Theory and Applications of Random Cellular Automata and Related Models

随机元胞自动机及相关模型的理论与应用

• 批准号: 9971543
• 负责人: David Griffeath
• 金额: $ 8.4万
• 依托单位: University of Wisconsin-Madison
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971543&HistoricalAwards=false
• 关键词: Theory; Applications; Random; Cellular; Automata

9971543Griffeath On the theoretical side, the present proposal will further develop the investigator's rigorous theory of threshold growth, and related dynamics, for deterministic and probabilistic cellular automata (CA). The theory of growth models has played a key role in the analysis of spatial complex systems, dating back to the middle of the century. During the present funding period, Griffeath will seek to further that theory by focusing on five main research projects: scaling laws for critical monotone solidification, shape theory for random perturbations of threshold growth, a prototype of spinodal clustering, quantitative analysis of 'small-world networks', and asymptotics for von Koch - type CA solidification. These studies are primarily theoretical, their goal being rigorous mathematical results. On the experimental side, Griffeath has begun to examine more complex interactions which exhibit a wide variety of intriguing self-organizational effects not present in the simplest "linear" spatial systems. Preliminary empirical modeling projects at present include digital biofilms, nonlinear spatial modeling of social interaction, CA pattern dynamics, and a simple CA paradigm for traffic jams. In broad strokes, the ongoing research of David Griffeath combines mathematical analysis and computer visualization in the study of complex spatial systems. The investigator exploits this interplay for the theoretical and empirical study of deterministic and random cellular automata, local lattice dynamics which serve as prototypes for spiral formation in excitable media, the morphology of crystals, and various other nonlinear processes such as nucleation, flocking, host-parasite interactions, dendritic growth, and phase separation by surface tension. The analysis of aggregation, shape, interfaces, and droplet interaction for such basic growth models sheds new light on many areas of applied science. The investigator's research contributes to the mathematical foundations of such processes.

9971543Griffeath在理论方面，本提案将进一步发展研究者的严格的阈值增长理论，以及相关的动力学，用于确定性和概率性的元胞自动机（CA）。 增长模型理论自世纪中期以来在空间复杂系统的分析中发挥了重要作用。 在目前的资助期间，Griffeath将寻求进一步的理论，重点是五个主要的研究项目：临界单调凝固的标度律，阈值增长的随机扰动的形状理论，一个原型的spinodal聚类，定量分析的“小世界网络”，和冯科赫型CA凝固的渐近性。 这些研究主要是理论性的，他们的目标是严格的数学结果。 在实验方面，Griffeath已经开始研究更复杂的相互作用，这些相互作用表现出各种各样的有趣的自组织效应，这些效应在最简单的“线性”空间系统中不存在。 目前，初步的实证建模项目包括数字生物膜，社会互动的非线性空间建模，CA模式动力学，以及一个简单的CA交通拥堵范例。从广义上讲，大卫·格里菲思正在进行的研究将数学分析和计算机可视化结合在复杂空间系统的研究中。 研究人员利用这种相互作用的理论和实证研究的确定性和随机元胞自动机，局部晶格动力学作为原型的螺旋形成在可激发的介质中，晶体的形态，以及各种其他非线性过程，如成核，植绒，主机寄生虫相互作用，树枝状生长，和表面张力相分离。 这些基本生长模型的聚集、形状、界面和液滴相互作用的分析为应用科学的许多领域提供了新的视角。 研究人员的研究有助于这些过程的数学基础。