THEORY OF IDEAL BOUNDARIES OF AN INFINITE NETWORK AND ITS APPLICATION

无限网络理想边界理论及其应用

• 批准号: 09640188
• 负责人: YAMASAKI M.
• 金额: $ 1.92万
• 依托单位: SHIMANE UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09640188/
• 关键词: Infinite networks; discrete Kuramochi function; discrete Kuramochi boundary; discrete Laplacian; Hardy's inequality; 固有値問題; Hardyの不等式; ヒルベルトネットワーク; 離散ラプラシアン; 倉持境界; 離散ポテンシャル

(1) Solutions of partial difference equations on an infinite network show some interesting behavior at the point at infinity of the network. The aim of this project is to characterize the point at infinity with the aid of potential theoretic tools and to study its effects for the study of the partial difference equations. We succeeded to construct a theory of discrete Kuramochi boundary comparable to the theory of discrete Martin boundary for random walks. Our theory is a discrete analogue to the theory of Kuramoch compactification of Riemann surfaces due to Kuramochi and Ohtsuka. After introducing a discrete Kuramochi function of a network, we define the Kuramochi compactification as the space on which the Kuramochi function can be extended continuously. We obtain results concerning a classification of Kuramochi boundary points and the behavior of Kuramochi potentials and SHS functions(2) As an application of our study, we give a kind of Hardy's inequality on finite networks. With some numerical experiments, we show that our estimation of the wighted minimum eigenvalue problem for the discrete Laplacian is more effective than the usual theory when the size of the net-work becomes large.

(1) 无限网络上的偏差分方程的解在网络的无穷远点处表现出一些有趣的行为。该项目的目的是借助势理论工具来表征无穷远点，并研究其对偏差分方程研究的影响。我们成功地构建了与随机游走的离散 Martin 边界理论相当的离散 Kuramochi 边界理论。我们的理论与 Kuramochi 和 Ohtsuka 提出的黎曼曲面的 Kuramoch 紧致化理论是离散的类比。在引入网络的离散Kuramochi函数之后，我们将Kuramochi紧化定义为Kuramochi函数可以连续扩展的空间。我们获得了有关Kuramochi边界点的分类以及Kuramochi势和SHS函数的行为的结果(2) 作为我们研究的应用，我们给出了有限网络上的一种Hardy不等式。通过一些数值实验，我们表明，当网络规模变大时，我们对离散拉普拉斯算子的加权最小特征值问题的估计比通常的理论更有效。

项目成果

村上温, 山崎稀嗣: "An introduction of Kuramochi boundary of an infinite netwok" Mem.Fac.Sci.Eng.Shimane Univ. Series B : Mathematical Science. 30. 57-89 (1997)

Atsushi Murakami、Maretsugu Yamazaki：“无限网络的仓持边界简介”Mem.Fac.Sci.Eng.Shimane Univ. Series B：数学科学。

相川弘明: "Capacity and Hausdorff content of certain enlarged sets" Mem.Fac.Sci.Eng.Shimane Univ. Series B : Mathematical Science. 30. 1-21 (1997)

Hiroaki Aikawa：“某些放大集的容量和 Hausdorff 内容”Mem.Fac.Sci.Eng.Shimane Univ. Series B：数学科学。

M.Yamasaki: "Extremum problems on a Hilbert network" Mem.Fac.Sci.Simane Univ.Ser.B : Mathematical Science. 31. 57-71 (1998)

M.Yamasaki：“希尔伯特网络上的极值问题”Mem.Fac.Sci.Simane Univ.Ser.B：数学科学。

村上温, 山崎稀嗣: "Discrete Kuramochi function" Proc.3rd.Intern.Conference on Difference Equation and Applications. (to Appear). (1999)

Atsushi Murakami、Maretsugu Yamazaki：“离散 Kuramochi 函数”Proc.3rd.Intern.Conference on Difference Equation and Applications（即将出现）。

Y.Shogenji and M.Yamasaki: "Hardy's inequality on finite networks" Mem.Fac.Sci.Shimane Univ.Ser.B : Mathematical Science. 32 (to appear). (1999)

Y.Shogenji 和 M.Yamasaki：“有限网络上的哈代不等式”Mem.Fac.Sci.Shimane Univ.Ser.B：数学科学。