Some Remarks on the Approximation of Transitional Density in Stochastic Differential Equations

Aijt-Sahalia (2002) introduced a method to estimate transitional probability densities of di®usion processes by means of Hermite expansions with coe±cients determined by means of Taylor series. This note describes a numerical procedure to ¯nd these coe±cients based on the calculation of moments. One advantage of this procedure is that it can be used e®ectively when the mathematical operations required to ¯nd closed-form expressions for these coe±cients are otherwise infeasible.

Computationally efficient numerical methods for time- and space-fractional Fokker–Planck equations

Fractional Fokker–Planck equations have been used to model several physical situations that present anomalous diffusion. In this paper, a class of time- and space-fractional Fokker–Planck equations (TSFFPE), which involve the Riemann–Liouville time-fractional derivative of order 1-α (α(0, 1)) and the Riesz space-fractional derivative (RSFD) of order μ(1, 2), are considered. The solution of TSFFPE is important for describing the competition between subdiffusion and Lévy flights. However, effective numerical methods for solving TSFFPE are still in their infancy. We present three computationally efficient numerical methods to deal with the RSFD, and approximate the Riemann–Liouville time-fractional derivative using the Grünwald method. The TSFFPE is then transformed into a system of ordinary differential equations (ODE), which is solved by the fractional implicit trapezoidal method (FITM). Finally, numerical results are given to demonstrate the effectiveness of these methods. These techniques can also be applied to solve other types of fractional partial differential equations.

One approach to finding evidence for the effectiveness of scientific visualisations in high school physics and chemistry education

Enormous amounts of money and energy are being devoted to the development, use and organisation of computer-based scientific visualisations (e.g. animations and simulations) in science education. It seems plausible that visualisations that enable students to gain visual access to scientific phenomena that are too large, too small or occur too quickly or too slowly to be seen by the naked eye, or to scientific concepts and models, would yield enhanced conceptual learning. When the literature is searched, however, it quickly becomes apparent that there is a dearth of quantitative evidence for the effectiveness of scientific visualisations in enhancing students’ learning of science concepts. This paper outlines an Australian project that is using innovative research methodology to gather evidence on this question in physics and chemistry classrooms.

Efficient solution of two-sided nonlinear space-fractional diffusion equations using fast Poisson preconditioners

We develop a fast Poisson preconditioner for the efficient numerical solution of a class of two-sided nonlinear space fractional diffusion equations in one and two dimensions using the method of lines. Using the shifted Gr¨unwald finite difference formulas to approximate the two-sided(i.e. the left and right Riemann-Liouville) fractional derivatives, the resulting semi-discrete nonlinear systems have dense Jacobian matrices owing to the non-local property of fractional derivatives. We employ a modern initial value problem solver utilising backward differentiation formulas and Jacobian-free Newton-Krylov methods to solve these systems. For efficient performance of the Jacobianfree Newton-Krylov method it is essential to apply an effective preconditioner to accelerate the convergence of the linear iterative solver. The key contribution of our work is to generalise the fast Poisson preconditioner, widely used for integer-order diffusion equations, so that it applies to the two-sided space fractional diffusion equation. A number of numerical experiments are presented to demonstrate the effectiveness of the preconditioner and the overall solution strategy.

Diophantine equations as a context for technology-enhanced training in conjecturing and proving

Solving indeterminate algebraic equations in integers is a classic topic in the mathematics curricula across grades. At the undergraduate level, the study of solutions of non-linear equations of this kind can be motivated by the use of technology. This article shows how the unity of geometric contextualization and spreadsheet-based amplification of this topic can provide a discovery experience for prospective secondary teachers and information technology students. Such experience can be extended to include a transition from a computationally driven conjecturing to a formal proof based on a number of simple yet useful techniques.

A method of lines approach for modelling saltwater intrusion in coastal aquifiers

The equations governing saltwater intrusion in coastal aquifers are complex. Backward Euler time stepping approaches are often used to advance the solution to these equations in time, which typically requires that small time steps be taken in order to ensure that an accurate solution is obtained. We show that a method of lines approach incorporating variable order backward differentiation formulas can greatly improve the efficiency of the time stepping process.

Political theory and theoretical physics

This paper explains, somewhat along a Simmelian line, that political theory may produce practical and universal theories like those developed in theoretical physics. The reasoning behind this paper is to show that the Element of Democracy Theory may be true by way of comparing it to Einstein’s Special Relativity – specifically concerning the parameters of symmetry, unification, simplicity, and utility. These parameters are what make a theory in physics as meeting them not only fits with current knowledge, but also produces paths towards testing (application). As the Element of Democracy Theory meets these same parameters, it could settle the debate concerning the definition of democracy. This will be shown firstly by discussing why no one has yet achieved a universal definition of democracy; secondly by explaining the parameters chosen (as in why these and not others confirm or scuttle theories); and thirdly by comparing how Special Relativity and the Element of Democracy match the parameters.

Political theory and theoretical physics [Revised]

This paper explains, somewhat along a Simmelian line, that political theory may produce practical and universal theories like those developed in theoretical physics. The reasoning behind this paper is to show that the Element of Democracy Theory may be true by way of comparing it to Einstein’s Special Relativity – specifically concerning the parameters of symmetry, unification, simplicity, and utility. These parameters are what make a theory in physics as meeting them not only fits with current knowledge, but also produces paths towards testing (application). As the Element of Democracy Theory meets these same parameters, it could settle the debate concerning the definition of democracy. This will be shown firstly by discussing why no one has yet achieved a universal definition of democracy; secondly by explaining the parameters chosen (as in why these and not others confirm or scuttle theories); and thirdly by comparing how Special Relativity and the Element of Democracy match the parameters.

Accurate series solutions for gravity-driven Stokes waves

In the past, high order series expansion techniques have been used to study the nonlinear equations that govern the form of periodic Stokes waves moving steadily on the surface of an inviscid fluid. In the present study, two such series solutions are recomputed using exact arithmetic, eliminating any loss of accuracy due to accumulation of round-off error, allowing a much greater number of terms to be found with confidence. It is shown that higher order behaviour of series generated by the solution casts doubt over arguments that rely on estimating the series’ radius of convergence. Further, the exact nature of the series is used to shed light on the unusual nature of convergence of higher order Pade approximants near the highest wave. Finally, it is concluded that, provided exact values are used in the series, these Pade approximants prove very effective in successfully predicting three turning points in both the dispersion relation and the total energy.

Political theory and theoretical physics

This paper argues, somewhat along a Simmelian line, that political theory may produce practical and universal theories like those developed in theoretical physics. The reasoning behind this paper is to show that the theory of ‘basic democracy’ may be true by way of comparing it to Einstein’s Special Relativity – specifically concerning the parameters of symmetry, unification, simplicity, and utility. These parameters are what make a theory in physics as meeting them not only fits with current knowledge, but also produces paths towards testing (application). As the theory of ‘basic democracy’ may meet these same parameters, it could settle the debate concerning the definition of democracy. This will be argued firstly by discussing what the theory of ‘basic democracy’ is and why it differs from previous work; secondly by explaining the parameters chosen (as in why these and not others confirm or scuttle theories); and thirdly by comparing how Special Relativity and the theory of ‘basic democracy’ may match the parameters.

A parallel computation of power system equations

Streaming SIMD Extensions (SSE) is a unique feature embedded in the Pentium III and P4 classes of microprocessors. By fully exploiting SSE, parallel algorithms can be implemented on a standard personal computer and a theoretical speedup of four can be achieved. In this paper, we demonstrate the implementation of a parallel LU matrix decomposition algorithm for solving power systems network equations with SSE and discuss advantages and disadvantages of this approach.

SSE based parallel solution for power systems network equations

Streaming SIMD Extensions (SSE) is a unique feature embedded in the Pentium III class of microprocessors. By fully exploiting SSE, parallel algorithms can be implemented on a standard personal computer and a theoretical speedup of four can be achieved. In this paper, we demonstrate the implementation of a parallel LU matrix decomposition algorithm for solving power systems network equations with SSE and discuss advantages and disadvantages of this approach.

Body composition and its relationship to metabolic risk factors in Asian children

Obesity is a major public health problem in both developed and developing countries. The body mass index (BMI) is the most common index used to define obesity. The universal application of the same BMI classification across different ethnic groups is being challenged due to the inability of the index to differentiate fat mass (FM) and fat�]free mass (FFM) and the recognized ethnic differences in body composition. A better understanding of the body composition of Asian children from different backgrounds would help to better understand the obesity�]related health risks of people in this region. Moreover, the limitations of the BMI underscore the necessity to use where possible, more accurate measures of body fat assessment in research and clinical settings in addition to BMI, particularly in relation to the monitoring of prevention and treatment efforts. The aim of the first study was to determine the ethnic difference in the relationship between BMI and percent body fat (%BF) in pre�]pubertal Asian children from China, Lebanon, Malaysia, the Philippines, and Thailand. A total of 1039 children aged 8�]10 y were recruited using a non�]random purposive sampling approach aiming to encompass a wide BMI range from the five countries. Percent body fat (%BF) was determined using the deuterium dilution technique to quantify total body water (TBW) and subsequently derive proportions of FM and FFM. The study highlighted the sex and ethnic differences between BMI and %BF in Asian children from different countries. Girls had approximately 4.0% higher %BF compared with boys at a given BMI. Filipino boys tended to have a lower %BF than their Chinese, Lebanese, Malay and Thai counterparts at the same age and BMI level (corrected mean %BF was 25.7�}0.8%, 27.4�}0.4%, 27.1�}0.6%, 27.7�}0.5%, 28.1�}0.5% for Filipino, Chinese, Lebanese, Malay and Thai boys, respectively), although they differed significantly from Thai and Malay boys. Thai girls had approximately 2.0% higher %BF values than Chinese, Lebanese, Filipino and Malay counterparts (however no significant difference was seen among the four ethnic groups) at a given BMI (corrected mean %BF was 31.1�}0.5%, 28.6�}0.4%, 29.2�}0.6%, 29.5�}0.6%, 29.5�}0.5% for Thai, Chinese, Lebanese, Malay and Filipino girls, respectively). However, the ethnic difference in BMI�]%BF relationship varied by BMI. Compared with Caucasians, Asian children had a BMI 3�]6 units lower for a given %BF. More than one third of obese Asian children in the study were not identified using the WHO classification and more than half were not identified using the International Obesity Task Force (IOTF) classification. However, use of the Chinese classification increased the sensitivity by 19.7%, 18.1%, 2.3%, 2.3%, and 11.3% for Chinese, Lebanese, Malay, Filipino and Thai girls, respectively. A further aim of the first study was to determine the ethnic difference in body fat distribution in pre�]pubertal Asian children from China, Lebanon, Malaysia, and Thailand. The skin fold thicknesses, height, weight, waist circumference (WC) and total adiposity (as determined by deuterium dilution technique) of 922 children from the four countries was assessed. Chinese boys and girls had a similar trunk�]to�]extremity skin fold thickness ratio to Thai counterparts and both groups had higher ratios than the Malays and Lebanese at a given total FM. At a given BMI, both Chinese and Thai boys and girls had a higher WC than Malays and Lebanese (corrected mean WC was 68.1�}0.2 cm, 67.8�}0.3 cm, 65.8�}0.4 cm, 64.1�}0.3 cm for Chinese, Thai, Lebanese and Malay boys, respectively; 64.2�}0.2 cm, 65.0�}0.3 cm, 62.9�}0.4 cm, 60.6�}0.3 cm for Chinese, Thai, Lebanese and Malay girls, respectively). Chinese boys and girls had lower trunk fat adjusted subscapular/suprailiac skinfold ratio compared with Lebanese and Malay counterparts. The second study aimed to develop and cross�]validate bioelectrical impedance analysis (BIA) prediction equations of TBW and FFM for Asian pre�]pubertal children from China, Lebanon, Malaysia, the Philippines, and Thailand. Data on height, weight, age, gender, resistance and reactance measured by BIA were collected from 948 Asian children (492 boys and 456 girls) aged 8�]10 y from the five countries. The deuterium dilution technique was used as the criterion method for the estimation of TBW and FFM. The BIA equations were developed from the validation group (630 children randomly selected from the total sample) using stepwise multiple regression analysis and cross�]validated in a separate group (318 children) using the Bland�]Altman approach. Age, gender and ethnicity influenced the relationship between the resistance index (RI = height2/resistance), TBW and FFM. The BIA prediction equation for the estimation of TBW was: TBW (kg) = 0.231�~Height2 (cm)/resistance (ƒ¶) + 0.066�~Height (cm) + 0.188�~Weight (kg) + 0.128�~Age (yr) + 0.500�~Sex (male=1, female=0) . 0.316�~Ethnicity (Thai ethnicity=1, others=0) �] 4.574, and for the estimation of FFM: FFM (kg) = 0.299�~Height2 (cm)/resistance (ƒ¶) + 0.086�~Height (cm) + 0.245�~Weight (kg) + 0.260�~Age (yr) + 0.901�~Sex (male=1, female=0) �] 0.415�~Ethnicity (Thai ethnicity=1, others=0) �] 6.952. The R2 was 88.0% (root mean square error, RSME = 1.3 kg), 88.3% (RSME = 1.7 kg) for TBW and FFM equation, respectively. No significant difference between measured and predicted TBW and between measured and predicted FFM for the whole cross�]validation sample was found (bias = �]0.1�}1.4 kg, pure error = 1.4�}2.0 kg for TBW and bias = �]0.2�}1.9 kg, pure error = 1.8�}2.6 kg for FFM). However, the prediction equation for estimation of TBW/FFM tended to overestimate TBW/FFM at lower levels while underestimate at higher levels of TBW/FFM. Accuracy of the general equation for TBW and FFM compared favorably with both BMI�]specific and ethnic�]specific equations. There were significant differences between predicted TBW and FFM from external BIA equations derived from Caucasian populations and measured values in Asian children. There were three specific aims of the third study. The first was to explore the relationship between obesity and metabolic syndrome and abnormalities in Chinese children. A total of 608 boys and 800 girls aged 6�]12 y were recruited from four cities in China. Three definitions of pediatric metabolic syndrome and abnormalities were used, including the International Diabetes Federation (IDF) and National Cholesterol Education Program (NCEP) definition for adults modified by Cook et al. and de Ferranti et al. The prevalence of metabolic syndrome varied with different definitions, was highest using the de Ferranti definition (5.4%, 24.6% and 42.0%, respectively for normal�]weight, overweight and obese children), followed by the Cook definition (1.5%, 8.1%, and 25.1%, respectively), and the IDF definition (0.5%, 1.8% and 8.3%, respectively). Overweight and obese children had a higher risk of developing the metabolic syndrome compared to normal�]weight children (odds ratio varied with different definitions from 3.958 to 6.866 for overweight children, and 12.640�]26.007 for obese children). Overweight and obesity also increased the risk of developing metabolic abnormalities. Central obesity and high triglycerides (TG) were the most common while hyperglycemia was the least frequent in Chinese children regardless of different definitions. The second purpose was to determine the best obesity index for the prediction of cardiovascular (CV) risk factor clustering across a 2�]y follow�]up among BMI, %BF, WC and waist�]to�]height ratio (WHtR) in Chinese children. Height, weight, WC, %BF as determined by BIA, blood pressure, TG, high�]density lipoprotein cholesterol (HDL�]C), and fasting glucose were collected at baseline and 2 years later in 292 boys and 277 girls aged 8�]10 y. The results showed the percentage of children who remained overweight/obese defined on the basis of BMI, WC, WHtR and %BF was 89.7%, 93.5%, 84.5%, and 80.4%, respectively after 2 years. Obesity indices at baseline significantly correlated with TG, HDL�]C, and blood pressure at both baseline and 2 years later with a similar strength of correlations. BMI at baseline explained the greatest variance of later blood pressure. WC at baseline explained the greatest variance of later HDL�]C and glucose, while WHtR at baseline was the main predictor of later TG. Receiver�]operating characteristic (ROC) analysis explored the ability of the four indices to identify the later presence of CV risk. The overweight/obese children defined on the basis of BMI, WC, WHtR or %BF were more likely to develop CV risk 2 years later with relative risk (RR) scores of 3.670, 3.762, 2.767, and 2.804, respectively. The final purpose of the third study was to develop age�] and gender�]specific percentiles of WC and WHtR and cut�]off points of WC and WHtR for the prediction of CV risk in Chinese children. Smoothed percentile curves of WC and WHtR were produced in 2830 boys and 2699 girls aged 6�]12 y randomly selected from southern and northern China using the LMS method. The optimal age�] and gender�]specific thresholds of WC and WHtR for the prediction of cardiovascular risk factors clustering were derived in a sub�]sample (n=1845) by ROC analysis. Age�] and gender�]specific WC and WHtR percentiles were constructed. The WC thresholds were at the 90th and 84th percentiles for Chinese boys and girls, respectively, with sensitivity and specificity ranging from 67.2% to 83.3%. The WHtR thresholds were at the 91st and 94th percentiles for Chinese boys and girls, respectively, with sensitivity and specificity ranging from 78.6% to 88.9%. The cut�]offs of both WC and WHtR were age�] and gender�]dependent. In conclusion, the current thesis quantifies the ethnic differences in the BMI�]%BF relationship and body fat distribution between Asian children from different origins and confirms the necessity to consider ethnic differences in body composition when developing BMI and other obesity index criteria for obesity in Asian children. Moreover, ethnicity is also important in BIA prediction equations. In addition, WC and WHtR percentiles and thresholds for the prediction of CV risk in Chinese children differ from other populations. Although there was no advantage of WC or WHtR over BMI or %BF in the prediction of CV risk, obese children had a higher risk of developing the metabolic syndrome and abnormalities than normal�]weight children regardless of the obesity index used.

New percentage body fat prediction equations for Japanese females

Anthropometry is a simple and cost-efficient method for the assessment of body composition. However prediction equations to estimate body composition using anthropometry should be ‘population-specific’. Most popular body composition prediction equations for Japanese females were proposed more than 40 years ago and there is some concern regarding their usefulness in Japanese females living today. The aim of this study was to compare percentage body fat (%BF) estimated from anthropometry and dual energy x-ray absorptiometry (DXA) to examine the applicability of commonly used prediction equations in young Japanese females. Body composition of 139 Japanese females aged between 18 and 27 years of age (BMI range: 15.1–29.1 kg/m2) was measured using whole-body DXA (Lunar DPX-LIQ) scans. From anthropometric measurements %BF was estimated using four equations developed from Japanese females. The results showed that the traditionally employed prediction equations for anthropometry significantly (p<0.01) underestimate %BF of young Japanese females and therefore are not valid for the precise estimation of body composition. New %BF prediction equations were proposed from the DXA and anthropometry results. Application of the proposed equations may assist in more accurate assessment of body fatness in Japanese females living today.