Mathematics and Statistics
http://hdl.handle.net/10523/96
2016-02-13T02:29:53ZFeatures 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.
2014-11-13T03:52:07ZEnvironmental stochasticity and density dependence in animal population models
http://hdl.handle.net/10523/5012
Environmental stochasticity and density dependence in animal population models
2006
Samaranayaka, Ariyapala Hattasinge (Ari)
Biological management of populations plays an indispensable role in all areas of population biology. In deciding between possible management options, one of the most important pieces of information required by population managers is the likely population status under possible management actions. Population dynamic models are the basic tool used in deriving this information. These models elucidate the complex processes underlying the population dynamics, and address the possible consequences/merits of management actions. These models are needed to guide the population towards desired/chosen management goals, and therefore allow managers to make informed decisions between alternative management actions.
The reliability that can be placed on inferences drawn from a model about the fate of a population is undoubtedly dependent on how realistically the model represents the dynamic process of the population. The realistic representation of population characteristics in models has proved to be somewhat of a thorn in the side of population biologists. This thesis focuses in particular on ways to represent environmental stochasticity and density dependence in population models.
Various approaches that are used in building environmental stochasticity into population models are reviewed. The most common approach represents the environmental variation by changes to demographic parameters that are assumed to follow a simple statistical distribution. For this purpose, a distribution is often selected on the basis of expert opinion, previous practice, and convenience. This thesis assesses the effect of this subjective choice of distribution on the model predictions, and develops some objective criteria for that selection based on ecological and statistical acceptability. The more commonly used distributions are compared as to their suitability, and some recommendations are made.
Density dependence is usually represented in population models by specifying one or more of the vital rates as a function of population density. For a number of reasons, a population-specific function cannot usually be selected based on data. The thesis develops some ecologically-motivated criteria for identifying possible function(s) that could be used for a given population by matching functional properties to population characteristics when they are known. It also identifies a series of properties that should be present in a general function which could be suitable for modelling a population when relevant population characteristics are unknown. The suitability of functions that are commonly chosen for such purposes is assessed on this basis.
I also evaluate the effect of the choice of a function on the resulting population trajectories. The case where the density dependence of one demographic rate is influenced by the density dependence of another is considered in some detail, as in some situations it can be modelled with little information in a relatively function-insensitive way.
The findings of this research will help in embedding characteristics of animal populations into population dynamics models more realistically. Even though the findings are presented in the context of slow-growing long-lived animal populations, they are more generally applicable in all areas of biological management.
2014-10-03T03:32:48ZNumerical Construction of Static Fluid Interfaces with the Embedding Formalism
http://hdl.handle.net/10523/4878
Numerical Construction of Static Fluid Interfaces with the Embedding Formalism
2014
Laing, Christopher James
This research project develops a mathematical and numerical framework for representing static fluid interfaces as embedded manifolds. A variational principle is developed for the embedding function of a smooth manifold, along with the necessary boundary and gauge conditions. The variational problem is solved by a combination of Finite Elements, constrained optimisation techniques, and original algorithms. The approach is applied to problems inspired by modern technological and scientific applications of static fluid interfaces, and the results are compared to exact solutions, experimental data, or other numerical methods, where possible. The impact of the numerical methods on the quality of the solution is discussed in de- tail, with reference to the boundary conditions, the gauge conditions, and the constraints.
2014-07-01T20:44:32ZUntangling Evolution
http://hdl.handle.net/10523/4802
Untangling Evolution
2014
Voorkamp, Joshua Stewart
Molecular biology makes extensive use of methods that can accurately estimate the evolutionary relationships between species, particularly when the evolutionary history can be represented on an `evolutionary tree'. However, attention is increasingly turning towards approaches that can be applied when the history might not be adequately represented by a tree.
This thesis presents computational methods that work towards the goal of inferring a `evolutionary network' directly from sequence data. The particular focus was on computational methods that are `fit agnostic' which refers to the idea that any method for evaluating how well a structure matches provided data could be used in place of the one used here. Given the underlying mechanisms used the evaluation methods that do well will be those that preserve the relationship that if a particular tree gives a good fit to the data then restricting the tree and data to a subset of taxa will also give a good fit. The final goal of achieving a network was not reached as there remain some problems and no obvious way yet to solve them. As a temporary placeholder in order to ascertain if the methods \emph{could} be applied to networks sets of evolutionary trees were used whereby the sets of evolutionary trees are to be interpreted as existing in some network which a future algorithm may work on directly. The methods and theory in this paper have thus been formulated so that there should be an analogue from these that are developed for sets of evolutionary trees to ones that can be applied to evolutionary networks.
The first chapters and bulk of the thesis concentrates on developing methods that can infer the history on this simpler structure and the last chapter is initial work on a method that is intended to allow extension of the previous methods to networks, although the new structure described turns up results that are interesting in their own right.
2014-05-11T23:30:24ZThe weak global dimension of Gaussian rings
http://hdl.handle.net/10523/4736
The weak global dimension of Gaussian rings
2014
Alhassan, Eman
A Prüfer domain is defined as an integral domain for which each nonzero finitely generated ideal is invertible or, equivalently, projective. In general, if R is a domain, we have the following equivalent statements:
(i) R is semihereditary (see §2.6 for the definition).
(ii) w.gl.dim(R) ≤ 1 (see §2.5 for the definition).
(iii) R_m is a chain domain for every 𝔪 ∈ Max(R) (see §1.2, §1.3 for definitions).
(iv) R is a Gaussian domain (see §3.1 for the definition).
(v) R is a Prüfer domain.
The definition of each class of ring featured in (i)–(v) can be extended to commutative rings which are not necessarily integral domains, i.e., which may have zero-divisors. However, there are examples showing that no two of (i)–(v) are equivalent in this more general setting. In fact we have strict implications
R is semihereditary ⇒ w.gl.dim(R) ≤ 1 ⇒ R_m is a chain ring for every maximal ideal 𝔪 of R ⇒ R is a Gaussian ring ⇒ (v) R is a Prüfer ring.
We concentrate our studies on the weak global dimension of Gaussian rings. The authors of [BaGl] proposed a conjecture that there were only three possibilities for the weak global dimension of a Gaussian ring, namely 0, 1 or ∞. We follow the authors of [DT] by referring to this as the Bazzoni–Glaz Conjecture. In [Gl2, Theorem 2.2], it is shown that if the Gaussian ring R is a reduced ring (i.e., R is a ring for which the zero element is the only nilpotent element) then R has weak global dimension at most 1, verifying the Conjecture in this case. The case of non-reduced Gaussian rings is given a great deal of attention in the 2011 preprint [DT]. The authors of [DT] prove the Conjecture is true using a number of concepts from homological algebra. We give details for some of these results and refer to articles and books for others. The proof in [DT] is quite long, involving several reduction steps to reach the final outcome.
In the Chapter 1 we introduce some well-established results from the ideal theory of commutative rings. For the most part, we will skip the proofs of these results and give references for them to the reader. This chapter is very important for the following chapters. We then look at some homological algebra definitions, results and methods in Chapter 2. In particular, this chapter will look at the weak global dimension of a ring R. The third chapter concentrates on the ideal structure of Gaussian rings, especially local Gaussian rings, detailing general properties of their internal structure. Finally, in Chapter 4, we give a detailed proof of the Bazzoni–Glaz Conjecture. This final lengthy chapter considers an all-inclusive number of cases of Gaussian rings and shows that the conjecture holds for every case.
2014-03-27T23:03:58Z