均匀度解释混沌及生态现象的数学依据

定义了任意分布函数的均匀度,在此基础上提出了“均匀度的独立性”公理和“均匀分布最均匀”的命题,阐明了均匀度与熵的相似性,从而揭示了均匀度解释混沌的内在机理.根据独立性公理,k步期望均匀度是存在的,可以用k步平均均匀度估计.由于任意分布函数F可以看作是F^-1对均匀分布的变换,数值计算表明,分布函数的非线性程度是导致k步平均均匀度降低的原因.熵是描述分布的不确定性程度的量,也描述样本的分散程度,样本分散程度意味着样本均匀程度,反之亦然,分布的确定性程度意味着样本的集中程度,样本的集中程度意味着样本的不均匀程度,反之亦然,这是熵与均匀度的共同点.生态学中,大多数格局是集聚格局是由存在于自然界中的非线性变换所致.

Mathematical foundation for interpreting chaos and ecological phenomenon with uniform index

The uniform index of arbitrary distribution function was first defined, then the axioms of ‘uniform index independence’ and proposition of ‘uniform distribution is the most uniform’ were put forward. The similarity between uniform index and entropy was clearly illustrated, and the inner mechanism of interpreting chaos with uniform index was revealed thereby. According to the independence axiom, k step expectation uniform index does exist, and can be estimated by k step average uniform index. As the arbitrary distribution function F can be regarded as a transformation of uniform distribution by F^-1, numerical calculation indicates the the nonlinear degree of distribution function is the direct cause for k step average uniform index reduction. Entropy is a varable to describe the uncertainty of distribution, meanwhile, it also describes the disperse degree of samples, which means the degree of uniformity and vice versa. Certainty of distribution signifies the concentration degree of samples, which also implies the non-uniformity and vice versa. This is the common ground of entropy and uniform index. In ecology, the fact of most structures are clustering is caused by the nonlinear transformation existing in the nature.