Cloaking for a quasi-linear elliptic partial differential equation

Tuhin Ghosh; Karthik Iyer

doi:10.3934/ipi.2018020

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April 2018, 12(2): 461-491. doi: 10.3934/ipi.2018020

Cloaking for a quasi-linear elliptic partial differential equation

Tuhin Ghosh ^1, and Karthik Iyer ^2,

1.	Jockey Club Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong, China
2.	Department of Mathematics, University of Washington, Seattle, WA 98195-4350, USA

Received June 2017 Revised September 2017 Published February 2018

In this article we consider cloaking for a quasi-linear elliptic partial differential equation of divergence type defined on a bounded domain in $\mathbb{R}^N$ for $N = 2, 3$. We show that a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular change of variables scheme. These approximate cloaks, though non-degenerate, are anisotropic. We also show, within the framework of homogenization, that it is possible to get isotropic regular approximate cloaks. This work generalizes to quasi-linear settings previous work on cloaking in the context of Electrical Impedance Tomography for the conductivity equation.

Keywords: Quasilinear elliptic, homogenization, cloaking, inverse problems.

Mathematics Subject Classification: Primary: 35R30; Secondary: 35J62.

Citation: Tuhin Ghosh, Karthik Iyer. Cloaking for a quasi-linear elliptic partial differential equation. Inverse Problems & Imaging, 2018, 12 (2) : 461-491. doi: 10.3934/ipi.2018020

References:

[1]	G. Allaire, Shape Optimization by the Homogenization Method, vol. 146 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. Google Scholar
[2]	H. Ammari, H. Kang, H. Lee and M. Lim, Enhancement of near cloaking using generalized polarization tensors vanishing structures. part Ⅰ: The conductivity problem, Communications in Mathematical Physics, 317 (2013), 253-266. doi: 10.1007/s00220-012-1615-8. Google Scholar
[3]	K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265. Google Scholar
[4]	K. Astala, L. Päivärinta and M. Lassas, Calderón's inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207-224. doi: 10.1081/PDE-200044485. Google Scholar
[5]	J. Bear and Y. Bachmat, Introduction to Modelling of Transport Phenomena in Porous Media, 1991.Google Scholar
[6]	A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 5 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, 1978. Google Scholar
[7]	L. Boccardo and F. Murat, Remarques sur l'homogénéisation de certains problémes quasi-linéaires, Portugal. Math., 41 (1982), 535-562 (1984). Google Scholar
[8]	R. M. Brown and G. A. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027. doi: 10.1080/03605309708821292. Google Scholar
[9]	A. -P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, 65-73. Google Scholar
[10]	P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities Forum Math. Pi, 4 (2016), e2, 28pp. doi: 10.1017/fmp.2015.9. Google Scholar
[11]	M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-9982-5. Google Scholar
[12]	D. Cioranescu and P. Donato, An Introduction to Homogenization, vol. 17 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1999. Google Scholar
[13]	Y. Deng, H. Liu and G. Uhlmann, Full and partial cloaking in electromagnetic scattering, Archive for Rational Mechanics and Analysis, 223 (2017), 265-299. doi: 10.1007/s00205-016-1035-6. Google Scholar
[14]	Y. Deng, H. Liu and G. Uhlmann, On regularized full-and partial-cloaks in acoustic scattering, Comm. Partial Differential Equations, 42 (2017), 821-851. doi: 10.1080/03605302.2017.1286673. Google Scholar
[15]	R. Fleury, F. Monticone and A. Alú, Invisibility and cloaking: Origins, present, and future perspectives, Phys. Rev. Applied, 4 (2015), 037001. doi: 10.1103/PhysRevApplied.4.037001. Google Scholar
[16]	N. Fusco and G. Moscariello, On the homogenization of quasilinear divergence structure operators, Ann. Mat. Pura Appl. (4), 146 (1987), 1-13. doi: 10.1007/BF01762357. Google Scholar
[17]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition. Google Scholar
[18]	A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Improvement of cylindrical cloaking with the shs lining, Opt. Express, 15 (2007), 12717-12734. Google Scholar
[19]	A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Full-wave invisibility of active devices at all frequencies, Comm. Math. Phys., 275 (2007), 749-789. doi: 10.1007/s00220-007-0311-6. Google Scholar
[20]	A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Electromagnetic wormholes via handlebody constructions, Comm. Math. Phys., 281 (2008), 369-385. doi: 10.1007/s00220-008-0492-7. Google Scholar
[21]	A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Isotropic transformation optics: Approximate acoustic and quantum cloaking, New Journal of Physics, 10 (2008), 115024. Google Scholar
[22]	A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Cloaking devices, electromagnetic wormholes, and transformation optics, SIAM Rev., 51 (2009), 3-33. doi: 10.1137/080716827. Google Scholar
[23]	A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Invisibility and inverse problems, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 55-97. doi: 10.1090/S0273-0979-08-01232-9. Google Scholar
[24]	A. Greenleaf, M. Lassas and G. Uhlmann, Anisotropic conductivities that cannot be detected by eit, Physiological measurement, 24. doi: 10.1088/0967-3334/24/2/353. Google Scholar
[25]	A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem, Math. Res. Lett., 10 (2003), 685-693. doi: 10.4310/MRL.2003.v10.n5.a11. Google Scholar
[26]	B. Haberman, Uniqueness in Calderón's problem for conductivities with unbounded gradient, Comm. Math. Phys., 340 (2015), 639-659. doi: 10.1007/s00220-015-2460-3. Google Scholar
[27]	B. Haberman and D. Tataru, Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. J., 162 (2013), 496-516. doi: 10.1215/00127094-2019591. Google Scholar
[28]	G. Hu and H. Liu, Nearly cloaking the elastic wave fields, J. Math. Pures Appl. (9), 104 (2015), 1045-1074. doi: 10.1016/j.matpur.2015.07.004. Google Scholar
[29]	V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994, Translated from the Russian by G. A. Yosifian [G. A. Iosifýan]. doi: 10.1007/978-3-642-84659-5. Google Scholar
[30]	A. Karageorghis and D. Lesnic, Steady-state nonlinear heat conduction in composite materials using the method of fundamental solutions, Comput. Methods Appl. Mech. Engrg., 197 (2008), 3122-3137. doi: 10.1016/j.cma.2008.02.011. Google Scholar
[31]	R. V. Kohn, H. Shen, M. S. Vogelius and M. I. Weinstein, Cloaking via change of variables in electric impedance tomography Inverse Problems, 24 (2008), 015016, 21pp. doi: 10.1088/0266-5611/24/1/015016. Google Scholar
[32]	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. Google Scholar
[33]	J. M. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112. doi: 10.1002/cpa.3160420804. Google Scholar
[34]	U. Leonhardt, Optical conformal mapping, Science, 312 (2006), 1777-1780. doi: 10.1126/science.1126493. Google Scholar
[35]	J. Li and J. B. Pendry, Hiding under the carpet: A new strategy for cloaking, Phys. Rev. Lett., 101 (2008), 203901. doi: 10.1103/PhysRevLett.101.203901. Google Scholar
[36]	Y. -H. Lin, Nearly cloaking for the elasticity system with residual stress, Asymptotic Analysis, 106 (2018), 1-23, arXiv: 1611.05151v2 doi: 10.3233/ASY-171439. Google Scholar
[37]	H. Liu and H. Sun, Enhanced near-cloak by FSH lining, J. Math. Pures Appl. (9), 99 (2013), 17-42. doi: 10.1016/j.matpur.2012.06.001. Google Scholar
[38]	H. Liu and G. Uhlmann, Regularized transformation-optics cloaking in acoustic and electromagnetic scattering, in Inverse problems and imaging, vol. 44 of Panor. Synth`eses, Soc. Math. France, Paris, 2015,111-136. Google Scholar
[39]	H. Liu and T. Zhou, On approximate electromagnetic cloaking by transformation media, SIAM J. Appl. Math., 71 (2011), 218-241. doi: 10.1137/10081112X. Google Scholar
[40]	J. Malík, On homogenization of a quasilinear elliptic equation connected with heat conductivity, URL http://www2.cs.cas.cz/mweb/download/publi/MaII2006.pdf.Google Scholar
[41]	A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96. doi: 10.2307/2118653. Google Scholar
[42]	J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782. doi: 10.1126/science.1125907. Google Scholar
[43]	Z. Ruan, M. Yan, C. W. Neff and M. Qiu, Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations, Phys. Rev. Lett., 99 (2007), 113903. doi: 10.1103/PhysRevLett.99.113903. Google Scholar
[44]	Z. Sun, On a quasilinear inverse boundary value problem, Math. Z., 221 (1996), 293-305. doi: 10.1007/BF02622117. Google Scholar
[45]	Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math., 119 (1997), 771-797, URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v119/119.4sun.pdf. doi: 10.1353/ajm.1997.0027. Google Scholar
[46]	Z. Sun and G. Uhlmann, Anisotropic inverse problems in two dimensions, Inverse Problems, 19 (2003), 1001-1010. doi: 10.1088/0266-5611/19/5/301. Google Scholar
[47]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169. doi: 10.2307/1971291. Google Scholar
[48]	L. Tartar, The General Theory of Homogenization, vol. 7 of Lecture Notes of the Unione Matematica Italiana, Springer-Verlag, Berlin; UMI, Bologna, 2009, A personalized introduction. doi: 10.1007/978-3-642-05195-1. Google Scholar
[49]	G. Uhlmann, Inverse problems: seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279. doi: 10.1007/s13373-014-0051-9. Google Scholar
[50]	A. Whittington, A. Hofmeister and P. Nabelek, Temperature-dependent thermal diffusivity of the earths crust and implications for magmatism, Nature, 458 (2009), 319-321. Google Scholar

show all references

References:

[1]	G. Allaire, Shape Optimization by the Homogenization Method, vol. 146 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. Google Scholar
[2]	H. Ammari, H. Kang, H. Lee and M. Lim, Enhancement of near cloaking using generalized polarization tensors vanishing structures. part Ⅰ: The conductivity problem, Communications in Mathematical Physics, 317 (2013), 253-266. doi: 10.1007/s00220-012-1615-8. Google Scholar
[3]	K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane, Ann. of Math. (2), 163 (2006), 265-299. doi: 10.4007/annals.2006.163.265. Google Scholar
[4]	K. Astala, L. Päivärinta and M. Lassas, Calderón's inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207-224. doi: 10.1081/PDE-200044485. Google Scholar
[5]	J. Bear and Y. Bachmat, Introduction to Modelling of Transport Phenomena in Porous Media, 1991.Google Scholar
[6]	A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, vol. 5 of Studies in Mathematics and its Applications, North-Holland Publishing Co., Amsterdam-New York, 1978. Google Scholar
[7]	L. Boccardo and F. Murat, Remarques sur l'homogénéisation de certains problémes quasi-linéaires, Portugal. Math., 41 (1982), 535-562 (1984). Google Scholar
[8]	R. M. Brown and G. A. Uhlmann, Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions, Comm. Partial Differential Equations, 22 (1997), 1009-1027. doi: 10.1080/03605309708821292. Google Scholar
[9]	A. -P. Calderón, On an inverse boundary value problem, in Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro, 1980), Soc. Brasil. Mat., Rio de Janeiro, 1980, 65-73. Google Scholar
[10]	P. Caro and K. M. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities Forum Math. Pi, 4 (2016), e2, 28pp. doi: 10.1017/fmp.2015.9. Google Scholar
[11]	M. Chipot, Elliptic Equations: An Introductory Course, Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks], Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-9982-5. Google Scholar
[12]	D. Cioranescu and P. Donato, An Introduction to Homogenization, vol. 17 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1999. Google Scholar
[13]	Y. Deng, H. Liu and G. Uhlmann, Full and partial cloaking in electromagnetic scattering, Archive for Rational Mechanics and Analysis, 223 (2017), 265-299. doi: 10.1007/s00205-016-1035-6. Google Scholar
[14]	Y. Deng, H. Liu and G. Uhlmann, On regularized full-and partial-cloaks in acoustic scattering, Comm. Partial Differential Equations, 42 (2017), 821-851. doi: 10.1080/03605302.2017.1286673. Google Scholar
[15]	R. Fleury, F. Monticone and A. Alú, Invisibility and cloaking: Origins, present, and future perspectives, Phys. Rev. Applied, 4 (2015), 037001. doi: 10.1103/PhysRevApplied.4.037001. Google Scholar
[16]	N. Fusco and G. Moscariello, On the homogenization of quasilinear divergence structure operators, Ann. Mat. Pura Appl. (4), 146 (1987), 1-13. doi: 10.1007/BF01762357. Google Scholar
[17]	D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition. Google Scholar
[18]	A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Improvement of cylindrical cloaking with the shs lining, Opt. Express, 15 (2007), 12717-12734. Google Scholar
[19]	A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Full-wave invisibility of active devices at all frequencies, Comm. Math. Phys., 275 (2007), 749-789. doi: 10.1007/s00220-007-0311-6. Google Scholar
[20]	A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Electromagnetic wormholes via handlebody constructions, Comm. Math. Phys., 281 (2008), 369-385. doi: 10.1007/s00220-008-0492-7. Google Scholar
[21]	A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Isotropic transformation optics: Approximate acoustic and quantum cloaking, New Journal of Physics, 10 (2008), 115024. Google Scholar
[22]	A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Cloaking devices, electromagnetic wormholes, and transformation optics, SIAM Rev., 51 (2009), 3-33. doi: 10.1137/080716827. Google Scholar
[23]	A. Greenleaf, Y. Kurylev, M. Lassas and G. Uhlmann, Invisibility and inverse problems, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 55-97. doi: 10.1090/S0273-0979-08-01232-9. Google Scholar
[24]	A. Greenleaf, M. Lassas and G. Uhlmann, Anisotropic conductivities that cannot be detected by eit, Physiological measurement, 24. doi: 10.1088/0967-3334/24/2/353. Google Scholar
[25]	A. Greenleaf, M. Lassas and G. Uhlmann, On nonuniqueness for Calderón's inverse problem, Math. Res. Lett., 10 (2003), 685-693. doi: 10.4310/MRL.2003.v10.n5.a11. Google Scholar
[26]	B. Haberman, Uniqueness in Calderón's problem for conductivities with unbounded gradient, Comm. Math. Phys., 340 (2015), 639-659. doi: 10.1007/s00220-015-2460-3. Google Scholar
[27]	B. Haberman and D. Tataru, Uniqueness in Calderón's problem with Lipschitz conductivities, Duke Math. J., 162 (2013), 496-516. doi: 10.1215/00127094-2019591. Google Scholar
[28]	G. Hu and H. Liu, Nearly cloaking the elastic wave fields, J. Math. Pures Appl. (9), 104 (2015), 1045-1074. doi: 10.1016/j.matpur.2015.07.004. Google Scholar
[29]	V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994, Translated from the Russian by G. A. Yosifian [G. A. Iosifýan]. doi: 10.1007/978-3-642-84659-5. Google Scholar
[30]	A. Karageorghis and D. Lesnic, Steady-state nonlinear heat conduction in composite materials using the method of fundamental solutions, Comput. Methods Appl. Mech. Engrg., 197 (2008), 3122-3137. doi: 10.1016/j.cma.2008.02.011. Google Scholar
[31]	R. V. Kohn, H. Shen, M. S. Vogelius and M. I. Weinstein, Cloaking via change of variables in electric impedance tomography Inverse Problems, 24 (2008), 015016, 21pp. doi: 10.1088/0266-5611/24/1/015016. Google Scholar
[32]	O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York-London, 1968. Google Scholar
[33]	J. M. Lee and G. Uhlmann, Determining anisotropic real-analytic conductivities by boundary measurements, Comm. Pure Appl. Math., 42 (1989), 1097-1112. doi: 10.1002/cpa.3160420804. Google Scholar
[34]	U. Leonhardt, Optical conformal mapping, Science, 312 (2006), 1777-1780. doi: 10.1126/science.1126493. Google Scholar
[35]	J. Li and J. B. Pendry, Hiding under the carpet: A new strategy for cloaking, Phys. Rev. Lett., 101 (2008), 203901. doi: 10.1103/PhysRevLett.101.203901. Google Scholar
[36]	Y. -H. Lin, Nearly cloaking for the elasticity system with residual stress, Asymptotic Analysis, 106 (2018), 1-23, arXiv: 1611.05151v2 doi: 10.3233/ASY-171439. Google Scholar
[37]	H. Liu and H. Sun, Enhanced near-cloak by FSH lining, J. Math. Pures Appl. (9), 99 (2013), 17-42. doi: 10.1016/j.matpur.2012.06.001. Google Scholar
[38]	H. Liu and G. Uhlmann, Regularized transformation-optics cloaking in acoustic and electromagnetic scattering, in Inverse problems and imaging, vol. 44 of Panor. Synth`eses, Soc. Math. France, Paris, 2015,111-136. Google Scholar
[39]	H. Liu and T. Zhou, On approximate electromagnetic cloaking by transformation media, SIAM J. Appl. Math., 71 (2011), 218-241. doi: 10.1137/10081112X. Google Scholar
[40]	J. Malík, On homogenization of a quasilinear elliptic equation connected with heat conductivity, URL http://www2.cs.cas.cz/mweb/download/publi/MaII2006.pdf.Google Scholar
[41]	A. I. Nachman, Global uniqueness for a two-dimensional inverse boundary value problem, Ann. of Math. (2), 143 (1996), 71-96. doi: 10.2307/2118653. Google Scholar
[42]	J. B. Pendry, D. Schurig and D. R. Smith, Controlling electromagnetic fields, Science, 312 (2006), 1780-1782. doi: 10.1126/science.1125907. Google Scholar
[43]	Z. Ruan, M. Yan, C. W. Neff and M. Qiu, Ideal cylindrical cloak: Perfect but sensitive to tiny perturbations, Phys. Rev. Lett., 99 (2007), 113903. doi: 10.1103/PhysRevLett.99.113903. Google Scholar
[44]	Z. Sun, On a quasilinear inverse boundary value problem, Math. Z., 221 (1996), 293-305. doi: 10.1007/BF02622117. Google Scholar
[45]	Z. Sun and G. Uhlmann, Inverse problems in quasilinear anisotropic media, Amer. J. Math., 119 (1997), 771-797, URL http://muse.jhu.edu/journals/american_journal_of_mathematics/v119/119.4sun.pdf. doi: 10.1353/ajm.1997.0027. Google Scholar
[46]	Z. Sun and G. Uhlmann, Anisotropic inverse problems in two dimensions, Inverse Problems, 19 (2003), 1001-1010. doi: 10.1088/0266-5611/19/5/301. Google Scholar
[47]	J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math. (2), 125 (1987), 153-169. doi: 10.2307/1971291. Google Scholar
[48]	L. Tartar, The General Theory of Homogenization, vol. 7 of Lecture Notes of the Unione Matematica Italiana, Springer-Verlag, Berlin; UMI, Bologna, 2009, A personalized introduction. doi: 10.1007/978-3-642-05195-1. Google Scholar
[49]	G. Uhlmann, Inverse problems: seeing the unseen, Bull. Math. Sci., 4 (2014), 209-279. doi: 10.1007/s13373-014-0051-9. Google Scholar
[50]	A. Whittington, A. Hofmeister and P. Nabelek, Temperature-dependent thermal diffusivity of the earths crust and implications for magmatism, Nature, 458 (2009), 319-321. Google Scholar

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