Mathematical Sciences: Number Theory and Probability

REU网站奖将支持六名本科生研究数论和概率论领域的问题。学生将沉浸在研究生院式的环境中，包括研讨会、个人研究和成果展示。研究课题包括：切眼功能场；整数环与质因数分解；唯一分解域；斐波那契序列;连分数;概率界与线性规划。在代数函数领域，学生的专题将处理类似于代数数领域的经典结果。特别令人感兴趣的将是发现关于二次互易的基本原理的类似物的努力。斐波那契数列模m的性质已成为最近几项研究的目标，并将继续研究已为某些特殊模类建立的周期关系的推广。概率领域的工作将侧重于在单个事件只有部分信息可用时获取联合概率的信息。精确解被概率界所取代。这些问题自然导致了多参数线性规划问题中有趣的问题，并给出了如何以有效的方式获得上述边界的答案。该项目将使用利哈伊谷独立学院协会的资源，包括两名教师顾问。预期在这项奖励下完成的工作将在1990年举行的摩拉维亚数学学生年会第四次会议上提出。

This REU site award will support work of six undergraduate students investigating problems in the areas of number theory and probability. Students will be immersed in a graduate school type setting involving seminars, individual research and presentation of results. Research topics include cyclotomic function fields; rings of integers and prime factorization; unique factorization domains; Fibonacci sequences; continued fraction; probability bounds and linear programming. In algebraic function fields, the students' projects will deal with analogues to classical results from algebraic number fields. Of particular interest will be efforts to discover analogues of fundamental principles regarding quadratic reciprocity. The properties of the Fibonacci sequence modulo m has been the target of several recent investigations and work will continue on extensions of periodicity relations which have been established for certain special classes of moduli. Work in the area of probability will focus on obtaining information on joint probabilities when only partial information is available on the individual events. Exact solutions are replaced by bounds on the probability. These questions lead naturally into interesting questions of multi-parametric linear programming problems which yield answers to questions of how one can obtain the above bounds in an efficient manner. The project will use resources from the Lehigh Valley Association of Independent Colleges including two faculty advisors. It is expected that work completed under this award will be presented at the fourth meeting of the Annual Moravian Mathematical Student Conference in 1990.