scholarly journals Combinatorial configurations, fractals, fractal dimension of combinatorial sets

2021 ◽  
pp. 170-174
Author(s):  
Nadija Tymofijeva
Keyword(s):  
Fractal Dimension ◽  
Fractal Structure ◽  
Self Similar ◽  
Combinatorial Configurations

Combinatorial configurations and their sets are considered. The definitions of these objects are given, recurrent combinatorial operators are introduced, with the help of which they are formed, and rules are formulated according to which their sets are ordered. The property of periodicity, which takes place in the generation of combinatorial configurations, is described. It follows from the recurrent way of their formation and ordering. The fractal structure of combinatorial sets is formed due to the described rules, in which the property of periodicity is used. Analysis of these structures shows that they are self-similar, both finite and infinite, which is characteristic of fractals. Their fractal dimension is introduced, which follows from the rules of generating combinatorial configurations and corresponds to the number of these objects in their set.

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.

Fractal and Convolutional Analysis for Deep Atmospheric Turbulence Using Machine Learning

2021 ◽  
Author(s):  
Nicholas Dudu ◽  
Arturo Rodriguez ◽  
Gael Moran ◽  
Jose Terrazas ◽  
Richard Adansi ◽  
...  
Keyword(s):  
Fractal Dimension ◽  
Atmospheric Turbulence ◽  
Isotropic Turbulence ◽  
Passive Scalar ◽  
Counting Method ◽  
Data Set ◽  
Box Counting ◽  
Box Counting Dimension ◽  
Self Similar ◽  
Box Counting Method

Abstract Atmospheric turbulence studies indicate the presence of self-similar scaling structures over a range of scales from the inertial outer scale to the dissipative inner scale. A measure of this self-similar structure has been obtained by computing the fractal dimension of images visualizing the turbulence using the widely used box-counting method. If applied blindly, the box-counting method can lead to misleading results in which the edges of the scaling range, corresponding to the upper and lower length scales referred to above are incorporated in an incorrect way. Furthermore, certain structures arising in turbulent flows that are not self-similar can deliver spurious contributions to the box-counting dimension. An appropriately trained Convolutional Neural Network can take account of both the above features in an appropriate way, using as inputs more detailed information than just the number of boxes covering the putative fractal set. To give a particular example, how the shape of clusters of covering boxes covering the object changes with box size could be analyzed. We will create a data set of decaying isotropic turbulence scenarios for atmospheric turbulence using Large-Eddy Simulations (LES) and analyze characteristic structures arising from these. These could include contours of velocity magnitude, as well as of levels of a passive scalar introduced into the simulated flows. We will then identify features of the structures that can be used to train the networks to obtain the most appropriate fractal dimension describing the scaling range, even when this range is of limited extent, down to a minimum of one order of magnitude.

scholarly journals Thermal and Wet Comfort of Fabrics Based on Fractal Dimension of Silicone Coating

2018 ◽  
Vol 13 (1) ◽  
pp. 155892501801300
Author(s):  
Yunlong Shi ◽  
Liang Wang ◽  
Wenhuan Zhang ◽  
Xiaoming Qian
Keyword(s):  
Fractal Dimension ◽  
Thermal Insulation ◽  
Coating Layer ◽  
Fractal Theory ◽  
Barrier Effect ◽  
Evaporative Resistance ◽  
Silicone Coating ◽  
Self Similar ◽  
Warmth Retention ◽  
Coated Fabrics

In this paper, thermal and wet comforts of silicone coated windbreaker shell jacket fabrics were studied. Both thermal insulation and evaporative resistance of fabric increased with an increase in coating area due to the barrier effect of the silicone coating layer. Moreover, the coated fabrics with self-similar structures showed different thermal insulation and evaporative resistance under the same total coating area. Fractal theory was used to explain this phenomenon. Optimal thermal-wet comfort properties were obtained when the fractal dimension (D=1.599) was close to the Golden Mean (1.618). When the fractal dimension of coating was lower than 1.599, fabric warmth retention was not high enough. In contrast, fabric evaporative resistance was beyond the value at which people would feel comfortable when the fractal dimension was greater than 1.599.

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.

scholarly journals Turbulence and magnetic fields in molecular clouds

1991 ◽  
Vol 147 ◽  
pp. 83-92
Author(s):  
R. N. Henriksen
Keyword(s):  
Magnetic Fields ◽  
Wavelet Analysis ◽  
Molecular Clouds ◽  
Fractal Structure ◽  
Compressible Turbulence ◽  
Dynamo Theory ◽  
First Results ◽  
Self Similar

in this paper I first review some of the simple structural concepts associated with compressible turbulence. In particular the hierarchical or self-similar fractal structure to be expected is formulated in a manner readily compared to the observations, and to previous work. In the next section I present the first results of a wavelet analysis on molecular clouds, which seem to comfirm the hierarchical scaling. I conclude with an extention of the theory to include magnetic fields. This latter theory represents an alternative to the more conventional dynamo theory.

ESTIMATING THE FRACTAL DIMENSION OF PLANTS USING THE TWO-SURFACE METHOD: AN ANALYSIS BASED ON 3D-DIGITIZED TREE FOLIAGE

Fractals ◽  
2006 ◽  
Vol 14 (03) ◽  
pp. 149-163 ◽  
Author(s):  
FRÉDÉRIC BOUDON ◽  
CHRISTOPHE GODIN ◽  
CHRISTOPHE PRADAL ◽  
OLIVIER PUECH ◽  
HERVÉ SINOQUET
Keyword(s):  
Fractal Dimension ◽  
Field Measurements ◽  
Three Dimensions ◽  
Total Leaf Area ◽  
Topological Information ◽  
Surface Method ◽  
Plant Foliage ◽  
Self Similar ◽  
Branching System ◽  
Peach Trees

In this paper, we present a method to estimate the fractal dimension of plant foliage in three dimensions (3D). This method is derived from the two-surface method introduced in the 90s to estimate the fractal dimension of tree species from field measurements on collections of trees. Here we adapted the method to individual plants. The multiscale topology and geometry of the plant must first be digitized in 3D. Then leafy branching systems of different sizes are constructed from the plant database, using the topological information. 3D convex envelops are then computed for each leafy branching system. The fractal dimension of the plant is finally estimated by comparing the total leaf area and the convex envelop area of these leafy modules. The method was assessed on a set of four peach trees entirely digitized at shoot scale. Results show that the peach trees have a marked self-similar foliage with fractal dimension close to 2.4.

scholarly journals A laboratory study of sediment deformation: stress heterogeneity and grain-size evolution

1996 ◽  
Vol 22 ◽  
pp. 167-175 ◽  
Author(s):  
Neal R. Iverson ◽  
Thomas S. Hooyer ◽  
Roger Leb. Hooke
Keyword(s):  
Grain Size ◽  
Fractal Dimension ◽  
Normal Stress ◽  
Size Distribution ◽  
Grain Size Distribution ◽  
Local Stress ◽  
Large Strains ◽  
Self Similar ◽  
Deformation Stress ◽  
Stress Heterogeneity

In shearing sediment beneath glaciers, networks of grains may transiently support shear and normal stresses that are larger than spatial averages. Consistent with studies of fault-gouge genesis, we hypothesize that crushing of grains in such networks is responsible for surrounding larger grains with smaller grains. At sufficiently large strains, this should minimize stress heterogeneity, favor intergranular sliding and abrasion rather than crushing, and result in a self-similar grain-size distribution.This hypothesis is tested with a ring-shear device that slowly shears a large annular sediment sample to high strains. Shearing and comminution of weak equigranular (2.0–3.3 mm) sediment resulted in a self-similar grain-size distribution with a fractal dimension that increased with shear strain toward a steady value of 2.85. This value is significantly larger than that of gouges produced purely by crushing, 2.6, but it is comparable to values for tilts thought to be deforming beneath modern glaciers, 2.8 to nearly 3.0. At low strains, under a steady mean normal stress of 84 kPa, variations in normal stress measured locally ranged in amplitude from 50 to 300 kPa with wavelengths that were 100 times larger than the initial grain diameter. Crushing of grains, observed through the transparent walls of the device, apparently caused the failure of grain networks. At shearing displacements ranging from 0.7 to 1.0 m, the amplitude of local stress fluctuations decreased abruptly. This change is attributed to fine sediment that distributed stresses more uniformly and caused grain networks to fail primarily by intergranular sliding rather than by crushing of grains. Sliding between grains apparently produced silt by abrasion and resulted in a fractal dimension that was higher than if there had been only crushing.A size distribution with a fractal dimension greater than 2.6 is probably a necessary but not sufficient condition for determining whether a basal till has been highly deformed. Stress heterogeneity in subglacial sediment that is shearing through its full thickness should contribute to the erosion of underlying rock.

A fractal continuum model of the pulmonary arterial tree

1992 ◽  
Vol 72 (6) ◽  
pp. 2225-2237 ◽  
Author(s):  
G. S. Krenz ◽  
J. H. Linehan ◽  
C. A. Dawson
Keyword(s):  
Fractal Structure ◽  
Cumulative Distribution ◽  
Total Resistance ◽  
Arterial Tree ◽  
Vascular Volume ◽  
Capillary Bed ◽  
Pulmonary Arterial ◽  
Self Similar ◽  
The Mean ◽  
Level Order

The extant morphometric data from the intrapulmonary arteries of dog, human, and cat lungs produce graphs of the log of the vessel number, (N) or length (l) in each level vs. the log of the mean diameter (D) in each level that are sufficiently linear to suggest that a scale-independent self-similar or fractal structure may underlie the observed relationships. These data can be correlated by the following formulas: Nj = a1Dj-beta 1, and lj = a2Dj beta 2, where j denotes the level (order or generation) number measured from the largest vessel at the entrance to the arterial tree to the smallest vessel at the entrance to the capillary bed. With the hemodynamic resistance (R) represented by Rj = 128 microliterj/(Nj pi Dj4) and the vascular volume (Q) by Qj = Nj pi Dj2lj/4, the continuous cumulative distribution of vascular resistance (Rcum) vs. cumulative vascular volume (Qcum) (where Rcum and Qcum represent the total resistance or volume, respectively, upstream from the jth level) can be calculated from [formula: see text] where r = Dj/Dj+1 is a constant independent of j. Analogous equations are developed for the inertance and compliance distributions, providing simple formulas to represent the hemodynamic consequences of the pulmonary arterial tree structure.

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
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