Numerical methods for second-order stochastic differential equations

We seek numerical methods for second‐order stochastic differential equations that reproduce the stationary density accurately for all values of damping. A complete analysis is possible for scalar linear second‐order equations (damped harmonic oscillators with additive noise), where the statistics are Gaussian and can be calculated exactly in the continuous‐time and discrete‐time cases. A matrix equation is given for the stationary variances and correlation for methods using one Gaussian random variable per timestep. The only Runge–Kutta method with a nonsingular tableau matrix that gives the exact steady state density for all values of damping is the implicit midpoint rule. Numerical experiments, comparing the implicit midpoint rule with Heun and leapfrog methods on nonlinear equations with additive or multiplicative noise, produce behavior similar to the linear case.

The impact of nonlinear exposure risk relationships on seasonal time series data: modelling danish neonatal birth anthropometric data

Background Birth weight and length have seasonal fluctuations. Previous analyses of birth weight by latitude effects identified seemingly contradictory results, showing both 6 and 12 monthly periodicities in weight. The aims of this paper are twofold: (a) to explore seasonal patterns in a large, Danish Medical Birth Register, and (b) to explore models based on seasonal exposures and a non-linear exposure-risk relationship. Methods Birth weight and birth lengths on over 1.5 million Danish singleton, live births were examined for seasonality. We modelled seasonal patterns based on linear, U- and J-shaped exposure-risk relationships. We then added an extra layer of complexity by modelling weighted population-based exposure patterns. Results The Danish data showed clear seasonal fluctuations for both birth weight and birth length. A bimodal model best fits the data, however the amplitude of the 6 and 12 month peaks changed over time. In the modelling exercises, U- and J-shaped exposure-risk relationships generate time series with both 6 and 12 month periodicities. Changing the weightings of the population exposure risks result in unexpected properties. A J-shaped exposure-risk relationship with a diminishing population exposure over time fitted the observed seasonal pattern in the Danish birth weight data. Conclusion In keeping with many other studies, Danish birth anthropometric data show complex and shifting seasonal patterns. We speculate that annual periodicities with non-linear exposure-risk models may underlie these findings. Understanding the nature of seasonal fluctuations can help generate candidate exposures.

On the computation and interpretation of auto- and cross-trispectra

This paper discusses the principal domains of auto- and cross-trispectra. It is shown that the cumulant and moment based trispectra are identical except on certain planes in trifrequency space. If these planes are avoided, their principal domains can be derived by considering the regions of symmetry of the fourth order spectral moment. The fourth order averaged periodogram will then serve as an estimate for both cumulant and moment trispectra. Statistics of estimates of normalised trispectra or tricoherence are also discussed.

Numerical analysis for a variable-order nonlinear cable equation

In this paper, a variable-order nonlinear cable equation is considered. A numerical method with first-order temporal accuracy and fourth-order spatial accuracy is proposed. The convergence and stability of the numerical method are analyzed by Fourier analysis. We also propose an improved numerical method with second-order temporal accuracy and fourth-order spatial accuracy. Finally, the results of a numerical example support the theoretical analysis.

Serum bone formation marker correlation with improved osseointegration in osteoporotic rats treated with simvastatin

Objective: Simvastatin has been shown to enhance osseointegration of pure titanium implants in osteoporotic rats. This study aimed to evaluate the relationship between the serum level of bone formation markers and the osseointegration of pure titanium implants in osteoporotic rats treated with simvastatin. Materials and methods: Fifty-four female Sprague Dawley rats, aged 3 months old, were randomly divided into three groups: Sham-operated group (SHAM; n=18), ovariectomized group (OVX; n=18), and ovariectomized with Simvastatin treatment group (OVX+SIM; n=18). Fifty-six days after ovariectomy, screw-shaped titanium implants were inserted into the tibiae. Simvastatin was administered orally at 5mg/kg each day after the placement of the implant in the OVX+SIM group. The animals were sacrificed at either 28 or 84 days after implantation and the undecalcified tissue sections were processed for histological analysis. Total alkaline phosphatase (ALP), bone specific alkaline phosphatase (BALP) and bone Gla protein (BGP) were measured in all animal sera collected at the time of euthanasia and correlated with the histological assessment of osseointegration. Results: The level of ALP in the OVX group was higher than the SHAM group at day 28, with no differences between the three groups at day 84. The level of BALP in the OVX+SIM group was significantly higher than both OVX and SHAM groups at days 28 and 84. Compared with day 28, the BALP level of all three groups showed a significant decrease at day 84. There were no significant differences in BGP levels between the three groups at day 28, but at day 84 the OVX+SIM group showed significantly higher levels than both the OVX and SHAM groups. There was a significant increase in BGP levels between days 28 and 84 in the OVX+SIM group. The serum bone marker levels correlated with the histological assessment showing reduced osseointegration in the OVX compared to the SHAM group which is subsequently reversed in the OVX+SIM group.

Higher-order spectral analysis of nonlinear ocean surface gravity waves

Higher-order spectral analysis is used to detect the presence of secondary and tertiary forced waves associated with the nonlinearity of energetic swell observed in 8- and 13-m water depths. Higher-order spectral analysis techniques are first described and then applied to the field data, followed by a summary of the results.

Higher-Order spectra of nonlinear polynomial models for Chua's circuit

Polynomial models are shown to simulate accurately the quadratic and cubic nonlinear interactions (e.g. higher-order spectra) of time series of voltages measured in Chua's circuit. For circuit parameters resulting in a spiral attractor, bispectra and trispectra of the polynomial model are similar to those from the measured time series, suggesting that the individual interactions between triads and quartets of Fourier components that govern the process dynamics are modeled accurately. For parameters that produce the double-scroll attractor, both measured and modeled time series have small bispectra, but nonzero trispectra, consistent with higher-than-second order nonlinearities dominating the chaos.

An analytical solution for diffusion and nonlinear uptake of oxygen in a spherical cell

We develop a new analytical solution for a reactive transport model that describes the steady-state distribution of oxygen subject to diffusive transport and nonlinear uptake in a sphere. This model was originally reported by Lin (Journal of Theoretical Biology, 1976 v60, pp449–457) to represent the distribution of oxygen inside a cell and has since been studied extensively by both the numerical analysis and formal analysis communities. Here we extend these previous studies by deriving an analytical solution to a generalized reaction-diffusion equation that encompasses Lin’s model as a particular case. We evaluate the solution for the parameter combinations presented by Lin and show that the new solutions are identical to a grid-independent numerical approximation.

A review of road traffic noise indicators and their correlation with the LA10(18hour)

This paper outlines a study to determine the correlation between the LA10(18hour) and other road traffic noise indicators. It is based on a database comprising of 404 measurement locations including 947 individual days of valid noise measurements across numerous circumstances taken between November 2001 and November 2007. This paper firstly discusses the need and constraints on the indicators and their nature of matching a suitable indicator to the various road traffic noise dynamical characteristics. The paper then presents a statistical analysis of the road traffic noise monitoring data, correlating various indicators with the LA10(18hour) statistical indicator and provides a comprehensive table of linear correlations. There is an extended analysis on relationships across the night time period. The paper concludes with a discussion on the findings.

A nonlinear PI and backstepping based controller for tractor-steerable trailer influenced by slip

Autonomous guidance of agricultural vehiclesis vital as mechanized farming production becomes more prevalent. It is crucial that tractor-trailers are guided with accuracy in both lateral and longitudinal directions, whilst being affected by large disturbance forces, or slips, owing to uncertain and undulating terrain. Successful research has been concentrated on trajectory control which can provide longitudinal and lateral accuracy if the vehicle moves without sliding, and the trailer is passive. In this paper, the problem of robust trajectory tracking along straight and circular paths of a tractor-steerable trailer is addressed. By utilizing a robust combination of backstepping and nonlinear PI control, a robust, nonlinear controller is proposed. For vehicles subjected to sliding, the proposed controller makes the lateral deviations and the orientation errors of the tractor and trailer converge to a neighborhood near the origin. Simulation results are presented to illustrate that the suggested controller ensures precise trajectory tracking in the presence of slip.

Patterns of correlation between vehicle occupant seat pressure and anthropometry

Seat pressure is known as a major factor of seat comfort in vehicles. In passenger vehicles, there is lacking research into the seat comfort of rear seat occupants. As accurate seat pressure measurement requires significant effort, simulation of seat pressure is evolving as a preferred method. However, analytic methods are based on complex finite element modeling and therefore are time consuming and involve high investment. Based on accurate anthropometric measurements of 64 male subjects and outboard rear seat pressure measurements in three different passenger vehicles, this study investigates if a set of parameters derived from seat pressure mapping are sensitive enough to differentiate between different seats and whether they correlate with anthropometry in linear models. In addition to the pressure map analysis, H-Points were measured with a coordinate measurement system based on palpated body landmarks and the range of H-Point locations in the three seats is provided. It was found that for the cushion, cushion contact area and cushion front area/force could be modeled by subject anthropometry,while only seatback contact area could be modeled based on anthropometry for all three vehicles. Major differences were found between the vehicles for other parameters.