【作者】 王恒亮；

【导师】 李书超；

【作者基本信息】 华中师范大学 ， 运筹学与控制论， 2007， 硕士

【摘要】 本文研究了有关矩阵积和式两方面的问题，一方面我们研究了具有一定约束条件的(0，-1)矩阵，(-1，1)矩阵，(0，-1，1)矩阵的积和式的极值问题；另一方面我们还刻划了非负整数矩阵积和式的上界，改进了前人的结果。具体来讲，本文首先给出了如下几类元素绝对值不超过1的整数矩阵的积和式极值，即(1)S(m×n，l)：具有l个0元的m×n阶(0，-1)矩阵；(2)F(m×n，k)：具有k个-1元的m×n阶(-1，1)矩阵；(3)G(m×n，k，l)：具有l个0元、k个-1元的m×n阶(0，-1，1)矩阵。对S(m×n，l)而言，给出了其相应的积和式的最大值、第二大值、最小值、第二小值以及相应极值条件下矩阵的构造特征；对F(m×n，k)而言，给出了其相应的积和式的最小值以及积和式达到最小值时矩阵的构造特征；对F(n×n，k)而言，给出了其相应的积和式的最大值，第二大值，第二最小值以及积和式达到最大值，第二大值，第二最小值时矩阵的构造特征；对G(m×n，k，l)而言，在满足条件max{k+l，m}≤n时，给出了其相应的积和式的最小值以及此时矩阵的构造特征；在满足条件k≤「(n-l)／2」且矩阵中的0元和-1元不在同一条线时，给出了其相应的积和式的最大值以及此时矩阵的构造特征。其次，本文刻划了非负整数矩阵积和式的上界，即将一个n×n阶(0，1)矩阵的积和式的上界问题推广到m×n阶矩阵，得到了一个关于m×n阶(0，1)矩阵的积和式的新上界。接下来，对具有一定约束条件的非负整数元的完全不可分解矩阵的积和式的上界问题作出更进一步的刻划，改进了积和式的已有上界。更多还原

【Abstract】 In this paper, we discuss the following problems of the permanents:1. Extremes of permanents for (0, -1)-matrix, (-1, 1)-matrix, (0, -1, 1)-matrix which have some restricted conditions;2. The upper bounds of permanents for some (0, 1)-matrices and fully indecomposable matrices with nonnegative integer entries.We firstly investigate the values of permanents of three classes of matrices:(1) S(m×n, 1) is the set of all m×n (0, -1)-matrices with exactly l 0’s;(2) F(m×n, k) is the set of all m×n (-1, 1)-matrices with exactly k -1’s;(3) G(m×n, k, l) is the set of all m×n (0, -1, 1)-matrix with exactly k -1’s and l 0’s.We determine the maximum, the second largest, the minimum, the second smallest values of the permanent function and find the matrices which attain some extremes values for S(m×n, l) and F(n×n, k). For G(m×n, k, l) we determine the minimum values of the permanent function and find the matrices which attain these extremes values under condition max{k + l, m}≤n; we also determine the maximum values of the permanent function and find the matrices which attain these extremes values for G(n×n, k, l) under the conditions: 0 and -1 are not on a same line and . Secondly, we give some new upper bounds of permanents for some (0, 1)-matrices and fully indecomposable matrices with nonnegative integer entries. Concretly, we generalize some upper bounds of the permanents for the n×n (0, 1)-matrices. We also genenralize some results of the permanents for fully indecomposable matrices with nonnegative integer entries to some new bounds. Some further research problems are also put forward in this paper.更多还原