scholarly journals The Solutions to the Uncertainty Problem of Urban Fractal Dimension Calculation

Entropy ◽  
2019 ◽  
Vol 21 (5) ◽  
pp. 453 ◽  
Author(s):  
Chen
Keyword(s):  
Fractal Dimension ◽  
Spatial Analysis ◽  
Fractal Geometry ◽  
Fractal Dimensions ◽  
Dimension Estimation ◽  
Scale Free ◽  
Factors Influencing ◽  
Calculation Results ◽  
Fractal Patterns ◽  
Main Factors

Fractal geometry provides a powerful tool for scale-free spatial analysis of cities, but the fractal dimension calculation results always depend on methods and scopes of the study area. This phenomenon has been puzzling many researchers. This paper is devoted to discussing the problem of uncertainty of fractal dimension estimation and the potential solutions to it. Using regular fractals as archetypes, we can reveal the causes and effects of the diversity of fractal dimension estimation results by analogy. The main factors influencing fractal dimension values of cities include prefractal structure, multi-scaling fractal patterns, and self-affine fractal growth. The solution to the problem is to substitute the real fractal dimension values with comparable fractal dimensions. The main measures are as follows. First, select a proper method for a special fractal study. Second, define a proper study area for a city according to a study aim, or define comparable study areas for different cities. These suggestions may be helpful for the students who take interest in or have already participated in the studies of fractal cities.

scholarly journals Spatial Measures of Urban Systems: from Entropy to Fractal Dimension

Entropy ◽  
2018 ◽  
Vol 20 (12) ◽  
pp. 991 ◽  
Author(s):  
Yanguang Chen ◽  
Linshan Huang
Keyword(s):  
Fractal Dimension ◽  
Spatial Analysis ◽  
Fractal Dimensions ◽  
Scale Dependence ◽  
Urban Systems ◽  
Scale Free ◽  
Spatial Entropy ◽  
Spatial Systems ◽  
Urban Patterns ◽  
Generalized Entropies

One type of fractal dimension definition is based on the generalized entropy function. Both entropy and fractal dimensions can be employed to characterize complex spatial systems such as cities and regions. Despite the inherent connection between entropy and fractal dimensions, they have different application scopes and directions in urban studies. This paper focuses on exploring how to convert entropy measurements into fractal dimensions for the spatial analysis of scale-free urban phenomena using the ideas from scaling. Urban systems proved to be random prefractal and multifractal systems. The spatial entropy of fractal cities bears two properties. One is the scale dependence: the entropy values of urban systems always depend on the linear scales of spatial measurement. The other is entropy conservation: different fractal parts bear the same entropy value. Thus, entropy cannot reflect the simple rules of urban processes and the spatial heterogeneity of urban patterns. If we convert the generalized entropies into multifractal spectrums, the problems of scale dependence and entropy homogeneity can be solved to a degree for urban spatial analysis. Especially, the geographical analyses of urban evolution can be simplified. This study may be helpful for students in describing and explaining the spatial complexity of urban evolution.

scholarly journals Differences in fractal patterns and characteristic periodicities between word salads and normal sentences: Interference of meaning and sound

PLoS ONE ◽  
2021 ◽  
Vol 16 (2) ◽  
pp. e0247133
Author(s):  
Jun Shimizu ◽  
Hiromi Kuwata ◽  
Kazuo Kuwata
Keyword(s):  
Fractal Dimension ◽  
Random Walk ◽  
Long Range ◽  
Mental State ◽  
Fractal Dimensions ◽  
Social Robots ◽  
Fractal Patterns

Fractal dimensions and characteristic periodicities were evaluated in normal sentences, computer-generated word salads, and word salads from schizophrenia patients, in both Japanese and English, using the random walk patterns of vowels.　In normal sentences, the walking curves were smooth with gentle undulations, whereas computer-generated word salads were rugged with mechanical repetitions, and word salads from patients with schizophrenia were unreasonably winding with meaningless repetitive patterns or even artistic cohesion. These tendencies were similar in both languages. Fractal dimensions between normal sentences and word salads of schizophrenia were significantly different in Japanese [1.19 ± 0.09 (n = 90) and 1.15 ± 0.08 (n = 45), respectively] and English [1.20 ± 0.08 (n = 91), and 1.16 ± 0.08 (n = 42)] (p < 0.05 for both). Differences in long-range (>10) periodicities between normal sentences and word salads from schizophrenia patients were predominantly observed at 25.6 (p < 0.01) in Japanese and 10.7 (p < 0.01) in English. The differences in fractal dimension and characteristic periodicities of relatively long-range (>10) presented here are sensitive to discriminate between schizophrenia and healthy mental state, and could be implemented in social robots to assess the mental state of people in care.

EXPLORING THE VULNERABILITY OF FRACTAL COMPLEX NETWORKS THROUGH CONNECTION PATTERN AND FRACTAL DIMENSION

Fractals ◽  
2019 ◽  
Vol 27 (06) ◽  
pp. 1950102
Author(s):  
DONG-YAN LI ◽  
XING-YUAN WANG ◽  
PENG-HE HUANG
Keyword(s):  
Fractal Dimension ◽  
Quantitative Research ◽  
Fractal Dimensions ◽  
Network Models ◽  
Small World ◽  
Brain Network ◽  
Complex Model ◽  
Scale Free ◽  
Connection Pattern ◽  
The Stability

The structure of network has a significant impact on the stability of the network. It is useful to reveal the effect of fractal structure on the vulnerability of complex network since it is a ubiquitous feature in many real-world networks. There have been many studies on the stability of the small world and scale-free models, but little has been down on the quantitative research on fractal models. In this paper, the vulnerability was studied from two perspectives: the connection pattern between hubs and the fractal dimensions of the networks. First, statistics expression of inter-connections between any two hubs was defined and used to represent the connection pattern of the whole network. Our experimental results show that statistic values of inter-connections were obvious differences for each kind of complex model, and the more inter-connections, the more stable the network was. Secondly, the fractal dimension was considered to be a key factor related to vulnerability. Here we found the quantitative power function relationship between vulnerability and fractal dimension and gave the explicit mathematical formula. The results are helpful to build stable artificial network models through the analysis and comparison of the real brain network.

scholarly journals Fractal Modeling and Fractal Dimension Description of Urban Morphology

Entropy ◽  
2020 ◽  
Vol 22 (9) ◽  
pp. 961 ◽  
Author(s):  
Yanguang Chen
Keyword(s):  
Fractal Dimension ◽  
Urban Area ◽  
Fractal Geometry ◽  
Urban Form ◽  
Characteristic Length ◽  
Fractal Theory ◽  
Urban Morphology ◽  
Scaling Analysis ◽  
Mathematical Methods ◽  
Scale Free

The conventional mathematical methods are based on characteristic length, while urban form has no characteristic length in many aspects. Urban area is a scale-dependence measure, which indicates the scale-free distribution of urban patterns. Thus, the urban description based on characteristic lengths should be replaced by urban characterization based on scaling. Fractal geometry is one powerful tool for the scaling analysis of cities. Fractal parameters can be defined by entropy and correlation functions. However, the question of how to understand city fractals is still pending. By means of logic deduction and ideas from fractal theory, this paper is devoted to discussing fractals and fractal dimensions of urban landscape. The main points of this work are as follows. Firstly, urban form can be treated as pre-fractals rather than real fractals, and fractal properties of cities are only valid within certain scaling ranges. Secondly, the topological dimension of city fractals based on the urban area is 0; thus, the minimum fractal dimension value of fractal cities is equal to or greater than 0. Thirdly, the fractal dimension of urban form is used to substitute the urban area, and it is better to define city fractals in a two-dimensional embedding space; thus, the maximum fractal dimension value of urban form is 2. A conclusion can be reached that urban form can be explored as fractals within certain ranges of scales and fractal geometry can be applied to the spatial analysis of the scale-free aspects of urban morphology.

Fractal analysis and imaging of the proximal nephron cell

1996 ◽  
Vol 270 (3) ◽  
pp. C953-C963 ◽  
Author(s):  
D. Welling ◽  
J. Urani ◽  
L. Welling ◽  
E. Wagner
Keyword(s):  
Fractal Dimension ◽  
Cross Sections ◽  
Three Dimensional ◽  
Fractal Dimensions ◽  
Cell Periphery ◽  
Proximal Tubule Cell ◽  
Three Dimensional Model ◽  
Cell Height ◽  
Fractal Patterns ◽  
Cell Base

Cells of the S1 proximal renal tubule were examined to determine whether their peculiar shapes are a result of certain constructs of fractal mathematics. Morphometric measurements of the cell perimeter were made at several levels of cell height by measuring the intercellular boundaries that appear on electron micrographs of tubule cross sections. When the measurements were made over a range of scale lengths, the fractal dimension, D, of the cell perimeter was found to increase from 1.3 near the cell apex to 1.78 near the cell base. The length of scale was found to range between 8 and 0.4 micron and to represent the approximate dimensions of actual cell processes. Fractal patterns that conformed to the measured parameters were then constructed from a fractal generator composed of budlike formations that originated near the cell apex and that increased in number and decreased in size with cell depth according to a fractal scaling. It was found that the fractal rule of keeping a constant relative scale could be maintained between budding processes but, to obtain patterns that resemble biological structure, the processes must be positioned randomly on the cell periphery. It is shown that when the relative sizes of the buds decrease exponentially and their numbers increase geometrically, the perimeter can grow to the correct length without overlap. This suggests that patterns of the cell periphery corresponding to different levels of cell height obey a law of scale but occur randomly in a way that increases to high fractal dimension or near plane-filling values at the cell base. The fractal patterns that correspond to the measured fractal dimensions can be assembled into a three-dimensional model that closely resembles the known shape of the proximal tubule cell.

FRACTAL DIMENSION ESTIMATION OF RGB COLOR IMAGES USING MAXIMUM COLOR DISTANCE

Fractals ◽  
2016 ◽  
Vol 24 (04) ◽  
pp. 1650040 ◽  
Author(s):  
XIN ZHAO ◽  
XINGYUAN WANG
Keyword(s):  
Fractal Dimension ◽  
Estimation Method ◽  
Fractal Dimensions ◽  
Color Images ◽  
Natural Images ◽  
Real World Data ◽  
Dimension Estimation ◽  
Fractal Images ◽  
Partition Method ◽  
High Degree

Natural images exhibit a high degree of complexity, randomness and irregularity in color and texture, however fractal can be an effective tool to describe various irregular phenomena in nature. Fractal dimensions are important because they can be defined in connection with real-world data, and they can be measured approximately by means of experiments. In this paper, we proposed a fractal dimension estimation method for RGB color images. In the proposed method, we present a hyper-surface partition method which considers the hyper-surface as continuous and divide the image into nonoverlapped blocks. We also defined a counting method in color domain. To validate the proposed method, experiments were carried on two types of color images: synthesized fractal images and natural RGB color images. The experimental results demonstrate that the proposed method is effective and efficient. The behaviors of the proposed method on the rescale images are also shown in the paper. And it can be performed as a reliable FD estimation approach for the RGB color images.

Influencing Factors Analysis for Failure Risk of Upstream Revetment of Existing Levees

2011 ◽  
Vol 250-253 ◽  
pp. 2843-2847 ◽  
Author(s):  
Yan Hong Gao ◽  
Jun Zhi Zhang
Keyword(s):  
Risk Analysis ◽  
Influencing Factors ◽  
Load Effect ◽  
Factors Analysis ◽  
Factors Influencing ◽  
Failure Risk ◽  
Analysis Theory ◽  
Proposed Model ◽  
Calculation Results ◽  
Main Factors

According to risk analysis theory, a model of analysis for failure risk of upstream revetment of existing levees is proposed in this paper. Based on the model, the load effect and generalized resistance for the failure risk of upstream revetment of existing levees are analyzed, and then the influencing factors of the failure risk of upstream revetment are studied. The calculation results show that the proposed model is workable and effective for analysis of the failure risk of upstream revetment of existing levees, and the main factors influencing failure risk of existing upstream revetment are the randomness of the existing effective thickness of upstream revetment and ratio mean of upstream slope of the existing levees.

scholarly journals MATHEMATICAL MODEL OF LIQUID FUEL COMBUSTION IN THE APPARATUS

2020 ◽  
Vol 2018 (1) ◽  
pp. 173-174
Author(s):  
Larisa Kondrat'eva ◽  
Ol'ga Sverdlova
Keyword(s):  
Mathematical Model ◽  
Oxygen Concentration ◽  
Liquid Fuel ◽  
Organic Liquid ◽  
Material Balance ◽  
Fuel Combustion ◽  
Combustion Model ◽  
Factors Influencing ◽  
Calculation Results ◽  
Main Factors

The article deals with the mathematical combustion model of organic liquid fuel in the technological furnace. The main factors influencing on the process are considered. The formula for the calculation of the oxygen concentration at the reactionary surface is obtained. The material balance of oxygen including equations for the diffusive and dense parts is compiled. The calculation results are shown in the diagrams.

scholarly journals A survey on fractal dimension for fractal structures

2016 ◽  
Vol 1 (2) ◽  
pp. 437-472 ◽  
Author(s):  
M. Fernández-Martínez
Keyword(s):  
Fractal Dimension ◽  
Fractal Geometry ◽  
Fractal Structures ◽  
Fractal Patterns ◽  
Self Similar

AbstractAlong the years, the foundations of Fractal Geometry have received contributions starting from mathematicians like Cantor, Peano, Hilbert, Hausdorff, Carathéodory, Sierpiński, and Besicovitch, to quote some of them. They were some of the pioneers exploring objects having self-similar patterns or showing anomalous properties with respect to standard analytic attributes. Among the new tools developed to deal with this kind of objects, fractal dimension has become one of the most applied since it constitutes a single quantity which throws useful information concerning fractal patterns on sets. Several years later, fractal structures were introduced from Asymmetric Topology to characterize self-similar symbolic spaces. Our aim in this survey is to collect several results involving distinct definitions of fractal dimension we proved jointly with Prof.M.A. Sánchez-Granero in the context of fractal structures.

Fractal Geometry Based Mathematical Model and Simulation of Permeability of Dehulled Rapeseed Cake

2010 ◽  
Vol 29-32 ◽  
pp. 269-274
Author(s):  
Xiao Zheng ◽  
Jing Zhou Wang ◽  
Guo Xiang Lin ◽  
Nong Wan ◽  
Don Ping He
Keyword(s):  
Fractal Dimension ◽  
Fractal Geometry ◽  
Fractal Dimensions ◽  
Image Analyzer ◽  
Rapeseed Cake ◽  
Box Counting ◽  
Model And Simulation ◽  
Pore Size Distributions ◽  
Cold Condition ◽  
Box Counting Method

In view of the fact that dehulled rapeseed cake formed under cold pressing condition is a fractal structure, the relation between the permeability and the pore fractal dimension of dehulled rapeseed cake has been investigated using fractal geometry. The microstructures of dehulled rapeseed cake under six pressing pressures are measured by using scanning electronic microscope and Image-pro image analyzer. The fractal dimensions of pore size distributions are measured by the box-counting method. Combining Hagen-Poiseulle equation with Darcy’s law for flow of fluid through porous media, the relational expression of fractal dimension and permeability has been developed to predicate the permeability of compressed dehulled rapeseed cake under cold condition. The permeability experiments of dehulled rapeseed cake are also carried out in order to validate the predication model proposed in this study. The value of mean relative error is 15.5%. A fairly good agreement is obtained in the case of high pressing pressures.

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