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The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals

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  • Mohsen Soltanifar

    (Continuing Studies Division, Population Data BC, University of Victoria, B364-3800 Finnerty Road, Victoria, BC V8P 5C2, Canada
    Real World Analytics, Cytel Canada Health Inc., 802-777 West Broadway, Vancouver, BC V5Z 1J5, Canada
    Biostatistics Division, Dalla Lana School of Public Health, University of Toronto, 620-155 College Street, Toronto, ON M5T 3M7, Canada)

Abstract

In this paper, we present a second partial solution for the problem of cardinality calculation of the set of fractals for its subcategory of the random virtual ones. Consistent with the deterministic case, we show that for the given quantities of the Hausdorff dimension and the Lebesgue measure, there are aleph-two virtual random fractals with, almost surely, a Hausdorff dimension of a bivariate function of them and the expected Lebesgue measure equal to the latter one. The associated results for three other fractal dimensions are similar to the case given for the Hausdorff dimension. The problem remains unsolved in the case of non-Euclidean abstract fractal spaces.

Suggested Citation

  • Mohsen Soltanifar, 2022. "The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals," Mathematics, MDPI, vol. 10(5), pages 1-11, February.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:5:p:706-:d:757149
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    References listed on IDEAS

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    1. Mohsen Soltanifar, 2021. "A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals," Mathematics, MDPI, vol. 9(13), pages 1-9, July.
    2. Frezza, Massimiliano & Bianchi, Sergio & Pianese, Augusto, 2021. "Fractal analysis of market (in)efficiency during the COVID-19," Finance Research Letters, Elsevier, vol. 38(C).
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