基于格图像的康托集分维与泛逻辑运算

对康托集的研究一直是分形领域的经典而又热点的课题。本文尝试用一种新的模式——“格图像”来研究康托集的分形特性,给出了康托集的格图像构造与分形维数计算方法,计算表明：康托集的自相似分维是格图像分维的特例,格图像分维是自相似分维的扩展。文章首次在分形领域引入泛逻辑的概念,给出了基于格图像的康托集的泛逻辑“与,或,非”运算模型,它不仅考虑了集合的代数列度大小。而且考虑了在参考格中的几何位置关系,这给分形图像的研究提供了一种新的思路,同时也拓展了泛逻辑学的应用领域。

The Fractal Dimension and Universal Logical Operation of Cantor-Set Based on Grid Image

The study of cantor-set is classical but heated subject in fractal field. This paper tries to study the fractal characteristic of cantor-set by a new model called "grid image", gives the method of construction of grid image and calculation of fractal dimension. The results show that the self-similar dimension is the specialty of grid image dimension, and the grid image dimension is the extended case of self-similar dimension. The paper imports the conception of universal logic firstly, gives the operation models of "NOT", "AND", and "OR" based on grid image. They take into account not only the values of algebraic measure but also the relationship of geometric location. This work provides a new thought-way for studying fractal image, and enlarges the applied fields of universal logic.