Research On The Model Of Quotient Fractal Based On The Theory Of Quotient Space And Its Application

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.