A Novel High Order Space-Time Spectral Method for the Time Fractional Fokker--Planck Equation

The fractional Fokker-Planck equation is an important physical model for simulating anomalous diffusions with external forces. Because of the non-local property of the fractional derivative an interesting problem is to explore high accuracy numerical methods for fractional differential equations. In this paper, a space-time spectral method is presented for the numerical solution of the time fractional Fokker-Planck initial-boundary value problem. The proposed method employs the Jacobi polynomials for the temporal discretization and Fourier-like basis functions for the spatial discretization. Due to the diagonalizable trait of the Fourier-like basis functions, this leads to a reduced representation of the inner product in the Galerkin analysis. We prove that the time fractional Fokker-Planck equation attains the same approximation order as the time fractional diffusion equation developed in [23] by using the present method. That indicates an exponential decay may be achieved if the exact solution is sufficiently smooth. Finally, some numerical results are given to demonstrate the high order accuracy and efficiency of the new numerical scheme. The results show that the errors of the numerical solutions obtained by the space-time spectral method decay exponentially.

Numeracy counts in the statistical reasoning equation

The relationship between mathematics and statistical reasoning frequently receives comment (Vere-Jones 1995, Moore 1997); however most of the research into the area tends to focus on maths anxiety. Gnaldi (Gnaldi 2003) showed that in a statistics course for psychologists, the statistical understanding of students at the end of the course depended on students’ basic numeracy, rather than the number or level of previous mathematics courses the student had undertaken. As part of a study into the development of statistical thinking at the interface between secondary and tertiary education, students enrolled in an introductory data analysis subject were assessed regarding their statistical reasoning ability, basic numeracy skills and attitudes towards statistics. This work reports on the relationships between these factors and in particular the importance of numeracy to statistical reasoning.

Investigating the impact of project definition clarity on project performance: Structural equation modeling study

Having a clear project definition is crucial for successful construction projects. It affects design quality, project communication between stakeholders and final project performance in terms of cost, schedule and quality. This study examines the relationship between project definition and final project performance through a structural equation model comprising 4 latent constructs and 6 path hypotheses using data from a questionnaire survey of 120 general contractors in the Malaysian construction industry. The results show that in the study population, all three items impact the project performance, but the link between design quality and project performance is indirect. Instead, the clarity of project definition affects project performance indirectly through design quality and project communication and design quality affects project performance indirectly through project communication. The primary contribution is to provide quantitative confirmation of the more general statements made in the literature from around the world and therefore adds to and consolidates existing knowledge. Practical implications derived from the finding are also proposed for various project stakeholders. Furthermore, as lack of the clarity of project definition is a very common occurrence in construction projects globally, these findings have important ramifications for all construction projects in expanding and clarifying existing knowledge on what is needed for the successful delivery of construction projects.

Development of height-weight based equation for assessment of body composition in Sri Lankan children

Objective To develop a height and weight based equation to estimate total body water (TBW) in Sri Lankan children. Methods Cross sectional descriptive study done involving 5–15 year old healthy children. Height and weight were measured. TBW was assessed using isotope dilution method (D2O) and fat free mass (FFM) calculated. Multiple regression analysis was used to develop prediction equation and validated using PRESS statistical technique. Height, weight and sex code (male=1; female=0) were used as prediction variables. Results This study provides height and weight equation for the prediction of TBW in Sri Lankan children. To the best of our knowledge there are no published height weight prediction equations validated on South Asian populations. Conclusion Results of this study need to be affirmed by more studies on other closely related populations by using multicomponent body composition.

Numerical analysis of a new space-time variable fractional order advection-dispersion equation

Many physical processes appear to exhibit fractional order behavior that may vary with time and/or space. The continuum of order in the fractional calculus allows the order of the fractional operator to be considered as a variable. In this paper, we consider a new space–time variable fractional order advection–dispersion equation on a finite domain. The equation is obtained from the standard advection–dispersion equation by replacing the first-order time derivative by Coimbra’s variable fractional derivative of order α(x)∈(0,1]α(x)∈(0,1], and the first-order and second-order space derivatives by the Riemann–Liouville derivatives of order γ(x,t)∈(0,1]γ(x,t)∈(0,1] and β(x,t)∈(1,2]β(x,t)∈(1,2], respectively. We propose an implicit Euler approximation for the equation and investigate the stability and convergence of the approximation. Finally, numerical examples are provided to show that the implicit Euler approximation is computationally efficient.

Spatially explicit structural equation modeling

Structural equation modeling (SEM) is a powerful statistical approach for the testing of networks of direct and indirect theoretical causal relationships in complex data sets with intercorrelated dependent and independent variables. SEM is commonly applied in ecology, but the spatial information commonly found in ecological data remains difficult to model in a SEM framework. Here we propose a simple method for spatially explicit SEM (SE-SEM) based on the analysis of variance/covariance matrices calculated across a range of lag distances. This method provides readily interpretable plots of the change in path coefficients across scale and can be implemented using any standard SEM software package. We demonstrate the application of this method using three studies examining the relationships between environmental factors, plant community structure, nitrogen fixation, and plant competition. By design, these data sets had a spatial component, but were previously analyzed using standard SEM models. Using these data sets, we demonstrate the application of SE-SEM to regularly spaced, irregularly spaced, and ad hoc spatial sampling designs and discuss the increased inferential capability of this approach compared with standard SEM. We provide an R package, sesem, to easily implement spatial structural equation modeling.

Assessment of body composition in Sri Lankan children: Validation of a bioelectrical impedance prediction equation

Objective: To develop bioelectrical impedance analysis (BIA) equations to predict total body water (TBW) and fat-free mass (FFM) of Sri Lankan children. Subjects/Methods: Data were collected from 5- to 15-year-old healthy children. They were randomly assigned to validation (M/F: 105/83) and cross-validation (M/F: 53/41) groups. Height, weight and BIA were measured. TBW was assessed using isotope dilution method (D2 O). Multiple regression analysis was used to develop preliminary equations and cross-validated on an independent group. Final prediction equation was constructed combining the two groups and validated by PRESS (prediction of sum of squares) statistics. Impedance index (height2/impedance; cm2/Ω), weight and sex code (male = 1; female = 0) were used as variables. Results: Independent variables of the final prediction equation for TBW were able to predict 86.3% of variance with root means-squared error (RMSE) of 2.1l. PRESS statistics was 2.1l with press residuals of 1.2l. Independent variables were able to predict 86.9% of variance of FFM with RMSE of 2.7 kg. PRESS statistics was 2.8 kg with press residuals of 1.4 kg. Bland Altman technique showed that the majority of the residuals were within mean bias±1.96 s.d. Conclusions: Results of this study provide BIA equation for the prediction of TBW and FFM in Sri Lankan children. To the best of our knowledge there are no published BIA prediction equations validated on South Asian populations. Results of this study need to be affirmed by more studies on other closely related populations by using multi-component body composition assessment.

A comment on the consistency of truncated nonlinear integral equation based theories of freezing

We report the results of two studies of aspects of the consistency of truncated nonlinear integral equation based theories of freezing: (i) We show that the self-consistent solutions to these nonlinear equations are unfortunately sensitive to the level of truncation. For the hard sphere system, if the Wertheim–Thiele representation of the pair direct correlation function is used, the inclusion of part but not all of the triplet direct correlation function contribution, as has been common, worsens the predictions considerably. We also show that the convergence of the solutions found, with respect to number of reciprocal lattice vectors kept in the Fourier expansion of the crystal singlet density, is slow. These conclusions imply great sensitivity to the quality of the pair direct correlation function employed in the theory. (ii) We show the direct correlation function based and the pair correlation function based theories of freezing can be cast into a form which requires solution of isomorphous nonlinear integral equations. However, in the pair correlation function theory the usual neglect of the influence of inhomogeneity of the density distribution on the pair correlation function is shown to be inconsistent to the lowest order in the change of density on freezing, and to lead to erroneous predictions. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.

A comment on the consistency of truncated nonlinear integral equation based theories of freezing

We report the results of two studies of aspects of the consistency of truncated nonlinear integral equation based theories of freezing: (i) We show that the self-consistent solutions to these nonlinear equations are unfortunately sensitive to the level of truncation. For the hard sphere system, if the Wertheim–Thiele representation of the pair direct correlation function is used, the inclusion of part but not all of the triplet direct correlation function contribution, as has been common, worsens the predictions considerably. We also show that the convergence of the solutions found, with respect to number of reciprocal lattice vectors kept in the Fourier expansion of the crystal singlet density, is slow. These conclusions imply great sensitivity to the quality of the pair direct correlation function employed in the theory. (ii) We show the direct correlation function based and the pair correlation function based theories of freezing can be cast into a form which requires solution of isomorphous nonlinear integral equations. However, in the pair correlation function theory the usual neglect of the influence of inhomogeneity of the density distribution on the pair correlation function is shown to be inconsistent to the lowest order in the change of density on freezing, and to lead to erroneous predictions. The Journal of Chemical Physics is copyrighted by The American Institute of Physics.

Analysis of Paschen Curves for air, N2 and SF6 Using the Townsend Breakdown Equation

It has been shown that it is possible to extend the validity of the Townsend breakdown criterion for evaluating the breakdown voltages in the complete pd range in which Paschen curves are available. Evaluation of the breakdown voltages for air (pd=0.0133 to 1400 kPa · cm), N2(pd=0.0313 to 1400 kPa · cm) and SF6 (pd=0.3000 to 1200 kPa · cm) has been done and in most cases the computed values are accurate to ±3% of the measured values. The computations show that it is also possible to estimate the secondary ionization coefficient ¿ in the pd ranges mentioned above.

Multinomial solutions of the beach equation

Exact multinomial solutions of the beach equation for shallow water waves on a uniformly sloping beach are found and related to solution of the same equation found earlier by other investigators, using integral transform techniques. The use of these solutions for a general initialvalue problem for the equation under investigation is briefly discussed.

Convenient construction of tetrahedral finite elements for the quasi-harmonic equation

It is shown that in the finite-element formulation of the general quasi-harmonic equation using tetrahedral elements, for every member of the element family there exists just one numerical universal matrix indpendent of the size, shape and material properties of the element. Thus the element matrix is conveniently constructed by manipulating this single matrix along with a set of reverse sequence codes at the same time accounting for the size, shape and material properties in a simple manner.

Multinomial solutions of the beach equation

Exact multinomial solutions of the beach equation for shallow water waves on a uniformly sloping beach are found and related to solution of the same equation found earlier by other investigators, using integral transform techniques. The use of these solutions for a general initialvalue problem for the equation under investigation is briefly discussed.

Some direct numerical approaches to the solution of the singular nonlinear transonic integral equation

The nonlinear singular integral equation of transonic flow is examined, noting that standard numerical techniques are not applicable in solving it. The difficulties in approximating the integral term in this expression were solved by special methods mitigating the inaccuracies caused by standard approximations. It was shown how the infinite domain of integration can be reduced to a finite one; numerical results were plotted demonstrating that the methods proposed here improve accuracy and computational economy.

Water-wave scattering by two submerged plane vertical barriers-Abel integral-equation approach

The classical problem of surface water-wave scattering by two identical thin vertical barriers submerged in deep water and extending infinitely downwards from the same depth below the mean free surface, is reinvestigated here by an approach leading to the problem of solving a system of Abel integral equations. The reflection and transmission coefficients are obtained in terms of computable integrals. Known results for a single barrier are recovered as a limiting case as the separation distance between the two barriers tends to zero. The coefficients are depicted graphically in a number of figures which are identical with the corresponding figures given by Jarvis (J Inst Math Appl 7:207-215, 1971) who employed a completely different approach involving a Schwarz-Christoffel transformation of complex-variable theory to solve the problem.