June  2020, 14(3): 463-487. doi: 10.3934/ipi.2020022

Uniqueness and stability for the recovery of a time-dependent source in elastodynamics

1. 

School of Mathematical Sciences, Nankai University, Tianjin, China, 300071

2. 

Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France

* Corresponding author

Received  July 2019 Revised  December 2019 Published  March 2020

This paper is concerned with inverse source problems for the time-dependent Lamé system in an unbounded domain corresponding to either the exterior of a bounded cavity or the full space $ {\mathbb{R}}^3 $. If the time and spatial variables of the source term can be separated with compact support, we prove that the vector valued spatial source term can be uniquely determined by boundary Dirichlet data in the exterior of a given cavity. Uniqueness and stability for recovering some class of time-dependent source terms are also obtained by using, respectively, partial and full boundary data.

Citation: Guanghui Hu, Yavar Kian. Uniqueness and stability for the recovery of a time-dependent source in elastodynamics. Inverse Problems and Imaging, 2020, 14 (3) : 463-487. doi: 10.3934/ipi.2020022
References:
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K. Aki and P. G. Richards, Quantitative Seismology, 2nd edition, University Science Books, Mill Valley: California, 2002.

[2]

H. AmmariE. BretinJ. Garnier and A. Wahab, Time-reversal algorithms in viscoelastic media, European J. Appl. Math., 24 (2013), 565-600.  doi: 10.1017/S0956792513000107.

[3] H. AmmariE. BretinJ. GarnierH. KangH. Lee and A. Wahab, Mathematical Methods in Elasticity Imaging, of Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2015.  doi: 10.1515/9781400866625.
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D. D. AngM. IkehataD. D. Trong and M. Yamamoto, Unique continuation for a stationary isotropic Lamé system with variable coefficients, Comm. Partial Differential Equations, 23 (1998), 371-385.  doi: 10.1080/03605309808821349.

[5]

J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.

[6]

G. Bao, G. Hu, Y. Kian and T. Yin, Inverse source problems in elastodynamics, Inverse Problems, 34 (2018), 045009. doi: 10.1088/1361-6420/aaaf7e.

[7]

G. BaoG. HuJ. Sun and T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263-301.  doi: 10.1016/j.matpur.2018.01.007.

[8]

M. Bellassoued, Unicité et contrôle pour le système de Lamé, ESAIM Control Optim. Calc. Var., 6 (2001), 561-592.  doi: 10.1051/cocv:2001123.

[9]

M. BellassouedO. Imanuvilov and M. Yamamoto, Inverse problem of determining the density and two Lamé coefficients by boundary data, SIAM J. Math. Anal., 40 (2008), 238-265.  doi: 10.1137/070679971.

[10]

M. Bellassoued and M. Yamamoto, Carleman estimate and inverse source problem for Biot's equations describing wave propagation in porous media, Inverse Problems, 29 (2013), 115002. doi: 10.1088/0266-5611/29/11/115002.

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M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.

[12]

M. Bellassoued and M. Yamamoto, Lipschitz stability in determining density and two Lamé coefficients, J. Math. Anal. Appl., 329 (2007), 1240-1259.  doi: 10.1016/j.jmaa.2006.06.094.

[13]

I. B. Aïcha, Stability estimate for hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010. doi: 10.1088/0266-5611/31/12/125010.

[14]

M. V. de HoopL. Oksanen and J. Tittelfitz, Uniqueness for a seismic inverse source problem modeling a subsonic rupture, Comm. Partial Differential Equations, 41 (2016), 1895-1917.  doi: 10.1080/03605302.2016.1240183.

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A. L. Buhgem, Multidimensional inverse problems of spectral analysis, (Russian) Dokl. Akad. Nauk SSSR, 284 (1985), 21-24.

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M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl., 114 (2018), 235-261.  doi: 10.1016/j.matpur.2017.12.003.

[17]

M. Choulli and M. Yamamoto, Some stability estimates in determining sources and coefficients, J. Inverse Ill-Posed Probl., 14 (2006), 355-373.  doi: 10.1515/156939406777570996.

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G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Vol. 219, Springer-Verlag, Berlin-New York, 1976.

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M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the cauchy problem for Maxwell and elasticity systems, in Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, Vol. XIV (Paris, 1997/1998), Stud. Math. Appl., 31, North-Holland, Amsterdam, 2002, 329–349. doi: 10.1016/S0168-2024(02)80016-9.

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M. Eller and D. Toundykov, A global Holmgren theorem for multidimensional hyperbolic partial differential equations, Appl. Anal., 91 (2012), 69-90.  doi: 10.1080/00036811.2010.538685.

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K. Fujishiro and Y. Kian, Determination of time dependent factors of coefficients in fractional diffusion equations, Math. Control Relat. Fields, 6 (2016), 251-269.  doi: 10.3934/mcrf.2016003.

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G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, 164, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-68545-6.

[23]

G. Hu, Y. Kian, P. Li and Y. Zhao, Inverse moving source problems in electrodynamics, Inverse Problems, 35 (2019), 075001. doi: 10.1088/1361-6420/ab1496.

[24]

G. HuY. Kian and Y. Zhao, Uniqueness to some inverse source problems for the wave equation in unbounded domains, Acta Math. Appl. Sin. Engl. Ser., 36 (2020), 134-150.  doi: 10.1007/s10255-020-0917-4.

[25]

M. IkehataG. Nakamura and M. Yamamoto, Uniqueness in inverse problems for the isotropic Lamé system, J. Math. Sci. Univ. Tokyo, 5 (1998), 627-692. 

[26]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lamé system and the application to an inverse problem, ESAIM Control Optim. Calc. Var., 11 (2005), 1-56.  doi: 10.1051/cocv:2004030.

[27]

O. Imanuvilov and M. Yamamoto, Carleman estimates for the three-dimensional nonstationary Lamé system and application to an inverse problem, in Control Theory of Partial Differential Equations, Lect. Notes Pure Appl. Math., 242, Chapman & Hall/CRC, Boca Raton, FL, 2005, 337–374. doi: 10.1201/9781420028317.

[28]

V. Isakov, Inverse Source Problems, Mathematical Surveys and Monographs, 34, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/surv/034.

[29]

V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.

[30]

V. Isakov, A nonhyperbolic Cauchy problem for $\square_b\square_c$ and its applications to elasticity theory, Comm. Pure Appl. Math., 39 (1986), 747-767.  doi: 10.1002/cpa.3160390603.

[31]

V. Isakov, Inverse obstacle problems, Inverse Problems, 25 (2009), 123002. doi: 10.1088/0266-5611/25/12/123002.

[32]

V. Isakov, On uniqueness of obstacles and boundary conditions from restricted dynamical and scattering data, Inverse Probl. Imaging, 2 (2008), 151-165.  doi: 10.3934/ipi.2008.2.151.

[33]

D. JiangY. Liu and M. Yamamoto, Inverse source problem for the hyperbolic equation with a time-dependent principal part, J. Differential Equations, 262 (2017), 653-681.  doi: 10.1016/j.jde.2016.09.036.

[34]

D. Jiang, Z. Li, Y. Liu and M. Yamamoto, Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations, Inverse Problems, 33 (2017), 055013. doi: 10.1088/1361-6420/aa58d1.

[35]

M. Kawashita, On the local-energy decay property for the elastic wave equation with Neumann boundary condition, Duke Math. J., 67 (1992), 333-351.  doi: 10.1215/S0012-7094-92-06712-3.

[36]

Y. Kian, Stability in the determination of a time-dependent coefficient for wave equations from partial data, J. Math. Anal. Appl., 436 (2016), 408-428.  doi: 10.1016/j.jmaa.2015.12.018.

[37]

Y. Kian, D. Sambou and E. Soccorsi, Logarithmic stability inequality in an inverse source problem for the heat equation on a waveguide, Appl. Anal., (2018). doi: 10.1080/00036811.2018.1557324.

[38]

S. Shumin, Carleman estimates for second-order hyperbolic systems in anisotropic cases and applications. Part I: Carleman estimates, Appl. Anal., 94 (2015), 2261-2286.  doi: 10.1080/00036811.2014.983486.

[39]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.

[40]

V. D. Kupradze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Vol. 25, North Holland, Amsterdam, 1979.

[41]

J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972.

[42]

J-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II, Dunod, Paris, 1968.

[43]

E. Malinnikova, The theorem on three spheres for harmonic differential forms, in Complex Analysis, Operators, and Related Topics, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000, 213–220.

[44] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[45]

N. S. Nadirašvili, A generalization of Hadamard's three circles theorem, Vestnik Moskov. Univ. Ser. I Mat. Meh., 31 (1976), 39-42. 

[46]

K. Rashedi and M. Sini, Stable recovery of the time-dependent source term from one measurement for the wave equation, Inverse Problems, 31 (2015), 105011. doi: 10.1088/0266-5611/31/10/105011.

[47] P. M. Shearer, Introduction to Seismology, 2nd edition, Cambridge University Press, New York, 2009.  doi: 10.1017/CBO9780511841552.
[48]

P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications, Trans. Amer. Math. Soc., 365 (2013), 5737-5758.  doi: 10.1090/S0002-9947-2013-05703-0.

[49]

M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by control method, Inverse Problems, 11 (1995), 481-496.  doi: 10.1088/0266-5611/11/2/013.

[50]

M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pure Appl., 78 (1999), 65-98.  doi: 10.1016/S0021-7824(99)80010-5.

[51]

M. Yamamoto and X. Zhang, Global Uniqueness and stability for an inverse wave source problem for less regular data, J. Math. Anal. Appl., 263 (2001), 479-500.  doi: 10.1006/jmaa.2001.7621.

show all references

References:
[1]

K. Aki and P. G. Richards, Quantitative Seismology, 2nd edition, University Science Books, Mill Valley: California, 2002.

[2]

H. AmmariE. BretinJ. Garnier and A. Wahab, Time-reversal algorithms in viscoelastic media, European J. Appl. Math., 24 (2013), 565-600.  doi: 10.1017/S0956792513000107.

[3] H. AmmariE. BretinJ. GarnierH. KangH. Lee and A. Wahab, Mathematical Methods in Elasticity Imaging, of Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2015.  doi: 10.1515/9781400866625.
[4]

D. D. AngM. IkehataD. D. Trong and M. Yamamoto, Unique continuation for a stationary isotropic Lamé system with variable coefficients, Comm. Partial Differential Equations, 23 (1998), 371-385.  doi: 10.1080/03605309808821349.

[5]

J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.

[6]

G. Bao, G. Hu, Y. Kian and T. Yin, Inverse source problems in elastodynamics, Inverse Problems, 34 (2018), 045009. doi: 10.1088/1361-6420/aaaf7e.

[7]

G. BaoG. HuJ. Sun and T. Yin, Direct and inverse elastic scattering from anisotropic media, J. Math. Pures Appl., 117 (2018), 263-301.  doi: 10.1016/j.matpur.2018.01.007.

[8]

M. Bellassoued, Unicité et contrôle pour le système de Lamé, ESAIM Control Optim. Calc. Var., 6 (2001), 561-592.  doi: 10.1051/cocv:2001123.

[9]

M. BellassouedO. Imanuvilov and M. Yamamoto, Inverse problem of determining the density and two Lamé coefficients by boundary data, SIAM J. Math. Anal., 40 (2008), 238-265.  doi: 10.1137/070679971.

[10]

M. Bellassoued and M. Yamamoto, Carleman estimate and inverse source problem for Biot's equations describing wave propagation in porous media, Inverse Problems, 29 (2013), 115002. doi: 10.1088/0266-5611/29/11/115002.

[11]

M. Bellassoued and M. Yamamoto, Carleman Estimates and Applications to Inverse Problems for Hyperbolic Systems, Springer Monographs in Mathematics, Springer, Tokyo, 2017. doi: 10.1007/978-4-431-56600-7.

[12]

M. Bellassoued and M. Yamamoto, Lipschitz stability in determining density and two Lamé coefficients, J. Math. Anal. Appl., 329 (2007), 1240-1259.  doi: 10.1016/j.jmaa.2006.06.094.

[13]

I. B. Aïcha, Stability estimate for hyperbolic inverse problem with time-dependent coefficient, Inverse Problems, 31 (2015), 125010. doi: 10.1088/0266-5611/31/12/125010.

[14]

M. V. de HoopL. Oksanen and J. Tittelfitz, Uniqueness for a seismic inverse source problem modeling a subsonic rupture, Comm. Partial Differential Equations, 41 (2016), 1895-1917.  doi: 10.1080/03605302.2016.1240183.

[15]

A. L. Buhgem, Multidimensional inverse problems of spectral analysis, (Russian) Dokl. Akad. Nauk SSSR, 284 (1985), 21-24.

[16]

M. Choulli and Y. Kian, Logarithmic stability in determining the time-dependent zero order coefficient in a parabolic equation from a partial Dirichlet-to-Neumann map. Application to the determination of a nonlinear term, J. Math. Pures Appl., 114 (2018), 235-261.  doi: 10.1016/j.matpur.2017.12.003.

[17]

M. Choulli and M. Yamamoto, Some stability estimates in determining sources and coefficients, J. Inverse Ill-Posed Probl., 14 (2006), 355-373.  doi: 10.1515/156939406777570996.

[18]

G. Duvaut and J.-L. Lions, Inequalities in Mechanics and Physics, Vol. 219, Springer-Verlag, Berlin-New York, 1976.

[19]

M. Eller, V. Isakov, G. Nakamura and D. Tataru, Uniqueness and stability in the cauchy problem for Maxwell and elasticity systems, in Nonlinear Partial Differential Equations and their Applications. Collège de France Seminar, Vol. XIV (Paris, 1997/1998), Stud. Math. Appl., 31, North-Holland, Amsterdam, 2002, 329–349. doi: 10.1016/S0168-2024(02)80016-9.

[20]

M. Eller and D. Toundykov, A global Holmgren theorem for multidimensional hyperbolic partial differential equations, Appl. Anal., 91 (2012), 69-90.  doi: 10.1080/00036811.2010.538685.

[21]

K. Fujishiro and Y. Kian, Determination of time dependent factors of coefficients in fractional diffusion equations, Math. Control Relat. Fields, 6 (2016), 251-269.  doi: 10.3934/mcrf.2016003.

[22]

G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, 164, Springer-Verlag, Berlin, 2008. doi: 10.1007/978-3-540-68545-6.

[23]

G. Hu, Y. Kian, P. Li and Y. Zhao, Inverse moving source problems in electrodynamics, Inverse Problems, 35 (2019), 075001. doi: 10.1088/1361-6420/ab1496.

[24]

G. HuY. Kian and Y. Zhao, Uniqueness to some inverse source problems for the wave equation in unbounded domains, Acta Math. Appl. Sin. Engl. Ser., 36 (2020), 134-150.  doi: 10.1007/s10255-020-0917-4.

[25]

M. IkehataG. Nakamura and M. Yamamoto, Uniqueness in inverse problems for the isotropic Lamé system, J. Math. Sci. Univ. Tokyo, 5 (1998), 627-692. 

[26]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimates for the non-stationary Lamé system and the application to an inverse problem, ESAIM Control Optim. Calc. Var., 11 (2005), 1-56.  doi: 10.1051/cocv:2004030.

[27]

O. Imanuvilov and M. Yamamoto, Carleman estimates for the three-dimensional nonstationary Lamé system and application to an inverse problem, in Control Theory of Partial Differential Equations, Lect. Notes Pure Appl. Math., 242, Chapman & Hall/CRC, Boca Raton, FL, 2005, 337–374. doi: 10.1201/9781420028317.

[28]

V. Isakov, Inverse Source Problems, Mathematical Surveys and Monographs, 34, American Mathematical Society, Providence, RI, 1990. doi: 10.1090/surv/034.

[29]

V. Isakov, Inverse Problems for Partial Differential Equations, Applied Mathematical Sciences, 127, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4899-0030-2.

[30]

V. Isakov, A nonhyperbolic Cauchy problem for $\square_b\square_c$ and its applications to elasticity theory, Comm. Pure Appl. Math., 39 (1986), 747-767.  doi: 10.1002/cpa.3160390603.

[31]

V. Isakov, Inverse obstacle problems, Inverse Problems, 25 (2009), 123002. doi: 10.1088/0266-5611/25/12/123002.

[32]

V. Isakov, On uniqueness of obstacles and boundary conditions from restricted dynamical and scattering data, Inverse Probl. Imaging, 2 (2008), 151-165.  doi: 10.3934/ipi.2008.2.151.

[33]

D. JiangY. Liu and M. Yamamoto, Inverse source problem for the hyperbolic equation with a time-dependent principal part, J. Differential Equations, 262 (2017), 653-681.  doi: 10.1016/j.jde.2016.09.036.

[34]

D. Jiang, Z. Li, Y. Liu and M. Yamamoto, Weak unique continuation property and a related inverse source problem for time-fractional diffusion-advection equations, Inverse Problems, 33 (2017), 055013. doi: 10.1088/1361-6420/aa58d1.

[35]

M. Kawashita, On the local-energy decay property for the elastic wave equation with Neumann boundary condition, Duke Math. J., 67 (1992), 333-351.  doi: 10.1215/S0012-7094-92-06712-3.

[36]

Y. Kian, Stability in the determination of a time-dependent coefficient for wave equations from partial data, J. Math. Anal. Appl., 436 (2016), 408-428.  doi: 10.1016/j.jmaa.2015.12.018.

[37]

Y. Kian, D. Sambou and E. Soccorsi, Logarithmic stability inequality in an inverse source problem for the heat equation on a waveguide, Appl. Anal., (2018). doi: 10.1080/00036811.2018.1557324.

[38]

S. Shumin, Carleman estimates for second-order hyperbolic systems in anisotropic cases and applications. Part I: Carleman estimates, Appl. Anal., 94 (2015), 2261-2286.  doi: 10.1080/00036811.2014.983486.

[39]

M. V. Klibanov, Inverse problems and Carleman estimates, Inverse Problems, 8 (1992), 575-596.  doi: 10.1088/0266-5611/8/4/009.

[40]

V. D. Kupradze, Three-dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity, Vol. 25, North Holland, Amsterdam, 1979.

[41]

J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer-Verlag, New York-Heidelberg, 1972.

[42]

J-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications, Vol. II, Dunod, Paris, 1968.

[43]

E. Malinnikova, The theorem on three spheres for harmonic differential forms, in Complex Analysis, Operators, and Related Topics, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000, 213–220.

[44] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000. 
[45]

N. S. Nadirašvili, A generalization of Hadamard's three circles theorem, Vestnik Moskov. Univ. Ser. I Mat. Meh., 31 (1976), 39-42. 

[46]

K. Rashedi and M. Sini, Stable recovery of the time-dependent source term from one measurement for the wave equation, Inverse Problems, 31 (2015), 105011. doi: 10.1088/0266-5611/31/10/105011.

[47] P. M. Shearer, Introduction to Seismology, 2nd edition, Cambridge University Press, New York, 2009.  doi: 10.1017/CBO9780511841552.
[48]

P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications, Trans. Amer. Math. Soc., 365 (2013), 5737-5758.  doi: 10.1090/S0002-9947-2013-05703-0.

[49]

M. Yamamoto, Stability, reconstruction formula and regularization for an inverse source hyperbolic problem by control method, Inverse Problems, 11 (1995), 481-496.  doi: 10.1088/0266-5611/11/2/013.

[50]

M. Yamamoto, Uniqueness and stability in multidimensional hyperbolic inverse problems, J. Math. Pure Appl., 78 (1999), 65-98.  doi: 10.1016/S0021-7824(99)80010-5.

[51]

M. Yamamoto and X. Zhang, Global Uniqueness and stability for an inverse wave source problem for less regular data, J. Math. Anal. Appl., 263 (2001), 479-500.  doi: 10.1006/jmaa.2001.7621.

Figure 1.  Radiation of a source in an inhomogeneous isotropic elastic medium in the exterior of a cavity. Suppose that the cavity $ D $ is known. The inverse problem is to determine the source term from the data measured on $ \partial B_R = \{x\in {\mathbb{R}}^3: |x| = R\} $
Figure 2.  Suppose that the data are collected on $ \omega $ and on $ \partial\Omega $. The inverse problem is to determine the value of $ g $ on $ \Omega $
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