Tropical Markov dynamics and Cayley cubic

We study the tropical version of Markov dynamics on the Cayley cubic, introduced by V.E. Adler and one of the authors. We show that this action is semi-conjugated to the standard action of $SL_2(\mathbb Z)$ on a torus, and thus is ergodic with the Lyapunov exponent and entropy given by the logarithm of the spectral radius of the corresponding matrix.Comment: Extended version, accepted for publication in "Integrable Systems and Algebraic Geometry" (Editors: R. Donagi, T. Shaska), Cambridge Univ. Press: LMS Lecture Notes Series, 201

Conway river and Arnold sail

We establish a simple relation between two geometric constructions in number theory: the Conway river of a real indefinite binary quadratic form and the Arnold sail of the corresponding pair of lines.Comment: Slightly modified version to appear in Arnold Mathematical Journa

Optical Modelling and Phylogenetic Analysis Provide Clues to the Likely Function of Corneal Nipple Arrays in Butterflies and Moths.

This is the final version. Available from MDPI via the DOI in this recordThe lenses in compound eyes of butterflies and moths contain an array of nipple-shaped protuberances, or corneal nipples. Previous work has suggested that these nipples increase light transmittance and reduce the eye glare of moths that are inactive during the day. This work builds on but goes further than earlier analyses suggesting a functional role for these structures including, for the first time, an explanation of why moths are attracted to UV light. Using a phylogenetic approach and 3D optical modelling, we show empirically that these arrays have been independently lost from different groups of moths and butterflies and vary within families. We find differences in the shape of nipples between nocturnal and diurnal species, and that anti-glow reflectance levels are different at different wave-lengths, a result thereby contradicting the currently accepted theory of eye glow for predator avoidance. We find that there is reduced reflectance, and hence greater photon absorption, at UV light, which is probably a reason why moths are attracted to UV. We note that the effective refractive index at the end of the nipples is very close to the refractive index of water, allowing almost all the species with nipples to see without distortion when the eye is partially or completely wet and providing the potential to keep eyes dry. These observations provide a functional explanation for these arrays. Of special interest is the finding that their repeated and independent loss across lepidopteran phylogeny is inconsistent with the explanation that they are being lost in the 'higher', more active butterflies

The HOSTS Survey for Exozodiacal Dust: Preliminary results and future prospects

[abridged] The presence of large amounts of dust in the habitable zones of nearby stars is a significant obstacle for future exo-Earth imaging missions. We executed an N band nulling interferometric survey to determine the typical amount of such exozodiacal dust around a sample of nearby main sequence stars. The majority of our data have been analyzed and we present here an update of our ongoing work. We find seven new N band excesses in addition to the high confidence confirmation of three that were previously known. We find the first detections around Sun-like stars and around stars without previously known circumstellar dust. Our overall detection rate is 23%. The inferred occurrence rate is comparable for early type and Sun-like stars, but decreases from 71% [+11%/-20%] for stars with previously detected mid- to far-infrared excess to 11% [+9%/-4%] for stars without such excess, confirming earlier results at high confidence. For completed observations on individual stars, our sensitivity is five to ten times better than previous results. Assuming a lognormal luminosity function of the dust, we find upper limits on the median dust level around all stars without previously known mid to far infrared excess of 11.5 zodis at 95% confidence level. The corresponding upper limit for Sun-like stars is 16 zodis. An LBTI vetted target list of Sun-like stars for exo-Earth imaging would have a corresponding limit of 7.5 zodis. We provide important new insights into the occurrence rate and typical levels of habitable zone dust around main sequence stars. Exploiting the full range of capabilities of the LBTI provides a critical opportunity for the detailed characterization of a sample of exozodiacal dust disks to understand the origin, distribution, and properties of the dust.Comment: To appear in SPIE Astronomical Telescopes + Instrumentation 2018 proceedings. Some typos fixed, one reference adde

Accreting Protoplanets in the LkCa 15 Transition Disk

Exoplanet detections have revolutionized astronomy, offering new insights into solar system architecture and planet demographics. While nearly 1900 exoplanets have now been discovered and confirmed, none are still in the process of formation. Transition discs, protoplanetary disks with inner clearings best explained by the influence of accreting planets, are natural laboratories for the study of planet formation. Some transition discs show evidence for the presence of young planets in the form of disc asymmetries or infrared sources detected within their clearings, as in the case of LkCa 15. Attempts to observe directly signatures of accretion onto protoplanets have hitherto proven unsuccessful. Here we report adaptive optics observations of LkCa 15 that probe within the disc clearing. With accurate source positions over multiple epochs spanning 2009 - 2015, we infer the presence of multiple companions on Keplerian orbits. We directly detect H{\alpha} emission from the innermost companion, LkCa 15 b, evincing hot (~10,000 K) gas falling deep into the potential well of an accreting protoplanet.Comment: 35 pages, 3 figures, 1 table, 9 extended data item