Fractal and Fractional

15 pages, 516 KiB

Open AccessArticle

Remarks on the Relationship Between Fractal Dimensions and Convergence Speed

by Jiaqi Qiu and Yongshun Liang

Fractal Fract. 2025, 9(5), 303; https://doi.org/10.3390/fractalfract9050303 (registering DOI) - 6 May 2025

This paper conducts an in-depth investigation into the fundamental relationship between the fractal dimensions and convergence properties of mathematical sequences. By concentrating on three representative classes of sequences, namely, the factorial-decay, logarithmic-decay, and factorial–exponential types, a comprehensive framework is established to link their [...] Read more.

This paper conducts an in-depth investigation into the fundamental relationship between the fractal dimensions and convergence properties of mathematical sequences. By concentrating on three representative classes of sequences, namely, the factorial-decay, logarithmic-decay, and factorial–exponential types, a comprehensive framework is established to link their geometric characteristics with asymptotic behavior. This study makes two significant contributions to the field of fractal analysis. Firstly, a unified methodology is developed for the calculation of multiple fractal dimensions, including the Box, Hausdorff, Packing, and Assouad dimensions, of discrete sequences. This methodology reveals how these dimensional quantities jointly describe the structures of sequences, providing a more comprehensive understanding of their geometric properties. Secondly, it is demonstrated that different fractal dimensions play distinct yet complementary roles in regulating convergence rates. Specifically, the Box dimension determines the global convergence properties of sequences, while the Assouad dimension characterizes the local constraints on the speed of convergence. The theoretical results presented herein offer novel insights into the inherent connection between geometric complexity and analytical behavior within sequence spaces. These findings have immediate and far-reaching implications for various applications that demand precise control over convergence properties, such as numerical algorithm design and signal processing. Notably, the identification of dimension-based convergence criteria provides practical and effective tools for the analysis of sequence behavior in both pure mathematical research and applied fields. Full article

(This article belongs to the Special Issue Fractal Functions: Theoretical Research and Application Analysis)

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17 pages, 313 KiB

Open AccessArticle

An Exploration of the Qualitative Analysis of the Generalized Pantograph Equation with the q-Hilfer Fractional Derivative

by R. Vivek, K. Kanagarajan, D. Vivek, T. D. Alharbi and E. M. Elsayed

Fractal Fract. 2025, 9(5), 302; https://doi.org/10.3390/fractalfract9050302 - 6 May 2025

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Fractal and Fractional

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