Abstract
We consider the scattering problem of line source electromagnetic waves using a multi-layered obstacle with a core, which may be a perfect conductor, a dielectric, or has an impedance surface. We formulate this problem in two dimensions and we prove some useful scattering relations. In particular, we state and prove a reciprocity principle and a general scattering theorem for line source waves for any possible positions of the source. These theorems can be used to approximate the far-field pattern in the low-frequency theory. Moreover, an optical theorem is recovered as a corollary of the general scattering theorem. Finally, we obtain a mixed reciprocity relation which can be used in proving the uniqueness results of the inverse scattering problems.
Keywords:
two-dimensional electromagnetic scattering; piecewise obstacle; line source waves; scattering relations MSC:
35P25; 45B05; 47A40
1. Introduction
This paper is concerned with relations for the time-harmonic electromagnetic scattering of a multi-layered scatterer. An obstacle of this type is made up of a nested body consisting of a finite number of homogeneous layers. On the surfaces of the layers the dielectric transmission conditions are imposed. The scatterer’s core may be a perfect conductor, a dielectric, or has an impedance surface. In the time-harmonic scattering theory [] some theorems between the scattered fields of two problems which correspond to two distinct incident waves for the same scatterer appear. The incident waves can have a plane or line source. These properties are referred to as scattering theorems. In this work, we state and prove some scattering theorems and, in particular, a reciprocity, a general, a mixed reciprocity, and an optical theorem for a multi-layered scatterer which is excited by line source waves.
The reciprocity principle for line sources connects the total fields in two layers. The total field at due to a source at relates the total field at due to a source at . A mixed reciprocity theorem associates the scattered field corresponding to a plane wave incident on a scatterer with the far-field pattern corresponding to a line source. The general theorem connects the far-field pattern generators (which are defined in [] Formula (15)) at and . The optical theorem for a plane incident field with direction of propagation states the total energy (the sum of the scattered and the absorption cross-section) that the obstacle removes for the incident field is proportional to the far-field pattern, which is valued in . In case of line source waves, the far-field pattern is replaced by a far-field pattern generator. This theorem is obtained from the general theorem for .
Using the reciprocity theorem, we can prove that the far-field operator corresponding to particular scattering problems is injective, with a dense range. These properties lead to a solution for the inverse scattering problems, Refs. [,,]. The general scattering theorem is utilized in order to found the eigenvalues and eigenfunctions of the far-field operator and then to solve the inverse scattering problem using the linear sampling method or the factorization method [,,]. The optical theorem can be used to evaluate the energy scattered by an obstacle, Refs. [,]. It is worth mentioning that these scattering relations can be mainly used in solving inverse scattering problems. In [] near-field inverse scattering problems for ellipsoid scatterers have been solved.
A multi-layered scatterer for acoustic waves in three dimensions is described in []. Three-dimensional scattering relations have been proved for various kinds of scatterers. Indicatively, we refer to [] for electromagnetic and [] for elastic waves. Reciprocity, scattering, and optical theorems for acoustic and electromagnetic waves have been proven by Twersky in [,,]. In view of these results, and by applying the low-frequency theory, he calculated an approximation of the real part of the far-field pattern. Reciprocity theorems for acoustic and electromagnetic waves are contained in the books [,] by Colton and Kress. In the book [] written by Dassios and Kleinman, reciprocity, general, and optical theorems have also been recorded. In [], Angell et al. have proven a reciprocity relation for electromagnetic scattering corresponding to a scatterer with an impedance boundary. Recently, a general optical theorem for an arbitrary multipole excitation is proven in [] and scattering cross sections are evaluated in [,]. Corresponding theorems for elastic waves have been proven in [] by using spherical coordinates.
Furthermore, in the case of three-dimensional multi-layered scatterers, scattering relations have been proven for acoustic waves in []. Scattering theorems for spherical waves have been proven in [] (acoustic and electromagnetic). Moreover, in [], a mixed reciprocity relation has been proven. In [] Potthast has studied an inverse acoustic scattering problem using a mixed reciprocity relation for a piecewise constant inhomogeneous medium.
Following the procedure which is described by Colton and Cakoni in [], the three dimensional electromagnetic scattering problem is reduced to a system of Helmholtz equations for a two-dimensional multi-layered scatterer. This kind of scatterer appears when a multi-layered infinitely long cylinder (all the layers have a common axis) is intersected vertically by a plane. Two-dimensional scattering theorems have been proven in [,]. In [], plane electromagnetic scattering is studied, whereas in [] line source elastic waves are studied. In [], a mixed reciprocity principle is proven in two dimensions.
In the majority of the papers, the source which generates the incident wave is located in the exterior of the scatterer. In cases where the source is located in the interior of the scatterer, there are some important applications. Specifically, the detection of buried objects in a layered background medium requires the solution of an inverse scattering problem in which the source is located in an interior layer. Moreover, for the study of seismic waves, it is necessary to put the sources inside the layered media. In [], inversion algorithms for determining the geometrical and physical characteristics of a spherical scatterer, which is excited by a line source, are developed. In [] an inverse scattering problem for an object which is buried in a layered medium is studied. There are also applications in medicine, Ref. [], such as implantation inside the human head for hyperthermia or biotelemetry purposes, Ref. [], results (scattering theorems) are used for a scatterer, which is either a layered ellipsoid or sphere with internal sources.
First, in Section 2, we formulate two-dimensional scattering problems for a multi-layered scatterer for three different imposed boundary conditions on its core. In Section 3, we state and prove a two-dimensional reciprocity principle in the case of line source waves. Defining the line source generator, a general scattering theorem in classical form is proven in Section 4. In both Section 3 and Section 4, a multi-layered scatterer is excited by two line source waves located at any two layers. In Section 5, a mixed theorem is proven when the scatterer is excited by a line source and a plane wave. In Section 6, we formulate an optical theorem for the corresponding scattering problems.
2. Formulation
We consider electromagnetic scattering by a piecewise homogeneous obstacle in with a -boundary . The exterior of the obstacle is an infinite homogeneous isotropic medium with electric permittivity , magnetic permeability , and vanishing conductivity. The interior of is divided by means of closed and nonintersecting -surfaces , , into layers , , with . The surface surrounds and there is one normal unit vector at each point of any surface pointing into . The region , within which lies the origin, is the core of the obstacle, which may be a dielectric, a perfect conductor, or an imperfect conductor. The layer is a homogeneous isotropic medium with electric permittivity , magnetic permeability , and vanishing conductivity. All the physical parameters and are positive constants. The geometry of the electromagnetic scattering problem in is presented in Figure 1.

Figure 1.
The geometry of the electromagnetic scattering problem in .
The total electric and magnetic fields and in satisfy the time-harmonic Maxwell equations [],
where is the angular frequency.
The corresponding scattered electromagnetic field , satisfies the Silver–Müller radiation condition
uniformly in all directions . The total electromagnetic field , in satisfies
where , is the incident wave field. All the fields , , , , , are divergence-free.
The transmission conditions on the surface for are
On the surface of the core, we impose either the perfect conductor boundary condition
or the dielectric transmission conditions
or the impedance boundary condition
where is the surface impedance.
In the present work, we are interested in proving scattering relations that both play an important role in studying inverse scattering problems, see Ref. []. Specifically, the general scattering theorem is applied to the computation of the far-field pattern in the low-frequency theory (see []). The study of these relations will be done in two dimensions for the line source waves of the following scattering problems. The first problem, which corresponds to the perfect conductor core, is defined by Equations (1)–(6). The second one, which corresponds to the dielectric transmission condition of the core, is defined by Equations (1)–(5), (7) and (8). The third one, which corresponds to the impedance boundary condition of the core, is defined by Equations (1)–(5) and (9).
We suppose that is an infinitely long cylinder which is oriented parallel to the -axis. The cross-section D of in the -plane will be referred to as a two-dimensional scatterer. We also suppose that , , are infinitely long cylindrical surfaces. The -plane intersects the surfaces at the -curves , see Figure 2.

Figure 2.
The geometry of the multi-layered scatterer in .
Following the process specified by Cakoni and Colton in Ref. [] (pp. 47 and 83), for the electric fields, we make the additional assumption that , where is the total field in . The incident field can be either a plane wave
where is the wave number in and is the incident direction with , [], or a line source wave, [],
where is the zero order Hankel function of the first kind. Using these assumptions and applying the Maxwell Equation (1), we obtain , , and the following equations in
Equations (12) and (13) imply that the -component of satisfies the Helmholtz equation
for for the perfect conductor or the impedance boundary condition on the core core and for the dielectric core. The symbol stands for the two-dimensional Laplace operator. The -component of the total electric field in is given by
where is the -component of the scattered field . Specifically, , satisfies the Sommerfeld radiation condition
The transmission condition Equations (4) and (5) are written in terms of the -component of the total electric field , , as
The boundary condition Equation (6) for the perfect conductor core is reformulated as
The transmission condition Equations (7) and (8) for the dielectric conductor are rewritten as
and for the impedance boundary condition Equation (9), we have
Summarizing the above analysis, we formulate the following three two-dimensional scattering problems. The first problem is defined by Equations (14)–(19) and is denoted by , the second one is defined by Equations (14)–(18), (20) and (21) and is denoted by , and the third one is defined by Equations (14)–(18) and (22) and is denoted by . The well-posedness of these problems can be obtained by extending the techniques of [].
In all the above problems, we assume that the multi-layered scatterer is excited by two line source waves located at any two layers. In case of the mixed reciprocity, the scatterer is excited by a line source and a plane wave.
We will denote by , and the dependence of the scattered field, the far-field pattern, and the total field in on the incident direction . For incident line source wave at , we utilize the fundamental solution
where is the zero-order Hankel function of the first kind. We will denote by , , and to represent the dependence of the total field in , , the scattered field, and the far-field pattern on the position of the source .
The scattered line source field has the following asymptotic behaviour in two dimensions
uniformly in all directions []. The far-field pattern is given by [],
In the rest of the paper, we will use Twersky’s notation [],
As we can see, the far-field pattern is expressed through the Twersky’s notation as
3. Reciprocity for Line Source Waves
For incident plane waves, similar scattering relations have already been proven in [] for three dimensions and in [] for two dimensions. Now, we will emphasize proving the corresponding relations for line source waves in two dimensions. For this purpose, we consider two line source waves at and , with . Then, we prove the following lemma.
Lemma 1.
Let be an incident line source wave at . Let be an incident line source wave at with corresponding scattered field and total field in . Then, it holds
where is a disc centered at with radius , is a disc centered at with radius , and is a disc centered at the origin with radius R surrounding the points and .
Proof.
Since and are solutions of the Helmholtz equation for and satisfy the Sommerfeld radiation condition, we obtain relation Equation (28).
Using the asymptotic forms, Ref. [],
as , applying the mean value theorem and letting we obtain relation Equation (29).
For relation Equation (30), we replace and we take
The first integral of the right-hand side of Equation (33) is equal to , due to relation Equation (29). For the last integral, taking into account the interior elliptic regularity, see Refs. [,,], we have
and by using the Cauchy–Schwarz inequality we conclude that
□
Remark 1.
Relation Equation (29) is also valid if is replaced by . In general, can be replaced by bounded fields.
Next, we prove the following reciprocity theorem for line source waves and multi-layered scatterers in two dimensions.
Theorem 1.
Let and be two incident line source waves at and , respectively. Then the corresponding total fields and for the scattering problem , , satisfy the reciprocity relation
Proof.
Due to the bilinearity of Equation (26), and taking into account tion , we have
We initially consider the case , i.e., the two sources are outside of the scatterer. For the first integral of the right-hand side of relation Equation (37), since and are regular solutions of the Helmholtz equation in , an application of the scalar Green’s second theorem gives
For the computation of the next integral, we consider two small disjoint discs and centered at and with radius and , respectively. We also consider a large disc centered at the origin with radius R surrounding the points , and the scatterer.
Since and are solutions of the Helmholtz equation for , we apply the scalar Green’s second theorem and obtain
Letting , , using Lemma 1, and taking into account that the last integral of Equation (39) is zero, due to the application of the scalar Green’s second theorem in and since and are regular solutions of the Helmholtz equations for , we obtain
Similarly, we obtain
Since , are radiating solutions of the Helmholtz equation in , we have
The total fields and , , are regular solutions of the Helmholtz equation in . Thus, the successive application of the scalar Green’s second theorem and the use of the transmission conditions on imply that
By using the imposed boundary conditions on for and , we directly obtain that the integral is equal to zero. For , we apply the transmission conditions, we use the scalar Green’s second theorem, and we obtain
From definition Equation (23) of the line source, we see that , and due to relation Equation (45), we have
Next, we consider the case , i.e., one source is inside and the other one is outside the scatterer.
Taking into account that , are solutions of the Helmholtz equation in D for and by using Lemma 1, we obtain
For the next integral of Equation (37), working as in the previous case, we obtain
Moreover, it holds
since the involved functions are radiating solutions of the Helmholtz equation in . For the total fields we have
The first integral of the right-hand side is equal to zero due to the imposed boundary and transmission conditions on the surface of the core. Applying Lemma 1, we obtain
We finally consider the case , i.e., both of the sources are inside the scatterer.
It holds
since and are radiating solutions of the Helmholtz equation. Moreover, and are radiating solutions of the Helmholtz equation, and thus we obtain
Concerning the total fields, by successively applying the scalar Green’s second theorem, we have
Due to the boundary and the transmission conditions on the core and via Lemma 1, we obtain
Remark 2.
If the positions of the line source waves are located at the same layer or both of them are outside the scatterer, i.e., , then the reciprocity relation Equation (36) is reformulated as
4. A General Scattering Theorem
As in the reciprocity theorem, we suppose that the scatterer D is excited by two line source waves. In what follows, denotes the complex conjugate of w and denotes the unit disc in .
Lemma 2.
Let be an incident line source wave at . For an incident line source wave at with corresponding scattered field and far-field pattern , we have
where is a disc centered at with radius , is a disc centered at with radius and is a disc centered at the origin with radius R surrounding the points and .
Proof.
In order to formulate a two-dimensional general scattering theorem for line source waves in the classical form, see for example the book [], we introduce a line source far-field pattern generator , [], as follows
By using this generator we are able to prove the following general scattering theorem in two dimensions.
Theorem 2.
Let and be two incident line source waves at and , respectively. Then for the scattering problem , , we have
where
and
Proof.
As in the proof of the reciprocity theorem, we use the following analysis:
For , the incident fields and are regular solutions of the Helmholtz equation in D. We apply the scalar Green’s second theorem in D and we take
As before, we consider the discs and and applying again the scalar Green’s second theorem, we obtain
The last integral of relation Equation (72) is equal to zero since and are regular solutions of the Helmholtz equation in . Letting , and by using Lemma 2, we obtain
Moreover, we have
Similarly, taking into account asymptotic relation Equation (24), we have
The total fields and , , are regular solutions of the Helmholtz equation in . Thus, the successive application of the scalar Green’s second theorem and the use of the transmission conditions on imply that
By using the boundary and transmission conditions on and applying the scalar Green’s second theorem, we obtain
where for (), () and for (). Therefore, we obtain
Substituting relation Equations (71), (73)–(75) and (78) in Equation (70), we take relation Equation (67).
For the case , taking into account that , are solutions of the Helmholtz equation in D for and by using Lemma 2, we obtain
The computation of the next two integrals can be done as before and we have
and
Moreover, it holds
For the total fields, we successively apply the scalar Green’s second theorem and we have
For the second term we use Lemma 2, we obtain
If both of the sources are in the interior of the scatterer, i.e., , then, taking into account Lemma 2, we obtain
Similarly, we consider the large disc which contains the scatterer and we apply the scalar Green’s second theorem on the space between and and by using Lemma 2, we have
Moreover, we have
Since and are regular solutions of the Helmholtz equation in and using Equation (24), we obtain
Using the transmission conditions, successively applying the scalar Green’s second theorem, considering the small discs and , and taking into account Lemma 2, we have
5. Mixed Reciprocity
A mixed reciprocity theorem, i.e., a theorem which connects the far-field pattern of a line source wave and the scattered field of a plane wave, will be proven in this section. This theorem plays a crucial role in studying inverse scattering problems [,,].
Theorem 3.
Let be an incident line source wave at and let be an incident plane wave with propagation direction . Then,
Proof.
Taking into account that and and due to the bilinearity of Equation (26), we again obtain the following analysis
Let . Then, and are regular solutions of the Helmholtz equation in D and, thus, the scalar Green’s second theorem gives
For the next integral of the right-hand side of Equation (91), we consider a small disc centered at with radius and a large disc centered at the origin with radius R surrounding the scatterer and the small disc . Applying the scalar Green’s second theorem for , and in the space between the curves , and , we obtain
Letting and , using Lemma 1, and taking into account that , are radiating solutions of the Helmholtz equation, we take
From the definition of the far-field pattern Equation (27), we have
The scattered fields are regular solutions of the Helmholtz equation which satisfy the Sommerfeld radiation condition and thus
The total fields and are regular solutions of the Helmholtz equation in , . Successively applying the scalar Green’s second theorem, using the transmission and boundary conditions, we obtain
Let . In this case, the source is in the interior of the scatterer and specifically in the layer . We consider the small disc , we use Lemma 1, and working as before we find
Since and , are radiating solutions of the Helmholtz equation, we have
As before, it holds
and
6. The Optical Theorem
As is well-known, the scattering cross-section constitutes a measure of the disturbance caused by the scatterer to the incident wave. For a line source wave at , we define, Ref. [],
Moreover, we define the absorption cross-section , given by
which expresses the total energy absorbed by the scatterer. In particular, the energy which is taken from the incident line source wave is adsorbed by the boundary of the scatterer in the impedance case.
Taking into account relation Equation (78), we have
Moreover, the extinction cross-section is defined by
and it describes the total power that the scatterer extracts from the incident line source wave either by radiation or by absorption. We are now in the position to prove an optical theorem in two dimensions.
Theorem 4.
Let be an incident line source wave and be the corresponding far-field pattern. Then, the extinction cross-section satisfies
7. A Special Case
We consider a dielectric with a perfect conductor core. In this case, the scattering theorems are simplified as follows. Corresponding relations can be found in [,,,,,].
- Reciprocity theorem:
- General scattering theorem:
- Mixed reciprocity theorem:
- Optical theorem:
8. Conclusions
In this work, we proved two-dimensional scattering relations for line source incident waves. We aim to use the derived results in a future work. Specifically, we are interested in computing the coefficients of the scattered field in low-frequency approximation theory and then we will be able to solve inverse scattering problems, which are referred to as spherical or ellipsoid scatterers. This can be mainly done through the general scattering theorem, see Ref. []. By using the mixed reciprocity theorem, we will study inverse scattering problems for multi-layered scatterers applying the Potthast line source method [,,].
Author Contributions
Conceptualization, C.E.A.; Methodology, P.R.; Validation, C.E.A. and P.R.; Formal analysis, C.E.A. and P.R.; Writing—original draft, P.R.; Supervision, C.E.A. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Not applicable.
Conflicts of Interest
The authors declare no conflict of interest.
References
- Dassios, G.; Kleinman, R. Low Frequency Scattering; Oxford University Press, Clarenton Press: Oxford, UK, 2000. [Google Scholar]
- Athanasiadis, C.; Martin, P.A.; Spyropoulos, A.; Stratis, I.G. Scattering relations for point sources: Acoustic and electromagnetic waves. J. Math. Phys. 2002, 43, 5683–5697. [Google Scholar] [CrossRef]
- Cakoni, F.; Colton, D. Qualitative Methods in Inverse Electromagnetic Scattering Theory; Springer: Berlin, Germany, 2005. [Google Scholar]
- Colton, D.; Kress, R. Inverse Acoustic and Electromagnetic Scattering Theory, 3rd ed.; Springer: New York, NY, USA, 2013. [Google Scholar]
- Liu, X.; Zhang, B.; Hu, G. Uniqueness in the inverse scattering problem in a piecewise homogeneous medium. Inverse Probl. 2010, 26, 015002. [Google Scholar] [CrossRef]
- Cakoni, F.; M’Barek, Fares; Haddar, H. Analysis of two linear sampling methods applied to electromagnetic imaging of buried objects. Inverse Probl. 2006, 22, 845–867. [Google Scholar] [CrossRef]
- Eremin, Y.A.; Wriedt, T. Generalization of the Optical Theorem to an Arbitrary Multipole Excitation of a Particle near a Transparent Substrate. Mathematics 2021, 9, 3244. [Google Scholar] [CrossRef]
- Athanasiadis, C.E.; Athanasiadou, E.S.; Zoi, S.; Arkoudis, I. Near-field inverse electromagnetic scattering problems for ellipsoids. Z. Angew. Math. Phys. 2019, 70, 73. [Google Scholar] [CrossRef]
- Athanasiadis, C.; Tsitsas, N. Scattering theorems for acoustic excitation of a layered obstacle by an interior point source. Stud. Appl. Math. 2007, 118, 397–418. [Google Scholar] [CrossRef]
- Dassios, G.; Kiriaki, K.; Polysos, D. On the scattering amplitudes for elastic waves. Z. Angew. Math. Phys. 1987, 38, 856–873. [Google Scholar] [CrossRef]
- Twersky, V. Certain transmissions and reflection theorems. J. Appl. Phys. 1954, 25, 859–862. [Google Scholar] [CrossRef]
- Twersky, V. On a general class of scattering problems. J. Math. Phys. 1962, 3, 716–723. [Google Scholar] [CrossRef]
- Twersky, V. Multiple scattering of electromagnetic waves by arbitrary configurations. J. Math. Phys. 1967, 8, 589–598. [Google Scholar] [CrossRef]
- Colton, D.; Kress, R. Integral Equation Methods in Scattering Theory; SIAM: Philadelphia, PA, USA, 2013. [Google Scholar]
- Angell, T.S.; Colton, D.; Kress, R. Far field patterns and inverse scattering problems for imperfectly conducting obstacles. Math. Proc. Camb. Phil. Soc. 1989, 106, 553–569. [Google Scholar] [CrossRef]
- Kalogeropoulos, A.; Tsitsas, N.L. Electromagnetic interactions of dipole distributions with a stratified medium: Power fluxes and scattering cross sections. Stud. Appl. Math. 2021, 148, 1040–1068. [Google Scholar] [CrossRef]
- Martin, P.A. Multiple scattering and scattering cross sections. J. Acoust. Soc. Am. 2018, 143, 995–1002. [Google Scholar] [CrossRef]
- Potthast, R. A new non-iterative singular sources method for the reconstruction of piecewise constant media. Numer. Math. 2004, 98, 703–730. [Google Scholar] [CrossRef]
- Athanasiadis, C.E.; Athanasiadou, E.S.; Roupa, P. On the far field patterns for electromagnetic scattering in two dimensions. Rep. Math. Phys. 2022, 89, 253–265. [Google Scholar] [CrossRef]
- Athanasiadis, C.E.; Athanasiadou, E.S.; Roupa, P. Scattering Relations for Two-Dimensional Electromagnetic Waves in Chiral Media. J. Appl. Math. Phys. 2022, 10, 1200–1216. [Google Scholar] [CrossRef]
- Athanasiadis, C.; Sevroglou, V.; Stratis, I.G. Scattering relations for point-generated dyadic fields in two-dimensional linear elasticity. Q. Appl. Math. 2006, 64, 695–710. [Google Scholar] [CrossRef]
- Guo, J.; Yan, G.; Cai, M. Multilayered Scattering Problem with Generalized Impedance Boundary Condition on the Core. J. Appl. Math. 2015, 2015, 195460. [Google Scholar] [CrossRef]
- Tsitsas, N.L.; Martin, P.A. Finding a source inside a sphere. Inverse Probl. 2012, 28, 015003. [Google Scholar] [CrossRef]
- Dassios, G.; Kariotou, F. Magnetoencephalography in ellipsoidal geometry. J. Math. Phys. 2003, 44, 220–241. [Google Scholar] [CrossRef]
- Kim, J.; Rahmat-Samii, Y. Implanted antennas inside a human body: Simulations, designs, and characterizations. IEEE Trans. Microw. Theory Tech. 2004, 52, 1934–1943. [Google Scholar] [CrossRef]
- Lazaridis, D.S.; Tsitsas, N.L. Detecting Line Sources inside Cylinders by Analytical Algorithms. Mathematics 2023, 11, 2935. [Google Scholar] [CrossRef]
- Arens, T. Scattering by Biperiodic Layered Media: The Integral Equation Approach. Habilitation Thesis, Karlsruhe Institute of Technology, Karlsruhe, Germany, 2010. [Google Scholar]
- Evans, L.C. Partial Differential Equations; American Mathematical Society: Providence, RI, USA, 1998. [Google Scholar]
- Morse, P.M.; Feshbach, H. Methods of Theoretical Physics; McGraw-Hill: New York, NY, USA, 1953; Volumes I and II. [Google Scholar]
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