scholarly journals Calculating Hausdorff Dimension in Higher Dimensional Spaces

Symmetry ◽  
2019 ◽  
Vol 11 (4) ◽  
pp. 564 ◽  
Author(s):  
Manuel Fernández-Martínez ◽  
Juan Luis García García Guirao ◽  
Miguel Ángel Sánchez-Granero
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Fractal Structure ◽  
General Setting ◽  
Higher Dimensional

In this paper, we prove the identity dim H(F) = d dim H(a?1(F)), where dim H denotesHausdorff dimension, F Rd, and a : [0, 1] ! [0, 1]d is a function whose constructive definition isaddressed from the viewpoint of the powerful concept of a fractal structure. Such a result standsparticularly from some other results stated in a more general setting. Thus, Hausdorff dimension ofhigher 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 calculationof the fractal dimension in applications by applying a procedure contributed by the authors inprevious works. It is also worth pointing out that our results generalize both Skubalska-Rafajłowiczand García-Mora-Redtwitz theorems.

A fractal process of hydrogen diffusion in a-Si:H with exponential energy distribution

2017 ◽  
Vol 31 (09) ◽  
pp. 1750060
Author(s):  
Harumi Hikita ◽  
Hirohisa Ishikawa ◽  
Kazuo Morigaki
Keyword(s):  
Brownian Motion ◽  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Energy Distribution ◽  
Exponential Distribution ◽  
Fractal Structure ◽  
Hydrogen Diffusion ◽  
Hydrogen Migration ◽  
Dimension Formula ◽  
Fractal Process

Hydrogen diffusion in a-Si:H with exponential distribution of the states in energy exhibits the fractal structure. It is shown that a probability [Formula: see text] of the pausing time [Formula: see text] has a form of [Formula: see text] ([Formula: see text]: fractal dimension). It is shown that the fractal dimension [Formula: see text]/[Formula: see text] ([Formula: see text]: hydrogen temperature, [Formula: see text]: a temperature corresponding to the width of exponential distribution of the states in energy) is in agreement with the Hausdorff dimension. A fractal graph for the case of [Formula: see text] is like the Cantor set. A fractal graph for the case of [Formula: see text] is like the Koch curves. At [Formula: see text], hydrogen migration exhibits Brownian motion. Hydrogen diffusion in a-Si:H should be the fractal process.

scholarly journals Effect of changes in the fractal structure of a littoral zone in the course of lake succession on the abundance, body size sequence and biomass of beetles

PeerJ ◽  
2018 ◽  
Vol 6 ◽  
pp. e5662
Author(s):  
Joanna Pakulnicka ◽  
Andrzej Zawal
Keyword(s):  
Fractal Dimension ◽  
Body Size ◽  
Fractal Structure ◽  
Lake Management ◽  
Open Water ◽  
Taxonomic Diversity ◽  
The Body ◽  
Water Beetles ◽  
Water Surface Area ◽  
Depth Analysis

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.

scholarly journals Charge accumulation in thunderstorm clouds: fractal dynamic model

2019 ◽  
Vol 127 ◽  
pp. 01001 ◽  
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.

ON THE MENDÈS FRANCE FRACTAL DIMENSION OF STRANGE ATTRACTORS: THE HÉNON CASE

2002 ◽  
Vol 12 (07) ◽  
pp. 1549-1563 ◽  
Author(s):  
M. PIACQUADIO ◽  
R. HANSEN ◽  
F. PONTA
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Strange Attractors ◽  
Planar Curves ◽  
Box Counting ◽  
Box Counting Dimension

We consider the Hénon attractor ℋ as a curve limit of continuous planar curves H(n), as n → ∞. We describe a set of tools for studying the Hausdorff dimension dim H of a certain family of such curves, and we adapt these tools to the particular case of the Hénon attractor, estimating its Hausdorff dimension dim H (ℋ) to be about 1.258, a number smaller than the usual estimates for the box-counting dimension of the attractor. We interpret this discrepancy.

THE CANTOR SET’S MULTI-FRACTAL SPECTRUM FORMED BY DIFFERENT PROBABILITY FACTORS IN MATHEMATICAL EXPERIMENT

Fractals ◽  
2017 ◽  
Vol 25 (01) ◽  
pp. 1750002 ◽  
Author(s):  
XUEZAI PAN ◽  
XUDONG SHANG ◽  
MINGGANG WANG ◽  
ZUO-FEI
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Statistical Physics ◽  
Computer Calculation ◽  
Fractal Dimensions ◽  
The Other ◽  
Mathematical Experiment ◽  
Maximal Width ◽  
Fractal Spectrum ◽  
Vertical Height

With the purpose of researching the changing regularities of the Cantor set’s multi-fractal spectrums and generalized fractal dimensions under different probability factors, from statistical physics, the Cantor set is given a mass distribution, when the mass is given with different probability ratios, the different multi-fractal spectrums and the generalized fractal dimensions will be acquired by computer calculation. The following conclusions can be acquired. On one hand, the maximal width of the multi-fractal spectrum and the maximal vertical height of the generalized fractal dimension will become more and more narrow with getting two probability factors closer and closer. On the other hand, when two probability factors are equal to 1/2, both the multi-fractal spectrum and the generalized fractal dimension focus on the value 0.6309, which is not the value of the physical multi-fractal spectrum and the generalized fractal dimension but the mathematical Hausdorff dimension.

Fractal structure and fractal dimension determination at nanometer scale

1999 ◽  
Vol 42 (9) ◽  
pp. 965-972 ◽  
Author(s):  
Yue Zhang ◽  
Qikai Li ◽  
Wuyang Chu ◽  
Chen Wang ◽  
Chunli Bai
Keyword(s):  
Fractal Dimension ◽  
Fractal Structure ◽  
Nanometer Scale

ON THE DYNAMICAL EVOLUTION OF THE FRACTAL STRUCTURE OF INTERSTELLAR MEDIUM

Fractals ◽  
1993 ◽  
Vol 01 (04) ◽  
pp. 908-916 ◽  
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.

FRACTAL STRUCTURE OF FINANCIAL HIGH FREQUENCY DATA

Fractals ◽  
2002 ◽  
Vol 10 (01) ◽  
pp. 13-18 ◽  
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.

FRACTAL DIMENSION OF A SPECIAL CONTINUOUS FUNCTION

Fractals ◽  
2018 ◽  
Vol 26 (04) ◽  
pp. 1850048 ◽  
Author(s):  
NING LIU ◽  
KUI YAO
Keyword(s):  
Fractal Dimension ◽  
Continuous Function ◽  
Hausdorff Dimension ◽  
Fractional Integral ◽  
Fractal Function

In this paper, we mainly construct a special fractal function defined on [Formula: see text] of unbounded variation by method of iteration, and prove that both Box and Hausdorff dimension of this function be 1. Further, we discuss the Riemann–Liouville fractional integral of this function. Finally, some numerical and graphic results are provided to characterize this special fractal function.

ON THE FRACTAL STRUCTURE OF ELECTRON AVALANCHES

Fractals ◽  
1993 ◽  
Vol 01 (04) ◽  
pp. 939-946 ◽  
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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