Flow of a micropolar fluid bounded by a stretching sheet

We consider boundary layer flow of a micropolar fluid driven by a porous stretching sheet. A similarity solution is defined, and numerical solutions using Runge-Kutta and quasilinearisation schemes are obtained. A perturbation analysis is also used to derive analytic solutions to first order in the perturbing parameter. The resulting closed form solutions involve relatively complex expressions, and the analysis is made more tractable by a combination of offline and online work using a computational algebra system (CAS). For this combined numerical and analytic approach, the perturbation analysis yields a number of benefits with regard to the numerical work. The existence of a closed form solution helps to discriminate between acceptable and spurious numerical solutions. Also, the expressions obtained from the perturbation work can provide an accurate description of the solution for ranges of parameters where the numerical approaches considered here prove computationally more difficult.

Numerical methods for fractional partial differential equations with Riesz space fractional derivatives

In this paper, we consider the numerical solution of a fractional partial differential equation with Riesz space fractional derivatives (FPDE-RSFD) on a finite domain. Two types of FPDE-RSFD are considered: the Riesz fractional diffusion equation (RFDE) and the Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second-order space derivative with the Riesz fractional derivative of order αset membership, variant(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first-order and second-order space derivatives with the Riesz fractional derivatives of order βset membership, variant(0,1) and of order αset membership, variant(1,2], respectively. Firstly, analytic solutions of both the RFDE and RFADE are derived. Secondly, three numerical methods are provided to deal with the Riesz space fractional derivatives, namely, the L1/L2-approximation method, the standard/shifted Grünwald method, and the matrix transform method (MTM). Thirdly, the RFDE and RFADE are transformed into a system of ordinary differential equations, which is then solved by the method of lines. Finally, numerical results are given, which demonstrate the effectiveness and convergence of the three numerical methods.

Generating rules with predicates, terms and variables from the pruned neural networks

Artificial neural networks (ANN) have demonstrated good predictive performance in a wide range of applications. They are, however, not considered sufficient for knowledge representation because of their inability to represent the reasoning process succinctly. This paper proposes a novel methodology Gyan that represents the knowledge of a trained network in the form of restricted first-order predicate rules. The empirical results demonstrate that an equivalent symbolic interpretation in the form of rules with predicates, terms and variables can be derived describing the overall behaviour of the trained ANN with improved comprehensibility while maintaining the accuracy and fidelity of the propositional rules.

Fundamental solution and discrete random walk model for time-space fractional diffusion equation

In this paper, we consider a time-space fractional diffusion equation of distributed order (TSFDEDO). The TSFDEDO is obtained from the standard advection-dispersion equation by replacing the first-order time derivative by the Caputo fractional derivative of order α∈(0,1], the first-order and second-order space derivatives by the Riesz fractional derivatives of orders β 1∈(0,1) and β 2∈(1,2], respectively. We derive the fundamental solution for the TSFDEDO with an initial condition (TSFDEDO-IC). The fundamental solution can be interpreted as a spatial probability density function evolving in time. We also investigate a discrete random walk model based on an explicit finite difference approximation for the TSFDEDO-IC.

Adaptive depth estimation in image based visual servo control of dynamic systems

This paper considers the question of designing a fully image based visual servo control for a dynamic system. The work is motivated by the ongoing development of image based visual servo control of small aerial robotic vehicles. The observed targets considered are coloured blobs on a flat surface to which the normal direction is known. The theoretical framework is directly applicable to the case of markings on a horizontal floor or landing field. The image features used are a first order spherical moment for position and an image flow measurement for velocity. A fully non-linear adaptive control design is provided that ensures global stability of the closed-loop system. © 2005 IEEE.

GYAN: A methodology for rule extraction from artificial neural networks

Artificial neural network (ANN) learning methods provide a robust and non-linear approach to approximating the target function for many classification, regression and clustering problems. ANNs have demonstrated good predictive performance in a wide variety of practical problems. However, there are strong arguments as to why ANNs are not sufficient for the general representation of knowledge. The arguments are the poor comprehensibility of the learned ANN, and the inability to represent explanation structures. The overall objective of this thesis is to address these issues by: (1) explanation of the decision process in ANNs in the form of symbolic rules (predicate rules with variables); and (2) provision of explanatory capability by mapping the general conceptual knowledge that is learned by the neural networks into a knowledge base to be used in a rule-based reasoning system. A multi-stage methodology GYAN is developed and evaluated for the task of extracting knowledge from the trained ANNs. The extracted knowledge is represented in the form of restricted first-order logic rules, and subsequently allows user interaction by interfacing with a knowledge based reasoner. The performance of GYAN is demonstrated using a number of real world and artificial data sets. The empirical results demonstrate that: (1) an equivalent symbolic interpretation is derived describing the overall behaviour of the ANN with high accuracy and fidelity, and (2) a concise explanation is given (in terms of rules, facts and predicates activated in a reasoning episode) as to why a particular instance is being classified into a certain category.

Soil carbon saturation: Linking concept and measurable carbon pools

The soil C saturation concept suggests a limit to whole soil organic carbon (SOC) accumulation determined by inherent physicochemical characteristics of four soil C pools: unprotected, physically protected, chemically protected, and biochemically protected. Previous attempts to quantify soil C sequestration capacity have focused primarily on silt and clay protection and largely ignored the effects of soil structural protection and biochemical protection. We assessed two contrasting models of SOC accumulation, one with no saturation limit (i.e., linear first-order model) and one with an explicit soil C saturation limit (i.e., C saturation model). We isolated soil fractions corresponding to the C pools (i.e., free particulate organic matter POM], microaggregate-associated C, silt- and clay-associated C, and non-hydrolyzable C) from eight long-term agroecosystern experiments across the United States and Canada. Due to the composite nature of the physically protected C pool, we firactioned it into mineral- vs. POM-associated C. Within each site, the number of fractions fitting the C saturation model was directly related to maximum SOC content, suggesting that a broad range in SOC content is necessary to evaluate fraction C saturation. The two sites with the greatest SOC range showed C saturation behavior in the chemically, biochemically, and some mineral-associated fractions of the physically protected pool. The unprotected pool and the aggregate-protected POM showed linear, nonsaturating behavior. Evidence of C saturation of chemically and biochemically protected SOC pools was observed at sites far from their theoretical C saturation level, while saturation of aggregate-protected fractions occurred in soils closer to their C saturation level.

Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by a Caputo fractional derivative, and the second order space derivative by a symmetric fractional derivative. First, a method of separating variables expresses the analytical solution of the TSS-FDE in terms of the Mittag--Leffler function. Second, we propose two numerical methods to approximate the Caputo time fractional derivative: the finite difference method; and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation

We consider a time and space-symmetric fractional diffusion equation (TSS-FDE) under homogeneous Dirichlet conditions and homogeneous Neumann conditions. The TSS-FDE is obtained from the standard diffusion equation by replacing the first-order time derivative by the Caputo fractional derivative and the second order space derivative by the symmetric fractional derivative. Firstly, a method of separating variables is used to express the analytical solution of the tss-fde in terms of the Mittag–Leffler function. Secondly, we propose two numerical methods to approximate the Caputo time fractional derivative, namely, the finite difference method and the Laplace transform method. The symmetric space fractional derivative is approximated using the matrix transform method. Finally, numerical results are presented to demonstrate the effectiveness of the numerical methods and to confirm the theoretical claims.

Alkaline hydrothermal treatment of titanate nanostructures

Since its initial proposal in 1998, alkaline hydrothermal processing has rapidly become an established technology for the production of titanate nanostructures. This simple, highly reproducible process has gained a strong research following since its conception. However, complete understanding and elucidation of nanostructure phase and formation have not yet been achieved. Without fully understanding phase, formation, and other important competing effects of the synthesis parameters on the final structure, the maximum potential of these nanostructures cannot be obtained. Therefore this study examined the influence of synthesis parameters on the formation of titanate nanostructures produced by alkaline hydrothermal treatment. The parameters included alkaline concentration, hydrothermal temperature, the precursor material‘s crystallite size and also the phase of the titanium dioxide precursor (TiO2, or titania). The nanostructure‘s phase and morphology was analysed using X-ray diffraction (XRD), Raman spectroscopy and transmission electron microscopy. X-ray photoelectron spectroscopy (XPS), dynamic light scattering (non-invasive backscattering), nitrogen sorption, and Rietveld analysis were used to determine phase, for particle sizing, surface area determinations, and establishing phase concentrations, respectively. This project rigorously examined the effect of alkaline concentration and hydrothermal temperature on three commercially sourced and two self-prepared TiO2 powders. These precursors consisted of both pure- or mixed-phase anatase and rutile polymorphs, and were selected to cover a range of phase concentrations and crystallite sizes. Typically, these precursors were treated with 5–10 M sodium hydroxide (NaOH) solutions at temperatures between 100–220 °C. Both nanotube and nanoribbon morphologies could be produced depending on the combination of these hydrothermal conditions. Both titania and titanate phases are comprised of TiO6 units which are assembled in different combinations. The arrangement of these atoms affects the binding energy between the Ti–O bonds. Raman spectroscopy and XPS were therefore employed in a preliminary study of phase determination for these materials. The change in binding energy from a titania to a titanate binding energy was investigated in this study, and the transformation of titania precursor into nanotubes and titanate nanoribbons was directly observed by these methods. Evaluation of the Raman and XPS results indicated a strengthening in the binding energies of both the Ti (2p3/2) and O (1s) bands which correlated to an increase in strength and decrease in resolution of the characteristic nanotube doublet observed between 320 and 220 cm.1 in the Raman spectra of these products. The effect of phase and crystallite size on nanotube formation was examined over a series of temperatures (100.200 �‹C in 20 �‹C increments) at a set alkaline concentration (7.5 M NaOH). These parameters were investigated by employing both pure- and mixed- phase precursors of anatase and rutile. This study indicated that both the crystallite size and phase affect nanotube formation, with rutile requiring a greater driving force (essentially �\harsher. hydrothermal conditions) than anatase to form nanotubes, where larger crystallites forms of the precursor also appeared to impede nanotube formation slightly. These parameters were further examined in later studies. The influence of alkaline concentration and hydrothermal temperature were systematically examined for the transformation of Degussa P25 into nanotubes and nanoribbons, and exact conditions for nanostructure synthesis were determined. Correlation of these data sets resulted in the construction of a morphological phase diagram, which is an effective reference for nanostructure formation. This morphological phase diagram effectively provides a .recipe book�e for the formation of titanate nanostructures. Morphological phase diagrams were also constructed for larger, near phase-pure anatase and rutile precursors, to further investigate the influence of hydrothermal reaction parameters on the formation of titanate nanotubes and nanoribbons. The effects of alkaline concentration, hydrothermal temperature, crystallite phase and size are observed when the three morphological phase diagrams are compared. Through the analysis of these results it was determined that alkaline concentration and hydrothermal temperature affect nanotube and nanoribbon formation independently through a complex relationship, where nanotubes are primarily affected by temperature, whilst nanoribbons are strongly influenced by alkaline concentration. Crystallite size and phase also affected the nanostructure formation. Smaller precursor crystallites formed nanostructures at reduced hydrothermal temperature, and rutile displayed a slower rate of precursor consumption compared to anatase, with incomplete conversion observed for most hydrothermal conditions. The incomplete conversion of rutile into nanotubes was examined in detail in the final study. This study selectively examined the kinetics of precursor dissolution in order to understand why rutile incompletely converted. This was achieved by selecting a single hydrothermal condition (9 M NaOH, 160 °C) where nanotubes are known to form from both anatase and rutile, where the synthesis was quenched after 2, 4, 8, 16 and 32 hours. The influence of precursor phase on nanostructure formation was explicitly determined to be due to different dissolution kinetics; where anatase exhibited zero-order dissolution and rutile second-order. This difference in kinetic order cannot be simply explained by the variation in crystallite size, as the inherent surface areas of the two precursors were determined to have first-order relationships with time. Therefore, the crystallite size (and inherent surface area) does not affect the overall kinetic order of dissolution; rather, it determines the rate of reaction. Finally, nanostructure formation was found to be controlled by the availability of dissolved titanium (Ti4+) species in solution, which is mediated by the dissolution kinetics of the precursor.

Electron diffusion and back reaction in dye-sensitized solar cells : the effect of nonlinear recombination kinetics

The electron collection efficiency in dye-sensitized solar cells (DSCs) is usually related to the electron diffusion length, L = (Dτ)^1/2, where D is the diffusion coefficient of mobile electrons and τ is their lifetime, which is determined by electron transfer to the redox electrolyte. Analysis of incident photon-to-current efficiency (IPCE) spectra for front and rear illumination consistently gives smaller values of L than those derived from small amplitude methods. We show that the IPCE analysis is incorrect if recombination is not first-order in free electron concentration, and we demonstrate that the intensity dependence of the apparent L derived by first-order analysis of IPCE measurements and the voltage dependence of L derived from perturbation experiments can be fitted using the same reaction order, γ ≈ 0.8. The new analysis presented in this letter resolves the controversy over why L values derived from small amplitude methods are larger than those obtained from IPCE data.

Robust designs for Poisson regression models

We consider the problem of how to construct robust designs for Poisson regression models. An analytical expression is derived for robust designs for first-order Poisson regression models where uncertainty exists in the prior parameter estimates. Given certain constraints in the methodology, it may be necessary to extend the robust designs for implementation in practical experiments. With these extensions, our methodology constructs designs which perform similarly, in terms of estimation, to current techniques, and offers the solution in a more timely manner. We further apply this analytic result to cases where uncertainty exists in the linear predictor. The application of this methodology to practical design problems such as screening experiments is explored. Given the minimal prior knowledge that is usually available when conducting such experiments, it is recommended to derive designs robust across a variety of systems. However, incorporating such uncertainty into the design process can be a computationally intense exercise. Hence, our analytic approach is explored as an alternative.

Robust designs for Poisson regression models

We consider the problem of how to construct robust designs for Poisson regression models. An analytical expression is derived for robust designs for first-order Poisson regression models where uncertainty exists in the prior parameter estimates. Given certain constraints in the methodology, it may be necessary to extend the robust designs for implementation in practical experiments. With these extensions, our methodology constructs designs which perform similarly, in terms of estimation, to current techniques, and offers the solution in a more timely manner. We further apply this analytic result to cases where uncertainty exists in the linear predictor. The application of this methodology to practical design problems such as screening experiments is explored. Given the minimal prior knowledge that is usually available when conducting such experiments, it is recommended to derive designs robust across a variety of systems. However, incorporating such uncertainty into the design process can be a computationally intense exercise. Hence, our analytic approach is explored as an alternative.

Linear models for endocytic transformations from live cell imaging

Endocytosis is the process by which cells internalise molecules including nutrient proteins from the extracellular media. In one form, macropinocytosis, the membrane at the cell surface ruffles and folds over to give rise to an internalised vesicle. Negatively charged phospholipids within the membrane called phosphoinositides then undergo a series of transformations that are critical for the correct trafficking of the vesicle within the cell, and which are often pirated by pathogens such as Salmonella. Advanced fluorescent video microscopy imaging now allows the detailed observation and quantification of these events in live cells over time. Here we use these observations as a basis for building differential equation models of the transformations. An initial investigation of these interactions was modelled with reaction rates proportional to the sum of the concentrations of the individual constituents. A first order linear system for the concentrations results. The structure of the system enables analytical expressions to be obtained and the problem becomes one of determining the reaction rates which generate the observed data plots. We present results with reaction rates which capture the general behaviour of the reactions so that we now have a complete mathematical model of phosphoinositide transformations that fits the experimental observations. Some excellent fits are obtained with modulated exponential functions; however, these are not solutions of the linear system. The question arises as to how the model may be modified to obtain a system whose solution provides a more accurate fit.

The role of soil characteristics on temperature sensitivity of soil organic matter

The uncertainty associated with how projected climate change will affect global C cycling could have a large impact on predictions of soil C stocks. The purpose of our study was to determine how various soil decomposition and chemistry characteristics relate to soil organic matter (SOM) temperature sensitivity. We accomplished this objective using long-term soil incubations at three temperatures (15, 25, and 35°C) and pyrolysis molecular beam mass spectrometry (py-MBMS) on 12 soils from 6 sites along a mean annual temperature (MAT) gradient (2–25.6°C). The Q10 values calculated from the CO2 respired during a long-term incubation using the Q10-q method showed decomposition of the more resistant fraction to be more temperature sensitive with a Q10-q of 1.95 ± 0.08 for the labile fraction and a Q10-q of 3.33 ± 0.04 for the more resistant fraction. We compared the fit of soil respiration data using a two-pool model (active and slow) with first-order kinetics with a three-pool model and found that the two and three-pool models statistically fit the data equally well. The three-pool model changed the size and rate constant for the more resistant pool. The size of the active pool in these soils, calculated using the two-pool model, increased with incubation temperature and ranged from 0.1 to 14.0% of initial soil organic C. Sites with an intermediate MAT and lowest C/N ratio had the largest active pool. Pyrolysis molecular beam mass spectrometry showed declines in carbohydrates with conversion from grassland to wheat cultivation and a greater amount of protected carbohydrates in allophanic soils which may have lead to differences found between the total amount of CO2 respired, the size of the active pool, and the Q10-q values of the soils.