SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3703-3719, June 2024. Abstract. In this paper, we study the nonlinear stability of a shear layer profile for Navier–Stokes equations near a boundary. More precisely, we investigate the effect of cubic interactions on the growth of the linear instability. In the case of the exponential profile, we obtain that the nonlinearity tames the linear instability. We thus conjecture that small perturbations grow until they reach a magnitude [math] only, forming small rolls in the critical layer near the boundary. The mathematical proof of this conjecture is open.
{"title":"Onset of Nonlinear Instabilities in Monotonic Viscous Boundary Layers","authors":"D. Bian, E. Grenier","doi":"10.1137/22m1505773","DOIUrl":"https://doi.org/10.1137/22m1505773","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3703-3719, June 2024. <br/> Abstract. In this paper, we study the nonlinear stability of a shear layer profile for Navier–Stokes equations near a boundary. More precisely, we investigate the effect of cubic interactions on the growth of the linear instability. In the case of the exponential profile, we obtain that the nonlinearity tames the linear instability. We thus conjecture that small perturbations grow until they reach a magnitude [math] only, forming small rolls in the critical layer near the boundary. The mathematical proof of this conjecture is open.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141190325","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3646-3678, June 2024. Abstract. In the last two decades, there has been significant progress in the understanding of ergodic properties of white-forced dissipative PDEs. The previous studies mostly focus on equations posed on bounded domains since they rely on different compactness properties and the discreteness of the spectrum of the Laplacian. In the present paper, we consider the damped complex Ginzburg–Landau equation on the real line driven by a white-in-time noise. Under the assumption that the noise is sufficiently nondegenerate, we establish the uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is based on coupling techniques combined with a generalization of Foiaş–Prodi estimate to the case of the real line and special space-time weighted estimates, which help to handle the behavior of solutions at infinity.
{"title":"Exponential Mixing for the White-Forced Complex Ginzburg–Landau Equation in the Whole Space","authors":"Vahagn Nersesyan, Meng Zhao","doi":"10.1137/23m1597150","DOIUrl":"https://doi.org/10.1137/23m1597150","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3646-3678, June 2024. <br/> Abstract. In the last two decades, there has been significant progress in the understanding of ergodic properties of white-forced dissipative PDEs. The previous studies mostly focus on equations posed on bounded domains since they rely on different compactness properties and the discreteness of the spectrum of the Laplacian. In the present paper, we consider the damped complex Ginzburg–Landau equation on the real line driven by a white-in-time noise. Under the assumption that the noise is sufficiently nondegenerate, we establish the uniqueness of stationary measure and exponential mixing in the dual-Lipschitz metric. The proof is based on coupling techniques combined with a generalization of Foiaş–Prodi estimate to the case of the real line and special space-time weighted estimates, which help to handle the behavior of solutions at infinity.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141549362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3605-3645, June 2024. Abstract. In this paper, we revisit the Melnikov’s persistency problem and illustrate that the Craig–Wayne–Bourgain method can be strengthened to obtain both the existence and linear stability of the invariant tori. The proof is free from the second Melnikov’s condition.
{"title":"On Linear Stability of KAM Tori via the Craig–Wayne–Bourgain Method","authors":"Xiaolong He, Jia Shi, Yunfeng Shi, Xiaoping Yuan","doi":"10.1137/22m1512958","DOIUrl":"https://doi.org/10.1137/22m1512958","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3605-3645, June 2024. <br/> Abstract. In this paper, we revisit the Melnikov’s persistency problem and illustrate that the Craig–Wayne–Bourgain method can be strengthened to obtain both the existence and linear stability of the invariant tori. The proof is free from the second Melnikov’s condition.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141152188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3588-3604, June 2024. Abstract. We investigate a linearized Calderón problem in a two-dimensional bounded simply connected [math] domain [math]. After extending the linearized problem for [math] perturbations, we orthogonally decompose [math] and prove Lipschitz stability on each of the infinite-dimensional [math] subspaces. In particular, [math] is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearized) Calderón problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmic type in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert–Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fréchet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general [math] perturbation onto the [math] spaces, hence reconstructing any [math] perturbation.
{"title":"Linearized Calderón Problem: Reconstruction and Lipschitz Stability for Infinite-Dimensional Spaces of Unbounded Perturbations","authors":"Henrik Garde, Nuutti Hyvönen","doi":"10.1137/23m1609270","DOIUrl":"https://doi.org/10.1137/23m1609270","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3588-3604, June 2024. <br/>Abstract. We investigate a linearized Calderón problem in a two-dimensional bounded simply connected [math] domain [math]. After extending the linearized problem for [math] perturbations, we orthogonally decompose [math] and prove Lipschitz stability on each of the infinite-dimensional [math] subspaces. In particular, [math] is the space of square-integrable harmonic perturbations. This appears to be the first Lipschitz stability result for infinite-dimensional spaces of perturbations in the context of the (linearized) Calderón problem. Previous optimal estimates with respect to the operator norm of the data map have been of the logarithmic type in infinite-dimensional settings. The remarkable improvement is enabled by using the Hilbert–Schmidt norm for the Neumann-to-Dirichlet boundary map and its Fréchet derivative with respect to the conductivity coefficient. We also derive a direct reconstruction method that inductively yields the orthogonal projections of a general [math] perturbation onto the [math] spaces, hence reconstructing any [math] perturbation.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141154001","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3559-3587, June 2024. Abstract. We investigate the retrieval of a binary time-frequency mask from a few observations of filtered white ambient noise. Confirming household wisdom in acoustic modeling, we show that this is possible by inspecting the average spectrogram of ambient noise. Specifically, we show that the lower quantile of the average of [math] masked spectrograms is enough to identify a rather general mask [math] with confidence at least [math], up to shape details concentrated near the boundary of [math]. As an application, the expected measure of the estimation error is dominated by the perimeter of the time-frequency mask. The estimator requires no knowledge of the noise variance, and only a very qualitative profile of the filtering window, but no exact knowledge of it.
{"title":"Estimation of Binary Time-Frequency Masks from Ambient Noise","authors":"José Luis Romero, Michael Speckbacher","doi":"10.1137/22m149805x","DOIUrl":"https://doi.org/10.1137/22m149805x","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3559-3587, June 2024. <br/> Abstract. We investigate the retrieval of a binary time-frequency mask from a few observations of filtered white ambient noise. Confirming household wisdom in acoustic modeling, we show that this is possible by inspecting the average spectrogram of ambient noise. Specifically, we show that the lower quantile of the average of [math] masked spectrograms is enough to identify a rather general mask [math] with confidence at least [math], up to shape details concentrated near the boundary of [math]. As an application, the expected measure of the estimation error is dominated by the perimeter of the time-frequency mask. The estimator requires no knowledge of the noise variance, and only a very qualitative profile of the filtering window, but no exact knowledge of it.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3537-3558, June 2024. Abstract. A new set of Hadamard series expansions for the Airy functions, [math] and [math], is presented. Previous Hadamard expansions were defined in terms of an infinite number of integration path subdivisions. Unlike the earlier expansions, the expansions in the present work originate in the splitting of the steepest descent into a number of segments that is not only finite but very small, and these segments are defined on the basis of the location of the branch points. One of the segments reaches to infinity, and this gives rise to the presence of upper incomplete gamma functions. This is one of the most important differences from the Hadamard series as defined in the work of R.B. Paris, where all the incomplete gamma functions are of the lower type. The interest of the new series expansion is twofold. First, it shows how to convert an asymptotic series into a convergent one with a finite splitting of the steepest descent path in a process that can be named “exactification.” Second, the inverse of the phase function that is part of the Laplace-type integral is Taylor-expanded around branch points to produce Puiseux series when necessary. Regarding their computational application, these series expansions require a relatively small number of terms for each of them to reach a very high precision.
{"title":"Extended Hadamard Expansions for the Airy Functions","authors":"Jose Luis Alvarez-Perez","doi":"10.1137/23m1599884","DOIUrl":"https://doi.org/10.1137/23m1599884","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3537-3558, June 2024. <br/>Abstract. A new set of Hadamard series expansions for the Airy functions, [math] and [math], is presented. Previous Hadamard expansions were defined in terms of an infinite number of integration path subdivisions. Unlike the earlier expansions, the expansions in the present work originate in the splitting of the steepest descent into a number of segments that is not only finite but very small, and these segments are defined on the basis of the location of the branch points. One of the segments reaches to infinity, and this gives rise to the presence of upper incomplete gamma functions. This is one of the most important differences from the Hadamard series as defined in the work of R.B. Paris, where all the incomplete gamma functions are of the lower type. The interest of the new series expansion is twofold. First, it shows how to convert an asymptotic series into a convergent one with a finite splitting of the steepest descent path in a process that can be named “exactification.” Second, the inverse of the phase function that is part of the Laplace-type integral is Taylor-expanded around branch points to produce Puiseux series when necessary. Regarding their computational application, these series expansions require a relatively small number of terms for each of them to reach a very high precision.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062566","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
P. Acampora, E. Cristoforoni, C. Nitsch, C. Trombetti
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3509-3536, June 2024. Abstract. We are interested in the thermal insulation of a bounded open set [math] surrounded by a set whose thickness is locally described by [math], where [math] is a nonnegative function defined on the boundary [math]. We study the problem in the limit for [math] going to zero using a first-order asymptotic development by [math]-convergence.
{"title":"On the Optimal Shape of a Thin Insulating Layer","authors":"P. Acampora, E. Cristoforoni, C. Nitsch, C. Trombetti","doi":"10.1137/23m1572544","DOIUrl":"https://doi.org/10.1137/23m1572544","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3509-3536, June 2024. <br/> Abstract. We are interested in the thermal insulation of a bounded open set [math] surrounded by a set whose thickness is locally described by [math], where [math] is a nonnegative function defined on the boundary [math]. We study the problem in the limit for [math] going to zero using a first-order asymptotic development by [math]-convergence.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062564","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3478-3508, June 2024. Abstract. In this paper, we study the temperature distribution of a body when the heat is transmitted only by radiation. The heat transmitted by convection and conduction is ignored. We consider the stationary radiative transfer equation in local thermodynamic equilibrium. We prove that the stationary radiative transfer equation coupled with the nonlocal temperature equation is well-posed in general geometries when emission-absorption or scattering of interacting radiation is considered. The emission-absorption and the scattering coefficients are assumed to be general, and they can depend on the frequency of radiation. We also establish an entropy production formula of the system which is used to prove the uniqueness of solutions for an incoming radiation with constant temperature.
{"title":"On the Temperature Distribution of a Body Heated by Radiation","authors":"Jin Woo Jang, Juan J. L. Velázquez","doi":"10.1137/23m1580917","DOIUrl":"https://doi.org/10.1137/23m1580917","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3478-3508, June 2024. <br/> Abstract. In this paper, we study the temperature distribution of a body when the heat is transmitted only by radiation. The heat transmitted by convection and conduction is ignored. We consider the stationary radiative transfer equation in local thermodynamic equilibrium. We prove that the stationary radiative transfer equation coupled with the nonlocal temperature equation is well-posed in general geometries when emission-absorption or scattering of interacting radiation is considered. The emission-absorption and the scattering coefficients are assumed to be general, and they can depend on the frequency of radiation. We also establish an entropy production formula of the system which is used to prove the uniqueness of solutions for an incoming radiation with constant temperature.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062561","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yue-Hong Feng, Haifeng Hu, Ming Mei, Yue-Jun Peng, Guo-Jing Zhang
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3452-3477, June 2024. Abstract. This paper concerns the relaxation time limits for the one-dimensional steady hydrodynamic model of semiconductors in the form of Euler–Poisson equations with sonic or nonsonic boundary. The sonic boundary is the critical and difficult case because of the degeneracy at the boundary and the formation of the boundary layers. In order to avoid the degeneracy of the second order derivatives, we technically introduce an invertible transform to the working equation. This guarantees that the remaining one order degeneracy becomes a good term since the transform used here is strictly increasing. Then we efficiently overcome the degenerate effect. When the relaxation time [math], we first show the strong convergence of the approximate solutions to their asymptotic profiles in [math] norm with the order [math]. When [math], the boundary layer appears because the boundary data are not equal to each other, and we further derive the uniform error estimates of the approximate solutions to their background profiles in [math] norm with the order [math] or [math] according to the different cases of boundary data. Unlike the methods adopted in the previous studies, we propose some altogether new techniques of the asymptotic limit analysis to successfully describe the width of the boundary layer, which is almost the order [math] provided [math]. These original approaches develop and improve the existing studies. Finally, some numerical simulations are carried out, which confirm our theoretical study, in particular, the appearance of boundary layers.
{"title":"Relaxation Time Limits of Subsonic Steady States for Hydrodynamic Model of Semiconductors with Sonic or Nonsonic Boundary","authors":"Yue-Hong Feng, Haifeng Hu, Ming Mei, Yue-Jun Peng, Guo-Jing Zhang","doi":"10.1137/23m1607490","DOIUrl":"https://doi.org/10.1137/23m1607490","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3452-3477, June 2024. <br/> Abstract. This paper concerns the relaxation time limits for the one-dimensional steady hydrodynamic model of semiconductors in the form of Euler–Poisson equations with sonic or nonsonic boundary. The sonic boundary is the critical and difficult case because of the degeneracy at the boundary and the formation of the boundary layers. In order to avoid the degeneracy of the second order derivatives, we technically introduce an invertible transform to the working equation. This guarantees that the remaining one order degeneracy becomes a good term since the transform used here is strictly increasing. Then we efficiently overcome the degenerate effect. When the relaxation time [math], we first show the strong convergence of the approximate solutions to their asymptotic profiles in [math] norm with the order [math]. When [math], the boundary layer appears because the boundary data are not equal to each other, and we further derive the uniform error estimates of the approximate solutions to their background profiles in [math] norm with the order [math] or [math] according to the different cases of boundary data. Unlike the methods adopted in the previous studies, we propose some altogether new techniques of the asymptotic limit analysis to successfully describe the width of the boundary layer, which is almost the order [math] provided [math]. These original approaches develop and improve the existing studies. Finally, some numerical simulations are carried out, which confirm our theoretical study, in particular, the appearance of boundary layers.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3430-3451, June 2024. Abstract. We consider sequences of quadratic nonlocal functionals, depending on a small parameter [math], that approximate the Dirichlet integral by a well-known result by Bourgain, Brezis, and Mironescu. Similarly to what is done for core-radius approximations to vortex energies in the case of the Dirichlet integral, we further scale such energies by [math] and restrict them to [math]-valued functions. We introduce a notion of convergence of functions to integral currents with respect to which such energies are equicoercive, and show the convergence to a vortex energy, similarly to the limit behavior of Ginzburg–Landau energies at the vortex scaling.
{"title":"Nonlocal-Interaction Vortices","authors":"Margherita Solci","doi":"10.1137/23m1563438","DOIUrl":"https://doi.org/10.1137/23m1563438","url":null,"abstract":"SIAM Journal on Mathematical Analysis, Volume 56, Issue 3, Page 3430-3451, June 2024. <br/> Abstract. We consider sequences of quadratic nonlocal functionals, depending on a small parameter [math], that approximate the Dirichlet integral by a well-known result by Bourgain, Brezis, and Mironescu. Similarly to what is done for core-radius approximations to vortex energies in the case of the Dirichlet integral, we further scale such energies by [math] and restrict them to [math]-valued functions. We introduce a notion of convergence of functions to integral currents with respect to which such energies are equicoercive, and show the convergence to a vortex energy, similarly to the limit behavior of Ginzburg–Landau energies at the vortex scaling.","PeriodicalId":51150,"journal":{"name":"SIAM Journal on Mathematical Analysis","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-05-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140936746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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