14,502 research outputs found

    Partitions of the polytope of Doubly Substochastic Matrices

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    In this paper, we provide three different ways to partition the polytope of doubly substochastic matrices into subpolytopes via the prescribed row and column sums, the sum of all elements and the sub-defect respectively. Then we characterize the extreme points of each type of convex subpolytopes. The relations of the extreme points of the subpolytopes in the three partitions are also given

    On Exponential Sums, Nowton identities and Dickson Polynomials over Finite Fields

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    Let Fq\mathbb{F}_{q} be a finite field, Fqs\mathbb{F}_{q^s} be an extension of Fq\mathbb{F}_q, let f(x)∈Fq[x]f(x)\in \mathbb{F}_q[x] be a polynomial of degree nn with gcd⁑(n,q)=1\gcd(n,q)=1. We present a recursive formula for evaluating the exponential sum βˆ‘c∈FqsΟ‡(s)(f(x))\sum_{c\in \mathbb{F}_{q^s}}\chi^{(s)}(f(x)). Let aa and bb be two elements in Fq\mathbb{F}_q with aβ‰ 0a\neq 0, uu be a positive integer. We obtain an estimate for the exponential sum βˆ‘c∈Fqsβˆ—Ο‡(s)(acu+bcβˆ’1)\sum_{c\in \mathbb{F}^*_{q^s}}\chi^{(s)}(ac^u+bc^{-1}), where Ο‡(s)\chi^{(s)} is the lifting of an additive character Ο‡\chi of Fq\mathbb{F}_q. Some properties of the sequences constructed from these exponential sums are provided also.Comment: 18 page

    On the maximum of the permanent of (I-A)

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    Let A be an n by n doubly substochastic matrix and denote {\sigma}(A) the sum of all elements of A. In this paper we give the upper bound of the permanent of (I-A) with respect to n and {\sigma}(A)

    Symmetric, Hankel-symmetric, and Centrosymmetric Doubly Stochastic Matrices

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    We investigate convex polytopes of doubly stochastic matrices having special structures: symmetric, Hankel symmetric, centrosymmetric, and both symmetric and Hankel symmetric. We determine dimensions of these polytopes and classify their extreme points. We also determine a basis of the real vector spaces generated by permutation matrices with these special structures
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