Mather sets for sequences of matrices and applications to the study of joint spectral radii

The joint spectral radius of a compact set of d-times-d matrices is defined ?to be the maximum possible exponential growth rate of products of matrices drawn from that set. In this article we investigate the ergodic-theoretic structure of those sequences of matrices drawn from a given set whose products grow at the maximum possible rate. This leads to a notion of Mather set for matrix sequences which is analogous to the Mather set in Lagrangian dynamics. We prove a structure theorem establishing the general properties of these Mather sets and describing the extent to which they characterise matrix sequences of maximum growth. We give applications of this theorem to the study of joint spectral radii and to the stability theory of discrete linear inclusions. These results rest on some general theorems on the structure of orbits of maximum growth for subadditive observations of dynamical systems, including an extension of the semi-uniform subadditive ergodic theorem of Schreiber, Sturman and Stark, and an extension of a noted lemma of Y. Peres. These theorems are presented in the appendix

A reappraisal of parameters for the putative planet PTFO 8-8695b and its potentially precessing parent star

Published photometry of fading events in the PTFO 8-8695 system is modelled using improved treatments of stellar geometry, surface intensities, and, particularly, gravity darkening, with a view to testing the planetary-transit hypothesis. Variability in the morphology of fading events can be reproduced by adopting convective-envelope gravity darkening, but near-critical stellar rotation is required. This leads to inconsistencies with spectroscopic observations; the model also predicts substantial photometric variability associated with stellar precession, contrary to observations. Furthermore, the empirical ratio of orbital to rotational angular momenta is at odds with physically plausible values. An exoplanet transiting a precessing, gravity-darkened star may not be the correct explanation of periodic fading events in this system

Emission-line stars in the LMC: the Armagh survey, and a metacatalogue

[Aims] Accurate astrometry is required to reliably cross-match 20th-century photographic catalogues against 21st-century digital surveys. The present work provides modern-era identifications and astrometry for the 801 emission-line objects "of stellar appearance" in the Armagh survey (the largest of its nature to date). [Methods] Targets have been individually identified in digital images using the Armagh Atlas and, in most cases, unambiguously matched to entries in the UCAC astrometric catalogues. [Results] Astrometry with sub-arcsecond precision is now available for all the major photographic spectroscopic surveys of the LMC. The results are used to compile an annotated metacatalogue of 1675 individual, spectroscopically identified candidate H-alpha-emission stars, including detailed cross-matching between catalogues, and resolving many (though not all) identification ambiguities in individual primary sources

The Competition Account of Achievement‐Value

A great achievement makes one’s life go better independently of its results, but what makes an achievement great? A simple answer is—its difficulty. I defend this view against recent, pressing objections by interpreting difficulty in terms of competitiveness. Difficulty is determined not by how hard the agent worked for the end but by how hard others would need to do in order to compete. Successfully reaching a goal is a valuable achievement because it is difficult, and it is difficult because it is competitive. Hence, both virtuosic performances and lucky successes can be valuable achievements

Improved astrometry for the Bohannan & Epps catalogue

Aims: Accurate astrometry is required to reliably cross-match 20th-century catalogues against 21st-century surveys. The present work aims to provide such astrometry for the 625 entries of the Bohannan & Epps (BE74) catalogue of H$\alpha$ emission-line stars. Methods: BE74 targets have been individually identified in digital images and, in most cases, unambiguously matched to entries in the UCAC4 astrometric catalogue. Results: Sub-arcsecond astrometry is now available for almost all BE74 stars. Several identification errors in the literature illustrate the perils of relying solely on positional coincidences using poorer-quality astrometry.Comment: 3 pages, 1 figure. Accepted in A&