分散因子的普遍性--余项的力量
Scattered Factor Universality -- The Power of the Remainder
摘要：
散射因子（圆）普适性由Barker等人于2020年首次引入。如果$w$的字母表中每个长度为$k$的词都作为分散因子出现在$w$中，则一个词$w$被称为$k$ -通用于某个自然数$k$;如果$w$的共轭是$k$-通用的，则它被称为循环$k$ -通用的。这里，如果通过删除$w$的部分从$w$获得$u$，即存在（可能为空）具有$w=v_1u_1v_2\cdots v_nu_nv_{n+1}$的字$v_1,\dots,v_{n+1}$，则字$u=u_1\cdots u_n$被称为$w$的分散因子。在这项工作中，我们证明了两个问题，左开放在上述文件中，即一个概括的主要定理之一，以任意字母和轻微修改的另一个定理，使我们的循环普遍性的普遍性。在路上，我们提出了深刻的见解的行为，其余的所谓的拱因式分解Hebrard时，重复的话被认为是。
英文摘要：
Scattered factor (circular) universality was firstly introduced by Barker et al. in 2020. A word $w$ is called $k$-universal for some natural number $k$, if every word of length $k$ of $w$'s alphabet occurs as a scattered factor in $w$; it is called circular $k$-universal if a conjugate of $w$ is $k$-universal. Here, a word $u=u_1\cdots u_n$ is called a scattered factor of $w$ if $u$ is obtained from $w$ by deleting parts of $w$, i.e. there exists (possibly empty) words $v_1,\dots,v_{n+1}$ with $w=v_1u_1v_2\cdots v_nu_nv_{n+1}$. In this work, we prove two problems, left open in the aforementioned paper, namely a generalisation of one of their main theorems to arbitrary alphabets and a slight modification of another theorem such that we characterise the circular universality by the universality. On the way, we present deep insights into the behaviour of the remainder of the so called arch factorisation by Hebrard when repetitions of words are considered.
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