scholarly journals A new fractal dimension: The topological Hausdorff dimension

2015 ◽  
Vol 274 ◽  
pp. 881-927 ◽  
Author(s):  
Richárd Balka ◽  
Zoltán Buczolich ◽  
Márton Elekes
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension

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 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.

scholarly journals Adsorption on Fractal Surfaces: A Non Linear Modeling Approach of a Fractional Behavior

2021 ◽  
Vol 5 (3) ◽  
pp. 65
Author(s):  
Vincent Tartaglione ◽  
Jocelyn Sabatier ◽  
Christophe Farges
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Nonlinear Model ◽  
Fractional Dynamics ◽  
Random Sequential Adsorption ◽  
Linear Modeling ◽  
High Coverage ◽  
Fractal Surfaces ◽  
Affine Model ◽  
Sequential Adsorption

This article deals with the random sequential adsorption (RSA) of 2D disks of the same size on fractal surfaces with a Hausdorff dimension 1<d<2. According to the literature and confirmed by numerical simulations in the paper, the high coverage regime exhibits fractional dynamics, i.e., dynamics in t−1/d where d is the fractal dimension of the surface. The main contribution this paper is that it proposes to capture this behavior with a particular class of nonlinear model: a driftless control affine model.

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.

The 2-adic derivatives and fractal dimension of Takagi-like function on 2-series field

2020 ◽  
Vol 23 (3) ◽  
pp. 875-885
Author(s):  
Bo Wu
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Box Dimension ◽  
Derivatives Of

AbstractIn this paper, we consider a Takagi-like function on 2-series field and give its 2-adic derivatives by applying Vladimirov operator. The 2-adic derivatives of Takagi-like function with order 0 < α < 1 exist and show some fractal feature. Furthermore, both box dimension and Hausdorff dimension of the graph of its derivatives are obtained and equal to 1 + α.

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.

THE AVERAGE FRACTAL DIMENSION AND PROJECTIONS OF MEASURES AND SETS IN Rn

Fractals ◽  
1995 ◽  
Vol 03 (04) ◽  
pp. 747-754 ◽  
Author(s):  
M. ZÄHLE
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Average Density ◽  
Higher Dimension ◽  
Orthogonal Projections ◽  
Analytic Set ◽  
Average Dimension ◽  
Packing Dimensions ◽  
Almost All ◽  
Local Average

In this note we introduce the concept of local average dimension of a measure µ, at x∈ℝn as the unique exponent where the lower average density of µ, at x jumps from zero to infinity. Taking the essential infimum or supremum over x we obtain the lower and upper average dimensions of µ, respectively. The average dimension of an analytic set E is defined as the supremum over the upper average dimensions of all measures supported by E. These average dimensions lie between the corresponding Hausdorff and packing dimensions and the inequalities can be strict. We prove that the local Hausdorff dimensions and the local average dimensions of µ at almost all x are invariant under orthogonal projections onto almost all m- dimensional linear subspaces of higher dimension. The corresponding global results for µ and E (which are known for Hausdorff dimension) follow immediately.

Fractal dimension: Limit capacity or Hausdorff dimension?

1990 ◽  
Vol 58 (10) ◽  
pp. 986-988 ◽  
Author(s):  
Christopher Essex ◽  
M. A. H. Nerenberg
Keyword(s):  
Fractal Dimension ◽  
Hausdorff Dimension ◽  
Limit Capacity
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