In this thesis we look at a number of properties related to Kazhdan's property (T), for a locally compact, metrisable, σ-compact group. For such a group, <i>G</i>, the following properties are equivalent. 1. Kazhdan's definition of property (T): the trivial representations is isolated in the unitary dual of <i>G</i> (with the Fell topology). 2. The group, <i>G</i>, is compactly generated and for every compact generating set, <i>K</i>, there is a positive constant, ε, such that if π is a unitary representation of <i>G</i> on a Hilbert space, cal H, and ζ is a unit vector in cal H such that vskip 0.7cmthen π fixes some non-zero vector in cal H. This is often taken as the definition of property (T). 3. Every conditionally negative type function on <i>G</i> is bounded. 4. For a discrete group, <i>G</i> is finitely generated and for every finite generating set, <i>K</i>, zero is an isolated point in the spectrum of the Laplacian. From 2 we can define the Kazhdan constant, the largest possible value of ε for a given <i>G</i> and <i>K</i>. In Chapter 2 we investigate how to calculate these constants. In Chapter 4 we look at the bound on conditionally negative type functions and use its existence to extend a result of A.Connes and V.Jones about the von Neumann algebras of property (T) groups. The first half of Chapter 3 examines the spectrum of the Laplacian for a discrete group and finite generating set and compares its least positive element to the Kazhdan constant. Non-compact property (T) groups are all non-amenable. However, the standard example of a non-amenable group <i>F</i>_2, does not have property (T). The second half of Chapter 3 looks at the spectrum of λ(Δ) for F_2 with various generating sets, where λ is the left regular representation of <i>G</i> on <i>l</i><SUP>2</SUP>(<i>G</i>). For any non-amenable group, the smallest element of Spλ(Δ) is positive. Chapter 5 is an attempt to extend various results about <i>F</i><SUB>2</SUB> to other non-amenable groups.

We introduce an order γ(γ&gt;1/2) strong scheme and an improved weak scheme for the numerical approximation of solutions to stochastic differential equations (SDEs), driven by N Weinner processes. The strong scheme, called the ¾ Scheme, which is dependent on a differently constructed Brownian path, involves the area terms to bring better asymptotic accuracy than any numerical method based on classic constructed N-Dimension Brownian path. We demonstrate how to construct such a Brownian path, besides how to subdivide the Brownian path to get a sequence of approximations which converges pathwise as h tends to 0. We prove that the convergence of such method is guaranteed if the time step size h tends to 0. We also present the Improved Weak Euler Scheme (IWES), whose sample error is much smaller than the classic Weak Euler Scheme’s. The method reduces computation load and the sample error, which is generated during the Monte-Carlo(MC) approximation, by balancing the times of Euler iteration and MC simulation. A further improved IWES can be achieved by reusing the Brownian path. We prove that the extra sample error from reusing Brownian path is negligible in the latter method.

Stochastic differential inclusions (SDIs) on <i>R^d</i>have been investigated in this thesis, <i>dx</i>(<i>t</i>) Î <i>a</i>(<i>t, x(t)</i>)<i>dt </i>+   (<i>t, x (t)d</i> where <i>a</i> is a maximal monotone mapping, <i>b</i> is a Lipschitz continuous function, and <i>w</i> is a Wiener process. The principal aim of this work is to present some new results on solvability and approximations of SDIs. Two methods are adapted to obtain our results: the method of minimization and the method of implicit approximation. We interpret the method of monotonicity as a method of constructing minimizers to certain convex functions. Under the monotonicity condition and the usual linear growth condition, the solutions are characterized as the minimizers of convex functionals, and are constructed via implicit approximations. Implicit numerical scheme is given and the result on the rate of convergence is also presented. The ideas of our work are inspired by N.V. Krylov, where stochastic differential equations (SDEs0 in <i>R^d</i> are solved by minimizing convex functions via Euler approximations. Furthermore, since the linear growth condition is too strong, an approach is proposed for truncating maximal monotone functions to get bounded maximal monotone functions. It is a technical challenge in this thesis. Thus the existence of solutions to SDIs is proved under essentially weaker growth condition than the linear growth. For a special case of SDEs, a few of recent results from [5] are generalized. Some existing results of the convergence by implicit numerical schemes are proved under the locally Lipschitz condition. We will show that under certain weaker conditions, if the drift coefficient satisfies one-sided Lipschitz and the diffusion coefficient is Lipschitz continuous, implicit approximations applied to SDEs, converge almost surely to the solution of SDEs. The rate of convergence we get is ¼.

We consider "minimal" vector fields on a surface S with genus <i>g. </i>These are non-degenerate vector fields with the minimal number of vanishing points that satisfy a set of technical conditions to exclude pathological cases. We show that a minimal vector field gives rise to a directed graph with 2<i>g</i> - 2 vertices such that each vertex has two edges entering and leaving it, a "dual" pair of circuit decompositions of equal size and a function that pairs up the circuits of this dual pair. Conversely, we show that given such a graph with a pair of circuit decompositions and such a function we can construct a unique minimal vector field. This correspondence enables us to classify these vector fields. The proof of the correspondence result requires several invariants, one each from graph theory and the topology of the surface. These invariants are, respectively, the directed graph G formed by the non-compact flowlines of the vector field and a neighbourhood of this graph. Invariants arising from the homology of the pair (S, <i>V</i>) are also discussed, where <i>V</i> is the set of vertices of the directed graph G. Further, we show how to construct all possible minimal vector fields from a given graph provided the graph satisfies certain natural properties and give an algorithm that identifies which circuit decompositions have a suitable dual. We obtain some new results on the Martin polynomial, a skein-type polynomial of graphs first identified by P. Martin (1977). Some other combinatorial results concerning polynomials and graphs are proved.

The Logical Framework (LF) is a formal system for the representation of logics and formal systems as theories within it, which was developed by Harper, Ilonscll and Plotkin. The LF consists of two components: the All-calculus, the A-calculus with dependent function space (or product) types; and the principle of judgements as types in which judgements, the units of inference in logics and other formal systems, are identified with the types of the All-calculus. This work is strongly influenced by the work of Martin-L6f on judgements in logic. The LF is suitable for mechanical implementation because the AlJ-calculus is decidable. We present a theory of search and logic programming for the Alt-calculus and consequently for logics which are adequately represented in the LF. The presentation of the All-calculus of Harper, Honsell and Plotkin is as a (linearized) natural deduction system. The search space induced by such a system is highly non-deterministic and so the first step is to define a Gcntzenized system, in which the natural deduction style II-elimination rule is replace by a sequent calculus style II-elimination rule, which is sound and complete with respect to the system of Harper, Honsell and Plotkin. The Gentzenized system thus provides the foundation for a metacalculus, possessing an important subformula property, for the inhabitation assertions of the A-calculus which forms a suitable basis for proof-search in the A-11-calculus. By exploiting the structure of a form of hypolhetica-general judgement, we arc able to obtain (complete) calculi based on certain forms of resolution rule, the search spaces of which are properly contained within that of our basic metacalculus. Furthermore, we are able to further constrain proof-search in these calculi by considering a certain uniform form for proofs. We develop a unification algorithm for the All-calculus by extending the work of Huet for the simply-typed A-calculus. We use this unification algorithm to allow us to eliminate non-deterministic choices of terms, when performing proof-search with the (II/) or resolution rules. This is done by the introduction of a class of universal variables which are later instantiated via the unification algorithm. When unification is used to instantiate universal variables well-formedness may not be preserved. However, by extending the work of Bibel and Wallen (for first-order classical, modal and inluitionistic logics) to the All-calculus we are able to obtain a theory which allows to accept as many such instantiations as possible by detecting when the derivation can be reordered in order to yield a well-formed proof. We extend the theory described above to provide a notion of logic programming by admitting universal variables in cndsequents: these are analogous to the logical variables of PROLOG. Finally, we provide a denota-tional semantics for our logic programs by performing a least fixed point construction in a collection of Herbrand interpretations - maps from |C(£)| to (families of) sets of terms in | 7 |, where C(£) is the syntactic category constructed out of the All-calculus with signature E, and 7 is the category of families of sets. This construction is similar to one of Miller, and exploits a Kripke-like satisfaction predicate. We characterize this construction in terms of the model theory of the All-calculus using the Yoneda functor.

The methods of Bailey, Eastwood and Graham for the parabolic invariant theory of conformal geometry are adapted to study the conformal polynomial invariants in the jets of differential forms, with analogous results being obtained. The methods of Bailey and Gover are then used to give the 'exceptional invariants'. These methods are extended to a different problem - that of the polynomial invariants in the jets of curves at a point, yielding complete results for a particular class of invariants. A construction was given by Graham, Jenne, Mason and Sparling of a set of conformally invariant, linear differential operators with leading term a power of the Laplacian, on general conformal manifolds. Their method involves the use of the 'ambient metric' construction. We give an alternative construction of most of these operators, using an invariant operator on the 'tractor bundle,' and describe the relationship between the tractor bundle and the ambient construction. We also relate these ideas to methods used by Wünsch to find some conformally invariant powers of the Laplacian. We introduce another parabolic geometry, not appearing previously in the literature, which we call contact-projective geometry. The flat model is sp(2<I>n</I> + 2,π) acting on π<SUP>2<I>n</I>+1</SUP>. The invariants of positively homogenous functions on the flat model are studied, using methods similar to those of the conformal case. We suggest a curved version of this geometry and describe the form of a tractor bundle - a vector bundle with connection and a skew-symmetric bi-linear form; and an ambient space - an affine manifold of one higher dimension equipped with a symplectic form.

Within the framework of categorical logic or categorical type theory, predicate logics and type theories are understood as fibrations with structure. Fibrations, or fibred categories, provide an abstract account of the notions of indexing and substitution. These notions are central to the interpretations of predicate logics and type theories with dependent types or polymorphism. In these systems, predicates/dependent types are indexed by the contexts which declare the types of their free variables, and there is an operation of substitution of terms for free variables. With this setting it is natural to give a category-theoretic account of certain logical issues in terms of fibrations. In this thesis we explore logical predicates for simply typed theories, induction principles for inductive data types, and indeterminate elements for fibrations in relation to polymorphic λ-calculi. The notions of logical predicate is a useful tool in the study of type theories like simply typed λ-calculus. For a categorical account of this concept, we are led to study certain structure of fibred categories. In particular, the kind of structure involved in the interpretation of simply typed λ-calculus, namely cartesian closure, is expressed in terms of adjunctions. Hence we are led to consider adjunctions between fibred categories. We give a characterisation of these adjunctions which allows us to provide categorical structure, given by adjunctions, to a fibred category using similar structure on its base and its fibres. By expressing the abovementioned categorical construction logically, in the internal language of a fibration, we can then account for the notion of logical predicate for a cartesian closed category. With a similar argument, we provide a categorical interpretation of the induction principle for inductive data types, given by initial algebras for endofunctors on a distributive category. We also consider the problem of adjoining indeterminate elements to fibrations.

No description available.

This thesis consists of three chapters, which are loosely linked by the concept of normal elements (an clement x of a ring R is normal if xR = R.\). In Chapter 1 we examine a class of rings known as Noetherian unique factorisation rings (Noetherian UFRs). These arc prime Noetherian rings in which every prime ideal of height one is generated by a normal element. The main result of this chapter is that if A is a commutative UFD and G is a polycyclic-by-finite group such that the group ring AC is a Noetherian UFR then the set C consisting of the elements of K which are regular modulo all of the height one prime ideals of R is an Ore set in R; we also describe some aspects of the structure of the rings obtained by localising AC at the set C. In Chapter 2 we study the process of localisation at prime ideals which are generated by a regular normalising sequence. Section 2.1 contains a summary of the theory of localisation at cliques in Noetherian rings; in 2.2 we prove some useful results related to this theory. In section 2.3 we study the properties of regular normalising sequences and identify the clique of a poly-(regular normal) prime ideal (i.e., a prime generated by a regular normalising sequence). In section 2.4 we define a class of rings called RL rings, which are the rings obtained by localising at the clique of a poly-(rcgular normal) prime ideal. We describe various properties of these rings: in particular, we calculate the classical Krull dimension and the global dimension of such a ring (T, say), and show that they are both equal to the length of any regular normalising sequence generating any maxima] ideal of T. In section 2.5, we apply homological methods to RL rings to study the heights of prime ideals. The main result here is a height formula, which states that if g is a finite-dimensional complex soluble Lie algebra and T is the ring obtained by localising at the clique of any prime ideal in U(g), the universal enveloping algebra of g, then we have ht(Q) + Kdim(T/Q; = Kdim (T) for every prime ideal Q of T. In Chapter 3, we consider a ring of skew Laurent polynomials T = R[S 1 ..... S jfl; a i ..... a „] with a j (. Aut R. In 3.1 and 3.2 a map of Laurent <f> is constructed from T to the ring 5 = R[x ^ polynomials in n central indcterminates x\ .... xn and it is shown that if R is commutative then this map induces an isomorphism between the lattice of two-sided ideals of T and the lattice of G-stable ideals of S, where G is the subgroup of Aut 5 generated by 01, ..., on; furthermore, under this isomorphism the prime ideals of 7" correspond to the F-prime ideals of 5, where F 3 G is a certain subset of End 5. In section 3.3 we show that with appropriate restrictions on R and G the ring T is normally separated; this yields information about the height of prime ideals in 7" and S, and enables us to prove in 3.4 that the ring T is often catenary.

No description available.