Two isoperimetric inequalities for the Sobolev constant

In this note we prove two isoperimetric inequalities for the sharp constant in the Sobolev embedding and its associated extremal function. The first such inequality is a variation on the classical Schwarz Lemma from complex analysis, similar to recent inequalities of Burckel, Marshall, Minda, Poggi-Corradini, and Ransford, while the second generalises an isoperimetric inequality for the first eigenfunction of the Laplacian due to Payne and Rayner.Comment: 11 page

MedZIM: Mediation analysis for Zero-Inflated Mediators with applications to microbiome data

The human microbiome can contribute to the pathogenesis of many complex diseases such as cancer and Alzheimer's disease by mediating disease-leading causal pathways. However, standard mediation analysis is not adequate in the context of microbiome data due to the excessive number of zero values in the data. Zero-valued sequencing reads, commonly observed in microbiome studies, arise for technical and/or biological reasons. Mediation analysis approaches for analyzing zero-inflated mediators are still lacking largely because of challenges raised by the zero-inflated data structure: (a) disentangling the mediation effect induced by the point mass at zero; and (b) identifying the observed zero-valued data points that are actually not zero (i.e., false zeros). We develop a novel mediation analysis method under the potential-outcomes framework to fill this gap. We show that the mediation effect of the microbiome can be decomposed into two components that are inherent to the two-part nature of zero-inflated distributions. The first component corresponds to the mediation effect attributable to a unit-change over the positive relative abundance and the second component corresponds to the mediation effect attributable to discrete binary change of the mediator from zero to a non-zero state. With probabilistic models to account for observing zeros, we also address the challenge with false zeros. A comprehensive simulation study and the applications in two real microbiome studies demonstrate that our approach outperforms existing mediation analysis approaches.Comment: Corresponding: Zhigang L

Mass-to-light ratio gradients in early-type galaxy haloes

Since the near future should see a rapidly expanding set of probes of the halo masses of individual early-type galaxies, we introduce a convenient parameter for characterising the halo masses from both observational and theoretical results: \dML, the logarithmic radial gradient of the mass-to-light ratio. Using halo density profiles from LCDM simulations, we derive predictions for this gradient for various galaxy luminosities and star formation efficiencies $\epsilon_{SF}$. As a pilot study, we assemble the available \dML\ data from kinematics in early-type galaxies - representing the first unbiassed study of halo masses in a wide range of early-type galaxy luminosities - and find a correlation between luminosity and \dML, such that the brightest galaxies appear the most dark-matter dominated. We find that the gradients in most of the brightest galaxies may fit in well with the LCDM predictions, but that there is also a population of fainter galaxies whose gradients are so low as to imply an unreasonably high star formation efficiency $\epsilon_{SF} > 1$. This difficulty is eased if dark haloes are not assumed to have the standard LCDM profiles, but lower central concentrations.Comment: 17 pages, 13 figures. Accepted for publication on MNRA