This thesis presents a computational method for constructing Siegel sets for the action of \Gamma = SU(n; 1;O) on HnC, where O is the ring of integers of an imaginary quadratic field with trivial class group. The thesis first presents a basic algorithm for computing Siegel sets and then considers practical improvements which can be made to this algorithm in order to decrease computation time. This improved algorithm is implemented in a C++ program called siegel, the source code for which is freely available at http://code.google.com/p/siegel/, and this program is used to compute explicit Siegel sets for the action of all applicable groups \Gamma on H2C and H3C.

In Chapter 1 we determine the minimal density of triangles in a tripartite graph with prescribed edge densities. This extends work of Bondy, Shen, Thomassé and Thomassen characterizing those edge densities guaranteeing the existence of a triangle in a tripartite graph. We also determine those edge densities guaranteeing a copy of a triangle or C5 in a tripartite graph. In Chapter 2 we describe Razborov's flag algebra method and apply this to Erdös' jumping hypergraph problem to find the first non-trivial regions of jumps. We also use Razborov's method to prove five new sharp Turan densities, by looking at six vertex 3-graphs which are edge minimal and not 2-colourable. We extend Razborov's method to hypercubes. This allows us to significantly improve the upper bound given by Thomason and Wagner on the number of edges in a C4-free subgraph of the hypercube. We also show that the vertex Turan density of a 3-cube with a single vertex removed is precisely 3/4. In Chapter 3 we look at problems for intersecting families of sets on graphs. We give a new bound for the size of G-intersecting families on a cycle, disproving a conjecture of Johnson and Talbot. We also extend this result to cross-intersecting families and to powers of cycles. Finally in Chapter 4 we disprove a conjecture of Hurlbert and Kamat that the largest trivial intersecting family of independent r-sets from the vertex set of a tree is centred on a leaf.

Following the work of Rees, Thomas and Barton on the divisibility properties of certain normal chern numbers some chem numbers of the Milnor-Novikov generators of the cobordism ring are examined. The divisibility properties, at least up to 2-torsion, of these chem numbers are computed and these properties are then used to construct the manifolds whose chern numbers realize the minimum divisibility. As an example of how these direct methods can be employed an observation of Libgobcr and Wood is verified and improved upon. Odd torsion is also examined. It is observed that a proof from the work of Barton and Rees is incomplete and that proof is duly completed. Symmetric functions are introduced to form natural coefficients for a formal sum of the chem numbers of a manifold. Using this construction a bound on the primes contained in the hcf of any chem number is obtained, where this bound is dependent upon the length of the partition which indexes the chern number. A systematic method for lowering this bound (often eliminating odd torsion completely) for particular examples is demonstrated. As a digression the link between chem numbers and symmetric functions is examined in its own right. In particular the combinatorial side is addressed through the generalization of a partition of a number to a partition of a set. The general case of an arbitrary chem number of an arbitrary cross product of projective spaces is considered in detail and a general formula is obtained using the language of the lattice of partitions of a set. Examples to demonstrate the viability of this approach are presented.

Inspired by an observation in a paper by Dipper and Donkin, we tackle the problem of defining a quantum analogue of <i>SL_n</i>in the Multiparameter Quantum Matrices setting when the quantum determinant is not central. We construct a candidate for this algebra in a natural way using the process of Noncommutative Dehomogenisation. We go on to show that the object defined has many appropriate properties for such an analogue and observe that our new algebra can also be obtained via a process known as <i>twisting. </i>Finally we see what our definition means in the particular case of Dipper-Donkin Quantum Matrices and also look at the Standard Quantum Matrices case. In Chapter 3 we move on to study, Quantum Skew-symmetric Matrices. We show that a q-Laplace expansion of q-Pfaffians holds and that the highest-length q-Pfaffian is central. Finally we show that a factor of Quantum Skew-symmetric Matrices is isomorphic to <i>G_q</i>(2,<i>n</i>). Quantum Skew-symmetric Matrices are also mentioned in a 1996 paper by Noumi. In Chapter 4 we recall his definition of the algebra and of q-Pfaffians. These definitions are different to those of Strickland. We show that these contrasting definitions are in fact the same when <i>q</i> is not a root of unity. Using Noumi’s definition we show that another Laplace-type expansion, the natural <i>q</i>-analogue of a classical result, holds for q-Pfaffians.

We study λ-models in a constructive setting. We present two novel ways of deriving λ-models. These two definitions make sense classically, but yield nothing of interest. The first extends the structure of a λ-model to its pace of singletons. These two models and all the models inbetween have the same equational theory. The second takes a full function space hierarchy and defines a λ-submodel whose universe consists of those points in the hierarchy that satisfy a logical relation. Call a model obtained in this way extension model. We prove that, given a 'classsical' λ-model, it is consistent with IZF that it be isomorphic to an extension model. Also, this extension model has the same equational theory as the full function space hierarchy from which it was obtained. We prove these claims by building a fairly simple model of IZF in which these statements hold. This set thoeretic model only depends on the cardinality of the original λ-model. We deduce that there is a model of IZF in which there exists a full function space hierarchy for every classical model such that the two have the same theory. We go on to explore the logic of the world where these λ-models exist.

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The objects of study in this thesis are symmetric products and spaces of algebraic cycles. The first new result concerns symmetric products and it describes the geometry of truncated symmetric products (or, in other terminology, symmetric products modulo 2). We prove that if <I>M</I> is a closed compact connected triangulable manifold, a necessary and sufficient condition for its symmetric products modulo 2 to be manifolds is that <I>M</I> is a circle. We also show that the symmetric products of the circle modulo 2 are homeomorphic to real projective spaces and give an interpretation of this homeomorphism as a real topological analogue of Vieta's theorem. The second result concerns the spaces of real algebraic cycles, first studied by T.K. Lam. We describe a method of calculating the homotopy groups of the spaces of real cycles with integral coefficients on projective spaces; we give an explicit formula for the groups which lie in the "stable range". The third result (or, rather, a group of results) is the construction of a quaternionic analogue of Lawson's theory of algebraic cycles. We define quaternionic objects as those, which are invariant (in the case of varieties) or equivalent (in the case of polynomials) with respect to a free involution on CP<I><SUP>2n</SUP></I><SUP>+1</SUP>, induced by the action of the quaternion <I>j</I> on H<I><SUP>n</SUP></I>. Basic properties of quaternionic algebraic cycles are studied; a rational "quaternionic suspension theorem" is proved and the spaces of quaternionic cycles with rational coefficients on CP<I><SUP>2n</SUP></I><SUP>+1</SUP> are described. We also present a method of calculating the Betti numbers of the spaces of quaternionic cycles of degree 2 and odd codimension on CP<SUP>∞</SUP>. Some other results that are included in the thesis are a twisted version of the Dold-Thom theorem and an interpretation of the Kuiper-Massey theorem via symmetric products. After the main results on quaternionic cycles were proved, the author learned that similar results were obtained by Lawson, Lima-Filho and Michelson. Their version of the quaternionic suspension theorem is stronger and requires more sophisticated machinery for the proof.

In this thesis we study functions of generators of uniformly bounded semigroups of operators on a Hilbert space. A recent paper of V.V. Peller considers polynomials in an operator T whose iterates (T")n > 0 form a uniformly bounded discrete semigroup. Upper bounds for the norm of a polynomial in T are obtained and both a representation of the Besov space B^-j in B(#) and a von Neumann-type inequality follow. After studying Peller's methods and results, we use a similar approach to study polynomials in two commuting power-bounded operators and obtain comparable norm estimates. These results require a characterisation of Hankel operators on H2(I12), the Hardy space of functions on the two-dimensional torus. We show that the class of such Hankel operators is isometrically isomorphic to the dual of a quotient of a Banach space of operator-valued functions, and we investigate conditions for a generalisation of Nchari's Theorem. Finally, in Chapter 5 we show that analogues of Peller's results hold for functions of the infinitesimal generator of a uniformly bounded, strongly continuous semigroup of operators. This requires a characterisation of the dual space of the injective tensor product L*(R)+)&Ll(R+) using conditional expectation operators, and an identification of the class of Hankel-type integral operator kernels with a subspace of the dual of H'(R).

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No description available.