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.

CONSTRUCTING CROSSOVER-FRACTALS USING INTRINSIC MODE FUNCTIONS

2010 ◽  
Vol 02 (04) ◽  
pp. 509-520 ◽  
Author(s):  
SY-SANG LIAW ◽  
FENG-YUAN CHIU
Keyword(s):  
Fractal Dimension ◽  
Real Time ◽  
Empirical Mode Decomposition ◽  
Fractal Dimensions ◽  
The Other ◽  
Partial Sums ◽  
Intrinsic Mode Functions ◽  
Displacement Method ◽  
Mode Decomposition ◽  
Time Sequences

Real nonstationary time sequences are in general not monofractals. That is, they cannot be characterized by a single value of fractal dimension. It has been shown that many real-time sequences are crossover-fractals: sequences with two fractal dimensions — one for the short and the other for long ranges. Here, we use the empirical mode decomposition (EMD) to decompose monofractals into several intrinsic mode functions (IMFs) and then use partial sums of the IMFs decomposed from two monofractals to construct crossover-fractals. The scale-dependent fractal dimensions of these crossover-fractals are checked by the inverse random midpoint displacement method (IRMD).

scholarly journals Comparative analysis of cranial suture complexity in the genus Caiman (Crocodylia, alligatoridae)

2000 ◽  
Vol 60 (4) ◽  
pp. 689-694 ◽  
Author(s):  
L. R. MONTEIRO ◽  
L. G. LESSA
Keyword(s):  
Fractal Dimension ◽  
Significant Interaction ◽  
Fractal Dimensions ◽  
The Other ◽  
Regional Differentiation ◽  
Cranial Sutures ◽  
Food Items ◽  
Facial Skull ◽  
Species Effect ◽  
Functional Stress

The variation in degrees of interdigitation (complexity) in cranial sutures among species of Caiman in different skull regions was studied by fractal analysis. Our findings show that there is a small species effect in the fractal dimension of cranial sutures, but most variation is accounted for by regional differentiation within the skull. There is also a significant interaction between species and cranial regions. The braincase sutures show higher fractal dimension than the facial skull sutures for all three species. The fractal dimension of nasal-maxilla suture is larger in Caiman latirostris than in the other species. The braincase sutures show higher fractal dimensions in C. sclerops than in the other species. The results suggest that different regions of the skull in caimans are under differential functional stress and the braincase sutures must counteract stronger disarticulation forces than the facial sutures. The larger fractal dimension shown by C. latirostris in facial sutures has probably a functional basis also. Caiman latirostris is known to have preferences for harder food items than the other species.

The fractal facets of turbulence

1986 ◽  
Vol 173 ◽  
pp. 357-386 ◽  
Author(s):  
K. R. Sreenivasan ◽  
C. Meneveau
Keyword(s):  
Fractal Dimension ◽  
Turbulent Flows ◽  
Fractal Dimensions ◽  
Dissipative Structures ◽  
The Other ◽  
Turbulent Shear ◽  
Constant Property ◽  
Turbulent Environment ◽  
Developed Turbulence ◽  
Self Similar

Speculations abound that several facets of fully developed turbulent flows are fractals. Although the earlier leading work of Mandelbrot (1974, 1975) suggests that these speculations, initiated largely by himself, are plausible, no effort has yet been made to put them on firmer ground by, resorting to actual measurements in turbulent shear flows. This work is an attempt at filling this gap. In particular, we examine the following questions: (a) Is the turbulent/non-turbulent interface a self-similar fractal, and (if so) what is its fractal dimension ? Does this quantity differ from one class of flows to another? (b) Are constant-property surfaces (such as the iso-velocity and iso-concentration surfaces) in fully developed flows fractals? What are their fractal dimensions? (c) Do dissipative structures in fully developed turbulence form a fractal set? What is the fractal dimension of this set? Answers to these questions (and others to be less fully discussed here) are interesting because they bring the theory of fractals closer to application to turbulence and shed new light on some classical problems in turbulence - for example, the growth of material lines in a turbulent environment. The other feature of this work is that it tries to quantify the seemingly complicated geometric aspects of turbulent flows, a feature that has not received its proper share of attention. The overwhelming conclusion of this work is that several aspects of turbulence can be described roughly by fractals, and that their fractal dimensions can be measured. However, it is not clear how (or whether), given the dimensions for several of its facets, one can solve (up to a useful accuracy) the inverse problem of reconstructing the original set (that is, the turbulent flow itself).

DISTINCT FRACTAL CHARACTERISTICS OF MONOMER AND MULTIMER PROTEINS

Fractals ◽  
2010 ◽  
Vol 18 (02) ◽  
pp. 207-214 ◽  
Author(s):  
DANA CRACIUN ◽  
ADRIANA ISVORAN ◽  
R. D. REISZ ◽  
N. M. AVRAM
Keyword(s):  
Surface Roughness ◽  
Fractal Dimension ◽  
Structural Characteristics ◽  
Fractal Dimensions ◽  
The Other ◽  
Mean Values ◽  
Surface Fractal Dimension ◽  
Surface Fractal ◽  
Fractal Characteristics ◽  
The Mean

Within this study we have calculated the surface fractal dimension (Ds) and the backbone fractal dimensions associated to the local folding (D1) and to the global folding (D2) for two unbiased sets of 50 proteins each, one for monomer and the other for homo- multimer proteins. The mean surface fractal dimension is Ds = 2.29 ± 0.02 for monomers and Ds = 2.21 ± 0.01 for multimers, the two means being significantly different. The mean backbone fractal dimensions associated to the local folding are D1 = 1.34 ± 0.14 for monomers and D1 = 1.33 ± 0.11 for multimers and those associated to the global folding are D2 = 1.33 ± 0.05 for monomers and D2 = 1.29 ± 0.04 for multimers, respectively. There are not significant differences between the mean values of the backbone fractal dimensions corresponding to monomers and multimers. These results suggest that there are different structural characteristics between monomer and multimer proteins only concerning their surface roughness, with multimers being smoother than monomers.

A Review on Fractals and Fracture, Part I: Calculating Fractal Dimensions by CAD Model

2011 ◽  
Vol 148-149 ◽  
pp. 818-821
Author(s):  
Asma A. Shariff ◽  
M. Hadi Hafezi
Keyword(s):  
Fractal Dimension ◽  
Fracture Surface ◽  
Fractal Geometry ◽  
Fractal Dimensions ◽  
Experimental Results ◽  
The Other ◽  
Cad Model ◽  
Cloud Of Points

The objective of this paper is to consider the use of fractal geometry as a tool for the study of non-smooth and discontinuous objects for which Euclidean coordinate is not able to fully describe their shapes. We categorized the methods for computing fractal dimension with a discussion into that. We guide readers up to the point they can dig into the literature, but with more advanced methods that researchers are developing. Considerations show that is necessary to understand the numerous theoretical and experimental results concerning searching of the conformality before evaluating the fractal dimension to our own objects. We suggested examining a cloud of points of growth of fracture surface at laboratory using CATIA - Digitized Shape Editor software in order to reconstruct the surface (CAD model). Then, the author carried out measurement/calculation of more accurate fractal dimension which are introduced by [1] in the other paper as Part II.

FRACTAL DIMENSION, PRIMES, AND THE PERSISTENCE OF MEMORY

2003 ◽  
Vol 06 (02) ◽  
pp. 241-249
Author(s):  
JOSEPH L. PE
Keyword(s):  
Number Theory ◽  
Fractal Dimension ◽  
Fractal Dimensions ◽  
Distinguishing Characteristic ◽  
Local Behavior ◽  
Remarkable Similarity ◽  
Recursive Procedures

Many sequences from number theory, such as the primes, are defined by recursive procedures, often leading to complex local behavior, but also to graphical similarity on different scales — a property that can be analyzed by fractal dimension. This paper computes sample fractal dimensions from the graphs of some number-theoretic functions. It argues for the usefulness of empirical fractal dimension as a distinguishing characteristic of the graph. Also, it notes a remarkable similarity between two apparently unrelated sequences: the persistence of a number, and the memory of a prime. This similarity is quantified using fractal dimension.

scholarly journals Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks

Gels ◽  
2021 ◽  
Vol 7 (2) ◽  
pp. 46
Author(s):  
Pedram Nasr ◽  
Hannah Leung ◽  
France-Isabelle Auzanneau ◽  
Michael A. Rogers
Keyword(s):  
Fractal Dimension ◽  
Crystallization Temperature ◽  
Cayley Tree ◽  
Fractal Dimensions ◽  
Self Similarity ◽  
Avrami Model ◽  
Cluster Aggregation ◽  
Box Counting ◽  
Self Assembled ◽  
Limited Diffusion

Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.

scholarly journals Dynamic characteristics and fractal representations of crack propagation of rock with different fissures under multiple impact loadings

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Bing Sun ◽  
Shun Liu ◽  
Sheng Zeng ◽  
Shanyong Wang ◽  
Shaoping Wang
Keyword(s):  
Fractal Dimension ◽  
Crack Propagation ◽  
Crack Growth ◽  
Impact Test ◽  
Fractal Theory ◽  
Dynamic Change ◽  
Fractal Dimensions ◽  
Peak Stress ◽  
Dynamic Mechanical Properties ◽  
Gravitational Potential Energy

AbstractTo investigate the influence of the fissure morphology on the dynamic mechanical properties of the rock and the crack propagation, a drop hammer impact test device was used to conduct impact failure tests on sandstones with different fissure numbers and fissure dips, simultaneously recorded the crack growth after each impact. The box fractal dimension is used to quantitatively analyze the dynamic change in the sandstone cracks and a fractal model of crack growth over time is established based on fractal theory. The results demonstrate that under impact test conditions of the same mass and different heights, the energy absorbed by sandstone accounts for about 26.7% of the gravitational potential energy. But at the same height and different mass, the energy absorbed by the sandstone accounts for about 68.6% of the total energy. As the fissure dip increases and the number of fissures increases, the dynamic peak stress and dynamic elastic modulus of the fractured sandstone gradually decrease. The fractal dimensions of crack evolution tend to increase with time as a whole and assume as a parabolic. Except for one fissure, 60° and 90° specimens, with the extension of time, the increase rate of fractal dimension is decreasing correspondingly.

scholarly journals A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1546
Author(s):  
Mohsen Soltanifar
Keyword(s):  
Hausdorff Dimension ◽  
Lebesgue Measure ◽  
Virtual World ◽  
Fractal Dimensions ◽  
Definition Of ◽  
The Given

How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.

LATTICE DYNAMICS OF THE BINARY APERIODIC CHAINS OF ATOMS I: FRACTAL DIMENSION OF PHONON SPECTRA

1995 ◽  
Vol 09 (12) ◽  
pp. 1429-1451 ◽  
Author(s):  
WŁODZIMIERZ SALEJDA
Keyword(s):  
Fractal Dimension ◽  
Lattice Dynamics ◽  
Nearest Neighbor ◽  
Average Distance ◽  
Local Maximum ◽  
Fractal Dimensions ◽  
Dimension Formula ◽  
Phonon Spectra ◽  
Wide Range ◽  
Silver Mean

The microscopic harmonic model of lattice dynamics of the binary chains of atoms is formulated and studied numerically. The dependence of spring constants of the nearest-neighbor (NN) interactions on the average distance between atoms are taken into account. The covering fractal dimensions [Formula: see text] of the Cantor-set-like phonon spec-tra (PS) of generalized Fibonacci and non-Fibonaccian aperiodic chains containing of 16384≤N≤33461 atoms are determined numerically. The dependence of [Formula: see text] on the strength Q of NN interactions and on R=mH/mL, where mH and mL denotes the mass of heavy and light atoms, respectively, are calculated for a wide range of Q and R. In particular we found: (1) The fractal dimension [Formula: see text] of the PS for the so-called goldenmean, silver-mean, bronze-mean, dodecagonal and Severin chain shows a local maximum at increasing magnitude of Q and R>1; (2) At sufficiently large Q we observe power-like diminishing of [Formula: see text] i.e. [Formula: see text], where α=−0.14±0.02 and α=−0.10±0.02 for the above specified chains and so-called octagonal, copper-mean, nickel-mean, Thue-Morse, Rudin-Shapiro chain, respectively.

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