The thesis concerns the (class) structure No of Conway’s surreal numbers. The main concern is the behaviour on No of some of the classical functions of real analysis, and a definition of integral for such functions. In the main texts on No, most definitions and proofs are done by transfinite recursion and induction on the complexity of elements. In the thesis I consider a general scheme of definition for functions on No, generalising those for sum, product and exponential. If a function has such a definition, and can live in a Hardy field, and satisfies some auxiliary technical conditions, one can obtain in No a substantial analogue of real analysis for that function. One example is the sign-change property, and this (applied to polynomials) gives an alternative treatment of the basic fact that No is real closed. I discuss the analogue for the exponential. Using these ideas one can define a generalization of Riemann integration (the indefinite integral falling under the recursion scheme). The new integral is linear, monotone, and satisfies integration by parts. For some classical functions (eg polynomials) the integral yields the traditional formulas of analysis. There are, however, anomalies for the exponential function. But one can show that the logarithm, defined as the inverse of the exponential, is the integral of 1/x as usual.

This thesis studies the problem of estimating the largest possible dimension of a linear space of real matrices under the assumption that every non-zero matrix in the space has (the same) fixed rank. The complex version of this problem has been studied by R. Westwick and J. Sylvester. Sylvester introduced a technique based on the theory of Chern classes for estimating the dimension from above. The question of determining the largest dimension of a linear space of maximal-rank real <I>n </I>x<I> n</I> matrices (or, equivalently, of determining the largest number of nonsingular <I>n </I>x<I> n</I> matrices all of whose non-trivial linear combinations are non-singular) was solved by J.F. Adams, P. Lax and R. Phillips. Their proof uses Adams' solution of the vector fields on spheres problem to show that the linear spaces constructed by J. Radon and A. Hurwitz are of the largest possible dimension under this hypothesis. A number of general results on the dimensions of linear spaces of fixed-rank real matrices, as well as related questions concerning linear spaces whose non-zero matrices have rank bounded below, are due to E. Rees and K.Y. Lam. The method used to provide upper bounds for the dimension is analogous to the complex case; here Stiefel-Whitney classes and K-theory are used for the calculations. Clifford Algebras are then used to construct spaces and so provide lower bounds for the dimension. We show how calculations with Stiefel-Whitney classes together with information about the existence of certain bilinear maps enable us to determine the dimensions of spaces of real <I>n</I> x<I> k</I> matrices of fixed-rank <I>k</I> for all <I>n</I> and <I>k</I> with <I>k</I> ≤ 9. The case of fixed-rank symmetric matrices is also investigated. The main result here is that every space of real symmetric <I>n</I> x<I> n</I> matrices of fixed rank 2<I>k</I> + 1 must have dimension 1.

This thesis is in two parts. In part I, after a review of the relevant theory, we study the semi-standard map with extra terms added. In the second part of the thesis we look at the "Julia set" of two dimensional maps.

The subject of this thesis is the development of ways to solve structured nonlinear nonconvex programming problems by a decomposition procedure. This thesis extends the existing decomposition methods for linear or convex problems to the nonconvex nonlinear case. The algorithms presented are in principle applicable to a general nonlinear problem, although in order to be efficient compared with a nondecomposed method a certain structure is highly advantageous. Two main ideas are explored. In the first augmented Lagrangians are employed to relax some key constraints of the subproblems, thus guaranteeing that they are feasible for all choices of complicating variables. The resulting formulation is then decomposed by a generalized Benders decomposition scheme, resulting in a three-level problem. As an alternative a more direct generalization of Benders decomposition is considered. The problem of infeasible subproblems is overcome here by using feasibility cuts that build up a local approximation of the (nonconvex) feasible region in the master problem. Apart from the issue of infeasible subproblems, there are various differences from the linear/convex case, which are addressed. The subproblem value functions are shown to be piecewise differentiable nonconvex functions, whose subgradients can in general be obtained as certain Lagrange multipliers at the solution of the subproblems. Efficient ways of obtaining first and second derivatives of the value function from the subproblems are derived. A bundle method is used to solve the master problems at the top and middle level of the decomposition. The bundle concept is extended to cope with nonconvex functions and to incorporate second order information of the value function as well as its subgradient. The resulting method is demonstrated to converge superlinearly. The proposed bundle method can also be used outside the decomposition framework to minimize a nonconvex nonsmooth function subject to smooth constraints.

This thesis contains an investigation of Coquand's Calculus of Construction, a basic impredicative Type Theory. We review syntactic properties of the calculus, in particular decidability of equality and type-checking, based on the equality-as-judgement presentation. We present a set-theoretic notion of model, CC-structures, and use this to give a new strong normalisation proof based on a modification of the realizability interpretation. An extension of the core calculus by inductive types is investigated and we show, using the example of infinite trees, how the realizability semantics and the strong normalisation argument can be extended to non-algebraic inductive types. We emphasise that our interpretation is sound for <I>large eliminations</I>, e.g. allows the definition of sets by recursion. Finally we apply the extended calculus to a non-trivial problem: the formalization of the strong normalisation argument for Girard's System F. This formal proof has been developed and checked using the LEGO system, which has been implemented by Randy Pollack. We include the LEGO files in the appendix.

In 1964, Tauer gave examples of countably many masas inside the hyperfinite II₁von Neumann factor <i>R.</i> These masas were shown to be pairwise non-conjugate in <i>R</i> using a length invariant for the normalisers of semi-regular masas. A class of masas, the <i>Tauer masas,</i> is introduced consisting of all those masas obtained using her basic method of construction. The main body of this thesis is then concerned with examining the properties of these Tauer masas. In particular, the concepts of singularity, strong singularity and the weak asymptotic homomorphism property coincide for Tauer masas, and all Tauer mass have Pukánszky invariant {1}. Modern methods for calculating von Neumann algebras generated by normalisers are used to examine Tauer’s original examples, leading to shorter proofs of all of her results. Her initial example of a singular masa is studied in further detail. A generalisation of her semi-regular masas leads to the construction of an uncountable family of semi-regular masas of infinite length inside <i>R. </i>Examination of the Jones index of inclusions of the iterated normaliser algebras demonstrates that no pair of these masas can be conjugate by an automorphism of <i>R.</i> Centralising sequences for <i>R</i> lying inside masas are examined, with examples given to show that singular masas can be found containing non-trivial centralising sequences. An invariant, Γ(<i>A</i>), for a masa inside a II₁ factor is introduced as the size of a maximal cut-down for which the resulting masa contains non-trivial centralising sequences. This invariant is then used to exhibit a <i>d_∞,</i>₂-continuous path of uncountably many strongly singular masas in <i>R</i> with the same Pukánszky invariant, no pair of which is conjugate by an automorphism of <i>R.</i> Various issues arising from these concepts are discussed, such as possible masas in <i>R_ω</i> and the relationship between <i>A</i>-valued centralising sequences and automorphisms of <i>R</i> fixing <i>A</i> pointwise. Possible connections between this relative automorphism group and the Pukánskzy invariant will also be touched upon.

In this thesis we study the coalescence of invariant curves in certain codimension-two bifurcations of families of mappings. This phenomenon has been described in considerable detail by A Chenciner [Bifurcations de points fixes elliptiques: I - Courbes invariants, Publ. Math. I.H.E.S. 61, p67-127, 1985] in families of mappings which exhibit the Hopf bifurcation with higher order degeneracy. We give a review of some of his reviews in Chapter 1. In Chapter 2 we study the coalescence of invariant curves in two-parameter families of mappings which exhibit the Hopf bifurcation with 1:2 strong resonance. That is, families of the form <I>f<SUB>μ,v</SUB></I> : <I>U </I>→ R<SUP>2</SUP>, where <I>U</I> is an open subset of R<SUP>2</SUP> containing the origin, such that <I>f<SUB>μ, v</SUB></I> (0) = 0 and <I>Df<SUB>0,0</SUB></I> (0) has -1 as a double eigenvalue. We obtain a first return map for <I>f<SUB>μ,v</SUB></I> on a suitably defined region which shows that the coalescence of invariant curves in this case occurs in much the same way as described by Chenciner. The derivation of the first return map is dependent on the value of a complex valued functional χ which depends analytically on the entire Taylor expansion of f<SUB>0,0</SUB>. We require χ<I> </I>(f<SUB>0,0</SUB>) - 0. In Chapter 3 we describe some of the properties of χ and develop numerical methods for estimating χ(<I>g</I>), where <I>g </I>is a polynomial of degree ≤ 2 with linear part (<SUP>-1</SUP><SUB>0 </SUB><SUP>-1</SUP><SUB>-1</SUB>). Finally we use the numerical estimation methods to investigate the set {<I>g : </I>χ<I>(g) </I>= 0}.

No description available.

The accurate representation of the stable boundary layer (SBL) is a key issue for weather prediction and climate models. The SBL exerts a crucial influence controlling heat, moisture and momentum fluxes between the surface and the rest of the atmosphere. Some of the world's most stably stratified boundary layers develop on the Antarctic continent. The British Antarctic Survey has observed the boundary layer at their Halley Station for the past several decades. Previous work investigating stable boundary layers has tended to take either a purely observational or purely modelling-based approach. In this thesis, a novel three-way methodology has been developed which uses the Halley observations, alongside single-column model (SCM) and large-eddy simulation (LES) techniques to examine two case studies. The LES and observations were first used together to establish the correct initial conditions and forcings for each case study. Very close agreement was generally achieved between the LES and observations, particularly for the first case study. This approach represents a powerful framework for verifying SCM and LES results against a range of in-situ observations. The choice of stability function is an important decision for column-based parameterizations of the SBL. Four schemes were tested in the SCM, providing persuasive evidence for the use of shorter-tailed stability functions. The LES data was also used to extract implied stability functions. These experiments reinforced the conclusion that shorter-tailed stability functions offered improved performance for the Antarctic stable boundary layer. The wind turning angle was defined as the difference between the geostrophic and near-surface wind directions. A slightly larger wind-turning angle was found with the LES and SCM results presented in this thesis, as compared to previous work. This difference might be explained by the shallowness of the boundary layers studied here. Finally, some investigations into the resolution sensitivity of the LES and SCM were conducted. Increases in resolution in the LES generally led to convergence towards the observations, with grid-convergence being qualitatively approached with a grid-length of 2 m. The use of enhanced vertical resolution yielded excellent agreement against the observations, with lower computational expense. Vertical resolution sensitivity tests were also conducted using the SCM. Limited sensitivity was found over the grid-length range explored here, with the main benefits being delivered close to the surface.

For a finite, totally ramified Galois extension L/K (of prime degree p) of local fields of characterstic p, we investigate the embedding dimension of the associated order, and the minimal number of generators over the associated order, for an arbitrary fractional ideal in L. This is intricately linked to the continued fraction expansion of s/p, where s is the ramification number of the extension. This investigation can be thought of as a generalisation of 'Local Module Structure in Positive Characteristic' (de Smit &amp; Thomas, Arch. Math 2007) - which was concerned with the rings of integers only - and also as a specific, worked example of the more general 'Scaffolds and Generalized Integral Galois Module Structure' (Byott &amp; Elder, arXiv:1308.2088[math.NT], 2013) - which deals with degree pk extensions, for some k, which admit a Galois scaffold. We also obtain necessary and sufficient conditions for the freeness of these ideals over their associated orders. We show these conditions agree with the analogous conditions in the characteristic 0 case, as described in 'Sur les ideaux d'une extension cyclique de degre premier d'un corps local' (Ferton, C.R. Acad. Sc. Paris, 1973).