The Fundamental Plane of Open Clusters

We utilize the data from the Apache Point Observatory Galactic Evolution Experiment-2 (APOGEE-2) in the fourteenth data release of the Sloan Digital Sky Survey (SDSS) to calculate the line-of-sight velocity dispersion $\sigma_{1D}$ of a sample of old open clusters (age larger than 100\,Myr) selected from the Milky Way open cluster catalog of Kharchenko et al. (2013). Together with their $K_s$ band luminosity $L_{K_s}$, and the half-light radius $r_{h}$ of the most probable members, we find that these three parameters show significant pairwise correlations among each other. Moreover, a fundamental plane-{\it like} relation among these parameters is found for the oldest open clusters (age older than 1\,Gyr), $L_{K_s}\propto\sigma_{1D}^{0.82\pm0.29}\cdot r_h^{2.19\pm0.52}$ with $rms \sim\, 0.31$\,mag in the $K_s$ band absolute magnitude. The existence of this relation, which deviates significantly from the virial theorem prediction, implies that the dynamical structures of the old open clusters are quite similar, when survived from complex dynamical evolution to age older than 1 Gyr.Comment: accepted publication for ApJ lette

Counterexamples in Scale Calculus

We construct counterexamples to classical calculus facts such as the Inverse and Implicit Function Theorems in Scale Calculus -- a generalization of Multivariable Calculus to infinite dimensional vector spaces in which the reparameterization maps relevant to Symplectic Geometry are smooth. Scale Calculus is a cornerstone of Polyfold Theory, which was introduced by Hofer-Wysocki-Zehnder as a broadly applicable tool for regularizing moduli spaces of pseudoholomorphic curves. We show how the novel nonlinear scale-Fredholm notion in Polyfold Theory overcomes the lack of Implicit Function Theorems, by formally establishing an often implicitly used fact: The differentials of basic germs -- the local models for scale-Fredholm maps -- vary continuously in the space of bounded operators when the base point changes. We moreover demonstrate that this continuity holds only in specific coordinates, by constructing an example of a scale-diffeomorphism and scale-Fredholm map with discontinuous differentials. This justifies the high technical complexity in the foundations of Polyfold Theory.Comment: published in PNAS, final versio

An Apparent Redshift Dependence of Quasar Continuum: Implication for Cosmic Dust Extinction?

We investigate the luminosity and redshift dependence of the quasar continuum by means of composite spectrum using a large non-BAL radio-quiet quasar sample drawn from the Sloan Digital Sky Survey. Quasar continuum slopes in the UV-Opt band are measured at two different wavelength ranges, i.e., $\alpha_{\nu12}$ ($1000\sim 2000 \rm\AA$) and $\alpha_{\nu24}$ ($2000 \sim 4000 \rm\AA$) derived from power law fitting. Generally, the UV spectra slope becomes harder (higher $\alpha_{\nu}$) towards higher bolometric luminosity. On the other hand, when quasars are further grouped into luminosity bins, we find both $\alpha_{\nu12}$ and $\alpha_{\nu24}$ show significant anti-correlation with redshift (i.e., quasar continuum becomes redder towards higher redshift). We suggest that the cosmic dust extinction is very likely the cause of this observed $\alpha_\nu-z$ relation. We build a simple cosmic dust extinction model to quantify the observed reddening tendency and find an effective dust density $n\sigma_v \sim 10^{-5}h~\rm Mpc^{-1}$ at $z<1.5$. The other possibilities that could produce such a reddening effect have also been discussed.Comment: 6 pages, 5 figures; published in ApJ