Effect of aluminum ion on sedimentation of kaolinite using anionic polyacrylamides as flocculant based on two-dimensional fractal dimension

Abstract. The need to assess the three-dimensional fractal dimension of fractal aggregates from the fractal dimension of two-dimensional projections is very frequent in geophysics, soil, and atmospheric sciences. However, a generally valid approach to relate the two- and three-dimensional fractal dimensions is missing, thus questioning the accuracy of the method used until now in practical applications. A mathematical approach developed for application to suspended aggregates made of cohesive sediment is investigated and applied here more generally to Diffusion-Limited Aggregates (DLA) and Cluster-Cluster Aggregates (CCA), showing higher accuracy in determining the three-dimensional fractal dimension compared to the method currently used.

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Fractal geometry of ammonoid sutures

Paleobiology ◽

10.1017/s0094837300013336 ◽

1995 ◽

Vol 21 (3) ◽

pp. 329-342 ◽

Cited By ~ 23

Author(s):

Timothy M. Lutz ◽

George E. Boyajian

Keyword(s):

Fractal Dimension ◽

Fractal Geometry ◽

Two Dimensional ◽

Evolutionary Time ◽

Linear Features ◽

Divider Method ◽

Richardson Method ◽

Components Analysis ◽

Box Method ◽

Range Of Variation

Interior chamber walls of ammonites range from smoothly undulating surfaces in some taxa to complex surfaces, corrugated on many scales, in others. The ammonite suture, which is the expression of the intersection of these walls on the exterior of the shell, has been used to assess anatomical complexity. We used the fractal dimension to measure sutural complexity and to investigate complexity over evolutionary time and showed that the range of variation in sutural complexity increased through time. In this paper we extend our analyses and consider two new parameters that measure the range of scales over which fractal geometry is a satisfactory metric of a suture. We use a principal components analysis of these parameters and the fractal dimension to establish a two-dimensional morphospace in which the shapes of sutures can be plotted and in which variations and evolution of suture morphology can be investigated. Our results show that morphospace coordinates of ammonitic sutures correspond to visually perceptible differences in suture shape. However, three main classes of sutures (goniatitic, ceratitic, and ammonitic) are not unambiguously discriminated in this morphospace. Interestingly, ammonitic sutures occupy a smaller morphospace than other suture types (roughly one-half of the morphospace of goniatitic and ceratitic sutures combined), and the space they occupied did not change dimensions from the Jurassic to the late Cretaceous.We also compare two methods commonly used to measure the fractal dimension of linear features: the Box method and the Richardson (or divider) method. Both methods yield comparable results for ammonitic sutures but the Richardson method yields more precise results for less complex sutures.

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Surface fractal dimension of two-dimensional percolation

Journal of Physics A General Physics ◽

10.1088/0305-4470/21/3/039 ◽

1988 ◽

Vol 21 (3) ◽

pp. 833-837 ◽

Cited By ~ 9

Author(s):

C Vanderzande

Keyword(s):

Fractal Dimension ◽

Two Dimensional ◽

Surface Fractal Dimension ◽

Surface Fractal

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Simulation of Aggregates with Point-Contacting Monomers in the Cluster–Dilute Regime. Part 1: Determining the Most Reliable Technique for Obtaining Three-Dimensional Fractal Dimension from Two-Dimensional Images

Aerosol Science and Technology ◽

10.1080/02786826.2010.520363 ◽

2011 ◽

Vol 45 (1) ◽

pp. 75-80 ◽

Cited By ~ 20

Author(s):

Rajan K. Chakrabarty ◽

Mark A. Garro ◽

Bruce A. Garro ◽

Shammah Chancellor ◽

Hans Moosmüller ◽

...

Keyword(s):

Fractal Dimension ◽

Three Dimensional ◽

Two Dimensional ◽

Reliable Technique ◽

Two Dimensional Images

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CHAOS AND FRACTAL OF THE GENERAL TWO-DIMENSIONAL QUADRATIC MAP

International Journal of Modern Physics B ◽

10.1142/s0217979208039927 ◽

2008 ◽

Vol 22 (20) ◽

pp. 3461-3471

Author(s):

XINGYUAN WANG

Keyword(s):

Fractal Dimension ◽

General Feature ◽

Strange Attractors ◽

Two Dimensional ◽

Control Parameters ◽

Self Similarity ◽

Boundary Equation ◽

Quadratic Map ◽

Fractal Images ◽

Stable Points

The nature of the stable points of the general two-dimensional quadratic map is considered analytically, and the boundary equation of the first bifurcation of the map in the parameter space is given out. The general feature of the nonlinear dynamic activities of the map is analyzed by the method of numerical computation. By utilizing the Lyapunov exponent as a criterion, this paper constructs the strange attractors of the general two-dimensional quadratic map, and calculates the fractal dimension of the strange attractors according to the Lyapunov exponents. At the same time, the researches on the fractal images of the general two-dimensional quadratic map make it clear that when the control parameters are different, the fractal images are different from each other, and these fractal images exhibit the fractal property of self-similarity.

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The Finite-Dimensional Uniform Attractors for the Nonautonomous g-Navier-Stokes Equations

Journal of Applied Mathematics ◽

10.1155/2009/150420 ◽

2009 ◽

Vol 2009 ◽

pp. 1-17 ◽

Cited By ~ 1

Author(s):

Delin Wu

Keyword(s):

Fractal Dimension ◽

Bounded Domain ◽

Stokes Equations ◽

Navier Stokes ◽

Two Dimensional ◽

Uniform Attractor ◽

Navier Stokes Equations ◽

Finite Dimensional ◽

Uniform Attractors

We consider the uniform attractors for the two dimensional nonautonomous g-Navier-Stokes equations in bounded domain . Assuming , we establish the existence of the uniform attractor in and . The fractal dimension is estimated for the kernel sections of the uniform attractors obtained.

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SCALING AND FINITE-SIZE EFFECTS FOR THE CRITICAL BACKBONE

Fractals ◽

10.1142/s0218348x03001689 ◽

2003 ◽

Vol 11 (supp01) ◽

pp. 19-27 ◽

Cited By ~ 6

Author(s):

M. BARTHELEMY ◽

S. V. BULDYREV ◽

S. HAVLIN ◽

H. E. STANLEY

Keyword(s):

Fractal Dimension ◽

Probability Distribution ◽

Size Effects ◽

Infinite System ◽

Finite Size ◽

Multifractal Spectrum ◽

System Size ◽

Two Dimensional ◽

High Current ◽

Finite Size Effects

In a first part, we study the backbone connecting two given sites of a two-dimensional lattice separated by an arbitrary distance r in a system of size L. We find a scaling form for the average backbone mass and we also propose a scaling form for the probability distribution P(MB) of backbone mass for a given r. For r ≈ L, P(MB) is peaked around LdB, whereas for r ≪ L, P(MB) decreases as a power law, [Formula: see text], with τB ≃ 1.20 ± 0.03. The exponents ψ and τB satisfy the relation ψ = dB(τB - 1), and ψ is the codimension of the backbone, ψ = d - dB. In a second part, we study the multifractal spectrum of the current in the two-dimensional random resistor network at the percolation threshold. Our numerical results suggest that in the infinite system limit, the probability distribution behaves for small i as P(i) ~ 1/i where i is the current. As a consequence, the moments of i of order q ≤ qc = 0 diverge with system size, and all sets of bonds with current values below the most probable one have the fractal dimension of the backbone. Hence we hypothesize that the backbone can be described in terms of only (i) blobs of fractal dimension dB and (ii) high current carrying bonds of fractal dimension going from d red to dB, where d red is the fractal dimension of the red bonds carrying the maximal current.

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Influence of Size on the Fractal Dimension of Dislocation Microstructure

Metals ◽

10.3390/met9040478 ◽

2019 ◽

Vol 9 (4) ◽

pp. 478

Author(s):

Yinan Cui ◽

Nasr Ghoniem

Keyword(s):

Fractal Dimension ◽

Three Dimensional ◽

Dislocation Dynamics ◽

Two Dimensional ◽

Discrete Dislocation Dynamics ◽

Discrete Dislocation ◽

Transmission Electron ◽

Dynamics Simulations ◽

2D And 3D ◽

Microscopy Images

Three-dimensional (3D) discrete dislocation dynamics simulations are used to analyze the size effect on the fractal dimension of two-dimensional (2D) and 3D dislocation microstructure. 2D dislocation structures are analyzed first, and the calculated fractal dimension ( n 2 ) is found to be consistent with experimental results gleaned from transmission electron microscopy images. The value of n 2 is found to be close to unity for sizes smaller than 300 nm, and increases to a saturation value of ≈1.8 for sizes above approximately 10 microns. It is discovered that reducing the sample size leads to a decrease in the fractal dimension because of the decrease in the likelihood of forming strong tangles at small scales. Dislocation ensembles are found to exist in a more isolated way at the nano- and micro-scales. Fractal analysis is carried out on 3D dislocation structures and the 3D fractal dimension ( n 3 ) is determined. The analysis here shows that ( n 3 ) is significantly smaller than ( n 2 + 1 ) of 2D projected dislocations in all considered sizes.

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Estimate of fractal dimension of rat tissues submitted to experimental protocols v2

10.17504/protocols.io.bxzqpp5w ◽

2021 ◽

Author(s):

Gabriel not provided Cao PhD ◽

Graciela Ottaviano ◽

Francisco Capani PhD

Keyword(s):

Fractal Dimension ◽

Structural Changes ◽

Rat Tissues ◽

Spatial Relationships ◽

Two Dimensional ◽

Two Dimensional Image ◽

Additional Information ◽

Mathematical Formulas ◽

Histological Images ◽

Important Additional Information

Those who are dedicated to the analysis of structural changes in tissues have tried, over time, to seek increasingly "more rigorous" methods to be able to detach themselves from the merely observational and subjective. That is, leaving aside the semi-quantitative scores based on scores that are given to the lesion in a tissue according to its degree of severity. The argument is that the final injury score will depend more on the subjectivity and experience of the observer. With the advent of digital images and programs for their analysis, the application of numerical methods for estimating changes in tissues was greatly facilitated. With them we do not completely suppress the observational, but, to a large extent and if we are rigorous, we can significantly reduce its influence. Thus, in two-dimensional images, we can make direct measurements such as the diameter and length of a gland, its surface, etc., always in previously calibrated systems. We can also estimate the dimensions of structures that are part of a tissue and the spatial relationships between them based on a two-dimensional image. In this case we will use stereology, which uses simple mathematical formulas, but is very time consuming for analysis. Now, structuralists have realized that the normal components of a tissue or a cell maintain certain spatial relationships and proportionality to each other, which also defines their shapes and textures (complexity), constituting the characteristic histological images of a kidney, liver, uterus, etc. Both the pathology and the functional adaptations alter these normal relationships, which wanted to be estimated through the application of the fractal dimension. The justification is that, when faced with a certain insult or stimulus, the tissue or organ responds “in toto”, not one part yes and another no. The single measurement of diameters, surfaces, etc., while complementary, was always thought to be incomplete because we were missing those changes in the relationships between tissue components or from one cell to another, which provide important additional information.

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