Numerical investigations of linear least squares methods for derivative estimation

The results of a numerical investigation into the errors for least squares estimates of function gradients are presented. The underlying algorithm is obtained by constructing a least squares problem using a truncated Taylor expansion. An error bound associated with this method contains in its numerator terms related to the Taylor series remainder, while its denominator contains the smallest singular value of the least squares matrix. Perhaps for this reason the error bounds are often found to be pessimistic by several orders of magnitude. The circumstance under which these poor estimates arise is elucidated and an empirical correction of the theoretical error bounds is conjectured and investigated numerically. This is followed by an indication of how the conjecture is supported by a rigorous argument.

The application of the statutory derivative action in Pt 2F.1A of the Corporations Act to companies in liquidation under Ch 5 : Well oiled machine or is Pt 2F.1A in need of a tune up?

The statutory derivative action was introduced in Australia in 2000. This right of action has been debated in the literature and introduced in a number of other jurisdictions as well. However, it is by no means clear that all issues have been resolved despite its operation in Australia for over 10 years. This article considers the application of Pt 2F.1A of the Corporations Act to companies in liquidation under Ch 5. It demonstrates that the application involves consideration of not only proper statutory interpretation but also policy matters around the role and the supervision by the court of a liquidator once a company has entered liquidation.

Spectral approximations to the fractional integral and derivative

In this paper, the spectral approximations are used to compute the fractional integral and the Caputo derivative. The effective recursive formulae based on the Legendre, Chebyshev and Jacobi polynomials are developed to approximate the fractional integral. And the succinct scheme for approximating the Caputo derivative is also derived. The collocation method is proposed to solve the fractional initial value problems and boundary value problems. Numerical examples are also provided to illustrate the effectiveness of the derived methods.

Numerical methods of the variable-order Rayleigh-Stokes problem for a heated generalized second grade fluid with fractional derivative

Rayleigh–Stokes problems have in recent years received much attention due to their importance in physics. In this article, we focus on the variable-order Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative. Implicit and explicit numerical methods are developed to solve the problem. The convergence, stability of the numerical methods and solvability of the implicit numerical method are discussed via Fourier analysis. Moreover, a numerical example is given and the results support the effectiveness of the theoretical analysis.

Application of kaolin amorphous derivative as a nitrogen carrier

Sandy soils have low nutrient holding capacity and high water conductivity. Consequently, nutrients applied as highly soluble chemical fertilisers are prone to leaching, particularly in heavily irrigated environments such as horticultural soils and golf courses. Amorphous derivatives of kaolin with high cation exchange capacity may be loaded with desired nutrients and applied as controlledrelease fertilisers. Kaolin is an abundant mineral, which can be converted to a meso-porous amorphous derivative (KAD) using facile chemical processes. KAD is currently being used to sequester ammonium from digester effluent in sewage treatment plants in a commercial environment. This material is also known in Australia by the trade name MesoLite. The ammonium-saturated form of KAD may be applied to soils as a nitrogen fertiliser. Up to 7% N can be loaded onto KAD by contacting it with high-ammonia concentration wastewater from sewerage treatment plants. This poster paper demonstrates plant uptake of nitrogen from KAD and compares its efficiency as a fertiliser with NH4SO4. Rye grass was grown in 1kg pots in a glass-house. Nitrogen was applied at a range of rates using NH4SO4 and two KAD materials carrying 7% and 3% nitrogen, respectively. All other nutrients were applied in adequate amounts. All treatments were replicated three times. Plants were harvested after four weeks. Dry mass and N concentrations were determined by standard methods. At all N application rates, ammonium-loaded KAD produced significantly higher plant mass than for NH4SO4. The lower fertiliser effectiveness of NH4SO4 is attributed to possible loss of some N through volatilisation. Of the two KAD types, the material with lower CEC value supported slightly higher plant yields. The KAD materials did not show any adverse effect on availability of trace elements, as evidenced by lack of deficiency symptoms and plant analyses. Clearly, nitrogen loaded on to KAD in the form of ammonium is likely to be protected from leaching, but is still available to plants. These data suggest that KAD-based fertilisers may be suitable substitutes for water soluble N, K and other cation fertilisers for leaching soils.

Someone else's boom but always our bust: Australia as a derivative economy, implications for regions

This paper examines the socio-economic impact of mineral and agricultural resource extraction on local communities and explores policy options for addressing them. An emphasis on the marketisation of services together with tight fiscal control has reinforced decline in many country communities in Australia and elsewhere. However, the introduction by the European Union of Regional Policy which emphasises ‘smart specialisation’ can enhance greatly the capacity of local people to generate decent livelihoods. For this to have real effect, the innovative state has to enable partnerships between communities, researchers and industry. For countries like Australia, this would be a substantive policy shift.

Facile synthesis, ex-vivo and in vitro screening of 3-sulfonamide derivative of 5-(4-chlorophenyl)-1-(2,4-dichlorophenyl)-4-methyl-1H-pyrazole-3-carboxylic acid piperidin-1-ylamide (SR141716) a potent CB1 receptor antagonist

Facile synthesis of biaryl pyrazole sulfonamide derivative of 5-(4-chlorophenyl)-1-(2,4-dichlorophenyl)-4-methyl-1H-pyrazole-3-carboxylic acid piperidin-1-ylamide (SR141716, 1) and an investigation of the effect of replacement of the –CO group in the compound 1 by the –SO2 group in the aminopiperidine region is reported. Primary ex-vivo pharmacological testing and in vitro screening of sulfonamide derivative 2 showed the loss of CB1 receptor antagonism.

Detailed analysis of a conservative difference approximation for the time fractional diffusion equation

Diffusion equations that use time fractional derivatives are attractive because they describe a wealth of problems involving non-Markovian Random walks. The time fractional diffusion equation (TFDE) is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order α ∈ (0, 1). Developing numerical methods for solving fractional partial differential equations is a new research field and the theoretical analysis of the numerical methods associated with them is not fully developed. In this paper an explicit conservative difference approximation (ECDA) for TFDE is proposed. We give a detailed analysis for this ECDA and generate discrete models of random walk suitable for simulating random variables whose spatial probability density evolves in time according to this fractional diffusion equation. The stability and convergence of the ECDA for TFDE in a bounded domain are discussed. Finally, some numerical examples are presented to show the application of the present technique.

Effective Cell-Centred Time-Domain Maxwell's Equations Numerical Solvers

This research work analyses techniques for implementing a cell-centred finite-volume time-domain (ccFV-TD) computational methodology for the purpose of studying microwave heating. Various state-of-the-art spatial and temporal discretisation methods employed to solve Maxwell's equations on multidimensional structured grid networks are investigated, and the dispersive and dissipative errors inherent in those techniques examined. Both staggered and unstaggered grid approaches are considered. Upwind schemes using a Riemann solver and intensity vector splitting are studied and evaluated. Staggered and unstaggered Leapfrog and Runge-Kutta time integration methods are analysed in terms of phase and amplitude error to identify which method is the most accurate and efficient for simulating microwave heating processes. The implementation and migration of typical electromagnetic boundary conditions. from staggered in space to cell-centred approaches also is deliberated. In particular, an existing perfectly matched layer absorbing boundary methodology is adapted to formulate a new cell-centred boundary implementation for the ccFV-TD solvers. Finally for microwave heating purposes, a comparison of analytical and numerical results for standard case studies in rectangular waveguides allows the accuracy of the developed methods to be assessed.

Implicit difference approximation for the time fractional diffusion equation

In this paper, we consider a time fractional diffusion equation on a finite domain. The equation is obtained from the standard diffusion equation by replacing the first-order time derivative by a fractional derivative (of order $0<\alpha<1$ ). We propose a computationally effective implicit difference approximation to solve the time fractional diffusion equation. Stability and convergence of the method are discussed. We prove that the implicit difference approximation (IDA) is unconditionally stable, and the IDA is convergent with $O(\tau+h^2)$, where $\tau$ and $h$ are time and space steps, respectively. Some numerical examples are presented to show the application of the present technique.