Dissertation (PhD)--Stellenbosch University, 2002. / ENGLISH ABSTRACT: In this thesis we construct a central confidence interval for a smooth scalar non-linear function of parameter vector f3 in a single general linear regression model Y = X f3 + c. We do this by first developing an Edgeworth expansion for the distribution function of a standardised point estimator. The confidence interval is then constructed in the manner discussed. Simulation studies reported at the end of the thesis show the interval to perform well in many small-sample situations. Central to the development of the Edgeworth expansion is our use of the index notation which, in statistics, has been popularised by McCullagh (1984, 1987). The contributions made in this thesis are of two kinds. We revisit the complex McCullagh Index Notation, modify and extend it in certain respects as well as repackage it in the manner that is more accessible to other researchers. On the new contributions, in addition to the introduction of a new small-sample confidence interval, we extend the theory of stochastic polynomials (SP) in three respects. A method, which we believe to be the simplest and most transparent to date, is proposed for deriving cumulants for these. Secondly, the theory of the cumulants of the SP is developed both in the context of Edgeworth expansion as well as in the regression setting. Thirdly, our new method enables us to propose a natural alternative to the method of Hall (1992a, 1992b) regarding skewness-reduction in Edgeworth expansions. / AFRIKAANSE OPSOMMING: In hierdie proefskrif word daar aandag gegee aan die konstruksie van 'n sentrale vertrouensinterval vir 'n gladde skalare nie-lineêre funksie van die parametervektor (3 in 'n enkele algemene lineêre regressiemodel y = X (3 + e.. Dit behels eerstens die ontwikkeling van 'n Edgeworth uitbreiding vir die verdelingsfunksie van 'n gestandaardiseerde puntberamer. Die vertrouensinterval word dan op grond van hierdie uitbreiding gekonstrueer. Simulasiestudies wat aan die einde van die proefskrif gerapporteer word, toon dat die voorgestelde interval goed vertoon in verskeie klein-steekproef gevalle. Die gebruik van indeksnotasie, wat in die statistiek deur McCullagh (1984, 1987) bekendgestel is, speel 'n sentrale rol in die ontwikkeling van die Edgeworth uitbreiding. Die bydrae wat in hierdie proefskrif gemaak word, is van 'n tweërlei aard. Die ingewikkelde Indeksnotasie van McCullagh word ondersoek, aangepas en ten opsigte van sekere aspekte uitgebrei. Die notasie word ook aangebied in 'n vorm wat dit hopelik meer toeganklik sal maak vir ander navorsers. Betreffende die bydrae wat gemaak word, word 'n nuwe klein-steekproef vertrouensinterval voorgestel, en word die teorie van stogastiese polinome (SP) ook in drie opsigte uitgebrei. 'n Metode word voorgestelom die kumulante van SP'e af te lei. Ons glo dat hierdie metode die duidelikste en eenvoudigste metode is wat tot dusver hiervoor voorgestel is. Tweedens word die teorie van die kumulante van SP'e ontwikkel binne die konteks van Edgeworth uitbreidings, sowel as die konteks van regressie. Derdens stelons nuwe metode ons in staat om 'n natuurlike alternatief voor te stel vir die metode van Hall (1992a, 1992b) vir die vermindering van skeefheid in Edgeworth uitbreidings.

Proposed designs for Very Large Floating Structures motivate us to understand water-wave propagation through arrays of hundreds, or possibly thousands, of floating structures. The water-wave problems we study are each formulated under the usual conditions of linear wave theory. We study the frequency-domain problem of water-wave propagation through a periodically arranged array of structures, which are solved using a variety of methods. In the first instance we solve the problem for a periodically arranged infinite array using the method of matched asymptotic expansions for both shallow and deep water; the structures are assumed to be small relative to the wavelength and the array periodicity, and may be fixed or float freely. We then solve the same infinite array problem using a numerical approach, namely the Rayleigh-Ritz method, for fixed cylinders in water of finite depth and deep water. No limiting assumptions on the size of the structures relative to other length scales need to be made using this method. Whilst we aren t afforded the luxury of explicit approximations to the solutions, we are able to compute diagrams that can be used to aid an investigation into negative refraction. Finally we solve the water-wave problem for a so-called strip array (that is, an array that extends to infinity in one horizontal direction, but is finite in the other), which allows us to consider the transmission and reflection properties of a water-wave incident on the structures. The problem is solved using the method of multiple scales, under the assumption that the evolution of waves in a horizontal direction occurs on a slower scale than the other time scales that are present, and the method of matched asymptotic expansions using the same assumptions as for the infinite array case.

In this thesis, we use asymptotic methods to solve problems of wave propagation through infinite and finite (only consider those that are finite in one direction) arrays of scatterers. Both two- and three-dimensional arrays are considered. We always assume the scatterer size is much smaller than both the wavelength and array periodicity. Therefore a small parameter is involved and then the method of matched asymptotic expansions is applicable. When the array is infinite, the elastic wave scattering in doubly-periodic arrays of cavity cylinders and acoustic wave scattering in triply-periodic arrays of arbitrary shape rigid scatterers are considered. In both cases, eigenvalue problems are obtained to give perturbed dispersion approximations explicitly. With the help of the computer-algebra package Mathematica, examples of explicit approximations to the dispersion relation for perturbed waves are given. In the case of finite arrays, we consider the multiple resonant wave scattering problems for both acoustic and elastic waves. We use the methods of multiple scales and matched asymptotic expansions to obtain envelope equations for infinite arrays and then apply them to a strip of doubly or triply periodic arrays of scatterers. Numerical results are given to compare the transmission wave intensity for different shape scatterers for acoustic case. For elastic case, where the strip is an elastic medium with arrays of cavity cylinders bounded by acoustic media on both sides, we first give numerical results when there is one dilatational and one shear wave in the array and then compare the transmission coefficients when one dilatational wave is resonated in the array for normal incidence. Key words: matched asymptotic expansions, multiple scales, acoustic scattering, elastic scattering, periodic structures, dispersion relation.

Robust algorithms for the approximation of functions are studied and developed in this thesis. Novel results and algorithms on piecewise polynomial interpolation, rational interpolation and best polynomial and rational approximations are presented. Algorithms for the extension of Chebfun, a software system for the numerical computation with functions, are described. These algorithms allow the construction and manipulation of piecewise smooth functions numerically with machine precision. Breakpoints delimiting subintervals are introduced explicitly, implicitly or automatically, the latter method combining recursive subdivision and edge detection techniques. For interpolation by rational functions with free poles, a novel method is presented. When the interpolation nodes are roots of unity or Chebyshev points the algorithm is particularly simple and relies on discrete Fourier transform matrices, which results in a fast implementation using the Fast Fourier Transform. The method is generalised for arbitrary grids, which requires the construction of polynomials orthogonal on the set of interpolation nodes. The new algorithm has connections with other methods, particularly the work of Jacobi and Kronecker, Berrut and Mittelmann, and Egecioglu and Koc. Computed rational interpolants are compared with the behaviour expected from the theory of convergence of these approximants, and the difficulties due to truncated arithmetic are explained. The appearance of common factors in the numerator and denominator due to finite precision arithmetic is characterised by the behaviour of the singular values of the linear system associated with the rational interpolation problem. Finally, new Remez algorithms for the computation of best polynomial and rational approximations are presented. These algorithms rely on interpolation, for the computation of trial functions, and on Chebfun, for the location of trial references. For polynomials, the algorithm is particularly robust and efficient, and we report experiments with degrees in the thousands. For rational functions, we clarify the numerical issues that affect its application.

Dislocation channel-veins and persistent slip bands (PSBs) are characteristic dislocation configurations that are of interest to both industry and academia. However, existing mathematical models are not adequate to describe the mechanism of the transition between these two states. In this thesis, a series of models are proposed to give a quantitative description to such a transition. The full problem has been considered from two angles. Firstly, the general motion and instabilities of arbitrary curved dislocations have been studied both analytically and numerically. Then the law of motion and local expansions are used to track the shapes of screw segments moving through channels, which are believed to induce dislocation multiplication by cross-slip. The second approach has been to investigate the collective behavior of a large number of dislocations, both geometrically necessary and otherwise. The traditional method of multiple scales does not apply well to describe the pile-up of two arrays of dislocations of opposite signs on a pair of neighbouring glide planes in two dimensional space. Certain quantities have to be more accurately defined under the multiple-scale coordinates to capture the much more localised resultant stress caused by these dislocation pairs. Through detailed calculations, one-dimensional dipoles can be homogenised to obtain some insightful results both on a local scale where the dipole pattern is the key diagnostic and on a macroscopic scale on which density variations are of most interest. Equilibria of dislocation dipoles in a two-dimensional regular lattice have been also studied. Some natural transitions between different patterns can be found as a result of geometrical instabilities.

Thesis (Ph.D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 1979. / MICROFICHE COPY AVAILABLE IN ARCHIVES AND ENGINEERING. / Includes bibliographical references. / by Khaled I. Yared. / Ph.D.

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No. of bitstreams: 2 Tese - Marc Alexandre Duarte Gigonzac - 2016.pdf: 12763622 bytes, checksum: 3479eadda35402525c2387337a3a0d69 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-02-02 / Fundação de Amparo à Pesquisa do Estado de Goiás - FAPEG / X-Fragile Syndrome (FXS) is the leading cause of inherited intellectual disability in the world and the second of genetic etiology, with an estimated prevalence of 1/4000 men and 1/8000 women. The most common molecular mechanism in SXF is due to changes in the expression of the FMR1 gene, located in Xq27.3, due to CGG trinucleotide expansions in the promoter region and subsequent methylation of the gene. In spite of presenting consistent clinical findings, they are not exclusive, and the existence of carriers of alteration in the FMR1 gene without apparent clinical manifestations makes it impossible to diagnose SXF based only on the evaluation. In the present study, a methodological proposal for the molecular diagnosis of X-Fragile Syndrome was developed from the methylation-specific triple amplification of the promoter region of the FMR1 gene combined with capillary electrophoresis. Thirty-four patients with clinical indication of SXF were referred to a laboratory of the public health network. After extraction and quantification of the DNA, the samples were amplified in an optimized protocol and the products submitted to 36cm capillary electrophoresis to verify the amount of CGG repeats and the degree of DNA methylation of each sample. Pre-mutation (3%) and six complete mutations (18%) were detected, all of which revealed a high degree of methylation. Considering the clinical signs commonly presented, the patients were also analyzed for the occurrence of Autism Spectrum Disorder (ASD), which shadowing and overlapping the SXF, verifying that 100% of the individuals with complete mutation presented the phenotype. Thus, it was possible to observe small behavioral differences in the patients analyzed, indicating a lighter clinical picture regarding aspects of social interaction and stereotypies. Thus, the new methodological proposal allows to effectively determine the CGG trinucleotide expansions in FMR1 allowing an assertive diagnosis of SXF for the families of patients attended in the public health network in Goiás. / A Síndrome do X-Frágil (SXF) é a principal causa de deficiência intelectual herdável no mundo e a segunda de etiologia genética, com uma prevalência estimada de 1/4000 homens e 1/8000 mulheres. O mecanismo molecular mais comum na SXF é decorrente de alterações na expressão do gene FMR1, localizado em Xq27.3, devido a expansões trinucleotídicas CGG na região promotora e subsequente metilação do gene. Apesar de apresentar achados clínicos consistentes, os mesmos não são exclusivos, e a existência de portadores de alteração no gene FMR1 sem manifestações clínicas aparentes impossibilitam o diagnóstico da SXF baseado apenas no exame físico. No presente estudo foi desenvolvido uma proposta metodológica para o diagnóstico molecular da Síndrome do X-Frágil a partir da amplificação tripla metilação-específica da região promotora do gene FMR1 combinada a eletroforese em capilar. Foram utilizados 34 pacientes com indicação clínica de SXF encaminhados para um laboratório da rede pública de saúde. Após extração e quantificação do DNA, as amostras foram amplificadas em protocolo otimizado e os produtos submetidos a eletroforese em capilar de 36cm para verificar a quantidade de repetições CGG e o grau de metilação do DNA de cada amostra. Foram detectadas uma pré-mutação (3%) e seis mutações completas (18%), sendo que todas estas últimas revelaram um alto grau de metilação. Considerando os sinais clínicos comumente apresentados, os pacientes foram também analisados para a ocorrência do Transtorno do Espectro do Autismo (TEA), que sombreia e se sobrepõe à SXF, verificando que 100% dos indivíduos com mutação completa apresentaram o fenótipo. Foi possível observar pequenas diferenças comportamentais nos pacientes analisados, indicando um quadro clínico mais leve quanto aos aspectos da interação social e das estereotipias. Sendo assim, a nova proposta metodológica permite determinar de forma eficaz as expansões trinucleotídicas CGG no FMR1 permitindo um diagnóstico assertivo da SXF para as famílias de pacientes atendidos na rede pública de saúde em Goiás.

The interactions between plants and their insect herbivores are one of the main generators of biological diversity. A fundamental process generating this outstanding diversity is diet expansion to novel host plants. During the last four decades scientists accumulated evidence showing that co-adaptation between plants and herbivores is a major process assembling plant-herbivore interactions. However, rescent research suggests that adaptation is not always a prerequisite to generate novel plant-herbivore interactions. Novel associations between plants and insect herbivores may be assembled by ecological fitting - an ecological process whereby herbivores colonize novel host plants as a result of the suites of preadapted traits that they carry at the time of colonization. A widespread assumption concerning the architecture of insect herbivore genotypes is the "Jack of all trades master of none" principle. This principle proposes that there is a trade-off in genotype performances between host plants. The main prediction of this principle is that genotype performance will be negatively correlated among hosts. Genotypes displaying high performance on a given host will perform poorly on other hosts. This constraint of adaptation to multiple host plants implies that diet specialization will be selected over generalization. Contrary to these theoretical expectations, in most cases, genotypes that perform well in one host will also perform well in other host plants. Positive correlations in cross-host performance represent ecological and evolutionary dynamics opposite to the "Jack of all trades" principle. In this scenario genotypes with high performance on one host plant also have high performance on other plants, promoting generalization. The predictions of the current theory on the assemblage of novel plant-herbivore interactions focus on the fact that most insect herbivores are specialists. However, to fully understand the processes underlying the assembly of novel plant-insect interactions, it is necessary to study diet expansions in both specialist and generalist insect herbivores. This dissertation was performed at La Selva Biological Station, a tropical rain forest in Costa Rica, Central America. We studied a group of neotropical herbivores, the "rolled-leaf beetles" (Cephaloleia, Chrysomelidae: Cassidinae) and their host plants, neotropical plants in the order Zingiberales. Cephaloleia beetles have evolved with neotropical Zingiberales for the last 40-60 MY. Four paleotropical and one South American members of the Zingiberales have been introduced to La Selva during the last decade. After these introductions, currently seven Cephaloleia beetles are expanding their diets to exotic Zingiberales. These incipient diet expansions represent an opportunity to understand: 1) the relative roles of adaptation vs ecological fitting on the demography and colonization success of novel hosts in generalist and specialist herbivores and 2) whether genotypic performance across original and novel hosts are negatively correlated, as predicted by the "Jack of all trades" principle, or genotype performances across original and novel hosts are positively correlated. For most of the experiments included in this dissertation, I focused on the performance of larvae and adults of two generalist (Cephaloleia belti, C. dilaticollis) and two specialist beetles (Cephaloleia dorsalis, C. placida) reared in the laboratory on native or exotic Zingiberales. Generalist and specialist species display similar responses when changing their diets to novel hosts. Larvae preferred and performed better in the original than in the novel hosts. Adults usually displayed the opposite pattern, i.e. higher preference for and longevity on the exotic than on the novel hosts. In most novel interactions between Cephaloleia beetles and exotic Zingiberales, larval performance required adaptation, but adult performance was pre-adapted to the novel hosts. Therefore, both adaptation and ecological fitting are playing a role during diet expansions to novel hosts. Vital rates estimated through experimental demography show that population growth is reduced on novel host plants for both generalist and specialist Cephaloleia. Although in some cases population growth on the novel hosts is negative, suggesting the potential outcome of extinction after colonization or source-sink dynamics, several beetle species displayed positive population growth in the novel host plants. Positive instantaneous population growth rates in novel hosts supports diet expansions without substantial initial evolutionary change through ecological fitting. In quantitative genetics experiments testing for cross-host genetic correlations in performance between the original and novel host plants, we did not find evidence for negative genetic correlations, as predicted by the "Jack of all trades" principle. Most genetic correlations in performance between original and novel hosts were either not significant or they were positive. These results represent very different ecological and evolutionary dynamics than those predicted by the "Jack of all trades" principle. In this case, genotypes with high performance on original hosts also displayed high performance on novel hosts, promoting generalization. In conclusion, interactions between Cephaloleia beetles and plants from the order Zingiberales are labile. In some cases diet expansions may occur without substantial evolutionary change. In addition the genetic architecture of genotypes promotes generalization during diet expansions to novel hosts.

It is shown that bounded solutions to semilinear elliptic Fuchsian equations obey complete asymptoic expansions in terms of powers and logarithms in the distance to the boundary. For that purpose, Schuze's notion of asymptotic type for conormal asymptotics close to a conical point is refined. This in turn allows to perform explicit calculations on asymptotic types - modulo the resolution of the spectral problem for determining the singular exponents in the asmptotic expansions.

We develop a cluster expansion in space-time for an infinite-dimensional system of interacting diffusions where the drift term of each diffusion depends on the whole past of the trajectory; these interacting diffusions arise when considering the Langevin dynamics of a ferromagnetic system submitted to a disordered external magnetic field.