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p-adic method in discrete mathematics and its application

离散数学中的p-adic方法及其应用

基本信息

批准号：
10640032
负责人：
KUDO Aichi
金额：
$ 1.54万
依托单位：
NAGASAKI UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1998
资助国家：
日本
起止时间：
1998 至 1999
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640032/
关键词：
Bernoulli number p-adic L-function p-adic zeta function p-adic Euler constant shortest path problems associative path routing problems duality theorem P進デデキント和一般ベルヌイ数結合型最適経路問題双対定理単位半群有向グラフ P進解析学

项目摘要

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.

本研究主要处理一部分p-adic分析离散优化问题。工藤研究了普通伯努利数和广义伯努利数的第一类特征的p进同余性质。 1．对于附加第一类狄利克雷特征的p进L函数，他用特征和改进了广义伯努利数的p进近似公式，并导出了精确的p进同余，将Ferrero-Greenberg公式作为p进极限。 2．对于普通伯努利数和具有Teichmuller特征幂的广义伯努利数，他获得了包含其次数的p进同余。作为应用，给出了p进欧拉常数的p进近似值和p进zeta函数在某些非正整数处的微分系数。 Maruyama研究了关联最优路径问题： 1．利用不变嵌入技术，推导出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，并提出了收敛于该解的序列。 2．对于每一个多目标路由问题，他都将另一个密切相关的问题联系起来，并推导出原问题和对偶问题之间的对偶定理。 3．通过求解两个相互关联的递归方程组，他同时找到了最短和最长的路径长度。他证明了系统解的存在性和唯一性。给出了解决一类最短路径问题的算法。