Permutation polynomials of the form (xp −x +δ)s +L(x)

Recently, several classes of permutation polynomials of the form (x2 + x + δ)s + x over F2m have been discovered. They are related to Kloosterman sums. In this paper, the permutation behavior of polynomials of the form (xp − x + δ)s + L(x) over Fpm is investigated, where L(x) is a linearized polynomial with coefficients in Fp. Six classes of permutation polynomials on F2m are derived. Three classes of permutation polynomials over F3m are also presented.

Manoeuvring control of fully-actuated marine vehicles : a Port-Hamiltonian system approach to tracking

This paper presents a novel control strategy for trajectory tracking of marine vehicles manoeuvring at low speed. The model of the marine vehicle is formulated as a Port-Hamiltonian system, and the tracking controller is designed using energy shaping and damping assignment. The controller guarantees global asymptotic stability and includes integral action for output variables with relative degree greater than one.

Genetic technologies and the regulation of reproductive decision-making in Australia

This article provides a critical analysis of the current Australian regulatory landscape at the interface between genetics and reproductive decision- making. The authors argue that a comparative analysis with other countries and international law and a contextual examination of the way law regulates concepts such as disease and health, abnormality and normality is necessary before we can develop appropriate policy and legislative responses in this area. Specific genetic testing technologies are considered including prenatal genetic testing, preimplantation genetic diagnosis and inheritable genetic modification. An increasing number of members of the Australian community are using genetic testing technologies when they decide to have a baby. The authors argue that as concepts of disease and health vary among members of the community and the potential to test for traits other than illness increases, a new tension arises between an ethic of individual choice and a role for government in regulating reproductive decision-making.

Statistical analysis of reduction in tensile strength of cotton strips as a measure of soil microbial activity

The cotton strip assay (CSA) is an established technique for measuring soil microbial activity. The technique involves burying cotton strips and measuring their tensile strength after a certain time. This gives a measure of the rotting rate, R, of the cotton strips. R is then a measure of soil microbial activity. This paper examines properties of the technique and indicates how the assay can be optimised. Humidity conditioning of the cotton strips before measuring their tensile strength reduced the within and between day variance and enabled the distribution of the tensile strength measurements to approximate normality. The test data came from a three-way factorial experiment (two soils, two temperatures, three moisture levels). The cotton strips were buried in the soil for intervals of time ranging up to 6 weeks. This enabled the rate of loss of cotton tensile strength with time to be studied under a range of conditions. An inverse cubic model accounted for greater than 90% of the total variation within each treatment combination. This offers support for summarising the decomposition process by a single parameter R. The approximate variance of the decomposition rate was estimated from a function incorporating the variance of tensile strength and the differential of the function for the rate of decomposition, R, with respect to tensile strength. This variance function has a minimum when the measured strength is approximately 2/3 that of the original strength. The estimates of R are almost unbiased and relatively robust against the cotton strips being left in the soil for more or less than the optimal time. We conclude that the rotting rate X should be measured using the inverse cubic equation, and that the cotton strips should be left in the soil until their strength has been reduced to about 2/3.

A survey in mathematics for industry: Two-timing and matched asymptotic expansions for singular perturbation problems

Following the derivation of amplitude equations through a new two-time-scale method [O'Malley, R. E., Jr. & Kirkinis, E (2010) A combined renormalization group-multiple scale method for singularly perturbed problems. Stud. Appl. Math. 124, 383-410], we show that a multi-scale method may often be preferable for solving singularly perturbed problems than the method of matched asymptotic expansions. We illustrate this approach with 10 singularly perturbed ordinary and partial differential equations. © 2011 Cambridge University Press.

Examples illustrating the use of renormalization techniques for singularly perturbed differential equations

With nine examples, we seek to illustrate the utility of the Renormalization Group approach as a unification of other asymptotic and perturbation methods.

Renormalization group interpretation of the Born and Rytov approximations

In this paper the method of renormalization group (RG) [Phys. Rev. E 54, 376 (1996)] is related to the well-known approximations of Rytov and Born used in wave propagation in deterministic and random media. Certain problems in linear and nonlinear media are examined from the viewpoint of RG and compared with the literature on Born and Rytov approximations. It is found that the Rytov approximation forms a special case of the asymptotic expansion generated by the RG, and as such it gives a superior approximation to the exact solution compared with its Born counterpart. Analogous conclusions are reached for nonlinear equations with an intensity-dependent index of refraction where the RG recovers the exact solution. © 2008 Optical Society of America.

Asymptotic minimax robust and misspecified Lorden quickest change detection for dependent stochastic processes

The quick detection of an abrupt unknown change in the conditional distribution of a dependent stochastic process has numerous applications. In this paper, we pose a minimax robust quickest change detection problem for cases where there is uncertainty about the post-change conditional distribution. Our minimax robust formulation is based on the popular Lorden criteria of optimal quickest change detection. Under a condition on the set of possible post-change distributions, we show that the widely known cumulative sum (CUSUM) rule is asymptotically minimax robust under our Lorden minimax robust formulation as a false alarm constraint becomes more strict. We also establish general asymptotic bounds on the detection delay of misspecified CUSUM rules (i.e. CUSUM rules that are designed with post- change distributions that differ from those of the observed sequence). We exploit these bounds to compare the delay performance of asymptotically minimax robust, asymptotically optimal, and other misspecified CUSUM rules. In simulation examples, we illustrate that asymptotically minimax robust CUSUM rules can provide better detection delay performance at greatly reduced computation effort compared to competing generalised likelihood ratio procedures.

Equilibrium and column studies of iron exchange with strong acid cation resin

The exchange of iron species from iron (III) chloride solutions with a strong acid cation resin has been investigated in relation to a variety of water and wastewater applications. A detailed equilibrium isotherm analysis was conducted wherein models such as Langmuir Vageler, Competitive Langmuir, Freundlich, Temkin, Dubinin Astakhov, Sips and Brouers-Sotolongo were applied to the experimental data. An important conclusion was that both the bottle-point method and solution normality used to generate the ion exchange equilibrium information influenced which sorption model fitted the isotherm profiles optimally. Invariably, the calculated value for the maximum loading of iron on strong acid cation resin was substantially higher than the value of 47.1 g/kg of resin which would occur if one Fe3+ ion exchanged for three “H+” sites on the resin surface. Consequently, it was suggested that above pH 1, various iron complexes sorbed to the resin in a manner which required less than 3 sites per iron moiety. Column trials suggested that the iron loading was 86.6 g/kg of resin when 1342 mg/L Fe (III) ions in water were flowed at 31.7 bed volumes per hour. Regeneration with 5 to 10 % HCl solutions reclaimed approximately 90 % of exchange sites.

A smooth transition logit model of the effects of deregulation in the electricity market

This paper introduces the smooth transition logit (STL) model that is designed to detect and model situations in which there is structural change in the behaviour underlying the latent index from which the binary dependent variable is constructed. The maximum likelihood estimators of the parameters of the model are derived along with their asymptotic properties, together with a Lagrange multiplier test of the null hypothesis of linearity in the underlying latent index. The development of the STL model is motivated by the desire to assess the impact of deregulation in the Queensland electricity market and ascertain whether increased competition has resulted in significant changes in the behaviour of the spot price of electricity, specifically with respect to the occurrence of periodic abnormally high prices. The model allows the timing of any change to be endogenously determined and also market participants' behaviour to change gradually over time. The main results provide clear evidence in support of a structural change in the nature of price events, and the endogenously determined timing of the change is consistent with the process of deregulation in Queensland.

Ion exchange treatment of saline solutions using lanxess S108H strong acid cation resin

Common to many types of water and wastewater is the presence of sodium ions which can be removed by desalination technologies, such as reverse osmosis and ion exchange. The focus of this investigation was ion exchange as it potentially offered several advantages compared to competing methods. The equilibrium and column behaviour of a strong acid cation (SAC) resin was examined for the removal of sodium ions from aqueous sodium chloride solutions of varying normality as well as a coal seam gas water sample. The influence of the bottle-point method to generate the sorption isotherms was evaluated and data interpreted with the Langmuir Vageler, Competitive Langmuir, Freundlich, and Dubinin-Astakhov models. With the constant concentration bottle point method, the predicted maximum exchange levels of sodium ions on the resin ranged from 61.7 to 67.5 g Na/kg resin. The general trend was that the lower the initial concentration of sodium ions in the solution, the lower the maximum capacity of the resin for sodium ions. In contrast, the constant mass bottle point method was found to be problematic in that the isotherm profiles may not be complete, if experimental parameters were not chosen carefully. Column studies supported the observations of the equilibrium studies, with maximum sodium loading of ca. 62.9 g Na/kg resin measured, which was in excellent agreement with the predictions of the data from the constant concentration bottle point method. Equilibria involving coal seam gas water were more complex due to the presence of sodium bicarbonate in solution, albeit the maximum loading capacity for sodium ions was in agreement with the results from the more simple sodium chloride solutions.

Methods for constructing uncertainty intervals for queries of Bayesian nets

In this paper the issue of finding uncertainty intervals for queries in a Bayesian Network is reconsidered. The investigation focuses on Bayesian Nets with discrete nodes and finite populations. An earlier asymptotic approach is compared with a simulation-based approach, together with further alternatives, one based on a single sample of the Bayesian Net of a particular finite population size, and another which uses expected population sizes together with exact probabilities. We conclude that a query of a Bayesian Net should be expressed as a probability embedded in an uncertainty interval. Based on an investigation of two Bayesian Net structures, the preferred method is the simulation method. However, both the single sample method and the expected sample size methods may be useful and are simpler to compute. Any method at all is more useful than none, when assessing a Bayesian Net under development, or when drawing conclusions from an ‘expert’ system.

Exact N-wave solutions of generalized Burgers equations

Exact N-wave solutions for the generalized Burgers equation u(t) + u(n)u(x) + (j/2t + alpha) u + (beta + gamma/x) u(n+1) = delta/2u(xx),where j, alpha, beta, and gamma are nonnegative constants and n is a positive integer, are obtained. These solutions are asymptotic to the (linear) old-age solution for large time and extend the validity of the latter so as to cover the entire time regime starting where the originally sharp shock has become sufficiently thick and the viscous effects are felt in the entire N wave.