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We conducted a longitudinal study involving 2 4 0 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. [ABSTRACT FROM AUTHOR]

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.

Porous materials are formed from dispersed particles of various shapes. The porous structure of such materials is determined by the shape and nature of the packing of their structural units. As a model for the analysis of the structure of porous materials, model-packing units were selected in the form of spherical particles and fibers. Separately, the pore structure in foam-like regular porous materials was considered. It is shown that the pore volume in such packages grows with the reduction of the coordination number of the packed spherical particles. To study this regularity, an analysis was made of the structural types of crystal lattices of various chemical substances. Examples of such packages are most metallic elements and all inert gases, with the exception of helium. These substances crystallize in simple structural types. All of them can be considered as packing of balls of identical radii. Such regularities are observed strictly up to the coordination number 4. With coordination number 3, a very openwork, loose package is formed. When forming mixed packages, the coordination number can have fractional values. A special group is represented by structures with coordination number <3. For such structures, an analysis was made of possible combinations of mixed structures with coordination numbers 3 and 2. On the basis of the analysis of the porosity data of various types of packages of spherical particles, an empirical relationship is obtained between the binding porosity and the coordination number of the corresponding package: α = α'∞ + B' / (n - n0) . An analysis of this dependence showed that there is a certain critical value of the average in the structure of the coordination number in the structures of disperse systems. Less than this value, there is no rigid structure of the framework of the material collected from individual particles. Below this coordination number, the particles are collected only into separate chain structures that are not related to each other, in which the coordination number 2. The task of analyzing the packing of fibrous particles is much more complicated than this analysis for spherical particles. The article presents data of computer modeling of the formation of such structures. An attempt was also made to model fibrous structures by analogy with an openwork packing of spherical particles. For such and other packing models, the corresponding empirical dependencies. Gel-like materials are characterized by openwork packing of particles forming a gel. They have a fractal and hierarchical nature of the structure. The value of the fractal dimension allows one to determine how the structure of the gel was organized. Analysis of the formation of possible structures in the sol-gel transition showed there is a limiting fractal dimension above which no other structure can be formed from a sol with given properties. This fractal dimension is the amount to which it tends to form a gel structure during the aggregation of the sol, and during the formation of the gel structure, after passing through the sol-gel transition point. An analysis is made of the structure of foam-like regular porous materials. This analysis showed that the state of foam with polyhedral cells is close to the equilibrium state, so these foams have greater stability than foams with spherical cells. Therefore, for the convenience of description, a cellular model based on existing regular polyhedra was applied to these materials. Because of this model, the dependence of the fraction of the solid phase in a porous foam-like solid material, constructed on the basis of cells having the form of various regular polyhedra. The densest structures are formed in the case of using tetrahedral and octahedral cells. The intermediate position is occupied by cubic cells representing a regular three-dimensional network of regular mutually perpendicular columns and bridges. [ABSTRACT FROM AUTHOR]

Using models of thermoassociated polymers, a cellular model of associated solutions is developed for cases where molecules of a solute are able to form from 2 to 6 bonds with each other. It is shown that solutions associated by weak intermolecular interactions in a certain range of concentrations can be considered as systems of flickering pseudo-polymeric macromolecules with a fractal structure. [ABSTRACT FROM AUTHOR]

In this paper, we explore the fractal dimension of Cone Beam Computed Tomography images to analyze the trabecular bone structure of healthy subjects. That quantity, computed throughout three distinct approaches, provided us accurate values of normality concerning the radiographic density of this kind of bones and will allow us to establish comparisons with respect to the fractal dimension from patients with different pathologies that may affect the density of trabecular bones. [ABSTRACT FROM AUTHOR]

In this paper, we highlight an intelligent system to properly construct a function between a pair of generalized-fractal spaces: the d-cube [0, 1] d endowed with its natural fractal structure, and the closed unit interval endowed with a natural-like fractal structure. Such a function allows the definition of space-filling curves by levels as well as the calculation of the box dimension of higher dimensional subsets through the box dimension of subsets on the real line. [ABSTRACT FROM AUTHOR]

In this study, we examined the fractal structure of the Nikkei225, HangSeng, Shanghai Stock Exchange and Straits Times Index of Singapore. Empirical analysis was performed via non-parametric, semi-parametric long memory tests and also fractal dimension calculations. In order to avoid spurious long memory features, besides the Detrended Fluctuations Analysis (DFA), we also used Smith’s (2005) modified GPH method. As for fractal dimension calculations, they were conducted via Box-Counting and Variation tests. According to the results, while there is no long memory property in log returns of any index, we found evidence for long memory properties in the volatility of the HangSeng, the Shanghai Stock Exchange and the Straits Times Index. However, we could not find any sign of long memory in the volatility of Nikkei225 index using either the DFA or modified GPH test. Fractal dimension analysis also demonstrated that all raw index prices have fractal structure properties except for the Nikkei225 index. These findings showed that the Nikkei225 index has the most efficient market properties among these markets.

Moran's Theorem is one of the milestones in Fractal Geometry. It allows the calculation of the similarity dimension of any (strict) self-similar set lying under the open set condition. Throughout a newfractal dimension we provide in the context of fractal structures, we generalize such a classical result for attractors which are required to satisfy no separation properties. [ABSTRACT FROM AUTHOR]

A fractal structure is a countable family of coverings which displays accurate information about the irregularities that a set presents when being explored with enough level of detail. It is worth noting that fractal structures become especially appropriate to provide new definitions of fractal dimension, which constitutes a valuablemeasure to test for chaos in dynamical systems. In this paper, we explore several approaches to calculate the fractal dimension of a subset with respect to a fractal structure. These models generalize the classical box dimension in the context of Euclidean subspaces from a discrete viewpoint. To illustrate the flexibility of the new models, we calculate the fractal dimension of a family of self-affine sets associated with certain discrete dynamical systems. [ABSTRACT FROM AUTHOR]

The main goal in this paper was to provide a novel chaos indicator based on a topological model which allows to calculate the fractal dimension of any curve. A fractal structure is a topological tool whose recursiveness becomes ideal to generalize the concept of fractal dimension. In this paper, we provide an algorithm to calculate a new fractal dimension specially designed for a parametrization of a curve or a random process, whose definition is made by means of fractal structures. As an application, we explore the use of this new concept of fractal dimension as a chaos indicator for dynamical systems, in a similar way to the classical maximal Lyapunov exponent. To illustrate it, we apply the new fractal dimension as an indicator to model the chaotic behavior of a satellite which is moving around a planet whose gravity field is approximated by the field of a point mass. [ABSTRACT FROM AUTHOR]

In this paper, we explain how to generate adequate pre-fractals in order to properly approximate attractors of iterated function systems on the real line within a priori known Hausdor dimension. To deal with, we have applied the classical Moran's Theorem, so we have been focused on nonoverlapping strict self-similar sets. This involves a quite significant hypothesis: the so-called open set condition. The main theoretical result contributed in this paper becomes quite interesting from a computational point of view, since in such a context, there is always a maximum level (of the natural fractal structure we apply in this work) that may be achieved. [ABSTRACT FROM AUTHOR]

Hausdor dimension, which is the oldest and also the most accurate model for fractal dimension, constitutes the main reference for any fractal dimension definition that could be provided. In fact, its definition is quite general, and is based on a measure, which makes the Hausdor model pretty desirable from a theoretical point of view. On the other hand, it turns out that fractal structures provide a perfect context where a new definition of fractal dimension could be proposed. Further, it has been already shown that both Hausdor and box dimensions can be generalized by some definitions of fractal dimension formulated in terms of fractal structures. Given this, and being mirrored in some of the properties satisfied by Hausdor dimension, in this paper we explore which ones are satisfied by the fractal dimension definitions for a fractal structure, that are explored along this work. [ABSTRACT FROM AUTHOR]