Integrated research of the Theory of infinite networks and their applications

无限网络理论及其应用综合研究

• 批准号: 13640214
• 负责人: YAMASAKI Maretsugu
• 金额: $ 0.7万
• 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: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640214/
• 关键词: infinite network; Hardy's inequality; eigen-value problem; discrete Laplacian; Martin boundary; Sierpinski Gadket; integro-difference equations of Voltera's type; complementarity problem; 離散ラプラシアン; 固有値問題; ソボレフ・ポアンカレ不等式; 変分法; 差分方程式

1. Inequalities on networks have played important roles in the theory of networks. We study several famous inequalities on networks such as Wirtinger's inequality, Hardy's inequality, Poincare-Sobolev's inequality and the strong isoperimetric inequality, etc. These inequalities are closely related to the smallest eigen value of weighted, discrete Laplacian. We discuss some relations between these inequalities and the potential-theoretic magnitude of the ideal boundary of an infinite network.2. Martin boundary plays important roles both in the theory of harmonic functions and the study of geometric figure of domains. We determine Martin boundary of a uniform John domain. Our results are applied to the study of Sierpinski Gasket.3. We study some properties of the solutions of integro-difference equations of Voltera's type. In particular, we focus the boundedness property and the periodicity of solutions.4. We attempt to obtain more useful information from the output of the verification method proposed by Alefeld, Chen and Potra. We use the Faekas lemma to check the nonexistence of solutions of inear complementarity problems.5. As an integrated research of the theory of infinite networks and their applications, we submit a lecture note related this field.

1。网络上的不平等现象在网络理论中起着重要作用。我们研究了几种著名的不平等现象，例如Wirtinger的不平等，Hardy的不平等，Poincare-Sobolev的不平等和强烈的等距不平等等。这些不平等与加权，离散的Laplacian的最小特征值密切相关。我们讨论了这些不平等现象与无限网络理想边界的潜在理论幅度之间的一些关系。2。马丁边界在谐波函数理论和域的几何图中都起着重要作用。我们确定统一约翰域的马丁边界。我们的结果应用于Sierpinski垫片的研究3。我们研究了伏特拉类型的全差分方程解的某些特性。特别是，我们将界限属性和解决方案的周期性重点4。我们尝试从Alefeld，Chen和Potra提出的验证方法的输出中获取更多有用的信息。我们使用Faekas引理来检查互补问题的不存在解决方案。5。作为对无限网络理论及其应用理论的综合研究，我们提交了与该领域有关的讲座。

项目成果

A,Murakami., M,Yamasaki.: "A Weighted Sobolev-Poicare's inequality on infinite networks, Mem. Fac. Sci. Shimane Univ. Ser. B"Mthematical Science. 34. 45-52 (2001)

A,Murakami., M,Yamasaki.：“无限网络上的加权 Sobolev-Poicare 不等式，Mem. Fac. Sci. Shimane Univ. Ser. B”数学科学。

古用哲夫: "Periodic colutions of Volterra difference equations and attractivity"Nonlinear Analysis. 47. 4013-4024 (2001)

Tetsuo Furuyo：“Volterra 差分方程和吸引力的周期解”非线性分析 47. 4013-4024 (2001)。

陳小君, 生源寺亨浩, 山崎稀嗣: "Verification for existence of solutions of linear complementarity problems"Linear Algebra and Its Applications. 324. 15-26 (2001)

陈晓君、成根寺时宏、山崎真理嗣：“线性互补问题解的存在性验证”线性代数及其应用 324. 15-26 (2001)。

H,Aikawa.: "Positive harmonic functions of finite order in a Denjoy type domain"Proc. Amer. Math. Soc., (to appear).

H，Aikawa：“Denjoy 型域中有限阶的正调和函数”Proc。

T,Furumochi.: "Periodic solutions of Volterra difference equations and attractiviry"Nonlinear Analysis. 47. 4013-4024 (2001)

T，Furumochi.：“Volterra 差分方程和吸引性的周期解”非线性分析。