Journal articles: 'Fractal measure'

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Mörters, Peter, and David Preiss. "Tangent measure distributions of fractal measures." Mathematische Annalen 312, no. 1 (September 1, 1998): 53–93. http://dx.doi.org/10.1007/s002080050212.

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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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Kadanoff, Leo P. "Fractal Singularities in a Measure and How to Measure Singularities on a Fractal." Progress of Theoretical Physics Supplement 86 (1986): 383–86. http://dx.doi.org/10.1143/ptps.86.383.

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SENGUPTA, KAUSHIK, and K. J. VINOY. "A NEW MEASURE OF LACUNARITY FOR GENERALIZED FRACTALS AND ITS IMPACT IN THE ELECTROMAGNETIC BEHAVIOR OF KOCH DIPOLE ANTENNAS." Fractals 14, no. 04 (December 2006): 271–82. http://dx.doi.org/10.1142/s0218348x06003313.

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In recent years, fractal geometries have been explored in various branches of science and engineering. In antenna engineering several of these geometries have been studied due to their purported potential of realizing multi-resonant antennas. Although due to the complex nature of fractals most of these previous studies were experimental, there have been some analytical investigations on the performance of the antennas using them. One such analytical attempt was aimed at quantitatively relating fractal dimension with antenna characteristics within a single fractal set. It is however desirable to have all fractal geometries covered under one framework for antenna design and other similar applications. With this objective as the final goal, we strive in this paper to extend an earlier approach to more generalized situations, by incorporating the lacunarity of fractal geometries as a measure of its spatial distribution. Since the available measure of lacunarity was found to be inconsistent, in this paper we propose to use a new measure to quantize the fractal lacunarity. We also demonstrate the use of this new measure in uniquely explaining the behavior of dipole antennas made of generalized Koch curves and go on to show how fundamental lacunarity is in influencing electromagnetic behavior of fractal antennas. It is expected that this averaged measure of lacunarity may find applications in areas beyond antennas.

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Falconer, Kenneth J., and Gerald A. Edgar. "Measure, Topology and Fractal Geometry." Mathematical Gazette 75, no. 472 (June 1991): 237. http://dx.doi.org/10.2307/3620293.

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Norton, Alec, and Gerald A. Edgar. "Measure, Topology, and Fractal Geometry." American Mathematical Monthly 99, no. 4 (April 1992): 378. http://dx.doi.org/10.2307/2324919.

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Tan, Teewoon, and Hong Yan. "The fractal neighbor distance measure." Pattern Recognition 35, no. 6 (June 2002): 1371–87. http://dx.doi.org/10.1016/s0031-3203(01)00125-x.

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Kolb, M. "Harmonic measure for fractal objects." Nuclear Physics B - Proceedings Supplements 5, no. 1 (September 1988): 129–34. http://dx.doi.org/10.1016/0920-5632(88)90027-8.

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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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SIMMONS, DAVID. "On interpreting Patterson–Sullivan measures of geometrically finite groups as Hausdorff and packing measures." Ergodic Theory and Dynamical Systems 36, no. 8 (July 21, 2015): 2675–86. http://dx.doi.org/10.1017/etds.2015.27.

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We provide a new proof of a theorem whose proof was sketched by Sullivan [Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math.149(3–4) (1982), 215–237], namely that if the Poincaré exponent of a geometrically finite Kleinian group $G$ is strictly between its minimal and maximal cusp ranks, then the Patterson–Sullivan measure of $G$ is not proportional to the Hausdorff or packing measure of any gauge function. This disproves a conjecture of Stratmann [Multiple fractal aspects of conformal measures; a survey. Workshop on Fractals and Dynamics (Mathematica Gottingensis, 5). Eds. M. Denker, S.-M. Heinemann and B. Stratmann. Springer, Berlin, 1997, pp. 65–71; Fractal geometry on hyperbolic manifolds. Non-Euclidean Geometries (Mathematical Applications (N.Y.), 581). Springer, New York, 2006, pp. 227–247].

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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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Oliva, A. I., E. Anguiano, and M. Aguilar. "Fractal dimension: measure of coating quality." Surface Engineering 15, no. 2 (April 1999): 101–4. http://dx.doi.org/10.1179/026708499101516344.

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Mitic, Vojislav, Goran Lazovic, Jelena Manojlovic, Wen-Chieh Huang, Mladen Stojiljkovic, Hans Facht, and Branislav Vlahovic. "Entropy and fractal nature." Thermal Science 24, no. 3 Part B (2020): 2203–12. http://dx.doi.org/10.2298/tsci191007451m.

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Existing, the biunivocal correspondents between the fractal nature and the nature discovered by fractals is the source and meeting point from those two aspects which are similar to the thermodynamically philosophical point of view. Sometimes we can begin from the end. We are substantial part of such fractals space nature. The mathematics fractal structures world have been inspired from nature and Euclidian geometry imagined shapes, and now it is coming back to nature serving it. All our analysis are based on several experimental results. The substance of the question regarding entropy and fractals could be analyzed on different ceramics and materials in general. We have reported the results based on consolidation BaTiO3- ceramics by the standard sintering technology, performed with BaTiO3 and different additives (MnCO3, CeO2, Bi2O3, Fe2O3, CaZrO3, Nb2O5, Er2O3, Yt2O3, Ho2O3). Thermodynamic principles are very important. Beside the energy and temperature, the entropy as a measure between the order and disorder (chaos) is very important parameter. In this paper, we establish the relation between the entropy and fractal that opens new frontiers with the goal to understand and establish the order-disorder relation.

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Hu, Xiaoyu, and S. James Taylor. "Fractal properties of products and projections of measures in ℝd." Mathematical Proceedings of the Cambridge Philosophical Society 115, no. 3 (May 1994): 527–44. http://dx.doi.org/10.1017/s0305004100072285.

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AbstractBorel measures in ℝd are called fractal if locally at a.e. point their Hausdorff and packing dimensions are identical. It is shown that the product of two fractal measures is fractal and almost all projections of a fractal measure into a lower dimensional subspace are fractal. The results rely on corresponding properties of Borel subsets of ℝd which we summarize and develop.

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El-Nabulsi, Rami Ahmad. "Superconductivity and nucleation from fractal anisotropy and product-like fractal measure." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2249 (May 2021): 20210065. http://dx.doi.org/10.1098/rspa.2021.0065.

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Superconductivity is analysed based on the product-like fractal measure approach with fractal dimension α introduced by Li and Ostoja-Starzewski in their attempt to explore anisotropic fractal elastic media. Our study shows the emergence of a massless state at the boundary of the superconductor and the simultaneous occurrence of isothermal and adiabatic processes in the superconductor depending on the position of the electrons. Several physical quantities were found to be position-dependent comparable with those arising in heavy doping and p–n junction. At the boundary of the superconductor, a shrinkage of the magnetic field was observed, leading to a scenario equivalent to the Meissner–Ochsenfeld effect. An enhancement of the London penetration depth is revealed and such an improvement was observed in pnictides at the onset of commensurate spin-density-wave order inside the superconducting phase at zero temperature. The Bardeen–Cooper–Schrieffer theory was also analysed and the appearance of zero-energy states is detected. Nucleation of superconductivity in a bulk was also studied. The system acts as a quantum damped harmonic oscillator and our analysis showed that type-I superconductivity occurs when κ < 2 / ( 1 + α ) , whereas type II occurs for κ > 2 / ( 1 + α ) , where κ is the Ginzburg–Landau parameter. The transition at the passage from the ‘genuine’ to the ‘intermediate’ type-I estimates 0.767767 < α ≤ 1 .

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Osman, Daniel, David Newitt, Alice Gies, Thomas Budinger, Vu Hao Truong, Sharmila Majumdar, and John Kinney. "Fractal Based Image Analysis of Human Trabecular Bone using the Box Counting Algorithm." Fractals 06, no. 03 (September 1998): 275–83. http://dx.doi.org/10.1142/s0218348x98000328.

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An image texture measure based on the box counting algorithm is evaluated for its potential to characterize human trabecular bone structure in medical images. Although bone images lack the self-similarity of theoretical fractals, bone images are candidates for characterization using fractal analysis because of their highly complex structure. The fractal based measure, herein called the box counting dimension (BCD), is an effective dimension, and does not imply an underlying fractal geometry. The importance of resolution in quantifying bone characteristics using the BCD is addressed. The relationship of BCD to standard measures of trabecular bone structure is also analyzed. To evaluate the variability of the BCD with change in resolution, the BCD is determined for two sections from each of seven 3D X-ray Tomographic Microscopy (XTM) images of human radius bone specimens, while the resolution is varied using lowpass filtering. An automated method of choosing the range of scales for the fractal analysis curve regression is used. The relationship of BCD to trabecular bone width and spacing is analyzed both for the XTM images and for simulated images representing idealized structures. The range of BCD values is 1.21–1.54. Variation in BCD over a range of resolutions is found to be small compared to the variation in BCD between different bone specimens. Maximum change in BCD over a large range of resolutions (17.60–176 microns per pixel) is 0.08. BCD decreases as space between trabeculae increases. Fractal based texture measures may potentially allow clinical monitoring of changes in bone structure — for example, using Magnetic Resonance Imaging at 150–200 micron resolution.

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Puente, Carlos E., Miguel M. López, Jorge E. Pinzón, and José M. Angulo. "The Gaussian distribution revisited." Advances in Applied Probability 28, no. 02 (June 1996): 500–524. http://dx.doi.org/10.1017/s000186780004859x.

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A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let f Θ,D :I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = f Θ,D (X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

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Puente, Carlos E., Miguel M. López, Jorge E. Pinzón, and José M. Angulo. "The Gaussian distribution revisited." Advances in Applied Probability 28, no. 2 (June 1996): 500–524. http://dx.doi.org/10.2307/1428069.

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A new construction of the Gaussian distribution is introduced and proven. The procedure consists of using fractal interpolating functions, with graphs having increasing fractal dimensions, to transform arbitrary continuous probability measures defined over a closed interval. Specifically, let X be any probability measure on the closed interval I with a continuous cumulative distribution. And let fΘ,D:I → R be a deterministic continuous fractal interpolating function, as introduced by Barnsley (1986), with parameters Θ and fractal dimension for its graph D. Then, the derived measure Y = fΘ,D(X) tends to a Gaussian for all parameters Θ such that D → 2, for all X. This result illustrates that plane-filling fractal interpolating functions are ‘intrinsically Gaussian'. It explains that close approximations to the Gaussian may be obtained transforming any continuous probability measure via a single nearly-plane filling fractal interpolator.

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Bright, D. S. "Particle-Like Fractal Images for Testing Algorithms that Measure Boundary Fractal Dimension." Microscopy and Microanalysis 8, S02 (August 2002): 1574–75. http://dx.doi.org/10.1017/s1431927602104454.

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SOLOMYAK, BORIS. "Measure and dimension for some fractal families." Mathematical Proceedings of the Cambridge Philosophical Society 124, no. 3 (November 1998): 531–46. http://dx.doi.org/10.1017/s0305004198002680.

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dos Santos, Leonardo, Alessandra Carvalho, Jair Leão, Paulo Neto, Tatijana Stosic, and Borko Stosic. "Fractal Measure and Microscopic Modeling of Osseointegration." International Journal of Periodontics & Restorative Dentistry 35, no. 6 (November 2015): 851–55. http://dx.doi.org/10.11607/prd.2324.

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DREMIN, I. M. "THE FRACTAL CORRELATION MEASURE FOR MULTIPLE PRODUCTION." Modern Physics Letters A 03, no. 14 (October 1988): 1333–35. http://dx.doi.org/10.1142/s0217732388001604.

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A new simple measure of multiple production processes is proposed. It takes into account an eventual inhomogeneity of particle positions on the (pseudo) rapidity scale and can be used to separate prominent spike (or ring) events.

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RAGHAVENDRA, B. S., and D. NARAYANA DUTT. "SIGNAL CHARACTERIZATION USING FRACTAL DIMENSION." Fractals 18, no. 03 (September 2010): 287–92. http://dx.doi.org/10.1142/s0218348x10004968.

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Fractal Dimensions (FD) are one of the popular measures used for characterizing signals. They have been used as complexity measures of signals in various fields including speech and biomedical applications. However, proper interpretation of such analyses has not been thoroughly addressed. In this paper, we study the effect of various signal properties on FD and interpret results in terms of classical signal processing concepts such as amplitude, frequency, number of harmonics, noise power and signal bandwidth. We have used Higuchi's method for estimating FDs. This study may help in gaining a better understanding of the FD complexity measure itself, and for interpreting changing structural complexity of signals in terms of FD. Our results indicate that FD is a useful measure in quantifying structural changes in signal properties.

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WU, JUNRU, and CHENGYUAN WANG. "FRACTAL STOKES’ THEOREM BASED ON INTEGRALS ON FRACTAL MANIFOLDS." Fractals 28, no. 01 (January 23, 2020): 2050010. http://dx.doi.org/10.1142/s0218348x20500103.

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In this paper, with the Hausdorff measure, the Hausdorff integral on fractal sets with one or lower dimension is firstly introduced via measure theory. Then the definition of the integral on fractal sets in [Formula: see text] is given. With the variable substitution theorem in the Riemann integral generalized to the integral on fractal sets, the integral on fractal manifolds is defined. As a result, with the generalization of Gauss’ theorem, Stokes’ theorem is generalized to the integral on fractal manifolds in [Formula: see text].

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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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Lombardini, Luca. "Fractional Perimeters from a Fractal Perspective." Advanced Nonlinear Studies 19, no. 1 (February 1, 2019): 165–96. http://dx.doi.org/10.1515/ans-2018-2016.

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AbstractThe purpose of this paper consists in a better understanding of the fractional nature of the nonlocal perimeters introduced in [L. Caffarelli, J.-M. Roquejoffre and O. Savin, Nonlocal minimal surfaces, Comm. Pure Appl. Math. 63 2010, 9, 1111–1144]. Following [A. Visintin, Generalized coarea formula and fractal sets, Japan J. Indust. Appl. Math. 8 1991, 2, 175–201], we exploit these fractional perimeters to introduce a definition of fractal dimension for the measure theoretic boundary of a set. We calculate the fractal dimension of sets which can be defined in a recursive way, and we give some examples of this kind of sets, explaining how to construct them starting from well-known self-similar fractals. In particular, we show that in the case of the von Koch snowflake{S\subseteq\mathbb{R}^{2}}this fractal dimension coincides with the Minkowski dimension. We also obtain an optimal result for the asymptotics as{s\to 1^{-}}of the fractional perimeter of a set having locally finite (classical) perimeter.

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VASS, JÓZSEF. "ON INTERSECTING IFS FRACTALS WITH LINES." Fractals 22, no. 04 (November 12, 2014): 1450014. http://dx.doi.org/10.1142/s0218348x14500145.

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IFS fractals — the attractors of Iterated Function Systems — have motivated plenty of research to date, partly due to their simplicity and applicability in various fields, such as the modeling of plants in computer graphics, and the design of fractal antennas. The statement and resolution of the Fractal-Line Intersection Problem is imperative for a more efficient treatment of certain applications. This paper intends to take further steps towards this resolution, building on the literature. For the broad class of hyperdense fractals, a verifiable condition guaranteeing intersection with any line passing through the convex hull of a planar IFS fractal is shown, in general ℝd for hyperplanes. The condition also implies a constructive algorithm for finding the points of intersection. Under certain conditions, an infinite number of approximate intersections are guaranteed, if there is at least one. Quantification of the intersection is done via an explicit formula for the invariant measure of IFS.

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Pikkujämsä, Sirkku M., Timo H. Mäkikallio, K. E. Juhani Airaksinen, and Heikki V. Huikuri. "Determinants and interindividual variation of R-R interval dynamics in healthy middle-aged subjects." American Journal of Physiology-Heart and Circulatory Physiology 280, no. 3 (March 1, 2001): H1400—H1406. http://dx.doi.org/10.1152/ajpheart.2001.280.3.h1400.

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Determinants and intersubject variations of fractal and complexity measures of R-R interval variability were studied in a random population of 200 healthy middle-aged women (age 51 ± 6 yr) and 189 men (age 50 ± 6 yr) during controlled conditions in the supine and sitting positions. The short-term fractal exponent (α1) was lower in women than men in both the supine (1.18 ± 0.20 vs. 1.12 ± 0.17, P < 0.01) and sitting position ( P < 0.001). Approximate entropy (ApEn), a measure of complexity, was higher in women in the sitting position (1.16 ± 0.17 vs. 1.07 ± 0.19, P < 0.001), but no gender-related differences were observed in ApEn in the supine position. Fractal and complexity measures were not related to any other demographic, laboratory, or lifestyle factors. Intersubject variations in a fractal measure, α1 (e.g., 1.15 ± 0.20 in the supine position, z value 1.24, not significant), and in a complexity measure, ApEn (e.g., 1.14 ± 0.18 in the supine position, z value 1.44, not significant), were generally smaller and more normally distributed than the variations in the traditional measures of heart rate variability (e.g., standard deviation of R-R intervals 49 ± 21 ms in the supine position, z value 2.53, P < 0.001). These results in a large random population sample show that healthy subjects express relatively little interindividual variation in the fractal and complexity measures of heart rate behavior and, unlike the traditional measures of heart rate variability, they are not related to lifestyle, metabolic, or demographic variables. However, subtle gender-related differences are also present in fractal and complexity measures of heart rate behavior.

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KULISH, Vladimir, and Vladimír HORÁK. "FORECASTING THE BEHAVIOR OF FRACTAL TIME SERIES: HURST EXPONENT AS A MEASURE OF PREDICTABILITY." Review of the Air Force Academy 14, no. 2 (December 8, 2016): 61–68. http://dx.doi.org/10.19062/1842-9238.2016.14.2.8.

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Escobar, Marco A., José R. Guzmán Sepúlveda, Jorge R. Parra Michel, and Rafael Guzmán Cabrera. "A proposal to measure the similarity between retinal vessel segmentations images." Nova Scientia 11, no. 22 (May 29, 2019): 224–45. http://dx.doi.org/10.21640/ns.v11i22.1872.

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Introduction: We propose a novel approach for the assessment of the similarity of retinal vessel segmentation images that is based on linking the standard performance metrics of a segmentation algorithm, with the actual structural properties of the images through the fractal dimension.Method: We apply our methodology to compare the vascularity extracted by automatic segmentation against manually segmented images.Results: We demonstrate that the strong correlation between the standard metrics and fractal dimension is preserved regardless of the size of the subimages analyzed.Discussion or Conclusion: We show that the fractal dimension is correlated to the segmentation algorithm’s performance and therefore it can be used as a comparison metric.

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Chen, Yanguang. "Two Sets of Simple Formulae to Estimating Fractal Dimension of Irregular Boundaries." Mathematical Problems in Engineering 2020 (February 26, 2020): 1–15. http://dx.doi.org/10.1155/2020/7528703.

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Irregular boundary lines can be characterized by fractal dimension, which provides important information for spatial analysis of complex geographical phenomena such as cities. However, it is difficult to calculate fractal dimension of boundaries systematically when image data are limited. An approximation estimation formula of boundary dimension based on square is widely applied in urban and ecological studies. But the boundary dimension is sometimes overestimated. This paper is devoted to developing a series of practicable formulae for boundary dimension estimation using ideas from fractals. A number of regular figures are employed as reference shapes, from which the corresponding geometric measure relations are constructed; from these measure relations, two sets of fractal dimension estimation formulae are derived for describing fractal-like boundaries. Correspondingly, a group of shape indexes can be defined. A finding is that different formulae have different merits and spheres of application, and the second set of boundary dimensions is a function of the shape indexes. Under condition of data shortage, these formulae can be utilized to estimate boundary dimension values rapidly. Moreover, the relationships between boundary dimension and shape indexes are instructive to understand the association and differences between characteristic scales and scaling. The formulae may be useful for the prefractal studies in geography, geomorphology, ecology, landscape science, and especially, urban science.

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Wang, Lei, Ya-Nan Bai, Ning Huang, and Qing-Guo Wang. "Fractal-Based Reliability Measure for Heterogeneous Manufacturing Networks." IEEE Transactions on Industrial Informatics 15, no. 12 (December 2019): 6407–14. http://dx.doi.org/10.1109/tii.2019.2901890.

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Lima, F. F., V. M. Oliveira, and M. A. F. Gomes. "A Galilean experiment to measure a fractal dimension." American Journal of Physics 61, no. 5 (May 1993): 421–22. http://dx.doi.org/10.1119/1.17234.

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Kumar, Sanjeev, Amod Kumar, Anjan Trikha, Sneh Anand, and Prashanth Gantla. "Higuchi fractal dimension as a measure of analgesia." International Journal of Medical Engineering and Informatics 4, no. 1 (2012): 66. http://dx.doi.org/10.1504/ijmei.2012.045304.

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La Torre, Davide, and Edward R. Vrscay. "A generalized fractal transform for measure-valued images." Nonlinear Analysis: Theory, Methods & Applications 71, no. 12 (December 2009): e1598-e1607. http://dx.doi.org/10.1016/j.na.2009.01.239.

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WANG, Ming-Hua. "The Hausdorff measure of a Sierpinski-like fractal." Hokkaido Mathematical Journal 36, no. 1 (February 2007): 9–19. http://dx.doi.org/10.14492/hokmj/1285766665.

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Devaney, Robert L. "Measure, Topology, and Fractal Geometry (Gerald A. Edgar)." SIAM Review 33, no. 4 (December 1991): 668–69. http://dx.doi.org/10.1137/1033153.

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Ondřej, Zindulka. "Strong measure zero and meager-additive sets through the prism of fractal measures." Commentationes Mathematicae Universitatis Carolinae 60, no. 1 (July 8, 2019): 131–55. http://dx.doi.org/10.14712/1213-7243.2015.277.

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Yeragani, V. K., K. Srinivasan, S. Vempati, R. Pohl, and R. Balon. "Fractal dimension of heart rate time series: an effective measure of autonomic function." Journal of Applied Physiology 75, no. 6 (December 1, 1993): 2429–38. http://dx.doi.org/10.1152/jappl.1993.75.6.2429.

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Previous studies suggested that heart rate (HR) time series may be more appropriately analyzed by nonlinear techniques because of the nonlinear nature of these data. In this study, we quantified the complexity of the HR time series, using fractal dimension, a previously described measure developed to study axonal growth, which quantifies the space-filling propensity and convolutedness of a waveform, and compared these results with another recently used measure, approximate entropy. Fractal dimension and approximate entropy of HR time series (unfiltered) correlate highly with each other and also with the high-frequency power (0.2–0.5 Hz) and, hence, appear to reflect vagal modulation of HR variability. These measures were also statistically more consistent and effective than measures of spectral analysis. Fractal dimension of the midfrequency time series of HR (filtered with a pass band of 0.05–0.15 Hz) also appears to be a statistically effective measure of relative sympathetic activity, especially in the standing posture.

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PARAMANATHAN, P., and R. UTHAYAKUMAR. "SIZE MEASURE RELATIONSHIP METHOD FOR FRACTAL ANALYSIS OF SIGNALS." Fractals 16, no. 03 (September 2008): 235–41. http://dx.doi.org/10.1142/s0218348x08003995.

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The fractal dimension of signals represents a powerful tool for analyzing the irregular behavior and state of the dynamical systems. Analysis of waveforms has been used to identify and distinguish specific complex patterns. A variety of algorithms are available for the computation of fractal dimension of waveforms. In this paper we evaluate the performance of our algorithm based on size measure relationship method, quantifying the synthetic waveforms and electroencephalographic signals. Compared to Katz's, Higuchi's and Petrosian's algorithm advantages of this method include greater speed and not affected by noise. The computation time for the algorithm suggested in this paper is much less than the other methods.

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McNamee, J. E. "Fractal perspectives in pulmonary physiology." Journal of Applied Physiology 71, no. 1 (July 1, 1991): 1–8. http://dx.doi.org/10.1152/jappl.1991.71.1.1.

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Like other organs that exchange substantial quantities of material with blood, the lung accommodates a large two-dimensional surface in a component three-dimensional volume. The lung's structure shows a resemblance to certain one- and two-dimensional mathematical functions that possess plane- and space-filling properties. When viewed from a conventional geometric perspective, many of the familiar forms and functions of pulmonary tissue appear to possess unusual qualities that defy explanation. Mathematically, they behave as though they had a fractional geometric dimension. This property is shared by a class of functions known as fractals. Fractals are described, and practical techniques are presented to measure the properties of the edges and surfaces of the lung. The consequences of fractal structure are also considered for the bronchial tree, pulmonary vasculature, and microcirculation. Insights arising from viewing the lung in this new perspective are summarized.

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Yu, Kanhua, Jian Gong, Yan Jing, Shuqian Liu, and Shihao Liang. "The Use of Fractal Theory Methods to Measure Growth Boundary, Planning and Control of Qinling and Bashan Mountainous Regions." Open House International 42, no. 3 (September 1, 2017): 116–19. http://dx.doi.org/10.1108/ohi-03-2017-b0024.

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Many cities of various types are distributed in the large area of mountainous regions in China. In these cities, there are acute contradictions between man and earth. Considering that the space growth mode of mountainous cities is widely different from that of flatland cities, the fractal method was adopted in the research aimed at demarcating the urban growth boundary of mountainous cities. The fractal features of the investigated mountainous cities in space were figured out via inference from their function, dimension, region, grade, and environment, and the fractal mode and conceptual framework of urban growth boundary of Qin-Ba mountainous region were constructed according to some concepts and methods such as fractal dimension, fractal network, and fractal order. In the research, the traditional urban growth boundary form-was decomposed into scattered points (point form), paths (linear form), and patches (plane form) to form the fractal theory units for the research of urban growth boundary, and the leading idea, procedure, and control method for “fractal demarcation of urban growth boundary” were established to provide strategies for demarcation of urban space growth boundary of Qin-Ba mountainous region.

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Meakin, Paul. "Scaling properties for the growth probability measure and harmonic measure of fractal structures." Physical Review A 35, no. 5 (March 1, 1987): 2234–45. http://dx.doi.org/10.1103/physreva.35.2234.

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Lubkin, Sharon R., Sarah E. Funk, and E. Helene Sage. "Quantifying Vasculature: New Measures Applied to Arterial Trees in the Quail Chorioallantoic Membrane." Journal of Theoretical Medicine 6, no. 3 (2005): 173–80. http://dx.doi.org/10.1080/10273660500264684.

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A wide variety of measures is currently in use in the morphometry of vascular systems. We introduce two additional classes of measures based on erosions and dilations of the image. Each measure has a clear biological interpretation in terms of the measured structures and their function. The measures are illustrated on images of the arterial tree of the quail chorioallantoic membrane (CAM). The new measures are correlated with widely-used measures, such as fractal dimension, but allow a clearer biological interpretation. To distinguish one CAM arterial tree from another, we propose reporting just three independent, uncorrelated numbers: (i) the fraction of tissue which is vascular (VF0, a pure ratio), (ii) a measure of the typical distance of the vascularized tissue to its vessels (CL, a length), and (iii) the flow capacity of the tissue (P, an area). An unusually largeCLwould indicate the presence of large avascular areas, a characteristic feature of tumor tissue.CLis inversely highly correlated with fractal dimension of the skeletonized image, but has a more direct biological interpretation.

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Li, Jun, and Martin Ostoja-Starzewski. "Fractal solids, product measures and fractional wave equations." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2108 (June 4, 2009): 2521–36. http://dx.doi.org/10.1098/rspa.2009.0101.

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This paper builds on the recently begun extension of continuum thermomechanics to fractal porous media that are specified by a mass (or spatial) fractal dimension D , a surface fractal dimension d and a resolution length scale R . The focus is on pre-fractal media (i.e. those with lower and upper cut-offs) through a theory based on a dimensional regularization, in which D is also the order of fractional integrals employed to state global balance laws. In effect, the governing equations are cast in forms involving conventional (integer order) integrals, while the local forms are expressed through partial differential equations with derivatives of integer order but containing coefficients involving D , d and R . This procedure allows a specification of a geometry configuration of continua by ‘fractal metric’ coefficients, on which the continuum mechanics is subsequently constructed. While all the derived relations depend explicitly on D , d and R , upon setting D =3 and d =2, they reduce to conventional forms of governing equations for continuous media with Euclidean geometries. Whereas the original formulation was based on a Riesz measure—and thus more suited to isotropic media—the new model is based on a product measure, making it capable of grasping local fractal anisotropy. Finally, the one-, two- and three-dimensional wave equations are developed, showing that the continuum mechanics approach is consistent with that obtained via variational energy principles.

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BOYARSKY, A., and Y. S. LOU. "A MATRIX METHOD FOR APPROXIMATING FRACTAL MEASURES." International Journal of Bifurcation and Chaos 02, no. 01 (March 1992): 167–75. http://dx.doi.org/10.1142/s021812749200015x.

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Let X be a bounded subset of Rn and let A be the Lebesgue measure on X. Let {X:τ1,…, τN} be an iterated function system (IFS) with attractor S. We associate probabilities p1,…, pN with τ1,…, τN, respectively. Let M(X) be the space of Borel probability measures on X, and let M: M(X)→M(X) be the Markov operator associated with the IFS and its probabilities given by: [Formula: see text] where A is a measurable subset of X. Then there exists a unique µ∈M (A) such that Mµ=µ; µ is referred to as the measure invariant under the iterated function system with the associated probabilities. The support of μ is the attractor S. We prove the existence of a sequence of step functions {fi}, which are the eigenvectors of matrices {Mi}, such that the measures {fidλ} converge weakly to µ. An algorithm is presented for the construction of Mi and an example is given.

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Zähle, U. "The Fractal Character of Localizable Measure-Valued Processes, III. Fractal Carrying Sets of Branching Diffusions." Mathematische Nachrichten 138, no. 1 (1988): 293–311. http://dx.doi.org/10.1002/mana.19881380121.

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Zähle, U. "The Fractal Character of Localizable Measure-Valued Processes. I — Random Measures on Product Spaces." Mathematische Nachrichten 136, no. 1 (1988): 149–55. http://dx.doi.org/10.1002/mana.19881360110.

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RAUT, SANTANU, and DHURJATI PRASAD DATTA. "ANALYSIS ON A FRACTAL SET." Fractals 17, no. 01 (March 2009): 45–52. http://dx.doi.org/10.1142/s0218348x09004156.

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The formulation of a new analysis on a zero measure Cantor set C(⊂I = [0,1]) is presented. A non-Archimedean absolute value is introduced in C exploiting the concept of relative infinitesimals and a scale invariant ultrametric valuation of the form log ε-1 (ε/x) for a given scale ε > 0 and infinitesimals 0 < x < ε, x ∈ I\C. Using this new absolute value, a valued (metric) measure is defined on C and is shown to be equal to the finite Hausdorff measure of the set, if it exists. The formulation of a scale invariant real analysis is also outlined, when the singleton {0} of the real line R is replaced by a zero measure Cantor set. The Cantor function is realized as a locally constant function in this setting. The ordinary derivative dx/dt in R is replaced by the scale invariant logarithmic derivative d log x/d log t on the set of valued infinitesimals. As a result, the ordinary real valued functions are expected to enjoy some novel asymptotic properties, which might have important applications in number theory and in other areas of mathematics.

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Ko, Ker-I. "On the computability of fractal dimensions and Hausdorff measure." Annals of Pure and Applied Logic 93, no. 1-3 (April 1998): 195–216. http://dx.doi.org/10.1016/s0168-0072(97)00060-2.

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Journal articles on the topic 'Fractal measure'

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