Sizes of pentagonal clusters in fullerenes

Stability and chemistry, both exohedral and endohedral, of fullerenes are critically dependent on the distribution of their obligatory 12 pentagonal faces. It is well known that there are infinitely many IPR-fullerenes and that the pentagons in these fullerenes can be at an arbitrarily large distance from each other. IPR-fullerenes can be described as fullerenes in which each connected cluster of pentagons has size 1. In this paper we study the combinations of cluster sizes that can occur in fullerenes and whether the clusters can be at an arbitrarily large distance from each other. For each possible partition of the number 12, we are able to decide whether the partition describes the sizes of pentagon clusters in a possible fullerene, and state whether the different clusters can be at an arbitrarily large distance from each other. We will prove that all partitions with largest cluster of size 5 or less can occur in an infinite number of fullerenes with the clusters at an arbitrarily large distance of each other, that 9 partitions occur in only a finite number of fullerene isomers and that 15 partitions do not occur at all in fullerenes

Multi-messenger observations of a binary neutron star merger

On 2017 August 17 a binary neutron star coalescence candidate (later designated GW170817) with merger time 12:41:04 UTC was observed through gravitational waves by the Advanced LIGO and Advanced Virgo detectors. The Fermi Gamma-ray Burst Monitor independently detected a gamma-ray burst (GRB 170817A) with a time delay of ~1.7 s with respect to the merger time. From the gravitational-wave signal, the source was initially localized to a sky region of 31 deg2 at a luminosity distance of 40+8-8 Mpc and with component masses consistent with neutron stars. The component masses were later measured to be in the range 0.86 to 2.26 Mo. An extensive observing campaign was launched across the electromagnetic spectrum leading to the discovery of a bright optical transient (SSS17a, now with the IAU identification of AT 2017gfo) in NGC 4993 (at ~40 Mpc) less than 11 hours after the merger by the One- Meter, Two Hemisphere (1M2H) team using the 1 m Swope Telescope. The optical transient was independently detected by multiple teams within an hour. Subsequent observations targeted the object and its environment. Early ultraviolet observations revealed a blue transient that faded within 48 hours. Optical and infrared observations showed a redward evolution over ~10 days. Following early non-detections, X-ray and radio emission were discovered at the transient’s position ~9 and ~16 days, respectively, after the merger. Both the X-ray and radio emission likely arise from a physical process that is distinct from the one that generates the UV/optical/near-infrared emission. No ultra-high-energy gamma-rays and no neutrino candidates consistent with the source were found in follow-up searches. These observations support the hypothesis that GW170817 was produced by the merger of two neutron stars in NGC4993 followed by a short gamma-ray burst (GRB 170817A) and a kilonova/macronova powered by the radioactive decay of r-process nuclei synthesized in the ejecta