Fractal structure and fractal dimension determination at nanometer scale

Dystrophic lakes undergo natural disharmonic succession, in the course of which an increasingly complex and diverse, mosaic-like pattern of habitats evolves. In the final seral stage, the most important role is played by a spreading Sphagnum mat, which gradually reduces the lake’s open water surface area. Long-term transformations in the primary structure of lakes cause changes in the structure of lake-dwelling fauna assemblages. Knowledge of the succession mechanisms in lake fauna is essential for proper lake management. The use of fractal concepts helps to explain the character of fauna in relation to other aspects of the changing complexity of habitats. Our 12-year-long study into the succession of water beetles has covered habitats of 40 selected lakes which are diverse in terms of the fractal dimension. The taxonomic diversity and density of lake beetles increase parallel to an increase in the fractal dimension. An in-depth analysis of the fractal structure proved to be helpful in explaining the directional changes in fauna induced by the natural succession of lakes. Negative correlations appear between the body size and abundance. An increase in the density of beetles within the higher dimension fractals is counterbalanced by a change in the size of individual organisms. As a result, the biomass is constant, regardless of the fractal dimension.

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Charge accumulation in thunderstorm clouds: fractal dynamic model

E3S Web of Conferences ◽

10.1051/e3sconf/201912701001 ◽

2019 ◽

Vol 127 ◽

pp. 01001 ◽

Cited By ~ 1

Author(s):

Tembulat Kumykov

Keyword(s):

Fractal Dimension ◽

Comparative Study ◽

Dynamic Model ◽

Model Equation ◽

Fractal Structure ◽

Analytic Solution ◽

Numerical Calculations ◽

Charge Accumulation ◽

Dynamic Charge ◽

The Relationship

The paper considers a fractal dynamic charge accumulation model in thunderstorm clouds in view of the fractal dimension. Analytic solution to the model equation has been found. Using numerical calculations we have shown the relationship between the charge accumulation and the medium with the fractal structure. A comparative study of thunderstorm electrification mechanisms have been performed.

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ON THE DYNAMICAL EVOLUTION OF THE FRACTAL STRUCTURE OF INTERSTELLAR MEDIUM

Fractals ◽

10.1142/s0218348x93000952 ◽

1993 ◽

Vol 01 (04) ◽

pp. 908-916 ◽

Cited By ~ 1

Author(s):

Z.Y. YUE ◽

B. ZHANG ◽

G. WINNEWISSER ◽

J. STUTZKI

Keyword(s):

Fractal Dimension ◽

Interstellar Medium ◽

Density Distribution ◽

Fractal Structure ◽

Initial Conditions ◽

Initial Density ◽

Dynamical Evolution ◽

Fractal Dimensions ◽

Compressible Turbulence ◽

Initial States

Two-dimensional compressible turbulence in a self-gravitating, magnetic interstellar medium is calculated as an initial value problem. It is shown that even if the initial density distribution is homogeneous and the initial velocity distribution contains only a few Fourier components, the nonlinear interaction among the Fourier components will generate more and more Fourier components and lead to a turbulent and fractal structure in the interstellar medium. The calculations are carried out for three different initial states. In order to see the time evolution, detailed density distributions and fractal dimensions of the density contours are calculated at three moments of time for each of the initial states. The results show that the fractal dimension remains almost the same (~1.4–1.5), although the detailed density distribution has changed considerably. The insensibility of the fractal dimension of density contours to both the initial conditions and the evolution time is in good agreement with observations of molecular clouds in the interstellar medium.

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FRACTAL STRUCTURE OF FINANCIAL HIGH FREQUENCY DATA

Fractals ◽

10.1142/s0218348x02001002 ◽

2002 ◽

Vol 10 (01) ◽

pp. 13-18 ◽

Cited By ~ 10

Author(s):

YOSHIAKI KUMAGAI

Keyword(s):

Time Series ◽

Fractal Dimension ◽

Transaction Costs ◽

Time Scale ◽

High Frequency ◽

Fractal Structure ◽

Extreme Values ◽

High Frequency Data ◽

Frequency Data ◽

Time Intervals

We propose a new method to describe scaling behavior of time series. We introduce an extension of extreme values. Using these extreme values determined by a scale, we define some functions. Moreover, using these functions, we can measure a kind of fractal dimension — fold dimension. In financial high frequency data, observations can occur at varying time intervals. Using these functions, we can analyze non-equidistant data without interpolation or evenly sampling. Further, the problem of choosing the appropriate time scale is avoided. Lastly, these functions are related to a viewpoint of investor whose transaction costs coincide with the spread.

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ON THE FRACTAL STRUCTURE OF ELECTRON AVALANCHES

Fractals ◽

10.1142/s0218348x9300099x ◽

1993 ◽

Vol 01 (04) ◽

pp. 939-946 ◽

Cited By ~ 1

Author(s):

Z. DONKÓ ◽

I. PÓCSIK

Keyword(s):

Fractal Dimension ◽

Power Law ◽

Inelastic Collision ◽

Fractal Structure ◽

Secondary Electrons ◽

Electron Multiplication ◽

Helium Gas ◽

Self Similar ◽

Collision Processes ◽

Electron Avalanches

The motion of electrons in helium gas in the presence of a homogeneous external electric field was studied. Moving between the two electrodes, the electrons participate in elastic and inelastic collision processes with gas atoms. In ionizing collisions, secondary electrons are also created and in this way self-similar electron avalanches build up. The statistical distribution of the fractal dimension and electron multiplication of electron avalanches was obtained based on the simulation of a large number of electron avalanches. The fractal dimension shows a power-law dependence on electron multiplication with an exponent of ≈0.33.

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The linkage between box-counting and geomorphic fractal dimensions in the fractal structure of river networks: the junction angle

Hydrology Research ◽

10.2166/nh.2020.082 ◽

2020 ◽

Vol 51 (6) ◽

pp. 1397-1408

Author(s):

Xianmeng Meng ◽

Pengju Zhang ◽

Jing Li ◽

Chuanming Ma ◽

Dengfeng Liu

Keyword(s):

Fractal Dimension ◽

Fractal Structure ◽

Fractal Dimensions ◽

River Networks ◽

Stream Length ◽

Box Counting ◽

Linear Relationships ◽

Box Counting Dimension ◽

Junction Angle

Abstract In the past, a great deal of research has been conducted to determine the fractal properties of river networks, and there are many kinds of methods calculating their fractal dimensions. In this paper, we compare two most common methods: one is geomorphic fractal dimension obtained from the bifurcation ratio and the stream length ratio, and the other is box-counting method. Firstly, synthetic fractal trees are used to explain the role of the junction angle on the relation between two kinds of fractal dimensions. The obtained relationship curves indicate that box-counting dimension is decreasing with the increase of the junction angle when geomorphic fractal dimension keeps constant. This relationship presents continuous and smooth convex curves with junction angle from 60° to 120° and concave curves from 30° to 45°. Then 70 river networks in China are investigated in terms of their two kinds of fractal dimensions. The results confirm the fractal structure of river networks. Geomorphic fractal dimensions of river networks are larger than box-counting dimensions and there is no obvious relationship between these two kinds of fractal dimensions. Relatively good non-linear relationships between geomorphic fractal dimensions and box-counting dimensions are obtained by considering the role of the junction angle.

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Fractal-Based Modeling and Spatial Analysis of Urban Form and Growth: A Case Study of Shenzhen in China

ISPRS International Journal of Geo-Information ◽

10.3390/ijgi9110672 ◽

2020 ◽

Vol 9 (11) ◽

pp. 672

Author(s):

Xiaoming Man ◽

Yanguang Chen

Keyword(s):

Fractal Dimension ◽

Urban Growth ◽

Urban Form ◽

Fractal Structure ◽

Southern China ◽

Growth Curves ◽

Northern China ◽

Logistic Function ◽

Urban Region ◽

The Common

Fractal dimension curves of urban growth can be modeled with sigmoid functions, including logistic function and quadratic logistic function. Different types of logistic functions indicate different spatial dynamics. The fractal dimension curves of urban growth in Western countries follow the common logistic function, while the fractal dimension growth curves of cities in northern China follow the quadratic logistic function. Now, we want to investigate whether other Chinese cities, especially cities in South China, follow the same rules of urban evolution and attempt to analyze the reasons. This paper is devoted to exploring the fractals and fractal dimension properties of the city of Shenzhen in southern China. The urban region is divided into four subareas using ArcGIS technology, the box-counting method is adopted to extract spatial datasets, and the least squares regression method is employed to estimate fractal parameters. The results show that (1) the urban form of Shenzhen city has a clear fractal structure, but fractal dimension values of different subareas are different; (2) the fractal dimension growth curves of all the four study areas can only be modeled by the common logistic function, and the goodness of fit increases over time; (3) the peak of urban growth in Shenzhen had passed before 1986 and the fractal dimension growth is approaching its maximum capacity. It can be concluded that the urban form of Shenzhen bears characteristics of multifractals and the fractal structure has been becoming better, gradually, through self-organization, but its land resources are reaching the limits of growth. The fractal dimension curves of Shenzhen’s urban growth are similar to those of European and American cities but differ from those of cities in northern China. This suggests that there are subtle different dynamic mechanisms of city development between northern and southern China.

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Fractal Features of Cracked Backfill Particle Size Distribution

Advanced Materials Research ◽

10.4028/www.scientific.net/amr.588-589.1894 ◽

2012 ◽

Vol 588-589 ◽

pp. 1894-1898

Author(s):

Yong Jian Zhu ◽

Dai Qiang Deng ◽

Ping Wang

Keyword(s):

Particle Size ◽

Fractal Dimension ◽

Particle Size Distribution ◽

Size Distribution ◽

Negative Correlation ◽

Fractal Structure ◽

Fine Grained ◽

Calculation Data ◽

Relationship Of

Based on the taking sample by geological drilling, combined with the fractal principle, analysis on the cracked backfill particle size of its fractal features and strength correlation. Even each backfill sand specimen particle size is difference, but calculation data shows that the particle size of each sand specimen has preferable fractal feature, the sand specimen particle size distribution has remarkable fractal structure by the linear fitted results of the sand specimens. The fractal relationship of strength and particle size distribution shows that with the increased of fractal dimension, the strength of backfill is decreased, that is to say there is negative correlation, the main cause is that the higher parameter D of the fractal dimension, the higher fine-grained content and more non-uniform of the particle size distribution, especially for the thinner full tailings, if properly increasing the content of slightly crude particles, the strength of backfill will be certainly improved to some extent.

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The Different Fractal Structure of Oxide Nanopowders Depending on Method of Production

Solid State Phenomena ◽

10.4028/www.scientific.net/ssp.271.124 ◽

2018 ◽

Vol 271 ◽

pp. 124-132 ◽

Cited By ~ 2

Author(s):

Vyacheslav V. Syzrantsev ◽

Ludmila S. Vikulina ◽

S.P. Bardakhanov ◽

Andrey V. Nomoev ◽

Natalya O. Kopanitsa ◽

...

Keyword(s):

Fractal Dimension ◽

Fractal Structure ◽

Structural Parameters ◽

Synthesis Method ◽

Qualitative And Quantitative ◽

Methods Of Synthesis ◽

Polymer Strength ◽

Internal Electron ◽

Method Of Production ◽

Different Characteristics

This study was undertaken to compare chemically identical nanoparticles that have been synthesed by different methods. The methodology applied allows the identification of different characteristics in the structure and surface parameters of nanoparticles. The study shows that the structural parameters of nanoparticles are to a great extend related to the conditions in which nanoparticles are formed. This is demonstrated through the comparison of three oxides and their different methods of synthesis. The results show that the method of synthesis defines the structure of the nanoparticles; the surface and qualitative and quantitative parameters of the crystaline phases, energy shifts and changes in the internal electron levels. The examples of nanoliquids and the associated polymer strength identify that the interaction of nanoparticles with the environment is also depends on the synthesis method. It is proposed that a fractal dimension may be used as a basic parameter to classify nanoparticles and predict the properties in their interaction with various media.

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MONTE CARLO STUDY OF THE XY-MODEL ON SIERPIŃSKI CARPET

International Journal of Modern Physics C ◽

10.1142/s0129183113500873 ◽

2014 ◽

Vol 25 (02) ◽

pp. 1350087 ◽

Cited By ~ 1

Author(s):

BOŽIDAR MITROVIĆ ◽

MICHELLE A. PRZEDBORSKI

Keyword(s):

Monte Carlo ◽

Fractal Dimension ◽

Infinite Order ◽

Fractal Structure ◽

Cluster Algorithm ◽

Monte Carlo Study ◽

Xy Model ◽

Sierpinski Carpet ◽

Bkt Transition ◽

Sierpiński Carpet

We have performed a Monte Carlo (MC) study of the classical XY-model on a Sierpiński carpet, which is a planar fractal structure with infinite order of ramification and fractal dimension 1.8928. We employed the Wolff cluster algorithm in our simulations and our results, in particular those for the susceptibility and the helicity modulus, indicate the absence of finite-temperature Berezinskii–Kosterlitz–Thouless (BKT) transition in this system.

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