On the inverse parabolicity of pdf equations in turbulent flows

The focus of the present work is the well-known feature of the probability density function (PDF) transport equations in turbulent flows-the inverse parabolicity of the equations. While it is quite common in fluid mechanics to interpret equations with direct (forward-time) parabolicity as diffusive (or as a combination of diffusion, convection and reaction), the possibility of a similar interpretation for equations with inverse parabolicity is not clear. According to Einstein's point of view, a diffusion process is associated with the random walk of some physical or imaginary particles, which can be modelled by a Markov diffusion process. In the present paper it is shown that the Markov diffusion process directly associated with the PDF equation represents a reasonable model for dealing with the PDFs of scalars but it significantly underestimates the diffusion rate required to simulate turbulent dispersion when the velocity components are considered.

Uniqueness for boundary value problems for second order finite difference equations

We establish maximum principles for second order difference equations and apply them to obtain uniqueness for solutions of some boundary value problems.

Analysis of Melanoma Onset: Assessing Familial Aggregation by Using Estimating Equations and Fitting Variance Components via Bayesian Random Effects Models

We investigate whether relative contributions of genetic and shared environmental factors are associated with an increased risk in melanoma. Data from the Queensland Familial Melanoma Project comprising 15,907 subjects arising from 1912 families were analyzed to estimate the additive genetic, common and unique environmental contributions to variation in the age at onset of melanoma. Two complementary approaches for analyzing correlated time-to-onset family data were considered: the generalized estimating equations (GEE) method in which one can estimate relationship-specific dependence simultaneously with regression coefficients that describe the average population response to changing covariates; and a subject-specific Bayesian mixed model in which heterogeneity in regression parameters is explicitly modeled and the different components of variation may be estimated directly. The proportional hazards and Weibull models were utilized, as both produce natural frameworks for estimating relative risks while adjusting for simultaneous effects of other covariates. A simple Markov Chain Monte Carlo method for covariate imputation of missing data was used and the actual implementation of the Bayesian model was based on Gibbs sampling using the free ware package BUGS. In addition, we also used a Bayesian model to investigate the relative contribution of genetic and environmental effects on the expression of naevi and freckles, which are known risk factors for melanoma.

Implicit stochastic Runge-Kutta methods for stochastic differential equations

In this paper we construct implicit stochastic Runge-Kutta (SRK) methods for solving stochastic differential equations of Stratonovich type. Instead of using the increment of a Wiener process, modified random variables are used. We give convergence conditions of the SRK methods with these modified random variables. In particular, the truncated random variable is used. We present a two-stage stiffly accurate diagonal implicit SRK (SADISRK2) method with strong order 1.0 which has better numerical behaviour than extant methods. We also construct a five-stage diagonal implicit SRK method and a six-stage stiffly accurate diagonal implicit SRK method with strong order 1.5. The mean-square and asymptotic stability properties of the trapezoidal method and the SADISRK2 method are analysed and compared with an explicit method and a semi-implicit method. Numerical results are reported for confirming convergence properties and for comparing the numerical behaviour of these methods.

Adaptive stepsize based on control theory for stochastic differential equations

The numerical solution of stochastic differential equations (SDEs) has been focussed recently on the development of numerical methods with good stability and order properties. These numerical implementations have been made with fixed stepsize, but there are many situations when a fixed stepsize is not appropriate. In the numerical solution of ordinary differential equations, much work has been carried out on developing robust implementation techniques using variable stepsize. It has been necessary, in the deterministic case, to consider the best choice for an initial stepsize, as well as developing effective strategies for stepsize control-the same, of course, must be carried out in the stochastic case. In this paper, proportional integral (PI) control is applied to a variable stepsize implementation of an embedded pair of stochastic Runge-Kutta methods used to obtain numerical solutions of nonstiff SDEs. For stiff SDEs, the embedded pair of the balanced Milstein and balanced implicit method is implemented in variable stepsize mode using a predictive controller for the stepsize change. The extension of these stepsize controllers from a digital filter theory point of view via PI with derivative (PID) control will also be implemented. The implementations show the improvement in efficiency that can be attained when using these control theory approaches compared with the regular stepsize change strategy. (C) 2004 Elsevier B.V. All rights reserved.

Numerical simulation of stochastic ordinary differential equations in biomathematical modelling

In this work we discuss the effects of white and coloured noise perturbations on the parameters of a mathematical model of bacteriophage infection introduced by Beretta and Kuang in [Math. Biosc. 149 (1998) 57]. We numerically simulate the strong solutions of the resulting systems of stochastic ordinary differential equations (SDEs), with respect to the global error, by means of numerical methods of both Euler-Taylor expansion and stochastic Runge-Kutta type. (C) 2003 IMACS. Published by Elsevier B.V. All rights reserved.

Numerical methods for strong solutions of stochastic differential equations: an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

A genetic estimation algorithm for parameters of stochastic ordinary differential equations

A generic method for the estimation of parameters for Stochastic Ordinary Differential Equations (SODEs) is introduced and developed. This algorithm, called the GePERs method, utilises a genetic optimisation algorithm to minimise a stochastic objective function based on the Kolmogorov-Smirnov statistic. Numerical simulations are utilised to form the KS statistic. Further, the examination of some of the factors that improve the precision of the estimates is conducted. This method is used to estimate parameters of diffusion equations and jump-diffusion equations. It is also applied to the problem of model selection for the Queensland electricity market. (C) 2003 Elsevier B.V. All rights reserved.

Gauge Poisson representations for birth/death master equations

Poisson representation techniques provide a powerful method for mapping master equations for birth/death processes -- found in many fields of physics, chemistry and biology -- into more tractable stochastic differential equations. However, the usual expansion is not exact in the presence of boundary terms, which commonly occur when the differential equations are nonlinear. In this paper, a gauge Poisson technique is introduced that eliminates boundary terms, to give an exact representation as a weighted rate equation with stochastic terms. These methods provide novel techniques for calculating and understanding the effects of number correlations in systems that have a master equation description. As examples, correlations induced by strong mutations in genetics, and the astrophysical problem of molecule formation on microscopic grain surfaces are analyzed. Exact analytic results are obtained that can be compared with numerical simulations, demonstrating that stochastic gauge techniques can give exact results where standard Poisson expansions are not able to.

Mine waste beach profile and flow resistance equations

Four mine waste beach longitudinal profile equations are compared theoretically and in statistical analyses of profile data from 64 field and laboratory beaches formed by mine tailings, co-disposed coal mine wastes, and sand. All four equations fit the profile data well. The best performing equation both accounts for particle sorting and satisfies hydraulic constraints, and the combination of assumptions underlying it is considered to best represent the processes occurring on mine waste beaches. Combining these assumptions with the Lacey normal equation leads to a variant of the Manning resistance equation. Features that it is desirable to incorporate in theoretical and numerical models of mine waste beaches are listed.

Existence and multiplicity results for quasilinear elliptic equations

The goal of this paper is to study the multiplicity of positive solutions of a class of quasilinear elliptic equations. Based on the mountain pass theorems and sub-and supersolutions argument for p-Laplacian operators, under suitable conditions on nonlinearity f (x, s), we show the following problem: -Delta(p)u = lambda f(x,u) in Omega, u/(partial derivative Omega) = 0, where Omega is a bounded open subset of R-N, N >= 2, with smooth boundary, lambda is a positive parameter and Delta(p) is the p-Laplacian operator with p > 1, possesses at least two positive solutions for large lambda.

Predictive regression equations and clinical uses of peripheral pulse timing characteristics in children

Studies have shown that increased arterial stiffening can be an indication of cardiovascular diseases like hypertension. In clinical practice, this can be detected by measuring the blood pressure (BP) using a sphygmomanometer but it cannot be used for prolonged monitoring. It has been established that pulse wave velocity (PWV) is a direct measure of arterial stiffening but its usefulness is hampered by the absence of non-invasive techniques to estimate it. Pulse transit time (PTT) is a simple and non-invasive method derived from PWV. However, limited knowledge of PTT in children is found in the present literature. The aims of this study are to identify independent variables that confound PTT measure and describe PTT regression equations for healthy children. Therefore, PTT reference values are formulated for future pathological studies. Fifty-five Caucasian children (39 male) aged 8.4 +/- 2.3 yr (range 5-12 yr) were recruited. Predictive equations for PTT were obtained by multiple regressions with age, vascular path length, BP indexes and heart rate. These derived equations were compared in their PWV equivalent against two previously reported equations and significant agreement was obtained (p < 0.05). Findings herein also suggested that PTT can be useful as a continuous surrogate BP monitor in children.

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. © 2004 Elsevier Inc. All rights reserved.

Quantum computing and polynomial equations over the finite field Z(2)

What is the computational power of a quantum computer? We show that determining the output of a quantum computation is equivalent to counting the number of solutions to an easily computed set of polynomials defined over the finite field Z(2). This connection allows simple proofs to be given for two known relationships between quantum and classical complexity classes, namely BQP subset of P-#P and BQP subset of PP.