AI 中文总结
本研究针对杂耍序列，提出两种扩展操作及对应条件，辅以示例与交互式可视化，丰富了有效siteswap序列的构造方法。
AI 中文摘要
在本注中，我们研究用于扩展杂耍（siteswap）序列的两个简单操作，称为首次投掷和末位接球。这两种构造源于杂耍者的自然问题：如何在不产生碰撞的情况下，在模式开头添加投掷或在末尾添加接球？利用着陆时间和置换检验，我们给出必要且充分条件，证明将这些构造应用于任意有效siteswap时，会产生另一个有效的siteswap。我们的分析辅以多个示例，相关扩展的交互式可视化可在[Para]中获取。
英文摘要
In this note, we study two simple operations for extending juggling (siteswap) sequences, called first throws and last catches. Both constructions come from a natural question for a juggler: how can one add throws at the beginning or catches at the end of a pattern without creating a collision? Using landing times and the permutation test, we give necessary and sufficient conditions under which these constructions, when applied to an arbitrary valid siteswap, produce another valid siteswap. We complement our analysis with several examples. An interactive visualization of these extensions is available in [Para].
