论文标题
双随机矩阵的普遍性
Universality for Doubly Stochastic Matrices
论文作者
论文摘要
我们表明，与随机矩阵或单一矩阵相比，任何有限的随机矩阵产生的一组条目都无处浓密。换句话说，即使是最弱的普遍性概念，也没有有限的偶然随机矩阵集合。 我们的证明是基于具有收敛性质的拓扑半群的定理。如果每种无限产品都收敛，则拓扑半群是收敛的。我们表明，在某些限制下，在紧凑而收敛的半群中，可以通过生成集和无限产品的限制直接生成每个有限生成的子群的关闭。
We show that the set of entries generated by any finite set of doubly stochastic matrices is nowhere dense, in contrast to the cases of stochastic matrices or unitary matrices. In other words, there is no finite universal set of doubly stochastic matrices, even with the weakest notion of universality. Our proof is based on a theorem for topological semigroups with the convergent property. A topological semigroup is convergent if every infinite product converges. We show that in a compact and convergent semigroup, under some restrictions, the closure of every finitely generated subsemigroup can be instead generated directly by the generating set and the limits of infinite products.
