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Journal articles on the topic 'Fractal Squares'
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ZHANG, YAN-FANG. "A LOWER BOUND OF TOPOLOGICAL HAUSDORFF DIMENSION OF FRACTAL SQUARES." Fractals 28, no. 06 (September 2020): 2050115. http://dx.doi.org/10.1142/s0218348x20501157.
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Given an integer [Formula: see text] and a digit set [Formula: see text], there is a self-similar set [Formula: see text] satisfying the set equation [Formula: see text]. This set [Formula: see text] is called a fractal square. By studying the line segments contained in [Formula: see text], we give a lower estimate of the topological Hausdorff dimension of fractal squares. Moreover, we compute the topological Hausdorff dimension of fractal squares whose nontrivial connected components are parallel line segments, and introduce the Latin fractal squares to investigate the question when the topological Hausdorff dimension of a fractal square coincides with its Hausdorff dimension.
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LAU, KA–SING, JUN JASON LUO, and HUI RAO. "Topological structure of fractal squares." Mathematical Proceedings of the Cambridge Philosophical Society 155, no. 1 (March 1, 2013): 73–86. http://dx.doi.org/10.1017/s0305004112000692.
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AbstractGiven an integer n ≥ 2 and a digit set ⊊ {0,1,. . .,n − 1}2, there is a self-similar set F ⊂ ℝ2 satisfying the set equation: F=(F+)/n. We call such F a fractal square. By studying a periodic extension H= F + ℤ2, we classify F into three types according to their topological properties. We also provide some simple criteria for such classification.
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Liang, Zhen, and Huo-Jun Ruan. "Gap sequences of fractal squares." Journal of Mathematical Analysis and Applications 472, no. 2 (April 2019): 1475–86. http://dx.doi.org/10.1016/j.jmaa.2018.12.003.
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RAO, FENG, XIAOHUA WANG, and SHENGYOU WEN. "ON THE TOPOLOGICAL CLASSIFICATION OF FRACTAL SQUARES." Fractals 25, no. 03 (May 11, 2017): 1750028. http://dx.doi.org/10.1142/s0218348x17500281.
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A fractal square is a nonempty compact set in the plane satisfying [Formula: see text], where [Formula: see text] is an integer and [Formula: see text] is nonempty. We give the topological classification of fractal squares with [Formula: see text] and [Formula: see text].
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LUO, JUN JASON, and JING-CHENG LIU. "ON THE CLASSIFICATION OF FRACTAL SQUARES." Fractals 24, no. 01 (March 2016): 1650008. http://dx.doi.org/10.1142/s0218348x16500080.
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In the previous paper [K. S. Lau, J. J. Luo and H. Rao, Topological structure of fractal squares, Math. Proc. Camb. Phil. Soc. 155 (2013) 73–86], Lau, Luo and Rao completely classified the topological structure of so called fractal square [Formula: see text] defined by [Formula: see text], where [Formula: see text]. In this paper, we further provide simple criteria for the [Formula: see text] to be totally disconnected, then we discuss the Lipschitz classification of [Formula: see text] in the case [Formula: see text], which is an attempt to consider non-totally disconnected sets.
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Xiao, Jian-Ci. "Fractal squares with finitely many connected components *." Nonlinearity 34, no. 4 (February 18, 2021): 1817–36. http://dx.doi.org/10.1088/1361-6544/abd611.
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Feng, Rao, Wang Xiaohua, and Zhu Yunjie. "Lipschitz equivalence of a pair of fractal squares." SCIENTIA SINICA Mathematica 48, no. 3 (May 24, 2017): 363. http://dx.doi.org/10.1360/n012016-00180.
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Ruan, Huo-Jun, and Yang Wang. "Topological invariants and Lipschitz equivalence of fractal squares." Journal of Mathematical Analysis and Applications 451, no. 1 (July 2017): 327–44. http://dx.doi.org/10.1016/j.jmaa.2017.02.012.
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Rezende, Veridiana, Mariana Moran, Thais Michele Mártires, and Fabricia Carvalho Paixão. "O Fractal Árvore Pitagórica e Diferentes Representações: uma Investigação com Alunos do Ensino Médio." Jornal Internacional de Estudos em Educação Matemática 11, no. 2 (September 11, 2018): 160. http://dx.doi.org/10.17921/2176-5634.2018v11n2p160-171.
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A Geometria dos Fractais em sala de aula propicia diversas possibilidades de estudos de conceitos matemáticos, incentiva o uso de recursos tecnológicos, proporciona surpresas pela beleza e complexidade dos fractais. Apresentamos neste texto uma investigação acerca das possibilidades do uso de diferentes registros de representação semiótica aliado à Geometria dos Fractais, no que concerne ao ensino de matemática. E, para exemplificar estas possibilidades em sala de aula, relatamos uma situação de ensino, que foi implementada com quinze (15) alunos do 3º ano do Ensino Médio de uma escola pública do interior do Paraná, relacionada à construção do fractal Árvore Pitagórica. Como procedimentos metodológicos elaboramos cinco tarefas associadas ao fractal Árvore Pitagórica que tiveram por finalidade explorar diversos conceitos matemáticos emergidos no processo de construção deste fractal, nas quais foram contempladas as representações figural, numérica, algébrica e linguagem natural. As análises dos registros mostram que a implementação das tarefas possibilitou aos alunos: o estudo de diversos elementos matemáticos tais como: áreas e perímetros de quadrados e triângulos, teorema de Pitágoras, ângulos, congruência de triângulos, frações, potências, números decimais entre outros; as construções figurais da Árvore Pitagórica por meio de diferentes representações e a visualização das principais características de um fractal, bem como a compreensão de seu processo de construção.Palavras-chave: Ensino de Matemática. Representação Semiótica. Geometria dos Fractais.AbstractThe geometry of the fractures in the classroom provides several possibilities for studying mathematical concepts, encourages the use of technological resources, provides surprises for the beauty and complexity of the fractals. We present in this text an investigation about the possibilities of the use of different registers of semiotic representation allied to the Geometry of the Fractais, in what concerns the teaching of mathematics. And, to exemplify these possibilities in the classroom, we report a teaching situation, which was implemented with fifteen (15) students of the 3rd year of High School in a public school in the interior of Paraná, related to the construction of the Pythagorean Tree fractal. As methodological procedures we elaborated five tasks associated to the fractal Pythagorean Tree that had as purpose to explore several mathematical concepts emerged in the process of construction of this fractal, in which figural, numerical, algebraic and natural language representations were contemplated. The analysis of the registers shows that the implementation of the tasks enabled the students to study several mathematical elements such as: areas and perimeters of squares and triangles, Pythagorean theorem, angles, congruence of triangles, fractions, powers, decimals, among others; the figurative constructions of the Pythagorean Tree by means of different representations and the visualization of the main characteristics of a fractal, as well as the understanding of its process of construction.Keywords: Mathematics Teaching. Semiotic Representation. Geometry of Fractions.
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PALAGALLO, JUDITH, and MARIA SALCEDO. "SYMMETRIES OF FRACTAL TILINGS." Fractals 16, no. 01 (March 2008): 69–78. http://dx.doi.org/10.1142/s0218348x08003806.
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Self-similar tilings of the plane can be generated by an iterative replacement property using squares and triangles. We show tilings with tiles that are topological disks and others whose tiles are connected but have an interior that is disconnected. In each example we note the symmetries of the individual tiles and describe how each tiling exhibits a property common to crystallographic tilings.
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Man, Xiaoming, and Yanguang Chen. "Fractal-Based Modeling and Spatial Analysis of Urban Form and Growth: A Case Study of Shenzhen in China." ISPRS International Journal of Geo-Information 9, no. 11 (November 13, 2020): 672. http://dx.doi.org/10.3390/ijgi9110672.
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Fractal dimension curves of urban growth can be modeled with sigmoid functions, including logistic function and quadratic logistic function. Different types of logistic functions indicate different spatial dynamics. The fractal dimension curves of urban growth in Western countries follow the common logistic function, while the fractal dimension growth curves of cities in northern China follow the quadratic logistic function. Now, we want to investigate whether other Chinese cities, especially cities in South China, follow the same rules of urban evolution and attempt to analyze the reasons. This paper is devoted to exploring the fractals and fractal dimension properties of the city of Shenzhen in southern China. The urban region is divided into four subareas using ArcGIS technology, the box-counting method is adopted to extract spatial datasets, and the least squares regression method is employed to estimate fractal parameters. The results show that (1) the urban form of Shenzhen city has a clear fractal structure, but fractal dimension values of different subareas are different; (2) the fractal dimension growth curves of all the four study areas can only be modeled by the common logistic function, and the goodness of fit increases over time; (3) the peak of urban growth in Shenzhen had passed before 1986 and the fractal dimension growth is approaching its maximum capacity. It can be concluded that the urban form of Shenzhen bears characteristics of multifractals and the fractal structure has been becoming better, gradually, through self-organization, but its land resources are reaching the limits of growth. The fractal dimension curves of Shenzhen’s urban growth are similar to those of European and American cities but differ from those of cities in northern China. This suggests that there are subtle different dynamic mechanisms of city development between northern and southern China.
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LI, HAO, JIAN HUANG, ANBO LE, QIN WANG, and LIFENG XI. "SCALE-FREE AND SMALL-WORLD PROPERTIES OF VAF FRACTAL NETWORKS." Fractals 24, no. 03 (August 30, 2016): 1650033. http://dx.doi.org/10.1142/s0218348x1650033x.
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In this paper, we investigate the vertical-affiliation-free (VAF) evolving networks whose node set is the basic squares in the process of generating the Sierpinski carpet and edge exists between any two nodes if and only if the corresponding basic squares intersect just on their boundary. Although the VAF networks gets rid of the hierarchial organizations produced naturally by the self-similar structures of fractals, we still prove that they are scale-free and have the small-world effect.
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Slater, P. B. "Fractal fits to Riemann zeros." Canadian Journal of Physics 85, no. 4 (April 1, 2007): 345–57. http://dx.doi.org/10.1139/p07-050.
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Wu and Sprung (Phys. Rev. E, 48, 2595 (1993)) reproduced the first 500 nontrivial Riemann zeros, using a one-dimensional local potential model. They concluded — as did van Zyl and Hutchinson (Phys. Rev. E, 67, 066211 (2003)) — that the potential possesses a fractal structure of dimension d = 3/2. We model the nonsmooth fluctuating part of the potential by the alternating-sign sine series fractal of Berry and Lewis A(x,γ). Setting d = 3/2, we estimate the frequency parameter (γ), plus an overall scaling parameter (σ) that we introduce. We search for that pair of parameters (γ,σ) that minimizes the least-squares fit Sn(γ,σ) of the lowest n eigenvalues — obtained by solving the one-dimensional stationary (nonfractal) Schrodinger equation with the trial potential (smooth plus nonsmooth parts) — to the lowest n Riemann zeros for n = 25. For the additional cases, we study, n = 50 and 75, we simply set σ = 1. The fits obtained are compared to those found by using just the smooth part of the Wu–Sprung potential without any fractal supplementation. Some limited improvement — 5.7261 versus 6.392 07 (n = 25), 11.2672 versus 11.7002 (n = 50), and 16.3119 versus 16.6809 (n = 75) — is found in our (nonoptimized, computationally bound) search procedures. The improvements are relatively strong in the vicinities of γ = 3 and (its square) 9. Further, we extend the Wu-Sprung semiclassical framework to include higher order corrections from the Riemann–von Mangoldt formula (beyond the leading, dominant term) into the smooth potential. PACS Nos.: 02.10.De, 03.65.Sq, 05.45.Df, 05.45.Mt
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Liu, Limin, and Yingying Cui. "European Option Based on Least-Squares Method under Non-Extensive Statistical Mechanics." Entropy 21, no. 10 (September 25, 2019): 933. http://dx.doi.org/10.3390/e21100933.
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This paper is devoted to the study of the pricing of European options under a non-Gaussian model. This model follows a non-extensive statistical mechanics which can better describe the fractal characteristics of price movement in the financial market. Moreover, we present a simple but precise least-square method for approximation and obtain a closed-form solution of the price of European options. The advantages of this technique are illustrated by numerical simulation, which shows that the least-squares method is better compared with Borland’s two methods in 2002 and 2004.
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Gan, Ben, Yi Jian Huang, and Gui Xia Zheng. "Prediction of Surface Roughness Profiles for Milling Process with Fractal Parameters Based on LS-SVM." Advanced Materials Research 97-101 (March 2010): 1186–93. http://dx.doi.org/10.4028/www.scientific.net/amr.97-101.1186.
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Least squares support vector machines (LS-SVM) were developed for the analysis and prediction of the relationship between the cutting conditions and the corresponding fractal parameters of machined surfaces in face milling operation. These models can help manufacturers to determine the appropriate cutting conditions, in order to achieve specific surface roughness profile geometry, and hence achieve the desired tribological performance (e.g. friction and wear) between the contacting surfaces. The input parameters of the LS-SVM are the cutting parameters: rotational speed, feed, depth of milling. The output parameters of the LS-SVM are the corresponding calculated fractal parameters: fractal dimension D and vertical scaling parameter G. The LS-SVM were utilized successfully for training and predicting the fractal parameters D and G in face milling operations. Moreover, Weierstrass-Mandelbrot（W–M ）fractal function was integrated with the LS-SVM in order to generate an artificially fractal predicted profiles at different milling conditions. The predicted profiles were found statistically similar to the actual measured profiles of test specimens and there is a relationship between the scale-independent fractal coefficients(D and G).
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Pouraimis, Georgios, Apostolos Kotopoulis, Basil Massinas, and Panayiotis Frangos. "Sea State Characterization Using Experimental One-Dimensional Radar Signatures and Fractal Techniques." Elektronika ir Elektrotechnika 27, no. 3 (June 28, 2021): 71–77. http://dx.doi.org/10.5755/j02.eie.28906.
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This paper presents a novel method of sea state characterization by using four criteria, which are applied to normalized experimental Synthetic Aperture Radar (SAR) one–dimensional signatures (range profiles), provided to our research group by SET 215 Working Group on “SAR radar techniques”. These criteria are the “Fractal Dimension”, “Fractal Length”, “Variance σ2”, and “Power Spectrum Density - Least Squares”. The “Fractal Dimension” and “Fractal Length” criteria, which appear to be the most important out of the four criteria, use the “blanket” technique to provide sea state characterization from SAR radar range profiles. It is based on the calculation of the area of a “blanket”, corresponding to the range profile under examination, and then on the calculation of the corresponding “Fractal Dimension” and “Fractal Length” of the range profile. The main idea concerning this proposed technique is the fact that normalized SAR radar range profiles, corresponding to different sea states, produce different values of “Fractal Dimension” and “Fractal Length” for all angles of incidence examined here. As a result, a sea state characterization technique for two different sea states (turbulent and calm sea) is presented in this paper.
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Ma, Ji-hua, and Yan-fang Zhang. "Topological Hausdorff dimension of fractal squares and its application to Lipschitz classification." Nonlinearity 33, no. 11 (October 12, 2020): 6053–71. http://dx.doi.org/10.1088/1361-6544/aba0c4.
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BROWNE, CAMERON. "ARTISTIC BOX TREES." Fractals 15, no. 03 (September 2007): 249–53. http://dx.doi.org/10.1142/s0218348x07003551.
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This short note describes a development of the traditional Pythagorean tree fractal that produces three-dimensional structures based on cubes rather than squares. Adding a 90° rotation at each bifurcation encourages three-dimensional growth, creating increasingly artistic shapes as the branching angle decreases.
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Katrinak, Karen A., Peter Rez, Paul R. Perkes, and Peter R. Buseck. "Fractal geometry of carbonaceous aerosol particles as determined using Transmission Electron Microscopy." Proceedings, annual meeting, Electron Microscopy Society of America 50, no. 1 (August 1992): 312–13. http://dx.doi.org/10.1017/s042482010012196x.
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Carbonaceous aggregates collected from the aerosol of Phoenix, Arizona have an irregular branched morphology. The aggregates resemble combustion soots and were probably emitted by motor vehicles. Fractal analysis provides a means of quantifying morphologic variations among aggregates and relating these variations to mechanisms of formation. Bright-field transmission electron microscope (TEM) images of 38 individual aggregates were recorded on negatives at magnifications of 15,000 to 200,000. The aggregates have maximum lengths ranging from 0.21 to 2.61 μm and are composed of interconnected spherules, each averaging 26 nm in diameter. The number of spherules in each aggregate ranges from 32 to 1842; the average number is 551. “The nesting squares” method of fractal analysis was applied to digital binary images to calculate the fractal dimension (D) of each aggregate.
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Chen, Yanguang, and Yuqing Long. "Spatial Signal Analysis Based on Wave-Spectral Fractal Scaling: A Case of Urban Street Networks." Applied Sciences 11, no. 1 (December 24, 2020): 87. http://dx.doi.org/10.3390/app11010087.
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A number of mathematical methods have been developed to make temporal signal analyses based on time series. However, no effective method for spatial signal analysis, which are as important as temporal signal analyses for geographical systems, has been devised. Nonstationary spatial and temporal processes are associated with nonlinearity, and cannot be effectively analyzed by conventional analytical approaches. Fractal theory provides a powerful tool for exploring spatial complexity and is helpful for spatio-temporal signal analysis. This paper is devoted to developing an approach for analyzing spatial signals of geographical systems by means of wave-spectrum scaling. The traffic networks of 10 Chinese cities are taken as cases for positive studies. Fast Fourier transform (FFT) and ordinary least squares (OLS) regression methods are employed to calculate spectral exponents. The results show that the wave-spectrum density distribution of all these urban traffic networks follows scaling law, and that the spectral scaling exponents can be converted into fractal dimension values. Using the fractal parameters, we can make spatial analyses for the geographical signals. The wave-spectrum scaling methods can be applied to both self-similar fractal signals and self-affine fractal signals in the geographical world. This study has implications for the further development of fractal-based spatiotemporal signal analysis in the future.
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Davoine, Franck, Etienne Bertin, and Jean-Marc Chassery. "An Adaptive Partition for Fractal Image Coding." Fractals 05, supp01 (April 1997): 243–56. http://dx.doi.org/10.1142/s0218348x97000796.
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In this paper we present a flexible partitioning scheme for fractal image compression, based on the Delaunay triangles. The aim is to have the advantage of triangular blocks over squares, in terms of adaptivity to the image content. In a first step, the triangulation is computed so that the triangles are more densely distributed in regions containing interesting features such as corners and edges, or so that they tend to run along the strong edges in the image. In a second step we merge adjacent triangles into quadrilaterals, in order to decrease the number of blocks. Quadrilaterals permit a reduction of the number of local contractive affine transformations composing the fractal transform, and thus to increase the compression ratio, while preserving the visual quality of the decoded image.
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He, Tao, Long Fei Cheng, Qing Hua Wu, Zheng Jia Wang, Lian Gen Yang, and Lang Yu Xie. "An Image Segmentation Calculation Based on Differential Box-Counting of Fractal Geometry." Applied Mechanics and Materials 719-720 (January 2015): 964–68. http://dx.doi.org/10.4028/www.scientific.net/amm.719-720.964.
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Differential box-counting of fractal geometry has been widely used in image processing.A method which uses the differential box-counting to segment the gathered images is discussed in this paper . It is to construct a three-dimensional gray space and use the same size boxes to contain the three dimensional space.The number of boxes needed to cover the entire image are calculated .Different sizes of boxes can receive different number of boxes, so least squares method is used to calculate the fractal dimension. According to the fractal dimension parameters, appropriate threshold is chose to segment the image by using binarization .From the handle case of bearing pictures can be seen that image segmentation based on differential box-counting method can get clear image segmentation .This method is easy to understand, to operate, and has important significance on computer image segmentation .
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Hammouda, Boualem. "Analysis of the Beaucage model." Journal of Applied Crystallography 43, no. 6 (October 1, 2010): 1474–78. http://dx.doi.org/10.1107/s0021889810033856.
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The Beaucage model is used to analyze small-angle scattering (SAS) data from fractal and particulate systems. It models the Guinier and Porod regions with a smooth transition between them and yields a radius of gyration and a Porod exponent. This model is an approximate form of an earlier polymer fractal model that has been generalized to cover a wider scope. The practice of allowing both the Guinier and the Porod scale factors to vary independently during nonlinear least-squares fits introduces undesired artefacts in the fitting of SAS data to this model. Such artefacts as well as an error in the original formulation of the model are discussed. This model is compared with other published models.
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BRINKS, RALPH. "A HYBRID ALGORITHM FOR THE SOLUTION OF THE INVERSE PROBLEM IN FRACTAL INTERPOLATION." Fractals 13, no. 03 (September 2005): 215–26. http://dx.doi.org/10.1142/s0218348x05002866.
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A new algorithm for the solution of the inverse problem concerning interpolatory iterated functions systems (IFS) is presented. It combines the advantages of the greedy least squares approach of Mazel and Hayes, and the self-affinity in the wavelet scalogram. Thus, the algorithm is computationally cheap and yields results with high accuracy.
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Zhu, Yuwen, and Haoyu Li. "A New Method for the Process Division and Effect Evaluation of Coagulation Based on Particle Size Fractal Dimension." Processes 6, no. 12 (November 23, 2018): 237. http://dx.doi.org/10.3390/pr6120237.
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To divide, control, and predict the effects of the coagulation process in water treatment, a characteristic analysis of the change in particle size distribution (particle number and fractal dimension) during aided coagulation with hydrated MnO2 was performed. The results showed that the process of coagulation could be divided into three characteristic stages based on the first derivative of the particle size fractal dimension. In the primary stage, most of the microflocs aggregated to form small flocs; in the growth stage, most of the small flocs aggregated to form large flocs; and in the stable stage, some large flocs broke apart and reformed. The first derivative of the particle size fractal dimension had a good linear relationship with the coagulation time in the primary stage and growth stage, and its slope had a power function relationship with the particle number in settled water; the first derivative could thus be used to evaluate the coagulation effect. In the stable stage, the rate of change in particle size fractal dimension fluctuated along the fitted line, and the mean residual sum of squares had a linear relation with the particle number in settled water; therefore, this parameter could be used as an indicator of the coagulation effect.
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Liang, Jiang, Yanqin Hu, and Hui Sun. "The Design Evaluation of the Green Space Layout of Urban Squares Based on Fractal Theory." Nexus Network Journal 15, no. 1 (December 22, 2012): 33–49. http://dx.doi.org/10.1007/s00004-012-0135-3.
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Zhao, Haosu, Bart Julien Dewancker, Feng Hua, Junping He, and Weijun Gao. "Restrictions of Historical Tissues on Urban Growth, Self-Sustaining Agglomeration in Walled Cities of Chinese Origin." Sustainability 12, no. 14 (July 21, 2020): 5849. http://dx.doi.org/10.3390/su12145849.
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This article uses a fractal observation to help delineate the constraints placed by multiple city walls on the growth of historical East Asian cities. By applying advanced technologies from economic geography and fractal indices, a staged scaling process within urban dimension coherence can be applied to both indices. In this study, a discovery is proposed based on the urban organism concept that is capable of indicating a proportional intra-urban structure from a fundamental wall-bounded urban element (local specificity) to other greater walled spatial properties (global variables). This local specificity potentially performs approximate scaling regularities, and spatially denotes an average historical threshold of urban growth for its overall size, with similar scaling law constraints. This finding involves territorial, urban planning, and ancient architectural perspectives, providing a historical and local response to the expansion of contemporary cities. By employing growing fractal estimation, data processing enables the logarithmic city size to be obtained by measuring each wall’s specific features using the Ordinary Least Squares (OLS) method. On the basis of two-dimensional allometric scaling patches, a spatial unfolding mechanism is utilized to reproduce these dynamic changes with city walls as a result of the human trajectories in time geography.
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Mather, George. "Aesthetic Image Statistics Vary with Artistic Genre." Vision 4, no. 1 (February 1, 2020): 10. http://dx.doi.org/10.3390/vision4010010.
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Research to date has not found strong evidence for a universal link between any single low-level image statistic, such as fractal dimension or Fourier spectral slope, and aesthetic ratings of images in general. This study assessed whether different image statistics are important for artistic images containing different subjects and used partial least squares regression (PLSR) to identify the statistics that correlated most reliably with ratings. Fourier spectral slope, fractal dimension and Shannon entropy were estimated separately for paintings containing landscapes, people, still life, portraits, nudes, animals, buildings and abstracts. Separate analyses were performed on the luminance and colour information in the images. PLSR fits showed shared variance of up to 75% between image statistics and aesthetic ratings. The most important statistics and image planes varied across genres. Variation in statistics may reflect characteristic properties of the different neural sub-systems that process different types of image.
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Xu, Fangzhou, Weidong Zhou, Yilin Zhen, Qi Yuan, and Qi Wu. "Using Fractal and Local Binary Pattern Features for Classification of ECOG Motor Imagery Tasks Obtained from the Right Brain Hemisphere." International Journal of Neural Systems 26, no. 06 (July 19, 2016): 1650022. http://dx.doi.org/10.1142/s0129065716500222.
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The feature extraction and classification of brain signal is very significant in brain–computer interface (BCI). In this study, we describe an algorithm for motor imagery (MI) classification of electrocorticogram (ECoG)-based BCI. The proposed approach employs multi-resolution fractal measures and local binary pattern (LBP) operators to form a combined feature for characterizing an ECoG epoch recording from the right hemisphere of the brain. A classifier is trained by using the gradient boosting in conjunction with ordinary least squares (OLS) method. The fractal intercept, lacunarity and LBP features are extracted to classify imagined movements of either the left small finger or the tongue. Experimental results on dataset I of BCI competition III demonstrate the superior performance of our method. The cross-validation accuracy and accuracy is 90.6% and 95%, respectively. Furthermore, the low computational burden of this method makes it a promising candidate for real-time BCI systems.
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Durán-Meza, G., J. López-García, and J. L. del Río-Correa. "The self-similarity properties and multifractal analysis of DNA sequences." Applied Mathematics and Nonlinear Sciences 4, no. 1 (June 29, 2019): 267–78. http://dx.doi.org/10.2478/amns.2019.1.00023.
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AbstractIn this work is presented a pedagogical point of view of multifractal analysis deoxyribonucleic acid (DNA) sequences is presented. The DNA sequences are formed by 4 nucleotides (adenine, cytosine, guanine, and tymine). Following Jeffrey’s paper we associated a simple contractive function to each nucleotide, and constructed the Hutchinson’s operator W, which was used to build covers of different sizes of the unitary square Q, thus Wk(Q) is a cover of Q, conformed by 4k squares Qk of size 2−k, as each Qk corresponds to a unique subsequence of nucleotides with length k : b1b2...bk. Besides, it is obtained the optimal cover Ck to the fractal F generated for each DNA sequence was obtained. We made a multifractal decomposition of Ck in terms of the sets Jα conformed by the Qk’s with the same value of the Holder exponent α, and determined f (α), the Hausdorff dimension of Jα, using the curdling theorem.
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Long, Yuqing, and Yanguang Chen. "Multifractal scaling analyses of urban street network structure: The cases of twelve megacities in China." PLOS ONE 16, no. 2 (February 18, 2021): e0246925. http://dx.doi.org/10.1371/journal.pone.0246925.
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Traffic networks have been proved to be fractal systems. However, previous studies mainly focused on monofractal networks, while complex systems are of multifractal structure. This paper is devoted to exploring the general regularities of multifractal scaling processes in the street network of 12 Chinese cities. The city clustering algorithm is employed to identify urban boundaries for defining comparable study areas; box-counting method and the direct determination method are utilized to extract spatial data; the least squares calculation is employed to estimate the global and local multifractal parameters. The results showed multifractal structure of urban street networks. The global multifractal dimension spectrums are inverse S-shaped curves, while the local singularity spectrums are asymmetric unimodal curves. If the moment order q approaches negative infinity, the generalized correlation dimension will seriously exceed the embedding space dimension 2, and the local fractal dimension curve displays an abnormal decrease for most cities. The scaling relation of local fractal dimension gradually breaks if the q value is too high, but the different levels of the network always keep the scaling reflecting singularity exponent. The main conclusions are as follows. First, urban street networks follow multifractal scaling law, and scaling precedes local fractal structure. Second, the patterns of traffic networks take on characteristics of spatial concentration, but they also show the implied trend of spatial deconcentration. Third, the development space of central area and network intensive areas is limited, while the fringe zone and network sparse areas show the phenomenon of disordered evolution. This work may be revealing for understanding and further research on complex spatial networks by using multifractal theory.
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CUI, GAOYUAN, BIN ZHANG, and RODRIGUES MARLENE. "TRAJECTORY SIMULATION OF BADMINTON ROBOT BASED ON FRACTAL BROWN MOTION." Fractals 28, no. 08 (August 10, 2020): 2040021. http://dx.doi.org/10.1142/s0218348x20400216.
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This paper focuses on the design of badminton robots, and designs high-precision binocular stereo vision synchronous acquisition system hardware and multithreaded acquisition programs to ensure the left and right camera exposure synchronization and timely reading of data. Aiming at specific weak moving targets, a shape-based Brown motion model based on dynamic threshold adjustment based on singular value decomposition is proposed, and a discriminative threshold is set according to the similarity between the background and the foreground to improve detection accuracy. The three-dimensional trajectory points are extended by Kalman filter and the kinematics equation of badminton is established. The parameters of the kinematics equation of badminton are solved by the method of least squares. Based on the fractal Brownian motion algorithm, a real-time robot pose estimation algorithm is proposed to realize the real-time accurate pose estimation of the robot. A PID control model for the badminton robot executive mechanism is established between the omnidirectional wheel speed and the robot’s translation and rotation movements to achieve the precise movement of the badminton robot. All the algorithms can meet the system’s requirements for real-time performance, realize the badminton robot’s simple hit to the ball, and prospect the future research direction.
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Hosseininia, M., M. H. Heydari, F. M. Maalek Ghaini, and Z. Avazzadeh. "A meshless technique based on the moving least squares shape functions for nonlinear fractal-fractional advection-diffusion equation." Engineering Analysis with Boundary Elements 127 (June 2021): 8–17. http://dx.doi.org/10.1016/j.enganabound.2021.03.003.
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Al Qurashi, Maysaa Mohamed. "Role of fractal-fractional operators in modeling of rubella epidemic with optimized orders." Open Physics 18, no. 1 (December 30, 2020): 1111–20. http://dx.doi.org/10.1515/phys-2020-0217.
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Abstract Fractal-fractional (FF) differential and integral operators having the capability to subsume features of retaining memory and self-similarities are used in the present research analysis to design a mathematical model for the rubella epidemic while taking care of dimensional consistency among the model equations. Infectious diseases have history in their transmission dynamics and thus non-local operators such as FF play a vital role in modeling dynamics of such epidemics. Monthly actual rubella incidence cases in Pakistan for the years 2017 and 2018 have been used to validate the FF rubella model and such a data set also helps for parameter estimation. Using nonlinear least-squares estimation with MATLAB function lsqcurvefit, some parameters for the classical and the FF model are obtained. Upon comparison of error norms for both models (classical and FF), it is found that the FF produces the smaller error. Locally asymptotically stable points (rubella-free and rubella-present) of the model are computed when the basic reproduction number { {\mathcal R} }_{0} is less and greater than unity and the sensitivity is investigated. Moreover, solution of the FF rubella system is shown to exist. A new iterative method is proposed to carry out numerical simulations which resulted in getting insights for the transmission dynamics of the rubella epidemic.
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Жихарев, Л., and L. Zhikharev. "Fractal Dimensionalities." Geometry & Graphics 6, no. 3 (November 14, 2018): 33–48. http://dx.doi.org/10.12737/article_5bc45918192362.77856682.
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One of the most important characteristics of a fractal is its dimensionality. In general, there are several options for mathematical definition of this value, but usually under the object dimensionality is understood the degree of space filling by it. It is necessary to distinguish the dimensionality of space and the dimension of multitude. Segment, square and cube are objects with dimensionality 1, 2 and 3, which can be in respective spaces: on a straight line, plane or in a 3D space. Fractals can have a fractional dimensionality. By definition, proposed by Bernois Mandelbrot, this fractional dimensionality should be less than the fractal’s topological dimension. Abram Samoilovich Bezikovich (1891–1970) was the author of first mathematical conclusions based on Felix Hausdorff (1868–1942) arguments and allowing determine the fractional dimensionality of multitudes. Bezikovich – Hausdorff dimensionality is determined through the multitude covering by unity elements. In practice, it is more convenient to use Minkowsky dimensionality for determining the fractional dimensionalities of fractals. There are also numerical methods for Minkowsky dimensionality calculation. In this study various approaches for fractional dimensionality determining are tested, dimensionalities of new fractals are defined. A broader view on the concept of dimensionality is proposed, its dependence on fractal parameters and interpretation of fractal sets’ structure are determined. An attempt for generalization of experimental dependences and determination of general regularities for fractals structure influence on their dimensionality is realized. For visualization of three-dimensional geometrical constructions, and plain evidence of empirical hypotheses were used computer models developed in the software for three-dimensional modeling (COMPASS, Inventor and SolidWorks), calculations were carried out in mathematical packages such as Wolfram Mathematica.
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Kermarrec, Gaël. "On Estimating the Hurst Parameter from Least-Squares Residuals. Case Study: Correlated Terrestrial Laser Scanner Range Noise." Mathematics 8, no. 5 (April 29, 2020): 674. http://dx.doi.org/10.3390/math8050674.
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Many signals appear fractal and have self-similarity over a large range of their power spectral densities. They can be described by so-called Hermite processes, among which the first order one is called fractional Brownian motion (fBm), and has a wide range of applications. The fractional Gaussian noise (fGn) series is the successive differences between elements of a fBm series; they are stationary and completely characterized by two parameters: the variance, and the Hurst coefficient (H). From physical considerations, the fGn could be used to model the noise of observations coming from sensors working with, e.g., phase differences: due to the high recording rate, temporal correlations are expected to have long range dependency (LRD), decaying hyperbolically rather than exponentially. For the rigorous testing of deformations detected with terrestrial laser scanners (TLS), the correct determination of the correlation structure of the observations is mandatory. In this study, we show that the residuals from surface approximations with regression B-splines from simulated TLS data allow the estimation of the Hurst parameter of a known correlated input noise. We derive a simple procedure to filter the residuals in the presence of additional white noise or low frequencies. Our methodology can be applied to any kind of residuals, where the presence of additional noise and/or biases due to short samples or inaccurate functional modeling make the estimation of the Hurst coefficient with usual methods, such as maximum likelihood estimators, imprecise. We demonstrate the feasibility of our proposal with real observations from a white plate scanned by a TLS.
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Zhong, Yang, Aiwen Lin, Lijie He, Zhigao Zhou, and Moxi Yuan. "Spatiotemporal Dynamics and Driving Forces of Urban Land-Use Expansion: A Case Study of the Yangtze River Economic Belt, China." Remote Sensing 12, no. 2 (January 15, 2020): 287. http://dx.doi.org/10.3390/rs12020287.
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It is important to analyze the expansion of an urban area and the factors that drive its expansion. Therefore, this study is based on Defense Meteorological Satellite Program Operational Linescan System (DMSP/OLS) night lighting data, using the landscape index, spatial expansion strength index, compactness index, urban land fractal index, elasticity coefficient, the standard deviation ellipse, spatial correlation analysis, and partial least squares regression to analyze the spatial and temporal evolution of urban land expansion and its driving factors in the Yangtze River Economic Belt (YREB) over a long period of time. The results show the following: Through the calculation of the eight landscape pattern indicators, we found that during the study period, the number of cities and towns and the area of urban built-up areas in the YREB are generally increasing. Furthermore, the variations in these landscape pattern indicators not only show more frequent exchanges and interactions between the cities and towns of the YREB, but also reflect significant instability and irregularity of the urbanization development in the YREB. The spatial expansion intensity indices of 1992–1999, 1999–2006, and 2006–2013 were 0.03, 0.16, and 0.34, respectively. On the whole, the urban compactness of the YREB decreased with time, and the fractal dimension increased slowly with time. Moreover, the long axis and the short axis of the standard deviation ellipse of the YREB underwent a small change during the inspection period. The spatial distribution generally showed the pattern of “southwest-north”. In terms of gravity shift, during the study period, the center of gravity moved from northeast to southwest. In addition, the Moran's I values for the four years of 1992, 1999, 2006, and 2013 were 0.451, 0.495, 0.506, and 0.424, respectively. Furthermore, by using correlation analysis, we find that the correlation coefficients between these four driving indicators and the urban expansion of the YREB were: 0.963, 0.998, 0.990 and 0.994, respectively. Through the use of partial least squares regression, we found that in 1992-2013, the four drivers of urban land expansion in the YREB were ranked as follows: gross domestic product (GDP), total fixed asset investment, urban population, total retail sales of consumer goods.
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Famoso, Nicholas A., and Edward Byrd Davis. "On the relationship between enamel band complexity and occlusal surface area in Equids (Mammalia, Perissodactyla)." PeerJ 4 (July 6, 2016): e2181. http://dx.doi.org/10.7717/peerj.2181.
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Enamel patterns on the occlusal surfaces of equid teeth are asserted to have tribal-level differences. The most notable example compares the Equini and Hipparionini, where Equini have higher crowned teeth with less enamel-band complexity and less total occlusal enamel than Hipparionini. Whereas previous work has successfully quantified differences in enamel band shape by dividing the length of enamel band by the square root of the occlusal surface area (Occlusal Enamel Index, OEI), it was clear that OEI only partially removes the effect of body size. Because enamel band length scales allometrically, body size still has an influence on OEI, with larger individuals having relatively longer enamel bands than smaller individuals. Fractal dimensionality (D) can be scaled to any level, so we have used it to quantify occlusal enamel complexity in a way that allows us to get at an accurate representation of the relationship between complexity and body size. To test the hypothesis of tribal-level complexity differences between Equini and Hipparionini, we digitally traced a sample of 98 teeth, one tooth per individual; 31 Hipparionini and 67 Equini. We restricted our sampling to the P3-M2 to reduce the effect of tooth position. After calculating theDof these teeth with the fractal box method which uses the number of boxes of various sizes to calculate theDof a line, we performed at-test on the individual values ofDfor each specimen, comparing the means between the two tribes, and a phylogenetically informed generalized least squares regression (PGLS) for each tribe with occlusal surface area as the independent variable andDas the dependent variable. The slopes of both PGLS analyses were compared using at-test to determine if the same linear relationship existed between the two tribes. Thet-test between tribes was significant (p< 0.0001), suggesting differentDpopulations for each lineage. The PGLS for Hipparionini was a positive but not significant (p= 0.4912) relationship betweenDand occlusal surface area, but the relationship for Equini was significantly negative (p= 0.0177).λwas 0 for both tests, indicating no important phylogenetic signal is present in the relationship between these two characters, thus the PGLS collapses down to a non-phylogenetic generalized least squares (GLS) model. Thet-test comparing the slopes of the regressions was not significant, indicating that the two lineages could have the same relationship betweenDand occlusal surface area. Our results suggest that the two tribes have the same negative relationship betweenDand occlusal surface area but the Hipparionini are offset to higher values than the Equini. This offset reflects the divergence between the two lineages since their last common ancestor and may have constrained their ability to respond to environmental change over the Neogene, leading to the differential survival of the Equini.
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Hearst, R. J., and P. Lavoie. "Decay of turbulence generated by a square-fractal-element grid." Journal of Fluid Mechanics 741 (February 17, 2014): 567–84. http://dx.doi.org/10.1017/jfm.2013.684.
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AbstractA novel square-fractal-element grid was designed in order to increase the downstream measurement range of fractal grid experiments relative to the largest element of the grid. The grid consists of a series of square fractal elements mounted to a background mesh with spacing$L_0 = 100\, {\rm mm}$. Measurements were performed in the region$3.5 \le x/L_0 \le 48.5$, which represents a significant extension to the$x/L_0 < 20$of previously reported square fractal grid measurements. For the region$x/L_0 \gtrsim 24$it was found that a power-law decay region following$\langle {q}^2 \rangle \sim (x - x_0)^m$exists with decay exponents of$m = -1.39$and$-1.37$at$\mathit{Re}_{L_0} = 57\, 000$and$65\, 000$, respectively. This agrees with decay values previously measured for regular grids ($-1 \gtrsim m \gtrsim -1.4$). The turbulence in the near-grid region,$x/L_0 < 20$, is shown to be inhomogeneous and anisotropic, in apparent contrast with previous fractal grid measurements. Nonetheless, power-law fits to the decay of turbulent kinetic energy in this region result in$m = -2.79$, similar to$m \approx -2.5$recently reported by Valente & Vassilicos (J. Fluid Mech., vol. 687, 2011, pp. 300–340) for space-filling square fractals. It was also found that$C_\epsilon $is approximately constant for$x/L_0 \ge 25$, while it grows rapidly for$x/L_0 < 20$. These results reconcile previous fractal-generated turbulence measurements with classical grid turbulence measurements.
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Anisimova, Elizaveta A. "Kazimir Malevich: Fractalization as a Way to Suprematism." Observatory of Culture, no. 6 (December 28, 2015): 66–73. http://dx.doi.org/10.25281/2072-3156-2015-0-6-66-73.
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The artistic methods of K. Malevich are discussed in the article in the light of the theory of fractals. It is claimed that the main creative ideas of the artist can be described using the notions of visual, semiotic and dynamical fractals. The author demonstrates that the fractal analysis allows us to designate the artistic works of K. Malevich as visual, semiotic and dynamical fractals, and the image fractalization provides us with some specific perception effects. The article investigates the fractality of the “Black Square” and the multidimensional fractality of architectons.
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Reddy, V. V., and N. V. S. N. Sarma. "Reactive impedance surface-based broadband circularly polarized Koch fractal boundary microstrip antenna." International Journal of Microwave and Wireless Technologies 8, no. 2 (December 22, 2014): 243–50. http://dx.doi.org/10.1017/s1759078714001421.
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A circularly polarized (CP) broadband antenna is proposed for wireless applications in the range of 2–3 GHz frequency. It consists of asymmetrical Koch fractal boundary patch over a reactive impedance surface (RIS) substrate. The simulations of single-layer Koch fractal antenna, dual layer with square and fractal RIS elements are carried out in a systematic way for broadband CP radiation and corresponding results are presented. For better CP characteristics, properties of fractal curves and dimensions of RIS elements are optimized. The antenna with fractal RIS iteration order one (iteration1) is experimentally studied. The 10-dB return loss bandwidth is 50.35%, whereas 3-dB axial ratio bandwidth is 7.45%, which indicate that by applying fractals concept to RIS technique, with a single probe feed, broadband CP radiation can be obtained.
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Zheng, Hong, and Hongfei Lu. "A least-squares support vector machine (LS-SVM) based on fractal analysis and CIELab parameters for the detection of browning degree on mango (Mangifera indica L.)." Computers and Electronics in Agriculture 83 (April 2012): 47–51. http://dx.doi.org/10.1016/j.compag.2012.01.012.
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SUCHARD, EYTAN H. "SQUARE FRACTAL ALGORITHM." Fractals 13, no. 01 (March 2005): 43–55. http://dx.doi.org/10.1142/s0218348x05002763.
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Spanning a planar graph the way D. Hilbert's curve does has various image processing and industrial applications. Spanning a planar graph by two disjoint curves with fractal properties has even more scientific and industrial uses. For example, given two liquids and an active osmosis through membrane between them, we would like to both cool the liquids and to find a cost-effective structure for the osmosis to occur. Another equivalent problem is to expose two liquids to light that passes through a transparent slab as the osmosis between them occurs. Two disjoint curves can be the answer for the required structure. Differences of lengths between the curves can also be useful. A fractal structure is obvious in the lungs, where osmosis of oxygen is vital. Fractal structures are often found in organic osmotic processes in Nature. In this article, a method for spanning a planar graph by two disjoint curves will be presented.
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Zhou, Tianqi, Chaodong Wu, Zhongkui Shi, Jialin Wang, Wen Zhu, Bo Yuan, and Disheng Yang. "Multi-Scale Quantitative Characterization of Pore Distribution Networks in Tight Sandstone by integrating FE-SEM, HPMI, and NMR with the Constrained Least Squares Algorithm." Energies 12, no. 18 (September 12, 2019): 3514. http://dx.doi.org/10.3390/en12183514.
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The goal of this study was to investigate the impacts of various sedimentary-diagenetic conditions on the macroscopic petrophysical parameters and microscopic pore structures of tight sandstones from the Lower Jurassic Badaowan Formation in the Southern Junggar Basin, China. Based on the traditional methods for establishing pore size distribution, including integrating the results of high-pressure mercury injection, nuclear magnetic resonance, and scanning electron microscopy, the constrained least squares algorithm was employed to automatically determine the porosity contributions of pore types with different origins. The results show that there are six genetic pore types: residual intergranular pores (RIPs), feldspar dissolution pores (FDPs), rock fragment dissolution pores (RFDPs), clay mineral intergranular pores (CIPs), intercrystalline pores of kaolinite (IPKs), and matrix pores (MPs). Four lithofacies were identified: the quartz cemented-dissolution facies (QCDF), carbonate cemented facies (CCF), authigenic clay mineral facies (ACMF), and matrix-caused tightly compacted facies (MCTF). Modified by limited dissolution, the QCDF with a high proportion of macropores (RIPs, FDPs, and RFDPs) exhibited a slightly higher porosity and considerably higher permeability than those of others. A large number of micropores (MPs, CIPs, and IPKs) in MCTF and ACMF led to slightly lower porosities but considerably lower permeabilities. Due to the tightly cemented carbonates in the CCF, its porosity reduced sharply, but the permeability of the CCF remained much higher those of the MCTF and ACMF. The results highlight that a high proportion of macropores with large radii and regular shapes provide more effective percolation paths than storage spaces. Nevertheless, micropores with small radii and complex pore structures have a limited contribution to flow capability. The fractal dimension analysis shows that a high proportion of MPs is the major reason for the heterogeneity in tight sandstones. The formation of larger macropores with smooth surfaces are more conductive for oil and gas accumulation.
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CRISTEA, LIGIA L., and PAUL SURER. "TRIANGULAR LABYRINTH FRACTALS." Fractals 27, no. 08 (December 2019): 1950131. http://dx.doi.org/10.1142/s0218348x19501317.
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We define and study a class of fractal dendrites called triangular labyrinth fractals. For the construction, we use triangular labyrinth pattern systems, consisting of two triangular patterns: a white and a yellow one. Correspondingly, we have two fractals: a white and a yellow one. The fractals studied here are self-similar, and fit into the framework of graph directed constructions. The main results consist in showing how special families of triangular labyrinth patterns systems, which are defined based on some shape features, can generate exactly three types of dendrites: labyrinth fractals where all nontrivial arcs have infinite length, fractals where all nontrivial arcs have finite length, or fractals where the only arcs of finite lengths are line segments parallel to a certain direction. We also study the existence of tangents to arcs. The paper is inspired by research done on labyrinth fractals in the unit square that have been studied during the last decade. In the triangular case, due to the geometry of triangular shapes, some new techniques and ideas are necessary in order to obtain the results.
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HUANG, LIN-SHAN, and YAN-GUANG CHEN. "A COMPARISON BETWEEN TWO OLS-BASED APPROACHES TO ESTIMATING URBAN MULTIFRACTAL PARAMETERS." Fractals 26, no. 01 (February 2018): 1850019. http://dx.doi.org/10.1142/s0218348x18500196.
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Multifractal theory provides a new spatial analytical tool for urban studies, but many basic problems remain to be solved. Among various pending issues, the most significant one is how to obtain proper multifractal dimension spectrums. If an algorithm is improperly used, the parameter spectrums will be abnormal. This paper is devoted to investigating two ordinary least squares (OLS)-based approaches for estimating urban multifractal parameters. Using empirical study and comparative analysis, we demonstrate how to utilize the adequate linear regression to calculate multifractal parameters. The OLS regression analysis has two different approaches. One is that the intercept is fixed to zero, and the other is that the intercept is not limited. The results of comparative study show that the zero-intercept regression yields proper multifractal parameter spectrums within certain scale range of moment order, while the common regression method often leads to abnormal multifractal parameter values. A conclusion can be reached that fixing the intercept to zero is a more advisable regression method for multifractal parameters estimation, and the shapes of spectral curves and value ranges of fractal parameters can be employed to diagnose urban problems. This research is helpful for scientists to understand multifractal models and apply a more reasonable technique to multifractal parameter calculations.
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Spivey, Alvin, and Anthony Vodacek. "Fourier Landscape Pattern Indices for Predicting South Carolina Watershed Fecal Coliform." Journal of Landscape Ecology 10, no. 1 (January 1, 2017): 20–34. http://dx.doi.org/10.1515/jlecol-2017-0007.
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AbstractExtending the Landscape Pattern Metric (LPM) model analysis in Smith et al. (2001) into a LPM decision model, decadal scale prediction of fecal coliform compromised South Carolina watersheds is developed. The model’s parameter variability identifies the greatest contributors to a compromised watershed’s prediction. The complete set of model parameters include Land Cover Land Use (LCLU) & slope,along stream proportion, Fourier Metric of Fragmentation (FMF), Fourier Metric of Proportion (FMP), and Least Squares Fourier Transform Fractal Dimension (LsFT). The 1992 National Land Cover Data (NLCD) Land Cover Land Use (LCLU) within fecal coliform compromised watersheds is used to train the model parameters, and the 2001 NLCD LCLU is used to test the LPM model. The most significant model parameters arealong stream bare rock LsFT,FMF between urban/recreational grasses and evergreen forests, andFMF between deciduous forests and high density residential areas. These metrics contribute significantly more than the bestproportiondescriptor:proportion of urban/recreational grasses. In training, the proposed model correctly identified 92 % of the compromised watersheds; while the Smith et al. (2001) model 94 % of the compromised watersheds were correctly identified. This study reveals the ability of Fourier metrics to interpret ecological processes, and the need for more appropriate landscape level models.
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Asadzadeh, Seyed Mohammad, and Roberto Galeazzi. "The predictive power of track dynamic response for monitoring ballast degradation in turnouts." Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit 234, no. 9 (November 4, 2019): 976–91. http://dx.doi.org/10.1177/0954409719883547.
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This paper develops a novel method for the prediction of the level of degradation of the ballast layer in railway turnouts by establishing a statistical mapping between the train-induced vertical track acceleration and the vertical track geometry. This is performed by exploiting the full-scale data provided by a loaded geometry car and the track-side sensing system over a two-year period. Irregularities in the vertical track geometry are analyzed and correlated with the level of ballast degradation by means of fractal analysis. Partial least squares regression is then employed to identify the components of the power spectral density of vertical track acceleration, which provided more information about the ballast degradation. The proposed method has a remarkable predictive power of the ballast degradation as evaluated both in terms of accuracy and consistency of the prediction. The reported results, based on the data from four different locations of the turnout before and after a ballast tamping event, show a mean absolute prediction error between 2% and 8% and a minimum correlation greater than 90%. The proposed method enables the continuous monitoring of the track substructure components by exploiting the vertical track acceleration as a supplementary source for the prediction of ballast degradation.
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Calatayud, A., V. Ferrando, F. Giménez, W. D. Furlan, G. Saavedra, and J. A. Monsoriu. "Fractal square zone plates." Optics Communications 286 (January 2013): 42–45. http://dx.doi.org/10.1016/j.optcom.2012.09.002.
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ZHANG, JUNHUAN, and JUN WANG. "FRACTAL DETRENDED FLUCTUATION ANALYSIS OF CHINESE ENERGY MARKETS." International Journal of Bifurcation and Chaos 20, no. 11 (November 2010): 3753–68. http://dx.doi.org/10.1142/s0218127410028082.
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In this paper, we analyze and compare long-range power-law correlations of returns, absolute returns, squared returns, cubed returns and square waved returns for sixteen individual stocks from the block of energy sources of Chinese stock market and five stock indices (Shanghai Composite Index, Shenzhen Component Index, Dow Jones Industrial Average index, Nasdaq Composite Index, the Standard and Poor's 500 Index) by using a detrended fluctuation analysis approach. The empirical evidence suggests that Shanghai Composite Index is very close to Shenzhen Component Index and Nasdaq, DJIA is very close to S&P 500 in all cases. And the exponent trends of the returns are close to that of square waved returns. Also, five indices deviate from other sixteen individual energy stocks in all cases except square waved returns. Further, there are long-range correlations and persistence in volatility series of absolute returns and squared returns. Moreover, we investigate the long-term memory of these returns by applying Lo's modified rescaled range statistic. We find that the China energy market exhibits fractal and persistence properties.