平衡单形
Balanced simplices
摘要：
加法元胞自动机是有限循环群$\mathbb{Z}/m\mathbb{Z}$中元素的无限多维数组集合上的线性映射。本文研究了加法元胞自动机生成的算术阵列轨道中的单形。我们证明了它们是平衡单形的一个来源，平衡单形是包含具有相同重数的$\mathbb{Z}/m\mathbb{Z}$的所有元素的单形。对于任何添加剂元胞自动机的维度$1$或更高，存在的无穷多平衡单纯的$\mathbb{Z}/m\mathbb{Z}$出现在这样的轨道，这表明，对于一个无限多的值$m$。本文详细研究了Pascal元胞自动机的一种特殊情况，即生成Pascal单形的元胞自动机，这种单形是将杨辉三角形推广到任意维的.
英文摘要：
An additive cellular automaton is a linear map on the set of infinite multidimensional arrays of elements in a finite cyclic group $\mathbb{Z}/m\mathbb{Z}$. In this paper, we consider simplices appearing in the orbits generated from arithmetic arrays by additive cellular automata. We prove that they are a source of balanced simplices, that are simplices containing all the elements of $\mathbb{Z}/m\mathbb{Z}$ with the same multiplicity. For any additive cellular automaton of dimension $1$ or higher, the existence of infinitely many balanced simplices of $\mathbb{Z}/m\mathbb{Z}$ appearing in such orbits is shown, and this, for an infinite number of values $m$. The special case of the Pascal cellular automata, the cellular automata generating the Pascal simplices, that are a generalization of the Pascal triangle into arbitrary dimension, is studied in detail.
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