W. Magnus represents a free group in a formal power series ring with no relations. We obtain power series representations for certain relatively free groups by putting various relations on the set of variables of the power series. Among those we obtain power series representations for are F/Fm (the free, nilpotent groups) , F/F" (the free metabelian group) F/(F')3(F3)', F/(F')3(F4)', F/[F", F] (the free centre by metabelian group), F/[F",F,F] (the free centre by centre by metabelian group), and F/[F",F,F,F](F')3. In the process it is shown that F"/[F",F] is free abelian and an explicit basis is given. This basis is used to derive a basis for [F",F]/[F",F,F] and various other subgroups of the group's, for which we obtain power series representations, are shown to be free abelian. We prove that all these groups mentioned above are residually torsion free nilpotent using their power series representations. W. Magnus has also proved that the so-called dimension subgroups and the lower central factors of the free group coincide. In Chapter 5 we present analogues of this result of Magnus for the groups F/F", F/(F')3(F3)' and F/(F')3(F4)' and in the process, compute the structure of the lower central factors of these three groups. We conclude with a contribution to a problem of Fox on the determination of certain ideals in the group ring of the free group.

In many works on Hausdorff Measure Theory it has been the practice to place certain restrictions on the measure functions used. These restrictions usually ensure both the monotonloity and the continuity of the functions The aim of the first four chapters of this thesis is to find conditions under which the restrictions of continuity and monotonicity may be relaxed. In the first chapter we deal with the monotonicity condition with respect to both measures and pre-measures. The second and third chapters are concerned with an investigation of the continuity condition with regard to measures and pre-measures, respectively. Then, having found conditions under which these restrictions may or may not be relaxed, we are able, in the fourth chapter, to generalize some known results to the case of discontinuous and non-mono tonic functions. Some of the results of the first four chapters prompted an Investigation of the properties of measures corresponding to sequences of measure functions, and this is incorporated in the fifth chapter. The main purpose of the final chapter is to determine whether or not some of the results of the earlier chapters may be extended to Hilbert space.

CHAPTER I The definition of all the measure functions used in the thesis. CHAPTER II The condition for a measure function to to be a Hausdorff diametral dimension function in p-dimensional real Euclidean space is first established. Then the fact that an analytical set of infinite Hausdorff diametral measure is then proved and the necessary and sufficient conditions for a subset of a set with Hausdorff diametral dimension function h(x) to have dimension function g(x) are established. CHAPTER III Conditions on the dimension function of the cartesian product of two one-dimensional sets whose dimension functions are known, are established. CHAPTER IV The proof of the existence of a plane set S with Hausdorff diametral dimension function x2[alpha] [equation], such that if S log 3 is translated through any distance in the plane then the intersection of S with itself translated has zero Hausdorff diametral measure with dimension function x2[alpha]/ CHAPTER V The two area measures are considered in two dimensional real Euclidean space only. The necessary and sufficient condition for a measure function to be a non-metric-area dimension function is established and the metric area measure of sets which are the cartesian products of intervals with linear sets is found. These a re used to deduce that non-metric-area measure is in fact non-metric. The condition for x[alpha] to he a metric area measure is also established. CHAPTER VI This deals with sets on the frontier of the unit circle. First the connection between the area measures and the generalized affine length is established. Then the triangle of minimum area covering a given total arc length is found and finally the necessary and sufficient condition for a measure function to be a Hausdopff diametral dimension function for such sets is found.

In this thesis we deal with the theory of unitary p-dilations of bounded operators on a Hilbert space H, as developed by Sz. Nagy and Foias, and a related functional on B(H), the algebra of bounded linear operators on H. In the first Chapter we consider the classes of operators possessing a unitary p-dilation, and obtain their basic properties , using an approach which adapts itself to a unified treatment. Next in Chapter 2, we examine the behaviour of the sequence of powers of an element p arbitrary and positive, and we show that the sequence converges to a non-negative limit, less than or equal to this is a generalization of a result by M.J. Crabb in which he considers the special case p=2. We then give an intrinsic characterization of the elements x in and obtain various results concerning the structure of operators which satisfy for some For every , the classes , turn out to be balanced, absorbing sets of operators which contain the zero operator, and hence a generalized Minkowski functional may be unambiguously defined on them by This functional, usually referred to in the literature as the Holbrook radius of T, plays a very important role in the study of unitary p-dilations, since the elements T of are characterized by [diagram]. The basic properties of the Holbrook radius for a bounded operator are studied in Chapter 3. A number of new results concerning the Holbrook radius of nilpotent operators of arbitrary index greater than 2 are obtained which enable us to have a clearer view of the general structure of the [mathematical symbol][rho] classes, in a unified framework.

No description available.

This thesis examines the completability of an incomplete set of m-row orthogonal Latin rectangles (MOLRm) from a set theoretical viewpoint. We focus on the case of two rows, i.e. MOLR2, and define its independence system (IS) and the associated clutter of bases, which is the collection of all MOLR2. Any such clutter gives rise to a unique clutter of circuits which is the collection of all minimal dependent sets. To decide whether an incomplete set of MOLR2 is completable, it suffices to show that it does not contain a circuit therefore full knowledge of the clutter of circuits is needed. For the IS associated with 2-row orthogonal Latin rectangles (OLR2) we establish a methodology based on the notion of an availability matrix to fully characterise the corresponding clutter of circuits.

Let F_n denote the free group of rank n with free basis X. The palindromic automorphism group PiA_n of F_n consists of automorphisms taking each member of X to a palindrome: that is, a word on X that reads the same backwards as forwards. We obtain finite generating sets for certain stabiliser subgroups of PiA_n. We use these generating sets to find an infinite generating set for the so-called palindromic Torelli group PI_n, the subgroup of PiA_n consisting of palindromic automorphisms inducing the identity on the abelianisation of F_n. Two crucial tools for finding this generating set are a new simplicial complex, the so-called complex of partial pi-bases, on which PiA_n acts, and a Birman exact sequence for PiA_n, which allows us to induct on n. We also obtain a rigidity result for automorphism groups of right-angled Artin groups. Let G be a finite simplicial graph, defining the right-angled Artin group A_G. We show that as A_G ranges over all right-angled Artin groups, the order of Out(Aut(A_G)) does not have a uniform upper bound. This is in contrast with extremal cases when A_G is free or free abelian: in these cases, |Out(Aut(A_G))| < 5. We prove that no uniform upper bound exists in general by placing constraints on the graph G that yield tractable decompositions of Aut(A_G). These decompositions allow us to construct explicit members of Out(Aut(A_G)).

This thesis is concerned with fluid-structure interaction problems using the immersed boundary method (IBM). Fluid-structure interaction problems can be classified into two categories: a remeshing approach and a fixed-grid approach. Both approaches consider the fluid and structure separately and then couple them together via suitable interface conditions. A common choice of remeshing approach is the Arbitrary-Eulerian-Lagrangian (ALE) technique. Whilst the ALE method is a good choice if deformations are small, it becomes computationally very expensive if deformations are large. In such a scenario, one turns to a fixed-grid approach. However, the issue with a fixed-grid approach is the enforcement of the interface conditions. An alternative to the remeshing and fixed-grid approach is the IBM. The IBM considers the immersed elastic structure to be part of the surrounding fluid by replacing the immersed structure with an Eulerian force density. Therefore, the interface conditions are enforced implicitly. This thesis applies the finite element immersed boundary method (IBM) to both Newtonian and Oldroyd-B viscoelastic fluids, where the fluid variables are approximated using the spectral element method (hence we name the method the spectral element immersed boundary method (SE-IBM)) and the immersed boundary variables are approximated using either the finite element method or the spectral element method. The IBM is known to suffer from area loss problems, e.g. when a static closed boundary is immersed in a fluid, the area contained inside the closed boundary decreases as the simulation progresses. The main source of error in such a scenario can be found in the spreading and interpolation phases. The aim of using a spectral element method is to improve the accuracy of the spreading and interpolation phases of the IBM. We illustrate that the SE-IBM can obtain better area conservation than the FE-IBM when a static closed boundary is considered. Also, the SE-IBM obtains higher order convergence of the velocity in the L2 and H1 norms, respectively. When the SE-IBM is applied to a viscoelastic fluid, any discontinuities which occur in either the velocity gradients or the pressure, introduce oscillations in the polymeric stress components. These oscillations are undesirable as they could potentially cause the numerics to break down. Finally, we consider a higher-order enriched method based on the extended finite element method (XFEM), which we call the eXtended Spectral Element Method (XSEM). When XSEM is applied to the SE-IBM with a viscoelastic fluid, the oscillations present in the polymeric stress components are greatly reduced.

This thesis is concerned with the analytic properties of arithmetic zeta functions, which remain largely conjectural at the time of writing. We will focus primarily on the most basic amongst them - meromorphic continuation and functional equation. Our weapon of choice is the so-called “mean-periodicity correspondence”, which provides a passage between nicely behaved arithmetic schemes and mean-periodic functions in certain functional spaces. In what follows, there are two major themes. 1. The comparison of the mean-periodicity properties of zeta functions with the much better known, but nonetheless conjectural, automorphicity properties of Hasse–Weil L functions. The latter of the two is a widely believed aspect of the Langlands program. In somewhat vague language, the two notions are dual to each other. One route to this result is broadly comparable to the Rankin-Selberg method, in which Fesenko’s “boundary function” plays the role of an Eisenstein series. 2. The use of a form of “lifted” harmonic analysis on the non-locally compact adele groups of arithmetic surfaces to develop integral representations of zeta functions. We also provide a more general discussion of a prospective theory of GL1(A(S)) zeta-integrals, where S is an arithmetic surface. When combined with adelic duality, we see that mean-periodicity may be accessible through further developments in higher dimensional adelic analysis. The results of the first flavour have some bearing on questions asked first by Langlands, and those of the second kind are an extension of the ideas of Tate for Hecke L-functions. The theorems proved here directly extend those of Fesenko and Suzuki on two-dimensional adelic analysis and the interplay between mean-periodicity and automorphicity.

In this thesis we study so-called two-grid hp-version discontinuous Galerkin finite element methods for the numerical solution of quasilinear partial differential equations. The two-grid method is constructed by first solving the nonlinear system of equations stemming from the discontinuous Galerkin finite element method on a coarse mesh partition; then, this coarse solution is used to linearise the underlying problem so that only a linear system is solved on a finer mesh. Solving the complex nonlinear problem on a coarse enough mesh should reduce computational complexity without adversely affecting the numerical error. We first focus on the a priori and a posteriori error estimation for a scalar second-order quasilinear elliptic PDEs of strongly monotone type with respect to a mesh-dependent energy norm. We then devise an hp-adaptive mesh refinement algorithm, using the a posteriori error estimator, to automatically refine both the coarse and fine meshes present in the two-grid method. We then perform numerical experiments to validate the algorithm and demonstrate the improvements from utilising a two-grid method in comparison to a standard (single-grid) approach. We also consider deviation of the energy norm based a priori and a posteriori error bounds for both the standard and two-grid discretisations of a quasi-Newtonian fluid flow problem of strongly monotone type. Numerical experiments are performed to validate these bounds. We finally consider the dual weighted residual based a posteriori error estimate for both the second-order quasilinear elliptic PDE and the quasi-Newtonian fluid flow problem with generic nonlinearities.