On a manifold with edge we construct a specific class of (edgedegenerate) elliptic differential operators. The ellipticity refers to the principal symbolic structure σ = (σψ, σ^) of the edge calculus consisting of the interior and edge symbol, denoted by σψ and σ^, respectively. For our choice of weights the ellipticity will not require additional edge conditions of trace or potential type, and the operators will induce isomorphisms between the respective edge spaces.

We give a brief survey on some new developments on elliptic operators on manifolds with polyhedral singularities. The material essentially corresponds to a talk given by the author during the Conference “Elliptic and Hyperbolic Equations on Singular Spaces”, October 27 - 31, 2008, at the MSRI, University of Berkeley.

We construct elliptic elements in the algebra of (classical pseudo-differential) operators on a manifold M with conical singularities. The ellipticity of any such operator A refers to a pair of principal symbols (σ0, σ1) where σ0 is the standard (degenerate) homogeneous principal symbol, and σ1 is the so-called conormal symbol, depending on the complex Mellin covariable z. The conormal symbol, responsible for the conical singularity, is operator-valued and acts in Sobolev spaces on the base X of the cone. The σ1-ellipticity is a bijectivity condition for all z of real part (n + 1)/2 − γ, n = dimX, for some weight γ. In general, we have to rule out a discrete set of exceptional weights that depends on A. We show that for every operator A which is elliptic with respect to σ0, and for any real weight γ there is a smoothing Mellin operator F in the cone algebra such that A + F is elliptic including σ1. Moreover, we apply the results to ellipticity and index of (operator-valued) edge symbols from the calculus on manifolds with edges.

The polarization behaviour for six species of Scarab beetles from the Chrysina genus is investigated with Mueller Matrix Spectroscopic Ellipsometer (MMSE). The m41 element of the matrix, which is related to the circular polarization behaviour, is analysed. The ellipticity, degree of polarization and azimuth angle are also presented to get a better understanding of the polarization effect. The measurements were done with a dual rotating compensator ellipsometer. The measured wavelength region was from 240 to 1000 nm and the angle of incidence from 25° to 75° in most of the cases. In general very high ellipticities (near circular) are reported. All specimens studied reflect both right- and left-handed polarized light. Depending on the species, two general types of polarization behaviour were observed. Chrysina macropus and Chrysina peruviana showed m41 values close to 0. Green stripes on Chrysina gloriosa showed similar polarization behaviour whereas gold stripes on the same beetle had much more pronounced m41 variations. Large m41 variations were also observed for Chrysina argenteola, Chrysina chrysargyrea and Chrysina resplendens. Four specimens of Chrysina resplendens show different m41 patterns suggesting differences in their structures.

Light reflected from the scarab beetles Cetonia aurata (C. aurata) and Chrysina gloriosa (C. gloriosa) has left-handed polarization. In this work the polarizing properties and structural characteristics of the cuticles of these two beetles are investigated with two different techniques: scanning electron microscopy (SEM) and Mueller-matrix spectroscopic ellipsometry (MMSE). SEM is used to get cross section images of the epicutucle and the endocuticle. Thicknesses around 18 μm were measured for both layers for C. aurata and between 12 and 16 μm for C. gloriosa. A layered structure is observed in both beetles. In addition, a cusp-like structure is also observed in C. gloriosa. MMSE showed left-handed near-circular polarization of light reflected on both beetles. For C. aurata this is observed in a narrow wavelength range (500-600 nm) and for C. gloriosa in a wider wavelength range (400-700 nm) when measured on golden areas of the cuticle. C. gloriosa also has green areas where the reflected light was linearly polarized. The results are used in regression modelling. A good model approximation was found for C. aurata for angles up to 60 whereas a good starting point for future work was reached for C. gloriosa.

The obstruction to the existence of Fredholm problems for elliptic differentail operators on manifolds with edges is a topological invariant of the operator. We give an explicit general formula for this invariant. As an application we compute this obstruction for geometric operators.

L'aléa sismique d'un site donné peut être largement influencé par des effets de site. Afin d'évaluer ces effets, la structure locale du sous-sol ainsi que les propriétés du champ d'onde doivent être étudiées. Les ondes de surface (ondes de Love et de Rayleigh) s'avèrent utiles, leurs propriétés (courbes de dispersion, ellipticité des ondes de Rayleigh) étant directement liées à la structure du sous-sol. Le paramètre clé pour l'identification du type d'onde est la polarisation. Dans la première partie de la thèse, de nouvelles méthodes pour l'estimation de paramètres de polarisation d'ondes de surface sont développées. Deux méthodes, DELFI et RayDec, estiment l'ellipticité des ondes de Rayleigh à partir d'enregistrements d'un seul capteur sismique. La troisième méthode, MUSIQUE, est basée sur la méthode MUSIC et utilise les enregistrements multi-composantes de réseaux sismiques afin de distinguer ondes de Love et ondes de Rayleigh et d'estimer leurs propriétés. Dans la deuxième partie de la thèse, une étude théorique de l'inversion de courbes d'ellipticité montre quelles parties de ces courbes véhiculent les informations importantes sur la structure du sol et comment l'inversion peut être améliorée. Le schéma d'inversion résultant est alors testé en l'appliquant à des données réelles mesurées pour 14 sites européens. Finalement, 22 séismes enregistrés par un réseau de capteurs dans la vallée de Santa Clara en Californie sont analysés par MUSIQUE. La répartition azimutale, les courbes de dispersion, la courbe d'ellipticité et les énergies des différents types d'ondes sont analysées et soulignent l'importance des ondes de surface diffractées dans le champ d'ondes enregistré. / The seismic hazard of a given site can be largely influenced by site effects. In order to estimate these effects, the local soil structure and the wave field properties have to be investigated. This can be done using surface waves (Love and Rayleigh waves) since their properties (dispersion curves, Rayleigh wave ellipticity) are closely linked to the soil structure. The key parameter for the correct wave type identification is the polarization. In a first part of this thesis, three new methods to estimate the polarization parameters of surface waves have been developed. Two methods, DELFI and RayDec, estimate the Rayleigh wave ellipticity using the recordings of a single seismic station. The third method, MUSIQUE, is a further developed version of the MUSIC algorithm, and uses seismic array recordings to discriminate between Love and Rayleigh waves and estimate their respective properties. In the second part of the thesis, a theoretical investigation of the inversion of ellipticity curves shows which parts of an ellipticity curve carry the important information on the soil structure and by which means the inversion can be disambiguated. The developed inversion scheme is then tested on real data measurements obtained at 14 different European sites. Finally, the seismological wave field in the Californian Santa Clara valley is investigated applying MUSIQUE to seismic array recordings for 22 earthquakes. The azimuthal energy repartition between the different wave types is investigated and the respective dispersion curves as well as the Rayleigh wave ellipticity curve are estimated. This study outlines the importance of diffracted surface waves in the recorded wave field. STAR

Este trabalho tem como objetivos projetar, construir e caracterizar experimentalmente metamateriais quirais bi-planares para faixa de micro-ondas com o objetivo de investigar a rotação de campo como transdutor em aplicações de biossensoriamento. As estruturas metamateriais estudadas foram primeiramente modeladas e caracterizadas com o uso de aplicativos computacionais comerciais baseadas no método dos elementos finitos, onde os parâmetros geométricos das estruturas foram definidos para operarem na faixa de 8GHz a 12GHz. A partir dos projetos das estruturas simuladas, os metamateriais foram construídos e caracterizados experimentalmente. No estágio de fabricação, foi desenvolvida uma técnica inédita de confecção de metamateriais em placas de circuito impresso flexível. Esta técnica, baseada no processo tradicional de fabricação de placas de circuito impresso, utiliza uma combinação de filmes finos de alumínio e folhas plásticas adesivas flexíveis formando, assim, uma folha metalizada fina e flexível. Além de ser tão eficiente quanto o método convencional, a abordagem proposta utiliza materiais de baixo custo, podendo ser adaptada a outros substratos de qualquer material, formato e espessura. Também foram projetadas e construídas placas de metamateriais com geometrias similares via método tradicional (FR4 e cobre) com o objetivo de validar os procedimentos propostos neste trabalho. A caracterização experimental dos metamateriais quirais foi realizada, por meio de antenas cornetas e analisador de redes. As propriedades efetivas do meio (quiralidade, ângulo de azimute, elipticidade, e índice de refração) foram obtidas a partir dos parâmetros-S do material, por meio de um método de extração de parâmetros. Como esperado, os resultados experimentais tiveram boa concordância com os obtidos numericamente. Por fim, foi proposta uma aplicação de metamaterial quiral utilizando suas características de rotação de campo como transdutor para a medida de glicose em amostras aquosas. No experimento proposto, o ângulo de azimute de uma onda plana polarizada que atravessa o material foi obtido numericamente e sua rotação de campo foi diretamente correlacionada com a concentração de glicose. Os resultados indicaram uma elevada sensibilidade mesmo para baixas concentrações de glicose, demonstrando que a rotação de campo pode realmente ser utilizada para este propósito. / This work focus on the design, fabrication and experimental characterization of bi-planar chiral metamaterials in the microwave frequency range aiming at investigating field rotation as a possible transductor for biosensig applications. First, the metamaterial structures were simulated with commercial-based finite element methods, where the structures geometric parameters were defined to operate in the range of 8-12 GHz. Next, the metamaterials were fabricated and characterized experimentally. We have also developed a new technique for making metamaterials on flexible printed circuit boards. This proposed technique, based on traditional processes of printed circuit boards fabrication, utilizes combination of thin aluminum films and adhesive flexible plastic foils, thus forming a thin, flexible metallic foil. In addition to being as efficient as the conventional method, the proposed approach utilizes low cost materials with low insertion loss that can be adapted to substrates of any material, shape and thickness. We have also designed and fabricated metamaterials with similar geometries via the traditional circuit board method (FR4 and copper) in order to validate the procedures proposed in this work. The experimental characterization of the chiral metamaterials was carried out with antenna horns and a network analyzer. The effective medium parameters (chirality, azimuth angle, ellipticity, and refractive index) were then obtained from the Sparameters of the material via a parameter extraction method. The experimental results were in good agreement with those obtained numerically. Finally, we proposed a new chiral metamaterial application using its field rotation characteristics as a transductor for measuring glucose concentration in aqueous solutions. In the proposed experiment, the azimuth angle of a polarized plane wave propagating through the material was numerically calculeted and its polarization field rotation was directly correlated with the glucose concentration. The results indicated a very high sensitivity even for very low glucose concentrations, indicating that field rotation can indeed be used for this purpose.

We establish elements of a new approach to ellipticity and parametrices within operator algebras on manifolds with higher singularities, only based on some general axiomatic requirements on parameter-dependent operators in suitable scales of spaes. The idea is to model an iterative process with new generations of parameter-dependent operator theories, together with new scales of spaces that satisfy analogous requirements as the original ones, now on a corresponding higher level. The "full" calculus involves two separate theories, one near the tip of the corner and another one at the conical exit to infinity. However, concerning the conical exit to infinity, we establish here a new concrete calculus of edge-degenerate operators which can be iterated to higher singularities. / Wir führen einen neuen Zugang ein zu Elliptizität und Parametrices in Operatorenalgebren auf Mannigfaltigkeiten mit höheren Singularitäten, nur basierend auf allgemeinen axiomatischen Voraussetzungen über parameter-abhängige Operatoren in geeigneten Skalen von Räumen. Die Idee besteht darin, ein iteratives Verfahren zu modellieren mit neuen Generationen von parameter-abhängigen Operatortheorien, zusammen mit neuen Skalen von Räumen, die analoge Voraussetzungen erfüllen wie die ursprünglichen Objekte, jetzt auf dem entsprechenden höheren Niveau. Der „volle“ Kalkül besteht aus zwei separaten Theorien, eine nahe der Spitze der Ecke und eine andere am konischen Ausgang nach Unendlich. Allerdings, bezüglich des konischen Ausgangs nach Unendlich, bauen wir hier einen neuen konkreten Kalkül von kanten-entarteten Operatoren auf, der für höhere Singularitäten iteriert werden kann.

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No. of bitstreams: 1 ribeiro_tas_me_guara.pdf: 20638194 bytes, checksum: 4ad859be9baee51c207bfe90cad14bce (MD5) Previous issue date: 2018-02-21 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Ocultações estelares de Chariklo em 2013 revelaram algo que era desconhecido até o momento: anéis de partículas em torno de um corpo celeste do Sistema Solar diferente de um planeta. Este fato despertou o interesse da comunidade científica a respeito de assuntos como, por exemplo, o processo de formação desses anéis, por quanto tempo eles existirão, qual a probabilidade de corpos como o de Chariklo possuírem anéis ou ainda como eles se mantêm estáveis ao redor de um pequeno objeto, se comparado aos planetas. Neste trabalho estamos interessados em determinar o modelo físico de Chariklo através da manutenção da estrutura atual dos anéis. Acreditamos que este centauro é um corpo semelhante a um elipsoide, entretanto não sabemos com exatidão as dimensões de seus semieixos físicos a, b e c. As razões entre os valores destes semieixos resultam em diferentes valores para os termos de achatamento e elipticidade de Chariklo, que podem ser representados pelos termos J 2 e C 22 em seu potencial gravitacional. Além de determinar os limites para a forma de Chariklo, queremos também estudar quais são os efeitos que o achatamento e elipticidade do corpo central provocam na dinâmica das partículas dos anéis. Para isso, fizemos simulações numéricas usando um integrador que leva em conta os termos relacionados aos coeficientes J 2 e C 22 . O sistema que integramos é composto por Chariklo sendo orbitado por uma partícula com aproximadamente a mesma distância em que estão os anéis, fazendo uso dos elementos orbitais osculadores. O objetivo dessas simulações foi reproduzir as características físicas dos anéis obtidas através das ocultações estelares, entretanto apenas conseguimos fazer essa reprodução usando valores de J 2 e C 22 muito pequenos, o que contradiz a hipótese do centauro ter diferenças significativas entre seus semieixos físicos. Para valores maiores de J 2 e C 22 as partículas descrevem órbitas com excentricidades muito altas, gerando uma grande variação no raio orbital. Diante disso, passamos a estudar o sistema fazendo uso dos elementos orbitais geométricos encontrados nos artigos de Borderies; Longaretti (1987), Longaretti; Borderies (1991), Borderies-Rappaport; Longaretti (1994) e Renner; Sicardy (2006). Estes trabalhos mostram que o uso de elementos orbitais osculadores, em simulações de partículas que orbitam corpos que são muito achatados, não é adequada. Assim sendo, utilizamos esses novos elementos em nossas simulações considerando apenas o termo J 2 e o efeito de altas excentricidades, que antes ocorria, foi corrigido. No entanto, ao adicionarmos o termo C 22 , a excentricidade das órbitas das partículas voltou a aumentar significativamente, efeito que havia ocorrido quando usamos elementos osculadores nas simulações considerando o termo J 2 . Então, fazemos uma discussão sobre uma possível forma de diminuir a excentricidade provocada pela elipticidade de Chariklo. Por fim, além do estudo sobre a dinâmica das partículas este trabalho conta com uma análise, usando seções de Poincaré, de uma possível ressonância responsável pela estabilidade dos anéis. / Chariklo’s stellar occultations in 2013 revealed something that was unknown to date: particle rings around a celestial body of the Solar system other than a planet. This fact has aroused the interest of the scientific community on issues such as the process of forming these rings, how long they will exist, how likely Chariklo bodies are to have rings or how stable they are around of a small object, compared to the planets. In this work we are interested in determining Chariklo’s physical model by maintaining the current structure of the rings. We believe that this centaur is a body similar to an ellipsoid, but we do not know exactly the dimensions of its physical axes a, b, and c. The ratios between the values of these semi axes result in different values for the Chariklo flattening and ellipticity terms which can be represented by the terms J2 and C22 in its gravitational potential. In addition to determining the limits for the Chariklo form, we also want to study the effects of flattening and ellipticity on the dynamics of ring particles. For this, we did numerical simulations using an integrator that takes into account the terms related to the coefficients J2 and C22. The system we integrate is composed of Chariklo being orbited by a particle approximately the same distance as the rings, making use of the orbital osculating elements. The objective of these simulations was to reproduce the physical characteristics of the rings obtained through stellar occultations, however we can only do this reproduction using values of J2 and C22 very small, which contradicts the hypothesis of the centaur to have significant differences between their physical semi axis. For values greater than J2 and C22 the particles describe orbits with very high eccentricities, generating a large variation in the orbital radius. Therefore, we proceed to study the system making use of the geometric orbital elements found in the Borderies and Longaretti (1987), Longaretti and Borderies (1991), Borderies-Rappaport and Longaretti (1994) and Renner and Sicardy (2006). These studies show that the use of orbital osculating elements in simulations of particles orbiting bodies that are very flattened is not adequate. Therefore, we use these new elements in our simulations considering only the term J2 and the effect of high eccentricity that before happened was corrected. However, in addition to adding the term C22, an eccentricity of the orbits of the particles has again increased significantly, an effect that has occurred when using osculating elements in the simulations considering the term J2. So we did discuss about a possible way to lessen the eccentricity brought about by Chariklo’s elipticity. Finally, besides the study on the dynamics of the particles, this work has an analysis, using sections of Poincaré, of a possible resonance responsible for the stability of the rings / 2016/03727-7