To represent real $m$-dimensional vectors, a positional vector system given by a non-singular matrix $M \in \mathbb{Z}^{m \times m}$ and a digit set $\mathcal{D} \subset \mathbb{Z}^m$ is used. If $m = 1$, the system coincides with the well known numeration system used to represent real numbers. We study some properties of the vector systems which are transformable from the case $m = 1$ to higher dimensions. We focus on algorithm for parallel addition and on systems allowing an eventually periodic representation of vectors with rational coordinates.
翻译:为了表示实际的$m$维向量,使用一个由非奇异矩阵$M \in \mathbb{Z}^{m \times m}$和数字集$\mathcal{D} \subset \mathbb{Z}^m$组成的位置向量系统。如果$m = 1$,则该系统与用于表示实数的众所周知数字系统相同。我们研究了从$m=1$到更高维度可以转换的向量系统的一些属性。我们专注于并行加法算法和允许有理坐标向量具有周期性表示的系统。