Quantifying spatial structure in experimental observations and agent-based simulations using pair-correlation functions

We define a pair-correlation function that can be used to characterize spatiotemporal patterning in experimental images and snapshots from discrete simulations. Unlike previous pair-correlation functions, the pair-correlation functions developed here depend on the location and size of objects. The pair-correlation function can be used to indicate complete spatial randomness, aggregation or segregation over a range of length scales, and quantifies spatial structures such as the shape, size and distribution of clusters. Comparing pair-correlation data for various experimental and simulation images illustrates their potential use as a summary statistic for calibrating discrete models of various physical processes.

Distinguishing between mean-field, moment dynamics and stochastic descriptions of birth-death-movement processes

Mathematical descriptions of birth–death–movement processes are often calibrated to measurements from cell biology experiments to quantify tissue growth rates. Here we describe and analyze a discrete model of a birth–death-movement process applied to a typical two–dimensional cell biology experiment. We present three different descriptions of the system: (i) a standard mean–field description which neglects correlation effects and clustering; (ii) a moment dynamics description which approximately incorporates correlation and clustering effects, and; (iii) averaged data from repeated discrete simulations which directly incorporates correlation and clustering effects. Comparing these three descriptions indicates that the mean–field and moment dynamics approaches are valid only for certain parameter regimes, and that both these descriptions fail to make accurate predictions of the system for sufficiently fast birth and death rates where the effects of spatial correlations and clustering are sufficiently strong. Without any method to distinguish between the parameter regimes where these three descriptions are valid, it is possible that either the mean–field or moment dynamics model could be calibrated to experimental data under inappropriate conditions, leading to errors in parameter estimation. In this work we demonstrate that a simple measurement of agent clustering and correlation, based on coordination number data, provides an indirect measure of agent correlation and clustering effects, and can therefore be used to make a distinction between the validity of the different descriptions of the birth–death–movement process.

Understanding mild acid pretreatment of sugarcane bagasse through particle scale modeling

Sugarcane bagasse is an abundant and sustainable resource, generated as a by-product of sugarcane milling. The cellulosic material within bagasse can be broken down into glucose molecules and fermented to produce ethanol, making it a promising feedstock for biofuel production. Mild acid pretreatment hydrolyses the hemicellulosic component of biomass, thus allowing enzymes greater access to the cellulosic substrate during saccharification. A particle-scale mathematical model describing the mild acid pretreatment of sugarcane bagasse has been developed, using a volume averaged framework. Discrete population-balance equations are used to characterise the polymer degradation kinetics, and diffusive effects account for mass transport within the cell wall of the bagasse. As the fibrous material hydrolyses over time, variations in the porosity of the cell wall and the downstream effects on the reaction kinetics are accounted for using conservation of volume arguments. Non-dimensionalization of the model equations reduces the number of parameters in the system to a set of four dimensionless ratios that compare the timescales of different reaction and diffusion events. Theoretical yield curves are compared to macroscopic experimental observations from the literature and inferences are made as to constraints on these “unknown” parameters. These results enable connections to be made between experimental data and the underlying thermodynamics of acid pretreatment. Consequently, the results suggest that data-fitting techniques used to obtain kinetic parameters should be carefully applied, with prudent consideration given to the chemical and physiological processes being modeled.

Separable and non-separable discrete wavelet transform based texture features and image classification of breast thermograms

Highly sensitive infrared cameras can produce high-resolution diagnostic images of the temperature and vascular changes of breasts. Wavelet transform based features are suitable in extracting the texture difference information of these images due to their scale-space decomposition. The objective of this study is to investigate the potential of extracted features in differentiating between breast lesions by comparing the two corresponding pectoral regions of two breast thermograms. The pectoral regions of breastsare important because near 50% of all breast cancer is located in this region. In this study, the pectoral region of the left breast is selected. Then the corresponding pectoral region of the right breast is identified. Texture features based on the first and the second sets of statistics are extracted from wavelet decomposed images of the pectoral regions of two breast thermograms. Principal component analysis is used to reduce dimension and an Adaboost classifier to evaluate classification performance. A number of different wavelet features are compared and it is shown that complex non-separable 2D discrete wavelet transform features perform better than their real separable counterparts.

Incorporating spatial correlations into multispecies mean-field models

In biology, we frequently observe different species existing within the same environment. For example, there are many cell types in a tumour, or different animal species may occupy a given habitat. In modelling interactions between such species, we often make use of the mean field approximation, whereby spatial correlations between the locations of individuals are neglected. Whilst this approximation holds in certain situations, this is not always the case, and care must be taken to ensure the mean field approximation is only used in appropriate settings. In circumstances where the mean field approximation is unsuitable we need to include information on the spatial distributions of individuals, which is not a simple task. In this paper we provide a method that overcomes many of the failures of the mean field approximation for an on-lattice volume-excluding birth-death-movement process with multiple species. We explicitly take into account spatial information on the distribution of individuals by including partial differential equation descriptions of lattice site occupancy correlations. We demonstrate how to derive these equations for the multi-species case, and show results specific to a two-species problem. We compare averaged discrete results to both the mean field approximation and our improved method which incorporates spatial correlations. We note that the mean field approximation fails dramatically in some cases, predicting very different behaviour from that seen upon averaging multiple realisations of the discrete system. In contrast, our improved method provides excellent agreement with the averaged discrete behaviour in all cases, thus providing a more reliable modelling framework. Furthermore, our method is tractable as the resulting partial differential equations can be solved efficiently using standard numerical techniques.

Data and measurement in Year 4 of the national curriculum : Mathematics

The activities introduced here were used in association with a research project in four Year 4 classrooms and are suggested as a motivating way to address several criteria for Measurement and Data in the Australian Curriculum: Mathematics. The activities involve measuring the arm span of one student in a class many times and then of all students once.

Perspectives on Reconceptualizing Early Mathematics Learning

This introductory section provides an overview of the different perspectives on reconceptualizing early mathematics learning. The chapters provide a broad scope in their topics and approaches to advancing young children’s mathematical learning. They incorporate studies that highlight the importance of pattern and structure across the curriculum, studies that target particular content such as statistics, early algebra, and beginning number, and studies that consider how technology and other tools can facilitate early mathematical development. Reconceptualizing the professional learning of teachers in promoting young children’s mathematics, including a consideration of the role of play, is also addressed. Although these themes are diffused throughout the chapters, we restrict our introduction to the core focus of each of the chapters.

Reconceptualizing Early Mathematics Learning : The Fundamental Role of Pattern and Structure

The Pattern and Structure Mathematics Awareness Program (PASMAP) was developed concurrently with the studies of AMPS and the development of the Pattern and Structure Assessment (PASA) interview. We summarize some early classroom-based teaching studies and describe the PASMAP that resulted. A large-scale two-year longitudinal study, Reconceptualizing Early Mathematics Learning (REML) resulted. We provide an overview of the REML study and discuss the consequences for our view of early mathematics learning. A purposive sample of four large primary schools, two in Sydney and two in Brisbane, representing 316 students from diverse socio-economic and cultural contexts, participated in an evaluation of the PASMAP intervention throughout the 2009 school year and a follow-up assessment in 2010. Two different mathematics programs were implemented: in each school, two Kindergarten teachers implemented the PASMAP and another two implemented their regular program. The study shows that both groups of students made substantial gains on the ‘I Can Do Maths’ standardized assessment and the PASA interview, but highly significant differences were found on the latter with PASMAP students outperforming the regular group on PASA scores. Qualitative analysis of students’ responses for structural development showed increased levels for the PASMAP students. Implications for pedagogy and curriculum are discussed.

Accelerating the Mathematics Learning of Low Socio-Economic Status Junior Secondary Students : an early report

The Accelerating the Mathematics Learning of Low Socio-Economic Status Junior Secondary Students project aims to address the issues faced by very underperforming mathematics students as they enter high school. Its aim is to accelerate learning of mathematics through a vertical curriculum to enable students to access Year 10 mathematics subjects, thus improving life chances. This paper reports upon the theory underpinning this project and illustrates it with examples of the curriculum that has been designed to achieve acceleration.

A RBF meshless approach for modeling a fractal mobile/immobile transport model

A fractional differential equation is used to describe a fractal model of mobile/immobile transport with a power law memory function. This equation is the limiting equation that governs continuous time random walks with heavy tailed random waiting times. In this paper, we firstly propose a finite difference method to discretize the time variable and obtain a semi-discrete scheme. Then we discuss its stability and convergence. Secondly we consider a meshless method based on radial basis functions (RBFs) to discretize the space variable. In contrast to conventional FDM and FEM, the meshless method is demonstrated to have distinct advantages: calculations can be performed independent of a mesh, it is more accurate and it can be used to solve complex problems. Finally the convergence order is verified from a numerical example which is presented to describe a fractal model of mobile/immobile transport process with different problem domains. The numerical results indicate that the present meshless approach is very effective for modeling and simulating fractional differential equations, and it has good potential in the development of a robust simulation tool for problems in engineering and science that are governed by various types of fractional differential equations.

A conceptual framework for the cultural integration of cooperative learning : a Thai primary mathematics education perspective

The Thailand education reform adopted cooperative learning to improve the quality of education. However, it has been reported that the introduction and maintenance of cooperative learning has been difficult and uncertain because of the cultural differences. The study proposed a conceptual framework developed based on making a connection between Thai cultures and cooperative learning elements, and implemented a small-scale research project in a Thai primary mathematics class with a teacher and thirty-two Grade 4 students. The results uncovered that the three components including preparation of teachers, instructional strategies and preparation of students can be vehicles for the culture integration in cooperative learning.

Lattice-free descriptions of collective motion with crowding and adhesion

Cell-to-cell adhesion is an important aspect of malignant spreading that is often observed in images from the experimental cell biology literature. Since cell-to-cell adhesion plays an important role in controlling the movement of individual malignant cells, it is likely that cell-to-cell adhesion also influences the spatial spreading of populations of such cells. Therefore, it is important for us to develop biologically realistic simulation tools that can mimic the key features of such collective spreading processes to improve our understanding of how cell-to-cell adhesion influences the spreading of cell populations. Previous models of collective cell spreading with adhesion have used lattice-based random walk frameworks which may lead to unrealistic results, since the agents in the random walk simulations always move across an artificial underlying lattice structure. This is particularly problematic in high-density regions where it is clear that agents in the random walk align along the underlying lattice, whereas no such regular alignment is ever observed experimentally. To address these limitations, we present a lattice-free model of collective cell migration that explicitly incorporates crowding and adhesion. We derive a partial differential equation description of the discrete process and show that averaged simulation results compare very well with numerical solutions of the partial differential equation.

The velocity synchronous discrete Fourier transform for order tracking in the field of rotating machinery

Diagnostics of rotating machinery has developed significantly in the last decades, and industrial applications are spreading in different sectors. Most applications are characterized by varying velocities of the shaft and in many cases transients are the most critical to monitor. In these variable speed conditions, fault symptoms are clearer in the angular/order domains than in the common time/frequency ones. In the past, this issue was often solved by synchronously sampling data by means of phase locked circuits governing the acquisition; however, thanks to the spread of cheap and powerful microprocessors, this procedure is nowadays rarer; sampling is usually performed at constant time intervals, and the conversion to the order domain is made by means of digital signal processing techniques. In the last decades different algorithms have been proposed for the extraction of an order spectrum from a signal sampled asynchronously with respect to the shaft rotational velocity; many of them (the so called computed order tracking family) use interpolation techniques to resample the signal at constant angular increments, followed by a common discrete Fourier transform to shift from the angular domain to the order domain. A less exploited family of techniques shifts directly from the time domain to the order spectrum, by means of modified Fourier transforms. This paper proposes a new transform, named velocity synchronous discrete Fourier transform, which takes advantage of the instantaneous velocity to improve the quality of its result, reaching performances that can challenge the computed order tracking.