This work is essentially a detailed version of notes published by the author (some in collaboration) in the “Comptes Rendus Hebdomadaires des Séances de l’Académie de Sciences, Série A (Paris)” in 1971 and early 1972 (numbers 272 and 274). The first chapter studies Lie algebras constituted by finite rank operators in a complex Hilbert space, and which correspond in infinite dimensions to the classical simple Lie algebras in finite dimensions. Completions of these with respect to the Schatten uniform crossnorms and to the weak topology provide the Banach-Lie algebras investigated in chapter II. As applications : homogeneous spaces of the corresponding Banach-Lie groups are studied in chapter III; the relationship between cohomologies of the groups, of their Lie algebras and of their classifying spaces is the subject of chapter IV. The main results are: Algebras of derivations, groups of automorphisms, classification of the real forms and cohomological computations for the Lie algebras introduced in chapters I and II. The methods used have been suggested by the theories of finite dimensional semi-simple real and complex Lie algebras, and of associative Banach algebras of linear operators. The results are related to the theory of L*-algebras (Schue, Balachandran) and to the investigations about the Lie structure of simple associative rings (Herstein, Martindale).

This thesis takes the construction (due to Grothendieck) of the projective fibred scheme of a coherent sheaf and iinvestigates certain aspects of its geometry, (and, in Chapter IV, topology). In Chapter I, after quoting the basic definitions, of the projective fibred scheme of ϵ with its projection π : ρϵ·→X and fundamental invertible sheaf ϴ(1) on ρϵ, and giving some simple illustrative examples, we turn (§§ 2,3) to some particular features of the geometry, notably the Fitting subschemes and the dominating component (and interactions between the two). Here and throughout most of the work we keep close to geometrical intuition by considering only coherent modules on locally-neotherian, reduced schemes. In an appendix we introduce some sheaves that are in: a sense universal for coherent sheaves on projective varieties. (These do not play any essential role in the rest of the thesis). Chapter II is concerned with the canonical homomorphism X, ϵ→π*ϴ (1) which is known to be an isomorphism when ϵ is locally-free. We extend this result to a larger class of sheaves and show, for example, that X is an isomorphism if ρϵ normal. In view of this result, and for general reasons, it is of interest to look for examples of smooth projective fibred schemes. This we do in Chapter III and show that a "generic" Module ϵ. of the type that locally has a resolution 0→0pu→θqu→∈u→0, (U, smooth where p(x) =0, has smooth ρϵ in a neighbourhood of ρϵ(x). Chapter IV considers the (singular) cohomology ring of PE when ϵ is a sheaf on a complex variety (with the classical topology). We include a discussion of the effect on cohomology of blowing-up a smooth variety with centre of smooth subvariety.

The theory of saturated formations introduced by Gaschütz 7 in 1963 is now an integral part of the study of finite soluble groups. Extensions of this theory have since been obtained by Stonehewer 18, for the class of periodic locally soluble groups with a normal locally nilpotent subgroup of finite index, and Tomlinson 21, for the class of periodic locally soluble FC -groups. In the latter case of course conjugacy of the various types of subgroups concerned is replaced by local conjugacy. Wehrfritz 23 also developed a theory of basis normalizers and Carter subgroups for the class of all homomorphic images of periodic soluble linear groups. Much of this work was unified in 1971 in paper by Gardiner, Hartley and Tomkinson 6. They introduced a class U periodic locally soluble groups and showed that it is possible to obtain a theory of saturated formations in any subclass of U which is closed under taking subgroups and homomorphic images. Their work covers all the previous theories except that for periodic locally soluble FC - groups •. It is our aim to show that with a 'good' Sylow structure and a 'well behaved' group of automorphisms which permutes the Sylow structure transitively, a theory of saturated formations can always be constructed. The approach which we shall discuss gives a theory which covers all the previous theories including that for periodic locally soluble FC -groups. This thesis is divided into two main parts and is organised as follows. In the next section we introduce our notation and terminology. In part one, which forms the bulk of the thesis, we shall define axiomatically what we mean by 'good' and 'well behaved', and in so doing introduce a class W of periodic locally soluble groups. We shall consider a fixed, but arbitrary, QS -closed subclass K of W and a saturated K -formation 7 satisfying certain conditions, and show that any K-group G possesses a unique A(G) - transitive class of 7-projectors, where A(G) is a 'well behaved' group of automorphisms of G. In the final section of part one we extend the work of Chambers 4 to KA -groups, and in particular we characterize the 7-normalizers of KA -groups by the covering/avoiding property. In part two, we introduce a class V of periodic locally soluble groups, where the automorphisms involved 'are the locally inner ones; this class properly contains the class of periodic locally soluble FC -groups. We define what we mean by a Fitting class7 of V-groups, and show that any V-group possesses a unique local .I conjugacy class of 7-injectors. This extends the work of Tomkinson 22, and we follow his techniques when proving the above. The final sections of part two concern normal Fitting classes of V-groups, where "re extend the work of Blessenohl and Gaschütz 2 and Lausch 15. ~ In particular we show that every non - trivial normal Fitting class of V-groups admits one and only one normal Fitting pair ( up to isomorphic Fitting pairs ).

Parasoluble groups have been defined by Wehrfritz as a generalization of nilpotent and supersoluble groups, and this thesis is concerned with the analogous Lie rings and Lie algebras. Most results are proved for Lie rings only, but in several cases the proofs for general Lie algebras are identical. Chapter 1 sets up notation and terminology and introduces the concepts of power derivations and quasi-centralizers. Chapter 2 defines the pparalyzer and stabilizer of a series as terms similar to those used in group theory. Also defined are quasicentral series, and parasoluble Lie rings as those with a finite quasicentral series. In chapter 3 it is shewn that under certain conditions the paralyzer of a finite series is parasoluble. This is always true for torsion-free Lie rings, but an example is given to shew that it is not true in general. Chapter 4 is concerned with paralyzers of ascending series and results are obtained which are generalizations of Lie ring analogues of some results of Hall and Hartley. Chapter 5 looks at the join problem for hypercyclic, parasoluble and supersoluble Lie rings. Chapter 6 is concerned with the class of soluble Lie rings in which all subideals are ideals. These are the Lie ring analogue of similarly defined groups of Robinson. Chapter 7 deals with local parasolubility and we shew that Lie rings which locally have a quasicentral series of bounded length are parasoluble. Chapter 8 employs some of the methods of the preceding chapters to obtain group-theoretic results, the main one being an improvement of a theorem of Hill.

This thesis considers constructing optimising distributions with applications in estimation and optimal design by exploring a class of mUltiplicative algorithms. Chapter 1 opens with an introduction to the area of linear design theory. It begins with an outline of a linear regression design problem including properties of the information matrix of the design. The second half of this chapter focuses on several design criteria and their properties. This part consists of two cases: when interest is in inference about all of the parameters of the model and when interest is in some of these parameters. The criteria include D-, A-, 0-, E-, D A-, L- (linear) and EA-optimality. Chapter 2 considers classes of optimisation problems. These include problems [labelled (PI), (P2)] in which the aim is to find an optimising distribution p. In examples of problem (P2) p is seen to define a distribution on a design space. Optimality conditions are determined for such optimisation problems. The emphasis is on a differential calculus approach in contrast to a Lagrangian one. An important tool is the directional derivative F</>{p, q} of a criterion function ¢(.) at p in the direction of q. The properties of </>{p, q} are studied, differentiability is expressed in terms of it, and further properties are considered when differentiability is defined. The chapter ends with providing some optimality theorems based on the results of the previous sections. Chapter 3 proposes a class of multiplicative algorithms for these problems. Iterations are of the form· p\r+1) ex p\r)f(x\r») where x\r) = d\r) or F(r) and .} }}' } } j d)r) = a¢/aPj while Ft) = F</>{p(r),ej} = d)r) - ~p~r)d~r) (a vertex directional ~ derivative) at p = p(r) and f(.) satisfies some suitable properties (positive and strictly increasing) and may depend on one or more free parameters. We refer to this as algorithm (3.1) [the label it is assigned]. These iterations neatly satisfy theconstraints of problems (PI), (P2). Some properties of this algorithm are demonstrated. Chapter 4 focuses on an estimation problem which in the first instance is a seeming generalisation of problem (PI). It is an example of an optimisation problem [labelled (P3) in chapter 2] with respect to variables which should be nonnegative and satisfy several linear constraints. However, it can be transformed to an example of problem (P2). The problem is that of determining maximum likelihood estimates under a hypothesis of marginal homogeneity for data in a square contingency table. The case of a 3 x 3 and of a 4 x 4 contingency table are considered. Chapter 5 investigates the performance of the above algorithm in constructing optimal designs by exploring a variety of choices of f(.) including a class of functions based on a distribution function. These investigations also explore various choices of the argument of f(.). Convergence of the above algorithm are compared for these choices of f(.) and it's argument. Convergence rates can also be controlled through judicious choice of free parameters. The work for this chapter along with the work in chapter 4 has appeared in Mandai and Torsney (2000a). Chapter 6 explores more objective choices of f(.). It mainly considers two approaches - approach I and approach II to improve convergence. In the first f(.) is based on a function h(.) which can have both positive and negative arguments. This approach is appropriate when taking Xj in f(xj) to be Pj , since these vertex directional derivatives being 'centred' on zero, take both positive and negative values. The second bases f(.) on a function g(.) defined only for positive arguments. This is appropriate when taking Xj to be dj if theRe partial derivatives are positive as in the case with design criteria. These enjoy improved convergence rates. Chapter 7 is devoted to a more powerful improvement - a 'clustering approach'. This idea emerges while running algorithm (3.1) in a design space which is a discretisation of a continuous space. It can be observed that 'clusters' start forming in early iterations of the above algorithm. Each cluster centres on a support point of the optimal design on the continuous space. The idea is that, at an appropriate iterate p(r), the single distribution p(r) should be replaced by conditional distributions within clusters and a marginal distribution across the clusters. This approach is formulated for a general regression problem and, then is explored through several regression models, namely, trigonometric, quadratic, cubic, quartic and a second-order model in two design variables. Improvements in convergence are seen considerably for each of these examples. Chapter 8 deals with the problem of finding an 'approximate' design maximising a criterion under a linear model subject to an equality constraint. The constraint is the equality of variances of the estimates of two linear functions (gT fl. and !l fl.) of the parameters of interest. The criteria considered are D-, D A- and A-optimality, where A = [g, QJT. Initially the Lagrangian is formulated but the Lagrange parameter is removed through a substitution, using linear equation theory, in an approach which transforms the constrained optimisation problem to a problem of maximising two functions (Q and G) of the design weights simultaneously. They have a common maximum of zero which is simultaneously attained at the constrained optimal design weights. This means that established algorithms for finding optimising distributions can be considered. The work for this chapter has appeared in Torsney and MandaI (2000). Chapter 9 concludes with a brief review of the main findings of the thesis and a discussion of potential future work on three topics: estimation problems, optimisation with respect to several distributions and constrained optimisation problems.

Viscoelastic flows are characterised by fast spatial and temporal variations in the solution featuring thin stress boundary near walls and stress concentrations in the vicinity of geometrical singularities. Resolving these fast variations of the fields in space and time is important for two reasons: (i) they affect the quantity of interest of the computation (e.g. drag force); and (ii) they are commonly believed to be associated with the numerical breakdown of the computation. Traditional discretisation methods such as finite differences or low-order finite elements require a large number of degrees of freedom to resolve these variations. Spectral methods enable this issue to be resolved by defining spatial expansions that are able to represent such variations with a smaller number of degrees of freedom. However, such methods are limited in terms of geometric flexibility. Recently, the spectral/hp element method (Karniadakis and Sherwin, 2005) has been developed in order to guarantee both spectral convergence, and geometric flexibility by allowing the use of quadrilateral and triangular elements. Our work is the first attempt to apply this method to viscoelastic free surface flows in arbitrary complex geometries. The conservation equations are solved in combination with the Oldroyd-B or Giesekus constitutive equation using the DEVSS-G/DG formulation. The combination of this formulation with a spectral element method is novel. A continuous approximation is employed for the velocity and discontinuous approximations for pressure, velocity gradient and polymeric stress. The conservation equations are discretised using the Galerkin method and the constitutive equation using a discontinuous Galerkin method to increase the stability of the approximation. The viscoelastic free surface is traced using an arbitrary Lagrangian Eulerian method. The performance of our scheme is demonstrated on the time-dependent Poiseuille flow in a channel, the flow around a cylinder and the die-swell problem.

Critical Care Units (CCUs) are one of the most complex and expensive of all medical resources and hospital managers are challenged to meet the demand for critical care services with adequate capacity. The pressure on critical care beds is continuously increasing as new medical equipment provides the opportunity to save more patients lives. It is therefore crucial that beds are managed well and used efficiently. This thesis describes two major projects, the first undertaken in conjunction with the CCU at the University Hospital of Wales in Cardiff (UHW); and the second with two CCUs from the Aneurin Bevan Health Board. In the first project data has been analysed to determine the flow of patients through the Unit. Admissions to CCUs were categorised under two headings: emergency, and elective. The length of stay in the CCU is heavily dependent on the admission category. In this thesis, both computer simulation and theoretical queueing models have been considered, which show how improvements in bed management may be achieved by considering these two categories of patients separately. The vast majority of previous literature in this field is concerned only with steady-state conditions, whereas in reality the processes are time-dependent. This thesis goes some way to addressing this deficiency. The second project relates to work undertaken with managers from the Royal Gwent Hospital in Newport and at the Nevill Hall Hospital in Abergavenny. Data from both hospitals have been analysed to define arrival and service processes. A state-dependent theoretical queueing model has been considered which has been used to investigate the significance of combining the two units. The model has been also utilised to advise on the number of beds the new combined unit should have in order to satisfy targets quoted by the hospital managers. In the final part of the thesis, consideration has been given to the impact of collaboration, or lack thereof, between hospitals using a game theoretical approach. The effect of patient diversion has been studied. To formally investigate the impact of patients transfers, a Markov chain model of the two CCUs has been set-up, each admitting two arrival streams: namely, their own patients and transfers from other hospital. Four different models were considered and for each model the effect of targets, demand and capacity were studied. The efficiency of a system which degrades due to selfish behaviour of its agents has been measured in terms of Price of Anarchy.

The work presented in this thesis concerns the problem of dynamic vehicle routing. The motivation for this is the increasing demands on transportation services to deliver fast, efficient and reliable service. Systems are now needed for dispatching transportation requests that arrive dynamically throughout the scheduling horizon. Therefore the focus of this research is the dynamic pickup and delivery problem with time windows, where requests are not completely known in advance but become available during the scheduling horizon. All requests have to be satisfied by a given fleet of vehicles and each request has a pickup and delivery location, along with a time window at which services can take place. To solve the DPDPTW, our algorithm is embedded in a rolling horizon framework, thus allowing the problem to be viewed as a series of static sub-problems. This research begins by considering the static variant of the problem. Both heuristic and metaheuristic methods are applied and an analysis is performed across a range of well-known instances. Results competitive with the state of the art are obtained. For the dynamic problem, investigations are performed to identify how requests arriving dynamically should be incorporated into the solution. Varying degrees of urgency and proportions of dynamic requests have been examined. Further investigations look at improving the solutions over time and identifying appropriate improvement heuristics. Again competitive results are achieved across a range of instances from the literature. This continually increasing area of research covers many real-life problems such as a health courier service. Here, the problem consists of the pickup and delivery of mail, specimens and equipment between hospitals, GP surgeries and health centres. Final research applies our findings to a real-life example of this problem, both for static schedules and a real-time 24/7 service.

The Restricted Euler equations, taken from the Vieillefosse model for the velocity gradient tensor, are re-investigated using data from direct numerical simulations of an intense event, rather than using data from forced simulations of homogeneous, isotropic turbulence. The goal is to develop ideas for extensions to turbulence models based on the RE equations that can handle these intense events. With this goal in mind, the new numerical data is compared against the evolution of the RE equations towards a finite time limit and its predictions on how ratios of the RE moments converge. The analysis starts by looking at distributions of the invariants in the R-Q phase space. From this, the analysis then compares the Vieillefosse equations to the full equations and notes that there is a significant change in behaviour around t = 0:5. It is suggested that this is associated with a change in ow topology due to the reconnection of vortex tubes in the flow field. To build a higher-order model, more terms from the full RE equations should be used, which is investigated by looking at the co-evolution of the second invariant Q and the third-order moments, Rw and Rs.

This thesis is an investigation into some of the lattice properties of the strong containment lattice (H, «) of Schunck classes and also of its important sublattice (D, «). The general aim is to characterise lattice properties of Schunck classes by avoidance class properties. Our main result, Theorem 8.5, is an avoidance class characterisation of those D-classes all of whose maximal ascending proper chains of Q-classes to S have the,same length. The problem extended to H is much more difficult but in Corollary 4.3 we describe an avoidance class condition for a Schunck class only to have chains of finite length to S. The lack of duality in H shows up clearly in section 3. The fascinating problem of deciding whether or not H is atomic is considered in section 9. Our results suggest that it probably is since any counterexample must be very complicated.