Let ω(m) be the number of distinct prime factors of m. A celebrated theorem of Erdös-Kac states that the quantity (ω(m)-loglog m)/√(loglog m) distributes normally. Let φ(m) be Euler's φ-function. Erdös and Pomerance proved that the quantity(ω(φ(m)-(1/2)(loglog m)^2)\((1/√(3)(loglog m)^(3/2)) also distributes normally. In this thesis, we prove these two results. We also prove a function field analogue of the Erdös-Pomerance Theorem in the setting of the Carlitz module.

We are interested in investigating the number of integral points on quadrics. First, we consider non-degenerate plane conic curves defined over Z. In particular we look at two types of conic sections: hyperbolas with two rational points at infinity, and ellipses. We give upper bounds for the number of integral points on such curves which depends on the number of divisors of the determinant of a given conic. Next we consider quadratic surfaces of the form q(x, y, z) = k, where k is an integer and q is a non-degenerate homogeneous quadratic form defined over Z. We give an upper bound for the number of integral points (x, y, z) with bounded height.

Let ω(m) be the number of distinct prime factors of m. A celebrated theorem of Erdös-Kac states that the quantity (ω(m)-loglog m)/√(loglog m) distributes normally. Let φ(m) be Euler's φ-function. Erdös and Pomerance proved that the quantity(ω(φ(m)-(1/2)(loglog m)^2)\((1/√(3)(loglog m)^(3/2)) also distributes normally. In this thesis, we prove these two results. We also prove a function field analogue of the Erdös-Pomerance Theorem in the setting of the Carlitz module.

We are interested in investigating the number of integral points on quadrics. First, we consider non-degenerate plane conic curves defined over Z. In particular we look at two types of conic sections: hyperbolas with two rational points at infinity, and ellipses. We give upper bounds for the number of integral points on such curves which depends on the number of divisors of the determinant of a given conic. Next we consider quadratic surfaces of the form q(x, y, z) = k, where k is an integer and q is a non-degenerate homogeneous quadratic form defined over Z. We give an upper bound for the number of integral points (x, y, z) with bounded height.

Roth's theorem, proved by Roth in 1953, states that when A is a subset of the integers [1,N] with A dense enough, A has a three term arithmetic progression (3-AP). Since then the bound originally given by Roth has been improved upon by number theorists several times. The theorem can also be generalized to finite abelian groups. In 1994 Meshulam worked on finding an upper bound for subsets containing only trivial 3-APs based on the number of components in a finite abelian group. Meshulam’s bound holds for finite abelian groups of odd order. In 2003 Lev generalised Meshulam’s result for almost all finite abelian groups. In 2009 Liu and Spencer generalised the concept of a 3-AP to a linear equation and obtained a similar bound depending on the number of components of the group. In 2011, Liu, Spencer and Zhao generalised the 3-AP to a system of linear equations. This thesis is an overview of these results.

The key feature of the Integrating Cavity Absorption Meter (ICAM) is that it produces an isotropic illumination of the liquid sample and thereby dramatically minimizes scattering effects. The ICAM can produce an effective optical path length up to several meters. As a consequence, it is capable of measuring absorption coefficients as low as 0.001 m-1. The early version of the ICAM was used previously to measure the absorption spectrum of pure water over the 380-700 nm range. To extend its range into the ultraviolet, several modifications have been completed. The preliminary tests showed that the modified ICAM was able to measure the absorption of pure water for the wavelength down to 300 nm. After extensive experimental investigation and analysis, we found that the absorption of Spectralon® (the highly diffusive and reflective material used to build the ICAM) has a higher impact on measurements of absorption in the UV range than we had expected. Observations of high values for pure water absorption in the UV, specifically between 300 and 360 nm, are a consequence of absorption by the Spectralon®. These results indicated that even more serious modifications were required (e.g. Spectralon® can not be used for a cavity in the UV). Consequently, we developed a new diffuse reflecting material and used fused silica powder (sub-micron level) sealed inside a quartz cell to replace the inner Spectralon® cavity of the ICAM. The new data is in excellent agreement with the Pope and Fry data (380-600 nm) and fills the gap between the 320 nm data of Quickenden and Irvin and 380 nm data of Pope and Fry. We present definitive results for the absorption spectrum of pure water between 300 and 600 nm.

Thesis (Ph. D.)--University of Hong Kong, 2000. / Includes bibliographical references (leaves 308-337).

This thesis is mainly concerned with an error analysis of numerical methods for two point boundary value problems. in particular for the method of collocation using polynomial and certain piecewise polynomial bases. As in previous work on strict error bounds an operator theoretical approach is taken. The setting for the theory and the principal results for later use are firstly considered. Then two types of 'a posteriori' error bounds are developed. These"bounds are made computable by relating the inverse of the approximating operator to the inverse of certain matrices formed in the actual application of the approximation method. The application of this theory to the numerical solution of linear two point boundary value problems is then considered. It is demonstrated how the differential equation can be split to fit into the setting required by the theory. It is also demonstrated how the global and the piecewise collocation method can be expressed in terms of a projection method applied to the operator equation. The conditions required by the theory are expressed in terms of continuity requirements on the coefficients of the differential equation and in terms of the distribution of the collocation points. In examining these bounds on a variety of problems. it is noticed that with some problems the conditions for applicability may not hold except for more points than one actually required to obtain a satisfactory solution. To improve the applicability. the theory is reconsidered with a different splitting of the differential equation. The method of collocation is expressed accordingly in terms of a new projection operator which is proved to have some nice properties in practice. This new approach is then compared with the original one and it is shown to be superior on various problems. By examining the inverse differential operator and the residual improved error bounds and estimates are shown to be obtainable. These estimates are tested in a large variety of examples and some graphs are presented to describe their behaviour in more detail. Finally these estimates are used to develop various adaptive mesh selection algorithms for solving two point boundary value problems. These strategies are tested and compared in several representative examples and some conclusions are drawn. The thesis concludes with a brief review of the work with an indication of possible improvements and extensions.

This thesis is mainly concerned with indirect numerical solution methods for linear two point boundary value problems. We concentrate particularly on problems with separated boundary conditions which have a 'dichotomy' property. We investigate the inter-relationship of various methods including some which have first appeared since the work for this thesis began. We examine the stability of these methods and in particular we consider circumstances in which the methods discussed give rise to well conditioned decoupling transformations. Empirical comparisons of some of the methods are described using a set of test problems including a number of ill conditioned problams. 'stiff' and marginally In the past the main method of error estimation has been to repaat the whole calculation. Here an alternative error estimation technique is proposed and a related iterative improvement method is considered. Although results for this are not completely conclusive we think they justify the need for further research on the method as it shows promise of being a novel and reliable practical method of solving both well conditioned and ill conditioned problems.

In this thesis we investigate results about convex combinations of unitaries in unital J B* -algebras, the Jordan algebra analogues of C* -algebras. After giving background material in chapter 1 we introduce the concept of unitary isotopes of J B* -algebras in chapter 2 and develop their theory including identification of their centre. We also show important unit aries for our results come from the polar decomposition of invertible elements. In chapter 3 we investigate which elements are self-adjoint in some isotope to start the development of the theory of convex combinations of two or more unitaries. This leads us in chapter 4 to introduce and give examples of a subclass of J B* -algebras in which the invertible elements are dense. We also show that extreme points of the unit ball sufficiently close to the invertibles must be unit aries and deduce that in the subclass, all extreme points are unitaries. In chapter 5 we look the relationships between the distance of an element to the invertibles or unit aries and special types of convex combinations, for example those of unit aries having all (but one) of the coefficients equal and those of unit ball elements only one of which need be unitary. Finally in chapter 6 we investigate some possible further developments and open problems