Mathematical Sciences: Random Access Approximations in Combinatorics and Number Theory

摘要研究组合学和数论中一些最基本的结构的概率性质，如集合和整数的划分，一致选择的整数分解为素数。他们应用从遗传学和随机排列研究中学到的技术，包括耦合、总变异距离和Wasserstein度量，用于观察部分或全部可能的成分大小、成分的大小偏倚排列、群体的抽样特性和Stein的方法。纯方法和应用方法之间的相互作用对双方都是有益的。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群体中抽样，这导致了大小偏差排列的概念，这简化了对涉及随机整数的大素数因子的纯数学问题的理解。在另一个方向上，证明了关于素数的结果，例如“总变异距离”的界限，这是三十年来统计学中关于有替换和没有替换的抽样比较的重要工作所预期的。提议的调查试图跨越纯粹数学和应用数学之间的界限，因为它涉及到遗传学，抽样和素数。

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.