Modelling water droplet movement on a leaf surface

The central aim for the research undertaken in this PhD thesis is the development of a model for simulating water droplet movement on a leaf surface and to compare the model behavior with experimental observations. A series of five papers has been presented to explain systematically the way in which this droplet modelling work has been realised. Knowing the path of the droplet on the leaf surface is important for understanding how a droplet of water, pesticide, or nutrient will be absorbed through the leaf surface. An important aspect of the research is the generation of a leaf surface representation that acts as the foundation of the droplet model. Initially a laser scanner is used to capture the surface characteristics for two types of leaves in the form of a large scattered data set. After the identification of the leaf surface boundary, a set of internal points is chosen over which a triangulation of the surface is constructed. We present a novel hybrid approach for leaf surface fitting on this triangulation that combines Clough-Tocher (CT) and radial basis function (RBF) methods to achieve a surface with a continuously turning normal. The accuracy of the hybrid technique is assessed using numerical experimentation. The hybrid CT-RBF method is shown to give good representations of Frangipani and Anthurium leaves. Such leaf models facilitate an understanding of plant development and permit the modelling of the interaction of plants with their environment. The motion of a droplet traversing this virtual leaf surface is affected by various forces including gravity, friction and resistance between the surface and the droplet. The innovation of our model is the use of thin-film theory in the context of droplet movement to determine the thickness of the droplet as it moves on the surface. Experimental verification shows that the droplet model captures reality quite well and produces realistic droplet motion on the leaf surface. Most importantly, we observed that the simulated droplet motion follows the contours of the surface and spreads as a thin film. In the future, the model may be applied to determine the path of a droplet of pesticide along a leaf surface before it falls from or comes to a standstill on the surface. It will also be used to study the paths of many droplets of water or pesticide moving and colliding on the surface.

Mixing colours : an ICT tool based on a semiotic framework for mathematical meaning-making about ratio and fractions

This paper reports on the research and development of an ICT tool to facilitate the learning of ratio and fractions by adult prisoners. The design of the ICT tool was informed by a semiotic framework for mathematical meaning-making. The ICT tool thus employed multiple semiotic resources including topological, typological, and social-actional resources. The results showed that individual semiotic resource could only represent part of the mathematical concept, while at the same time it might signify something else to create a misconception. When multiple semiotic resources were utilised the mathematical ideas could be better learnt.

Mathematical modelling of muscle recruitment and function in the lumbar spine

Low back pain is an increasing problem in industrialised countries and although it is a major socio-economic problem in terms of medical costs and lost productivity, relatively little is known about the processes underlying the development of the condition. This is in part due to the complex interactions between bone, muscle, nerves and other soft tissues of the spine, and the fact that direct observation and/or measurement of the human spine is not possible using non-invasive techniques. Biomechanical models have been used extensively to estimate the forces and moments experienced by the spine. These models provide a means of estimating the internal parameters which can not be measured directly. However, application of most of the models currently available is restricted to tasks resembling those for which the model was designed due to the simplified representation of the anatomy. The aim of this research was to develop a biomechanical model to investigate the changes in forces and moments which are induced by muscle injury. In order to accurately simulate muscle injuries a detailed quasi-static three dimensional model representing the anatomy of the lumbar spine was developed. This model includes the nine major force generating muscles of the region (erector spinae, comprising the longissimus thoracis and iliocostalis lumborum; multifidus; quadratus lumborum; latissimus dorsi; transverse abdominis; internal oblique and external oblique), as well as the thoracolumbar fascia through which the transverse abdominis and parts of the internal oblique and latissimus dorsi muscles attach to the spine. The muscles included in the model have been represented using 170 muscle fascicles each having their own force generating characteristics and lines of action. Particular attention has been paid to ensuring the muscle lines of action are anatomically realistic, particularly for muscles which have broad attachments (e.g. internal and external obliques), muscles which attach to the spine via the thoracolumbar fascia (e.g. transverse abdominis), and muscles whose paths are altered by bony constraints such as the rib cage (e.g. iliocostalis lumborum pars thoracis and parts of the longissimus thoracis pars thoracis). In this endeavour, a separate sub-model which accounts for the shape of the torso by modelling it as a series of ellipses has been developed to model the lines of action of the oblique muscles. Likewise, a separate sub-model of the thoracolumbar fascia has also been developed which accounts for the middle and posterior layers of the fascia, and ensures that the line of action of the posterior layer is related to the size and shape of the erector spinae muscle. Published muscle activation data are used to enable the model to predict the maximum forces and moments that may be generated by the muscles. These predictions are validated against published experimental studies reporting maximum isometric moments for a variety of exertions. The model performs well for fiexion, extension and lateral bend exertions, but underpredicts the axial twist moments that may be developed. This discrepancy is most likely the result of differences between the experimental methodology and the modelled task. The application of the model is illustrated using examples of muscle injuries created by surgical procedures. The three examples used represent a posterior surgical approach to the spine, an anterior approach to the spine and uni-lateral total hip replacement surgery. Although the three examples simulate different muscle injuries, all demonstrate the production of significant asymmetrical moments and/or reduced joint compression following surgical intervention. This result has implications for patient rehabilitation and the potential for further injury to the spine. The development and application of the model has highlighted a number of areas where current knowledge is deficient. These include muscle activation levels for tasks in postures other than upright standing, changes in spinal kinematics following surgical procedures such as spinal fusion or fixation, and a general lack of understanding of how the body adjusts to muscle injuries with respect to muscle activation patterns and levels, rate of recovery from temporary injuries and compensatory actions by other muscles. Thus the comprehensive and innovative anatomical model which has been developed not only provides a tool to predict the forces and moments experienced by the intervertebral joints of the spine, but also highlights areas where further clinical research is required.

Investigating functional thinking in the elementary classroom : foundations of early algebraic reasoning

This paper examines the development of student functional thinking during a teaching experiment that was conducted in two classrooms with a total of 45 children whose average age was nine years and six months. The teaching comprised four lessons taught by a researcher, with a second researcher and classroom teacher acting as participant observers. These lessons were designed to enable students to build mental representations in order to explore the use of function tables by focusing on the relationship between input and output numbers with the intention of extracting the algebraic nature of the arithmetic involved. All lessons were videotaped. The results indicate that elementary students are not only capable of developing functional thinking but also of communicating their thinking both verbally and symbolically.

Reunitising hundredths: prototypic and nonprototypic representations

This paper reports on a study in which 29 Year 6 students (selected from the top 30% of 176 Year 6 students) were individually interviewed to explore their ability to reunitise hundredths as tenths (Behr, Harel, Post & Lesh, 1992) when represented by prototypic (PRO) and nonprototypic (NPRO) models. The results showed that 55.2% of the students were able to unitise both models and that reunitising was more successful with the PRO model. The interviews revealed that many of these students had incomplete, fragmented or non-existent structural knowledge of the reunitising process and often relied on syntactic clues to complete the tasks. The implication for teaching is that instruction should not be limited to PRO representations of the part/whole notion of fraction and that the basic structures (equal parts, link between name and number of equal parts) of the part/whole notion needs to be revisited often.

Mathematical modelling in the early school years

In this article we explore young children's development of mathematical knowledge and reasoning processes as they worked two modelling problems (the Butter Beans Problem and the Airplane Problem). The problems involve authentic situations that need to be interpreted and described in mathematical ways. Both problems include tables of data, together with background information containing specific criteria to be considered in the solution process. Four classes of third-graders (8 years of age) and their teachers participated in the 6-month program, which included preparatory modelling activities along with professional development for the teachers. In discussing our findings we address: (a) Ways in which the children applied their informal, personal knowledge to the problems; (b) How the children interpreted the tables of data, including difficulties they experienced; (c) How the children operated on the data, including aggregating and comparing data, and looking for trends and patterns; (c) How the children developed important mathematical ideas; and (d) Ways in which the children represented their mathematical understandings.

Skydweller and representations of the adolescent crisis : group identity versus alienation

This document contains a creative work – the text of a young adult novel, Skydweller – and an exegesis discussing the ways in which identity and the adolescent crisis of group identity versus alienation are represented in young adult science fiction/fantasy novels.

Learning spatial concepts from RatSLAM representations

RatSLAM is a biologically-inspired visual SLAM and navigation system that has been shown to be effective indoors and outdoors on real robots. The spatial representation at the core of RatSLAM, the experience map, forms in a distributed fashion as the robot learns the environment. The activity in RatSLAM’s experience map possesses some geometric properties, but still does not represent the world in a human readable form. A new system, dubbed RatChat, has been introduced to enable meaningful communication with the robot. The intention is to use the “language games” paradigm to build spatial concepts that can be used as the basis for communication. This paper describes the first step in the language game experiments, showing the potential for meaningful categorization of the spatial representations in RatSLAM.

Experience mapping : producing spatially continuous environment representations using RatSLAM

RatSLAM is a system for vision-based Simultaneous Localisation and Mapping (SLAM) inspired by models of the rodent hippocampus. The system can produce stable representations of large complex environments during robot experiments in both indoor and outdoor environments. These representations are both topological and metric in nature, and can involve multiple representations of the same place as well as discontinuities. In this paper we describe a new technique known as experience mapping that can be used online with the RatSLAM system to produce world representations known as experience maps. These maps group together multiple place representations and are spatially continuous. A number of experiments have been conducted in simulation and a real world office environment. These experiments demonstrate the high degree to which experience maps are representative of the spatial arrangement of the environment.

Exploring visual features through Gabor representations for facial expression detection

Gabor representations have been widely used in facial analysis (face recognition, face detection and facial expression detection) due to their biological relevance and computational properties. Two popular Gabor representations used in literature are: 1) Log-Gabor and 2) Gabor energy filters. Even though these representations are somewhat similar, they also have distinct differences as the Log-Gabor filters mimic the simple cells in the visual cortex while the Gabor energy filters emulate the complex cells, which causes subtle differences in the responses. In this paper, we analyze the difference between these two Gabor representations and quantify these differences on the task of facial action unit (AU) detection. In our experiments conducted on the Cohn-Kanade dataset, we report an average area underneath the ROC curve (A`) of 92.60% across 17 AUs for the Gabor energy filters, while the Log-Gabor representation achieved an average A` of 96.11%. This result suggests that small spatial differences that the Log-Gabor filters pick up on are more useful for AU detection than the differences in contours and edges that the Gabor energy filters extract.

Predicting active material utilisation in LiFePO4 electrodes using a multi-scale mathematical model

A mathematical model is developed to simulate the discharge of a LiFePO4 cathode. This model contains 3 size scales, which match with experimental observations present in the literature on the multi-scale nature of LiFePO4 material. A shrinking-core is used on the smallest scale to represent the phase-transition of LiFePO4 during discharge. The model is then validated against existing experimental data and this validated model is then used to investigate parameters that influence active material utilisation. Specifically the size and composition of agglomerates of LiFePO4 crystals is discussed, and we investigate and quantify the relative effects that the ionic and electronic conductivities within the oxide have on oxide utilisation. We find that agglomerates of crystals can be tolerated under low discharge rates. The role of the electrolyte in limiting (cathodic) discharge is also discussed, and we show that electrolyte transport does limit performance at high discharge rates, confirming the conclusions of recent literature.