Inscribing or Circumscribing a Histogon to a Convex Polygon

We consider two optimization problems of approximating a convex polygon, one by a largest inscribed histogon and the other by a smallest circumscribed histogon. An axis-aligned histogon is an axis-aligned rectilinear polygon such that every horizontal edge has an integer length. A histogon of orientation ? is a copy of an axis-aligned histogon rotated by ? in counterclockwise direction. The goal is to find a largest inscribed histogon and a smallest circumscribed histogon over all orientations in [0,?). Depending on whether the horizontal width of a histogon is predetermined or not, we consider several different versions of the problem and present exact algorithms. These optimization problems belong to shape analysis, classification, and simplification, and they have applications in various cost-optimization problems

6MapNet: Representing soccer players from tracking data by a triplet network

Although the values of individual soccer players have become astronomical, subjective judgments still play a big part in the player analysis. Recently, there have been new attempts to quantitatively grasp players' styles using video-based event stream data. However, they have some limitations in scalability due to high annotation costs and sparsity of event stream data. In this paper, we build a triplet network named 6MapNet that can effectively capture the movement styles of players using in-game GPS data. Without any annotation of soccer-specific actions, we use players' locations and velocities to generate two types of heatmaps. Our subnetworks then map these heatmap pairs into feature vectors whose similarity corresponds to the actual similarity of playing styles. The experimental results show that players can be accurately identified with only a small number of matches by our method.Comment: 12 pages, 4 figures, In 8th Workshop on Machine Learning and Data Mining for Sports Analytics (MLSA21

Radial Dependence of the Proto-Globular Cluster Contribution to the Milky Way Formation

Recent interpretation of the color$-$magnitude diagrams of the Milky Way (MW) bulge has suggested that the observed double red-clump feature can be a natural consequence of He-enhanced stellar populations in the MW bulge. This implies that globular clusters (GCs), where the He-enhanced second-generation (SG) stars can be efficiently created, are the most likely candidate contributors of He-rich stars to the MW bulge. We extend this idea to the Galactic inner halo and investigate the fraction of the SG stars as a function of the Galactocentric distance. We use bluer blue-horizontal branch (bBHB) stars, which are assumed to be originated from He-rich SG populations, as proxies of SG stars, and find that the fraction of bBHB stars increases with decreasing Galactocentric distance. Simulations of the GC evolution in the MW tidal field qualitatively support the observed trend of bBHB enhancement in the inner halo. In these simulations, the increasing tidal force with decreasing Galactocentric distance leads to stripping of stars not only from the outskirts but also from the central regions of GCs, where SG stars are more abundant. We discuss the implication and prospect of our findings concerning the formation history of the bulge and inner halo of the MW.Comment: 11 pages, 4 figures, accepted for publication in the ApJ