The study of zeta functions is one of the primary aspects of modern number theory. Hecke was the first to prove that the Dedekind zeta function of any algebraic number field has an analytic continuation over the whole plane and satisfies a simple functional equation. He soon realized that his method would work, not only for Dedekind zeta functions and L–series, but also for a zeta function formed with a new type of ideal character which, for principal ideals depends not only on the residue class of the number(representing the principal ideal) modulo the conductor, but also on the position of the conjugates of the number in the complex field. He then showed that these “Hecke” zeta functions satisfied the same type of functional equation as the Dedekind zeta function, but with a much more complicated factor. In his doctoral thesis, John Tate replaced the classical notion of zeta function, as a sum over integral ideals of a certain type of ideal character, by the integral over the idele group of a rather general weight function times an idele character which is trivial on field elements. He derived a Poisson Formula for general functions over the adeles, summed over the discrete subgroup of field elements. This was then used to give an analytic continuation for all of the generalized zeta functions and an elegant functional equation was established for them. The mention of the Poisson Summation Formula immediately reminds one of the Theta function and the proof of the functional equation for the Riemann zeta function. The two proofs share close analogues with the functional equation for the Theta function now replaced by the number theoretic Riemann–Roch Theorem. Translating the results back into classical terms one obtains the Hecke functional equation, together with an interpretation of the complicated factor in it as a product of certain local factors coming form the archimedean primes and the primes of the conductor. This understanding of Tate’s results in the classical framework essentially boils down to constructing the generalized weight function and idele group characters which are trivial on field elements. This is facilitated by the understanding of the local zeta functions. We explicitly compute in both cases, the local and the global, illustrating the working of the ideas in a concrete setup. I have closely followed Tate’s original thesis in this exposition.

Orientador: Giorgio Basilici / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-07-31T21:55:14Z (GMT). No. of bitstreams: 1 Cesar_PauloHenriqueTavares_M.pdf: 3736356 bytes, checksum: 3d8dd5ffaf2d7b3591926eadf6fdeb18 (MD5) Previous issue date: 2001 / Resumo: O presente trabalho é uma análise vertical de depósitos marinhos profundos, cujos processos deposicionais se dá por fluxos gravitaticionais, na Unidade Apiúna, na Bacia do Itajaí, Estado de Santa Catarina. Os dados estão dispostos em 5 seções, das quais as 4 primeiras são correlacionáveis, com aproximadamente 200 metros de espessura total, 524 camadas e 102 transições de fácies. Em que a média das camadas é de 25 cm. Através da transformada de Fourier, no domínio da freqüência obteve-se os seguintes dados: nenhum ciclo na seção 1; 2 ciclos na seção 2; 11 ciclos na seção 3; 6 ciclos na seção 4; 4 ciclos na seção 5 e 16 ciclos nas seções 1-4, que puderam ser correlacionáveis. As freqüências (ciclos), foram definidas com o auxílio da análise visual da seções. Com relação aos ciclos de afinamento ou engrossamento ascendente (megassequências), os resultados foram: engrossamento ascendente: 2 na seção 2, 6 na seção 3, 2 na seção 4 e 18 na seçôes 1-4, enquanto que o afinamento ascendente:3 na seção 3,3 na seção 4,3 na seção 5 e 10 nas seções 1-4. Existe diferença entre a análise visual e a função da freqüência, porém as megassequências, foram definidas segundo a análise visual. A análise de Markov corrobora que as principais transições estão inseridas num contexto de lençóis de areia, proximais e distais, em que ocorre a intercalação de arenitos fino a médio, intercalados com finas camadas de arenito com siltito, respectivamente fácies D e E. Diante de tais observações foi possível refutar a inclusão da Unidade Apiúna, num clássico depósito de leques profundos supridos por canyon, pois como largamente disposto na literatura, este depósitos apresentam seqüências de engrossamento ou afinamento ascendente (thinning e thickening upward), o que se expressa nas 5 seções de maneira bastante tímida. Quanto aos trends de granulometria, que pode refletir que o mecanismo de deposição, o fluxo de detritos (debris flow) é dominante, em detrimento de correntes de turbidez (turbidite currents), isto porque não é comum a gradação ascendente nas camadas de arenito, sendo somente no topo de algumas camadas que ocorrem este último fenômeno. Foi possível diferenciar quatro associações de fácies, numeradas de 1 a 4 a seguir: depósitos de slope, depósitos de lençóis de areia proximais, depósitos de lençóis de areia distais e finalmente depósitos de canal-dique marginal / Abstract: This work is a vertical analysis of deepwater deposits outcrops, whose depositional processes occurs by gravity flows in the Apiúna Unit, Itajaí Basin, Santa Catarina State. The data are organized in 5 sections where each section is one outcrop. The sections from 1 to 4 have mutual correlation, with almost 200 m of total thickness, 534 beds and 102 facies transitions. Through Fourier transform, in the frequency domain, 16 sequences was obtained, defined by visual analysis of thickness beds. The results for the sequences of thinning and thickening upward were: 10 of thickening upward and 6 of thinning upward. The vertical trends of fining upward are often randomic. The use of Markov Chain's tooI confirmed the main facies transitions which are located in a context of sheet sands, proximal and distal, in which occurs the superposition of fine to medium (D facies) sandstones with the thin beds of sandstone and siltstone (E facies ). According to the data above, it was possible to refuse the Apiuna Unit as a classical canyon-fed deposits of deep water fan, because of absence of better defined and low abundant sequences of thinning and thickening upward. The debris flow is the main depositional processes, while turbidity currents occur in the upper part of flow in some sandstone beds. This configuration is showed in the beds with the trend of fining upward. It was possible to identify four facies associations, namely: slope deposits; proximal sheet sands; distal sheet sands and channel-levee / Mestrado / Mestre em Ciências e Engenharia de Petróleo

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Frontal sinus variation has been used in forensic anthropology to aid in positive identification since the 1920s. As radiographic technology has evolved, so has the quality and quantity of data that radiologists and anthropologists can collect during an individual’s lifetime. This has led to new methodology when comparing antemortem and postmortem radiographic images. The current study aims to look at frontal sinus morphology and dimensional variations on computed tomography (CT) scans, as these are currently the most commonly collected images showing the frontal sinus in its entirety used in clinics in the United States. This study assessed 307 individuals for the morphological analysis and 325 individuals in the dimensional analysis. These individuals represented females and males from Asian, African, European, and Latin American derived groups. It is hypothesized that frontal sinus shape variations will cluster based on assigned sex and population affinities. Similarly, it is hypothesized that dimensional variation, specifically the maximum height, maximum width, and maximum depth, will show statistically significant clustering based on assigned sex and population affinities. The frontal sinus outlines from the CT images were transferred into SHAPE v1.3 in order to run an elliptical Fourier analysis. The dimensional data was measured directly from the images using a MicroDicom viewer. All of the statistical analyses, including Pearson’s Chi-squared and ANOVA tests, were run in R studio. Results indicated that morphologically there is no statistically significant clustering based on assigned sex or population affinity. However, there was statistically significant clustering dimensionally when tested against both assigned sex and population affinity using an ANOVA, indicating that the interactive effects of sexual dimorphism and population affinity influence the dimensions but not the shape of the frontal sinus. These results add to the foundational knowledge that practitioners have surrounding frontal sinus indicating that assigned sex and population affinity have impact on the approximate dimensions of this structure while these variables do not have a statistically significant effect on morphological variation. The results also speak to the idiosyncratic nature of the frontal sinus and bolster confidence using morphological variations as a mean to individuate.

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In this Bachelor's thesis we present a few results about the absolute convergence of Fourier series, followed by an example of a differentiable function whose Fourier series does not converge absolutely. In the end we provide a suggestion for future work on generalizing the given example, and we briefly discuss an issue that has not been given much attention in the existing literature on the subject.

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A thesis submitted to the School of Graduate Studies in partial fulfilment of the requirements of the Master of Applied Science degree / This thesis investigated structure-function relationships of materials designed to mimic the plant cell wall by comparing their mechanical properties measured using strain-induced buckling methods. The plant cell wall mimics are submicrometer-thick films composed of cellulose nanocrystals (CNCs) and various types of xyloglucan (XG), a common plant hemicellulose. Our goal was to establish links between film composition/architecture and elastic modulus, to better understand the interactions between plant cell wall components and their influence on mechanical properties. Three buckling methods for measuring mechanical properties of supported films were compared. All methods involved compressing a thin film deposited onto a shape memory polymer or an elastomeric substrate, through thermal shrinking or mechanical compression, respectively. Two thermal shrinking methods (constrained in one axis or unconstrained) and one compression method (using a mechanical strain stage) were used. Based on the mismatch of mechanical properties between the film and the substrate, the rigid thin film “buckles” upon compression to dissipate strain. The resulting surface wrinkle sizes are characteristic of the mechanical properties of the thin film. A Fourier analysis algorithm with Gaussian curve fitting was optimized to extract wrinkle sizes accurately and reproducibly from microscopy images to reliably quantify the elastic moduli of thin films. To select the most precise strain-induced buckling method, model layer-by-layer (LbL) thin films composed of CNCs and polyethylene imine were tested. All three buckling methods precisely quantified the elastic moduli of the films and helped us build connections between the mechanical properties and the film composition. Elastic moduli determined were 15-44 GPa (depending on composition) and films up to 350 nm-thick were tested. Based on sensitivity analyses, however, unconstrained thermal shrinking proved to be the most robust method for calculating the elastic modulus. We believe these buckling methods may find widespread use in the characterization and surface structuring of thin films for applications in biosensors, flexible electronics, point-of-care diagnostics, and for studying plant cell wall mimics. Using the unconstrained thermal shrinking method, plant cell wall mimics were constructed using LbL thin film assembly with various concentrations of CNCs and XG. Three types of XG were compared: (1) unmodified XG, (2) XG with a fraction of the galactosyl residues removed (degalactosylated), and (3) a fragmented lower molecular weight XG. It was inferred that molecular weight impacts the stiffness of XG-CNC based on adsorption conformation of XG onto CNCs, where lower molecular weight XG results in a higher modulus film (27 ± 1 vs. 19 ± 2 GPa). As well, saccharide residues of XG, specifically galactosyl, impact XG’s capacity for self-association and interaction with CNCs, because saccharide residues hinder association through their glucan backbone. This is evidenced by the higher elastic moduli calculated for degalactosylated XG-CNC films (75 ± 6, GPa), compared to native XG-CNC films (19 ± 2 GPa). This work highlights the importance of material structure as it relates to overall performance and therefore function in natural systems, such as the plant cell wall. These studies contribute to a greater understanding of the mechanical properties of the plant cell wall and serve as a basis to extend bio-based and biomimetic materials to applications such as drug delivery, packaging, and coatings. / Thesis / Master of Applied Science (MASc) / The plant cell wall boasts impressive mechanical properties, balancing seemingly opposing properties of structural strength with flexibility. These natural materials have been a source of inspiration for new material design, but the phenomena that govern interactions between components and how their structures translate into function, have yet to be fully understood. In this work, we have constructed thin multilayered films to mimic the plant cell wall, composed of cellulose nanocrystals (rod-shaped nanoparticles derived from plant cellulose) and xyloglucan (a common hemicellulose “glue”). When the films on flexible supports are compressed, they buckle into wrinkled surface patterns that can be used to calculate their mechanical properties. This investigation compares three buckling methods and supports the notion that the mechanical performance of the plant cell wall is strongly dependent on the structure of the different components and the way they interact.