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AimsIt is generally accepted that post-MI heart failure (HF) changes a variety of aspects of sarcoplasmic reticular Ca2+ fluxes but for some aspects there is disagreement over whether there is an increase or decrease. The commonest statistical approach is to treat data collected from each cell as independent, even though they are really clustered with multiple likely similar cells from each heart. In this study, we test whether this statistical assumption of independence can lead the investigator to draw conclusions that would be considered erroneous if the analysis handled clustering with specific statistical techniques (hierarchical tests).Methods and resultsCa2+ transients were recorded in cells loaded with Fura-2AM and sparks were recorded in cells loaded with Fluo-4AM. Data were analysed twice, once with the common statistical approach (assumption of independence) and once with hierarchical statistical methodologies designed to allow for any clustering. The statistical tests found that there was significant hierarchical clustering. This caused the common statistical approach to underestimate the standard error and report artificially small P values. For example, this would have led to the erroneous conclusion that time to 50% peak transient amplitude was significantly prolonged in HF.Spark analysis showed clustering, both within each cell and also within each rat, for morphological variables. This means that a three-level hierarchical model is sometimes required for such measures. Standard statistical methodologies, if used instead, erroneously suggest that spark amplitude is significantly greater in HF and spark duration is reduced in HF.ConclusionCa2+ fluxes in isolated cardiomyocytes show so much clustering that the common statistical approach that assumes independence of each data point will frequently give the false appearance of statistically significant changes. Hierarchical statistical methodologies need a little more effort, but are necessary for reliable conclusions. We present cost-free simple tools for performing these analyses.

We study the global stability of charged dust grains orbiting an axisymmetric planet. The corotating magnetic field and induced electric field are described in an inertial frame using the magnetic stream function and a two-dimensional effective potential U e , parametrized by the conserved angular momentum p f . Stable equilibria form the centers of complex potential wells, so that a particle which loses local stability can still be trapped globally. Applying the results to Saturn, we find that nonequatorial "halo" orbits are dominated by positively charged grains. PACS numbers: 96.30.Wr, 45.20.Jj, 96.35.Kx

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A general method is presented for analytically calculating linear stability limits for symplectic maps of arbitrary dimension in terms of the coefficients of the characteristic polynomial and the Krein signatures. Explicit results are given for dimensions 4, 6, and 8. The codimension and unfolding are calculated for all cases having a double eigenvalue on the unit circle. The results are applicable to many physical problems, including the restricted three-body problem and orbital stability in particle accelerators.