p-adic method in discrete mathematics and its application

在本研究中，主要研究了部分p进分析离散优化问题。Kudo研究了普通伯努利数和广义伯努利数的p进同余性质的第一类特征。对于第一类Dirichlet特征的p进l函数，他用字符和改进了广义伯努利数的p进近似公式，导出了一个p进精确同余，并给出了Ferrero-Greenberg公式作为p进极限。对于普通伯努利数和具有Teichmuller次幂特征的广义伯努利数，他得到了包含其次的p进同余。作为应用，给出了p进欧拉常数的p进逼近值和p进zeta函数在非正整数处的微分系数。Maruyama研究了关联最优路径问题。利用不变嵌入技术，导出了一类关联最短路径问题的参数化递推方程，证明了其解的唯一性，并给出了一个收敛于该解的序列。对于每一个多目标路径问题，他都将另一个密切相关的问题联系起来，并推导出了原始问题与对偶问题之间的对偶定理。通过求解两个相互关联的递归方程，他同时找到了最短和最长的路径长度。他证明了该系统解的存在唯一性。给出了求解一类最短路径问题的一种算法。

In this research, a part of p-adic analysis discrete optimization problems were mainly treated.A. Kudo investigated p-adic congruence properties of ordinary and generalized Bernoulli numbers for the character of the first kind.1. For p-adic L-function attached to Dirichlet character of the first kind, he improved a p-adic approximation formula of generalized Bernoulli number by character sum, and derived an exact p-adic congruence which gives the Ferrero-Greenberg formula as p-adic limit.2. For ordinary Bernoulli number and generalized Bernoulli number with a power of Teichmuller character, he obtained a p-adic congruence containing its degree. As application, p-adic approximative values of p-adic Euler constant and defferential coefficients of p-adic zeta function at some non-positive integers are given.Y. Maruyama investigated associative optimal path problems.1. Using an invariant imbedding technique, he derived a parameterized recursive equation for the class of associative shortest path problems, proved the uniqueness of the solution of it and proposed a sequence which converges to the solution.2. For every multiobjective routing problem, he associated another closely related problem and derived a duality theorem between the primal problem and the dual one.3. Solving a system of two interrelated recursive equations, he found both the shortest and the longest path lengths simultaneously. He proved the existence and uniqueness of the solution of the system. An algorithm which solves the class of shortest path problems was given.