scholarly journals After notes on self-similarity exponent for fractal structures

Open Physics ◽  
2017 ◽  
Vol 15 (1) ◽  
pp. 440-448 ◽  
Author(s):  
Manuel Fernández-Martínez ◽  
Manuel Caravaca Garratón
Keyword(s):  
Fractal Dimension ◽  
Broad Spectrum ◽  
Fractal Structure ◽  
Similarity Index ◽  
Random Processes ◽  
The Self ◽  
Sample Function ◽  
Fractal Structures ◽  
Self Similarity ◽  
Image Set

AbstractPrevious 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.

Fractal Characteristic of Electrical Trees Grown in Silicone Rubber under Environmental Stress

2017 ◽  
Vol 7 (3) ◽  
pp. 1628
Author(s):  
M. S. Mohd Fua’ad ◽  
M.H. Ahmad ◽  
Y. Z. Arief ◽  
N. A. Ahmad
Keyword(s):  
Fractal Dimension ◽  
Fractal Structure ◽  
The Self ◽  
Fractal Structures ◽  
Self Similarity ◽  
Fractal Characteristic ◽  
Electrical Tree ◽  
Breakdown Phenomenon ◽  
Electrical Trees ◽  
Electrical Treeing

<div class="WordSection1"><p>One of the degradations of insulation is in the form of electrical treeing in which classified as a pre-breakdown phenomenon of electrical insulation. The electrical tree is commonly forming in the shape of tree-like or root-like which may have fractal structures. Due to this fractal structure, electrical treeing formation and patterns are analysed via fractal dimension and lacunarity to study the self-similarity patterns of electrical treeing. Many types of research have been conducted to study the fractal dimension and lacunarity of electrical treeing to fully understand the electrical tree mechanism and characteristics. However, fractal and lacunarity structures of</p></div>

SELF-SIMILARITY IN COMPLEX NETWORKS: FROM THE VIEW OF THE HUB REPULSION

2013 ◽  
Vol 27 (28) ◽  
pp. 1350201 ◽  
Author(s):  
HAIXIN ZHANG ◽  
XIN LAN ◽  
DAIJUN WEI ◽  
SANKARAN MAHADEVAN ◽  
YONG DENG
Keyword(s):  
Fractal Dimension ◽  
Complex Systems ◽  
Complex Networks ◽  
Fractal Structure ◽  
The Self ◽  
New Method ◽  
Self Similarity ◽  
Nature And Society ◽  
Real Complex

Complex networks are widely used to model the structure of many complex systems in nature and society. Recently, fractal and self-similarity of complex networks have attracted much attention. It is observed that hub repulsion is the key principle that leads to the fractal structure of networks. Based on the principle of hub repulsion, the metric in complex networks is redefined and a new method to calculate the fractal dimension of complex networks is proposed in this paper. Some real complex networks are investigated and the results are illustrated to show the self-similarity of complex networks.

scholarly journals A Community-Structure-Based Method for Estimating the Fractal Dimension, and its Application to Water Networks for the Assessment of Vulnerability to Disasters

2021 ◽  
Vol 35 (4) ◽  
pp. 1197-1210
Author(s):  
C. Giudicianni ◽  
A. Di Nardo ◽  
R. Greco ◽  
A. Scala
Keyword(s):  
Fractal Dimension ◽  
Community Structure ◽  
Distribution Systems ◽  
Water Distribution ◽  
Scaling Law ◽  
Vulnerability Index ◽  
The Self ◽  
Node Degree ◽  
Specific Response ◽  
Self Similarity

AbstractMost real-world networks, from the World-Wide-Web to biological systems, are known to have common structural properties. A remarkable point is fractality, which suggests the self-similarity across scales of the network structure of these complex systems. Managing the computational complexity for detecting the self-similarity of big-sized systems represents a crucial problem. In this paper, a novel algorithm for revealing the fractality, that exploits the community structure principle, is proposed and then applied to several water distribution systems (WDSs) of different size, unveiling a self-similar feature of their layouts. A scaling-law relationship, linking the number of clusters necessary for covering the network and their average size is defined, the exponent of which represents the fractal dimension. The self-similarity is then investigated as a proxy of recurrent and specific response to multiple random pipe failures – like during natural disasters – pointing out a specific global vulnerability for each WDS. A novel vulnerability index, called Cut-Vulnerability is introduced as the ratio between the fractal dimension and the average node degree, and its relationships with the number of randomly removed pipes necessary to disconnect the network and with some topological metrics are investigated. The analysis shows the effectiveness of the novel index in describing the global vulnerability of WDSs.

A conservation law, entropy principle and quantization of fractal dimensions in hadron interactions

2018 ◽  
Vol 33 (10) ◽  
pp. 1850057 ◽  
Author(s):  
I. Zborovský
Keyword(s):  
Conservation Law ◽  
Fractal Structure ◽  
Fractal Dimensions ◽  
The Self ◽  
Hadron Structure ◽  
Self Similarity ◽  
Similarity Variable ◽  
Variable Formula ◽  
Entropy Principle ◽  
Fragmentation Processes

Fractal self-similarity of hadron interactions demonstrated by the [Formula: see text]-scaling of inclusive spectra is studied. The scaling regularity reflects fractal structure of the colliding hadrons (or nuclei) and takes into account general features of fragmentation processes expressed by fractal dimensions. The self-similarity variable [Formula: see text] is a function of the momentum fractions [Formula: see text] and [Formula: see text] of the colliding objects carried by the interacting hadron constituents and depends on the momentum fractions [Formula: see text] and [Formula: see text] of the scattered and recoil constituents carried by the inclusive particle and its recoil counterpart, respectively. Based on entropy principle, new properties of the [Formula: see text]-scaling concept are found. They are conservation of fractal cumulativity in hadron interactions and quantization of fractal dimensions characterizing hadron structure and fragmentation processes at a constituent level.

A NEW SPECTRAL–SPATIAL JOINTED HYPERSPECTRAL IMAGE CLASSIFICATION APPROACH BASED ON FRACTAL DIMENSION ANALYSIS

Fractals ◽  
2019 ◽  
Vol 27 (05) ◽  
pp. 1950079
Author(s):  
JUNYING SU ◽  
YINGKUI LI ◽  
QINGWU HU
Keyword(s):  
Fractal Dimension ◽  
Image Classification ◽  
Classification Accuracy ◽  
Hyperspectral Image ◽  
Spatial Domain ◽  
Spectral Domain ◽  
Hyperspectral Image Classification ◽  
Self Similarity ◽  
Classification Approach ◽  
Image Set

To maximize the advantages of both spectral and spatial information, we introduce a new spectral–spatial jointed hyperspectral image classification approach based on fractal dimension (FD) analysis of spectral response curve (SRC) in spectral domain and extended morphological processing in spatial domain. This approach first calculates the FD image based on the whole SRC of the hyperspectral image and decomposes the SRC into segments to derive the FD images with each SRC segment. These FD images based on the segmented SRC are composited into a multidimensional FD image set in spectral domain. Then, the extended morphological profiles (EMPs) are derived from the image set through morphological open and close operations in spatial domain. Finally, all these EMPs and FD features are combined into one feature vector for a probabilistic support vector machine (SVM) classification. This approach was demonstrated using three hyperspectral images in urban areas of the university campus and downtown area of Pavia, Italy, and the Washington DC Mall area in the USA, respectively. We assessed the potential and performance of this approach by comparing with PCA-based method in hyperspectral image classification. Our results indicate that the classification accuracy of our proposed method is much higher than the accuracies of the classification methods based on the spectral or spatial domain alone, and similar to or slightly higher than the classification accuracy of PCA-based spectral–spatial jointed classification method. The proposed FD approach also provides a new self-similarity measure of land class in spectral domain, a unique property to represent hyperspectral self-similarity of SRC in hyperspectral imagery.

scholarly journals Rock Mechanical Property Influenced by Inhomogeneity

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Yunliang Tan ◽  
Dongmei Huang ◽  
Ze Zhang
Keyword(s):  
Mechanical Property ◽  
Fractal Dimension ◽  
Rock Type ◽  
Fractal Dimensions ◽  
Rock Material ◽  
The Self ◽  
Self Similarity ◽  
Red Sandstone ◽  
Variation Patterns ◽  
Rock Microstructure

In order to identify the microstructure inhomogeneity influence on rock mechanical property, SEM scanning test and fractal dimension estimation were adopted. The investigations showed that the self-similarity of rock microstructure markedly changes with the scanned microscale. Different rocks behave in different fractal dimension variation patterns with the scanned magnification, so it is conditional to adopt fractal dimension to describe rock material. Grey diabase and black diabase have high suitability; red sandstone has low suitability. The suitability of fractal-dimension-describing method for rocks depends on both investigating scale and rock type. The homogeneities of grey diabase, black diabase, grey sandstone, and red sandstone are 7.8, 5.7, 4.4, and 3.4, separately; their average fractal dimensions of microstructure are 2.06, 2.03, 1.72, and 1.40 correspondingly, so the homogeneity is well consistent with fractal dimension. For rock material, the stronger brittleness is, the less profile fractal dimension is. In a sense, brittleness is an image of rock inhomogeneity in macroscale, while profile fractal dimension is an image of rock inhomogeneity in microscale. To combine the test of brittleness with the estimation of fractal dimension with condition will be an effective approach for understanding rock failure mechanism, patterns, and behaviours.

scholarly journals A COMPARISON OF THREE HURST EXPONENT APPROACHES TO PREDICT NASCENT BUBBLES IN S&P500 STOCKS

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1750006 ◽  
Author(s):  
M. FERNÁNDEZ-MARTÍNEZ ◽  
M. A. SÁNCHEZ-GRANERO ◽  
M. J. MUÑOZ TORRECILLAS ◽  
BILL MCKELVEY
Keyword(s):  
Time Series ◽  
Hurst Exponent ◽  
Price Change ◽  
Similarity Index ◽  
The Self ◽  
Stock Index ◽  
Fluctuation Analysis ◽  
Market Behavior ◽  
Self Similarity ◽  
Efficient Market

Since the pioneer contributions due to Vandewalle and Ausloos, the Hurst exponent has been applied by econophysicists as a useful indicator to deal with investment strategies when such a value is above or below [Formula: see text], the Hurst exponent of a Brownian motion. In this paper, we hypothesize that the self-similarity exponent of financial time series provides a reliable indicator for herding behavior (HB) in the following sense: if there is HB, then the higher the price, the more the people will buy. This will generate persistence in the stocks which we shall measure by their self-similarity exponents. Along this work, we shall explore whether there is some connections between the self-similarity exponent of a stock (as a HB indicator) and the stock’s future performance under the assumption that the HB will last for some time. With this aim, three approaches to calculate the self-similarity exponent of a time series are compared in order to determine which performs best to identify the transition from random efficient market behavior to HB and hence, to detect the beginning of a bubble. Generalized Hurst Exponent, Detrended Fluctuation Analysis, and GM2 algorithms have been tested. Traditionally, researchers have focused on identifying the beginning of a crash. We study the beginning of the transition from efficient market behavior to a market bubble, instead. Our empirical results support that the higher (respectively the lower) the self-similarity index, the higher (respectively the lower) the mean of the price change, and hence, the better (respectively the worse) the performance of the corresponding stock. This would imply, as a consequence, that the transition process from random efficient market to HB has started. For experimentation purposes, S&P500 stock Index constituted our main data source.

A Modified Box-Counting Method to Estimate the Fractal Dimensions

2011 ◽  
Vol 58-60 ◽  
pp. 1756-1761 ◽  
Author(s):  
Jie Xu ◽  
Giusepe Lacidogna
Keyword(s):  
Fractal Dimension ◽  
Fractal Dimensions ◽  
The Self ◽  
Whole Body ◽  
Counting Method ◽  
Border Effect ◽  
Self Similarity ◽  
Box Counting ◽  
Self Similar ◽  
Box Counting Method

A fractal is a property of self-similarity, each small part of the fractal object is similar to the whole body. The traditional box-counting method (TBCM) to estimate fractal dimension can not reflect the self-similar property of the fractal and leads to two major problems, the border effect and noninteger values of box size. The modified box-counting method (MBCM), proposed in this study, not only eliminate the shortcomings of the TBCM, but also reflects the physical meaning about the self-similar of the fractal. The applications of MBCM shows a good estimation compared with the theoretical ones, which the biggest difference is smaller than 5%.

SELF SIMILARITY IN SWELLING SYSTEMS: FRACTAL PROPERTIES OF PEAT

Fractals ◽  
1994 ◽  
Vol 02 (01) ◽  
pp. 45-52 ◽  
Author(s):  
A. V. NEIMARK ◽  
E. ROBENS ◽  
K. K. UNGER ◽  
Yu. M. VOLFKOYICH
Keyword(s):  
Fractal Dimension ◽  
Water Adsorption ◽  
The Self ◽  
Self Similarity ◽  
Sphagnum Peat ◽  
Fractal Properties ◽  
Wide Range ◽  
Capillary Equilibrium ◽  
Scale Range ◽  
Self Similar

Sphagnum peat gives an example of a swelling system with a self-similar structure in sufficiently wide range of scales. The surface fractal dimension, dfs, has been calculated by means of thermodynamic method on the basis of water adsorption and capillary equilibrium measurements. This method makes possible the exploration of the self-similarity in the scale range over at least 4 decimal orders of magnitude from 1 nm to 10 μm. In a sample explored, two ranges of fractality have been observed: dfs ≈ 2.55 in the range 1.5–80 nm and dfs ≈ 2.42 in the range 0.25–9 µm.

scholarly journals On the self-similarity index of 𝑝-adic analytic pro-𝑝 groups

2022 ◽  
Vol 0 (0) ◽  
Author(s):  
Francesco Noseda ◽  
Ilir Snopce
Keyword(s):  
Rooted Tree ◽  
Similarity Index ◽  
The Self ◽  
Self Similarity ◽  
Similar Action ◽  
3 Dimensional ◽  
Self Similar

Abstract Let 𝑝 be a prime. We say that a pro-𝑝 group is self-similar of index p k p^{k} if it admits a faithful self-similar action on a p k p^{k} -ary regular rooted tree such that the action is transitive on the first level. The self-similarity index of a self-similar pro-𝑝 group 𝐺 is defined to be the least power of 𝑝, say p k p^{k} , such that 𝐺 is self-similar of index p k p^{k} . We show that, for every prime p ⩾ 3 p\geqslant 3 and all integers 𝑑, there exist infinitely many pairwise non-isomorphic self-similar 3-dimensional hereditarily just-infinite uniform pro-𝑝 groups of self-similarity index greater than 𝑑. This implies that, in general, for self-similar 𝑝-adic analytic pro-𝑝 groups, one cannot bound the self-similarity index by a function that depends only on the dimension of the group.

Sign in / Sign up

Export Citation Format

Share Document