The star cluster mass--galactocentric radius relation: Implications for cluster formation

Whether or not the initial star cluster mass function is established through a universal, galactocentric-distance-independent stochastic process, on the scales of individual galaxies, remains an unsolved problem. This debate has recently gained new impetus through the publication of a study that concluded that the maximum cluster mass in a given population is not solely determined by size-of-sample effects. Here, we revisit the evidence in favor and against stochastic cluster formation by examining the young ($\lesssim$ a few $\times 10^8$ yr-old) star cluster mass--galactocentric radius relation in M33, M51, M83, and the Large Magellanic Cloud. To eliminate size-of-sample effects, we first adopt radial bin sizes containing constant numbers of clusters, which we use to quantify the radial distribution of the first- to fifth-ranked most massive clusters using ordinary least-squares fitting. We supplement this analysis with an application of quantile regression, a binless approach to rank-based regression taking an absolute-value-distance penalty. Both methods yield, within the $1\sigma$ to $3\sigma$ uncertainties, near-zero slopes in the diagnostic plane, largely irrespective of the maximum age or minimum mass imposed on our sample selection, or of the radial bin size adopted. We conclude that, at least in our four well-studied sample galaxies, star cluster formation does not necessarily require an environment-dependent cluster formation scenario, which thus supports the notion of stochastic star cluster formation as the dominant star cluster-formation process within a given galaxy.Comment: ApJ, in press, 39 pages in AAS preprint format, 10 multi-panel figures (some reduced in size to match arXiv compilation routines

Extremal properties of the first eigenvalue and the fundamental gap of a sub-elliptic operator

We consider the problems of extreming the first eigenvalue and the fundamental gap of a sub-elliptic operator with Dirichlet boundary condition, when the potential $V$ is subjected to a $p$-norm constraint. The existence results for weak solutions, compact embedding theorem and spectral theory for sub-elliptic equation are given. Moreover, we provide the specific characteristics of the corresponding optimal potential function

A Novel Vector-Field-Based Motion Planning Algorithm for 3D Nonholonomic Robots

This paper focuses on the motion planning for mobile robots in 3D, which are modelled by 6-DOF rigid body systems with nonholonomic kinematics constraints. We not only specify the target position, but also bring in the requirement of the heading direction at the terminal time, which gives rise to a new and more challenging 3D motion planning problem. The proposed planning algorithm involves a novel velocity vector field (VF) over the workspace, and by following the VF, the robot can be navigated to the destination with the specified heading direction. In order to circumvent potential collisions with obstacles and other robots, a composite VF is designed by composing the navigation VF and an additional VF tangential to the boundary of the dangerous area. Moreover, we propose a priority-based algorithm to deal with the motion coupling issue among multiple robots. Finally, numerical simulations are conducted to verify the theoretical results

Extended main-sequence turnoffs in the double cluster hh and χ\chi Persei: The complex role of stellar rotation

Using {\sl Gaia} Data Release 2 photometry, we report the detection of extended main-sequence turnoff (eMSTO) regions in the color--magnitude diagrams (CMDs) of the $\sim 14$ Myr-old double clusters $h$ and $\chi$ Persei (NGC 869 and NGC 884). We find that stars with masses below $\sim$1.3 $M_{\odot}$ in both $h$ and $\chi$ Persei populate narrow main sequences (MSs), while more massive stars define the eMSTO, closely mimicking observations of young Galactic and Magellanic Cloud clusters (with ages older than $\sim$30 Myr). Previous studies based on clusters older than $\sim$30 Myr find that rapidly rotating MS stars are redder than slow rotators of similar luminosity, suggesting that stellar rotation may be the main driver of the eMSTO. By combining photometry and projected rotational velocities from the literature of stars in $h$ and $\chi$ Persei, we find no obvious relation between the rotational velocities and colors of non-emission-line eMSTO stars, in contrast with what is observed in older clusters. Similarly to what is observed in Magellanic Cloud clusters, most of the extremely rapidly rotating stars, identified by their strong H$\alpha$ emission lines, are located in the red part of the eMSTOs. This indicates that stellar rotation plays a role in the color and magnitude distribution of MSTO stars. By comparing the observations with simulated CMDs, we find that a simple population composed of coeval stars that span a wide range of rotation rates is unable to reproduce the color spread of the clusters' MSs. We suggest that variable stars, binary interactions, and stellar rotation affect the eMSTO morphology of these very young clusters.Comment: 14 pages, 12 figures, ApJ accepte

Some controllability results of a class of N-dimensional parabolic equations with internal single-point degeneracy

This paper investigates the controllability of a class of $N$-dimensional degenerate parabolic equations with interior single-point degeneracy. We employ the Galerkin method to prove the existence of solutions for the equations. The analysis is then divided into two cases based on whether the degenerate point $x=0$ lies within the control region $\omega_0$ or not. For each case, we establish specific Carleman estimates. As a result, we achieve null controllability in the first case $0\in\omega_0$ and unique continuation and approximate controllability in the second case $0\notin\omega_0$