谢尔宾斯基分形天线对应分形网络的本征时间及相关问题研究

彭雨露; PENG Yulu; 黄煜可; HUANG Yuke

doi:10.48014/bcam.20250408002

谢尔宾斯基分形天线对应分形网络的本征时间及相关问题研究

Research on the Eigentime and Related Problems of the Corresponding Fractal Network of Sierpiński Fractal Antenna

中国应用数学通报 2025年第3卷第2期页码[10-18] 下载全文[0.8MB]

摘要

本文研究了基于谢尔宾斯基分形天线对应分形网络中随机游走的本征时间及相关问题。访问时间或命中时间H (i, j) 是从节点i出发随机游走到节点j的期望时间。网络G的本征时间定义为H (i, j) 对于任意i, j∈G的数学期望。经典的本征时间计算时, 直接将返回本身的时间记为H (i, i) =0。但在研究电子在分形天线对应分形网络中的随机游走行为时, 考虑电子离开某节点后、首次返回该节点的时间 (自返时间) 是必要的。本文采用了两种研究方法———基于谱理论的特征值方法和基于随机过程理论的马尔可夫链方法———得到修正的本征时间, 并通过一个实例展示了经典的本征时间与修正的本征时间之间的差异正好来源于自返时间。本文展示了分形网络在现代通信中的应用。研究结果有望为分形天线的设计和性能优化提供理论基础。

Abstract

This paper studies the eigentime and related issues in the fractal network corresponding to the Sierpiński fractal antenna. The access time or hitting time H (i, j) is the expected number of steps before node j is visited, starting from node i . The eigentime of G is the expectation of H (i, j) for all i, j ∈G . Classical eigenvalue method let H (i, i) =0. However, when studying the random walk behavior. of electrons in fractal networks corresponding to fractal antennas, it is necessary to consider the time when electrons leave a certain node and first return to that node (self return time) . This paper adopts two research methods-the eigenvalue method based on spectral theory and the Markov chain method based on stochastic process theory-to obtain the modified intrinsic time, and demonstrates through an example that the difference between the classical intrinsic time and the modified intrinsic time is precisely due to the self return time. This paper shows the applications of fractal networks in modern communications. The research results are expected to provide a theoretical basis for the design and performance optimization of fractal antennas.

DOI

10.48014/bcam.20250408002

文章类型

研究性论文

收稿日期

2025-04-08

接收日期

2025-05-07

出版日期

2025-06-28

关键词

谢尔宾斯基分形天线, 分形网络, 随机游走, 本征时间, 马尔可夫链

Keywords

Sierpiński fractal antenna, fractal network, random walk, eigentime, Markov Chain

作者

彭雨露^1,2, 黄煜可^1,2,*

Author

PENG Yulu^1,2, HUANG Yuke^1,2,*

所在单位

1. 北京邮电大学数学科学学院, 北京 102206
2. 数学与信息网络教育部重点实验室 (北京邮电大学) , 北京 102206

Company

1. School of Mathematical Sciences, Beijing University of Posts and Telecommunications (BUPT) , Beijing 102206, China
2. Key Laboratory of Mathematics and Information Networks (BUPT) , Ministry of Education, Beijing 102206, China

浏览量

131

下载量

基金项目

This work is supported by the Young Talent Fund of Association for Science and Technology in Shaanxi,China,under grant No.20230513.

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引用本文

彭雨露, 黄煜可. 谢尔宾斯基分形天线对应分形网络的本征时间及相关问题研究[J]. 中国应用数学通报, 2025, 3(2): 10-18.

Citation

PENG Yulu, HUANG Yuke. Research on the eigentime and related problems of the corresponding fractal network of Sierpiński Fractal Antenna[J]. Bulletin of Chinese Applied Mathematics, 2025, 3(2): 10-18.