24,071 research outputs found
Spectral statistics of molecular resonances in erbium isotopes: How chaotic are they?
We perform a comprehensive analysis of the spectral statistics of the
molecular resonances in Er and Er observed in recent ultracold
collision experiments [Frisch et al., Nature {\bf 507}, 475 (2014)] with the
aim of determining the chaoticity of this system. We calculate different
independent statistical properties to check their degree of agreement with
random matrix theory (RMT), and analyze if they are consistent with the
possibility of having missing resonances. The analysis of the short-range
fluctuations as a function of the magnetic field points to a steady increase of
chaoticity until G. The repulsion parameter decreases for higher
magnetic fields, an effect that can be interpreted as due to missing
resonances. The analysis of long-range fluctuations allows us to be more
quantitative and estimate a fraction of missing levels. Finally, a
study of the distribution of resonance widths provides additional evidence
supporting missing resonances of small width compared with the experimental
magnetic field resolution. We conclude that further measurements with increased
resolution will be necessary to give a final answer to the problem of missing
resonances and the agreement with RMT.Comment: 9 pages, 6 figure
A Cautionary Note on Generalized Linear Models for Covariance of Unbalanced Longitudinal Data
Missing data in longitudinal studies can create enormous challenges in data analysis when coupled with the positive-definiteness constraint on a covariance matrix. For complete balanced data, the Cholesky decomposition of a covariance matrix makes it possible to remove the positive-definiteness constraint and use a generalized linear model setup to jointly model the mean and covariance using covariates (Pourahmadi, 2000). However, this approach may not be directly applicable when the longitudinal data are unbalanced, as coherent regression models for the dependence across all times and subjects may not exist. Within the existing generalized linear model framework, we show how to overcome this and other challenges by embedding the covariance matrix of the observed data for each subject in a larger covariance matrix and employing the familiar EM algorithm to compute the maximum likelihood estimates of the parameters and their standard errors. We illustrate and assess the methodology using real data sets and simulations
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