A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals
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 Ha...
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MDPI AG
2021-07-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/13/1546 |
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doaj-b235c31577c449c792a06c5ff981ffa92021-07-15T15:41:39ZengMDPI AGMathematics2227-73902021-07-0191546154610.3390/math9131546A Generalization of the Hausdorff Dimension Theorem for Deterministic FractalsMohsen Soltanifar0Biostatistics Division, Dalla Lana School of Public Health, University of Toronto, 620-155 College Street, Toronto, ON M5T 3M7, CanadaHow 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.https://www.mdpi.com/2227-7390/9/13/1546cantor setfractalsHausdorff dimensioncontinuumaleph-two |
| collection |
DOAJ |
| language |
English |
| format |
Article |
| sources |
DOAJ |
| author |
Mohsen Soltanifar |
| spellingShingle |
Mohsen Soltanifar A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals Mathematics cantor set fractals Hausdorff dimension continuum aleph-two |
| author_facet |
Mohsen Soltanifar |
| author_sort |
Mohsen Soltanifar |
| title |
A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals |
| title_short |
A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals |
| title_full |
A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals |
| title_fullStr |
A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals |
| title_full_unstemmed |
A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals |
| title_sort |
generalization of the hausdorff dimension theorem for deterministic fractals |
| publisher |
MDPI AG |
| series |
Mathematics |
| issn |
2227-7390 |
| publishDate |
2021-07-01 |
| description |
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. |
| topic |
cantor set fractals Hausdorff dimension continuum aleph-two |
| url |
https://www.mdpi.com/2227-7390/9/13/1546 |
| work_keys_str_mv |
AT mohsensoltanifar ageneralizationofthehausdorffdimensiontheoremfordeterministicfractals AT mohsensoltanifar generalizationofthehausdorffdimensiontheoremfordeterministicfractals |
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1721298895541108736 |