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Timekeeping
Published in Jill L. Baker, Technology of the Ancient Near East, 2018
Timekeeping was and continues to be based on the progression of the moon, sun, and other celestial bodies. Astronomers, those who regularly observed and recorded these movements, were the timekeepers of the ancient world. The Sumerians and Babylonians in Mesopotamia were among the first organized astronomers. As early as 3000–2000 bce, they recorded constellations, stars, planets, comets, lunar eclipses, and other astronomical events. They named the constellations they identified, including Leo, Taurus, Scorpius, Auriga, Gemini, Capricorn, and Sagittarius. The Babylonians created the zodiac to mark the twelve constellations traversed by the sun, moon, and planets. The MUL.APIN is a catalog of stars and constellations and a sort of “program” for predicting the heliacal rising and setting of planets, the length of daylight as measured by a water clock, gnomon, shadows, and intercalations (the combination of lunar and solar calendars). It primarily discussed the rising and setting of constellations in relation to the calendar, and determined when to add an extra month (Britton 2002:23, 25). The earliest copy of the MUL.APIN dates to ca. 686 bce; however, it is speculated that the original was probably completed ca. 1000 bce (Rochberg 1999; Reiner 1999; Brack-Bernsen 1999).
Trajectory tracking of a bouncing ball in a triangular billiard by unfolding and folding the billiard table
Published in International Journal of Control, 2022
Laura Menini, Corrado Possieri, Antonio Tornambè
As shown in Chapter 7 of Weeks (2001) the mapping from the interior of the hexagon to the surface of the hexagonal torus, that is is bijective. Furthermore, by construction, trajectories defined over the surface of the hexagonal torus are not subject to neither position nor velocity jumps and hence are continuously differentiable and piece-wise twice continuously differentiable. As an example, Figure 13 depicts the trajectory represented in Figure 10 over the six-billiards table over the hexagonal torus.However, by describing the equations of motion on such a manifold, the corresponding dynamics are nonlinear. A wholly similar approach can be used to deal with the finite billiards table considered in Remark 4.1. Namely, the eight-billiards table depicted in Figure 11(a) can be mapped to a torus by firstly stretching one of the side of the wall (so to avoid that multiple points collapse to a single axis) and then gluing opposite sides; see Figure 14. On the other hand, the twelve-billiards table shown in Figure 11(b) can be mapped to a hexagonal torus as shown in Figure 12.
Dissimilarity measure of local structure in inorganic crystals using Wasserstein distance to search for novel phosphors
Published in Science and Technology of Advanced Materials, 2021
Shota Takemura, Takashi Takeda, Takayuki Nakanishi, Yukinori Koyama, Hidekazu Ikeno, Naoto Hirosaki
Figure 1 schematically shows the calculation of the Wasserstein distance between a cubic and a square antiprism that distorts the helix angle 45 degrees from cubic keeping its height as an example. When a cubic structure is expressed as the bag in Equation (1), the bag consists of eight center-ligand distances, which are normalized to 1 and the 28 normalized ligand-ligand distances. The normalized ligand-ligand distances are twelve , twelve , and four 2. In the case of square antiprism, the eight center-ligand distances are the same for cubic, except 28 ligand-ligand distances differ from the cubic one. The ligand-ligand distances are eight , eight , four , and eight . The distributions of cubic and square antiprism in Equation (2) are indicated in Figure 1 as a histogram, where the arrows denote the optimal transport and the numbers next to the arrows are the transport cost. W is calculated as the sum of the products of the transport distance and the transport cost. In this case, W is 0.094.