有限个数列相加后盒维数估计及对应级数的敛散性判断
Estimating the Box Dimension of the Sum of a Finite Number of Sequences and Judging the Convergence or Divergence of the Corresponding Series
DOI: 10.12677/pm.2024.1412412, PDF,    国家自然科学基金支持
作者: 孙玺轩*, 梁永顺：南京理工大学数学与统计学院，江苏 南京
关键词: 盒维数；单调数列；级数敛散性；The Box Dimension； Monotone Sequence； Convergence or Divergence of Series
摘要: 本文主要讨论两个严格单调正项数列加法运算后盒维数与原先两个数列盒维数的关系，并将其推广到有限个数列进行加法运算的情形。在原先研究基础上，讨论加法运算后数列对应级数的敛散性，并得到了一些初步的研究成果。
Abstract: In this paper, we mainly solve the relationship between the box dimension of two strictly monotone positive series after the addition operation and the box dimension of the original two series. Then extend the conclusion to the addition operation of a finite number of sequences. On the basis of the previous research, the convergence or divergence of the corresponding series of the sequences after the addition operation is discussed. And some preliminary research results are obtained.
文章引用：孙玺轩, 梁永顺. 有限个数列相加后盒维数估计及对应级数的敛散性判断[J]. 理论数学, 2024, 14(12): 112-119. https://doi.org/10.12677/pm.2024.1412412

参考文献

[1] 马瑞华. Mn元素多重分形分析[J]. 应用数学进展, 2020(4): 560-564.
[2] 江南. 盒维数概念的历史演变[J]. 自然辩证法研究, 2020, 36(10): 84-90.
[3] Wang, Y., Han, Y.Q. and Mao, E. (1996) Fractal Dimensions Studying of Random Sequence. Proceedings of Third International Conference on Signal Processing (ICSP’96), Beijing, 18 October 1996, 257-260. [Google Scholar] [CrossRef] 
[4] Xiao, Y., Chen, R., Shen, R., Sun, J. and Xu, J. (1995) Fractal Dimension of Exon and Intron Sequences. Journal of Theoretical Biology, 175, 23-26. [Google Scholar] [CrossRef] [PubMed]
[5] Koçak, S. and Azcan, H. (1993) Fractal Dimensions of Some Sequences of Real Numbers. Turkish Journal of Mathematics, 17, 298-304.
[6] 张云, 梁永顺. 一类单调递减正项数列的BOX维数估计[J]. 理论数学, 2024, 14(7): 275-283.
[7] Yu, B., Liang, Y. and Liu, J. (2024) The Intrinsic Connection between the Fractal Dimension of a Real Number Sequence and Convergence or Divergence of the Series Formed by It. Journal of Mathematical Analysis and Applications, 539, Article ID: 128485. [Google Scholar] [CrossRef] 
[8] 华东师范大学数学科学学院. 数学分析(下册) [M]. 第5版. 北京: 高等教育出版社, 2019: 3.
[9] Liang, Y.S. (2010) Box Dimensions of Riemann-Liouville Fractional Integrals of Continuous Functions of Bounded Variation. Nonlinear Analysis: Theory, Methods & Applications, 72, 4304-4306. [Google Scholar] [CrossRef] 
[10] Verma, M. and Priyadarshi, A. (2023) Graphs of Continuous Functions and Fractal Dimensions. Chaos, Solitons & Fractals, 172, 113513. [Google Scholar] [CrossRef] 
[11] Yu, B. and Liang, Y. (2022) Estimation of the Fractal Dimensions of the Linear Combination of Continuous Functions. Mathematics, 10, Article 2154. [Google Scholar] [CrossRef] 
[12] Liu, P., Yu, B. and Liang, Y. (2024) A Note on Fractal Dimensions of Graphs of Certain Continuous Functions. Chaos, Solitons & Fractals, 188, Article ID: 115413. [Google Scholar] [CrossRef] 
[13] Achour, R., Li, Z., Selmi, B. and Wang, T. (2024) General Fractal Dimensions of Graphs of Products and Sums of Continuous Functions and Their Decompositions. Journal of Mathematical Analysis and Applications, 538, Article ID: 128400. [Google Scholar] [CrossRef] 
[14] 刘沛智, 杜雨萌, 叶子秦, 梁永顺. 关于某些连续函数线性组合的分形维数估计[J]. 理论数学, 2022, 12(4): 540-548.
[15] Yu, B. and Liang, Y. (2022) On the Lower and Upper Box Dimensions of the Sum of Two Fractal Functions. Fractal and Fractional, 6, Article 398. [Google Scholar] [CrossRef] 
[16] Falconer, K.J. (2014) Fractal Geometry: Mathematical Foundations and Applications. 3rd Edition, John Wiley Sons Ltd.

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