Bibliography: pages 108-110. / When it became apparent that many varieties of algebras do not satisfy the Amalgamation Property, George Grätzer introduced the concept of an amalgamation class of a variety . The bulk of this dissertation is concerned with the amalgamation classes of residually small lattice varieties, with an emphasis on lattice varieties that are finitely generated. Our main concern is whether the amalgamation classes of such varieties are elementary classes or not. Chapters 0 and 1 provide a more detailed guide and summary of new and known results to be found in this dissertation. Chapter 2 is concerned with a cofinal sub-class of the amalgamation class of a residually small lattice variety, namely the class of absolute retracts, and completely characterizes the absolute retracts of finitely generated lattice varieties. Chapter 3 explores the strong connection between amalgamation and congruence extension properties in residually small lattice varieties. In Chapter 4, we investigate the closure of the amalgamation class under finite products. Chapter 5 is concerned with the amalgamation class of the variety generated by the pentagon. We prove that this amalgamation class is not an elementary class, but that, surprisingly, the class of all bounded members of the amalgamation class is a finitely axiomatizable Horn class. Chapters 6 and 7 introduce two techniques for proving that the amalgamation class of a residually small lattice variety is not an elementary class, and we give many examples. Finally, in Chapter 8, we look at the amalgamation classes of some residually large varieties, namely those generated by a finite dimensional simple lattice.

Bibliography: pages 139-145. / This thesis examines consistency conditions for fluid solutions of the field equations of general relativity. The exact non-linear dynamic equations for a generic irrotational dust spacetime are consistent. To analyse conditions characterizing pure gravity waves, linearization instability in general relativity and consistency of the so-called "silent universes", further exact conditions are imposed locally on irrotational dust. These are classified into Class II conditions, which change evolution equations into constraint equations, and Class I and III conditions, which do not doso-rather they add a new constraint, leaving the propagation equations unchanged in form. Class I conditions are imposed on terms in the constraint equations, while Class II and III conditions are imposed on terms in the evolution equations. In the Class I case it is shown that for irrotational dust space times the divergence-free magnetic Weyl tensor and the divergence-free electric Weyl tensor (necessary conditions for gravity waves interacting with matter), both imply integrability conditions in the exact non-linear case. The integrability conditions for the divergence-free magnetic Weyl tensor are identically satisfied in the linearized perturbation case, but are non-trivial in the exact non-linear case. This leads to a linearization instability in these models. The integrability conditions for the divergence-free electric Weyltensor are non-trivial in both the linear and non-linear cases. The Class II case focuses on irrotational silent cosmological dust models characterized by vanishing magnetic Weyl tensor and vanishing electric Weyl tensor. In both these models there exist a series of integrability conditions that need to be satisfied. Integrability conditions for the zero magnetic Weyl tensor condition hold identically for linearized case, but are non-trivial in the exact non-linear case. Thus there is also a linearization instability. The zero electric Weyl tensor condition leads to a chain of non-trivial integrability conditions in both the linear and non-linear cases. Because of the complexity of the integrability conditions, it is highly unlikely that there is a large class of models in both the silent zero magnetic Weyl tensor case and the silent zero electric Weyl tensor case.

The questions I ask myself are generally all along the lines of "so where did all this structure come from?". I hoped that work in the CMB and its cosmological implications would give me insight into this. It is an adventure that is still young. I began my PhD with an investigation of some formal aspects of Ehlers-Ellis Relativistic Kinetic Theory in mind { the implications of the truncation conditions found in the exact theory. I ended up trying to calculate CMB anisotropies as an application of this beautiful and somewhat purist formalism. The Ehler-Ellis (1+3) Lagrangian approach to General Relativity (GR) and Relativistic Kinetic Theory (RKT) are apparently not well known nor well used and have only recently begun to show advantages over the more usual ADM and Bardeen perturbative approaches to astrophysical cosmology when combined with the Ellis Bruni perturbation theory.

Includes abstract. / Includes bibliographical references (p.122-132) and index. / A few decades ago, Kaplansky raised the question whether unital linear invertibility preserving maps between unital algebras are Jordan homomorphisms. This question is still unanswered, and the progress that has been made has mainly been in the context of Banach algebras, including C*-algebras and von Neumann algebras. Let M be a von Neumann algebra with a faithful semifinite normal trace τ , and M~ the algebra of τ-measurable operators (measurable for short) affiliated with M. The algebra M~ can be endowed with a topology Уcm, called the topology of convergence in measure, such that M~ becomes a complete metrizable topological *-algebra in which M is dense. One of the aims of this thesis is to find answers to Kaplansky’s question in the context of algebras of measurable operators.

Includes bibliographical references. / A generic feature of most inflationary scenarios is the generation of primordial perturbations. Ordinarily, such perturbations can interact with a weak magnetic field in a plasma, resulting in a wide range of phenomena, such as the parametric excitation of plasma waves by gravitational waves. This mechanism has been studied in different contexts in the literature, such as the possibility of indirect detection of gravitational waves through electromagnetic signatures of the interaction. In this work, we consider this concept in the particular case of magnetic field amplification. Specifically, we use non-linear gauge-in variant perturbation theory to study the interaction of a primordial seed magnetic field with density and gravitational wave perturbations in an almost Friedmann-Lemaıtre-Robertson- Walker (FLRW) spacetime with zero spatial curvature. We compare the effects of this coupling under the assumptions of poor conductivity, perfect conductivity and the case where the electric field is sourced via the coupling of velocity perturbations to the seed field in the ideal magnetohydrodynamic (MHD) regime, thus generalizing, improving on and correcting previous results. We solve our equations for long wavelength limits and numerically integrate the resulting equations to generate power spectra for the electromagnetic field variables, showing where the modes cross the horizon. We find that the interaction can seed Electric fields with non-zero curl and that the curl of the electric field dominates the power spectrum on small scales, in agreement with previous arguments. The second focus area of the thesis is the development a stable high order mesh refinement scheme for the solution of hyperbolic partial differential equations. It has now become customary in the field of numerical relativity to couple high order finite difference schemes to mesh refinement algorithms. This approach combines the efficiency of local mesh refinement with the robustness and accuracy of higher order methods. To this end, different modifications of the standard Berger-Oliger adaptive mesh refinement a logarithm have been proposed. In this work we present a new fourth order convergent mesh refinement scheme with sub- cycling in time for numerical relativity applications. One of the distinctive features of our algorithm is that we do not use buffer zones to deal with refinement boundaries, as is currently done in the literature, but explicitly specify boundary data for refined grids instead. We argue that the incompatibility of the standard mesh refinement algorithm with higher order Runge Kutta methods is a manifestation of order reduction phenomena which is caused by inconsistent application of boundary data in the refined grids. Indeed, a peculiar feature of high order explicit Runge Kutta schemes is that they behave like low order schemes when applied to hyperbolic problems with time dependent Dirichlet boundary conditions. We present a new algorithm to deal with this phenomenon and through a series of examples demonstrate fourth order convergence. Our scheme also addresses the problem of spurious reflections that are generated when propagating waves cross mesh refinement boundaries. We introduce a transition zone on refined levels within which the phase velocity of propagating modes is allowed to decelerate in order to smoothly match the phase velocity of coarser grids. We apply the method to test problems involving propagating waves and show a significant reduction in spurious reflections.

Bibliography: pages 113-119. / In Chapter One we develop a basis for studying higher degree alternating forms. The concepts and results we present are mostly obvious analogues of Harrison's treatment of higher degree symmetric forms. We explain antisymmetrization; discuss the derivative of an alternating form and its corresponding anticommutative polynomial; define alternating spaces and their direct sum; establish decomposition and cancellation results for alternating spaces; and construct a Witt-Grothendieck group of alternating spaces. In Chapter Two we discuss hyperbolic alternating space. We compute the centre, algebraic isometry group and its corresponding Lie algebra, and prove a descent result. There are important parallels with Keet's results for hyperbolic symmetric spaces, as well as significant differences, especially in the methods we employ. In Chapter Three we develop a framework for the study of two aspects of forms of general Young symmetry type: their hyperbolics, and a generalization of the Weil-Siegel duality between symmetric and alternating bilinear forms. We introduce notions like nondegeneracy, derivative of a form, and derivative and integral symmetry types, and are then able to construct a hyperbolic space which is cofinal for spaces equipped with a form of the same symmetry type, and show that symmetry types are Siegel duals in our generalized sense if they have the same derivative symmetry type. In Chapter Four we present a few results and observations concerning nondegeneracytype conditions on symmetric forms. These include: an extension of Harrison's proof that nonsingularity implies nonzero Hessian to forms of arbitrary degree; a discussion of s-nondegeneracy and s-regularity; and a relation between a strong nondegeneracy condition on forms of even degree and the catalecticant, a classical invariant.

In the solution of the boundary value problems of mathematical physics in a separable 3-dimensional coordinate system, the shape of the boundary of the space may be such that the Green's function of the second order differential operator can be expanded as an infinite series of orthogonal functions. In many coordinate systems (such as the spherical, spheroidal and some cyclidal systems) these expansions are given in terms of Legendre associated functions of integral order and degree. Starting with Dougall's identities for Legendre associated functions of non-integral degree, new identities for infinite series of Legendre associated functions of integral degree are derived. Uniform convergence of each new identity is investigated in detail. The direct applicability of these identities is demonstrated by using them to verify theorems satisfied by the Dirichlet Green's function of the infinite half-space and of the interior of the prolate hemispheroid. The results and techniques are then generalized, and a sufficient condition found under which a generalized orthogonal function which satisfies Dougall's identity will also satisfy the new identity. This theorem is applied to the Legendre associated function, the generalized Legendre associated function and to the Jacobi function.

Includes bibliographical references. / In this thesis we consider four different topics in the field of cosmology, namely, black hole topology, the averaging problem, the effect of surface terms on the dynamics of classical and quantum fields, and the generation of an open universe through inflation with random initial conditions. It should be mentioned that while the research for this thesis was being done, no large effort was made to pursue a single theme. One reason for the diversity of the topics in this thesis is that the results which came out of this research were not always the results which were expected to be found when the investigation was started. Another reason for looking at several topics is simply that once a problem has been solved, then it is natural to move on to another problem which has not yet been solved. For those readers who value that a thesis is centered around a single unifying theme, let me mention that each of the four topics in this thesis are indeed related. Namely, each topic which we discuss focuses on an aspect of the global dynamics of the universe, in a situation where this is non-trivially different from the local dynamics. The non-trivial relation between global and local dynamics is rarely addressed in cosmology. Partially this is because of the difficulties which arise when one considers a realistic universe with infinitely many coupled degrees of freedom. Hence, it is a common practice to rely on simplifications which reduce the number of degrees of freedom, or the couplings between them. Further, there are few direct observations which probe the large-scale dynamics of the universe, or none at all, depending on the length scale and the type of cosmological model which one considers. As a consequence, there is a considerable freedom in choosing a priori assumptions or simplifications in the field of cosmology, without being able to falsify the validity thereof. For instance, when we analyse the relation between field perturbations at spatial infinity and perturbations here and now, we assume that quantum field theory, as we know it, is valid everywhere between here and spatial infinity. Although one cannot avoid making certain fundamental assumptions, the type of simplifications which are adopted in a calculation plays a less fundamental role. It is the objective of this thesis to improve our understanding of the large scale dynamics of the universe by showing rigorously what one can and what one cannot derive from certain fundamental assumptions. Interestingly, our results are often quite different from the results which are based on the same assumptions, but which involve certain commonly made simplifications as well.

Includes abstract. / Includes bibliographical references (p. 171-185). / This thesis concerns higher order perturbations of the standard model of cosmology. The theme is addressed in two distinct research areas. The first area deals with linear perturbations of Bianchi type I model filled with dust whose flow is irrotational, and which is an analogue to second order perturbations about the standard model. We investigate both density perturbations and gravitational waves in the shear dominated and the matter dominated regimes. We find that whereas the analysis of the perturbations in the matter dominated regime recovers the standard FLRW results, the analysis of perturbations in the shear dominated regime reveals that density perturbations and gravitational waves decouple only when the background shear is locally rotational symmetric.

Bibliography: pages 111-117.