Fractal dimension for fractal structures

Fernández–Martínez, M.; Sánchez-Granero, M.A.

doi:10.1016/j.topol.2013.10.010

Cited by 56 publications

(43 citation statements)

References 6 publications

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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]).…”

Section: Theoretical Models For Fractal Dimension Based On Fractal Stmentioning

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.…”

Section: Counterexamples Regarding Theoretical Properties For a Fractmentioning

confidence: 99%

“…Hence, δ(C i , Γ n ) = c n i , for all n ∈ N. In addition to that, let us consider the natural fractal structure on each space C i as a self-similar set, denoted by Γ i : i = 1, 2. Then easy calculations lead to (or apply [14,Theorem 4.19], as well):…”

Section: Some Differences Between Fractal Dimensions I and Iimentioning

confidence: 99%

“…In fact, such a natural fractal structure, which was first described in [14,Definition 3.1], is locally finite, starbase and induces the usual topology.…”

Section: Introductionmentioning

confidence: 99%

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Counterexamples in theory of fractal dimension for fractal structures

Fernández–Martínez¹,

Nowak

Sánchez-Granero

2016

Chaos, Solitons & Fractals

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Fractal dimension constitutes the main tool to test for fractal patterns in Euclidean contexts. For this purpose, it is always used the box dimension, since it is easy to calculate, though the Hausdorff dimension, which is the oldest and also the most accurate fractal dimension, presents the best analytical properties. Additionally, fractal structures provide an appropriate topological context where new models of fractal dimension for a fractal structure could be developed in order to generalize the classical models of fractal dimension. In this paper, we provide some counterexamples regarding these new models of fractal dimension in order to show the reader how they behave mathematically with respect to the classical models, and also to point out which features of such models can be exploited to powerful effect in applications.

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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]).…”

Section: Theoretical Models For Fractal Dimension Based On Fractal Stmentioning

confidence: 99%

Section: Counterexamples Regarding Theoretical Properties For a Fractmentioning

confidence: 99%

Section: Some Differences Between Fractal Dimensions I and Iimentioning

confidence: 99%

“…In fact, such a natural fractal structure, which was first described in [14,Definition 3.1], is locally finite, starbase and induces the usual topology.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Counterexamples in theory of fractal dimension for fractal structures

Fernández–Martínez¹,

Nowak

Sánchez-Granero

2016

Chaos, Solitons & Fractals

Self Cite

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Particle fractal dimension and total phosphorus of soil in a typical watershed of Yangtze River, China

Liu

2014

Environ Earth Sci

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Phosphorous (P) is an essential nutrient in soil which can lead to eutrophication when eroding into rivers or lakes. As a new method to represent soil properties, fractal theory was applied to study soil characteristics and related process. To understand the correlation of total soil P (TSP) and fractal dimension (D) with the effect of soil depth and slope gradient, soil grid samples were collected at 0-10, 10-20 and 20-40 cm depths in Yingwugou watershed, Yangtze River, China. The physico-chemical properties of the samples were analyzed to determine the TSP content and D. Fractal scaling theory was then applied to establish the relationship between D and TSP. With silt as the dominant particle size, the D values of the study area were in the range of 2.42-2.55. On the flat ground, more fine particles (clay and silt) were reserved and on the contrary relatively high content of coarse sand was retained on steep slope. D values decreased with slope and soil depth increased. The average TSP contents of the three layers were 0.65, 0.63 and 0.57 g/kg for 0-10, 10-20 and 20-40 cm, respectively, showing a decreasing tendency with increased soil depth. With increasing slope gradient, the TSP contents decreased first and then increased slowly. The TSP contents were positively related to silt content (P \ 0.05) and clay content (P \ 0.05) and negatively associated with sand contents (P \ 0.01). There was an one-dimensional linear relationship (n = 530, R 2 = 0.16, P \ 0.01) between TSP content and D values. The correlation would improve when concrete slope was taken into account. To better understand and control soil erosion, the relationship between D value and soil nutrients deserves more attention.

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