Studies on integrable cellular automata

我研究了(1)有限域上的离散可积方程和(2)初等元胞自动机的分类。(1) (i)用有限域中的值代替参数，可以唯一地确定有限域中和无穷域中因变量的值。由于这一性质，我得到了有限域上（广义）离散KdV方程的时间演化和N个孤子解。（ii）定义了近似良好约简（AGR）的概念，它是通过奇点约束检验的一种算法模拟，并证明了离散painlevel ii方程（dPII）具有AGR。通过AGR，我得到了dPII在有限域上的时间演化和解。我对qPII、qPIII、qPIV和dKdV有类似的结果。(2)提出了用奇异值分解对初等元胞自动机进行分类的方法。虽然第4类与Wolfram分类的第3类相似，但可以通过上述方法区分它们。

I investigated (1) discrete integrable equations over finite fields and (2) a classification of an elementary cellular automaton.(1) (i) Replacing a parameter with a value in finite fields, one can uniquely determine the values of the dependent variables in finite fields and infinity. Thanks to this property, I obtained the time evolution and N soliton solutions of the (generalized) discrete KdV equations over finite fields. (ii) I defined the notion of almost good reduction (AGR), which is an arithmetic analogue of passing the singularity confinement test, and proved that the discrete Painleve II equation (dPII) has AGR. By AGR, I obtained the time evolution and a solution of the dPII over finite fields. I had the similar results for qPII, qPIII, qPIV and dKdV.(2) I suggested a classification of an elementary cellular automaton by singular value decomposition. Though class 4 resembles class 3 of Wolfram's classification, one can distinguish them by using the above method.