论文标题
随机映射的组件和周期
Components and Cycles of Random Mappings
论文作者
论文摘要
映射$ \ {1,2，...，n \} \ rightarrow \ {1,2，...，n \} $的每个连接组件包含一个唯一的周期。可以通过延迟微分方程或逆laplace变换来概率研究最大的此类组件。最长的这样的循环同样接受了两种方法：我们为其长度找到了一个（显然是新的）密度公式。还检查了约束的含义 - 完全存在一个组件。例如，最长周期的平均长度是$（0.7824 ...）\ sqrt n $，但对于特殊情况，它是$（0.7978 ...）\ sqrt n $，差异小于$ 2 \％$。
Each connected component of a mapping $\{1,2,...,n\}\rightarrow\{1,2,...,n\}$ contains a unique cycle. The largest such component can be studied probabilistically via either a delay differential equation or an inverse Laplace transform. The longest such cycle likewise admits two approaches: we find an (apparently new) density formula for its length. Implications of a constraint -- that exactly one component exists -- are also examined. For instance, the mean length of the longest cycle is $(0.7824...)\sqrt n$ in general, but for the special case, it is $(0.7978...)\sqrt n$, a difference of less than $2\%$.
