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TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Transmission Eigenvalues in Inverse Scattering Theory

Fioralba Cakoni

Department of Mathematical Sciences University of Delaware Newark, DE 19716, USA email: cakoni@math.udel.edu

Research supported by a grant from AFOSR and NSF

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Scattering by an Inhomogeneous Media

u

s

D

i

u

∆us + k2us = 0 in Rm \ D ∇ · A∇u + k2nu = 0 in D u = us + ui in ∂D ν · A∇u = ν · ∇(us + ui) in ∂D lim

r→∞ r

m−1 2

∂us ∂r − ikus

• = 0

A, n represent the inhomogeneuos media, here meant in a general sense. Question: Is there an incident wave ui that does not scatter? The answer to this question leads to the transmission eigenvalue problem.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Transmission Eigenvalues

If there exists a nontrivial solution to the homogeneous interior transmission problem ∆v + k2v = 0 in D ∇ · A∇w + k2nw = 0 in D w = v

• n

∂D ν · A∇w = ν · ∇v

• n

∂D such that v can be extended outside D as a solution to the Helmholtz equation ˜ v, then the scattered field due to ˜ v as incident wave is identically zero. Values of k for which this problem has non trivial solution are referred to as transmission eigenvalues.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Transmission Eigenvalues

In general such an extension of v does not exits! Since superposition of plane waves so-called Herglotz wave functions vg(x) :=

• Ω

eikx·dg(d)ds(d), Ω := {d : |d| = 1}

• r superposition of point sources

Sϕ(x) :=

• Λ

ϕ(y)Φ(x, y)dsy, Λ is a surface in Rm \ D where Φ(x, y) is the fundamental solution of the Helmholtz equation, are dense in

• v ∈ H(D) : ∆v + k2v = 0

in D

• ,

at a transmission eigenvalue there is an incident field that produces arbitrarily small scattered field (here H(D) is either L2(D) or H1(D)).

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Motivation

Two important issues: Real transmission eigenvalues can be determined from the scattering data. Transmission eigenvalues carry information about material properties. Therefore, transmission eigenvalues can be used to quantify the presence of abnormalities inside homogeneous media and use this information to test the integrity of materials. How are real transmission eigenvalues seen in the scattering data?

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Measurements

Since transmission eigenvalues correspond to "non scattering" frequencies, at a transmission eigenvalue the (modified) measurement operator fails to be injective. Exploring this, gives a way to see transmission eigenvalues in the scattering data. To fix our ideas consider the far field operator F : L2(Ω) → L2(Ω) (Fg)(ˆ x) :=

• Ω

u∞(ˆ x, d, k)g(d)ds(d),

• S = I +

ik √ 2πk e−iπ/4F

• where u∞(ˆ

x, d, k) is the far field of the scattered field us(x, d, k) due to a plane wave ui(x) = eikx·d, for ˆ x, d ∈ Ω, and k ∈ [k0, k1] and the far field equation (Fg)(ˆ x) = Φ∞(ˆ x, z, k), g ∈ L2(Ω)

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Computation of Real TE

Assume that z ∈ D and δ > 0 is the measurement noise level. Let gz,δ,k be the Tikhonov regularized solution of the far field equation, i.e the unique minimizer F δg − Φ∞(· , z)2

L2(Ω) + ǫ(δ)g2 L2(Ω),

ǫ(δ) → 0 as δ → 0 If k is not a transmission eigenvalue then lim

δ→0 vgz,δ,k H(D)

exists. Arens, Inv. Probs. (2004), Arens-Lechleiter, Int. Eqn. Appl. (2009) If k is a transmission eigenvalue then lim

δ→0 vgz,δ,k H(D) = ∞.

Cakoni-Colton-Haddar, Comp. Rend. Math. 2010

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Computation of Real TE

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 5 10 15 20 25 30

Wave number k Norm of the Herglotz kernel

A composite plot of gzi L2(Ω) against k for 25 random points zi ∈ D

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 5 10 15

Wave number k Average norm of the Herglotz kernels

The average of gzi L2(Ω)

• ver all choices of zi ∈ D.

Computation of the transmission eigenvalues from the far field equation for the unit square D.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Historical Overview

The transmission eigenvalue problem in scattering theory was introduced by Kirsch (1986) and Colton-Monk (1988) Research was focused on the discreteness of transmission eigenvalues for variety of scattering problems: Colton-Kirsch-Päivärinta (1989) – many more ..... In the above work, it is always assumed that either n − 1 > 0 or 1 − n > 0 in D (may be zero at the boundary ∂D). The first proof of existence of at least one transmission eigenvalues for large enough contrast is due to Päivärinta-Sylvester (2009). The existence of an infinite set of transmission eigenvalues is proven by Cakoni-Gintides-Haddar (2010) under only assumption that either n − 1 > 0 or 1 − n > 0. The existence has been extended to other scattering problems by Kirsch (2009), Cakoni-Haddar (2010) Cakoni-Kirsch (2010), Bellis-Cakoni-Guzina (2011), Cossonniere (Ph.D. thesis) etc.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Historical Overview, cont.

Cakoni-Colton-Haddar (2010) and Cossonniere-Haddar (2011) have studied the case when inside media there are subregions with the same material properties as the background. Hitrik-Krupchyk-Ola-Päivärinta (2010), in a series of papers have extended the transmission eigenvalue problem to a more general class of differential operators with constant coefficients. Finch has connected the discreteness of the transmission spectrum to a uniqueness question in thermo-acoustic imaging for which n − 1 can change sign. Sylvester (2012) has shown that the set of transmission eigenvalues is at most discrete if A = I and n − 1 is positive (or negative) only in a neighborhood of ∂D but otherwise could change sign inside D. A similar result is obtained by Bonnet Ben Dhia - Chesnel - Haddar (2011) using T-coercivity and Lakshtanov-Vainberg (to appear), for the case A − I and n − 1 keep fixed sign in a neighborhood at the boundary ∂D.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Transmission Eigenvalue Problem

Recall the transmission eigenvalue problem (set A = I) ∆w + k2nw = 0 in D ∆v + k2v = 0 in D w = v

• n

∂D ∂w ∂ν = ∂v ∂ν

• n

∂D It is a nonstandard eigenvalue problem

• D
• ∇w · ∇ ψ − k2n(x)wψ
• dx =
• D
• ∇v · ∇φ − k2v φ
• dx

If n = 1 the interior transmission problem is degenerate If ℑ(n) > 0 in D, there are no real transmission eigenvalues.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Transmission Eigenvalue Problem

Let u = w − v, we have that ∆u + k2nu = k2(n − 1)v. Applying (∆ + k2), the transmission eigenvalue problem can be written for u ∈ H2

0(D) as an eigenvalue problem for the fourth order

equation: (∆ + k2) 1 n − 1(∆ + k2n)u = 0 i.e. in the variational form

• D

1 n − 1(∆u + k2nu)(∆ϕ + k2ϕ) dx = 0 for all ϕ ∈ H2

0(D)

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Transmission Eigenvalues

Assuming n real valued and letting k2 := τ, the variational formulation leads to the eigenvalue problem for a quadratic pencil operator u − τK1u + τ 2K2u = 0, u ∈ H2

0(D)

with selfadjoint compact operators K1 = T −1/2T1T −1/2 and K2 = T −1/2T2T −1/2 where (Tu, ϕ)H2(D) =

• D

1 n − 1∆u ∆ϕ dx coercive (T1u, ϕ)H2(D) = −

• D

1 n − 1 (∆u ϕ + nu ∆ϕ) dx (T2u, ϕ)H2(D) =

• D

n n − 1u ϕ dx non-negative.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Transmission Eigenvalues

The transmission eigenvalue problem can be transformed to the eigenvalue problem in H2

0(D) × H2 0(D)

(K − ξI)U = 0, U =

• u

τK 1/2

2

u

• ,

ξ := 1 τ for the non-selfadjoint compact operator K :=

• K1

−K 1/2

2

K 1/2

2

• .

However from here one can see that the transmission eigenvalues form a discrete set with +∞ as the only possible accumulation point. Note: in the special case of spherically stratified media it is possible to prove existence of complex transmission eigenvalues, Leung-Colton (to appear).

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Media with Voids

How to approach the transmission eigenvalue problem. We first consider the case A = I.

• D

D D

• We assume n ∈ L∞(D), n > 0

n = 1 in D0 ⊂ D and n − 1 > 0 or 1 − n > 0 in D \ D0. Literature: Cakoni-Colton-Haddar, The interior transmission problem for regions with cavities, SIAM J. Math. Analysis 42, no 1, 145-162 (2010). Cakoni-Gintides-Haddar, The existence of an infinite discrete set

• f transmission eigenvalues, SIAM J. Math. Analysis, 42, no 1,

237-255 (2010). Cossonniere-Haddar (2011) have investigated this problem for Maxwell’s equation.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Transmission Eigenvalue Problem

Recall the transmission eigenvalue problem, and to fix our ideas assume n − 1 > 0 ∆w + k2nw = 0 in D ∆v + k2v = 0 in D w = v

• n

∂D ∂w ∂ν = ∂v ∂ν

• n

∂D Let u = w − v ∈ H2(D). Then

• ∆ + k2n
• 1

n − 1

• ∆ + k2

u = 0 in D \ D0

• ∆ + k2

u = 0 in D0, and u = 0 and ∂u ∂ν = 0

• n ∂D.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Transmission Eigenvalue Problem

V0(D, D0, k) := {u ∈ H2

0(D) such that ∆u + k2u = 0 in D0}.

The transmission eigenvalue problem reads, for ψ ∈ V0(D, D0, k)

• D\D0

1 n − 1

• ∆ + k2

u

• ∆ + k2 ¯

ψ dx + k2

• D\D0

(∆u + k2u) ¯ ψ dx = 0 It is possible to obtain lower bounds for transmission eigenvalues.

• D\D0

1 n − 1|∆u + k2nu|2 dx − k4

• D\D0

n|u|2 dx + k2

• D\D0

|∇u|2 dx −k4

• D0

|u|2 dx + k2

• D0

|∇u|2 dx = 0 Then u = 0 as long as k2 < λ1(D) supD n, where λ1(D) is the first Dirichlet eigenvalue of −∆ in D.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Transmission Eigenvalue Problem

Ak(u, ψ) + Bk(u, ψ) = 0 for all ψ ∈ V0(D, D0, k). Ak : V0(D, D0, k) → V0(D, D0, k) is self-adjoint and positive definite. Bk : V0(D, D0, k) → V0(D, D0, k) is self-adjoint and compact. k is a transmission eigenvalue if and only if the operator Ak + Bk

• r

Ik + A−1/2

k

BkA−1/2

k

: V0(D, D0, k) → V0(D, D0, k) has a nontrivial kernel where Ik is the identity operator on V0(D, D0, k). Obviously, a transmission eigenvalue has finite multiplicity.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Transmission Eigenvalue Problem

To avoid dealing with function spaces depending on k we introduce the orthogonal projection operator Pk from H2

0(D) onto V0(D, D0, k)

and the corresponding injection Rk : V0(D, D0, k) → H2

0(D). Then k is

a transmission eigenvalue if and only if the operator I + Tk : H2

0(D) → H2 0(D)

has nontrivial kernel, where Tk := RkA−1/2

k

BkA−1/2

k

Pk is compact. Discreteness: Unfortunately Pk is not analytic in a neighborhood

• f real axis. We build an analytic operator that mimics the effect
• f projection which allows us to build an analytic extension of Ak

and Bk on H2

0(D). The result follows for the Analytic Fredholm

Theory. Existence: Consider the auxiliary eigenvalue problem (I + λ(k)Tk)u = 0

• n

H2

0(D)

and find solutions to λ(k) = 1.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Transmission Eigenvalues

(I + λ(k)Tk)u = 0

• n

H2

0(D)

For a fixed k > 0 there exists an increasing sequence of eigenvalues λj(k)j≥1 such that λj(k) → +∞ as j → ∞. Thanks to max-min principle λj(k) depend continuously on k ∈ [0, ∞). Hence, if there exists two positive constants k0 > 0 and k1 > 0 such that I + Tk0 is positive on H2

0(D),

I + Tk1 is non positive on a m dimensional subspace of H2

0(D)

then each of the equations λj(k) = 1 for j = 1, . . . , m, has at least one solution in [k0, k1] meaning that there exists m transmission eigenvalues (counting multiplicity) within the interval [k0, k1].

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Existence-Inequalities

There exists an infinite set of real transmission eigenvalues accumulating at +∞. If k1(D0, n(x)) is the first eigenvalue, then for a fixed D we have: The Faber Krahn inequality 0 < λ1(D) supD n ≤ k(D0, n(x)). Monotonicity with respect to the index of refraction k1(D0, n(x)) ≤ k1(D0, ˜ n(x)), ˜ n(x) ≤ n(x). Monotonicity with respect to voids k1(D0, n(x)) ≤ k1(˜ D0, n(x)), D0 ⊂ ˜ D0. where λ1(D) is the first Dirichlet eigenvalue of −∆ in D. Similar results can be obtained for 1 − n > 0.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Numerical Example: Media with Voids

Maxwell’s Equations due to A. Cossonniere, Ph.D. Thesis – appeared in Cossonniere-Haddar, SIMA, (2011)

3 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Wave number k LSM Integral equations Exact transmission eigenvalues

Figure 6.20: Sphere of radius 1 and index of refraction .

The sphere with radius 1 and N = 4I k1 = 3.16

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Numerical Example: Media with Voids

3.2 3.25 3.3 3.35 3.4 0.5 1 1.5 2 2.5 Wave number k LSM Integral equations

The sphere with radius 1, N = 4I containing a cubic cavity k1 = 3.33, i.e. k1 is shifted to the right

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The Case of n − 1 Changing Sign

Recently, progress has been made in the case of the contrast n − 1 changing sign inside D with state of the art result by Sylvester, SIMA (2012). Roughly speaking he shows that transmission eigenvalues form a discrete (possibly empty) set provided n − 1 has fixed sign

• nly in a neighborhood of ∂D. There are two aspects in the proof:

Fredholm property. Sylvester consideres the problem in the form ∆u+k2nu = k2(n − 1)v, ∆v+k2v = 0, u ∈ H2

0(D), v ∈ H1(D)

and uses the concept of upper-triangular compact operators. This property can also be obtained via variational formulation or integral equation formulation, Haddar-Cossoniere, (to appear). Find a k that is not a transmission eigenvalues. This requires careful estimates for the solution inside D in terms of its values in a neighborhood of ∂D. The existence of transmission eigenvalues under such weaker assumptions is still open.

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Absorbing-Dispersive Media

Cakoni-Colton-Haddar, Inverse Problems (2012) has initiated the study of the existence transmission eigenvalues for the case of absorbing media. ∆w + k2 ǫ1 + i γ1 k

• w = 0

in D ∆v + k2 ǫ0 + i γ0 k

• v = 0

in D where ǫ0 ≥ α0 > 0, ǫ1 ≥ α1 > 0, γ0 ≥ 0, γ1 ≥ 0 are bounded functions. For the corresponding spherically stratified case there exits infinitely many (complex) transmission eigenvalues. It is possible for real transmission eigenvalues to exit for some combinations of media and background.

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Absorbing-Dispersive Media

In the general case: The set of transmission eigenvalues k ∈ C in the right half plane is discrete, provided ǫ1(x) − ǫ0(x) > 0. Using the stability of a finite set of eigenvalues for closed

• perators one can show that if supD(γ0 + γ1) is small enough

there exists at least ℓ > 0 transmission eigenvalues each in a small neighborhood of the first ℓ real transmission eigenvalues corresponding to γ0 = γ1 = 0. For the case of ǫ0, ǫ1, γ0, γ1 constant, we have identified eigenvalue free zones in the complex plane The existence of transmission eigenvalues for general media if absorption is present is still open.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Anisotropic Media

The corresponding transmission eigenvalue problem is to find v, w ∈ H1(D) such that ∇ · A∇w + k2nw = 0 in D ∆v + k2v = 0 in D w = v

• n

∂D ν · A∇w = ν · ∇v

• n

∂D. This transmission eigenvalue problem has a more complicated nonlinear structure than quadratic. The existence has been shown in Cakoni-Gintides-Haddar, SIAM J.

• Math. Anal. (2010) and Cakoni-Kirsch, IJCSM (2010).

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Existence of Transmission Eigenvalues

Set u = w − v ∈ H1

0(D). Find v = vu by solving a Neuman type

problem: For every ψ ∈ H1(D)

• D

(A − I)∇v · ∇ψ − k2(n − 1)vψ dx =

• D

A∇u · ∇ψ − k2nuψ dx. Having u → vu, we require that v := vu satisfies ∆v + k2v = 0. Thus we define Lk : H1

0(D) → H1 0(D)

(Lku, φ)H1

0(D) =

• D

∇vu · ∇φ − k2vu · φ dx, φ ∈ H1

0(D).

Then the transmission eigenvalue problem is equivalent to Lku = 0 in H1

0(D)

which can be written (I + L−1/2 CkL−1/2 )u = 0 in H1

0(D)

L0 self-adjoint positive definite and Ck self-adjoint compact.

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Existence of Transmission Eigenvalues

If n(x) ≡ 1 and the contrast A − I is either positive or negative in D then there exists an infinite discrete set of real transmission eigenvalues accumulating at +∞. If the contrasts A − I and n − 1 have the opposite fixed sign, then there exists an infinite discrete set of real transmission eigenvalues accumulating at +∞. If the contrasts A − I and n − 1 have the same fixed sign, then there exits at least one real transmission eigenvalue providing that n is small enough.

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Discreteness of Transmission Eigenvalues

The strongest result on the discreteness of transmission eigenvalues for this problem is due to Bonnet Ben Dhia - Chesnel - Haddar, Comptes Rendus Math. (2011) (using the concept of ⊤- coercivity ). In particular, the discreteness of transmission eigenvalues is proven under either one of the following assumptions (weaker than for the existence): Either A − I > 0 or A − I < 0 in D, and

• D

(n − 1) dx = 0 or n ≡ 1. The contrasts A − I and n − 1 have the same fixed sign only in a neighborhood of the boundary ∂D.

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Numerical Example: Homogeneous Anisotropic Media

Take n ≡ 1. Find an isotropic-homogenous media a0 that has the first transmission eigenvalue the same as the (measured) first transmission eigenvalue for the unknown anisotropic media. We consider D to be the unit square [−1/2, 1/2] × [−1/2, 1/2] and A1 = 2 8

• A2 =

6 8

• A2r =
• 7.4136

−0.9069 −0.9069 6.5834

• Matrix

Eigenvalues a∗, a∗ Predicted a0 Aiso 4, 4 4.032 A1 2, 8 5.319 A2 6, 8 7.407 A2r 6, 8 6.896 Cakoni-Colton-Monk-Sun, Inverse Problems, (2010)

TE and Scattering Theory Isotropic Media Anisotropic Media Open Problems

Numerical Examples

Examples for Anisotropic Maxwell’s Equations. D is a sphere of radius 1. The matrix N = ǫ/ǫ0 is relative electric permittivity. The relative magnetic permeability is one.

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.145 1.15 1.155 1.16 1.165 1.17 1.175 1.18

Perturbation parameter a Lowest real transmission eigenvalue N=diag([16−a,16,16+a]) N=diag([16,16,16+a])

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2.2 2.25 2.3 2.35 2.4 2.45 2.5 2.55 2.6 2.65 2.7

Perturbation parameter a Lowest real transmission eigenvalue N=diag([5−a,5,5+a]) N=diag([5,5,5+a])

Perturbation of N = 16I Perturbation of N = 5I

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Numerical Examples (cont)

We first compute the trans- mission eigenvalues for anisotropic N from mea- sured scattering data. Then compute the isotropic n0 that has the same first transmission eigenvalue as the measured one.

16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 17 1.135 1.14 1.145 1.15 1.155 1.16 1.165 1.17 1.175 1.18

Isotropic Permittivity N

Lowest real transmission eigenvalue

Lowest transmission eigenvalue against n0 (isotropic)

N k1,D,N(x) n0 diag([15.5, 16, 16.5]) 1.163 16.33 diag([15, 16, 17]) 1.151 16.65 diag([16, 16, 16.5]) 1.161 16.38 diag([16, 16, 17]) 1.146 16.77

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Numerical Examples (cont)

The same procedure can be carried out at lower N as well (the lowest transmission eigenvalue increases) N k1,D,N(x) n0 diag([4.5, 5, 5.5]) 2.442 5.339 diag([4, 5, 6]) 2.302 5.631 diag([5, 5, 5.5]) 2.410 5.397 diag([5, 5, 6]) 2.245 5.778

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 Isotropic Permittivity N Lowest real transmission eigenvalue

Lowest transmission eigenvalue against n0 (isotropic)

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Open Problems

Can the existence of real transmission eigenvalues for non-absorbing media be established if the assumptions on the sign of the contrast are weakened? Do complex transmission eigenvalues exists for general non-absorbing media? Do real transmission eigenvalues exist for absorbing media? What would the necessary conditions be on the contrasts that guaranty the discreteness of transmission eigenvalues? Can Faber-Krahn type inequalities be established for the higher eigenvalues? Can an inverse spectral problem be developed for the general transmission eigenvalue problem? (Completeness of eigen-solutions?) Cakoni - Haddar, Transmission Eigenvalues in Inverse Scattering Theory, in Inside Out 2, Uhlmann edt. MSRI Publication (to appear).