This work is an examination of various aspects of twisted vertex operator representations of Kac-Moody algebras. It starts with an introduction to Kac-Moody algebras and string theories, including a discussion of the propagation of strings on orbifolds. String interactions in a subclass of such models naturally involve twisted vertex operators. The centrally extended loop algebra realization of Kac-Moody algebras is used to explain why the inequivalent gradations of basic representations of Kac-Moody algebras g(^r) associated with g are in one-to-one correspondence with the conjugacy classes of the automorphism group of the root system, aut Ф(_g).The structure of the automorphism groups of the simple Lie algebra root systems are examined. A method of classifying the conjugacy classes of the Weyl groups is explained and then extended to cover the whole automorphism group in cases where there are additional Dynkin diagram symmetries. All possible automorphisms, a, that have the property that det (1 – σ(^r)) ≠ 0, r = 1, ….. , n - 1 where n is the order of a, are determined. Such automorphisms lead to interesting orbifold models in which some of the calculations are simplified. A thorough exposition of the twisted vertex operator representation is given including a detailed explanation of the zero-mode Hilbert space and the construction of the required cocycle operators. The relation of the vacuum degeneracy to the number of fixed subspace singularities in the orbifold construction is discussed. Explicit examples of twisted vertex operators and their associated cocycles are given. Finally it is shown how the twisted and an alternative shifted vertex operator representation of the same gradation may be identified. This is used to determine the invariant subalgebras of the gradations along with the vacuum degeneracies and conformal weights of the representations. The results of calculations for inequivalent gradations of the simply laced exceptional algebras are given.

The central objects of study in this thesis are quantized coordinate algebras. These algebras originated in the 1980s in the work of Drinfeld and Jumbo and are noncommutative analogues of coordinate rings of algebraic varieties. The organic nature by which these algebras arose is of great interest to algebraists. In particular, investigating ring theoretic properties of these noncommutative algebras in comparison to the properties already known about their classical (commutative) counterparts proves to be a fruitful process. The prime spectrum of an algebra has always been seen as an important key to understanding its fundamental structure. The search for prime spectra is a central focus of this thesis. Our focus is mainly on Quantum Grassmannian subalgebras of quantized coordinate rings of Matrices of size m x n (denoted Oq(Mm;n)). Quantum Grassmannians of size m x n are denoted Gq(m; n) and are the subalgebras generated by the maximal quantum minors of Oq(Mm;n). In Chapter 2 we look at the simplest interesting case, namely the 2 x 4 Quantum Grassmannian (Gq(2; 4)), and we identify the H-primes and automorphism group of this algebra. Chapter 3 begins with a very important result concerning the dehomogenisation isomorphism linking Gq(m; n) and Oq(Mm;n¡m). This result is applied to help to identify H-prime spectra of Quantum Grassmannians. Chapter 4 focuses on identifying the number of H-prime ideals in the 2xn Quan- tum Grassmannian. We show the link between Cauchon fillings of subpartitions and H-prime ideals. In Chapter 5, we look at methods of ordering the generating elements of Quantum Grassmannians and prove the result that Quantum Grassmannians are Quantum Graded Algebras with a Straightening Law is maintained on using one of these alternative orderings. Chapter 6 looks at the Poisson structure on the commutative coordinate ring, G(2; 4) encoded by the noncommutative quantized algebra Gq(2; 4). We describe the symplectic ideals of G(2; 4) based on this structure. Finally in Chapter 7, we present an analysis of the 2 x 2 Reflection Equation Algebra and its primes. This algebra is obtained from the quantized coordinate ring of 2 x 2 matrices, Oq(M2;2).

Anderson and Putnam, and Kellendonk discovered methods of defining a C*- algebra on a noncommutative space associated with a tiling. The method employed was to use Renault's theory of groupoid C*-algebras of an equivalence relation on the tiling metric space. C*-algebras of a tiling have two purposes, on one hand they reveal information about the long range order of the tiling and on the other hand they provide interesting examples of C*-algebras. However, the two constructions do not include tilings such as the pinwheel tiling, with tiles appearing in an infinite number of orientations. We rectify this deficiency, with many interesting results appearing in the process.

M.Sc. / One of the earliest results (1955) in the theory of derivations is the celebrated theorem of I. M. Singer and J. Wermer [14] which asserts that every bounded derivation on a commutative Banach algebra has range contained in the radical. However, they immediately conjectured that their result will still hold if the boundedness condition was dropped. This conjecture of Singer and Wermer was confirmed only in 1988, by M. P. Thomas [23], when he showed that every derivation (bounded or unbounded) on a commutative Banach algebra has range contained in the radical. But it is not known whether an analogue of the Kleinecke-Shirokov Theorem holds for everywhere defined unbounded derivation.

This thesis presents an investigation into the properties of various algebras of trees. In particular, we study the influence that the structure of a tree algebra has on the solution of algorithmic problems about trees in that algebra. The investigation is conducted within the framework provided by the Bird-Meertens formalism, a calculus for the construction of programs by equational reasoning from their specifications. We present three different tree algebras: two kinds of binary tree and a kind of general tree. One of the binary tree algebras, called "hip trees", is new. Instead of being built with a single ternary operator, hip trees are built with two binary operators which respectively add left and right children to trees which do not already have them; these operators enjoy a kind of associativity property. Each of these algebras brings with it with a class of "structure-respecting" functions called catamorphisms; the definition of a catamorphism and a number of its properties come for free from the definition of the algebra, because the algebra is chosen to be initial in a class of algebras induced by a (cocontinuous) functor. Each algebra also brings with it, but not for free, classes of "structure-preserving" functions called accumulations. An accumulation is a function that preserves the shape of a structured object such as a tree, but replaces each element of that object with some catamorphism applied to some of the other elements. The two classes of accumulation that we study are the "upwards" and "downwards" accumulations, which pass information from the leaves of a tree towards the root and from the root towards the leaves, respectively. Upwards and downwards accumulations turn out to be the key to the solution of many problems about trees. We derive accumulation-based algorithms for a number of problems; these include the parallel prefix algorithm for the prefix sums problem, algorithms for bracket matching and for drawing binary and general trees, and evaluators for decorating parse trees according to an attribute grammar.

A dissertation submitted to the Faculty of Science, University of the Witwatersrand, Johannesburg, in fulfilment of requirements for the degree of Master of Science. August 8, 2014. / We study the nonlinear models arising in heat transfer in extended surfaces (fins) and in solid slab (hot body). Here thermal conductivity, internal generation and heat transfer coefficient are temperature dependent. As such the models are rendered nonlinear. We employ Lie point symmetry techniques to analyse these models. Firstly we employ Lie point symmetry methods and determine the exact solutions for heat transfer in fins of spherical geometry. These solutions are compared with the solutions of heat transfer in fins of rectangular and radial geometries. Secondly, we consider models describing heat transfer in a hot body, for example, a plane wall. We then employ the preliminary group classification methods to determine the cases of the arbitrary function for which the principal Lie algebra is extended by one. Furthermore we the exact solutions.

Lau Yuk Hong. / Bibliography: leaves 63-64 / Thesis (M.Ph.)--Chinese University of Hong Kong, 1984

by Ng Yin Fun. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2000. / Includes bibliographical references (leaves 50-51). / Abstracts in English and Chinese. / Acknowledgments --- p.i / Abstract --- p.ii / Introduction --- p.3 / Chapter 1 --- Preliminaries --- p.4 / Chapter 1.1 --- Hilbert C*-modules --- p.4 / Chapter 2 --- Self-dual Hilbert C*-modules --- p.14 / Chapter 2.1 --- Self-duality --- p.14 / Chapter 2.2 --- Self-duality and some related concepts --- p.22 / Chapter 2.3 --- A criterion of self-duality of HA --- p.23 / Chapter 3 --- Hilbert W*-modules --- p.25 / Chapter 3.1 --- Extension of the inner product to --- p.25 / Chapter 3.2 --- Extension of operators to --- p.33 / Chapter 3.3 --- Self-dual Hilbert W*-modules --- p.36 / Chapter 3.4 --- Some equivalent conditions for a Hilbert W*-module to be self-dual --- p.43 / Bibliography --- p.50

Neste trabalho realizamos um estudo de algebras, co algebras e algebras de Hopf, introduzindo estas no c~oes e algumas de suas propriedades e exemplos. Al em disso, aprofundamos o estudo apresentando as algebras de Hopf quase cocomutativas e quasitriangulares, demonstrando que a ant poda dessas algebras cumpre certas condi c~oes. / In this work we present a study of algebras, coalgebras and Hopf algebras, introducing examples and some of its properties. Also, we expand this study presenting almost cocommutative and quasitriangular Hopf algebras, showing that the antipode of these algebras satis es determined conditions.

Thesis. 1975. M.S.--Massachusetts Institute of Technology. Dept. of Mathematics. / Includes bibliographical references. / M.S.