Mixing by cutting-and-shuffling can be understood and predicted using dynamical systems based tools and techniques. In existing studies, mixing is generated by maps that repeat the same cut-and-shuffle process at every iteration, in a "fixed" manner. However, mixing can be greatly improved by varying the cut-and-shuffle parameters at each step, using a "variable" approach. To demonstrate this approach, we show how to optimize mixing by cutting-and-shuffling on the one-dimensional line interval, known as an interval exchange transformation (IET). Mixing can be significantly improved by optimizing variable protocols, especially for initial conditions more complex than just a simple two-color line interval. While we show that optimal variable IETs can be found analytically for arbitrary numbers of iterations, for more complex cutting-and-shuffling systems, computationally expensive numerical optimization methods would be required. Furthermore, the number of control parameters grows linearly with the number of iterations in variable systems. Therefore, optimizing over large numbers of iterations is generally computationally prohibitive. We demonstrate an ad hoc approach to cutting-and-shuffling that is computationally inexpensive and guarantees the mixing metric is within a constant factor of the optimum. This ad hoc approach yields significantly better mixing than fixed IETs which are known to produce weak-mixing, because cut pieces never reconnect. The heuristic principles of this method can be applied to more general cutting-and-shuffling systems.

翻译：使用动态系统工具和技术可以理解和预测使用剪切和打拼混合的方法。 在现有的研究中,混合是由在每一次迭代中以“固定”的方式重复同样的剪切和打拼过程的地图产生的。 但是,通过使用“可变”的方法,每一步的剪切和打拼参数不同,混合可以大大改进。为了展示这一方法,我们展示了如何在单维线间隔上通过剪切和打拼来优化混合,称为间距交换转换(IET)。在现有的研究中,混合可以通过优化可变协议而得到显著改进,特别是对于初始条件比简单的两色线间隔更复杂的情况。虽然我们表明,对任意迭接、更复杂的剪切和打混合系统而言,可以通过分析方式进行最优化的混合。此外,控制参数的数量随着变换系统中的迭接数的增多而直线增长。因此,优化大量交错的混合方法一般是难以计算出来的,特别是对于初始的两色线段线段的初始条件。我们展示的是,这种最优的易变换的 I,因为这个固定的计算方法是最精确的压和最精确的计算方法,因为这个最精确的方法是最精确的压的计算方法是最精确的。