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“…It turns out that both fractal dimensions I & II do generalize the box dimension in the context of Euclidean spaces equipped with their natural fractal structures (see [14,Theorem 4.7]).…”

confidence: 99%

“…It turns out that both fractal dimensions I & II do generalize the box dimension in the context of Euclidean spaces equipped with their natural fractal structures (see [14,Theorem 4.7]).…”

confidence: 99%

“…In fact, as we will show next, there exist countable Euclidean subsets (equipped with an induced natural fractal structure), whose fractal dimensions I, II, III & IV are non-zero. Recall that fractal dimensions I and III are someway expected to not satisfy such a property, since their description is similar to the box dimension (in the case of fractal dimension I), or at least, they generalize it in the context of Euclidean spaces equipped with their natural fractal structures (as it happens with fractal dimension III, see [13,Theorem 4.15], but also with fractal dimension I, see [14,Theorem 3.5] Proof. Let us consider X = [0, 1], F = Q ∩ X, and Γ be the natural fractal structure on X, whose levels are defined as in Eq.…”

confidence: 99%

“…Mandelbrot (1967) first defined fractal dimension (D) as the set in which the Hausdorff dimension was strictly greater than the natural dimension and used fractal theory on particle morphology. Since it supplies information concerning the complexity of a given set (Fernández-Martínez and Sánchez-Granero 2013), D can be understood as a box-counting, self-similar dimension with plentiful applications in the field of soil science. With increasing research and use of the fractal method (Albanese et al 2007;Prosperini and Perugini 2008;Hamid et al 2013;Oldrich et al 2013), it has been developed as a mature computation method to characterize soil.…”

confidence: 99%