The geometry of probability generating functions

概率生成函数的几何

• 批准号: 1209117
• 负责人: Robin Pemantle
• 金额: $ 33万
• 依托单位: University of Pennsylvania
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1209117&HistoricalAwards=false
• 关键词: geometry; probability; generating; functions

The first part of this work concerns the estimation of Taylor coefficients of generating functions with a pole or other singularity along a complex algebraic variety. The relation between the zeros of the polynomial Q and the coefficients of P/Q have been demonstrated in a series of works from 2000 to 2011. Via Cauchy's integral representation, estimating coefficients is reduced to residue integration along the zero set of Q. There are two main open problems addressed in this proposal. In the case where the zero set of Q is smooth, we consider the effective computation of the homology class of the chain of integration with respect to a basis of saddle point contours. In the case where the zero set of Q is singular, only quadratic singularities have as yet been worked out; we consider here two examples of higher degree. The second part of the proposal concerns the relation between the zeros of Q and the coefficients of Q itself. This work is part of the emerging field of stable function theory. The main goal here is to extend the Borcea-Branden-Liggett theory of strong Rayleigh distributions to random variables taking more than two values. A third part of this proposal concerns problems in discrete probability theory, including locating the critical points of a random polynomial or power series, and a problem on totally anisotropic percolation.The broad impact of this research lies in computational infrastructure for computation in combinatorial applications. Generating functions are used in many areas of pure and applied mathematics: random graph theory, theory of algorithms, statistical physics, discrete probability theory, population biology and genetics to name a few. Estimating the coefficients of a generating function is a crucial step in any application of generating function techniques. This work, and particularly the part dealing with automated computation, allows users to compute without having to develop ad hoc methods in each case. Broad impacts of the second and third parts of the proposal include further understanding of negative dependence among random variables. There are also educational outcomes from work on curriculum development and K-12 education enabled by this grant.

本文的第一部分研究了具有极点或其他沿着复代数簇奇点的生成函数的泰勒系数的估计。 多项式Q的零点与P/Q的系数之间的关系已经在2000年至2011年的一系列工作中得到了证明。 通过柯西积分表示，估计系数被简化为沿Q的零集沿着的留数积分。 这项建议主要涉及两个尚未解决的问题。 在零集Q是光滑的情况下，我们考虑了积分链关于鞍点轮廓基的同调类的有效计算。 在零集的Q是奇异的情况下，只有二次奇点尚未制定;我们在这里考虑两个例子，更高的程度。该提案的第二部分涉及Q的零点与Q本身的系数之间的关系。 这项工作是稳定函数理论的新兴领域的一部分。 这里的主要目标是将Borcea-Branden-Liggett强瑞利分布理论扩展到取两个以上值的随机变量。 该建议的第三部分涉及离散概率论中的问题，包括定位随机多项式或幂级数的临界点，以及完全各向异性渗流问题，这项研究的广泛影响在于组合应用中的计算基础设施。 生成函数被用于许多领域的纯数学和应用数学：随机图论，算法理论，统计物理，离散概率论，人口生物学和遗传学仅举几例。 估计母函数的系数是任何母函数技术应用中的关键步骤。这项工作，特别是处理自动计算的部分，允许用户计算，而不必在每种情况下开发特设的方法。 提案第二和第三部分的广泛影响包括进一步理解随机变量之间的负相关性。 这项赠款还使课程开发和K-12教育工作取得了教育成果。