Mathematical aesthetic principles and nonintegrable systems

摘要

This article is an outline of the talks given by Muraskin and is a summary of his book "Mathematical Aesthetic Principles/Nonintegrable Systems" published by World Scientific Press in 1995, as well as many articles published by him, and also includes some additional observations. The discussion presents a study of a set of mathematical principles that can be classified as "aesthetic" and shows that these principles can be cast into a set of nonlinear equations. The system of equations is nonintegrable in general. New techniques to handle the nonintegrability feature are discussed. We then illustrate how this system of equations leads to sinusoidal solutions, sine within sine solutions, the phenomenon known as beats, random type oscillations, two and three-dimensional lattices, as well as multiwave packet systems. The sinusoidal solutions occur when the arbitrary data associated with the equations cause the equations to be linearized. The sinusoidal behavior totally disappears once the integrability equations are satisfied, illustrating how important the nonintegrability concept is to the development.