Thesis (M. Phil.)--University of Hong Kong, 2007. / Also available online.

Made available in DSpace on 2014-06-11T19:30:27Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-07-20Bitstream added on 2014-06-13T20:00:38Z : No. of bitstreams: 1 mello_mv_dr_sjrp_parcial.pdf: 191324 bytes, checksum: 834d46b5c37971622ceb90534e435e2c (MD5) Bitstreams deleted on 2014-08-22T14:57:09Z: mello_mv_dr_sjrp_parcial.pdf,Bitstream added on 2014-08-22T15:02:10Z : No. of bitstreams: 1 000697077.pdf: 803410 bytes, checksum: da262ae1b32f853d9d5b7460be7943f5 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Neste trabalho, estudamos propriedades dos zeros de polinômi os ortogonais do tipo Sobolev . Provam os resultados sobre entrelaçamento, monotonicidade e assintótica. Fornecemos, também , condições s necessárias e/ou suficientes para os zeros dos polinômios {Sn}n≥0, gerados pela fórmula Sn(x) = Pn(x) + an−1Pn−1(x), ou Sn(x) −bn−1Sn−1(x) = Pn(x), on d e {Pn}n≥0 é um a sequência de polinômios ortogonais, ser em todos reais / We study various properti s of the zeros of Sobolev typ e orthogonal polynomials. Results on interacing, monotonicity and asymptotic are proved . We also provide general necessary and/or sufficient con ditions in order to the zeros of the polynomials {Sn}n≥0, generated by the formulae Sn(x) = Pn(x) + an−1Pn−1(x), or Sn(x) −bn−1Sn−1(x) = Pn(x), where {Pn}n≥0 is a sequence of orthogon al polynomials, are all real

Factorial design has played a prominent role in the field of experimental design because of its richness in both theory and application. It explores the factorial effects by allowing the arrangement of efficient and economic experimentation, among which orthogonal design, uniform design and some other factorial designs have been widely used in various scientific investigations. The main contribution of this thesis shows the recent advances in the classification and selection of orthogonal designs. Design isomorphism is essential to the classification, selection and construction of designs. It also covers various popular design criteria as necessary conditions, such connection has led to a rapid growth of research on the novel approaches for either detecting the non-isomorphism or identifying the isomorphism. But further classification of non-isomorphic designs has received little attention, and hence remains an open question. It motivates us to propose the degree of isomorphism, as a more general view of isomorphism, for classifying non-isomorphic subclasses in orthogonal designs, and develop the column-wise identification framework accordingly. Selecting designs in sequential experiments is another concern. As a well-recognized strategy for improving the initial design, fold-over techniques have been widely applied to construct combined designs with better property in a certain sense. While each fold-over method has been comprehensively studied, there is no discussion on the comparison of them. It is the motivation behind our survey on the existing fold-over methods in view of statistical performance and computational complexity. The thesis involves five chapters and it is organized as follows. In the beginning chapter, the underlying statistical models in factorial design are demonstrated. In particular, we introduce orthogonal design and uniform design associated with commonly-used criteria of aberration and uniformity. In Chapter 2, the motivation and previous work of design isomorphism are reviewed. It attempts to explain the evolution of strategies from identification methods to detection methods, especially when the superior efficiency of the latter has been gradually appreciated by the statistical community. In Chapter 3, the concepts including the degree of isomorphism and pairwise distance are proposed. It allows us to establish the hierarchical clustering of non-isomorphic orthogonal designs. By applying the average linkage method, we present a new classification of L 27 (3 13 ) with six different clusters. In Chapter 4, an efficient algorithm for measuring the degree of isomorphism is developed, and we further extend it to a general framework to accommodate different issues in design isomorphism, including the detection of non-isomorphic designs, identification of isomorphic designs and the determination of non-isomorphic subclass for unclassified designs. In Chapter 5, a comprehensive survey of the existing fold-over techniques is presented. It starts with the background of these methods, and then explores the connection between the initial designs and their combined designs in a general framework. The dictionary cross-entropy loss is introduced to simplify a class of criteria that follows the dictionary ordering from pattern into scalar, it allows the statistical performance to be compared in a more straightforward way with visualization

Please read the abstract in the front of this document. / Thesis (PhD)--University of Pretoria, 2009. / Mathematics and Applied Mathematics / unrestricted

Indiana University-Purdue University Indianapolis (IUPUI) / We consider orthogonal polynomials P_n satisfying orthogonality relations where the measure of orthogonality is, in general, a complex-valued Borel measure supported on subsets of the complex plane. In our consideration we will focus on measures of the form d\mu(z) = \rho(z) dz where the function \rho may depend on other auxiliary parameters. Much of the asymptotic analysis is done via the Riemann-Hilbert problem and the Deift-Zhou nonlinear steepest descent method, and relies heavily on notions from logarithmic potential theory.

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Introduction. Statement of problem: The primary purpose of this study is to indicate the manner in which a Boundary Value problem in Physics leads to the solution in generalized Fourier Series. The conditions to be met in problems of this sort are generally the Partial Differential Equation and several unique physical conditions which are imposed on the distribution sought after. The problem is solved when a mathematical solution of the Differential Equation is found which satisfies all of the restrictions levied by the physical considerations. The secondary purpose of this study is to obtain a view of the generalized problem which leads ultimately to the Sturm-Liouville theory.

No / Wavelets are constructed from the basis sets of their parent scaling functions of the two-scale dilation equation (1). Whereas orthogonal wavelets come from one orthogonal basis set, the biorthogonal wavelets project from different basis sets. Each basis set is correspondingly weighted to form filters, either highpass or lowpass, which form the constituents of quadrature mirror filter (QMF) banks. Consequently, these filters can be used to design wavelets, the differently weighted parameters contributing respective wavelet properties which influence the performance of the transforms in applications, for example multicarrier modulation. This study investigated applications for onward multicarrier modulation applications. The results show that the optimum choice of particular wavelet adopted in digital multicarrier communication signal processing may be quite different from choices in other areas of wavelet applications, for example image and video compression.

The Givens orthogonalization algorithm is an efficient alternative to the normal equations method for solving many adjustment problems in photogrammetry. The Givens method is one of a class of methods for solving linear systems known generally as orthogonalization or QR methods. It allows for sequential processing and greatly simplifies the computation of statistics on the observations and residuals. The underlying reason for these advantages is the immediate availability of the orthogonal Q matrix, which is computed as the data are processed and is intimately related to the statistics needed for blunder detection. One of these statistics, the F statistic computed from externally studentized residuals, is both easily obtained and well-suited for blunder detection. The Givens method requires nearly four times the number of computations as compared to the normal equations approach in order to reach a solution. However, depending on the size of the problem, blunder detection through the normal equations requires far more computer time than is required when starting with a Givens decomposition. The method allows a user to review intermediate results, test residuals and modify the solution without having to compute a full solution. Adjustments of a level net and a single-photo resection are used to demonstrate the method. Because of the advantage in computational time, the Givens method is superior to the normal equations approach when rigorous blunder detection is required. / Master of Science