双随机单本征值问题的计算方法
The Doubly Stochastic Single Eigenvalue Problem: A Computational Approach
摘要：
确定$DS_n$（作为$n$乘$n$双随机阵的特征值出现的复数）的问题已经成为一段时间的研究目标。完美-米尔斯基区域$PM_n$包含在$DS_n$中，并且已知对于$n \leq 4$来说确切地是$DS_n$，但是对于$n = 5$来说严格地包含在$DS_n$中。在这里，我们提出了一个边界猜想，断言$DS_n$的边界是通过（或单个）置换矩阵的凸组合的特征值来实现的。我们提出了一种方法来有效地计算部分的$DS_n$，并获得计算结果，支持边界猜想。我们也给出了证据，$DS_n$等于$PM_n$某些$n > 5$。
英文摘要：
The problem of determining $DS_n$, the complex numbers that occur as an eigenvalue of an $n$-by-$n$ doubly stochastic matrix, has been a target of study for some time. The Perfect-Mirsky region, $PM_n$, is contained in $DS_n$, and is known to be exactly $DS_n$ for $n \leq 4$, but strictly contained within $DS_n$ for $n = 5$. Here, we present a Boundary Conjecture that asserts that the boundary of $DS_n$ is achieved by eigenvalues of convex combinations of pairs of (or single) permutation matrices. We present a method to efficiently compute a portion of $DS_n$, and obtain computational results that support the Boundary Conjecture. We also give evidence that $DS_n$ is equal to $PM_n$ for certain $n > 5$.
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