A genome linkage scan was carried out using a resource flock of 1029 sheep in six half-sib families. The families were offspring of sires derived by crossing divergent lines of sheep selected for response to challenge with the intestinal parasitic nematode Trichostrongylus colubriformis. All animals in the resource flock were phenotypically assessed for worm resistance soon after weaning using a vaccination/challenge regime. After correcting for fixed effects using a least squares linear model the faecal egg count data obtained following the first challenge and the faecal egg count data obtained after the second challenge were designated Trait 1 and Trait 2, respectively. A total of 472 lambs drawn from the phenotypic extremes of the Trait 2 faecal egg count distribution were genotyped with a panel of 133 microsatellite markers covering all 26 sheep autosomes. Detection of quantitative trait loci (QTL) for each of the faecal egg count traits was determined using interval analysis with the Animap program with recombination rates between markers derived from an existing marker map. No chromosomal regions attained genome-wide significance for QTL influencing either of the traits. However, one region attained chromosome-wide significance and five other regions attained point-wise significance for the presence of QTL affecting parasite resistance.

Abstract. 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.Key words. damped harmonic oscillators with noise, stationary distribution, stochastic RungeKutta methods, implicit midpoint rule, multiplicative noise AMS subject classifications. 60-08, 65C30 DOI. 10.1137/050646032 1. Introduction. Newton's second law of motion relates force to acceleration. Consequently, second-order differential equations are common in scientific applications, in the guise of "Langevin," "Monte Carlo," "molecular," or "dissipative particle" dynamics [1,2,3], and the study of methods for second-order ordinary differential equations is one of the most mature branches of numerical analysis [4]. The most exciting advances in recent decades have been the development of symplectic methods, capable of exactly preserving an energy-like quantity over very long times [5], and their extension to stochastic systems [6]. In the stochastic setting, the long-time dynamics of a typical physical system is governed by fluctuation-dissipation, so that the amount of time spent in different regions of phase space can be calculated from the stationary density [7]. This density can have a relatively simple explicit expression even when the dynamics is highly nonlinear [8]. Numerical methods replace continuous-time with discrete-time dynamics, generating values at times t 0 , t 1 , . . . . Usually t n+1 − t n is a fixed number Δt. The criterion for a good numerical method that will be examined in this work is that its discrete-time dynamics has a stationary density as close as possible to that of the continuous-time system.The differential equations describing second-order systems contain a parameter known as damping. The stationary density is independent of damping, but dynamical quantities, and the usefulness of numerical algorithms, are strongly dependent on it. In the infinite-damping limit, the system becomes first order. The limit of zero damping, on the other hand, corresponds to Hamiltonian systems, where symplectic methods can be applied. The aim in this paper is to devise methods capable of accurately reproducing the stationary density for all positive values of damping.

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