No description available.

Shear flows have a significant impact on the dynamics in various astrophysical objects, including accretion discs and stellar interiors. Due to observational limitations the complex dynamics in stellar interiors that result in turbulent motions, mixing processes, and magnetic field generation, are not entirely understood. It is therefore necessary to investigate the inevitable small-scale dynamics via numerical calculations. In particular a thin region with strong shear at the base of the convection zone in the Sun, the tachocline, is believed to play an important role in the Sun's interior dynamics and magnetic field generation. Velocity measurements suggest a stable tachocline. However, helioseismology can only provide large-scale time-averaged measurements, so small scale turbulent motions can still be present. Therefore, studying the stability of shear flows and their non-linear evolution in a fully compressible polytropic atmosphere provides a fundamental understanding of potential motion in stellar interiors and is the main focus of this thesis. To commence the investigations a linear stability analysis of a stratified system in a two-dimensional Cartesian geometry is performed to study the effect of compressibility and thermal diffusivity on the stability threshold. In addition, this first investigation provides a reference for subsequent non-linear calculations. Focusing on a local model of unstable shear flows, direct numerical calculations are used to first compare numerical forcing methods to sustain a shear flow against viscous dissipation; and then to study the effect of key parameters on the saturated quasi-steady regime. Finally, magnetic fields are included and the full set of MHD equations is solved to study a potential kinematic dynamo in shear-driven turbulence.

A new methodology and notation for the definition and construction of non-convex polyhedra is described. It uses constructive solid geometry to provide the geometric data for the fundamental cells and each composite structure is defined by a combination of such fundamental cells. To demonstrate the general principles behind this methodology we have used the construction techniques to produce crystalline structures. A minimal definition notation is used to construct a discrete surface representation in a standardised format which is used alongside a novel construction algorithm to develop an elaborate data structure representation, referred to as a polygonal mesh. The complexity of explicit description of non-convexity in discrete surface polyhedra is overcome using an extension to the minimal definition notation which permits the description of such polyhedra as a list of predominately convex fundamental cells. Non-convexity in the composite polyhedron will introduce new lines which are not part of the definition but are an artefact of the construction method. The display algorithm must explicitly depict these lines on the view plane of the chosen graphic medium. We have implemented two different methods for generating the final display. The first is a completely new display algorithm that pre-processes the data model to divide the discrete facet surfaces of the non-convex polyhedron into convex regions bounded by line segments. This subdivision renders the additional lines explicit in the data structure model. The second is the quad-tree display algorithm which uses quad-tree encryption to compute the projected image. Pre-processing is unnecessary since the interaction between the component cells is automatically generated during display, and the additional boundary lines will be explicitly drawn on the view plane. A database of fundamental cells and cell combinations is incorporated into the system. The construction algorithm allows interactive definition of an individual cell and its storage as cell polygonal mesh in the database. Thereafter compound structures can be generated either by interactive construction or retrieval of fundamental cells and cell combinations from this database. Any new combination of cells can be retained within the database. Retrieval of information from the database is achieved by reference to various key words, and the underlying database system performs the procedures necessary to maintain the dependencies.

Inverse semigroups provide a natural way to encode combinatorial data from geometric settings. Examples of this occur in both geometry and topology, where the data comes in the form of partial bijections that preserve the topology, and operator algebras, where the partial bijections encode *-subsemigroups of partial isometries of Hilbert space. In this thesis we explore the connections between these two pictures within the backdrop of coarse geometry. The first collection of results is concerned primarily with inverse semigroups and their C*-algebras. We give a construction of a six term sequence of C*-algebras connecting the semigroup C*-algebra to that of a naturally associated group C*-algebra. This result is a generalisation of the ideas of Pimsner and Voiculescu, who were concerned with computing K-theory groups associated to actions of groups. We outline how to connect this picture, via groupoids, to that of a partial translation algebra of Brodzki, Niblo andWright, and further consider applications of these sequences to computations of certain K-groups associated with group and semigroup C*-algebras. Secondly, we give an account of the coarse Baum-Connes conjecture associated to a uniformly discrete bounded geometry metric space and rephrase the conjecture in terms of groupoids and their C*-algebras that can naturally be associated to a metric space. We then consider the well known counterexamples to this conjecture, giving a unifying framework for their study in terms of groupoids and a new conjecture for metric spaces that we call the boundary coarse Baum-Connes conjecture. Generalising a result of Willett and Yu we prove this conjecture for certain classes of expanders including those of large girth by constructing a partial action of a discrete group on such spaces.

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Vertex enumeration (VE) algorithms explore the problem of listing some or all solutions that lie at corners of a convex polyhedron defined by a set of linear inequalities. Many algorithms have been developed for general polytopes. The most successful of these, from both an empirical and theoretical viewpoint, are based on pivoting. Dyer [24] gives an algorithm for simple polytopes which runs in time O (mn^2) per vertex. In this thesis we concentrate on the VE problem for certain special classes of polyhedron. We also address the problem of (approximately) counting the vertices without listing them. Pivoting algorithms rely on the correspondence between vertices and feasible bases and are consequently inefficient in the presence of a high degree of degeneracy such as frequently occurs in network polyhedra. Provan [79] gives a high-level description of a VE algorithm for such polyhedra which has running time that is quadratic in the number of vertices. W e describe an implementation of Provan's algorithm, present some computational results on transportation and assignment polytopes and discuss some practical difficulties with the algorithm. We then present an algorithmic description of a VE method via the dual Fourier-Motzkin (F-M) elimination method. One of the difficulties with F-M is that the number of constraints introduced in eliminating variables grows exponentially; we show that, for linear inequality systems with at most two variables per constraint, denoted LI(2), the growth is exponential in the number of variables but linear in the number of constraints. We go on to prove results which characterize the basis structure for LI(2) and dual LI(2) systems and hence develop a new pivoting algorithm for enumerating vertices of polyhedra associated with dual LI(2) systems. Counting the vertices of general polyhedra is #P-complete [24] and thus approximate counting procedures are of interest. In particular, some fpras for counting vertices of polyhedra associated with 0-1 Permanent, Down Sets, Independent Sets, 0-1 Knapsack Problems, 2 by n transportation problems, matroids and matchings in a non-bipartite graph are developed.

We are interested in calculating the Hilbert series of a polarised orbifold (X;D) (that is D is an ample divisor on an orbifold X). Indeed, its numerical data is encoded in its Hilbert series, so that calculating this sometimes gives us information about the ring, notably possible generators and relations, using the Hilbert syzygies theorem. Vaguely, we have PX(t) = Num/Denom where Num is given by the relations and syzygies of R and Denom is given by the generators. Thus in particular we hope that we can use the numerical data of the ring to deduce possible explicit constructions. A reasonable goal is therefore to calculate the Hilbert series of a polarised (X;D); we write it in closed form, where each term corresponds to an orbifold stratum, is Gorenstein symmetric and with integral numerator of "short support". The study of the Hilbert series where the singular locus has dimension at most 1 leads to questions about more general rational functions of the form __N___ II(1-tai) with N integral and symmetric. We prove various parsings in terms of the poles at the Uai ; each individual term is Gorenstein symmetric, with integral numerator of "short support" and geometrically corresponds to some orbifold locus. Chapters 1 and 2 are expository material: Chapter 1 is basic introductory material whilst in Chapter 2 we explain the Hilbert series parsing in the isolated singularity case, as solved in Buckley et al. [2013] and Zhou [2011] and go over worked examples for practice. Chapter 3 uses the structure of the parsing in the isolated case and the expected structure in the non-isolated case to discuss generalisations to arbitrary rational functions with symmetry and poles only at certain roots of unity. We prove some special cases. Chapter 4 discusses the Hilbert series parsing in the curve orbifold locus case in a more geometrical setting. Chapter 5 discusses further generalisations and issues. In particular we discuss how the strategies used in Chapter 3 could work in a more general section, and the non symmetric case.

Given a bialgebra B we present a unifying approach to deformations of associative algebras A with a left B-module algebra structure. Special deformations of the comultiplication of B yield universal deformation formulas, i.e. define deformations of the multiplicative structure for all B-module algebras A. This allows to derive known formulas of Moyal-Vey (1949) and Coll-Gerstenhaber-Giaquinto (1989) from a more general point of view.

Bibliography : leaves 115-119 / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Thesis (Ph.D.)--University of Adelaide, Dept. of Pure Mathematics, 1994

Evangelos E. Milios. / Originally presented as author's thesis (Ph. D.--Massachusetts Institute of Technology), 1986. / Bibliography: p. 216-219. / Supported in part by the Advanced Research Projects Agency monitored by ONR under contract no. N00014-81-K-0742 Supported in part by the National Science Foundation under grant ECS-8407285