This thesis is primarily concerned with the rank of elliptic curves over the rationals. It establishes a new long exact sequence of groups related to an elliptic curve <i>E </i>over <i>Q. </i> The sequence appears to have strong connections with the conjectures of Birch and Swinnerton-Dyer. Using Haar and Tamagawa measures in the case of the field <i>Q, </i>the algebraic rank and the analytic rank respectively are shown to be closely related to the first two groups. This approach also offers explanations, but not proofs, of the constants of the conjectures. The first part of the thesis ends with a discussion of the possible generalisation of the approach to a number field <i>K. </i> The second part of the thesis considers conductor related bounds on rank and also develops a theory of the distribution of rank for the set of all elliptic curves over <i>Q, </i>ordered by the size of coefficients. Published experimental results are compared with this theory. I conclude with algorithmic methods of identifying high rank elliptic curves.

Well-bounded operators of type (B) are the building blocks for trigonometrically well-bounded, polar and AC operators. We examine the relationship between polar and AC operators of type (B) and explore the concepts of bounded variation and absolute continuity for functions defined on annuli. In 1973, Hunt, Muckenhoupt and Wheeden showed that the Hilbert transform is a bounded operator on a weighted <I>L<SUP>p</SUP></I> space precisely when the weight satisfies the <I>A<SUB>p</SUB></I> condition. This result is proved independently for <I>L<SUP>p</SUP></I> spaces over the reals, the circle and the integers. We investigate the inter-relationships between these three theorems and show that the theorems for the reals and the integers are equivalent.

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We shall present a theory of fractional calculus for generalised functions on (0,∞) and use this theory as a basis for extensions to some related areas. In the first section, appropriate spaces of test-functions and generalised functions on (0,∞) are introduced and the properties of operators of fractional calculus obtained relative to these spaces. Applications are given to hypergeometric integral equations, Hankel transforms and dual integral equations of Titchmarsh type. In the second section, the Mellin transform is used to define fractional powers of a very general class of operators. These definitions include standard operators as special cases. Of particular interest are powers of differential operators of Bessel or hyper-Bessel type which are related to integral operators with special functions, notably G-functions, as kernels. In the third section, we examine operators whose Mellin multipliers involve products and/or quotients of Γ-functions. There is a detailed study of the range and invertibility of such operators in weighted L<SUP>P</SUP>-spaces and in appropriate spaces of smooth functions. The Laplace and Stieltjes transforms give two particular examples. In the final section, we show how our theory of fractional calculus on (0,∞) can be used to develop a corresponding theory on IR<SUP>n</SUP> in the presence of radial symmetry. In this framework the mapping properties of multidimensional radial integrals and Riesz potentials are obtained very precisely.

This thesis is divided into two distinct and independent parts. Part 1 concerns the Approximation Property (a.p.) and Radon Nikodym Property (RNP) in Banach Spaces. In Chapter 1 we outline the importance of the a.p. and produce examples of Banach Spaces without the a.p. by modifying a construction due to Szankowski. These spaces are closed subspaces of Rp direct sums of finite dimensional Rq spaces (1 < q < p < -), so with p < 2 we obtain Banach spaces of cotype 2 without the a.p. - this was unknown. In Chapter 2 we discuss the IMP proving in Theorem 2.9 the characterisation in terms of dentable subsets due to Rieffel and Huff (among others), of Banach spaces with the RNP. In theorem 2.18 we prove that dual spaces with the a.p. and RNP have the metric approximation property, obtaining as corollaries results of Grothendieck. We introduce p- nuclear and p- integral maps between Banach spaces E and F and prove in theorem 2.26 that, if E* has the RNP, all p- integral maps are p- nuclear, and in theorem 2.29 that, if F has the RNP all integral maps are nuclear. This extends work of Grothendieck, Perrson and Pietsch. Part 2 concerns the prediction theory of doubly stationary processes. In Chapter 3 we outline the basic prediction theory, and state, for the absolutely continuous case, Helson and Lowdenslager's characterisation, for a weight function w and an irrational a, of a process as type 1, 2 or 3. We give an example of a process of type 2, for all irrational a. In Chapter 4 we obtain in Theorem 4.10 an exact analogue of Helson and Szego's result, viz. that the past and future of a process are at positive angle if and only if dµ = wda , w = exp(u + v), where u, v are real L~ functions with null < ~ . We introduce a class of functions - BMO(a) functions, analogous to BMO functions, and prove BMO(a) is the dual of H1 (a) and {u + v u, v c L (a)} = BMO(a) in Theorems 4.19 and 4.20.

A holomorphic Penrose transform is described for Hitchin's correspondence between complex Einstein-Weyl spaces and "minitwistor" spaces, leading to isomorphisms between the sheaf cohomologies of holomorphic line bundles on a minitwistor space and the solution spaces of some conformally invariant field equations on the corresponding Einstein-Weyl space. The Penrose transforms for complex Euclidean 3-space and complex hyperbolic 3-space, two examples which have preferred Riemannian metrics, are explicitly discussed before the treatment of the general case. The non-holomorphic Penrose transform of Bailey, Eastwood and Singer, which translates holomorphic data on a complex manifold to data on a <I>smooth</I> manifold, using the notion of involutive cohomology, is reviewed and applied to the non-holomorphic twistor correspondences of four homogenous spaces: Euclidean 3-space, hyperbolic 3-space, Euclidean 5-space (considered as the space of trace-free symmetric 3 x 3 matrices) and the space of non-degenerate real conics in complex projective plane. The complexified holomorphic twistor correspondences of the last two cases turn out to be examples of more general correspondence between complex surfaces with rational curves of self-intersection number 4 and their moduli spaces.

This thesis explores three ways in which linear logic may be used to define a specification language for Petri nets, by giving precise correspondences, at different levels, between linear logic and Petri nets. Firstly, we define categories NC by analogy with de Paiva's dialectica categories GC. The category NSet has as objects the elementary Petri nets and morphisms refinement maps. We show that GC induces in NC sufficient structure for NC to be a sound model of linear logic. We demonstrate the computational significance of the net constructors induced by the interpretations in NSet of the linear connectives ®, A,-o,© and (-)-1-. Our framework unifies several existing approaches to categories of nets, and gives a model of full linear logic based on nets. Secondly, we show that the possible evolutions of a net generate a quan-tale. Quantales are algebraic models of linear logic. Further, we show that certain restrictions on nets, including being safe or bounded, rise as subquantales induced by suitable conuclei. This approach allows us to give a sound semantics for linear logic using sets of markings of a given net. Thus the probability of certain assertions in linear logic corresponds to properties of nets. Thirdly, we define a semantics for a fragment of linear logic £„ in terms of nets, by giving a partial function from formulae of linear logic to nets. This semantics is complete and sound where defined. Further, we show that whenever a net N can evolve to a net N', there is a canonical proof in to that the formula interpreted by N entails the formula interpreted by N'. A canonical proof expresses the causal dependencies of a net in a precise way, using the (Cut) rule. This approach allows us to use the techniques of proof theory to study the evolution of nets.

In this thesis are presented extensions to the theory of completely bounded operators and their relation to the Haagerup tensor product. The canonical map from the Haagerup tensor product of a von Neumann algebra with itself to the space of completely bounded operators on it is considered. A trace-like map on the space of completely bounded operators on the hyperfinite type II<SUB>1</SUB> factor is constructed and is shown to yield an estimate for the Jones index when applied to a conditional expectation.

The first chapter introduces some of the important concepts and structures associated with integrability, and includes a brief overview of some of the applications of integrable systems and their reductions in field theory. Chapter 2 describes the scaling similarity reductions of the Sawada-Kotera, fifth-order KdV, and Kaup-Kupershmidt equations. Similarity solutions of these evolution equations satisfy certain ODEs which are naturally viewed as fourth-order analogues of the Painlevé transcendents; they may also be written as non-autonomous Hamiltonian systems, which are time-dependent generalizations of the integrable Hénon-Heiles systems. The solutions to these systems are encoded into a tau-function, and Bäcklund transformations are presented which allow the construction of rational solutions and some other special solutions. The third chapter is concerned with the motion of the poles of singular solutions (especially rational solutions) of the NLS equation. It is demonstrated that the linear problem for NLS admits an analogue of the well-known Crum transformation for Schrödinger operators, leading to the construction of a sequence of rational solutions. The poles and zeros of these rational solutions are found to satisfy constrained Calogero-Moser equations, and some other singular solutions are also considered. Much use is made of Hirota's bilinear formalism, as well as a trilinear form for NLS related to its reduction from the KP hierarchy. The final chapter deals with soliton solutions of the <I>A<SUB>n</SUB></I><SUP>(1)</SUP> affine Toda field theories. By writing the soliton tau-functions as determinants of a particular form, these solutions are related to the hyperbolic spin Ruijsenaars-Schneider system. These results generalize the connection between the ordinary (non-spin) Ruijsenaars-Schneider model and the soliton solutions of the sine-Gordon equation.

The level spacing distribution of the two dimensional harmonic oscillator is investigated. By obtaining an explicit expression for the spacings, it is observed that the distribution is unstable under the semi-classical limit <I>h → </I>0. By defining a suitable average, a distribution stable under <I>h → </I>0 is obtained. Exact expressions are obtained for values of oscillator frequency ratio including the golden mean, 1/2, 1/5 and 1/e. Comparisons are made between these analytic results and the numerical ones in the paper of Berry and Tabor [1]. For a certain class of ratio, including the case 1/e, a delta function is found for the averaged spacing distribution. This is a fractal set shown to have Hausdorff dimension ½ as a subset of possible ratios. The case of generic frequency ratio is also studied for which a closed formula is found. Comments on the distribution follow. For the harmonic oscillator of general dimension <I>n</I> it is shown that the initial value of the level spacing distribution is (<I>n</I>)<SUP>-1</SUP>. Reasons for the conjecture that the distribution will be (monotonically) decreasing are also given. By employing a method used in the system above, it is shown for the particle in a two dimensional box, with certain possible box dimensions, that the spacing distribution is distinct from the Poisson one associated with the system.