significativas em relação ao índice de placa (IP) entre tabagistas e não tabagistas, desde o exame basal até o último exame da fase de MPP (24 meses). O índice gengival comportou-se de forma semelhante ao de placa, sem diferenças entre tabagistas e não tabagistas ao longo do tratamento e da fase de MPP. O sangramento à sondagem (SS) diminuiu significantemente ao longo do tempo do estudo e uma maior diminuição ocorreu nas bolsas de PS de >6mm. A média de profundidade de sondagem (PS) para não tabagistas não apresentou diferença estatística entre os grupos, bem como a perda de inserção (PI). Conclusão: O presente estudo mostrou que o hábito do tabagismo não afetou a resposta ao tratamento periodontal e a manutenção dos resultados ao longo de 24 meses. / Objective: To compare the effect to treatment of smokers and non-smokers during periodic preventive maintenance (MPP) over a period of 2 years. Materials and Methods: Fifty-one patients with moderate to advanced periodontitis, 22 smokers (mean age 48.3 + 8.1, 54.5% males, 6.4 + 5.2 missing teeth, mean time of tobacco smoking of 21.6 +11.4 years, mean of cigarettes smoked per day 12.9 +8.1) and 29 non-smokers (mean age of 54.1+9.4; 69% of males, 7.0+4.4 teeth missing) were treated according to a non-surgical protocol. After the therapeutic phase, the patients started the MPP phase. Periodontal examinations, oral hygiene instruction according to individual needs, and the interventions of MPP were carried out in quarterly consultations. Models of generalized estimation equations were applied to evaluate the impact of smoking on the response to periodontal treatment and maintenance over time. Results: There were no significant differences in the plaque index (PI) between smokers and non-smokers from baseline to the last exam of the MPP phase (24 months). Gingival index was similar to the plaque index, no differences between smokers and non-smokers throughout the treatment and MPP phase. Bleeding on probing (BOP) decreased significantly over the study time and a larger decrease occurred in probing depth (PD) pockets of > 6mm. The mean probing depth (PD) for non-smokers did not present statistical difference between groups, as well as clinical attachment level (CAL). Conclusion: The present study showed that the smoking habit did not affect the response to periodontal treatment and the maintenance of the results over the period of 24 months.

Demonstramos o seguinte teorema: Seja M uma variedade Riemanniana compacta, conexa e sem bordo. Dados um endomorismo f : M ightarrow M, uma função contínua \\phi: M ightarrow R e \\epsilon > 0, então existe um endomorísmo \\tilde f : M ightarrow M tal que d(f; \\tide f) = \\max_{x \\in M} d(f(x); \\tilde f(x)) < \\epsilon, e existe uma medida \\phi-maximizante para \\tilde f que está suportada em uma orbita periodica. Este teorema e uma generalização dos resultados obtidos por S. Addas-Zanatta e F. Tal. / We prove the following theorem: Let M be a bondaryless, compact and connected Riemannian Manifold. Given an endomorphism f on M, a continuous function \\phi : M ightarrow R and \\epsilon > 0, then there exist an endomorphism \\tilde f on M with d(f; \\tilde f) < \\epsilon such that, some \\phi-maximizing measure for \\tilde f is supported on a periodic orbit. This theorem is a generalization of the results obtained by S. Addas-Zanatta and F. Tal.

We investigate the existences and qualitative properties of periodic solutions of the following two classes of nonlinear differential equations: I) (Special) Relativistic Pendulum Equations (RPEs); II) (2-coupled) Gross-Pitaevskii Equations (GPEs). The pendulum equation describes the motion of a pendulum. According to Special Relativity, which was published by A. Einstein in 1905, causality is more fundamental than constant time-space, thus time will ow slower and space will distort to keep causality if the speed of motion is near the speed of light. In such high speed situations, the pendulum equation needs to be revised due to Special Relativity. The revised equation is called RPE. Our result answers some open questions about the existence of multiple periodic solutions for RPEs. GPEs are sometimes called coupled nonlinear schrodinger equations. the Schrodinger equation is the fundamental equation of Quantum Mechanics which is the \exotic" probabilistic fundamental physics law of the \micro" world { the world of atoms and molecules. A well-known physicist and Nobel laureate, R. Feynman, said \I think I can safely say that nobody understands quantum mechanics." which indicates the physical/ philosophical difficulty of interpretations. It raises paradoxical problems such the well-known Schrodinger's Cat. Setting aside these difficult, if we combine Special Relativity and Quantum Mechanics as a many-body system, then we have Quantum Field Theory (QFT) which is more deterministic, and governs even elementary particle physics. GPEs are also related to QFT. For example, superconductivity and Bose Einstein Condensates (BEC). These phenomena in condensed matter physics can be thought of as the emergence of the mysterious micro world physics at \macro" level. We study these equations from the viewpoint of mathematical interest. It is generally difficult to solve nonlinear differential equations. It is also generally difficult even to prove the existence of solutions. Although we show there exist solutions, we still do not know how to solve the differential equations analytically. Variational Methods (or Calculus of Variations) are useful tools to show there exist solutions of differential equations. The idea is to convert the problem of solving equations into the problem of finding critical points (i.e. minimum/maximum points or saddle points) of a functional, and each critical point can generally correspond to a weak solution. However, it is also generally difficult to find out such critical points because we look for critical points in an infinite-dimensional functions space. Thus many advanced mathematical theories or tools have been developed and used for decades in nonlinear analysis. We use some topological theories. From information of the functional's shape, these theories deduce if there exists a critical point, or how many critical points exist. The key of these theories is to use the symmetry of the equations. We also investigate bifurcation structures for II), i.e. the connection structures between the solutions. By linearizations which look at the equations \locally," we reduce the problem in the infinite dimension to one in a finite dimension. Furthermore, it allows us to apply Morse Theory, which connects between local and global aspects of the functional's information. In several cases, we show that there are infinitely many bifurcation points that give rise to global bifurcation branches.

In this work we deal with the nonlinear o.d.e u"+ku = εu³ which represents a spring-mass system with no damping but perturbed by external force εu³. We want to know how the spring constant k and the perturbed term act on the equation. So we study this equation by the way: (I) u" + ku = 0 (II)u" = u³ (III) u" + ku = εu³ During the period of calculating, we find that k, ε and energy constant E(0) play important roles in the properties of the solutions of the equation. Finally we give the relation about them.

Recent years have seen a growing interest in a new kind of periodic structures called ``metamaterials'. These new artificial materials exhibit many new appealing properties, not found in nature, and open many new possibilities in the domain of antenna design. This thesis describes efficient numerical tools and methods for the analysis of infinite and finite periodic structures. A numerical simulation code based on the Method of Moments has been developed for the study of both large phased arrays and periodic metamaterials made of metal and/or dielectrics. It is shown how fast infinite-array simulations can be used in a first instance to approximately describe the fields radiated by large antenna arrays or compute transmission and reflection properties of metamaterials. These infinite-array simulations rely on efficient computation schemes of the doubly periodic Green’s function and of its gradient. A technique based on eigenmode analysis is also described, that allows to efficiently compute the dispersion curves of periodic structures. Accounting for the finiteness of real structures is possible in good approximation thanks to a finite-by-infinite array approach. Moreover, the excitation of large finite periodic structures by a single (non periodic) source can be studied by using a combination of the Array Scanning Method with a windowing technique. All these techniques were validated numerically on several examples and it is finally shown how they can be combined to design high gain antennas, based on metamaterial superstrates excited by a slotted waveguide. The proposed design method relies on the separation of the whole structure in two different problems. An interior problem is used to optimize the input impedance of the antenna, while the radiation pattern can be optimized in the exterior problem.

Lately, there has been an increasing interest in studying the propagation of electromagnetic waves in periodic dielectric structures (photonic crystals). Like the electron propagation in semiconductors, these structures are represented by band diagrams in which gaps can be found where the electromagnetic propagation is forbidden. Much effort is dedicated to find structures that can prohibit the propagation of light in all directions. This effect could lead to light localization. / Singapore-MIT Alliance (SMA)

Dengue (pronounced den'guee) Fever (DF) and Dengue Hemorrhagic Fever (DHF), collectively known as "dengue," are mosquito-borne, potentially mortal, flu-like viral diseases that affect humans worldwide. Transmitted to humans by the bite of an infected mosquito, dengue is caused by any one of four serotypes, or antigen-specific viruses. In this thesis, both the spatial and temporal dynamics of dengue transmission are investigated. Different chapters present new models while building on themes of previous chapters. In Chapter 2, we explore the temporal dynamics of dengue viral transmission by presenting and analyzing an ODE model that combines an SIR human host- with a multi-stage SI mosquito vector transmission system. In the case where the juvenile populations are at carrying capacity, juvenile mosquito mortality rates are sufficiently small to be absorbed by juvenile maturation rates, and no humans die from dengue, both the analysis and numerical simulations demonstrate that an epidemic will persist if the oviposition rate is greater than the adult mosquito death rate. In Chapter 3, we present and analyze a non-autonomous, non-linear ODE system that incorporates seasonality into the modeling of the transmission of the dengue virus. We derive conditions for the existence of a threshold parameter, the basic reproductive ratio, denoting the expected number of secondary cases produced by a typically infective individual. In Chapter 4, we present and analyze a non-linear system of coupled reaction-diffusion equations modeling the virus' spatial spread. In formulating our model, we seek to establish the existence of traveling wave solutions and to calculate spread rates for the spatial dissemination of the disease. We determine that the epidemic wave speed increases as average annual, and in our case, winter, temperatures increase. In Chapter 5, we present and analyze an ODE model that incorporates two serotypes of the dengue virus and allows for the possibility of both primary and secondary infections with each serotype. We obtain an analytical expression for the basic reproductive number, R_0, that defines it as the maximum of the reproduction numbers for each strain/serotype of the virus. In each chapter, numerical simulations are conducted to support the analytical conclusions.