关键词:
Group theory
摘要:
In the real world, there are many phenomena related to the execution order of events. For example, in a 100 metres, several athletes start at the same starting command. In the downhill skiing event, athletes compete in sequence to decide the winner. Another example is that for a rigid body, if it first moves 10cm forward and then 10cm to the left, or first moves 10cm to the left and then 10cm forward, the total moving distance is 20 cm in both cases. However, when the rigid body has its center of mass fixed in a 3-D coordinate system, if it first rotates 90 degrees in the positive direction of the Y-axis and then 90 degrees in the positive direction of the X-axis, the final posture is different from that when it first rotates 90 degrees in the positive direction of the X-axis and then 90 degrees in the positive direction of the Y-axis. The execution of events serves as the foundation for many disciplines such as physics, computer science, and applied mathematics. Do the athletes in 100 metres start strictly at the same time? Why can we determine the 1st-place finisher in both head-to-head competitions and sequential competitions in speed sports? Why do some events yield the same results when their execution order is changed, while others produce different results? What algebraic properties does the execution order of events have? In this paper, we introduce algebraic theories such as set theory and group theory into the analysis of event execution order. We propose concepts like "optional intervals event" and "sequential operation", summarize their algebraic properties and draw Cayley tables. Then, we use these new concepts to answer the aforementioned questions. Based on these efforts, we offer new interpretations for certain physical phenomena and computer application scenarios. Finally, we present other issues derived from this paradigm. These concepts can deepen our understanding of motion and find applications in areas such as event arrangement, physical simula