We present high spatial resolution midand far-infrared images of the Vega debris disk obtained with the Multiband Imaging Photometer for Spitzer (MIPS). The disk is well resolved and its angular size is much larger than found previously. The radius of the disk is at least 43 (330 AU), 70 (543 AU), and 105 (815 AU) in extent at 24, 70 and 160 μm, respectively. The disk images are circular, smoot...

Introduction: Machine learning (ML) applications for studying asteroid resonant dynamics are a relatively new field of study. Results from several different approaches currently available asteroids interacting with the z 2 , 1 M1:2, and ν 6 resonances. However, one challenge when using ML to databases produced by these studies is that there often severe imbalance ratio between number in librati...

In an edge modification problem one has to change the edge set of a given graph as little as possible so as to satisfy a certain property. We prove the NP-hardness of a variety of edge modification problems with respect to some well-studied classes of graphs. These include perfect, chordal, chain, comparability, split and asteroidal triple free. We show that some of these problems become polyno...