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A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

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  • Mohsen Soltanifar

    (Biostatistics Division, Dalla Lana School of Public Health, University of Toronto, 620-155 College Street, Toronto, ON M5T 3M7, Canada
    Real World Analytics, Cytel Canada Health Inc., 802-777 West Broadway, Vancouver, BC V5Z 1J5, Canada)

Abstract

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

Suggested Citation

  • Mohsen Soltanifar, 2021. "A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals," Mathematics, MDPI, vol. 9(13), pages 1-9, July.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:13:p:1546-:d:586815
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    References listed on IDEAS

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    1. Eugen Anitas, 2014. "Small-angle scattering from fat fractals," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 87(6), pages 1-7, June.
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    Cited by:

    1. Mohsen Soltanifar, 2022. "The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals," Mathematics, MDPI, vol. 10(5), pages 1-11, February.

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