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和无穷域中的因变量的值。利用这一性质，得到了有限域上（广义）离散KdV方程的时间演化和N孤子解。(ii)我定义了几乎好约化（AGR）的概念，这是通过奇点限制测试的算术模拟，并证明了离散Painleve II方程（dPII）具有AGR。利用AGR方法，得到了有限域上dPII的时间演化和一个解。我对qPII、qPIII、qPIV和dKdV的结果相似。(2)我建议用奇异值分解对初等元胞自动机进行分类。虽然第4类类似于Wolfram分类的第3类，但可以使用上述方法区分它们。

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.