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doi: 10.3934/dcdsb.2021135
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On a generalized diffusion problem: A complex network approach

Laboratoire de Mathématiques Appliquées du Havre, Université du Havre Normandie, FR CNRS 3335, Institut des Systèmes Complexes de Normandie, 25, rue Philippe Lebon, BP 1123, 76063 Le Havre, Normandie, France

* Corresponding author: Guillaume Cantin

Received  February 2020 Revised  April 2021 Early access May 2021

In this paper, we propose a new approach for studying a generalized diffusion problem, using complex networks of reaction-diffusion equations. We model the biharmonic operator by a network, based on a finite graph, in which the couplings between nodes are linear. To this end, we study the generalized diffusion problem, establishing results of existence, uniqueness and maximal regularity of the solution via operator sums theory and analytic semigroups techniques. We then solve the complex network problem and present sufficient conditions for the solutions of both problems to converge to each other. Finally, we analyze their asymptotic behavior by establishing the existence of a family of exponential attractors.

Citation: Guillaume Cantin, Alexandre Thorel. On a generalized diffusion problem: A complex network approach. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021135
References:
[1]

B. AmbrosioM. Aziz-Alaoui and V. Phan, Large time behaviour and synchronization of complex networks of reaction–diffusion systems of Fitzhugh–Nagumo type, IMA Journal of Applied Mathematics, 84 (2019), 416-443.  doi: 10.1093/imamat/hxy064.  Google Scholar

[2]

M. Aziz-Alaoui, Synchronization of chaos, Encyclopedia of Mathematical Physics, Elsevier, 5 (2006), 213-226.  doi: 10.1016/B0-12-512666-2/00105-X.  Google Scholar

[3]

I. BelykhM. HaslerM. Lauret and H. Nijmeijer, Synchronization and graph topology, International Journal of Bifurcation and Chaos, 15 (2005), 3423-3433.  doi: 10.1142/S0218127405014143.  Google Scholar

[4]

A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, volume 151., Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-12905-0.  Google Scholar

[5]

B. Bialecki, A fourth order finite difference method for the Dirichlet biharmonic problem, Numerical Algorithms, 61 (2012), 351-375.  doi: 10.1007/s11075-012-9536-3.  Google Scholar

[6]

G. Cantin, Non identical coupled networks with a geographical model for human behaviors during catastrophic events, International Journal of Bifurcation and Chaos, 27 (2017), 1750213, 21pp. doi: 10.1142/S0218127417502133.  Google Scholar

[7]

G. CantinN. Verdière and M. Aziz-Alaoui, Large time dynamics in complex networks of reaction-diffusion systems applied to a panic model, IMA Journal of Applied Mathematics, 84 (2019), 974-1000.  doi: 10.1093/imamat/hxz022.  Google Scholar

[8]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[9]

G. ChenZ. Li and P. Lin, A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow, Advances in Computational Mathematics, 29 (2008), 113-133.  doi: 10.1007/s10444-007-9043-6.  Google Scholar

[10]

L. CherfilsA. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, DCDS-B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.  Google Scholar

[11]

D. S. Cohen and J. D. Murray, A generalized diffusion model for growth and dispersal in a population, Journal of Mathematical Biology, 12 (1981), 237-249.  doi: 10.1007/BF00276132.  Google Scholar

[12]

G. Da Prato, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Apple.(9), 54 (1975), 305-387.   Google Scholar

[13]

L. De Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rendiconti del Seminario Matematico della Università di Padova, 34 (1964), 205–223.  Google Scholar

[14]

V. Dolean, P. Jolivet and F. Nataf, An Introduction to Domain Decomposition Methods, Society for Industrial and Applied Mathematics, 2015. doi: 10.1137/1.9781611974065.ch1.  Google Scholar

[15]

G. Dore, $L^p$ regularity for abstract differential equations, In H. Komatsu, editor, Functional Analysis and Related Topics, 1991, pages 25–38, Berlin, Heidelberg, 1993. Springer Berlin Heidelberg. doi: 10.1007/BFb0085472.  Google Scholar

[16]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Mathematische Zeitschrift, 196 (1987), 189-201.  doi: 10.1007/BF01163654.  Google Scholar

[17]

M. Ebenbeck and H. Garcke, On a Cahn–Hilliard–Brinkman model for tumor growth and its singular limits, SIAM Journal on Mathematical Analysis, 51 (2019), 1868-1912.  doi: 10.1137/18M1228104.  Google Scholar

[18]

L. Ehrlich, Solving the biharmonic equation as coupled finite difference equations, SIAM Journal on Numerical Analysis, 8 (1971), 278-287.  doi: 10.1137/0708029.  Google Scholar

[19]

A. GiorginiA. Miranville and R. Temam, Uniqueness and regularity for the Navier–Stokes–Cahn–Hilliard system, SIAM Journal on Mathematical Analysis, 51 (2019), 2535-2574.  doi: 10.1137/18M1223459.  Google Scholar

[20]

R. Glowinski and O. Pironneau, Numerical methods for the first biharmonic equation and for the two-dimensional stokes problem, SIAM Review, 21 (1979), 167-212.  doi: 10.1137/1021028.  Google Scholar

[21]

M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bulletin of the american mathematical society, 43 (2006), 305-364.  doi: 10.1090/S0273-0979-06-01108-6.  Google Scholar

[22]

P. Grisvard, Équations différentielles abstraites, Annales scientifiques de l'Ecole Normale Superieure, 2 (1969), 311-395.   Google Scholar

[23]

P. Grisvard, Spazi di tracce e applicazioni, Rend. Mat., 5 (1972), 657-729.   Google Scholar

[24]

M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169. Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.  Google Scholar

[25]

D. Hilhorst, M. Mimura and R. Weidenfeld, On a reaction-diffusion system for a population of hunters and farmers, In Free Boundary Problems, Springer, 2003,189–196. Google Scholar

[26]

N. Katzourakis and T. Pryer, On the numerical approximation of $p$-biharmonic and $\infty$-biharmonic functions, Numerical Methods for Partial Differential Equations, 35 (2019), 155-180.  doi: 10.1002/num.22295.  Google Scholar

[27]

H. Komatsu, Fractional powers of operators, Pacific Journal of Mathematics, 19 (1966), 285-346.  doi: 10.2140/pjm.1966.19.285.  Google Scholar

[28]

R. LabbasK. LemrabetS. Maingot and A. Thorel, Generalized linear models for population dynamics in two juxtaposed habitats, Discrete & Continuous Dynamical Systems - A, 39 (2019), 2933-2960.  doi: 10.3934/dcds.2019122.  Google Scholar

[29]

R. LabbasS. MaingotD. Manceau and A. Thorel, On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain, Journal of Mathematical Analysis and Applications, 450 (2017), 351-376.  doi: 10.1016/j.jmaa.2017.01.026.  Google Scholar

[30]

R. Labbas and M. Moussaoui, On the resolution of the heat equation with discontinuous coefficients, Semigroup Forum, 60 (2000), 187-201.  doi: 10.1007/s002339910013.  Google Scholar

[31]

F. Lara Ochoa, A generalized reaction diffusion model for spatial structure formed by motile cells, Biosystems, 17 (1984), 35-50.  doi: 10.1016/0303-2647(84)90014-5.  Google Scholar

[32]

H. Matano, Asymptotic behavior of solutions of semilinear heat equations on S1, In Nonlinear Diffusion Equations and Their Equilibrium States II, 139–162, Math. Sci. Res. Inst. Publ., 13, Springer, New York, 1988. doi: 10.1007/978-1-4613-9608-6_8.  Google Scholar

[33]

D. Matthes and J. Zinsl, Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type, Nonlinear Analysis, 159 (2017), 316-338.  doi: 10.1016/j.na.2016.12.002.  Google Scholar

[34]

J. Morgan, Global existence for semilinear parabolic systems, SIAM journal on mathematical analysis, 20 (1989), 1128-1144.  doi: 10.1137/0520075.  Google Scholar

[35]

A. Novick-Cohen, Sur une classe d'espaces d'interpolation, Publications Mathématiques de l'I.H.É.S., 19 (1964), 5–68. Google Scholar

[36]

A. Novick-Cohen, On Cahn-Hilliard type equations, Nonlinear Analysis: Theory, Methods & Applications, 15 (1990), 797-814.  doi: 10.1016/0362-546X(90)90094-W.  Google Scholar

[37]

A. Okubo, Diffusion and Ecological Problems: Mathematical Models, Biomathematics, 10. Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[38]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan Journal of Mathematics, 78 (2010), 417-455.  doi: 10.1007/s00032-010-0133-4.  Google Scholar

[39]

J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Mathematische Zeitschrift, 203 (1990), 429-452.  doi: 10.1007/BF02570748.  Google Scholar

[40]

B. Rink and J. Sanders, Coupled cell networks: Semigroups, Lie algebras and normal forms, Transactions of the American Mathematical Society, 367 (2015), 3509-3548.  doi: 10.1090/S0002-9947-2014-06221-1.  Google Scholar

[41]

W. Ruan, Wavefront solutions of degenerate quasilinear reaction–diffusion systems with mixed quasi-monotonicity, Nonlinear Analysis, 182 (2019), 75-96.  doi: 10.1016/j.na.2018.12.003.  Google Scholar

[42]

J. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, Lecture Notes in Math., 1221 (1985), 195-222.  doi: 10.1007/BFb0099115.  Google Scholar

[43]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, volume 258., Springer Science & Business Media, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[44]

J. Stephenson, Single cell discretizations of order two and four for biharmonic problems, Journal of Computational Physics, 55 (1984), 65-80.  doi: 10.1016/0021-9991(84)90015-9.  Google Scholar

[45]

J. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics, 22. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4899-7278-1.  Google Scholar

[46]

A. Thorel, Operational approach for biharmonic equations in $L^p$-spaces, Journal of Evolution Equations, 20 (2020), 631-657.  doi: 10.1007/s00028-019-00536-2.  Google Scholar

[47]

A. Toselli and O. Widlund, Domain Decomposition Methods–Algorithms and Theory, Springer Series in Computational Mathematics, 34. Springer-Verlag, Berlin, 2005. doi: 10.1007/b137868.  Google Scholar

[48]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

[49]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

[50]

J. Zhang, An explicit fourth-order compact finite difference scheme for three-dimensional convection–diffusion equation, Communications in Numerical Methods in Engineering, 14 (1998), 209-218.  doi: 10.1002/(SICI)1099-0887(199803)14:3<209::AID-CNM139>3.0.CO;2-P.  Google Scholar

show all references

References:
[1]

B. AmbrosioM. Aziz-Alaoui and V. Phan, Large time behaviour and synchronization of complex networks of reaction–diffusion systems of Fitzhugh–Nagumo type, IMA Journal of Applied Mathematics, 84 (2019), 416-443.  doi: 10.1093/imamat/hxy064.  Google Scholar

[2]

M. Aziz-Alaoui, Synchronization of chaos, Encyclopedia of Mathematical Physics, Elsevier, 5 (2006), 213-226.  doi: 10.1016/B0-12-512666-2/00105-X.  Google Scholar

[3]

I. BelykhM. HaslerM. Lauret and H. Nijmeijer, Synchronization and graph topology, International Journal of Bifurcation and Chaos, 15 (2005), 3423-3433.  doi: 10.1142/S0218127405014143.  Google Scholar

[4]

A. Bensoussan and J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, volume 151., Springer-Verlag, Berlin, 2002. doi: 10.1007/978-3-662-12905-0.  Google Scholar

[5]

B. Bialecki, A fourth order finite difference method for the Dirichlet biharmonic problem, Numerical Algorithms, 61 (2012), 351-375.  doi: 10.1007/s11075-012-9536-3.  Google Scholar

[6]

G. Cantin, Non identical coupled networks with a geographical model for human behaviors during catastrophic events, International Journal of Bifurcation and Chaos, 27 (2017), 1750213, 21pp. doi: 10.1142/S0218127417502133.  Google Scholar

[7]

G. CantinN. Verdière and M. Aziz-Alaoui, Large time dynamics in complex networks of reaction-diffusion systems applied to a panic model, IMA Journal of Applied Mathematics, 84 (2019), 974-1000.  doi: 10.1093/imamat/hxz022.  Google Scholar

[8]

R. S. Cantrell and C. Cosner, Spatial Ecology via Reaction-Diffusion Equations, Wiley Series in Mathematical and Computational Biology. John Wiley & Sons, Ltd., Chichester, 2003. doi: 10.1002/0470871296.  Google Scholar

[9]

G. ChenZ. Li and P. Lin, A fast finite difference method for biharmonic equations on irregular domains and its application to an incompressible Stokes flow, Advances in Computational Mathematics, 29 (2008), 113-133.  doi: 10.1007/s10444-007-9043-6.  Google Scholar

[10]

L. CherfilsA. Miranville and S. Zelik, On a generalized Cahn-Hilliard equation with biological applications, DCDS-B, 19 (2014), 2013-2026.  doi: 10.3934/dcdsb.2014.19.2013.  Google Scholar

[11]

D. S. Cohen and J. D. Murray, A generalized diffusion model for growth and dispersal in a population, Journal of Mathematical Biology, 12 (1981), 237-249.  doi: 10.1007/BF00276132.  Google Scholar

[12]

G. Da Prato, Sommes d'opérateurs linéaires et équations différentielles opérationnelles, J. Math. Pures Apple.(9), 54 (1975), 305-387.   Google Scholar

[13]

L. De Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rendiconti del Seminario Matematico della Università di Padova, 34 (1964), 205–223.  Google Scholar

[14]

V. Dolean, P. Jolivet and F. Nataf, An Introduction to Domain Decomposition Methods, Society for Industrial and Applied Mathematics, 2015. doi: 10.1137/1.9781611974065.ch1.  Google Scholar

[15]

G. Dore, $L^p$ regularity for abstract differential equations, In H. Komatsu, editor, Functional Analysis and Related Topics, 1991, pages 25–38, Berlin, Heidelberg, 1993. Springer Berlin Heidelberg. doi: 10.1007/BFb0085472.  Google Scholar

[16]

G. Dore and A. Venni, On the closedness of the sum of two closed operators, Mathematische Zeitschrift, 196 (1987), 189-201.  doi: 10.1007/BF01163654.  Google Scholar

[17]

M. Ebenbeck and H. Garcke, On a Cahn–Hilliard–Brinkman model for tumor growth and its singular limits, SIAM Journal on Mathematical Analysis, 51 (2019), 1868-1912.  doi: 10.1137/18M1228104.  Google Scholar

[18]

L. Ehrlich, Solving the biharmonic equation as coupled finite difference equations, SIAM Journal on Numerical Analysis, 8 (1971), 278-287.  doi: 10.1137/0708029.  Google Scholar

[19]

A. GiorginiA. Miranville and R. Temam, Uniqueness and regularity for the Navier–Stokes–Cahn–Hilliard system, SIAM Journal on Mathematical Analysis, 51 (2019), 2535-2574.  doi: 10.1137/18M1223459.  Google Scholar

[20]

R. Glowinski and O. Pironneau, Numerical methods for the first biharmonic equation and for the two-dimensional stokes problem, SIAM Review, 21 (1979), 167-212.  doi: 10.1137/1021028.  Google Scholar

[21]

M. Golubitsky and I. Stewart, Nonlinear dynamics of networks: The groupoid formalism, Bulletin of the american mathematical society, 43 (2006), 305-364.  doi: 10.1090/S0273-0979-06-01108-6.  Google Scholar

[22]

P. Grisvard, Équations différentielles abstraites, Annales scientifiques de l'Ecole Normale Superieure, 2 (1969), 311-395.   Google Scholar

[23]

P. Grisvard, Spazi di tracce e applicazioni, Rend. Mat., 5 (1972), 657-729.   Google Scholar

[24]

M. Haase, The Functional Calculus for Sectorial Operators, Operator Theory: Advances and Applications, 169. Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.  Google Scholar

[25]

D. Hilhorst, M. Mimura and R. Weidenfeld, On a reaction-diffusion system for a population of hunters and farmers, In Free Boundary Problems, Springer, 2003,189–196. Google Scholar

[26]

N. Katzourakis and T. Pryer, On the numerical approximation of $p$-biharmonic and $\infty$-biharmonic functions, Numerical Methods for Partial Differential Equations, 35 (2019), 155-180.  doi: 10.1002/num.22295.  Google Scholar

[27]

H. Komatsu, Fractional powers of operators, Pacific Journal of Mathematics, 19 (1966), 285-346.  doi: 10.2140/pjm.1966.19.285.  Google Scholar

[28]

R. LabbasK. LemrabetS. Maingot and A. Thorel, Generalized linear models for population dynamics in two juxtaposed habitats, Discrete & Continuous Dynamical Systems - A, 39 (2019), 2933-2960.  doi: 10.3934/dcds.2019122.  Google Scholar

[29]

R. LabbasS. MaingotD. Manceau and A. Thorel, On the regularity of a generalized diffusion problem arising in population dynamics set in a cylindrical domain, Journal of Mathematical Analysis and Applications, 450 (2017), 351-376.  doi: 10.1016/j.jmaa.2017.01.026.  Google Scholar

[30]

R. Labbas and M. Moussaoui, On the resolution of the heat equation with discontinuous coefficients, Semigroup Forum, 60 (2000), 187-201.  doi: 10.1007/s002339910013.  Google Scholar

[31]

F. Lara Ochoa, A generalized reaction diffusion model for spatial structure formed by motile cells, Biosystems, 17 (1984), 35-50.  doi: 10.1016/0303-2647(84)90014-5.  Google Scholar

[32]

H. Matano, Asymptotic behavior of solutions of semilinear heat equations on S1, In Nonlinear Diffusion Equations and Their Equilibrium States II, 139–162, Math. Sci. Res. Inst. Publ., 13, Springer, New York, 1988. doi: 10.1007/978-1-4613-9608-6_8.  Google Scholar

[33]

D. Matthes and J. Zinsl, Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type, Nonlinear Analysis, 159 (2017), 316-338.  doi: 10.1016/j.na.2016.12.002.  Google Scholar

[34]

J. Morgan, Global existence for semilinear parabolic systems, SIAM journal on mathematical analysis, 20 (1989), 1128-1144.  doi: 10.1137/0520075.  Google Scholar

[35]

A. Novick-Cohen, Sur une classe d'espaces d'interpolation, Publications Mathématiques de l'I.H.É.S., 19 (1964), 5–68. Google Scholar

[36]

A. Novick-Cohen, On Cahn-Hilliard type equations, Nonlinear Analysis: Theory, Methods & Applications, 15 (1990), 797-814.  doi: 10.1016/0362-546X(90)90094-W.  Google Scholar

[37]

A. Okubo, Diffusion and Ecological Problems: Mathematical Models, Biomathematics, 10. Springer-Verlag, Berlin-New York, 1980.  Google Scholar

[38]

M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey, Milan Journal of Mathematics, 78 (2010), 417-455.  doi: 10.1007/s00032-010-0133-4.  Google Scholar

[39]

J. Prüss and H. Sohr, On operators with bounded imaginary powers in Banach spaces, Mathematische Zeitschrift, 203 (1990), 429-452.  doi: 10.1007/BF02570748.  Google Scholar

[40]

B. Rink and J. Sanders, Coupled cell networks: Semigroups, Lie algebras and normal forms, Transactions of the American Mathematical Society, 367 (2015), 3509-3548.  doi: 10.1090/S0002-9947-2014-06221-1.  Google Scholar

[41]

W. Ruan, Wavefront solutions of degenerate quasilinear reaction–diffusion systems with mixed quasi-monotonicity, Nonlinear Analysis, 182 (2019), 75-96.  doi: 10.1016/j.na.2018.12.003.  Google Scholar

[42]

J. Rubio de Francia, Martingale and integral transforms of Banach space valued functions, Lecture Notes in Math., 1221 (1985), 195-222.  doi: 10.1007/BFb0099115.  Google Scholar

[43]

J. Smoller, Shock Waves and Reaction-Diffusion Equations, volume 258., Springer Science & Business Media, 1994. doi: 10.1007/978-1-4612-0873-0.  Google Scholar

[44]

J. Stephenson, Single cell discretizations of order two and four for biharmonic problems, Journal of Computational Physics, 55 (1984), 65-80.  doi: 10.1016/0021-9991(84)90015-9.  Google Scholar

[45]

J. Thomas, Numerical Partial Differential Equations: Finite Difference Methods, Texts in Applied Mathematics, 22. Springer-Verlag, New York, 1995. doi: 10.1007/978-1-4899-7278-1.  Google Scholar

[46]

A. Thorel, Operational approach for biharmonic equations in $L^p$-spaces, Journal of Evolution Equations, 20 (2020), 631-657.  doi: 10.1007/s00028-019-00536-2.  Google Scholar

[47]

A. Toselli and O. Widlund, Domain Decomposition Methods–Algorithms and Theory, Springer Series in Computational Mathematics, 34. Springer-Verlag, Berlin, 2005. doi: 10.1007/b137868.  Google Scholar

[48]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

[49]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.  Google Scholar

[50]

J. Zhang, An explicit fourth-order compact finite difference scheme for three-dimensional convection–diffusion equation, Communications in Numerical Methods in Engineering, 14 (1998), 209-218.  doi: 10.1002/(SICI)1099-0887(199803)14:3<209::AID-CNM139>3.0.CO;2-P.  Google Scholar

Figure 1.  Splitting of a rectangular domain $ \Omega $ into a grid of sub-domains $ \omega_1, \, \dots, \, \omega_n $ in the dimension case $ N = 2 $
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