Parametric homogeneity and non-classical self-similarity. I. Mathematical background

Acta Mechanica

Volumn 131, Issue 1-2, 1998, Pages 27-45

  Borodich, F M ^a,b  

^a Glasgow Caledonian University ^*

^b GLASGOW CALEDONIAN UNIVERSITY   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

FRACTALS; FUNCTIONS;

NON-CLASSICAL SELF-SIMILARITY; PARAMETRIC HOMOGENEITY;

MECHANICS;

EID: 0032290779     PISSN: 00015970     EISSN: None     Source Type: Journal    
DOI: 10.1007/BF01178243     Document Type: Article

References (38)

1. Alexandroff, P. S., Kolmogorov, A. N.: Introduction in theory of sets and theory of functions. Moscow: OGIZ 1948. (German translation: Alexandroff, P. S.: Einführung in die Mengenlehre und die Theorie der reellen Funktionen. Berlin: VEB Deutscher Verlag der Wissenschaften 1971.)
2. Alexandroff, P. S., Kolmogorov, A. N.: Introduction in theory of sets and theory of functions. Moscow: OGIZ 1948. (German translation: Alexandroff, P. S.: Einführung in die Mengenlehre und die Theorie der reellen Funktionen. Berlin: VEB Deutscher Verlag der Wissenschaften 1971.)
3. Arnold, V. I.: Ordinary differential equations, 3rd ed. Moscow: Nauka 1984.
4. Ausloos, M., Berman, D. H.: A multivariate Weierstrass-Mandelbrot function. Proc. R. Soc. London Ser. A 400, 331-350 (1985).
5. Bak, P.: The devil's straircase. Phys. Today 39, 38-45 (1986).
6. Barenblatt, G. I.: Similarity, self-similarity, and intermediate asymptotics. New York: Consultants Bureau 1979.
7. Barenblatt, G. I.: Dimensional analysis. New York: Gordon and Breach 1987.
8. Barenblatt, G. I., Zeldovich, Ya. B.: Self-similar solutions as intermediate asymptotics. Annu. Rev. Fluid Mech. 4, 285-312 (1972).
9. Berry, M. V., Lewis, Z. V.: On the Weierstrass-Mandelbrot fractal functions. Proc. R. Soc. London Ser. A 370, 519-521 (1980).
10. Besicovitch, A. S.: Analysis of continuous functions in connection with the question about their differentiability. Math. Sbornik (Moscow Math. Soc.) 31, 529-556 (1924) (in Russian).
11. Besicovitch, A. S., Ursel, H. D.: Sets of fractional dimensions (V): on dimensional numbers of some continuous curves. J. London Math. Soc. 12, 18-25 (1937).
12. Birkhoff, G.: Hydrodynamics, study in logic, fact and similitude. Princeton: Princeton University Press 1960.
13. Borodich, F. M.: Self-similarity properties of static and dynamic contact between a punch and anisotropic media. In: Abstracts of Reports on 18th Int. Congress of Theoretical and Appl. Mech. Haifa: Elsevier 1992.
14. Borodich, F. M.: Similarity properties of discrete contact between a fractal punch and an elastic medium. C. R. Ac. Sci. (Paris), Ser. II. 316, 281-286 (1993).
15. Borodich, F. M.: Personal communication to A. G. Khovanskii (1993).
16. Borodich, F. M.: Some applications of the fractal parametric-homogeneous functions. Fractals 2, 311-314 (1994).
17. Borodich, F. M.: Similarity in the problem of discrete contact between fractal surfaces. In: Fractal reviews in the natural and applied sciences (Novak, M. M., ed.) pp. 113-120. London: Chapman and Hall 1995.
18. Borodich, F. M.: Parametric-homogeneous functions, similarity, and fractal function graphs. Glasgow Caledonian University, Technical Report TR/MAT/FMB/95-35, pp. 1-55. Glasgow 1995.
19. Borodich, F. M.: Applications of parametric-homogeneous functions to problems of interaction between fractal surfaces. ZAMM 76, 61-62 (1996).
20. Bridgman, P. W.: Dimensional analysis. New Haven: Yale University Press 1931.
21. Falconer, K. J.: Fractal geometry: mathematical foundations and applications. Chichester: John Wiley 1990.
22. Feder, J.: Fractals. New York: Plenum Press 1988.
23. Hardy, G. H.: Weierstrass' non-differentiable function. Trans. Am. Math. Soc. 17, 301-325 (1916).
24. Hille, E., Tamarkin, J. D.: Remarks on a known example of a monotone continuous function. Am. Math. Monthly 36, 255-264 (1929).
25. Hu, T. Y., Lau, K. S.: Fractal dimensions and singularities of the Weierstrasstype functions. Trans. Am. Math. Soc. 335, 649-665 (1993).
26. Klein, F.: Vorlesungen über die Entwicklung der Mathematik im 19. Jahrhundert. Teil 1. Berlin: Springer 1926.
27. Kolmogorov, A. N., Fomin, S. V.: Introductory real analysis. Englewood Cliffs: Prentice-Hall 1970.
28. Kolmogorov, A. N., Fomin, S. V.: Elements of the theory of functions and functional analysis. Moscow: Nauka 1976 (in Russian).
29. Mandelbrot, B. B.: Fractals: form, chance, and dimension. San Francisco: Freeman 1977.
30. Mauldin, R. D., Williams, S. C.: On the Hausdorff dimension of some graphs. Trans. Am. Math. Soc. 298, 793-803 (1986).
31. Mosolov, A. B.: Singular fractal functions and mesoscopic effects in mechanics. Chaos Solitons Fractals 4, 2093-2102 (1994).
32. Ovsiannikov, L. I.: Group analysis of differential equations. London: Academic Press 1982.
33. Sedov, L. I.: Similarity and dimensional methods in mechanics. New York: Academic Press 1959.
34. Shafarevich, I. R.: Basic notions of algebra. In: Encyclopaedia of mathematical sciences; vol. 11, Algebra 1. (Kostrikin, A. I., Shafarevich, I. R., eds.) Berlin: Springer 1990.
35. Tricot, C.: Curves and fractal dimension. Berlin Heidelberg New York Tokyo: Springer 1995.
36. Turbin, A. F., Pratsevityi, N. V.: Fractal sets, functions, distributions. Kiev: Naukova Dumka 1992 (in Russian).
37. van der Waerden, B. L.: Ein einfaches Beispiel einer nichtdifferenzierbaren stetigen Funktion. Math. Z. 32, 474-475 (1930).
38. Weyl, H.: Symmetry. Princeton: Princeton University Press 1952.