Mathematical Sciences: Topics in Applied Asymptotic Analysis

该方案由应用渐近分析中的几个主题组成。这些主题在一般的关于渐近的书籍中还没有被系统地涵盖，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石之一，但没有现成的关于渐近的资料讨论过它的应用和分支。组合学中的某些问题，特别是随机游走理论，可以用渐近方法来解释；然而，这些都没有得到统一的治疗。大多数书都对微分方程进行了广泛的处理，但同样重要的差分方程需要自己的研究方法。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，但尚未得到应有的重视。要研究的主题是由这些缺陷驱动的：醉汉的行走和卷积序列；非线性差分方程解的稳定性与收敛性概率方法在应用渐近分析问题中的应用矩阵数值反演中的误差分布以及采用多址信道的通信系统中信息恢复中出现的更新方程。科学家通常对“宏观”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internet的用户数量继续以某种明确定义的方式增长，那么登录系统所需的时间究竟会发生什么变化？摄入人体系统的药物在一段较长时间后达到其目标的可能性有多大？一种特定的农业害虫及其赖以为生的植物的种群在一段较长的时间内是如何相互关联的？一个城市未来重大刑事犯罪的增长是否强烈依赖于当前轻微刑事犯罪的增长率？一个国家的人口在有足够的粮食供应的情况下，会继续以灾难性的速度增长吗？还是存在人口增长的自我限制特征来阻止这种情况的发生？通常用来模拟这种情况的科学数学语言被称为差分方程理论。在这个研究项目中，我们打算开发新的方法来求解这些方程，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. ***