Mathematical Sciences: Random Access Approximations in Combinatorics and Number Theory

数学科学：组合数学和数论中的随机存取近似

• 批准号: 9626412
• 负责人: Richard Arratia
• 金额: $ 4万
• 依托单位: University of Southern California
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-07-01 至 1998-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9626412&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Random; Access; Approximations

9626412 Arratia/Tavare ABSTRACT The investigators study the probabilistic properties of some of the most basic structures in combinatorics and number theory, such as partitions of sets and integers, and the factorization of uniformly chosen integers into primes. They apply techniques learned from the study of genetics and random permutations, including couplings, the total variation distance and Wasserstein metric for processes observing some or all of the possible component sizes, size-biased permutations of the components, sampling properties for populations, and Stein's method. The interaction between pure and applied method can be fruitful for both. Techniques motivated by the needs and intuitions from each field can illuminate the other. For example, the applied study of genetics naturally involves sampling from populations, which leads to the notion of a size- biased permutation, which simplifies the understanding of a question from pure math, involving the large prime factors of a random integer. In the other direction, results proved about primes, such as bounds on the "total variation distance", anticipated by three decades important work in statistics on the comparison of sampling with and without replacement. The proposed investigation attempts to straddle the boundary between pure and applied math as it relates to genetics, sampling, and prime numbers.

9626412 Aratia/Tavare摘要研究人员研究组合学和数论中一些最基本结构的概率性质，例如集合和整数的划分，以及将均匀选择的整数因式分解为素数。他们应用了从遗传学和随机排列的研究中学到的技术，包括耦合、观察部分或全部可能组件大小的过程的总变异距离和Wasserstein度量、组件的大小偏向排列、总体的抽样特性和Stein方法。纯方法和应用方法之间的互动对双方都是有益的。由每个领域的需求和直觉驱动的技术可以照亮另一个领域。例如，遗传学的应用研究自然涉及从总体中抽样，这导致了大小偏向排列的概念，这简化了对纯数学问题的理解，涉及随机整数的大素数因子。在另一个方向上，关于素数的结果得到了证明，例如“总变异距离”的界，这是统计学上三十年来关于有和没有替换抽样比较的重要工作所预期的。拟议的调查试图跨越纯数学和应用数学之间的界限，因为它与遗传学、抽样和质数有关。