We consider the quantum dynamics of a neutral atom Bose-Einstein condensate in a double-well potential, including many-body hard-sphere interactions. Using a mean-field factorization we show that the coherent oscillations due to tunneling are suppressed when the number of atoms exceeds a critical value. An exact quantum solution, in a two-mode approximation, shows that the mean-field solution is modulated by a quantum collapse and revival sequence. ͓S1050-2947͑97͒01406-6͔

A method to estimate reference intervals which depend on gestational age is described. The approach is based on standard statistical techniques, so implementation should be straightforward with most statistical packages. The reference limits are defined by simple formulae which may be plotted as smooth curves. The method is illustrated with analyses of three real datasets arising from different aspects of fetal monitoring.

The age-speci®c reference interval is an important screening tool in medicine. Put crudely, an individual whose value of a variable of interest lies outside certain extreme centiles may be suspected of abnormality. We propose a parametric method for constructing such intervals. It provides smooth centile curves and explicit formulae for the centile estimates and for standard deviation (SD) scores (age-standardized values). Each parameter of an exponential±normal or modulus±exponential±normal density is modelled as a fractional polynomial function of age. Estimation is by maximum likelihood. These three-and four-parameter models involve transformations of the data towards normality which remove non-normal skewness and/or kurtosis. Fractional polynomials provide more¯exible curve shapes than do conventional polynomials. The method easily accommodates binary covariates facilitating, for example, parsimonious modelling of ageand sex-speci®c centile curves. A method of calculating precision pro®les for centile estimates is proposed. Goodness of ®t is assessed by using Q±Q-plots and Shapiro±Wilk W-tests of the SD scores, and likelihood ratio tests of the parameters of an enlarged model. Four substantial real data sets are used to illustrate the method. Comparisons are made with the semiparametric LMS method of Cole and Green.

Objective To create reliable reference ranges and calculate Z scores for fetal head ultrasound biometry Design A prospective, cross-sectional study.Setting Obstetric clinics (outpatient and delivery units) at the University Hospital of Zurich.Sample The study data were obtained from 6557 pregnant women.Methods Only the first ultrasound examination between 12 and 42 weeks of each fetus with exactly established gestational age was used for analysis. No exclusions were made on the grounds of small-for-date birthweight, prematurity or other events several weeks after the examination. Separate regression models were fitted to estimate the mean and standard deviation at each gestational age for each parameter.Results A total of 6217 fetal head biparietal diameters and 5510 occipito-frontal diameters were measured. Both head circumference and cephalic index were derived in 5462 cases where both biparietal diameter and occipito-frontal diameter could be measured on the same fetus. The centile charts, tables and regression formulae for biparietal and occipito-frontal diameters, head circumference and cephalic index are presented. An application to calculate 2 scores was developed using Excel (Microsoft Corporation, USA) and macros are presented in detail in the Figure 8 footnote. The comparison of our charts with those of the two most recent studies revealed almost no differences in biparietal diameter centiles. In one publication, occipito-frontal diameter charts, and in another, head circumference charts were different from the current study.Conclusions We have presented centile charts, tables and regression formulae for fetal head ultrasound biometry derived from a large and minimally selected sample size in a carefully designed cross-sectional study. Complete tables and regression formulae to calculate reference ranges and Z scores are presented for use in computer-aided evaluation of fetal ultrasound biometry.using a large sample size which is evenly distributed from 12 to 42 weeks of pregnancy.

The age‐specific reference interval is a commonly used screening tool in medicine. It involves estimation of extreme quantile curves (such as the 5th and 95th centiles) of a reference distribution of clinically normal individuals. It is crucial that models used to estimate such intervals fit the data extremely well. However, few procedures to assess goodness‐of‐fit have been proposed in the literature, and even fewer have been evaluated systematically. Here we consider procedures based on the distribution of the Z‐scores (standardized residuals) from a model and on Pearson χ2 statistics for observed and expected counts in groups defined by age and the estimated reference centile curves. Two of the procedures (Q and grid tests) are mainly inferential, whereas the third (permutation bands and B‐tests) is essentially graphical. We obtain approximations to the null distributions of several relevant test statistics and examine their size and power for a range of models based on real data sets. We recommend Q‐tests in all situations where Z‐scores are available since they are general, simple to calculate and usually have the highest power among the three classes of test considered. For the cases considered the grid tests are always inferior to the Q‐ and B‐ tests. Copyright © 2000 John Wiley & Sons, Ltd.