Much of weighted polynomial approximation originated with the famous Bernstein qualitative approximation problem of 1910/11. The classical Bernstein approximation problem seeks conditions on the weight functions \V such that the set of functions {W(x)Xn};;"=l is fundamental in the class of suitably weighted continuous functions on R, vanishing at infinity. Many people worked on the problem for at least 40 years. Here we present a short survey of techniques and methods used to prove Markov and Bernstein inequalities as they underlie much of weighted polynomial approximation. Thereafter, we survey classical techniques used to prove Jackson theorems in the unweighted setting. But first we start, by reviewing some elementary facts about orthogonal polynomials and the corresponding weight function on the real line. Finally we look at one of the processes (If approximation, the Lagrange interpolation and present the most recent results concerning mean convergence of Lagrange interpolation for Freud and Erdos weights. / Andrew Chakane 2018

MSc., Faculty of Science, University of the Witwatersrand, 2011 / We try to develop a collective field theory of single matrix models by using the formalism of Jevicki and Sakita in [1], with Schur polynomials as our collective fields. Field operators and the relation for the change of variables required to obtain the collective field Hamiltonian are found using group representation theory.

A thesis submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of the requirements for the degree of Doctor of Philosophy. Johannesburg, 2016. / The convenient pictorial descriptions of the half-BPS and near-BPS sectors of the AdS=CFT equivalent theories of N = 4, D = 4 super Yang-Mills and D = 10 Type IIB superstring theory on AdS5 S5 are exploited in this thesis by using Schur polynomials labelled by Young diagrams as a basis for the gauge invariant operators in the eld theory. We use a \Fourier transform" on these operators to construct asymptotic eigenstates of the dilatation operator, the spectrum of which agrees precisely with the rst two leading order terms in the smallcoupling expansion of the exact result determined by symmetry. Motivated by the geometric description of the systems of open strings with magnon excitations to which the operators are dual, we propose a simple and minimal all-loop expression that interpolates between anomalous dimensions computed in the gauge theory and energies computed in the string theory. The connection to the string theory result provides the insight necessary to understand the interpretation of our Gauss graphs in the magnon language. Symmetry determines the two-body scattering matrix for the magnons up to a phase, and it is demonstrated that integrability is spoiled by the boundary conditions on the open strings. The Schur polynomial construction is then applied to the study of closed strings on a class of half- BPS excitations of the AdS5 S5 background. The string theory predictions for the magnon energies are again reproduced by calculating the anomalous dimensions of particular linear combinations of our operators. Group theoretic quantities which can be read o the Young diagram labels provide the correct modi cation of terms in the dilatation action to account for the energies of magnons at di erent radii on the LLM plane. The representation theory implies a natural splitting of the full symmetry group - the distinction between what is the background and what is the excitation is accomplished in the choice of the subgroup and representations used to construct the operator. Connecting the descriptions utilised in obtaining these results is expected to allow the construction of operators dual to general open string con gurations on the class of backgrounds considered. / GR 2016

Orientador: Dimitar Kolev Dimitrov / Coorientador: Cleonice Fátima Braccialli / Banca: Roberto Andreani / Banca: Luis Gustavo Nonato / Banca: Eliana Xavier Linhares de Andrade / Banca: German Jesus Lozada Cruz / Resumo: 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 / Abstract: 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 / Doutor

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Typescript. Includes vita and abstract. Includes bibliographical references (leaves 98-99). Also available online in Scholars' Bank; and in ProQuest, free to University of Oregon users.

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.