Mathematics and Statistics
http://hdl.handle.net/10523/96
2016-05-13T10:03:35ZStudies 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.
2016-05-12T04:28:33ZEquilibrium 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.
2016-05-09T22:04:21ZKMS 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.
2016-04-06T21:22:34ZFlat Embeddings of Genetic and Distance Data
http://hdl.handle.net/10523/6286
Flat Embeddings of Genetic and Distance Data
2016
Balvočiūtė, Monika
The idea of displaying data in the plane is very attractive in many different fields of research. This thesis will focus on distance-based phylogenetics and multidimensional scaling (MDS). Both types of method can be viewed as a high-dimensional data reduction to pairwise distances and visualization of the data based on these distances. The difference between phylogenetics and multidimensional scaling is that the first one aims at finding a network or a tree structure that fits the distances, whereas MDS does not fix any structure and objects are simply placed in a low-dimensional space so that distances in the solution fit distances in the input as good as possible.
Chapter 1 provides an introduction to the phylogenetics and multidimensional scaling. Chapter 2 focuses on the theoretical background of flat split systems (planar split networks). We prove equivalences between flat split systems, planar split networks and loop-free acyclic oriented matroids of rank three. The latter is a convenient mathematical structure that we used to design the algorithm for computing planar split networks that is described in Chapter 3. We base our approach on the well established agglomerative algorithms Neighbor-Joining and Neighbor-Net. In Chapter 4 we introduce multidimensional scaling and propose a new method for computing MDS plots that is based on the agglomerative approach and spring embeddings. Chapter 5 presents several case studies that we use to compare both of our methods and some classical agglomerative approaches in the distance-based phylogenetics.
2016-03-17T22:02:52ZFeatures of written proofs from New Zealand IMO students
http://hdl.handle.net/10523/5953
Features of written proofs from New Zealand IMO students
2015
Norrish, Jordana Susan Lea
This research investigated the linguistic features of written mathematical proofs, and partial proofs, from a small group of secondary school mathematics students in New Zealand. The linguistic features included are outlined below. The students were part of a training camp for the purpose of selecting six students to represent New Zealand at the International Mathematical Olympiad. Micro-level and macro-level linguistic features of the students’ writing were analysed through a sociocultural lens. Using this lens, language was viewed as being influenced by cultural, social, and situational factors (Moschkovich, 2007) and the students’ language was observed in a naturally occurring context. Furthermore, during the training and selection camp, the tutors and lecturers were viewed as experienced members of the mathematical community of practice (Wenger, 1998).
The different linguistic features investigated were: personal pronouns, tense, causal connectives, abbreviations, mathematical equations and expressions, and argumentation. Personal pronouns and tense can indicate people’s views about the nature of mathematics (Burton & Morgan, 2000) as well as their perceptions about how people should talk about mathematics. They can also indicate the degree of generality (Rowland, 1999) involved in the author’s reasoning. Causal connectives serve to connect the parts of the reasoning to form a coherent argument. Where different connectives have a similar meaning, the choice of connectives by the author can indicate the language patterns of the community of practice. Abbreviations are also an interesting linguistic feature which can reflect the taken-as-shared sociomathematical norms (Yackel & Cobb, 1996) of a particular community of practice. Abbreviating words or phrases indicates that the author believes the reader will be able to understand and decode the abbreviations through mutually accepted knowledge and practices. The language patterns of the community of practice are further reflected by the density of mathematical equations and argument structures present in a piece of writing.
Students’ written examination answers from the conclusion of the training camp and the six students’ answers at the International Mathematical Olympiad were the main source of data collected. Furthermore, lecture sessions and solution sharing sessions were video-taped and transcribed, and field notes taken in order to understand the situation, teaching methods, and taken-as-shared socio-mathematical norms (Yackel & Cobb, 1996) during the training camp. Quantitative methods were employed to analyse the different linguistic features and link these to the training camp community of practice as well as the conventions of the wider mathematical community. These methods included descriptive statistics, chi square testing and the use of Fisher’s Exact Test. Significant chi square results were followed up with a post hoc Cramér’s V calculation in order to determine the strength of the association between variables. The Benjamini and Hochberg (1995) procedure was used to control the false discovery rate, and any results remaining significant after this procedure were followed up further with odds ratio and confidence interval calculations.
For these students, the training camp community includes the other students attending the camp (who were not selected for the IMO), and the former IMO competitors and university lecturers who mentored them at the camp. The wider mathematical community includes textbooks, journal papers, university lectures and so on. Results indicate associations between some linguistic features and both Topic and Score. Results also indicate that these students have accommodated some of the conventions from the training camp and the wider mathematical community. For example, often these students expressed themselves using the personal pronoun we or no pronoun at all, combined with the present tense, which is typical of textbooks and journal articles (Burton & Morgan, 2000). The examination responses provided rich data with numerous aspects to explore. There are several features yet to be investigated with this data set, and also several ways to extend and enhance this research to other settings.
This research has developed a profile of written proofs in the New Zealand IMO setting as well as investigating the features of written proofs associated with success. It has also investigated the influence of the training camp community of practice as well as the wider mathematical community of practice. This study has addressed the need for more research on the mathematical communication within the secondary school age group, and has also addressed the call from previous mathematics education researchers to investigate the wider context of mathematical communication, rather than just one aspect in isolation.
2015-10-14T19:52:31ZEffect of chemokine superdiffusion on leucocyte chemotaxis
http://hdl.handle.net/10523/5749
Effect of chemokine superdiffusion on leucocyte chemotaxis
2015
Jalilzadeh, Aidin
Chemotaxis is the major cytotaxic mechanism that leads the movement of phagocytes in the tissue towards the harmful agents. Loss of phagocytes ability to track and respond to danger signals can lead to chronic infections, sepsis or even death. This thesis examines the consequences of anomalous diffusion of chemokines on the chemotaxis of phagocytes in the event of acute inflammatory responses.
The main driver of any chemotactic system is the corresponding chemo-attractant, which is the role given to chemokines. Allowing anomalous (fractional) diffusion with the tail index of $0<\alpha<2$, leads to the front propagation rate proportional to $t^{1/\alpha}$: faster than the traditional Gaussian spread ($t^{1/2}$). Moreover, fractional chemokine concentration profiles obey power laws, which results in slower tail decays leading to heavy tails; whereas in the Gaussian scenario tail decays are exponential and rapid.
Changing the morphology of chemokine profile over the domain will affect all other entities that depend on chemokine concentration: the likes of tactic motility, sensitivity and velocity. Our study aims at understanding the influences of chemokine gradient field variations on phagocyte chemotaxis and hence on the acute inflammatory response. We show various circumstances in which normally diffusing chemokines fail to recruit adequate phagocytes and more importantly this behaviour stays the same even if the source of chemokine production is multiplied by several orders of magnitude. Another challenge is to insure the presence of an optimum number of phagocytes in the tissue, which is governed by a timely initiation of infiltration.
Overall, we observe differences in the outcomes of the inflammatory responses of the two different diffusion schemes. The consideration of fractional diffusion enables us to give new interpretation of how signals spread in the heterogeneous tissues and why in some cases the traditional Gaussian mechanism may fall short.
2015-06-29T03:22:34ZStatistical Methods for the Analysis of Copy Number Variation
http://hdl.handle.net/10523/5434
Statistical Methods for the Analysis of Copy Number Variation
2015
Nguyen, Tan Hoang
Copy number variation (CNV) is a type of genomic structural variation which has been associated with disease risk in humans and with trait variation in agricultural species. This type of variation has also been implicated in adaptive natural selection. Advances in next-generation sequencing (NGS) technologies facilitate the determination of CNV at specific loci. In this study, computational approaches based on NGS data have been proposed and applied to specific genomic loci. Firstly, a read-depth based method was developed specifically for the complex FCGR genetic locus. The pipeline was used to measure copy number at the FCGR3A/3B genes, and identified SNPs associated with CNV (tag SNPs) at this locus. Next, this method was modified and applied to two highly copy-number variable regions, CCL3L1 and DEFB103A. The new pipeline determined putative boundaries for CNV in these two regions, and reported CN genotype for both genes. This methodology was also used to identify novel polymorphic regions on chromosome 17 of the human genome. Next, evidence of selective pressure at two loci, CCL3L1 and FCGR3B, was investigated using tag SNP and CN information from the modified pipeline. Finally, an integrated framework of read-depth and split-read based approaches was developed to pinpoint breakpoints of CNV events occurring across samples.
2015-01-22T01:22:34ZGrünwald-type approximations and boundary conditions for one-sided fractional derivative operators
http://hdl.handle.net/10523/5216
Grünwald-type approximations and boundary conditions for one-sided fractional derivative operators
2014
Sankaranarayanan, Harish
The focus of this thesis is two-fold. The first part investigates higher order numerical schemes for one-dimensional fractional-in-space partial differential equations in 𝐿₁(ℝ). The approximations for the (space) fractional derivative operators are constructed using a shifted Grünwald-Letnikov fractional difference formula. Rigorous error and stability analysis of the Grünwald-type numerical schemes for space-time discretisations of the associated Cauchy problem are carried out using (Fourier) multiplier theory and semigroup theory. The use of a transference principle facilitates the generalisation of the results from the 𝐿₁-setting to any function space where the translation (semi) group is strongly continuous. Furthermore, the results extend to the case when the fractional derivative operator is replaced by the fractional power of a (semi) group generator on an arbitrary Banach space. The second part is dedicated to the study of certain fractional-in-space partial differential equations associated with (truncated) Riemann-Liouville and first degree Caputo fractional derivative operators on Ω:= [(0, 1)]. The boundary conditions encoded in the domains of the fractional derivative operators dictate the inclusion or exclusion of the end points of Ω. Elaborate technical constructions and detailed error analysis are carried out to show convergence of Grünwald-type approximations to fractional derivative operators on 𝑋 = C₀(Ω) and L₁[0, 1]. The wellposedness of the associated Cauchy problem on 𝑋 is established using the approximation theory of semigroups. The culmination of the thesis is the result which shows convergence in the Skorohod topology of the well understood stochastic processes associated with Grünwald-type approximations to the processes governed by the corresponding fractional-in-space partial differential equations.
2014-11-13T03:52:07Z