Mathematics and Statistics
http://hdl.handle.net/10523/96
Sun, 30 Apr 2017 14:47:25 GMT2017-04-30T14:47:25ZAnalogues of Leavitt path algebras for higher-rank graphs
http://hdl.handle.net/10523/7279
Analogues of Leavitt path algebras for higher-rank graphs
2017
Pangalela, Yosafat Eka Prasetya
Directed graphs and their higher-rank analogues provide an intuitive framework to study a class of C*-algebras which we call graph algebras. The theory of graph algebras has been developed by a number of researchers and also influenced other branches of mathematics: Leavitt path algebras and Cohn path algebras, to name just two.
Leavitt path algebras for directed graphs were developed independently by two groups of mathematicians using different approaches. One group, which consists of Ara, Goodearl and Pardo, was motivated to give an algebraic framework of graph algebras. Meanwhile, the motivation of the other group, which consists of Abrams and Aranda Pino, is to generalise Leavitt's algebras, in which the name Leavitt comes from. Later, Abrams and now with Mesyan introduced the notion of Cohn path algebras for directed graphs. Interestingly, both Leavitt path algebras and Cohn path algebras for directed graphs can be viewed as algebraic analogues of C*-algebras of directed graphs.
In 2013, Aranda Pino, J. Clark, an Huef and Raeburn introduced a higher-rank version of Leavitt path algebras which we call Kumjian-Pask algebras. At their first appearance, Kumjian-Pask algebras were only defined for row-finite higher-rank graphs with no sources. Clark, Flynn and an Huef later extended the coverage by also considering locally convex row-finite higher-rank graphs. On the other hand, Cohn path algebras for higher rank graphs still remained a mystery.
This thesis has two main goals. The first aim is to introduce Kumjian-Pask algebras for a class of higher-rank graphs called finitely-aligned higher-rank graphs. This type of higher-rank graph covers both row-finite higher-rank graphs with no sources and locally convex row-finite higher-rank graphs. Therefore, we give a generalisation of the existing Kumjian-Pask algebras. We also establish the graded uniqueness theorem and the Cuntz-Krieger uniqueness theorem for Kumjian-Pask algebras of finitely-aligned higher-rank graphs.
The second aim is to introduce a higher-rank analogue of Cohn path algebras. We then study the relationship between Kumjian-Pask algebras and Cohn path algebras and use this to investigate properties of Cohn path algebras. Finally, we establish a uniqueness theorem for Cohn path algebras.
Thu, 06 Apr 2017 20:59:28 GMThttp://hdl.handle.net/10523/72792017-04-06T20:59:28ZC*-algebras generated by semigroups of partial isometries
http://hdl.handle.net/10523/7277
C*-algebras generated by semigroups of partial isometries
2017
Tolich, Ilija
This thesis examines the C*-algebras associated to semigroups of partial isometries. There are many interesting examples of C*-algebras generated by families of partial isometries, for example the C*-algebras associated to directed graphs and the C*-algebras associated to inverse semigroups.
In 1992 Nica introduced a class of partially ordered groups called quasi-lattice ordered groups, and studied the C*-algebras generated by semigroups of isometries satisfying a covariance condition. We have adapted Nica's construction for semigroups of partial isometries associated to what we call doubly quasi-lattice ordered groups. For each doubly quasi-lattice ordered group we construct two algebras: a concretely defined reduced algebra, and a universal algebra generated by a covariant family of partial isometries. We examine when representations of the universal algebra are faithful, and this gives rise to a notion of amenability for doubly quasi-lattice ordered groups.
We prove several recognition theorems for amenability. In particular, we prove that the universal and reduced algebras are isomorphic if and only if the doubly quasi-lattice ordered group is amenable. Further, we prove that if there is an order preserving homomorphism from a doubly quasi-lattice ordered group to an amenable group, then the quasi-lattice ordered group is amenable and the associated universal algebra is nuclear.
Thu, 06 Apr 2017 04:29:56 GMThttp://hdl.handle.net/10523/72772017-04-06T04:29:56ZThe behaviour of sea ice in ocean waves
http://hdl.handle.net/10523/7000
The behaviour of sea ice in ocean waves
1993
Meylan, Michael
The entry of ocean waves from the open sea into pack ice is a feature of the marginal ice zone which has important consequences for navigation and the construction of offshore structures in ice-infested seas. In turn it is largely the action of waves which creates the marginal ice zone as it is the wave action which is responsible for the floe size distribution within the ice cover.
In this thesis a two-dimensional model for the behaviour of a single ice floe in ocean waves is developed using a Green's function formulation. This model allows us to calculate the reflection and transmission coefficients of a single floe. It predicts that there will be frequency-dependent critical floe lengths at which the reflection is zero, analogous to electromagnetic wave propagation through a homogeneous slab. It is also found that floe bending increases as a function of floe length until a critical length is reached, above which the strain is essentially constant. The model
is successfully validated, at least for elastic sheets floating on water, by experiments performed on a polypropylene sheet. The single floe theory may also be synthesized approximately by an extension of the model developed by Fox and Squire [1990, 1991] for the interaction of waves with a semi-infinite sheet. This acts as an independent check on both theories.
The solution for a single floe may be extended to many floes as a full solution or as an approximate solution. It is shown that the approximate solution is sufficiently accurate in nearly all situations. This allows the development of a simple model for ocean wave propagation through a cover composed of many discrete floes. This model predicts that a field of pack ice will low pass filter incoming ocean waves. The model also predicts that there will be a narrowing of directional spectra with propagation through an ice cover.
Finally the model is extended so that the surge response, a frequently measured property of ice floes, may be predicted. The surge response agrees with that found by Rattier [1992] and is a strong function of floe length.
A different model for the motion of a single floe developed by Shen and Ackley [1991] is also investigated. This model is applicable to small ice floes and is related to Morrisons equation which is used extensively in problems of offshore structures. The Shen and Ackley model is shown to predict that in most physical cases all floes will tend to the same drift velocity which will be a function almost exclusively of wave amplitude.
Tue, 06 Dec 2016 22:14:43 GMThttp://hdl.handle.net/10523/70002016-12-06T22:14:43ZTowards a Fast Bayesian Climate Reconstruction
http://hdl.handle.net/10523/6909
Towards a Fast Bayesian Climate Reconstruction
2016
Green, Peter
To understand global climate prior to the availability of widespread instrumental data, we need to reconstruct temperatures using natural proxies such as tree rings. For reconstructions of a temperature field with multiple proxies, the currently preferred method is RegEM (Schneider, 2001). However, this method has problems with speed, convergence, and interpretation.
In this thesis we show how one variant of RegEM can be replaced by the monotone EM algorithm (Liu, 1999). This method is much faster, especially in suitably designed pseudoproxy simulation experiments.
Multi-proxy reconstructions can be large, with thousands of variables and millions of parameters. We describe how monotone EM can be implemented efficiently for problems on this scale.
RegEM has been interpreted in a Bayesian context as a multivariate normal model with an inverse Wishart prior. We extend this interpretation, noting the empirical Bayesian aspects, the implications of the prior for the variance loss problem, and using posterior predictive checks for model criticism.
The Bayesian interpretation leads us to suggest a novel prior. Simulated reconstructions with this prior show promising performance against the usual prior, particularly in terms of low sensitivity to the tuning parameter.
Wed, 09 Nov 2016 03:35:47 GMThttp://hdl.handle.net/10523/69092016-11-09T03:35:47ZThe Numerical Initial Boundary Value Problem for the Generalised Conformal Field Equations in General Relativity
http://hdl.handle.net/10523/6852
The Numerical Initial Boundary Value Problem for the Generalised Conformal Field Equations in General Relativity
2016
Stevens, Christopher Zane
The purpose of this work is to develop for the first time a general framework for the Initial Boundary Value Problem (IBVP) of the Generalised Conformal Field Equations (GCFE). At present the only investigation toward obtaining such a framework was given in the mid 90's by Friedrich at an analytical level and is only valid for Anti-de Sitter space-time. There have so far been no numerical explorations into the validity of building such a framework.
The GCFE system is derived in the space-spinor formalism and Newman and Penrose's eth-calculus is imposed to obtain proper spin-weighted equations. These are then rigorously tested both analytically and numerically to confirm their correctness. The global structure of the Schwarzschild, Schwarzschild-de Sitter and Schwarzschild-Anti-de Sitter space-times are numerically reproduced from an IVP and for the first time, numerical simulations that incorporate both the singularity and the conformal boundary are presented.
A framework for the IBVP is then given, where the boundaries are chosen as arbitrary time-like conformal geodesics and where the constraints propagate on (at least) the numerical level. The full generality of the framework is verified numerically for gravitational perturbations of Minkowski and Schwarzschild space-times. A spin-frame adapted to the geometry of future null infinity is developed and the expressions for the Bondi-mass and the Bondi-time given by Penrose and Rindler are generalised. The Bondi-mass is found to equate to the Schwarzschild-mass for the standard Schwarzschild space-time and the famous Bondi-Sachs mass loss is reproduced for the gravitationally perturbed case.
Wed, 19 Oct 2016 02:48:28 GMThttp://hdl.handle.net/10523/68522016-10-19T02:48:28ZStudies of spacetimes with spatial topologies S3 and S1 X S2
http://hdl.handle.net/10523/6451
Studies of spacetimes with spatial topologies S3 and S1 X S2
2016
Escobar , Leon
The purpose of this work is to introduce a new analytical and numerical approach to the treatment of the initial value problem for the vacuum Einstein field equations on spacetimes with spatial topologies S3 or S1 × S2 and symmetry groups U(1) or U(1)×U(1). The general idea consists of taking advantage of the action of the symmetry group U(1) to rewrite those spacetimes as a principal fiber bundle, which is trivial for S1 × S2 but not for S3. Thus, the initial value problem in four dimensions can be reduced to a three-dimensional initial value problem for a certain manifold with spatial topology S2. Furthermore, we avoid coordinate representations that suffer from coordinate singularities for S2 by expressing all the fields in terms of the spin-weighted spherical harmonics.
We use the generalized wave map formalism to reduce the vacuum Einstein field equations on a manifold with three spatial dimensions to a system of quasilinear wave equations in terms of generalized gauge source functions with well-defined spin-weights. As a result, thanks to the fully tensorial character of these equations, the system of evolution equations can be solved numerically using a 2 + 1-pseudo-spectral approach based on a spin-weighted spherical harmonic transform. In this work, however, we apply our infrastructure to the study of Gowdy symmetric spacetimes, where thanks to the symmetry group U(1) × U(1), the system of hyperbolic equations obtained from the vacuum Einstein field equations can be reduced to a 1+1-system of partial differential equations. Therefore, we introduce an axial symmetric spin-weighted transform that provides an efficient treatment of axially symmetric functions in S2 by reducing the complexity of the general transform.
To analyse the consistency, accuracy, and feasibility of our numerical infrastructure, we reproduce an inhomogeneous cosmological solution of the vacuum Einstein field quations with spatial topology S3 . In addition, we consider two applications of our infrastructure. In the first one, we numerically explore the behaviour of Gowdy S1 × S2 spacetimes using our infrastructure. In particular, we study the behaviour of some geometrical quantities to investigate the behaviour of those spacetimes when approach a future singularity. As a second application, we conduct a systematic investigation on the non-linear instability of the Nariai spacetime and the asymptotic behaviour of its perturbations.
Thu, 12 May 2016 04:28:33 GMThttp://hdl.handle.net/10523/64512016-05-12T04:28:33ZEquilibrium States on Toeplitz Algebras
http://hdl.handle.net/10523/6444
Equilibrium States on Toeplitz Algebras
2016
Afsar, Zahra
This thesis describes the equilibrium states (the KMS states) of dynamical systems arising from local homeomorphisms. It has two main components. First, we consider a local homeomorphism on a compact space and the associated Hilbert bimodule. This Hilbert bimodule has both a Toeplitz algebra and a Cuntz-Pimsner algebra, which is a quotient of the Toeplitz algebra. Both algebras carry natural gauge actions of the circle, and hence one can obtain natural dynamics by lifting these actions to actions of the real numbers. We study KMS states of these dynamics at, above, and below a certain critical value. For inverse temperature larger than the critical value, we find a large simplex of KMS states on the Toeplitz algebra. For the Cuntz-Pimsner algebra, the KMS states all have inverse temperatures below the critical value. Our results for the Cuntz-Pimsner algebra overlap with recent work of Thomsen, but our proofs are quite different. At the critical value, we build a KMS state of the Toeplitz algebra which factors through the Cuntz-Pimsner algebra.
To understand KMS states below the critical value, we study the backward shift on the infinite path space of an ordinary directed graph. Merging our results for the Cuntz-Pimsner algebra of shifts with the recent work about KMS states of the graph algebras, we show that Thomsen's bounds on of the possible inverse temperature of KMS states are sharp.
In the second component, we consider a family of *-commuting local homeomorphisms on a compact space and build a compactly aligned product system of Hilbert bimodules (in the sense of Fowler). This product system also has two interesting algebras, the Nica-Toeplitz algebra and the Cuntz-Pimsner algebra. For these algebras, the gauge action is an action of a higher-dimensional torus, and there are many possible dynamics obtained by composing with different embeddings of the real line in the torus.
We use the techniques from the first component of the thesis to study the KMS states for these dynamics. For large inverse temperature, we describe the simplex of the KMS states on the Nica-Toeplitz algebra. To study KMS states for smaller inverse temperature, we consider a preferred dynamics for which there is a single critical inverse temperature, which we can normalise to be 1. We then find a KMS1 state for the Nica-Toeplitz algebra which factors through the Cuntz-Pimsner algebra. We then illustrate our results by considering different backward shifts on the infinite path space of some higher-rank graphs.
Mon, 09 May 2016 22:04:21 GMThttp://hdl.handle.net/10523/64442016-05-09T22:04:21ZKMS states of graph algebras with a generalised gauge dynamics
http://hdl.handle.net/10523/6346
KMS states of graph algebras with a generalised gauge dynamics
2016
McNamara, Richard Stuart Charles
The goal of this thesis is to study the KMS states of graph algebras with a generalised gauge dynamics.
We start by studying the KMS states of the Toeplitz algebra and graph algebra of a finite directed graph, each with a generalised gauge dynamics. We characterise the KMS states of the Toeplitz algebra and find an isomorphism between measures and KMS states at large inverse temperatures. When the graph is strongly connected we can describe all of the KMS states, and we get a unique KMS state at the critical inverse temperature. Viewing the graph algebra as a quotient of the Toeplitz algebra we describe the KMS states of the graph algebra.
Next we study the KMS states of graph algebras for row-finite infinite directed graphs with no sources and the gauge action. We characterise the KMS states of the Toeplitz algebra and discuss KMS states at large inverse temperatures. We then show that problems occur at the critical inverse temperature.
Lastly we study the KMS states of the Toeplitz algebras and graph algebras for higher-rank graphs with a generalised gauge dynamics, using the same method as we did for finite graphs. We finish by studying the preferred dynamics of the system, where we get our best results.
Wed, 06 Apr 2016 21:22:34 GMThttp://hdl.handle.net/10523/63462016-04-06T21:22:34Z