The principle of least effort (PLE) is believed to be a universal rule for living systems. Its application to the derivation of the power law probability distributions of living systems has long been challenging. Recently, a measure of efficiency was proposed as a tool of deriving Zipf’s and Pareto’s laws directly from the PLE. This work is a further investigation of this efficiency measure from a mathematical point of view. The aim is to get further insight into its properties and usefulness as a metric of performance. We address some key mathematical properties of this efficiency such as its sign, uniqueness and robustness. We also look at the relationship between this measure and other properties of the system of interest such as inequality and uncertainty, by introducing a new method for calculating nonnegative continuous entropy.

中文翻译：

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研究效率度量作为将最小努力原则应用于推导 ZipF 和帕累托定律的工具

最小努力原则（PLE）被认为是生命系统的普遍规则。长期以来，它在生命系统幂律概率分布的推导中的应用一直具有挑战性。最近，提出了一种效率度量，作为直接从 PLE 推导出 Zipf 和 Pareto 定律的工具。这项工作是从数学角度对这种效率度量的进一步研究。目的是进一步了解其作为性能指标的属性和有用性。我们解决了这种效率的一些关键数学特性，例如它的符号、唯一性和鲁棒性。我们还通过引入一种计算非负连续熵的新方法来研究该度量与相关系统的其他属性（例如不等式和不确定性）之间的关系。