Mathematical Sciences: Universally Good Averaging Sequences in Ergodic Theory

数学科学：遍历理论中普遍良好的平均序列

• 批准号: 9500577
• 负责人: Mate Wierdl
• 金额: $ 6.05万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-06-01 至 1998-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9500577&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Universally; Good; Averaging

9500577 Wierdl The investigator will work --- possibly in collaboration with several other researchers --- on questions in almost everywhere convergence that come out of some recent results in this area. Some of these questions concern the almost sure convergence of averages of measurements made on a stochastic process at a random sequence of times that is chosen in advance. Other questions concern subsequence ergodic theorems for subsequences coming from members of Hardy fields, along bases for the integers, and along sequences with big gaps. Ergodic theory grew out of statistical mechanics, the statistical description of matter. This latter means, for example, that instead of describing the behavior of each individual water-molecule in a cup of water, one is satisfied with finding the AVERAGE speed, energy etc. of the molecules. But then the fundamental question arises: HOW can we measure the average speed or energy. It is clearly impossible to measure the speed of each individual molecule and then take the mean of the data. The ergodic theorem says that it is enough to select a SINGLE molecule, measure its speed in each second, and if we make enough measurements and take the average of the data the number will be basically the average speed of ALL the molecules in the cup of water. This amazing theorem has one drawback: it requires that the measurements are taken EXACTLY at every second. But in practice, the measurements might be made at,say, 1.1, 2.4, 2.9 .... seconds instead of at 1, 2, 3 ... seconds. Obviously, we would like to know whether we still can compute accurately the average speed from the measured data. The proposed work addresses this VERY difficult question from various point of view (depending on WHAT causes the imperfect timing of measurements). ***

小行星9500577 调查员将与几个人合作， 其他研究者-关于几乎处处收敛的问题 这是最近在这个领域的一些研究成果。其中一些 问题涉及的平均值几乎肯定收敛 在随机序列上对随机过程进行的测量， 时间是提前选择的。其他问题涉及子序列遍历定理的连续性来自成员的哈代领域，沿着 整数的基，和具有大间隔的沿着序列。 遍历理论起源于统计力学， 物质的描述。例如，后者意味着， 描述杯子里每个水分子的行为 对于水，人们满意的是找到平均速度，能量等。 的分子。但随之而来的根本问题是：我们如何 测量平均速度或能量。 很明显，测量每个个体的速度是不可能的 分子，然后取数据的平均值。的 遍历定理说，它足以选择 单个分子，测量其每秒的速度，如果我们进行足够的测量并取数据的平均值， 这个数基本上是所有分子的平均速度 一杯水。这个惊人的定理有一个缺点：它要求， 每秒钟都进行精确的测量。但实际上， 可以在例如1.1、2.4、2.9....处进行测量。秒 而不是1 2 3秒很明显，我们想知道 我们是否还能准确地计算出 测量数据。拟议的工作解决了这一非常困难的问题 从不同的角度来看问题（取决于什么原因， 不完美的测量时间）。 ***