As our abilities to utilise high performance computing to theoretically probe many astrophysical systems increases, a genuine need to relate to real systems becomes ever more important. Here, Saturn s rings can be used as a nearby laboratory to investigate in real time many astrophysical processes. One such system is the narrow F ring and its interaction with its inner shepherd moon Prometheus. Through numerical modelling and direct observations of the in-situ spacecraft Cassini we find new and exciting dynamics. These might help explain some of the asymmetries witnessed in the distribution of embedded moonlets and azimuthal ring brightness known to exist within the F ring. Spatially we find asymmetry in the Prometheus induced channel edges with regards to density, velocity and acceleration variations of ring particles. Channel edges that show fans (embedded moonlets) are also the locations of highly localised increases in densities, velocity and acceleration changes where opposing edges are considerably less localised in their distribution. As a result of the highly localised nature of the velocity and acceleration changes chaotic fluctuations in density were witnessed. However, this could seek to work in favour of creating coherent objects at this channel edge as density increases were significantly large. Thus, density here had a greater chance of being enhanced beyond the local Roche density. Accompanied with these dynamics was the discovery of a non-zero component to vorticity in the perturbed area of the F ring post encounter. By removal of the background Keplerian flow we find that encounters typically created a large scale rotation of ~10,000 km^2. Within this area a much more rich distribution of local rotations is also seen located in and around the channel edges. Although the real F ring and our models are non-hydrodynamical in nature the existence of a curl in the velocity vector field in the perturbed region could offer some interesting implications for those systems that are gas rich.

This dissertation is devoted to our synthetic studies towards the total synthesis of the natural product Nakadomarin A, and Manzamine A using the Pauson-Khand reaction as the key step. Chapter 1 reviews past work using Pauson-Khand technology. Chapter 2 reviews the N-alkyl piperidine family of natural products. Chapter 3 reviews published total syntheses of Manzamine A and Nakadomarin A. Chapter 4 explores our work using the Pauson-Khand reaction to form the ABC rings of Nakadomarin A and subsequent B ring expansion to form the ABC ring core of Manzamine A. Chapter 5 explores our approaches to the furan portion, as well as, our approaches to the macrocyclic F ring. Finally Chapter 6 contains the description of the experiments performed along with relevant analytical data. / text

We study ideals of pointfree function rings. In particular, we study the lattices of z-ideals and d-ideals of the ring RL of continuous real-valued functions on a completely regular frame L. We show that the lattice of z-ideals is a coherently normal Yosida frame; and the lattice of d-ideals is a coherently normal frame. The lattice of z-ideals is demonstrated to be atly projectable if and only if the ring RL is feebly Baer. On the other hand, the frame of d-ideals is projectable precisely when the frame is cozero-complemented. These ideals give rise to two functors as follows: Sending a frame to the lattice of these ideals is a functorial assignment. We construct a natural transformation between the functors that arise from these assignments. We show that, for a certain collection of frame maps, the functor associated with z-ideals preserves and re ects the property of having a left adjoint. A ring is called a UMP-ring if every maximal ideal in it is the union of the minimal prime ideals it contains. In the penultimate chapter we give several characterisations for the ring RL to be a UMP-ring. We observe, in passing, that if a UMP ring is a Q-algebra, then each of its ideals when viewed as a ring in its own right is a UMP-ring. An example is provided to show that the converse fails. Finally, piggybacking on results in classical rings of continuous functions, we show that, exactly as in C(X), nth roots exist in RL. This is a consequence of an earlier proposition that every reduced f-ring with bounded inversion is the ring of fractions of its bounded part relative to those elements in the bounded part which are units in the bigger ring. We close with a result showing that the frame of open sets of the structure space of RL is isomorphic to L. / Mathematical Sciences / Mathematics / D.Phil. (Mathematics)

We study ideals of pointfree function rings. In particular, we study the lattices of z-ideals and d-ideals of the ring RL of continuous real-valued functions on a completely regular frame L. We show that the lattice of z-ideals is a coherently normal Yosida frame; and the lattice of d-ideals is a coherently normal frame. The lattice of z-ideals is demonstrated to be atly projectable if and only if the ring RL is feebly Baer. On the other hand, the frame of d-ideals is projectable precisely when the frame is cozero-complemented. These ideals give rise to two functors as follows: Sending a frame to the lattice of these ideals is a functorial assignment. We construct a natural transformation between the functors that arise from these assignments. We show that, for a certain collection of frame maps, the functor associated with z-ideals preserves and re ects the property of having a left adjoint. A ring is called a UMP-ring if every maximal ideal in it is the union of the minimal prime ideals it contains. In the penultimate chapter we give several characterisations for the ring RL to be a UMP-ring. We observe, in passing, that if a UMP ring is a Q-algebra, then each of its ideals when viewed as a ring in its own right is a UMP-ring. An example is provided to show that the converse fails. Finally, piggybacking on results in classical rings of continuous functions, we show that, exactly as in C(X), nth roots exist in RL. This is a consequence of an earlier proposition that every reduced f-ring with bounded inversion is the ring of fractions of its bounded part relative to those elements in the bounded part which are units in the bigger ring. We close with a result showing that the frame of open sets of the structure space of RL is isomorphic to L. / Mathematical Sciences / D.Phil. (Mathematics)

In his study of spectra of f-rings via pointfree topology, Banaschewski [6] considers lattices of l-ideals, radical l-ideals, and saturated l-ideals of a given f-ring A. In each case he shows that the lattice of each of these kinds of ideals is a coherent frame. This means that it is compact, generated by its compact elements, and the meet of any two compact elements is compact. This will form the basis of our main goal to show that the lattice-ordered rings studied in [6] are coherent frames. We conclude the dissertation by revisiting the d-elements of Mart nez and Zenk [30], and characterise them analogously to d-ideals in commutative rings. We extend these characterisa-tions to algebraic frames with FIP. Of necessity, this will require that we reappraise a great deal of Banaschewski's work on pointfree spectra, and that of Mart nez and Zenk on algebraic frames. / Mathematical Sciences / M. Sc. (Mathematics)

The first few chapters of the dissertation will catalogue what is known regarding z-ideals in commutative rings with identity. Some special attention will be paid to z-ideals in function rings to show how the presence of the topological description simplifies z-covers of arbitrary ideals. Conditions in an f-ring that ensure that the sum of z-ideals is a z-ideal will be given. In the latter part of the dissertation I will generalise a result in higher order z-ideals and introduce a notion of higher order d-ideals / Mathematical Sciences / M. Sc. (Mathematics)