Global level set structure of ultradiscrete integrable systems

我们发现并证明了一个新的组合数学公式。这就是周期可积细胞自动机水平集配分函数的费米子公式。在经典可积系统中，我们还成功地构造了一类特殊的离散极限--热带周期户田格中的交换相流（广义时间演化），这是户田格的一种特殊的离散极限.相流是重要的，以确定该动力系统的相空间的结构。

We found and proved a new mathematical formula in the field of combinatorics. That is called a fermionic formula for partition functions associated with level set of a periodic integrable cellular automaton. By inventing a cellular-automaton-like description in terms of two colored strips, we also succeeded to construct a family of commuting phase flows (generalized time evolutions) in tropical periodic Toda lattice, a special discrete limit of well known Toda lattice in the field of classical integrable systems. The phase flows are important to determine the structure of the phase space of this dynamical system.