RADIO PHYSICS AND RADIO ASTRONOMY

PACS numbers: 05.45.Df ,05.45.Tp

Purpose: Currently, there is a tendency to “fractalize” the science. Radiophysics is no exception. The subject of this work is a review of the basic ideas of “fractalization”, the mathematical foundations of modern fractal methods for describing and exploring the world. The purpose of the work is to present the basic concepts, definitions and relationships of the modern theory of fractals, as well as the classification and analysis of existing numerical characteristics of fractals.

Design/methodology/approch: The methods of constructing geometric monofractals and multifractals are considered. A comparative characteristic of the methods for assessing the dimension of physical fractals is given. Examples of physical fractals are given.

Findings: In the development of the “fractalization” of science, 4 stages are distinguished: the era of “monsters”, the preparatory stage, the stage of formation and development, the modern stage. For the correct description of fractals, the Hausdorff–Besicovitch dimension, which can also take noninteger values, is used. The following fractal classifications are considered: mathematical and physical, geometric and algebraic, mono- and multifractals, regular and stochastic, homogeneous and heterogeneous. It has been demonstrated that the fractal dimension of objects can be both fractional and integer, it is important that the fractal dimension should be greater than their topological dimension. The equality of the fractal dimensions of two objects does not imply the similarity of their structure. When describing thick fractals as regular monofractals, instead of the Hausdorff–Besicovitch dimension, the scaling exponents are used.

Conclusions: The mathematical foundations of the theory of fractals, used in the modern theoretical radiophysics, are presented.

Manuscript submitted 26.09.2019

Radio phys. radio astron. 2020, 25(1): 3-77

REFERENCES

1. GOUYET, J.-F., 1996. Physics and Fractal Structures. New York, USA: Springer-Verlag.

2. MANDELBROT, B.,  1975. Les Objets Fractals: Forme, Hasard et Dimension. Paris, France: Flammarion.

3. GOROBETS, YU. I., KUCHKO, A. M. and VAVILOVA, I. B., 2008. Fractal Geometry in Natural Science. Textbook. Kyiv, Ukraine: Naukova Dumka Publ. (in Ukrainian).

4. TARASOV, V. E.,  2011. Fractional Dynamics. Applications of Fractal Calculus to Dynamics of Particles, Fields and Media. New York, USA: Springer.

5. OLDHAM, K. B. and SPANIER, J.,  1974. The Fractional Calculus. Theory and Applications of Differentiation and Integration to Arbitrary Order. San Diego, USA: Academic Press.

6. ROSS, B., ed, 1975. Fractional Calculus and Its Applications. Berlin, Germany: Springer-Verlag.

7. MILLER, K. and ROSS, B.,  1993. An Introduction to the Fractional Calculus and Fractional Differential Equations. New Jersey: Wiley-Interscience.

8. HILFER, R.,  ed. 2000. Applications of Fractional Calculus in Physics. Singapore, New Jersey, et al.: World Scientific Publ. DOI: https://doi.org/10.1142/3779

9. NAKHUSHEV, A. M., 2003. Fractional calculus and its applications. Moscow, Russia: Fizmatlit Publ. (in Russian).

10. BAYIN, S.,  2006. Mathematical Methods in Science and Engineering. New Jersey: Wiley-Interscience. DOI: https://doi.org/10.1002/0470047429

11. HIBSCHWEILER, R. and MACGREGOR, T. H.,  2006. Fractional Cauchy Transforms. Boca Raton: Chapman & Hall/CRC. DOI: https://doi.org/10.1201/9781420034875

12. KILBAS, A. A., SRIVASTAVA, H. M. and TRUJILLO, J. J.,  2006. Theory and Applications of Fractional Differential Equations. New York: Elsevier.

13. SABATIER, J., AGRAWAL, O. P. and TENREIRO MACHADO, J. A., eds., 2007. Advances in Fractional Calculus. Theoretical Developments and Applications in Physics and Engineering. New York: Springer. DOI: https://doi.org/10.1007/978-1-4020-6042-7

14. VASILIYEV, V. V. and SIMAK, L. A., 2008. Fractal Calculus and Approximative Methods in Dynamical Systems Modelling. Kyiv, Ukraine: NAS of Ukraine Publ. (in Russian).

15. UCHAYKIN, V. V., 2008. The Method of Fractional Derivatives. Ul’yanovsk, Russia: Artishok Publ. (in Russian).

16. SAICHEV, A. I. and WOYCZYNSKI, W. A., 1997. Distributions in the Physical and Engineering Sciences: Distributional and Fractal Calculus, Integral Transforms and Wavelets. Boston: Birkhäuser.

17. GIL’MUTDINOV, A. K., USHAKOV, P. A. and EL-KHARAZI, R., 2017. Fractal Elements and their Applications.Cham,Switzerland: Springer Int. Publ.

18. GIL’MUTDINOV, A. K., ed., 2010. Fractals and Fractional Operators. Kazan’, Russia: Fan Publ. of Academy of Sciences of RT. (in Russian).

19. NAKAYAMA, T. and YAKUBO, K., 2010. Fractal Concepts in Condensed Matter Physics. Berlin, Heidelberg: Springer-Verlag.

20. POTAPOV, A. A., 2002. Fractals in Radio Physics and Radar. Moscow, Russia: Logos Publ. (in Russian).

21. POTAPOV, A. A., 2005. Fractals in Radio Physics and Radar. Sample Topology. Moscow, Russia: Universitetskaya Kniga Publ. (in Russian).

22. TURBIN, A. F. and PRATSEVITYI, N. V., 1992. Fractal Sets, Functions, Distributions. Kyiv, Ukraine: Naukova Dumka Publ. (in Russian).

23. ZEL’DOVICH, YA. B. and SOKOLOV, D. D., 1985. Fractals, Self-Similarity, Intermediate Asymptotic. Uspekhi fizicheskikh nauk. vol. 146, is. 3, pp. 493–506. (in Russian). DOI: https://doi.org/10.3367/UFNr.0146.198507d.0493

24. PEINTGEN, H.-O. and SAUPE, D., eds., 1988. The Science of Fractal Images. New York: Springer-Verlag.

25. MANDELBROT, B., 1989. Les Objets Fractals: Forme, Hasard et Dimension. Parise: Flammarion.

26. MANDELBROT, B. B., 1977. Fractals: Form, Chance and Dimension. San Francisco: W. H. Freeman and Company.

27. MANDELBROT, B. B., 1982. The Fractal Geometry of Nature. New York: W. H. Freeman and Company.

28. MANDELBROT, B. B., 2010. Les Objets Fractals: Forme, Hasard et Dimension. Paris: Flammarion.

29. MANDELBROT, B. B., 2002. The Fractal Geometry of Nature. Moscow, Russia: Institut komp’yuternykh issledovaniy Publ. (in Russian).

30. BARYSHEV, Y. and TEERIKORPI, P.,  2002. Discovery of Cosmic Fractals. New Jersey: World Scientific Publ. DOI: https://doi.org/10.1142/4896

31. CROWNOVER, R. M., 1995. Introduction to Fractals and Chaos. Boston: Jones and Bartlett Publ.

32. CROWNOVER, R. M., 2000. Fractals and chaos in dynamic systems. Fundamentals of theory. Moscow, Russia: Postmarket Publ. (in Russian).

33. BULAT, A. F. and DYRDA, V. I., 2005. Frcatals in Geomechanics. Kyiv, Ukraine: Naukova Dumka Publ. (in Russian).

34. HAUSDORFF, F., 1918. Dimension und äußeres Maß. Math. Ann. vol. 79, is. 1-2, pp. 157–179. DOI: https://doi.org/10.1007/BF01457179

35. THIM, J., 2003. Continuous Nowhere Differentiable Functions. Master’s Thesis. Lulea, Sweden: Lulea University of Technology Publ.

36. MASSOPUST, P. R., 1994. Fractal Functions, Fractal Surfaces and Wavelets. San Diego, New York et al.: Academic Press. DOI: https://doi.org/10.1016/B978-0-08-092458-8.50003-4

37. BANDT, C., BARNSLEY, M., DEVANEY, R., FALCONER, K. J., KANNAN, V. and VINOD KUMAR, P. B., eds., 2014. Fractals, Wavelets and their Applications: Contributions from the International Conference and Workshop on Fractals and Wavelets (Springer Proceedings in Mathematics & Statistics). Switzerland: Springer Int. Publ. DOI: https://doi.org/10.1007/978-3-319-08105-2

38. DU BOIS-REIMOND, P. 1875. Versuch einer Classification der willkürlichen Functionen reeller Argumente nach ihren Aenderungen in den kleinsten Intervallen. Journal für die reine und angewandte Mathematik. vol. 79, pp. 21–37. DOI: https://doi.org/10.1515/crll.1875.79.21

39. POTAPOV, A. A., GULYAEV, Yu. V., NIKITOV, S. A., PAKHOMOV, A. A. and GERMAN, V. A., 2008. The Newest Methods of Image Processing. Moscow, Russia: FIZMATLIT Publ. (in Russian).

40. CELLÉRIER, M. C., 1890. Note sur les principes fondamentaux de l’analyse. Darboux Bull., 14. pp. 142–160.

41. BRZHECHKA, V. F., 1949. About Bolzano (On the centenary of the death of the Czech mathematician Bernard Bolzano). Uspekhi matematicheskikh nauk. vol. 4, is. 2(30), pp. 15–21. (in Russian).

42. MOON, F., 1990. Chaotic Oscillations: An Introductory Course for Scientists and Engineers. Moscow, Russia: Mir Publ. (in Russian).

43. MCCAULEY, J. L., 1993. Chaos, Dynamics and Fractals. Cambridge: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511564154

44. LASOTA, A. and MACKEY, M. C., 1994. Chaos, Fractals, and Noise: Stochastic Aspects of Dynamics. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-4612-4286-4

45. KUZNETSOV, S. P., 2001. Dynamical Chaos. Saratov, Russia: Saratov State University Publ. (in Russian).

46. MOON, F. C., 2004. Chaotic and Fractal Dynamics. An Introduction for Applied Scientists and Engineers. Weinheim: Wiley-VCH Verlag.

47. MOON, F. C., 2004. Chaotic Vibrations. An Introduction for Applied Scientists and Engineers. New Jersey: Wiley-Interscience. DOI: https://doi.org/10.1002/3527602844

48. MANDELBROT, B. B., 2004. Fractals and Chaos. The Mandelbrot Sets and Beyond. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-4757-4017-2

49. PEITGEN, H.-O., JURGENS, H. and SAUPE, D., 2004. Chaos and Fractals. New Frontiers of Science. Second Edition. New York: Springer-Verlag. DOI: https://doi.org/10.1007/b97624

50. SZEMPLINSKA-STUPNICKA, W., 2004. Chaos. Bifurcations and Fractals Around Us. A Brief Introduction. New Jersey: World Scientific. DOI: https://doi.org/10.1142/5419

51. GRINCHENKO, V. V., MATSYPURA, V. T. and SNARSKYI, A. A., 2010. Introduction to Nonlinear Dynamics. Chaos and Fractals. Moscow, Russia: LKI Publ. (in Russian).

52. FELDMAN, D. P., 2012. Chaos and Fractals. An Elementary Introduction. Oxford: Oxford University Press. DOI: https://doi.org/10.1093/acprof:oso/9780199566433.001.0001

53. PICKOVER, C. A., ed., 1998. Chaos and Fractals: A Computer Graphical Journey. Ten Year Compilation of Advanced Research. Amsterdam, Lausanne et al.: Elsevier.

54. GULICK, D., 2012. Encounters with Chaos and Fractals. College Park, USA: University of Maryland Publ.

55. CRILLY, A. J., EARNSHAW, R. A. and JONES, H., eds., 1991. Fractals and Chaos. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-4612-3034-2

56. PEITGEN, H.-O., JÜRGENS, H., SAUPE, D., MALETSKY, E., PERCIANTE, T. and YUNKER, L., 1992. Fractals for the Classroom: Strategic Activities. Volume Two. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-4757-5276-2

57. PEITGEN, H.-O., JÜRGENS, H. and SAUPE, D., 1992. Fractals for the Classroom. Part One. Introduction to Fractals and Chaos. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-4757-2172-0

58. SCHEINERMAN, E. R., 1995. Invitation to Dynamical Systems. Upper Saddle River, NJ: Prentice Hall.

59. LESNE, A., 1995. Méthodes de renormalisation: Phénomènes critiques – Chaos – Structures fractales. Paris: Eyrolles Sciences.

60. KLAGES, R., 2007. Microscopic Chaos, Fractals and Transport in Nonequilibrium Statistical Mechanics. New Jersey, London et al.: World Scientific Publ. DOI: https://doi.org/10.1142/5945

61. MELIN, P. and CASTILLO, O., 2002. Modelling, Simulation and Control of Non-linear Dynamical Systems: An Intelligent Approach Using Soft Computing and Fractal Theory. Boca Raton, London et al.: Taylor and Francis Publ. DOI: https://doi.org/10.1201/9781420024524

62. KIVOTIDES, D., 2012. The Impact of Kinematic Simulations on Quantum Turbulence Theory. In: F. C. G. A. NICOLLEAU, C. CAMBON, J.-M. REDONDO, J. C. VASSILICOS, M. REEKS and A. F. NOWAKOWSKI, eds. New Approaches in Modeling Multiphase Flows and Dispersion in Turbulence, Fractal Methods and Synthetic Turbulence. Dordrecht, Heidelberg et al.: Springer, pp. 1–8.

63. GAPONOV-GREKHOV, A. V. and RABINOVICH, M. I., 1992. Nonlinearities in Action: Oscillations, Chaos, Order, Fractals. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-642-75292-6

64. LESNE, A., 1998. Renormalization Methods: Critical Phenomena, Chaos, Fractal Structures. Chichester: John Wiley & Sons.

65. CASTILLO, O. and MELIN, P., 2003. Soft Computing and Fractal Theory for Intelligent Manufacturing. Heidelberg: Physica-Verlag. DOI: https://doi.org/10.1007/978-3-7908-1766-9

66. FLAKE, G. W., 1998. The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation. Cambridge, MA: MIT Press.

67. SCHROEDER, M., 1991. Fractals, Chaos, Power Laws: Minutes from an Infinite. New York: W. H. Freeman and Company. DOI: https://doi.org/10.1063/1.2810323

68. SCHROEDER, M., 2001. Fractals, Chaos, Power Laws. Endless Paradise Miniatures. Izhevsk, Russia: NITS “Regulyarnaya i haoticheskaya dinamika” Publ. (in Russian).

69. PEANO, G., 1890. Sur une courbe, qui remplit toute une aire plane. Math. Ann. vol. 36, no. 1, pp. 157–160. (in French). DOI: https://doi.org/10.1007/BF01199438

70. VON KOCH, H., 1904. Sur une courbe continue sans tangente obtenue par une construction géométrique élémentaire. Arkiv för Matematik Astronomy, och Fisyc. vol. 1, pp. 681–702.

71. SMITH, H. J. S., 1874. On the integration of discontinuous functions. Proc. London Math. Soc., vol. s1-6, is. 1, pp. 140–153. DOI: https://doi.org/10.1112/plms/s1-6.1.140

72. DU BOIS-REYMOND, P., 1880. Der Beweis des Fundamentalsatzes der Integra lrechnung. Math. Ann. vol. 16, is. 1, pp. 115–128. DOI: https://doi.org/10.1007/BF01459233

73. VOLTERRA, V., 1881. Alcune osservazioni sulle funzioni punteggiate discontinue [Some observations on point-wise discontinuous function]. Giornale di Matematiche. vol. 19, pp. 76–86. (in Italian).

74. CANTOR, G., 1883. Über unendliche, lineare Punktmannigfaltigkeiten V. [On infinite, linear point-manifolds (sets), Part 5]. Math. Ann., vol. 21, is. 4, pp. 545–591. (in German). DOI: https://doi.org/10.1007/BF01446819

75. PERRIN, J., 1909. Movement brownien et réalité moléculaires. Annales de chimie et de physique. vol. 18, no. 8, pp. 5–114.

76. PERRIN, J., 1913. Les Atomies. Paris: Librairie Feléx Alcan. DOI: https://doi.org/10.1051/jcp/1910080057

77. GAZALE, M., 2002. Gnomon. From Pharaohs to Fractals. Moscow-Izhevsk, Russia: Institut komp’yuternykh issledovaniy Publ. (in Russian).

78. BARNSLEY, M. F., 1988. Fractals Everywhere. Boston: Academic Press Publ.

79. FAUVEL, J., FLOOD, R. and WILSON, R., eds., 2006. Music and Mathematics: From Pythagoras to Fractals. Oxford: Oxford University Press.

80. MANDELBROT, B. B., 1997. Fractals and Scaling in Finance: Discontinuity, Concentration, Risk. Selecta Volume E. New York: Springer-Verlag.

81. YANOVSKY, V. V., 2003. The Emergence of a New Paradigm in Physics. Universitates. no. 3, pp. 32–47. (in Russian).

82. YANOVSKY, V. V., 2006. Lectures on Nonlinear Phenomena. Volume 1. Kharkiv, Ukraine: Institut monokristallov Publ. (in Russian).

83. LOSA, G. A., MERLINI, D., NONNENMACHER, T. F. and WEIBEL, E. R., eds., 2005. Fractals in Biology and Medicine. Volume IV. Basel: Birkhäuser Verlag. DOI: https://doi.org/10.1007/3-7643-7412-8

84. MANDELBROT, B., 2004. Fractals, Case and Finance. Moscow-Izhevsk, Russia: NITS “Regulyarnaya i haoticheskaya dinamika” Publ. (in Russian).

85. PEINTGEN, H.-O. and RICHTER, P. H., 1993. The Beauty of Fractals. Images of Complex Dynamic Systems. Moscow, Russia: Mir Publ. (in Russian).

86. MANDELBROT, B., 2009. Fractals and Chaos. Mandelbrot Sets and Other Wonders. Moscow-Izhevsk, Russia: Institut komp’yuternykh issledovaniy Publ. (in Russian).

87. DEMENOK, S. L., 2012. Just Fractal. St. Petersburg, Russia: Strata Publ. (in Russian).

88. WICKS, K. R., 1991. Fractals and Hyperspaces. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/BFb0089156

89. BANDT, C., GRAF, S. and ZÄHLE, M., eds., 1995. Fractal Geometry and Stochastics. Basel: Birkhäuser Verlag. DOI: https://doi.org/10.1007/978-3-0348-7755-8

90. BANDT, C., FALCONER, K. and ZÄHLE, M., eds., 2015. Fractal Geometry and Stochastics V. Basel: Birkhäuser Verlag. DOI: https://doi.org/10.1007/978-3-319-18660-3

91. BANDT, C., GRAF, S. and ZÄHLE, M., eds., 2000. Fractal Geometry and Stochastics II. Basel: Birkhaäser Verlag. DOI: https://doi.org/10.1007/978-3-0348-8380-1

92. BANDT, C., MOSKO, U. and ZÄHLE, M., eds., 2004. Fractal Geometry and Stochastics III. Basel: Birkhäuser Verlag. DOI: https://doi.org/10.1007/978-3-0348-7891-3

93. PRZYTYCKI, F. and URBAŃSKI, M., 2010. Conformal Fractals: Ergodic Theory Methods. Cambridge: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9781139193184

94. BLEI, R., 2001. Analysis in Integer and Fractional Dimensions. Cambridge: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511543012

95. KIGAMI, J., 2001. Analysis on Fractals. Cambridge: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511470943

96. LOWEN, S. B. and TEICH, M. C., 2005. Fractal-Based Point Processes. Hoboken, New Jersey: John Wiley & Sons. DOI: https://doi.org/10.1002/0471754722

97. LÉVY-VÉHEL, J. and LUTTON, E., eds., 2005. Fractals in Engineering. New Trends in Theory and Applications. London: Springer-Verlag. DOI: https://doi.org/10.1007/b137729

98. AFRAIMOVICH, V., UGALDE, E. and URIAS, J., 2006. Fractal Dimensions for Poincare Recurrences, Volume 2. Amsterdam: Elsevier. DOI: https://doi.org/10.1016/S1574-6917(06)02001-0

99. JORGENSEN, P. E. T., 2006. Analysis and Probability: Wavelets, Signals, Fractals. New York: Springer-Verlag.

100. LAPIDUS, M. L. and VAN FRANKENHUIJSEN, M., 2013. Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings. New York: Springer-Verlag. https://doi.org/10.1007/978-1-4614-2176-4

101. LIPSCOMB, S. L., 2009. Fractals and Universal Spaces in Dimension Theory. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-0-387-85494-6

102. BARRAL, J. and SEURET, S., eds., 2010. Recent Developmentsin Fractals and Related Fields. Boston: Birkhäuser. DOI: https://doi.org/10.1007/978-0-8176-4888-6

103. ROSENBERG, E., 2018. A Survey of Fractal Dimensions of Networks. Cham, Switzerland: Springer Int. Publ.

104. KIRILLOV, A. A., 2013. A Tale of Two Fractals. Basel: Birkhäuser. DOI: https://doi.org/10.1007/978-0-8176-8382-5

105. LINDSTRØM, T., 1990. Brownian Motion on Nested Fractals. Mem. Am. Math. Soc. vol. 83, no. 420, pp. 1–128. DOI: https://doi.org/10.1090/memo/0420

106. CHEN, G. and HUANG, Y., 2011. Chaotic Maps. Dynamics, Fractals, and Rapid Fluctuations. San Rafael, USA: Morgan and Claypool Publ. DOI: https://doi.org/10.2200/S00373ED1V01Y201107MAS011

107. EDGAR, G. A., ed., 2004. Classics on Fractals. Boulder: Westview Press.

108. MAZZOLA, G., MILMEISTER, G. and WEISSMANN, J., 2005. Comprehensive Mathematics for Computer Scientists 2. Berlin, Heidelberg: Springer-Verlag.

109. WEINBERGER, S., 2005. Computers, Rigidity, and Moduli. The Large-Scale Fractal Geometry of Riemannian Moduli Space. Princeton, New Jersey: Princeton University Press.

110. STRICHARTZ, R. S ., 2006. Differential Equations on Fractals. A Tutorial. Princeton and Oxford: Princeton University Press. DOI: https://doi.org/10.1515/9780691186832

111. MAYER, V., SKORULSKI, B. and URBAŃSKI, M., 2011. Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-642-23650-1

112. LAPIDUS, M. L. and VAN FRANKENHUIJSEN, M., eds., 2004. Fractal Geometry and Applications: A Jubilee of Benoit Mandelbrot. Part 1.: Analysis, Number Theory, and Dynamical Systems. Providence, RL: American Mathematical Society Publ. DOI: https://doi.org/10.1090/pspum/072.1

113. CARFI, D., LAPIDUS, M. L., PEARSE, E. P. J. and VAN FRANKENHUIJSEN, M., eds., 2013. Fractal Geometry and Dynamical Systems in Pure and Applied Mathematics II: Fractals in Applied Mathematics. Providence, RL: American Mathematical Society Publ. DOI: https://doi.org/10.1090/conm/601

114. LAPIDUS, M. L. and VAN FRANKENHUIJSEN, M., 2000. Fractal Geometry and Number Theory: Complex Dimensions of Fractal Strings and Zeros of Zeta Functions. Basel: Birkhäuser. DOI: https://doi.org/10.1007/978-1-4612-5314-3_3

115. LAPIDUS, M., and ŽUBRINIĆ, D., 2017. Fractal Zeta Functions and Fractal Drums: Higher-Dimensional Theory of Complex Dimensions. New York: Springer Int. Publ. DOI: https://doi.org/10.1007/978-3-319-44706-3

116. TRIEBEL, H., 1997. Fractals and Spectra: Related to Fourier Analysis and Function Spaces. Basel: Birkhäuser. DOI: https://doi.org/10.1007/978-3-0348-0034-1_3

117. GRABNER, P. and WOESS, W., eds., 2003. Fractals in Graz 2001: Analysis – Dynamics – Geometry – Stochastics. Basel: Birkhäuser. DOI: https://doi.org/10.1007/978-3-0348-8014-5

118. BISHOP, C. J. and PERES ., 2016. Fractals in Probability and Analysis. Cambridge: Cambridge University Press. DOI: https://doi.org/10.1017/9781316460238

119. DAVID, G. and SEMMES, S., 1997. Fractured Fractals and Broken Dreams: Self-Similar Geometry through Metric and Measure. Oxford: Clarendon Press.

120. BARRAL, J. and SEURET, S., eds., 2013. Further developments in fractals and related fields: mathematical foundations and connections. Basel: Birkhäuser. DOI: https://doi.org/10.1007/978-0-8176-8400-6

121. FENG, D.-J. and LAU, K.-S., eds., 2014. Geometry and Analysis of Fractals: Hong Kong, December 2012. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-662-43920-3

122. MATTILA, P., 1995. Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability. Cambridge: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511623813

123. SU, W., 2018. Harmonic Analysis and Fractal Analysis over Local fields and Applications. Singapore: World Scientific Publ.

124. PESIN, Y. and CLIMENHAGA, V., 2009. Lectures on Fractal Geometry and Dynamical Systems. New York: American Mathematical Society Publ. DOI: https://doi.org/10.1090/stml/052

125. YAMAGUTI, M., HATA, M., KIGAMI, J. and HUDSON, K., 1997. Mathematics of Fractals. Providence, RL: American Mathematical Society Publ. DOI: https://doi.org/10.1090/mmono/167

126. LI, C., WU, Y. and YE, R., eds., 2013. Recent Advances in Applied Nonlinear Dynamics with Numerical Analysis: Fractional Dynamics, Network Dynamics, Classical Dynamics and Fractal Dynamics with their Numerical Simulations. Singapore: World Scientific Publ.

127. DOBRUSHIN, R. L. and KUSUOKA, S., 1993. Statistical Mechanics and Fractals. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/BFb0074238

128. FALCONER, K. J., 1986. The geometry of fractal sets. Cambridge: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511623738

129. KALMYKOV, Y. P., COFFEE, W. T. and RICE, S. A., eds., 2006. Fractals, Diffusion, and Relaxation in Disordered Complex Systems: Advances in Chemical Physics, Part A, Volume 133. New Jersey: Wiley-Interscience.

130. KOZLOV, G. V. and YANOVSKII, YU. G., 2015. Fractal mechanics of polymers: chemistry and physics of complex polymeric materials. Toronto: Apple Academic Press. DOI: https://doi.org/10.1201/b17730

131. BURDE, A. and HAVLIN, S., eds., 1996. Fractals and Disordered Systems. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-642-84868-1

132. TAKAYASU, H., 1990. Fractals in the physical sciences. Manchester, New York: Manchester University Press.

133. PIETRONERO, L. and TOSATTI, E., eds., 1986. Fractals in Physics. Amsterdam, Oxford et al.: North-Holland.

134. AMANN, A., CEDERBAUM, L. and GANS, W., eds., 1988. Fractals, Quasicrystals, Chaos, Knots and Algebraic Quantum Mechanics. Dordrecht: Kluwer Academic Press. DOI: https://doi.org/10.1007/978-94-009-3005-6

135. MEAKIN, P., 1998. Fractals, Scaling and Growth Far from Equilibrium. Cambridge: Cambridge University Press.

136. PIETRONERO, L., ed., 1989. Fractals’ Physical Origin and Properties. New York: Springer Science. DOI: https://doi.org/10.1007/978-1-4899-3499-4

137. STAUFFER, D. and STANLEY, H. E., 1996. From Newton to Mandelbrot: A Primer in Theoretical Physics with Fractals for the Macintosh (R). Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-662-11782-8_5

138. STANLEY, H. E. and OSTROWSKY, N., eds. 1986. On Growth and Form: Fractal and Non-fractal Patterns in Physics. Dordrecht: Martinus Nijhoff Publishers.

139. PICKOVER, C. A., ed., 1995. The Pattern Book: Fractals, Art and Nature. Singapore, New Jersey et al.: World Scientific Publ. DOI: https://doi.org/10.1142/2052

140. NOVAK, M. M., ed., 2004. Thinking in Patterns. Fractals and Related Phenomena in Nature. New Jersey, London et al.: World Scientific Publ.

141. HECK, A. and PERDANG, J. M., eds., 1991. Applying Fractals in Astronomy. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-540-47582-8

142. LESMOIR-GORDON, N., ed., 2010. The Colours of Infinity: The Beauty and Power of Fractals. London: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-84996-486-9

143. LUNG, C. W. and MARCH, N. H., 1999. Mechanical Properties of Metals: Atomistic and Fractal Continuum Approaches. Singapore, New Jersey et al.: World Scientific Publ. DOI: https://doi.org/10.1142/3064

144. AI-AKAIDI, M., 2004. Fractal Speech Processing. Cambridge: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511754548

145. BARNSLEY, M. F., SAUPE, D. and VRSCAY, E. R., eds., 2002. Fractals in Multimedia. New York, Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-4684-9244-6

146. BUNDE, A. and HAVLIN, S., eds., 1994. Fractals in Science. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-662-11777-4

147. ADDISON, P. S., 1997. Fractals and Chaos. An Illustrated Course. Bristol, Philadelphia: IOP Publishing Ltd. DOI: https://doi.org/10.1201/9780849384431

148. BIRDI, K. S., 1993. Fractals in Chemistry, Geochemistry, and Biophysics: An Introduction. New York: Springer Science. DOI: https://doi.org/10.1007/978-1-4899-1124-7

149. KOZLOV, G. V., DOBLIN, I. V. and ZAIKOV, G. E., 2013. The Fractal Physical Chemistry of Polymer Solutions and Melts. Toronto, New Jersey: Apple Academic Press. DOI: https://doi.org/10.1201/b16305

150. KOZLOV, G. V., MIKITAEV, A. K. and ZAIKOV, G. E., 2013. The Fractal Physics of Polymer Synthesis. Toronto, New Jersey: Apple Academic Press. DOI: https://doi.org/10.1201/b16303

151. KAANDORP, J. A., 1994. Fractal Modelling, Growth and Form in Biology. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-642-57922-6

152. LIEBOVITCH, L. S., 1998. Fractals and Chaos: Simplified for the Life Sciences. Oxford: Oxford University Press.

153. ISAYEVA, V. V., KARETIN, YU. A., CHERNYSHOV, A. V. and SHKURATOV, D. YU., 2004. Fractals and Chaos in Biological Morphogenesis. Vladivostok, Russia: Institut biologii morya DVO RAN Publ. (in Russian).

154. DI IEVA, A., ed., 2016. The Fractal Geometry of the Brain. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-4939-3995-4

155. BRAMBILA, F., ed., 2017. Fractal Analysis. Applications in Health Sciences and Social Sciences. Rijeka, Croatia: InTech. DOI: https://doi.org/10.5772/intechopen.68898

156. SADANA, A., 2005. Fractal Binding and Dissociation Kinetics for Different Biosensor Applications. Amsterdam, Boston et al.: Elsevier. DOI: https://doi.org/10.1016/B978-044451945-0/50005-4

157. BASSINGTHWAIGHTE, J. B., LIEBOVICH, L. S. and WEST, B. J., 1994. Fractal Physiology. Oxford, New York et al.: Oxford University Press.

158. WEST, B. J., 2013. Fractal Physiology and Chaos in Medicine. Singapore: World Scientific Publ. DOI: https://doi.org/10.1142/8577

159. WEST, B., 2010. Fractal Physiology and the Fractional Calculus: A Perspective. Front. Physiol. vol. 1, id. 12. DOI: https://doi.org/10.3389/fphys.2010.00012

160. KUMAR, D., ARJUNAN, S. P. and ALIAHMAD, B., 2017. Fractals: Application in Biological Signalling and Image Processing. Boca Raton: CRC Press. DOI: https://doi.org/10.1201/9781315165868

161. NONNENMACHER, T. F., LOSA, G. A. and WEIBEL, E. R., eds., 1994. Fractals in Biology and Medicine. Basel: Birkhäuser. DOI: https://doi.org/10.1007/978-3-0348-8501-0

162. LOSA, G. A., MERLINI, D., NONNENMACHER, T. F. and WEIBEL, E. R., eds., 2002. Fractals in Biology and Medicine. Volume 3. Basel, Boston, Berlin: Birkhäuser. DOI: https://doi.org/10.1007/978-3-0348-8119-7

163. TAKAHASHI, T., 2014. Microcirculation in Fractal Branching Networks. Japan:: Springer. DOI: https://doi.org/10.1007/978-4-431-54508-8

164. DEWEY, T. G., 1997. Fractals in Molecular Biophysics. Oxford, New York: Oxford University Press.

165. SENESI, N. and WILKINSON, K. J., eds., 2008. Biophysical Chemistry of Fractal Structures and Processes in Environmental Systems. New York: John Wiley & Sons Inc. DOI: https://doi.org/10.1002/9780470511206

166. SADANA, A., 2003. Biosensors: Kinetic of Binding and Dissociation Using Fractal. Amsterdam: Elsever. DOI: https://doi.org/10.1016/B978-044451512-4/50053-6

167. SADANA, A. and SADANA, N., 2008. Fractal Analysis of the Binding and Dissociation Kinetics for Different Analytes on biosensor Surfaces. Amsterdam: Elsevier. DOI: https://doi.org/10.1016/B978-044453010-3.50001-3

168. BANERJI, A., 2013. Fractal Symmetry of Protein Exterior. Basel: Springer. DOI: https://doi.org/10.1007/978-3-0348-0654-1

169. ADDISON, P. S., 2002. The Illustrated Wavelet Transform Handbook. Introductory Theory and Applications in Science, Engineering, Medicine and Finance. Bristol, Philadelphia: IOP Publishing Ltd. DOI: https://doi.org/10.1201/9781420033397

170. IONESCU, C. M., 2013. The Human Respiratory System: An Analysis of the Interplay between Anatomy, Structure, Breathing and Fractal Dynamics. London: Springer-Verlag.

171. BARABÁSI, A.-L., and STANLEY, H. E., 1995. Fractal Concept in Surface Growth. Cambridge: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511599798

172. QUADFEUL, S.-A., ed., 2012. Fractal Analysis and Chaos in Geosciences. Rijeka, Croatia: InTech Press.

173. TURCOTTE, D. L., 1997. Fractals and Chaos in Geology and Geophysics. Cambridge: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9781139174695

174. KRUHL, J. Y., ed., 1994. Fractals and Dynamic Systems in Geoscience. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-662-07304-9

175. BARTON, C. C. and LA POINTE, P. R., eds., 1995. Fractals in Petroleum Geology and Earth Processes. Bosnon, MA: Springer. DOI: https://doi.org/10.1007/978-1-4615-1815-0

176. BARTON, C. C. and LA POINTE, P. R., eds., 1995. Fractals in the Earth Sciences. New York: Springer. DOI: https://doi.org/10.1007/978-1-4899-1397-5

177. DAUPHINÉ, A., 2012. Fractal Geography. London, Hoboken: John Wiley & Sons. DOI: https://doi.org/10.1002/9781118603178

178. CHANDRASEKHAR, E., DIMRI, V. P. and GARDE, V. M., 2014. Wavelets and Fractals in Earth System Sciences. Boca Raton, London, New York: CRC Press. DOI: https://doi.org/10.1201/b16046

179. DIMRI, V. P., 2016. Fractal Solutions for Understanding Complex Systems in Earth Sciences. Cham, Heidelberg, at al.: Springer Int. Publ. DOI: https://doi.org/10.1007/978-3-319-24675-8

180. DIMRI, V. P., ed., 2005. Fractal Behavior of the Earth System. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/b137755

181. DIMRI, V. P., SRIVASTAVA, R. P. and VEDANTI, N., 2012. Fractal Models in Exploration Geophysics: Applications to Hydrocarbon Reservoirs. Oxford, UK: Elsevier Science & Technology. DOI: https://doi.org/10.1016/B978-0-08-045158-9.00003-8

182. GHANBARIAN, B. and HUNT, A. G., eds., 2017. Fractals: Concepts and Applications in Geosciences. Boca Raton, London, New York: CRC Press. DOI: https://doi.org/10.1201/9781315152264

183. SCHOLZ, C. H. and MANDELBROT, B. B., eds., 1989. Fractals in Geophysics. Basel: Birkhäuser. DOI: https://doi.org/10.1007/978-3-0348-6389-6

184. LUKK, A. A., DESHCHEREVSKIY, A. V., SIDORIN, A. YA. and SIDORIN, I. A., 1996. Variations of Geophysical Fields as a Manifestation of Deterministic Chaos in a Fractal Medium. Moscow, Russia: OIFZ RAN Publ. (in Russian).

185. BAVEYE, P., PARLANGE, J.-Y. and STEWART, B. A., eds., 1998. Fractals in Soil Science. Boca Raton: CRC Press.

186. MCNUTT, B., 2000. The Fractal Structure of Data Reference: Applications to the Memory Hierarchy. Boston, Dorbreht, London: Kluver Academic Publ.

187. FARMER, M. E., 2014. Application of Chaos and Fractals to Computer Vision. Sharjah: Bentham Sci. Publ. Ltd. DOI: https://doi.org/10.2174/97816080590031140101

188. MANDELBROT, B. and HUDSON, R. L., 2006. (Un)obedient Markets: a Fractal Revolution in Finance. Moscow, Russia: Vil’yams Publ. (in Russian).

189. CRILLY, A. J., EARNSHOW, R. and JONES, H., eds., 1993. Applications of Fractals and Chaos: The Shape of Things. Heidelberg: Springer-Verlag.

190. MANDELBROT, B. B. and HUDSON, R. L., 2006. The Misbehavior of Markets. A fractal View of Risk, Ruin and Reward. New York: Basic Books.

191. EGLASH, R., 1999. African Fractals. Modern Computing and Indigenous Design. New Brunswick, NJ: Rutgers University Press. DOI: https://doi.org/10.1007/978-1-4612-0843-3

192. BATTY, M. and LONGLEY, P., 1994. Fractal Cities: A Geometry of Form and Function. London, San Diego et al.: Academic Press.

193. BOVILL, C., 1996. Fractal Geometry in Architechture and Design. Basel: Birkhäuser.

194. KULAK, M. I., 2002. Fractal Mechanics of Materials. Minsk, Belarus: Vysheyshaya shkola Publ. (in Russian).

195. GARDNER, M., 1992. Fractal Music, Hypercards and more...: Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman and Company.

196. ENCARNACAO, J. L., PEITGEN, H.-O., SAKAS, G. and ENGLERT, G., eds., 1992. Fractal Geometry and Computer Graphics. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-642-95678-2

197. FRAME, M. and URRY, A., 2016. Fractal Worlds: Grown, Built, and Imagined. New Haven, London: Yale University Press.

198. FRANTZ, M. and CRANNELL, A., 2011. Viewpoints: Mathematical Perspective and Fractal Geometry in Art. Princeton, Oxford: Princeton University Press. DOI: https://doi.org/10.1515/9781400839056

199. NIKOLAYEVA, E. V., 2014. Fractals of Urban Culture. St. Petersburg, Russia: Strata Publ.

200. MARKS-TARLOW, T., 2008. Psyche's Veil. Psychotherapy, Fractals and Complexity. London, New York: Routledge, Taylor and Francis Group,.

201. WARNECKE, H.-J., 1993. The Fractal Company: A Revolution in Corporate Culture. Berlin, Heidelberg: Springer-Verlag.

202. HOVERSTADT, P., 2008. The Fractal Organization: Creating sustainable organizations with the Viable System Model. Chichester, UK: John Wiley & Sons, Ltd.

203. ROBBINS, B., 2011. Microscope: A Fractal Role-playing Game of Epic Histories. New York: Lame Mage Productions.

204. KRÁL, F., 2014. Social Invisibility and Diasporas in Anglophone Literature and Culture: The Fractal Gaze. Basingstoke,UK: Palgrave Macmillan. DOI: https://doi.org/10.1057/9781137401397

205. GHOSH, B., SINHA, S. and KARTIKEYAN, M. V., 2014. Fractal Apertures in Waveguides, Conducting Screens and Cavities: Analysis and Design. Switzerland: Springer Int. Publ. DOI: https://doi.org/10.1007/978-3-319-06535-9

206. JADCZYK, A., 2014. Quantum Fractals: From Heisenberg’s Uncertainty to Barnsley’s Fractality. Hackensack, NJ; London, UK: World Scientific Publ. DOI: 
https://doi.org/10.1142/8992

207. FEDER, J., 1991. Fractals. Moscow, Russia: Mir Publ. (in Russian).

208. LI, J. M., LU, L., LAI, M. O. and RALPH, B., 2003. Image-Based Fractal Description of Microstructures. New York: Springer.

209. MANDELBROT, B. B., 1986. Self-affiine fractal sets. In: L. PIETRONERO and E. TOSATTI, eds. Fractals in Physics. Amsterdam, Oxford et al.: North-Holland, pp. 3–28. DOI: https://doi.org/10.1016/B978-0-444-86995-1.50004-4

210. MOROZOV, A. D., 2002. Introduction in Fractal Theory. Moscow–Izhevsk, Russia: Institut komp’yuternikh issledovaniy Publ. (in Russian).

211. FALCONER, K., 2003. Fractal Geometry. Mathematical Foundations and Applications. Second Edition. Chichester, UK: John Wiley & Sons, Ltd. DOI: https://doi.org/10.1002/0470013850

212. FALCONER, K., 2014. Fractal Geometry. Mathematical Foundations and Applications. Third Edition. Chichester, UK: John Wiley & Sons, Ltd.

213. FALCONER, K., 1997. Techniques in Fractal Geometry. Chichester, UK: John Wiley & Sons, Ltd.

214. FISHER, Y., 1995. Fractal Image Compression. Theory and Application. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-4612-2472-3

215. MANDELBROT, B. B., 1999. Multifractals and 1/f Noise. wild self-Affinity in Physics (1963–1976). New York: Springer-Verlag.

216. SEURONT, L., 2010. Fractals and Multifractals in Ecology and Aquatic Science. Boca Raton, London, New York: CRC Press. DOI: https://doi.org/10.1201/9781420004243

217. GOLTZ, C., 1997. Fractal and Chaotic Properties of Earthquakes. Berlin, Heidelberg: Springer-Verlag. DOI: https://doi.org/10.1007/BFb0028315

218. PASYNKOV, B. A. and FEDORCHUK, V. V., 1984. Topology and Dimension Theory. Moscow, Russia: Znaniye Publ. (in Russian).

219. EFREMOVICH, V. A., 1966. Basic Topological Concepts. In: P. S. ALEKSANDROV, ed. Encyclopedia of Elementary Mathematics. Vol. 5. Geometry. Moscow, Russia: Nauka Publ., pp. 476–556. (in Russian).

220. GUREVICH, V. and WALLMAN, G., 1948. Dimension Theory. Moscow, Russia: Inostrannaya Literatura Publ. (in Russian).

221. EDGAR, G., 2008. Measure, Topology, and Fractal Geometry. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-0-387-74749-1

222. CHULICHKOV, A. I., 2003. Mathematical Methods of Nonlinear Dynamics. Moscow, Russia: FIZMATLIT Publ.

223. LE MÉHAUTÉ, A., 1991. Fractal geometries, theory and applications. Boca Raton: CRC Press.

224. DEVANEY, R. L. and KEEN, L., eds., 1989. Chaos and Fractals. The Mathematics Behind the Computer Graphics. Providence, RI: American Mathematical Society Publ. DOI: https://doi.org/10.1090/psapm/039

225. TRICOT, C., 1981. Douze definitions de la densité logarithmique. C. R. Acad. Sci. Paris Sér. I Math. vol. 293, pp. 549–552.

226. ABRY, P., GONÇALVES, P. and LÉVY VÉHEL, J., eds., 2009. Scaling, Fractals and Wavelets. New York: John Wiley & Sons. DOI: https://doi.org/10.1002/9780470611562

227. FRANCESCHETTI, G. and RICCIO, D., 2007. Scattering, Natural Surfaces and Fractals. New York: Elsevier. DOI: https://doi.org/10.1016/B978-012265655-2/50000-3

228. PASHCHENKO, R. E., 2005. Fundamentals of the Theory of Fractal Signal Formation. Kharkiv, Ukraine: NEO EkoPerspectiva Publ. (in Russian).

229. KOROLENKO, P. V., MAGANOVA, M. S. and MESNYANKIN, A. V., 2004. Innovative Methods for the Analysis of Stochastic Processes and Structures in Optics. Fractal and Multifractal Methods, Wavelet Transforms. Tutorial. Moscow, Russia: NIIYaF MGU Publ. (in Russian).

230. MALLAT, S., 2005. Wavelets in Signal Processing. Moscow, Russia: Mir Publ. (in Russian).

231. PRUSINKIEWICZ, P. and HANAN, J., 1989. Lindenmayer systems, Fractals, and Plants. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-4757-1428-9

232. FEDER, J., 1988. Fractals. New York and London: Springer. DOI: https://doi.org/10.1007/978-1-4899-2124-6

233. DANILOV, YU. A., 2006. Lectures on Nonlinear Dynamics: Elementary Introduction. Moscow, Russia: KomKniga Publ. (in Russian).

234. GULICK, D. and SCOTT, J., eds., 2010. The Beauty of Fractals: Six Different Views. New York: The Mathematical Association of America Publ. DOI: https://doi.org/10.1017/CBO9780883859711

235. LAZORENKO, O. V. and CHERNOGOR, L. F., 2019. Nonlinear Radiophysics: A Collection of Tasks. Kharkiv, Ukraine: V. N. Karazin KhNU Publ. (in Russian).

236. MISHRA, J. and MISHRA, S. N., eds. 2007. L-System Fractals. Amsterdam, Boston et al.: Elsevier.

237. BARNSLEY, M. F. and DEMKO, S., 1985. Iterated Function Systems and the Global Construction of Fractals. Proc. R. Soc. Lond. A. vol. 399, is. 1817, pp. 243–275. DOI: https://doi.org/10.1098/rspa.1985.0057

238. BARNSLEY, M. F., ELTON, J. H. and HARDIN, D. P., 1989. Recurrent Iterated Function Systems Fractal Approximation. Constr. Approx. vol. 5, is. 1, pp. 3–31. DOI: https://doi.org/10.1007/BF01889596

239. BARNSLEY, M. F., 2006. Superfractals. New York: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9781107590168

240. BARNSLEY, M. and SLOAN, A., 1988. A Better Way to Compress Images. Byte. vol. 13, is. 1, pp. 215–223.

241. WELSTEAD, S., 1999. Fractal and wavelet Image Compression Techniques. Belligham, WA: SPIE Optical Engineering Press. DOI: https://doi.org/10.1117/3.353798

242. BARNSLEY, M. F., 1989. The Desktop Fractal Design Handbook. Bosnon, San Diego et al.: Academic Press.

243. BARNSLEY, M., HEGLAND, M. and MASSOPUST, P., 2016. Self-referential functions. arXiv:1610.01369v1 [math.CA].

244. MASSOPUST, P. R., 2016. Fractal Functions, Fractal Surfaces and Wavelets. Amsterdam, Boston et al.: Academic Press. DOI: https://doi.org/10.1016/B978-0-12-804408-7.00009-6

245. LEVY VEHEL, J., LUTTON, E. and TRICOT, C., eds., 1997. Fractals in Engineering: From Theory to Industrial Applications. London: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-4471-0995-2

246. PEITGEN, H.-O., JÜRGENS, H., SAUPE, D., MALETSKY, E., PERCIANTE, T. and YUNKER, L., 1991. Fractals for the Classroom: Strategic Activities. Volume One. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-4613-9047-3

247. PEITGEN, H.-O., JÜRGENS, H. and SAUPE, D., 1992. Fractals for the Classroom. Part Two. Complex Systems and Mandelbrot Set. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-1-4612-4406-6

248. KATUNIN, A., 2017. A Concise Introduction to Hypercomplex Fractals. Boca Raton, London, New York: CRC Press. DOI: https://doi.org/10.1201/b22103

249. OSTASHKOV, V. N., 2011. Fractal Dialogs. Tyumen’, Russia: TyumGNGU Publ. (in Russian).

250. STOYAN, D. and STOYAN, H., 1994. Fractals, Random Shapes and Point Fields: Methods of Geometrical Statistics. Chichester: John Wiley & Sons.

251. SMIRNOV, B. M., 1991. Physics of Fractal Clusters. Moscow, Russia: Nauka Publ. (in Russian).

252. FAMILY, F. and VICSEK, T., eds., 1991. Dynamics of Fractal Surfaces. Singapore, New Jersey et al.: World Scientific Publ. DOI: https://doi.org/10.1142/1452

253. TONG, H., ed., 1993. Dimension Estimation and Models. Singapore, New Jersey et al.: World Scientific Publ. DOI: https://doi.org/10.1142/1986

254. HOLSCHNEIDER, M., 1995. Wavelets: An Analysis Tool. Oxford: Calderon Press.

255. KOLMOGOROV, A. N. and TIKHOMIROV, V. M., 1959. ε-entropy and εcapacity of sets in function spaces. Uspekhi Mat. Nauk. vol. 14, is. 2(86), pp. 3–86.

256. MINKOWSKI, H., 1901. Über die Begriffe Länge, Oberfläche und Volumen. Jahresbericht der Deutschen Mathematiker-Vereinigung. vol. 9, is. 1, pp. 115–121.

257. BOULIGAND, G., 1928. Ensembles impropres et nombre dimensionnel. Bull. Sci. Math. vol. 52, is. 2, pp. 320–344, 361–376.

258. LAM, L., 1998. Nonlinear Physics for Beginners. Fractals, Chaos, Solitons, Pattern Formation, Cellular Automata and Complex Systems. Singapore, New Jersey et al.: World Scientific Publ. DOI: https://doi.org/10.1142/1037

259. HELMBERG, G., 2007. Getting Acquainted with Fractals. Berlin: Walter de Gruyter. DOI: https://doi.org/10.1515/9783110190922

260. FALCONER, K., 2013. Fractals: A Very Short Introduction. Oxford: Oxford University Press. DOI: https://doi.org/10.1093/actrade/9780199675982.001.0001

261. KANTZ, H. and SCHREIBER, T., 2003. Nonlinear Time Series Analysis. New York: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511755798

262. HILBORN, R. C., 2000. Chaos and Nonlinear Dynamics. An Introduction for Scientists and Engineers. New York: Oxford University Press.

263. PAVLOV, A. N., 2008. Complex Signal Analysis Methods: Study Guide. Saratov, Russia: Nauchnaya kniga Publ. (in Russian).

264. GRASSBERGER, P. and PROCACCIA, I., 1983. Measuring the strangeness of strange attractors. Physica D. vol. 9, is. 1-2, pp. 189–208. DOI: https://doi.org/10.1016/0167-2789(83)90298-1

265. GRASSBERGER, P., 1983. Generalized dimensions of strange attractors. Phys. Lett. A. vol. 97, is. 6, pp. 227–230. DOI: https://doi.org/10.1016/0375-9601(83)90753-3

266. HENTSCHEL, H. G. E. and PROCACCIA, I., 1983. The infinite number of generalized dimensions of fractals and strange attractors. Physica D. vol. 8, is. 3, pp. 435–444. DOI: https://doi.org/10.1016/0167-2789(83)90235-X

267. RÉNYI, A., 1961. On measures of entropy and information. In: Proceedings of the Fourth Berkeley Symp. on Math. Statist. and Prob. Vol. 1. Univ. of Calif. Press. pp. 547–561.

268. BECKER, K.-H. and DÖRFLER, M., 1989. Dynamical Systems and Fractals. Computer Graphics Experiments in Pascal. New York: Cambridge University Press. DOI: https://doi.org/10.1017/CBO9780511663031

269. BOLOTIN, YU. L., TUR, A. V. and YANOVSKY, V. V., 2005. Constructive Chaos. Kharkiv, Ukraine: Institut monokristallov Publ. (in Russian).

270. CHERNOGOR, L. F., 2010. Nonlinear Radiophysics. Textbook. Kharkiv, Ukraine: V. N. Karazin KhNU Publ. (in Russian).

271. IL’YASHENKO, YU. S., 2005. Attractors and their Fractal Dimension. Moscow, Russia: MTsNMO Publ.

272. SORNETTE, D., 2006. Critical Phenomena in Natural Sciences: Chaos, Fractals, Selforganization and Disorder: Concepts and Tools. Berlin, Heidelberg: Springer-Verlag.

273. BEN-AVRAHAM, D. and HAVLIN, S., 2004. Diffusion and Reactions in Fractals and Disordered Systems. Cambridge: Cambridge University Press.

274. ALEXANDER, S. and ORBACH, R., 1982. Density of states on fractals: “ fractons ”. J. Phys. (Paris) Lett. vol. 43, is. 17, pp. 625–631. DOI: https://doi.org/10.1051/jphyslet:019820043017062500

275. ROY, A. and SOOD, A. K., 1995. Fracton dimension of porous silicon as determined by low-frequency Raman scattering. Solid State Commun. vol. 93, is. 12, pp. 995–998. DOI: https://doi.org/10.1016/0038-1098(94)00919-8

276. MANDELBROT, B., 1985. Self-Affine Fractals and Fractal Dimension. Phys. Scr. vol. 32, is. 4, pp. 257–260. DOI: https://doi.org/10.1088/0031-8949/32/4/001

277. RICHARDSON, L. F., 1961. The Problem of Contiguity: An Appendix to Statistics of Deadly Quarrels. Gen. Syst. vol. 6, pp. 139–187.

278. MANDELBROT, B., 1967. How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension. Science. vol. 156, is. 3775, pp. 636–638. DOI: https://doi.org/10.1126/science.156.3775.636

279. KAYE, B. H., 1994. A Random Walk Through Fractal Dimensions. Weinheim, New York et al.: VCH. DOI: https://doi.org/10.1002/9783527615995

280. STONE, E. C. and MINER, E. D., 1981. Voyager I Encounter with the Saturnian system. Science. vol. 212, is. 4491, pp. 159–163. DOI: https://doi.org/10.1126/science.212.4491.159

281. HARTER, W. G and PATTERSON, C. W., 1979. Theory of hyperfine and superfine levels in symmetric polyatomic molecules. Trigonal and tetrahedral molecules: Elementary spin-½ cases in vibronic ground states. Phys. Rev. A. vol. 19, is. 6, pp. 2277–2303. DOI: https://doi.org/10.1103/PhysRevA.19.2277

282. YU, F. T. S. and JUTAMULIA, S., eds., 1996. Optical Storage and Retrieval: Memory: Neural Networks, and Fractals. New York, Basel, Hong Kong: Marsel Dekker, Ink.

283. OKOROKOV, V. A. and SANDRAKOVA, E. V., 2009. Fractals in Fundamental Physics. Fractal Properties of Multiple Particle Formation and Sample Topology. Moscow, Russia: MIFI Publ. (in Russian).

284. KROEGER, H., 2000. Fractal geometry in quantum mechanics, field theory and spin systems. Phys. Rep. vol. 323, is. 2, pp. 81–181. DOI: https://doi.org/10.1016/S0370-1573(99)00051-4

285. NOTTALE, L., 1993. Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity. Singapore, New Jersey et al.: World Scientific Publ. DOI: https://doi.org/10.1142/1579

286. CARPINTERI, A. and MAINARDI, F., eds., 1997. Fractals and Fractional Calculus in Continuum Mechanics. Wien: Springer-Verlag. DOI: https://doi.org/10.1007/978-3-7091-2664-6

287. SAKAI, Y. and VASSILICOS, C., eds., 2016. Fractal Flow Design: How to Design Bespoke Turbulence and Why. New York: Springer Int. Publ. DOI: https://doi.org/10.1007/978-3-319-33310-6

288. CELLO, G. and MALAMUD, B. D., eds., 2006. Fractal Analysis for Natural Hazards Geological Sosiety. Special Publication No. 261. London: Geological Society Publ.

289. SMIRNOV, B. M., 1991. Fractal Tangle is a New State of Matter. Uspekhi fizicheskih nauk. vol. 161, is. 8, pp. 141–153. (in Russian). DOI: https://doi.org/10.3367/UFNr.0161.199108e.0141

290. ANDERS, A., 2008. Cathodic Arcs: From Fractal Spots to Energetic Condensation. New York: Springer-Verlag. DOI: https://doi.org/10.1007/978-0-387-79108-1

291. WORNELL, G., 1996. Signal Processing with Fractals: A Wavelet-Based Approach. Englwood Cliffs, NJ: Prentice Hall PTR.

292. GROMOV, YU. YU., ZEMSKOY, N. A., IVANOVA, O. G., LAGUTIN, A. V. and TYUTYUNIK, V. M., 2007. Fractal Analysis and Processes in Computer Networks: Tutorial. Tambov, Russia: TGTU Publ. (in Russian).

293. MARAGOS, P. and POTAMIANOS, A., 1999. Fractal dimensions of speech sounds: Computation and application to automatic speech recognition. J. Acoust. Soc. Am. vol. 105, is. 3, pp. 1925–1932. DOI: https://doi.org/10.1121/1.426738

294. MOGILEVSKY, E. I., 2001. Fractals on the Sun. Moscow, Russia: FIZMATLIT Publ. (in Russian).

295. PASHCHENKO, R. E., ed., 2