Localization on snowflake domains

Fractals

Volumn 15, Issue 3, 2007, Pages 255-272

  Daudert, Britta ^a   Lapidus, Michel L ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Entropy; Fractal geometry; Localization; Random walks; Snowflake domain

Indexed keywords

COMPUTATIONAL GEOMETRY; ENTROPY; FRACTALS; GRAPH THEORY;

FRACTAL GEOMETRY; RANDOM WALKS; SNOWFLAKE DOMAIN;

SNOW;

EID: 34548781400     PISSN: 0218348X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218348X0700354X     Document Type: Article

References (22)

1. K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications (John Wiley and Sons, London, 1990).
2. M. V. Berry, Distribution of modes in fractal resonators, in Structural Stability in Physics, eds. W. Güttinger and H. Eikemeier (Springer-Verlag, Berlin, 1979), pp. 51-53.
3. M. V. Berry, Some geometric aspects of wave motion: wavefront dislocations, diffraction catatrophes, diffractals, in Geometry of the Laplace Operator, Proc. Sympos. Pure Math., Vol. 36 (American Mathematics Society Providence, R.I., 1980), pp. 13-38.
4. J. Brossard and R. Carmona, Can one hear the dimension of a fractal?, Commun. Math. Phys. 104 (1986) 103-122.
5. M. L. Lapidus, Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture, Trans. Am. Math. Soc. 325 (1991) 465-529.
6. M. L. Lapidus, Spectral and fractal geometry: from the Weyl-Berry conjecture for the vibrations of fractal drums to the Riemann zeta-function, in Differential Equations and Mathematical Physics, ed. C. Bennewitz, Proc. 4th UAB Int. Conf., March 1990 Birmingham (Academic Press, New York, 1992), pp. 151-182.
7. M. L. Lapidus, Vibrations of fractal drums, the Riemann hypothesis, waves in fractal media, and the Weyl-Berry conjecture, in Ordinary and Partial Differential Equations, eds. B. D. Sleeman and R. J. Jarvis, Vol. IV, Proc. 12th Int. Conf., June 1992, Dundee, Scotland, UK), Pitman Research Notes in Mathematics Series, Vol. 289 (Longman Scientific and Technical, London, 1993), pp. 126-209.
8. M. L. Lapidus, J. W. Neuberger, R. J. Renka and C. A. Griffith, Snowflake harmonics and computer graphics: numerical computation of spectra on fractal drums, Int. J. Bifurc. Chaos 6 (1996) 1185-1210.
9. M. L. Lapidus and M. M. H. Pang, Eigenfunctions of the Koch snowflake domain, Commun. Math. Phys. 172 (1995) 359-376.
10. B. Sapoval, Experimental observation of local modes in fractal drums, Physica D 38 (1989) 296-298.
11. C. Even, S. Russ, V. Repain, P. Pieranski and B. Sapoval, Localization in fractal drums: an experimental study, Phys. Rev. Lett. 83 (1999) 726-729.
12. B. Sapoval and T. Gobron, Vibrations of strongly irregular or fractal resonators, Phys. Rev. E 47 (1993) 3013-3024.
13. B. Sapoval, T. Gobron and A. Margolina, Vibrations of fractal drums, Phys. Rev. Lett. 67 (1991) 2974-2977.
14. L. C. Evans, Partial differential equations, in Graduate Studies in Mathematics, Vol. 19 (American Mathematics Society. Providence, R.I., 1998).
15. E. B. Davies, Eigenvalue stability bounds via weighted Sobolev spaces, Math. Z. 214 (1993) 357-371.
16. M. M. H. Pang, Approximation of ground state eigenfuncts on the snowflake region, Bull. Lond. Math. Soc. 28 (1996) 488-494.
17. M. M. H. Pang, Approximation of ground state eigenvalues of Dirichlet laplacians, Bull. Lond. Math. Soc. 29 (1997) 720-730.
18. F. Wegner, Bounds on the density of states in disordered systems, Z. Phys. B 44 (1981) 9-15.
19. R. J. Bell and P. Dean, Atomic vibrationsin vitreous silica, Disc. Faraday Soc. 50 (1970) 55-61.
20. I. Simonsen, K. A. Eriksen, S. Maslov and K. Sneppen, Diffusion on complex networks: a way to probe their large-scale topological structures, Physica A 336 (2004) 163-173.
21. I. Simonsen, Diffusion and networks: a powerful combination!, Physica A 357 (2005) 317-330.
22. A. Graham, Nonnegative Matrices and Applicable Topics in Linear Algebra (John Wiley & Sons, New York, 1987).