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Infinitely many solutions and Morse index for non-autonomous elliptic problems
Analysis on hybrid fractals
1. | University of Connecticut, 341 Mansfield Road, Storrs CT 06269-1009, USA |
2. | The University of Chicago, 5747 S. Ellis Avenue, Chicago, IL 60637, USA |
3. | University of California Berkeley, 2333 College Ave, Berkeley, CA 94704, USA |
4. | Cornell University, 310 Malott Hall, Ithaca NY 14853, USA |
5. | University of California Berkeley, 2301 Durant Avenue, Berkeley CA 94704, USA |
We introduce hybrid fractals as a class of fractals constructed by gluing several fractal pieces in a specific manner and study energy forms and Laplacians on them. We consider in particular a hybrid based on the $ 3 $-level Sierpinski gasket, for which we construct explicitly an energy form with the property that it does not "capture" the $ 3 $-level Sierpinski gasket structure. This characteristic type of energy forms that "miss" parts of the structure of the underlying space is investigated in the more general framework of finitely ramified cell structures. The spectrum of the associated Laplacian and its asymptotic behavior in two different hybrids is analyzed theoretically and numerically. A website with further numerical data analysis is available at http://www.math.cornell.edu/~harry970804/.
References:
[1] |
B. Adams, S. A. Smith, R. S. Strichartz and A. Teplyaev, The spectrum of the Laplacian on the pentagasket, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, 1–24. |
[2] |
E. Akkermans, O. Benichou and G. V. Dunne,
A. Teplyaev and R. Voituriez, Spatial log-periodic oscillations of first-passage observables in fractals, Phys. Rev. E, 86 (2012), 061125.
|
[3] |
E. Akkermans, G. V. Dunne and E. Levy, Wave propagation in one-dimension: Methods and applications to complex and fractal structures, Optics of Aperiodic Structures - Fundamentals and Device Applications, L. Dal Negro (Ed), Pan Stanford Press, 2014, 407–449. |
[4] |
P. Alonso Ruiz, U. Freiberg and J. Kigami,
Completely symmetric resistance forms on the stretched Sierpiński gasket, J. of Fractal Geometry, 5 (2018), 227-277.
doi: 10.4171/JFG/61. |
[5] |
P. Alonso-Ruiz, D. J. Kelleher and A. Teplyaev,
Energy and Laplacian on Hanoi-type fractal quantum graphs, Journal of Physics A: Mathematical and Theoretical, 49 (2016), 165206.
doi: 10.1088/1751-8113/49/16/165206. |
[6] |
Patricia Alonso Ruiz, Power dissipation in fractal Feynman-Sierpinski AC circuits, J. Math. Phys., 58 (2017), 073503, 16.
doi: 10.1063/1.4994197. |
[7] |
N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst and A. Teplyaev,
Vibration modes of 3 n -gaskets and other fractals, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 015101.
doi: 10.1088/1751-8113/41/1/015101. |
[8] |
M. Begué, T. Kalloniatis and R. S. Strichartz, Harmonic functions and the spectrum of the Laplacian on the Sierpinski carpet, Fractals, 21 (2013), 1350002, 32.
doi: 10.1142/S0218348X13500023. |
[9] |
O. Ben-Bassat, R. S. Strichartz and A. Teplyaev,
What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal., 166 (1999), 197-217.
doi: 10.1006/jfan.1999.3431. |
[10] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. |
[11] |
J. P. Chen, L. G. Rogers, L. Anderson, U. Andrews, A. Brzoska, A. Coffey, H. Davis, L. Fisher, M. Hansalik, S. Loew and A. Teplyaev, Power dissipation in fractal AC circuits, J. Phys. A, 50 (2017), 325205, 20.
doi: 10.1088/1751-8121/aa7a66. |
[12] |
Y. Chen, H. Gu, R. S. Strichartz and Z. Zhou, Hybrid fractals, Available at http://www.math.cornell.edu/~harry970804/, 2017. |
[13] |
M. Fukushima and T. Shima,
On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35.
doi: 10.1007/BF00249784. |
[14] |
B. M. Hambly, V. Metz and A. Teplyaev,
Self-similar energies on post-critically finite self-similar fractals, J. London Math. Soc., 74 (2006), 93-112.
doi: 10.1112/S002461070602312X. |
[15] |
B. M. Hambly and S. O. G. Nyberg,
Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem, Proc. Edinb. Math. Soc., 46 (2003), 1-34.
doi: 10.1017/S0013091500000730. |
[16] |
M. Hata,
On some properties of set-dynamical systems, Proc. Japan Acad. Ser. A Math. Sci., 61 (1985), 99-102.
|
[17] |
M. Hinz,
Sup-norm-closable bilinear forms and Lagrangians, Ann. Mat. Pura Appl., 195 (2016), 1021-1054.
doi: 10.1007/s10231-015-0503-1. |
[18] |
M. Hinz and A. Teplyaev, Closability, regularity, and approximation by graphs for separable bilinear forms, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 441 (2015), no. Veroyatnost$\prime$ i Statistika. 22,299–317.
doi: 10.1007/s10958-016-3149-7. |
[19] |
M. Ionescu, L. G. Rogers and A. Teplyaev,
Derivations and Dirichlet forms on fractals, J. Funct. Anal., 263 (2012), 2141-2169.
doi: 10.1016/j.jfa.2012.05.021. |
[20] |
J. Kigami,
A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290.
doi: 10.1007/BF03167882. |
[21] |
_____, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721–755.
doi: 10.2307/2154402. |
[22] |
_____, Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511470943. |
[23] |
_____, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc., 216 (2012), ⅵ+132. |
[24] |
J. Kigami and M. L. Lapidus,
Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys., 158 (1993), 93-125.
|
[25] |
R. D. Mauldin and S. C. Williams,
Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829.
doi: 10.2307/2000940. |
[26] |
J. Murai,
Diffusion processes on mandala, Osaka J. Math., 32 (1995), 887-917.
|
[27] |
L. G. Rogers and A. Teplyaev,
Laplacians on the basilica Julia sets, Commun. Pure Appl. Anal., 9 (2010), 211-231.
doi: 10.3934/cpaa.2010.9.211. |
[28] |
R. S. Strichartz,
Taylor approximations on Sierpinski gasket type fractals, J. Funct. Anal., 174 (2000), 76-127.
doi: 10.1006/jfan.2000.3580. |
[29] |
_____, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006, A tutorial. |
[30] |
R. S. Strichartz and J. Zhu, Spectrum of the Laplacian on the Vicsek set "with no loose ends", Fractals, 25 (2017), 1750062, 15.
doi: 10.1142/S0218348X17500621. |
[31] |
A. Teplyaev,
Harmonic coordinates on fractals with finitely ramified cell structure, Canad. J. Math., 60 (2008), 457-480.
doi: 10.4153/CJM-2008-022-3. |
show all references
References:
[1] |
B. Adams, S. A. Smith, R. S. Strichartz and A. Teplyaev, The spectrum of the Laplacian on the pentagasket, Fractals in Graz 2001, Trends Math., Birkhäuser, Basel, 2003, 1–24. |
[2] |
E. Akkermans, O. Benichou and G. V. Dunne,
A. Teplyaev and R. Voituriez, Spatial log-periodic oscillations of first-passage observables in fractals, Phys. Rev. E, 86 (2012), 061125.
|
[3] |
E. Akkermans, G. V. Dunne and E. Levy, Wave propagation in one-dimension: Methods and applications to complex and fractal structures, Optics of Aperiodic Structures - Fundamentals and Device Applications, L. Dal Negro (Ed), Pan Stanford Press, 2014, 407–449. |
[4] |
P. Alonso Ruiz, U. Freiberg and J. Kigami,
Completely symmetric resistance forms on the stretched Sierpiński gasket, J. of Fractal Geometry, 5 (2018), 227-277.
doi: 10.4171/JFG/61. |
[5] |
P. Alonso-Ruiz, D. J. Kelleher and A. Teplyaev,
Energy and Laplacian on Hanoi-type fractal quantum graphs, Journal of Physics A: Mathematical and Theoretical, 49 (2016), 165206.
doi: 10.1088/1751-8113/49/16/165206. |
[6] |
Patricia Alonso Ruiz, Power dissipation in fractal Feynman-Sierpinski AC circuits, J. Math. Phys., 58 (2017), 073503, 16.
doi: 10.1063/1.4994197. |
[7] |
N. Bajorin, T. Chen, A. Dagan, C. Emmons, M. Hussein, M. Khalil, P. Mody, B. Steinhurst and A. Teplyaev,
Vibration modes of 3 n -gaskets and other fractals, Journal of Physics A: Mathematical and Theoretical, 41 (2008), 015101.
doi: 10.1088/1751-8113/41/1/015101. |
[8] |
M. Begué, T. Kalloniatis and R. S. Strichartz, Harmonic functions and the spectrum of the Laplacian on the Sierpinski carpet, Fractals, 21 (2013), 1350002, 32.
doi: 10.1142/S0218348X13500023. |
[9] |
O. Ben-Bassat, R. S. Strichartz and A. Teplyaev,
What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal., 166 (1999), 197-217.
doi: 10.1006/jfan.1999.3431. |
[10] |
G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, Mathematical Surveys and Monographs, vol. 186, American Mathematical Society, Providence, RI, 2013. |
[11] |
J. P. Chen, L. G. Rogers, L. Anderson, U. Andrews, A. Brzoska, A. Coffey, H. Davis, L. Fisher, M. Hansalik, S. Loew and A. Teplyaev, Power dissipation in fractal AC circuits, J. Phys. A, 50 (2017), 325205, 20.
doi: 10.1088/1751-8121/aa7a66. |
[12] |
Y. Chen, H. Gu, R. S. Strichartz and Z. Zhou, Hybrid fractals, Available at http://www.math.cornell.edu/~harry970804/, 2017. |
[13] |
M. Fukushima and T. Shima,
On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35.
doi: 10.1007/BF00249784. |
[14] |
B. M. Hambly, V. Metz and A. Teplyaev,
Self-similar energies on post-critically finite self-similar fractals, J. London Math. Soc., 74 (2006), 93-112.
doi: 10.1112/S002461070602312X. |
[15] |
B. M. Hambly and S. O. G. Nyberg,
Finitely ramified graph-directed fractals, spectral asymptotics and the multidimensional renewal theorem, Proc. Edinb. Math. Soc., 46 (2003), 1-34.
doi: 10.1017/S0013091500000730. |
[16] |
M. Hata,
On some properties of set-dynamical systems, Proc. Japan Acad. Ser. A Math. Sci., 61 (1985), 99-102.
|
[17] |
M. Hinz,
Sup-norm-closable bilinear forms and Lagrangians, Ann. Mat. Pura Appl., 195 (2016), 1021-1054.
doi: 10.1007/s10231-015-0503-1. |
[18] |
M. Hinz and A. Teplyaev, Closability, regularity, and approximation by graphs for separable bilinear forms, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 441 (2015), no. Veroyatnost$\prime$ i Statistika. 22,299–317.
doi: 10.1007/s10958-016-3149-7. |
[19] |
M. Ionescu, L. G. Rogers and A. Teplyaev,
Derivations and Dirichlet forms on fractals, J. Funct. Anal., 263 (2012), 2141-2169.
doi: 10.1016/j.jfa.2012.05.021. |
[20] |
J. Kigami,
A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290.
doi: 10.1007/BF03167882. |
[21] |
_____, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721–755.
doi: 10.2307/2154402. |
[22] |
_____, Analysis on Fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511470943. |
[23] |
_____, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc., 216 (2012), ⅵ+132. |
[24] |
J. Kigami and M. L. Lapidus,
Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys., 158 (1993), 93-125.
|
[25] |
R. D. Mauldin and S. C. Williams,
Hausdorff dimension in graph directed constructions, Trans. Amer. Math. Soc., 309 (1988), 811-829.
doi: 10.2307/2000940. |
[26] |
J. Murai,
Diffusion processes on mandala, Osaka J. Math., 32 (1995), 887-917.
|
[27] |
L. G. Rogers and A. Teplyaev,
Laplacians on the basilica Julia sets, Commun. Pure Appl. Anal., 9 (2010), 211-231.
doi: 10.3934/cpaa.2010.9.211. |
[28] |
R. S. Strichartz,
Taylor approximations on Sierpinski gasket type fractals, J. Funct. Anal., 174 (2000), 76-127.
doi: 10.1006/jfan.2000.3580. |
[29] |
_____, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006, A tutorial. |
[30] |
R. S. Strichartz and J. Zhu, Spectrum of the Laplacian on the Vicsek set "with no loose ends", Fractals, 25 (2017), 1750062, 15.
doi: 10.1142/S0218348X17500621. |
[31] |
A. Teplyaev,
Harmonic coordinates on fractals with finitely ramified cell structure, Canad. J. Math., 60 (2008), 457-480.
doi: 10.4153/CJM-2008-022-3. |























level 1 | level 2 | level 3 | level 4 | level 5 | level 6 | level 7 |
43.20, 1 | 33.98, 1 | 33.74, 1 | 33.69, 1 | 33.68, 1 | 33.68, 1 | 33.68, 1 |
57.19, 2 | 47.86, 2 | 49.44, 2 | 49.79, 2 | 49.88, 2 | 49.90, 2 | 49.91, 2 |
135.55, 2 | 104.23, 2 | 117.75, 2 | 121.25, 2 | 122.12, 2 | 122.34, 2 | 122.40, 2 |
149.54, 1 | 126.28, 1 | 184.79, 1 | 202.29, 1 | 206.86, 1 | 208.02, 1 | 208.31, 1 |
257.04, 1 | 207.30, 1 | 217.81, 1 | 220.21, 1 | 220.80, 1 | 220.95, 1 | |
265.30, 2 | 289.42, 2 | 331.89, 2 | 341.94, 2 | 344.43, 2 | 345.05, 2 | |
1881.37, 2 | 397.92, 2 | 474.25, 2 | 495.84, 2 | 501.41, 2 | 502.81, 2 | |
1881.41, 1 | 466.78, 1 | 584.16, 1 | 615.23, 1 | 623.24, 1 | 625.26, 1 | |
3103.35, 2 | 476.77, 1 | 700.76, 1 | 768.64, 1 | 786.02, 1 | 790.39, 1 | |
3103.74, 1 | 514.87, 2 | 787.80, 2 | 869.46, 2 | 890.54, 2 | 895.85, 2 | |
3259.84, 1 | 1607.46, 2 | 1089.04, 2 | 1264.93, 2 | 1309.12, 2 | 1320.11, 2 | |
3260.17, 2 | 1607.46, 1 | 1130.98, 1 | 1328.95, 1 | 1379.76, 1 | 1392.49, 1 | |
4883.03, 2 | 2150.98, 1 | 1455.32, 1 | 1653.59, 1 | 1662.94, 1 | 1664.78, 1 | |
4883.03, 1 | 2150.99, 2 | 1481.31, 2 | 1679.55, 2 | 1686.86, 2 | 1688.66, 2 | |
4889.90, 1 | 2314.95, 2 | 1666.05, 2 | 1861.84, 1 | 1948.37, 1 | 1970.24, 1 | |
4889.90, 2 | 2314.96, 1 | 1675.70, 1 | 1870.76, 2 | 1961.04, 2 | 1983.08, 2 | |
5383.38, 3 | 4046.57, 2 | 1812.72, 2 | 2405.81, 1 | 2492.48, 1 | 2510.96, 1 | |
4046.57, 1 | 1814.01, 1 | 2447.98, 2 | 2565.41, 2 | 2591.25, 2 | ||
6047.56, 1 | 2042.00, 1 | 2740.18, 2 | 2847.39, 2 | 2876.06, 2 | ||
6047.56, 2 | 2042.44, 2 | 2949.41, 1 | 3148.07, 1 | 3178.77, 1 | ||
6264.63, 2 | 2557.42, 1 | 3012.62, 1 | 3148.76, 1 | 3203.72, 1 | ||
6264.63, 1 | 2557.63, 2 | 3256.00, 2 | 3438.70, 2 | 3485.04, 2 | ||
9253.61, 3 | 2967.71, 2 | 3571.97, 2 | 3954.97, 2 | 4055.68, 2 | ||
9551.29, 3 | 2967.75, 1 | 3674.09, 1 | 4019.24, 1 | 4108.50, 1 | ||
9552.10, 3 | 5053.44, 3 | 4260.92, 1 | 4843.37, 2 | 4972.61, 2 | ||
67729.76, 3 | 7462.73, 3 | 4270.27, 2 | 4923.03, 1 | 5092.29, 1 | ||
83789.93, 3 | 7514.45, 3 | 4962.44, 1 | 5503.63, 1 | 5592.27, 1 | ||
83790.87, 3 | 9156.79, 3 | 5022.27, 2 | 5738.82, 2 | 5866.46, 2 | ||
111725.04, 3 | 11142.55, 3 | 5527.03, 2 | 6202.97, 2 | 6437.89, 2 |
level 1 | level 2 | level 3 | level 4 | level 5 | level 6 | level 7 |
43.20, 1 | 33.98, 1 | 33.74, 1 | 33.69, 1 | 33.68, 1 | 33.68, 1 | 33.68, 1 |
57.19, 2 | 47.86, 2 | 49.44, 2 | 49.79, 2 | 49.88, 2 | 49.90, 2 | 49.91, 2 |
135.55, 2 | 104.23, 2 | 117.75, 2 | 121.25, 2 | 122.12, 2 | 122.34, 2 | 122.40, 2 |
149.54, 1 | 126.28, 1 | 184.79, 1 | 202.29, 1 | 206.86, 1 | 208.02, 1 | 208.31, 1 |
257.04, 1 | 207.30, 1 | 217.81, 1 | 220.21, 1 | 220.80, 1 | 220.95, 1 | |
265.30, 2 | 289.42, 2 | 331.89, 2 | 341.94, 2 | 344.43, 2 | 345.05, 2 | |
1881.37, 2 | 397.92, 2 | 474.25, 2 | 495.84, 2 | 501.41, 2 | 502.81, 2 | |
1881.41, 1 | 466.78, 1 | 584.16, 1 | 615.23, 1 | 623.24, 1 | 625.26, 1 | |
3103.35, 2 | 476.77, 1 | 700.76, 1 | 768.64, 1 | 786.02, 1 | 790.39, 1 | |
3103.74, 1 | 514.87, 2 | 787.80, 2 | 869.46, 2 | 890.54, 2 | 895.85, 2 | |
3259.84, 1 | 1607.46, 2 | 1089.04, 2 | 1264.93, 2 | 1309.12, 2 | 1320.11, 2 | |
3260.17, 2 | 1607.46, 1 | 1130.98, 1 | 1328.95, 1 | 1379.76, 1 | 1392.49, 1 | |
4883.03, 2 | 2150.98, 1 | 1455.32, 1 | 1653.59, 1 | 1662.94, 1 | 1664.78, 1 | |
4883.03, 1 | 2150.99, 2 | 1481.31, 2 | 1679.55, 2 | 1686.86, 2 | 1688.66, 2 | |
4889.90, 1 | 2314.95, 2 | 1666.05, 2 | 1861.84, 1 | 1948.37, 1 | 1970.24, 1 | |
4889.90, 2 | 2314.96, 1 | 1675.70, 1 | 1870.76, 2 | 1961.04, 2 | 1983.08, 2 | |
5383.38, 3 | 4046.57, 2 | 1812.72, 2 | 2405.81, 1 | 2492.48, 1 | 2510.96, 1 | |
4046.57, 1 | 1814.01, 1 | 2447.98, 2 | 2565.41, 2 | 2591.25, 2 | ||
6047.56, 1 | 2042.00, 1 | 2740.18, 2 | 2847.39, 2 | 2876.06, 2 | ||
6047.56, 2 | 2042.44, 2 | 2949.41, 1 | 3148.07, 1 | 3178.77, 1 | ||
6264.63, 2 | 2557.42, 1 | 3012.62, 1 | 3148.76, 1 | 3203.72, 1 | ||
6264.63, 1 | 2557.63, 2 | 3256.00, 2 | 3438.70, 2 | 3485.04, 2 | ||
9253.61, 3 | 2967.71, 2 | 3571.97, 2 | 3954.97, 2 | 4055.68, 2 | ||
9551.29, 3 | 2967.75, 1 | 3674.09, 1 | 4019.24, 1 | 4108.50, 1 | ||
9552.10, 3 | 5053.44, 3 | 4260.92, 1 | 4843.37, 2 | 4972.61, 2 | ||
67729.76, 3 | 7462.73, 3 | 4270.27, 2 | 4923.03, 1 | 5092.29, 1 | ||
83789.93, 3 | 7514.45, 3 | 4962.44, 1 | 5503.63, 1 | 5592.27, 1 | ||
83790.87, 3 | 9156.79, 3 | 5022.27, 2 | 5738.82, 2 | 5866.46, 2 | ||
111725.04, 3 | 11142.55, 3 | 5527.03, 2 | 6202.97, 2 | 6437.89, 2 |
level 1 | level 2 | level 3 | level 4 | level 5 | level 6 | level 7 | |
Eigenvalue | 43.2000 | 33.9771 | 33.7412 | 33.6913 | 33.6794 | 33.6764 | 33.6756 |
Difference | 0.2359 | 0.0499 | 0.0119 | 0.0030 | 0.0008 | ||
Eigenvalue | 57.1907 | 47.8622 | 49.4401 | 49.7929 | 49.8791 | 49.9005 | 49.9058 |
Difference | 1.5779 | 0.3528 | 0.0862 | 0.0214 | 0.0053 | ||
Eigenvalue | 135.5477 | 104.2339 | 117.748 | 121.248 | 122.1248 | 122.3441 | 122.3989 |
Difference | 13.5141 | 3.5000 | 0.8768 | 0.2193 | 0.0548 | ||
Eigenvalue | 149.5385 | 126.2839 | 184.7948 | 202.2888 | 206.8625 | 208.0185 | 208.3083 |
Difference | 58.5109 | 17.4940 | 4.5737 | 1.156 | 0.2853 | ||
Eigenvalue | 257.0447 | 207.2981 | 217.8053 | 220.2108 | 220.8016 | 220.9487 | |
Difference | 49.7466 | 10.5072 | 2.4055 | 0.5908 | 0.1471 | ||
Eigenvalue | 265.302 | 289.4171 | 331.8853 | 341.9429 | 344.4302 | 345.0504 | |
Difference | 42.4682 | 10.0576 | 2.4873 | 0.6202 |
level 1 | level 2 | level 3 | level 4 | level 5 | level 6 | level 7 | |
Eigenvalue | 43.2000 | 33.9771 | 33.7412 | 33.6913 | 33.6794 | 33.6764 | 33.6756 |
Difference | 0.2359 | 0.0499 | 0.0119 | 0.0030 | 0.0008 | ||
Eigenvalue | 57.1907 | 47.8622 | 49.4401 | 49.7929 | 49.8791 | 49.9005 | 49.9058 |
Difference | 1.5779 | 0.3528 | 0.0862 | 0.0214 | 0.0053 | ||
Eigenvalue | 135.5477 | 104.2339 | 117.748 | 121.248 | 122.1248 | 122.3441 | 122.3989 |
Difference | 13.5141 | 3.5000 | 0.8768 | 0.2193 | 0.0548 | ||
Eigenvalue | 149.5385 | 126.2839 | 184.7948 | 202.2888 | 206.8625 | 208.0185 | 208.3083 |
Difference | 58.5109 | 17.4940 | 4.5737 | 1.156 | 0.2853 | ||
Eigenvalue | 257.0447 | 207.2981 | 217.8053 | 220.2108 | 220.8016 | 220.9487 | |
Difference | 49.7466 | 10.5072 | 2.4055 | 0.5908 | 0.1471 | ||
Eigenvalue | 265.302 | 289.4171 | 331.8853 | 341.9429 | 344.4302 | 345.0504 | |
Difference | 42.4682 | 10.0576 | 2.4873 | 0.6202 |
level 7 | |
Dirichlet | Neumann |
113759984105.32153, 3 | 25411878638.230247, 3 |
114102841006, 6 | 114102841006, 6 |
114788072698, 18 | 114788072698, 18 |
116839632421, 54 | 116839632421, 54 |
122950383156,162 | 122950383156,162 |
140734901210,243 | 140734901210,243 |
140736477695,243 | 140736477626,243 |
187655171187, 3 | 186882957201, 3 |
187659146198, 6 | 187659146198, 6 |
187667196655, 18 | 187667196655, 18 |
187692182839, 54 | 187692182839, 54 |
187775569901,162 | 187775569901,162 |
188131568230,243 | 188131568230,243 |
188147174195, 3 | 188147026223, 3 |
188147176156, 6 | 188147176156, 6 |
188147180228, 18 | 188147180228, 18 |
188147193778, 54 | 188147193778, 54 |
188147252150,162 | 188147252150,162 |
197109853834,243 | 197109853834,243 |
197130271198,243 | 197130271198,243 |
197130271465,243 | 197130271465,243 |
295258340787, 3 | 294738682923, 3 |
295259492723, 6 | 295259492723, 6 |
295261805573, 18 | 295261805573, 18 |
295268816917, 54 | 295268816917, 54 |
295290531946,162 | 295290531946,162 |
295362581347,243 | 295362581347,243 |
295362609295,243 | 295362609295,243 |
295673500703,243 | 295673500703,243 |
295673630427,486 | 295673630427,486 |
325512681630,729 | 325512681630,729 |
level 7 | |
Dirichlet | Neumann |
113759984105.32153, 3 | 25411878638.230247, 3 |
114102841006, 6 | 114102841006, 6 |
114788072698, 18 | 114788072698, 18 |
116839632421, 54 | 116839632421, 54 |
122950383156,162 | 122950383156,162 |
140734901210,243 | 140734901210,243 |
140736477695,243 | 140736477626,243 |
187655171187, 3 | 186882957201, 3 |
187659146198, 6 | 187659146198, 6 |
187667196655, 18 | 187667196655, 18 |
187692182839, 54 | 187692182839, 54 |
187775569901,162 | 187775569901,162 |
188131568230,243 | 188131568230,243 |
188147174195, 3 | 188147026223, 3 |
188147176156, 6 | 188147176156, 6 |
188147180228, 18 | 188147180228, 18 |
188147193778, 54 | 188147193778, 54 |
188147252150,162 | 188147252150,162 |
197109853834,243 | 197109853834,243 |
197130271198,243 | 197130271198,243 |
197130271465,243 | 197130271465,243 |
295258340787, 3 | 294738682923, 3 |
295259492723, 6 | 295259492723, 6 |
295261805573, 18 | 295261805573, 18 |
295268816917, 54 | 295268816917, 54 |
295290531946,162 | 295290531946,162 |
295362581347,243 | 295362581347,243 |
295362609295,243 | 295362609295,243 |
295673500703,243 | 295673500703,243 |
295673630427,486 | 295673630427,486 |
325512681630,729 | 325512681630,729 |
Level 0(Q) | Level 1(Q) | Level 2(Q) | Hanoi Attractor | ||||
Ev. | Renorm. ev. | Ev. | Renorm. ev. | Ev. | Renorm. ev. | Ev. | Mult. |
10.247 | 44.402 | 8.578 | 37.173 | 7.896 | 34.216 | 33.676 | 1 |
13.627 | 59.051 | 12.266 | 53.153 | 11.424 | 49.506 | 49.906 | 2 |
41.306 | 178.992 | 32.951 | 142.786 | 30.030 | 130.132 | 122.399 | 2 |
59.750 | 258.918 | 54.613 | 236.657 | 51.955 | 225.139 | 208.308 | 1 |
75.686 | 327.975 | 57.438 | 248.897 | 52.592 | 227.897 | 220.949 | 1 |
107.259 | 464.788 | 89.685 | 388.635 | 83.999 | 363.995 | 345.050 | 2 |
156.406 | 677.761 | 132.033 | 572.143 | 122.324 | 530.069 | 502.813 | 2 |
213.693 | 926.002 | 172.604 | 747.953 | 156.876 | 679.794 | 625.255 | 1 |
217.180 | 941.113 | 192.661 | 834.863 | 186.323 | 807.398 | 790.386 | 1 |
280.562 | 1215.767 | 232.571 | 1007.807 | 218.448 | 946.610 | 895.853 | 2 |
358.903 | 1555.247 | 320.370 | 1388.268 | 1320.110 | 2 | ||
400.372 | 1734.945 | 343.876 | 1490.131 | 1392.494 | 1 |
Level 0(Q) | Level 1(Q) | Level 2(Q) | Hanoi Attractor | ||||
Ev. | Renorm. ev. | Ev. | Renorm. ev. | Ev. | Renorm. ev. | Ev. | Mult. |
10.247 | 44.402 | 8.578 | 37.173 | 7.896 | 34.216 | 33.676 | 1 |
13.627 | 59.051 | 12.266 | 53.153 | 11.424 | 49.506 | 49.906 | 2 |
41.306 | 178.992 | 32.951 | 142.786 | 30.030 | 130.132 | 122.399 | 2 |
59.750 | 258.918 | 54.613 | 236.657 | 51.955 | 225.139 | 208.308 | 1 |
75.686 | 327.975 | 57.438 | 248.897 | 52.592 | 227.897 | 220.949 | 1 |
107.259 | 464.788 | 89.685 | 388.635 | 83.999 | 363.995 | 345.050 | 2 |
156.406 | 677.761 | 132.033 | 572.143 | 122.324 | 530.069 | 502.813 | 2 |
213.693 | 926.002 | 172.604 | 747.953 | 156.876 | 679.794 | 625.255 | 1 |
217.180 | 941.113 | 192.661 | 834.863 | 186.323 | 807.398 | 790.386 | 1 |
280.562 | 1215.767 | 232.571 | 1007.807 | 218.448 | 946.610 | 895.853 | 2 |
358.903 | 1555.247 | 320.370 | 1388.268 | 1320.110 | 2 | ||
400.372 | 1734.945 | 343.876 | 1490.131 | 1392.494 | 1 |
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