Higher-Rank Numerical Ranges and Compression Problems

We consider higher-rank versions of the standard numerical range for matrices. A central motivation for this investigation comes from quantum error correction. We develop the basic structure theory for the higher-rank numerical ranges, and give a complete description in the Hermitian case. We also consider associated projection compression problems.Comment: 14 pages, 3 figures, to appear in Linear Algebra and its Application

Effect of Sinusoidal Surface Roughness and Energy on the Orientation of Cylinder-Forming Block Copolymer Thin Films

We explore the relative stability of three possible orientations of cylinder-forming di-block copolymer on a sinusoidally corrugated substrate. The cylinders can be aligned either parallel to the substrate, with their long axis being oriented along or orthogonal to the corrugation trenches, or perpendicular to the substrate. Using self-consistent field theory, we investigate the influence of substrate roughness and surface preference on the phase transition between the three orientations. When the substrate preference, $u$, towards one of components is small, increasing the substrate roughness induces a phase transition from parallel to perpendicular cylindrical phase. However, when $u$ is large, the parallel orientation is more stable than the perpendicular one. Within this parallel phase, increasing the substrate roughness leads to a transition of cylinder orientation changing from being orthogonal to parallel to the trench long axis. Increasing the substrate preference leads to an opposite transition from parallel to orthogonal to the trenches. Furthermore, we predict that the perpendicular cylinder phase is easier to be obtained when the unidirectional corrugation is along the longer unit vector of the hexagonal packing than when it is along the shorter unit vector. Our results qualitatively agree with previous experiments, and contribute towards applications of the cylinder-forming block copolymer in nanotechnology.Comment: 9 pages, 7 figure

A Comet Active Beyond the Crystallization Zone

We present observations showing in-bound long-period comet C/2017 K2 (PANSTARRS) to be active at record heliocentric distance. Nucleus temperatures are too low (60 K to 70 K) either for water ice to sublimate or for amorphous ice to crystallize, requiring another source for the observed activity. Using the Hubble Space Telescope we find a sharply-bounded, circularly symmetric dust coma 10$^5$ km in radius, with a total scattering cross section of $\sim$10$^5$ km$^2$. The coma has a logarithmic surface brightness gradient -1 over much of its surface, indicating sustained, steady-state dust production. A lack of clear evidence for the action of solar radiation pressure suggests that the dust particles are large, with a mean size $\gtrsim$ 0.1 mm. Using a coma convolution model, we find a limit to the apparent magnitude of the nucleus $V >$ 25.2 (absolute magnitude $H >$ 12.9). With assumed geometric albedo $p_V$ = 0.04, the limit to the nucleus circular equivalent radius is $<$ 9 km. Pre-discovery observations from 2013 show that the comet was also active at 23.7 AU heliocentric distance. While neither water ice sublimation nor exothermic crystallization can account for the observed distant activity, the measured properties are consistent with activity driven by sublimating supervolatile ices such as CO$_2$, CO, O$_2$ and N$_2$. Survival of supervolatiles at the nucleus surface is likely a result of the comet's recent arrival from the frigid Oort cloud.Comment: 20 pages, 3 figures, 2 tables, published on Astrophysical Journal Letters, 847:L19 (5pp), 2017 October

Coma Anisotropy and the Rotation Pole of Interstellar Comet 2I/Borisov

Hubble Space Telescope observations of interstellar comet 2I/Borisov near perihelion show the ejection of large (>~100 um) particles at <~9 m/s speeds, with estimated mass-loss rates of ~35 kg/s. The total mass loss from comet Borisov corresponds to loss of a surface shell on the nucleus only ~0.4 m thick. This shell is thin enough to be susceptible to past chemical processing in the interstellar medium by cosmic rays, meaning that the ejected materials cannot necessarily be considered as pristine. Our high-resolution images reveal persistent asymmetry in the dust coma, best explained by a thermal lag on the rotating nucleus causing peak mass loss to occur in the comet nucleus afternoon. In this interpretation, the nucleus rotates with an obliquity of 30 deg (pole direction RA = 205 deg and Dec. = 52 deg). The subsolar latitude varied from -35 deg (southern solstice) at the time of discovery to 0 deg (equinox) in 2020 January, suggesting the importance of seasonal effects. Subsequent activity likely results from regions freshly activated as the northern hemisphere is illuminated for the first time.Comment: 8 pages, 7 figure

Fragment Dynamics in Active Asteroid 331P/Gibbs

We present a dynamical analysis of the fragmented active asteroid 331P/Gibbs. Using archival images taken by the Hubble Space Telescope from 2015 to 2018, we measured the astrometry of the primary and the three brightest (presumably the largest) components. Conventional orbit determination revealed a high-degree of orbital similarity between the components. We then applied a fragmentation model to fit the astrometry, obtaining key parameters including the fragmentation epochs and separation velocities. Our best-fit models show that Fragment B separated from the primary body at a speed of $\sim$1 cm s$^{-1}$ between 2011 April and May, whereas two plausible scenarios were identified for Fragments A and C. The former split either from the primary or from Fragment B, in 2011 mid-June at a speed of $\sim$8 cm s$^{-1}$, and the latter split from Fragment B either in late 2011 or between late 2013 and early 2014, at a speed of $\sim$0.7-0.8 cm s$^{-1}$. The results are consistent with rotational disruption as the mechanism causing the cascading fragmentation of the asteroid, as suggested by the rapid rotation of the primary. The fragments constitute the youngest known asteroid cluster, providing us with a great opportunity to study asteroid fragmentation and formation of asteroid clusters.Comment: 13 pages, 5 figures, 4 tables. Accepted by A