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Bending-torsion moments in thin multi-structures in the context of nonlinear elasticity
March
2020, 19(3): 1795-1845.
doi: 10.3934/cpaa.2020073
The mathieu differential equation and generalizations to infinite fractafolds
| 1. | Department of Mathematics, Cornell University, Ithaca, NY 14853, USA |
| 2. | Department of Mathematics, Indiana University Bloomington, Bloomington, IN 47405, USA |
| 3. | Department of Mathematics, The University of Hong Kong, Hong Kong |
| 4. | Department of Mathematics, Cornell University, Ithaca, NY 14853, USA |
One of the well-studied equations in the theory of ODEs is the Mathieu differential equation. A common approach for obtaining solutions is to seek solutions via Fourier series by converting the equation into an infinite system of linear equations for the Fourier coefficients. We study the asymptotic behavior of these Fourier coefficients and discuss the ways in which to numerically approximate solutions. We present both theoretical and numerical results pertaining to the stability of the Mathieu differential equation and the properties of solutions. Further, based on the idea of using Fourier series, we provide a method in which the Mathieu differential equation can be generalized to be defined on the infinite Sierpinski gasket. We discuss the stability of solutions to this fractal differential equation and describe further results concerning properties and behavior of these solutions.
Mathematics Subject Classification:
Primary: 34D23; Secondary: 28A80.
Citation:
Shiping Cao, Anthony Coniglio, Xueyan Niu, Richard H. Rand, Robert S. Strichartz. The mathieu differential equation and generalizations to infinite fractafolds. Communications on Pure & Applied Analysis,
2020, 19
(3)
: 1795-1845.
doi: 10.3934/cpaa.2020073
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References:
| [1] |
N. Asai, D. Cai, Y. Ikebe and Y. Miyazaki,
The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application, Linear Algebra Appl., 241 (1996), 599-618.
doi: 10.1016/0024-3795(95)00699-0. |
| [2] |
J. Avron and B. Simon,
The asymptotics of the gap in the mathieu equation, Ann. Physics, 134 (1981), 76-84.
doi: 10.1016/0003-4916(81)90005-1. |
| [3] |
O. Ben-Bassat, R. 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. |
| [4] |
K. Dalrymple, R. Strichartz and J. Vinson,
Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284.
doi: 10.1007/BF01261610. |
| [5] |
K. Falconer, Fractal Geometry, John Wiley & Sons, Chichester, 2014. |
| [6] |
M. Fukushima and T. Shima,
On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35.
doi: 10.1007/BF00249784. |
| [7] |
A. Gil, J. Segura and N. Temme, Numerical Methods for Special Functions, vol. 99, Society for Industrial and Applied Mathematics, 2007.
doi: 10.1137/1.9780898717822. |
| [8] |
B. Hambly and T. Kumagai,
Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc. (3), 78 (1999), 431-458.
doi: 10.1112/S0024611599001744. |
| [9] |
H. Hochstadt,
On the width of the instability intervals of the mathieu equation, SIAM J. Math. Anal., 15 (1984), 105-107.
doi: 10.1137/0515005. |
| [10] |
J. Hutchinson,
Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.
doi: 10.1512/iumj.1981.30.30055. |
| [11] |
M. Ionescu, L. Rogers and R. Strichartz,
Pseudo-differential operators on fractals and other metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 1159-1190.
doi: 10.4171/RMI/752. |
| [12] |
J. Kigami,
A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290.
doi: 10.1007/BF03167882. |
| [13] |
J. Kigami,
Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755.
doi: 10.2307/2154402. |
| [14] |
J. Kigami, Analysis on Fractals, vol. 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511470943. |
| [15] |
D. Levy and J. Keller,
Instability intervals of Hill's equation, Comm. Pure Appl. Math., 16 (1963), 469-476.
doi: 10.1002/cpa.3160160406. |
| [16] |
W. Loud,
Stability regions for Hill's equation, J. Differential Equations, 19 (1975), 226-241.
doi: 10.1016/0022-0396(75)90003-0. |
| [17] |
R. Rand, Mathieu's Equation, International Centre for Mechanical Sciences, 2016. Google Scholar |
| [18] |
H. Ruan and R. Strichartz,
Covering maps and periodic functions on higher dimensional Sierpinski gaskets, Canad. J. Math., 61 (2009), 1151-1181.
doi: 10.4153/CJM-2009-054-5. |
| [19] |
J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, vol. 2, New York: Interscience Publishers, 1950. |
| [20] |
R. Strichartz,
Fractals in the large, Can. J. Math., 50 (1996), 638-657.
doi: 10.4153/CJM-1998-036-5. |
| [21] |
R. Strichartz,
Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355 (2003), 4019-4043.
doi: 10.1090/S0002-9947-03-03171-4. |
| [22] |
R. Strichartz, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006.
Google Scholar
|
| [23] |
R. Strichartz,
Periodic and almost periodic functions on infinite Sierpinski gaskets, Canad. J. Math., 61 (2009), 1182-1200.
doi: 10.4153/CJM-2009-055-9. |
| [24] |
R. Strichartz and A. Teplyaev,
Spectral analysis on infinite Sierpiński fractafolds, J. Anal. Math., 116 (2012), 255-297.
doi: 10.1007/s11854-012-0007-5. |
| [25] |
A. Teplyaev,
Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567.
doi: 10.1006/jfan.1998.3297. |
| [26] |
M. Weinstein and J. Keller,
Asymptotic behavior of stability regions for Hill's equation, J. Appl. Math., 47 (1987), 941-958.
doi: 10.1137/0147062. |
show all references
References:
| [1] |
N. Asai, D. Cai, Y. Ikebe and Y. Miyazaki,
The eigenvalue problem for infinite complex symmetric tridiagonal matrices with application, Linear Algebra Appl., 241 (1996), 599-618.
doi: 10.1016/0024-3795(95)00699-0. |
| [2] |
J. Avron and B. Simon,
The asymptotics of the gap in the mathieu equation, Ann. Physics, 134 (1981), 76-84.
doi: 10.1016/0003-4916(81)90005-1. |
| [3] |
O. Ben-Bassat, R. 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. |
| [4] |
K. Dalrymple, R. Strichartz and J. Vinson,
Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284.
doi: 10.1007/BF01261610. |
| [5] |
K. Falconer, Fractal Geometry, John Wiley & Sons, Chichester, 2014. |
| [6] |
M. Fukushima and T. Shima,
On a spectral analysis for the Sierpiński gasket, Potential Anal., 1 (1992), 1-35.
doi: 10.1007/BF00249784. |
| [7] |
A. Gil, J. Segura and N. Temme, Numerical Methods for Special Functions, vol. 99, Society for Industrial and Applied Mathematics, 2007.
doi: 10.1137/1.9780898717822. |
| [8] |
B. Hambly and T. Kumagai,
Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc. (3), 78 (1999), 431-458.
doi: 10.1112/S0024611599001744. |
| [9] |
H. Hochstadt,
On the width of the instability intervals of the mathieu equation, SIAM J. Math. Anal., 15 (1984), 105-107.
doi: 10.1137/0515005. |
| [10] |
J. Hutchinson,
Fractals and self-similarity, Indiana Univ. Math. J., 30 (1981), 713-747.
doi: 10.1512/iumj.1981.30.30055. |
| [11] |
M. Ionescu, L. Rogers and R. Strichartz,
Pseudo-differential operators on fractals and other metric measure spaces, Rev. Mat. Iberoam., 29 (2013), 1159-1190.
doi: 10.4171/RMI/752. |
| [12] |
J. Kigami,
A harmonic calculus on the Sierpiński spaces, Japan J. Appl. Math., 6 (1989), 259-290.
doi: 10.1007/BF03167882. |
| [13] |
J. Kigami,
Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (1993), 721-755.
doi: 10.2307/2154402. |
| [14] |
J. Kigami, Analysis on Fractals, vol. 143 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, 2001.
doi: 10.1017/CBO9780511470943. |
| [15] |
D. Levy and J. Keller,
Instability intervals of Hill's equation, Comm. Pure Appl. Math., 16 (1963), 469-476.
doi: 10.1002/cpa.3160160406. |
| [16] |
W. Loud,
Stability regions for Hill's equation, J. Differential Equations, 19 (1975), 226-241.
doi: 10.1016/0022-0396(75)90003-0. |
| [17] |
R. Rand, Mathieu's Equation, International Centre for Mechanical Sciences, 2016. Google Scholar |
| [18] |
H. Ruan and R. Strichartz,
Covering maps and periodic functions on higher dimensional Sierpinski gaskets, Canad. J. Math., 61 (2009), 1151-1181.
doi: 10.4153/CJM-2009-054-5. |
| [19] |
J. Stoker, Nonlinear Vibrations in Mechanical and Electrical Systems, vol. 2, New York: Interscience Publishers, 1950. |
| [20] |
R. Strichartz,
Fractals in the large, Can. J. Math., 50 (1996), 638-657.
doi: 10.4153/CJM-1998-036-5. |
| [21] |
R. Strichartz,
Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc., 355 (2003), 4019-4043.
doi: 10.1090/S0002-9947-03-03171-4. |
| [22] |
R. Strichartz, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006.
Google Scholar
|
| [23] |
R. Strichartz,
Periodic and almost periodic functions on infinite Sierpinski gaskets, Canad. J. Math., 61 (2009), 1182-1200.
doi: 10.4153/CJM-2009-055-9. |
| [24] |
R. Strichartz and A. Teplyaev,
Spectral analysis on infinite Sierpiński fractafolds, J. Anal. Math., 116 (2012), 255-297.
doi: 10.1007/s11854-012-0007-5. |
| [25] |
A. Teplyaev,
Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal., 159 (1998), 537-567.
doi: 10.1006/jfan.1998.3297. |
| [26] |
M. Weinstein and J. Keller,
Asymptotic behavior of stability regions for Hill's equation, J. Appl. Math., 47 (1987), 941-958.
doi: 10.1137/0147062. |
Figure 2.
Curves corresponding to expansions of larger periods: curves with $ 2\pi $ -periodic solutions (black solid and black dashed), curves with $ 4\pi $ -periodic solutions (orange solid and orange dashed), curves with $ 8\pi $ -periodic solutions (red dashed) and curves with $ 16\pi $ -periodic solutions (blue dashed)
Figure 3.
The width of the $ 5 $ th stable band
Figure 4.
The width of first $ 5 $ stable bands
Figure 6.
The triangle area corresponding to $ R_{w_1},R_{w_2},R_{w_3}, $ and $ R_{w_4} $ . The green area is the stable region, and shaded area is the unstable region
Figure 7.
Probabilities $ P_i $ and the fitting curve
Figure 8.
Normalized solutions corresponding to $ p(\sin t,0) $ (solid black), $ p(\sin t,5) $ (red), $ p(\sin t,10) $ (orange), $ p(\sin t,20) $ (green), $ p(\sin t,40) $ (blue), $ p(\sin t,80) $ (purple)
Figure 9.
Normalized solutions corresponding to $ p(\sin 2t,0) $ (solid black), $ p(\sin 2t,5) $ (red), $ p(\sin 2t,10) $ (orange), $ p(\sin 2t,20) $ (green), $ p(\sin 2t,40) $ (blue), $ p(\sin 2t,80) $ (purple)
Figure 10.
Normalized solutions corresponding to $ p(\sin 3t,0) $ (solid black), $ p(\sin 3t,5) $ (red), $ p(\sin 3t,10) $ (orange), $ p(\sin 3t,20) $ (green), $ p(\sin 3t,40) $ (blue), $ p(\sin 3t,80) $ (purple)
Figure 11.
t-position of maximal points, with fitting curve $ t = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $
Figure 12.
t-position of the second maximal points, with fitting curve $ t = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $
Figure 13.
t-position of the third maximal points on curve $ p(\sin 3t, \varepsilon) $ , with fitting curve $ t = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $ , $ a = 9.567,b = 23.07,c = 22.86,d = 40.46 $
Figure 14.
u-position of the second maximal points on curve $ p(\sin 2t, \varepsilon) $ , with fitting curve $ t = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^2+d \varepsilon+e} $ . $ a = 0.8393,b = -0.5638,c = 5.566,d = -0.1973,e = 5.571 $
Figure 15.
u-position of the second and third maximal points corresponding to curve $ p(\sin 3t, \varepsilon) $ , with fitting curve $ u = \frac{a \varepsilon^3+b \varepsilon^2+c \varepsilon+d}{ \varepsilon^3+e \varepsilon^2+f \varepsilon+g} $
Figure 16.
Normalized solutions corresponding to $ p(\cos 0t, \varepsilon) $ , with $ \varepsilon = 0,1,2,3,4,5,10,20,40,80,160 $
Figure 17.
Normalized solutions corresponding to $ p(\cos t, \varepsilon) $ , with $ \varepsilon = 0,1,2,3,4,5,10 $ in the left graph, and $ \varepsilon = 10,20,40,80,160 $ in the right graph
Figure 18.
Normalized solutions corresponding to $ p(\cos 2t, \varepsilon) $ , with $ \varepsilon = 0,1,2,3,4,5,6 $ in the left, and $ \varepsilon = 6,7,8,9,10,20,30,40,60,80,100,160 $ in the right
Figure 19.
The $ u $ coordinate of minimum points of solutions for points $ p(\cos 0, \varepsilon) $ , with $ \varepsilon = 0,1,\cdots, 200 $ . Fitting curve $ u = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $ , with $ a = -0.02171,b = 0.2895,c = 0.2289,d = 0.2895 $
Figure 20.
The $ t $ coordinate of minimum points of solutions for points $ p(\cos t, \varepsilon) $ , with $ \varepsilon = 2,\cdots, 200 $ . Fitting curve $ y = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $ , with $ a = 241,9,b = 1284,c = 310.8,d = 177.1 $
Figure 21.
The $ u $ coordinate of minimum points of solutions for points $ p(\cos t, \varepsilon) $ , with $ \varepsilon = 0,1,2,\cdots, 200 $ . Fitting curve $ u = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^2+d \varepsilon+e} $ , with $ a = 0.8687,b = -1.631,c = 0.8498,d = -1.72 $ and $ e = 0.8497 $
Figure 22.
The $ u $ coordinate of minimum points of solutions for points $ p(\cos t, \varepsilon) $ , with $ \varepsilon = 0,1,2,\cdots, 200 $ . Fitting curve $ u = \frac{a \varepsilon+b}{ \varepsilon^2+c \varepsilon+d} $ , with $ a = 0.2495,b = -4.991,c = -3.037,d = 6.722 $
Figure 23.
The $ t $ coordinate of maximum points of solutions for points $ p(\cos 2t, \varepsilon) $ , with $ \varepsilon = 5,6,\cdots, 200 $ . Fitting curve $ t = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^2+d \varepsilon+e} $ , with $ a = 0.5572,b = 88.5,c = -3.009,d = 55.35 $ and $ e = -157.7 $
Figure 24.
The $ t $ coordinate of the second peak of solutions for points $ p(\cos 2t, \varepsilon) $ , with $ \varepsilon = 5,6,\cdots, 200 $ . Fitting curve $ t = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^3+d \varepsilon^2+e \varepsilon+f} $ , with $ a = 269.3,b = 4745,c = -9715,d = 641.3,e = -2040 $ and $ f = -6017 $
Figure 25.
The $ t $ coordinate of the second peak of solutions for points $ p(\cos 2t, \varepsilon) $ , with $ \varepsilon = 1,2,\cdots, 200 $ . Fitting curve $ t = \frac{a \varepsilon^3+b \varepsilon^2+c \varepsilon+d}{ \varepsilon^3+e \varepsilon^2+f \varepsilon+g} $ , with $ a = 0.7931,b = -4.642,c = 6.596,d = 13.45,e = --5.508,f = 9.829 $ and $ g = 13.44 $
Figure 26.
The $ t $ coordinate of the second peak of solutions for points $ p(\cos 2t, \varepsilon) $ , with $ \varepsilon = 1,2,\cdots, 200 $ . Fitting curve $ y = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^2+e \varepsilon+f} $ , with $ a = -0.006875,b = 0.7009,c = -8.537,d = -2.457,e = 8.857 $
Figure 27.
The $ t $ coordinate of the second peak of solutions for points $ p(\cos 2t, \varepsilon) $ , with $ \varepsilon = 5,6,\cdots, 200 $ . Fitting curve $ y = \frac{a \varepsilon^2+b \varepsilon+c}{ \varepsilon^3+d \varepsilon^2+e \varepsilon+f} $ , with $ a = 0.2268,b = -25.1,c = 587.7,d = 4.775,e = -156 $ and $ f = 1002 $
Figure 28.
Normalized solutions corresponding to $ p(\sin \frac12t, \varepsilon) $ , with $ \varepsilon = 0,1,2,3,4,5,10,20,40,80,160 $
Figure 29.
Normalized solutions corresponding to $ p(\sin\frac32t, \varepsilon) $ , with $ \varepsilon = 0,5,10,20,40,80,160 $
Figure 30.
Normalized solutions corresponding to $ p(\cos \frac12t, \varepsilon) $ , with $ \varepsilon = 0,1,2,3,4,5,10,20,40,80,160 $
Figure 31.
Normalized solutions corresponding to $ p(\cos \frac32t, \varepsilon) $ , with $ \varepsilon = 0,1,2,3,4,5,10,20,40,80,160 $
Figure 32.
The Sierpinski Gasket
Figure 33.
Approximating graphs $ \Gamma_0,\Gamma_1,\Gamma_2 $
Figure 34.
Transition curves for version 1, 5-series. The left picture shows transition curves of a single matrix corresponding to generation of birth $ 0 $ . The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
Figure 35.
Transition curves for version 1, 6-series. The left picture shows transition curves of a single matrix corresponding to generation of birth $ 0 $ . The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
Figure 36.
Transition curves for version 2, 5-series. The left picture shows transition curves of single matrix corresponding to generation of birth $ 0 $ . The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
Figure 37.
Transition curves for version 2, 6-series. The left picture shows transition curves of single matrix corresponding to generation of birth $ 0 $ . The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
Figure 38.
Transition curves for version 3, 5-series. The left picture shows transition curves of single matrix corresponding to generation of birth $ 0 $ . The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
Figure 39.
Transition curves for version 3, 6-series. The left picture shows transition curves of single matrix corresponding to generation of birth $ 0 $ . The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
Figure 40.
Transition curves for version 4, 5-series. The left picture shows transition curves of single matrix corresponding to generation of birth $ 0 $ . The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
Figure 41.
Transition curves for version 4, 6-series. The left picture shows transition curves of single matrix corresponding to generation of birth $ 0 $ . The right picture shows transition curves of generation of birth 0 (red), -1 (blue), -2 (green), -3 (orange), -4 (black)
Figure 42.
Initial values of the eigenfunctions with birth of generation $ 2 $ and initial eigenvalues $ 5 $ (left) and $ 6 $ (right)
Figure 43.
Solutions on the first curve for Version 1, initial eigenvalue $ 5 $ , with $ \varepsilon = 1000,2000,3000 $
Figure 44.
Solutions on the first curve for Version 1, initial eigenvalue $ 5 $ , with $ \varepsilon = 10000,11000,12000 $
Figure 45.
Solutions on the first curve for Version 1, initial eigenvalue $ 5 $ , with $ \varepsilon = 20000,25000,30000 $
Figure 46.
Solutions on the first curve for Version 1, initial eigenvalue $ 6 $ , with $ \varepsilon = 1000,2000,3000 $
Figure 47.
Solutions on the first curve for Version 1, initial eigenvalue $ 6 $ , with, with $ \varepsilon = 10000,11000,12000 $
Figure 48.
Solutions on the first curve for Version 1, initial eigenvalue $ 6 $ , with $ \varepsilon = 20000,25000,30000 $
Figure 49.
The position of peaks for version 1, 5 series
Figure 50.
The position of peaks for version 1, 6-series
Table 1.
The probability $ P_i $ 's
| 1 | 0.625056436387445 |
| 2 | 0.784428425813594 |
| 3 | 0.845139663995868 |
| 4 | 0.878143787704672 |
| 5 | 0.899154589232086 |
| 6 | 0.913800056779179 |
| 7 | 0.924632580597566 |
| 8 | 0.932988858616457 |
| 9 | 0.939640860147421 |
| 10 | 0.945067300763657 |
| 11 | 0.949581603650156 |
| 12 | 0.953397960289362 |
| 13 | 0.956667925443240 |
| 14 | 0.959502140424685 |
| 15 | 0.961982968671620 |
| 16 | 0.964174006346465 |
| 17 | 0.966133547589692 |
| 18 | 0.967239011961280 |
| 1 | 0.625056436387445 |
| 2 | 0.784428425813594 |
| 3 | 0.845139663995868 |
| 4 | 0.878143787704672 |
| 5 | 0.899154589232086 |
| 6 | 0.913800056779179 |
| 7 | 0.924632580597566 |
| 8 | 0.932988858616457 |
| 9 | 0.939640860147421 |
| 10 | 0.945067300763657 |
| 11 | 0.949581603650156 |
| 12 | 0.953397960289362 |
| 13 | 0.956667925443240 |
| 14 | 0.959502140424685 |
| 15 | 0.961982968671620 |
| 16 | 0.964174006346465 |
| 17 | 0.966133547589692 |
| 18 | 0.967239011961280 |
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