text:
\section{Introduction} \label{sec:intro}

If a dielectric material is coated by a meta-material with a negative dielectric constant, then a source outside the structure may cause an anomalous localized resonance by which the source may be cloaked. This phenomenon was first discovered in \cite{MN_PRSA_06, NMM_94} when the core and shell are concentric disks and the source is single or multiple polarizable dipoles. It was shown in \cite{MN_PRSA_06} that there is a critical radius determined by radii of the core and the shell such that if the dipole source is located inside the radius, then cloaking by anomalous localized resonance (CALR) takes place, and if the dipole source is located outside the radius, then CALR does not take place.

In a recent work \cite{ACKLM}, quantitatively precise analysis of CALR is presented on circular coated structure with arbitrary sources. Among the findings of the paper is that CALR is a spectral phenomenon occurring at the limit point of eigenvalues of the Neumann-Poincar\'e (NP) operator defined on concentric disks. In fact, by Plemelj's symmetrization principle, the NP operator can be realized as a symmetric operator \cite{KPS}. Since it is compact on concentric circles, the NP operator has real eigenvalues converging to $0$ (exponentially fast), and there CALR takes place. On a single disk $0$ is the only eigenvalue of the NP operator (other than $1/2$ of multiplicity $1$). So CALR does not take place. But by coating the disk with another disk, the boundary becomes two circles and the NP eigenvalues are perturbed and converges to $0$, and hence CALR does take place on the coated structure. In fact, CALR may take place on a single inclusion (not a coated inclusion). For example, the NP eigenvalues on ellipses are not $0$ and converge to $0$, CALR takes place on ellipses \cite{AK-JMAA-16}. There is yet another approach to CALR using the variational method, for which we refer to \cite{KLSW-CMP-14}.

Since CALR is a spectral phenomenon at the limit point of eigenvalues of the NP operator, it is interesting to look into CALR in the context other than electro-statics where the notion of the NP operator is defined. In this regard the system of elasto-statics is particularly interesting since the elastic NP operator is not compact even on smooth boundaries. However, it is proved in the recent paper \cite{AJKKY} that elastic NP eigenvalues on smooth boundaries of two-dimensional domains, whose Lam\'e parameters are $(\Gl, \Gm)$, accumulate at either $k_0$ or $-k_0$ where
\beq\label{kzero}
k_0= \frac{\Gm}{2 \left( \Gl + 2 \Gm \right)}.
\eeq

Using this spectral property of the elastic NP operator CALR for the linear isotropic elasticity is considered in \cite{AJKKY} when the background is the usual isotropic elastic material with the pair of Lam\'e constants $(\Gl, \Gm)$ and the inclusion has
\beq\label{meta}
  (\tilde{\Gl}, \tilde{\Gm}) = (c + i \Gd) (\Gl, \Gm)
\eeq
as its Lam\'e constants, where $c$ is a negative constant and $\Gd$ is a small positive constant representing dissipation. The asymptotics of NP eigenvalues are derived and it is proved that CALR occurs on ellipses if the constant $c$ in \eqnref{meta} satisfies
\beq\label{ccond}
z(c):= \frac{c+1}{2(1-c)} = \pm k_0.
\eeq
We emphasize that the condition \eqnref{ccond} can be satisfied only when $c$ is negative since $0<k_0 <1/2$. The elastic NP eigenvalues on two dimensional disks are
\beq\label{NPeigendisk}
  \frac{1}{2}, \quad - \frac{\Gl}{2 (2 \Gm + \Gl)}, \quad \pm k_0,
\eeq
where the first two have multiplicity one, while the last two are of infinite multiplicities, and CALR does not occur since $\pm k_0$ are the elastic NP eigenvalues (not limit points). One may expect that like the electro-static case if we coat the disk by an concentric circle, then the NP eigenvalues in \eqnref{NPeigendisk} are perturbed from $\pm k_0$ (while converging to them) and CALR may occur.

The purpose of this paper is to confirm it. To do so, we suppose that the disk $D_i$ is included in another concentric disk $D_e$ where the Lam\'e parameters are given by
\beq\label{total_lame_const}
  (\Gl^\Gd, \Gm^\Gd) =
  \begin{cases}
    (\Gl, \Gm), & \text{ in } \Rbb^2 \setminus \ol{D_e}, \\
    (c + i \delta) (\Gl, \Gm), & \text{ in } D_e \setminus \ol{D_i}, \\
    (\Gl, \Gm), & \text{ in } D_i.
  \end{cases}
\eeq
Here $c$ is a negative constant and $\Gd$ is a parameter which will be sent to $0$. We show that the elastic NP eigenvalues on the boundary of $D_e \setminus \ol{D_i}$ deviate from $\pm k_0$, but accumulate at $\pm k_0$. We actually show that the NP operator (with respect to the inner product on $H^{-1/2}$-space defined by the single layer potential) can be realized as a series of block matrices, and then using perturbation theory of eigenvalues we derive asymptotic formula of the convergence. We then show that CALR occurs if $c$ satisfies \eqnref{ccond}. It is worth emphasizing that the critical radii (for cloaking) when $z(c)=k_0$ and when $z(c)=-k_0$ are different, which is due to the different behavior of NP eigenvalues near $k_0$ and $-k_0$.

A few comments are in order before completing Introduction. The major emphasis of this paper is on the spectral property of the elastic NP operator on two circles. Once we obtain the asymptotic behavior of eigenvalues, the rest of arguments are parallel to those in \cite{ACKLM} even though they are more involved since a system of equations are being dealt with. Existence of elastic meta-materials satisfying \eqnref{meta} is out of the scope of our expertise. In this regard, we mention that the method of this paper works only when \eqnref{meta} is satisfied. If, for example, only one of the parameters is negative, the method may not work.

While this work is in progress, we were informed by Hongyu Liu that their work \cite{LL_SIMA-16} was completed. There it is proved CALR occurs on the annulus structure in elasticity when the source function $\Bf$ (see \eqnref{negative_indx_elasticity_problem}) is supported in a circle. We emphasize that the method of this paper is completely different from that of \cite{LL_SIMA-16}: their method uses the variational one similar to that of \cite{KLSW-CMP-14} while our method uses spectral properties of the NP operator. The method of this paper reveals more clearly the spectral nature of anomalous localized resonance.

This paper is organized as follows. Section \ref{sec:prelim} is to formulate the problem using layer potentials and define the elastic NP operator. Section \ref{sec:NP} is to derive asymptotics of the NP eigenvalues near $\pm k_0$. Occurrence of CALR is proved in section \ref{sec:CALR}. Appendix is to provide proofs of two technical identities.

\section{Layer potential formulation of the problem} \label{sec:prelim}

\subsection{Layer potentials for the Lam\'e system}\label{sec:layer}

Let us first recall definitions of the layer potential and the NP operator related to the Lam\'e system (see, for example, \cite{AK07}). With the pair of Lam\'{e} constants $(\Gl, \Gm)$ satisfying $\Gm > 0$ and $\Gl + \Gm > 0$, the isotropic elasticity tensor $\Cbb = \left( C_{ijkl} \right)_{i, j, k, l = 1}^2$ is defined by
\beq
  C_{ijkl} := \Gl \Gd_{ij} \Gd_{kl} + \Gm \left( \Gd_{ik} \Gd_{jl} + \Gd_{il} \Gd_{jk} \right). \nonumber
\eeq
Then the corresponding Lam\'{e} system of elasticity equations is defined to be $\Lcal_{\Gl, \Gm}:= \nabla \cdot \Cbb \hatna$
where $\hatna$ is the symmetric gradient, namely,
$$
\hatna \Bu := \frac{1}{2} \left( \nabla \Bu + \nabla \Bu^T \right) \quad (T \mbox{ for transpose}).
$$

Let $\GO$ be a connected bounded domain with the Lipschitz boundary in $\Rbb^2$. The single layer potential of the density function $\BGvf$ on $\p \GO$ associated with the Lam\'{e} operator $\Lcal_{\Gl, \Gm}$ is defined by
\begin{align*}
  \BS_{\p \GO} [\BGvf] (\Bx) & := \int_{\p \GO} \BGG(\Bx - \By) \BGvf(\By) d\Gs(\By), \quad \Bx \in \Rbb^2,
\end{align*}
where $d\Gs$ is the arc length of $\p \GO$ and $\BGG(\Bx) = \left( \GG_{ij}(\Bx) \right)_{i, j = 1}^2$ is the Kelvin matrix of the fundamental solution to the Lam\'{e} system in $\Rbb^2$, namely,
\beq
  \GG_{ij}(\Bx) = \frac{\Ga_1}{2 \pi} \Gd_{ij} \ln{\left\vert \Bx \right\vert} - \frac{\Ga_2}{2 \pi} \frac{x_i x_j}{\left\vert \Bx \right\vert^2},
\eeq
with
\beq
  \Ga_1 = \frac{1}{2} \left( \frac{1}{\Gm} + \frac{1}{2 \Gm + \Gl} \right) \ \text{ and } \ \Ga_2 = \frac{1}{2} \left( \frac{1}{\Gm} - \frac{1}{2 \Gm + \Gl} \right). \nonumber
\eeq

The (elasto-static) NP operator on $\p\GO$ is defined by
\beq
  \BK_{\p \GO}^* [\BGvf] (\Bx) := \mbox{p.v.} \int_{\p \GO} \p_{\Gv_\Bx} \BGG(\Bx - \By) \BGvf(\By) d\Gs(\By) \quad \text{a.e. } \Bx \in \p \GO. \nonumber
\eeq
Here $\mbox{p.v.}$ stands for the Cauchy principal value, the conormal derivative on $\p \GO$ associated with $\nabla \cdot \Cbb \hatna$ is defined to be
\beq
  \p_\Gv \Bu := ( \Cbb \hatna \Bu ) \Bn = \Gl \left( \nabla \cdot \Bu \right) \Bn + 2 \Gm ( \hatna \Bu ) \Bn \quad \text{ on } \p \GO,
\eeq
where $\Bn$ is the outward unit normal to $\p \GO$, and $\p_{\Gv_\Bx} \BGG(\Bx - \By)$ is defined by
\beq
  \p_{\Gv_\Bx} \BGG(\Bx - \By) \Bb = \p_{\Gv_\Bx} \left( \BGG(\Bx - \By) \Bb \right)
\eeq
for any constant vector $\Bb \in \Rbb^2$.

Let $H^{1 / 2}(\p \GO)^2$ be the usual $L^2$-Sobolev space of order $1 / 2$ and $H^{- 1 / 2}(\p \GO)^2$ be its dual space with respect to $L^2$-pairing $\la \cdot, \cdot \ra$.
Let $\GY$ be the subspace of $H^{- 1 / 2}(\p \GO)^2$ spanned by
\beq\label{Psi}
\begin{bmatrix} 1 \\ 0 \end{bmatrix},  \quad \begin{bmatrix} 0 \\ 1 \end{bmatrix}, \quad \begin{bmatrix} y \\ - x\end{bmatrix}.
\eeq
We denote by $H^{-1/2}_\GY(\p \GO)$ the annihilator of $\GY$, namely, the collection of all $\BGvf \in H^{-1/2}(\p \GO)^2$ such that $\la \BGvf, \Bf \ra = 0$ for all $\Bf \in \GY$.

The NP operator $\BK_{\p \GO}^*$ is bounded on $H^{-1 / 2}(\p \GO)^2$, and maps $H^{-1 / 2}_\GY(\p \GO)$ into itself. If $\Gl \notin [-1/2, 1/2)$, then $\Gl I - \BK_{\p \GO}^*$ is invertible on $H^{-1 / 2}(\p \GO)^2$ (and on $H^{-1 / 2}_\GY(\p \GO)$). The NP operator is related to the single layer potential through the following jump formula (see \cite{AK07, DKV88}):
\begin{align}
  \left. \p_\Gv \BS_{\p \GO} [\BGvf] \right|_\pm & = \left( \pm \frac{1}{2} I + \BK_{\p \GO}^* \right) [\BGvf] \quad \text{a.e. on } \p \GO, \label{single_jump_formula}
\end{align}
where the subscripts $\pm$ denote the limits (to $\p\GO$) from outside and inside of $\p \GO$. Even if $\BK_{\p \GO}^*$ is not self-adjoint with respect to the usual $L^2$-inner product, it can be realized as a self-adjoint operator on $H^{-1 / 2}_\GY(\p \GO)$. In fact, it is proved in \cite{AJKKY} (following the discovery of \cite{KPS}) that the product $(\cdot, \cdot)_*$, defined by
\beq
  \left( \BGvf, \Bpsi \right)_{*} := - \la \BGvf, {\BS}_{\p\GO} [\Bpsi] \ra,
\eeq
is actually an inner product on $H^{-1 / 2}_\GY(\p \GO)$ which yields a norm equivalent to the usual $H^{-1/2}$-norm. Then $\BK_{\p \GO}^*$ is  self-adjoint with respect to this new inner product thanks to the Plemelj's symmetrization principle:
$$
\BS_{\p\GO} \BK_{\p\GO}^* = \BK_{\p\GO} \BS_{\p\GO},
$$
where $\BK_{\p \GO}$ is the $L^2$-adjoint operator of $\BK_{\p \GO}^*$. It is well-known (see, for example, \cite{DKV88}) that $\BK_{\p \GO}^*$ is not a compact operator on $H^{-1 / 2}(\p \GO)$ even if $\p\GO$ is smooth. However, it is proved in \cite{AJKKY} that the spectrum of $\BK_{\p \GO}^*$ consists of pure point spectrum converging to $\pm k_0$, which is defined in \eqnref{kzero}.

\subsection{The CALR problem on coated disks}

To formulate the problem on coated disks, let $D_i := \{ |\Bx| < r_i \}$ and $D_e := \{ |\Bx| < r_e\}$, $0 < r_i < r_e$, and write $\GG_i=\p D_i$ and $\GG_e=\p D_e$. The distribution of Lam\'e parameters are given by \eqnref{total_lame_const}, and the elasticity tensor $\Cbb^\Gd = ( C_{ijkl}^\Gd )_{i,j,k,l=1}^2$ is given by
\beq
  C_{ijkl}^\Gd := \Gl^\Gd \Gd_{ij} \Gd_{kl} + \Gm^\Gd \left( \Gd_{ik} \Gd_{jl} + \Gd_{il} \Gd_{jk} \right).
\eeq
We then consider the following elasticity equation:
\beq\label{negative_indx_elasticity_problem}
    \nabla \cdot \Cbb^{\Gd} \hatna \Bu^{\Gd} = \Bf \quad \mbox{ in } \Rbb^2
\eeq
with the decaying condition $\Bu^\Gd(\Bx) \to {\bf 0}$ as $|\Bx| \to \infty$, where $\Bu^\Gd = ( u_1^\Gd, u_2^\Gd)^T$ and the source $\Bf$ is compactly supported in $\Rbb^2 \setminus \ol{D_e}$ which satisfies the following conservation law:
\beq
  \int_{\Rbb^2} \Bf = {\bf 0}. \label{conservation}
\eeq

Let $\Bu^\Gd$ be the solution to \eqnref{negative_indx_elasticity_problem} and define
\beq
  E(\Bu^\Gd):= \int_{D_e \setminus D_i} \Gl | \nabla \cdot \Bu^\Gd |^2 + 2 \Gm |\hatna \Bu^\Gd |^2.
\eeq
Here and afterwards, $\BA : \BB$ for two matrices $\BA = \left( a_{ij} \right)$ and $\BB = \left( b_{ij} \right)$ denotes $\sum_{i, j} a_{ij} b_{ij}$, and $|\BA|^2=\BA:\BA$. Then CALR is characterized by the following two conditions:
\begin{itemize}
\item the blow up of the dissipated energy on the annulus:
\beq\label{blow_up_diss_energy}
    E^\Gd := \Gd E(\Bu^\Gd) \to \infty \quad \text{ as } \Gd \to 0,
\eeq
\item the boundedness of the solution $\Bu^\Gd(\Bx)$ far away from the structure; more precisely, there is a radius $a > r_e$ such that $\vert \Bu^\Gd(\Bx) \vert < C$ for some $C > 0$ on $\vert \Bx \vert > a$ as $\Gd \to 0$.
\end{itemize}
It is worth mentioning that $E^\Gd$ is the imaginary part of the total energy, namely,
$$
E^\Gd = \Im \int_{\Rbb^2} \Cbb^\Gd \hatna\Bu^\Gd : \hatna\Bu^\Gd .
$$

\subsection{Layer potential formulation of the CALR problem}

We now formulate the CALR problem \eqnref{negative_indx_elasticity_problem} using layer potentials and define the NP operator for the problem. The formulation here is parallel to that in \cite{ACKLM} for the electro-static case. So we omit proofs.

Let us fix notation first: let $\Hscr^* = H^{-1 / 2}(\GG_i)^2 \times H^{-1 / 2}(\GG_e)^2$ and  $\Hscr^*_\GY:= H^{-1/2}_\GY(\GG_i)^2 \times H^{-1/2}_\GY (\GG_e)^2$. Let $\BF$ be the Newtonian potential of $\Bf$:
\beq
  \BF(\Bx) := \int_{\Rbb^2} \BGG(\Bx - \By) \Bf(\By) d\By, \quad \Bx \in \Rbb^2. \label{Fdef}
\eeq
Note that $\BF$ satisfies $\Lcal_{\Gl, \Gm} \BF = \Bf$ in $\Rbb^2$ and $\BF(\Bx) \to {\bf 0}$ as $\vert \Bx \vert \to \infty$ since $\Bf$ satisfies \eqref{conservation}. Let $\BS_{\GG_i}$ and $\BS_{\GG_e}$ be the single layer potentials on $\GG_i$ and $\GG_e$, respectively, with respect to the Lam\'e parameters $(\Gl, \Gm)$.
Then we seek the solution $\Bu^\Gd$ of \eqnref{negative_indx_elasticity_problem} in the following form:
\beq\label{rep_sol}
  \Bu^\delta(\Bx) = \BF(\Bx) + \BS_{\GG_i} [\BGvf_i^\Gd] (\Bx) + \BS_{\GG_e} [\BGvf_e^\Gd] (\Bx)
\eeq
for some $(\BGvf_i^\Gd ,\BGvf_e^\Gd) \in \Hscr^*_\GY$. We emphasize that the solution can be represented as \eqnref{rep_sol} because the Lam\'e parameters of the shell is of the form $(c + i \Gd) (\Gl, \Gm)$. If they are not of the form, then the single layer potential with different Lam\'e parameter should be used, and arguments of this paper may not be applied.

The transmission conditions to be satisfied by $\Bu^\Gd$ on $\GG_i$ and $\GG_e$ are
\beq
  \begin{cases}
    \left( c + i \Gd \right) \left. \p_\Gv \Bu^\Gd \right|_+ =\left. \p_\Gv \Bu^\Gd \right|_- & \text{ on } \GG_i, \\
    \left. \p_\Gv \Bu^\Gd \right|_+ = \left( c + i \Gd \right) \left. \p_\Gv \Bu^\Gd \right|_- & \text{ on } \GG_e.
  \end{cases} \nonumber
\eeq
So $(\BGvf_i^\Gd, \BGvf_e^\Gd)$ is the solution to the following system of integral equations:
\beq
  \begin{cases}
    & \left( c + i \Gd \right) \left. \p_{\Gv_i} \BS_{\GG_i} [\BGvf_i^\Gd] \right|_+ - \left. \p_{\Gv_i} \BS_{\GG_i} [\BGvf_i^\Gd] \right|_- + \left(c - 1 + i \Gd \right) \p_{\Gv_i} \BS_{\GG_e} [\BGvf_e^\Gd] \\
    &\qquad\qquad\qquad = \left(1 - c - i \Gd \right) \p_{\Gv_i} \BF \quad \text{ on } \GG_i, \\
    & \left(1 - c - i \Gd \right) \p_{\Gv_e} \BS_{\GG_i} [\BGvf_i^\Gd] + \left. \p_{\Gv_e} \BS_{\GG_e} [\BGvf_e^\Gd] \right|_+ - \left( c + i \Gd \right) \left. \p_{\Gv_e} \BS_{\GG_e} [\BGvf_e^\Gd] \right|_- \\
    &\qquad\qquad\qquad = \left(c - 1 + i \Gd \right) \p_{\Gv_e} \BF  \quad \text{ on } \GG_e,
  \end{cases} \nonumber
\eeq
where $\p_{\Gv_i}$ and $\p_{\Gv_e}$ are the conormal derivatives on $\GG_i$ and $\GG_e$, respectively.
Using the jump formula \eqref{single_jump_formula}, the above integral equations are written as
\beq
  \begin{bmatrix}
    - z_\Gd I + \BK_{\GG_i}^* & \p_{\Gv_i} \BS_{\GG_e} \\
    \p_{\Gv_e} \BS_{\GG_i} & z_\Gd I + \BK_{\GG_e}^*
  \end{bmatrix}
  \begin{bmatrix}
    \BGvf_i^\Gd \\
    \BGvf_e^\Gd
  \end{bmatrix}
  = -
  \begin{bmatrix}
    \p_{\Gv_i} \BF \\
    \p_{\Gv_e} \BF
  \end{bmatrix}, \label{raw_matrix_equation}
\eeq
where
\beq\label{zd}
  z_\Gd = \frac{1 + c + i \Gd}{2 \left( 1 - c \right) - 2 i \Gd}.
\eeq
We emphasize that $\p_{\Gv_i} \BF \in H^{-1/2}_\Psi(\GG_i)$ and $\p_{\Gv_e} \BF \in H^{-1/2}_\Psi(\GG_e)$ since $\Lcal_{\Gl, \Gm} \BF=0$ in $D_e$.

Following section~2 in \cite{ACKLM} we define $\Kbb^*: \Hscr^* \to \Hscr^*$ as
\beq\label{Kbbdef}
  \Kbb^* :=
  \begin{bmatrix}
    - \BK_{\GG_i}^* & - \p_{\Gv_i} \BS_{\GG_e} \\
    \p_{\Gv_e} \BS_{\GG_i} & \BK_{\GG_e}^*
  \end{bmatrix}.
\eeq
Then the equation \eqref{raw_matrix_equation} can be rewritten as
\beq
  \left( z_\Gd I + \Kbb^* \right) [{\bf \GF}^\Gd] = \BG, \label{int_equation}
\eeq
where
\beq
  {\bf \GF}^\Gd :=
  \begin{bmatrix}
    \BGvf_i^\Gd \\
    \BGvf_e^\Gd
  \end{bmatrix},
  \quad
  \BG :=
  \begin{bmatrix}
    \p_{\Gv_i} \BF \\
    - \p_{\Gv_e} \BF
  \end{bmatrix}. \nonumber
\eeq
The operator $\Kbb^*$ is the NP operator associated with two circles.

Next, we define $\Sbb$ on $\Hscr^*$ as
\beq\label{Sbbdef}
  \Sbb :=
  \begin{bmatrix}
    \BS_{\GG_i} & \BS_{\GG_e} \\
    \BS_{\GG_i} & \BS_{\GG_e}
  \end{bmatrix}.
\eeq
We emphasize that the $\BS_{\GG_e}$ in the first row is defined to be $\BS_{\GG_e}[\BGvf]|_{\GG_i}$ and $\BS_{\GG_i}$ in the second row is $\BS_{\GG_i}[\BGvf]|_{\GG_e}$. Define, for ${\bf \GF}, {\bf \GY} \in \Hscr^*$,
\beq
  \left( {\bf \GF}, {\bf \GY} \right)_{*} := -\left\la {\bf \GF},  \Sbb [{\bf \GY}] \right\ra, \label{inner_product_Hscr}
\eeq
where $\left\la \cdot, \cdot \right\ra$ denotes the pairing of $L^2(\GG_i)^2 \times L^2(\GG_e)^2$.
Following the same arguments in subsection~3.3 of \cite{ACKLM}, one can show that $( \cdot, \cdot )_{*}$ is actually an inner product on $\Hscr_\GY^*$, and $\Kbb^*$ is a self-adjoint operator with respect to this inner product. Let $\| \cdot \|_*$ be the norm induced by the new inner product. One can show in the same way as in \cite{AJKKY} that this norm is equivalent to the usual $H^{-1/2}$-norm.

Let $\Bu^\Gd$ be the solution to \eqnref{negative_indx_elasticity_problem} given in the form \eqnref{rep_sol}. Using Green's formula, namely,
$$
  \int_{\p \GO} \Bu \cdot \p_\Gv \Bv d\sigma = \int_\GO \Bu \cdot \Lcal_{\Gl, \Gm} \Bv + \int_\GO \Gl \left( \nabla \cdot \Bu \right) \left( \nabla \cdot \Bv \right) + 2 \Gm \hatna \Bu : \hatna \Bv,
$$
one can see that
$$
\int_{D_e \setminus D_i} \Gl | \nabla \cdot (\BS_{\GG_i} [\BGvf_i^\Gd] + \BS_{\GG_e} [\BGvf_e^\Gd] ) |^2 + 2 \Gm | \hatna (\BS_{\GG_i} [\BGvf_i^\Gd] + \BS_{\GG_e} [\BGvf_e^\Gd] ) |^2 = \| \BGF^\Gd \|_*^2.
$$
So, we obtain the following lemma
\begin{lemma}\label{lem:Eudel}
There are constants $C_1$ and $C_2$ such that
\beq
C_1 (\| \BGF^\Gd \|_*^2 -1) \le E(\Bu^\Gd) \le C_2 (\| \BGF^\Gd \|_*^2 +1).
\eeq
\end{lemma}

\section{Asymptotics of NP eigenvalues}\label{sec:NP}

In this section we represent the NP operator as block matrices acting on finite dimensional subspaces, and derive asymptotics of the NP eigenvalues. To do so, we first recall some computations in \cite{AJKKY}.

\subsection{Computations on a disk}

Let $\GG$ be the circle of radius $r_0$. For an integer $m$ let
\beq
\BGvf_m = \begin{bmatrix} \cos m\Go \\
    \sin m\Go \end{bmatrix} \quad\mbox{and}\quad
\widetilde \BGvf_m = \begin{bmatrix} - \sin m\Go \\
    \cos m\Go \end{bmatrix}.
\eeq
The following computations are from \cite{AJKKY}. For $\Bx=(r\cos\Go, r \sin \Go)$,
if $m=1$, then
\beq\label{oneGvf}
- \BS_{\GG} [\BGvf_1](\Bx) =
\begin{cases}
\ds \frac{\Ga_1 - \Ga_2}{2} r \BGvf_1(\Go), & \quad r < r_0,\\
\ds \frac{\Ga_1 - \Ga_2}{2} \frac{r_0^2}{r} \BGvf_1(\Go), & \quad r > r_0.
\end{cases}
\eeq
If $m \ge 2$, then
\beq\label{twoGvf}
- \BS_{\GG} [\BGvf_m](\Bx) =
  \begin{cases}
    \ds \frac{\Ga_1}{2 m} \frac{r^{ m}}{r_0^{m-1}} \BGvf_m(\Go) + \frac{\Ga_2}{2} \left( \frac{r_0^2}{r} - r \right) \frac{r^{m-1}}{r_0^{m-1}} \BGvf_{-m+2}(\Go), & \quad r < r_0, \\
    \ds \frac{\Ga_1}{2 m} \frac{r_0^{m + 1}}{r^{m}} \BGvf_m(\Go), & \quad r > r_0.
  \end{cases}
\eeq
If $m \le -1$, then
\beq\label{negaGvf}
- \BS_{\GG} [\BGvf_m](\Bx) =
  \begin{cases}
    \ds \frac{\Ga_1}{2|m|} \frac{r^{|m|}}{r_0^{|m| - 1}} \BGvf_m(\Go), & \quad r < r_0, \\
    \ds \frac{\Ga_1}{2|m|} \frac{r_0^{|m|+1}}{r^{|m|}} \BGvf_m(\Go) + \frac{\Ga_2}{2} \left( r - \frac{r_0^2}{r} \right) \frac{r_0^{|m|+1}}{r^{|m|+1}} \BGvf_{-m+2}, & \quad r > r_0.
  \end{cases}
\eeq

It is also shown that if $m \ge 2$, then
\beq\label{posiwideGvf}
- \BS_{\GG} [\widetilde\BGvf_m](\Bx) =
  \begin{cases}
  \ds \frac{\Ga_1}{2m} \frac{r^m}{r_0^{m-1}} \widetilde\BGvf_m(\Go) - \frac{\Ga_2}{2} \left( \frac{r_0^2}{r} - r \right) \frac{r^{m-1}}{r_0^{m-1}} \widetilde\BGvf_{-m+2}(\Go), & \quad r < r_0, \\
    \ds \frac{\Ga_1}{2m} \frac{r_0^{m + 1}}{r^{m}}\widetilde\BGvf_m(\Go) , & \quad r > r_0.
  \end{cases}
\eeq
If $m \le -1$, then
\beq\label{negawideGvf}
- \BS_{\GG} [\widetilde\BGvf_m](\Bx) =
  \begin{cases}
      \ds \frac{\Ga_1}{2|m|} \frac{r^{|m|}}{r_0^{|m|-1}} \widetilde\BGvf_m(\Go), & \quad r < r_0, \\
      \ds \frac{\Ga_1}{2|m|} \frac{r_0^{|m|+1}}{r^{|m|}} \widetilde\BGvf_m(\Go) - \frac{\Ga_2}{2} \left( r - \frac{r_0^2}{r} \right) \frac{r_0^{|m|+1}}{r^{|m|+1}} \widetilde\BGvf_{-m+2}(\Go), & \quad r > r_0 .
  \end{cases}
\eeq

As consequences of these computations, eigenvalues of $\BK^*_{\GG}$ are obtained in \cite{AJKKY}:
\beq\label{NPeigen1}
\BK^*_{\GG}[\BGvf_m]= -k_0 \BGvf_m \ (m <0), \quad  \BK^*_{\GG}[\widetilde \BGvf_m]= -k_0 \widetilde\BGvf_m \ (m <0),
\eeq
and
\beq
\BK^*_{\GG}[\BGvf_1]= - \frac{\Gl}{2 (2 \Gm + \Gl)} \BGvf_1, \quad \BK^*_{\GG}[\widetilde \BGvf_m]= k_0 \widetilde\BGvf_m \ (m \ge 2).
\eeq
Even though it is irrelevant to this work, we mention for the completeness's sake that
$$
\BK^*_{\GG}[\widetilde \BGvf_1]= \frac{1}{2} \widetilde\BGvf_1.
$$

\subsection{Block matrix structure of the inner product}

For $n=0, 1,2, \ldots$, we define
\begin{align}
  \BGF_{n, 1} := \begin{bmatrix} \BGvf_{n+1} \\ {\bf 0} \end{bmatrix}, \quad
  \BGF_{n, 2} := \begin{bmatrix} \BGvf_{-n+1} \\ {\bf 0} \end{bmatrix}, \quad
  \BGF_{n, 3} := \begin{bmatrix} {\bf 0} \\ \BGvf_{n+1} \end{bmatrix}, \quad
  \BGF_{n, 4} := \begin{bmatrix} {\bf 0} \\ \BGvf_{-n+1} \end{bmatrix}.
\end{align}
Here ${\bf 0}$ denotes two-dimensional zero vector, and so $\BGF_{n, j}$ are four-dimensional vector-valued functions.
We then define finite dimensional subspaces $\Vscr_n$ of $\Hcal^*$ by
\beq
  \Vscr_n :=
  \mbox{span} \left\{ \BGF_{n, 1}, \BGF_{n, 2}, \BGF_{n, 3}, \BGF_{n, 4} \right\}, \quad n \geq 0.
\eeq
It is worth mentioning that $\Vscr_0$ is of two dimensions and spanned by $\BGF_{0,1}$ and $\BGF_{0,3}$.

Similarly, we define
\begin{align}
  \widetilde \BGF_{n, 1} := \begin{bmatrix} \widetilde \BGvf_{n+1} \\ {\bf 0} \end{bmatrix}, \quad
  \widetilde \BGF_{n, 2} := \begin{bmatrix} \widetilde \BGvf_{-n+1} \\ {\bf 0} \end{bmatrix}, \quad
  \widetilde \BGF_{n, 3} := \begin{bmatrix} {\bf 0} \\ \widetilde \BGvf_{n+1} \end{bmatrix}, \quad
  \widetilde \BGF_{n, 4} := \begin{bmatrix} {\bf 0} \\ \widetilde \BGvf_{-n+1} \end{bmatrix},
\end{align}
and define
\beq
  \widetilde \Vscr_n=
    \mbox{span} \{ \widetilde \BGF_{n, 1}, \widetilde \BGF_{n, 2}, \widetilde \BGF_{n, 3}, \widetilde \BGF_{n, 4} \}, \quad n \geq 0.
\eeq

Using \eqnref{oneGvf}-\eqnref{negawideGvf} in the previous subsection, one can see that
\beq\label{inclusion}
\Sbb(\Vscr_n) \subset \Vscr_n \quad\mbox{and}\quad \Sbb(\widetilde\Vscr_n) \subset \widetilde\Vscr_n \quad\mbox{for each } n.
\eeq
For example, according to the definition \eqnref{Sbbdef}, $\Sbb[\BGF_{n, 1}]$ is given by
$$
\Sbb[\BGF_{n, 1}]=
  \begin{bmatrix}
    \BS_{\GG_i} [\BGvf_{n+1}] |_{\GG_i} \\
    \nm
    \BS_{\GG_i} [\BGvf_{n+1}] |_{\GG_e}
  \end{bmatrix}.
$$
So, using \eqnref{oneGvf}-\eqnref{negaGvf}, one can see that $\Sbb[\BGF_{n, 1}] \in \Vscr_n$.

Moreover, one can derive the block matrix representing $-\Sbb$ on each $\Vscr_n$ or $\widetilde\Vscr_n$. In the following, that $-\Sbb$ is represented by a block matrix $(a_{ij})_{i,j=1}^4$ on $\Vscr_n$ means
$$
-\Sbb[\BGF_{n,i}] = \sum_{j} a_{ij} \BGF_{n,j}, \quad i=1, \ldots, 4.
$$
In fact, they are given as follows. We do not include details of derivations since they are straight-forward, but tedious.
Set $\Gr := r_i / r_e$.
\begin{itemize}
\item[(i)] On $\Vscr_n$, $n \ge 2$,
\beq\label{Sblock}
-\Sbb =
  \begin{bmatrix}
    \frac{\Ga_1}{2 \left( n + 1 \right)} r_i & 0 &  \frac{\Ga_1}{2 \left( n + 1 \right)} r_i \Gr^n & 0 \\
    0 & \frac{\Ga_1}{2 \left( n - 1 \right)} r_i & \frac{\Ga_2}{2}(\frac{r_e^2}{r_i} - r_i) \Gr^n & \frac{\Ga_1}{2 \left( n - 1 \right)} \frac{r_e^2}{r_i} \Gr^n \\
    \frac{\Ga_1}{2 \left( n + 1 \right)} \frac{r_i^2}{r_e} \Gr^n &  \frac{\Ga_2}{2} \left( r_e - \frac{r_i^2}{r_e} \right) \Gr^n &  \frac{\Ga_1}{2 \left( n + 1 \right)} r_e & 0 \\
    0  &  \frac{\Ga_1}{2 \left( n - 1 \right)} r_e \Gr^n  & 0 &  \frac{\Ga_1}{2 \left( n - 1 \right)} r_e
  \end{bmatrix} .
\eeq

\item[(ii)] On $\widetilde{\Vscr}_n$, $n \ge 2$,
\beq\label{wideSblock}
  -\Sbb =
  \begin{bmatrix}
    \frac{\Ga_1}{2 \left( n + 1 \right)} r_i & 0 &  \frac{\Ga_1}{2 \left( n + 1 \right)} r_i \Gr^n & 0 \\
    0 &  \frac{\Ga_1}{2 \left( n - 1 \right)} r_i & -\frac{\Ga_2}{2} \left( \frac{r_e^2}{r_i} - r_i \right) \Gr^n &  \frac{\Ga_1}{2 \left( n - 1 \right)} \frac{r_e^2}{r_i} \Gr^n \\
    \frac{\Ga_1}{2 \left( n + 1 \right)} \frac{r_i^2}{r_e} \Gr^n & -\frac{\Ga_2}{2} \left( r_e - \frac{r_i^2}{r_e} \right) \Gr^n & \frac{\Ga_1}{2 \left( n + 1 \right)} r_e & 0 \\
    0 & \frac{\Ga_1}{2 \left( n - 1 \right)} r_e \Gr^n  & 0 & \frac{\Ga_1}{2 \left( n - 1 \right)} r_e
  \end{bmatrix}.
\eeq
\end{itemize}
We mention that only the cases when $n \ge 2$ are presented above. It is because the cases of large $n$ matter for our analysis.

As a consequence we obtain the following lemma.

\begin{lemma}\label{lem:decomp_Hscr}
\begin{itemize}
\item[(i)] It holds that $\Vscr_m \perp \Vscr_n$ and $\widetilde{\Vscr}_m \perp \widetilde{\Vscr}_n$ if $m\neq n$, and $\Vscr_m \perp \widetilde{\Vscr}_n$ for all $m$ and $n$, with respect to the inner product $( \cdot, \cdot )_{*}$.
\item[(ii)] For $j=1, \ldots, 4$,
\beq\label{phinest1}
( \BGF_{n, j}, \BGF_{n, j} )_{*} \approx \frac{1}{n}, \quad ( \widetilde\BGF_{n, j}, \widetilde\BGF_{n, j} )_{*} \approx \frac{1}{n}  .
\eeq
\item[(iii)] If $i \neq j$, then
\beq\label{phinest2}
      \left| ( \BGF_{n, i}, \BGF_{n, j} )_{*} \right| \lesssim \Gr^n, \quad \left| ( \widetilde\BGF_{n, i}, \widetilde\BGF_{n, j} )_{*} \right| \lesssim \Gr^n .
\eeq
  \end{itemize}
\end{lemma}
In the above and afterwards $( \BGF_{n, j}, \BGF_{n, j} )_{*} \approx \frac{1}{n}$ mean that there are constants $C_1$ and $C_2$ independent of $n$ such that
$$
\frac{C_1}{n} \le \left( \BGF_{n, j}, \BGF_{n, j}\right)_{*} \le \frac{C_2}{n},
$$
and $| ( \BGF_{n, i}, \BGF_{n, j} )_{*} | \lesssim \Gr^n$ means that there is a constant $C$ independent of $n$ such that
$$
\left| ( \BGF_{n, i}, \BGF_{n, j} )_{*} \right| \leq C \Gr^n.
$$

\noindent {\sl Proof of Lemma \ref{lem:decomp_Hscr}}.
The item (i) is an immediate consequence of \eqnref{inclusion}. For (ii) and (iii), we note that $\la \BGF_{n, i}, \BGF_{n, j}\ra_{L^2}=0$ if $i \neq j$, and $\la \BGF_{n, j}, \BGF_{n, j}\ra_{L^2}$ is $2\pi r_i$ if $j=1,2$, and $2\pi r_e$ if $j=3,4$. We then observe that the diagonal elements of the matrices in \eqnref{Sblock} and \eqnref{wideSblock} are of order $n^{-1}$, and off-diagonal elements are of order $\Gr^n$. So, \eqnref{phinest1} and \eqnref{phinest2} follow.
\qed

\subsection{Block matrix structure of the NP operator}

We now represent the NP operator $\Kbb^*$ on $\Vscr_n$ and $\widetilde\Vscr_n$.
According to the definition \eqnref{Kbbdef}, $\Kbb^*[\BGF_{n, 1}]$, for examples, is given by
$$
\Kbb^* [\BGF_{n, 1}] =
  \begin{bmatrix}
    - \BK_{\GG_i}^* [\BGvf_{n+1}] |_{\GG_i} \\
    \nm
    \p_{\Gv_e} \BS_{\GG_i} [\BGvf_{n+1}] |_{\GG_e}
  \end{bmatrix}.
$$
So, using \eqnref{twoGvf} and \eqnref{NPeigen1}, one can compute $\Kbb^* [\BGF_{n, 1}]$. In this way, one can show that
\beq\label{Kinclusion}
\Kbb^*(\Vscr_n) \subset \Vscr_n \quad\mbox{and}\quad \Kbb^*(\widetilde\Vscr_n) \subset \widetilde\Vscr_n \quad\mbox{for each } n.
\eeq
We also obtain the following lemma for the block matrix representation of the NP operator. We again omit the proof since it is straight-forward, but tedious.

\begin{lemma}\label{lem:NPblock}
The NP-operator $\Kbb^*$ admits the following matrix representation:
\begin{itemize}
\item[(i)] On $\Vscr_n$, $n \ge 2$,
\beq\label{J+Mn}
  \Kbb^* = J + M_n,
\eeq
where
\begin{equation*}
  J = k_0
  \begin{bmatrix}
    -1 && 0 && 0 && 0 \\
    \ds 0 && 1 && 0 && 0 \\
    \ds 0 && 0 && 1 && 0 \\
    \ds 0 && 0 && 0 && -1
  \end{bmatrix}
\end{equation*}
and
\begin{equation*}
  M_n = \Gr^n
  \begin{bmatrix}
    0 & 0 & \Gm \Ga_2 & 0 \\
    0 & 0 & - \left( n - 1 \right) \Gm \Ga_2 \left( 1 - \Gr^{-2} \right) & \Gm \Ga_1 \Gr^{-2} \\
    \Gm \Ga_1 \Gr^2 & - \left( n + 1 \right) \Gm \Ga_2 \left( \Gr^2 - 1 \right) & 0 & 0 \\
    0 & \Gm \Ga_2 & 0 & 0
  \end{bmatrix}.
\end{equation*}

\item[(ii)] On $\widetilde{\Vscr_n}$, $n \ge 2$:
\begin{equation*}
  \Kbb^* = J + \widetilde M_n,
\end{equation*}
where
\begin{equation*}
  \widetilde M_n = \Gr^n
  \begin{bmatrix}
    0 & 0 & \Gm \Ga_2 & 0 \\
    0 & 0 & \left( n - 1 \right) \Gm \Ga_2 \left( 1 - \Gr^{- 2} \right) & \Gm \Ga_1 \Gr^{- 2} \\
    \Gm \Ga_1 \Gr^2 & \left( n + 1 \right) \Gm \Ga_2 \left( \Gr^2 - 1 \right) & 0 & 0 \\
    0 & \Gm \Ga_2 & 0 & 0
  \end{bmatrix}.
\end{equation*}
\end{itemize}
\end{lemma}

\subsection{Asymptotics of NP eigenvalues}

It turns out that the exact expression of the eigenvalues of $\Kbb^*$ on $\Vscr_n$, or of the matrix $J+ M_n$ in \eqnref{J+Mn}, is extremely lengthy and complicated. However, for analysis of this paper, it is enough
to have their asymptotic behavior as $n \to \infty$, which we obtain in this subsection using the perturbation theory.

To investigate the asymptotic behavior of the eigenvalues, it is more convenient to express $\Kbb^*= J + M_n$ on $\Vscr_n$ as follows:
\begin{equation*}
  \Kbb^* = P_n + \Gr^n Q,
\end{equation*}
where
\begin{equation*}
  P_n =
  \begin{bmatrix}
    -k_0 & 0 & 0 & 0 \\
    \ds 0 & k_0 & -n \rho^n \mu\alpha_2 (1-\rho^{-2}) & 0 \\
    \ds 0 & -n \rho^n \mu\alpha_2(\rho^2-1) & k_0 & 0 \\
    \ds 0 & 0 & 0 & -k_0
  \end{bmatrix},
\end{equation*}
and
\begin{equation*}
  Q =
  \begin{bmatrix}
    0 & 0 & \Gm \Ga_2 & 0 \\
    0 & 0 & \Gm \Ga_2 \left( 1 - \Gr^{-2} \right) & \Gm \Ga_1 \Gr^{-2} \\
    \Gm \Ga_1 \Gr^2 & - \Gm \Ga_2 \left( \Gr^2 - 1 \right) & 0 & 0 \\
    0 & \Gm \Ga_2 & 0 & 0
  \end{bmatrix}.
\end{equation*}
It is worth emphasizing that $Q$ is independent of $n$.

It is easy to find exact eigenvalues and eigenfunctions of $P_n$. In fact, they are given as follows:
\begin{align}
&\Gl_{n,1}^0 = -k_0, \qquad\qquad\qquad\qquad\qquad \quad \BE_{n,1}^0=[ 1, 0, 0, 0]^T,
\\
&\Gl_{n,2}^0 = +k_0 -n \rho^n \Gm \Ga_2(\Gr-\Gr^{-1}), \qquad \BE_{n,2}^0=[0,1,\Gr,0]^T,  \label{BEn2} \\
&\Gl_{n,3}^0 = +k_0 +n \rho^n \Gm \Ga_2(\Gr-\Gr^{-1}), \qquad \BE_{n,3}^0=[0,-1,\Gr,0]^T,
\\
&\Gl_{n,4}^0 = -k_0, \qquad \qquad\qquad\qquad\quad\quad \quad \BE_{n,4}^0= [ 0, 0, 0, 1]^T.
\end{align}
Here and afterwards, $\BE_{n,j}^0$ written as a vector in $\Rbb^4$ actually represents a four dimensional vector-valued function. For example, $\BE_{n,2}^0$ in \eqnref{BEn2} represents $\BGF_{n,2} + \Gr \BGF_{n,3}$.

Note that, if $n$ is large, the matrix $\Gr^n Q$ becomes a small perturbation matrix. So we can derive asymptotic formula for eigenvalues using  standard arguments of the eigenvalue perturbation theory (see, for example, section XII of \cite{RS}).
We only mention that since eigenvalues $\Gl_2^0$ and $\Gl_3^0$ are simple, we apply non-degenerate perturbation theory. For
the cases of $\Gl_1^0=\Gl_4^0$, we apply the degenerate perturbation theory.

Let $\Gl_{n,j}$ and $\BE_{n,j}$ $(n \ge 1, j=1,2,3,4)$
be the eigenvalues and eigenvectors of $\mathbb{K}^*$, respectively. Then we have
\begin{align}
&\Gl_{n,1} = -k_0 - \rho^{2n} \frac{\mu^2\alpha_1\alpha_2 \rho^4}{k_0(1+\rho^2)}  +O(\rho^{3n}), \qquad \ \BE_{n,1}=\BE_{n,1}^0+O(n\rho^n),
\label{Gln1} \\
&\Gl_{n,2} = +k_0 -n \rho^n \Gm \Ga_2(\Gr-\Gr^{-1})+ O(\rho^n), \qquad \BE_{n,2}=\BE_{n,2}^0+O(\rho^n),
\label{Gln2} \\
&\Gl_{n,3}= +k_0 +n \rho^n \Gm \Ga_2(\Gr-\Gr^{-1})+ O(\rho^n), \qquad \BE_{n,3}=\BE_{n,3}^0+O(\rho^n),
\label{Gln3} \\
&\Gl_{n,4} = -k_0 - \rho^{2n} \frac{\mu^2\alpha_1\alpha_2 \rho^{-2}}{k_0(1+\rho^2)}  +O(\rho^{3n}),\qquad \BE_{n,4}=\BE_{n,4}^0+O(n\rho^n). \label{Gln4}
\end{align}

Similarly, we can also obtain the asymptotic behavior of eigenvalues of $\mathbb{K}^*$ on $\widetilde{\Vscr}_n$ as follows
\begin{align}
&\widetilde{\Gl}_{n,1} = -k_0 - \rho^{2n} \frac{\mu^2\alpha_1\alpha_2 \rho^4}{k_0(1+\rho^2)}  +O(\rho^{3n}),
\label{tildeGln1} \\
&\widetilde\Gl_{n,2}= +k_0 + n \rho^n \Gm \Ga_2(\Gr-\Gr^{-1}) +O(\rho^n),
\\
& \widetilde\Gl_{n,3} = +k_0 - n \rho^n \Gm \Ga_2(\Gr-\Gr^{-1}) + O(\rho^n),
\\
&\widetilde{\Gl}_{n,4} = -k_0 - \rho^{2n} \frac{\mu^2\alpha_1\alpha_2 \rho^{-2}}{k_0(1+\rho^2)}  +O(\rho^{3n}), \label{tildeGln4}
\end{align}
and the corresponding eigenvectors are
\begin{align}
&\widetilde{\BE}_{n,1}=[1, 0, 0, 0]^T+O(n\rho^n), \label{tildeBE1}
\\
&\widetilde{\BE}_{n,2}=[0,1,\rho,0]^T+O(\rho^n),
\\
&\widetilde{\BE}_{n,3}=[0,-1,\rho,0]^T+O(\rho^n),
\\
&\widetilde{\BE}_{n,4}=[0, 0, 0, 1]^T+O(n\rho^n). \label{tildeBE4}
\end{align}
We emphasize that all eigenvalues in \eqnref{Gln1}--\eqnref{tildeGln4} converge to either $k_0$ or $-k_0$.

In view of \eqnref{phinest1}, \eqnref{phinest2}, \eqnref{Gln1}--\eqnref{Gln4}, and \eqnref{tildeBE1}--\eqnref{tildeBE4}, we obtain the following lemma.

\begin{lemma}\label{eigen_inner_estim}
We have
\begin{align}
( \BE_{n, j}, \BE_{n, j} )_{*} & \approx n^{-1}, \quad ( \widetilde\BE_{n, j}, \widetilde\BE_{n, j})_{*} \approx n^{-1} \quad \mbox{for } 1 \leq j \leq 4,  \\
|( \BE_{n, j}, \BGF_{n, j} )_{*}| & \approx  n^{-1}, \quad  |( \widetilde\BE_{n, j}, \widetilde\BGF_{n, j} )_{*}| \approx  n^{-1} \quad \mbox{for } j=1,4, \\
| ( \BE_{n, j}, \BGF_{n, k} )_{*} | & \lesssim \Gr^n, \quad | ( \widetilde\BE_{n, j}, \widetilde\BGF_{n, k} )_{*} | \lesssim \Gr^n \quad \mbox{for }  j=1,4, \, k\neq j, \\
|( \BE_{n, j}, \BGF_{n, k} )_{*}|  & \approx  n^{-1}, \quad |( \widetilde\BE_{n, j}, \widetilde\BGF_{n, k} )_{*}|  \approx  n^{-1} \quad \mbox{for }j=2,3, \,k=2,3, \\
| ( \BE_{n, j}, \BGF_{n, k} )_{*} | & \lesssim \Gr^n, \quad | ( \widetilde\BE_{n, j}, \widetilde\BGF_{n, k} )_{*} | \lesssim \Gr^n \quad \mbox{for }  j=2,3, \,k=1,4.
\end{align}
\end{lemma}

\section{CALR}\label{sec:CALR}

\subsection{Estimates of the solution to the integral equation}

We now look into the integral equation \eqref{int_equation}. We first observe that since $\BG \in \Hscr_\GY^*$, $\BG$ is orthogonal to $\widetilde{\Vscr}_0$ in particular. So $\BG$ is uniquely represented as
$$
  \BG= \sum_{n = 0}^\infty \BG_n + \sum_{n = 1}^\infty \widetilde{\BG}_n, \quad \BG_n \in \Vscr_n, \ \widetilde{\BG}_n \in \widetilde{\Vscr}_n.
$$
It may be helpful to mention that $\BG_1$ has no component of $\BGF_{1,2}$ and $\BGF_{1,4}$ since $\BG$ is perpendicular to constant vectors. The solution $\BGF^\Gd$ to \eqref{int_equation} is given by
\beq
  {\bf \GF}^\delta= \sum_{n = 0}^\infty \BGF_n^\Gd + \sum_{n=1}^\infty \widetilde{\BGF}_n^\Gd, \label{BGFdelta}
\eeq
where $\BGF_n^\Gd \in \Vscr_n$ and $\widetilde{\BGF}_n^\Gd \in \widetilde{\Vscr}_n$ are, respectively, the solutions to
the finite dimensional equations
\beq \label{Int_eqn_mat}
  (z_{\Gd} I + \Kbb^*) \BGF_n^\Gd = \BG_n \ \text{ on } \Vscr_n, \quad (z_{\Gd} I + \Kbb^*) \widetilde{\BGF}_n^\Gd = \widetilde{\BG}_n \ \text{ on }
  \widetilde{\Vscr}_n.
\eeq

Let us consider the first equation in the above.
Since $\BE_{n,j}$, $1 \le j \le 4$, is an orthogonal basis for $\Vscr_n$, we have
$$
\BGF_n^\Gd = \sum_{j=1}^4 \frac{(\BG_n,
\BE_{n,j})_*}{(\BE_{n,j},\BE_{n,j})_* (z_{\Gd} + \Gl_{n,j})}
\BE_{n,j},
$$
and hence
\beq\label{BGFn1}
\| \BGF_n^\Gd \|_*^2 = \sum_{j=1}^4 \frac{|(\BG_n,\BE_{n,j})_*|^2}{\| \BE_{n,j} \|_*^2 |z_{\Gd} + \Gl_{n,j}|^2}.
\eeq

Note that $z_\Gd \to z(c)$ as $\Gd \to 0$ where $z(c)$ is defined in \eqnref{ccond}.
On the other hand, $\Gl_{n,j}$ approaches to either $k_0$ or $-k_0$ as $n \to \infty$. So, if $z(c) \neq \pm k_0$, then $|z_\Gd + \Gl_{n,j}| \ge C$ for some constant $C$ for all sufficiently large $n$ if $\Gd$ is small. So the norm given in \eqnref{BGFn1} does not blow up. So, we assume that $z(c)$ is either $k_0$ or $-k_0$.

Suppose that $z(c)=k_0$. If $j=2,3$, then we infer from \eqnref{Gln2} and \eqnref{Gln3} that $|z_\Gd + \Gl_{n,j}| \ge C$ for some constant $C$ independent of $n$. So, we have
\beq\label{BGFn2}
\sum_{j=2,3} \frac{|(\BG_n,\BE_{n,j})_*|^2}{\| \BE_{n,j} \|_*^2 |z_{\Gd} + \Gl_{n,j}|^2} \lesssim \| \BG_n \|_*^2.
\eeq
If $j=1,4$, then we infer from \eqnref{Gln1} and \eqnref{Gln4} that
\beq\label{BGFn3}
|z_\Gd + \Gl_{n,j}|^2 \approx \Gd^2 + \rho^{4n},
\eeq
and hence
\beq\label{BGFn4}
\sum_{j=1,4} \frac{|(\BG_n,\BE_{n,j})_*|^2}{\| \BE_{n,j} \|_*^2 |z_{\Gd} + \Gl_{n,j}|^2} \approx \sum_{j=1,4} \frac{|(\BG_n,\BE_{n,j})_*|^2}{\| \BE_{n,j} \|_*^2 (\Gd^2 + \rho^{4n})} .
\eeq

Since $\BG_n \in \Vscr_n$ and $\widetilde \BG_n \in \widetilde \Vscr_n$, we have
\beq\label{BGndef}
\BG_n = \sum_{k=1}^4 g_{n,k} \BGF_{n,k}, \quad \widetilde \BG_n = \sum_{k=1}^4 \widetilde g_{n,k} \widetilde \BGF_{n,k},
\eeq
for some constants $g_{n,k}$ and $\widetilde g_{n,k}$. Since $(\BG_{n}, \BE_{n,1})_{*} = \sum_{k=1}^4 g_{n,k}(\BGF_{n,k}, \BE_{n,1})_{*}$, one can see from Lemma \ref{eigen_inner_estim} that
\beq\label{BGFn5}
\frac{|g_{n,1}|^2}{n^2} - \rho^{2n} \sum_{k \neq 1} |g_{n,k}|^2 \lesssim |(\BG_{n}, \BE_{n,1})_{*}|^2 \lesssim \frac{|g_{n,1}|^2}{n^2} + \rho^{2n} \sum_{k \neq 1} |g_{n,k}|^2.
\eeq
Likewise, we have
\beq\label{BGFn6}
\frac{|g_{n,4}|^2}{n^2} - \rho^{2n} \sum_{k \neq 4} |g_{n,k}|^2 \lesssim |(\BG_{n}, \BE_{n,4})_{*}|^2 \lesssim \frac{|g_{n,4}|^2}{n^2} + \rho^{2n} \sum_{k \neq 4} |g_{n,k}|^2.
\eeq
It then follows from Lemma \ref{eigen_inner_estim} and \eqnref{BGFn4}--\eqnref{BGFn6} that
\begin{align}
&\frac{n^{-1} (|g_{n,1}|^2+ |g_{n,4}|^2) - n\rho^{2n} (|g_{n,2}|^2+ |g_{n,3}|^2) }{\Gd^2 + \rho^{4n}} \lesssim \nonumber \\
& \sum_{j=1,4} \frac{|(\BG_n,\BE_{n,j})_*|^2}{\| \BE_{n,j} \|_*^2 |z_{\Gd} + \Gl_{n,j}|^2} \lesssim \frac{n^{-1} (|g_{n,1}|^2+ |g_{n,4}|^2) + n\rho^{2n} (|g_{n,2}|^2+ |g_{n,3}|^2) }{\Gd^2 + \rho^{4n}}. \label{BGFn7}
\end{align}
So, we obtain from \eqnref{BGFn1}, \eqnref{BGFn2}, and \eqnref{BGFn7} that
\begin{align}
&\frac{n^{-1} (|g_{n,1}|^2+ |g_{n,4}|^2) - n\rho^{2n} (|g_{n,2}|^2+ |g_{n,3}|^2) }{\Gd^2 + \rho^{4n}} - \| \BG_n \|_*^2 \lesssim \nonumber \\
& \| \BGF_n^\Gd \|_*^2 \lesssim
\frac{n^{-1} (|g_{n,1}|^2+ |g_{n,4}|^2) + n\rho^{2n} (|g_{n,2}|^2+ |g_{n,3}|^2) }{\Gd^2 + \rho^{4n}} + \| \BG_n \|_*^2 . \label{BGFn8}
\end{align}

One can estimate $\| \widetilde\BGF_n^\Gd \|_*^2$ in a similar way to obtain
\begin{align}
&\frac{n^{-1} (|\widetilde g_{n,1}|^2+ |\widetilde g_{n,4}|^2) - n\rho^{2n} (|\widetilde g_{n,2}|^2+ |\widetilde g_{n,3}|^2) }{\Gd^2 + \rho^{4n}} - \| \widetilde \BG_n \|_*^2 \lesssim \nonumber \\
& \| \widetilde \BGF_n^\Gd \|_*^2 \lesssim
\frac{n^{-1} (|\widetilde g_{n,1}|^2+ |\widetilde g_{n,4}|^2) + n\rho^{2n} (|\widetilde g_{n,2}|^2+ |\widetilde g_{n,3}|^2) }{\Gd^2 + \rho^{4n}} + \| \widetilde \BG_n \|_*^2 . \label{BGFn9}
\end{align}

Let, for ease of notation,
\beq\label{Indef}
I_n:= |g_{n,1}|^2+ |g_{n,4}|^2 +|\widetilde g_{n,1}|^2+ |\widetilde g_{n,4}|^2
\eeq
and
\beq\label{IIndef}
II_n:= |g_{n,2}|^2+ |g_{n,3}|^2 +|\widetilde g_{n,2}|^2+ |\widetilde g_{n,3}|^2.
\eeq
Since
$$
\sum_{n=0}^\infty \| \BG_n \|_*^2 + \sum_{n=1}^\infty \| \widetilde \BG_n \|_*^2 = \| \BG \|_*^2,
$$
we infer by summing \eqnref{BGFn8} and \eqnref{BGFn9} over $n$ that there are constants $C_1$ and $C_2$ independent of $\Gd$ such that
\beq\label{BGFn10}
C_1 \left( \sum_{n=1}^\infty \frac{n^{-1} I_n - n\rho^{2n} II_n}{\Gd^2 + \rho^{4n}} - 1 \right) \le \| \BGF^\Gd \|_*^2 \le C_2 \left( \sum_{n=1}^\infty \frac{n^{-1} I_n + n\rho^{2n} II_n}{\Gd^2 + \rho^{4n}} + 1 \right).
\eeq

If $z(c)=-k_0$, then we obtain in a similar way that
\beq\label{BGFn11}
C_1 \left( \sum_{n=1}^\infty \frac{n^{-1} II_n - n\rho^{2n} I_n}{\Gd^2 + n^2\rho^{2n}} - 1 \right) \le \| \BGF^\Gd \|_*^2 \le C_2 \left( \sum_{n=1}^\infty \frac{n^{-1} II_n + n\rho^{2n} I_n}{\Gd^2 + n^2\rho^{2n}} + 1 \right).
\eeq
It is worth emphasizing that the quantity $n(\Gd^2 + n^2\rho^{2n})$ in the denominator is different from that in \eqnref{BGFn10}. This discrepancy is caused by the different asymptotic behaviors of eigenvalues near $k_0$ and $-k_0$ as shown in \eqnref{Gln1}--\eqnref{tildeGln4}.

In summary, we obtain the following proposition from Lemma \ref{lem:Eudel}, \eqnref{BGFn10} and \eqnref{BGFn11}.

\begin{prop}\label{prop:Eudel}
There are constants $C_1$ and $C_2$ independent of $\Gd$ such that
\begin{itemize}
\item[(i)] if $z(c)=k_0$, then
\beq\label{Eudelone}
C_1 \left( \sum_{n=1}^\infty \frac{n^{-1} I_n - n\rho^{2n} II_n}{\Gd^2 + \rho^{4n}} - 1 \right) \le E(\Bu^\Gd) \le C_2 \left( \sum_{n=1}^\infty \frac{n^{-1} I_n + n\rho^{2n} II_n}{\Gd^2 + \rho^{4n}} + 1 \right),
\eeq
\item[(ii)] if $z(c)=-k_0$, then
\beq\label{Eudeltwo}
C_1 \left( \sum_{n=1}^\infty \frac{n^{-1} II_n - n\rho^{2n} I_n}{\Gd^2 + n^2\rho^{2n}} - 1 \right) \le E(\Bu^\Gd) \le C_2 \left( \sum_{n=1}^\infty \frac{n^{-1} II_n + n\rho^{2n} I_n}{\Gd^2 + n^2\rho^{2n}} + 1 \right).
\eeq
\end{itemize}
\end{prop}

\subsection{Resonance by dipole sources}\label{sec:dipole}

In this section we assume that the source function $\Bf$ in \eqnref{negative_indx_elasticity_problem} is given by the dipole-type function, namely,
\beq\label{dipole}
  \Bf (\Bx) = \Bb^T \nabla_{\Bx} \left(
    \begin{bmatrix}
      \Gd_{\Bz}(\Bx) & 0 \\
      0 & \Gd_{\Bz}(\Bx)
    \end{bmatrix} \Ba \right),
\eeq
where $\Bz = (z_1, z_2)^T$ is the location of the dipole outside $D_e$, and $\Ba = (a_1, a_2)^T$, $\Bb = (b_1, b_2)^T$ are constant vectors. With this source function we obtain the following theorem using Proposition \ref{prop:Eudel}. In what follows, the notation $A \sim B$ ($A$ and $B$ are two quantities depending on $\Gd$) indicates that there are constants $C_1$ and $C_2$ independent of $\Gd$ such that $C_1 \le A/B \le C_2$.

\begin{theorem} \label{thm:res+}
Let $\Bf$ be given by \eqref{dipole}, and let $\rho_\Bz = r_e/|\Bz|$. It holds that
\begin{itemize}
\item[(i)] if $z(c) = k_0$, then
\beq\label{itemone}
E(\Bu^\Gd) \sim \left| \ln \Gd \right|^3 \Gd^{\frac{\ln \rho_\Bz}{\ln \rho} - 2} \quad \text{as } \Gd \to 0,
\eeq

\item[(ii)] if $z(c) = -k_0$, then
\beq\label{itemtwo}
E(\Bu^\Gd) \sim \left| \ln \Gd \right|^{-1-\frac{2\ln\rho_\Bz}{\ln \rho}} \Gd^{\frac{2\ln \rho_\Bz}{\ln \rho} - 2} \quad \text{as } \Gd \to 0.
\eeq
\end{itemize}
\end{theorem}

As an immediate consequence we obtain the following corollary.

\begin{cor}\label{cor:res+}
Suppose that $\Bf$ is given by \eqref{dipole}.
\begin{itemize}
\item[(i)] If $z(c) = k_0$, let $r_*:=r_e^2 / r_i$. As $\Gd \to 0$,
\beq
\Gd E(\Bu^\Gd) \to
\begin{cases}
\infty \quad &\mbox{if } r_e < |\Bz| \le r_* , \\
0 \quad &\mbox{if } |\Bz| > r_*.
\end{cases}
\eeq
\item[(ii)] If $z(c) = -k_0$, let $r_{**}:=\sqrt{r_e^3 / r_i}$. As $\Gd \to 0$,
\beq
\Gd E(\Bu^\Gd) \to
\begin{cases}
\infty \quad &\mbox{if } r_e < |\Bz| < r_{**} , \\
0 \quad &\mbox{if } |\Bz| \ge r_{**}.
\end{cases}
\eeq
\end{itemize}
\end{cor}

\noindent{\sl Proof of Theorem \ref{thm:res+}}.
Since $\Bf$ is assumed to be of the form \eqref{dipole}, $\BF$ defined by \eqnref{Fdef} is given by
\beq
  \BF(\Bx) = \Bb^T \nabla_{\Bx} \left( \BGG(\Bx - \Bz) \Ba \right) = - \Bb^T \nabla_\Bz \left( \BGG(\Bx - \Bz) \Ba \right).
\eeq
Let $g_{m, j}$ and $\widetilde g_{m, j}$, $j = 1,2,3,4$, be the coefficients of $\BG= (\p_{\Gv_i} \BF , - \p_{\Gv_e} \BF)^T$ as
defined by \eqnref{BGndef}. It turns out that $g_{n, j}$ and $\widetilde g_{n, j}$ satisfy
\beq\label{gnone}
  g_{n, 1}, \widetilde g_{n, 1} \approx \left( \frac{r_i}{|\Bz|} \right)^n, \quad
  g_{n, 2}, \widetilde g_{n, 2} \approx n^2 \left( \frac{r_i}{|\Bz|} \right)^n,
\eeq
and
\beq\label{gntwo}
  g_{n, 3}, \widetilde g_{n, 3} \approx \left( \frac{r_e}{|\Bz|} \right)^n, \quad
  g_{n, 4}, \widetilde g_{n, 4} \approx n^2 \left( \frac{r_e}{|\Bz|} \right)^n.
\eeq
We include proofs of these estimates in Appendix \ref{proofs}.

Suppose that $z(c)=k_0$. Then we infer from \eqnref{Indef}, \eqnref{IIndef}, \eqnref{gnone} and \eqnref{gntwo} that
\beq
I_n \approx n^4 \Gr_\Bz^{2n} \quad\mbox{and}\quad II_n \approx \Gr_\Bz^{2n}.
\eeq
It then follows from \eqnref{Eudelone} that
\begin{equation*}
E(\Bu^\Gd) \sim \sum_{n = 1}^\infty \frac{n^3 \Gr_\Bz^{2 n}}{\Gd^2 + \Gr^{4 n}}.
\end{equation*}

For small $\Gd$, let $N$ be a positive integer such that
\beq\label{Ndef}
  \Gr^{2 N} \leq \Gd \leq \Gr^{2 \left( N - 1 \right)}.
\eeq
Note that $N \sim |\ln \Gd|$.
Then we have
\begin{equation*}
E(\Bu^\Gd) \sim \sum_{n < N} + \sum_{n \geq N} \sim \sum_{n < N} \frac{n^3 \Gr_\Bz^{2 n}}{\Gr^{4 n}} + \sum_{n > N} \frac{1}{\Gd^2} {n^3 \Gr_\Bz^{2 n}} =: S_1^\Gd + S_2^\Gd.
\end{equation*}
Since
$$
S_1^\Gd \sim \int_1^N y^3 e^{2 y \ln (\Gr_\Bz / \Gr^2) } \, dy ,
$$
we have
\begin{equation*}
S_1^\Gd \sim N^3 e^{2 N \ln (\Gr_\Bz / \Gr^2)} \sim \left| \ln \Gd \right|^3 \Gd^{\frac{ \ln \Gr_\Bz}{\ln \Gr}-2}.
\end{equation*}
Likewise, since
\begin{equation*}
S_2^\Gd \sim \Gd^{- 2} \int_N^\infty y^3 e^{2 y \ln \Gr_\Bz} \, dy ,
\end{equation*}
we have
\begin{equation*}
S_2^\Gd \sim \Gd^{-2} N^3 e^{2N \ln \Gr_\Bz}\sim \left| \ln \Gd \right|^3 \Gd^{\frac{ \ln \Gr_\Bz}{\ln \Gr}-2} .
\end{equation*}
This proves \eqnref{itemtwo}

If $z(c)=-k_0$, we have
\beq
E(\Bu^\Gd) \sim \sum_{n = 1}^\infty \frac{\Gr_\Bz^{2 n}}{n(\Gd^2 + n^2 \rho^{2 n})}.
\eeq
In this case we choose $N$, instead of \eqnref{Ndef}, so that
$$
N\rho^N \leq \delta \leq (N-1)\rho^{N-1}.
$$
Then we have $\delta/N\sim e^{N\ln\rho}$ and $N\sim |\ln\delta|$. Then, in a way similar to above, one can show \eqnref{itemtwo}.
\qed

\subsection{Cloaking due to ALR}

Recall that
\begin{equation*}
  \Bu^\Gd = \BF + \BS_{\GG_i} [\BGvf_i^\Gd] + \BS_{\GG_e} [\BGvf_e^\Gd].
\end{equation*}
In this section, we show that
\beq\label{V_bdd}
  \big| \Bu^\Gd(\Bx) \big| \leq C, \quad \left\vert \Bx \right\vert = r > r_0,
\eeq
for some $r_0 > r_e$.
Note that the above boundedness means that CALR happens, since we have already proved the energy $E^\Gd$ blows up when a point source located inside the crical radius.

For simplicity, we only consider $\BS_{\GG_e} [\BGvf_e^\Gd]$.
The potential $\BS_{\GG_i} [\BGvf_i^\Gd]$ can be estimated in a similar way.
Let us write the solutions of the equations \eqref{Int_eqn_mat} using $\BGF_{n, j}$ and $\widetilde \BGF_{n, j}$.
\beq
\BGF_n^\Gd = \sum_{j=0}^4 \frac{g_{n,j}}{z_\Gd + \Gl_{n,j}} \BGF_{n,j}, \quad \widetilde\BGF_n^\Gd = \sum_{j=1}^4 \frac{\widetilde g_{n,j}}{z_\Gd + \widetilde \Gl_{n,j}} \widetilde \BGF_{n,j}, \nonumber
\eeq
where $g_{n,j}$ and $\widetilde g_{n,j}$ are given by \eqref{BGndef}.
Then, from \eqref{BGFdelta}, we have
\begin{align*}
  \BGvf_e^\Gd & = \sum_{n=0}^\infty \left\{ \frac{g_{n,3} \BGvf_{n+1}}{z_\Gd + \Gl_{n,3}} + \frac{\widetilde g_{n,3} \widetilde \BGvf_{n+1}}{z_\Gd + \widetilde \Gl_{n,3}} \right\} + \sum_{n=1}^\infty \left\{ \frac{g_{n,4} \BGvf_{-n+1}}{z_\Gd + \Gl_{n,4}} + \frac{\widetilde g_{n,4} \widetilde \BGvf_{-n+1}}{z_\Gd + \widetilde \Gl_{n,4}} \right\} \\
  & =: \BGvf_{e,3}^\Gd + \widetilde \BGvf_{e,3}^\Gd + \BGvf_{e,4}^\Gd + \widetilde \BGvf_{e,4}^\Gd.
\end{align*}
Hence, we have
\beq \label{BSBGvf_e}
\BS_{\GG_e}[\BGvf_e^\Gd] = \BS_{\GG_e}[\BGvf_{e,3}^\Gd] + \BS_{\GG_e}[\widetilde \BGvf_{e,3}^\Gd] + \BS_{\GG_e}[\BGvf_{e,4}^\Gd] + \BS_{\GG_e}[\widetilde \BGvf_{e,4}^\Gd].
\eeq

Suppose $z(c) = k_0$.
Let us estimate $\BS_{\GG_e}[\BGvf_{e,4}^\Gd]$ in \eqref{BSBGvf_e}.
The other terms can be estimated in the same manner.
From \eqref{negaGvf}, we have
\beq\label{1000}
|\BS_{\GG_e} [\BGvf_{e, 4}^\Gd] (\Bx)| \lesssim \sum_{n=1}^\infty \frac{|g_{n,4}|}{\Gd + \Gr^{2n}} \frac{r_e^n}{r^{n-1}} \lesssim \sum_{n=1}^\infty n^2 \left(\frac{r_e^2}{r_i}\right)^{2n} \left(\frac{1}{r |\Bz|}\right)^{n-1}.
\eeq
Therefore, for $|\Bx| = r > r_e^3 / r_i^2$, we have $|\BS_{\GG_e}[\BGvf_{e,4}^\Gd](\Bx)| < C$ for some $C > 0$.

Next, suppose $z(c) = - k_0$.
For the same reason, we only estimate $\BS_{\GG_e}[\BGvf_{e,3}^\Gd]$.
From \eqref{twoGvf}, we have
\beq\label{1001}
|\BS_{\GG_e} [\BGvf_{e, 3}^\Gd] (\Bx)| \lesssim \sum_{n=1}^\infty \frac{|g_{n,3}|}{\Gd + n \Gr^n} \frac{r_e^n}{r^{n-1}} \lesssim \sum_{n=1}^\infty \frac{1}{n} \left(\frac{r_e^3}{r_i}\right)^n \left(\frac{1}{r |\Bz|}\right)^{n-1}.
\eeq
Therefore, for $|\Bx| = r > r_e^2 / r_i$, we have $|\BS_{\GG_e}[\BGvf_{e,3}^\Gd](\Bx)| < C$ for some $C > 0$.

Since $\BF(\Bx) \to {\bf 0}$ as $|\Bx| \to \infty$, \eqnref{1000} and \eqnref{1001} together with Theorem~\ref{thm:res+} yield the following theorem.
\begin{theorem}
  Suppose that $\Bf$ be given by \eqref{dipole}.
  \begin{itemize}
  \item[(i)] If $z(c) = k_0$, then CALR occurs with the critical radius $r_* = r_e^2 / r_i$; more precisely, $E^\Gd=\Gd E(\Bu^\Gd) \to \infty$ as $\delta \to 0$ if $r_e < |\Bz| \le r_*$, and there is a constant $C$ such that
    \begin{equation*}
      \big\vert \Bu^\Gd(\Bx) \big\vert < C \quad \text{for } \left\vert \Bx \right\vert = r > r_e^3 / r_i^2 \quad \text{as } \Gd \to 0.
    \end{equation*}
  \item[(ii)] If $z(c) = - k_0$, then CALR occurs with the critical radius $r_{**} = \sqrt{r_e^3 / r_i}$; more precisely, $E^\Gd \to \infty$ as $\Gd \to 0$ if $r_e < |\Bz| < r_{**}$, and there is a constant $C$ such that
    \begin{equation*}
      \big\vert \Bu^\Gd(\Bx) \big\vert < C \quad \text{for } \left\vert \Bx \right\vert = r > r_e^2 / r_i \quad \text{as } \Gd \to 0.
    \end{equation*}
  \end{itemize}
\end{theorem}

redpajama_set_name: RedPajamaArXiv


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text:
Wednesday, 13 November 2019, 09:02 HKT/SGT
Source: Fujitsu Ltd
Cray and Fujitsu Partner to Power Supercomputing in the Exascale Era
Companies to bring first high-bandwidth memory Arm processor to market
TOKYO, Nov 13, 2019 - (JCN Newswire) - Global supercomputer leader Cray, a Hewlett Packard Enterprise company (NYSE: HPE), and leading Japanese information and communication technology company Fujitsu (TSE: 6702), today announced a partnership to offer high-performance technologies for the exascale era. Under the alliance agreement, Cray is developing the first-ever commercial supercomputer powered by the Fujitsu A64FX Arm-based processor with high-memory bandwidth (HBM) and supported on the proven Cray CS500 supercomputer architecture and programming environment. Initial customers include Los Alamos National Laboratory, Oak Ridge National Laboratory, RIKEN Center for Computational Science, Stony Brook University, and the University of Bristol.
As part of this new partnership, Cray and Fujitsu will explore engineering collaboration, co-development, and joint go-to-market in an effort to meet customer demand in the supercomputing space.
"Our partnership with Fujitsu means customers now have a broader choice of processor technology to address their pressing computational needs," said Fred Kohout, senior vice president and CMO at Cray, a Hewlett Packard Enterprise company. "We are delivering the development-to-deployment experience customers have come to expect from Cray, including exploratory development to the Cray Programming Environment (CPE) for Arm processors to optimize performance and scalability with additional support for Scalable Vector Extensions and high bandwidth memory."
Cray customers are leaders in their respective fields and often look for opportunities to gain the next edge in performance. The new Fujitsu processor is unique in that it is the first processor to deliver HBM and Arm Scalable Vector Extensions (SVE). HBM2 provides transfer speeds that are significantly faster than DDR4 giving the A64FX a maximum theoretical memory bandwidth greater than 1 terabyte per second (TB/s), and support for Arm SVE provides improved performance for artificial intelligence and analytics.
The Cray CS500 system can apply this compute power to a wide range of HPC and AI workloads while still delivering hallmark features of Arm-based systems with high parallelization, low power consumption, and high reliability.
"It's a pleasure to partner with Cray on building technologies for the next era of computing," said Takeshi Horie, Corporate Executive Officer, Vice Head of Service Platform Business Group at Fujitsu. "Both companies have a strong legacy of supercomputing and vector processing. The A64FX Arm processor was designed to empower a wide range of data-intensive applications and is the world's first CPU to adopt the SVE of the Armv8-A instruction set architecture, specifically extended for supercomputers."
The Cray supercomputer powered by Fujitsu A64FX Arm will be available through Cray to customers in mid-2020.
Supporting Quotes
Los Alamos National Laboratory (LANL) - "The most demanding computing work at LANL involves sparse, irregular, multi-physics, multi-link-scale, highly resolved, long-running 3D simulations," said Gary Grider, deputy division leader, HPC Division at LANL. "There are few existing architectures that currently serve this workload well. We are excited to see a potential solution and are happy to be helping prove this Cray and Fujitsu technology is a viable alternative for this need. Having this type of capability will be quite complementary to other resources in the NNSA computing complex."
Oak Ridge National Laboratory (ORNL) - "The U.S. Department of Energy is the world's leader in high-performance computing in support of its missions, which include ensuring America's prosperity by addressing energy and environmental challenges through transformative science and technology solutions," said Jeff Nichols, Associate Lab Director for Computing and Computational Sciences at Oak Ridge National Laboratory. "This new system will enable our scientists to experiment with possible system architectures for the exascale era and advance the DOE mission."
RIKEN Center for Computational Science (R-CCS) - "The new Fujitsu A64FX processor, one of the fastest Arm processors in the world and to be made commercially available from Cray and Fujitsu, also underpins 'Fugaku,' Japan's next-generation flagship supercomputer at our RIKEN Center for Computational Science," said Satoshi Matsuoka, Director of R-CCS. "We are extremely interested how A64FX on Cray CS500 machines with the Cray Programming Environment will excel in performance and hope it will have a major impact on high-performance computing, especially as converged workloads take precedence in the field. We look forward to working with Cray and our partner Fujitsu to further enhance the software ecosystem for high-performance Arm."
Stony Brook University - "Using the latest technologies is critical to advancing the frontiers of research, whether in industry or academia, and we're looking forward to putting our Cray-Fujitsu system, named Ookami, which is Japanese for wolf, into production," said Robert Harrison, principal investigator and project director for Stony Brook University. "Funded by the National Science Foundation, the new system will boost national competitiveness and enable U.S. researchers to achieve extremely high performance for a wide range of applications. Memory-bandwidth-intensive applications will be especially accelerated by the ultrahigh-bandwidth memory while still being able to employ familiar and successful multi-core programming models."
University of Bristol - "We've been on a journey toward Arm-based supercomputing and the new Cray and Fujitsu Arm system will bring us closer to that reality," said Professor Simon McIntosh-Smith, professor of high-performance computing at the University of Bristol. "We are looking forward to innovating and pushing the boundaries of what Arm technologies are capable of and witnessing the tremendous performance that will come about on the Cray A64FX Arm supercomputer. To that end, we are planning a new system, Isambard 2, which expects to incorporate these new technologies and make them available to UK scientists later in 2020."
Arm - "As the Arm Neoverse ecosystem continues to expand, we're seeing the growing demand for the performance and innovation Arm-based processors deliver," said Chris Bergey, senior vice president, and general manager, Infrastructure Line of Business, Arm. "Fujitsu and Cray have leveraged this innovation and achieved a significant milestone being the first high-bandwidth memory supercomputing system in the market, and Arm is proud to be part of it."
About Cray Inc.
Cray, a Hewlett Packard Enterprise company, combines computation and creativity so visionaries can keep asking questions that challenge the limits of possibility. Drawing on more than 45 years of experience, Cray develops the world's most advanced supercomputers, pushing the boundaries of performance, efficiency and scalability. Cray continues to innovate today at the convergence of data and discovery, offering a comprehensive portfolio of supercomputers, high-performance storage, data analytics, and artificial intelligence solutions. Go to www.cray.com for more information.
Fujitsu is the leading Japanese information and communication technology (ICT) company, offering a full range of technology products, solutions, and services. Approximately 132,000 Fujitsu people support customers in more than 100 countries. We use our experience and the power of ICT to shape the future of society with our customers. Fujitsu Limited (Code: 6702) reported consolidated revenues of 4.0 trillion yen (the US $36 billion) for the fiscal year ended March 31, 2019. For more information, please see www.fujitsu.com.
Cray Media:
Diana Brodskiy
Phone:Phone: +1-415-306-6199
E-mail: E-mail: [email protected]
Company: Cray Inc.
Fujitsu Limited
Public and Investor Relations
URL: www.fujitsu.com/global/news/contacts/
Sectors: Enterprise IT
Fujitsu Ltd Links
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Dec 18, 2019 14:24 HKT/SGT
Fujitsu Honored with IEEE Milestone for High Electron Mobility Transistor
Fujitsu Launches New PRIMERGY Model with High-Speed Virtual Switch for 5G Communications in Japan
Dec 6, 2019 17:20 HKT/SGT
Fujitsu Launches SAP Solution to Support Mass Customization in Engineer-to-Order Production
Fujitsu Successfully Grows Diamond Film to Boost Heat Dissipation Efficiency of GaN HEMT
Fujitsu Takes Key Step Toward Implementing SAP S/4HANA Manufacturing for Production Engineering and Operations to Streamline Aircraft Production at Kawasaki Heavy Industries
Fujitsu Begins Shipping Supercomputer Fugaku
Fujitsu Laboratories and Aichi Cancer Center in Japan Sign Comprehensive Joint Research Agreement to Drive Advances in Cancer Genomic Medicine with AI Technology
Fujitsu Formulates 'Fujitsu Future Insights - Digital Transformation in Retail'

redpajama_set_name: RedPajamaCommonCrawl


text:
Gliadel Implant
GLIADEL Implants
Full PI
Consumer medicine information (CMI) leaflet
Please read this leaflet carefully before you start using GLIADEL Implants.
Download CMI (PDF)
Download Large Text CMI (PDF)
This leaflet answers some common questions about GLIADEL. It does not contain all the available information that is known about GLIADEL. It does not take the place of talking to your doctor or pharmacist.
All medicines have risks and benefits. Your doctor has weighed the risks of treatment with GLIADEL against the benefits they expect it will have for you.
If you have any concerns about treatment with this medicine, ask your doctor or pharmacist.
Keep this leaflet as you may want to read it again after your GLIADEL treatment.
WHAT GLIADEL IS USED FOR
GLIADEL is used to treat some types of brain cancers after removal by surgery.
GLIADEL works by killing cancer cells and/or stopping cancer cells from growing and multiplying.
GLIADEL belongs to a group of medicines called antineoplastic or cytotoxic medicines. You may also hear these being called chemotherapy medicines.
Ask your doctor if you have any questions about why GLIADEL treatment has been prescribed for you.
GLIADEL is not addictive.
BEFORE YOU ARE GIVEN GLIADEL
When you must not be given it
You must not be given GLIADEL if you are breastfeeding or plan to breastfeed.
It is recommended that you do not breast-feed while undergoing GLIADEL treatment, as it may pass into the breast milk and therefore there is a possibility that the breast-fed baby may be affected.
Do not use GLIADEL to treat children.
There is no experience with the use of GLIADEL in children.
If you are not sure whether you should have GLIADEL, talk to your doctor or pharmacist.
Tell your doctor or pharmacist if you have any allergies to:
any other medicines
any other substances, such as foods, preservatives or dyes.
Tell your doctor if you are pregnant or intend to become pregnant.
Like most medicines used to treat cancer, GLIADEL is not recommended for use during pregnancy. If there is need to consider GLIADEL during your pregnancy, your doctor will discuss with you the benefits and risks of using it.
If you have not told your doctor about any of the above, tell them before you have GLIADEL.
Tell your doctor or pharmacist if you are taking any other medicines, including any that you buy without a prescription from your pharmacy, supermarket or health food shop.
HOW GLIADEL IS GIVEN
How much is given
Your doctor will decide what dose you will receive. This will depend on your condition and other factors.
Ask your doctor if you want to know more about the dose of GLIADEL you receive.
How it is given
The GLIADEL Implants are placed inside your head during the surgery to remove your tumor. The usual number of implants is up to eight (61.6 mg of carmustine), however this number may be reduced by your doctor depending on your tumor size.
How long is it given
You are only given GLIADEL Implants once when your tumor is removed during surgery. The implants will then slowly dissolve away.
If you are given too much (overdose)
Your doctor will know the correct dose for you, so the chances of overdose are highly unlikely.
AFTER YOU HAVE BEEN GIVEN GLIADEL
If you become pregnant after being given GLIADEL tell you doctor immediately.
If you are about to be started on any new medicine tell your doctor that you have had GLIADEL implanted.
Tell any other doctors, dentists and pharmacists who are treating you that you have had GLIADEL Implants.
In particular, GLIADEL Implants will be seen on brain scans so warn any doctors before you have any brain scans after the GLIADEL has been implanted.
If you are about to be started on any new medicine, tell your doctor, dentist or pharmacist that you have had GLIADEL implanted.
Tell your doctor, nurse or pharmacist as soon as possible if you do not feel well after being given GLIADEL Implants.
GLIADEL helps most people with brain tumors, but like other chemotherapeutic medicines, it may have unwanted side effects. All medicines can have side effects. Sometimes they are serious, most of the time they are not. You may need medical treatment if you get some of the side effects.
Ask your doctor or pharmacist to answer any questions you may have
If any of the following happen, tell your doctor or pharmacist immediately:
pain – head, neck, chest or back
confusion or stupor
fits or seizures
weakness or paralysis of one side of the body
unusual tiredness, sleepiness or weakness
pain or burning feeling when urinating
shortness of breath or difficulty in breathing
oral thrush – sore creamy yellow raised patches in the mouth
difficulty in talking
diarrhoea or constipation
problems seeing
depression, problems sleeping, abnormal thinking
clumsiness and lack of coordination
Most of these side effects are usually mild to moderate. However, they may also be signs of a more serious side effect. You may need medical attention
If any of the following happen, tell your doctor or pharmacist immediately or go to Accident and Emergency at your nearest hospital:
pneumonia – chest infection with cough and yellow/green sputum
a blood clot in your leg or severe chest pain with a shortness of breath
decreased level of consciousness
These are serious side effects. You may need urgent medical attention or hospitalisation. These side effects are rare.
Other side effects not listed above may also occur in some patients. Tell your doctor if you notice anything else that is making you feel unwell.
GLIADEL implants are stored in the freezer at or below -20°C in the pharmacy department of your hospital.
GLIADEL DESCRIPTION
GLIADEL Implants are an off-white to pale yellow flat disc approximately 1.45 cm in diameter and 1 mm thick.
GLIADEL Implants contain 7.7 mg of carmustine as the active ingredient.
The other ingredient in GLIADEL Implants is an inactive copolymer called Polifeprosan 20. This copolymer consists of two polymers - carboxypropane and sebacic acid.
Eisai Australia Pty. Ltd.
Level 2, 437 St. Kilda Rd.
Melbourne, VIC, 3004 Australia
ARTG Number 77283
Date of revision: 5 November 2015
1 Name of Medicine
Carmustine.
6.7 Physicochemical Properties
Carmustine is a light yellow powder. It is highly soluble in alcohol and lipids and is poorly soluble in water. Its melting point range is 29.5-32.0°C and it has a molecular weight of 214.06.
Polifeprosan 20 consists of poly[bis(p-carboxyphenoxy) propane: sebacic acid] in a 20:80 molar ratio and is used to control the local delivery of carmustine. Carmustine is homogeneously distributed in the copolymer matrix.
Carmustine is a nitrosourea. The chemical name of carmustine is 1,3-bis (2-chloroethyl)-1-nitrosourea (or BCNU).
Chemical structure.
The structural formula for carmustine is:
CAS number.
154-93-8.
2 Qualitative and Quantitative Composition
Each Gliadel implant contains 7.7 mg of carmustine.
For the full list of excipients, see Section 6.1 List of Excipients.
3 Pharmaceutical Form
Implant.
Gliadel Implants are sterile, off-white to pale yellow implants approximately 1.45 cm in diameter and 1 mm thick.
5 Pharmacological Properties
5.1 Pharmacodynamic Properties
Mechanism of action.
Gliadel is designed to deliver carmustine directly into the surgical cavity created after tumour resection. On exposure to the aqueous environment of the cavity, the anhydride bonds in the copolymer are hydrolysed, releasing carmustine, carboxyphenoxypropane, and sebacic acid. The carmustine released from Gliadel diffuses into the surrounding brain tissue and produces an antineoplastic effect by alkylating DNA and RNA.
Carmustine has been shown to degrade both spontaneously and metabolically. The production of an alkylating moiety, hypothesised to be chloroethyl carbonium ion, leads to the formation of DNA cross-links.
The tumouricidal activity of Gliadel is dependent on release of carmustine to the tumour cavity in concentrations sufficient for effective cytotoxicity.
More than 70% of the copolymer degrades by three weeks. The metabolic disposition and excretion of the monomers differ. Carboxyphenoxypropane is predominantly eliminated by the kidney and sebacic acid (an endogenous fatty acid) is metabolised by the liver and expired as CO2 in animals.
Clinical trials.
Primary surgery.
A randomised, double blind, placebo controlled clinical trial was conducted in adult patients with newly diagnosed high grade malignant glioma undergoing initial craniotomy for tumour resection. The trial determined the safety and efficacy of Gliadel implants plus surgery and radiation therapy compared to placebo implants plus surgery and radiation therapy. Two hundred and forty patients with newly-diagnosed malignant glioma were enrolled. The most common tumour type was glioblastoma multiforme (GBM) (n = 207), followed by anaplastic oligoastrocytoma (n = 11), anaplastic oligodendroglioma (n = 11), and anaplastic astrocytoma (n = 2). Gliadel wafers were implanted at the time of the surgery in 120 patients and placebo were implanted in 120 patients. The majority of patients received 6-8 wafers. The majority of patients (93/120, 77.5% in the Gliadel group and 98/120, 81.7% in the placebo group) with newly diagnosed malignant glioma received a standard course of radiotherapy (55 to 60 Gy) typically starting 3 weeks after surgery. There were 17 patients (14.2%) in the Gliadel group and 12 patients (10.0%) in the placebo group who received systemic chemotherapy during the study. All six patients with anaplastic oligodendroglioma received chemotherapy within 30 days of Gliadel implantation. Patients were followed for at least three years or until death. Only one patient was lost to follow-up. Median survival increased from 11.6 months with placebo to 13.9 months with Gliadel (p-value < 0.05, log rank test). The hazard ratio for Gliadel treatment was 0.73 (95% CI: 0.56-0.95). (See Figure 1.)
When only patients with glioblastoma multiforme were included in the analysis, the hazard ratio with Gliadel treatment was 0.78 (95% CI: 0.59-1.03, p = 0.08, log-rank test).
Surgery for recurrent disease.
A randomised, double blind, placebo controlled clinical trial was conducted in adult patients with recurrent malignant glioma. This trial determined the safety and efficacy of Gliadel implants plus surgery compared to placebo implants plus surgery. Ninety-five percent of the patients treated with Gliadel had 7-8 wafers implanted. Chemotherapy was withheld at least four weeks (six weeks for nitrosoureas) prior to and two weeks after surgery in patients undergoing reoperation for malignant glioma.
In 222 patients with recurrent malignant glioma who had failed initial surgery and radiation therapy, the six month survival rate after repeat surgery increased from 47% (53/112) for patients receiving placebo to 60% (66/110) for patients treated with Gliadel. Median survival increased by 33%, from 24 weeks (5.5 months) with placebo to 32 weeks (7.4 months) with Gliadel treatment. In patients with GBM, the six-month survival rate increased from 36% (26/73) with placebo to 56% (40/72) with Gliadel treatment. Median survival of GBM patients increased by 41% from 20 weeks (4.6 months) with placebo to 28 weeks (6.4 months) with Gliadel treatment. In patients with pathologic diagnoses other than GBM at the time of surgery for tumour recurrence, Gliadel produced no survival prolongation. (See Figures 2 and 3.)
5.2 Pharmacokinetic Properties
The absorption, distribution, metabolism, and excretion of the copolymer in humans are unknown. Carmustine concentrations delivered by Gliadel in human brain tissue have not been determined. Plasma levels of carmustine after Gliadel implantation were not determined. Animal studies suggest that nearly all the carmustine in the implants is released within 7 days of implantation.
Following an intravenous infusion of carmustine at doses ranging from 30 to 170 mg/m2, the average terminal half-life, clearance, and steady-state volume of distribution were 22 minutes, 56 mL/min/kg, and 3.25 L/kg, respectively.
Approximately 60% of the intravenous 200 mg/m2 dose of 14C-carmustine was excreted in the urine over 96 hours and 6% was expired as CO2.
Gliadel implants are biodegradable in the human brain when placed into the cavity after tumour resection. The rate of biodegradation is variable from patient to patient. During the biodegradation process, an implant remnant may be observed on brain imaging scans or at reoperation even though extensive degradation of all components has occurred. Data obtained from review of CT scans obtained 49 days after implantation of Gliadel demonstrated that images consistent with implants were visible to varying degrees in the scans of 11 of 18 patients. Data obtained at reoperation and autopsies have demonstrated implant remnants up to 232 days after Gliadel implantation.
Implant remnants removed at reoperation from two patients with recurrent malignant glioma, one at 64 days and the second at 92 days after implantation, were analysed for content. Table 3 presents the results of analyses completed on these remnants.
The implant remnants consisted mostly of water and monomeric components with minimal detectable carmustine present.
5.3 Preclinical Safety Data
Genotoxicity.
No mutagenicity studies have been conducted with Gliadel. However, such studies have been conducted with carmustine, the active component of Gliadel, when administered systemically. Carmustine was mutagenic in vitro and clastogenic in vitro and in vivo.
Carcinogenicity.
No carcinogenicity studies have been conducted with Gliadel. However, such studies have been conducted with carmustine, the active component of Gliadel, when administered systemically. Carcinogenicity studies were conducted in rats and mice, with the data indicating an increased incidence of pulmonary neoplasms in rats and possible tumorigenic activity in mice.
4 Clinical Particulars
4.1 Therapeutic Indications
Gliadel is indicated in newly-diagnosed high-grade malignant glioma patients as an adjunct to surgery and radiation.
Gliadel is also indicated for use as adjunct to surgery to prolong survival in patients with recurrent glioblastoma multiforme (GBM) for whom surgical resection is indicated.
4.3 Contraindications
Gliadel is contraindicated in patients with a history of hypersensitivity to carmustine or any of the excipients of Gliadel (see Section 6.1 List of Excipients).
Gliadel Implant is also contraindicated in breastfeeding mothers.
4.4 Special Warnings and Precautions for Use
Patients undergoing craniotomy for malignant glioma and implantation of Gliadel should be monitored closely for known complications of craniotomy which include seizures/ convulsions, intra-cranial infections, abnormal wound healing and brain oedema (see Section 4.8 Adverse Effects (Undesirable Effects)). Cases of intracerebral mass effect unresponsive to corticosteroids have been described in patients treated with Gliadel, including one case leading to brain herniation. Pneumocephalus has been reported, commonly, following surgery involving Gliadel administration.
Changes of the walls of cerebral blood vessels located close to Gliadel wafer, including cases of aneurysms leading to cerebral bleeding several months after Gliadel wafer implantation, have been described. Implantation of Gliadel wafers adjacent to large cerebral vessels should be avoided.
Communication between the surgical resection cavity and the ventricular system should be avoided to prevent the implants from migrating into the ventricular system and causing obstructive hydrocephalus. If a communication larger than the diameter of a wafer exists, it should be closed at operation prior to implantation.
Imaging studies.
Computed tomography (CT) and magnetic resonance imaging (MRI) of the head may demonstrate enhancement in the brain tissue surrounding the resection cavity after placement of Gliadel implants. This enhancement may represent oedema and inflammation caused by Gliadel or tumour progression.
Recommendations for safe handling.
Implants should be handled by personnel wearing surgical gloves because exposure to carmustine can cause severe burning and hyperpigmentation of the skin. Use of double gloves is recommended and the outer gloves should be discarded into a dedicated biohazard waste container after use. A surgical instrument dedicated to the handling of the implants should be used for implant placement. If repeat neurosurgical intervention is required, any implant or implant remnant should be handled as a potentially cytotoxic agent and discarded into the biohazard container.
Each implant is wrapped in a double foil system for use in operating theatres, with the outer sachet acting as a nonsterile overwrap and the inner sachet only being sterile.
Use in the elderly.
Limited data are available in patients aged ≥ 75 years. Gliadel should be used with caution in such patients.
Paediatric use.
The safety and efficacy of Gliadel in children has not been established.
Effects on laboratory tests.
4.5 Interactions with Other Medicines and Other Forms of Interactions
Interactions of Gliadel with other drugs have not been formally evaluated.
The short and long-term toxicity profiles of Gliadel when given in conjunction with chemotherapy have not been fully explored. Gliadel, when given in conjunction with radiotherapy, does not appear to have any short-term or chronic toxicities.
Systemic chemotherapy.
Some patients (n = 6) in the clinical trials have received systemic chemotherapy. Chemotherapy was withheld at least four weeks (six weeks for nitrosoureas) prior to and two weeks after surgery in patients undergoing reoperation for malignant glioma.
Radiotherapy.
Some patients in the clinical trials have received external beam radiation therapy (n = 36). External beam radiation therapy was initiated no sooner than three weeks after Gliadel implantation. Of the 36 patients who received Gliadel at initial surgery for newly diagnosed malignant glioma followed by external beam radiation therapy, 3/15 (20%) in one study and 11/21 (52%) in the other study experienced new or worsened seizures.
4.6 Fertility, Pregnancy and Lactation
Effects on fertility.
No impairment of fertility studies have been conducted with Gliadel.
However, such studies have been conducted with carmustine, the active component of Gliadel, when administered systemically. Carmustine has been shown to affect fertility in male rats, with conception rate being reduced in females mated to treated males, as carmustine can cause testicular degeneration.
(Category D)
There are no studies in either pregnant women or laboratory animals on Gliadel, but the active component, carmustine, when administered systemically, can cause foetal harm when administered to a pregnant woman and has been shown to be embryotoxic in rats and rabbits and teratogenic in rats.
Although it is preferable that Gliadel is not used in pregnancy, if it is, or if the patient becomes pregnant after Gliadel implantation, the patient must be warned of the potential hazard to the foetus.
It is not known if either carmustine, carboxyphenoxypropane or sebacic acid are excreted in human milk. Since some drugs are excreted in human milk and because of the potential for serious adverse reactions from carmustine in nursing infants, women being treated with Gliadel should not breastfeed their infants. For women who wish to breastfeed their infants, Gliadel is contraindicated.
4.8 Adverse Effects (Undesirable Effects)
Adverse reactions for the trials are described in Tables 1 and 2.
The following data are some of the most frequently occurring adverse events observed in 5% or more of the newly diagnosed malignant glioma patients during the trial. (See Table 1.)
The following postoperative adverse events were observed in 4% or more of the patients receiving Gliadel at recurrent surgery. Except for nervous system effects, where there is a possibility that the placebo wafers could have been responsible, only events more common in the Gliadel group are listed. These adverse events were either not present preoperatively or worsened postoperatively during the follow-up period. The follow-up period was up to 71 months. (See Table 2.)
The following four categories of adverse events are possibly related to treatment with Gliadel. The frequency with which they occurred in the randomised trials along with descriptive detail is provided below.
1. Seizures.
In the initial surgery trial, the incidence of seizures was 33.3% in patients receiving Gliadel and 37.5% in patients receiving placebo. Grand mal seizures occurred in 5% of Gliadel-treated patients and 4.2% of placebo-treated patients. The incidence of seizures within the first 5 days after wafer implantation was 2.5% in the Gliadel group and 4.2% in the placebo group. The time from surgery to the onset of the first postoperative seizure did not differ between the Gliadel and placebo treated patients.
In the surgery for recurrent disease trial, the incidence of postoperative seizures was 19% in both patients receiving Gliadel and placebo. In this study, 12/22 (54%) of patients treated with Gliadel and 2/22 (9%) of placebo patients experienced the first new or worsened seizure within the first five postoperative days. The median time to onset of the first new or worsened postoperative seizure was 3.5 days in patients treated with Gliadel and 61 days in placebo patients.
2. Brain oedema.
In the initial surgery trial, brain oedema was noted in 22.5% of patients treated with Gliadel and in 19.2% of patients treated with placebo. Development of brain oedema with mass effect (due to tumour recurrence, intracranial infection or necrosis) may necessitate reoperation and, in some cases, removal of Gliadel or its remnants.
3. Healing abnormalities.
The following healing abnormalities have been reported in clinical trials of Gliadel: wound dehiscence, delayed wound healing, subdural, subgaleal or wound effusions, and cerebrospinal fluid leak. In the initial surgery trial, healing abnormalities occurred in 15.8% of Gliadel-treated patients and in 11.7% of placebo recipients. Cerebrospinal fluid leaks occurred in 5% of Gliadel recipients and 0.8% of those given placebo.
During surgery, a water-tight dural closure should be obtained to minimise the risk of cerebrospinal fluid leak.
In the surgery for recurrent disease trial, the incidence of healing abnormalities was 14% in Gliadel-treated patients and 5% in patients receiving placebo.
4. Intracranial infection.
In the initial surgery trial, the incidence of brain abscess or meningitis was 5% in patients treated with Gliadel and 6% in patients receiving placebo. In the recurrent setting, the incidence of brain abscess or meningitis was 4% in patients treated with Gliadel and 1% in patients receiving placebo.
The following adverse events, not listed in Tables 1 to 2 above, were reported in less than 4% but at least 1% of patients treated with Gliadel in all studies. The events listed were either not present preoperatively or worsened postoperatively. Whether Gliadel caused these events cannot be determined.
Body as a whole.
Peripheral oedema (2%); neck pain (2%); accidental injury (1%); back pain (1%); allergic reaction (1%); asthenia (1%); chest pain (1%); sepsis (1%).
Cardiovascular system.
Hypertension (3%); hypotension (1%).
Digestive system.
Diarrhoea (2%); constipation (2%); dysphagia (1%); gastrointestinal haemorrhage (1%); faecal incontinence (1%).
Haemic and lymphatic system.
Thrombocytopaenia (1%); leukocytosis (1%).
Metabolic and nutritional disorders.
Hyponatraemia (3%); hyperglycaemia (3%); hypokalaemia (1%).
Musculoskeletal system.
Infection (1%).
Nervous system.
Hydrocephalus (3%); depression (3%); abnormal thinking (2%); ataxia (2%); dizziness (2%); insomnia (2%); monoplegia (2%); coma (1%); amnesia (1%); diplopia (1%); paranoid reaction (1%). In addition, cerebral haemorrhage and cerebral infarct were each reported in less than 1% of patients treated with Gliadel.
Respiratory system.
Infection (2%); aspiration pneumonia (1%).
Skin and appendages.
Rash (2%).
Special senses.
Visual field defect (2%); eye pain (1%).
Urogenital system.
Urinary incontinence (2%).
In a published clinical study, cyst formation after Gliadel Implant treatment has been reported. This reaction occurred in 10% of the patients observed in the study, however, the formation of cyst is possible after resection of a malignant glioma.
Reporting suspected adverse effects.
Reporting suspected adverse reactions after registration of the medicinal product is important.
It allows continued monitoring of the benefit-risk balance of the medicinal product. Healthcare professionals are asked to report any suspected adverse reactions at http://www.tga.gov.au/reporting-problems.
4.2 Dose and Method of Administration
Each Gliadel implant contains 7.7 mg of carmustine, resulting in a dose of 61.6 mg when eight implants are placed in the tumour resection cavity. It is recommended that a maximum of eight implants be placed if the size and shape of the resection cavity allows it. Otherwise, use the maximum number of implants possible. No more than eight implants should be used per surgical procedure, as there is no clinical experience with this dose.
Handling and disposal.
Gliadel implants should be handled with care (see Section 4.4 Special Warnings and Precautions for Use on safe handling).
The sachets containing Gliadel should be delivered to the operating room and remain unopened until ready to place the implants into the resection cavity. The outside surface of the outer foil sachet is not sterile and must be removed under sterile conditions from the sterile inner sachet prior to implantation.
Instructions for opening sachet containing Gliadel Implant.
1. To remove the sterile inner sachet from the outer sachet, locate the folded corner and slowly pull in an outward motion.
2. Do not pull in a downward motion rolling knuckles over the sachet. This may exert pressure on the implant and cause it to break.
3. Remove the inner sachet by grabbing hold of the crimped edge and pulling upward.
4. To open the inner sachet, gently hold the crimped edge and cut in an arc-like fashion around the implant.
5. To remove the Gliadel implant, gently grasp the implant with the aid of forceps and place it onto a designated sterile field.
Once the tumour is resected, tumour pathology is confirmed and haemostasis is obtained, up to eight Gliadel Implants may be placed to cover as much of the resection cavity as possible. Slight overlapping of the implants is acceptable. Implants broken in half may be used, but implants broken in more than two pieces should be discarded in a biohazard container.
Oxidised regenerated cellulose (Surgicel) may be placed over the implants to secure them against the cavity surface. After placement of the implants, the resection cavity should be irrigated and the dura closed in a water-tight fashion.
4.7 Effects on Ability to Drive and Use Machines
The effects of this medicine on a person's ability to drive and use machines were not assessed as part of its registration.
4.9 Overdose
For information on the management of overdose, contact the Poison Information Centre on 131126 (Australia).
There is no clinical experience with use of more than eight Gliadel implants per surgical procedure.
7 Medicine Schedule (Poisons Standard)
S4.
6 Pharmaceutical Particulars
6.1 List of Excipients
The implants contain 192.3 mg of a biodegradable polyanhydride copolymer (Polifeprosan 20).
6.2 Incompatibilities
Incompatibilities were either not assessed or not identified as part of the registration of this medicine.
6.3 Shelf Life
In Australia, information on the shelf life can be found on the public summary of the Australian Register of Therapeutic Goods (ARTG). The expiry date can be found on the packaging.
6.4 Special Precautions for Storage
Gliadel must be stored at or below -20°C (deep freeze). Gliadel implants have been demonstrated to be stable at either -18°C or -20°C.
Unopened outer sachets may be kept at a temperature of not more than 22°C for a maximum of six hours. Refreezing of sachets is allowed if they have not been opened and have been kept for a maximum of 6 hours at a temperature of not more than 22°C. Gliadel Implants should be used within 30 days when refrozen.
6.5 Nature and Contents of Container
Each implant is packaged in two aluminium foil laminate sachets. The inner sachet is sterile and is designed to maintain product sterility and protect the product from moisture. The outer sachet is a peelable overwrap.
The outside surface of the outer sachet is not sterile.
Each box of Gliadel contains eight implants representing a single dose.
6.6 Special Precautions for Disposal
Any unused implants or implant remnant should be handled as a potentially cytotoxic agent and discarded into a biohazard container.
Summary Table of Changes


Ricky Gervais has a key message for men everywhere.
LONDON — March 8 may be International Women's Day, but — if Twitter's anything to go by — it's also a day where certain men like to club together and moan about being hard done by and stuff.
Every year, nestled among the inspirational quotes and images, are hoards of tweets like this one.
#InternationalWomensDay Wondering why there isn't a international men's day.
This year, Ricky Gervais took to Twitter to put people straight.
It's #InternationalWomensDay. Don't embarrass yourself by asking why there's no international men's day. There is. Nov 19th. You're welcome.
He's not the only one saying it, either.
Hello, you rhetorical bastards asking "why isn't there a day for men?" - I hope you enjoy next November 19th. Happy #InternationalWomensDay!
All in all, this person may have put it best.


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Q: How does one dynamically agree on a SenderCompID and TargetCompID in quickfixj? Suppose that I'm running a web-based exchange and a new customer signs up. I give that customer a new CompID and now I want to add that CompID to my acceptor. Do I have to restart my acceptor or can that be done dynamically?
It doesn't seem very scalable to have to agree on the CompID ahead of time and then sticking it in a configuration somewhere.
I've seen some examples which supposedly do that, but they seem to simply change the configuration to pretty much ignore the Sender/Target CompID's by setting them to *. Any other ways to do that?

A: Take a look at the DynamicAcceptorSessionProvider and how it's used in the Executor example. The Executor example reads session "templates" from the settings file and associates these with the acceptor endpoint IP address. An acceptor template is identified by the "AcceptorTemplate=true" setting. Any of the session ID fields, including the FIX version, can be wildcarded, not just the SenderCompID.
See the executor_dynamic.cfg for an example of how session templates are defined. In this configuration, both the SenderCompID and TargetCompID are wildcarded and templates for various FIX versions are defined (to specify the appropriate data dictionary file).
When a message arrives for an unknown session ID, the DynamicAcceptorSessionProvider will try to match the session ID with a template's wildcarded sessionID. If it finds a match, the new session will be created automatically and added to the acceptor as if it had been explicitly defined in the settings. The new session will have the settings defined in the acceptor session template.
The DynamicAcceptorSessionProvider is an AcceptorSessionProvider implementation that uses the settings file to support dynamic session creation. However, you could also implement a custom AcceptorSessionProvider that could dynamically create sessions based on database information, for example.

A: I think you want to fix the TargetCompID and send a unique SenderCompID for each user to use.  Then you set the SenderCompID in your acceptor config to *.  This will allow the acceptance of all SenderCompIDs.  However you will need to have somewhere in your code the facility to check each order against your database of users.  If the user(SenderCompID) is not valid then you will send a reject message.  Many implementations will also use tunnels and whitelisted IP addresses as well to make sure no unwanted connections are coming in.

redpajama_set_name: RedPajamaStackExchange


West Gallery church music is my abiding interest, and takes up much of my time in retirement. It was the music of town and country churches during the period c1700-1850. It would have been familiar to many churchgoers (i.e. most of the population). It is different in both style and function from the now more familiar cathedral style, typified by composers such as Maurice Green, William Boyce and Samuel Wesley. Much of the repertoire consists of settings of metrical psalms. These are often rather lively and florid, which is one of the reasons why the Victorians systematically suppressed the style during the mid-19th Century.
I enjoy reading the novels of Thomas Hardy, who was himself a West Gallery musician, and who describes the genre in his novels and poems. In 1991 my son Oliver gave me for Christmas a cassette entitled Music of Hardy's Wessex by The Mellstock Band, which includes much West Gallery music. I immediately became interested, and joined the West Gallery Music Association (WGMA).
In 1995 I joined Chiltern West Gallery Quire, and in February 1997 I founded London Gallery Quire. I participate in many West Gallery activities, often as conductor or workshop leader. I lead an annual week-end West Gallery courses at Little Benslow Hills, Hitchin, and at Farncombe Estate. In 2006 I was elected to the committee of the WGMA, and in 2013 became Hon. Secretary.
I have researched and edited West Gallery music since 1993, when I started investigating the Colby Manuscripts in the Isle of Man. This work culminated in the award of a PhD by the University of Liverpool in 2002. Together with other research I have now some 850 editions of West Gallery settings, many available through Roding Music.
Ye Boundless Realms of Joy published by the Centre for Manx Studies. This is culled from my PhD thesis, The Colby Manuscripts: West Gallery Church Music from the Southern Isle of Man (Abstract).
Your Voices Raise published by Roding Music and issued by London Gallery Quire, of whose repertoire it is an anthology of favourites.