This thesis deals with generic determinacy and the number of equilibria for infinite dimensional economies. Our work could be seen as an infinite-dimensional analogue of Dierker and Dierker (1972) by characterising equilibria of an economy as a zero of the aggregate excess demand and studying its transversality. In this case, we can use extensions of the Sard-Smale theorem. Assuming separable utilities we give a new proof of generic determinacy of equilibria. We define regular price systems in this setting and show that an economy is regular if and only if its associated excess demand function only has regular equilibrium prices. We also define the infinite equilibrium manifold à la Balasko and show that it has the structure of a Banach manifold. We provide conditions that guarantee global uniqueness of equilibria for smooth infinite economies. We do this by introducing to the economic literature the notion of Z-Rothe vector fields that will allow us to construct an index theorem à la Dierker (1972); this shows that the number of equilibria is odd and in particular gives a new proof of existence. Extending the finite dimensional results of Balasko (1988), we characterise the equilibrium manifold as a covering space of the set of economies and we study global conditions under which the natural projection map is a diffeomorphism. We finally study the effects that critical equilibria have on the global invertibility of the natural projection map.

In this thesis a simple, stochastic, interacting particle system – the zero-range process (ZRP) – is studied with various analytical and numerical methods. in particular, the application of the ZRP and some of its generalisations to complex networks is focused upon. The ZRP is a hopping particle model where particles hop between sites of a lattice under certain rules that depend only on the properties of the site from which the particles hop – hence the name zero-range. Through its simplicity the steady state of the ZRP can be solved, even for nonequilibrium dynamics, and yet despite its simplicity it can exhibit interesting phenomena such as condensation transitions, where a finite fraction of the total particles in the system will condense onto a single site of the lattice. Firstly, interesting finite-size effects surrounding the condensation transition in a one-dimensional, driven version of the ZRP are studied. These take the form of discrepancies in the current-density diagram between finite and infinite systems, with the finite behaviour resembling that seen in real traffic data. Following this, direct applications of the ZRP to complex networks, and interesting phenomena arising from the specifics of the applications, are studied. The ZRP is applied as a model of networks and is found capable of reproducing power-law degree distributions, as observed in many real networks, at the critical point of the condensation transition. The degree is the number of connections a component of the network has. This model is then generalised to include creation and annihilation of particles or links, and this is found to exhibit critical behaviour – namely power-law particle and degree distributions – in a region of the parameter space, rather than at a critical point. The full phase diagram of this system is investigated, revealing low density and high density phases as well as subdivisions of the critical phase.

The main idea running throughout the thesis is that of a normal element. An element in a ring is normal if the one sided ideal it generates is actually two sided. In the first half of the thesis (Chapters 2, 3 and 4), our aim is to study certain quantum coordinate algebras. We are particularly interested in two subalgebras of the coordinate ring of quantum matrices, namely quantum Grassmannians and quantum Flag varieties. A basis is constructed for each of these algebras, from which we calculate their Gelfand-Kirillov dimensions. A well known result in the classical theory is that the dehomogenisation of the coordinate ring of the <i>m </i>x <i>n </i>Grassmannian at the 'rightmost' minor is isomorphic to the coordinate ring of the <i>m </i>x (<i>n </i>- <i>m</i>) matrices. The commutative notion of dehomogenisation does not immediately pass over to non-commutative theory. However, in the graded case, we show that it is possible to define non-commutative dehomogenisation at a regular normal homogeneous element, <i>x</i>, of degree 1 by considering a subring of the localisation of the ring at that element. The relationship between the prime spectrum of the original ring and that of the dehomogenisation is considered and a homeomorphism between the graded primes in the ring (not containing <i>x</i>) and a certain subset of primes in the dehomogenisation is constructed. Turning our attention back to quantum Grassmannians, we obtain the desired result, that the dehomogenisation of the quantum Grassmannians at the 'rightmost' minor is isomorphic to the quantum matrices. The smallest instructive example of a quantum Grassmannian is the 2 x 4 quantum Grassmannian, <i>G_q</i>(2, 4) and we restrict our attention to this case. By presenting the algebra as a factor ring of an iterated Ore extension, we see that <i>G_q</i>(2,4) is Auslander Gorenstein and Cohen Macaulay. We conjecture that this is also true in the general case. Finally, we consider the graded prime spectrum of <i>G_q</i>(2,4).

No description available.

No description available.

The propagation of waves along an elastic layer has long been an area of active research since the later part of the 19th century. Many contributions have already been made to the study of wave propagation in a linear isotropic elastic layer and multi-layered (composites) structures, with traction-free boundary conditions on the upper and lower surfaces. However, for other types of boundary conditions the problem is significantly more involved, especially when multi-layer structures are considered. The aim of this thesis is to perform a complete asymptotic analysis of the dispersion relations for a symmetric three-layer laminate subject to free, fixed, and fixed-free face boundary conditions. The second goal is to construct appropriate asymptotic models for these boundary conditions. Chapters 2 and 3 are devoted to the study of a single-layer laminate subject to all three types of boundary conditions, while chapters 4-10 discuss the case of a three-layer structure. Chapter 4 is concerned with the derivation of the dispersion relation for an unstressed 3-layer laminate with free faces. The symmetry of the laminate allows one to consider separately symmetric and antisymmetric motion. The associated asymptotic models for long wave low and high frequency, symmetric and anti-symmetric motions are presented in Chapters 5 and 6. Chapter 7 contains the derivation and discussion of the dispersion relation for a threelayer laminate with fixed faces, resulting in appropriate asymptotic models for long wave motion in Chapter 8. Finally, in Chapters 9 and 10, the dispersion relation for 3- layer plate with one free and one fixed faces is derived and analysed using appropriated asymptotic models for low and high frequency motion.

Theories of dependent types have been proposed as a foundation of constructive mathematics and as a framework in which to construct certified programs. In these applications an important role is played by identity types which internalise equality and therefore are essential for accommodating proofs and programs in the same formal system. This thesis attempts to reconcile the two different ways that type theories deal with identity types. In extensional type theory the propositional equality induced by the identity types is identified with definitional equality, i.e. conversion. This renders type-checking and well-formedness of propositions undecidable and leads to non-termination in the presence of universes. In intensional type theory propositional equality is coarser than definitional equality, the latter being confined to definitional expansion and normalisation. Then type-checking and well-formedness are decidable, and this variant is therefore adopted by most implementations. However, the identity type in intensional type theory is not powerful enough for formalisation of mathematics and program development. Notably, it does not identify pointwise equal functions (functional extensionality) and provides no means of redefining equality on a type as a given relation, i.e. quotient types. We call such capabilities extensional concepts. Other extensional concepts of interest are uniqueness of proofs and more specifically of equality proofs, subset types, and propositional extensionality---the identification of equivalent propositions. In this work we investigate to what extent these extensional concepts may be added to intensional type theory without sacrificing decidability and existence of canonical forms. The method we use is the translation of identity types into equivalence relations defined by induction on the type structure. In this way type theory with extensional concepts can be understood as a high-level language for working with equivalence relations instead of equality. Such translations of type theory into itself turn out to be best described using categorical models of type theory. We thus begin with a thorough treatment of categorical models with particular emphasis on the interpretation of type-theoretic syntax in such models. We then show how pairs of types and predicates can be organised into a model of type theory in which subset types are available and in which any two proofs of a proposition are equal. This model has applications in the areas of program extraction from proofs and modules for functional programs. For us its main purpose is to clarify the idea of syntactic translations via categorical model constructions. The main result of the thesis consists of the construction of two models in which functional extensionality and quotient types are available. In the first one types are modelled by types together with proposition-valued partial equivalence relations. This model is rather simple and in addition provides subset types and propositional extensionality. However, it does not furnish proper dependent types such as vectors or matrices. We try to overcome this disadvantage by using another model based on families of type-valued equivalence relations which is however much more complicated and validates certain conversion rules only up to propositional equality. We illustrate the use of these models by several small examples taken from both formalised mathematics and program development. We also establish various syntactic properties of propositional equality including a proof of the undecidability of typing in extensional type theory and a correspondence between derivations in extensional type theory and terms in intensional type theory with extensional concepts added. Furthermore we settle affirmatively the hitherto open question of the independence of unicity of equality proofs in intensional type theory which implies that the addition of pattern matching to intensional type theory does not yield a conservative extension.

No description available.

Cloud imagery is not currently used in numerical weather prediction (NWP) to extract the type of dynamical information that experienced forecasters have extracted subjectively for many years. For example, rapidly developing mid-latitude cyclones have characteristic signatures in the cloud imagery that are most fully appreciated from a sequence of images rather than from a single image. The Met Office is currently developing a technique to extract dynamical development information from satellite imagery using their full incremental 4D-Var (four-dimensional variational data assimilation) system. We investigate a simplified form of this technique in a fully nonlinear framework. We convert information on the vertical wind field, w(z), and profiles of temperature, T(z, t), and total water content, qt (z, t), as functions of height, z, and time, t, to a single brightness temperature by defining a 2D (vertical and time) variational assimilation testbed. The profiles of w, T and qt are updated using a simple vertical advection scheme. We define a basic cloud scheme to obtain the fractional cloud amount and, when combined with the temperature field, we convert this information into a brightness temperature, having developed a simple radiative transfer scheme. With the exception of some matrix inversion routines, all our code is developed from scratch. Throughout the development process we test all aspects of our 2D assimilation system, and then run identical twin experiments to try and recover information on the vertical velocity, from a sequence of observations of brightness temperature. This thesis contains a comprehensive description of our nonlinear models and assimilation system, and the first experimental results.

The number of meta-analysis of diagnostic studies is increasing and the models which deal with the summary receiver operating characteristic (SROC) have become quite popular. Many of these models have reached considerable statistical complexity, required expertise and knowledge. Here" a model named the proportional hazard model (PHM) is developed. The PHM model has a simple form and is easy to interpret. There is only one parameter of interest 0, which is called the diagnostic accuracy and has the interpretation that the smaller 0 is, the higher the diagnostic accuracy.