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Автор: Grafiati

Опубліковано: 30 травня 2022

Оновлено: 31 травня 2022

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1

Bavykin, O. B., S. V. Plaksin, and O. F. Vycheslavova. "Fractal analysis of surface profile of machine parts using measuring device MarSurf XR20." Izvestiya MGTU MAMI 8, no. 3-2 (April 10, 2014): 9–13. http://dx.doi.org/10.17816/2074-0530-67608.

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Анотація:

This paper proposes a method of fractal analysis of surface profile, based on joint application of the measuring device MarSurf XR20, spreadsheet editor Microsoft Excel and computer programs Fractan and MarWin. The results of surface profile fractal analysis obtained by using the proposed method are presented.

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2

Bavykin, Oleg, Tatyana Levina, Vladlena Matrosova, Anatoly Klochkov, and Vitaliy Enin. "Use of fractal analysis to evaluate the surface quality of agricultural machinery parts." BIO Web of Conferences 17 (2020): 00189. http://dx.doi.org/10.1051/bioconf/20201700189.

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The research of the determination of the fractal characteristics of the surface of a material proposes the use of a stationary profilograph and a computer program for calculating the Hurst exponent. The low accuracy of fractal analysis using the well-known computer program Fractan is revealed. A computer program developed in VBA for the fractal analysis of the time series is described. The high accuracy of the algorithms for calculating the Hurst exponent incorporated in this program is shown.

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3

Knowles, Asante James. "Fractal Philosophy: Grounding the Nature of the Mind with Fractals." NeuroQuantology 17, no. 8 (August 25, 2019): 19–23. http://dx.doi.org/10.14704/nq.2019.17.8.2799.

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4

Gavrishev, Aleksei A. "Application of nonlinear dynamics methods for quantitative and qualitative evaluation of properties of 2D models of S-chaos." Journal of Applied Informatics 16, no. 91 (February 26, 2021): 125–43. http://dx.doi.org/10.37791/2687-0649-2021-16-1-125-143.

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Анотація:

In this article, based on the mathematical, numerical and computer modeling carried out by the combined application of E&F Chaos, Past, Fractan, Visual Recurrence Analysis, Eviews Student Version Lite programs, some of the well-known 2D models of S-chaos are modeled, the data obtained are studied using nonlinear dynamics methods and the fact of their relation or non-relation to chaotic (quasi-chaotic) processes is established. As a result, it was found that the time diagrams obtained for the studied 2D models of S-chaos have a complex noise-like appearance and are continuous in the time domain. The resulting spectral diagrams have both a complex noise-like and regular appearance and are continuous in the spectral regions. The obtained values of BDS-statistics show that some of the time implementations can be attributed to chaotic (quasi-chaotic) processes. Also, the obtained values of BDS-statistics show that the studied 2D models of S-chaos have a property characteristic of classical chaotic (quasi-chaotic) processes: the slightest change in the initial conditions leads to the generation of a new set of signals. The obtained values of the lower bound of the KS-entropy show that the studied models also have the properties of chaotic (quasi-chaotic). Taking into account the conducted research and data from known works [1–5], it is possible to conclude that 2D models of S-chaos can relate to chaotic (quasi-chaotic) processes.

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5

Sander, Evelyn, Leonard M. Sander, and Robert M. Ziff. "Fractals and Fractal Correlations." Computers in Physics 8, no. 4 (1994): 420. http://dx.doi.org/10.1063/1.168501.

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6

DA CRUZ, WELLINGTON. "FRACTONS AND FRACTAL STATISTICS." International Journal of Modern Physics A 15, no. 24 (September 30, 2000): 3805–28. http://dx.doi.org/10.1142/s0217751x00002317.

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Fractons are anyons classified into equivalence classes and they obey specific fractal statistics. The equivalence classes are labeled by a fractal parameter or Hausdorff dimension h. We consider this approach in the context of the fractional quantum Hall effect (FQHE) and the concept of duality between such classes, defined by [Formula: see text] shows us that the filling factors for which the FQHE were observed just appear into these classes. A connection between equivalence classes h and the modular group for the quantum phase transitions of the FQHE is also obtained. A β-function is defined for a complex conductivity which embodies the classes h. The thermodynamics is also considered for a gas of fractons (h,ν) with a constant density of states and an exact equation of state is obtained at low-temperature and low-density limits. We also prove that the Farey sequences for rational numbers can be expressed in terms of the equivalence classes h.

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7

Southern, B. W., and A. R. Douchant. "Phonon-Fracton Crossover on Fractal Lattices." Physical Review Letters 55, no. 17 (October 21, 1985): 1808. http://dx.doi.org/10.1103/physrevlett.55.1808.

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8

Southern, B. W., and A. R. Douchant. "Phonon-Fracton Crossover on Fractal Lattices." Physical Review Letters 55, no. 9 (August 26, 1985): 966–68. http://dx.doi.org/10.1103/physrevlett.55.966.

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9

Yakubo, Kousuke, Minoru Nakano, and Tsuneyoshi Nakayama. "Fracton decay in nonlinear fractal systems." Physica B: Condensed Matter 219-220 (April 1996): 351–53. http://dx.doi.org/10.1016/0921-4526(95)00742-3.

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10

Moreles Vázquez, Uriel Octavio. "Los fractales." Acta Universitaria 13 (September 1, 2003): 19–22. http://dx.doi.org/10.15174/au.2003.249.

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11

Pelous, J., R. Vacher, T. Woignier, J. L. Sauvajol, and E. Courtens. "Scaling phonon-fracton dispersion laws in fractal aerogels." Philosophical Magazine B 59, no. 1 (January 1989): 65–74. http://dx.doi.org/10.1080/13642818908208446.

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12

Bąk, Z. "Fracton Excitations in the Magnetic "Net Fractal" Systems." Acta Physica Polonica A 113, no. 1 (January 2008): 541–44. http://dx.doi.org/10.12693/aphyspola.113.541.

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13

DA CRUZ, WELLINGTON, and ROSEVALDO DE OLIVEIRA. "FRACTAL INDEX, CENTRAL CHARGE AND FRACTONS." Modern Physics Letters A 15, no. 31 (October 10, 2000): 1931–39. http://dx.doi.org/10.1142/s0217732300002206.

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Анотація:

We introduce the notion of fractal index associated with the universal class h of particles or quasiparticles, termed fractons which obey specific fractal statistics. A connection between fractons and conformal field theory (CFT)-quasiparticles is established taking into account the central charge c[ν] and the particle-hole duality ν↔1/ν, for integer-value ν of the statistical parameter. In this way, we derive the Fermi velocity in terms of the central charge as [Formula: see text]. The Hausdorff dimension h which labeled the universal classes of particles and the conformal anomaly are therefore related. Following another route, we also established a connection between Rogers dilogarithm function, Farey series of rational numbers and the Hausdorff dimension.

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14

TAYLOR, S. J. "Fractals: The Geometry of Fractal Sets." Science 229, no. 4720 (September 27, 1985): 1381. http://dx.doi.org/10.1126/science.229.4720.1381.

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15

Uchaikin, V. V. "Fractal walk and walk on fractals." Technical Physics 49, no. 7 (July 2004): 929–32. http://dx.doi.org/10.1134/1.1778871.

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16

Mandelbrot, Benoit B. "Self-Affine Fractals and Fractal Dimension." Physica Scripta 32, no. 4 (October 1, 1985): 257–60. http://dx.doi.org/10.1088/0031-8949/32/4/001.

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17

COURTENS, E. "FRACTONS IN STRUCTURAL FRACTALS." Le Journal de Physique Colloques 24, no. C4 (April 1989): C4–143—C4–144. http://dx.doi.org/10.1051/jphyscol:1989422.

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18

Wang, Zi-dan, and Chang-de Gong. "The Spin-wave Fracton Excitations in Periodic Fractal Structure." Communications in Theoretical Physics 11, no. 2 (March 1989): 225–30. http://dx.doi.org/10.1088/0253-6102/11/2/225.

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19

MacDonald, Mairi, and Naeem Jan. "Fractons and the fractal dimension of proteins." Canadian Journal of Physics 64, no. 10 (October 1, 1986): 1353–55. http://dx.doi.org/10.1139/p86-239.

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Анотація:

We present numerical results and theoretical arguments to support the model introduced by Stapleton–Helman to describe the fractal and fracton dimensionalities of a protein. We postulate that the model, which is a self-avoiding walk with massless bonds, may have limited applicability in satisfying the known bonding requirement of proteins but a folding model may have the desired characteristics.

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20

Panagiotopoulos, P. D. "Fractals and fractal approximation in structural mechanics." Meccanica 27, no. 1 (1992): 25–33. http://dx.doi.org/10.1007/bf00453000.

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21

Bershadskii, A. G. "Inverse cascade in fractal turbulence (vortex-fractons)." Journal of Engineering Physics and Thermophysics 62, no. 2 (February 1992): 182–87. http://dx.doi.org/10.1007/bf00851784.

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22

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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23

Panagiotopoulos, P. D., and O. Panagouli. "Mechanics on fractal bodies. Data compression using fractals." Chaos, Solitons & Fractals 8, no. 2 (February 1997): 253–67. http://dx.doi.org/10.1016/s0960-0779(96)00105-1.

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24

Ishii, Tadao. "Relaxational Fractons in Hopping Conduction on Fractal Lattice." Journal of the Physical Society of Japan 61, no. 3 (March 15, 1992): 924–30. http://dx.doi.org/10.1143/jpsj.61.924.

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25

He, Qiu-Yan, Yi-Fei Pu, Bo Yu, and Xiao Yuan. "A class of fractal-chain fractance approximation circuit." International Journal of Electronics 107, no. 10 (February 24, 2020): 1588–608. http://dx.doi.org/10.1080/00207217.2020.1727030.

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26

Bottaccio, M., M. Montuori, and L. Pietronero. "Fractals vs. halos: Asymptotic scaling without fractal properties." Europhysics Letters (EPL) 66, no. 4 (May 2004): 610–16. http://dx.doi.org/10.1209/epl/i2003-10244-6.

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27

CAWLEY, R. "The Hard Science of Fractals: Fractal Growth Phenomena." Science 245, no. 4920 (August 25, 1989): 873. http://dx.doi.org/10.1126/science.245.4920.873.

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28

Wang, Xiao-Bing, Jian-Xin Li, Qing Jiang, Zhe-Hua Zhang, and De-Cheng Tian. "Effect of fractons in superconductors with fractal structure." Physical Review B 49, no. 14 (April 1, 1994): 9778–81. http://dx.doi.org/10.1103/physrevb.49.9778.

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29

Mottola, Horacio A. "Fractals and fractal-like concepts in chemical analysis." TrAC Trends in Analytical Chemistry 9, no. 9 (October 1990): 297–302. http://dx.doi.org/10.1016/0165-9936(90)85045-9.

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30

Assis, Thiago Albuquerque de, José Garcia Vivas Miranda, Fernando de Brito Mota, Roberto Fernandes Silva Andrade, and Caio Mário Castro de Castilho. "Geometria fractal: propriedades e características de fractais ideais." Revista Brasileira de Ensino de Física 30, no. 2 (2008): 2304.1–2304.10. http://dx.doi.org/10.1590/s1806-11172008000200005.

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Анотація:

Descobertas recentes revelam que modelos matemáticos euclidianos, de há muito estabelecidos e que procuram reproduzir a geometria da natureza, às vezes se apresentam incompletos e, em determinadas situações, inadequados. Especificamente, muitas das formas encontradas na natureza não são círculos, triângulos, esferas, icosaedros ou retângulos. Enfim, não são simples curvas, superfícies ou sólidos, conforme definidos na geometria clássica de Euclides (300 a.C), cujos teoremas possuem lugar de destaque nos textos de geometria. Neste trabalho apresenta-se uma breve e elementar, mas que busca ser consistente, discussão sobre algumas definições e aplicações relacionadas à geometria fractal, em particular fractais ideais. Caracterizaremos alguns fractais auto-similares que, por sua importância histórica ou riqueza de características, constituem exemplos ilustrativos "clássicos" de propriedades de fractais, propriedades estas que muitas vezes aparecem dispersas numa literatura mais especializada. Mostra-se, por construção, que suas medidas de comprimento, área e volume, nas dimensões euclidianas usuais, dão margem a resultados contraditórios. Estes podem ser explicados pelo fato de que tais objetos só podem ser adequadamente mensurados em espaços de dimensão fracionária.

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31

Tymofijeva, Nadija. "Combinatorial configurations, fractals, fractal dimension of combinatorial sets." Physico-mathematical modelling and informational technologies, no. 33 (September 6, 2021): 170–74. http://dx.doi.org/10.15407/fmmit2021.33.170.

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Combinatorial configurations and their sets are considered. The definitions of these objects are given, recurrent combinatorial operators are introduced, with the help of which they are formed, and rules are formulated according to which their sets are ordered. The property of periodicity, which takes place in the generation of combinatorial configurations, is described. It follows from the recurrent way of their formation and ordering. The fractal structure of combinatorial sets is formed due to the described rules, in which the property of periodicity is used. Analysis of these structures shows that they are self-similar, both finite and infinite, which is characteristic of fractals. Their fractal dimension is introduced, which follows from the rules of generating combinatorial configurations and corresponds to the number of these objects in their set.

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32

Aaron, S., Z. Conn, Robert Strichartz, and H. Yu. "Hodge-de Rham theory on fractal graphs and fractals." Communications on Pure and Applied Analysis 13, no. 2 (October 2013): 903–28. http://dx.doi.org/10.3934/cpaa.2014.13.903.

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33

Vacher, René, and Eric Courtens. "Phonon-Fracton Crossover and Self-Similarity in Real Fractals." Physica Scripta T29 (January 1, 1989): 239–43. http://dx.doi.org/10.1088/0031-8949/1989/t29/046.

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34

Debnath *, Lokenath. "A brief historical introduction to fractals and fractal geometry." International Journal of Mathematical Education in Science and Technology 37, no. 1 (January 15, 2006): 29–50. http://dx.doi.org/10.1080/00207390500186206.

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35

Stapleton, H. J. "Comment on "Fractons and the Fractal Structure of Proteins"." Physical Review Letters 54, no. 15 (April 15, 1985): 1734. http://dx.doi.org/10.1103/physrevlett.54.1734.

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36

Herrmann, Hans J. "Comment on "Fractons and the Fractal Structure of Proteins"." Physical Review Letters 56, no. 22 (June 2, 1986): 2432. http://dx.doi.org/10.1103/physrevlett.56.2432.

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37

Butenko, A. V., V. M. Shalaev, and M. I. Stockman. "Fractals: giant impurity nonlinearities in optics of fractal clusters." Zeitschrift f�r Physik D Atoms, Molecules and Clusters 10, no. 1 (March 1988): 81–92. http://dx.doi.org/10.1007/bf01425583.

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38

He, Qiu-Yan, Yi-Fei Pu, Bo Yu, and Xiao Yuan. "Scaling Fractal-Chuan Fractance Approximation Circuits of Arbitrary Order." Circuits, Systems, and Signal Processing 38, no. 11 (May 9, 2019): 4933–58. http://dx.doi.org/10.1007/s00034-019-01117-x.

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39

Tsallis, Constantino, and Roger Maynard. "On the geometrical interpretation of fraction and fractal dimensionality." Physics Letters A 129, no. 2 (May 1988): 118–20. http://dx.doi.org/10.1016/0375-9601(88)90081-3.

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40

Aerts, Diederik, Marek Czachor, and Maciej Kuna. "Crystallization of space: Space-time fractals from fractal arithmetic." Chaos, Solitons & Fractals 83 (February 2016): 201–11. http://dx.doi.org/10.1016/j.chaos.2015.12.004.

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41

Жихарев and L. Zhikharev. "Generalization to Three-Dimensional Space Fractals of Pythagoras and Koch. Part I." Geometry & Graphics 3, no. 3 (November 30, 2015): 24–37. http://dx.doi.org/10.12737/14417.

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Анотація:

Fractals are geometric objects, each part of which is similar to the whole object, so that if we take a part and increase its size to the size of the whole object, it would be impossible to notice a difference. In other words, fractals are sets having scale invariance. In mathematics, they are associated primarily with non-differentiable functions. The concept of "fractal" (from the Latin "Fractus" meaning «broken») had been introduced by Benoit Mandelbrot (1924–2010), French and American mathematician, physicist, and economist. Mandelbrot had found that seemingly arbitrary fluctuations in price of goods have a certain tendency to change: it turned out that daily fluctuations are symmetrical with long-term price fluctuations. In fact, Benoit Mandelbrot applied his recursive (fractal) method to solve the problem. Since the last quarter of the nineteenth century, a large number of fractal curves and flat objects have been created; and methods for their application have been developed. From geometrical point of view, the most interesting fractals are "Koch snowflake" and "Pythagoras Tree". Two classes of analogues of the volumetric fractals were created with modern three-dimensional modeling program: "Fractals of growth” – like Pythagoras Tree, “Fractals of separation” – like Koch snowflake; the primary classification was developed, their properties were studied. Empiric data was processed with basic arithmetic calculations as well as with computer software. Among other things, for fractals of separation the task was to create an object with an infinite surface area, which in the future might acquire great importance for the development of the chemical and other industries.

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42

Mitić, Vojislav V., Vesna Paunović, Goran Lazović, Ljubiša Kocić, and Hans Fecht. "Fractal dimension of fractals tensor product ferroelectric ceramic materials frontiers." Ferroelectrics 535, no. 1 (October 26, 2018): 114–19. http://dx.doi.org/10.1080/00150193.2018.1474653.

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43

Banks, David L. "The Beauty of Fractals and the Science of Fractal Images." CHANCE 2, no. 1 (January 1989): 47–49. http://dx.doi.org/10.1080/09332480.1989.11882330.

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44

Cullion, Rebecca, Deborah Pence, James Liburdy, and Vinod Narayanan. "Void Fraction Variations in a Fractal-Like Branching Microchannel Network." Heat Transfer Engineering 28, no. 10 (October 2007): 806–16. http://dx.doi.org/10.1080/01457630701378184.

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45

Wang, Zidan. "The Density of Vibrational States and Fracton Dimensionality for a Periodic Fractal Lattice." physica status solidi (b) 150, no. 1 (November 1, 1988): K15—K17. http://dx.doi.org/10.1002/pssb.2221500140.

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46

Maryenko, N. І., and O. Yu Stepanenko. "Fractal analysis of anatomical structures linear contours: modified Caliper method vs Box counting method." Reports of Morphology 28, no. 1 (February 23, 2022): 17–26. http://dx.doi.org/10.31393/morphology-journal-2022-28(1)-03.

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Анотація:

Fractal analysis estimates the metric dimension and complexity of the spatial configuration of different anatomical structures. This allows the use of this mathematical method for morphometry in morphology and clinical medicine. Two methods of fractal analysis are most often used for fractal analysis of linear fractal objects: the Box counting method (Grid method) and the Caliper method (Richardson’s method, Perimeter stepping method, Ruler method, Divider dimension, Compass dimension, Yard stick method). The aim of the research is a comparative analysis of two methods of fractal analysis – Box counting method and author's modification of Caliper method for fractal analysis of linear contours of anatomical structures. A fractal analysis of three linear fractals was performed: an artificial fractal – a Koch snowflake and two natural fractals – the outer contours of the pial surface of the human cerebellar vermis cortex and the cortex of the cerebral hemispheres. Fractal analysis was performed using the Box counting method and the author's modification of the Caliper method. The values of the fractal dimension of the artificial linear fractal (Koch snowflakes) obtained by the Caliper method coincide with the true value of the fractal dimension of this fractal, but the values of the fractal dimension obtained by the Box counting method do not match the true value of the fractal dimension. Therefore, fractal analysis of linear fractals using the Caliper method allows you to get more accurate results than the Box counting method. The values of the fractal dimension of artificial and natural fractals, calculated using the Box counting method, decrease with increasing image size and resolution; when using the Caliper method, fractal dimension values do not depend on these image parameters. The values of the fractal dimension of linear fractals, calculated using the Box counting method, increase with increasing width of the linear contour; the values calculated using the Caliper method do not depend on the contour line width. Thus, for the fractal analysis of linear fractals, preference should be given to the Caliper method and its modifications.

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47

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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Lara Galicia, Aline. "Fractales en Arqueología: aplicación en la pintura rupestre de sitios del México prehispánico." Virtual Archaeology Review 4, no. 8 (November 20, 2015): 80. http://dx.doi.org/10.4995/var.2013.4323.

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<p>This study exposed analysis fractal by the dimension D in the rock art. The Fractals is a new tool in the software application and digital photography<br />for the recognition of raw materials, the study of its form, degradation, specific parts of paint, and providing new data the virtual reconstruction.<br />The analyzes focus on various pictorial sets of the region Mezquital Valley, Hidalgo, Mexico; region where are concentrated more than 100 caves, comparing 74 images to analyze the various pictorial traditions.<br />The structure fractal describes the dimension of the represented objects, not only according to its space where the figure is placed, but the possibility of being a fraction in which all the surfaces can be observed (rock, paint).</p>

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Cherny, A. Yu, E. M. Anitas, V. A. Osipov, and A. I. Kuklin. "Scattering from surface fractals in terms of composing mass fractals." Journal of Applied Crystallography 50, no. 3 (June 1, 2017): 919–31. http://dx.doi.org/10.1107/s1600576717005696.

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It is argued that a finite iteration of any surface fractal can be composed of mass-fractal iterations of the same fractal dimension. Within this assertion, the scattering amplitude of a surface fractal is shown to be a sum of the amplitudes of the composing mass fractals. Various approximations for the scattering intensity of surface fractals are considered. It is shown that small-angle scattering (SAS) from a surface fractal can be explained in terms of a power-law distribution of sizes of objects composing the fractal (internal polydispersity), provided the distance between objects is much larger than their size for each composing mass fractal. The power-law decay of the scattering intensityI(q) ∝ q^{D_{\rm s}-6}, where 2 <Ds< 3 is the surface-fractal dimension of the system, is realized as a non-coherent sum of scattering amplitudes of three-dimensional objects composing the fractal and obeying a power-law distribution dN(r) ∝r−τdr, withDs= τ − 1. The distribution is continuous for random fractals and discrete for deterministic fractals. A model of the surface deterministic fractal is suggested, the surface Cantor-like fractal, which is a sum of three-dimensional Cantor dusts at various iterations, and its scattering properties are studied. The present analysis allows one to extract additional information from SAS intensity for dilute aggregates of single-scaled surface fractals, such as the fractal iteration number and the scaling factor.

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50

LAPIDUS, MICHEL L. "FRACTALS AND VIBRATIONS: CAN YOU HEAR THE SHAPE OF A FRACTAL DRUM?" Fractals 03, no. 04 (December 1995): 725–36. http://dx.doi.org/10.1142/s0218348x95000643.

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We study various aspects of the question “Can one hear the shape of a fractal drum?”, both for “drums with fractal boundary” (or “surface fractals”) and for “drums with fractal membrane” (or “mass fractals”).

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