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Journal articles on the topic 'Fractal dimensions'
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1
Chen, Yanguang. "Characterizing Growth and Form of Fractal Cities with Allometric Scaling Exponents." Discrete Dynamics in Nature and Society 2010 (2010): 1–22. http://dx.doi.org/10.1155/2010/194715.
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Fractal growth is a kind of allometric growth, and the allometric scaling exponents can be employed to describe growing fractal phenomena such as cities. The spatial features of the regular fractals can be characterized by fractal dimension. However, for the real systems with statistical fractality, it is incomplete to measure the structure of scaling invariance only by fractal dimension. Sometimes, we need to know the ratio of different dimensions rather than the fractal dimensions themselves. A fractal-dimension ratio can make an allometric scaling exponent (ASE). As compared with fractal dimension, ASEs have three advantages. First, the values of ASEs are easy to be estimated in practice; second, ASEs can reflect the dynamical characters of system's evolution; third, the analysis of ASEs can be made through prefractal structure with limited scale. Therefore, the ASEs based on fractal dimensions are more functional than fractal dimensions for real fractal systems. In this paper, the definition and calculation method of ASEs are illustrated by starting from mathematical fractals, and, then, China's cities are taken as examples to show how to apply ASEs to depiction of growth and form of fractal cities.
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CHEN, YAN-GUANG. "FRACTAL TEXTURE AND STRUCTURE OF CENTRAL PLACE SYSTEMS." Fractals 28, no. 01 (February 2020): 2050008. http://dx.doi.org/10.1142/s0218348x20500085.
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The boundaries of central place models proved to be fractal lines, which compose fractal texture of central place networks. However, the fractal texture cannot be verified by empirical analyses based on observed data. On the other hand, fractal structure of central place systems in the real world can be empirically confirmed by positive studies, but there are no corresponding models. The spatial structure of classic central place models bears Euclidean dimension [Formula: see text] rather than fractal dimensions [Formula: see text]. This paper is devoted to deriving structural fractals of central place models from the textural fractals. The method is theoretical deduction based on the dimension rules of fractal sets. The main results and findings are as follows. First, the central place fractals were formulated by the [Formula: see text] numbers and [Formula: see text] numbers. Second, three structural fractal models were constructed for central place systems according to the corresponding fractal dimensions. Third, the classic central place models proved to comprise Koch snowflake curve, Sierpinski space filling curve, and Gosper snowflake curve. Moreover, the traffic principle plays a leading role in urban and rural settlements evolution. A conclusion was reached that the textural fractal dimensions of central place models can be converted into the structural fractal dimensions and vice versa, and the structural dimensions can be directly used to appraise human settlement distributions in reality. Thus, the textural fractals can be indirectly employed to characterize the systems of human settlements.
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LIAW, SY-SANG, and FENG-YUAN CHIU. "CONSTRUCTING CROSSOVER-FRACTALS USING INTRINSIC MODE FUNCTIONS." Advances in Adaptive Data Analysis 02, no. 04 (October 2010): 509–20. http://dx.doi.org/10.1142/s1793536910000598.
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Real nonstationary time sequences are in general not monofractals. That is, they cannot be characterized by a single value of fractal dimension. It has been shown that many real-time sequences are crossover-fractals: sequences with two fractal dimensions — one for the short and the other for long ranges. Here, we use the empirical mode decomposition (EMD) to decompose monofractals into several intrinsic mode functions (IMFs) and then use partial sums of the IMFs decomposed from two monofractals to construct crossover-fractals. The scale-dependent fractal dimensions of these crossover-fractals are checked by the inverse random midpoint displacement method (IRMD).
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Chen, Yanguang. "Fractal Modeling and Fractal Dimension Description of Urban Morphology." Entropy 22, no. 9 (August 30, 2020): 961. http://dx.doi.org/10.3390/e22090961.
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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.
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Iversen, P. O., and G. Nicolaysen. "High correlation of fractals for regional blood flows among resting and exercising skeletal muscles." American Journal of Physiology-Heart and Circulatory Physiology 269, no. 1 (July 1, 1995): H7—H13. http://dx.doi.org/10.1152/ajpheart.1995.269.1.h7.
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The regional blood flow distributions within single skeletal muscles are markedly uneven both at rest and during exercise hyperemia. Fractals adequately describe this perfusion heterogeneity in the resting lateral head of the gastrocnemius muscle as well as in the myocardium. Recently, we provided evidence that the fractal dimension for the blood flow distributions in this resting muscle was strongly correlated with that of the myocardium in the same rabbit. Prompted by this hitherto unknown observation, we have now examined 1) whether fractals also describe perfusion distributions within muscles with a varying metabolic activity, and 2) whether the fractal dimensions for blood flow distributions to these muscles were correlated. We used pentobarbital-anesthetized rabbits and cats. The regional distributions of blood flow within various skeletal muscles were estimated by microsphere trapping. The data unequivocally showed that the perfusion distributions could be described with fractals both in resting and in exercising muscle in both species, the corresponding fractal dimensions ranging from 1.36 to 1.41. The fractal dimensions were markedly correlated (r2 ranged from 0.82 to 0.88) when both various resting and resting plus exercising muscles were compared in the same animal. This surprising finding of high correlations for the fractal dimensions among various muscles within one animal provides a novel characteristic of blood flow heterogeneity.
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Sreenivasan, K. R., and C. Meneveau. "The fractal facets of turbulence." Journal of Fluid Mechanics 173 (December 1986): 357–86. http://dx.doi.org/10.1017/s0022112086001209.
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Speculations abound that several facets of fully developed turbulent flows are fractals. Although the earlier leading work of Mandelbrot (1974, 1975) suggests that these speculations, initiated largely by himself, are plausible, no effort has yet been made to put them on firmer ground by, resorting to actual measurements in turbulent shear flows. This work is an attempt at filling this gap. In particular, we examine the following questions: (a) Is the turbulent/non-turbulent interface a self-similar fractal, and (if so) what is its fractal dimension ? Does this quantity differ from one class of flows to another? (b) Are constant-property surfaces (such as the iso-velocity and iso-concentration surfaces) in fully developed flows fractals? What are their fractal dimensions? (c) Do dissipative structures in fully developed turbulence form a fractal set? What is the fractal dimension of this set? Answers to these questions (and others to be less fully discussed here) are interesting because they bring the theory of fractals closer to application to turbulence and shed new light on some classical problems in turbulence - for example, the growth of material lines in a turbulent environment. The other feature of this work is that it tries to quantify the seemingly complicated geometric aspects of turbulent flows, a feature that has not received its proper share of attention. The overwhelming conclusion of this work is that several aspects of turbulence can be described roughly by fractals, and that their fractal dimensions can be measured. However, it is not clear how (or whether), given the dimensions for several of its facets, one can solve (up to a useful accuracy) the inverse problem of reconstructing the original set (that is, the turbulent flow itself).
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Lutz, Neil. "Fractal Intersections and Products via Algorithmic Dimension." ACM Transactions on Computation Theory 13, no. 3 (September 30, 2021): 1–15. http://dx.doi.org/10.1145/3460948.
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Algorithmic fractal dimensions quantify the algorithmic information density of individual points and may be defined in terms of Kolmogorov complexity. This work uses these dimensions to bound the classical Hausdorff and packing dimensions of intersections and Cartesian products of fractals in Euclidean spaces. This approach shows that two prominent, fundamental results about the dimension of Borel or analytic sets also hold for arbitrary sets.
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MCGINLEY, PATTON, ROBIN G. SMITH, and JEROME C. LANDRY. "FRACTAL DIMENSIONS OF MYCOSIS FUNGOIDES." Fractals 02, no. 04 (December 1994): 493–501. http://dx.doi.org/10.1142/s0218348x94000715.
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Prior to an investigation of early diagnosis of mycosis fungoides (MF) using fractal geometry, we set out to see if MF lesions are fractal in nature. We analyzed three aspects of MF lesions: the dermoepidermal profile of photomicrographs of patch stage lesions and normal skin, the perimeter of patch and plaque stage lesions, and the size distribution of patch and plaque lesions on the skin surface. The perimeter of plaque lesions was measured on close-up photographs by the divider walk method using various step sizes. Based on the perimeter values, the fractal dimension was determined. The dermoepidermal profile of MF patch lesions was analyzed by the divider walk method for self-affine fractals. The size distribution of MF patch and plaque lesions was determined by counting the number of patch and plaque lesions with an area greater than or equal to a specific size A on scaled photographs of a 19.6 cm × 19.6 cm affected region. A plot of number of lesions with area greater than or equal to A vs. lesion area on log-log paper allows the detection of a power-law distribution, indicative of one type of self-similar fractals. The dermoepidermal profile of patch stage lesions and normal skin was found to be self-affine fractals. Global measurements of normal thin skin and of patch stage lesions were distinct. All observed patch and plaque lesion area distributions were a fractal set. The perimeter of non-confluent plaque lesions was not fractal. This work revealed fractal dimensions in two aspects of MF lesions. Further investigation of application of fractal geometry to the diagnosis and staging of MF is planned.
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Zhang, Pei-Lin, Bing Li, Shuang-Shan Mi, Ying-Tang Zhang, and Dong-Sheng Liu. "Bearing Fault Detection Using Multi-Scale Fractal Dimensions Based on Morphological Covers." Shock and Vibration 19, no. 6 (2012): 1373–83. http://dx.doi.org/10.1155/2012/438789.
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Vibration signals acquired from bearing have been found to demonstrate complicated nonlinear characteristics in literature. Fractal geometry theory has provided effective tools such as fractal dimension for characterizing the vibration signals in bearing faults detection. However, most of the natural signals are not critical self-similar fractals; the assumption of a constant fractal dimension at all scales may not be true. Motivated by this fact, this work explores the application of the multi-scale fractal dimensions (MFDs) based on morphological cover (MC) technique for bearing fault diagnosis. Vibration signals from bearing with seven different states under four operations conditions are collected to validate the presented MFDs based on MC technique. Experimental results reveal that the vibration signals acquired from bearing are not critical self-similar fractals. The MFDs can provide more discriminative information about the signals than the single global fractal dimension. Furthermore, three classifiers are employed to evaluate and compare the classification performance of the MFDs with other feature extraction methods. Experimental results demonstrate the MFDs to be a desirable approach to improve the performance of bearing fault diagnosis.
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XIA, YUXUAN, JIANCHAO CAI, WEI WEI, XIANGYUN HU, XIN WANG, and XINMIN GE. "A NEW METHOD FOR CALCULATING FRACTAL DIMENSIONS OF POROUS MEDIA BASED ON PORE SIZE DISTRIBUTION." Fractals 26, no. 01 (February 2018): 1850006. http://dx.doi.org/10.1142/s0218348x18500068.
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Fractal theory has been widely used in petrophysical properties of porous rocks over several decades and determination of fractal dimensions is always the focus of researches and applications by means of fractal-based methods. In this work, a new method for calculating pore space fractal dimension and tortuosity fractal dimension of porous media is derived based on fractal capillary model assumption. The presented work establishes relationship between fractal dimensions and pore size distribution, which can be directly used to calculate the fractal dimensions. The published pore size distribution data for eight sandstone samples are used to calculate the fractal dimensions and simultaneously compared with prediction results from analytical expression. In addition, the proposed fractal dimension method is also tested through Micro-CT images of three sandstone cores, and are compared with fractal dimensions by box-counting algorithm. The test results also prove a self-similar fractal range in sandstone when excluding smaller pores.
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CHEN, WEN-SHIUNG, and SHANG-YUAN YUAN. "SOME FRACTAL DIMENSION ESTIMATE ALGORITHMS AND THEIR APPLICATIONS TO ONE-DIMENSIONAL BIOMEDICAL SIGNALS." Biomedical Engineering: Applications, Basis and Communications 14, no. 03 (June 25, 2002): 100–108. http://dx.doi.org/10.4015/s1016237202000152.
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Fractals can model many classes of time-series data. The fractal dimension is an important characteristic of fractals that contains information about their geometrical structure at multiple scales. The covering methods are a class of efficient approaches, e.g., box-counting (BC) method, to estimate the fractal dimension. In this paper, the differential box-counting (DBC) approach, originally for 2-D applications, is modified and applied to 1-D case. In addition, two algorithms, called 1-D shifting-DBC (SDBC-1D) and 1-D scanning-BC (SBC-1D), are also proposed for 1-D signal analysis. The fractal dimensions for 1-D biomedical pulse and ECG signals are calculated.
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Semkow, Thomas M. "Neighborhood Volume for Bounded, Locally Self-Similar Fractals." Fractals 05, no. 01 (March 1997): 23–33. http://dx.doi.org/10.1142/s0218348x97000048.
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We derive the formulas for neighborhood volume (Minkowski volume in d-dimensions) for fractals which have a curvature bias and are thus bounded. Both local surface fractal dimension and local mass fractal dimension are included as well as a radius of the neighborhood volume comparable with the size of the fractal. We consider two types of the neighborhood volumes: simplified and generalized, as well as the volumes below and above the fractal boundary. The formulas derived are generalizations of the equations for isotropic unbounded fractals. Based on the simplified-volume concept, we establish the procedure for calculating a distribution of physical quantities on bounded fractals and apply it to the distribution of trace elements in soil particles. Using the concept of the generalized volume, we show how an expectation value of a physical process can be calculated on bounded fractals, and apply it to the radon emanation from solid particles.
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Hermanowicz, S. W., U. Schindler, and P. Wilderer. "Fractal structure of biofilms: new tools for investigation of morphology." Water Science and Technology 32, no. 8 (October 1, 1995): 99–105. http://dx.doi.org/10.2166/wst.1995.0273.
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Fractal dimension was used to describe morphology of a biofilm. Images of biofilm sections were obtained with a confocal laser scanning microscope and were further enhanced using image analysis software. Fractal dimensions were estimated from the slopes of cross-correlation functions. Two geometric scales with different fractal dimensions were identified in the biofilm. Small scale biomass clusters (< 5 μm) had fractal dimensions close to the topological dimension while the fractal dimensions of larger aggregates were considerably smaller. Anisotropic morphology was also detected by the difference of fractal dimensions and was possibly related to the direction of water flow.
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Meng, Xianmeng, Pengju Zhang, Jing Li, Chuanming Ma, and Dengfeng Liu. "The linkage between box-counting and geomorphic fractal dimensions in the fractal structure of river networks: the junction angle." Hydrology Research 51, no. 6 (October 15, 2020): 1397–408. http://dx.doi.org/10.2166/nh.2020.082.
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Abstract In the past, a great deal of research has been conducted to determine the fractal properties of river networks, and there are many kinds of methods calculating their fractal dimensions. In this paper, we compare two most common methods: one is geomorphic fractal dimension obtained from the bifurcation ratio and the stream length ratio, and the other is box-counting method. Firstly, synthetic fractal trees are used to explain the role of the junction angle on the relation between two kinds of fractal dimensions. The obtained relationship curves indicate that box-counting dimension is decreasing with the increase of the junction angle when geomorphic fractal dimension keeps constant. This relationship presents continuous and smooth convex curves with junction angle from 60° to 120° and concave curves from 30° to 45°. Then 70 river networks in China are investigated in terms of their two kinds of fractal dimensions. The results confirm the fractal structure of river networks. Geomorphic fractal dimensions of river networks are larger than box-counting dimensions and there is no obvious relationship between these two kinds of fractal dimensions. Relatively good non-linear relationships between geomorphic fractal dimensions and box-counting dimensions are obtained by considering the role of the junction angle.
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Li, Chang, Liqiang Sima, Guoqiong Che, Wang Liang, Anjiang Shen, Qingxin Guo, and Bing Xie. "Vug and fracture characterization and gas production prediction by fractals: Carbonate reservoir of the Longwangmiao Formation in the Moxi-Gaoshiti area, Sichuan Basin." Interpretation 8, no. 3 (July 23, 2020): SL159—SL171. http://dx.doi.org/10.1190/int-2019-0260.1.
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A comprehensive knowledge of the development and connectivity of fractures and vugs in carbonate reservoirs plays a key role in reservoir evaluation, ultimately affecting the gas prediction of this kind of heterogeneous reservoir. The carbonate reservoirs with fractures and vugs that are well developed in the Longwangmiao Formation, Sichuan Basin are selected as a research target, with the fractal dimension calculated from the full-bore formation microimager (FMI) image proposed to characterize the fractures and vugs. For this purpose, the multipoint statistics algorithm is first used to reconstruct a high-resolution FMI image of the full borehole wall. And then, the maximum class-variance method (the Otsu method) realizes the automatic threshold segmentation of the FMI image and acquisition of the binary image, which accurately characterizes the fractures and vugs. Finally, the fractal dimension is calculated by the box dimension algorithm, with its small value difference enlarged to obtain a new fractal parameter ([Formula: see text]). The fractal dimensions for four different kinds of reservoirs, including eight subdivided models of vugs and fractures, show that the fractal dimension can characterize the development and the connectivity of fractures and vugs comprehensively. That is, the more developed that the fractures and vugs are, the better the connectivity will be, and simultaneously the smaller that the values of the fractal dimensions are. The fractal dimension is first applied to the gas production prediction by means of constructing a new parameter ([Formula: see text]) defined as a multiple of the effective thickness ([Formula: see text]), porosity (Por), and fractal dimension ([Formula: see text]). The field examples illustrate that the fractal dimensions can effectively characterize the fractures and vugs in the heterogeneous carbonate reservoir and predict its gas production. In summary, the fractals expand the characterization method for the vugs and fractures in carbonate reservoirs and extend its new application in gas production prediction.
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Chen. "The Solutions to the Uncertainty Problem of Urban Fractal Dimension Calculation." Entropy 21, no. 5 (April 30, 2019): 453. http://dx.doi.org/10.3390/e21050453.
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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.
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JÄRVENPÄÄ, ESA, MAARIT JÄRVENPÄÄ, ANTTI KÄENMÄKI, HENNA KOIVUSALO, ÖRJAN STENFLO, and VILLE SUOMALA. "Dimensions of random affine code tree fractals." Ergodic Theory and Dynamical Systems 34, no. 3 (January 30, 2013): 854–75. http://dx.doi.org/10.1017/etds.2012.168.
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AbstractWe study the dimension of code tree fractals, a class of fractals generated by a set of iterated function systems. We first consider deterministic affine code tree fractals, extending to the code tree fractal setting the classical result of Falconer and Solomyak on the Hausdorff dimension of self-affine fractals generated by a single iterated function system. We then calculate the almost sure Hausdorff, packing and box counting dimensions of a general class of random affine planar code tree fractals. The set of probability measures describing the randomness includes natural measures in random $V$-variable and homogeneous Markov constructions.
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Fioravanti, Stefano, and Daniele D. Giusto. "Estimation of qth-Order Fractal Dimensions." Fractals 05, supp01 (April 1997): 257–69. http://dx.doi.org/10.1142/s0218348x97000802.
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The paper deals with the theory of qth-order fractal dimensions and its application to texture analysis. In particular, the state-of-the-art regarding the fractal dimension estimation for characterizing textures is presented. After, the insufficiency of the single fractal dimension is proven and the qth order fractal dimensions are introduced to overcome such drawback. The multifractality spectrum function D(q) is described, a novel algorithm for estimating such dimensions is then proposed, and its use in digital-image processing is addressed. Results on real SAR image textures are reported and discussed.
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Hagerhall, Caroline M., Thorbjörn Laike, Richard P. Taylor, Marianne Küller, Rikard Küller, and Theodore P. Martin. "Investigations of Human EEG Response to Viewing Fractal Patterns." Perception 37, no. 10 (January 1, 2008): 1488–94. http://dx.doi.org/10.1068/p5918.
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Owing to the prevalence of fractal patterns in natural scenery and their growing impact on cultures around the world, fractals constitute a common feature of our daily visual experiences, raising an important question: what responses do fractals induce in the observer? We monitored subjects' EEG while they were viewing fractals with different fractal dimensions, and the results show that significant effects could be found in the EEG even by employing relatively simple silhouette images. Patterns with a fractal dimension of 1.3 elicited the most interesting EEG, with the highest alpha in the frontal lobes but also the highest beta in the parietal area, pointing to a complicated interplay between different parts of the brain when experiencing this pattern.
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Griffen, Dana T., and Kim R. Sullivan. "X-Ray Powder Diffraction Patterns as Random Fractals." Advances in X-ray Analysis 39 (1995): 739–46. http://dx.doi.org/10.1154/s0376030800023193.
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The concept of fractals (or fractional dimensions), although known earlier, was formalized theoretically and given that name in 1967 by Mandelbrot. A fractal is some object, whether mathematically constructed or observed in the physical world, that exhibits scale in variance—that is, it looks essentially the same at all scales, or over some range of scales. Objects that exhibit fractal geometry and that are measured in the same units in both the x and y directions are said to be self-similar; geologic examples of self-similar fractals are a rocky coastline and a topographic contour, for which the east-west and north-south coordinates of any point are expressed in, say, meters or kilometers. Objects that exhibit fractal geometry but which are measured in different units along x and y are said to be selfaffine fractals; an example of a self-affine fractal is a topographic profile, in which the horizontal dimension is measured in kilometers and the vertical dimension is measured in meters.
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MEI, MAOFEI, BOMING YU, JIANCHAO CAI, and LIANG LUO. "A HIERARCHICAL MODEL FOR MULTI-PHASE FRACTAL MEDIA." Fractals 18, no. 01 (March 2010): 53–64. http://dx.doi.org/10.1142/s0218348x1000466x.
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The size distributions of solid particles and pores in porous media are approximately hierarchical and statistically fractals. In this paper, a model for single-phase fractal media is constructed, and the analytical expressions for area, fractal dimension and distribution function for solid particles are derived. The distribution function of solid particles obtained from the proposed model is in good agreement with available experimental data. Then, a model for approximate two-phase fractal media is developed. Good agreement is found between the predicted fractal dimensions for pore space from the two-phase fractal medium model and the existing measured data. A model for approximate three-phase fractal media is also presented by extending the obtained two-phase model.
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MITINA, OLGA V., and FREDERICK DAVID ABRAHAM. "THE USE OF FRACTALS FOR THE STUDY OF THE PSYCHOLOGY OF PERCEPTION: PSYCHOPHYSICS AND PERSONALITY FACTORS, A BRIEF REPORT." International Journal of Modern Physics C 14, no. 08 (October 2003): 1047–60. http://dx.doi.org/10.1142/s0129183103005182.
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The present article deals with perception of time (subjective assessment of temporal intervals), complexity and aesthetic attractiveness of visual objects. The experimental research for construction of functional relations between objective parameters of fractals' complexity (fractal dimension and Lyapunov exponent) and subjective perception of their complexity was conducted. As stimulus material we used the program based on Sprott's algorithms for the generation of fractals and the calculation of their mathematical characteristics. For the research 20 fractals were selected which had different fractal dimensions that varied from 0.52 to 2.36, and the Lyapunov exponent from 0.01 to 0.22. We conducted two experiments: (1) A total of 20 fractals were shown to 93 participants. The fractals were displayed on the screen of a computer for randomly chosen time intervals ranging from 5 to 20 s. For each fractal displayed, the participant responded with a rating of the complexity and attractiveness of the fractal using ten-point scale with an estimate of the duration of the presentation of the stimulus. Each participant also answered the questions of some personality tests (Cattell and others). The main purpose of this experiment was the analysis of the correlation between personal characteristics and subjective perception of complexity, attractiveness, and duration of fractal's presentation. (2) The same 20 fractals were shown to 47 participants as they were forming on the screen of the computer for a fixed interval. Participants also estimated subjective complexity and attractiveness of fractals. The hypothesis on the applicability of the Weber–Fechner law for the perception of time, complexity and subjective attractiveness was confirmed for measures of dynamical properties of fractal images.
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Wang, J., and S. Ogawa. "Fractal Analysis Of Colors And Shapes For Natural And Urbanscapes URBANSCAPES." ISPRS - International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences XL-7/W3 (April 30, 2015): 1431–38. http://dx.doi.org/10.5194/isprsarchives-xl-7-w3-1431-2015.
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Fractal analysis has been applied in many fields since it was proposed by Mandelbrot in 1967. Fractal dimension is a basic parameter of fractal analysis. According to the difference of fractal dimensions for images, natural landscapes and urbanscapes could be differentiated, which is of great significance. In this paper, two methods were used for two types of landscape images to discuss the difference between natural landscapes and urbanscapes. Traditionally, a box-counting method was adopted to evaluate the shape of grayscale images. On the other way, for the spatial distributions of RGB values in images, the fractal Brownian motion (fBm) model was employed to calculate the fractal dimensions of colour images for two types of landscape images. From the results, the fractal dimensions of natural landscape images were lower than that of urbanscapes for both grayscale images and colour images with two types of methods. Moreover, the spatial distributions of RGB values in images were clearly related with the fractal dimensions. The results indicated that there was obvious difference (about 0.09) between the fractal dimensions for two kinds of landscapes. It was worthy to mention that when the correlation coefficient is 0 in the semivariogram, the fractal dimension is 2, which means that when the RGB values are completely random for their locations in the colour image, the fractal dimension becomes 3. Two kinds of fractal dimensions could evaluate the shape and the color distributions of landscapes and discriminate the natural landscapes from urbanscapes clearly.
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Shen, Hua, Yu Jun Cai, and Xin Pan. "Fractal Study on Blank Residue of Mould Cavity." Advanced Materials Research 418-420 (December 2011): 1734–38. http://dx.doi.org/10.4028/www.scientific.net/amr.418-420.1734.
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A mathematical model of fractal interpolation surface is introuduced in this paper. A calculation to fractal dimension of the messy point data is proposed and used to get appropriate vertical scale factor to construct precise blank residue model. Fractal dimensions of different fractal interpolation surfaces with different vertical scale factor are different. The relationship between vertical scale factor and fractal dimension is obtained through calculating fractal dimensions of tested surfaces which are created by fractal interpolation with different vertical scale factor.
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Soltanifar, Mohsen. "A Generalization of the Hausdorff Dimension Theorem for Deterministic Fractals." Mathematics 9, no. 13 (July 1, 2021): 1546. http://dx.doi.org/10.3390/math9131546.
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How many fractals exist in nature or the virtual world? In this paper, we partially answer the second question using Mandelbrot’s fundamental definition of fractals and their quantities of the Hausdorff dimension and Lebesgue measure. We prove the existence of aleph-two of virtual fractals with a Hausdorff dimension of a bi-variate function of them and the given Lebesgue measure. The question remains unanswered for other fractal dimensions.
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Huang, Dongmei, Xikun Chang, Yunliang Tan, Kai Fang, and Yanchun Yin. "From rock microstructure to macromechanical properties based on fractal dimensions." Advances in Mechanical Engineering 11, no. 3 (March 2019): 168781401983636. http://dx.doi.org/10.1177/1687814019836363.
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Basic rock mechanical parameters, that is, the uniaxial compressive strength σc and elastic modulus E, have close relationships with the fractal dimension and inhomogeneity. Scanning electronic microscopy and fractal dimension calculations are applied to four different rock types (mudstone, sandstone, limestone, and basalt) in order to investigate the relationships between the rock mechanical properties, fractal dimensions, and homogeneity. The results show that the fractal dimension of each rock type fluctuates as the scanning electronic microscopy magnification increases. Rocks with different uniaxial compressive strength and elastic modulus values possess different self-similarity properties, and when the uniaxial compressive strength or elastic modulus increases, the fractal dimension of the rock microstructure decreases. The rock homogeneity is consistent with the fractal dimension, that is, the higher the homogeneity is, the larger the fractal dimension. Generally, homogeneity refers to the macroscale, and fractal dimension refers to the microscale. Overall, this research provides an innovative and effective approach for researching the mechanical behavior of rocks through a combination of uniaxial compression tests, homogeneity, and fractal dimensions.
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TATOM, FRANK B. "THE RELATIONSHIP BETWEEN FRACTIONAL CALCULUS AND FRACTALS." Fractals 03, no. 01 (March 1995): 217–29. http://dx.doi.org/10.1142/s0218348x95000175.
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The general relationship between fractional calculus and fractals is explored. Based on prior investigations dealing with random fractal processes, the fractal dimension of the function is shown to be a linear function of the order of fractional integro-differentiation. Emphasis is placed on the proper application of fractional calculus to the function of the random fractal, as opposed to the trail. For fractional Brownian motion, the basic relations between the spectral decay exponent, Hurst exponent, fractal dimension of the function and the trail, and the order of the fractional integro-differentiation are developed. Based on an understanding of fractional calculus applied to random fractal functions, consideration is given to an analogous application to deterministic or nonrandom fractals. The concept of expressing each coordinate of a deterministic fractal curve as a “pseudo-time” series is investigated. Fractional integro-differentiation of such series is numerically carried out for the case of quadric Koch curves. The resulting time series produces self-similar patterns with fractal dimensions which are linear functions of the order of the fractional integro-differentiation. These curves are assigned the name, fractional Koch curves. The general conclusion is reached that fractional calculus can be used to precisely change or control the fractal dimension of any random or deterministic fractal with coordinates which can be expressed as functions of one independent variable, which is typically time (or pseudo-time).
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HAMBLY, B. M., JUN KIGAMI, and TAKASHI KUMAGAI. "Multifractal formalisms for the local spectral and walk dimensions." Mathematical Proceedings of the Cambridge Philosophical Society 132, no. 3 (May 2002): 555–71. http://dx.doi.org/10.1017/s0305004101005618.
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We introduce the concepts of local spectral and walk dimension for fractals. For a class of finitely ramified fractals we show that, if the Laplace operator on the fractal is defined with respect to a multifractal measure, then both the local spectral and walk dimensions will have associated non-trivial multifractal spectra. The multifractal spectra for both dimensions can be calculated and are shown to be transformations of the original underlying multifractal spectrum for the measure, but with respect to the effective resistance metric.
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Liu, Zheng, and Xiao Mei Liu. "Fractal Characteristics of Primary Phase Morphology in Semisolid A356 Alloy." Advanced Materials Research 535-537 (June 2012): 936–40. http://dx.doi.org/10.4028/www.scientific.net/amr.535-537.936.
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Semisolid A356 alloy was prepared by low superheat pouring and slightly electro- magnetic stirring(LSPSES). The fractal dimensions of primary phase morphology in semisolid A356 alloy were researched by the calculating program written to calculate the fractal dimensions of box-counting in the image of primary phase morphology in semisolid A356 alloy. The results indicated that the primary phase morphology in the alloy was characterized by fractal dimension, and the morphology obtained by the different processing parameters had the different fractal dimension. The morphology at the different position of ingot had the different fractal dimensions, which reflected the effect of solidified conditions at different position in the same ingot on the morphology in the alloy. Solidification of the alloy was a course of change in fractal dimension.
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Nasr, Pedram, Hannah Leung, France-Isabelle Auzanneau, and Michael A. Rogers. "Supramolecular Fractal Growth of Self-Assembled Fibrillar Networks." Gels 7, no. 2 (April 14, 2021): 46. http://dx.doi.org/10.3390/gels7020046.
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Complex morphologies, as is the case in self-assembled fibrillar networks (SAFiNs) of 1,3:2,4-Dibenzylidene sorbitol (DBS), are often characterized by their Fractal dimension and not Euclidean. Self-similarity presents for DBS-polyethylene glycol (PEG) SAFiNs in the Cayley Tree branching pattern, similar box-counting fractal dimensions across length scales, and fractals derived from the Avrami model. Irrespective of the crystallization temperature, fractal values corresponded to limited diffusion aggregation and not ballistic particle–cluster aggregation. Additionally, the fractal dimension of the SAFiN was affected more by changes in solvent viscosity (e.g., PEG200 compared to PEG600) than crystallization temperature. Most surprising was the evidence of Cayley branching not only for the radial fibers within the spherulitic but also on the fiber surfaces.
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Górski, A. Z., M. Stróż, P. Oświȩcimka, and J. Skrzat. "Accuracy of the box-counting algorithm for noisy fractals." International Journal of Modern Physics C 27, no. 10 (August 29, 2016): 1650112. http://dx.doi.org/10.1142/s0129183116501126.
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The box-counting (BC) algorithm is applied to calculate fractal dimensions of four fractal sets. The sets are contaminated with an additive noise with amplitude [Formula: see text]. The accuracy of calculated numerical values of the fractal dimensions is analyzed as a function of [Formula: see text] for different sizes of the data sample. In particular, it has been found that even in case of pure fractals ([Formula: see text]) as well as for tiny noise ([Formula: see text]) one has considerable error for the calculated exponents of order 0.01. For larger noise the error is growing up to 0.1 and more, with natural saturation limited by the embedding dimension. This prohibits the power-like scaling of the error. Moreover, the noise effect cannot be cured by taking larger data samples.
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BACCHI, O. O. S., K. REICHARDT, and N. A. VILLA NOVA. "FRACTAL SCALING OF PARTICLE AND PORE SIZE DISTRIBUTIONS AND ITS RELATION TO SOIL HYDRAULIC CONDUCTIVITY." Scientia Agricola 53, no. 2-3 (May 1996): 356. http://dx.doi.org/10.1590/s0103-90161996000200027.
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Fractal scaling has been applied to soils, both for void and solid phases, as an approach to characterize the porous arrangement, attempting to relate particle-size distribution to soil water retention and soil water dynamic properties. One important point of such an analysis is the assumption that the void space geometry of soils reflects its solid phase geometry, taking into account that soil pores are lined by the full range of particles, and that their fractal dimension, which expresses their tortuosity, could be evaluated by the fractal scaling of particle-size distribution. Other authors already concluded that although fractal scaling plays an important role in soil water retention and porosity, particle-size distribution alone is not sufficient to evaluate the fractal structure of porosity. It is also recommended to examine the relationship between fractal properties of solids and of voids, and in some special cases, look for an equivalence of both fractal dimensions. In the present paper data of 42 soil samples were analyzed in order to compare fractal dimensions of pore-size distribution, evaluated by soil water retention curves (SWRC) of soils, with fractal dimensions of soil particle-size distributions (PSD), taking the hydraulic conductivity as a standard variable for the comparison, due to its relation to tortuosity. A new procedure is proposed to evaluate the fractal dimension of pore-size distribution. Results indicate a better correlation between fractal dimensions of pore-size distribution and the hydraulic conductivity for this set of soils, showing that for most of the soils analyzed there is no equivalence of both fractal dimensions. For most of these soils the fractal dimension of particle-size distribution does not indicate properly the pore trace tortuosity. A better equivalence of both fractal dimensions was found for sandy soils.
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LIANG, YONG-SHUN. "PROGRESS ON ESTIMATION OF FRACTAL DIMENSIONS OF FRACTIONAL CALCULUS OF CONTINUOUS FUNCTIONS." Fractals 27, no. 05 (August 2019): 1950084. http://dx.doi.org/10.1142/s0218348x19500841.
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In this paper, fractal dimensions of fractional calculus of continuous functions defined on [Formula: see text] have been explored. Continuous functions with Box dimension one have been divided into five categories. They are continuous functions with bounded variation, continuous functions with at most finite unbounded variation points, one-dimensional continuous functions with infinite but countable unbounded variation points, one-dimensional continuous functions with uncountable but zero measure unbounded variation points and one-dimensional continuous functions with uncountable and non-zero measure unbounded variation points. Box dimension of Riemann–Liouville fractional integral of any one-dimensional continuous functions has been proved to be with Box dimension one. Continuous functions on [Formula: see text] are divided as local fractal functions and fractal functions. According to local structure and fractal dimensions, fractal functions are composed of regular fractal functions, irregular fractal functions and singular fractal functions. Based on previous work, upper Box dimension of any continuous functions has been proved to be no less than upper Box dimension of their Riemann–Liouville fractional integral. Fractal dimensions of Riemann–Liouville fractional derivative of certain continuous functions have been investigated elementary.
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Xu, Tingbao, Ian D. Moore, and John C. Gallant. "Fractals, fractal dimensions and landscapes — a review." Geomorphology 8, no. 4 (December 1993): 245–62. http://dx.doi.org/10.1016/0169-555x(93)90022-t.
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HU, BOWEN, J. G. WANG, ZHONGQIAN LI, and HUIMIN WANG. "EVOLUTION OF FRACTAL DIMENSIONS AND GAS TRANSPORT MODELS DURING THE GAS RECOVERY PROCESS FROM A FRACTURED SHALE RESERVOIR." Fractals 27, no. 08 (December 2019): 1950129. http://dx.doi.org/10.1142/s0218348x19501299.
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Previous studies ignore the evolutions of pore microstructure parameters (pore diameter fractal dimension [Formula: see text] and tortuosity fractal dimension [Formula: see text]) but these evolutions may significantly impact the gas transport during gas extraction. In order to investigate these evolutions of fractal dimension properties during gas extraction, following four aspects are studied. Firstly, surface diffusion in adsorbed multilayer is modeled for fractal shale matrix. Our new matrix permeability model considers the slip flow, Knudsen diffusion and surface diffusion. This model is verified by experimental data. Secondly, a new fracture permeability model is proposed based on fractal theory and the coupling of viscous flow and Knudsen diffusion. Thirdly, the multilayer adsorption and these permeability models are introduced into the equations of gas flow and reservoir deformation. Finally, sensitivity analysis is performed to determine the key factors on fractal dimension evolution. The results show that the multilayer adsorption can accurately describe the adsorption properties of real shale reservoir. Shale reservoir deformation and gas desorption govern the evolutions of fractal dimensions. The multilayer adsorption and adsorbed gas porosity [Formula: see text] play an important role in the evolutions of fractal dimensions during gas extraction. The monolayer saturated adsorption volume [Formula: see text] is the most sensitive parameter affecting the evolution of fractal dimensions. Therefore, the effects of gas adsorption on the evolution of fractal dimensions cannot be neglected in shale reservoirs.
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Fu, Pan, Wei Lin Li, and Li Qin Zhu. "Cutting Tool Wear Monitoring Based on Wavelet Denoising and Fractal Theory." Applied Mechanics and Materials 48-49 (February 2011): 349–52. http://dx.doi.org/10.4028/www.scientific.net/amm.48-49.349.
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Monitoring of metal cutting tool wear for turning is a very important economical consideration in automated manufacturing. In the process of turning, the vibration signals of cutting tool become more and more irregular with the increase of tool wear. The degree of tool wear can indirectly be determined according to the change of vibration signals of cutting tool. In order to quantitatively describe this change, the wavelet and fractal theory were introduced into the cutting tool wear monitoring area. To eliminate the effect of noise on fractal dimension, the wavelet denoising method was used to reduce the noise of original signals. Then, the fractal dimensions were got from the denoised signals, including box dimension, information dimension, and correlation dimension. The relationship between these fractal dimensions and tool wear was studied. Use these fractal dimensions as the status indicator of tool wear condition. The experiments result demonstrates that wavelet denoise method can efficiently eliminate the effect of noise, and the change of fractal dimensions can represent the condition of tool wear.
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TARASOV, VASILY E. "ELECTROMAGNETIC FIELDS ON FRACTALS." Modern Physics Letters A 21, no. 20 (June 28, 2006): 1587–600. http://dx.doi.org/10.1142/s0217732306020974.
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Fractals are measurable metric sets with non-integer Hausdorff dimensions. If electric and magnetic fields are defined on fractal and do not exist outside of fractal in Euclidean space, then we can use the fractional generalization of the integral Maxwell equations. The fractional integrals are considered as approximations of integrals on fractals. We prove that fractal can be described as a specific medium.
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ZHANG, DONGSHENG, ASHOK SAMAL, and JAMES R. BRANDLE. "A METHOD FOR ESTIMATING FRACTAL DIMENSION OF TREE CROWNS FROM DIGITAL IMAGES." International Journal of Pattern Recognition and Artificial Intelligence 21, no. 03 (May 2007): 561–72. http://dx.doi.org/10.1142/s0218001407005090.
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A new method for estimating fractal dimension of tree crowns from digital images is presented. Three species of trees, Japanese yew (Taxus cuspidata Sieb & Zucc), Hicks yew (Taxus × media), and eastern white pine (Pinus strobus L.), were studied. Fractal dimensions of Japanese yew and Hicks yew range from 2.26 to 2.70. Fractal dimension of eastern white pine range from 2.14 to 2.43. The difference in fractal dimension between Japanese yew and eastern white pine was statistically significant at 0.05 significance level as was the difference in fractal dimension between Hicks yew and eastern white pine. On average, the greater fractal dimensions of Japanese yew and Hicks yew were possibly related to uniform foliage distribution within their tree crowns. Therefore, fractal dimension may be useful for tree crown structure classification and for indexing tree images.
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PAN, XUEZAI, XUDONG SHANG, MINGGANG WANG, and ZUO-FEI. "THE CANTOR SET’S MULTI-FRACTAL SPECTRUM FORMED BY DIFFERENT PROBABILITY FACTORS IN MATHEMATICAL EXPERIMENT." Fractals 25, no. 01 (February 2017): 1750002. http://dx.doi.org/10.1142/s0218348x17500025.
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With the purpose of researching the changing regularities of the Cantor set’s multi-fractal spectrums and generalized fractal dimensions under different probability factors, from statistical physics, the Cantor set is given a mass distribution, when the mass is given with different probability ratios, the different multi-fractal spectrums and the generalized fractal dimensions will be acquired by computer calculation. The following conclusions can be acquired. On one hand, the maximal width of the multi-fractal spectrum and the maximal vertical height of the generalized fractal dimension will become more and more narrow with getting two probability factors closer and closer. On the other hand, when two probability factors are equal to 1/2, both the multi-fractal spectrum and the generalized fractal dimension focus on the value 0.6309, which is not the value of the physical multi-fractal spectrum and the generalized fractal dimension but the mathematical Hausdorff dimension.
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MILOŠEVIĆ, NEBOJŠA T., DUŠAN RISTANOVIĆ, JOVAN B. STANKOVIĆ, and RADMILA GUDOVIĆ. "FRACTAL ANALYSIS OF DENDRITIC ARBORISATION PATTERNS OF STALKED AND ISLET NEURONS IN SUBSTANTIA GELATINOSA OF DIFFERENT SPECIES." Fractals 15, no. 01 (March 2007): 1–7. http://dx.doi.org/10.1142/s0218348x07003411.
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Through analysis of the morphology of dendritic arborisation of neurons from the substantia gelatinosa of dorsal horns from four different species, we have established that two types of cells (stalked and islet) are always present. The aim of the study was to perform the intra- and/or inter-species comparison of these two neuronal populations by fractal analysis, as well as to clarify the importance of the fractal dimension as an objective and usable morphological parameter. Fractal analysis was carried out adopting the box-counting method. We have shown that the mean fractal dimensions for the stalked cells are significantly different between species. The same is true for the mean fractal dimensions of the islet cells. Still, no significant differences were found for the fractal dimensions of the stalked and islet cells within a particular species. The human species has shown as the only exception where fractal dimensions of these two types of cells differ significantly. This study shows once more that the fractal dimension is a useful and sensitive morphological descriptor of neuronal structures and differences between them.
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ZHANG, XIN-MIN, L. RICHARD HITT, BIN WANG, and JIU DING. "SIERPIŃSKI PEDAL TRIANGLES." Fractals 16, no. 02 (June 2008): 141–50. http://dx.doi.org/10.1142/s0218348x08003934.
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We generalize the construction of the ordinary Sierpiński triangle to obtain a two-parameter family of fractals we call Sierpiński pedal triangles. These fractals are obtained from a given triangle by recursively deleting the associated pedal triangles in a manner analogous to the construction of the ordinary Sierpiński triangle, but their fractal dimensions depend on the choice of the initial triangles. In this paper, we discuss the fractal dimensions of the Sierpiński pedal triangles and the related area ratio problem, and provide some computer-generated graphs of the fractals.
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VINOY, K. J., JOSE K. ABRAHAM, and V. K. VARADAN. "IMPACT OF FRACTAL DIMENSION IN THE DESIGN OF MULTI-RESONANT FRACTAL ANTENNAS." Fractals 12, no. 01 (March 2004): 55–66. http://dx.doi.org/10.1142/s0218348x04002288.
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During the last few decades, fractal geometries have found numerous applications in several fields of science and engineering such as geology, atmospheric sciences, forest sciences, physiology and electromagnetics. Although the very fractal nature of these geometries have been the impetus for their application in many of these areas, a direct quantifiable link between a fractal property such as dimension and antenna characteristics has been elusive thus far. In this paper, the variations in the input characteristics of multi-resonant antennas based on generalizations of Koch curves and fractal trees are examined by numerical simulations. Schemes for such generalizations of these geometries to vary their fractal dimensions are presented. These variations are found to have a direct influence on the primary resonant frequency, the input resistance at this resonance, and ratios resonant frequencies of these antennas. It is expected that these findings would further enhance the popularity of the study of fractals.
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OUYANG, PEICHANG, HUA YI, ZHIYUN DENG, XUAN HUANG, and TAO YU. "BOUNDARY DIMENSIONS OF FRACTAL TILINGS." Fractals 23, no. 04 (December 2015): 1550035. http://dx.doi.org/10.1142/s0218348x15500358.
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A fractal tiling is a tiling which possesses self-similarity and the boundary of which is a fractal. In this paper, we investigate the boundary dimension of [Formula: see text]- and [Formula: see text]-tilings. We first derive an explicit recursion formula for the boundary edges of [Formula: see text]- and [Formula: see text]-tilings. Then we present an analytical expression for their fractal boundary dimensions using matrix methods. Results indicate that, as [Formula: see text] increases, the boundaries of [Formula: see text]- and [Formula: see text]-tilings will degenerate into general Euclidean curves. The method proposed in this paper can be extended to compute the boundary dimensions of other kinds of fractal tilings.
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Shuai, Yang, Liu Wenbai, and Liu Hongwei. "Quantitative Analysis of the Relationship Between Shear Strength and Fractal Dimension of Solidified Dredger Fill with Different Fly Ash Content Under Monotonic Shear." Polish Maritime Research 25, s2 (August 1, 2018): 132–38. http://dx.doi.org/10.2478/pomr-2018-0084.
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Abstract The dredger fill of Shanghai Hengsha Island Dongtan is solidified by curing agents with different fly ash content, and the shear strength index of solidified dredger fill is measured by the direct shear test. The microscopic images of solidified dredger fill are obtained by using SEM. The microscopic images are processed and analyzed by using IPP, and the fractal dimension including particle size fractal dimension Dps, aperture fractal dimension Dbs and particle surface fractal dimension Dpr is calculated by fractal theory. The quantitative analysis of the relationship between shear strength index and fractal dimension of solidified dredger fill is done. The research results show that the internal friction angle and the cohesion are closely related to the fly ash content λ and the curing period T, and the addition of fly ash can improve the effect of curing agent; There is no obvious linear relationship between the internal friction angle and the three fractal dimensions; The smaller particle surface fractal dimension Dpr and particle size fractal dimension Dps, the larger aperture fractal dimension Dbs, the greater the cohesion, and the cohesion has a good linear relationship with three fractal dimensions, and the correlation coefficient R2 is above 0.91.
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YU, BOMING. "FRACTAL DIMENSIONS FOR MULTIPHASE FRACTAL MEDIA." Fractals 14, no. 02 (June 2006): 111–18. http://dx.doi.org/10.1142/s0218348x06003155.
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The simple expressions for the fractal dimensions of multiphase fractal media are derived and are found to be a function of porosity, phase content, ratio of the maximum to minimum pore sizes. There is no any empirical constant in the proposed fractal dimensions. For the three-phase fractal porous medium or unsaturated porous medium, the fractal dimensions are found to be meaningful only in certain ranges of saturation Sw, i.e. Sw > S min for wetting phase and Sw < S max for non-wetting phase for a given porosity, based on real porous media for requirements from both fractal theory and experimental observations. The present analysis of the fractal dimensions is verified by a comparison with the existing experimental measurements. It allows for the analysis of transport properties such as permeability, thermal dispersion, and conductivities (both thermal and electrical) in multiphase fractal media by the proposed model.
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PACHEPSKY, YA A., L. P. KORSUNSKAIA, and M. HAJNOS. "FRACTAL PARAMETERS OF SOIL PORE SURFACE AREA UNDER A DEVELOPING CROP." Fractals 04, no. 01 (March 1996): 97–104. http://dx.doi.org/10.1142/s0218348x96000121.
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Fractal parameters of soils has become increasingly important in understanding and quantifying transport and adsorption phenomena in soils. It is not known how soil plant development may affect fractal characteristics of soil pores. We estimated pore surface area fractal parameters from mercury porosimetry data on gray forest soil before and during crop development, in samples both containing and not containing soil carbohydrates known to be important structure-forming agents. Two distinct intervals with different fractal dimensions were found in the range of pore radii from 4 nm to 1 μm. This could be attributed to differences in mineral composition of soil particles of different sizes. The interval of the smallest radii had the highest average fractal dimension close to 3. Smaller surface area fractal dimensions corresponding to low surface irregularity were found in the next interval of radii. The plant development affected neither fractal dimensions nor the cutoff values of soil samples. The carbohydrate oxidation caused a significant increase in the fractal dimension in the interval of larger radii, but did not affect fractal dimension in the interval of small radii. The cutoff values decreased after carbohydrate oxidation.
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YERAGANI, Vikram K., E. SOBOLEWSKI, V. C. JAMPALA, Jerald KAY, Suneetha YERAGANI, and Gina IGEL. "Fractal dimension and approximate entropy of heart period and heart rate: awake versus sleep differences and methodological issues." Clinical Science 95, no. 3 (September 1, 1998): 295–301. http://dx.doi.org/10.1042/cs0950295.
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1.Investigations that assess cardiac autonomic function include non-linear techniques such as fractal dimension and approximate entropy in addition to the common time and frequency domain measures of both heart period and heart rate. This article evaluates the differences in using heart rate versus heart period to estimate fractal dimensions and approximate entropies of these time series. 2.Twenty-four-hour ECG was recorded in 23 normal subjects using Holter records. Time series of heart rate and heart period were analysed using fractal dimensions, approximate entropies and spectral analysis for the quantification of absolute and relative heart period variability in bands of ultra low (< 0.0033 ;Hz), very low (0.0033–0.04 ;Hz), low (0.04–0.15 ;Hz) and high (0.15–0.5 ;Hz) frequency. 3.Linear detrending of the time series did not significantly change the fractal dimension or approximate entropy values. We found significant differences in the analyses using heart rate versus heart period between waking up and sleep conditions for fractal dimensions, approximate entropies and absolute spectral powers, especially for the power in the band of 0.0033–0.5 ;Hz. Log transformation of the data revealed identical fractal dimension values for both heart rate and heart period. Mean heart period correlated significantly better with fractal dimensions and approximate entropies of heart period than did corresponding heart rate measures. 4.Studies using heart period measures should take the effect of mean heart period into account even for the analyses of fractal dimension and approximate entropy. As the sleep–awake differences in fractal dimensions and approximate entropies are different between heart rate and heart period, the results should be interpreted accordingly.
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GRANSTAM, Sven-Olof, Bengt FELLSTRÖM, and Lars LIND. "Influences of carvedilol treatment on the effects of acetylcholine on regional haemodynamics in the spontaneously hypertensive rat." Clinical Science 95, no. 3 (September 1, 1998): 303–9. http://dx.doi.org/10.1042/cs0950303.
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1.Investigations that assess cardiac autonomic function include non-linear techniques such as fractal dimension and approximate entropy in addition to the common time and frequency domain measures of both heart period and heart rate. This article evaluates the differences in using heart rate versus heart period to estimate fractal dimensions and approximate entropies of these time series. 2.Twenty-four-hour ECG was recorded in 23 normal subjects using Holter records. Time series of heart rate and heart period were analysed using fractal dimensions, approximate entropies and spectral analysis for the quantification of absolute and relative heart period variability in bands of ultra low (< 0.0033 ;Hz), very low (0.0033–0.04 ;Hz), low (0.04–0.15 ;Hz) and high (0.15–0.5 ;Hz) frequency. 3.Linear detrending of the time series did not significantly change the fractal dimension or approximate entropy values. We found significant differences in the analyses using heart rate versus heart period between waking up and sleep conditions for fractal dimensions, approximate entropies and absolute spectral powers, especially for the power in the band of 0.0033–0.5 ;Hz. Log transformation of the data revealed identical fractal dimension values for both heart rate and heart period. Mean heart period correlated significantly better with fractal dimensions and approximate entropies of heart period than did corresponding heart rate measures. 4.Studies using heart period measures should take the effect of mean heart period into account even for the analyses of fractal dimension and approximate entropy. As the sleep–awake differences in fractal dimensions and approximate entropies are different between heart rate and heart period, the results should be interpreted accordingly.
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Xie, Yan Shi, Jian Wen Yin, Kai Xuan Tan, Liang Chen, Yang Hu, and Yong Hui Li. "Fractal Structure of Au Geochemical Field for Shuikoushan Orefield in Hunan Province, China and its Application to Mineralization." Advanced Materials Research 1092-1093 (March 2015): 1398–401. http://dx.doi.org/10.4028/www.scientific.net/amr.1092-1093.1398.
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The fractal measure on Au geochemical field of Mawangtang and Xinmengshan in Shuikoushan Pb-Zn-Au polymetallic ore field, Hunan, China was achieved by projective covering method in this paper. The results show a bifractal relation for Au Geochemical field which includes a textural fractal dimension (D1) at small scale and a structural fractal dimension (D2) at large scale with average breakpoint 86.0m which may be look as the movement scale of ore-forming fluid. All of fractal dimensions were between 2 to 3, D1 was 2.0011 and D2 was 2.0001 at Mawangtang as well as D1 was 2.4466 and D2 was 2.0408 at Xinmengshan respectively. The fractal dimensions appear the textural fractal dimensions were larger than their structural fractal dimensions indicate that the evolution of ore-forming fluid more complex than background value of this ore field. And what’s more, the fractal values of Mawangtang were larger than Xinmengshan may result from the mineralization with the former not only control by the overthrust structure and fold the same as the latter but also had a closed relationship with the acid to mafic magmatism.
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Feng, Zhi Xin, Yu Jun Cai, and Zhen Li. "Experimental Research on the Relation between Processing Parameters and Fractal Dimensions of the Machined Surface." Advanced Materials Research 602-604 (December 2012): 2011–16. http://dx.doi.org/10.4028/www.scientific.net/amr.602-604.2011.
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In the finishing machining, processing parameters directly affects the quality of machined surface. While the micro topography of the surface maps with the fractal dimensions. The paper researched the relation between processing parameters and fractal dimensions through the experiments in order to obtain the relationship. Through the design of orthogonal rotation experiment, determine the experimental parameters. Then measure the experimental sample for the surface and calculate the fractal dimensions using the box counting method. In the end, obtain the functional relation of processing parameters and fractal dimensions by regression method. The results show that the specific processing parameters of the surface can predict the fractal dimension by this formula. And this provides reference for selection of processing parameters.