Estimating Two-Stage Models for Genetic Influences on Alcohol, Tobacco or Drug Use Initiation and Dependence Vulnerability in Twin and Family Data

Genetic research on risk of alcohol, tobacco or drug dependence must make allowance for the partial overlap of risk-factors for initiation of use, and risk-factors for dependence or other outcomes in users. Except in the extreme cases where genetic and environmental risk-factors for initiation and dependence overlap completely or are uncorrelated, there is no consensus about how best to estimate the magnitude of genetic or environmental correlations between Initiation and Dependence in twin and family data. We explore by computer simulation the biases to estimates of genetic and environmental parameters caused by model misspecification when Initiation can only be defined as a binary variable. For plausible simulated parameter values, the two-stage genetic models that we consider yield estimates of genetic and environmental variances for Dependence that, although biased, are not very discrepant from the true values. However, estimates of genetic (or environmental) correlations between Initiation and Dependence may be seriously biased, and may differ markedly under different two-stage models. Such estimates may have little credibility unless external data favor selection of one particular model. These problems can be avoided if Initiation can be assessed as a multiple-category variable (e.g. never versus early-onset versus later onset user), with at least two categories measurable in users at risk for dependence. Under these conditions, under certain distributional assumptions., recovery of simulated genetic and environmental correlations becomes possible, Illustrative application of the model to Australian twin data on smoking confirmed substantial heritability of smoking persistence (42%) with minimal overlap with genetic influences on initiation.

Controlling the complex Lorenz equations by modulation

We demonstrate that a system obeying the complex Lorenz equations in the deep chaotic regime can be controlled to periodic behavior by applying a modulation to the pump parameter. For arbitrary modulation frequency and amplitude there is no obvious simplification of the dynamics. However, we find that there are numerous windows where the chaotic system has been controlled to different periodic behaviors. The widths of these windows in parameter space are narrow, and the positions are related to the ratio of the modulation frequency of the pump to the average pulsation frequency of the output variable. These results are in good agreement with observations previously made in a far-infrared laser system.

On positive solutions of nonlinear elliptic equations involving concave and critical nonlinearities

We study the existence of nonnegative solutions of elliptic equations involving concave and critical Sobolev nonlinearities. Applying various variational principles we obtain the existence of at least two nonnegative solutions.

Boundary value problems for systems of difference equations associated with systems of second-order ordinary differential equations

We study difference equations which arise as discrete approximations to two-point boundary value problems for systems of second-order ordinary differential equations. We formulate conditions which guarantee a priori bounds on first differences of solutions to the discretized problem. We establish existence results for solutions to the discretized boundary value problems subject to nonlinear boundary conditions. We apply our results to show that solutions to the discrete problem converge to solutions of the continuous problem in an aggregate sense. (C) 2002 Elsevier Science Ltd. All rights reserved.

Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations

We study the continuous problem y"=f(x,y,y'), xc[0,1], 0=G((y(0),y(1)),(y'(0), y'(1))), and its discrete approximation (y(k+1)-2y(k)+y(k-1))/h(2) =f(t(k), y(k), v(k)), k = 1,..., n-1, 0 = G((y(0), y(n)), (v(1), v(n))), where f and G = (g(0), g(1)) are continuous and fully nonlinear, h = 1/n, v(k) = (y(k) - y(k-1))/h, for k =1,..., n, and t(k) = kh, for k = 0,...,n. We assume there exist strict lower and strict upper solutions and impose additional conditions on f and G which are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. We show that the discrete approximation also has solutions which approximate solutions of the continuous problem and converge to the solution of the continuous problem when it is unique, as the grid size goes to 0. Homotopy methods can be used to compute the solution of the discrete approximation. Our results were motivated by those of Gaines.

Estimating and evaluating the statistics of gapped local-alignment scores

We present a novel maximum-likelihood-based algorithm for estimating the distribution of alignment scores from the scores of unrelated sequences in a database search. Using a new method for measuring the accuracy of p-values, we show that our maximum-likelihood-based algorithm is more accurate than existing regression-based and lookup table methods. We explore a more sophisticated way of modeling and estimating the score distributions (using a two-component mixture model and expectation maximization), but conclude that this does not improve significantly over simply ignoring scores with small E-values during estimation. Finally, we measure the classification accuracy of p-values estimated in different ways and observe that inaccurate p-values can, somewhat paradoxically, lead to higher classification accuracy. We explain this paradox and argue that statistical accuracy, not classification accuracy, should be the primary criterion in comparisons of similarity search methods that return p-values that adjust for target sequence length.

Supersymmetric t-J Gaudin models and KZ equations

Supersymmetric t-J Gaudin models with open boundary conditions are investigated by means of the algebraic Bethe ansatz method. Off-shell Bethe ansatz equations of the boundary Gaudin systems are derived, and used to construct and solve the KZ equations associated with sl (2\1)((1)) superalgebra.

Predictor-corrector methods of Runge-Kutta type for stochastic differential equations

In this paper we construct predictor-corrector (PC) methods based on the trivial predictor and stochastic implicit Runge-Kutta (RK) correctors for solving stochastic differential equations. Using the colored rooted tree theory and stochastic B-series, the order condition theorem is derived for constructing stochastic RK methods based on PC implementations. We also present detailed order conditions of the PC methods using stochastic implicit RK correctors with strong global order 1.0 and 1.5. A two-stage implicit RK method with strong global order 1.0 and a four-stage implicit RK method with strong global order 1.5 used as the correctors are constructed in this paper. The mean-square stability properties and numerical results of the PC methods based on these two implicit RK correctors are reported.

A variable stepsize implementation for stochastic differential equations

Stochastic differential equations (SDEs) arise from physical systems where the parameters describing the system can only be estimated or are subject to noise. Much work has been done recently on developing higher order Runge-Kutta methods for solving SDEs numerically. Fixed stepsize implementations of numerical methods have limitations when, for example, the SDE being solved is stiff as this forces the stepsize to be very small. This paper presents a completely general variable stepsize implementation of an embedded Runge Kutta pair for solving SDEs numerically; in this implementation, there is no restriction on the value used for the stepsize, and it is demonstrated that the integration remains on the correct Brownian path.

Evaluation of line-transect sampling for estimating koala abundance in the Pine Rivers Shire, south-east Queensland

Distance sampling using line transects has not been previously used or tested for estimating koala abundance. In July 2001, a pilot survey was conducted to compare the use of line transects with strip transects for estimating koala abundance. Both methods provided a similar estimate of density. On the basis of the results of the pilot survey, the distribution and abundance of koalas in the Pine Rivers Shire, south-east Queensland, was determined using line-transect sampling. In total, 134 lines (length 64 km) were used to sample bushland areas. Eighty-two independent koalas were sighted. Analysis of the frequency distribution of sighting distances using the software program DISTANCE enabled a global detection function to be estimated for survey sites in bushland areas across the Shire. Abundance in urban parts of the Shire was estimated from densities obtained from total counts at eight urban sites that ranged from 26 to 51 ha in size. Koala abundance in the Pine Rivers Shire was estimated at 4584 (95% confidence interval, 4040-5247). Line-transect sampling is a useful method for estimating koala abundance provided experienced koala observers are used when conducting surveys.

Difference equations in Banach spaces

Difference equations which discretely approximate boundary value problems for second-order ordinary differential equations are analysed. It is well known that the existence of solutions to the continuous problem does not necessarily imply existence of solutions to the discrete problem and, even if solutions to the discrete problem are guaranteed, they may be unrelated and inapplicable to the continuous problem. Analogues to theorems for the continuous problem regarding a priori bounds and existence of solutions are formulated for the discrete problem. Solutions to the discrete problem are shown to converge to solutions of the continuous problem in an aggregate sense. An example which arises in the study of the finite deflections of an elastic string under a transverse load is investigated. The earlier results are applied to show the existence of a solution; the sufficient estimates on the step size are presented. (C) 2003 Elsevier Science Ltd. All rights reserved.

Selecting methods to solve multi-well master equations

There are several competing methods commonly used to solve energy grained master equations describing gas-phase reactive systems. When it comes to selecting an appropriate method for any particular problem, there is little guidance in the literature. In this paper we directly compare several variants of spectral and numerical integration methods from the point of view of computer time required to calculate the solution and the range of temperature and pressure conditions under which the methods are successful. The test case used in the comparison is an important reaction in combustion chemistry and incorporates reversible and irreversible bimolecular reaction steps as well as isomerizations between multiple unimolecular species. While the numerical integration of the ODE with a stiff ODE integrator is not the fastest method overall, it is the fastest method applicable to all conditions.