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.

排列是对象的排列，以某种顺序：甲板上的纸牌，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来理解对排列的理解的研究，特别是涉及大量对象的排列。组合剂和概率的界面上的强大新技术可以大规模研究排列的集合的行为。具有其他大型组合物体的经验导致数学家期望他们以某些可预测的方式行事。结果，当这种排列在本质上出现并且没有预测的属性时，科学家就会知道寻求隐藏的解释。例如，这涉及研究基因对染色体的重要性的研究，以及对大型随机排列的合奏行为的了解可能会告诉遗传学家何时寻找模式。新发现使得更多地了解大型置换的本质。特别是，所有尺寸的排列空间都可以通过极限对象来完成，这是具有均匀边缘的单位正方形上的概率度量。通过最大化某个熵积分，可以原则上找到几乎所有具有给定特性的排列的极限对象。提议者及其合作者将同时使用分析和实验技术来利用这一变异原则。一个重要的目标，几乎没有开始，是绘制具有两个固定图案密度的排列的景观，并确定每个可行点的相应排列的数量和特征。