Achievable accuracy of parameter estimation for multidimensional NMR experiments

A fundamental issue in NMR spectroscopy is the estimation of parameters such as the Larmor frequencies of nuclei, J coupling constants, and relaxation rates. The Cramer-Rao lower bound provides a method to assess the best achievable accuracy of parameter estimates resulting from an unbiased estimation procedure. We show how the Cramer-Rao lower bound can be calculated for data obtained from multidimensional NMR experiments. The Cramer-Rao lower bound is compared to the variance of parameter estimates for simulated data using a least-squares estimation procedure. It is also shown how our results on the Cramer-Rao lower bound can be used to analyze whether an experimental design can be improved to provide experimental data which can result in parameter estimates with higher accuracy. The concept of nonuniform averaging in the indirect dimension is introduced and studied in connection with nonuniform sampling of the data.     

Infinite-dimensional gauge groups and special nonlinear gravitons

A gauge theory in flat space—time, in which the gauge algebra is the (infinite-dimensional) algebra of vector fields on a surface, determines a curved space—time metric. This note deals with some completely integrable examples, concentrating on the N → ∞ limit of the Euler—Arnol'd equations [geodesics on SO(N)]. In this case, the metric turns out to be flat, which points the way to a coordinate transformation that solves the original equations.     

Calibration procedure for gel permeation chromatography

A novel procedure is presented for calibrating a gel permeation chromatograph by using broad polymer fractions. The limit in accuracy is set by the accuracy of number-average and weight-average molecular weight measurements and the reproducibility of the chromatograph. In the particular calibration for linear polyethylene described here, low molecular weight fractions were prepared by nitric acid oxidation of annealed single crystals. The calibration is then of particular value in morphological studies of polyethylene. The influence in the calibration of polar endgroups on the polymer, as produced in these oxidized samples is discussed.     

Gel permeation chromatographic studies of the degradation of polyethylene with fuming nitric acid. II. Bulk polyethylene

Preliminary results are reported on the use of gel permeation chromatography in morphological studies of bulk polymers and fibres. Several samples of bulk isotropic and drawn polyethylene were analyzed by gel permeation chromatography following nitric acid oxidation. In all cases suitable samples showed two peaks in the molecular weight distribution, suggesting a qualitative similarity with results for single crystals. It is concluded that the present data are consistent with chain folding in bulk polymers, both in the isotropic and oriented states, with a less degree of regularity than exists in single crystals.     

Gel permeation chromatographic studies of the degradation of polyethylene with fuming nitric acid. I. Single crystals

This paper describes the application of gel permeation chromatography to the morphology of polymer single crystals. Various types of single crystals of polyethylene were etched with fuming nitric acid, and the molecular weight distributions of the degraded fragments determined. The crystal preparations studied were monolayer crystals grown from xylene solution at 85°C and multilayer crystals grown at 84°C and 70°C. In all cases peaks in the molecular weight distribution were observed corresponding to single and double transverses of the molecular chains through the lamellae. By using the chromatograph calibration described in a previous paper, the position of these peaks were compared and correlated with previous estimates of lamellar thickness from low-angle x-ray measurements. The relative positions of the peaks provide information regarding the nature of the fold surface. The results are found to be consistent with a model in which the majority of the molecules are tightly folded.     

Backward photoproduction of ϱ° and f between 2.8 GeV and 4.8 GeV

A measurement of backward photoproduction of charged pion pairs on protons is reported. The pion pair mass spectrum shows strong ?° and f production. Data are presented on the u and s dependence for ?^O and f, together with the decay polarisation for the ?^O.