【作者】 毛军军；

【导师】 张铃；

【作者基本信息】 安徽大学 ， 计算机应用， 2006， 博士

【摘要】 自从Benoit B.Mandelbrot的《Fractal Geometry of Nature》一书于1982年出版后，分形受到了各行各业人士的关注，在许多科学领域得到了一定的应用。但是严格而且正式地定义分形是一件非常复杂而且困难的事情，要准确地反映千姿百态的分形现象及其丰富多彩的特征更不是一件易事。正是由于这个“瓶颈”，进入21世纪，分形的研究停滞不前。 1990年，张钹、张铃两位教授提出商空间粒度计算理论。商空间理论作为一种问题求解的方法，其坚实的理论基础，多侧面、多角度、多层次的问题求解模式，是描述现象和解决问题的强有力工具。Zhang指出：“人类智能的一个公认特点，就是人们能从极不相同的粒度上观察和分析同一问题。人们不仅能在不同粒度的世界上进行问题求解，而且能够很快地从一个粒度世界跳到另一个粒度的世界，往返自如，毫无困难。”商空间粒度计算理论重点研究不同粒度世界之间的互相转换、互相依存的关系和结构。 本文在深入研究商空间粒度理论的基础上，讨论分形和商空间粒度的关系，结合商空间理论和分形特征，建立商分形模型。然后将其应用于生物信息学中的蛋白质结构研究和经济学中的城市发展和规模分布分析。主要工作包括： 1．建立商分形模型。 本文首先从分形空间的度量距离出发，证明了分形的生成对应着模糊相似关系，而模糊相似关系可以改造为模糊等价关系，模糊等价关系又对应着一簇通常等价关系链，每一簇通常等价关系链对应一个商空间的有序链——分层递阶结构。这样，从理论上保证了新模型的存在性和合理性。具体模型的构建方法是：给定函数迭代系统IFS(W)，通过分形映射W定义分划，形成有序商集链，接着在商集链上引入距离，证明该距离空间是完备紧致距离空间，从而形成有序分形商空间链。这就形成了商分形模型。然后，我们讨论了分形映射、分形、商分形之间的关系，从侧面说明商空间理论中引入结构的重要性。最后，我们论述了新建模型的唯一性。 2．商分形模型的描述。 本文讨论五个参数主要想既反映模型的分形特征(如：维数，测度，密度)，又表达模型的商空间思想(如粒度，细度)。其中：盒维数、覆盖维数描述静态更多还原

【Abstract】 Since the Fractal Geometry of Nature was published by Benoit B. Mandelbrot in 1982, fractals not only have been attracting more attention from every walk of life but also have been applied to kinds of science field. However it is hard to define the concept of fractal exactly and the describing of the rich and colorful fractal phenomena with preciseness is more difficult. As a result of these bottlenecks, the step of the researching on fractal has been in logjam in 21 century.In 1990, Professor Zhang Bo and Zhang Ling proposed the quotient space theory. As a method of problem solving, which based on substantial theory, considering the problem from different aspects and multi-hierarchy in the process of problem solving, it is a kind of powerful tool in that it can decrease the difficulty of the problem and reduce the computational cost. Professor Zhang points out: "A well recognized feature of human intelligence is that human can observe and analyze the same problem at different granularity and moves back and forth among different granularity world with no difficulty." Therefore, the emphases of their research work are the framework and relationship of mutual-conversion and interdependent of different granularity world.In this thesis, the relationship between quotient space theory and fractal geometry is discussed and a new model which combines the character of granularity and fractal is put forward, some application of the model in bioinformatics and economics are given, main works and results include:1.Propose the quotient-fractal model.Beginning with the distance measure of fractal space, it is proved that the formation of fractals corresponds to fuzzy similar relation. A fuzzy similar relation can be rebuilt to fuzzy equivalence relation, a fuzzy equivalence relation come into being a cluster of general equivalence relation, furthermore, a cluster of general equivalence relation form a hierarchical structure. So the existence and rationality of new model are determined. Next step, we establish ideographic model. Given a function system IFS (W), the quotient chain can be formed, and then the distance is induced on the quotient chain, moreover the metric space is proved to be complete and compact. As a result, a hierarchical structure builds. That is to say, quotient-fractal model is conducted. At last the relationship among fractals, fractal mapping and quotient-fractal model are discussed.2. Describe quotient-fractal model.Five parameters are used not only to represent the fractal feature but also to manifest granularity of quotient-fractal model. Dimension is proposed to picture complexity and abnormity of system; Measure is discussed to describe cubage of system; Density is put forward to depict the degree of assembling in system; Granularity of model is studied to explain the capacity of information; fineness is brought to portray relationship among different granularities.3. Apply quotient-fractal model to researching on the structure of protein.更多还原