The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals
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, the...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/5/706 |
| _version_ | 1797474518043721728 |
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| author | Mohsen Soltanifar |
| author_facet | Mohsen Soltanifar |
| author_sort | Mohsen Soltanifar |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-09T20:31:15Z |
| format | Article |
| id | doaj.art-967d742638764fe0ba8d31cc857aaaab |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T20:31:15Z |
| publishDate | 2022-02-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-967d742638764fe0ba8d31cc857aaaab2023-11-23T23:22:25ZengMDPI AGMathematics2227-73902022-02-0110570610.3390/math10050706The Second Generalization of the Hausdorff Dimension Theorem for Random FractalsMohsen Soltanifar0Continuing Studies Division, Population Data BC, University of Victoria, B364-3800 Finnerty Road, Victoria, BC V8P 5C2, CanadaIn 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.https://www.mdpi.com/2227-7390/10/5/706random fractalsfat fractal perculationHausdorff dimensionpacking dimensionAssouad dimensionbox dimension |
| spellingShingle | Mohsen Soltanifar The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals Mathematics random fractals fat fractal perculation Hausdorff dimension packing dimension Assouad dimension box dimension |
| title | The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals |
| title_full | The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals |
| title_fullStr | The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals |
| title_full_unstemmed | The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals |
| title_short | The Second Generalization of the Hausdorff Dimension Theorem for Random Fractals |
| title_sort | second generalization of the hausdorff dimension theorem for random fractals |
| topic | random fractals fat fractal perculation Hausdorff dimension packing dimension Assouad dimension box dimension |
| url | https://www.mdpi.com/2227-7390/10/5/706 |
| work_keys_str_mv | AT mohsensoltanifar thesecondgeneralizationofthehausdorffdimensiontheoremforrandomfractals AT mohsensoltanifar secondgeneralizationofthehausdorffdimensiontheoremforrandomfractals |