Formation of fractal structures and the development of its simulation algorithms

分形图案形成的研究因其普遍性和一般性而备受物理学家的关注。为了揭示分形图案形成的物理起源，在计算机模拟等转换方法方面还没有成功。这门学科还有许多未解决的问题。本研究的目的是利用新的数值技术揭示分形图案形成的物理根源。我们的目标是(1)自由处理10^7×10^7矩阵和(2)将我们的数值算法应用于量子和经典分形结构。我们的方法是独特的，不同于其他传统的方法，将特征值问题映射到晶格振动问题。应用这种方法，我们成功地为我们的课题取得了新的成果。

The research on the formation of fractal patterns attracts much attention of physicists according to its universality and generality. To reveal the physical origin of the formation of fractal patterns, it has not been successful in terms of convertional approaches such as computer simulations. There has been numerous unsolved problems for this subject. The goal of the present research is to reveal the the physical origin of fractal pattern formation by means of employing quite new numerical techniques. Our goal is (1) to handle freely 10^7×10^7 matrices and (2) to apply our numerical algorithm to quantum and classical fractal structures. Our method is unique, different from other conventional methods, to map the eigenvalue problem to the problem of lattice vibrations. By applying this method, we have succeeded to obtain new results for our subject.