Mathematical Sciences: Ergodic Phenomena in Differential Dynamics

数学科学：微分动力学中的遍历现象

• 批准号: 9706794
• 负责人: Yakov Sinai
• 金额: $ 25.25万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-05-01 至 2001-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9706794&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Ergodic; Phenomena; Differential

Abstract Sinai 9706794 Sinai will work in three directions: 1. Qualitative behavior of solutions of some quasi-linear systems with or without randomness - in particular, pressure-less equations of gas dynamics, the creation of singularities of types of shocks, and concentration of masses on submanifolds of small dimensions; 2. Twist maps with strong statistical properties - in particular, the construction of twist maps with positive Lyapunov exponents; and 3. Problems related to quantum chaos and ensembles of random matrices - in particular, the universality of distributions found earlier by Wigner, Dyson and others. This grant is to fund the study of the long term behavior of dynamical systems and stochastic processes. Many important physical phenomena, as varied as the behavior of the stock market, lines in the supermarket, and data being received from a distant transmission source, exhibit locally erratic behavior which can be understood better by looking at the long term average behavior instead. The study of such stochastic processes and their connections with one another, as well as their connections in terms of their fine structure with other less random processes, is a central one in modern mathematics. It is only through such studies that we will be able to understand better and thus be able to control or predict the evolution of the physical phenomena which this mathematics can model.

西奈半岛9706794将朝着三个方向工作：1.一些具有或不具有随机性的拟线性系统的解的定性行为--特别是无压气体动力学方程、激波类型的奇异性的产生以及质量在小维子流形上的集中；2.具有强统计性质的扭曲映射-特别是具有正Lyapunov指数的扭曲映射的构造；以及3.与量子混沌和随机矩阵系综有关的问题-特别是Wigner、Dyson和其他人早些时候发现的分布的普适性。这笔资金用于研究动力系统和随机过程的长期行为。许多重要的物理现象，如股票市场的行为、超市里的队伍以及从远程传输源接收的数据，都表现出局部的不稳定行为，而通过查看长期平均行为可以更好地理解这些行为。研究这种随机过程及其相互之间的联系，以及它们与其他较不随机的过程的精细结构之间的联系，是现代数学中的一个中心问题。只有通过这样的研究，我们才能更好地理解，从而能够控制或预测这种数学可以模拟的物理现象的演变。