Mathematical Sciences: Topics in Applied Asymptotic Analysis

9625341 WIMP这个拟议的程序由应用渐近分析的几个主题组成。这些主题大体上没有在涉及渐近性的一般书籍中系统地涵盖，但在应用科学和工程中具有重要意义。例如，中心极限定理是概率论的基础之一，但还没有现成的关于渐近性的资料讨论它的应用和分支。组合学中的某些问题，特别是随机游动理论，可以用渐近方法来阐明；但这些问题并没有得到统一的处理。大多数书籍都对微分方程式有广泛的论述，但同样重要的差分方程式需要自己的攻击方法。更新方程是在计算机科学和工程研究中出现的，偶尔会被处理，但还没有得到应有的重视。要研究的问题是由这些缺陷引起的：醉汉行走和卷积序列；非线性差分方程解的稳定性和收敛；概率方法在应用渐近分析问题中的应用；矩阵数值求逆中的误差分布；以及在使用多址信道的通信系统中恢复信息时产生的更新方程。科学家通常对“大范围”发生的事情感兴趣。这样的例子在日常生活中比比皆是。例如，如果互联网的用户数量继续以某种明确的方式增长，那么登录系统所需的时间到底会发生什么？一种被吸入人体系统的药物在较长一段时间后达到目标的可能性有多大？在很长一段时间内，一种特定农业害虫的种群与其赖以生存的植物之间是如何相关的？在一个严重依赖于当前轻微刑事犯罪率的城市，未来重大刑事犯罪的增长是否会出现？在粮食供应充足的情况下，一个国家的人口是否会继续以灾难性的速度增长，或者人口增长是否存在自我限制的特征，以防止这种情况发生？经常用来模拟这种情况的科学数学语言被称为差分方程组理论。在这个研究项目中，我们打算开发新的方法来攻击这些方程，从而帮助解决这些方程所描述的一些实际问题。***

9625341 Wimp This proposed program consists of several topics in applied asymptotic analysis. These topics by and large have not been systematically covered in the general books dealing with asymptotics, yet are of great significance in applied science and engineering. For instance, the central limit theorem is one of the keystones of probability theory, yet no readily available source on asymptotics has discussed its applications and ramifications. Certain problems in combinatorics, in particular, the theory of random walks, can be elucidated by asymptotic methods; yet these have received no unified treatment. Most books have an extensive treatment of differential equations, but difference equations, which are just as important, require their own methods of attack. Renewal equations, which arise in investigations in computer science and engineering, have been occasionally treated but have not yet received the attention they deserve. The topics to be investigated are motivated by these deficiencies: the drunkard's walk and convolution sequences; the stability and convergence of the solutions of non-linear difference equations; the application of probabilistic methods to problems in applied asymptotic analysis; error distribution in the numerical inversion of matrices; and renewal equations arising in the recovery of information in communications systems employing multiaccess channels. %%% Scientists are often interested in what happens "in the large." Examples of this abound in everyday life. For instance, if the number of users of the internet continues to grow in some well-defined fashion, what precisely will happen to the time required to log onto the system? What is the likelihood that a medication ingested into the human system will reach its target after an extended period of time? How are the populations of a given agricultural pest and the plants on which it feeds related over an extended period of time? Is the future growth in major criminal offenses in a city strongly dependent on the current rate of minor criminal offenses? Will the population of a country, given sufficient food supply, continue to grow at a disastrous rate, or are there self-limiting features of population growth that will prevent this from happening? The scientific mathematical language often used to model such situations is called the theory of difference equations. In this research program, we intend to develop new methods for attacking such equations, and hence to help resolve some practical issues which these equations describe. ***