Advances in Mathematical Methods for Electromagnetics
2: Macquarie University, NSW, Australia
This book covers recent achievements in the area of advanced analytical and associated numerical methods as applied to various problems arising in all branches of electromagnetics. The unifying theme is the application of advanced or novel mathematical techniques to produce analytical solutions or effective analyticalnumerical methods for computational electromagnetics addressing more general problems. Each chapter contains an outline of its topic, discusses its scientific context and importance, describes approaches to date, gives an exposition of the author's approach to the problem tackled in the chapter, describes the results, and concludes with a discussion of the range or class of problems where the techniques described work most appropriately and effectively. Intended primarily for researchers in the fields of electrical engineering, mathematics, physics and related disciplines, the book offers systematic and thorough coverage of this complex topic. It is hoped that the book will help to stimulate further investigation and discussion of the important problems in electromagnetics within this research community.
Inspec keywords: computational electromagnetics; electromagnetic wave diffraction; waveguide theory; electromagnetic wave scattering
Other keywords: analytical regularization; dispersive material systems; elongated crosssection; wave equation; electromagnetic surface waves scattering; complex canonical diffraction problems; geophysical media; electromagnetics integral equations; uniform complexsource beams; volume singular integral equations; generalized functions; dielectricwedge Fourier series; line source radiation; electromagnetics mathematical methods; twodimensional arbitrary open cavities; discrete electromagnetics; hybrid raymode techniques; quantum optics; biological media; resonance scattering; Huygens' principle; complex environments electromagnetics; rightangled dielectric wedge; analytical regularization method; inhomogeneous media; farfield pattern; imperfectly conducting canonical bodies; high frequency techniques; sphericalmultipole analysis; circular cylinders arrays; randomly located obstacles; Green's theorem; WienerHopf Fredholm factorization technique; Multiple scattering; function theoretic methods; waveguides scattering; elliptic cylinder; topological photonics; quasiHelmholtz projectors; integral representation theory; WienerHopf analysis; beambased local diffraction tomography; bounded piecewise homogeneous domains
Subjects: Electromagnetic waves: theory; Waveguide and cavity theory; General electrical engineering topics; Textbooks; Electromagnetic wave propagation
 Book DOI: 10.1049/SBEW528E
 Chapter DOI: 10.1049/SBEW528E
 ISBN: 9781785613845
 eISBN: 9781785613852
 Page count: 748
 Format: PDF

Front Matter
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Introduction
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This Chapter provides an Introduction to the book entitled Advances in Mathematical Methods for Electromagnetics. Background information is presented and a summary of each chapter is given. The purpose of this book is to cover recent achievements in the area of advanced analytical and associated numerical methods as applied to problems arising in various branches of electromagnetics. The unifying theme is the application of advanced or novel mathematical techniques to produce analytical solutions (to canonical problems) or effective analyticalnumerical methods for computational electromagnetics addressing more general problems.

1 New insights in integral representation theory for the solution of complex canonical diffraction problems
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The problem of scattering by twodimensional (2D) and threedimensional (3D) canonical objects with imperfectly conducting surfaces requires some particular efforts on the representation of scattered and incident fields, and we present here some remarkable aspects of them in complex situations and their applications. We begin with the study of 2D problems. The SommerfeldMaliuzhinets integral and its inversion in the spectral domain of complex angles has opened a new way of investigation on diffraction by a wedgeshaped domain.

2 Scattering of electromagnetic surface waves on imperfectly conducting canonical bodies
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An imperfectly conducting surface may support surface waves provided appropriate impedance boundary conditions (Leontovich conditions) are satisfied. Electromagnetic surface waves propagate along an impedance surface and interact with its singular points such as edges or conical vertices giving rise to the reflection and transmission of such surface waves as well as to those diffracted into the space surrounding the canonical body. In this work, we discuss a mathematical approach describing some physical processes dealing with the diffraction of surface waves by canonical singularities like wedges and cones. We develop a mathematically justified theory of such processes with the attention centred on diffraction of a skew incident surface wave at the edge of an impedance wedge. Questions of excitation of the electromagnetic surface waves by a Hertzian dipole are also addressed as well as the Geometrical Optics laws of reflection and transmission of a surface wave across the edge of an impedance wedge.

3 Dielectricwedge Fourier series
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Diffraction of a magnetically polarized incident field by a dielectric wedge has been considered. The diffraction problem has been reduced to an integral equation which has been solved by iterations in the form of a converging series. The error of the nthorder iteration has been explicitly calculated. Furthermore, the chapter has introduced a new Fourier series of orthogonal BesselHankel modes. These modes behave as Bessel functions near the tip of the wedge and are outgoing Hankel functions for large values of kr. The scattered magnetic field represented by the Fourier series satisfies the radiation condition at infinity and is finite at the tip of the wedge. The Fourier series has been constructed by a mapping of the far field onto the Fourier series. The error in approximating the exact solution of the diffraction problem by n terms of the Fourier series has been explicitly evaluated. The first term in the Fourier series behaves as the Bessel function Js(kgr) with r^{s} being the static field singularity at the tip of the wedge. The theory has been tested numerically, and examples of field plots have been presented.

4 Green's theorem, Green's functions and Huygens' principle in discrete electromagnetics
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Green's theorem and Green's functions are applied to general dynamical systems on discrete structures in discrete time. The discrete structures may arise from spatial discretisations of continuum fields described by PDEs but are not restricted to this source and may be purely graphtheoretical in origin. The case of Maxwell's equations, spatially discretised by finitedifferences on a simplicial cell complex, is introduced as an example leading to coupled discrete potentials that each satisfies a secondorder discrete dynamical system, one on vertices and the other on edges. The main result is a generic expression (4.95) for the secondorder discretetime Green operator G as a polynomial in the system operator F whose order m is equal to the time step. The discretetime matrix elements G_{tf} (m) in this expression are exactly computable in a finite number of arithmetic steps when the operator F has finite adjacency measure. The typical candidate for F is the discrete Laplacian L, which has adjacency 1. A fully discrete form of Huygens' principle is obtained for this system, which predicts the field in an exterior region D from its values at earlier times in the cutset S consisting of pairs of elements, one in D and one in D, using explicit timestepping. The representation of Huygens' principle in this form is a superposition of expanding wavelets each radiated from an element of the cutset S. While the general setting for the description of the dynamical system is a cell complex with Hodge dual, deletion of the metric Hodge data from the Laplacian weights leaves behind a topological dynamical system, that is essentially exploring the connectivity of a graph in discrete time steps.

5 The concept of generalized functions and universal properties of the Green's functions associated with the wave equation in bounded piecewise homogeneous domains
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The concepts of generalized functions and derivatives in the sense of distribution have many properties which differ from those pertinent to classical functions. Therefore, the question that follows is of crucial importance: Are the differential equations of the classical physics valid in the sense of distribution? It is worthwhile to remark here that an affirmative response to this question does not only mean that the derivatives have to be computed as distributions but also the field components themselves involve singular parts which cannot be expressed in terms of classical functions.

6 Elliptic cylinder with a strongly elongated crosssection: highfrequency techniques and function theoretic methods
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In this chapter, we apply the function theoretic methods developed by Andronov to calculate the asymptotic currents on elliptic cylinders with a strongly elongated crosssection to more general cylindrical configurations where the crosssection is a truncated ellipse or composed of arcs of ellipses. Our objective is to establish the formulas for the asymptotic currents taking into account the diffraction by the edges.

7 Highfrequency hybrid ray–mode techniques
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Rays and modes are two alternative descriptions for electromagnetic field, and they are regarded as Fourier transform pairs that have complementary convergence properties. Accordingly, a modal description would be preferable for an internal structure, while a ray description would be appropriate for an exterior region. When a scatterer contains an open waveguide cavity structure, one wants to retain the fast converging description, namely, rays for the exterior and modes for the interior regions, respectively, and the conversion should be considered at the opening of the structure. This raymode conversion between them may be found from a mathematical formula called the Poisson summation formula, and this technique is found to be one of the powerful tools for analyzing highfrequency scattering problems. Some application examples, such as plane wave diffraction from a slit on a conducting screen and a trough on a ground, are given to show the validity of this technique.

8 Scattering and diffraction of scalar and electromagnetic waves using sphericalmultipole analysis and uniform complexsource beams
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The chapter deals with a sphericalmultipole expansion of scalar or electromagnetic fields in the presence of a semiinfinite elliptic cone and a uniform complexsource beam (CSB) as the incident field. The analysis is performed in spheroconal coordinates that can be understood as generalized spherical coordinates. The corresponding coordinate surfaces include the elliptic cone, the plane angular sector, and the wedge. As the uniform CSB paraxially approximates a Gaussian beam, its waist represents a localized inhomogeneous plane wave. Hence, it is possible to exclusively illuminate a desired area of the structurefor instance the tip of the coneby a localized plane wave. Differently from using a homogeneous plane wave as the incident field in that case the resulting sphericalmultipole series converges even if the scattered far field is evaluated. Therefore, the proposed technique should allow to extract diffraction and scattering characteristics of any desired part of the scattering objects without applying series transformations as it has been found to be necessary in the case of a homogeneous plane wave.

9 Changes in the farfield pattern induced by rounding the corners of a scatterer: dependence upon curvature
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In this chapter, we rigorously examine the differences in the farfield patterns (9.1) for an Epolarised scatterer with a single corner. The scattering of a plane wave by such a scatterer is formulated in Section 9.1, and an appropriate integral equation for the surface distribution on the obstacle is given. As a motivation for the analysis to follow, a brief discussion of numerical results is given in Section 9.2. In Section 9.3, the lemniscate (having a rightangled corner) and its rounded counterpart is used as a test case to establish analytic bounds for the maximum difference in the far field. An integral equation is obtained for the difference in the surface distributions on each obstacle; its approximate solution is shown to be O((kp)^{2/3}), as kp > 0 (Theorem 9.1). It then follows that the nondimensionalised farfield difference √κ∥μ_{ο}μ_{ρ}∥_{∞ }is O((kp)^{4/3}), as kp > 0 (Theorem 9.2).

10 Radiation from a line source at the vertex of a rightangled dielectric wedge
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We have shown that the use of the double WienerHopf technique involving the use of two complex variables and the complicated concomitant problems of factorization in the two complex variables is not necessary for the problem of the diffiaction by a rightangled dielectric wedge. The simpler approach of using directly the double complex Laplace or Fourier transform is sufficient to derive the basic transformed double integral equation that can be approximately solved as a Neumann series solution.

11 Wiener–Hopf analysis of the diffraction by a thin material strip
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The analysis of wave scattering and diffraction problems involving canonical objects is one of the important subjects in electromagnetic theory and radar cross section (RCS) studies. Various analytical and numerical methods have been developed so far and the scattering problems have been investigated for many kinds of two and threedimensional structures. Among a number of analysis methods, the WienerHopf technique is known as a rigorous, functiontheoretic approach for electromagnetic wave problems related to canonical geometries. In this chapter, we shall consider a thin material strip that is important from both the theoretical and engineering viewpoints, and analyze the electromagnetic wave diffraction by means of the Wiener Hopf technique. It is shown that our final solutions are valid over a broad frequency range. Numerical examples are presented for various physical parameters, and the far field scattering characteristics are discussed in detail. Some comparisons with other existing methods are also given.

12 TheWiener–Hopf Fredholm factorization technique to solve scattering problems in coupled planar and angular regions
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The WienerHopf (WH) technique is a very powerful tool in the spectral domain to solve field problems in the presence of discontinuities. This chapter introduces a generalization of this technique that allows to study geometries where coupled planar and angular region are present. Since exact solutions with closed form factorizations are available only in few cases, most of the problems require an alternative approximate technique as the Fredholm factorization. The Fredholm factorization reduces the solution of WH equations to a system of Fredholm integral equations (FIE) of second kind amenable of very efficient numerical solution. The deduction of the FIE is presented in this chapter for a relatively simple novel problem. The numerical solution of FIEs provides an analytical element of the spectra, which in general is not sufficient to evaluate the different components of the diffracted field. To obtain the whole spectrum of the unknowns, analytical continuations and recursive equations deduced by the WH equations are presented. The work ends with a short description of the numerical simulations for the novel scattering problem.

13 On the analytical regularization method in scattering and diffraction
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This chapter describes the principal ideas and more advanced techniques of the analytical regularization method (ARM) and the semiinversion procedure (SIP) particularly as they apply to the integral and integrodifferential equations that arise in scattering and diffraction problems in electromagnetics. The purpose of this chapter is to present a clear and unified treatment of the ARM and SIP approaches that draws on some recent developments using the language of pseudodifferential operators, while clarifying its rigorous application to boundary value problems (BVP) of electromagnetics.

14 Resonance scattering of Epolarized plane waves by twodimensional arbitrary open cavities: spectrum of complex eigenvalues
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The scattering of acoustic and electromagnetic waves by open cavities usually causes a resonant response, arising when the frequency of excitation approaches one of the frequencies belonging to the spectrum of eigenvalues of such a cavity. In our investigation we will focus on investigation of resonant objects widely used in practice, representing twodimensional (2D) hollow cylinders of arbitrary shape with longitudinal slits. In the context of propagating waves, these objects are treated as waveguides; in the context of standing waves, as resonators.

15 Numerical solutions of integral equations for electromagnetics
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Integral equations have proven their popularity for the electromagnetic analysis of radiation and scattering problems. The workhorse equations are the electricfield integral equation (EFIE) and the magneticfield integral equation (MFIE) [1,2]. The development of these equations will be reviewed in the following for perfectly conducting targets and homogeneous dielectric targets. When applied to certain closed surfaces, the original equations exhibit uniqueness difficulties at frequencies where the target surface coincides with a resonant cavity [3]. In addition, the original EFIE and MFIE also fail under certain circumstances for electrically small bodies. Alternate integral equations were proposed to remedy those situations, and these will also be summarized in the following section. In addition, we describe the numerical solution of these equations and report the progress made in recent years associated with the use of hierarchical vector basis functions, and the recent use of singular basis functions.

16 Electromagnetic modelling at arbitrarily low frequency via the quasiHelmholtz projectors
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We have identified the sources of the different problems plaguing the EFIE at low frequencies in both the frequency and the TD, as well as their traditional cures. Despite their apparent effectiveness, these techniques have been shown to have a limited applicability because they introduce their own set of problems which include the high computational burden of the LS decomposition and its effect on the highrefinement conditioning of the FDEFIE and the numerical instabilities introduced by the treatment of the TDEFIE. Techniques leveraging qH projectors, immune from the aforementioned sideeffects, have been introduced to address the different aspects of the lowfrequency breakdown of the FD formulation and of the large time step breakdown of its TD counterpart. In case of the FD, using projectors allows the same rescaling of the solenoidal and nonsolenoidal parts of the RWG space as traditional LS, but it has the added benefits of not requiring identification of the global loops of the structure as well as not introducing any further highrefinement illconditioning. In the TD case, the projectors are still used to separate the loop and star parts of the discretized space, but this separation is used to apply the correct derivative and integrative terms to the different parts of the operators. Coupled with an adequate mixed timediscretization scheme, this technique fully addresses the lowfrequency limitations of the TDEFIE. Along with presenting these purely theoretical concepts, we have provided implementation related hints, allowing the techniques presented in this chapter to be reliably and readily implemented into existing solvers. Finally, while we have addressed their lowfrequency breakdown, both EFIE formulations still suffer from a highrefinement breakdown. While in standard lowfrequency scenarios, a curing of lowfrequency issues may suffice, for more pathological cases techniques addressing both breakdowns may be required. Strategies based on qH projectors and Calderon identities have recently been introduced for the frequency and TD formulations [23,40] and should be used in this case.

17 Resistive and thin dielectric disk antennas with axially symmetric excitation analyzed using the method of analytical regularization
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We present accurate numerical method and computational results in the modeling of the axisymmetric electromagnetic wave scattering by resistive and thin dielectric circular diskonsubstrate antennas, simulated by using the twosided generalized boundary conditions (GBC) and singular integral equations (SIE).The numerical analysis is based on the method of analytical regularization (MAR) exploiting a Galerkin method (GM) with judicious basis functions, which convert SIE into a Fredholm secondkind infinite matrix equation. This guarantees convergence and enables one to compute the solution with controlled accuracy even near sharp resonances if they are present. Numerical results demonstrate the effect of the disk losses and transparency on the antenna bandwidth and radiation efficiency.

18 Scattering and guiding problems of electromagnetic waves in inhomogeneous media by improved Fourier series expansion method
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Scattering and guiding problems of electromagnetic waves in inhomogeneous media such as photonic crystals are very interesting in many areas of physics and engineering in nanoscale optical technology. However, the conventional method that is based on the Fourier series expansion method in inhomogeneous media cannot be applied to profiles with discontinuity. Our new method that is called IFEM (improved Fourier series expansion method) can be applied to profiles with discontinuity for scattering and guiding problems.

19 Methods and fast algorithms for the solution of volume singular integral equations
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We will consider two families of problems of electromagnetics: (I) the medium in a finite 3D domain Q is characterized by a dielectric permittivity tensor function e' with the permittivity being constant outside Q and the permeability constant everywhere; (II) the same domain Q lies over a perfectly conducting plane. The problem is to find the electromagnetic field excited in the medium by an external field. These problems can be reduced to volume singular integral equations (VSIEs) with respect to the electric field in domain Q. Also, we study the spectrum of the operators of integral equations.

20 Herglotz functions and applications in electromagnetics
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Herglotz functions inevitably appear in pure mathematics, mathematical physics, and engineering with a wide range of applications. In particular, they are the pertinent functions to model passive systems, and thus appear in modeling of electromagnetic phenomena in circuits, antennas, materials, and scattering. In this chapter, we review the basic theory of Herglotz functions and its applications to determine sum rules and physical bounds for passive systems.

21 Scattering and guidance by layered cylindrically periodic arrays of circular cylinders
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In this chapter, we discuss a selfcontained semianalytical approach for electromagnetic radiation, scattering and guidance by cylindrical arrays composed of circular rods periodically distributed along concentrically layered circular rings. Generalization of the problem to the eccentrically layered circular rings is straightforward. The rods can be dielectrics, perfect conductor, air holes, metals or magnetized ferrites. The method uses the transition matrix (Tmatrix) of a circular rod in isolation, the reflection and transmission matrices of a cylindrical array based on the cylindrical harmonics expansion and the generalized reflection and transmission matrices for a cylindrically layered structure.

22 Analytical and numerical solution techniques for forward and inverse scattering problems in waveguides
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Investigations of the wave propagation in waveguides with inhomogeneous dielectric inclusions are performed using advanced analytical and numerical methods. A particular attention is paid to the analysis of benchmark problems: forward and inverse scattering by parallelplane dielectric diaphragms and layered structures placed in a rectangular waveguide with perfectly conducting walls. A detailed study of the properties of transmitted fields using the theory of functions of one or several complex variables leads to the development of a technique aimed at reconstructing permittivity of the inclusions from the transmission coefficient. The unique solvability of this inverse problems is proved by establishing the domains where the transmission coefficient is a onetoone function ofpermittivity and other problem parameters. The obtained results are used to improve finitedifference time domain (FDTD) numerical solution to the corresponding forward and inverse scattering problems for Maxwell's equations in waveguides and create specifically developed codes. In general, the chapter contributes to the electromagnetic wave theory with several new findings and modified theoretical approaches.

23 Beambased local diffraction tomography
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In this chapter, the authors present a beambased local diffractiontomography (DT) inversion scheme as an alternative to the conventional Green's function or planewave inversion schemes. The beam waves are used here as basis functions for local phasespace processing of the data and for local backpropagation and reconstruction.

24 Modal expansions in dispersive material systems with application to quantum optics and topological photonics
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The objective of this chapter is to highlight that for lossless material platforms formed by arbitrary inhomogeneous bianisotropic and possibly nonreciprocal materials, the natural modes of oscillation form, indeed, a complete set of expansion functions. Based on our recent work, it is proven that the Maxwell equations in dispersive systems can always be reduced to a generalized dynamical problem whose time evolution is described by a Hermitian operator. The effects of material dispersion are taken into account by introducing additional variables that may model the internal degrees of freedom of the material. With such a result, we construct formal expansions of the electromagnetic field in terms of the normal modes, and in particular it is highlighted that the modal expansion coefficients are not unique. The developed theory is used to obtain a modal expansion of the system Green function.

25 Multiple scattering by a collection of randomly located obstacles distributed in a dielectric slab
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Multiple scattering of electromagnetic waves by a discrete collection of scatterers is a wellstudied subject, and many excellent treatments are found in the literature. The deterministic analysis of the scattering problem in this chapter is an extension of the problems treated previosly. Moreover, the present analysis generalizes the established results in two previous papers to a geometry with a more general background material, which is practical for a controlled experimental verification of the final result. The transmitted and reflected intensities are conveniently represented as a sum of two termsthe coherent and the incoherent contribution. In this chapter, we focus on the analysis of the coherent term. The chapter is organized as follows. In Section 25.2, the geometry of the multiple electromagnetic scattering problem is given, and in Section 25.3, the main tool to solve the problemthe integral representationis introduced. The integral representations are exploited in the various homogeneous regions of the problem in Section 25.4, and the appropriate expansions of the surface fields are introduced in Section 25.5. The final goal of the chapter is to calculate the transmitted and reflected coherent fields of the problem.

26 Electromagnetics of complex environments applied to geophysical and biological media
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One of the important areas of research is the study of electromagnetic and related wave theories, which have a wide range of practical applications in complex environments, such as microwave remote sensing of the Earth, object detection and imaging in clutter, medical optics and ultrasound imaging, characterization of metamaterials and composite and porous media, and communication through complex clutter environments. This chapter gives a review of wave theories applied to imaging in geophysical and biological media, including imaging through air turbulence and particulate matter, imaging nearocean rough surfaces and communication and signal processing in clutter, coherence in multiple scattering and super resolution, timereversal (TR) imaging, radiative transfer, waves in porous media, seismic CODA waves, and the memory effect.

27 Innovative tools for SI units in solving various problems of electrodynamics
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The goal of this chapter is rather ambitious: it proposes to "reboot" the standard system of Maxwell's equations in SI units in a new, much simpler, format, but executed within the SI metric system, as well.

Back Matter
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