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.

Vibration-rotation spectra of 13C containing acetylene: anharmonic resonances

High resolution vibration-rotation spectra of 13C2H2 were recorded in a number of regions from 2000 to 5200 cm−1 at Doppler or pressure limited resolution. In these spectral ranges cold and hot bands involving the bending-stretching combination levels have been analyzed up to high J values. Anharmonic quartic resonances for the combination levels ν1 + mν4 + nν5, ν2 + mν4 + (n + 2) ν5 and ν3 + (m − 1) ν4 + (n + 1) ν5 have been studied, and the l-type resonances within each polyad have been explicitly taken into account in the analysis of the data. The least-squares refinement provides deperturbed values for band origins and rotational constants, obtained by fitting rotation lines only up to J ≈ 20 with root mean square errors of ≈ 0.0003 cm−1. The band origins allowed us to determine a number of the anharmonicity constants xij0.

Simultaneous analysis of v1, v4, 2v2, v2+v5, and 2v5 bands of 12CH3F

The Fourier-transform spectrum of CH3F from 2800 to 3100 cm−1, obtained by Guelachvili in Orsay at a resolution of about 0.003 cm−1, was analyzed. The effective Hamiltonian used contained all symmetry allowed interactions up to second order in the Amat-Nielsen classification, together with selected third-order terms, amongst the set of nine vibrational basis functions represented by the states ν1(A1), ν4(E), 2ν2(A1), ν2 + ν5(E), 2ν50(A1), and 2ν5±2(E). A number of strong Fermi and Coriolis resonances are involved. The vibrational Hamiltonian matrix was not factorized beyond the requirements of symmetry. A total of 59 molecular parameters were refined in a simultaneous least-squares analysis to over 1500 upper-state energy levels for J ≤ 20 with a standard deviation of 0.013 cm−1. Although the standard deviation remains an order of magnitude greater than the precision of the measurements, this work breaks new ground in the simultaneous analysis of interacting symmetric top vibrational levels, in terms of the number of interacting vibrational states and the number of parameters in the Hamiltonian.

The method of predicate least squares refinement

Parameters to be determined in a least squares refinement calculation to fit a set of observed data may sometimes usefully be `predicated' to values obtained from some independent source, such as a theoretical calculation. An algorithm for achieving this in a least squares refinement calculation is described, which leaves the operator in full control of the weight that he may wish to attach to the predicate values of the parameters.

Heterosis for yield and its physiological determinants in wheat

Heterosis in hybrid wheat varieties produced using a chemical hybridising agent was assessed in field experiments. Hyno Esta and its parents were compared in factorial combinations of four-seed rates (25-300 seeds m(-2)) and two nitrogen fertilizer rates (0 and 200 kg N ha(-1)) in 2001/02 and again in 2002/03. Hyno Rista and Hyno Renta and their parents were compared at two-seed rates in 2001/02. Hyno Rista and its parents were added factorially to the Hyno Esta experiment in 2002/03, while Hyno Renta and Hybred and their parents were compared at two seed rates in 2002/03. Mid parent heterosis for grain yield was found in three hybrids and two of these showed high parent heterosis. High parent heterosis in Hyno Esta over a range of sowing densities was mostly exhibited in total biomass but also, in one of two years, in harvest index. High parent heterosis in Hyno Renta was associated more with harvest index than with biomass. The heterosis for biomass in Hyno Esta resulted from greater interception of photosynthetically active radiation (PAR) than the male parent, with better radiation use efficiency than the female parent. In both seasons Hyno Esta achieved grain numbers per ear at least as high as the high parent for this trait (Audace), and combined this with mean grain weights at least as heavy as the high parent for mean grain weight (Estica). Much of the increased biomass and grain yield in the hybrid came late in the season as high parent heterosis was expressed for both maximum grain filling rate and grain filling duration. Heterosis was higher when nitrogen was applied than when withheld; only greater at lower seed rates when expressed in proportionate terms (e.g. as a percentage of the parents), rather than in absolute terms (e.g. t ha(-1)); and greater in the year with the cooler and wetter summer.

Extreme host plant conservatism during at least 20 million years of host plant pursuit by oak gallwasps

Diversification of insect herbivores is often associated with coevolution between plant toxins and insect countermeasures, resulting in a specificity that restricts host plant shifts. Gall inducers, however, bypass plant toxins and the factors influencing host plant associations in these specialized herbivores remain unclear. We reconstructed the evolution of host plant associations in Western Palaearctic oak gallwasps (Cynipidae: Cynipini), a species-rich lineage of specialist herbivores on oak (Quercus). (1) Bayesian analyses of sequence data for three genes revealed extreme host plant conservatism, with inferred shifts between major oak lineages (sections Cerris and Quercus) closely matching the minimum required to explain observed diversity. It thus appears that the coevolutionary demands of gall induction constrain host plant shifts, both in cases of mutualism (e.g., fig wasps, yucca moths) and parasitism (oak gallwasps). (2) Shifts between oak sections occurred independently in sexual and asexual generations of the gallwasp lifecycle, implying that these can evolve independently. (3) Western Palaearctic gallwasps associated with sections Cerris and Quercus diverged at least 20 million years ago (mya), prior to the arrival of oaks in the Western Palaearctic from Asia 5-7 mya. This implies an Asian origin for Western Palaearctic gallwasps, with independent westwards range expansion by multiple lineages.

Fitting models for the joint action of two drugs using SAS(R)

Combinations of drugs are increasingly being used for a wide variety of diseases and conditions. A pre-clinical study may allow the investigation of the response at a large number of dose combinations. In determining the response to a drug combination, interest may lie in seeking evidence of synergism, in which the joint action is greater than the actions of the individual drugs, or of antagonism, in which it is less. Two well-known response surface models representing no interaction are Loewe additivity and Bliss independence, and Loewe or Bliss synergism or antagonism is defined relative to these. We illustrate an approach to fitting these models for the case in which the marginal single drug dose-response relationships are represented by four-parameter logistic curves with common upper and lower limits, and where the response variable is normally distributed with a common variance about the dose-response curve. When the dose-response curves are not parallel, the relative potency of the two drugs varies according to the magnitude of the desired effect and the models for Loewe additivity and synergism/antagonism cannot be explicitly expressed. We present an iterative approach to fitting these models without the assumption of parallel dose-response curves. A goodness-of-fit test based on residuals is also described. Implementation using the SAS NLIN procedure is illustrated using data from a pre-clinical study. Copyright © 2007 John Wiley & Sons, Ltd.