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首页> 外文期刊>Information Theory, IEEE Transactions on >Book Inequalities
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Book Inequalities

机译：图书不平等

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摘要

Information theoretical inequalities have strong ties with polymatroids and their representability. A polymatroid is entropic if its rank function is given by the Shannon entropy of the subsets of some discrete random variables. The book is a special iterated adhesive extension of a polymatroid with the property that entropic polymatroids have -page book extensions over an arbitrary spine. We prove that every polymatroid has an -page book extension over a single element and over an all-but-one-element spine. Consequently, for polymatroids on four elements, only book extensions over a two-element spine should be considered. Matúš proved that the Zhang–Yeung inequalities characterize polymatroids on four elements which have such a two-page book extension. The -page book inequalities, defined in this paper, are conjectured to characterize polymatroids on four elements which have -page book extensions over a two-element spine. We prove that the condition is necessary; consequently, every book inequality is an information inequality on four random variables. Using computer-aided multiobjective optimization, the sufficiency of the condition is verified up to nine-page book extensions.
机译：信息理论上的不平等与多类拟物及其可表示性密切相关。如果多类拟阵的秩函数由某些离散随机变量的子集的Shannon熵给出，则它是熵的。这本书是一种特殊的迭代方法，适用于多类动物，它具有熵多类动物在任意脊柱上都具有页扩展名的特性。我们证明，每个多类拟态机器人在单个元素和仅一个元素的书脊上都有一页书的扩展名。因此，对于四个元素上的多类拟似体，仅应考虑两元素脊柱上的书本扩展。 Matúš证明张-杨不等式在四个元素上具有多两面体的特征，该元素具有两页的扩展名。推测本文定义的页书不等式是为了刻画在两个元素的书脊上具有页书扩展名的四个元素上的拟拟阵。我们证明条件是必要的；因此，每本书的不平等都是关于四个随机变量的信息不平等。使用计算机辅助的多目标优化，可以验证条件的充分性，最多可扩展至9页。

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