This paper is a study of T-sets of normed linear spaces. Geometrical properties of normed linear spaces are developed in terms of intersection properties shared by a subcollection of T-sets of the space and in terms of special spanning properties shared by each T-set of a subcollection of T-sets of the space. A characterization of the extreme points of the unit ball of the dual of a normed linear space is given in terms of the T-sets of the space. Conditions on the collection of T-sets of a normed linear space are determined so that the normed linear space has the property that extreme points of the unit ball of the dual space map canonically to extreme points of the unit ball of the third dual space.

by Chi-keung Ng. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1991. / Includes bibliographical references. / Introduction --- p.1 / Chapter Chapter 0. --- Preliminary --- p.4 / Chapter 0.1 --- Topological vector spaces / Chapter 0.2 --- Ordered vector spaces / Chapter 0.3 --- Ordered normed spaces / Chapter 0.4 --- Ordered topological vector spaces / Chapter 0.5 --- Ordered bornological vector spaces / Chapter Chapter 1. --- Results on Ordered Normed Spaces --- p.23 / Chapter 1.1 --- Results on e∞-spaces and e1-spaces / Chapter 1.2 --- Complemented subspaces of ordered normed spaces / Chapter 1.3 --- Half injections and Half surjections / Chapter 1.4 --- Strict quotients and strict subspaces / Chapter Chapter 2. --- Helley's Selection Theorem and Local Reflexivity Theorem of order type --- p.55 / Chapter 2.1 --- Helley's selection theorem of order type / Chapter 2.2 --- Local reflexivity theorem of order type / Chapter Chapter 3. --- Operator Modules and Ideal Cones --- p.68 / Chapter 3.1 --- Operator modules and ideal cones / Chapter 3.2 --- Space cones and space modules / Chapter 3.3 --- Injectivity and surjectivity / Chapter 3. 4 --- Dual and pre-dual / Chapter Chapter 4. --- Topologies and Bornologies Defined by Operator Modules and Ideal Cones --- p.95 / Chapter 4.1 --- Generalized polars / Chapter 4.2 --- Topologies and bornologies defined by β and ε / Chapter 4. 3 --- Injectivity and generating topologies / Chapter 4.4 --- Surjectivity and generating bornologies / Chapter 4.5 --- The solid property and the generating topologies / Chapter 4.6 --- The solid property and the generating bornologies / Chapter Chapter 5. --- Semi-norms and Bounded disks defined by Operator Modules and Ideal Cones --- p.129 / Chapter 5.1 --- Results on semi-norms / Chapter 5.2 --- Results on bounded disks / References --- p.146 / Notations --- p.149

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 83-89). Also available on the Internet.

Thesis (Ph. D.)--University of Missouri-Columbia, 2003. / Typescript. Vita. Includes bibliographical references (leaves 83-89). Also available on the Internet.

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract. / Master of Mathematical Sciences

"A natural process in examining properties of Banach spaces is to see if a Banach space can be decomposed into simpler Banach spaces; in other words, to see if a Banach space has complemented subspaces. This thesis concentrates on three main aspects of this problem: norm of projections of a Banach space onto its finite dimensional subspaces; a class of Banach spaces, each of which has a large number of infinite dimensional complemented subspaces; and methods of finding Banach spaces which have uncomplemented subspaces, where the subspaces and the quotient spaces are chosen as well-known classical sequence spaces (finding non-trivial twisted sums)." --Abstract. / Master of Mathematical Sciences

The purpose of this paper is to present the results of a study of linear spaces with special emphasis of linear transformations, norms, and inner products.

This paper is a study of linear operators defined on normed linear spaces. A basic knowledge of set theory and vector spaces is assumed, and all spaces considered have real vector spaces. The first chapter is a general introduction that contains assumed definitions and theorems. Included in this chapter is material concerning linear functionals, continuity, and boundedness. The second chapter contains the proofs of three fundamental theorems of linear analysis: the Open Mapping Theorem, the Hahn-Banach Theorem, and the Uniform Boundedness Principle. The third chapter is concerned with applying some of the results established in earlier chapters. In particular, the concepts of compact operators and Schauder bases are introduced, and a proof that an operator is compact if and only if its adjoint is compact is included. This chapter concludes with a proof of an important application of the Open Mapping Theorem, namely, the Closed Graph Theorem.

<p>Statistical models often rely on several assumptions including distributional assumptions on outcome variables and relational assumptions where we model the relationship between outcomes and independent variables. Further assumptions are also made depending on the complexity of the data and the model being used. Model diagnostics is, therefore, a crucial component of any model fitting problem. Residuals play important roles in model diagnostics. Residuals are not only used to check adequacy of model fit, but they also are excellent tools to validate model assumptions as well as identify outliers and influential observations. Residuals in univariate models are studied extensively and are routinely used for model diagnostics. In multivariate models residuals are not commonly used to assess model fit, although a few approaches have been proposed to check multivariate normality. However, in the analysis of longitudinal data, the resulting residuals are correlated and are not normally distributed. It is, therefore, not clear as to how ordinary residuals can be used for model diagnostics. Under sufficiently large sample size, a transformation of ordinary residuals are proposed to check the normality assumption. The transformation is based solely on removing correlation among the residuals. However, we show that these transformed residuals fail in the presence of model mis-specification. In this thesis, we investigate residuals in the analysis of longitudinal data. We consider ordinary residuals, Fitzmaurice’s transformed (uncorrelated) residuals as well as von Rosen’s decomposed residuals. Using simulation studies, we show how the residuals behave under multivariate normality and when this assumption is violated. We also investigate their properties under correct fitting as well as wrongly fitted models. Finally, we propose new residuals by transforming von Rosen’s decomposed residuals. We show that these residuals perform better than Fitzmourice’s transformed residuals in the presence of model mis-specification. We illustrate our approach using two real data sets.</p> / Master of Science (MSc)

Cette thèse est composée de deux parties : l’une traite des propriétés combinatoires de mots infinis et l’autre des problèmes de colorations des graphes.La première partie du manuscrit concerne les structures régulières dans les mots apériodiques infinis, à savoir les sous-séquences arithmétiques et les premiers retours complets.Nous étudions la fonction qui donne la longueur maximale d’une sous-séquence arithmétique monochromatique (une progression arithmétique) en fonction de la différence commune d pour une famille de mots morphiques uniformes, qui inclut le mot de Thue-Morse. Nous obtenons la limite supérieure explicite du taux de croissance de la fonction et des emplacements des progressions arithmétiques de longueurs maximales et de différences d. Pour étudier des sous-séquences arithmétiques périodiques dans des mots infinis, nous définissons la notion d'indice arithmétique et obtenons des bornes supérieures et inférieures sur le taux de croissance de la fonction donnant l’indice arithmétique dans la même famille de mots.Dans la même veine, une autre question concerne l’étude de deux nouvelles fonctions de complexité de mots infinis basées sur les notions de mots ouverts et fermés. Nous dérivons des formules explicites pour les fonctions de complexité ouverte et fermée pour un mot d'Arnoux-Rauzy sur un alphabet de cardinalité finie.La seconde partie de la thèse traite des colorations parfaites (des partitions équitables) de graphes infinis de degré borné. Nous étudions les graphes de Caley de groupes additifs infinis avec un ensemble de générateurs fixé. Nous considérons le cas où l'ensemble des générateurs est composé d'entiers de l'intervalle [-n, n], et le cas où les générateurs sont des entiers impairs de [-2n-1, 2n+1], où n est un entier positif. Pour les deux familles de graphes, nous obtenons une caractérisation complète des colorations parfaites à deux couleurs / The content of the thesis is comprised of two parts: one deals with combinatorial properties of infinite words and the other with graph coloring problems.The first main part of the manuscript concerns regular structures in infinite aperiodic words, such as arithmetic subsequences and complete first returns.We study the function that outputs the maximal length of a monochromatic arithmetic subsequence (an arithmetic progression) as a function of the common difference d for a family of uniform morphic words, which includes the Thue-Morse word. We obtain the explicit upper bound on the rate of growth of the function and locations of arithmetic progressions of maximal lengths and difference d. To study periodic arithmetic subsequences in infinite words we define the notion of an arithmetic index and obtain upper and lower bounds on the rate of growth of the function of arithmetic index in the same family of words.Another topic in this direction involves the study of two new complexity functions of infinite words based on the notions of open and closed words. We derive explicit formulae for the open and closed complexity functions for an Arnoux-Rauzy word over an alphabet of finite cardinality.The second main part of the thesis deals with perfect colorings (a.k.a. equitable partitions) of infinite graphs of bounded degree. We study Caley graphs of infinite additive groups with a prescribed set of generators. We consider the case when the set of generators is composed of integers from the interval [-n,n], and the case when the generators are odd integers from [-2n-1,2n+1], where n is a positive integer. For both families of graphs, we obtain a complete characterization of perfect 2-colorings