RUI: Percolation Model

RUI：渗透模型

• 批准号: 0405150
• 负责人: Yu Zhang
• 金额: --
• 依托单位: University of Colorado at Colorado Springs
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405150&HistoricalAwards=false
• 关键词: RUI; Percolation; Model

0405150Zhang Percolative models form a very active subfield of probability theory. They have drawn much of their inspiration and motivation from physics, biology and computer science. They have also been used heavily in physics, biology, industry, and computer science. This project focuses on the following percolation questions: General percolation model: (a) The power laws and scaling relations. (b) The central limit theorems for the incipient infinite cluster. (c) Percolation on arbitrary words. (d) On the infinite differentiability of the right edge in oriented percolation. First passage percolation: (a) Non-differentiability of the time constant. (b) The large deviations. The research makes use of probability theorems (martingale, Markov chains, CLT theorem, correlation inequalities), graph theorems (duality, lattice surfaces, planarity), combinatorics (partition lattice, renormalization) and functional analysis (complex variables). The project has a broad impact on research, education and dissemination to advance scientific and technological understanding in the following ways: First, percolation theory is one of most important subjects in mathematics. In fact, issues and open questions in percolation theory are easy to state, but whose solutions are apparently difficult and require innovative methods. Furthermore, percolation theory is also directly applicable in biology, industry, and computer science. Zhang presents six traditional problems in this field and proposed new methods to solve them. Second, the proposal promotes teaching experiences for Zhang's undergraduate students in areas of engineering, computer science and mathematics. It also offers an opportunity for the students to conduct research activities on percolation problems. Moreover, the proposal also enhances scientific and technological understanding. Since percolation theory involves numerical computations and simulations, technological advances in computing science are required.

张氏渗流模型是概率论中一个非常活跃的分支。他们的灵感和动力主要来自物理学、生物学和计算机科学。它们还被广泛应用于物理、生物、工业和计算机科学。本项目主要研究以下渗流问题：一般渗流模型：(A)幂定律和标度关系。(B)初始无限团簇的中心极限定理。(C)对任意词语的渗漏。(D)关于定向渗流中右边缘的无限可微性。首次通过渗流：(A)时间常数的不可微性。(B)偏差较大。这项研究使用了概率理论(鞅、马尔可夫链、CLT定理、相关不等式)、图论(对偶、格子曲面、平面性)、组合学(分拆格、重整化)和泛函分析(复变量)。该项目对研究、教育和传播产生了广泛的影响，在以下方面促进了对科学技术的理解：第一，渗流理论是数学中最重要的学科之一。事实上，渗流理论中的问题和开放问题很容易陈述，但解决这些问题显然很困难，需要创新的方法。此外，渗流理论也直接适用于生物学、工业和计算机科学。张提出了这一领域的六个传统问题，并提出了解决这些问题的新方法。其次，该提案促进了张的本科生在工程、计算机科学和数学领域的教学经验。它还为学生提供了一个开展渗流问题研究活动的机会。此外，该建议还增进了对科学技术的理解。由于渗流理论涉及数值计算和模拟，因此需要计算科学的技术进步。