Complex modelling in the primary/middle school years

The world’s increasing complexity, competitiveness, interconnectivity, and dependence on technology generate new challenges for nations and individuals that cannot be met by continuing education as usual (Katehi, Pearson, & Feder, 2009). With the proliferation of complex systems have come new technologies for communication, collaboration, and conceptualisation. These technologies have led to significant changes in the forms of mathematical and scientific thinking that are required beyond the classroom. Modelling, in its various forms, can develop and broaden children’s mathematical and scientific thinking beyond the standard curriculum. This paper first considers future competencies in the mathematical sciences within an increasingly complex world. Next, consideration is given to interdisciplinary problem solving and models and modelling. Examples of complex, interdisciplinary modelling activities across grades are presented, with data modelling in 1st grade, model-eliciting in 4th grade, and engineering-based modelling in 7th-9th grades.

Optimising container transfers at multimodal terminals

The use of containers have greatly reduced handling operations at ports and at all other transfer points, thus increasing the efficiency and speed of transportation. This was done in an attempt to cut down the cost of maritime transport, mainly by reducing cargo handling and costs, and ships' time in port by speeding up handling operations. This paper discusses the major factors influencing the transfer efficiency of seaport container terminals. A network model is designed to analyse container progress in the system and applied to a seaport container terminal. The model presented here can be seen as a decision support system in the context of investment appraisal of multimodal container terminals. (C) 2000 Elsevier Science Ltd.

Velocity jump processes with proliferation

Cell invasion involves a population of cells that migrate along a substrate and proliferate to a carrying capacity density. These two processes, combined, lead to invasion fronts that move into unoccupied tissues. Traditional modelling approaches based on reaction–diffusion equations cannot incorporate individual–level observations of cell velocity, as information propagates with infinite velocity according to these parabolic models. In contrast, velocity jump processes allow us to explicitly incorporate individual–level observations of cell velocity, thus providing an alternative framework for modelling cell invasion. Here, we introduce proliferation into a standard velocity–jump process and show that the standard model does not support invasion fronts. Instead, we find that crowding effects must be explicitly incorporated into a proliferative velocity–jump process before invasion fronts can be observed. Our observations are supported by numerical and analytical solutions of a novel coupled system of partial differential equations, including travelling wave solutions, and associated random walk simulations.

Tracking structural development through data modelling in highly able Grade 1 students

A 3-year longitudinal study Transforming Children’s Mathematical and Scientific Development integrates, through data modelling, a pedagogical approach focused on mathematical patterns and structural relationships with learning in science. As part of this study, a purposive sample of 21 highly able Grade 1 students was engaged in an innovative data modelling program. In the majority of students, representational development was observed. Their complex graphs depicting categorical and continuous data revealed a high level of structure and enabled identification of structural features critical to this development.

STI-GMaS : an open-source environment for simulation of sexually-transmitted infections

Background Sexually-transmitted pathogens often have severe reproductive health implications if treatment is delayed or absent, especially in females. The complex processes of disease progression, namely replication and ascension of the infection through the genital tract, span both extracellular and intracellular physiological scales, and in females can vary over the distinct phases of the menstrual cycle. The complexity of these processes, coupled with the common impossibility of obtaining comprehensive and sequential clinical data from individual human patients, makes mathematical and computational modelling valuable tools in developing our understanding of the infection, with a view to identifying new interventions. While many within-host models of sexually-transmitted infections (STIs) are available in existing literature, these models are difficult to deploy in clinical/experimental settings since simulations often require complex computational approaches. Results We present STI-GMaS (Sexually-Transmitted Infections – Graphical Modelling and Simulation), an environment for simulation of STI models, with a view to stimulating the uptake of these models within the laboratory or clinic. The software currently focuses upon the representative case-study of Chlamydia trachomatis, the most common sexually-transmitted bacterial pathogen of humans. Here, we demonstrate the use of a hybrid PDE–cellular automata model for simulation of a hypothetical Chlamydia vaccination, demonstrating the effect of a vaccine-induced antibody in preventing the infection from ascending to above the cervix. This example illustrates the ease with which existing models can be adapted to describe new studies, and its careful parameterisation within STI-GMaS facilitates future tuning to experimental data as they arise. Conclusions STI-GMaS represents the first software designed explicitly for in-silico simulation of STI models by non-theoreticians, thus presenting a novel route to bridging the gap between computational and clinical/experimental disciplines. With the propensity for model reuse and extension, there is much scope within STI-GMaS to allow clinical and experimental studies to inform model inputs and drive future model development. Many of the modelling paradigms and software design principles deployed to date transfer readily to other STIs, both bacterial and viral; forthcoming releases of STI-GMaS will extend the software to incorporate a more diverse range of infections.

Problem Solving with Delphi

The purpose of the book is to use Delphi as a vehicle to introduce some fundamental algorithms and to illustrate several mathematical and problem-solving techniques. This book is therefore intended to be more of a reference for problem-solving, with the solution expressed in Delphi. It introduces a somewhat eclectic collection of material, much of which will not be found in a typical book on Pascal or Delphi. Many of the topics have been used by the author over a period of about ten years at Bond University, Australia in various subjects from 1993 to 2003. Much of the work was connected with a data structures subject (second programming course) conducted variously in MODULA-2, Oberon and Delphi, at Bond University, however there is considerable other, more recent material, e.g., a chapter on Sudoku.

Structuring representations of continuous data through investigations in grade 1

The study of data modelling with elementary students involves the analysis of a developmental process beginning with children’s investigations of meaningful contexts: visualising, structuring, and representing data and displaying data in simple graphs (English, 2012; Lehrer & Schauble, 2005; Makar, Bakker, & Ben-Zvi, 2011). A 3-year longitudinal study investigated young children’s data modelling, integrating mathematical and scientific investigations. One aspect of this study involved a researcher-led teaching experiment with 21 mathematically able Grade 1 students. The study aimed to describe explicit developmental features of students’ representations of continuous data...

Investigation of the effects of the fatty acid profile on fuel properties using a Multi-Criteria Decision Analysis

The structural features of fatty acids in biodiesel, including degree of unsaturation, percentage of saturated fatty acids and average chain length, influence important fuel properties such as cetane number, iodine value, density, kinematic viscosity, higher heating value and oxidation stability. The composition of fatty acid esters within the fuel should therefore be in the correct ratio to ensure fuel properties are within international biodiesel standards such as ASTM 6751 or EN 14214. This study scrutinises the influence of fatty acid composition and individual fatty acids on fuel properties. Fuel properties were estimated based on published equations, and measured according to standard procedure ASTM D6751 and EN 14214 to confirm the influences of the fatty acid profile. Based on fatty acid profile-derived calculations, the cetane number of the microalgal biodiesel was estimated to be 11.6, but measured 46.5, which emphasises the uncertainty of the method used for cetane number calculation. Multi-criteria decision analysis (MCDA), PROMETHEE-GAIA, was used to determine the influence of individual fatty acids on fuel properties in the GAIA plane. Polyunsaturated fatty acids increased the iodine value and had a negative influence on cetane number. Kinematic viscosity was negatively influenced by some long chain polyunsaturated fatty acids such as C20:5 and C22:6 and some of the more common saturated fatty acids C14:0 and C18:0. The positive impact of average chain length on higher heating value was also confirmed in the GAIA plane

Influence of LRP5 polymorphisms on normal variation in BMD

Genetic studies based on cohorts with rare and extreme bone phenotypes have shown that the LRP5 gene is an important genetic modulator of BMD. Using family-based and case-control approaches, this study examines the role of the LRP5 gene in determining normal population variation of BMD and describes significant association and suggestive linkage between LRP5 gene polymorphisms and BMD in >900 individuals with a broad range of BMD. Introduction: Osteoporosis is a common, highly heritable condition determined by complex interactions of genetic and environmental etiologies. Genetic factors alone can account for 50-80% of the interindividual variation in BMD. Mutations in the LRP5 gene on chromosome 11q12-13 have been associated with rare syndromes characterized by extremely low or high BMD, but little is known about the contribution of this gene to the development of osteoporosis and determination of BMD in a normal population. Materials and Methods: To examine the entire spectrum of low to high BMD, 152 osteoporotic probands, their families (597 individuals), and 160 women with elevated BMD (T score > 2.5) were recruited. BMD at the lumbar spine, femoral neck, and hip were measured in each subject using DXA. Results: PAGE sequencing of the LRP5 gene revealed 10 single nucleotide polymorphisms (SNPs), 8 of which had allele frequencies of >5%, in exons 8, 9, 10, 15, and 18 and in introns 6, 7, and 21. Within families, a strong association was observed between an SNP at nucleotide C171346A in intron 21 and total hip BMD (p < 1 × 10-5 in men only, p = 0.0019 in both men and women). This association was also observed in comparisons of osteoporotic probands and unrelated elevated BMD in women (p = 0.03), along with associations with markers in exons 8 (C135242T, p = 0.007) and 9 (C141759T, p = 0.02). Haplotypes composed of two to three of the SNPs G121513A, C135242T, G138351A, and C141759T were strongly associated with BMD when comparing osteoporotic probands and high BMD cases (p < 0.003). An SNP at nucleotide C165215T in exon 18 was linked to BMD at the lumbar spine, femoral neck, and total hip (parametric LOD scores = 2.8, 2.5, and 2.2 and nonparametric LOD scores = 0.3, 1.1, and 2.2, respectively) but was not genetically associated with BMD variation. Conclusion: These results show that common LRP5 polymorphisms contribute to the determination of BMD in the general population.

Continuum percolation with discs having a distribution of radii

It is shown that for continuum percolation with overlapping discs having a distribution of radii, the net areal density of discs at percolation threshold depends non-trivially on the distribution, and is not bounded by any finite constant. Results of a Monte Carlo simulation supporting the argument are presented.

Orbital diamagnetism of a charged Brownian particle undergoing a birth-death process

Considers the magnetic response of a charged Brownian particle undergoing a stochastic birth-death process. The latter simulates the electron-hole pair production and recombination in semiconductors. The authors obtain non-zero, orbital diamagnetism which can be large without violating the Van Leeuwen theorem (1921).

Calculating the bound spectrum by path summation in action-angle variables

The density of states n(E) is calculated for a bound system whose classical motion is integrable, starting from an expression in terms of the trace of the time-dependent Green function. The novel feature is the use of action-angle variables. This has the advantages that the trace operation reduces to a trivial multiplication and the dependence of n(E) on all classical closed orbits with different topologies appears naturally. The method is contrasted with another, not applicable to integrable systems except in special cases, in which quantization arises from a single closed orbit which is assumed isolated and the trace taken by the method of stationary phase.

Expansion of high pressure gas into air - A more realistic blast wave model

In this paper, we consider a more realistic model of a spherical blast wave of moderate strength. An arbitrary number of terms for the series solution in each of the regions behind the main shock - the expansion region, the nearly uniform region outside the main expansion and the region between the contact surface and the main shock, have been generated and matched across the boundaries. We then study the convergence of the solution by using Pade approximation. It constitutes a genuine analytic solution for a moderately strong explosion, which, however, does not involve a secondary shock. The pressure distribution behind the shock however shows some significant changes in the location of the tail of the rarefaction and the interface, in comparison to the planar problem. The theory developed for the spherical blasts is also extended to cylindrical blasts. The results are compared with the numerical solution.