Dissertations / Theses on the topic 'Fields of Research – 230000 Mathematical Sciences'
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1
Rubin, Sasha. "Automatic structures." Thesis, University of Auckland, 2004. http://hdl.handle.net/2292/1262.
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This thesis investigates structures that are presentable by finite automata working synchronously on tuples of finite words. The emphasis is on understanding the expressiveness and limitations of automata in this setting. In particular, the thesis studies the classification of classes of automatic structures, the complexity of the isomorphism problem, and the relationship between definability and recognisability.
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Marshall, T. H. (Timothy Hamilton). "Hyperbolic Geometry and Reflection Groups." Thesis, University of Auckland, 1994. http://hdl.handle.net/2292/2140.
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The n-dimensional pseudospheres are the surfaces in Rn+l given by the equations x12+x22+...+xk2-xk+12-...-xn+12=1(1 ≤ k ≤ n+1). The cases k=l, n+1 give, respectively a pair of hyperboloids, and the ordinary n-sphere. In the first chapter we consider the pseudospheres as surfaces h En+1,k, where Em,k=Rk x (iR)m-k, and investigate their geometry in terms of the linear algebra of these spaces. The main objects of investigation are finite sequences of hyperplanes in a pseudosphere. To each such sequence we associate a square symmetric matrix, the Gram matrix, which gives information about angle and incidence properties of the hyperplanes. We find when a given matrix is the Gram matrix of some sequence of hyperplanes, and when a sequence is determined up to isometry by its Gram matrix. We also consider subspaces of pseudospheres and projections onto them. This leads to an n-dimensional cosine rule for spherical and hyperbolic simplices. In the second chapter we derive integral formulae for the volume of an n-dimensional spherical or hyperbolic simplex, both in terms of its dihedral angles and its edge lengths. For the regular simplex with common edge length γ we then derive power series for the volume, both in u = sinγ/2, and in γ itself, and discuss some of the properties of the coefficients. In obtaining these series we encounter an interesting family of entire functions, Rn(p) (n a nonnegative integer and pεC). We derive a functional equation relating Rn(p) and Rn-1(p). Finally we classify, up to isometry, all tetrahedra with one or more vertices truncated, for which the dihedral angles along the edges formed by the truncatons. are all π/2, and the remaining dihedral angles are all sub-multiples of π. We show how to find the volumes of these polyhedra, and find presentations and small generating sets for the orientation-preserving subgroups of their reflection groups. For particular families of these groups, we find low index torsion free subgroups, and construct associated manifolds and manifolds with boundary In particular, we find a sequence of manifolds with totally geodesic boundary of genus, g≥2, which we conjecture to be of least volume among such manifolds.
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Ben-Tal, Alona. "A Study of Symmetric Forced Oscillators." Thesis, University of Auckland, 2001. http://hdl.handle.net/2292/2323.
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In this thesis we study a class of symmetric forced oscillators modeled by non-linear ordinary differential equations. Solutions for this class of systems can be symmetric or non-symmetric. When a symmetric periodic solution loses its stability as a physical parameter is varied, and two non-symmetric periodic solutions appear, this is called a symmetry breaking bifurcation. In a symmetry increasing bifurcation two conjugate chaotic attractors (i.e.,attractors which are related to each other by the symmetry) collide and form a larger symmetric chaotic attractor. Symmetry can also be restored via explosions where, as a physical parameter is varied, two conjugate attractors (chaotic or periodic) which do not intersect are suddenly embedded in one symmetric attractor. In this thesis we show that all these apparently distinct bifurcations can be realized by a single mechanism in which two conjugate attractors collide with a symmetric limit set. The same mechanism seems to operate for at least some bifurcations involving non-attracting limit sets. We illustrate this point with examples of symmetry restoration in attracting and non-attracting sets found in the forced Duffing oscillator and in a power system. Symmetry restoration in the power system is associated with a phenomenon known as ferroresonance. The study of the ferroresonance phenomenon motivated this thesis. Part of this thesis is devoted to studying one aspect of the ferroresonance phenomenon the appearance of a strange attractor with a band-like structure. This attractor was called previously a 'pseudo-periodic' attractor. Some methods for analyzing the non-autonomous systems under study are shown. We construct three different maps which highlight different features of symmetry restoring bifurcations. One map in particular captures the symmetry of a solution by sampling it every half the period of the forcing. We describe a numerical method to construct a bifurcation diagram of periodic solutions and present a non-standard approach for converting the forced oscillator to an autonomous system.
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Barton, Bill 1948. "Ethnomathematics: Exploring Cultural Diversity in Mathematics." Thesis, University of Auckland, 1996. http://hdl.handle.net/2292/2332.
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This thesis provides a new conceptualisation of ethnomathematics which avoids some of the difficulties which emerge in the literature. In particular, work has been started on a philosophic basis for the field. There is no consistent view of ethnomathematics in the literature. The relationship with mathematics itself has been ignored, and the philosophical and theoretical background is missing. The literature also reveals the ethnocentricity implied by ethnomathematics as a field of study based in a culture which has mathematics as a knowledge category. Two strategies to over come this problem are identified: universalising the referent of ‘mathematics’ so that it is the same as “knowledge-making”; or using methodological techniques to minimise it. The position of ethnomathematics in relationship to anthropology, sociology, history, and politics is characterised on a matrix. A place for ethnomathematics is found close the anthropology of mathematics, but the aim of anthropology is to better understand culture in general, while ethnomathematics aims to better understand mathematics. Anthropology, however, contributes its well-established methodologies for overcoming ethnocentricity. The search for a philosophical base finds a Wittgensteinian orientation which enables culturally based ‘systems of meaning’ to gain credibility in mathematics. A definition is proposed for ethnomathematics as the study of mathematical practices within context. Four types of ethnomathematical activity are identified: descriptive, archaeological, mathematising, and analytical activity. The definition also gives rise to a categorisation of ethnomathematical work along three dimensions: the closeness to conventional mathematics; the historical time; and the type of host culture. The mechanisms of interaction between mathematical practices are identified, and the imperialistic growth of mathematics is explained. Particular features of ethnomathematical theory are brought out in a four examples. By admitting the legitimacy of other viewpoints, ethnomathematics opens mathematics to new creative forces. Within education, ethnomathematics provides new choices, and turns cultural conflict into a useful tool for teaching. Mathematical activity exists in a variety of contexts. Learning mathematics involves being aware of, and integrating, diverse concepts. Ethnomathematics expands mathematical horizons, so that cultural diversity becomes a richer contributor to the cultural structures which humans use to understand their world.
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Curran, James Michael. "Forensic Applications of Bayesian Inference to Glass Evidence." Thesis, University of Auckland, 1996. http://hdl.handle.net/2292/2454.
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The role of the scientist in the courtroom has come under more scrutiny this century than ever before. As a consequence, scientists must constantly look for ways to improve the validity of the evidence they deliver. It is here that the professional statistician can provide assistance. The use of statistics in the courtroom and in forensic science is not new, but until recently has not been common either. Statistics can provide objectivity to subjective assessments and strengthen a case for the prosecution or the defence, but only if is used correctly. The aim of this thesis is to enhance and replace the existing technology used in statistical analysis and presentation of trace evidence, i.e. all non-genetic evidence (hairs, fibres, glass, paint, etc.) and transfer problems.
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Budden, Stephen Mark. "Knots and quandles." Thesis, University of Auckland, 2009. http://hdl.handle.net/2292/5292.
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Quandles were introduced to Knot Theory in the 1980s as an almost complete algebraic invariant for knots and links. Like their more basic siblings, groups, they are difficult to distinguish so a major challenge is to devise means for determining when two quandles having different presentations are really different. This thesis addresses this point by studying algebraic aspects of quandles. Following what is mainly a recapitulation of existing work on quandles, we firstly investigate how a link quandle is related to the quandles of the individual components of the link. Next we investigate coset quandles. These are motivated by the transitive action of the operator, associated and automorphism group actions on a given quandle, allowing techniques of permutation group theory to be used. We will show that the class of all coset quandles includes the class of all Alexander quandles; indeed all group quandles. Coset quandles are used in two ways: to give representations of connected quandles, which include knot quandles; and to provide target quandles for homomorphism invariants which may be useful in enabling one to distinguish quandles by counting homomorphisms onto target quandles. Following an investigation of the information loss in going from the fundamental quandle of a link to the fundamental group, we apply our techniques to calculations for the figure eight knot and braid index two knots and involving lower triangular matrix groups. The thesis is rounded out by two appendices, one giving a short table of knot quandles for knots up to six crossings and the other a computer program for computing the homomorphism invariants.
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Korobeinikov, Andrei. "Stability and bifurcation of deterministic infectious disease models." Thesis, University of Auckland, 2001. http://wwwlib.umi.com/dissertations/fullcit/3015611.
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Autonomous deterministic epidemiological models are known to be asymptotically stable. Asymptotic stability of these models contradicts observations. In this thesis we consider some factors which were suggested as able to destabilise the system. We consider discrete-time and continuous-time autonomous epidemiological models. We try to keep our models as simple as possible and investigate the impact of different factors on the system behaviour. Global methods of dynamical systems theory, especially the theory of bifurcations and the direct Lyapunov method are the main tools of our analysis. Lyapunov functions for a range of classical epidemiological models are introduced. The direct Lyapunov method allows us to establish their boundedness and asymptotic stability. It also helps investigate the impact of such factors as susceptibles' mortality, horizontal and vertical transmission and immunity failure on the global behaviour of the system. The Lyapunov functions appear to be useful for more complicated epidemiological models as well. The impact of mass vaccination on the system is also considered. The discrete-time model introduced here enables us to solve a practical problem-to estimate the rate of immunity failure for pertussis in New Zealand. It has been suggested by a number of authors that a non-linear dependence of disease transmission on the numbers of infectives and susceptibles can reverse the stability of the system. However it is shown in this thesis that under biologically plausible constraints the non-linear transmission is unable to destabilise the system. The main constraint is a condition that disease transmission must be a concave function with respect to the number of infectives. This result is valid for both the discrete-time and the continuous-time models. We also consider the impact of mortality associated with a disease. This factor has never before been considered systematically. We indicate mechanisms through which the disease-induced mortality can affect the system and show that the disease-induced mortality is a destabilising factor and is able to reverse the system stability. However the critical level of mortality which is necessary to reverse the system stability exceeds the mortality expectation for the majority of human infections. Nevertheless the disease-induced mortality is an important factor for understanding animal diseases. It appears that in the case of autonomous systems there is no single factor able to cause the recurrent outbreaks of epidemics of such magnitudes as have been observed. It is most likely that in reality they are caused by a combination of factors.Subscription resource available via Digital Dissertations
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Liao, Kecheng. "The AP integral." Thesis, University of Auckland, 1993. http://hdl.handle.net/2292/1969.
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This study attempts to develop the theory surrounding a controlled convergence theorem in the setting of AP integration. First, after a brief review, we spend some time on developing concepts and definitions. We are able to use these concepts to successfully present an initial control convergent theorem. Then we study our conditions more profoundly and find a number of equivalences and inequivalences between them. We are able to weaken some of the standard hypotheses significantly. The Riesz type approach to our integral has been included in our theory. Appropriately enough, we are able to use some of our approaches to extend the theory of the Henstock-Kurweil integral.
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Zhang, Xingyou. "Dynamics and numerics of generalised Euler equations : a thesis submitted to Massey University in partial fulfillment of the requirements for the degree of Ph.D in Mathematics." Massey University, 2008. http://hdl.handle.net/10179/980.
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This thesis is concerned with the well-posedness, dynamical properties and numerical treatment of the generalised Euler equations on the Bott-Virasoro group with respect to the general Hk metric , k 2. The term “generalised Euler equations” is used to describe geodesic equations on Lie groups, which unifies many differential equations and has found many applications in such as hydrodynamics, medical imaging in the computational anatomy, and many other fields. The generalised Euler equations on the Bott-Virasoro group for k = 0, 1 are well-known and intensively studied— the Korteweg-de Vries equation for k = 0 and the Camassa-Holm equation for k = 1. Unlike these, the equations for k 2, which we call the modified Camassa-Holm (mCH) equation, is not known to be integrable. This distinction motivates the study of the mCH equation. In this thesis, we derive the mCH equation and establish the short time existence of solutions, the well-posedness of the mCH equation, long time existence, the existence of the weak solutions, both on the circle S and R, and three conservation laws, show some quite interesting properties, for example, they do not lead to the blowup in finite time, unlike the Camassa-Holm equation. We then consider two numerical methods for the modified Camassa-Holm equation: the particle method and the box scheme. We prove the convergence result of the particle method. The numerical simulations indicate another interesting phenomenon: although mCH does not admit blowup in finite time, it admits solutions that blow up (which means their maximum value becomes infinity) at infinite time, which we call weak blowup. We study this novel phenomenon using the method of matched asymptotic expansion. A whole family of self-consistent blowup profiles is obtained. We propose a mechanism by which the actual profile is selected that is consistent with the simulations, but the mechanism is only partly supported by the analysis. We study the four particle systems for the mCH equation finding numerical evidence both for the non-integrability of the mCH equations and for the existence of the fourth integral. We also study the higher dimensional case and obtain the short time existence and well-posedness for the generalised Euler equation in the two dimension case.
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Harper, Sharleen Anne. "Mathematical models for dispersal of aerosol droplets in an agricultural setting : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand." Massey University, 2008. http://hdl.handle.net/10179/814.
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Agrichemical spray drift is an issue of concern for the orcharding industry. Shelterbelts surrounding orchard blocks can significantly reduce spray drift by intercepting droplets from the airflow. At present, there is little information available with which to predict drift deposits downwind, particularly in the case of a fully-sheltered orchard block. In this thesis, we develop a simple mathematical model for the transport of airborne drifting spray droplets, including the effects of droplet evaporation and interception by a shelterbelt. The object is for the model to capture the major features of the droplet transport, yet be simple enough to determine an analytic solution, so that the deposit on the ground may be easily calculated and the effect of parameter variations observed. We model the droplet transport using an advection-dispersion equation, with a trapping term added to represent the shelterbelt. In order to proceed analytically, we discretise the shelterbelt by dividing it into a three-dimensional array of blocks, with the trapping in each block concentrated to the point at its centre. First, we consider the more straightforward case where the droplets do not evaporate; solutions are presented in one, two and three dimensions, along with explicit expressions for the total amount trapped and the deposit on the ground. With evaporation, the model is more difficult to solve analytically, and the solutions obtained are nestled in integral equations which are evaluated numerically. In both cases, examples are presented to show the deposition profile on the ground downwind of the shelterbelt, and the corresponding reduction in deposit from the same scenario without the shelterbelt.
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Yee, Thomas William. "The Analysis of binary data in quantitative plant ecology." Thesis, University of Auckland, 1993. http://hdl.handle.net/2292/1973.
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The analysis of presence/absence data of plant species by regression analysis is the subject of this thesis. A nonparametric approach is emphasized, and methods which take into account correlations between species are also considered. In particular, generalized additive models (GAMs) are used, and these are applied to species’ responses to greenhouse scenarios and to examine multispecies interactions. Parametric models are used to estimate optimal conditions for the presence of species and to test several niche theory hypotheses. An extension of GAMs called vector GAMs is proposed, and they provide a means for proposing nonparametric versions of the following models: multivariate regression, the proportional and nonproportional odds model, the multiple logistic regression model, and bivariate binary regression models such as bivariate probit model and the bivariate logistic model. Some theoretical properties of vector GAMs are deduced from those pertaining to ordinary GAMs, and its relationship with the generalized estimating equations (GEE) approach elucidated.Whole document restricted, but available by request, use the feedback form to request access.
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Byard, Kevin. "Qualified difference sets : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand." Massey University, 2009. http://hdl.handle.net/10179/1204.
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Qualified difference sets are a class of combinatorial configuration. The sets are related to the residue difference sets that were first discussed in detail in 1953 by Emma Lehmer. Qualified difference sets consist of a set of residues modulo an integer v and they possess attractive properties that suggest potential applications in areas such as image formation, signal processing and aperture synthesis. This thesis outlines the theory behind qualified difference sets and gives conditions for the existence and nonexistence of these sets in various cases. A special case of the qualified difference sets is the qualified residue difference sets. These consist of the set of nth power residues of certain types of prime. Necessary and sufficient conditions for the existence of qualified residue difference sets are derived and the precise conditions for the existence of these sets are given for n = 2, 4 and 6. Qualified residue difference sets are proved nonexistent for n = 8, 10, 12, 14 and 18. A generalisation of the qualified residue difference sets is introduced. These are the qualified difference sets composed of unions of cyclotomic classes. A cyclotomic class is defined for an integer power n and the results of an exhaustive computer search are presented for n = 4, 6, 8, 10 and 12. Two new families of qualified difference set were discovered in the case n = 8 and some isolated systems were discovered for n = 6, 10 and 12. An explanation of how qualified difference sets may be implemented in physical applications is given and potential applications are discussed.
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O'Neale, Dion Robert James. "Preservation of phase space structure in symplectic integration : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand." Massey University, 2009. http://hdl.handle.net/10179/1127.
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This thesis concerns the study of geometric numerical integrators and how they preserve phase space structures of Hamiltonian ordinary differential equations. We examine the invariant sets of differential equations and investigate which numerical integrators preserve these sets, and under what conditions. We prove that when periodic orbits of Hamiltonian differential equations are discretized by a symplectic integrator they are preserved in the numerical solution when the integrator step size is not resonant with the frequency of the periodic orbit. The preservation of periodic orbits is the result of a more general theorem which proves preservation of lower dimensional invariant tori from dimension zero (fixed points) up to full dimension (the same as the number of degrees of freedom for the differential equation). The proof involves first embedding the numerical trajectory in a non-autonomous flow and then applying a KAM type theorem for flows to achieve the result. This avoids having to prove a KAM type theorem directly for the symplectic map which is generally difficult to do. We also numerically investigate the break up of periodic orbits when the integrator's step size is resonant with the frequency of the orbit. We study the performance of trigonometric integrators applied to highly oscillatory Hamiltonian differential equations with constant frequency. We show that such integrators may not be as practical as was first thought since they suffer from higher order resonances and can perform poorly at preserving various properties of the di fferential equation. We show that, despite not being intended for such systems, the midpoint rule performs no worse than many of the trigonometric integrators, and indeed, better than some. Lastly, we present a numerical study of a Hamiltonian system consisting of two magnetic moments in an applied magnetic field. We investigate the effect of both the choice of integrator and the choice of coordinate system on the numerical solutions of the system. We show that by a good choice of integrator (in this case the generalised leapfrog method) one can preserve phase space structures of the system without having to resort to a change of coordinates that introduce a coordinate singularity.
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Cooper, Haydn. "Two generator discrete groups of isometries and their representation : a thesis presented in partial fulfillment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand." Massey University, 2008. http://hdl.handle.net/10179/973.
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Let M Φ and Mψ be elements of PSL(2,C) representing orientation preserving isometries on the upper half-space model of hyperbolic 3-space Φ and ψ respectively. The parameters β = tr2(M Φ) - 4, β1 = tr2(Mψ) - 4, γ = tr[M Φ,Mψ] - 2, determine the discrete group (Φ ,ψ) uniquely up to conjugacy whenever γ ≠ 0. This thesis is concerned with explicitly lifting this parameterisation of (Φ , ψ) to PSO(1, 3) realised as a discrete 2 generator subgroup of orientation preserving isometries on the hyperboloid model of hyperbolic 3-space. We particularly focus on the case where both Φ and ψ are elliptic.
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Aryal, Jagannath. "Optimisation of a buyer’s sourcing strategy in the mixed auction/direct supply of New Zealand wool." Lincoln University, 2009. http://hdl.handle.net/10182/1548.
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The New Zealand Wool Industry (NZWI) contributes over a billion dollars a year to NZ gross output. However, this industry is at a crossroads and the incumbent practitioners are looking for ways to increase the value of the New Zealand wool clip. The value of the industry to the economy is directly related to the price which buyers are prepared to pay for wool, primarily as a result of the marketing approaches used, physical parameters of wool as well as intra and inter-fibre competition. The inflation adjusted price has steadily decreased over recent years and understanding of its dynamics is a fundamental problem for the stakeholders. Among the stakeholders, buyers / exporters, heavily involved in the process of price formation currently face a real time problem of sourcing strong wool from two parallel but different marketing systems operated simultaneously – auction and direct supply. The underlying mathematics which governs the decision making of buyers on the price dynamics in these sourcing options is poorly understood. This study developed system models for price formation in both auction and direct supply sourcing and an associated optimization model for the buyer / exporter of the New Zealand wool clip. All three of these models were original and none appear to have been described previously. It is hoped that these three models will be of quite general utility and also be useful therefore for other agricultural commodities that are traded simultaneously via auction and direct supply. The average price for a given wool type, which is the output from this new modelling exercise is precisely what is required as input data for solving the minimization problem in wool blending models.
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Zhou, Zhongkun. "Data analysis and preliminary model development for an odour detection system based on the behaviour of trained wasps." Diss., Lincoln University, 2008. http://hdl.handle.net/10182/654.
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Microplitis croceipes, one of the nectar feeding parasitoid wasps, has been found to associatively learn chemical cues through feeding. The experiments on M. croceipes are performed and recorded by a Sony camcorder in the USDA-ARS Biological Control Laboratory in Tifton, GA, USA. The experimental videos have shown that M. croceipes can respond to Coffee odour in this study. Their detection capabilities and the behaviour of M. croceipes with different levels of coffee odours were studied. First, the data that are related to trained M. croceipes behaviour was extracted from the experimental videos and stored in a Microsoft Excel database. The extracted data represent the behaviour of M. croceipes trained to 0.02g and then exposed to 0.001g, 0.005g, 0.01g, 0.02g and 0.04g of coffee. Secondly, indices were developed to uniquely characterise the behaviour of trained M. croceipes under different coffee concentrations. Thirdly, a preliminary model and its parameters were developed to classify the response of trained wasps when exposed to these five different coffee odours. In summary, the success of this thesis demonstrates the usefulness of data analysis for interpreting experimental data, developing indices, as well as understanding the design principles of a simple model based on trained wasps.
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Schliep, Klaus Peter. "Some applications of statistical phylogenetics : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Biomathematics at Massey University." Massey University, 2009. http://hdl.handle.net/10179/931.
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The increasing availability of molecular data means that phylogenetic studies nowadays often use datasets which combine a large number of loci for many different species. This leads to a trade-off. On the one hand more complex models are preferred to account for heterogeneity in evolutionary processes. On the other hand simple models that can answer biological questions of interest that are easy to interpret and can be computed in reasonable time are favoured. This thesis focuses on four cases of phylogenetic analysis which arise from this conflict. - It is shown that edge weight estimates can be non-identifiable if the data are simulated under a mixture model. Even if the underlying process is known the estimation and interpretation may be difficult due to the high variance of the parameters of interest. - Partition models are commonly used to account for heterogeneity in data sets. Novel methods are presented here which allow grouping of genes under similar evolutionary constraints. A data set, containing 14 genes of the chloroplast from 19 anciently diverged species is used to find groups of co-evolving genes. The prospects and limitations of such methods are discussed. - Penalised likelihood estimation is a useful tool for improving the performance of models and allowing for variable selection. A novel approach is presented that uses pairwise dissimilarities to visualise the data as a network. It is further shown how penalised likelihood can be used to decrease the variance of parameter estimates for mixture and partition models, allowing a more reliable analysis. Estimates for the variance and the expected number of parameters of penalised likelihood estimates are derived. - Tree shape statistics are used to describe speciation events in macroevolution. A new tree shape statistic is introduced and the biases of different cluster methods on tree shape statistics are discussed.
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Grievink, Liat Shavit. "Lineage specific evolution and phylogenetic analysis : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Biomathematics at Massey University, Palmerston North, New Zealand." Massey University, 2009. http://hdl.handle.net/10179/1048.
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Phylogenetic models generally assume a homogeneous, time reversible, stationary process. These assumptions are often violated by the real, far more complex, evolutionary process. This thesis is centered on non-homogeneous, lineage-specific, properties of molecular sequences. It consist several related but independent studies. LineageSpecificSeqgen, an extension to the Seq-Gen program, which allows generation of sequences with changes in the proportion of variable sites, is introduced. This program is then used in a simulation study showing that changes in the proportion of variable sites can hinder tree estimation accuracy, and that tree reconstruction under the best-fit model chosen using a relative test can result in a wrong tree. In this case, the less commonly used absolute model-fit was a better predictor of tree estimation accuracy. This study found that increased taxon sampling of lineages that have undergone a change in the proportion of variable sites was critical for accurate tree reconstruction and that, in contrast to some earlier findings, the accuracy of maximum parsimony is adversely affected by such changes. This thesis also addresses the well-known long-branch attraction artifact. A nonparametric bootstrap test to identify changes in the substitution process is introduced, validated, and applied to the case of Microsporidia, a highly reduced intracellular parasite. Microsporidia was first thought to be an early branching eukaryote, but is now believed to be sister to, or included within, fungi. Its apparent basal eukaryote position is considered a result of long-branch attraction due to an elevated evolutionary rate in the microsporidian lineage. This study shows that long-branch estimates and basal positioning of Microsporidia both correlate with increased proportions of radical substitutions in the microsporidian lineage. In simulated data, such increased proportions of radical substitutions leads to erroneous long-branch estimates. These results suggest that the long microsporidian branch is likely to be a result of an increased proportion of radical substitutions on that branch, rather than increased evolutionary rate per se. The focus of the last study is the intriguing case of Mesostigma, a fresh water green alga for which contradicting phylogenetic relationships were inferred. While some studies placed Mesostigma within the Streptophyta lineage (which includes land plants), others placed it as the deepest green algae divergence. This basal positioning is regarded as a result of long-branch attraction due to poor taxon sampling. Reinvestigation of a 13- taxon mitochondrial amino acid dataset and a sub-dataset of 8 taxa reveals that site sampling, and in particular the treatment of missing data, is just as important a factor for accurate tree reconstruction as taxon sampling. This study identifies a difficulty in recreating the long-branch attraction observed for the 8-taxon dataset in simulated data. The cause is likely to be the smaller number of amino acid characters per site in simulated data compared to real data, highlighting the fact that there are properties of the evolutionary process that are yet to be accurately modeled.
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Nguyen, Lan K. "Dynamical modelling of feedback gene regulatory networks." Diss., Lincoln University, 2009. http://hdl.handle.net/10182/1340.
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Living cells are made up of networks of interacting genes, proteins and other bio-molecules. Simple interactions between network components in forms of feedback regulations can lead to complex collective dynamics. A key task in cell biology is to gain a thorough understanding of the dynamics of intracellular systems and processes. In this thesis, a combined approach of mathematical modelling, computational simulation and analytical techniques, has been used to obtain a deeper insight into the dynamical aspects of a variety of feedback systems commonly encountered in cells. These systems range from model system with detailed available molecular knowledge to general regulatory motifs with varying network structures. Deterministic as well as stochastic modelling techniques have been employed, depending primarily on the specific questions asked. The first part of the thesis focuses on dissecting the principles behind the regulatory design of the Tryptophan Operon system in Escherichia coli. It has evolved three negative feedback loops, namely repression, attenuation and enzyme inhibition, as core regulator mechanisms to control the intracellular level of tryptophan amino acid, which is taken up for protein synthesis. Despite extensive experimental knowledge, the roles of these seemingly redundant loops remain unclear from a dynamical point of view. We aim to understand why three loops, rather than one, have evolved. Using a large-scale perturbation/response analysis through modelling and simulations and novel metrics for transient dynamics quantification, it has been revealed that the multiple negative feedback loops employed by the tryptophan operon are not redundant. In fact, they have evolved to concertedly give rise to a much more efficient, adaptive and stable system, than any single mechanism would provide. Since even the full topology of feedback interactions within a network is insufficient to determine its behavioural dynamics, other factors underlying feedback loops must be characterised to better predict system dynamics. In the second part of the thesis, we aim to derive these factors and explore how they shape system dynamics. We develop an analytical approach for stability and bifurcation analysis and apply it to class of feedback systems commonly encountered in cells. Our analysis showed that the strength and the Hill coefficient of a feedback loop play key role in determining the dynamics of the system carrying the loop. Not only that, the position of the loop was also found to be crucial in this decision. The analytical method we developed also facilitates parameter sensitivity analysis in which we investigate how the production and degradation rates affect system dynamics. We find that these rates are quite different in the way they shape up system behaviour, with the degradation rates exhibiting a more intricate manner. We demonstrated that coupled-loop systems display greater complexity and a richer repertoire of behaviours in comparison with single-loop ones. Different combinations of the feedback strengths of individual loops give rise to different dynamical regimes. The final part of the thesis aims to understand the effects of molecular noise on dynamics of specific systems, in this case the Tryptophan Operon. We developed two stochastic models for the system and compared their predictions to those given by the deterministic model. By means of simulations, we have shown that noise can induce oscillatory behaviour. On the other hand, incorporating noise in an oscillatory system can alter the characteristics of oscillation by shifting the bifurcation point of certain parameters by a substantial amount. Measurement of fluctuations reveals that that noise at the transcript level is most significant while noise at the enzyme level is smallest. This study highlights that noise should not be neglected if we want to obtain a complete understanding of the dynamic behaviour of cells.
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Kean, J. M. "Metapopulation theory in practice." Lincoln University, 1999. http://hdl.handle.net/10182/1372.
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A metapopulation is defined as a set of potential local populations among which dispersal may occur. Metapopulation theory has grown rapidly in recent years, but much has focused on the mathematical properties of metapopulations rather than their relevance to real systems. Indeed, barring some notable exceptions, metapopulation theory remains largely untested in the field. This thesis investigates the importance of metapopulation structure in the ‘real world’, firstly by building additional realism into metapopulation models, and secondly through a 3-year field study of a real metapopulation system. The modelling analyses include discrete-and continuous-time models, and cover single species, host-parasitoid, and disease-host systems, with and without stochasticity. In all cases, metapopulation structure enhanced species persistence in time, and often allowed long-term continuance of otherwise non-persistent interactions. Spatial heterogeneity and patterning was evident whenever local populations were stochastic or deterministically unstable in isolation. In metapopulations, the latter case often gave rise to self-organising spatial patterns. These were composed of spiral wave fronts (or ‘arcs of infection’ in disease models) of different sizes, and were related to the stability characteristics of local populations as well as the dispersal rates. There was no evidence for self-organising spatial patterns in the host-parasitoid system studied in the field (the weevil Sitona discoideus and its braconid parasitoid Microctonus aethiopoides), and a new model for the interaction suggested that this is probably due to the strong host density-dependence and stabilising parasitism acting on local populations. Dispersal may be important because of very high mortality in dispersing weevils, which may be related to the scarcity of their host plant in the landscape. If this is the case, the model suggested that local weevil density may be sensitive to the area of crop grown. Stochastic models showed that species in suitably large metapopulations may persist for very long times at relatively low overall density and with very low incidence of density-dependence. This suggests that metapopulation processes may explain a general inability to detect density-dependence in many real populations, and may also play an important part in the persistence of rare species. For host-parasitoid metapopulation models, persistence often depended on the way in which they were initialised. Initial conditions corresponding to a biological control release were the least likely to persist, and the maximum host suppression observed in this case was 84%, as compared with 60% for the corresponding non-spatial models and >90% often observed in the field. Metapopulation structure also allowed persistence of ‘boom-bust’ disease models, although the dynamics of these were particularly dependent on assumptions about what happens to disease classes at very low densities. Models assuming infinitely divisible units of density, models incorporating a non-zero extinction threshold, and individual-based models all gave differing results in terms of disease persistence and rate of spatial spread. Fitting models to overall metapopulation dynamics often gave misleading results in terms of underlying local dynamics, emphasising the need to sample real populations at an appropriate scale when seeking to understand their behaviour.
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Brown, Samuel David James. "Molecular systematics and colour variation of Carpophilus species (Coleoptera: Nitidulidae) of the South Pacific." Diss., Lincoln University, 2009. http://hdl.handle.net/10182/1430.
Full textAbstract:
The sap beetle genus Carpophilus Stephens (Coleoptera: Nitidulidae) is a large genus consisting of over 200 species and are found worldwide. Several species are important pests of crops and stored products, and are frequently intercepted as part of biosecurity operations. The genus is poorly known taxonomically, and there are several species groups that are challenging to identify by morphological methods. In particular, two species found across the Pacific, C. maculatus Murray and C. oculatus Murray are frequently confused with each other. These two species are similar in size and colour, but differ primarily by the shape of the colour pattern on their elytra. However, this colour pattern is highly variable within both species, leading to ambiguity in the indentification of these species. Within C. oculatus, three subspecies have been described based on differences in the male genitalia and pronotal punctation: C. o. oculatus and C. o. gilloglyi Dobson are distributed widely across the Pacific, while C. o. cheesmani Dobson is known only from Vanuatu. A search of literature records and specimen collections revealed 32 species of Carpophilus recorded from the Pacific region. In addition there remain several unidentified specimens representing at least four species, two of which will be described subsequent to this research. A number of species recorded in the literature may have been misidentified, and these require further field collections and inspection of museum specimens to confirm their presence in the Pacific. To test the validity of the subspecies of C. oculatus, and its distinctiveness from C. maculatus, a phylogeny of available specimens of Carpophilus was inferred from one mitochondrial gene (cytochrome c oxidase subunit I (COI)), and two nuclear genes (28S ribsomal RNA (28S) and the internal transcribed spacer 2 (ITS2)). These data show large genetic distances between the three subspecies of C. oculatus of 7-12%. Given these distances are similar to those between other species in the genus, this indicates these subspecies may be elevated to full species. The data also consistently support a monophyletic relationship between C. o. oculatus and C. o. gilloglyi. Nuclear genes also support C. o. cheesmani as part of a clade with the other subspecies, but these relationships are unresolved in COI. Carpophilus maculatus was not supported as being the sister taxon of the C. o. oculatus and C. o. gilloglyi clade. Other relationships within Carpophilus were unresolved, possibly due to a combination of incomplete taxon sampling, and saturation of substitutions within the COI gene. Phylogeographic analysis of specimens collected from several localities within the range of C. oculatus showed that, with only one exception, there were no shared haplotypes between archipelagoes. This result suggests it may be possible to determine the provenence of intercepted specimens, providing further information regarding potential invasion pathways. A degree of geographic structuring was also present within C. o. gilloglyi, being separated into a western clade found in Fiji and Rotuma and an eastern clade distributed from the Kermadec Islands and Tonga to French Polynesia. This separation was most profound in COI data, with a mean pairwise distance between the clades of 7%. ITS2 data also demonstrates a degree of differentiation between the two clades, based on differences in the insertions and deletions between the clades. The variability in the shape and colour of the elytral pattern of C. oculatus was also investigated. Colour was quantified using a method based on Red-Green-Blue (RGB) colour values derived from digital photographs, while an outline analysis of the elytral pattern was conducted using elliptic Fourier analysis (EFA). Principal Components Analysis of the RGB values and EFA coefficients showed no clear separation between subspecies, nor were any trends correlated with host fruit or collection localities. Variation at all levels and all measures studied in this thesis show that this geographic region and this genus of beetles offer intruiging insights into speciation, biogeography and biological invasions. There is much scope for further research on the causes and consequences of this variation and the lives of these interesting insects.
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Connell, R. J. "Unstable equilibrium : modelling waves and turbulence in water flow." Diss., Lincoln University, 2008. http://hdl.handle.net/10182/592.
Full textAbstract:
This thesis develops a one-dimensional version of a new data driven model of turbulence that uses the KL expansion to provide a spectral solution of the turbulent flow field based on analysis of Particle Image Velocimetry (PIV) turbulent data. The analysis derives a 2nd order random field over the whole flow domain that gives better turbulence properties in areas of non-uniform flow and where flow separates than the present models that are based on the Navier-Stokes Equations. These latter models need assumptions to decrease the number of calculations to enable them to run on present day computers or super-computers. These assumptions reduce the accuracy of these models. The improved flow field is gained at the expense of the model not being generic. Therefore the new data driven model can only be used for the flow situation of the data as the analysis shows that the kernel of the turbulent flow field of undular hydraulic jump could not be related to the surface waves, a key feature of the jump. The kernel developed has two parts, called the outer and inner parts. A comparison shows that the ratio of outer kernel to inner kernel primarily reflects the ratio of turbulent production to turbulent dissipation. The outer part, with a larger correlation length, reflects the larger structures of the flow that contain most of the turbulent energy production. The inner part reflects the smaller structures that contain most turbulent energy dissipation. The new data driven model can use a kernel with changing variance and/or regression coefficient over the domain, necessitating the use of both numerical and analytical methods. The model allows the use of a two-part regression coefficient kernel, the solution being the addition of the result from each part of the kernel. This research highlighted the need to assess the size of the structures calculated by the models based on the Navier-Stokes equations to validate these models. At present most studies use mean velocities and the turbulent fluctuations to validate a models performance. As the new data driven model gives better turbulence properties, it could be used in complicated flow situations, such as a rock groyne to give better assessment of the forces and pressures in the water flow resulting from turbulence fluctuations for the design of such structures. Further development to make the model usable includes; solving the numerical problem associated with the double kernel, reducing the number of modes required, obtaining a solution for the kernel of two-dimensional and three-dimensional flows, including the change in correlation length with time as presently the model gives instant realisations of the flow field and finally including third and fourth order statistics to improve the data driven model velocity field from having Gaussian distribution properties. As the third and fourth order statistics are Reynolds Number dependent this will enable the model to be applied to PIV data from physical scale models. In summary, this new data driven model is complementary to models based on the Navier-Stokes equations by providing better results in complicated design situations. Further research to develop the new model is viewed as an important step forward in the analysis of river control structures such as rock groynes that are prevalent on New Zealand Rivers protecting large cities.
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Paulin, M. G. (Michael Geoffrey). "A mathematical and comparative study on cerebellar control of vestibular reflexes." Thesis, University of Auckland, 1985. http://hdl.handle.net/2292/2041.
Full textAbstract:
The first aim of this thesis is an introduction to some basic aspects of multivariate control theory which are relevant to the question of how the brain controls movements. A regulator is a device which forces a system to follow a specified trajectory in the presence of perturbations which might cause it to diverge from that trajectory. Regulation involves constructing an additional control input which depends upon the difference between the actual system state and the desired state. This requires the construction of a state estimate from raw data about system input and output. For effective state estimation, the sensor input gain to the state estimator needs to be time-varying. Under certain assumptions, the appropriate input gain can be specified analytically. The feedback regulation signal can then be constructed as a function of the state estimate. For effective regulation, the gain of the feedback function has to vary during maneuvers. Under certain assumptions an appropriate feedback gain can be specified analytically. The state observer input gain equations have a simple relationship to the feedback gain equations, so that gain specification is essentially the same task in each case. Cerebellar research has been dominated for the past 25 years by the theories of James Albus and David Marr. These mathematicians proposed similar models in which certain synapses in the cerebellar cortex are continuously modified by experience in such a way that movements which are consistently repeated under a given set of circumstances come to be performed automatically by the cerebellum. Much experimental work has focussed on the role of the vestibulo-cerebellum in fine control and learning of the vestibulo-ocular reflex. The state of the art along this line is formally described by Fujita's adaptive filter model of the cerebellar cortex. In chapter 4 it is shown that a basic feature of Fujita's model is inconsistent with available evidence. The 'Tensorial theory of brain function' is discussed in chapter 5. This is a novel theory of brain function which has been used in an attempt two explain cerebellar function. The attempt is a failure, based on sophistocated misconceptions and flawed by poor reasoning and clumsy analysis. The approach serves to confuse rather than clarify the question of cerebellar function. The final chapter of the first part of the thesis presents a basis for a new approach to cerebellar function based on the engineering theory of control of multivariate dynamical systems. It is proposed that the cerebellum is involved in movement regulation by controlling the gains of brainstem motor pathways, and in mapping the animal's environment by controlling the gains of sensory inputs to the midbrain. While learning undoubtedly does occur in the cerebellar cortex, this is not specifically a 'learning device', as commonly conceived. The second part of the thesis is concerned with the development and application of a method of system identification for characterising the dynamics of the vestibulo-ocular reflex and its components in an elasmobranch. The chosen method involves pulse-rate modulated bilateral electrical stimulation of the horizontal semicircular canal ampullary nerves. This produces a synthetic vestibulo-ocular reflex in a stationary preparation. The stimulus pattern is a pseudorandom binary sequence of pulse rates, so that cross-correlation of the stimulus pattern with the response signal gives a Unit Impulse Response dynamic signature for the system. Computer software for signal generation, recording, analysis and display was written by the author. The identification system was applied first to characterise the dynamics of the eye movement response to horizontal canal ampullary nerve stimulation, and compare this to the dynamics of the eye motor plant alone. The eye motor preparation acts as a first order low-pass filter with a time constant of about 0.2 seconds (16°C), while the ampullary preparation acts as a second order low-pass filter with a dominant time constant of about 0.75 seconds (16°C). Central pathways of the elasmobranch vestibulo-ocular reflex extend the time constant of the motor plant by a factor of 3-4, as in other animals. Eye movements predicted by fitted linear models accurately mimic eye movements recorded during experiments, suggesting both that central pathways of the reflex operate normally during this somewhat un-naturally evoked response and that the identification procedure is effective. Furthermore, combination of the ampullary nerve to eye movement transfer function obtained in this study, with head rotation to ampullary nerve transfer functions obtained by other workers, gives a consistent picture of elasmobranch vestibulo-ocular reflex function predicting compensatory eye movements in the band 0.2 - 4.0 Hz., and perhaps higher. The identification method has also been applied to produce models of vestibulocerebellar Purkinje cell dynamics during electrically evoked vestibular eye movements. Linear identification gives a poor characterisation of Purkinje cell activity during the high frequency vestibulo-ocular reflex. This is incompatible with linear phase-compensator models of the cerebellar cortex, but consistent with the reflex gain modulation theory of cerebellar function advocated in the first part of the thesis.
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Zhao, Yue. "Modelling avian influenza in bird-human systems : this thesis is presented in the partial fulfillment of the requirement for the degree of Masters of Information Science in Mathematics at Massey University, Albany, New Zealand." Massey University, 2009. http://hdl.handle.net/10179/1145.
Full textAbstract:
In 1997, the first human case of avian influenza infection was reported in Hong Kong. Since then, avian influenza has become more and more hazardous for both animal and human health. Scientists believed that it would not take long until the virus mutates to become contagious from human to human. In this thesis, we construct avian influenza with possible mutation situations in bird-human systems. Also, possible control measures for humans are introduced in the systems. We compare the analytical and numerical results and try to find the most efficient control measures to prevent the disease.
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Xie, Zhi. "Modelling genetic regulatory networks: a new model for circadian rhythms in Drosophila and investigation of genetic noise in a viral infection process." Phd thesis, Lincoln University. Agriculture and Life Sciences Division, 2007. http://theses.lincoln.ac.nz/public/adt-NZLIU20070712.144258/.
Full textAbstract:
In spite of remarkable progress in molecular biology, our understanding of the dynamics and functions of intra- and inter-cellular biological networks has been hampered by their complexity. Kinetics modelling, an important type of mathematical modelling, provides a rigorous and reliable way to reveal the complexity of biological networks. In this thesis, two genetic regulatory networks have been investigated via kinetic models.
In the first part of the study, a model is developed to represent the transcriptional regulatory network essential for the circadian rhythms in Drosophila. The model incorporates the transcriptional feedback loops revealed so far in the network of the circadian clock (PER/TIM and VRI/PDP1 loops). Conventional Hill functions are not used to describe the regulation of genes, instead the explicit reactions of binding and unbinding processes of transcription factors to promoters are modelled. The model is described by a set of ordinary differential equations and the parameters are estimated from the in vitro experimental data of the clocks components. The simulation results show that the model reproduces sustained circadian oscillations in mRNA and protein concentrations that are in agreement with experimental observations. It also simulates the entrainment by light-dark cycles, the disappearance of the rhythmicity in constant light and the shape of phase response curves resembling that of experimental results. The model is robust over a wide range of parameter variations. In addition, the simulated E-box mutation, perS and perL mutants are similar to that observed in the experiments. The deficiency between the simulated mRNA levels and experimental observations in per01, tim01 and clkJrk mutants suggests some differences in the model from reality. Finally, a possible function of VRI/PDP1 loops is proposed to increase the robustness of the clock.
In the second part of the study, the sources of intrinsic noise and the influence of extrinsic noise are investigated on an intracellular viral infection system. The contribution of the intrinsic noise from each reaction is measured by means of a special form of stochastic differential equation, the chemical Langevin equation. The intrinsic noise of the system is the linear sum of the noise in each of the reactions. The intrinsic noise arises mainly from the degradation of mRNA and the transcription processes. Then, the effects of extrinsic noise are studied by means of a general form of stochastic differential equation. It is found that the noise of the viral components grows logarithmically with increasing noise intensities. The system is most susceptible to noise in the virus assembly process. A high level of noise in this process can even inhibit the replication of the viruses.
In summary, the success of this thesis demonstrates the usefulness of models for interpreting experimental data, developing hypotheses, as well as for understanding the design principles of genetic regulatory networks.
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Stewart, Sepideh. "Understanding linear algebra concepts through the embodied, symbolic and formal worlds of mathematical thinking." 2008. http://hdl.handle.net/2292/2912.
Full textAbstract:
Linear algebra is one of the first advanced mathematics courses that students encounter at university level. The transfer from a primarily procedural or algorithmic school approach to an abstract and formal presentation of concepts through concrete definitions, seems to be creating difficulty for many students who are barely coping with procedural aspects of the subject. This research proposes applying APOS theory, in conjunction with Tall’s three worlds of embodied, symbolic and formal mathematics, to create a framework in order to examine the learning of a variety of linear algebra concepts by groups of first and second year university students. The aim is to investigate the difficulties in understanding some linear algebra concepts and to propose potential paths for preventing them. As part of this research project several case studies were conducted where groups of first and second year students were exposed to teaching and learning some introductory linear algebra concepts based on the framework and expressed their thinking through their involvements in tests, interviews and concept maps. The results suggest that the students had limited understanding of the concepts, they struggled to recognise the concepts in different registers, and their lack of ability in linking the major concepts became apparent. However, they also revealed that those with more representational diversity had more overall understanding of the concepts. In particular the embodied introduction of the concept proved a valuable adjunct to their thinking. Since difficulties with learning linear algebra by average students are universally acknowledged, it is anticipated that this study may provide suggestions with the potential for widespread positive consequences for learning.
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Delos, Santos Alan Gil Gutierrez. "An investigation of students’ understanding and representation of derivative in a graphic calculator-mediated teaching and learning environment." 2006. http://hdl.handle.net/2292/2161.
Full textAbstract:
This research is a collective case study that investigates the constitutive relationship between students' representational competences and mathematical understanding of derivative. Its goal was to describe the representational abilities characterising different ways of knowing, and these were categorised as procedure-oriented, process-oriented, object-oriented, concept-oriented and versatile. The study was conducted in four Form 7 classrooms, in their Mathematics with Calculus classes, where graphic calculators were used in the teaching and learning of derivative. The choice of the context was based on the belief that the use of graphic calculators might encourage a multi-representational approach to teaching, and support the development of students' multi-representational way of thinking. The research data were collected both from teachers and their students. These data comprise teacher and student interviews, with the first interviews conducted before their lessons on derivative and the second after the lesson. The students were also given pre-lesson and post-lesson tests in order to triangulate student data. A Representational Framework of Knowing Derivative was constructed as an analysis tool, and used to explore students' representational abilities and ways of knowing. From the analysis, the students' cognitive processes were construed, together with the nature of their representational, cognitive and conceptual schemas. The representational framework of knowing was later refined to present an empirically-based theoretical framework that bridges the gap between what was theorised and what was observed. The results of the study suggest that the relationship between students' ways of knowing and their representational abilities is mediated by the following factors: (i) the students' interpretation of the mathematical notion; (ii) the representational nature of their interpretations of derivative, and the representational aspects in their problem solving activities; and (iii) the nature of the representational links that they have formed between procedures, processes, objects and sub-concepts that were construed to constitute their conceptual and cognitive schemas of derivative. With regard to the use of the graphic calculator, this research has noted a possible contribution of the graphic calculator in the development of students' multi-representational ways of thinking and learning.
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Rattenbury, Nicolette. "Almost Runge-Kutta methods for stiff and non-stiff problems." Thesis, 2005. http://hdl.handle.net/2292/2276.
Full textAbstract:
Ordinary differential equations arise frequently in the study of the physical world. Unfortunately many cannot be solved exactly. This is why the ability to solve these equations numerically is important. Traditionally mathematicians have used one of two classes of methods for numerically solving ordinary differential equations. These are linear multistep methods and Runge–Kutta methods. General linear methods were introduced as a unifying framework for these traditional methods. They have both the multi-stage nature of Runge–Kutta methods as well as the multi-value nature of linear multistep methods. This extremely broad class of methods, besides containing Runge–Kutta and linear multistep methods as special cases, also contains hybrid methods, cyclic composite linear multistep methods and pseudo Runge–Kutta methods. In this thesis we present a class of methods known as Almost Runge–Kutta methods. This is a special class of general linear methods which retains many of the properties of traditional Runge–Kutta methods, but with some advantages. Most of this thesis concentrates on explicit methods for non-stiff differential equations, paying particular attention to a special fourth order method which, when implemented in the correct way, behaves like order five. We will also introduce low order diagonally implicit methods for solving stiff differential equations.
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Oates, Greg. "Integrated technology in the undergraduate mathematics curriculum : a case study of computer algebra systems." 2009. http://hdl.handle.net/2292/4533.
Full textAbstract:
The effective integration of technology into the teaching and learning of mathematics remains one of the critical challenges facing tertiary mathematics, which has traditionally been slow to respond to technological innovation. This thesis reveals that the term integration is widely used in the literature with respect to technology and the curriculum, although its meaning can vary substantially, and furthermore, the term is seldom well defined. A review of the literature provides the basis for a survey of undergraduate mathematics educators, to determine their use of technology, their views of what an Integrated Technology Mathematics Curriculum (ITMC) may resemble, and how it may be achieved. Responses to this survey, and factors identified in the literature, are used to construct a taxonomy of integrated technology. The taxonomy identifies six defining characteristics of an ITMC, each with a number of associated elements. A visual model using radar diagrams is developed to compare courses against the taxonomy, and to identify aspects needing attention in individual courses. T Evidence from an observational study of initiatives to introduce Computer Algebra Systems into undergraduate mathematics courses at The University of Auckland, firstly using CAS-calculators and latterly computer software, is examined against the taxonomy. A number of critical issues influencing the integration of these technologies are identified. These include mandating technology use in official departmental policy, attention to congruency and fairness in assessment, re-evaluating the value of topics in the curriculum, re-establishing the goals of undergraduate courses, and developing the pedagogical technical knowledge of teaching staff. The thesis concludes that effective integration of technology in undergraduate mathematics requires a recognition of, and comprehensive attention to, the interdependence of the taxonomy components. An integrated, holistic approach, which aims for curricular congruency across all elements of the taxonomy, provides the basis for a more consistent, effective and sustainable ITMC.
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Holden, Jennifer Kay. "New methods for analysing generalised linear models with applications to epidemiology." 2001. http://hdl.handle.net/2292/2061.
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Whole document restricted, see Access Instructions file below for details of how to access the print copy.The aim of capture-recapture methods in epidemiology is to accurately estimate the total number of people who have a specific disease. These methods were first developed by ecologists interested in estimating the total population size of various animal species. Capture-recapture methods have a relatively short history, at least in terms of application to epidemiological data sets. If applied correctly they can be of great benefit, and are invaluable for planning and resource allocation. The aim of this thesis is to enhance the existing methods used in epidemiological capture-recapture problems. This research explores new methods for analysing generalised linear models, within the capture-recapture framework, with applications to epidemiology. In particular, we critically examine two New Zealand data sets. We compare two small sample adjustments for capture-recapture methods, and find that the Evans and Bonett adjustment can be a useful tool for sparse data. We employ stratified capture-recapture analyses to alleviate problems with heterogeneity and reporting patterns. In addition, we consider a type of cost-benefit analysis for the reporting sources. Two proposed methods of internal validity analysis are scrutinised. We find that one of these is counter-intuitive and of no use, however, the other method may be of some use in at least indicating the direction of any bias in the capture-recapture estimates. We use simulation to explore the effects of errors on patient records, and find that even relatively small percentages of errors can affect estimates dramatically. We conclude with a study of the optimal number of sources to use in epidemiological capture-recapture analyses. We argue that using three sources is not necessarily optimal, and that using four sources is also entirely manageable. This thesis outlines a strategy for analysing epidemiological data sets using capture-recapture methods, and includes aspects of model fitting and selection, cost-benefit analysis, diagnostic checking through simulations of the effects of record errors, and the effects of collapsing lists, as well as a critical check of the capture-recapture assumptions. This investigation demonstrates the potential of capture-recapture methods to provide accurate estimates of the size of various disease populations.
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Sheu, Ru-Shuo. "Queueing and Storage Control Models." 2002. http://hdl.handle.net/2292/2316.
Full textAbstract:
Whole document restricted at the request of the authorThis thesis is divided into two parts. The first part is about the control of a special queueing network which has two service nodes in tandem on each service channel. With capacity at each service node being finite, we compare som different control policies to find the admission and routing policies that minimise the blocking rate in the queueing system. We obtain limit theorems as the number of channels becomes large. The stochastic optimization technique we apply here is the Lagrangian method, using the Complementary Slackness Conditions to choose the optimal action. In the second part we consider two reservoir control problems. In the first, the cost function is a single simple linear function, and the second has two different cost functions and the choice of them forms a finite-state Markov chain. We find the optimal policies to determine how many units of water should be released from the reservoir under these two different models. We model the reservoir as a Markov decision process. The policy-iteration algorithm and the value-iteration algorithm are the main methods we apply in this part. In both problems we apply stochastic optimization techniques. The reservoir model uses a standard Markov decision process model, with the associated methods of policy-iteration and value-iteration to find the optimal state-dependant policy. In the routing problem we also interested in state-dependent policies, but here we wish to look at the system in the limit as the number of queues becomes large, so we can no longer us the technique of Markov decision processes. We look, instead, at the limiting deterministic problem to find the optimal policy.
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Smith, Thomasin Ann. "Mathematical modelling of underground flow processes in hydrothermal eruptions : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Palmerston North, New Zealand." 2000. http://hdl.handle.net/10179/1303.
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This thesis reports on a study of underground fluid flow and boiling processes which take place in hydrothermal eruptions. A conceptual model is presented for the eruptive process and a laboratory scale physical model confirming the effectiveness of this process is described. A mathematical formulation of the underground flow problem is given for two fluid flow regimes: two-phase homogeneous mixture (HM) flow and separable two-phase (SP) flow. Solutions to the system of equations obtained are solved under the simplifying assumptions of two-dimensional steady isothermal flow and transient non-isothermal horizontal flow. The main contribution of the study on steady isothermal flows is a description of how the ground flow may recover following a hydrothermal eruption. A numerical technique developed for plotting the streamlines in this case (and verified against analytic results) may also have applications in solving the steady non-isothermal flow problem. The main contribution of the study on the transient horizontal flow problem is a comparison of the differing predictions of HM and SP flow. The rate at which a boiling front progresses through a porous medium and the degree of boiling which occurs is described for each fluid flow regime. A set of horizontal physical experiments and numerical simulations have also been carried out for comparison with the mathematical model. Qualitative results for these three models agree. Suggestions given for improvements to the design of the physical experiment provide a basis for future study into the type of flow which occurs in hydrothermal eruptions
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Song, Qin Sarah. "Microarray-based gene set analysis in cancer studies." 2008. http://hdl.handle.net/2292/4239.
Full textAbstract:
This work addresses the development and application of gene set analysis methods to problems in microarray-based data sets. The work consists of three parts. In the first part a gene set analysis method (PCOT2) is developed. It utilizes inter-gene correlation to detect significant alteration in gene sets across experimental conditions. The second part is focused on the exploration of correlation-based gene sets in conjunction with the application of the PCOT2 testing method in the investigation of biological mechanisms underlying breast cancer recurrence. In the third part, statistical models for analyzing combined microarray-based expression and genomic copy number data are developed. In addition, an analysis which incorporates tumour subgroups is shown to provide more accurate prognosis assessment for breast cancer patients.
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Wang, Ting. "Statistical models for earthquakes incorporating ancillary data : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Palmerston North, New Zealand." 2010. http://hdl.handle.net/10179/1486.
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This thesis consists of two parts. The first part proposes a new model – the Markov-modulated Hawkes process with stepwise decay (MMHPSD) to investigate the seismicity rate. The MMHPSD is a self-exciting process which switches among different states, in each of which the process has distinguishable background seismicity and decay rates. Parameter estimation is developed via the expectation maximization algorithm. The model is applied to data from the Landers earthquake sequence, demonstrating that it is useful for modelling changes in the temporal patterns of seismicity. The states in the model can capture the behavior of main shocks, large aftershocks, secondary aftershocks and a period of quiescence with different background rates and decay rates. The state transitions can then explain the seismicity rate changes and help indicate if there is any seismicity shadow or relative quiescence. The second part of this thesis develops statistical methods to examine earthquake sequences possessing ancillary data, in this case groundwater level data or GPS measurements of deformation. For the former, signals from groundwater level data at Tangshan Well, China, are extracted for the period from 2002 to 2005 using a moving window method. A number of different statistical techniques are used to detect and quantify coseismic responses to P, S, Love and Rayleigh wave arrivals. The P phase arrivals appear to trigger identifiable oscillations in groundwater level, whereas the Rayleigh waves amplify the water level movement. Identifiable coseismic responses are found for approximately 40 percent of magnitude 6+ earthquakes worldwide. A threshold in the relationship between earthquake magnitude and well–epicenter distance is also found, satisfied by 97% of the identified coseismic responses, above which coseismic changes in groundwater level at Tangshan Well are most likely. A non-linear filter measuring short-term deformation rate changes is introduced to extract signals from GPS data. For two case studies of a) deep earthquakes in central North Island, New Zealand, and b) shallow earthquakes in Southern California, a hidden Markov model (HMM) is fitted to the output from the filter. Mutual information analysis indicates that the state having the largest variation of deformation rate contains precursory information that indicates an elevated probability for earthquake occurrence.
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Mitchell, Stuart Anthony. "Operational forest harvest scheduling optimisation: a mathematical model and solution strategy." 2004. http://hdl.handle.net/2292/1761.
Full textAbstract:
This thesis describes the Operational Harvest Scheduling (OHS) problem and develops an algorithm that solves instances of the problem. The solution to an OHS problem is an Operational Harvest Schedule (OHS). An OHS: ² assigns forest harvesting crews to locations within a forest in the short-term (4-8 weeks); ² instructs crews to harvest specific log-types and allocates these log-types to customers; ² maximises profitability while meeting customer demand. The OHS problem is modelled as a Mixed Integer Linear Program (MILP). The formulation given in this thesis differs significantly from previous literature, especially with regard to the construction of the problem variables. With this novel formulation, the problem can be solved using techniques developed in previous work on aircraft crew scheduling optimisation (Ryan 1992). These techniques include constraint branching and column generation. The concept of relaxed integer solutions is introduced. A traditional integer solution to the OHS problem will require harvesting crews to move between harvesting locations at the end of a week. However, a relaxed integer solution allows crews to move at any time during a week. This concept allows my OHS model to more effectively model the practical problem. The OHS model is formulated for New Zealand and Australian commercial forestry operations,though the model could be applied to other intensively managed production forests. Three case studies are developed for two companies. These case studies show improvements in profitability over manual solution methods and a significant improvement in the ability to meet demand restrictions. The optimised solutions increased profit (revenue less harvesting and transportation costs) by between 3-7%, while decreasing the total value of excess or shortfall logs by between 15-86%.
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McKubre-Jordens, Maarten Nicolaas. "Minimising weighted mean distortion : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand." 2009. http://hdl.handle.net/10179/1279.
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There has been considerable recent interest in geometric function theory, nonlinear partial differential equations, harmonic mappings, and the connection of these to minimal energy phenomena. This work explores Nitsche's 1962 conjecture concerning the nonexistence of harmonic mappings between planar annuli, cast in terms of distortion functionals. The connection between the Nitsche problem and the famous Grötzsch problem is established by means of a weight function. Traditionally, these kinds of problems are investigated in the class of quasiconformal mappings, and the assumption is usually made a priori that solutions preserve various symmetries. Here the conjecture is solved in the much wider class of mappings of finite distortion, symmetry-preservation is proved, and ellipticity of the variational equations concerning these sorts of general problems is established. Furthermore, various alternative interpretations of the weight function introduced herein lead to an interesting analysis of a much wider variety of critical phenomena -- when the weight function is interpreted as a thickness, density or metric, the results lead to a possible model for tearing or breaking phenomena in material science. These physically relevant critical phenomena arise, surprisingly, out of purely theoretical considerations.
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Godfrey, A. Jonathan R. "Dealing with sparsity in genotype x environment analyses : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University, Palmerston North, New Zealand." 2004. http://hdl.handle.net/10179/1616.
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Researchers are frequently faced with the problem of analyzing incomplete and often unbalanced genotype-by-environment (GxE) matrices which arise as a trials programme progresses over seasons. The principal data for this investigation, arising from a ten year programme of onion trials, has less than 2,300 of the 49,200 combinations from the 400 genotypes and 123 environments. This 'sparsity' renders standard GxE methodology inapplicable. Analysis of this data to identify onion varieties that suit the shorter, hotter days of tropical and subtropical locations therefore presented a unique challenge. Removal of some data to form a complete GxE matrix wastes information and is consequently undesirable. An incomplete GxE matrix can be analyzed using the additive main effects and multiplicative interaction (AMMI) model in conjunction with the EM algorithm but proved unsatisfactory in this instance. Cluster analysis has been commonly used in GxE analyses, but current methods are inadequate when the data matrix is incomplete. If clustering is to be applied to incomplete data sets, one of two routes needs to be taken: either the clustering procedure must be modified to handle the missing data, or the missing entries must be imputed so that standard cluster analysis can be performed. A new clustering method capable of handling incomplete data has been developed. 'Two-stage clustering', as it has been named, relies on a partitioning of squared Euclidean distance into two independent components, the GxE interaction and the genotype main effect. These components are used in the first and second stages of clustering respectively. Two-stage clustering forms the basis for imputing missing values in a GxE matrix, so that a more complete data array is available for other GxE analyses. 'Two-stage imputation' estimates unobserved GxE yields using inter-genotype similarities to adjust observed yield data in the environment in which the yield is missing. This new imputation method is transferrable to any two-way data situation where all observations are measured on the same scale and the two factors are expected to have significant interaction. This simple, but effective, imputation method is shown to improve on an existing method that confounds the GxE interaction and the genotype main effect. Future development of two-stage imputation will use a parameterization of two-stage clustering in a multiple imputation process. Varieties recommended for use in a certain environment would normally be chosen using results from similar environments. Differing cluster analysis approaches were applied, but led to inconsistent environment clusterings. A graphical summary tool, created to ease the difficulty in identifying the differences between pairs of clusterings, proved especially useful when the number of clusters and clustered observations were high. 'Cluster influence diagrams' were also used to investigate the effects the new imputation method had on the qualitative structure of the data. A consequence of the principal data's sparsity was that imputed values were found to be dependent on the existence of observable inter-genotype relationships, rather than the strength of these observable relationships. As a result of this investigation, practical recommendations are provided for limiting the detrimental effects of sparsity. Applying these recommendations will enhance the future ability of two-stage imputation to identify those onion varieties that suit tropical and subtropical locations.
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Abdul, Latif Nurul Syaza. "Analysis of a dynamical system of animal growth and composition : a thesis submitted in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University, Albany, New Zealand." 2010. http://hdl.handle.net/10179/1654.
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This thesis investigates the analysis of the extended model of animal growth proposed by Oliviera et al (personal communication, July 2009). This mechanistic model of animal growth based on a detailed representation of energy dynamics focussing on the interaction between four compartment of body composition; nutrient level, fat content, visceral protein and non-visceral protein. The model is mathematically analysed and the behaviour of the model for different feeding level is examined. The animal growth model exhibits thresholds typical of nonlinear systems and multiple stable steady states which have distinct basins of stability which depend on the value of the large number of physiologically-determined parameters. These have not been previously explored theoretically and these are done in this thesis. The model demonstrates richer behaviour where path-following techniques are used to explore the distribution in parameter space of the varying phenomenology.
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Lam, Heung Yeung. "q-series in number theory and combinatorics : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand." 2006. http://hdl.handle.net/10179/1477.
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Srinivasa Ramanujan (1887-1920) was one of the world's greatest mathematical geniuses. He work extensively in a branch of mathematics called "q-series". Around 1913, he found an important formula which now is known as Ramanujan's 1ψ1summation formula. The aim of this thesis is to investigate Ramanujan's 1ψ1summation formula and explore its applications to number theory and combinatorics. First, we consider several classical important results on elliptic functions and then give new proofs of these results using Ramanujan's 1ψ1 summation formula. For example, we will present a number of classical and new solutions for the problem of representing an integer as sums of squares (one of the most celebrated in number theory and combinatorics) in this thesis. This will be done by using q-series and Ramanujan's 1ψ1 summation formula. This in turn will give an insight into how Ramanujan may have proven many of his results, since his own proofs are often unknown, thereby increasing and deepening our understanding of Ramanujan's work.
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McLeod, Kristin. "Risk analysis of a flatfish stock complex : a thesis presented in partial fulfilment of the requirements for the degree of Master of Science in Mathematics at Massey University." 2010. http://hdl.handle.net/10179/1533.
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The New Zealand Ministry of Fisheries relies on fishery assessments to determine suitable catch quotas for exploited fisheries. Currently, 628 fish stocks are managed in New Zealand using the Quote Management System, which includes the 8 com- mercial flatfish species caught within the Exclusive Economic Zone. These eight species of flatfish, which includes four species of flounder, two species of sole, brill and turbot, are currently managed using a combined catch quota. Since these eight species are managed using a common catch quota, there is concern that some of the individual species may be under or over-fished. This thesis describes work involving the flatfish species caught in the FLA3 man- agement area, around the south island of New Zealand. The FLA3 management area contains three key species: New Zealand sole, lemon sole, and sand flounder. Due to the nature and limitations of the data available, simple biomass dynamic models were applied to these species. The maximum likelihood and Bayesian goodness of fit techniques were used to estimate the model parameters. Three models were used: the Fox model, the Schaefer model and the Pella-Tomlinson model with m = 3. As a mathematical/statistical exercise, these models were used to conduct a risk analysis to analyse the advantages and disadvantages of six management options for setting a TACC. However, because of issues over the way that the parameter K has been modelled (due to necessity caused by the lack of data), this should not be seen as an appropriate method for estimating the fish stock. Conclusions were drawn from the results regarding suitable future action for the assessment and management of flatfish stock in FLA3.
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Lim, Leng Leng. "Modelling of volcanic ashfall : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand." 2006. http://hdl.handle.net/10179/1472.
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Modelling of volcanic ashfall has been attempted by volcanologists but very little work has been done by mathematicians. In this thesis we show that mathematical models can accurately describe the distribution of particulate materials that fall to the ground following an eruption. We also report on the development and analysis of mathematical models to calculate the ash concentration in the atmosphere during ashfall after eruptions. Some of these models have analytical solutions. The mathematical models reported on in this thesis not only describe the distribution of ashfall on the ground but are also able to take into account the effect of variation of wind direction with elevation. In order to model the complexity of the atmospheric flow, the atmosphere is divided into horizontal layers. Each layer moves steadily and parallel to the ground: the wind velocity components, particle settling speed and dispersion coefficients are assumed constant within each layer but may differ from layer to layer. This allows for elevation-dependent wind and turbulence profiles, as well as changing particle settling speeds, the last allowing the effects of the agglomeration of particles to be taken into account.
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Stoyanov, Stefan Kremenov. "An investigation of the methods for estimating usual dietary intake distributions : a thesis presented in partial fulfillment of the requirements for the degree of Master of Applied Statistics at Massey University, Albany, New Zealand." 2008. http://hdl.handle.net/10179/767.
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The estimation of the distribution of usual intake of nutrients is important for developing nutrition policies as well as for etiological research and educational purposes. In most nutrition surveys only a small number of repeated intake observations per individual are collected. Of main interest is the longterm usual intake which is defined as long-term daily average intake of a dietary component. However, dietary intake on a single day is a poor estimate of the individual’s long-term usual intake. Furthermore, the distribution of individual intake means is also a poor estimator of the distribution of usual intake since usually there is large within-individual compared to between-individual variability in the dietary intake data. Hence, the variance of the mean intakes is larger than the variance of the usual intake distribution. Essentially, the estimation of the distribution of long-term intake is equivalent to the estimation of a distribution of a random variable observed with measurement error. Some of the methods for estimating the distributions of usual dietary intake are reviewed in detail and applied to nutrient intake data in order to evaluate their properties. The results indicate that there are a number of robust methods which could be used to derive the distribution of long-term dietary intake. The methods share a common framework but differ in terms of complexity and assumptions about the properties of the dietary consumption data. Hence, the choice of the most appropriate method depends on the specific characteristics of the data, research purposes as well as availability of analytical tools and statistical expertise.
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Heller, Florian. "Adjusting the parameter estimation of the parentage analysis software MasterBayes to the presence of siblings : a thesis presented in partial fulfillment of the requirements for the degree of Master of Applied Statistics at Massey University, Albany, New Zealand." 2009. http://hdl.handle.net/10179/1375.
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Parentage analysis is concerned with the estimation of a sample’s pedigree structure, which is often essential knowledge for estimating population parameters of animal species, such as reproductive success. While it is often easy to relate one parent to an offspring simply by observation, the second parent remains frequently unknown. Parentage analysis uses genotypic data to estimate the pedigree, which then allows inferring the desired parameters. There are several software applications available for parentage analysis, one of which is MasterBayes, an extension to the statistical software package R. MasterBayes makes use of behavioural, phenotypic, spatial and genetic data, providing a Bayesian approach to simultaneously estimate pedigree and population parameters of interest, allowing for a range of covariate models. MasterBayes however assumes the sample to be a randomly collected from the population of interest. Often however, collected data will come from nests or otherwise from groups that are likely to contain siblings. If siblings are present, the assumption of a random population sample is not met anymore and as a result, the parameter variance will be underestimated. This thesis presents four methods to adjust MasterBayes’ parameter estimate to the presence of siblings, all of which are based on the pedigree structure, as estimated by MasterBayes. One approach, denoted as DEP, provides a Bayesian estimate, similar to MasterBayes’ approach, but incorporating the presence of siblings. Three further approaches, denoted as W1, W2 and W3, apply importance sampling to re-weight parameter estimates obtained from MasterBayes and DEP. Though fully satisfying adjustment of the estimate’s variance is only achieved at nearly perfect pedigree assignment, the presented methods do improve MasterBayes’ parameter estimation in the presence of siblings considerably, when the pedigree is uncertain. DEP and W3 show to be the most successful adjustment methods, providing comparatively accurate, though yet underestimated variances for small family sizes. W3 is the superior approach when the pedigree is highly uncertain, whereas DEP becomes superior when about half of all parental assignments are correct. Large family sizes introduce to all approaches a tendency to underestimate the parameter variance, the degree of underestimation depending on the certainty of pedigree. Additionally, the importance sampling schemes provide at large uncertainty of pedigree comparatively good estimates of the parameter’s expected values, where the non importance sampling approaches severely fail.
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Alexander, D. L. J. "Convergence rates of stochastic global optimisation algorithms with backtracking : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Statistics at Massey University." 2004. http://hdl.handle.net/10179/1613.
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A useful measure of quality of a global optimisation algorithm such as simulated annealing is the length of time it must be run to reach a global optimum within a certain accuracy. Such a performance measure assists in choosing and tuning algorithms. This thesis proposes an approach to obtaining such a measure through successive approximation of a generic stochastic global optimisation algorithm with a sequence of stochastic processes culminating in backtracking adaptive search. The overall approach is to approximate the progress of an optimisation algorithm with that of a model process, backtracking adaptive search. The known convergence rate of the model then provides an estimator of the unknown convergence rate of the original algorithm. Parameters specifying this model are chosen based on observation of the optimisation algorithm. The optimisation algorithm may first be approximated with a time-inhomogeneous Markovian process defined on the problem range. The distribution of the number of iterations to convergence for this averaged range process is shown to be identical with that of the original process. This process is itself approximated by a time-homogeneous Markov process in the range, the asymptotic averaged range process. This approximation is defined for all Markovian optimisation algorithms and a weak condition under which its convergence time closely matches that of the original algorithm is developed. The asymptotic averaged range process is of the same form as backtracking adaptive search, the final stage of approximation. Backtracking adaptive search is an optimisation algorithm which generalises pure adaptive search and hesitant adaptive search. In this thesis the distribution of the number of iterations for which the algorithm runs in order to reach a sufficiently extreme objective function level is derived. Several examples of backtracking adaptive search on finite problems are also presented, including special cases that have received attention in the literature. Computational results of the entire approximation framework are reported for several examples. The method can be applied to any optimisation algorithm to obtain an estimate of the time required to obtain solutions of a certain quality. Directions for further work in order to improve the accuracy of such estimates are also indicated.
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Mann, Joanne L. "Modelling infectious disease epidemiology and vaccination impact : a thesis presented in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand." 2009. http://hdl.handle.net/10179/1085.
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This thesis presents mathematical models for the dynamics of vaccine preventable diseases, specifically looking at the New Zealand situation. Through the use of integral and differential equations, we develop models and compare the results of these to known data. Using game theory analysis we determine and compare the proportion of the population that needs to be vaccinated in order to minimise the expected costs to the individuals in the population and to the community. Two different scenarios and methods are considered, where the effects of vaccination last only one epidemic cycle (using an integral equation method) and where vaccination is effective over an entire lifetime (using a differential equation method). For both scenarios, we find that the minimum cost for the individuals is reached when a lower proportion of the population is vaccinated than needed for the minimum cost to the community. We then elaborate on the integral equation method to produce a model for repeated epidemics of measles in a population, where a discrete mapping is used to include the year to year demographics of the population. The results of this model show a different epidemic pattern then that produced from a differential equation model, with numerical problems encountered. From here on, we use differential equation models in our analysis. A critique and extension to an existing model for the dynamics of the hepatitis B virus is presented, with discussion on the appropriateness of the model’s construct for predicting the incidence of infection. Alternative differential equation models for hepatitis B virus and immunisation that include splitting the population into age groups with nonhomogeneous mixing are presented. The results of these models are compared with the known data on incidence of infection and carriage in New Zealand, showing how affective different immunisation schedules may have been. Differential equation models are then presented for meningococcal B virus epidemiology in New Zealand, with the models incorporating different features of the virus until the best model is found that fits the New Zealand data. Each model is compared with the known incidence of infection, with the population being either treated as a whole or split into age groups with non-homogeneous mixing. The effect of vaccination is included in this model so that we can explore the future of the infection in the population, and how best to tackle any future epidemics. The model shows that the current vaccination campaign was the best solution for controlling the epidemic, but there will be epidemics in the future that will need subsequent vaccination campaigns to limit the number of infections.
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Hall, Alistair John. "Steady size distributions in cell populations : a thesis presented in partial fulfilment of the requirements for the degree of Doctor Philosophy in Mathematics at Massey University." 1991. http://hdl.handle.net/10179/1730.
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In any population of cells, individual cells grow for some period of time and then divide into two or more parts, called daughters. To describe this process mathematically, we need to specify functions describing the growth rate, size at division, and proportions into which each cell divides. In this thesis, it is assumed that the growth rate of a cell can be determined precisely from its size, but that both its size at division and the proportions into which it divides may be described stochastically, by probability density functions whose parameters are dependent on cell size and age (or birth-size). Special cases are also considered where all cells with the same birth-size divide at the same size, or where all cells divide exactly in half. We consider a population of cells growing and dividing steadily, such that the total cell population is increasing, but the proportion of cells in any size class remains constant. In Chapter 1, equations are derived which need to be solved in order to deduce the shape of the steady size distribution (or steady size/age or size/birth-size distributions) from any given growth rate and probability distributions describing the division rate and division proportions. In the general case, a Fredholm-type integral equation is obtained, but if the probability of cell division depends on cell size only (i.e. not age or birth-size), and all cells divide into equal-sized daughters, then we obtain a functional differential equation. In two special cases, the resulting equations simplify considerably, and it is these cases which are explored further in this thesis. The first is where the probability of a cell dividing in any instant of time is a constant, independent of cell age or size. In Chapter 2, the functional differential equation resulting when cells divide into equal-sized daughters is solved for the special case where the growth rate is constant, and in an appendix the case where the growth rate is described by a power law is dealt with. The second case which simplifies is where the time-independent part of the growth rate of a cell is proportional to cell size. This case is particularly important, as it is a good first-order approximation to the real cell growth rate in some structured tissues, and in some bacteria. The special case in which this leads to a functional differential equation is discussed in Chapter 3, and the integral equation arising in the general case is dealt with in Chapter 4. Finally, the conditions under which the integral operator in Chapter 4 will be both square-integrable and non-factorable are discussed in Chapter 5. It is shown that if these conditions are satisfied then a unique, stable, steady size distribution will exist.
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Wang, Cynthia Dewi. "The linear wave response of a single and a periodic line-array of floating elastic plates: a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand." 2004. http://hdl.handle.net/10179/1625.
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We propose an improved technique to calculate the linear response of a single and multiple plates models due to ocean waves. The single plate model is the basis for the multiple plates model which we take to be a periodic array of identical plates. For the single plate model we solve the plate displacement by the Finite Element Method (FEM) and the water potential by the Boundary Element Method (BEM). The displacement is expanded in terms of the basis functions of the FEM. The boundary integral equation representing the potential is approximated by these basis functions. The resulting integral operator involving the free-surface Green's function is solved using an elementary integration scheme. Results are presented for the single plate model. We then use the same technique to solve for the periodic array of plates problem because the single and the periodic array plates model differ only in the expression of the Green's function. For the periodic array plate model the boundary integral equation for the potential involves a periodic Green's function which can be obtained by taking an infinite sum of the free-surface Green's function for the single plate model. The solution for the periodic array plate is derived in the same way as the single plate model. From this solution we then calculate the waves scattered by this periodic array.
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Miller, Steven Duncan. "Stochastic modelling of rat invasions among islands in the New Zealand archipelago." 2008. http://hdl.handle.net/2292/3375.
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This project was formulated with the purpose of advancing knowledge of the invasion dynamics of rats within archipelagos in New Zealand. The concentration on islands reflected the conservation focus of this project - islands are the last refuges for many native New Zealand species that cannot survive in the wild on the mainland. This project can be divided into four areas: 1. Data collection: There was no intent for innovation here, but a deeper understanding of the environments in which rats are born, breed, migrate, and die was developed. 2. Development of tools for data exploration: • A user-friendly point-and-click graphical interface for the R program was designed to allow any user to easily explore simple genetic characteristics of the data. • A novel method for exploring the genetic similarity between individuals was developed and showcased with real data, proving successful in cases of both high and low genetic differentiation, and in detecting likely individual migrants. 3. Improvement of a method for estimating migration: • An attempt was made to improve the Markov chain Monte Carlo procedure underlying this method. • The migration model used by the method was significantly improved, so that it could cope with any level of migration. Previously, results from situations where migration rates were high were invalid. 4. Investigated topics of ecological interest: • Field measurements of rats were used to show that Norway rats tend to have larger masses than ship rats, southern rats are generally larger than northern rats, but the effect on mass of living on an island as opposed to the mainland depends on the latitude. It was also shown that relative tail length is a good species discriminator. • Multiple paternity was confirmed for both Norway and ship rats. This breeding characteristic might form part of the explanation for why rats are such successful invaders. During the project, case studies involving rats on Big South Cape Island, Great Barrier Island and in the Bay of Islands were used to highlight the methods developed, and provided some unexpected and fascinating results.
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Elvin, Amanda Jane. "Pattern formation in a neural field model : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Auckland, New Zealand." 2008. http://hdl.handle.net/10179/717.
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In this thesis I study the effects of gap junctions on pattern formation in a neural field model for working memory. I review known results for the base model (the “Amari model”), then see how the results change for the “gap junction model”. I find steady states of both models analytically and numerically, using lateral inhibition with a step firing rate function, and a decaying oscillatory coupling function with a smooth firing rate function. Steady states are homoclinic orbits to the fixed point at the origin. I also use a method of piecewise construction of solutions by deriving an ordinary differential equation from the partial integro-differential formulation of the model. Solutions are found numerically using AUTO and my own continuation code in MATLAB. Given an appropriate level of threshold, as the firing rate function steepens, the solution curve becomes discontinuous and stable homoclinic orbits no longer exist in a region of parameter space. These results have not been described previously in the literature. Taking a phase space approach, the Amari model is written as a four-dimensional, reversible Hamiltonian system. I develop a numerical technique for finding both symmetric and asymmetric homoclinic orbits. I discover a small separate solution curve that causes the main curve to break as the firing rate function steepens and show there is a global bifurcation. The small curve and the global bifurcation have not been reported previously in the literature. Through the use of travelling fronts and construction of an Evans function, I show the existence of stable heteroclinic orbits. I also find asymmetric steady state solutions using other numerical techniques. Various methods of determining the stability of solutions are presented, including a method of eigenvalue analysis that I develop. I then find both stable and transient Turing structures in one and two spatial dimensions, as well as a Type-I intermittency. To my knowledge, this is the first time transient Turing structures have been found in a neural field model. In the Appendix, I outline numerical integration schemes, the pseudo-arclength continuation method, and introduce the software package AUTO used throughout the thesis.
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Lee, Arier Chi-Lun. "Random effects models for ordinal data." 2009. http://hdl.handle.net/2292/4544.
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One of the most frequently encountered types of data is where the response variables are measured on an ordinal scale. Although there have been substantial developments in the statistical techniques for the analysis of ordinal data, methods appropriate for repeatedly assessed ordinal data collected from field experiments are limited. A series of biennial field screening trials for evaluating cultivar resistance of potato to the disease, late blight, caused by the fungus Phytophthora infestans (Mont.) de Bary has been conducted by the New Zealand Institute of Crop and Food Research since 1983. In each trial, the progression of late blight was visually assessed several times during the planting season using a nine-point ordinal scale based on the percentage of necrotic tissues. As for many other agricultural field experiments, spatial differences between the experimental units is one of the major concerns in the analysis of data from the potato late blight trial. The aim of this thesis is to construct a statistical model which can be used to analyse the data collected from the series of potato late blight trials. We review existing methodologies for analysing ordinal data with mixed effects particularly those methods in the Bayesian framework. Using data collected from the potato late blight trials we develop a Bayesian hierarchical model for the analyses of repeatedly assessed ordinal scores with spatial effects, in particular the time dependence of the scores assessed on the same experimental units was modelled by a sigmoid logistic curve. Data collected from the potato late blight trials demonstrated the importance of spatial effects in agricultural field trials. These effects cannot be neglected when analysing such data. Although statistical methods can be refined to account for the complexity of the data, appropriate trial design still plays a central role in field experiments.Accompanying dataset is at http://hdl.handle.net/2292/5240