# American Institute of Mathematical Sciences

November  2016, 10(4): 977-1006. doi: 10.3934/ipi.2016029

## Team organization may help swarms of flies to become invisible in closed waveguides

 1 INRIA/Centre de mathématiques appliquées, École Polytechnique, Université Paris-Saclay, Route de Saclay, 91128 Palaiseau, France 2 Faculty of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya naberezhnaya, 7-9, 199034, St. Petersburg, Russian Federation

Received  November 2015 Revised  July 2016 Published  October 2016

We are interested in a time harmonic acoustic problem in a waveguide containing flies. The flies are modelled by small sound soft obstacles. We explain how they should arrange to become invisible to an observer sending waves from $-\infty$ and measuring the resulting scattered field at the same position. We assume that the flies can control their position and/or their size. Both monomodal and multimodal regimes are considered. On the other hand, we show that any sound soft obstacle (non necessarily small) embedded in the waveguide always produces some non exponentially decaying scattered field at $+\infty$ for wavenumbers smaller than a constant that we explicit. As a consequence, for such wavenumbers, the flies cannot be made completely invisible to an observer equipped with a measurement device located at $+\infty$.
Citation: Lucas Chesnel, Sergei A. Nazarov. Team organization may help swarms of flies to become invisible in closed waveguides. Inverse Problems & Imaging, 2016, 10 (4) : 977-1006. doi: 10.3934/ipi.2016029
##### References:
 [1] A. Alù, M. G. Silveirinha and N. Engheta, Transmission-line analysis of $\varepsilon$-near-zero-filled narrow channels, Phys. Rev. E, 78 (2008), 016604. Google Scholar [2] T. Arens, D. Gintides and A. Lechleiter, Direct and inverse medium scattering in a three-dimensional homogeneous planar waveguide, SIAM J. Appl. Math., 71 (2011), 753-772. doi: 10.1137/100806333.  Google Scholar [3] A. Bendali, P. H. Cocquet and S. Tordeux, Approximation by multipoles of the multiple acoustic scattering by small obstacles in three dimensions and application to the foldy theory of isotropic scattering, Arch. Ration. Mech. Anal., 219 (2016), 1017-1059. doi: 10.1007/s00205-015-0915-5.  Google Scholar [4] A.-S. Bonnet-Ben Dhia, L. Chesnel and S. A. Nazarov, Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions, Inverse Problems, 31 (2015), 045006, 24pp. doi: 10.1088/0266-5611/31/4/045006.  Google Scholar [5] A.-S. Bonnet-Ben Dhia, E. Lunéville, Y. Mbeutcha and S. A. Nazarov, A method to build non-scattering perturbations of two-dimensional acoustic waveguides, Math. Methods Appl. Sci., 2015. Google Scholar [6] A.-S. Bonnet-Ben Dhia and S. A. Nazarov, Obstacles in acoustic waveguides becoming "invisible'' at given frequencies, Acoust. Phys., 59 (2013), 633-639. Google Scholar [7] A.-S. Bonnet-Ben Dhia, S. A. Nazarov and J. Taskinen, Underwater topography "invisible'' for surface waves at given frequencies, Wave Motion, 57 (2015), 129-142. doi: 10.1016/j.wavemoti.2015.03.008.  Google Scholar [8] L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation, Inverse Problems, 24 (2008), 015018,20pp. doi: 10.1088/0266-5611/24/1/015018.  Google Scholar [9] F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory inverse problems and applications, Inside Out 60, 2013. Google Scholar [10] G. Cardone, S. A. Nazarov and C. Perugia, A gap in the essential spectrum of a cylindrical waveguide with a periodic aperturbation of the surface, Math. Nachr., 283 (2010), 1222-1244. doi: 10.1002/mana.200910025.  Google Scholar [11] G. Cardone, S. A. Nazarov and K. Ruotsalainen, Asymptotic behaviour of an eigenvalue in the continuous spectrum of a narrowed waveguide, Sb. Math., 203 (2012), 153-182. doi: 10.1070/SM2012v203n02ABEH004217.  Google Scholar [12] M. Cassier and C. Hazard, Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: mathematical justification of the Foldy-Lax model, Wave Motion, 50 (2013), 18-28. doi: 10.1016/j.wavemoti.2012.06.001.  Google Scholar [13] H. Chen, C. T. Chan and P. Sheng, Transformation optics and metamaterials, Nat. Mater., 9 (2010), 387-396. doi: 10.1038/nmat2743.  Google Scholar [14] M. Cheney, The linear sampling method and the music algorithm, Inverse Problems, 17 (2001), 591-595. doi: 10.1088/0266-5611/17/4/301.  Google Scholar [15] L. Chesnel, X. Claeys and S. A. Nazarov, Spectrum of a diffusion operator with coefficient changing sign over a small inclusion, Z. Angew. Math. Phys., 66 (2015), 2173-2196. doi: 10.1007/s00033-015-0559-1.  Google Scholar [16] L. Chesnel, N. Hyvönen and S. Staboulis, Construction of indistinguishable conductivity perturbations for the point electrode model in electrical impedance tomography, SIAM J. Appl. Math., 75 (2015), 2093-2109. doi: 10.1137/15M1006404.  Google Scholar [17] X. Claeys, On the theoretical justification of Pocklington's equation, Math. Models Meth. App. Sci., 19 (2009), 1325-1355. doi: 10.1142/S0218202509003802.  Google Scholar [18] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003.  Google Scholar [19] D. Colton, M. Piana and R. Potthast, A simple method using morozov's discrepancy principle for solving inverse scattering problems, Inverse Problems, 13 (1997), 1477-1493. doi: 10.1088/0266-5611/13/6/005.  Google Scholar [20] S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith and J. Pendry, Full-wave simulations of electromagnetic cloaking structures, Phys. Rev. E, 74 (2006), 036621. doi: 10.1103/PhysRevE.74.036621.  Google Scholar [21] B. Edwards, A. Alù, M. G. Silveirinha and N. Engheta, Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects, J. Appl. Phys., 105 (2009), 044905. doi: 10.1063/1.3074506.  Google Scholar [22] D. V. Evans, M. McIver and R. Porter, Transparency of structures in water waves, In Proceedings of 29th International Workshop on Water Waves and Floating Bodies, 2014. Google Scholar [23] R. Fleury and A. Alù, Extraordinary sound transmission through density-near-zero ultranarrow channels, Phys. Rev. Lett., 111 (2013), 055501. doi: 10.1103/PhysRevLett.111.055501.  Google Scholar [24] L. L. Foldy, The multiple scattering of waves. i. general theory of isotropic scattering by randomly distributed scatterers, Phys. Rev., 67 (1945), 107-119. doi: 10.1103/PhysRev.67.107.  Google Scholar [25] Y. Fu, Y. Xu and H. Chen, Additional modes in a waveguide system of zero-index-metamaterials with defects, Scientific Reports, 4 (2014), 1-7. doi: 10.1038/srep06428.  Google Scholar [26] 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 [27] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Birkhäuser, 2006.  Google Scholar [28] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, volume 31, Amer. Math. Soc., 1957.  Google Scholar [29] I. V. Kamotskii and S. A. Nazarov, Spectral problems in singularly perturbed domains and selfadjoint extensions of differential operators, In Proceedings of the St. Petersburg Mathematical Society, Vol. VI, volume 199 of Amer. Math. Soc. Transl. Ser. 2, pages 127-181. Amer. Math. Soc., Providence, RI, 2000. doi: 10.1090/trans2/199/04.  Google Scholar [30] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, reprint of the corr. print. of the 2nd ed. 1980 edition, 1995.  Google Scholar [31] A. Kirsch, The music-algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306.  Google Scholar [32] V. A. Kondratiev, Boundary-value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., 16 (1967), 209-292.  Google Scholar [33] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.  Google Scholar [34] J.-L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Dunod, 1968. Google Scholar [35] P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles, Number 107. Cambridge University Press, 2006. doi: 10.1017/CBO9780511735110.  Google Scholar [36] V. G. Maz'ya, S. A. Nazarov and B. A. Plamenevskiĭ, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Birkhäuser, Basel, 2000.  Google Scholar [37] S. A. Nazarov, The polynomial property of self-adjoint elliptic boundary-value problems and an algebraic description of their attributes, Russ. Math. Surv., 54 (1999), 947-1014. doi: 10.1070/rm1999v054n05ABEH000204.  Google Scholar [38] S. A. Nazarov, Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains, In Sobolev Spaces in Mathematics II, Springer, 9 (2009), 261-309. doi: 10.1007/978-0-387-85650-6_12.  Google Scholar [39] S. A. Nazarov, Opening of a gap in the continuous spectrum of a periodically perturbed waveguide, Mathematical Notes, 87 (2010), 738-756. doi: 10.1134/S0001434610050123.  Google Scholar [40] S. A. Nazarov, Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide, Theor. Math. Phys., 167 (2011), 606-627. doi: 10.1007/s11232-011-0046-6.  Google Scholar [41] S. A. Nazarov, Eigenvalues of the laplace operator with the Neumann conditions at regular perturbed walls of a waveguide, J. Math. Sci., 172 (2011), 555-588. doi: 10.1007/s10958-011-0206-0.  Google Scholar [42] S. A. Nazarov, Trapped waves in a cranked waveguide with hard walls, Acoust. Phys., 57 (2011), 764-771. Google Scholar [43] S. A. Nazarov, The asymptotic analysis of gaps in the spectrum of a waveguide perturbed with a periodic family of small voids, J. Math. Sci., New York, 186 (2012), 247-301. doi: 10.1007/s10958-012-0985-y.  Google Scholar [44] S. A. Nazarov, Enforced stability of an eigenvalue in the continuous spectrum of a waveguide with an obstacle, Comput. Math. and Math. Phys., 52 (2012), 448-464. doi: 10.1134/S096554251203013X.  Google Scholar [45] S. A. Nazarov, Enforced stability of a simple eigenvalue in the continuous spectrum of a waveguide, Funct. Anal. Appl., 47 (2013), 195-209.  Google Scholar [46] S. A. Nazarov and B. A. Plamenevskiĭ, Elliptic problems in domains with piecewise smooth boundaries,, volume 13 of Expositions in Mathematics, ().   Google Scholar [47] V. C. Nguyen, L. Chen and K. Halterman, Total transmission and total reflection by zero index metamaterials with defects, Phys. Rev. Lett., 105 (2010), 233908. doi: 10.1103/PhysRevLett.105.233908.  Google Scholar [48] A. Ourir, A. Maurel and V. Pagneux, Tunneling of electromagnetic energy in multiple connected leads using $\varepsilon$-near-zero materials, Opt. Lett., 38 (2013), 2092-2094. Google Scholar [49] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951. Google Scholar [50] A. G. Ramm, Wave Scattering By Small Bodies of Arbitrary Shapes, Springer, 2005. doi: 10.1142/9789812701206.  Google Scholar [51] C. W. Therrien, Discrete Random Signals and Statistical Signal Processing, Prentice Hall, Englewood Cliffs, NJ, 1992. Google Scholar

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##### References:
 [1] A. Alù, M. G. Silveirinha and N. Engheta, Transmission-line analysis of $\varepsilon$-near-zero-filled narrow channels, Phys. Rev. E, 78 (2008), 016604. Google Scholar [2] T. Arens, D. Gintides and A. Lechleiter, Direct and inverse medium scattering in a three-dimensional homogeneous planar waveguide, SIAM J. Appl. Math., 71 (2011), 753-772. doi: 10.1137/100806333.  Google Scholar [3] A. Bendali, P. H. Cocquet and S. Tordeux, Approximation by multipoles of the multiple acoustic scattering by small obstacles in three dimensions and application to the foldy theory of isotropic scattering, Arch. Ration. Mech. Anal., 219 (2016), 1017-1059. doi: 10.1007/s00205-015-0915-5.  Google Scholar [4] A.-S. Bonnet-Ben Dhia, L. Chesnel and S. A. Nazarov, Non-scattering wavenumbers and far field invisibility for a finite set of incident/scattering directions, Inverse Problems, 31 (2015), 045006, 24pp. doi: 10.1088/0266-5611/31/4/045006.  Google Scholar [5] A.-S. Bonnet-Ben Dhia, E. Lunéville, Y. Mbeutcha and S. A. Nazarov, A method to build non-scattering perturbations of two-dimensional acoustic waveguides, Math. Methods Appl. Sci., 2015. Google Scholar [6] A.-S. Bonnet-Ben Dhia and S. A. Nazarov, Obstacles in acoustic waveguides becoming "invisible'' at given frequencies, Acoust. Phys., 59 (2013), 633-639. Google Scholar [7] A.-S. Bonnet-Ben Dhia, S. A. Nazarov and J. Taskinen, Underwater topography "invisible'' for surface waves at given frequencies, Wave Motion, 57 (2015), 129-142. doi: 10.1016/j.wavemoti.2015.03.008.  Google Scholar [8] L. Bourgeois and E. Lunéville, The linear sampling method in a waveguide: A modal formulation, Inverse Problems, 24 (2008), 015018,20pp. doi: 10.1088/0266-5611/24/1/015018.  Google Scholar [9] F. Cakoni and H. Haddar, Transmission eigenvalues in inverse scattering theory inverse problems and applications, Inside Out 60, 2013. Google Scholar [10] G. Cardone, S. A. Nazarov and C. Perugia, A gap in the essential spectrum of a cylindrical waveguide with a periodic aperturbation of the surface, Math. Nachr., 283 (2010), 1222-1244. doi: 10.1002/mana.200910025.  Google Scholar [11] G. Cardone, S. A. Nazarov and K. Ruotsalainen, Asymptotic behaviour of an eigenvalue in the continuous spectrum of a narrowed waveguide, Sb. Math., 203 (2012), 153-182. doi: 10.1070/SM2012v203n02ABEH004217.  Google Scholar [12] M. Cassier and C. Hazard, Multiple scattering of acoustic waves by small sound-soft obstacles in two dimensions: mathematical justification of the Foldy-Lax model, Wave Motion, 50 (2013), 18-28. doi: 10.1016/j.wavemoti.2012.06.001.  Google Scholar [13] H. Chen, C. T. Chan and P. Sheng, Transformation optics and metamaterials, Nat. Mater., 9 (2010), 387-396. doi: 10.1038/nmat2743.  Google Scholar [14] M. Cheney, The linear sampling method and the music algorithm, Inverse Problems, 17 (2001), 591-595. doi: 10.1088/0266-5611/17/4/301.  Google Scholar [15] L. Chesnel, X. Claeys and S. A. Nazarov, Spectrum of a diffusion operator with coefficient changing sign over a small inclusion, Z. Angew. Math. Phys., 66 (2015), 2173-2196. doi: 10.1007/s00033-015-0559-1.  Google Scholar [16] L. Chesnel, N. Hyvönen and S. Staboulis, Construction of indistinguishable conductivity perturbations for the point electrode model in electrical impedance tomography, SIAM J. Appl. Math., 75 (2015), 2093-2109. doi: 10.1137/15M1006404.  Google Scholar [17] X. Claeys, On the theoretical justification of Pocklington's equation, Math. Models Meth. App. Sci., 19 (2009), 1325-1355. doi: 10.1142/S0218202509003802.  Google Scholar [18] D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Problems, 12 (1996), 383-393. doi: 10.1088/0266-5611/12/4/003.  Google Scholar [19] D. Colton, M. Piana and R. Potthast, A simple method using morozov's discrepancy principle for solving inverse scattering problems, Inverse Problems, 13 (1997), 1477-1493. doi: 10.1088/0266-5611/13/6/005.  Google Scholar [20] S. A. Cummer, B.-I. Popa, D. Schurig, D. R. Smith and J. Pendry, Full-wave simulations of electromagnetic cloaking structures, Phys. Rev. E, 74 (2006), 036621. doi: 10.1103/PhysRevE.74.036621.  Google Scholar [21] B. Edwards, A. Alù, M. G. Silveirinha and N. Engheta, Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects, J. Appl. Phys., 105 (2009), 044905. doi: 10.1063/1.3074506.  Google Scholar [22] D. V. Evans, M. McIver and R. Porter, Transparency of structures in water waves, In Proceedings of 29th International Workshop on Water Waves and Floating Bodies, 2014. Google Scholar [23] R. Fleury and A. Alù, Extraordinary sound transmission through density-near-zero ultranarrow channels, Phys. Rev. Lett., 111 (2013), 055501. doi: 10.1103/PhysRevLett.111.055501.  Google Scholar [24] L. L. Foldy, The multiple scattering of waves. i. general theory of isotropic scattering by randomly distributed scatterers, Phys. Rev., 67 (1945), 107-119. doi: 10.1103/PhysRev.67.107.  Google Scholar [25] Y. Fu, Y. Xu and H. Chen, Additional modes in a waveguide system of zero-index-metamaterials with defects, Scientific Reports, 4 (2014), 1-7. doi: 10.1038/srep06428.  Google Scholar [26] 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 [27] A. Henrot, Extremum Problems for Eigenvalues of Elliptic Operators, Birkhäuser, 2006.  Google Scholar [28] E. Hille and R. S. Phillips, Functional Analysis and Semi-Groups, volume 31, Amer. Math. Soc., 1957.  Google Scholar [29] I. V. Kamotskii and S. A. Nazarov, Spectral problems in singularly perturbed domains and selfadjoint extensions of differential operators, In Proceedings of the St. Petersburg Mathematical Society, Vol. VI, volume 199 of Amer. Math. Soc. Transl. Ser. 2, pages 127-181. Amer. Math. Soc., Providence, RI, 2000. doi: 10.1090/trans2/199/04.  Google Scholar [30] T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, Berlin, reprint of the corr. print. of the 2nd ed. 1980 edition, 1995.  Google Scholar [31] A. Kirsch, The music-algorithm and the factorization method in inverse scattering theory for inhomogeneous media, Inverse Problems, 18 (2002), 1025-1040. doi: 10.1088/0266-5611/18/4/306.  Google Scholar [32] V. A. Kondratiev, Boundary-value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc., 16 (1967), 209-292.  Google Scholar [33] N. S. Landkof, Foundations of Modern Potential Theory, Springer-Verlag, New York-Heidelberg, 1972. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180.  Google Scholar [34] J.-L. Lions and E. Magenes, Problèmes Aux Limites non Homogènes et Applications, Dunod, 1968. Google Scholar [35] P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles, Number 107. Cambridge University Press, 2006. doi: 10.1017/CBO9780511735110.  Google Scholar [36] V. G. Maz'ya, S. A. Nazarov and B. A. Plamenevskiĭ, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, Birkhäuser, Basel, 2000.  Google Scholar [37] S. A. Nazarov, The polynomial property of self-adjoint elliptic boundary-value problems and an algebraic description of their attributes, Russ. Math. Surv., 54 (1999), 947-1014. doi: 10.1070/rm1999v054n05ABEH000204.  Google Scholar [38] S. A. Nazarov, Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains, In Sobolev Spaces in Mathematics II, Springer, 9 (2009), 261-309. doi: 10.1007/978-0-387-85650-6_12.  Google Scholar [39] S. A. Nazarov, Opening of a gap in the continuous spectrum of a periodically perturbed waveguide, Mathematical Notes, 87 (2010), 738-756. doi: 10.1134/S0001434610050123.  Google Scholar [40] S. A. Nazarov, Asymptotic expansions of eigenvalues in the continuous spectrum of a regularly perturbed quantum waveguide, Theor. Math. Phys., 167 (2011), 606-627. doi: 10.1007/s11232-011-0046-6.  Google Scholar [41] S. A. Nazarov, Eigenvalues of the laplace operator with the Neumann conditions at regular perturbed walls of a waveguide, J. Math. Sci., 172 (2011), 555-588. doi: 10.1007/s10958-011-0206-0.  Google Scholar [42] S. A. Nazarov, Trapped waves in a cranked waveguide with hard walls, Acoust. Phys., 57 (2011), 764-771. Google Scholar [43] S. A. Nazarov, The asymptotic analysis of gaps in the spectrum of a waveguide perturbed with a periodic family of small voids, J. Math. Sci., New York, 186 (2012), 247-301. doi: 10.1007/s10958-012-0985-y.  Google Scholar [44] S. A. Nazarov, Enforced stability of an eigenvalue in the continuous spectrum of a waveguide with an obstacle, Comput. Math. and Math. Phys., 52 (2012), 448-464. doi: 10.1134/S096554251203013X.  Google Scholar [45] S. A. Nazarov, Enforced stability of a simple eigenvalue in the continuous spectrum of a waveguide, Funct. Anal. Appl., 47 (2013), 195-209.  Google Scholar [46] S. A. Nazarov and B. A. Plamenevskiĭ, Elliptic problems in domains with piecewise smooth boundaries,, volume 13 of Expositions in Mathematics, ().   Google Scholar [47] V. C. Nguyen, L. Chen and K. Halterman, Total transmission and total reflection by zero index metamaterials with defects, Phys. Rev. Lett., 105 (2010), 233908. doi: 10.1103/PhysRevLett.105.233908.  Google Scholar [48] A. Ourir, A. Maurel and V. Pagneux, Tunneling of electromagnetic energy in multiple connected leads using $\varepsilon$-near-zero materials, Opt. Lett., 38 (2013), 2092-2094. Google Scholar [49] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies, no. 27, Princeton University Press, Princeton, N. J., 1951. Google Scholar [50] A. G. Ramm, Wave Scattering By Small Bodies of Arbitrary Shapes, Springer, 2005. doi: 10.1142/9789812701206.  Google Scholar [51] C. W. Therrien, Discrete Random Signals and Statistical Signal Processing, Prentice Hall, Englewood Cliffs, NJ, 1992. Google Scholar
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