Strategies for comparing decimal numbers with the same whole-number part

The strategies employed by 130 Grade 5 Brisbane students in comparing decimal numbers which have the same whole-number part were compared with those identified in similar studies conducted in the USA, France and Israel. Three new strategies were identified. Similar to USA results, the most common comparison errors stemmed from the incorrect whole-number strategy in which length is confused with size. The findings of this present study tend to support Resnick et al.’s (1989) hypothesis that the introduction of decimal-fraction recording before common-fraction recording seems to promote better comparison of decimal numbers.

Construction of multiplicative abstract schema for decimal-number numeration

This paper reports on an intervention study planned to help Year 6 students construct the multiplicative structure underlying decimal-number numeration. Three types of intervention were designed from a numeration model developed from a large study of 173 Year 6 students’ decimal-number knowledge. The study found that students could acquire multiplicative structure as an abstract schema if instruction took account of prior knowledge as informed by the model.

Fractions, reunitisation and the number-line representation

Centre for Mathematics and Science Education, QUT, Brisbane, Australia This paper reports on a study in which Years 6 and 10 students were individually interviewed to determine their ability to unitise and reunitise number lines used to represent mixed numbers and improper fractions. Only 16.7% of the students (all Year 6) were successful on all three tasks and, in general, Year 6 students outperformed Year 8 students. The interviews revealed that the remaining students had incomplete, fragmented or non-existent structural knowledge of mixed numbers and improper fractions, and were unable to unitise or reunitise number lines. The implication for teaching is that instruction should focus on providing students with a variety of fraction representations in order to develop rich and flexible schema for all fraction types (mixed numbers, and proper and improper fractions).

Bayesian hybrid algorithms and models : implementation and associated issues

This thesis addresses computational challenges arising from Bayesian analysis of complex real-world problems. Many of the models and algorithms designed for such analysis are ‘hybrid’ in nature, in that they are a composition of components for which their individual properties may be easily described but the performance of the model or algorithm as a whole is less well understood. The aim of this research project is to after a better understanding of the performance of hybrid models and algorithms. The goal of this thesis is to analyse the computational aspects of hybrid models and hybrid algorithms in the Bayesian context. The first objective of the research focuses on computational aspects of hybrid models, notably a continuous finite mixture of t-distributions. In the mixture model, an inference of interest is the number of components, as this may relate to both the quality of model fit to data and the computational workload. The analysis of t-mixtures using Markov chain Monte Carlo (MCMC) is described and the model is compared to the Normal case based on the goodness of fit. Through simulation studies, it is demonstrated that the t-mixture model can be more flexible and more parsimonious in terms of number of components, particularly for skewed and heavytailed data. The study also reveals important computational issues associated with the use of t-mixtures, which have not been adequately considered in the literature. The second objective of the research focuses on computational aspects of hybrid algorithms for Bayesian analysis. Two approaches will be considered: a formal comparison of the performance of a range of hybrid algorithms and a theoretical investigation of the performance of one of these algorithms in high dimensions. For the first approach, the delayed rejection algorithm, the pinball sampler, the Metropolis adjusted Langevin algorithm, and the hybrid version of the population Monte Carlo (PMC) algorithm are selected as a set of examples of hybrid algorithms. Statistical literature shows how statistical efficiency is often the only criteria for an efficient algorithm. In this thesis the algorithms are also considered and compared from a more practical perspective. This extends to the study of how individual algorithms contribute to the overall efficiency of hybrid algorithms, and highlights weaknesses that may be introduced by the combination process of these components in a single algorithm. The second approach to considering computational aspects of hybrid algorithms involves an investigation of the performance of the PMC in high dimensions. It is well known that as a model becomes more complex, computation may become increasingly difficult in real time. In particular the importance sampling based algorithms, including the PMC, are known to be unstable in high dimensions. This thesis examines the PMC algorithm in a simplified setting, a single step of the general sampling, and explores a fundamental problem that occurs in applying importance sampling to a high-dimensional problem. The precision of the computed estimate from the simplified setting is measured by the asymptotic variance of the estimate under conditions on the importance function. Additionally, the exponential growth of the asymptotic variance with the dimension is demonstrated and we illustrates that the optimal covariance matrix for the importance function can be estimated in a special case.

Exponential asymptotics with multiple stokes lines

When asymptotic series methods are applied in order to solve problems that arise in applied mathematics in the limit that some parameter becomes small, they are unable to demonstrate behaviour that occurs on a scale that is exponentially small compared to the algebraic terms of the asymptotic series. There are many examples of physical systems where behaviour on this scale has important effects and, as such, a range of techniques known as exponential asymptotic techniques were developed that may be used to examinine behaviour on this exponentially small scale. Many problems in applied mathematics may be represented by behaviour within the complex plane, which may subsequently be examined using asymptotic methods. These problems frequently demonstrate behaviour known as Stokes phenomenon, which involves the rapid switches of behaviour on an exponentially small scale in the neighbourhood of some curve known as a Stokes line. Exponential asymptotic techniques have been applied in order to obtain an expression for this exponentially small switching behaviour in the solutions to orginary and partial differential equations. The problem of potential flow over a submerged obstacle has been previously considered in this manner by Chapman & Vanden-Broeck (2006). By representing the problem in the complex plane and applying an exponential asymptotic technique, they were able to detect the switching, and subsequent behaviour, of exponentially small waves on the free surface of the flow in the limit of small Froude number, specifically considering the case of flow over a step with one Stokes line present in the complex plane. We consider an extension of this work to flow configurations with multiple Stokes lines, such as flow over an inclined step, or flow over a bump or trench. The resultant expressions are analysed, and demonstrate interesting implications, such as the presence of exponentially sub-subdominant intermediate waves and the possibility of trapped surface waves for flow over a bump or trench. We then consider the effect of multiple Stokes lines in higher order equations, particu- larly investigating the behaviour of higher-order Stokes lines in the solutions to partial differential equations. These higher-order Stokes lines switch off the ordinary Stokes lines themselves, adding a layer of complexity to the overall Stokes structure of the solution. Specifically, we consider the different approaches taken by Howls et al. (2004) and Chap- man & Mortimer (2005) in applying exponential asymptotic techniques to determine the higher-order Stokes phenomenon behaviour in the solution to a particular partial differ- ential equation.

Modelling of some biological materials using continuum mechanics

Continuum mechanics provides a mathematical framework for modelling the physical stresses experienced by a material. Recent studies show that physical stresses play an important role in a wide variety of biological processes, including dermal wound healing, soft tissue growth and morphogenesis. Thus, continuum mechanics is a useful mathematical tool for modelling a range of biological phenomena. Unfortunately, classical continuum mechanics is of limited use in biomechanical problems. As cells refashion the �bres that make up a soft tissue, they sometimes alter the tissue's fundamental mechanical structure. Advanced mathematical techniques are needed in order to accurately describe this sort of biological `plasticity'. A number of such techniques have been proposed by previous researchers. However, models that incorporate biological plasticity tend to be very complicated. Furthermore, these models are often di�cult to apply and/or interpret, making them of limited practical use. One alternative approach is to ignore biological plasticity and use classical continuum mechanics. For example, most mechanochemical models of dermal wound healing assume that the skin behaves as a linear viscoelastic solid. Our analysis indicates that this assumption leads to physically unrealistic results. In this thesis we present a novel and practical approach to modelling biological plasticity. Our principal aim is to combine the simplicity of classical linear models with the sophistication of plasticity theory. To achieve this, we perform a careful mathematical analysis of the concept of a `zero stress state'. This leads us to a formal de�nition of strain that is appropriate for materials that undergo internal remodelling. Next, we consider the evolution of the zero stress state over time. We develop a novel theory of `morphoelasticity' that can be used to describe how the zero stress state changes in response to growth and remodelling. Importantly, our work yields an intuitive and internally consistent way of modelling anisotropic growth. Furthermore, we are able to use our theory of morphoelasticity to develop evolution equations for elastic strain. We also present some applications of our theory. For example, we show that morphoelasticity can be used to obtain a constitutive law for a Maxwell viscoelastic uid that is valid at large deformation gradients. Similarly, we analyse a morphoelastic model of the stress-dependent growth of a tumour spheroid. This work leads to the prediction that a tumour spheroid will always be in a state of radial compression and circumferential tension. Finally, we conclude by presenting a novel mechanochemical model of dermal wound healing that takes into account the plasticity of the healing skin.

Critical analysis of high particle number emissions from accelerating compressed natural gas buses

Compressed natural gas (CNG) engines are thought to be less harmful to the environment than conventional diesel engines, especially in terms of particle emissions. Although, this is true with respect to particulate matter (PM) emissions, results of particle number (PN) emission comparisons have been inconclusive. In this study, results of on-road and dynamometer studies of buses were used to derive several important conclusions. We show that, although PN emissions from CNG buses are significantly lower than from diesel buses at low engine power, they become comparable at high power. For diesel buses, PN emissions are not significantly different between acceleration and operation at steady maximum power. However, the corresponding PN emissions from CNG buses when accelerating are an order of magnitude greater than when operating at steady maximum power. During acceleration under heavy load, PN emissions from CNG buses are an order of magnitude higher than from diesel buses. The particles emitted from CNG buses are too small to contribute to PM10 emissions or contribute to a reduction of visibility, and may consist of semivolatile nanoparticles.

Characterization of particle number concentrations and PM2.5 in a school : influence of outdoor air pollution on indoor air

Background, Aim and Scope The impact of air pollution on school children’s health is currently one of the key foci of international and national agencies. Of particular concern are ultrafine particles which are emitted in large quantities, contain large concentrations of toxins and are deposited deeply in the respiratory tract. Materials and methods In this study, an intensive sampling campaign of indoor and outdoor airborne particulate matter was carried out in a primary school in February 2006 to investigate indoor and outdoor particle number (PN) and mass concentrations (PM2.5), and particle size distribution, and to evaluate the influence of outdoor air pollution on the indoor air. Results For outdoor PN and PM2.5, early morning and late afternoon peaks were observed on weekdays, which are consistent with traffic rush hours, indicating the predominant effect of vehicular emissions. However, the temporal variations of outdoor PM2.5 and PN concentrations occasionally showed extremely high peaks, mainly due to human activities such as cigarette smoking and the operation of mower near the sampling site. The indoor PM2.5 level was mainly affected by the outdoor PM2.5 (r = 0.68, p<0.01), whereas the indoor PN concentration had some association with outdoor PN values (r = 0.66, p<0.01) even though the indoor PN concentration was occasionally influenced by indoor sources, such as cooking, cleaning and floor polishing activities. Correlation analysis indicated that the outdoor PM2.5 was inversely correlated with the indoor to outdoor PM2.5 ratio (I/O ratio) (r = -0.49, p<0.01), while the indoor PN had a weak correlation with the I/O ratio for PN (r = 0.34, p<0.01). Discussion and Conclusions The results showed that occupancy did not cause any major changes to the modal structure of particle number and size distribution, even though the I/O ratio was different for different size classes. The I/O curves had a maximum value for particles with diameters of 100 – 400 nm under both occupied and unoccupied scenarios, whereas no significant difference in I/O ratio for PM2.5 was observed between occupied and unoccupied conditions. Inspection of the size-resolved I/O ratios in the preschool centre and the classroom suggested that the I/O ratio in the preschool centre was the highest for accumulation mode particles at 600 nm after school hours, whereas the average I/O ratios of both nucleation mode and accumulation mode particles in the classroom were much lower than those of Aitken mode particles. Recommendations and Perspectives The findings obtained in this study are useful for epidemiological studies to estimate the total personal exposure of children, and to develop appropriate control strategies for minimizing the adverse health effects on school children.

Exposure to particle number, surface area and PM concentrations in pizzerias

The aim of this work was to quantify exposure to particles emitted by wood-fired ovens in pizzerias. Overall, 15 microenvironments were chosen and analyzed in a 14-month experimental campaign. Particle number concentration and distribution were measured simultaneously using a Condensation Particle Counter (CPC), a Scanning Mobility Particle Sizer (SMPS), an Aerodynamic Particle Sizer (APS). The surface area and mass distributions and concentrations, as well as the estimation of lung deposition surface area and PM1 were evaluated using the SMPS-APS system with dosimetric models, by taking into account the presence of aggregates on the basis of the Idealized Aggregate (IA) theory. The fraction of inhaled particles deposited in the respiratory system and different fractions of particulate matter were also measured by means of a Nanoparticle Surface Area Monitor (NSAM) and a photometer (DustTrak DRX), respectively. In this way, supplementary data were obtained during the monitoring of trends inside the pizzerias. We found that surface area and PM1 particle concentrations in pizzerias can be very high, especially when compared to other critical microenvironments, such as the transport hubs. During pizza cooking under normal ventilation conditions, concentrations were found up to 74, 70 and 23 times higher than background levels for number, surface area and PM1, respectively. A key parameter is the oven shape factor, defined as the ratio between the size of the face opening in respect

Development of a composite line source emission model for traffic interrupted microenvironments and its application in particle number emissions at a bus station

A composite line source emission (CLSE) model was developed to specifically quantify exposure levels and describe the spatial variability of vehicle emissions in traffic interrupted microenvironments. This model took into account the complexity of vehicle movements in the queue, as well as different emission rates relevant to various driving conditions (cruise, decelerate, idle and accelerate), and it utilised multi-representative segments to capture the accurate emission distribution for real vehicle flow. Hence, this model was able to quickly quantify the time spent in each segment within the considered zone, as well as the composition and position of the requisite segments based on the vehicle fleet information, which not only helped to quantify the enhanced emissions at critical locations, but it also helped to define the emission source distribution of the disrupted steady flow for further dispersion modelling. The model then was applied to estimate particle number emissions at a bi-directional bus station used by diesel and compressed natural gas fuelled buses. It was found that the acceleration distance was of critical importance when estimating particle number emission, since the highest emissions occurred in sections where most of the buses were accelerating and no significant increases were observed at locations where they idled. It was also shown that emissions at the front end of the platform were 43 times greater than at the rear of the platform. Although the CLSE model is intended to be applied in traffic management and transport analysis systems for the evaluation of exposure, as well as the simulation of vehicle emissions in traffic interrupted microenvironments, the bus station model can also be used for the input of initial source definitions in future dispersion models.