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“…Such a result is interesting in itself since it enables Hausdorff dimension to be used in computational applications involving fractal dimension. To deal with this, the algorithm contributed in ( [12], Section 3.1) becomes the key to estimating Hausdorff dimension in 1-dimensional subsets.…”

confidence: 99%

“…Such a result is interesting in itself since it enables Hausdorff dimension to be used in computational applications involving fractal dimension. To deal with this, the algorithm contributed in ( [12], Section 3.1) becomes the key to estimating Hausdorff dimension in 1-dimensional subsets.…”

confidence: 99%

“…Throughout this paper, we have been focused on calculating the fractal dimension of a subset F ⊆ Y in terms of the fractal dimension of its pre-image Theorems 7,9,12,16,17,19,21,and 23). In this section, we state a powerful result (c.f.…”

confidence: 99%

“…The last step is to provide three models for a fractal dimension definition (with respect to any fractal structure) following the spirit of the Hausdorff dimension. Thus, while the first one is specially interesting, since its description is made in terms of finite coverings (which allowed the authors to contribute the first known overall algorithm to calculate the Hausdorff dimension of any compact Euclidean subset, see [17]), the remaining definitions become close approaches to classical Hausdorff dimension. Hence, upcoming fractal dimension V becomes a discrete version of the latter, whereas fractal dimension VI is proposed in terms of δ-covers.…”

confidence: 99%

“…However, though its definition is made in terms of finite coverings, it holds that both fractal dimension IV and Hausdorff dimension are equal for compact Euclidean subspaces (see [16,Theorem 3.13 & Corollary 3.14 (2)]). This fact allowed the authors to provide the first known procedure to calculate the Hausdorff dimension in practical applications (see [17,Algorithm 3.1]). It is worth mentioning that fractal dimension IV becomes also an intermediate model between the classical fractal dimension definitions (for additional details, we refer the reader to [16,Remark 3.15]).…”

confidence: 99%