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14 pages, 1 figure
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magnetic Schrodinger operators, quantum Hall effect, edge states
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\begin{document} 
\markboth{Edge states for a magnetic Hamiltonian}
{Edge states for a magnetic Hamiltonian}

%TITLE PAGE

%

\title{Propagating Edge States for a Magnetic Hamiltonian}

\author{Stephan De Bi\`evre\\ UFR de Math\'ematiques et UMR AGAT\\Universit\'e
des Sciences et Technologies de Lille\\ 59655 Villeneuve d'Ascq Cedex
France\\e-mail: debievre@gat.univ-lille1.fr 
\and 
Joseph V. Pul\'e\\ 
Department of Mathematical Physics\\
National University of Ireland, Dublin\\ 
(University College Dublin)\\
Belfield, Dublin 4, Ireland\\
e-mail: Joe.Pule@ucd.ie }

%
%\date{\today}
\date{February 23, 1999}

%\date{1998}

\maketitle



\begin{abstract}
We study the quantum mechanical motion of a charged particle moving in a 
half plane ($x>0$) subject to a uniform constant magnetic field $B$ directed
along the
$z$-axis and to an arbitrary impurity potential $W_B$, assumed to be weak in the
sense that $||W_B||_\infty < \delta  B$, for some $\delta$ small enough. 
We show rigorously a phenomenon pointed out by Halperin in 
his work on the quantum Hall effect, namely the existence of current
carrying and extended edge
states in such a situation. More precisely, we show that there exist
states propagating with a speed of size $B^{1/2}$ in
the $y$-direction, no matter how fast $W_B$ fluctuates. 
 As a result of this, we
obtain that the spectrum of the Hamiltonian is purely absolutely
continuous  in a spectral interval of size $\gamma B$ (for some $\gamma <1$)
between the Landau levels of the unperturbed system (i.e. the system without
edge or potential), so that the corresponding eigenstates are extended.  
\end{abstract}
\newpage
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
It is well known that a classical charged particle, constrained to a plane and 
subjected to a perpendicular magnetic field will move along physical boundaries
when those are present. In the case of a particle moving in a half plane
($x>0, y\in\RR$), it is easy to see that the circular trajectories that are at
a distance less than $\sqrt E / B$ from the edge will bounce of it in such a way
that the particle speeds alongside the edge with a velocity of the order of
$\sqrt E$, where $E$ denotes the energy of the particle. If, on the other hand,
the centre of the trajectory is too far from the edge, it will not affect the
motion of the particle.  



If, as one would expect, this picture is to carry over to the quantum
mechanical situation, then an initial state localized close to the edge in a
region of  size $B^{-1/2}$ -- an {\it edge state} -- should move ballistically
along the edge with a speed of order $\sqrt B$: here we used that the lowest
Landau level, in absence of the edge, is of order $B$. On the other hand,
although states further away from the edge -- {\it bulk states} --  should, due
to the uncertainty principle, not remain completely localized in the
$y$-direction, as in the classical case, they should nevertheless move much
more slowly than the edge states. This picture has long  been known to be
correct, but as a preparation for the case when an impurity potential is
present, we give a precise statement of the above properties in  Corollary
\ref{edgebulk}. We consider the Hamiltonian 
\begin{equation} 
H_0 =\frac{1}{2}p_x^2 +
\frac{1}{2}(p_y-Bx)^2,\label{1.a}
\end{equation}
with a Dirichlet boundary condition at $x=0$. 
Corresponding to each Landau band, we introduce the notion of 
$H_0$-invariant
{\it edge} and {\it bulk} spaces, with the following properties.
The
$y$-component of the velocity, given by $i[H_0,Y]$, is of order $\sqrt B$ on
an edge space, whereas it is exponentially small in $B$ on a bulk space
(Corollary \ref{edgebulk}). Furthermore, states belonging to the edge spaces
are negligeably small at distances much larger than the magnetic length
scale $1/\sqrt{B}$, reflecting the intuitively clear 
fact that the presence of the edge makes itself
felt only in a region of size $1/\sqrt B$ from the edge. In this sense the
edge states are quasi one-dimensional. The eigenfunctions of the restriction
of $H_0$ to the edge spaces are extended along the entire edge.

The existence of non-localized current-carrying quasi one-dimensional 
edge states plays a role in
certain theories of the quantum Hall effect \cite{h} (see \cite{gp}, 
\cite{w} and \cite{fs} for further details). It is therefore
of importance  to understand if such states exist in systems exhibiting a
quantized Hall resistance.  This is
argued to be the case  in  \cite{h}, in the case when the full Hamiltonian
is obtained by  adding a  weak impurity
potential to $H_0$. In other words, such potentials are not supposed 
to destroy the edge states existing in the free case.  
A very simple and rigorous proof of this statement is given in the 
present paper.  A
weak potential is a potential 
$W_B\in L^\infty(\RR_+\times\RR)$ satisfying 
$\delta_B \equiv ||W_B||_\infty < \frac{1}{2} B$. Since the distance between
successive Landau levels equals $B$, such a potential can not close the gaps 
between the Landau levels of the infinite system without a edge, even though its
size can be of order $B$: in this sense it is weak. It can however fluctuate
arbitrarily fast, and in particular on the magnetic length scale, which is of
order $1/\sqrt B$ (about $50-100$ Angstrom in realistic situations):
 this is important since, as explained in \cite{gp}, the weak
impurity potential is created by impurities at a distance of order $1/\sqrt B$
or less of the layer and can vary rapidly on this length scale.  As a typical
form for $W_B$ we can keep in mind a potential of the type
$$
W_B = \delta B \sum_{i\in\ZZ^+\times\ZZ} u_i(B^\alpha(\vec{x}- 
\frac{i}{B^\beta}))
$$
for some compactly supported site-potentials $u_i$ and exponents
$\alpha\geq 0$ and $\beta \geq 1/2$.


For weak potentials, we show that, in a spectral interval of size $B$
 between the Landau
levels, there are no bound states and that the speed in the $y$-direction is
still of order $\sqrt B$. 
As a consequence, we
obtain that in the same spectral interval, the spectrum is 
absolutely continuous, implying the corresponding eigenstates are extended. 


The results described above in the case when no impurity potential is
present have been known for a long time and
can be obtained by studying explicitly the spectrum and eigenfunctions 
of
$H_0$, since it is an explicitly solvable Hamiltonian. Such an approach would
however not easily extend to the case when an impurity potential is added. 
Instead, we show in Proposition \ref{poscom2} that 
the magnitude of the speed in the 
$y$-direction
is strictly positive on the spectral subspaces corresponding to suitable 
spectral intervals between Landau levels. Such a positive commutator estimate is
then shown to be stable under perturbations in section 3, yielding the main
results via the virial theorem and the Mourre theory of positive commutators. 
(Theorem \ref{mainth}).



The idea that positive commutator methods and the virial theorem can be used to
obtain information about magnetic Hamiltonians in the presence of boundaries was
first proposed in \cite{mmp}. They consider a model with a {\it soft} edge, 
modeled
by a positive potential  $V$, supported on the negative axis and steeply rising
from $0$, and prove the absence of eigenvalues  in certain regions between the
Landau levels in this case. The conjugate operator used in this approach is the
quantum observable $C_y$ corresponding  to the $y$-coordinate of the centre of
the classical circular orbit: $C_y= y-(p_x/B)$. Classically this is indeed a
monotonic function of time for orbits close to the edge, since the Poisson
bracket $\{C_y, H\}=\frac{1}{B}(\partial_x V+\partial_x W)<0$ in that region,
provided the impurity potential $W$ has a small enough derivative.  


In the present paper, we deal with the problem with a hard edge, as described
before.  We use the
$y$-coordinate itself as a conjugate operator, proving that 
the speed in the $y$-direction, $i[H,Y]$,  is strictly negative on edge states. 
This is marginally surprising since it is not true classically, but it
 turns out
to be extremely simple to understand in terms of the band structure
of the free Hamiltonian $H_0$.  
Using $y$ also has the important advantage of not introducing
derivatives of the potential in the commutator, as is the case when using
$C_y$, and therefore eliminating the need to control
their size. In addition, it renders the interpretation of the results in terms 
of propagation along the edge more transparent. On the down side, it is not 
obvious
the
present method will adapt itself easily to cases where the edge is not straight. 


Let us  point out that we could treat the soft edge in the same way.
It seems however that this model does not lend itself to an analysis of the
high field regime, which is important for the quantum Hall effect. 
In that case, the particles will, even in the lowest Landau
level, penetrate deeply into the region $x<0$, so that there is an effective
edge around those values of $x$ where $V(x)\sim B$, where $V$ is the edge
potential. The high field behaviour of the speed, for example, will then depend
crucially on the precise shape of the edge, and this is not satisfactory.
We will therefore not deal any further with the soft edge in the following.

The results of \cite{mmp}  on the soft edge have recently been extended
\cite{fgw1} to a proof of
absence of singular continuous spectrum in suitable intervals
between the Landau levels, using the same conjugate operator as in \cite{mmp} 
to prove a positive commutator estimate. While 
finishing the present work, we learned that those results were further extended,
still using $C_y$ as a conjugate operator,
to the case of the {\it hard} edge in \cite{fgw2}. A result comparable to
our Theorem \ref{mainth} is proven there, but under the additional 
assumptions that both the first and second derivatives of the impurity
potential are small, so that rapid fluctuations in the impurity potential are
no longer allowed. In addition, our proof
is  technically
considerably less complicated partially because, in the model with a hard edge,
 the operator $C_y$ is symmetric but not self-adjoint, leading to complications
in applying the Mourre theory of positive commutators. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%SECTION 2

\section{The free Hamiltonian: edge and bulk spaces}
To study $H_0$ in (\ref{1.a}), we first use the translational invariance
in the $y$-direction to write
\begin{equation}
H_0=\int^\oplus_{\RR}dk \  H(k),\quad\hbox{with}\quad H(k) =
-\frac{1}{2}\frac{d^2}{dx^2} + \frac{1}{2}(k-Bx)^2, \ \ \ x>0,
\end{equation}
acting on $L^2(\RR_+\times \RR, dxdk)$, 
$k$ being the Fourier transform variable conjugate to $y$. 
We recall that $H_0$ is essentially self-adjoint on 
the space of functions $\varphi\in C_0^\infty (\overline{\RR_+} \times \RR)$ vanishing on the boundary \cite{he}.
\par

The spectrum of $H(k)$ consists of isolated non-degenerate 
eigenvalues $E_n(k)$, $n\in {\NN}_0$, with {\sl normalized} eigenfunctions 
$\varphi_n(x,k)$. We
will write ${\cal H}_n$ for the $n^{\hbox{th}}$ {\sl band space}, namely the 
space consisting of
vectors of the form $f(k)\varphi_n(x,k), f\in L^2(\RR, dk)$. This is an
$H_0$-invariant subspace of $L^2(\RR_+\times \RR, dxdk)$; we shall on
occasion view it as a subspace of  $L^2(\RR_+\times\RR, dxdy)$ as well, with the 
same notation. To understand the behaviour of
the $E_n(k)$ and the $\varphi_n(x,k)$, and in particular their dependence on
$B$, we introduce the following scaling:
\begin{equation}
\tilde x = \sqrt B x, \ \tilde y= \sqrt B y, \ H_0=B{\tilde H_0},\  
{\tilde H_0}=-\frac{1}{2}\frac{\partial^2}{\partial\tilde x^2} +
\frac{1}{2}(\frac{1}{i}\frac{\partial}{\partial\tilde y}-\tilde x)^2.
\end{equation}
Note that, strictly speaking, $H_0$ is unitarily equivalent to $B{\tilde H_0}$, 
not 
equal to it, but since the unitary transformation is just the rescaling
of the variables, we allow ourselves this slight abuse of notation. Again
\begin{equation}
{\tilde H_0}=\int^\oplus_{\RR} d\kappa\ {\tilde H}(\kappa),
\quad \hbox{with}\quad {\tilde H}(\kappa) =
-\frac{1}{2}\frac{d^2}{d\tilde x^2} 
+ \frac{1}{2}(\kappa-\tilde x)^2, \ \ \ \ \tilde x>0.
\end{equation}

Here $\kappa$ is the Fourier transform variable conjugate to $\tilde y$, so
that $ky=\kappa\tilde y$ and hence $k=\sqrt B \kappa$. The spectrum of
${\tilde H}(\kappa)$ consists of isolated eigenvalues $\alpha_n(\kappa)$. The 
normalized eigenfunctions
of ${\tilde H}(\kappa)$, at each fixed $\kappa$, $\tilde \varphi_n (\cdot, 
\kappa)$ are given by
\bea
\tilde \varphi_n (\tilde x, \kappa)=C_n 
D_{\alpha_n(\kappa)-1/2}(\sqrt2({\tilde x}-\kappa)),
\eea
where $D_{\alpha-1/2}$ is the Whittaker function (\cite{as} p686) with parameter 
$\alpha$ and $\alpha_n(\kappa)$ is determined by the boundary condition 
\be
D_{\alpha_n(\kappa)-1/2}(-\sqrt2\kappa)=0.
\label{2.b}
\ee
One can check that the eigenvalues $\alpha_n(\kappa)$ 
are smooth functions of $\kappa$. The other properties of the
$\alpha_n(\kappa)$ that we shall be needing are collected in the following 
Lemma:

\begin{lem} \label{alphan} 
\begin{itemize}
\item[(i)] $\alpha_n(0)=2n +3/2,
\qquad{\displaystyle \alpha'_n(0)=
-\frac{(2n+2)!}{n!(n+1)!\pi^{1/2}2^{2n}}},$
\item[(ii)]$\alpha'_n(\kappa)=
-\frac{1}{2}|\tilde \varphi_n'(0,\kappa)|^2<0,$
\item[(iii)] $\alpha_{n}(\kappa)>(n+\half)$ and there exist $C_n>0$ 
so that, for all $\kappa\geq 0$, 
$$
\alpha_{n}(\kappa) -(n+\frac{1}{2}) \leq 
C_n \exp -\frac{1}{4}(\kappa -\sqrt n)^2.
$$
\item[(iv)]  For all $n\in \NN_0$ and all $\epsilon>0$, there exist 
positive constants $X_n$, $K_{n,\epsilon}, C_{n,\epsilon}$ so that, for all
$\kappa\geq K_{n,\epsilon}$ and all $\tilde x\in[0,X_n]$,
$$
|\tilde\varphi_n(\tilde x, \kappa)|^2\leq
C_{n,\epsilon}\exp\left \{-\half (1-\epsilon)(\tilde x - \kappa)^2\right \},
$$
and
$$ 
|\alpha_n'(\kappa)| \leq C_{n,\epsilon}\exp \left \{-\half 
(1-\epsilon)\kappa^2\right \}.
$$
\item[(v)] For all $\kappa <0$, $|\alpha_n'(\kappa)|> |\kappa|$ 
and
$\lim_{\kappa\to-\infty}\alpha_n(\kappa) =\infty$.
\end{itemize}
\end{lem}

The proof of this lemma, which uses only  standard techniques of Schr\"odinger 
operator theory, is postponed to Section \ref{proof}.  Numerical computation of
the $\alpha_n(\kappa)$ indicates that they are convex functions of 
$\kappa$ (see Figure 1). In addition, it seems that
$$
\alpha_{n+1}(\kappa) - \alpha_n(\kappa) >1,
$$
a result that is  intuitively clear, at least for $\kappa>0$. 
We have not been able to prove these last two results and shall not use them.  
If
they are true, the statements of our results below can be simplified somewhat. 

%
%%%%%%%%%%%%%%%%%FIGURE

\begin{figure}[hbt] 

\begin{center}\includegraphics[width=10cm]{eigen.eps}

%%%%%%%%%%%%%%%%%%\end{center}\vspace{-25mm}

\end{center}

%%%%%%%%%%%%%%%%%%%%%%%%%%\vspace{-25mm}

\caption{$\alpha_n(\kappa)$ for $n=0,1,2,3$}

\label{ }

\end{figure}

%

\par
It is clear from the above that the spectrum of ${\tilde H}_0$ and hence of
$H_0$ is absolutely continuous and fills the entire half-axis from $1/2$ to
infinity. The bands $E_n(k)$ can now be written
\begin{equation}
E_n(k) = B\alpha_n(\frac{k}{\sqrt B}).
\end{equation}

Writing $\varphi_n (x, k)$ for the normalized eigenfunctions
of ${H}(k)$ (at each fixed $k$), we have 
\begin{equation}
\varphi_n(x,k) = B^{1/4} \tilde\varphi_n(\sqrt B x, \frac{k}{\sqrt B}).
\label{eigenf}\end{equation}
These simple observations will now allow us to define within
${\cal H}_n$   {\it  edge spaces} and  {\it bulk spaces} as follows. We define,
for each
$\sigma>0,\gamma>0$:
\begin{eqnarray}
{\cal H}_{n,e}(\sigma, \gamma) &\cong& L^2(]-\infty, \sigma B^\gamma], dk)
\subset {\cal H}_n,\\
{\cal H}_{n,b}(\sigma, \gamma) &\cong& L^2([ \sigma B^\gamma, \infty), dk)
\subset {\cal H}_n,\\
{\cal H}_n& =& {\cal H}_{n,e} \oplus {\cal H}_{n,b}.
\end{eqnarray}
Note that these spaces are $H_0$ invariant. We will call ${\cal H}_{n,e}(\sigma,
\gamma)$ an edge space for all
$\gamma\leq 1/2$ and ${\cal H}_{n,b}(\sigma, \gamma)$ a bulk space for all
$\gamma>1/2$. 
For a different approach to the definition of bulk and edge spaces,
in the case of a bounded geometry, we refer to \cite{av}.

To understand those definitions, recall first that
a standard  stationary phase argument shows
that $-\partial_k E_n(k_0)=-\sqrt B\alpha_n'(k/{\sqrt B})$
 is the  group speed in the $y$-direction
of a wave packet $f(k)\varphi_n(x, k)$ with the support of $f$ close to $k_0$.
If $k_0$ is inside an interval $(-\infty, k_B]$ where $k_B$ is of order $\sqrt B$ 
or smaller,  the wave packet belongs to 
the edge space ${\cal H}_{n,e}(\sigma, 1/2)$ and it follows from Lemma
\ref{alphan}  that such a wave packet speeds along the edge in the $y$ direction
with a velocity of order $\sqrt B$. In addition, it follows from standard
exponential estimates on the eigenfunctions $\tilde\varphi_n$ (as in the proof of Lemma \ref{alphan}) that in this
case the wave packet is exponentially small for $x$ much bigger than $1/\sqrt
B$.  If, on the other hand, $k_0$ belongs to an interval of the form $[k_B,\infty[$
with $k_B$ of order $B^\gamma, \gamma >\half$, then the group velocity is 
exponentially small in $B$ (see Lemma \ref{alphan} (iv)). In addition, if 
$f(k)\varphi_n(x,k)\in {\cal H}_{n,b}(\sigma, \gamma)$, with $\gamma>\half$, then 
Lemma \ref{alphan}(iv) immediately implies that 
$$
\int_0^{\frac{1}{\sqrt B}} \int_{-\infty}^{\infty}|f(k)\varphi_n(x,k)|^2
dkdx \leq C_{n,\epsilon} \exp -(1-\epsilon)(\sigma^2 B^{2\gamma -1}-1),
$$
so that the wave packet is exponentially small in the region 
$0\leq x\leq \frac{1}{\sqrt B}$ close to the edge.
We note also that  the spectrum of
$H_0$ restricted to a bulk space ${\cal H}_{n,b}(\sigma, \gamma)$ is an
exponentially small interval (in $B$) just above the $n$th Landau level (Lemma \ref{alphan} (iii)),
that we will refer to as the bulk spectrum. The spectrum of $H_0$ restricted to
an edge space ${\cal H}_{n,e}(\sigma, 1/2)$ -- {\em the edge spectrum} -- is on
the other hand of the form 
$[B(n+\frac{1}{2} + c_\sigma),\infty)$. In particular, it fills up an interval 
of
size
$B$ below the $(n+1)$th Landau level, including the latter. 

To give a formulation of the above statements that is at once more precise and
does not use the band structure of the Hamiltonian $H_0$, so that it has a
chance to pass to the perturbed Hamiltonian, we now turn  to the statement
and proof of
 a {\it positive commutator estimate}. We will show that the speed $V_y =
i[H_0, Y]$, where the operator $Y$ is multiplication by $y$,
is strictly negative away from the Landau levels. This is the content of the 
following proposition, which will be
generalized to the perturbed Hamiltonian in the next section.  
%%%
\par
Let $L_n=(n+1/2,n+3/2]$ be the $n$th Landau band when $B=1$.
Let
\be
\theta_n(\kappa,n',n'')
=\cases{| \alpha_{n'}(\kappa)- \alpha_{n''}(\kappa)|& if both 
$\alpha_{n'}(\kappa)$ and $\alpha_{n''}(\kappa)$ are in $L_n$,\cr
1 & otherwise. \cr}
\ee
We let $\delta_0=1$ and for 
$n \geq 1$ we let
\be
\delta_n=\inf_{{n'\neq n''}\atop{n',n''\leq n}}
\inf_\kappa \theta_n(\kappa,n',n'').
\label{2.c}
\ee
Note that for $n'>n$, $\alpha_{n'}(\kappa)>n+3/2$ for all $\kappa$, 
so that in (\ref{2.c}) it is not necessary to have the restriction
$n',n''\leq n$. From Lemma \ref{alphan} {\it(i)} we see that $\alpha_0(0)=3/2$ 
and $\alpha_1(0)=7/2$ and from {\it(ii)} we see that $\alpha_n(\kappa)$ is 
strictly decreasing in $\kappa$.
Therefore for $\kappa \leq 0$ only $\alpha_0(\kappa)$ can be in $L_1$ while for
$\kappa > 0$ only $\alpha_1(\kappa)$ can be in $L_1$.
Definition (\ref{2.c}) then  implies that
$\delta_1=1$ also.  As mentioned before, Fig.1 suggests that  $\delta_n=1$ for
all $n$. We will only prove $\delta_n>0$, as follows. 


\par

Since $\alpha_{n'}(\kappa)\to \infty$ as $\kappa \to -\infty$ for all
$n'$, there exists $\kappa_n<0$ such that  for all $\kappa<\kappa_n$,
$\alpha_{n'}(\kappa)>n+3/2$ for $n'\leq n$. 
Also there exists $\kappa'_n>0$ such that
for all $\kappa>\kappa'_n$, $\alpha_{n'}(\kappa)<n+1/2$ for $n'< n$. 
Thus for $n'\neq n''$, $n',n''\leq n$, $\theta_n(\kappa,n',n'')=1$ for $\kappa$ 
outside the compact interval 
$[\kappa_n,\kappa'_n]$ and therefore $\delta_n>0$.
\par\noindent
\begin{pro}\label{poscom2}
Let $\Delta \subset L_n $ be a closed interval with $|\Delta|<\delta_n$. Let 
\be
\nu_-(\Delta)= \inf_{\{n',\kappa \ |\ \alpha_{n'}(\kappa)\in \Delta \}}| 
\alpha'_{n'}(\kappa)| 
>0
\ee
and
\be
\nu_+(\Delta)= \sup_{\{n',\kappa \ |\ \alpha_{n'}(\kappa)\in \Delta \}}| 
\alpha'_{n'}(\kappa)| 
>0.
\ee
If ${\tilde Y}$ is multiplication by ${\tilde y}$ and ${\tilde P}_0(\Delta)$
is the spectral projection of ${\tilde H}_0$ onto $\Delta$, then
\begin{equation}
\nu_-(\Delta){\tilde P}_0(\Delta)\leq
{\tilde P}_0(\Delta) i[{\tilde Y}, {\tilde H}_0]{\tilde P}_0(\Delta)\leq
\nu_+(\Delta){\tilde P}_0(\Delta),
\end{equation}
and consequently, for $\psi$ with $||\psi||=1$ in the range of ${\tilde 
P}_0(\Delta)$
\begin{equation}
-\nu_+(\Delta)t \leq \langle\psi_t, {\tilde Y} \psi_t\rangle - \langle\psi_0, 
{\tilde Y}
\psi_0\rangle \leq -\nu_-(\Delta)t.
\end{equation} 
\end{pro}
\noindent{\bf Proof:} We have that 
$\alpha_{n'}^{-1}(\Delta)\cap\alpha_{n''}^{-1}(\Delta) =\emptyset $ 
if $n'\neq n''$, since $|\Delta|< \delta_n$.
Thus for any $\psi$ we can write
\be
{\tilde P}_0(\Delta)\psi(\tilde x, \kappa)
= \sum_{n'=0}^n\psi_{n'}(\tilde x, \kappa),
\ee
where 
\be
\psi_{n'}({\tilde x}, \kappa)=\beta_{n'}(\kappa)
{\bf 1}_{\alpha_{n'}^{-1}(\Delta)}(\kappa)
{\tilde\varphi}_{n'}({\tilde x}, \kappa)
\ee
and
\be
\beta_{n'}(\kappa) =\int_{\RR_+} \overline{\psi(\tilde x,\kappa)}\ \tilde
\varphi_{n'}(\tilde x,\kappa) dx .
\ee
Since $i[{\tilde Y}, {\tilde H}_0] = {\tilde x}-p_{\tilde y}$ , 
it is clear that $\langle \psi_{n'}, i[{\tilde Y}, {\tilde H}_0]
\psi_{n''}\rangle=0$ if $n'\not=n''$ since the supports of $\psi_{n'}$ and of 
$\psi_{n''}$ 
are disjoint in the $\kappa$ variable. On the other hand,
\be
\langle \psi_{n'}, i[{\tilde Y}, {\tilde H}_0]\psi_{n'}\rangle
=\int_{\alpha_{n'}^{-1}(\Delta)}d\kappa\
|\beta_{n'}(\kappa)|^2\int_{\RR_+}d{\tilde x}  ({\tilde
x}-\kappa)|\varphi_{n'}({\tilde x},\kappa)|^2 ,
\ee
and, by the Feynman-Hellman theorem
$$
\int_{\RR_+}d{\tilde x} ({\tilde x}-\kappa)|{\tilde\varphi}_{n'}({\tilde 
x},\kappa)|^2
= -\alpha'_{n'}(\kappa) = |\alpha'_{n'}(\kappa)|.
$$
The 
proposition is now immediate. 
\qed
\par
Using the scaling behaviour in $B$ we now have the following Corollary:
\begin{cor}\label{edgebulk}\mbox{}\hfil\break 
(i) Let $n\in\NN$ be fixed and let 
$\Delta\subset ((n+\frac{1}{2})B, (n+\frac{3}{2})B]$ be a closed interval
with $|\Delta|<\delta_n B$. Then
\be
\sqrt B\nu_-(B^{-1}\Delta)P_0(\Delta)\leq
P_0(\Delta) i[Y, H_0]P_0(\Delta)\leq
\sqrt B\nu_+(B^{-1}\Delta)P_0(\Delta).
\ee
where $B^{-1}\Delta=\{E/B \ |\ E\in \Delta\}$. 
\par
\noindent (ii) For all $n\in\NN$, for all $\sigma>0$ there exists a 
constant $C_{n, \sigma}>0$ so that for all 
$\psi \in {\cal H}_{n,e}(\sigma,1/2)$ and for all $B$
\begin{equation}
\langle\psi, i[Y,  H_0]\psi\rangle 
\geq\sqrt B \inf_{\kappa\leq \sigma }|\alpha'_n(\kappa)|\ ||\psi ||^2
 \ >C_{n,\sigma}\sqrt B ||\psi ||^2.
\end{equation}
(iii)  Let $\epsilon>0$. Then for all $n\in{\NN}_0$, 
for all $\sigma>0$ there exists a constant $C_{n,\sigma,\epsilon}>0$ so that for
all $B$ and for all $\psi \in {\cal H}_{n,b}(\sigma,1/2+\epsilon)$
\begin{equation}
\langle\psi, i[Y, H_0]\psi\rangle\leq
\sqrt B \sup_{\kappa\geq \sigma B^\epsilon }|\alpha'_n(\kappa)|\ ||\psi ||^2
\ < C_{n,\sigma,\epsilon}\sqrt B\exp\left
\{-\half(1-\epsilon)\sigma^2B^{2\epsilon}\right \}||\psi||^2.
\end{equation}
\end{cor}
\noindent{\bf Proof:} This is now an immediate consequence of Lemma
\ref{alphan} and of the proof of Proposition \ref{poscom2}.\qed

\begin{rmk} Parts (ii) and (iii) of the corollary state that the speed
in the y direction is at least of order $\sqrt B$  for any edge state and
at most of order  $\exp-B^{2\epsilon}$ for any bulk state.  
\end{rmk}




%%%%%%%%%%%%%%%%%%%%SECTION 3

\section{Adding a weak impurity potential}
We now consider the Hamiltonian 
$$
H=H_0 + W_B
$$
where $W_B\in L^\infty(\RR_+\times \RR, dxdy)$ is a real potential satisfying 
$||W_B||_\infty \leq AB$ where $A<\infty$ is independent of $B$.  
Let
$$
{\tilde H}={\tilde H}_0 + \tilde W_B ; \tilde W_B(\tilde x, \tilde y)
=B^{-1}W_B(\frac{\tilde x}{\sqrt B}, \frac{\tilde y}{\sqrt B})
$$
and let ${\tilde P}(\cdot)$ denote the spectral family of ${\tilde H}$.
\par
Our main result is then the
following theorem, which should be compared to Proposition 2.1.
For $\lambda <1$, let $L_n^\lambda = (n+1/2+\lambda, n+3/2]$.
Let 
\be
\nu(n,\lambda)=\nu_-(L_n^\lambda)
= \inf\{| \alpha'_{n'}(\kappa)| \ |\  n'\leq n,\
n+1/2+\lambda<\alpha_{n'}(\kappa)\leq n+3/2 \}  >0.
\label{nunlambda}\ee
\begin{thm}\label{mainth} Let
$n\in\NN$ be fixed. Let
$\lambda$,
$\lambda' >0$ with
$\lambda +\lambda'<1$ and let 
$L_n^{\lambda,\lambda'}= (n+1/2+\lambda, n+3/2-\lambda')$.
There exists $\delta(n,\lambda,\lambda')>0$ such that if 
$||W_B||_\infty<\delta(n,\lambda,\lambda') B$
and
$\epsilon<\delta(n,\lambda,\lambda')$,  then, for all $\alpha \in L_n^{\lambda,\lambda'}$, for the
interval 
$\Delta \equiv (\alpha-\epsilon, \alpha+\epsilon)$,
\be
{\tilde P}(\Delta) i[{\tilde Y}, {\tilde H}]{\tilde P}(\Delta)\geq
\half \nu(n, \lambda/2){\tilde P}(\Delta).
\label{3.d}
\ee
Consequently if $||W_B||_\infty<\delta(n,\lambda, \lambda')B$, then
\be
\sigma_{\rm sing}({\tilde H})\cap L_n^{\lambda,\lambda'} =\emptyset.
\label{3.e}
\ee
\end{thm}
\par 
Clearly we can give a scaled up version of this theorem:
\begin{cor}\label{maincor}Let $n\in\NN$ be fixed and let $\lambda$, $\lambda' 
>0$ with
$\lambda +\lambda'<1$. There exists $\delta(n,\lambda,\lambda')>0$ 
such that if $||W_B||_\infty<\delta(n,\lambda,\lambda') B$, then
\be
\sigma_{\rm sing}(H)\cap (B(n+1/2+\lambda), B(n+3/2-\lambda')) =\emptyset.
\ee
\end{cor}
It is useful to have the following variant of Theorem \ref{mainth}. Here 
we fix a bound on $||W_B||_\infty/B$ and give the dependence on this bound 
of the endpoints $a$,$b$ of the interval $(a,b)\subset L_n$, such that 
$(a,b)$ contains only absolutely continuous spectrum.
\begin{thm}\label{varth} Let
$n\in\NN$ be fixed. 
Suppose that 
$||W_B||_\infty<\delta B$ where $\delta<1/2$.
Let
$\lambda_n$,
$\lambda_n' \in (0,1/2)$ be such that
$\min(\lambda_n,\delta_n) \nu(n,\lambda_n/2)^2 > 2^9(n+2)\delta$ and 
\break
$\min(\lambda_n',\delta_n) \nu(n,1/4)^2 > 2^9(n+2)\delta$
then 
for all $\alpha \in L_n^{\lambda_n,\lambda_n'}$, there exists an interval 
$\Delta $ containing $\alpha$ such that
\be
{\tilde P}(\Delta) i[{\tilde Y}, {\tilde H}]{\tilde P}(\Delta)\geq
\half \nu(n, \lambda_n/2){\tilde P}(\Delta).
\label{3.f}
\ee
Therefore
\be
\sigma_{\rm sing}({\tilde H})\cap L_n^{\lambda_n,\lambda_n'} =\emptyset.
\label{3.g}
\ee
\end{thm}
\par 
Note that, given $\delta$, no $\lambda_n$ and $\lambda_n'$ satisfying the 
conditions of the theorem might exist. Nevertheless, it is clear that for 
sufficiently small $\delta$, the above results guarantee the existence of an 
interval of absolutely continuous spectrum between the Landau levels.
The scaled up version of this theorem is then:
\begin{cor}\label{varcor} Under the conditions of Theorem \ref{varth},
\be
\sigma_{\rm sing}(H)\cap (B(n+1/2+\lambda_n), B(n+3/2-\lambda_n')) 
=\emptyset.
\ee
\end{cor}
\par\noindent
{\bf Proof of Theorem \ref{mainth}:}\ Note first that 
$i[\tilde Y, \tilde H_0] = i [\tilde Y, \tilde H]$,
so that the result would follow from Proposition 2.1 if we could
replace ${\tilde P}(\Delta)$ by ${\tilde P}_0(\Delta)$. This
can indeed be achieved with a few tricks and at not too high 
a cost, provided
one replaces the interval $\Delta$ by an auxiliary one $\Delta'$,
that is larger but for which $\nu(\Delta')$ is not too small. 
Let $\sigma\equiv \min(\lambda, \lambda', \delta_n)/4$, where 
$\delta_n$ is
defined in (\ref{2.c}), and let
$\Delta'$  be the interval $[\alpha-\sigma, \alpha+\sigma]\subset
L^{\lambda/2,\lambda'/2}_n$.
Let $\Delta$
be the interval $[\alpha-\epsilon,\alpha+\epsilon]$,
where $\epsilon\leq\sigma$. 
Let $\psi\in {\tilde P}(\Delta){\cal H}$.
Then, recalling that $A \leq ||\tilde W_B||_\infty$,
$$
||({\tilde H}_0 - \alpha)\psi||\leq ||({\tilde H}-\alpha){\tilde P}
(\Delta)\psi||
+A||\psi||\leq(\epsilon + A)||\psi||.
$$
Hence
\be
||{\tilde P}_0(\Delta^{'c})\psi||\leq
||\frac{1}{{\tilde H}_0-\alpha}\tilde P_0(\Delta^{'c})||
\ ||({\tilde H}_0-\alpha)\psi||\leq
{\sigma}^{-1}(\epsilon + A)||\psi||,
\label{3.c}
\ee
since
$\hbox{min}\{|\lambda - \alpha|\ |\ \lambda\in\Delta'^{c}\}\geq \sigma.$ 
Clearly
\begin{eqnarray}
i\langle\psi, [\tilde Y, \tilde H]\psi\rangle
&\geq &i\langle \tilde P_0(\Delta')\psi,[\tilde Y, \tilde H_0]
\tilde P_0(\Delta')\psi\rangle\nonumber\\
&\ & \hskip 1cm -2||[\tilde Y, \tilde H_0]\tilde P_0 (\Delta^{'c})\psi||\ ||\psi||.
\label{3.a}
\end{eqnarray}
The required positivity will come from the first term, 
so we only have to control the last one. We find
\begin{eqnarray*}
||[{\tilde Y}, {\tilde H}_0]{\tilde P}_0(\Delta^{'c})\psi|| 
&\leq&2 \langle
{\tilde P}_0(\Delta^{'c})\psi, {\tilde H}_0 {\tilde P}_0
(\Delta^{'c})\psi\rangle^{1/2}\\
&\leq&
2||{\tilde H}_0{\tilde P}_0(\Delta^{'c})\psi||^{1/2} 
||{\tilde P}_0(\Delta^{'c})\psi||^{1/2}.
\end{eqnarray*}
But
\begin{eqnarray*}
||{\tilde H}_0{\tilde P}_0(\Delta^{'c})\psi||^{1/2}
&\leq& (||{\tilde H}\psi|| + A||\psi||)^{1/2}\\
&\leq&( n+3/2+A)^{1/2}||\psi||^{1/2}\\
&\leq&( n+2)^{1/2}||\psi||^{1/2},
\end{eqnarray*}
if $A\leq 1/2$. Therefore
$$
||[{\tilde Y}, {\tilde H}_0]{\tilde P}_0(\Delta^{'c})\psi||\leq 
2( n+2)^{1/2}{\sigma}^{-1/2}(\epsilon +A)^{1/2}||\psi||.
$$
Inserting this into (\ref{3.a}) yields 
$$
\hskip-2cm i\langle \psi, 
[{\tilde Y}, {\tilde H}]\psi\rangle 
\geq i\langle {\tilde P}_0(\Delta')\psi,
[{\tilde Y}, {\tilde H}_0] 
{\tilde P}_0(\Delta')\psi\rangle
$$
\be
 \hskip2cm - 4( n+2)^{1/2}{\sigma}^{-1/2}(\epsilon +A)^{1/2}||\psi||^2.

 \label{3.j}\ee
On the other hand, since $|\Delta'|=2\sigma <\delta_n$, 
Proposition 2.1 states that
$$
i\langle {\tilde P}_0(\Delta')\psi, [{\tilde Y}, 
{\tilde H}_0]{\tilde P}_0(\Delta')\psi\rangle \geq
\nu(\Delta')||{\tilde P}_0(\Delta')\psi||^2
\geq
\nu(n,\lambda/2)||{\tilde P}_0(\Delta')\psi||^2,
$$
where $\nu(n,\lambda/2)$ is defined in (\ref{nunlambda}). 
Inserting this into (\ref{3.j}) and using (\ref{3.c}) together with the observation that
$
||\psi||^2=||{\tilde P}_0(\Delta')\psi||^2+||{\tilde P}_0(\Delta'^{c})\psi||^2,
$
 yields
\be
 i\langle \psi,[{\tilde Y}, {\tilde H}]\psi\rangle 
\geq \nu(n,\frac{\lambda}{2})
\bigl[1-\left(\frac{(\epsilon+A)^2}{\sigma^2}+
\frac{4(n+2)^{1/2}(\epsilon +A)^{1/2}}{{\sigma}^{1/2}
\nu(n,\frac{\lambda}{2})}\right)\bigr]||\psi||^2.
\label{ineq}\ee
Let $\delta(n,\lambda,
\lambda') =\min\left (\frac{\sigma \nu(n,\lambda/2)^2}{
2^9(n+2)}, 
\frac {\sigma}{4}, \frac{1}{2}\right)$. 
Then if $A<\delta$ and $\epsilon <\delta$, one has that 
$4( n+2)^{1/2}{\sigma}^{-1/2}(\epsilon +A)^{1/2}
<\frac{1}{4}\nu(n,\lambda/2)$ and $((\epsilon + A)/\sigma)^2\leq 1/4$, so that
 the first statement in the theorem
follows. To prove  (\ref{3.e}) 
it is now sufficient to use (\ref{3.d}) and to apply the Mourre theory of
positive commutators. For a textbook treatment, we refer to  \cite{abg}; see
also \cite{gg} for a concise review of the domain questions involved. The latter 
are trivial in the present case. Indeed, the commutator $[H_0, Y] = [H, Y]$ is 
obviously relatively $H_0$ bounded, the domain of the Hamiltonian is invariant 
under the unitary group $\exp is Y$ and the second commutator $[[H_0,Y],Y]$ is
bounded.
\qed



%%%%%%%%%%%%%%%%%%%%%PROOF OF LEMMA


\section{Proof of Lemma \ref{alphan} }\label{proof}
\par
Part (i) follows from a computation using standard
properties of the Hermite polynomials.  To prove (ii), we write $V_\kappa(\tilde
x) = \half (\tilde x - \kappa)^2$ and use the Feynman-Hellman formula  to write
(see \cite{hd})
\begin{eqnarray*}
\alpha'_n(\kappa)&=& -\int_0^\infty V_\kappa'(\tilde x) 
\tilde\varphi_n^2(\tilde x, \kappa) \ d\tilde x\\
&=&2\int_0^\infty V_\kappa (\tilde x) \tilde \varphi_n (\tilde x, \kappa) 
\tilde \varphi_n'(\tilde x, \kappa) \ d\tilde x\\
&=& \int_0^\infty \varphi_n'' (\tilde x, \kappa)\varphi_n' (\tilde x, \kappa) 
\ d\tilde x + 2\alpha_n(\kappa) 
\int_0^\infty \varphi_n(\tilde x, \kappa)\varphi_n' (\tilde x, \kappa) \ d\tilde x,
\end{eqnarray*}
from which the result follows.  Note that by uniqueness $\varphi_n'(0)$ cannot be zero. 
For (iii), we will use a perturbative argument, treating
the Dirichlet boundary condition at $0$ as a perturbation. We note first that, 
by
the min-max principle,  $\alpha_n(\kappa)> n+\half$. Now, let $h_n$ denote the
$n$th  Hermite function and let $h_{n,\kappa}(x)=h_n(x-\kappa)$. 
Let $\theta$ be a smooth function such that $\theta(x)=0$ for $x\leq 0$ and
$\theta(x)=1$ for $x\geq 1$ We compute
$$
(\tilde H (\kappa) - (n+\half))\theta h_{n,\kappa}=[\tilde H(\kappa), 
\theta]h_{n,\kappa}=
\half(-\theta''-2i\theta'p)h_{n,\kappa}.
$$
Now, since the supports of $\theta'$ and $\theta''$ are contained
in $[0,1]$, and since 
$$
||\theta'p h_n||^2 = \langle h_{n,\kappa}, [p,\theta'^2] p h_{n,\kappa}\rangle +
\langle \theta'^2h_{n,\kappa}, p^2h_{n,\kappa}\rangle,
$$
one easily concludes there exists a constant $C_n$ so that
$$
||(\tilde H(\kappa) - (n+\half))\theta h_{n,\kappa}||
\leq C_n 
||{\bf 1}_{[0,1]} h_{n,\kappa}||^{\half},
$$
where ${\bf 1}_{[0,1]}$ denotes the characteristic function
of $[0,1]$. Standard properties of the Hermite functions then imply that, for 
$\kappa$ large enough
$$
||(\tilde H(\kappa) - (n+\half))\theta h_{n,\kappa}||
\leq C_n
\exp-\frac{1}{4}(\kappa-\sqrt n)^2.
$$
This shows that, for $n$ fixed, 
$$
{\rm dist}(\sigma(\tilde H(\kappa)),(n+\half))
\leq 2C_n
\exp-\frac{1}{4}(\kappa-\sqrt n)^2.
$$
For $n=0$, $|\alpha_0(\kappa)-\half|={\rm dist}(\sigma(\tilde H(\kappa)),
\half)\leq 2C_0\exp-\frac{1}{4}(\kappa)^2$, since $\alpha_1(\kappa)>3/2$, and 
(iii) 
then follows by induction on $n$.
We now turn to the proof of (iv). This only involves a rather straightforward
application of the standard method for proving exponential decay estimates on
eigenfunctions in a classically  forbidden region (see, for example \cite{a,
he}). With $V_\kappa(\tilde x)\equiv\half (\tilde x-\kappa)^2$ as before,
we first define, for all $\kappa>0$, $0<x_n(\kappa)<\kappa$ by
$V_\kappa(x_n(\kappa))=\alpha_n(\kappa)$. 
Clearly, for all
$0\leq
\tilde x\leq x_n(\kappa)$, $\tilde\varphi_n(\tilde x, \kappa)$ and 
$\tilde\varphi_n''(\tilde x, \kappa)$ have the same sign, which we can 
assume to be strictly positive. 
Also, on the same region $\varphi_n'(\tilde x,
\kappa)>0$.  As a result, for any $a\in[0,x_n(\kappa)-2]$, one has
$$
|\tilde\varphi_n(a,\kappa)|^2\leq \int_a^{a+1}
|\tilde\varphi_n(y,\kappa)|^2 dy.
$$
Let, for $0\leq \tilde x\leq x_n(\kappa)$,
$$
f_n(\tilde x,\kappa) = \int_{\tilde x}^{x_n(\kappa)}
\sqrt{2(V_\kappa(y)-\alpha_n(\kappa)+1)} dy.
$$
Note that
$$
\half f'_n(\tilde x,\kappa)^2 - 
(V_\kappa(\tilde x)-\alpha_n(\kappa)+1)=0.
$$
We introduce $\eta_n(\tilde x, \kappa)$, a smooth characteristic function of the
interval $[0, x_n(\kappa)-2]$, with ${\rm supp}\ \eta_n'\subset[x_n(\kappa)-2,
x_n(\kappa)-1]$.  Then
\bea
\int_a^{a+1} |\tilde\varphi_n(y,\kappa)|^2 dy &\leq&
\exp-2f_n(a,\kappa)\int_a^{a+1}\exp 2f_n(y,\kappa) |\tilde\varphi_n
(y, \kappa)|^2 dy\non \\
&\leq &\exp-2f_n(a,\kappa)
\langle\psi_n,(V_\kappa-\alpha_n(\kappa)-\half {f'}_n^2) \psi_n\rangle,
\eea
where $\psi_n = \eta_n (\exp f_n) \tilde\varphi_n$.  
A simple computation shows
$$
(\exp f_n)(H-\alpha_n)(\exp-f_n) \psi_n= \frac{p^2}{2}\psi_n + (V_\kappa
-\alpha_n-\half {f'}_n^2)\psi_n + \half(f'_n\frac{d}{d\tilde x} + 
\frac{d}{d\tilde x}f'_n)\psi_n,
$$
so that
$$
\hbox{\rm Re}\langle\psi_n, (\exp f_n)(H-\alpha_n)(\exp-f_n)\psi_n\rangle
\geq\langle \psi_n, (V_\kappa-\alpha_n-\half {f'}_n^2)\psi_n\rangle.
$$
On the other hand
$$
\hbox{\rm Re}\langle\psi_n, (\exp f_n)(H-\alpha_n)(\exp-f_n)\psi_n\rangle =
\hbox{\rm Re}\langle\tilde\varphi_n, (\exp 2f_n)
\eta_n\half[p^2,\eta_n]\tilde\varphi_n\rangle.
$$
Consequently,
$$
|\tilde \varphi_n(a,\kappa)|^2\leq (\exp-2f_n(a,\kappa))
\hbox{\rm Re}\langle\tilde\varphi_n, W_n\tilde\varphi_n\rangle,
$$
where
$$
W_n =(\exp 2f_n)\ \eta_n \half [p^2, \eta_n]= 
-\half (\exp 2f_n)(\eta_n \eta_n'' +2i \eta_n \eta_n' p),
$$
so that
$$
\hbox{\rm Re} W_n =
-\half \exp 2f_n(\eta_n \eta_n'' - 2f_n' \eta_n\eta_n' - (\eta_n \eta_n')').
$$
It follows from the support properties of $\eta_n'$ and the definition of $f_n$ 
that
there exists a constant $C_n$ so that for all $\kappa$, one has $|\hbox{\rm Re} 
W_n|\leq C_n$. As a result, a simple computation shows that, for all 
$\epsilon$, 
there exists  constants $C_{n,\epsilon}, K_{n,\epsilon}$ so that for all 
$\kappa>K_{n,\epsilon}$, and for all $0\leq \tilde x < x_n(\kappa)-2$
$$
|\tilde\varphi_n(\tilde x,\kappa)|^2\leq C_{n,\epsilon} \exp\left \{ - 
\half(1-\epsilon)(\kappa-\tilde x)^2\right\}.
$$
This proves the first statement of (iv). To prove the second estimate, it is
clear from part (ii) that we need to prove the above estimate holds for 
$|\tilde\varphi_n'(\tilde x,\kappa)|^2$ as well. If $\chi$ is a smooth 
characteristic function of the interval $[0,1]$ with support in $[0,2]$, one 
has, using the eigenvalue equation and two partial integrations that
\begin{eqnarray*}
 |\tilde\varphi_n'(0,\kappa)|^2&\leq& \int_0^1|\tilde\varphi_n'(\tilde 
x,\kappa)|^2 d\tilde x \\
&\leq& \int_0^\infty \chi(\tilde x)  
\tilde\varphi_n'(\tilde x,\kappa)^2\leq C \int_0^1|\tilde\varphi_n(\tilde 
x,\kappa)|^2 d\tilde x,
\end{eqnarray*}
from which the result follows. The last part of the Lemma is an immediate 
consequence of the Feynman-Hellman formula.
\qed

\noindent {\bf Acknowledgements:} The authors would like to thank Forbairt,
the Royal Irish Academy, the CNRS and the Minist\`ere des Affaires Etrang\`eres 
for their financial support. They also thank J.Fr\"ohlich, G.M.Graf and
J.Walcher for bringing their results to their attention prior to publication.
\newpage
%%%%%%%%%%%%%%%%%%%Bibliograpy%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{aAaAaA} \baselineskip=10pt
\bibitem[1]{h} B.I. Halperin, Phys. Rev. {\bf B38}, 2185-2190 (1982). 

\bibitem[2]{gp} S. M. Girvin and R.E. Prange, 
{\em The quantum Hall effect},
Editors, Springer Verlag, 1987.

\bibitem[3]{w} X. G. Wen, Phys. Rev. B {\bf 43}, 11025 (1991).

\bibitem[4]{fs} J. Fr\"ohlich and U. M. Studer, Rev. Mod. Phys 
{\bf 65}, 733 (1993)
.

\bibitem[5]{mmp} N. Macris, P.A. Martin and J.V. Pul\'e, 
{\em On edge states in
semi-infinite quantum Hall systems}, preprint 1998 (to appear in J. Phys. A). 

\bibitem[6]{fgw1} J. Fr\"ohlich, G.M. Graf, and J. Walcher, private 
communication, 1998.

\bibitem[7]{fgw2} J. Fr\"ohlich, G.M. Graf, and J. Walcher, {\em On the 
extended nature of edge states of Quantum Hall Hamiltonians}, preprint 
march 1999.

\bibitem[8]{he} B. Helffer, 
{\em Semi-classical analysis for the Schr\"odinger
operator and applications}, Lecture Notes in Mathematics 1336, 
Springer-Verlag, 1988.


\bibitem[9]{as} M. Abramowitz and I.A. Stegun: 
{\em Handbook of Mathematical Functions}, 
Dover Publications - New York, 1965.

\bibitem[10]{av} E. Akkermans, J.E. Avron, R. Narevich
and R. Seiler, {\em Boundary conditions
for bulk and edge states in  quantum Hall systems}, preprint 1998.

\bibitem[11]{abg} W. Amrein, A. Boutet de Monvel and V. Georgescu,
{\em $C_0$-groups, Commutator Methods and Spectral Theory of N-Body 
Hamiltonians}, Birkh\"auser, Basel-Boston-Berlin, 1996. 

\bibitem[12]{gg} V. Georgescu and C. G\'erard, {\em On the virial theorem in 
quantum mechanics}, preprint 1998 mp-arc 98-744.

\bibitem[13]{hd} M. Dauge and B. Helffer, J. Diff. Eqns. 104, 2, 243-262 (1993).

\bibitem[14]{a} S. Agmon, {\em Lectures on exponential decay of
eigenfunctions of second order elliptic equations}, Princeton
University Press, 1982.


\end{thebibliography}




\end{document}


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S
NP 899 2481 m 907 2481 l
S
NP 932 2481 m 907 2481 l
S
NP 932 2481 m 940 2481 l
S
NP 964 2481 m 940 2481 l
S
NP 964 2481 m 972 2481 l
S
NP 996 2481 m 972 2481 l
S
NP 996 2481 m 1004 2481 l
S
NP 1029 2481 m 1004 2481 l
S
NP 1029 2481 m 1037 2481 l
S
NP 1061 2481 m 1037 2481 l
S
NP 1061 2481 m 1069 2481 l
S
NP 1093 2481 m 1069 2481 l
S
NP 1093 2481 m 1101 2481 l
S
NP 1126 2481 m 1101 2481 l
S
NP 1126 2481 m 1134 2481 l
S
NP 1158 2481 m 1134 2481 l
S
NP 1158 2481 m 1166 2481 l
S
NP 1190 2481 m 1166 2481 l
S
NP 1190 2481 m 1198 2481 l
S
NP 1223 2481 m 1198 2481 l
S
NP 1223 2481 m 1230 2481 l
S
NP 1255 2481 m 1230 2481 l
S
NP 1255 2481 m 1263 2481 l
S
NP 1287 2481 m 1263 2481 l
S
NP 1287 2481 m 1295 2481 l
S
NP 1319 2481 m 1295 2481 l
S
NP 1319 2481 m 1327 2481 l
S
NP 1352 2481 m 1327 2481 l
S
NP 1352 2481 m 1360 2481 l
S
NP 1384 2481 m 1360 2481 l
S
NP 1384 2481 m 1392 2481 l
S
NP 1416 2481 m 1392 2481 l
S
NP 1416 2481 m 1424 2481 l
S
NP 1449 2481 m 1424 2481 l
S
NP 1449 2481 m 1457 2481 l
S
NP 1481 2481 m 1457 2481 l
S
NP 1481 2481 m 1489 2481 l
S
NP 1513 2481 m 1489 2481 l
S
NP 1513 2481 m 1521 2481 l
S
NP 1546 2481 m 1521 2481 l
S
NP 1546 2481 m 1554 2481 l
S
NP 1578 2481 m 1554 2481 l
S
NP 1578 2481 m 1586 2481 l
S
NP 1610 2481 m 1586 2481 l
S
NP 1610 2481 m 1618 2481 l
S
NP 1643 2481 m 1618 2481 l
S
NP 1643 2481 m 1650 2481 l
S
NP 1675 2481 m 1650 2481 l
S
NP 1675 2481 m 1683 2481 l
S
NP 1707 2481 m 1683 2481 l
S
NP 1707 2481 m 1715 2481 l
S
NP 1739 2481 m 1715 2481 l
S
NP 1739 2481 m 1747 2481 l
S
NP 1772 2481 m 1747 2481 l
S
NP 1772 2481 m 1780 2481 l
S
NP 1804 2481 m 1780 2481 l
S
NP 1804 2481 m 1812 2481 l
S
NP 1836 2481 m 1812 2481 l
S
NP 1836 2481 m 1844 2481 l
S
NP 1869 2481 m 1844 2481 l
S
NP 1869 2481 m 1877 2481 l
S
NP 1901 2481 m 1877 2481 l
S
NP 1901 2481 m 1909 2481 l
S
NP 1933 2481 m 1909 2481 l
S
NP 1933 2481 m 1941 2481 l
S
NP 1966 2481 m 1941 2481 l
S
NP 1966 2481 m 1974 2481 l
S
NP 1998 2481 m 1974 2481 l
S
NP 1998 2481 m 2006 2481 l
S
NP 2030 2481 m 2006 2481 l
S
NP 2030 2481 m 2038 2481 l
S
NP 2063 2481 m 2038 2481 l
S
NP 2063 2481 m 2070 2481 l
S
NP 2095 2481 m 2070 2481 l
S
NP 2095 2481 m 2103 2481 l
S
NP 2127 2481 m 2103 2481 l
S
NP 2127 2481 m 2135 2481 l
S
NP 2159 2481 m 2135 2481 l
S
NP 2159 2481 m 2167 2481 l
S
NP 2192 2481 m 2167 2481 l
S
NP 2192 2481 m 2200 2481 l
S
NP 2224 2481 m 2200 2481 l
S
NP 2224 2481 m 2232 2481 l
S
NP 2256 2481 m 2232 2481 l
S
NP 2256 2481 m 2264 2481 l
S
NP 2289 2481 m 2264 2481 l
S
NP 2289 2481 m 2297 2481 l
S
NP 2321 2481 m 2297 2481 l
S
NP 2321 2481 m 2329 2481 l
S
NP 2353 2481 m 2329 2481 l
S
NP 2353 2481 m 2361 2481 l
S
NP 2386 2481 m 2361 2481 l
S
NP 2386 2481 m 2394 2481 l
S
NP 2418 2481 m 2394 2481 l
S
NP 2418 2481 m 2426 2481 l
S
NP 2450 2481 m 2426 2481 l
S
NP 2450 2481 m 2458 2481 l
S
NP 2483 2481 m 2458 2481 l
S
NP 2483 2481 m 2490 2481 l
S
NP 2515 2481 m 2490 2481 l
S
NP 2515 2481 m 2523 2481 l
S
NP 2547 2481 m 2523 2481 l
S
NP 2547 2481 m 2555 2481 l
S
NP 2580 2481 m 2555 2481 l
S
NP 2580 2481 m 2587 2481 l
S
NP 2612 2481 m 2587 2481 l
S
NP 2612 2481 m 2620 2481 l
S
NP 2644 2481 m 2620 2481 l
S
NP 2644 2481 m 2652 2481 l
S
NP 2676 2481 m 2652 2481 l
S
NP 2676 2481 m 2684 2481 l
S
NP 2709 2481 m 2684 2481 l
S
NP 2709 2481 m 2717 2481 l
S
NP 2741 2481 m 2717 2481 l
S
NP 2741 2481 m 2749 2481 l
S
NP 2773 2481 m 2749 2481 l
S
NP 2773 2481 m 2781 2481 l
S
NP 2806 2481 m 2781 2481 l
S
NP 2806 2481 m 2814 2481 l
S
NP 2838 2481 m 2814 2481 l
S
NP 2838 2481 m 2846 2481 l
S
NP 2870 2481 m 2846 2481 l
S
NP 2870 2481 m 2878 2481 l
S
NP 2903 2481 m 2878 2481 l
S
NP 2903 2481 m 2910 2481 l
S
NP 2935 2481 m 2910 2481 l
S
NP 2935 2481 m 2943 2481 l
S
NP 2967 2481 m 2943 2481 l
S
NP 2967 2481 m 2975 2481 l
S
NP 3000 2481 m 2975 2481 l
S
NP 3000 2481 m 3007 2481 l
S
NP 3032 2481 m 3007 2481 l
S
NP 3032 2481 m 3040 2481 l
S
NP 3064 2481 m 3040 2481 l
S
NP 3064 2481 m 3072 2481 l
S
NP 3096 2481 m 3072 2481 l
S
NP 3096 2481 m 3104 2481 l
S
NP 3129 2481 m 3104 2481 l
S
NP 3129 2481 m 3137 2481 l
S
NP 3161 2481 m 3137 2481 l
S
NP 3161 2481 m 3169 2481 l
S
NP 3193 2481 m 3169 2481 l
S
NP 3193 2481 m 3201 2481 l
S
NP 3226 2481 m 3201 2481 l
S
NP 3226 2481 m 3234 2481 l
S
NP 3258 2481 m 3234 2481 l
S
NP 3258 2481 m 3266 2481 l
S
NP 3290 2481 m 3266 2481 l
S
NP 3290 2481 m 3298 2481 l
S
NP 3323 2481 m 3298 2481 l
S
NP 3323 2481 m 3331 2481 l
S
NP 3355 2481 m 3331 2481 l
S
NP 3355 2481 m 3363 2481 l
S
NP 3387 2481 m 3363 2481 l
S
NP 3387 2481 m 3395 2481 l
S
NP 3420 2481 m 3395 2481 l
S
NP 3420 2481 m 3427 2481 l
S
NP 3452 2481 m 3427 2481 l
S
NP 3452 2481 m 3460 2481 l
S
NP 3484 2481 m 3460 2481 l
S
NP 3484 2481 m 3492 2481 l
S
NP 3516 2481 m 3492 2481 l
S
NP 3516 2481 m 3524 2481 l
S
NP 3549 2481 m 3524 2481 l
S
NP 3549 2481 m 3557 2481 l
S
NP 3581 2481 m 3557 2481 l
S
NP 3581 2481 m 3589 2481 l
S
NP 3613 2481 m 3589 2481 l
S
NP 3613 2481 m 3621 2481 l
S
NP 3646 2481 m 3621 2481 l
S
NP 3646 2481 m 3654 2481 l
S
NP 3678 2481 m 3654 2481 l
S
NP 3678 2481 m 3686 2481 l
S
NP 3710 2481 m 3686 2481 l
S
NP 3710 2481 m 3718 2481 l
S
NP 3743 2481 m 3718 2481 l
S
NP 3743 2481 m 3751 2481 l
S
NP 3775 2481 m 3751 2481 l
S
NP 3775 2481 m 3783 2481 l
S
NP 3807 2481 m 3783 2481 l
S
NP 3807 2481 m 3815 2481 l
S
NP 3840 2481 m 3815 2481 l
S
NP 3840 2481 m 3847 2481 l
S
NP 3872 2481 m 3847 2481 l
S
NP 3872 2481 m 3880 2481 l
S
NP 3904 2481 m 3880 2481 l
S
NP 3904 2481 m 3912 2481 l
S
NP 3936 2481 m 3912 2481 l
S
NP 3936 2481 m 3944 2481 l
S
NP 3969 2481 m 3944 2481 l
S
NP 3969 2481 m 3977 2481 l
S
NP 4001 2481 m 3977 2481 l
S
NP 4001 2481 m 4009 2481 l
S
NP 4033 2481 m 4009 2481 l
S
NP 4033 2481 m 4041 2481 l
S
NP 4066 2481 m 4041 2481 l
S
NP 4066 2481 m 4074 2481 l
S
NP 4098 2481 m 4074 2481 l
S
NP 4098 2481 m 4106 2481 l
S
NP 4130 2481 m 4106 2481 l
S
NP 4130 2481 m 4138 2481 l
S
NP 4163 2481 m 4138 2481 l
S
NP 4163 2481 m 4171 2481 l
S
NP 4195 2481 m 4171 2481 l
S
NP 4195 2481 m 4203 2481 l
S
NP 4227 2481 m 4203 2481 l
S
NP 4227 2481 m 4235 2481 l
S
NP 4260 2481 m 4235 2481 l
S
NP 4260 2481 m 4267 2481 l
S
NP 4292 2481 m 4267 2481 l
S
NP 4292 2481 m 4300 2481 l
S
NP 4324 2481 m 4300 2481 l
S
NP 4324 2481 m 4332 2481 l
S
NP 4356 2481 m 4332 2481 l
S
NP 4356 2481 m 4364 2481 l
S
NP 4389 2481 m 4364 2481 l
S
NP 4389 2481 m 4397 2481 l
S
NP 4421 2481 m 4397 2481 l
S
NP 576 1339 m 601 1339 l
S
NP 609 1339 m 601 1339 l
S
NP 609 1339 m 633 1339 l
S
NP 641 1339 m 633 1339 l
S
NP 641 1339 m 665 1339 l
S
NP 673 1339 m 665 1339 l
S
NP 673 1339 m 697 1339 l
S
NP 706 1339 m 697 1339 l
S
NP 706 1339 m 730 1339 l
S
NP 738 1339 m 730 1339 l
S
NP 738 1339 m 762 1339 l
S
NP 770 1339 m 762 1339 l
S
NP 770 1339 m 794 1339 l
S
NP 803 1339 m 794 1339 l
S
NP 803 1339 m 827 1339 l
S
NP 835 1339 m 827 1339 l
S
NP 835 1339 m 859 1339 l
S
NP 867 1339 m 859 1339 l
S
NP 867 1339 m 891 1339 l
S
NP 899 1339 m 891 1339 l
S
NP 899 1339 m 924 1339 l
S
NP 932 1339 m 924 1339 l
S
NP 932 1339 m 956 1339 l
S
NP 964 1339 m 956 1339 l
S
NP 964 1339 m 988 1339 l
S
NP 996 1339 m 988 1339 l
S
NP 996 1339 m 1021 1339 l
S
NP 1029 1339 m 1021 1339 l
S
NP 1029 1339 m 1053 1339 l
S
NP 1061 1339 m 1053 1339 l
S
NP 1061 1339 m 1085 1339 l
S
NP 1093 1339 m 1085 1339 l
S
NP 1093 1339 m 1118 1339 l
S
NP 1126 1339 m 1118 1339 l
S
NP 1126 1339 m 1150 1339 l
S
NP 1158 1339 m 1150 1339 l
S
NP 1158 1339 m 1182 1339 l
S
NP 1190 1339 m 1182 1339 l
S
NP 1190 1339 m 1214 1339 l
S
NP 1223 1339 m 1214 1339 l
S
NP 1223 1339 m 1247 1339 l
S
NP 1255 1339 m 1247 1339 l
S
NP 1255 1339 m 1279 1339 l
S
NP 1287 1339 m 1279 1339 l
S
NP 1287 1339 m 1311 1339 l
S
NP 1319 1339 m 1311 1339 l
S
NP 1319 1339 m 1344 1339 l
S
NP 1352 1339 m 1344 1339 l
S
NP 1352 1339 m 1376 1339 l
S
NP 1384 1339 m 1376 1339 l
S
NP 1384 1339 m 1408 1339 l
S
NP 1416 1339 m 1408 1339 l
S
NP 1416 1339 m 1441 1339 l
S
NP 1449 1339 m 1441 1339 l
S
NP 1449 1339 m 1473 1339 l
S
NP 1481 1339 m 1473 1339 l
S
NP 1481 1339 m 1505 1339 l
S
NP 1513 1339 m 1505 1339 l
S
NP 1513 1339 m 1538 1339 l
S
NP 1546 1339 m 1538 1339 l
S
NP 1546 1339 m 1570 1339 l
S
NP 1578 1339 m 1570 1339 l
S
NP 1578 1339 m 1602 1339 l
S
NP 1610 1339 m 1602 1339 l
S
NP 1610 1339 m 1634 1339 l
S
NP 1643 1339 m 1634 1339 l
S
NP 1643 1339 m 1667 1339 l
S
NP 1675 1339 m 1667 1339 l
S
NP 1675 1339 m 1699 1339 l
S
NP 1707 1339 m 1699 1339 l
S
NP 1707 1339 m 1731 1339 l
S
NP 1739 1339 m 1731 1339 l
S
NP 1739 1339 m 1764 1339 l
S
NP 1772 1339 m 1764 1339 l
S
NP 1772 1339 m 1796 1339 l
S
NP 1804 1339 m 1796 1339 l
S
NP 1804 1339 m 1828 1339 l
S
NP 1836 1339 m 1828 1339 l
S
NP 1836 1339 m 1861 1339 l
S
NP 1869 1339 m 1861 1339 l
S
NP 1869 1339 m 1893 1339 l
S
NP 1901 1339 m 1893 1339 l
S
NP 1901 1339 m 1925 1339 l
S
NP 1933 1339 m 1925 1339 l
S
NP 1933 1339 m 1958 1339 l
S
NP 1966 1339 m 1958 1339 l
S
NP 1966 1339 m 1990 1339 l
S
NP 1998 1339 m 1990 1339 l
S
NP 1998 1339 m 2022 1339 l
S
NP 2030 1339 m 2022 1339 l
S
NP 2030 1339 m 2054 1339 l
S
NP 2063 1339 m 2054 1339 l
S
NP 2063 1339 m 2087 1339 l
S
NP 2095 1339 m 2087 1339 l
S
NP 2095 1339 m 2119 1339 l
S
NP 2127 1339 m 2119 1339 l
S
NP 2127 1339 m 2151 1339 l
S
NP 2159 1339 m 2151 1339 l
S
NP 2159 1339 m 2184 1339 l
S
NP 2192 1339 m 2184 1339 l
S
NP 2192 1339 m 2216 1339 l
S
NP 2224 1339 m 2216 1339 l
S
NP 2224 1339 m 2248 1339 l
S
NP 2256 1339 m 2248 1339 l
S
NP 2256 1339 m 2281 1339 l
S
NP 2289 1339 m 2281 1339 l
S
NP 2289 1339 m 2313 1339 l
S
NP 2321 1339 m 2313 1339 l
S
NP 2321 1339 m 2345 1339 l
S
NP 2353 1339 m 2345 1339 l
S
NP 2353 1339 m 2378 1339 l
S
NP 2386 1339 m 2378 1339 l
S
NP 2386 1339 m 2410 1339 l
S
NP 2418 1339 m 2410 1339 l
S
NP 2418 1339 m 2442 1339 l
S
NP 2450 1339 m 2442 1339 l
S
NP 2450 1339 m 2474 1339 l
S
NP 2483 1339 m 2474 1339 l
S
NP 2483 1339 m 2507 1339 l
S
NP 2515 1339 m 2507 1339 l
S
NP 2515 1339 m 2539 1339 l
S
NP 2547 1339 m 2539 1339 l
S
NP 2547 1339 m 2571 1339 l
S
NP 2580 1339 m 2571 1339 l
S
NP 2580 1339 m 2604 1339 l
S
NP 2612 1339 m 2604 1339 l
S
NP 2612 1339 m 2636 1339 l
S
NP 2644 1339 m 2636 1339 l
S
NP 2644 1339 m 2668 1339 l
S
NP 2676 1339 m 2668 1339 l
S
NP 2676 1339 m 2701 1339 l
S
NP 2709 1339 m 2701 1339 l
S
NP 2709 1339 m 2733 1339 l
S
NP 2741 1339 m 2733 1339 l
S
NP 2741 1339 m 2765 1339 l
S
NP 2773 1339 m 2765 1339 l
S
NP 2773 1339 m 2798 1339 l
S
NP 2806 1339 m 2798 1339 l
S
NP 2806 1339 m 2830 1339 l
S
NP 2838 1339 m 2830 1339 l
S
NP 2838 1339 m 2862 1339 l
S
NP 2870 1339 m 2862 1339 l
S
NP 2870 1339 m 2894 1339 l
S
NP 2903 1339 m 2894 1339 l
S
NP 2903 1339 m 2927 1339 l
S
NP 2935 1339 m 2927 1339 l
S
NP 2935 1339 m 2959 1339 l
S
NP 2967 1339 m 2959 1339 l
S
NP 2967 1339 m 2991 1339 l
S
NP 3000 1339 m 2991 1339 l
S
NP 3000 1339 m 3024 1339 l
S
NP 3032 1339 m 3024 1339 l
S
NP 3032 1339 m 3056 1339 l
S
NP 3064 1339 m 3056 1339 l
S
NP 3064 1339 m 3088 1339 l
S
NP 3096 1339 m 3088 1339 l
S
NP 3096 1339 m 3121 1339 l
S
NP 3129 1339 m 3121 1339 l
S
NP 3129 1339 m 3153 1339 l
S
NP 3161 1339 m 3153 1339 l
S
NP 3161 1339 m 3185 1339 l
S
NP 3193 1339 m 3185 1339 l
S
NP 3193 1339 m 3218 1339 l
S
NP 3226 1339 m 3218 1339 l
S
NP 3226 1339 m 3250 1339 l
S
NP 3258 1339 m 3250 1339 l
S
NP 3258 1339 m 3282 1339 l
S
NP 3290 1339 m 3282 1339 l
S
NP 3290 1339 m 3314 1339 l
S
NP 3323 1339 m 3314 1339 l
S
NP 3323 1339 m 3347 1339 l
S
NP 3355 1339 m 3347 1339 l
S
NP 3355 1339 m 3379 1339 l
S
NP 3387 1339 m 3379 1339 l
S
NP 3387 1339 m 3411 1339 l
S
NP 3420 1339 m 3411 1339 l
S
NP 3420 1339 m 3444 1339 l
S
NP 3452 1339 m 3444 1339 l
S
NP 3452 1339 m 3476 1339 l
S
NP 3484 1339 m 3476 1339 l
S
NP 3484 1339 m 3508 1339 l
S
NP 3516 1339 m 3508 1339 l
S
NP 3516 1339 m 3541 1339 l
S
NP 3549 1339 m 3541 1339 l
S
NP 3549 1339 m 3573 1339 l
S
NP 3581 1339 m 3573 1339 l
S
NP 3581 1339 m 3605 1339 l
S
NP 3613 1339 m 3605 1339 l
S
NP 3613 1339 m 3638 1339 l
S
NP 3646 1339 m 3638 1339 l
S
NP 3646 1339 m 3670 1339 l
S
NP 3678 1339 m 3670 1339 l
S
NP 3678 1339 m 3702 1339 l
S
NP 3710 1339 m 3702 1339 l
S
NP 3710 1339 m 3734 1339 l
S
NP 3743 1339 m 3734 1339 l
S
NP 3743 1339 m 3767 1339 l
S
NP 3775 1339 m 3767 1339 l
S
NP 3775 1339 m 3799 1339 l
S
NP 3807 1339 m 3799 1339 l
S
NP 3807 1339 m 3831 1339 l
S
NP 3840 1339 m 3831 1339 l
S
NP 3840 1339 m 3864 1339 l
S
NP 3872 1339 m 3864 1339 l
S
NP 3872 1339 m 3896 1339 l
S
NP 3904 1339 m 3896 1339 l
S
NP 3904 1339 m 3928 1339 l
S
NP 3936 1339 m 3928 1339 l
S
NP 3936 1339 m 3961 1339 l
S
NP 3969 1339 m 3961 1339 l
S
NP 3969 1339 m 3993 1339 l
S
NP 4001 1339 m 3993 1339 l
S
NP 4001 1339 m 4025 1339 l
S
NP 4033 1339 m 4025 1339 l
S
NP 4033 1339 m 4058 1339 l
S
NP 4066 1339 m 4058 1339 l
S
NP 4066 1339 m 4090 1339 l
S
NP 4098 1339 m 4090 1339 l
S
NP 4098 1339 m 4122 1339 l
S
NP 4130 1339 m 4122 1339 l
S
NP 4130 1339 m 4154 1339 l
S
NP 4163 1339 m 4154 1339 l
S
NP 4163 1339 m 4187 1339 l
S
NP 4195 1339 m 4187 1339 l
S
NP 4195 1339 m 4219 1339 l
S
NP 4227 1339 m 4219 1339 l
S
NP 4227 1339 m 4251 1339 l
S
NP 4260 1339 m 4251 1339 l
S
NP 4260 1339 m 4284 1339 l
S
NP 4292 1339 m 4284 1339 l
S
NP 4292 1339 m 4316 1339 l
S
NP 4324 1339 m 4316 1339 l
S
NP 4324 1339 m 4348 1339 l
S
NP 4356 1339 m 4348 1339 l
S
NP 4356 1339 m 4381 1339 l
S
NP 4389 1339 m 4381 1339 l
S
NP 4389 1339 m 4413 1339 l
S
NP 4421 1339 m 4413 1339 l
S
NP 576 4764 m 584 4764 l
S
NP 609 4764 m 584 4764 l
S
NP 609 4764 m 616 4764 l
S
NP 641 4764 m 616 4764 l
S
NP 641 4764 m 649 4764 l
S
NP 673 4764 m 649 4764 l
S
NP 673 4764 m 681 4764 l
S
NP 706 4764 m 681 4764 l
S
NP 706 4764 m 713 4764 l
S
NP 738 4764 m 713 4764 l
S
NP 738 4764 m 746 4764 l
S
NP 770 4764 m 746 4764 l
S
NP 770 4764 m 778 4764 l
S
NP 803 4764 m 778 4764 l
S
NP 803 4764 m 810 4764 l
S
NP 835 4764 m 810 4764 l
S
NP 835 4764 m 843 4764 l
S
NP 867 4764 m 843 4764 l
S
NP 867 4764 m 875 4764 l
S
NP 899 4764 m 875 4764 l
S
NP 899 4764 m 907 4764 l
S
NP 932 4764 m 907 4764 l
S
NP 932 4764 m 939 4764 l
S
NP 964 4764 m 939 4764 l
S
NP 964 4764 m 972 4764 l
S
NP 996 4764 m 972 4764 l
S
NP 996 4764 m 1004 4764 l
S
NP 1029 4764 m 1004 4764 l
S
NP 1029 4764 m 1036 4764 l
S
NP 1061 4764 m 1036 4764 l
S
NP 1061 4764 m 1069 4764 l
S
NP 1093 4764 m 1069 4764 l
S
NP 1093 4764 m 1101 4764 l
S
NP 1126 4764 m 1101 4764 l
S
NP 1126 4764 m 1133 4764 l
S
NP 1158 4764 m 1133 4764 l
S
NP 1158 4764 m 1166 4764 l
S
NP 1190 4764 m 1166 4764 l
S
NP 1190 4764 m 1198 4764 l
S
NP 1223 4764 m 1198 4764 l
S
NP 1223 4764 m 1230 4764 l
S
NP 1255 4764 m 1230 4764 l
S
NP 1255 4764 m 1263 4764 l
S
NP 1287 4764 m 1263 4764 l
S
NP 1287 4764 m 1295 4764 l
S
NP 1319 4764 m 1295 4764 l
S
NP 1319 4764 m 1327 4764 l
S
NP 1352 4764 m 1327 4764 l
S
NP 1352 4764 m 1359 4764 l
S
NP 1384 4764 m 1359 4764 l
S
NP 1384 4764 m 1392 4764 l
S
NP 1416 4764 m 1392 4764 l
S
NP 1416 4764 m 1424 4764 l
S
NP 1449 4764 m 1424 4764 l
S
NP 1449 4764 m 1456 4764 l
S
NP 1481 4764 m 1456 4764 l
S
NP 1481 4764 m 1489 4764 l
S
NP 1513 4764 m 1489 4764 l
S
NP 1513 4764 m 1521 4764 l
S
NP 1546 4764 m 1521 4764 l
S
NP 1546 4764 m 1553 4764 l
S
NP 1578 4764 m 1553 4764 l
S
NP 1578 4764 m 1586 4764 l
S
NP 1610 4764 m 1586 4764 l
S
NP 1610 4764 m 1618 4764 l
S
NP 1643 4764 m 1618 4764 l
S
NP 1643 4764 m 1650 4764 l
S
NP 1675 4764 m 1650 4764 l
S
NP 1675 4764 m 1683 4764 l
S
NP 1707 4764 m 1683 4764 l
S
NP 1707 4764 m 1715 4764 l
S
NP 1739 4764 m 1715 4764 l
S
NP 1739 4764 m 1747 4764 l
S
NP 1772 4764 m 1747 4764 l
S
NP 1772 4764 m 1779 4764 l
S
NP 1804 4764 m 1779 4764 l
S
NP 1804 4764 m 1812 4764 l
S
NP 1836 4764 m 1812 4764 l
S
NP 1836 4764 m 1844 4764 l
S
NP 1869 4764 m 1844 4764 l
S
NP 1869 4764 m 1876 4764 l
S
NP 1901 4764 m 1876 4764 l
S
NP 1901 4764 m 1909 4764 l
S
NP 1933 4764 m 1909 4764 l
S
NP 1933 4764 m 1941 4764 l
S
NP 1966 4764 m 1941 4764 l
S
NP 1966 4764 m 1973 4764 l
S
NP 1998 4764 m 1973 4764 l
S
NP 1998 4764 m 2006 4764 l
S
NP 2030 4764 m 2006 4764 l
S
NP 2030 4764 m 2038 4764 l
S
NP 2063 4764 m 2038 4764 l
S
NP 2063 4764 m 2070 4764 l
S
NP 2095 4764 m 2070 4764 l
S
NP 2095 4764 m 2103 4764 l
S
NP 2127 4764 m 2103 4764 l
S
NP 2127 4764 m 2135 4764 l
S
NP 2159 4764 m 2135 4764 l
S
NP 2159 4764 m 2167 4764 l
S
NP 2192 4764 m 2167 4764 l
S
NP 2192 4764 m 2200 4764 l
S
NP 2224 4764 m 2200 4764 l
S
NP 2224 4764 m 2232 4764 l
S
NP 2256 4764 m 2232 4764 l
S
NP 2256 4764 m 2264 4764 l
S
NP 2289 4764 m 2264 4764 l
S
NP 2289 4764 m 2296 4764 l
S
NP 2321 4764 m 2296 4764 l
S
NP 2321 4764 m 2329 4764 l
S
NP 2353 4764 m 2329 4764 l
S
NP 2353 4764 m 2361 4764 l
S
NP 2386 4764 m 2361 4764 l
S
NP 2386 4764 m 2393 4764 l
S
NP 2418 4764 m 2393 4764 l
S
NP 2418 4764 m 2426 4764 l
S
NP 2450 4764 m 2426 4764 l
S
NP 2450 4764 m 2458 4764 l
S
NP 2483 4764 m 2458 4764 l
S
NP 2483 4764 m 2490 4764 l
S
NP 2515 4764 m 2490 4764 l
S
NP 2515 4764 m 2523 4764 l
S
NP 2547 4764 m 2523 4764 l
S
NP 2547 4764 m 2555 4764 l
S
NP 2580 4764 m 2555 4764 l
S
NP 2580 4764 m 2587 4764 l
S
NP 2612 4764 m 2587 4764 l
S
NP 2612 4764 m 2620 4764 l
S
NP 2644 4764 m 2620 4764 l
S
NP 2644 4764 m 2652 4764 l
S
NP 2676 4764 m 2652 4764 l
S
NP 2676 4764 m 2684 4764 l
S
NP 2709 4764 m 2684 4764 l
S
NP 2709 4764 m 2716 4764 l
S
NP 2741 4764 m 2716 4764 l
S
NP 2741 4764 m 2749 4764 l
S
NP 2773 4764 m 2749 4764 l
S
NP 2773 4764 m 2781 4764 l
S
NP 2806 4764 m 2781 4764 l
S
NP 2806 4764 m 2813 4764 l
S
NP 2838 4764 m 2813 4764 l
S
NP 2838 4764 m 2846 4764 l
S
NP 2870 4764 m 2846 4764 l
S
NP 2870 4764 m 2878 4764 l
S
NP 2903 4764 m 2878 4764 l
S
NP 2903 4764 m 2910 4764 l
S
NP 2935 4764 m 2910 4764 l
S
NP 2935 4764 m 2943 4764 l
S
NP 2967 4764 m 2943 4764 l
S
NP 2967 4764 m 2975 4764 l
S
NP 3000 4764 m 2975 4764 l
S
NP 3000 4764 m 3007 4764 l
S
NP 3032 4764 m 3007 4764 l
S
NP 3032 4764 m 3040 4764 l
S
NP 3064 4764 m 3040 4764 l
S
NP 3064 4764 m 3072 4764 l
S
NP 3096 4764 m 3072 4764 l
S
NP 3096 4764 m 3104 4764 l
S
NP 3129 4764 m 3104 4764 l
S
NP 3129 4764 m 3136 4764 l
S
NP 3161 4764 m 3136 4764 l
S
NP 3161 4764 m 3169 4764 l
S
NP 3193 4764 m 3169 4764 l
S
NP 3193 4764 m 3201 4764 l
S
NP 3226 4764 m 3201 4764 l
S
NP 3226 4764 m 3233 4764 l
S
NP 3258 4764 m 3233 4764 l
S
NP 3258 4764 m 3266 4764 l
S
NP 3290 4764 m 3266 4764 l
S
NP 3290 4764 m 3298 4764 l
S
NP 3323 4764 m 3298 4764 l
S
NP 3323 4764 m 3330 4764 l
S
NP 3355 4764 m 3330 4764 l
S
NP 3355 4764 m 3363 4764 l
S
NP 3387 4764 m 3363 4764 l
S
NP 3387 4764 m 3395 4764 l
S
NP 3420 4764 m 3395 4764 l
S
NP 3420 4764 m 3427 4764 l
S
NP 3452 4764 m 3427 4764 l
S
NP 3452 4764 m 3460 4764 l
S
NP 3484 4764 m 3460 4764 l
S
NP 3484 4764 m 3492 4764 l
S
NP 3516 4764 m 3492 4764 l
S
NP 3516 4764 m 3524 4764 l
S
NP 3549 4764 m 3524 4764 l
S
NP 3549 4764 m 3556 4764 l
S
NP 3581 4764 m 3556 4764 l
S
NP 3581 4764 m 3589 4764 l
S
NP 3613 4764 m 3589 4764 l
S
NP 3613 4764 m 3621 4764 l
S
NP 3646 4764 m 3621 4764 l
S
NP 3646 4764 m 3653 4764 l
S
NP 3678 4764 m 3653 4764 l
S
NP 3678 4764 m 3686 4764 l
S
NP 3710 4764 m 3686 4764 l
S
NP 3710 4764 m 3718 4764 l
S
NP 3743 4764 m 3718 4764 l
S
NP 3743 4764 m 3750 4764 l
S
NP 3775 4764 m 3750 4764 l
S
NP 3775 4764 m 3783 4764 l
S
NP 3807 4764 m 3783 4764 l
S
NP 3807 4764 m 3815 4764 l
S
NP 3840 4764 m 3815 4764 l
S
NP 3840 4764 m 3847 4764 l
S
NP 3872 4764 m 3847 4764 l
S
NP 3872 4764 m 3880 4764 l
S
NP 3904 4764 m 3880 4764 l
S
NP 3904 4764 m 3912 4764 l
S
NP 3936 4764 m 3912 4764 l
S
NP 3936 4764 m 3944 4764 l
S
NP 3969 4764 m 3944 4764 l
S
NP 3969 4764 m 3976 4764 l
S
NP 4001 4764 m 3976 4764 l
S
NP 4001 4764 m 4009 4764 l
S
NP 4033 4764 m 4009 4764 l
S
NP 4033 4764 m 4041 4764 l
S
NP 4066 4764 m 4041 4764 l
S
NP 4066 4764 m 4073 4764 l
S
NP 4098 4764 m 4073 4764 l
S
NP 4098 4764 m 4106 4764 l
S
NP 4130 4764 m 4106 4764 l
S
NP 4130 4764 m 4138 4764 l
S
NP 4163 4764 m 4138 4764 l
S
NP 4163 4764 m 4170 4764 l
S
NP 4195 4764 m 4170 4764 l
S
NP 4195 4764 m 4203 4764 l
S
NP 4227 4764 m 4203 4764 l
S
NP 4227 4764 m 4235 4764 l
S
NP 4260 4764 m 4235 4764 l
S
NP 4260 4764 m 4267 4764 l
S
NP 4292 4764 m 4267 4764 l
S
NP 4292 4764 m 4300 4764 l
S
NP 4324 4764 m 4300 4764 l
S
NP 4324 4764 m 4332 4764 l
S
NP 4356 4764 m 4332 4764 l
S
NP 4356 4764 m 4364 4764 l
S
NP 4389 4764 m 4364 4764 l
S
NP 4389 4764 m 4396 4764 l
S
NP 4421 4764 m 4396 4764 l
S
NP 576 3623 m 600 3623 l
S
NP 609 3623 m 600 3623 l
S
NP 609 3623 m 633 3623 l
S
NP 641 3623 m 633 3623 l
S
NP 641 3623 m 665 3623 l
S
NP 673 3623 m 665 3623 l
S
NP 673 3623 m 697 3623 l
S
NP 706 3623 m 697 3623 l
S
NP 706 3623 m 730 3623 l
S
NP 738 3623 m 730 3623 l
S
NP 738 3623 m 762 3623 l
S
NP 770 3623 m 762 3623 l
S
NP 770 3623 m 794 3623 l
S
NP 803 3623 m 794 3623 l
S
NP 803 3623 m 827 3623 l
S
NP 835 3623 m 827 3623 l
S
NP 835 3623 m 859 3623 l
S
NP 867 3623 m 859 3623 l
S
NP 867 3623 m 891 3623 l
S
NP 899 3623 m 891 3623 l
S
NP 899 3623 m 923 3623 l
S
NP 932 3623 m 923 3623 l
S
NP 932 3623 m 956 3623 l
S
NP 964 3623 m 956 3623 l
S
NP 964 3623 m 988 3623 l
S
NP 996 3623 m 988 3623 l
S
NP 996 3623 m 1020 3623 l
S
NP 1029 3623 m 1020 3623 l
S
NP 1029 3623 m 1053 3623 l
S
NP 1061 3623 m 1053 3623 l
S
NP 1061 3623 m 1085 3623 l
S
NP 1093 3623 m 1085 3623 l
S
NP 1093 3623 m 1117 3623 l
S
NP 1126 3623 m 1117 3623 l
S
NP 1126 3623 m 1150 3623 l
S
NP 1158 3623 m 1150 3623 l
S
NP 1158 3623 m 1182 3623 l
S
NP 1190 3623 m 1182 3623 l
S
NP 1190 3623 m 1214 3623 l
S
NP 1223 3623 m 1214 3623 l
S
NP 1223 3623 m 1247 3623 l
S
NP 1255 3623 m 1247 3623 l
S
NP 1255 3623 m 1279 3623 l
S
NP 1287 3623 m 1279 3623 l
S
NP 1287 3623 m 1311 3623 l
S
NP 1319 3623 m 1311 3623 l
S
NP 1319 3623 m 1343 3623 l
S
NP 1352 3623 m 1343 3623 l
S
NP 1352 3623 m 1376 3623 l
S
NP 1384 3623 m 1376 3623 l
S
NP 1384 3623 m 1408 3623 l
S
NP 1416 3623 m 1408 3623 l
S
NP 1416 3623 m 1440 3623 l
S
NP 1449 3623 m 1440 3623 l
S
NP 1449 3623 m 1473 3623 l
S
NP 1481 3623 m 1473 3623 l
S
NP 1481 3623 m 1505 3623 l
S
NP 1513 3623 m 1505 3623 l
S
NP 1513 3623 m 1537 3623 l
S
NP 1546 3623 m 1537 3623 l
S
NP 1546 3623 m 1570 3623 l
S
NP 1578 3623 m 1570 3623 l
S
NP 1578 3623 m 1602 3623 l
S
NP 1610 3623 m 1602 3623 l
S
NP 1610 3623 m 1634 3623 l
S
NP 1643 3623 m 1634 3623 l
S
NP 1643 3623 m 1667 3623 l
S
NP 1675 3623 m 1667 3623 l
S
NP 1675 3623 m 1699 3623 l
S
NP 1707 3623 m 1699 3623 l
S
NP 1707 3623 m 1731 3623 l
S
NP 1739 3623 m 1731 3623 l
S
NP 1739 3623 m 1763 3623 l
S
NP 1772 3623 m 1763 3623 l
S
NP 1772 3623 m 1796 3623 l
S
NP 1804 3623 m 1796 3623 l
S
NP 1804 3623 m 1828 3623 l
S
NP 1836 3623 m 1828 3623 l
S
NP 1836 3623 m 1860 3623 l
S
NP 1869 3623 m 1860 3623 l
S
NP 1869 3623 m 1893 3623 l
S
NP 1901 3623 m 1893 3623 l
S
NP 1901 3623 m 1925 3623 l
S
NP 1933 3623 m 1925 3623 l
S
NP 1933 3623 m 1957 3623 l
S
NP 1966 3623 m 1957 3623 l
S
NP 1966 3623 m 1990 3623 l
S
NP 1998 3623 m 1990 3623 l
S
NP 1998 3623 m 2022 3623 l
S
NP 2030 3623 m 2022 3623 l
S
NP 2030 3623 m 2054 3623 l
S
NP 2063 3623 m 2054 3623 l
S
NP 2063 3623 m 2087 3623 l
S
NP 2095 3623 m 2087 3623 l
S
NP 2095 3623 m 2119 3623 l
S
NP 2127 3623 m 2119 3623 l
S
NP 2127 3623 m 2151 3623 l
S
NP 2159 3623 m 2151 3623 l
S
NP 2159 3623 m 2183 3623 l
S
NP 2192 3623 m 2183 3623 l
S
NP 2192 3623 m 2216 3623 l
S
NP 2224 3623 m 2216 3623 l
S
NP 2224 3623 m 2248 3623 l
S
NP 2256 3623 m 2248 3623 l
S
NP 2256 3623 m 2280 3623 l
S
NP 2289 3623 m 2280 3623 l
S
NP 2289 3623 m 2313 3623 l
S
NP 2321 3623 m 2313 3623 l
S
NP 2321 3623 m 2345 3623 l
S
NP 2353 3623 m 2345 3623 l
S
NP 2353 3623 m 2377 3623 l
S
NP 2386 3623 m 2377 3623 l
S
NP 2386 3623 m 2410 3623 l
S
NP 2418 3623 m 2410 3623 l
S
NP 2418 3623 m 2442 3623 l
S
NP 2450 3623 m 2442 3623 l
S
NP 2450 3623 m 2474 3623 l
S
NP 2483 3623 m 2474 3623 l
S
NP 2483 3623 m 2507 3623 l
S
NP 2515 3623 m 2507 3623 l
S
NP 2515 3623 m 2539 3623 l
S
NP 2547 3623 m 2539 3623 l
S
NP 2547 3623 m 2571 3623 l
S
NP 2580 3623 m 2571 3623 l
S
NP 2580 3623 m 2603 3623 l
S
NP 2612 3623 m 2603 3623 l
S
NP 2612 3623 m 2636 3623 l
S
NP 2644 3623 m 2636 3623 l
S
NP 2644 3623 m 2668 3623 l
S
NP 2676 3623 m 2668 3623 l
S
NP 2676 3623 m 2700 3623 l
S
NP 2709 3623 m 2700 3623 l
S
NP 2709 3623 m 2733 3623 l
S
NP 2741 3623 m 2733 3623 l
S
NP 2741 3623 m 2765 3623 l
S
NP 2773 3623 m 2765 3623 l
S
NP 2773 3623 m 2797 3623 l
S
NP 2806 3623 m 2797 3623 l
S
NP 2806 3623 m 2830 3623 l
S
NP 2838 3623 m 2830 3623 l
S
NP 2838 3623 m 2862 3623 l
S
NP 2870 3623 m 2862 3623 l
S
NP 2870 3623 m 2894 3623 l
S
NP 2903 3623 m 2894 3623 l
S
NP 2903 3623 m 2927 3623 l
S
NP 2935 3623 m 2927 3623 l
S
NP 2935 3623 m 2959 3623 l
S
NP 2967 3623 m 2959 3623 l
S
NP 2967 3623 m 2991 3623 l
S
NP 3000 3623 m 2991 3623 l
S
NP 3000 3623 m 3023 3623 l
S
NP 3032 3623 m 3023 3623 l
S
NP 3032 3623 m 3056 3623 l
S
NP 3064 3623 m 3056 3623 l
S
NP 3064 3623 m 3088 3623 l
S
NP 3096 3623 m 3088 3623 l
S
NP 3096 3623 m 3120 3623 l
S
NP 3129 3623 m 3120 3623 l
S
NP 3129 3623 m 3153 3623 l
S
NP 3161 3623 m 3153 3623 l
S
NP 3161 3623 m 3185 3623 l
S
NP 3193 3623 m 3185 3623 l
S
NP 3193 3623 m 3217 3623 l
S
NP 3226 3623 m 3217 3623 l
S
NP 3226 3623 m 3250 3623 l
S
NP 3258 3623 m 3250 3623 l
S
NP 3258 3623 m 3282 3623 l
S
NP 3290 3623 m 3282 3623 l
S
NP 3290 3623 m 3314 3623 l
S
NP 3323 3623 m 3314 3623 l
S
NP 3323 3623 m 3347 3623 l
S
NP 3355 3623 m 3347 3623 l
S
NP 3355 3623 m 3379 3623 l
S
NP 3387 3623 m 3379 3623 l
S
NP 3387 3623 m 3411 3623 l
S
NP 3420 3623 m 3411 3623 l
S
NP 3420 3623 m 3443 3623 l
S
NP 3452 3623 m 3443 3623 l
S
NP 3452 3623 m 3476 3623 l
S
NP 3484 3623 m 3476 3623 l
S
NP 3484 3623 m 3508 3623 l
S
NP 3516 3623 m 3508 3623 l
S
NP 3516 3623 m 3540 3623 l
S
NP 3549 3623 m 3540 3623 l
S
NP 3549 3623 m 3573 3623 l
S
NP 3581 3623 m 3573 3623 l
S
NP 3581 3623 m 3605 3623 l
S
NP 3613 3623 m 3605 3623 l
S
NP 3613 3623 m 3637 3623 l
S
NP 3646 3623 m 3637 3623 l
S
NP 3646 3623 m 3670 3623 l
S
NP 3678 3623 m 3670 3623 l
S
NP 3678 3623 m 3702 3623 l
S
NP 3710 3623 m 3702 3623 l
S
NP 3710 3623 m 3734 3623 l
S
NP 3743 3623 m 3734 3623 l
S
NP 3743 3623 m 3767 3623 l
S
NP 3775 3623 m 3767 3623 l
S
NP 3775 3623 m 3799 3623 l
S
NP 3807 3623 m 3799 3623 l
S
NP 3807 3623 m 3831 3623 l
S
NP 3840 3623 m 3831 3623 l
S
NP 3840 3623 m 3863 3623 l
S
NP 3872 3623 m 3863 3623 l
S
NP 3872 3623 m 3896 3623 l
S
NP 3904 3623 m 3896 3623 l
S
NP 3904 3623 m 3928 3623 l
S
NP 3936 3623 m 3928 3623 l
S
NP 3936 3623 m 3960 3623 l
S
NP 3969 3623 m 3960 3623 l
S
NP 3969 3623 m 3993 3623 l
S
NP 4001 3623 m 3993 3623 l
S
NP 4001 3623 m 4025 3623 l
S
NP 4033 3623 m 4025 3623 l
S
NP 4033 3623 m 4057 3623 l
S
NP 4066 3623 m 4057 3623 l
S
NP 4066 3623 m 4090 3623 l
S
NP 4098 3623 m 4090 3623 l
S
NP 4098 3623 m 4122 3623 l
S
NP 4130 3623 m 4122 3623 l
S
NP 4130 3623 m 4154 3623 l
S
NP 4163 3623 m 4154 3623 l
S
NP 4163 3623 m 4187 3623 l
S
NP 4195 3623 m 4187 3623 l
S
NP 4195 3623 m 4219 3623 l
S
NP 4227 3623 m 4219 3623 l
S
NP 4227 3623 m 4251 3623 l
S
NP 4260 3623 m 4251 3623 l
S
NP 4260 3623 m 4283 3623 l
S
NP 4292 3623 m 4283 3623 l
S
NP 4292 3623 m 4316 3623 l
S
NP 4324 3623 m 4316 3623 l
S
NP 4324 3623 m 4348 3623 l
S
NP 4356 3623 m 4348 3623 l
S
NP 4356 3623 m 4380 3623 l
S
NP 4389 3623 m 4380 3623 l
S
NP 4389 3623 m 4413 3623 l
S
NP 4421 3623 m 4413 3623 l
S
thin setlinewidth
/thin 10 def
/medium 14 def
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medium setlinewidth
[] 0 setdash   0.007 G
NP 835 5751 m 832 5765 l
831 5771 l
S
NP 823 5808 m 831 5771 l
S
NP 823 5808 m 817 5832 l
S
NP 813 5851 m 817 5832 l
S
NP 813 5851 m 803 5894 l
803 5894 l
S
NP 867 5608 m 861 5636 l
859 5647 l
S
NP 852 5679 m 859 5647 l
S
NP 852 5679 m 845 5709 l
S
NP 842 5722 m 845 5709 l
S
NP 842 5722 m 835 5751 l
S
NP 899 5469 m 891 5507 l
888 5523 l
S
NP 881 5550 m 888 5523 l
S
NP 881 5550 m 873 5585 l
S
NP 871 5593 m 873 5585 l
S
NP 871 5593 m 867 5608 l
S
NP 932 5332 m 931 5334 l
931 5335 l
S
NP 922 5377 m 931 5335 l
S
NP 922 5377 m 917 5397 l
S
NP 911 5421 m 917 5397 l
S
NP 911 5421 m 902 5461 l
S
NP 901 5464 m 902 5461 l
S
NP 901 5464 m 899 5469 l
S
NP 964 5198 m 963 5205 l
962 5208 l
S
NP 953 5248 m 962 5208 l
S
NP 953 5248 m 947 5271 l
S
NP 942 5291 m 947 5271 l
S
NP 942 5291 m 932 5332 l
S
NP 996 5066 m 994 5076 l
993 5080 l
S
NP 984 5119 m 993 5080 l
S
NP 984 5119 m 978 5144 l
S
NP 973 5162 m 978 5144 l
S
NP 973 5162 m 964 5198 l
S
NP 1029 4937 m 1027 4947 l
1025 4951 l
S
NP 1016 4990 m 1025 4951 l
S
NP 1016 4990 m 1010 5015 l
S
NP 1005 5033 m 1010 5015 l
S
NP 1005 5033 m 996 5066 l
S
NP 1061 4811 m 1059 4818 l
1059 4821 l
S
NP 1049 4861 m 1059 4821 l
S
NP 1049 4861 m 1042 4886 l
S
NP 1038 4904 m 1042 4886 l
S
NP 1038 4904 m 1029 4937 l
S
NP 1093 4687 m 1093 4688 l
1093 4689 l
S
NP 1082 4731 m 1093 4689 l
S
NP 1082 4731 m 1076 4754 l
S
NP 1071 4774 m 1076 4754 l
S
NP 1071 4774 m 1061 4811 l
S
NP 1126 4566 m 1116 4602 l
1111 4622 l
S
NP 1105 4645 m 1111 4622 l
S
NP 1105 4645 m 1093 4687 l
S
NP 1158 4447 m 1151 4473 l
1147 4487 l
S
NP 1140 4516 m 1147 4487 l
S
NP 1140 4516 m 1129 4555 l
S
NP 1128 4559 m 1129 4555 l
S
NP 1128 4559 m 1126 4566 l
S
NP 1190 4331 m 1187 4344 l
1185 4351 l
S
NP 1175 4387 m 1185 4351 l
S
NP 1175 4387 m 1166 4419 l
S
NP 1163 4430 m 1166 4419 l
S
NP 1163 4430 m 1158 4447 l
S
NP 1223 4217 m 1211 4258 l
1204 4282 l
S
NP 1199 4301 m 1204 4282 l
S
NP 1199 4301 m 1190 4331 l
S
NP 1255 4106 m 1249 4128 l
1245 4142 l
S
NP 1236 4171 m 1245 4142 l
S
NP 1236 4171 m 1224 4213 l
S
NP 1223 4214 m 1224 4213 l
S
NP 1223 4214 m 1223 4217 l
S
NP 1287 3997 m 1287 3999 l
1286 4000 l
S
NP 1274 4042 m 1286 4000 l
S
NP 1274 4042 m 1265 4071 l
S
NP 1261 4085 m 1265 4071 l
S
NP 1261 4085 m 1255 4106 l
S
NP 1319 3891 m 1313 3913 l
1309 3928 l
S
NP 1300 3956 m 1309 3928 l
S
NP 1300 3956 m 1287 3997 l
S
NP 1352 3787 m 1340 3827 l
1331 3855 l
S
NP 1326 3870 m 1331 3855 l
S
NP 1326 3870 m 1319 3891 l
S
NP 1384 3686 m 1381 3698 l
1378 3706 l
S
NP 1367 3741 m 1378 3706 l
S
NP 1367 3741 m 1354 3781 l
S
NP 1353 3784 m 1354 3781 l
S
NP 1353 3784 m 1352 3787 l
S
NP 1416 3588 m 1409 3611 l
1403 3630 l
S
NP 1395 3655 m 1403 3630 l
S
NP 1395 3655 m 1384 3686 l
S
NP 1449 3491 m 1437 3525 l
1428 3553 l
S
NP 1423 3568 m 1428 3553 l
S
NP 1423 3568 m 1416 3588 l
S
NP 1481 3398 m 1467 3439 l
1455 3474 l
S
NP 1452 3482 m 1455 3474 l
S
NP 1452 3482 m 1449 3491 l
S
NP 1513 3306 m 1512 3310 l
1511 3313 l
S
NP 1497 3353 m 1511 3313 l
S
NP 1497 3353 m 1482 3395 l
S
NP 1482 3396 m 1482 3395 l
S
NP 1482 3396 m 1481 3398 l
S
NP 1546 3217 m 1543 3224 l
1541 3230 l
S
NP 1528 3267 m 1541 3230 l
S
NP 1528 3267 m 1513 3306 l
S
NP 1546 5816 m 1536 5851 l
1531 5870 l
S
NP 1525 5894 m 1531 5870 l
S
NP 1578 3131 m 1575 3138 l
1573 3144 l
S
NP 1559 3181 m 1573 3144 l
S
NP 1559 3181 m 1546 3217 l
S
NP 1578 5697 m 1571 5722 l
1567 5736 l
S
NP 1560 5765 m 1567 5736 l
S
NP 1560 5765 m 1549 5804 l
S
NP 1548 5808 m 1549 5804 l
S
NP 1548 5808 m 1546 5816 l
S
NP 1610 3047 m 1609 3051 l
1607 3056 l
S
NP 1592 3095 m 1607 3056 l
S
NP 1592 3095 m 1578 3131 l
S
NP 1610 5580 m 1607 5593 l
1605 5600 l
S
NP 1595 5636 m 1605 5600 l
S
NP 1595 5636 m 1586 5668 l
S
NP 1583 5679 m 1586 5668 l
S
NP 1583 5679 m 1578 5697 l
S
NP 1643 2965 m 1643 2965 l
1642 2965 l
S
NP 1626 3008 m 1642 2965 l
S
NP 1626 3008 m 1610 3047 l
S
NP 1643 5466 m 1631 5507 l
1624 5531 l
S
NP 1619 5550 m 1624 5531 l
S
NP 1619 5550 m 1610 5580 l
S
NP 1675 2886 m 1660 2922 l
1643 2965 l
S
NP 1675 5354 m 1668 5377 l
1664 5392 l
S
NP 1656 5421 m 1664 5392 l
S
NP 1656 5421 m 1644 5462 l
S
NP 1643 5464 m 1644 5462 l
S
NP 1643 5464 m 1643 5466 l
S
NP 1707 2809 m 1696 2836 l
1681 2871 l
S
NP 1678 2879 m 1681 2871 l
S
NP 1678 2879 m 1675 2886 l
S
NP 1707 5244 m 1706 5248 l
1705 5251 l
S
NP 1693 5291 m 1705 5251 l
S
NP 1693 5291 m 1684 5322 l
S
NP 1681 5334 m 1684 5322 l
S
NP 1681 5334 m 1675 5354 l
S
NP 1739 2735 m 1733 2750 l
1724 2771 l
S
NP 1714 2793 m 1724 2771 l
S
NP 1714 2793 m 1707 2809 l
S
NP 1739 5137 m 1732 5162 l
1727 5179 l
S
NP 1719 5205 m 1727 5179 l
S
NP 1719 5205 m 1707 5244 l
S
NP 1772 2662 m 1771 2664 l
1770 2666 l
S
NP 1752 2707 m 1770 2666 l
S
NP 1752 2707 m 1739 2735 l
S
NP 1772 5032 m 1772 5033 l
1771 5033 l
S
NP 1758 5076 m 1771 5033 l
S
NP 1758 5076 m 1749 5106 l
S
NP 1745 5119 m 1749 5106 l
S
NP 1745 5119 m 1739 5137 l
S
NP 1804 2592 m 1791 2621 l
1772 2662 l
S
NP 1804 4930 m 1799 4947 l
1795 4959 l
S
NP 1785 4990 m 1795 4959 l
S
NP 1785 4990 m 1772 5032 l
S
NP 1836 2525 m 1832 2535 l
1823 2552 l
S
NP 1811 2578 m 1823 2552 l
S
NP 1811 2578 m 1804 2592 l
S
NP 1836 4830 m 1826 4861 l
1819 4884 l
S
NP 1812 4904 m 1819 4884 l
S
NP 1812 4904 m 1804 4930 l
S
NP 1869 2460 m 1853 2492 l
1836 2525 l
S
NP 1869 4732 m 1855 4774 l
1844 4808 l
S
NP 1840 4818 m 1844 4808 l
S
NP 1840 4818 m 1836 4830 l
S
NP 1901 2396 m 1896 2405 l
1886 2425 l
S
NP 1874 2448 m 1886 2425 l
S
NP 1874 2448 m 1869 2460 l
S
NP 1901 4636 m 1898 4645 l
1896 4653 l
S
NP 1883 4688 m 1896 4653 l
S
NP 1883 4688 m 1869 4731 l
1869 4731 l
1869 4732 l
S
NP 1933 2336 m 1919 2362 l
1901 2396 l
S
NP 1933 4543 m 1928 4559 l
1923 4573 l
S
NP 1913 4602 m 1923 4573 l
S
NP 1913 4602 m 1901 4636 l
S
NP 1966 2277 m 1942 2319 l
1933 2336 l
S
NP 1966 4452 m 1958 4473 l
1952 4492 l
S
NP 1943 4516 m 1952 4492 l
S
NP 1943 4516 m 1933 4543 l
S
NP 1998 2221 m 1991 2233 l
1967 2274 l
S
NP 1966 2276 m 1967 2274 l
S
NP 1966 2276 m 1966 2277 l
S
NP 1998 4364 m 1989 4387 l
1981 4409 l
S
NP 1974 4430 m 1981 4409 l
S
NP 1974 4430 m 1966 4452 l
S
NP 2030 2167 m 2016 2190 l
1998 2221 l
S
NP 2030 4277 m 2021 4301 l
2013 4324 l
S
NP 2005 4344 m 2013 4324 l
S
NP 2005 4344 m 1998 4364 l
S
NP 2063 2114 m 2042 2147 l
2030 2167 l
S
NP 2063 4193 m 2054 4214 l
2046 4237 l
S
NP 2038 4258 m 2046 4237 l
S
NP 2038 4258 m 2030 4277 l
S
NP 2095 2064 m 2069 2104 l
2063 2114 l
S
NP 2095 4112 m 2088 4128 l
2081 4147 l
S
NP 2071 4171 m 2081 4147 l
S
NP 2071 4171 m 2063 4193 l
S
NP 2127 2016 m 2126 2018 l
2118 2030 l
S
NP 2097 2061 m 2118 2030 l
S
NP 2097 2061 m 2095 2064 l
S
NP 2127 4032 m 2123 4042 l
2118 4054 l
S
NP 2105 4085 m 2118 4054 l
S
NP 2105 4085 m 2095 4112 l
S
NP 2159 1971 m 2157 1975 l
2127 2016 l
S
NP 2159 3955 m 2159 3956 l
2159 3957 l
S
NP 2141 3999 m 2159 3957 l
S
NP 2141 3999 m 2127 4032 l
S
NP 2192 1927 m 2188 1932 l
2159 1971 l
S
NP 2192 3880 m 2177 3913 l
2159 3955 l
S
NP 2224 1885 m 2221 1889 l
2192 1927 l
S
NP 2224 3807 m 2215 3827 l
2203 3855 l
S
NP 2196 3870 m 2203 3855 l
S
NP 2196 3870 m 2192 3880 l
S
NP 2224 5816 m 2212 5851 l
2203 5880 l
S
NP 2198 5894 m 2203 5880 l
S
NP 2256 1845 m 2256 1845 l
2224 1885 l
S
NP 2256 3737 m 2255 3741 l
2252 3747 l
S
NP 2235 3784 m 2252 3747 l
S
NP 2235 3784 m 2224 3807 l
S
NP 2256 5721 m 2256 5722 l
2256 5723 l
S
NP 2241 5765 m 2256 5723 l
S
NP 2241 5765 m 2229 5802 l
S
NP 2227 5808 m 2229 5802 l
S
NP 2227 5808 m 2224 5816 l
S
NP 2256 1845 m 2257 1845 l
S
NP 2289 1808 m 2257 1845 l
S
NP 2289 3669 m 2275 3698 l
2256 3737 l
S
NP 2289 5629 m 2286 5636 l
2284 5642 l
S
NP 2271 5679 m 2284 5642 l
S
NP 2271 5679 m 2256 5721 l
S
NP 2321 1772 m 2293 1802 l
2289 1808 l
S
NP 2321 3602 m 2316 3611 l
2308 3629 l
S
NP 2295 3655 m 2308 3629 l
S
NP 2295 3655 m 2289 3669 l
S
NP 2321 5539 m 2317 5550 l
2314 5560 l
S
NP 2301 5593 m 2314 5560 l
S
NP 2301 5593 m 2289 5629 l
S
NP 2353 1738 m 2332 1759 l
2321 1772 l
S
NP 2353 3538 m 2338 3568 l
2321 3602 l
S
NP 2353 5451 m 2349 5464 l
2344 5476 l
S
NP 2333 5507 m 2344 5476 l
S
NP 2333 5507 m 2321 5539 l
S
NP 2386 1706 m 2374 1716 l
2353 1738 l
S
NP 2386 3477 m 2383 3482 l
2376 3496 l
S
NP 2360 3525 m 2376 3496 l
S
NP 2360 3525 m 2353 3538 l
S
NP 2386 5366 m 2381 5377 l
2377 5390 l
S
NP 2365 5421 m 2377 5390 l
S
NP 2365 5421 m 2353 5451 l
S
NP 2386 1706 m 2412 1681 l
S
NP 2418 1675 m 2412 1681 l
S
NP 2418 3417 m 2405 3439 l
2386 3477 l
S
NP 2418 5283 m 2414 5291 l
2411 5301 l
S
NP 2397 5334 m 2411 5301 l
S
NP 2397 5334 m 2386 5366 l
S
NP 2450 1647 m 2420 1673 l
2418 1675 l
S
NP 2450 3359 m 2429 3396 l
2418 3417 l
S
NP 2450 5201 m 2449 5205 l
2447 5210 l
S
NP 2431 5248 m 2447 5210 l
S
NP 2431 5248 m 2418 5283 l
S
NP 2483 1620 m 2469 1630 l
2450 1647 l
S
NP 2483 3304 m 2479 3310 l
2467 3331 l
S
NP 2454 3353 m 2467 3331 l
S
NP 2454 3353 m 2450 3359 l
S
NP 2483 5123 m 2466 5162 l
2450 5201 l
S
NP 2483 1620 m 2501 1606 l
S
NP 2515 1595 m 2501 1606 l
S
NP 2515 3251 m 2505 3267 l
2483 3304 l
S
NP 2515 5046 m 2502 5076 l
2486 5115 l
S
NP 2484 5119 m 2486 5115 l
S
NP 2484 5119 m 2483 5123 l
S
NP 2547 1571 m 2524 1587 l
2515 1595 l
S
NP 2547 3200 m 2531 3224 l
2515 3251 l
S
NP 2547 4971 m 2539 4990 l
2528 5015 l
S
NP 2520 5033 m 2528 5015 l
S
NP 2520 5033 m 2515 5046 l
S
NP 2547 1571 m 2571 1555 l
S
NP 2580 1549 m 2571 1555 l
S
NP 2580 3151 m 2559 3181 l
2547 3200 l
S
NP 2580 4899 m 2577 4904 l
2574 4910 l
S
NP 2558 4947 m 2574 4910 l
S
NP 2558 4947 m 2547 4971 l
S
NP 2612 1529 m 2587 1544 l
2580 1549 l
S
NP 2612 3104 m 2588 3138 l
2580 3151 l
S
NP 2612 4829 m 2597 4861 l
2580 4899 l
S
NP 2612 1529 m 2632 1517 l
S
NP 2644 1509 m 2632 1517 l
S
NP 2644 3059 m 2618 3095 l
2612 3104 l
S
NP 2644 4761 m 2637 4774 l
2626 4798 l
S
NP 2617 4818 m 2626 4798 l
S
NP 2617 4818 m 2612 4829 l
S
NP 2676 1492 m 2659 1501 l
2644 1509 l
S
NP 2676 3016 m 2649 3051 l
2644 3059 l
S
NP 2676 4696 m 2658 4731 l
2644 4761 l
S
NP 2676 1492 m 2687 1487 l
S
NP 2709 1476 m 2687 1487 l
S
NP 2709 2975 m 2682 3008 l
2676 3016 l
S
NP 2709 4632 m 2702 4645 l
2687 4674 l
S
NP 2680 4688 m 2687 4674 l
S
NP 2680 4688 m 2676 4696 l
S
NP 2709 1476 m 2738 1462 l
S
NP 2741 1460 m 2738 1462 l
S
NP 2741 2936 m 2716 2965 l
2709 2975 l
S
NP 2741 4571 m 2724 4602 l
2709 4632 l
S
NP 2773 1447 m 2747 1458 l
2741 1460 l
S
NP 2773 2899 m 2752 2922 l
2741 2936 l
S
NP 2773 4511 m 2771 4516 l
2764 4529 l
S
NP 2747 4559 m 2764 4529 l
S
NP 2747 4559 m 2741 4571 l
S
NP 2773 1447 m 2784 1444 l
S
NP 2806 1435 m 2784 1444 l
S
NP 2806 2864 m 2791 2879 l
2773 2899 l
S
NP 2806 4454 m 2795 4473 l
2773 4511 l
S
NP 2806 1435 m 2829 1427 l
S
NP 2838 1423 m 2829 1427 l
S
NP 2838 2831 m 2832 2836 l
2806 2864 l
S
NP 2838 4399 m 2819 4430 l
2806 4454 l
S
NP 2870 1413 m 2863 1415 l
2838 1423 l
S
NP 2838 2831 m 2851 2818 l
S
NP 2870 2800 m 2851 2818 l
S
NP 2870 4347 m 2845 4387 l
2838 4399 l
S
NP 2870 1413 m 2872 1412 l
S
NP 2903 1404 m 2872 1412 l
S
NP 2903 2770 m 2877 2793 l
2870 2800 l
S
NP 2903 4296 m 2899 4301 l
2882 4328 l
S
NP 2872 4344 m 2882 4328 l
S
NP 2872 4344 m 2870 4347 l
S
NP 2903 1404 m 2912 1402 l
S
NP 2935 1395 m 2912 1402 l
S
NP 2935 2743 m 2926 2750 l
2903 2770 l
S
NP 2935 4247 m 2928 4258 l
2903 4296 l
S
NP 2935 5866 m 2920 5894 l
S
NP 2935 1395 m 2951 1393 l
S
NP 2967 1388 m 2951 1393 l
S
NP 2935 2743 m 2947 2733 l
S
NP 2967 2717 m 2947 2733 l
S
NP 2967 4201 m 2957 4214 l
2935 4247 l
S
NP 2967 5800 m 2963 5808 l
2955 5825 l
S
NP 2941 5851 m 2955 5825 l
S
NP 2941 5851 m 2935 5866 l
S
NP 2967 1388 m 2990 1384 l
S
NP 3000 1381 m 2990 1384 l
S
NP 3000 2693 m 2980 2707 l
2967 2717 l
S
NP 3000 4157 m 2988 4171 l
2967 4201 l
S
NP 3000 5737 m 2984 5765 l
2967 5800 l
S
NP 3000 1381 m 3029 1376 l
S
NP 3032 1375 m 3029 1376 l
S
NP 3000 2693 m 3020 2680 l
S
NP 3032 2671 m 3020 2680 l
S
NP 3032 4114 m 3020 4128 l
3000 4157 l
S
NP 3032 5675 m 3030 5679 l
3025 5688 l
S
NP 3007 5722 m 3025 5688 l
S
NP 3007 5722 m 3000 5737 l
S
NP 3064 1370 m 3050 1372 l
3032 1375 l
S
NP 3064 2650 m 3042 2664 l
3032 2671 l
S
NP 3064 4074 m 3054 4085 l
3032 4114 l
S
NP 3064 5616 m 3053 5636 l
3032 5675 l
S
NP 3064 1370 m 3066 1370 l
S
NP 3096 1365 m 3066 1370 l
S
NP 3064 2650 m 3081 2641 l
S
NP 3096 2631 m 3081 2641 l
S
NP 3096 4036 m 3091 4042 l
3064 4074 l
S
NP 3096 5559 m 3077 5593 l
3064 5616 l
S
NP 3096 1365 m 3101 1366 l
S
NP 3129 1362 m 3101 1366 l
S
NP 3129 2614 m 3115 2621 l
3096 2631 l
S
NP 3096 4036 m 3124 4005 l
S
NP 3129 4000 m 3124 4005 l
S
NP 3129 5505 m 3128 5507 l
3123 5514 l
S
NP 3102 5550 m 3123 5514 l
S
NP 3102 5550 m 3096 5559 l
S
NP 3129 1362 m 3136 1362 l
S
NP 3161 1358 m 3136 1362 l
S
NP 3129 2614 m 3136 2611 l
S
NP 3161 2598 m 3136 2611 l
S
NP 3161 3966 m 3129 3999 l
3129 4000 l
S
NP 3161 5452 m 3153 5464 l
3129 5505 l
S
NP 3161 1358 m 3171 1359 l
S
NP 3193 1355 m 3171 1359 l
S
NP 3161 2598 m 3186 2587 l
S
NP 3193 2583 m 3186 2587 l
S
NP 3193 3934 m 3170 3956 l
3161 3966 l
S
NP 3193 5402 m 3180 5421 l
3161 5452 l
S
NP 3193 1355 m 3205 1356 l
S
NP 3226 1353 m 3205 1356 l
S
NP 3226 2570 m 3206 2578 l
3193 2583 l
S
NP 3226 3904 m 3215 3913 l
3193 3934 l
S
NP 3226 5354 m 3209 5377 l
3193 5402 l
S
NP 3226 1353 m 3239 1354 l
S
NP 3258 1351 m 3239 1354 l
S
NP 3226 2570 m 3232 2569 l
S
NP 3258 2559 m 3232 2569 l
S
NP 3226 3904 m 3245 3888 l
S
NP 3258 3876 m 3245 3888 l
S
NP 3258 5308 m 3238 5334 l
3226 5354 l
S
NP 3258 1351 m 3273 1352 l
S
NP 3290 1349 m 3273 1352 l
S
NP 3258 2559 m 3276 2554 l
S
NP 3290 2548 m 3276 2554 l
S
NP 3290 3849 m 3264 3870 l
3258 3876 l
S
NP 3290 5264 m 3269 5291 l
3258 5308 l
S
NP 3290 1349 m 3307 1350 l
S
NP 3323 1347 m 3307 1350 l
S
NP 3290 2548 m 3318 2540 l
S
NP 3323 2538 m 3318 2540 l
S
NP 3323 3825 m 3320 3827 l
3290 3849 l
S
NP 3323 5222 m 3301 5248 l
3290 5264 l
S
NP 3323 1347 m 3340 1349 l
S
NP 3355 1346 m 3340 1349 l
S
NP 3355 2530 m 3335 2535 l
3323 2538 l
S
NP 3323 3825 m 3325 3823 l
S
NP 3355 3802 m 3325 3823 l
S
NP 3355 5183 m 3335 5205 l
3323 5222 l
S
NP 3355 1346 m 3373 1347 l
S
NP 3387 1345 m 3373 1347 l
S
NP 3355 2530 m 3358 2530 l
S
NP 3387 2523 m 3358 2530 l
S
NP 3387 3781 m 3383 3784 l
3355 3802 l
S
NP 3387 5145 m 3371 5162 l
3355 5183 l
S
NP 3387 1345 m 3406 1346 l
S
NP 3420 1344 m 3406 1346 l
S
NP 3387 2523 m 3396 2522 l
S
NP 3420 2516 m 3396 2522 l
S
NP 3387 3781 m 3390 3780 l
S
NP 3420 3762 m 3390 3780 l
S
NP 3420 5110 m 3410 5119 l
3387 5145 l
S
NP 3420 1344 m 3439 1346 l
S
NP 3452 1343 m 3439 1346 l
S
NP 3420 2516 m 3434 2516 l
S
NP 3452 2511 m 3434 2516 l
S
NP 3420 3762 m 3446 3749 l
S
NP 3452 3745 m 3446 3749 l
S
NP 3420 5110 m 3449 5079 l
S
NP 3452 5077 m 3449 5079 l
S
NP 3452 1343 m 3472 1345 l
S
NP 3484 1342 m 3472 1345 l
S
NP 3452 2511 m 3471 2510 l
S
NP 3484 2506 m 3471 2510 l
S
NP 3484 3729 m 3460 3741 l
3452 3745 l
S
NP 3484 5045 m 3452 5076 l
3452 5077 l
S
NP 3484 1342 m 3505 1344 l
S
NP 3516 1342 m 3505 1344 l
S
NP 3484 2506 m 3507 2504 l
S
NP 3516 2502 m 3507 2504 l
S
NP 3484 3729 m 3496 3725 l
S
NP 3516 3715 m 3496 3725 l
S
NP 3516 5016 m 3497 5033 l
3484 5045 l
S
NP 3516 1342 m 3537 1344 l
S
NP 3549 1342 m 3537 1344 l
S
NP 3516 2502 m 3543 2500 l
S
NP 3549 2498 m 3543 2500 l
S
NP 3516 3715 m 3543 3705 l
S
NP 3549 3702 m 3543 3705 l
S
NP 3549 4989 m 3548 4990 l
3516 5016 l
S
NP 3549 1342 m 3570 1344 l
S
NP 3581 1341 m 3570 1344 l
S
NP 3549 2498 m 3578 2496 l
S
NP 3581 2495 m 3578 2496 l
S
NP 3581 3691 m 3560 3698 l
3549 3702 l
S
NP 3549 4989 m 3550 4989 l
S
NP 3581 4964 m 3550 4989 l
S
NP 3581 1341 m 3602 1343 l
S
NP 3613 1341 m 3602 1343 l
S
NP 3581 2495 m 3613 2492 l
3613 2492 l
S
NP 3581 3691 m 3587 3690 l
S
NP 3613 3681 m 3587 3690 l
S
NP 3613 4941 m 3604 4947 l
3581 4964 l
S
NP 3613 1341 m 3635 1343 l
S
NP 3646 1341 m 3635 1343 l
S
NP 3646 2490 m 3620 2492 l
3613 2492 l
S
NP 3613 3681 m 3628 3678 l
S
NP 3646 3672 m 3628 3678 l
S
NP 3613 4941 m 3621 4937 l
S
NP 3646 4920 m 3621 4937 l
S
NP 3646 1341 m 3667 1343 l
S
NP 3678 1341 m 3667 1343 l
S
NP 3646 2490 m 3647 2490 l
S
NP 3678 2488 m 3647 2490 l
S
NP 3646 3672 m 3668 3668 l
S
NP 3678 3664 m 3668 3668 l
S
NP 3678 4900 m 3671 4904 l
3646 4920 l
S
NP 3678 1341 m 3699 1343 l
S
NP 3710 1341 m 3699 1343 l
S
NP 3678 2488 m 3680 2489 l
S
NP 3710 2487 m 3680 2489 l
S
NP 3678 3664 m 3708 3658 l
S
NP 3710 3657 m 3708 3658 l
S
NP 3678 4900 m 3682 4899 l
S
NP 3710 4882 m 3682 4899 l
S
NP 3710 1341 m 3732 1343 l
S
NP 3743 1340 m 3732 1343 l
S
NP 3710 2487 m 3713 2488 l
S
NP 3743 2486 m 3713 2488 l
S
NP 3743 3651 m 3722 3655 l
3710 3657 l
S
NP 3710 4882 m 3734 4872 l
S
NP 3743 4866 m 3734 4872 l
S
NP 3743 1340 m 3764 1343 l
S
NP 3775 1340 m 3764 1343 l
S
NP 3743 2486 m 3746 2487 l
S
NP 3775 2485 m 3746 2487 l
S
NP 3743 3651 m 3745 3651 l
S
NP 3775 3646 m 3745 3651 l
S
NP 3775 4852 m 3754 4861 l
3743 4866 l
S
NP 3775 1340 m 3796 1343 l
S
NP 3807 1340 m 3796 1343 l
S
NP 3775 2485 m 3779 2486 l
S
NP 3807 2484 m 3779 2486 l
S
NP 3775 3646 m 3781 3647 l
S
NP 3807 3642 m 3781 3647 l
S
NP 3775 4852 m 3783 4850 l
S
NP 3807 4839 m 3783 4850 l
S
NP 3807 1340 m 3829 1343 l
S
NP 3840 1340 m 3829 1343 l
S
NP 3807 2484 m 3812 2485 l
S
NP 3840 2483 m 3812 2485 l
S
NP 3807 3642 m 3816 3643 l
S
NP 3840 3638 m 3816 3643 l
S
NP 3807 4839 m 3827 4834 l
S
NP 3840 4828 m 3827 4834 l
S
NP 3840 1340 m 3861 1343 l
S
NP 3872 1340 m 3861 1343 l
S
NP 3840 2483 m 3844 2485 l
S
NP 3872 2483 m 3844 2485 l
S
NP 3840 3638 m 3851 3640 l
S
NP 3872 3635 m 3851 3640 l
S
NP 3840 4828 m 3871 4818 l
S
NP 3872 4818 m 3871 4818 l
S
NP 3872 1340 m 3893 1343 l
S
NP 3904 1340 m 3893 1343 l
S
NP 3872 2483 m 3877 2485 l
S
NP 3904 2483 m 3877 2485 l
S
NP 3872 3635 m 3885 3637 l
S
NP 3904 3633 m 3885 3637 l
S
NP 3904 4809 m 3873 4818 l
3872 4818 l
S
NP 3904 1340 m 3926 1343 l
S
NP 3936 1340 m 3926 1343 l
S
NP 3904 2483 m 3909 2484 l
S
NP 3936 2482 m 3909 2484 l
S
NP 3904 3633 m 3919 3635 l
S
NP 3936 3631 m 3919 3635 l
S
NP 3904 4809 m 3910 4809 l
S
NP 3936 4802 m 3910 4809 l
S
NP 3936 1340 m 3958 1343 l
S
NP 3969 1340 m 3958 1343 l
S
NP 3936 2482 m 3942 2484 l
S
NP 3969 2482 m 3942 2484 l
S
NP 3936 3631 m 3953 3633 l
S
NP 3969 3629 m 3953 3633 l
S
NP 3936 4802 m 3949 4801 l
S
NP 3969 4795 m 3949 4801 l
S
NP 3969 1340 m 3990 1343 l
S
NP 4001 1340 m 3990 1343 l
S
NP 3969 2482 m 3974 2484 l
S
NP 4001 2482 m 3974 2484 l
S
NP 3969 3629 m 3986 3631 l
S
NP 4001 3628 m 3986 3631 l
S
NP 3969 4795 m 3986 4794 l
S
NP 4001 4790 m 3986 4794 l
S
NP 4001 1340 m 4022 1343 l
S
NP 4033 1340 m 4022 1343 l
S
NP 4001 2482 m 4007 2484 l
S
NP 4033 2482 m 4007 2484 l
S
NP 4001 3628 m 4019 3630 l
S
NP 4033 3627 m 4019 3630 l
S
NP 4001 4790 m 4023 4788 l
S
NP 4033 4785 m 4023 4788 l
S
NP 4033 1340 m 4055 1343 l
S
NP 4066 1340 m 4055 1343 l
S
NP 4033 2482 m 4039 2484 l
S
NP 4066 2482 m 4039 2484 l
S
NP 4033 3627 m 4052 3629 l
S
NP 4066 3626 m 4052 3629 l
S
NP 4033 4785 m 4059 4783 l
S
NP 4066 4781 m 4059 4783 l
S
NP 4066 1340 m 4087 1343 l
S
NP 4098 1340 m 4087 1343 l
S
NP 4066 2482 m 4071 2484 l
S
NP 4098 2482 m 4071 2484 l
S
NP 4066 3626 m 4085 3628 l
S
NP 4098 3625 m 4085 3628 l
S
NP 4066 4781 m 4095 4779 l
S
NP 4098 4778 m 4095 4779 l
S
NP 4098 1340 m 4119 1343 l
S
NP 4130 1340 m 4119 1343 l
S
NP 4098 2482 m 4104 2484 l
S
NP 4130 2482 m 4104 2484 l
S
NP 4098 3625 m 4118 3628 l
S
NP 4130 3625 m 4118 3628 l
S
NP 4098 4778 m 4130 4775 l
4130 4775 l
S
NP 4130 1340 m 4152 1343 l
S
NP 4163 1340 m 4152 1343 l
S
NP 4130 2482 m 4136 2484 l
S
NP 4163 2482 m 4136 2484 l
S
NP 4130 3625 m 4151 3627 l
S
NP 4163 3624 m 4151 3627 l
S
NP 4163 4773 m 4134 4774 l
4130 4775 l
S
NP 4163 1340 m 4184 1343 l
S
NP 4195 1340 m 4184 1343 l
S
NP 4163 2482 m 4168 2484 l
S
NP 4195 2482 m 4168 2484 l
S
NP 4163 3624 m 4183 3627 l
S
NP 4195 3624 m 4183 3627 l
S
NP 4163 4773 m 4164 4773 l
S
NP 4195 4771 m 4164 4773 l
S
NP 4195 1340 m 4216 1343 l
S
NP 4227 1340 m 4216 1343 l
S
NP 4195 2482 m 4201 2484 l
S
NP 4227 2481 m 4201 2484 l
S
NP 4195 3624 m 4216 3627 l
S
NP 4227 3624 m 4216 3627 l
S
NP 4195 4771 m 4197 4772 l
S
NP 4227 4769 m 4197 4772 l
S
NP 4227 1340 m 4248 1344 l
S
NP 4260 1340 m 4248 1344 l
S
NP 4227 2481 m 4233 2484 l
S
NP 4260 2481 m 4233 2484 l
S
NP 4227 3624 m 4248 3627 l
S
NP 4260 3623 m 4248 3627 l
S
NP 4227 4769 m 4230 4770 l
S
NP 4260 4768 m 4230 4770 l
S
NP 4260 1340 m 4281 1344 l
S
NP 4292 1340 m 4281 1344 l
S
NP 4260 2481 m 4265 2484 l
S
NP 4292 2481 m 4265 2484 l
S
NP 4260 3623 m 4281 3626 l
S
NP 4292 3623 m 4281 3626 l
S
NP 4260 4768 m 4263 4770 l
S
NP 4292 4767 m 4263 4770 l
S
NP 4292 1340 m 4313 1344 l
S
NP 4324 1340 m 4313 1344 l
S
NP 4292 2481 m 4298 2484 l
S
NP 4324 2481 m 4298 2484 l
S
NP 4292 3623 m 4313 3626 l
S
NP 4324 3623 m 4313 3626 l
S
NP 4292 4767 m 4296 4769 l
S
NP 4324 4767 m 4296 4769 l
S
NP 4324 1340 m 4345 1344 l
S
NP 4356 1340 m 4345 1344 l
S
NP 4324 2481 m 4330 2484 l
S
NP 4356 2481 m 4330 2484 l
S
NP 4324 3623 m 4345 3626 l
S
NP 4356 3623 m 4345 3626 l
S
NP 4324 4767 m 4329 4768 l
S
NP 4356 4766 m 4329 4768 l
S
NP 4356 1340 m 4377 1344 l
S
NP 4389 1340 m 4377 1344 l
S
NP 4356 2481 m 4362 2484 l
S
NP 4389 2481 m 4362 2484 l
S
NP 4356 3623 m 4378 3626 l
S
NP 4389 3623 m 4378 3626 l
S
NP 4356 4766 m 4361 4768 l
S
NP 4389 4766 m 4361 4768 l
S
NP 4389 1340 m 4410 1344 l
S
NP 4421 1340 m 4410 1344 l
S
NP 4389 2481 m 4394 2484 l
S
NP 4421 2481 m 4394 2484 l
S
NP 4389 3623 m 4410 3626 l
S
NP 4421 3623 m 4410 3626 l
S
NP 4389 4766 m 4394 4768 l
S
NP 4421 4765 m 4394 4768 l
S
    0 G
medium setlinewidth
NP 1858 666 m 1858 5999 l
S
thin setlinewidth
NP 1826 769 m 1890 769 l
S
NP 1842 997 m 1874 997 l
S
NP 1842 1225 m 1874 1225 l
S
NP 1842 1454 m 1874 1454 l
S
NP 1842 1682 m 1874 1682 l
S
NP 1826 1910 m 1890 1910 l
S
NP 1842 2139 m 1874 2139 l
S
NP 1842 2367 m 1874 2367 l
S
NP 1842 2595 m 1874 2595 l
S
NP 1842 2823 m 1874 2823 l
S
NP 1826 3052 m 1890 3052 l
S
NP 1842 3280 m 1874 3280 l
S
NP 1842 3508 m 1874 3508 l
S
NP 1842 3737 m 1874 3737 l
S
NP 1842 3965 m 1874 3965 l
S
NP 1826 4193 m 1890 4193 l
S
NP 1842 4422 m 1874 4422 l
S
NP 1842 4650 m 1874 4650 l
S
NP 1842 4878 m 1874 4878 l
S
NP 1842 5107 m 1874 5107 l
S
NP 1826 5335 m 1890 5335 l
S
NP 1842 5563 m 1874 5563 l
S
NP 1842 5792 m 1874 5792 l
S
medium setlinewidth
NP 500 769 m 4500 769 l
S
thin setlinewidth
NP 576 726 m 576 811 l
S
NP 705 747 m 705 790 l
S
NP 833 747 m 833 790 l
S
NP 961 747 m 961 790 l
S
NP 1089 747 m 1089 790 l
S
NP 1217 726 m 1217 811 l
S
NP 1345 747 m 1345 790 l
S
NP 1473 747 m 1473 790 l
S
NP 1602 747 m 1602 790 l
S
NP 1730 747 m 1730 790 l
S
NP 1986 747 m 1986 790 l
S
NP 2114 747 m 2114 790 l
S
NP 2242 747 m 2242 790 l
S
NP 2371 747 m 2371 790 l
S
NP 2499 726 m 2499 811 l
S
NP 2627 747 m 2627 790 l
S
NP 2755 747 m 2755 790 l
S
NP 2883 747 m 2883 790 l
S
NP 3011 747 m 3011 790 l
S
NP 3140 726 m 3140 811 l
S
NP 3268 747 m 3268 790 l
S
NP 3396 747 m 3396 790 l
S
NP 3524 747 m 3524 790 l
S
NP 3652 747 m 3652 790 l
S
NP 3780 726 m 3780 811 l
S
NP 3908 747 m 3908 790 l
S
NP 4037 747 m 4037 790 l
S
NP 4165 747 m 4165 790 l
S
NP 4293 747 m 4293 790 l
S
NP 4421 726 m 4421 811 l
S
medium setlinewidth
/Courier findfont 115 scalefont setfont
(0) dup stringbbox 769 exch sub exch 1826 exch sub exch m show
(1) dup stringbbox 2 idiv 1910 exch sub exch 1826 exch sub exch m show
(2) dup stringbbox 2 idiv 3052 exch sub exch 1826 exch sub exch m show
(3) dup stringbbox 2 idiv 4193 exch sub exch 1826 exch sub exch m show
(4) dup stringbbox 2 idiv 5335 exch sub exch 1826 exch sub exch m show
/Symbol findfont 115 scalefont setfont
(a) dup stringbbox 2 idiv 3384 exch sub exch 1698 exch sub exch m show
/Courier findfont 115 scalefont setfont
(-2) dup stringbbox 726 exch sub exch 2 idiv 576 exch sub exch m show
(-1) dup stringbbox 726 exch sub exch 2 idiv 1217 exch sub exch m show
(1) dup stringbbox 726 exch sub exch 2 idiv 2499 exch sub exch m show
(2) dup stringbbox 726 exch sub exch 2 idiv 3140 exch sub exch m show
(3) dup stringbbox 726 exch sub exch 2 idiv 3780 exch sub exch m show
(4) dup stringbbox 726 exch sub exch 2 idiv 4421 exch sub exch m show
/Symbol findfont 115 scalefont setfont
(k) dup stringbbox 555 exch sub exch 2 idiv 3179 exch sub exch m show

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end
%%Trailer
%%EOF
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