Thesis (M.Sc.) - Carleton University, 2006. / Includes bibliographical references (p. 64-67). Also available in electronic format on the Internet.

A research report submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in partial fulfilment of the degree of Master of Science / The importance of Pade approximation has been increasingly recognized in ' recent years. The first convergence result of Pade approximants valid for general meromorphic functions was obtained by de Montessus de Ballore in 1902. He proved that when a function f has precisely n poles in I z 1< R, then the (n+ 1)th column in thePade table of f converges to f in I z J< R. (Abbreviation abstract) / Andrew Chakane 2019

In this study, new Tsai-type 1/1 quasicrystal approximants (ACs) in the RE-Au-SM systems (RE = Gd, Tb, Ho, Yb; SM = Si, Ge) were synthesized using high temperature synthesis techniques such as self-flux, arc-melting-annealing and novel arc-melting-self-flux methods. The syntheses not only provided appropriate samples for the intended structural and physical property measurements but could also be adapted to other systems, especially where crystal growth is a challenge. The newly developed arc-melting-self-flux method uses a temperature program that oscillates near the nucleation and melting points of the intended phase in order to obtain large single crystals. Self-flux methods employed to synthesize Ho-Au-Si and Tb-Au-Si ACs using a precursor alloy ≈Au79Si21 resulted in 100 mm3 and 8 mm3 single crystals, respectively. The crystal structures of the compounds are determined by either one or combinations of the following diffraction techniques; single crystal x-ray, powder x-ray, powder neutron and single crystal neutron diffraction methods.  The crystal structure refinements indicated that the compounds are essentially iso-structural with the prototype Tsai-type 1/1 approximant crystal, YbCd6. In the present compounds there are some structural variations at the cluster center and in the so-called cubic interstices located at (¼, ¼, ¼). For the current ACs; either thermoelectric, magnetic or both properties were investigated. The measured properties were understood further by correlating the properties with the atomic structures of the ACs. Significant differences are observed in the thermoelectric properties, particularly on the lattice thermal conductivities (Kphonon) of Gd-Au-Si, Gd-Au-Ge and Yb-Au-Ge ACs. The difference is attributed to the presence of chemical and positional disorder. Magnetic susceptibility and specific heat measurements revealed ferromagnetic transitions at low temperatures, Tc ≈ 22.5 K for Gd-Au-Si and Tc ≈ 13.1 K for Gd-Au-Ge. For a Tb-Au-Si AC with 14 % central-Tb occupancy, a ferrimagnetic-like transition was observed at Tc ≈ 9 K. Later, it was noted that the Tc and other magnetic properties depend on the occupancy of the central-RE site. Consistent decrease of Tc with increasing central-Tb occupancy is observed. The dependency of magnetic behavior with central-RE occupancy was clarified by solving the magnetic structure of the Tb-Au-Si AC.

<p>This thesis is focused on the synthesis and structural determination of a series of quasicrystal approximants in the <i>RE</i>₁₃Zn_~58 (<i>RE</i> = Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu), Ce₁₃Cd_~58and <i>RE</i>Cd₆ (<i>RE</i> = Tb, Ho, Er, Tm and Lu) systems. Structural studies were performed by single crystal X-ray diffraction methods.</p><p>The <i>RE</i>Cd₆ phases are 1/1 cubic approximants to the stable icosahedral quasicrystals. The structure of quasicrystals cannot be solved by conventional crystallography due to the lack of periodicity in three-dimensional space. On the other hand, quasicrystal approximants exhibit long-range order, which makes their structural determination possible by standard methods. Since they are believed to display the same local arrangement as the quasicrystals, the structures of the quasicrystal approximants play a key role in understanding quasicrystals.</p><p>The <i>RE</i>₁₃Zn_~58 and Ce₁₃Cd_~58 quasicrystal approximants show measurable compositional variations coupled to subtle structural differences. The crystal structures are generally rather more complex than previously reported, and exhibit a number of different ordering and disordering modes.</p>

This thesis is focused on the synthesis and structural determination of a series of quasicrystal approximants in the RE13Zn~58 (RE = Ce, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu), Ce13Cd~58 and RECd6 (RE = Tb, Ho, Er, Tm and Lu) systems. Structural studies were performed by single crystal X-ray diffraction methods. The RECd6 phases are 1/1 cubic approximants to the stable icosahedral quasicrystals. The structure of quasicrystals cannot be solved by conventional crystallography due to the lack of periodicity in three-dimensional space. On the other hand, quasicrystal approximants exhibit long-range order, which makes their structural determination possible by standard methods. Since they are believed to display the same local arrangement as the quasicrystals, the structures of the quasicrystal approximants play a key role in understanding quasicrystals. The RE13Zn~58 and Ce13Cd~58 quasicrystal approximants show measurable compositional variations coupled to subtle structural differences. The crystal structures are generally rather more complex than previously reported, and exhibit a number of different ordering and disordering modes.

The alternation between labial and velar fricatives (e.g. [x]uego fuego ‘fire’) and labial and velar approximants (e.g. a[ɣ]uelo abuelo ‘grandfather’) frequently co-occur in disparate Spanish dialects (Colombia, New Mexico, El Salvador, Ecuador, Chile, among others). I hypothesize that these alternations are triggered by the perceptual similarity between such variants in the context of [u] and [w]. I further hypothesize that the spread of these variables to the upper layers of society is prevented by formal education, since orthography can block sound change. Although the labio-velar alternations have been observed before, there are few experimental studies addressing their acoustic and perceptual motivations. Yet, the only way to understand the mechanisms of sound variation and change is to analyze the physical, acoustic and perceptual characteristics of the sounds involved. This dissertation uniquely combines three methodologies of data elicitation in order to achieve a better understanding of the alternations. Vernacular speech was collected through sociolinguistic interviews. Contextually controlled target words were elicited via a picture naming task. Finally, the hypothesis that the alternations were driven by the perceptual similarity between the sounds was tested via an AX discrimination test. The sociolinguistic data was correlated with the results from the perception experiment to determine whether more variation in speech correlates with higher rates of confusion in perception. The results reveal that Education and Following Context are two of the most powerful factor groups that influence the alternations. The alternation is almost exclusively found before the diphthongs [we, wi], and in stressed syllables. Knowing the orthography plays an important role in blocking the diffusion of this perceptually driven variation. The same factors affecting the variation in sociolinguistics interviews were found to be significant in increasing the confusion between [f] ~ [x] and [β] ~ [ɣ] in the perception experiment. The acoustic analysis (centre of gravity and F2 at vowel onset), however, did not support the hypothesis regarding the similarity of labial and velar fricatives and approximants.

The alternation between labial and velar fricatives (e.g. [x]uego fuego ‘fire’) and labial and velar approximants (e.g. a[ɣ]uelo abuelo ‘grandfather’) frequently co-occur in disparate Spanish dialects (Colombia, New Mexico, El Salvador, Ecuador, Chile, among others). I hypothesize that these alternations are triggered by the perceptual similarity between such variants in the context of [u] and [w]. I further hypothesize that the spread of these variables to the upper layers of society is prevented by formal education, since orthography can block sound change. Although the labio-velar alternations have been observed before, there are few experimental studies addressing their acoustic and perceptual motivations. Yet, the only way to understand the mechanisms of sound variation and change is to analyze the physical, acoustic and perceptual characteristics of the sounds involved. This dissertation uniquely combines three methodologies of data elicitation in order to achieve a better understanding of the alternations. Vernacular speech was collected through sociolinguistic interviews. Contextually controlled target words were elicited via a picture naming task. Finally, the hypothesis that the alternations were driven by the perceptual similarity between the sounds was tested via an AX discrimination test. The sociolinguistic data was correlated with the results from the perception experiment to determine whether more variation in speech correlates with higher rates of confusion in perception. The results reveal that Education and Following Context are two of the most powerful factor groups that influence the alternations. The alternation is almost exclusively found before the diphthongs [we, wi], and in stressed syllables. Knowing the orthography plays an important role in blocking the diffusion of this perceptually driven variation. The same factors affecting the variation in sociolinguistics interviews were found to be significant in increasing the confusion between [f] ~ [x] and [β] ~ [ɣ] in the perception experiment. The acoustic analysis (centre of gravity and F2 at vowel onset), however, did not support the hypothesis regarding the similarity of labial and velar fricatives and approximants.

A multipoint Padë approximant, R, to a function of Stieltjes1 type is determined.The function R has numerator of degree n-l and denominator of degree n.The 2n interpolation points must belong to the region where f is analytic,and if one non-real point is amongst the interpolation points its complex-conjugated point must too.The problem is to characterize R and to find some convergence results as n tends to infinity. A certain kind of continued fraction expansion of f is used.From a characterization theorem it is shown that in each step of that expansion a new function, g, is produced; a function of the same type as f. The function g is then used,in the second step of the expansion,to show that yet a new function of the same type as f is produced. After a finite number of steps the expansion is truncated,and the last created function is replaced by the zero function.It is then shown,that in each step upwards in the expansion a rational function is created; a function of the same type as f.From this it is clear that the multipoint Padê approximant R is of the same type as f.From this it is obvious that the zeros of R interlace the poles, which belong to the region where f is not analytical.Both the zeros and the poles are simple. Since both f and R are functions of Stieltjes ' type the theory of Hardy spaces can be applied (p less than one ) to show some error formulas.When all the interpolation points coincide ( ordinary Padé approximation) the expected error formula is attained. From the error formula above it is easy to show uniform convergence in compact sets of the region where f is analytical,at least wien the interpolation points belong to a compact set of that region.Convergence is also shown for the case where the interpolation points approach the interval where f is not analytical,as long as the speed qî approach is not too great. / digitalisering@umu

Thesis (M. App. Sc.)--Carleton University, 2002. / Includes bibliographical references (p. 85-89). Also available in electronic format on the Internet.

Orientador: Maria Amélia Novais Schleicher / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica e Instituto de Geociências / Made available in DSpace on 2018-08-18T17:07:35Z (GMT). No. of bitstreams: 1 Mondini_Debora_M.pdf: 2991739 bytes, checksum: 04c2af83e74d4f531ac6c278f2a0f1dc (MD5) Previous issue date: 2011 / Resumo: Em três dimensões, os métodos de migração baseados na resolução da equação da onda unidirecional, além de enfrentar problemas para imagear refletores com fortes mergulhos e tratar ondas evanescentes, ainda são computacionalmente caros. Para os problemas de imagear refletores com forte mergulho e ondas evanescentes, nessa dissertação, usamos a aproximação em série de Padé complexa. Pelo fato da resolução do problema tridimensional ser computacionalmente cara, ao longo dos anos várias técnicas foram elaboradas com o objetivo de reduzir os custos e ainda manter a qualidade do método de migração que se estiver usando. Uma técnica comumente utilizada é o splitting. Nosso objetivo com esse trabalho é testar os operadores de migração usando a aproximação em série de Padé complexa, a técnica de splitting em duas ou quatro direções alternadas, bem como o termo de correção de Li. Para o caso de splitting em apenas duas direções, enfrentamos o problema de anisotropia numérica, ou seja, o operador de migração age de forma diferente em direções diferentes, resultando em grandes erros de posicionamento. Para corrigir esse problema usamos a correção de Li. Sem alterar a migração FD 2D, a correção de Li é uma extrapolação do campo residual por um deslocamento de fase. Quando o splitting é aplicado em quatro direções (nas coordenadas horizontais e nas diagonais) de forma alternada ainda podemos enfrentar problemas de anisotropia numérica e consequentemente mau posicionamento dos refletores muito inclinados. Por isso, testamos a aplicação da correção de Li para este caso. Nessa dissertação, comparamos os resultados obtidos pela técnica de migração FD, os testes foram realizados em um meio homogêneo e nos dados sintéticos 3D SEG-EAGE / Abstract: In three dimensions, migration methods based on solving the one-way wave equation, besides facing problems to handle evanescent waves and to image steep dip reflectors, are still computationally expensive. For the problems of imaging steep dip reflectors and treat evanescent waves, in this dissertation, we use the complex Padé approximation. Because solving three dimensional problems is computationally expensive, several techniques have been developed in order to reduce costs and still maintain the quality of the migration method. A commonly used technique is splitting. Our goal with this study is to test the migration operators using the complex Padé approximation, the technique of splitting into two or four alternating directions, as well as the Li correction term. For the case of splitting in two directions only, we face the problem of numerical anisotropy, i.e., the migration operator acts differently in different directions, resulting in a mispositioning of the reflectors in the situation where the strike direction of the reflector is far off the migration planes. To correct for this problem we use the Li correction. Without changing 2D FD migration, Li correction is an extrapolation of the residual wave field by a phase shift. When splitting is applied in four directions (the horizontal coordinates and the diagonals) alternately we can still face problems of numerical anisotropy and consequently mispositioning of steep dip reflectors. Because of that, we also tested the application of the Li correction. In this dissertation, we compare the results obtained by the FD migration technique. The tests were conducted in a homogeneous media and synthetic 3D data in SEG-EAGE / Mestrado / Reservatórios e Gestão / Mestre em Ciências e Engenharia de Petróleo