Article

# Fractal Dimension for Fractal Structures

Authors:
• Centro Universitario de la Defensa, San Javier, Spain
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## Abstract

The main goal of this paper is to provide a generalized definition of fractal dimension for any space equipped with a fractal structure. This novel theory generalizes the classical box-counting dimension theory on the more general context of GF-spaces. In this way, if we select the so-called natural fractal structure on any Euclidean space, then the box-counting dimension becomes just a particular case. This idea allows to consider a wide range of fractal structures to calculate the effective fractal dimension for any subset of this space. Unlike it happens with the classical theory of fractal dimension, the new definitions we provide may be calculated in contexts where the box-counting one can have no sense or cannot be calculated. Nevertheless, the new models can be computed for any space admitting a fractal structure, just as easy as the box-counting dimension in empirical applications.

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... Accepted manuscript to appear in FRACTALS Definition 2.1 (c.f. Definition 3.1 in [8]). The levels of the natural fractal structure on R d are given by ...
... (2) (c.f. Definition 3.3 in [8]) Let Γ be a fractal structure on X and N n (F ) the number of elements in Γ n that intersect F . Then the (lower/upper) fractal dimension I of F is defined in terms of the following (lower/upper) limit: ...
... Theorem 2.3 (c.f. [8], Theorem 3.5). Let F be a subset of R d and Γ the natural fractal structure induced on F . ...
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We conducted a longitudinal study involving 240 patients grouped according to the classification of periodontal diseases agreed in the World Workshop by the different groups of specialists gathered there. We proceed to select images of Cone Beam Computed Tomography (CBCT) that were used to perform a study of bone density through a precise algorithm allowing an accurate calculation of the fractal dimensions of such images. A detailed anthropometric analysis was also carried out. Our objective was to demonstrate that there exists a direct relationship between either the loss of bone or the changes related to its height and diameter and the variations in bone density. Our results highlight significant differences among the initial and moderate periodontal groups with respect to both the control and the periodontal groups, where patients experience a severe and controlled periodontal disease. We conclude that there is a variation in the architecture of patients with periodontal disease that have an acute component and have not been treated or their treatment is not effective and their bone loss does not slow down.
... Both allow non-integer values for dimension of sets, such as Cantor sets (see [20]). The computation, or estimation, of the Hausdorff dimension of the graph of functions is extensively dealt with in the literature: e.g. for Julia sets of functions [26], Takagi's function [3], Weierstrass-type functions [16,17,6,28], self-affine functions [8] and fractal functions [23,24,29]. The goal of this paper is to investigate the dimension of graphs for a class of functions which are solutions of systems of iterative functional equations (see [5]). ...
... Fractal structures play a fundamental role in obtaining the Hausdorff dimension (see [15]) and other non-integer dimensions (see [24]). ...
... For some sets it is reasonable to define its fractal natural structure. Fernandez-Martinez and Sanchez-Granero defined the natural fractal structure of any Euclidean space with a bi-uniform countable family of coverings (see [24]). Definition 7. The natural fractal structure on the Euclidean space R d is defined as the countable family of coverings Γ = {Γ n : n ∈ N}, whose levels are given by ...
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We consider systems of non-affine iterative functional equations. From the constructive form of the solutions, recently established by the authors, representations of these systems in terms of symbolic spaces as well as associated fractal structures are constructed. These results are then used to derive upper bounds both for the appropriate fractal dimension and the corresponding Hausdorff dimension of solutions. Using the formalism of iterated function systems, we obtain a sharp result on the Hausdorff dimension in terms of the corresponding fractal structures. The connections of our results with related objects known in the literature, including Girgensohn functions, fractal interpolation functions and Weierstrass functions, are established.
... In other words, we shall guarantee the existence of a map a : X ! Y satisfying some desirable properties allowing to achieve the identity dim (F) = d · dim (a 1 (F)), where F ✓ Y, a 1 (F) ✓ X, and dim refers to fractal dimensions I, II, III, IV, and V (introduced in previous works by the authors, c.f. [8][9][10]), as well as the classical fractal dimensions, namely, both box and Hausdorff dimensions. The nature of both spaces X and Y will be unveiled along each section in this paper. ...
... Definition 1 (c.f. Definition 3.1 in [9]). The natural fractal structure on the Euclidean space R d is given by the countable family of coverings G = {G n : n 2 N} with levels defined as ...
... The fractal dimension models for a fractal structure involved in this paper, namely, fractal dimensions I, II, III, IV, and V, were introduced previously by the authors (c.f. [8][9][10]) and proved to generalize both box and Hausdorff dimensions in the Euclidean setting (c.f. ( [9], Theorem 3.5, Theorem 4.7), ( [8], Theorem 4.15), ( [10], Theorem 3.13)) through their natural fractal structures (c.f. ( [9], Definition 3.1)). Thus, they become ideal candidates to explore the fractal nature of subsets. ...
Article
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In this paper, we prove the identity $\h(F)=d\cdot \h(\alpha^{-1}(F))$, where $\h$ denotes Hausdorff dimension, $F\subseteq \R^d$, and $\alpha:[0,1]\to [0,1]^d$ is a function whose constructive definition is addressed from the viewpoint of the powerful concept of a fractal structure. Such a result stands particularly from some other results stated in a more general setting. Thus, Hausdorff dimension of higher dimensional subsets can be calculated from Hausdorff dimension of $1-$dimensional subsets of $[0,1]$. As a consequence, Hausdorff dimension becomes available to deal with the effective calculation of the fractal dimension in applications by applying a procedure contributed by the authors in previous works. It is also worth pointing out that our results generalize both Skubalska-Rafaj{\l}owicz and Garc{\'{i}}a-Mora-Redtwitz theorems.
... The main goal in this paper is to calculate the (more awkward) fractal dimension of objects contained in Y in terms of the (easier to be calculated) fractal dimension of subsets of X through an appropriate function α : X → Y . In other words, we shall guarantee the existence of a map α : X → Y satisfying some desirable properties allowing to achieve the identity dim (F ) = d · dim (α −1 (F )), where F ⊆ Y, α −1 (F ) ⊆ X, and dim refers to fractal dimensions I, II, III, IV, and V (introduced in previous works by the authors, c.f. [5,6,7]), as well as the classical fractal dimensions, namely, both box and Hausdorff dimensions. The nature of both spaces X and Y will be unveiled along each section in this paper. ...
... Definition 2.1 (c.f. Definition 3.1 in [6]). The natural fractal structure on the Euclidean space R d is given by the countable family of coverings Γ = {Γ n : n ∈ N} with levels defined as ...
... The fractal dimension models for a fractal structure involved along this paper, namely, fractal dimensions I, II, III, IV, and V, were introduced previously by the authors (c.f. [5,6,7]) and proved to generalize both box and Hausdorff (1) ( [11]) If X = R d , then the (lower/upper) box dimension of F is defined through the (lower/upper) limit ...
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... The main goal in this paper is to calculate the (more awkward) fractal dimension of objects contained in Y in terms of the (easier to be calculated) fractal dimension of subsets of X through an appropriate function α : X → Y . In other words, we shall guarantee the existence of a map α : X → Y satisfying some desirable properties allowing to achieve the identity dim (F ) = d · dim (α −1 (F )), where F ⊆ Y, α −1 (F ) ⊆ X, and dim refers to fractal dimensions I, II, III, IV, and V (introduced in previous works by the authors, c.f. [5,6,7]), as well as the classical fractal dimensions, namely, both box and Hausdorff dimensions. The nature of both spaces X and Y will be unveiled along each section in this paper. ...
... Definition 2.1 (c.f. Definition 3.1 in [6]). The natural fractal structure on the Euclidean space R d is given by the countable family of coverings Γ = {Γ n : n ∈ N} with levels defined as ...
... The fractal dimension models for a fractal structure involved along this paper, namely, fractal dimensions I, II, III, IV, and V, were introduced previously by the authors (c.f. [5,6,7]) and proved to generalize both box and Hausdorff (1) ( [11]) If X = R d , then the (lower/upper) box dimension of F is defined through the (lower/upper) limit ...
Preprint
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In this paper, we prove the identity $\dim_{\textrm H}(F)=d\cdot \dim_{\textrm H}(\alpha^{-1}(F))$, where $\dim_{\textrm H}$ denotes Hausdorff dimension, $F\subseteq \mathbb{R}^d$, and $\alpha:[0,1]\to [0,1]^d$ is a function whose constructive definition is addressed from the viewpoint of the powerful concept of a fractal structure. Such a result stands particularly from some other results stated in a more general setting. Thus, Hausdorff dimension of higher dimensional subsets can be calculated from Hausdorff dimension of $1-$dimensional subsets of $[0,1]$. As a consequence, Hausdorff dimension becomes available to deal with the effective calculation of the fractal dimension in applications by applying a procedure contributed by the authors in previous works. It is also worth pointing out that our results generalize both Skubalska-Rafaj\l{}owicz and Garc\'{\i}a-Mora-Redtwitz theorems.
... In fact, the new models can be applied to calculate the fractal dimension of any space admitting a fractal structure as easy as the boxdimension in empirical applications. The results contained in this section were first contributed in [26]. ...
... Remark 2. (c.f. [26,Remark 2.5]) Another suitable description concerning the levels of such a fractal structure is as follows: ...
... (c.f. [26,Example 2]) The natural fractal structure on the Sierpiński gasket as a self-similar set is the countable family of coverings Γ Γ Γ = {Γ n } n∈N , where Γ 1 is the union of three equilateral "triangles" with sides equal to 1/2, Γ 2 consists of the union of 3 2 equilateral "triangles" with sides equal to 1/2 2 , and in general, Γ n is the union of 3 n equilateral "triangles" whose sides are equal to 1/2 n for each natural number n. In addition, this is a finite starbase fractal structure. ...
Article
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Along the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Carathéodory, Sierpinski, and Besicovitch, to quote some of them. They were some of the pioneers exploring objects having self-similar patterns or showing anomalous properties with respect to standard analytic attributes. Among the new tools developed to deal with this kind of objects, fractal dimension has become one of the most applied since it constitutes a single quantity which throws useful information concerning fractal patterns on sets. Several years later, fractal structures were introduced from Asymmetric Topology to characterize self-similar symbolic spaces. Our aim in this survey is to collect several results involving distinct definitions of fractal dimension we proved jointly with Prof. M.A. Sánchez-Granero in the context of fractal structures.
... The concept of fractal [48] is very important for the study of non-linear objects. Fractal dimension is an important approach to study fractal, which includes information about the complexity of fractal objects [49]. Hausdorff dimension is one of the oldest and most important fractal dimensions, it gave a new form to the usual concepts of length and area, and it formed the basic theoretical model of other fractal dimensions. ...
... The four positions of the yellow circle are where the two sequences differ Fig. 10 The FFT of P1 and P2. The x-coordinate means the i-th amino acid, and the y-coordinate is FFT using second level range of scales [49]. In order to apply Box counting dimension to digital image processing more conveniently, scholars also put forward Minkowski dimension [50]. ...
Article
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Background Amino acid property-aware phylogenetic analysis (APPA) refers to the phylogenetic analysis method based on amino acid property encoding, which is used for understanding and inferring evolutionary relationships between species from the molecular perspective. Fast Fourier transform (FFT) and Higuchi’s fractal dimension (HFD) have excellent performance in describing sequences’ structural and complexity information for APPA. However, with the exponential growth of protein sequence data, it is very important to develop a reliable APPA method for protein sequence analysis. Results Consequently, we propose a new method named FFP, it joints FFT and HFD. Firstly, FFP is used to encode protein sequences on the basis of the important physicochemical properties of amino acids, the dissociation constant, which determines acidity and basicity of protein molecules. Secondly, FFT and HFD are used to generate the feature vectors of encoded sequences, whereafter, the distance matrix is calculated from the cosine function, which describes the degree of similarity between species. The smaller the distance between them, the more similar they are. Finally, the phylogenetic tree is constructed. When FFP is tested for phylogenetic analysis on four groups of protein sequences, the results are obviously better than other comparisons, with the highest accuracy up to more than 97%. Conclusion FFP has higher accuracy in APPA and multi-sequence alignment. It also can measure the protein sequence similarity effectively. And it is hoped to play a role in APPA’s related research.
... With the rapid development of nano technology, the more and more attention is obtained from the people on the electrospun nanofibres non-woven fabrics, especially for its unique property. However, as its structure is very complex and irregular, the use of fractal theory provides a predictable research direction [1][2][3] . In this paper, the electrospun nanofibres non-woven fabrics image that is captured by the electron microscope through digital picture processing technique is treated through digital image processing technology (Visual C++ and Matlab program), and then we use the method of mathematical morphology to calculate the diameter distribution of electrospun nanofibres and calculate the fractal dimension [6] of electrospun nanofibres according to fractal theory. ...
... Regarding study of non-woven fabrics fractal properties, with study by box dimension method on fractal property of melt-blown nonwoven fabric, we acquire a series of valuable conclusions [1][2][3] . However, as the obvious fiber distribution of electrospun nanofibres non-woven fabrics, it is evidently not appropriate to use the box dimension. ...
Conference Paper
distribution; correlation fractal dimension. Abstract: In this paper, samples are first designed and made based on the orthogonal experiment law. Afterward, image gray processing and image binarization processing are performed on the electrospun nanofibres non-woven fabrics image that is captured by the electron microscope through digital picture processing technique, and then is refined with fiber structure through the image thinning algorithm and the diameter distribution of the simple can be obtained through calculating the average diameter of the fabric distributed in various position. Subsequently, the mathematical treatment is performed to the sample's diameter distribution. G-P algorithm is used to calculate the correlation fractal dimension of the diameter distribution. After acquired with fractal dimension, the mapping relation between the fractal dimension and physical properties of filtration efficiency can be verified through the analysis of numerical relation between the fractal feature of electrospun nanofiber nonwoven fabric and the properties of filtration efficiency, which furthermore provides another scale to the evaluation of the electrostatic spinning nano fiber non-woven cloth.
... Fractal dimension (FD) measures the roughness and texture of a self-similar figure [26]. Some biologic structures show complexity and self-similarity that can be described by FD [27,28]. ...
Article
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Purpose: To elucidate the role of atrial anatomical remodeling in atrial fibrillation (AF), we proposed an automatic method to extract and analyze morphological characteristics in left atrium (LA), left atrial appendage (LAA) and pulmonary veins (PVs) and constructed classifiers to evaluate the importance of identified features. Methods: The LA, LAA and PVs were segmented from contrast computed tomography images using either a commercial software or a self-adaptive algorithm proposed by us. From these segments, geometric and fractal features were calculated automatically. To reduce the model complexity, a feature selection procedure is adopted, with the important features identified via univariable analysis and ensemble feature selection. The effectiveness of this approach is well illustrated by the high accuracy of our models. Results: Morphological features, such as LAA ostium dimensions and LA volume and surface area, statistically distinguished (p<0.01) AF patients or AF with LAA filling defects (AF(def+)) patients among all patients. On the test set, the best model to predict AF among all patients had an area under the receiver operating characteristic curve (AUC) of 0.91 (95% CI, 0.8–1) and the best model to predict AF(def+) among all patients had an AUC of 0.92 (95% CI, 0.81–1). Conclusion: This study automatically extracted and analyzed atrial morphology in AF and identified atrial anatomical remodeling that statistically distinguished AF or AF(def+). The importance of identified atrial morphological features in characterizing AF or AF(def+) was validated by corresponding classifiers. This work provides a good foundation for a complete computer-assisted diagnostic workflow of predicting the occurrence of AF or AF(def+).
... Second, this work focused on n-dimensional Eulidean spaces as relatively tangible case of fractal settings. It is plausible to establish and evaluate these results for more abstract settings [44]. ...
Article
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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.
... Hausdorff dimension is the best way to measure fractal dimension of a bounded subset of R n since it considers all the possible coverings (of a given diameter) that the bounded subset may admit, and it possesses better analytical properties than the box dimension. We refer to the works of Fernández-Martínez et al. [14][15][16][17] who proposed a way to deal with the calculation of Hausdorff dimension in applications for compact Euclidean subsets including the higher dimensional case in more general settings. Their approach combines both theoretical results along with techniques from machine learning, thus leading to the first-known attempt to calculate Hausdorff dimension in applications. ...
Article
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Fractals are geometric shapes and patterns that may repeat their geometry at smaller or larger scales. It is well established that fractals can describe shapes and surfaces that cannot be represented by the classical Euclidean geometry. An eclectic survey of fractals is presented in two parts encompassing applications of fractals in a variety of diverse and innovative fields. The goal of the first part is to focus on the glossary of fractals, their mathematical description, aesthetic, artistic, and architectural applications, while the second part is focused on engineering, industry, commercial, and futuristic applications of fractals.
... An object exhibiting self-similar structures at different length scales is known as a fractal. Several systems in nature are fractal, including biological tissue samples from different organs [1][2][3][4][5]. Biological tissues have spatial heterogeneity in their mass density distribution and are a self-similar structure. ...
Article
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Fractal dimension, a measure of self-similarity in a structure, is a powerful physical parameter for the characterization of structural property of many partially filled disordered materials. Biological tissues are fractal in nature and reports show a change in self-similarity associated with the progress of cancer, resulting in changes in their fractal dimensions. Here, we report that fractal dimension measurement is a potential technique for the detection of different stages of cancer using transmission optical microscopy. Transmission optical microscopy of a thin tissue sample produces intensity distribution patterns proportional to its refractive index pattern, representing its mass density distribution. We measure fractal dimension detection of different cancer stages and find its universal feature. Many deadly cancers are difficult to detect in their early to different stages due to the hard-to-reach location of the organ and/or lack of symptoms until very late stages. To study these deadly cancers, tissue microarray (TMA) samples containing different stages of cancers are analyzed for pancreatic, breast, colon, and prostate cancers. The fractal dimension method correctly differentiates cancer stages in progressive cancer, raising possibilities for a physics-based accurate diagnosis method for cancer detection.
... coefficient and small shortest path length), (iii) the self-similar property is often found [12] (A network is believed to be self-similar when the small portion is similar to the whole part under a length-scale transformation), and (iv) fractal phenomena are generally observed [13] (A fractal network is defined as a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reducedsize copy of the whole). In particular, as a class of intriguing mathematical models, fractal networks that show graceful fractal feature have been widely discussed [14]- [25]. ...
Article
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More attention has been paid to research of various kinds of fractals due to a great number of applications in different fields over the past years. In this paper, we present stochastic generalized Vicsek fractal networks to model underlying structures on many hyperbranched polymers. Then, we consider random walks, the widely-studied dynamical behavior, on stochastic models built in order to better understand structural properties. Specifically, we analytically derive the closed-form solution to mean first-passage time for random walks, which is a quantity that allows one to better understand the underlying structure of model in question, using a more effective approach compared with the typical computational methods including spectral technique. At the same time, some other fundamental structural parameters including Kirchhoff index and fractal dimension are obtained analytically as well. The results suggest that the fractal characteristic makes the underlying structure of network more loose, and thus leads the efficiency of delivering information in a random-walk-based manner to become lower. Finally, we conduct extensive simulations to demonstrate that theoretical analysis and numerical simulations are in perfect agreement with each other.
... Fractal dimensions for fractal structures. The fractal dimension models for a fractal structure involved in this paper, i.e., fractal dimensions III and IV, were first introduced in [14,15], and could be understood as subsequent models from those explored in [16]. It is worth pointing out that they allowed generalizing both box dimension (c.f. ...
... Moreover, for the case of open problem #3, the cardinality of the set of all distinctive fractals in R n is independent from their determinitic or random nature. The work limitation -as in the deterministic case-is its limited generalizability for more generalized and abstract fractal structures and their fractal dimensions [27]. ...
Preprint
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 Hausdorff dimension and Lebesgue measure, there are aleph-two virtual random fractals with almost surely Hausdorff dimension of a bivariate function of them and the expected Lebesgue measure equal to the later one. The associated results for three other fractal dimensions are similar to the case given for the Hausdorff dimension. The problem remains unsolved for the case of non-Euclidean abstract fractal spaces.
... The fractal research method, as an optimized structure of nature, is suitable for nonlinear and irregular complex shapes and phenomena and is helping cast off the past shackles of having to solve nonlinear problems with linear geometry. With the help of selfsimilarity and fractal dimensions (parameters to measure irregularity), the effectiveness of the space occupied by complex shapes can be characterized [27]. Through fractal theory, the scientific configuration of urban elements and the maximization of potential spatial benefits can be realized under limited resources; therefore, introducing fractal theory to study the space complexity of park green space will help continuously optimize the structure of green space and push its development toward maximum efficiency so that the park system can be continuously optimized in a limited space. ...
Article
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This paper applies fractal theory to research of green space in megacity parks due to the lack of a sufficient qualitative description of the scale structure of park green space, a quantifiable evaluation system, and operable planning methods in traditional studies. Taking Beijing, Shanghai, Guangzhou, and Shenzhen as examples, GIS spatial analysis technology and the Zipf model are used to calculate the fractal dimension (q), the goodness of fit (R2), and the degree of difference (C) to deeply interpret the connotation of indicators and conduct a comparative analysis between cities to reveal fractal characteristics and laws. The research results show that (1) the fractal dimension is related to the complexity of the park green space system; (2) the fractal dimension characterizes the hierarchical iteration of the park green space to a certain extent and reflects the internal order of the scale distribution; (3) the scale distribution of green space in megacity parks deviates from the ideal pyramid configuration; and (4) there are various factors affecting the scale structure of park green space, such as natural base conditions, urban spatial structure, and the continuation of historical genes working together. On this basis, a series of targeted optimization strategies are proposed.
... For instance, we can consider the Renyi dimension (with special cases, such as the Minkowski dimension [16], Information dimension [22], Correlation dimension [23]), the Higuchi dimension [24], the Lyapunov dimension [25], Packing dimension [26], the Assouad dimension [27] and the generalization of the Hausdorff dimension [28]. Third, a more rigorous and comprehensive method is to investigate the existence problem of fractals for the generalized fractal space equipped with a fractal structure and the generalized fractal dimension [29]. Finally, the existence result in this work is limited to deterministic fractals constructed by their associated deterministic recursive processes. ...
Article
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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.
... Thus, while the definition of former is based on measurement (from a theoretical approach), the latter becomes easier to be empirically calculated or estimated (from the viewpoint of applications). In this way, most empirical applications involving fractal dimension have been carried out in the context of Euclidean spaces through the box dimension [109,110]. Due to the features of ease of utilized simple algorithms and relatively high accuracy, the box-counting method may be one of the most widely utilized methods to evaluate the fractal dimension of complex surfaces [111][112][113]. Some backgrounds of the box-counting method for calculating fractal dimension of a structure can be found in Refs [114,115]. ...
Article
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Cement-based materials are widely utilized in infrastructure. The main product of hydrated products of cement-based materials is calcium silicate hydrate (C-S-H) gels that are considered as the binding phase of cement paste. C-S-H gels in Portland cement paste account for 60–70% of hydrated products by volume, which has profound influence on the mechanical properties and durability of cement-based materials. The preparation method of C-S-H gels has been well documented, but the quality of the prepared C-S-H affects experimental results; therefore, this review studies the preparation method of C-S-H under different conditions and materials. The progress related to C-S-H microstructure is explored from the theoretical and computational point of view. The fractality of C-S-H is discussed. An evaluation of the mechanical properties of C-S-H has also been included in this review. Finally, there is a discussion of the durability of C-S-H, with special reference to the carbonization and chloride/sulfate attacks.
... Geometry plays an important role in relativity, cosmology, field theory, thermodynamics, and classical mechanics. 1 Mandelbrot characterized the geometry of nature as "fractal", where patterns repeat across multiple levels of scale generating irregular boundaries such as coastlines, mountains, the Koch snowflake, Romanesco broccoli, the human brain, branching tree limbs and leaf veins, feathers, mountains, tumors, blood vessels, planet, etc. [2][3][4] Fractals provide accurate modeling of irregular phenomena that occur in economics, biophysics, engineering, chemistry, biology, thin films, etc. [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] Fractals are characterized by non-integer, "fractal" dimensions, which exceed their topological dimension, like the Cantor set, Weierstrass function, (1) By recognizing spacetime as fractal. [42][43][44][45] (2) Or by recognizing phenomena as fractal, within traditional differentiable spacetime. ...
Article
Fractal calculus generalizes ordinary calculus, offering a way to differentiate otherwise non-differentiable domains and phenomena. This paper discusses the equilibrium and non-equilibrium statistical mechanics involving fractal structure, as well as fractal temperature in the partition function.
... Fractals are shapes and objects, their fractal dimensions exceed topological dimensions and were seen in many physical models [1][2][3]. Physicists and mathematicians have begun to investigate the territory of applications of fractals with ever new development speedily taking place in the field of statistical and nonlinear physics including hydrology, heat conduction and diffusion, polymer physics, biophysics and thermodynamics, Brownian random walks with memory, modeling dispersion and turbulence, transfer equation in a medium with fractal geometry, kinetic theories, far-from equilibrium statistical models manifesting scale invariance and scaling processes, plasma physics, financial time series, signal processing, fluctuations in solids, reservoir engineering, environmental geophysics, geophysical fluid dynamics, to ecology and climatology and so on [1][2][3][4][5][6][7]. The axiomatic framework was suggested to define Laplace operator on fractional spaces and applied in sciences. ...
Article
We study the dynamics of particles in cold electron plasma medium based on two dissimilar approaches: the fractional actionlike varia-tional and fractal calculus approaches. In each case, the corresponding Boltzmann and Vlasov-Boltzmann equations were derived. Although the mathematical relationships between fractional calculus and fractal calculus were established in the literature, it was revealed throughout this study that the corresponding physics for its approach is quite different. Each model is characterized by its corresponding Boltzmann and Vlasov-Boltzmann equations which describe dissimilar dynamics and gives rise to unrelated Bohm-Gross formulas for electron plasma waves (dispersion relations) and different group velocities connected to the numerical ranges of the matching fractional parameter. Several consequences were obtained and discussed accordingly.
... Even though the faults can be extracted using automatic or semiautomatic methods, e.g., point clouds, images, and geophysical data [20][21][22][23], the qualitative description also restricts the scale and significance of tectonics. Recently, as one of the nonlinear sciences, fractal theory [24][25][26] offered geologists new insights to explore complexity and regularity by using the fractal dimension [4,[27][28][29][30][31][32][33]. The fractal geometry of lineaments helps geologists to understand very complex systems, which feature very organized patterns in seemingly chaotic phenomena [4,5]. ...
Article
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The distribution and characteristics of geological lineaments in areas with active faulting are vital for providing a basis for regional tectonic identification and analyzing the tectonic significance. Here, we extracted the lineaments in the Qianhe Graben, an active mountainous area on the southwest margin of Ordos Block, China, by using the tensor voting algorithm after comparing them with the segment tracing algorithm (STA) and LINE algorithm in PCI Geomatica Software. The main results show that (1) the lineaments in this area are mostly induced by the active fault events with the main trending of NW-SE, (2) the box dimensions of all lineaments, NW-SE trending lineaments, and NE-SW trending lineaments are 1.60, 1.48, and 1.44 (R 2 > 0.9), respectively, indicating that the faults exhibit statistical self-similarity, and (3) the lineaments have multifractal characteristics according to the mass index τ(q), generalized fractal dimension D(q), fractal width (Δα = 2.25), fractal spectrum shape (f(α) is a unimodal left-hook curve), and spectrum width (Δf = 1.21). These results are related to the tectonic activity in this area, where a higher tectonic activity leads to more lineaments being produced and a higher fractal dimension. All of these results suggest that such insights can be beneficial for providing potential targets in reconstructing the tectonic structure of the area and trends of plate movement.
... [28][29][30] Many physical processes in science with the fractal structures have been studied by many researchers. [31][32][33][34][35][36][37][38][39][40] Analysis on fractals is formulated using various methodologies such as fractional spaces, fractional calculus, probability theory, measure theory, harmonic analysis and nonstandard analysis. [41][42][43][44][45][46][47][48][49][50][51][52][53][54][55][56] The Laplace operator in the fractional spaces was defined and applied to model the physical processes involving disordered mediums and fractal structure. ...
Article
In this study, Einstein's field equations are derived based on two dissimilar frameworks: the first is based on the concepts of "fractional velocity" and "fractal action" motivated by Calcagni's approach to fractional spacetime while the second is derived based on fractal calculus which is a generalization of ordinary calculus that include fractal sets and curves. The fractional theory displays a breakdown of Lorentz invariance. It was observed that a spatially dependent cosmological constant emerges in the fractional theory. A connection between the fractional order parameter and the dimensionless parameter $\gamma$ arising in the parameterized post-Newtonian (PPN) formalism is observed. A confrontation with very long-baseline radio interferometry targeting quasars 3C273 and 3C279 is done which proves that the fractional order parameter is within the range $0.99980 <\alpha < 1.00004$. Moreover, emergence of quantum Hawking radiation is realized in the theory supporting Hawking's best calculations that black holes are not black. Nevertheless based on the fractal calculus approach, there is a conservation of the Lorentz invariance and absence of spatially-dependent cosmological constant. The theory depends on the fractal order $0<\beta<1$ and gives rise a fractal Schwarzschild radius of the massive body greater than the conventional radius besides a fractal Hawking's temperature less than the standard one. However, the confrontation with radio interferometry targeting quasars 3C273 and 3C279 gives $0.999997444 \beta< 1.0000128$.
... discrete scale symmetry, and fractal dimension which can be a real number. It is well known that many processes in nature can be represented by fractals; such types can be found anywhere in nature [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]. The analysis on fractals was constructed using various techniques, such as harmonic analysis, probabilistic methods, measure theory, fractional calculus, fractional spaces, and time-scale calculus [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. ...
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In this article, the price adjustment equation has been proposed and studied in the frame of fractal calculus which plays an important role in market equilibrium. Fractal time has been recently suggested by researchers in physics due to the self-similar properties and fractional dimension. We investigate the economic models from the viewpoint of local and non-local fractal Caputo derivatives. We derive some novel analytical solutions via the fractal Laplace transform. In fractal calculus, a useful local fractal derivative is a generalized local derivative in the standard computational sense, and the non-local fractal Caputo fractal derivative is a generalization of the non-local fractional Caputo derivative. The economic models involving fractal time provide a new framework that depends on the dimension of fractal time. The suggested fractal models are considered as a generalization of standard models that present new models to economists for fitting the economic data. In addition, we carry out a comparative analysis to understand the advantages of the fractal calculus operator on the basis of the additional fractal dimension of time parameter, denoted by α, which is related to the local derivative , and we also indicate that when this dimension is equal to 1, we obtain the same results in the standard fractional calculus as well as when α and the nonlocal memory effect parameter, denoted by γ, of the nonlocal fractal derivative are both equal to 1, we obtain the same results in the standard calculus. Keywords:Fractal calculus, the fractal market equation, the local fractal Laplace transform , the nonlocal fractal Laplace transform.
... For instance, we can consider the Renyi dimension (with special cases, such as the Minkowski dimension [9], Information dimension [15], Correlation dimension [16]), the Higuchi dimension [17], the Lyapunov dimension [18], Packing dimension [19], the Assouad dimension [20] and the generalization of the Hausdorff dimension [21]. Third, a more rigorous and comprehensive method is to investigate the existence problem of fractals for the generalized fractal space equipped with a fractal structure and the generalized fractal dimension [22]. Finally, the existence result in this work is limited to deterministic fractals constructed by their associated deterministic recursive processes. ...
Preprint
How many fractals exist in nature or the virtual world? In this work, 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 beth-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the givenLebesgue measure. The question remains unanswered for other fractal dimensions.
... The growth velocity was measured by analyzing the video recordings, and a subsequent box-counting method was used to determine the fractal dimension by a linear approximation of 13 measured values f dim = −log(N)/log (ε), which provides information about the complexity or fractality of a system [11]. All calculations were performed with Fiji [12]/FracLac (BoxCounting), and 13 different sized boxes (ε = 1-64 pixels) were used to scan the number of boxes (N) containing the PEDOT structure in a binary representation of the pictures. ...
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Controlling the growth of conductive polymers via electrolysis enables defined surface modifications and can be used as a rapid prototyping process. In this study, the controlled dendritic growth of poly(3,4-ethylenedioxythiophene) (PEDOT) in a two-electrode setup was investigated by pulsed voltage-driven electropolymerization of the precursor EDOT and a low concentration of tetrabutylammonium perchlorate dissolved in acetonitrile. Rapid growth of different polymeric shapes was reliably achieved by varying the reduction voltage and duty factor. The obtained structures were optically examined and quantified using fractal dimensions. Their shapes ranged from solid coatings over branched fractals to straight fibers without requiring any template. These rapid and controllable electropolymerization processes were further combined to increase conductor complexity.
... Sierpinski triangle fractal dimension is equal to 1.58 and Koch fractal dimension is 1.26 [13]. Fractal dimension is calculated with the following equation [14]: ...
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Fractal shapes has unusual properties. These unique features will affect antenna parameters when designed in fractal shapes. Two fractal shapes combined together to generate new fractal shape dipole antenna. Seirpinski and modified Koch fractal shapes allow this antenna to operate at too far apart frequencies lies in X and K band. Fractal dimension of modified Koch is found to be 1.08 which led the antenna to be electrically small. This is explaining the resonant points at higher frequencies. Uniting X and K band in single antenna will make the possibility of combining the applications of these two bands in one device. Good results have been obtained from calculating antenna parameters.
... Sierpinski triangle fractal dimension is equal to 1.58 and Koch fractal dimension is 1.26 [13]. Fractal dimension is calculated with the following equation [14]: ...
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Fractal shapes has unusual properties. These unique features will affect antenna parameters when designed in fractal shapes. Two fractal shapes combined together to generate new fractal shape dipole antenna. Seirpinski and modified Koch fractal shapes allow this antenna to operate at too far apart frequencies lies in X and K band. Fractal dimension of modified Koch is found to be 1.08 which led the antenna to be electrically small. This is explaining the resonant points at higher frequencies. Uniting X and K band in single antenna will make the possibility of combining the applications of these two bands in one device. Good results have been obtained from calculating antenna parameters.
... Below, the natural fractal structure on each 55 Euclidean space R d , which stands as a particular case 56 from Definition 3.1 in [7], is described. ...
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... The structural properties of the ferrite powder were investigated by X-ray diffraction (XRD; Proker D8) with CuK α radiation of wavelength 1.5481Åat room temperature. The fractal dimensions were calculated using the box counting method [27,28]. Highquality images of the ferrite tree are taken and converted to monochromatic contours in order to calculate the fractal dimension using box counting method [29]. ...
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We report a tree fractal growth of ferrite nanoparticles prepared by Citrate-Gel Auto-Combustion method. We compared the growth pattern of CuFe2O4, Cr2FeO4, CdFe2O4, MgFe2O4, and Li2Fe3O5. The ferrite 3D growth was found to follow Family-Vicsek fractal growth in which the next added particle is looking for the best 3D orientation to minimize its surface free energy. The nanoparticles position in the sites of the growing tree forms a pattern that depends on the temperature, particle size, the orientation of the first seed particle and the particle to particle interaction forces. The results showed that the fractal arrangement is more preferred in the thermal growth of nanoparticles.
... Next, we theoretically connect fractal dimension III with classical de nitions of fractal dimension and fractal dimension II (c.f. [8,Section 4]), as well. ...
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Previous works have highlighted the suitability of the concept of fractal structure, which derives from asymmetric topology, to propound generalized definitions of fractal dimension. The aim of the present article is to collect some results and approaches allowing to connect the self-similarity index and the fractal dimension of a broad spectrum of random processes. To tackle with, we shall use the concept of induced fractal structure on the image set of a sample curve. The main result in this paper states that given a sample function of a random process endowed with the induced fractal structure on its image, it holds that the self-similarity index of that function equals the inverse of its fractal dimension.
... The fractal dimension models for a fractal structure involved along this paper, i.e., fractal dimensions III and IV, have been explored in previous works by the authors (c.f. [9,11]) and can be considered as subsequent models from those studied in [10]. It is worth pointing out that they allowed to generalize both box dimension (c.f. ...
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Chapter
This paper describes the selection of parameters of an Evolutionary Algorithm (EA) suitable for optimising the genotype of a fractal model of phenotypically realistic structures. To achieve the proposed goal an EA is implemented as a metaheuristic search tool to find the coefficients of the transformation matrices of an Iterated Function System (IFS) which then generates regular fractal patterns. Fractal patterns occur throughout nature, a striking example being the fern patterns modelled by Barnsley. Thus the algorithm is evaluated using the IFS for the fern fractal using the EA-evolved parameters.
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The complex microstructure of the surrounding porous rock of a tunnel has a significant influence on tunnel leakage. However, traditional tunnel leakage models are rarely able to quantify the interactions between the microstructure and macroscopic parameters of the surrounding porous rock of a tunnel, and the influence of the microstructure evolution on tunnel leakage. In order to quantify the contribution of the surrounding rock microstructure to leakage, a model capable of investigating the interaction between the evolution of microstructure and tunnel leakage behavior under the action of multi-physical fields is established based on fractal theory. The model couples the evolution of microstructure with the stress effect, adsorption–desorption effect, water pressure effect and surrounding rock deformation during tunnel leakage. The accuracy of the model is verified against experimental results, and the contribution of the surrounding porous rock pore size and pore distribution to tunnel water leakage is investigated. This study provides a theoretical basis and technical guidance for a reasonably accurate study of the tunnel leakage mechanism.
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We give a generalization of Mersenne hybrid numbers. We find the Binet formula, the generating function, the sum, the character, the norm and the vector representation of the generalizition. We obtain some relations among this generalizition and well known hybrid numbers. Then we present some important identities, Cassini, Catalan, Vajda, D’ocagne, Honsberger for the generalizition.
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Coastlines are irregular in nature having (random) fractal geometry and are formed by various natural activities. Fractal dimension is a measure of degree of geometric irregularity present in the coastline. A novel multicore parallel processing algorithm is presented to calculate the fractal dimension of coastline of Australia. The reliability of the coastline length of Australia is addressed by recovering the power law from our computational results. For simulations, the algorithm is implemented on a parallel computer for multi-core processing using the QGIS software, R -programming language and Python codes.
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The mining of stratigraphically low coal seams in North China-type coalfields is subject to water inrush from the underlying Ordovician limestone aquifer. The water-inrush coefficient method that is currently used for the evaluation of the water-inrush risk has inherent shortcomings, because it takes into account only the aquifer head pressure and the aquiclude thickness. Therefore, an improved water-inrush coefficient (IWIC) model is proposed. Based on the normalized water-inrush parameter, water-resisting parameter and structural parameter, the IWIC model is established using a linear weighting method. The first-order weights of each parameter are determined by the analytic hierarchy process, and the second-order weights are determined by the trapezoidal fuzzy number technique. Contour maps of the water-inrush risk calculated with the IWIC model are then obtained. The water-inrush risk grades are classified by thresholds derived via the Jenks natural breaks technique. The IWIC model is applied to the Longgu coal mine, as a typical coal mine in China, to evaluate the water-inrush risk of the lower four coal seams (L4CS). The evaluation results show that the risk of water inrush in the L4CS can be divided into five grades: safe, slightly safe, slightly dangerous, dangerous, and extremely dangerous. Overall, the L4CS mining in the Longgu coal mine is seriously threatened by the underlying Ordovician limestone aquifer. As the depth increases, the risk of water inrush increases from the No. 151 to No. 182 coal seams. Among the L4CS, No. 17 and No. 182 have the highest grade of water-inrush risk, and it is proposed that these two coal seams should not be mined to prevent water-inrush accidents.
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Along this talk, we shall deal with a classical problem in Fractal Geometry consisting of the calculation of the similarity dimension of self-similar sets. Clasically, the open set condition has been understood as the right separation condition for IFS-attractors since it becomes a sufficient (though not necessary) condition allowing to easily calculate their similarity dimensions. However, it depends on an external open set. Our contribution consists of a novel separation condition for self-similar sets we shall characterize in terms of the natural fractal structure which any IFS-attractor can be endowed with. We justify that such a separation condition is weaker than the strong open set condition and allows to prove some Moran’s type theorems. For additional details, we refer the reader to M.A. Sánchez-Granero and M. Fernández-Martínez, "Irreducible fractal structures for Moran's type theorems".
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One of the milestones in Fractal Geometry is the so-called Moran’s Theorem, which allows the calculation of the similarity dimension of any strict self-similar set under the open set condition. In this paper, we contribute a generalized version of the Moran’s theorem, which does not require the $$\mathrm{OSC}$$ to be satisfied by the similitudes that give rise to the corresponding attractor. To deal with, two generalized versions for the classical fractal dimensions, namely, the box and the Hausdorff dimensions, are explored in terms of fractal structures, a kind of uniform spaces.
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In this paper we use fractal structures to study self-similar sets and self-similar symbolic spaces. We show that these spaces have a natural fractal structure, justifying the name of fractal structure, and we characterize self-similar symbolic spaces in terms of fractal structures. We prove that self-similar symbolic spaces can be characterized in a similar way, in the form, to the definition of classical self-similar sets by means of iterated function systems. We also study when a self-similar symbolic space is a self-similar set. Finally, we study relations between fractal structures with “pieces” homeomorphic to the space and different concepts of self-homeomorphic spaces. Along the paper, we propose several methods in order to construct self-similar sets and self-similar symbolic spaces from a geometrical approach. This allows to construct these kind of spaces in a very easy way.
Chapter
"...a blend of erudition (fascinating and sometimes obscure historical minutiae abound), popularization (mathematical rigor is relegated to appendices) and exposition (the reader need have little knowledge of the fields involved) ...and the illustrations include many superb examples of computer graphics that are works of art in their own right." Nature
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We present some results concerning fractals generated by an iterated function system in the infinite dimensional space of continuous functions on a compact interval. Namely, we approximate the fractal via a finite approximant set and project this approximant set in two dimensions, in order to “draw” a picture of it.
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Facial feature extraction is an important step in many applications such as human face recognition, video conferencing, surveillance systems, human computer interfacing etc. The eye is the most important facial feature. A reliable and fast method for locating the eye pairs in an image is vital to many practical applications. A new method for locating eye pairs based on valley field detection and measurement of fractal dimensions is proposed. Possible eye candidates in an image with a complex background are identified by valley field detection. The eye candidates are then grouped to form eye pairs if their local properties for eyes are satisfied. Two eyes are matched if they have similar roughness and orientation as represented by fractal dimensions. A modified approach to estimating fractal dimensions that is less sensitive to lighting conditions and provides information about the orientation of an image under consideration is proposed. Possible eye pairs are further verified by comparing the fractal dimensions of the eye-pair window and the corresponding face region with the respective means of the fractal dimensions of the eye-pair windows and the face regions. The means of the fractal dimensions are obtained based on a number of facial images in a database. Experiments have shown that this approach is fast and reliable
Quasi-uniformities: Reconciling domains with metric spaces Mathematical Foundations of Programming Language Semantics, 3rd Workshop Current address: Area of Geometry and Topology, Faculty of Science, Universidad de Almería, 04071 Almería, Spain E-mail address: misanche@ual.es and fmm124@ual
• M B Smyth
M.B. Smyth, Quasi-uniformities: Reconciling domains with metric spaces,, in: M. Main, et al. (Eds.), Mathematical Foundations of Programming Language Semantics, 3rd Workshop, Tulane, 1987, in: Lecture Notes Computer Science, vol. 298, Springer, Berlin, 1988, 236-253. Current address: Area of Geometry and Topology, Faculty of Science, Universidad de Almería, 04071 Almería, Spain E-mail address: misanche@ual.es and fmm124@ual.es
Sánchez-Granero, archimedeanly quasimetrizable spaces
• F G Arenas
F.G. Arenas and M.A. Sánchez-Granero, archimedeanly quasimetrizable spaces, Rend. Istit. Mat. Univ. Trieste, Suppl. Vol. XXX (1999), pp. 21-30.
• K Falconer
• Fractal Geometry
K. Falconer, Fractal Geometry. Mathematical Foundations and Applications, John Wiley & Sons, 1990.
Thus, there exists a non-empty bounded open subset O ⊆ R d such that i∈I f i (O) ⊂ O, with f i (O) ∩ f j (O) = ∅ for all i, j ∈ I such that i = j. Moreover, O ∩ K = ∅, so we can take x ∈ O ∩ K ⊂ O. Since O is an open set, then there exists ε > 0 such that B := B(x, ε) ⊂ O. n (B) = ∅
• Proof
• First
Proof. First, since K ⊆ R d is a self-similar set, then we have that the open set condition is equivalent to the strong open set condition (see Remark 4.17). Thus, there exists a non-empty bounded open subset O ⊆ R d such that i∈I f i (O) ⊂ O, with f i (O) ∩ f j (O) = ∅ for all i, j ∈ I such that i = j. Moreover, O ∩ K = ∅, so we can take x ∈ O ∩ K ⊂ O. Since O is an open set, then there exists ε > 0 such that B := B(x, ε) ⊂ O. n (B) = ∅, with ω n, u n ∈ I n such that ω n = u n. Thus, let us show it for n + 1. Indeed, let ω n+1, u n+1 ∈ I n+1. Then we can distinguish the two following cases: