Mathematical Sciences: Structure and Rigidity of Graphs with Applications to Network Models of Materials

数学科学：图的结构和刚性及其在材料网络模型中的应用

• 批准号: 9412017
• 负责人: Deborah Franzblau
• 金额: $ 6.79万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1994
• 资助国家: 美国
• 起止时间: 1994-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9412017&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Structure; Rigidity; Graphs

9412017 Franzblau In this project, the principal investigator aims to obtain the following results: (1) an implementation of a practical combinatorial algorithm to compute bounds on degrees of freedom of a network (in 3 or more dimensions), (2) new combinatorial conditions for rigidity or simple formulas for degrees of freedom in families of graphs, (3) new, easily computed measures of medium-range order in network models of glasses. The methods employed include those of combinatorial optimization, graph theory, and discrete algorithm design. Problems on network models are addressed largely through computer experiments, and make use of algorithms developed and implemented by the investigator. This project has two related aims, and is intended to contribute new results both to mathematics and materials science. The first aim is to address key open problems in the mathematical theory of rigidity, including computing the degrees of freedom of a network (also called a graph). This theory has a long history, which includes the work of Maxwell (1864) on determining whether a "scaffold" made of rigid bars and movable joints is itself rigid. The second aim is to address basic issues on network models of solids; such models (also called ball-and-stick models), in which points represent atoms and connections between points represent chemical bonds, are often studied to better understand the properties of both crystalline solids and glassy materials. One focus is to characterize the relationship between the structure of a network model and its rigidity, and the other is to find useful measures which capture this network structure. The work therefore leads to a better understanding of materials properties.

小行星9412017 在本项目中，主要研究者旨在获得以下结果：（1）计算网络自由度界限的实用组合算法的实现（在3维或更多维中），（2）刚性的新组合条件或图族中自由度的简单公式，（3）新的，在眼镜的网络模型中容易计算的中程有序度。 所采用的方法包括组合优化，图论和离散算法设计。 网络模型的问题主要通过计算机实验来解决，并利用研究人员开发和实施的算法。 该项目有两个相关的目标，旨在为数学和材料科学贡献新成果。 第一个目标是解决刚性数学理论中的关键开放问题，包括计算网络（也称为图）的自由度。 这个理论有很长的历史，其中包括麦克斯韦（1864）的工作，确定是否“脚手架”制成的刚性杆和活动关节本身是刚性的。第二个目标是解决固体网络模型的基本问题;这种模型（也称为球棒模型），其中点代表原子，点之间的连接代表化学键，通常被研究以更好地理解结晶固体和玻璃质材料的性质。 一个重点是描述网络模型的结构与其刚性之间的关系，另一个重点是找到捕获这种网络结构的有用度量。 因此，这项工作可以更好地了解材料的特性。