This thesis is concerned with exact solutions to various massive field theories in 1+1 dimensions. Two approaches are described. The first, abstract and non-Lagrangian, relies on the considerable understanding that there now is of massless two-dimensional field theories. A perturbative scheme can be developed within which various exact statements may be made. Chapter 1 contains a review of this technique, together with some work applying it in various simple situations. The particular structures studied turn out to have a deep connection with certain Lie algebras, a fact which is discussed in the concluding three sections of the chapter. A complementary approach is to study specific, classically integrable, lagrangians in the hope that their quantum versions will also permit an exact treatment. Motivated to some extent by the findings of chapter 1, the remainder of the thesis is devoted to a particular class of models known as affine Toda field theories. Mixtures of perturbative and non-perturbative ideas are employed. The non-perturbative elements are to be found in analytic S-matrix theory, reviewed in chapter 2, while various features of the classical theory necessary for a perturbative quantum treatment are derived in chapter 3. Making use of this information, chapter 4 proposes exact expressions for the S-matrices for a large subset of the Toda theories, which are then checked in perturbation theory. Finally, the relevance or otherwise of the Toda S-matrices to the perturbations of massless theories studied in chapter 1 is discussed, and some possible directions for future work are mentioned.

There are many applications of Lie algebras to theoretical physics. This thesis is a study of some new mathematical structures which also are applicable to current physical ideas. The structures studied are Lie algebras of infinite dimension and the deformations of Lie algebras known as quantum algebras. The approach is algebraic, although physical applications are indicated. Chapter 1 The mathematics of finite and infinite dimensional Lie algebras is reviewed, together with an indication of well established uses in physics. The terms and notation used in the rest of the thesis are introduced. Chapter 2 Explicit examples of new infinite dimensional algebras of a type related to the algebras of conformal transformations on arbitrary genus Riemann surfaces are given. The relationship of these algebras to the Virasoro algebra is discussed. Chapter 3 The sine algebra is introduced and its relationship to the Moyal bracket discussed. The finite Lie algebras are given in a trigonometric basis. The many applications of the Moyal algebra are reviewed. Chapter 4 An original proof of the uniqueness of the Moyal algebra is presented. It is shown that the Moyal bracket is the most general Lie bracket of functions of two variables, and thus that the underlying associative star product is unique. It follows that all 2-index Lie algebras correspond to the Moyal algebra in some basis. Chapter 5 Quantum deformations of Lie algebras, or quantum algebras, are introduced. The many deformations of su(2) are described and the associativity conditions are discussed. Some new higher dimensional and infinite dimensional quantum algebras are given. Chapter 6 Quantum groups are discussed as groups of transformations of the quantum plane. Higher dimensional quantum groups and quantum supergroups are also described.

This thesis investigates the two dimensional, integrable field theories known as the affine Toda field theories, which are based on the Kac-Moody algebras with zero central extension. In particular, the construction of static solitons in these theories and their topological charges are considered. Following a general overview of the affine Toda theories and the Kac-Moody structure which underlies them, the construction of solitons in the a(_n)((^1)) theory using Hirota's method, originally used by HoUowood, is generalized and extended to the remaining theories. The soliton masses are calculated and general expressions presented for the twisted as well as the untwisted theories. The major results of this work concern the calculation of topological charge, one of the infinite number of conserved quantities that each theory possesses. Firstly, the a(_n)((^1)) model is considered. An expression for the number of charges associated with each soliton, as well as a general expression for the charges themselves, is constructed. The previously alluded to connection between the charges and the associated fundamental representations is proven showing that the charges are, in general, a subset of the weights lying in these representations. For the a(_n)((^1)) theory, the charges associated with each soliton can be derived from just one by making use of the cyclic symmetry of the model's extended Dynkin diagram. Further, the action of this symmetry on the set of charges is synonymous with the action of a Coxeter element. It is found that the ordering of the Weyl reflections which make up this element is important (except when the end-point solitons are considered) - the familiar "black-white" ordering doesn't work. The multisolitons of the theory are considered and it is shown that when the individual solitons are sufficiently well separated their topological charges simply add together. Multi-solitons can be constructed having topological charge equal to each of the simple roots, and can therefore be used to construct further solitons filling the entire weight lattice. Next, the topological charges of the remaining aflttne Toda theories are investigated. For the infinite series of algebras the number of topological charges and expressions for the charges themselves are derived. For the remaining cases, the charges are calculated explicitly. This thesis concludes with some comments on more recent work into the theory of quantum solitons and considers further lines of enquiry.

This thesis examines the relationship between the L-series of an elliptic curve evaluated at s = 2 and the image of the regulator map when the curve is defined over a real quadratic field with narrow class number one, thus providing numerical evidence for Beilinson's conjecture. In doing so it provides a practical formula for calculating the L-series for modular elliptic curves over real quadratic fields, and in outline for more general totally real fields, and also provides numerical evidence for the generalization of the Taniyarna-Weil-Shimura conjecture to real quadratic fields.

Let r denote a finite group and R a commutative ring. Factorizability theories seek to describe similarities between the local structure of R1-modules M and N, where M and N are related by, for example, being isomorphic when tensored up with Q. In the first three chapters of this thesis, we define two families of factorizability theories, the invariance and coinvariance factorizability theories. We will consider three members of these families. We demonstrate that monomial invariance factorizability is equivalent to monomial factorizability as defined in [19]. We go on to consider the two cyclic cases. We demonstrate that the weak cyclic invariance factorizability theory is strict and is identical to the weak cyclic coinvariance factorizability theory. We also demonstrate that the strong cyclic invariance factorizability theory and the strong cyclic coinvariance factorizability theory are not identical but are equivalent. In chapters 4 and 5, we discuss C.M.M. F-functors over R. Thus we find relations which can simplify the calculation of the invariance and coinvariance factorizability theories. An index of the less well known definitions used in this thesis is included as an appendix.

In this thesis my emphasis is on the resolution of the singularities of fibre products of Arithmetic Surfaces. In chapter one as an introduction to my thesis some elementary concepts related to regular and singular points are reviewed and the concept of tangent cone is defined for schemes over a discrete valuation ring. The concept of arithmetic surfaces is introduced briefly in the end of this chapter. In chapter 2 my new procedures namely the procedure of Mojgan(_1) and the procedure of Mahtab(_2) and a new operator called Moje are introduced. Also the concept of tangent space is defined for schemes over a discrete valuation ring. In chapter 3 the singularities of schemes which are the fibre products of some surfaces with ordinary double points are resolved. It is done in two different methods. The results from both methods are consistent. In chapter 4, I have tried to resolve the singularities of a special class of arithmetic three-folds, namely those which are the fibre product of two arithmetic surfaces, which were very helpful to achieve my final results about the resolution of singularities of fibre products of the minimal regular models of Tate. Chapter 5 includes my final results which are about the resolution of singularities of the fibre product of two minimal regular models of Tate.

This thesis is an account of work carried out at the Department of Mathematics, Durham University, between October 1979 and September 1982. A method of approximating functions in regions of the complex plane is given. Although it is not, in general, a near minimax approximation it is shown that it can give good results. A review of approximation in the complex plane is given in Chapter 1.Chapter 2 contains the basic properties of Chebyshev polynomials and the Chebyshev series, together with methods for calculating the coefficients in the series. The maximum error, over a complex domain, of a truncated Chebyshev series is investigated in Chapter 3 and Chapter 4 shows how the Bessel functions of the first and second kinds of integer order could be approximated over the entire complex plane. Numerical calculations were performed on the NUMAC IBM370.168 computer.

We consider probability measures on a space S(^A) (where S and A are countable and the σ-field is the natural one) which are Markov random fields with respect to a given neighbour relation ~ on A. In particular, we study the set G(II) of Markov random fields corresponding to a given Markov specification II, i.e. to a consistent family of "Markov" conditional probability distributions associated with the finite subsets of A. First, we review the relation between II and G(II). We consider also the representation of II by a family of interaction functions associated with the simplices of the graph (A,~) , together with some related problems. The rest of the thesis is concerned with the case where (A,~) is a tree. We define Markov chains on and consider their relation to the wider class of Markov random fields. We then derive analytical methods for the study of the set M(II) of Markov chains in G(II). These results are applied to homogeneous Markov specifications on regular infinite trees. Finally, we consider Markov specifications which are either attractive or repulsive with respect to a total ordering on S. For these we obtain quite strong results, including an exact condition for G(II) to contain precisely one element. We thereby generalise results obtained by Preston and Spitzer for binary S.

This thesis is principally concerned with Arthur Cayley's work on Invariant Theory, but also considers his contribution to matrix algebra and other algebraic systems, drawing on sources including unpublished letters between Cayley and his contemporary, J. J. Sylvester. The history of modern linear algebra and Cayley's part in its development has been extensively researched in the last decade by Thomas Hawkins. However, little has been written on Cayley's contribution to Invariant Theory, a subject to which he constantly reverted over a period of fifty years. In comparison, his work on Matrix Theory was a minor interest. The focal points in Cayley's passage through Invariant theory are investigated with reference being made, inter alia, to his correspondence with J. J. Sylvester which affords special insights into both the development of this Theory and the nature of their collaboration. Where appropriate, particulars of Sylvester's own work are given. Biographical details are included where these are believed to be unpublished or otherwise not generally available. A survey of Cayley's mathematical thought is offered in so far as it may be determined from his scattered remarks. Cayley pursued his algebraic researches on two distinct levels. First, he absorbed himself in calculation which led him to the combinatorial aspects of Invariant Theory and, secondly, he displayed a remarkable proclivity for systemisation, although this expressed itself in the classification of specific forms rather than in the development of an abstract theory as with the German algebraists. The basic text contains four chapters on Cayley's work in approximate chronological order followed by a final chapter on his general mathematical thinking. The Appendices include a statistical survey of his work, a bibliography of manuscripts, including, of course, his letters to Sylvester and a number of, little known photographs associated with Cayley and his times.

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