The generalised wreath product of permutation groups, due to Dixon, Fournelle and Silcock, is studied in this thesis. Under nice conditions this turns out to be a good generalisation of the permutational wreath product. We will explain precisely what we mean by nice. The centre of the generalised wreath product is determined and we look at the centraliser of certain elements of the group. The remainder of the thesis is concerned with looking to answer the question: given a class <i>X</i>, can we find necessary and sufficient conditions for the generalised wreath product to lie in <i>X</i>? We consider the class of abelian groups; nilpotent groups; locally nilpotent groups; <i>ZA </i>groups; residually nilpotent groups; locally boundedly nilpotent groups; bounded Engel groups; soluble groups; and locally soluble groups.

No description available.

Calorons, or periodic instantons, are anti-self-dual (ASD) connections on <i>S¹</i> x?³, and form an intermediate case between instantons (ASD connections on ?⁴) and monopoles (translation invariant instantons). Complete constrictions of instantons and monopoles have been found: there is a complete construction of instantons from algebraic data, the ADHM construction, due to Atiyah and others; while Nahm gave a construction of monopoles from solutions to a system of ODEs known as Nahm's equation. Both these constructions can be thought of as generalizations of a correspondence between ASD connections on the 4-torus, and ASD connections over the dual 4-torus, originally due to Mukai and Braam-Baal. This correspondence, often called the 'Nahm transform', is invertible and the inverse of the transform is the transform itself. Given an ASD connection on the 4-tours it is defined in terms of the kernel of a family of Dirac operators parameterized by the dual torus. The aim of this thesis is to generalize the Nahm transform to the caloron case. In particular, our approach is via analysis of these families of Dirac operators rather than via twistor theory. We start by exploring topological aspects of calorons, and boundary conditions. These are needed to ensure that the Dirac operators that define the Nahn transform are Fredholm. Our main innovation is to regard ?³ as the interior of the closed 3-ball <i>B³</i>, and to stipulate fixed behaviour on the boundary, rather than imposing asymptotic boundary conditions. The boundary conditions for calorons can be stated as follows: give a bundle on <i>S¹</i> x <i>B³</i> we fix some gauge <i>f</i> on the boundary, and we require that in the gauge <i>f, </i>a <i>U </i>(<i>n</i>)caloron must resemble the pull-back of a <i>U </i>(<i>n</i>) monopole. There is a topological obstruction to extending <i>f </i>to the interior of <i>S¹</i> x <i>B³</i>, which we call the 'instanton charge' of the caloron.

This thesis provides an in-depth study of the properties of pseudo-distributive laws, and as one of its applications, a unified framework to model substitution and variable binding for various different types of contexts; in particular, the construction presented in this thesis for modelling substitution unifies that for Cartesian contexts as in the work by Flore et al, and that for linear contexts by Tanaka. The main mathematical result of the thesis is the proof that, given a pseudo-monad <i>S</i> on a 2-category C, the 2-category of pseudo-distributive laws of <i>S</i> over pseudo-endofunctors on C and that of liftings of pseudo-endofunctors on C to the 2-category of the pseudo-algebras of <i>S </i>are equivalent. The proof for the non-pseudo case, i.e., a version for ordinary categories and monads, is given in detail as a prelude to the proof of the pseudo-case, followed by some investigation into the relation between distributive laws and Kleisli categories. Our analysis of pseudo-distributive laws is then extended to pseudo-distributivity over pseudo-endofunctors and over pseudo-natural transformations and modifications. The natural bimonoidal structures on the 2-category of pseudo-distributive laws and that of (pseudo)-liftings are also investigated as part of the proof of the equivalence. Fiore et al., and Tanaka take the free cocartesian category on 1 and the free symmetric monoidal category on 1 respectively as a category of contexts and then consider its presheaf category to construct abstract models for binding and substitution. In this thesis a model that unifies these two and other variations is constructed by using the presheaf category on a small category with structure that models contexts. Such structures for contexts are given as pseudo-monads <i>S </i>on <i>Cat, </i>and presheaf categories are given as the cocompletion (partial) pseudo-monad <i>T </i>on <i>Cat, </i>therefore our analysis of pseudo-distributive laws is applied to the combination of a pseudo-monad for contexts with the cocompletion pseudo-monad <i>T. </i>The existence of such pseudo-distributive laws leads to a natural monoidal structure that models substitution. This follows from the second main mathematical result of the thesis, the framework for such monoidal structures, which is given in terms of pseudo-strengths of pseudo-monads on <i>Cat </i>and the monoidal structures induced by them. We first prove that a pseudo-distributive law of <i>S </i>over <i>T </i>renders the composite <i>TS </i>to be again a pseudo-monad, from which it follows that the category <i>TS1 </i>has a monoidal structure, which, in our examples, models the substitution.

The purpose of this thesis is to implement Bayesian methods to solve theoretical and practical statistical problems in the selection of animals for breeding. The thesis is therefore mainly on the calculation of posterior distributions of variance components and functions of them, and the construction of optimum Bayesian selection methods for a single quantitative trait and multiple traits. Half-sib family structures are considered throughout, although the theory considered is more general in its application. Conventional and Bayesian methods for variance components estimation are reviewed from an animal breeding point of view, with emphasis on balanced data, but unbalanced data are also discussed. In Bayesian statistics the necessary integrations in several dimensions are usually difficult to perform by analytical means. A Gibbs sampling approach, which yields output readily translated into required inference summaries, is applied to integrations using suitable families of prior distributions. Gibbs sampling output is then used to develop appropriate graphical methods for summarising posterior distributions of genetic and phenotypic parameters, and to calculate the posterior expectations of breeding values and the expected progress using different selection procedures. The selection of farm animals for breeding is treated as a decision problem in which the utility of choosing a given number of individuals is assumed to be proportional to the sum of the corresponding breeding values. The Bayesian selection procedure in this case is contrasted with conventional procedures based on point estimates of parameters including a method based on modified parameter estimates known as <I>bending</I>.

A hierarchical triple system consists of two bodies forming a binary system and a third body on a wider orbit. The evolution of the eccentricity of an initially circular inner binary of a hierarchical triple system with well separated components is examined. Systems with different mass ratios and orbital characteristics (e.g. inclination) are investigated and theoretical formulae are derived for each case. The derivation of these formulae is based on the expansion of the rate of change of the eccentric vector in terms of the orbital period ratio of the two binaries using first order perturbation theory. Some elements from secular theory are used wherever necessary. Special cases are also discussed (e.g. secular resonances). The validity of the results is tested by integrating the full equations of motion numerically and the agreement is satisfactory. The stability of hierarchical triple systems with initially circular and coplanar orbits and small initial period ratio is also examined. Mean motion resonances are found to play an important role in the dynamics of the system. Special reference to the 3 : 1 and 4 : 1 resonances is made and a theoretical criterion for the 3 : 1 resonance is developed. A more general stability criterion (applicable in principle to other resonances besides 3 : 1) is obtained through a canonical transformation of an averaged Hamiltonian, and comparison is made with other results on the subject.

This work contains several developments in the area of numerical solution of pathwise solutions to stochastic differential equations (SDE's). In the first chapter we define and motivate pathwise solutions and give a brief survey of numerical methods for approximating them. The main key to enlarging the scope of numerical methods for SDE's is a good representation of Brownian paths. A binary tree structure is an essential tool in Chapter Two, which presents a general method for solution of SDE's using variable time steps. In the case of a general SDE, improvement of the order of convergence compared with standard methods, demands generation of the Lévy area integrals. Chapter Three presents a method of random generation of the Lévy area for a Brownian path in <I>IR</I><SUP>2</SUP>. The method is based on Marsaglia's rectangle-wedge-tail method for fast generation of normally distributed deviates. Since the solution of an SDE generally depends on an infinite sequence of iterated integrals of the driving noise, it makes sense to examine these integrals and the algebraic relations between them. In Chapter Four, it is shown how known facts about shuffle algebras can be used to get a better understanding of stochastic iterated integrals of Ito and Stratonovich type and obtain practical algebraic bases for these two sets. We use the algebra to calculate moments of stochastic integrals, needed when calculating moments of error during numerical solutions of SDE's. The work on the generation of area integrals, described in Chapter Three, gives rise to general questions about the generation of random deviates, some of which are addressed in the last two chapters. In Chapter Five, we present a polynomial-time algorithm for finding the partition, into rectangles or triangles, of certain types of region in <I>IR</I><SUP>2</SUP>, that has the lowest entropy.

This thesis contains results of a study in kernel density estimation, Bayesian inference and random effects models, with application to forensic problems. Estimation of the Bayes' factor in a forensic science problem involved the derivation of predictive distributions in non-standard situations. The distribution of the values of a characteristic of interest among different items in forensic science problems is often non-Normal. Background, or training, data were available to assist in the estimation of the distribution for measurements on cat and dog hairs. An informative prior, based on the kernel method of density estimation, was used to derive the appropriate predictive distributions. The training data may be considered to be derived from a random effects model. This was taken into consideration in modelling the Bayes' factor. The usual assumption of the random factor being Normally distributed is unrealistic, so a kernel density estimate was used as the distribution of the unknown random factor. Two kernel methods were employed: the ordinary and adaptive kernel methods. The adaptive kernel method allowed for the longer tail, where little information was available. Formulae for the Bayes' factor in a forensic science context were derived assuming the training data were grouped or not grouped (for example, hairs from one cat would be thought of as belonging to the same group), and that the within-group variance was or was not known. The Bayes' factor, assuming known within-group variance, for the training data, grouped or not grouped, was extended to the multivariate case. The method was applied to a practical example in a bivariate situation. Similar modelling of the Bayes' factor was derived to cope with a particular form of mixture data. Boundary effects were also taken into consideration. Application of kernel density estimation to make inferences about the variance components under the random effects model was studied. Employing the maximum likelihood estimation method, it was shown that the between-group variance and the smoothing parameter in the kernel density estimation were related. They were not identifiable separately. With the smoothing parameter fixed at some predetermined value, the within-and between-group variance estimates from the proposed model were equivalent to the usual ANOVA estimates. Within the Bayesian framework, posterior distribution for the variance components, using various prior distributions for the parameters were derived incorporating kernel density functions. The modes of these posterior distributions were used as estimates for the variance components. A Student-t within a Bayesian framework was derived after introduction of a prior for the smoothing prameter. Two methods of obtaining hyper-parameters for the prior were suggested, both involving empirical Bayes methods. They were a modified leave-one-out maximum likelihood method and a method of moments based on the optimum smoothing parameter determined from Normality assumption.

No description available.

This is the first study in which the effects of initial conditions (e.g. rheology and material properties), boundary conditions and fault growth properties on fault size scaling are explicitly considered. I use a 2D finite element code to generate kilometre-scale, conjugate, normal faults in cross-section under a range of boundary conditions. The deforming material is modelled with a strain-softening, non-healing, Von Mises rheology with Gaussian heterogeneity in yield strength distributed randomly throughout the mesh. Faults are not defined <i>a priori. </i>Consequently the evolution of geologically realistic structures in the model can be attributed to the physical principles involved, not to a pre-defined geometry. Numerical modelling results indicate that initial conditions and boundary conditions control which growth processes dominate at a particular place and time. Thus, they are also control fault size scaling. Both power law and non-power law distribution types emerged spontaneously, and the power law distributions showed a range of values of <i>c</i> between 0.53 and 1.27. In each simulation, the exponent <i>c</i> of the fault size cumulative frequency distribution was observed to decrease with increasing extension; partly due to coalescence, but also because larger faults grew disproportionately faster than smaller ones. The dependence of <i>c</i> on total strain was weak and easily masked by other contributing factors. The exponent <i>c</i> systematically decreased as heterogeneity decreased and strength loss on failure increased. Most significantly, simulations with statistically identical material properties but different random heterogeneity in space gave power law distributions with as much variation in <i>c</i> as was observed in experiments with different material properties and different total strains. This result implies that extrapolating information about fault size scaling from one area to an adjacent area is inadvisable, even if the regions have the same lithologies and tectonic histories.