The effect of incremental replacement of wheat with soya hulls in diets for Jersey cows on lactational performance, diet digestibility and feeding behaviour

Eight Jersey cows were used in two balanced 4 x 4 Latin Squares to investigate the effects of replacement of dietary starch with non-forage fibre on productivity, diet digestibility and feeding behaviour. Total-mixed rations consisted of maize silage, grass silage and a soyabean meal-based concentrate mixture, each at 250g/kg DM, with the remaining 250g consisting of cracked wheat/soya hulls (SH) in the ratios of 250:0, 167:83; 83:167 and 0:250 g, respectively, for treatments SH0, SH83, SH167 and SH250. Starch concentrations were 302, 248, 193 and 140g/kg DM, and NDF concentrations were 316, 355, 394 and 434g/kg DM, for treatments SHO, SH83, SH167 and SH250, respectively. Total eating time increased (p < 0.05) as SH inclusion increased, but total rumination time was unaffected. Digestibility of DM, organic matter and starch declined (p < 0.01) as SH inclusion increased, whilst digestibility of NDF and ADF increased (p < 0.01). Dry-matter intake tended to decline with increasing SH, whilst bodyweight, milk yield and fat and lactose concentrations were unaffected by treatment. Milk protein concentration decreased (p < 0.01) as SH level increased. Feed conversion efficiency improved (p < 0.05) as SH inclusion rose, but it was not possible to determine whether this was due to the increased fibre levels alone, or the favourable effect on rumen fermentation of decreasing starch levels. (c) 2006 Elsevier B.V. All rights reserved.

Statistical isomorphism of three-level factional factorial designs

From a statistician's standpoint, the interesting kind of isomorphism for fractional factorial designs depends on the statistical application. Combinatorially isomorphic fractional factorial designs may have different statistical properties when factors are quantitative. This idea is illustrated by using Latin squares of order 3 to obtain fractions of the 3(3) factorial. design in 18 runs.

Effects of 3-nitrooxypropanol on methane emission,digestion, and energy and nitrogen balance of lactating dairy cows

The objective was to measure effects of 3-nitrooxypropanol (3NP) on methane production of lactating dairy cows and any associated changes in digestion and energy and nitrogen metabolism. Six Holstein-Friesian dairy cows in mid-lactation were fed twice daily a total mixed ration with maize silage as the primary forage source. Cows received 1 of 3 treatments using an experimental design based on two 3 × 3 Latin squares with 5-wk periods. Treatments were a control placebo or 500 or 2,500 mg/d of 3NP delivered directly into the rumen, via the rumen fistula, in equal doses before each feeding. Measurements of methane production and energy and nitrogen balance were obtained during wk 5 of each period using respiration calorimeters and digestion trials. Measurements of rumen pH (48 h) and postprandial volatile fatty acid and ammonia concentrations were made at the end of wk 4. Daily methane production was reduced by 3NP, but the effects were not dose dependent (reductions of 6.6 and 9.8% for 500 and 2,500 mg/d, respectively). Dosing 3NP had a transitory inhibitory effect on methane production, which may have been due to the product leaving the rumen in liquid outflow or through absorption or metabolism. Changes in rumen concentrations of volatile fatty acids indicated that the pattern of rumen fermentation was affected by both doses of the product, with a decrease in acetate:propionate ratio observed, but that acetate production was inhibited by the higher dose. Dry matter, organic matter, acid detergent fiber, N, and energy digestibility were reduced at the higher dose of the product. The decrease in digestible energy supply was not completely countered by the decrease in methane excretion such that metabolizable energy supply, metabolizable energy concentration of the diet, and net energy balance (milk plus tissue energy) were reduced by the highest dose of 3NP. Similarly, the decrease in nitrogen digestibility at the higher dose of the product was associated with a decrease in body nitrogen balance that was not observed for the lower dose. Milk yield and milk fat concentration and fatty acid composition were not affected but milk protein concentration was greater for the higher dose of 3NP. Twice-daily rumen dosing of 3NP reduced methane production by lactating dairy cows, but the dose of 2,500 mg/d reduced rumen acetate concentration, diet digestibility, and energy supply. Further research is warranted to determine the optimal dose and delivery method of the product. Key words: 3-nitrooxypropanol, methane, digestion, rumen, dairy cow

Approximate Gauss-Newton methods for nonlinear least squares problems

The Gauss–Newton algorithm is an iterative method regularly used for solving nonlinear least squares problems. It is particularly well suited to the treatment of very large scale variational data assimilation problems that arise in atmosphere and ocean forecasting. The procedure consists of a sequence of linear least squares approximations to the nonlinear problem, each of which is solved by an “inner” direct or iterative process. In comparison with Newton’s method and its variants, the algorithm is attractive because it does not require the evaluation of second-order derivatives in the Hessian of the objective function. In practice the exact Gauss–Newton method is too expensive to apply operationally in meteorological forecasting, and various approximations are made in order to reduce computational costs and to solve the problems in real time. Here we investigate the effects on the convergence of the Gauss–Newton method of two types of approximation used commonly in data assimilation. First, we examine “truncated” Gauss–Newton methods where the inner linear least squares problem is not solved exactly, and second, we examine “perturbed” Gauss–Newton methods where the true linearized inner problem is approximated by a simplified, or perturbed, linear least squares problem. We give conditions ensuring that the truncated and perturbed Gauss–Newton methods converge and also derive rates of convergence for the iterations. The results are illustrated by a simple numerical example. A practical application to the problem of data assimilation in a typical meteorological system is presented.

On integral equation and least squares methods for scattering by diffraction gratings

In this paper we consider the scattering of a plane acoustic or electromagnetic wave by a one-dimensional, periodic rough surface. We restrict the discussion to the case when the boundary is sound soft in the acoustic case, perfectly reflecting with TE polarization in the EM case, so that the total field vanishes on the boundary. We propose a uniquely solvable first kind integral equation formulation of the problem, which amounts to a requirement that the normal derivative of the Green's representation formula for the total field vanish on a horizontal line below the scattering surface. We then discuss the numerical solution by Galerkin's method of this (ill-posed) integral equation. We point out that, with two particular choices of the trial and test spaces, we recover the so-called SC (spectral-coordinate) and SS (spectral-spectral) numerical schemes of DeSanto et al., Waves Random Media, 8, 315-414 1998. We next propose a new Galerkin scheme, a modification of the SS method that we term the SS* method, which is an instance of the well-known dual least squares Galerkin method. We show that the SS* method is always well-defined and is optimally convergent as the size of the approximation space increases. Moreover, we make a connection with the classical least squares method, in which the coefficients in the Rayleigh expansion of the solution are determined by enforcing the boundary condition in a least squares sense, pointing out that the linear system to be solved in the SS* method is identical to that in the least squares method. Using this connection we show that (reflecting the ill-posed nature of the integral equation solved) the condition number of the linear system in the SS* and least squares methods approaches infinity as the approximation space increases in size. We also provide theoretical error bounds on the condition number and on the errors induced in the numerical solution computed as a result of ill-conditioning. Numerical results confirm the convergence of the SS* method and illustrate the ill-conditioning that arises.

The method of least squares used to invert an orbit problem

Six parameters uniquely describe the orbit of a body about the Sun. Given these parameters, it is possible to make predictions of the body's position by solving its equation of motion. The parameters cannot be directly measured, so they must be inferred indirectly by an inversion method which uses measurements of other quantities in combination with the equation of motion. Inverse techniques are valuable tools in many applications where only noisy, incomplete, and indirect observations are available for estimating parameter values. The methodology of the approach is introduced and the Kepler problem is used as a real-world example. (C) 2003 American Association of Physics Teachers.