论文标题
支持极端双随机阵列
Support of extremal doubly stochastic arrays
论文作者
论文摘要
如果其每行的条目的总和为$ m $，并且每列的条目的总和为$ n $，则带有非负条目的$ n \ times m $数组被称为双随机性。所有$ n \ times m $ doubly随机阵列的集合是一个有限的极端点的凸多角形。本文的主要结果是所有极端$ n \ times m $ doubly随机阵列的支撑可能大小。特别是，我们证明了$ n \ times m $ doubly随机阵列的最小尺寸为$ n + m- \ gcd（n，m）$。此外，对于$ m = kn+1 $，我们还表征了极端阵列支撑的结构。
An $n \times m$ array with nonnegative entries is called doubly stochastic if the sum of its entries at each row is $m$ and at each column is $n$. The set of all $n \times m$ doubly stochastic arrays is a convex polytope with finitely many extremal points. The main result of this paper characterizes the possible sizes of the supports of all extremal $n \times m$ doubly stochastic arrays. In particular we prove that the minimal size of the support of an $n \times m$ doubly stochastic array is $n + m - \gcd(n,m)$. Moreover, for $m=kn+1$ we also characterize the structure of the support of the extremal arrays.
