Large Permutations

大排列

• 批准号: 1600116
• 负责人: Peter Winkler
• 金额: $ 27万
• 依托单位: Dartmouth College
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-06-01 至 2020-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600116&HistoricalAwards=false
• 关键词: Large; Permutations

A permutation is an arrangement of objects in some order: cards in a deck, atoms in a linear polymer, physical or historical events in a chronology. This award supports research into the understanding of permutations from a mathematical perspective, in particular permutations involving large numbers of objects. Powerful new techniques at the interface of combinatorics and probability allow the large-scale study of the behavior of ensembles of permutations. Experience with other large combinatorial objects has led mathematicians to expect them to behave in certain predictable ways. As a result, when such permutations arise in nature and do not have the predicted properties, scientists know to seek a hidden explanation. For example, this involves the study of the significance of the order of genes on a chromosome, and the knowledge of the behavior of ensembles of large random permutations may tell geneticists when to look for a pattern.New discoveries have made it possible to learn much more about the nature of large permutations. In particular, the space of permutations of all sizes can be completed by limit objects that are probability measures on the unit square with uniform marginals. By maximizing a certain entropy integral, one can, in principle, find the limit object that describes nearly all permutations with certain given properties. The proposer and his collaborators will be using both analytic and experimental techniques to exploit this variational principle. One important objective, barely begun, is to plot the landscape of permutations with two fixed pattern densities and determine, for each feasible point, the number and character of the corresponding permutations.

排列是物体按某种顺序的排列：一副牌中的卡片，线性聚合物中的原子，年表中的物理或历史事件。该奖项支持从数学角度理解排列的研究，特别是涉及大量对象的排列。在组合学和概率学的界面上，强大的新技术允许对排列集合的行为进行大规模的研究。其他大型组合对象的经验使数学家们期望它们以某些可预测的方式表现。因此，当这种排列出现在自然界中并且不具有预测的属性时，科学家们知道寻找隐藏的解释。例如，这涉及到研究染色体上基因顺序的重要性，而大随机排列的集合行为的知识可以告诉遗传学家何时寻找一种模式，新的发现使人们有可能更多地了解大排列的本质。特别是，所有大小的置换空间都可以由极限对象来完成，这些极限对象是具有均匀边缘的单位正方形上的概率测度。通过最大化某个熵积分，原则上可以找到描述几乎所有具有某些给定性质的排列的极限对象。提议者和他的合作者将使用分析和实验技术来利用这个变分原理。一个重要的目标，刚刚开始，是绘制景观的排列有两个固定的模式密度和确定，为每个可行的点，相应的排列的数量和字符。