标题： 齐普夫定律，1/f噪声和分形层次结构 显示英文标题

标题： Zipf's law, 1/f noise, and fractal hierarchy

摘要： 分形、1/f噪声、齐普夫定律以及大规模灾难性事件的发生是各个学科中典型的普遍经验观察结果，这些现象无法在特定科学领域内发展的参考体系中得到理解。 所有这些观察结果都与标度定律有关，并在科学界引起了广泛的研究兴趣。 然而，这些标度现象之间的内在关系仍然是有待研究的悬而未决的问题。 在本文中，理论推导和数学实验被用来揭示分形模式、1/f噪声和齐普夫分布之间的类比关系。 首先，多分形过程在经验上遵循广义的齐普夫定律。 其次，1/f谱在数学形式上与齐普夫定律相同。 第三，1/f谱和齐普夫定律都可以转化为自相似层次结构。 第四，分形、1/f谱、齐普夫定律以及大规模灾难性事件的发生可以用类似的指数定律和幂律来描述。 自相似层次结构是一种更一般的框架或结构，可用于涵盖或统一物理和社会系统中的不同标度现象和规则，如城市、河流、地震、分形、1/f噪声和等级规模分布。 层次结构上的数学定律可以为我们提供一个整体的观点来看待复杂性，如自组织临界性（SOC）。 显示英文摘要

摘要： Fractals, 1/f noise, Zipf's law, and the occurrence of large catastrophic events are typical ubiquitous general empirical observations across the individual sciences which cannot be understood within the set of references developed within the specific scientific domains. All these observations are associated with scaling laws and have caused a broad research interest in the scientific circle. However, the inherent relationships between these scaling phenomena are still pending questions remaining to be researched. In this paper, theoretical derivation and mathematical experiments are employed to reveal the analogy between fractal patterns, 1/f noise, and the Zipf distribution. First, the multifractal process follows the generalized Zipf's law empirically. Second, a 1/f spectrum is identical in mathematical form to Zipf's law. Third, both 1/f spectra and Zipf's law can be converted into a self-similar hierarchy. Fourth, fractals, 1/f spectra, Zipf's law, and the occurrence of large catastrophic events can be described with similar exponential laws and power laws. The self-similar hierarchy is a more general framework or structure which can be used to encompass or unify different scaling phenomena and rules in both physical and social systems such as cities, rivers, earthquakes, fractals, 1/f noise, and rank-size distributions. The mathematical laws on the hierarchical structure can provide us with a holistic perspective of looking at complexity such as self-organized criticality (SOC).