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The logarithmic method and the solution to the TP2-completion problem.

机译：对数方法和TP2补全问题的解决方案。

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摘要

A matrix is called TP2 if all 1-by-1 and 2-by-2 minors are positive. A partial matrix is one with some of its entries specified, while the remaining, unspecified, entries are free to be chosen. A TP2-completion, of a partial matrix T, is a choice of values for the unspecified entries of T so that the resulting matrix is TP2. The TP2-completion problem asks which partial matrices have a TP2-completion. A complete solution is given here. It is shown that the Bruhat partial order on permutations is the inverse of a certain natural partial order induced by TP2 matrices and that a positive matrix is TP2 if and only if it satisfies certain inequalities induced by the Bruhat order. The Bruhat order on permutations is generalized to a partial order, GBr, on nonnegative matrices, and the concept of majorization is generalized to a partial order, DM, on nonnegative matrices. It is shown that these two partial orders are inverses of each other on the set of nonnegative matrices. Using this relationship and the Hadamard exponential transform on nonnegative matrices, explicit conditions for TP2-completability of a given partial matrix are given. It is shown that an m-by- n partial TP2 matrix T is TP2-completable if and Y onl if tijspecified taijij≥1 for every matrix A = (aij) ∈ Mm,n having (1) aij specified aij = 0 if tij is unspecified; (2) each row sum and each column sum of A is zero; and (3) 1≤i≤p1≤j≤ qaij≥0 , for all (p, q) ∈ {1, 2, ... , m} x {1, 2, ... , n}.;However, there may be infinitely many such conditions, and some of them may be obtainable from others. In order to find a set of minimal conditions, the theory of cones and generators, and the logarithmic method are used. It is shown that the set of matrices used in the exponents of the inequalities forms a finitely generated cone with integral generators. This gives finitely many polynomial inequalities on the specified entries of a partial matrix of given pattern as conditions for TP2-completability. A computational scheme for explicitly finding the generators is given and the combinatorial structure of TP2-completable pattern is investigated.
机译：如果所有1-by-1和2-by-2未成年人均为正，则该矩阵称为TP2。部分矩阵是其中一些条目已指定的矩阵，而其余未指定的条目则可以自由选择。部分矩阵T的TP2补全是T的未指定条目的值的选择，因此所得矩阵为TP2。 TP2完成问题询问哪些子矩阵具有TP2完成。这里给出了一个完整的解决方案。结果表明，置换的Bruhat偏序是TP2矩阵引起的某些自然偏序的逆，并且当且仅当正矩阵满足由Bruhat阶引起的某些不等式时，正矩阵才是TP2。关于置换的Bruhat阶在非负矩阵上一般化为部分阶GBr，而在非负矩阵上将泛化的概念一般化为部分阶DM。结果表明，这两个偏阶在非负矩阵集合上彼此相反。利用这种关系和对非负矩阵的Hadamard指数变换，给出了给定部分矩阵TP2可完成性的明确条件。结果表明，对于每个矩阵A =（aij）∈Mm，n，如果tij指定的taijij≥1，则m×n局部TP2矩阵T是TP2完全的，如果tij指定的taijij≥1，则tij指定的aiji = 0未指定； （2）A的每一行和每一列之和为零；和（3）对于所有（p，q）∈{1，2，...，m} x {1,2，...，n}，1≤i≤p1≤j≤qaij≥0; ，可能有无限多个这样的条件，并且其中一些条件可以从其他条件获得。为了找到一组最小条件，使用了锥和生成器的理论以及对数方法。结果表明，在不等式指数中使用的矩阵集与积分生成器一起形成了有限生成的圆锥。这给定模式的部分矩阵的指定条目上的有限多项式不等式，作为TP2可完成性的条件。给出了一种用于显式查找生成器的计算方案，并研究了TP2可替换模式的组合结构。

著录项

  • 作者

    Nasserasr, Shahla.;

  • 作者单位

    The College of William and Mary.;

  • 授予单位 The College of William and Mary.;
  • 学科 Mathematics.;Theoretical Mathematics.
  • 学位 Ph.D.
  • 年度 2010
  • 页码 93 p.
  • 总页数 93
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类
  • 关键词

  • 入库时间 2022-08-17 11:36:56

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