Department of Mathematics, Ohio State University, Columbus, Ohio 43210, USA.
Chaos. 2009 Sep;19(3):033126. doi: 10.1063/1.3207822.
We propose a new mathematical framework to formulate scale structures of general systems. Stack equations characterize a system in terms of accumulative scales. Their behavior at each scale level is determined independently without referring to other levels. Most standard geometries in mathematics can be reformulated in such stack equations. By involving interaction between scales, we generalize stack equations into scale equations. Scale equations are capable to accommodate various behaviors at different scale levels into one integrated solution. On contrary to standard geometries, such solutions often reveal eccentric scale-dependent figures, providing a clue to understand multiscale nature of the real world. Especially, it is suggested that the Gaussian noise stems from nonlinear scale interactions.
我们提出了一个新的数学框架来构建一般系统的尺度结构。栈方程根据累积尺度来描述系统。它们在每个尺度水平上的行为是独立确定的，不涉及其他水平。数学中的大多数标准几何形状都可以用这样的栈方程重新表述。通过在尺度之间引入相互作用，我们将栈方程推广为尺度方程。尺度方程能够将不同尺度水平的各种行为纳入一个综合的解中。与标准几何形状不同，这样的解通常会揭示出偏心的尺度相关图形，为理解现实世界的多尺度性质提供了线索。特别是，有人提出，高斯噪声源于非线性尺度相互作用。
