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75.10.-b, 75.60.-d, 05.50.+q
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Quantum spin model, Heisenberg XXZ chain, kink states,  
domain wall, spectral gap
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\begin{document}

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\preprint{UC Davis Math 2002-07}

%Title of paper
\title{The Ferromagnetic Heisenberg {\sf XXZ} chain in a pinning field}

% repeat the \author .. \affiliation  etc. as needed
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\author{Pierluigi Contucci}
\email[]{contucci@dm.unibo.it}
%\homepage[]{Your web page}
%\thanks{}
%\altaffiliation{}
\affiliation{Dipartimento de Matematica \\
Universit\`a di Bologna \\
40127 Bologna, Italy}
%
\author{Bruno Nachtergaele}
\email[]{bxn@math.ucdavis.edu}
%
\author{Wolfgang L.~Spitzer}
\email[]{spitzer@math.ucdavis.edu}
\affiliation{Department of Mathematics \\
\normalsize University of California \\
\normalsize Davis, CA 95616-8633 USA}
%\homepage[]{Your web page}
%\thanks{}
%\altaffiliation{}

\date{\today}

\begin{abstract}
We investigate the effect of a magnetic field supported at a
single lattice site on the low-energy spectrum of the
ferromagnetic Heisenberg {\sf XXZ} chain.  Such fields, caused by
impurities, can modify the low-energy spectrum significantly by
pinning certain excitations, such as kink and droplet states. We
distinguish between different boundary conditions  (or sectors),
the direction and also the strength of the magnetic field. E.g.,
with a magnetic field in the $z$-direction applied at the origin
and $++$ boundary conditions, there is a critical field strength
$B_c$ (which depends on the anisotropy of the Hamiltonian  and the
spin value) with the following properties: for $B < B_c$ there is a
unique ground state with a gap, at the  critical value, $B_c$,
there are infinitely many (droplet) ground states with gapless
excitations, and for $B>B_c$ there is again a unique ground state
but now belonging to the continuous spectrum. In contrast, any
magnetic field with a non-vanishing component in the $xy$-plane
yields a unique ground state, which, depending on the boundary
conditions, is either an (anti)kink, or an (anti)droplet state.
For such fields, i.e., {\it not} aligned with the $z$-axis,  excitations
always have a gap and we obtain a rigorous lower bound for that
gap.
\end{abstract}

% insert suggested PACS numbers in braces on next line
\pacs{75.10.-b, %General theory and models of magnetic ordering
75.60.-d, %Domain effects, magnetization curves, and hysteresis
05.50.+q %Lattice theory and statistics 
}
% insert suggested keywords - APS authors don't need to do this
\keywords{Quantum spin model, Heisenberg {\sf XXZ} chain, kink states, 
domain wall, spectral gap}

%\maketitle must follow title, authors, abstract, \pacs, and \keywords
\maketitle

\section{Introduction}

The quantum spin $j$ Heisenberg {\sf XXZ} chain has been the focus
of intensive studies in recent years. The spin 1/2 chain by itself
has connections with a surprising variety of interesting
mathematical structures, such as quantum groups \cite{PS}, vertex
algebras \cite{JM}, and fundamental  problems in combinatorics
\cite{Kup,BdGN}, to name just a few. Of course, the interest in
the {\sf XXZ} model is not limited to mathematics. In 1995,
Alcaraz, Salinas, and Wreszinski \cite{ASW} and, independently,
Gottstein and Werner \cite {GW} discovered that, with suitable
boundary terms, the ferromagnetic {\sf XXZ} chain possesses a
family of kink ground states which describe a domain wall of
finite thickness (these domain walls are exponentially localized,
with a width depending on the anisotropy parameter $\Delta$, which
diverges as $\Delta\downarrow 1$). Moreover, it was shown in
\cite{ASW}, that similar states exist for the spin $j$ model for
arbitrary $j$ and in all dimensions.

From the physical point of view, the discovery of Giant
Magnetoresistance and its connection with transport properties in
the presense of magnetic domain walls has also  spurred renewed
interest in the microscopic description of domain walls
\cite{DCS,GBB,JPRT,KY1,KY2,KY3,MNSO,NN,TF,WCJADN}. Of particular
relevance are low-lying excitations associated with  them. Koma
and Nachtergaele discovered that, although the {\sf XXZ} model has
a gap in its spectrum above the trivial translation invariant
ground  states, gapless excitations exists associated with
diagonal domain walls (11, 111, ...) in two or more dimensions
\cite{KN2}. The scaling behavior of these excitations was recently
determined in \cite{BCNS} and \cite{CapMart1}, \cite{CapMart2}.

It is interesting to note that the kink and antikink ground states
were discovered by a careful study of the {\sf XXZ} chain with the
special boundary conditions that make the spin 1/2 model
$SU_q(2)$-symmetric. Although this quantum group symmetry is
destroyed for $j>1/2$ or $d>1$, interface ground states exist in
general. In one dimension it has been proven rigorously that no
other ground states exist, in the sense of local stability, no
matter what boundary conditions are considered \cite{Mat,KN3}. In
the last reference it was also proved that the {\sf XXX} chain
does not have domain wall ground states that are stable in the
infinite-volume limit.

There is an obvious need for a clear understanding of excitations near
magnetic interfaces in order to develop more accurate models of electron
scattering at such interfaces. It has been noted, however,  that pinning of
interfaces by impurities may have to be taken into account  as well
\cite{JPRT}. Here, we study the {\sf XXZ} chain perturbed at one site by a
magnetic field as a caricature model for a pinned domain wall. Admittedly,
the one-dimensional nature of the model restricts its direct applicability to
experimental situations. We will see however that the low-lying spectrum
of this model of pinned interfaces exhibits a number of interesting features
that we expect will carry over, {\em mutatis mutandis}, to the two- and
three-dimensional case.

Let us briefly summarize our main results. We consider the
perturbation $\vec{B}\cdot \vec{S}_y$, i.e. a magnetic field at
the site $y$, to the standard spin-$j$, ferromagnetic {\sf XXZ}
Hamiltonian chain with easy-axis anisotropy defined in (\ref{H_0
bare}). As a way to impose boundary conditions, we also add
magnetic fields in the $z$-direction at the boundary spins. First,
consider boundary fields in the negative $z$-direction at both
ends, which we will refer to as ($++$) boundary conditions (b.c.),
indicating that they favor the spins at the ends to point in the
positive $z$-direction.

If the perturbation at the interior site $y$ has a component
orthogonal to the $z$-direction, we find that the ground state is
unique and describes a {\em droplet state}, i.e., the
magnetization is reduced from its maximum possible value in some
neighborhood of $y$. Strictly speaking, the magnetization is
reduced everywhere in the chain, but by an amount that decays
exponentially fast away from $y$.

However, when the field is in the $z$-direction, then there is a
critical value $B_c$ such that for $B<B_c$, the all spin-up state
is the ground state. At $B=B_c$, there are infinitely many ground
states which are droplet states describing domains of negative
magnetization of arbitrary size embedded in a environment of
positive magnetization. For stronger fields, $B>B_c$, the magnetic
field selects the all spin-down state as its ground state. This is
illustrated in Figure \ref{fig1}.

\begin{figure}[h]
\resizebox{3in}{2in}{\includegraphics{figure.eps}}
\caption{\label{fig1} Ground state energy (thick line) in a
magnetic field in the $z$-direction and $(++)$ or $(--)$ b.c.. The
dotted lines indicated droplet states which describe excitations
except at one value of the field strength, $B_c$, given in Proposition
\ref{droplet sector}. Also see Figure \ref{fig5}.}
\end{figure}

The ground state picture is simpler when we impose ($+-$) b.c.,
i.e. fields in opposite directions at the boundary spins. For
boundary fields of magnitude (\ref{DefA}), and without a
perturbation in the interior, we then have a set of {\em kink
states} as the ground state, one for every possible value of the
magnetization \cite{ASW,GW}. In that case, any non-zero field
$\vec{B}$ at an interior site $y$ selects a unique ground state.
If $\vec{B}$ is parallel to the $z$-direction the ground state is
in the continuous spectrum, and hence there are excitations of
arbitrary small energy. If there is a non-vanishing component of
$\vec{B}$ in the $xy$-plane, the unique ground state is separated
by a gap form the rest of the spectrum. The unique ground state is
a kink state centered at a position which we calculate. We also
obtain an estimate for the gap.

In Section 2, we define the model and find the set of ground
states. Section 3 is devoted to the study of the gap in the spin
$1/2$ case. Some less illuminating calculations are presented in
three appendices.

\section{The model and its ground states}

The main lesson to be learned from the proof of completeness of
the list of ground states of the infinite ferromagnetic XXZ chain
(cf.~\cite{KN1}), is that one only needs to study finite chain
Hamiltonians with very simple b.c. This remains true if we add a
bounded perturbation with finite support to the Hamiltonian, e.g.,
a magnetic field at one site. These simple b.c. are fields in the
$z$-direction with either equal or opposite sign which we will
introduce shortly. The spin $j$ {\sf XXZ} Hamiltonian without
boundary terms and with anisotropy parameter $\D>1$ is defined on
the finite chain labeled by the integers from $a$ to $b$ by
 \be\label{H_0 bare}
    H_0^{} = -\sum_{x=a}^{b-1} \left[\frac{1}{\D}(S^1_x S^1_{x+1}
    + S^2_x S^2_{x+1})+ S^3_x S^3_{x+1} - j^2{\bf 1}\right]  ,
 \ee where the copies of the spin operators at position $x$,
$S_x^1,S_x^2,S_x^3$ satisfy the usual commutation relations:
 \be
 [S_x^\a,S_x^\b]= i\,\e^{\a \b \g}\,S_x^\g .
 \ee
The anisotropy $\D>1$ has been put in front of the ${\sf XX}$ part
so that we can easily take the Ising limit $\D\to\infty$. Unless
otherwise stated, we set $a=1$. $H_0$ is non-negative, and the two
translation invariant all spin-up/down states are the ground
states of $H_0$. It will be convenient to separate them by adding
the equal-sign boundary fields, $jA(S_1^3+S_b^3)$ with
\be\label{DefA} A = A(\D)= \sqrt{1-\D^{-2}}, 
\ee and define the {\it droplet} and {\it antidroplet} Hamiltonian
\bea
   H_0^{++}& = &H_0^{} - jA(S^3_{1} + S^3_{b} - 2j{\bf 1}) ,
 \\\label{} \nonumber
 H_0^{--} &=& H_0^{} + jA(S^3_{1} + S^3_{b} + 2j{\bf 1}) .
 \eea
For convenience we have normalized the ground state energy to 0.
By reflecting all $S^3_x$ into$-S^3_x$ the two Hamiltonians,
$H^{++}$ and $H^{--}$ are unitarily equivalent and we study only
study $H_0^{++}$.

Additional ground states emerge when we add opposite-sign boundary
terms. It turns out that precisely for the fields $\pm jA(S^3_{1}
- S^3_{b})$ one discovers the full set of new ground states.
Therefore, we define the {\it kink} and {\it antikink} Hamiltonian
 \bea\label{H_0 kink}
 H_0^{+-}& = &H_0^{} - jA(S^3_{1} - S^3_{b}) ,
 \\ \label{H_0 antikink}
 H_0^{-+} &=& H_0^{} + jA(S^3_{1} - S^3_{b}) .
 \eea
Again, by spin reflection, the kink and anti-kink Hamiltonians are
unitarily equivalent. Let us define
 \bea \label{h^{+-}}
 h_{x x+1}^{+-} &=&-\mfr{1}{\D}(S^1_x S^1_{x+1} + S^2_x S^2_{x+1}) -
 S^3_x S^3_{x+1} + j^2 {\bf 1} \nonumber
 \\
 &&- j A(S^3_{x} - S^3_{x+1}) ,
 \eea
and
 \bea \label{h^{++}} h_{x x+1}^{-+}
 &=&-\mfr{1}{\D}(S^1_x S^1_{x+1} + S^2_x S^2_{x+1})
 - S^3_x S^3_{x+1} + j^2 {\bf 1}
 \nonumber
 \\
 &&+j A(S^3_{x} - S^3_{x+1}) .
 \eea
In terms of these interactions terms we may write
 \be
 H_0^{+-} = \sum_{x=1}^{b-1} h_{x x+1}^{+-},
 \ee
and
 \be
 H_0^{-+} = \sum_{x=1}^{b-1} h_{x x+1}^{-+} .
 \ee
This will be used in Section 3.

It is useful to introduce another parameter, $0<q<1$, such that
$q+q^{-1}=2\D$. The Hamiltonians defined in (\ref{H_0 kink}) and
(\ref{H_0 antikink}) in the spin 1/2 case, commute with a
representation of $SU_q(2)$. For $j>1/2$, the only obvious
conserved quantity is the total $S^3$-component, which commutes
with all the Hamiltonians defined above.

In the following we first deal with the kink Hamiltonian,
$H_0^{+-}$. There is a unique ground state for each value (sector)
$m$ of $S^3$, all of which have the same energy 0. The
eigenvalues, $m_x$, of $S_x^3$ are in  $\{-j,-j+1,\ldots,j-1,j\}$,
such that the total $S^3$-component takes the values
$m=\sum_{x\in[1,b]} m_x$. The eigenvectors of $S^3$ are denoted by
$|(m_x)\rangle$, and we have $S_y^3|(m_x)\rangle =
m_y|(m_x)\rangle$. Further, let
 $$
 w_m=\sqrt{2j\choose m+j}.
 $$
The unique ground states in the respective $S^3$-sectors are
called kink states which were found by Alcaraz, Salinas and
Wreszinski~\cite{ASW}. They are given by
 \be
 \psi_m = \sum_{\{m_x\}} \prod_{x=1}^b q^{-x(j-m_x)}w_{m_x}|(m_x)\rangle ,
 \ee
As the $\psi_m$ are not normalized, we also define
$\phi_m=\psi_m/\|\psi_m\|$. The sum over $\{m_x\}$ is restricted
to combinations such that $\sum m_x = m $. If $m=\pm jb$ is
maximal/minimal, then the state $\phi_m$ is the all spin-up/down
state, i.e. the magnetization profile in the $z$-direction,
$\langle\phi_m |S_x^3|\phi_m\rangle = \pm j$.
%We refer to the span of kink states with $|m| < jb$ as the {\it kink sector}.
Both from a physical and mathematical point of view, the infinite
chain limit is the most interesting case. Clearly, more care has
to be taken when using an infinite volume Hamiltonian. The natural
tool is the {\sf GNS} representation. In our case here, there are
four (unitarily) inequivalent representations on Hilbert spaces,
which are also called {\it sectors}. These are the {\it (anti)kink
sectors} containing the infinite volume (anti)kink ground states,
and the {\it (anti)droplet sectors} with the all spin (down)up
ground state. As mentioned above, for the infinite volume limit it
is sufficient to study the effect of the perturbation for the
finite chain Hamiltonians, $H_0^{++}$ and $H_0^{+-}$. For more
details, we refer to \cite{KN1}.

For our purposes it will be very convenient to define the states
 \be
 \psi(z) = C \sum_{|m|\le jb} z^{jb-m} \psi_{m}.
 \ee
where $C$ is a normalization constant. They are product states,
i.e.
 \be
 \label{product}\psi(z) = \bigotimes_{x=1}^b \chi_x(z),
 \ee
with
 \be \label{chi}
 \chi_x(z) = (1+|z|^2q^{-2x})^{-j}
 \sum_{m_x=-j}^j (zq^{-x})^{j-m_x} w_{m_x} |(m_x)\rangle .
 \ee

The same construction can be carried through for the antikink
Hamiltonian. We denote the corresponding states for the antikink
Hamiltonian by $\tilde{\psi}(z)$. If we let
 \bea\lefteqn{\tilde{\chi}_x(z)}
 \\
 &=&(1+|z|^2q^{-2x})^{-j} \sum_{m_x=-j}^j (zq^{-x})^{j-m_x}  w_{m_x}
    |(-m_x)\rangle ,\nonumber
 \eea
then $\tilde{\psi}(z) = \bigotimes_{x=1}^b \tilde{\chi}_x(z)$ are
ground states of $H^{-+}$.

Now, let $\vec{B}=(B_1,B_2,B_3)$ be a magnetic field vector with
(real) parameters, and $V=\vec{B}\cdot \vec{S} = B_1 S^1 + B_2 S^2
+B_3 S^3$. Then the eigenvalues of $V$ are $\|\vec{B}\|\cdot m$
with $m= -j,-j+1,\ldots,j$. Define
 \bea H^{+-}(\vec{B})&=&H_0^{+-} + \vec{B}\cdot \vec{S}_y ,
 \\
 H^{++}(\vec{B})&=&H_0^{++} + \vec{B}\cdot \vec{S}_y .
 \eea
In the study of the spectrum of $H^{+-}(\vec{B})$ and
$H^{-+}(\vec{B})$, it is important to distinguish two cases:
$B_1^2+B_2^2>0$ and $B_1^2+B_2^2=0$. The ground state in these two
cases is described in Propositions \ref{prop:kink-1} and
\ref{prop:kink-2}, and the Remarks \ref{rem:1}.

\begin{prop}[Kink sector, $B_1^2+B_2^2>0$] \label{prop:kink-1}
Let $1\le y\le b$.
Then, the ground state of $H^{+-}(\vec{B})$ is the state $\psi(z)$
of eq (\ref{product}) with
$z=-j\frac{\|\vec{B}\|+B_3}{B_1-iB_2}\,q^{y}$. Its energy is
$-j\|\vec{B}\|$.
\end{prop}
The proof is a combination of previously known results and a
straightforward calculation. See Appendix A.

\begin{remarks}\label{rem:1}
\begin{enumerate}
\item For simplicity, let us assume that $B_1^2+B_2^2=1$. If
$B_3=0$, then the ground state, $\psi(z=-j(B_1+iB_2)\,q^{y})$, is a
kink state (exponentially) localized at the magnetic field at $y$;
among the spanning set of ground states $\psi(z)$, the
perturbation picks the one which is most localized at $y$. If
$B_3\neq0$, the extra term $B_3 S_y^3$ has the effect of shifting
the kink from $y$ by the distance $|\log_q{\sqrt{1+B_3^2}+B_3}|$
to the left if $B_3>0$, and to the right (by the same distance) if
$B_3<0$.

\item The proof also shows that the state $\psi(z)$ with
$z=+j\frac{\|\vec{B}\| + B_3}{B_1-iB_2}\,q^{y}$ is an eigenstate
of $H^{+-}(\vec{B})$ with energy $+j\|\vec{B}\|$. We see this (it
is obviously linear if the field is in the $x$ or $y$ direction
only, see Figure \ref{fig2}) branch ascending from $|B_3|/2$ in
Figure \ref{fig2}-\ref{fig2.5}. In Figure \ref{fig3} it has too
high an energy to be among the plotted lowest eigenvalues.

\item Of special interest is the second-lowest eigenvalue, in
particular, whether there is a gap uniformly in the number of
sites, $b$, and how it depends on $\vec{B}$ and the anisotropy
$\D$. This will be treated in Section 3. We can extend a method
\cite{Na1} which was first applied to prove a gap for $H^{+-}_0$.

\item We discuss now qualitatively the low energy spectrum, and
assume for simplicity that $j=1/2$ and $B_2=0$. It was proven in
\cite{KN1} that the gap above the $b+1$ ground states of the
unperturbed Hamiltonian, $H_0^{+-}$, is equal to
$1-\cos{(\pi/b)}\D^{-1}$, which tends to $1-\D^{-1}$ in the
infinite chain limit.

If $B_3=0$, then there are $b$ eigenvalues (recall, $b$ is the
length of the chain) of $H^{+-}(B,0,0)$ descending from 0. This
can be seen as follows. We introduce the function
$N(B)=\chi_{(-\infty,0]}\left(H^{+-}(B,0,0) - \frac{1}{2}B{\bf
1}\right)$ counting the number of non-positive eigenvalues; here
$\chi_{(-\infty,0]}$ is the characteristic function of
$(-\infty,0]$. $N(B)$ is monotonically increasing, and equal to
$b+1$ for $|B| < 1-\D^{-1} + {\cal O}(b^{-1})$; this is guaranteed
by the gap above the ground states. In item \ref{item5} below we
calculate the lowest energy state, $\psi_e$, descending from
$1-\D^{-1}$ at $B=0$ with energy equal to
$1-\D^{-1}-\frac{1}{2}|B|$ (up to ${\cal O}(b^{-1})$). This state
is `parallel' to the ground state energy and intersects with the
state in item 2 at $|B|=1-\cos{(\pi/b)}\D^{-1}$, see Figure
\ref{fig2}.

We can say more about the average of these lowest $b+1$
eigenvalues by recalling the Min-max Principle, namely that their
average is a concave function in $B$. By symmetry  ($B\to-B$) its
(left and right) derivative is always negative and less than
$1/2$. Since the average is 0 at $B=0$ it continues to be
negative. In the infinite volume limit there is an infinite number
of ground states of $H_0^{+-}$, and eigenvalues for
$H^{+-}(B,0,0)$ have to accumulate at some value between
$-\frac{1}{2}|B|$ and $1-\D^{-1}-\frac{1}{2}|B|$, see Figure
\ref{fig2}, and in case $B_3\not=0$, see Figures
\ref{fig2.5}-\ref{fig3}.

\item If $B_3\not=0$, then there appear to be $b-y+1$ eigenvalues
descending from $-|B_3|/2$; $b-y+1$ is the number of kinks to the
left of $y$, see Figure \ref{fig2.5}. The next eigenvalues depend
on $B_3$. If $|B_3|<1-\D^{-1}$, then the next lowest $y$
eigenvalues descend from $-|B_3|/2$. If $|B_3|>1-\D^{-1}$, then
the state $\psi_e$ appears to be the next lowest in energy, see
Figure \ref{fig3}.

\item \label{item5} We can calculate the state mentioned in the
previous two items. Let $\psi_e=\sum_{x=1}^b a_x
S_x^-|\!\!\uparrow\rangle$ be the first excited state of
$H_0^{+-}$ in the one-overturned spin-sector. The coefficients
$a_x$ are a solution to the discrete Laplace equation (see
\cite{KN3}, or cf.~Appendix C by setting $B=0$). The energy of
$\psi_e$ is equal to the gap $1-\D^{-1}+{\cal O}(b^{-1})$. Now,
define
 $$
 \psi_e(z)=\sum_{n=0}^{b} z^n (S_q^- )^n \psi_e =\sum_{x=1}^b a_x
 S_x^- \psi(z).
 $$
By choosing $z$ as in Proposition \ref{prop:kink-1}, we obtain the
equation
 $$
 H^{+-}(\vec{B})\psi_e(z)=\left(1-\D^{-1}
 -\mfr{1}{2}\|\vec{B}\|\right)\psi_e(z) +{\cal
 O}\left(b^{-1}\right).
 $$
In Figure \ref{fig2}, this is the straight line parallel to the
ground state energy.
\end{enumerate}
\end{remarks}

\begin{figure}[t]
\resizebox{3.5in}{3in}{\includegraphics{kink2.eps}}
\caption{\label{fig2} Energy of the lowest 16 eigenvalues for the
spin $1/2$ kink Hamiltonian for the field $(B,0,0)$ on a chain of
13 sites and $\D=2.25$. Notice the ground state energy (straight
line downwards from 0), the gap to the 2nd eigenvalue, the energy
$\frac{1}{2}|B|$, and the energy $1-\D^{-1}-|B|/2$ of $\psi_e$. }
\end{figure}

\begin{figure}[t]
\resizebox{3.5in}{3in}{\includegraphics{kink2.5.eps}}
\caption{\label{fig2.5} Energy of the lowest 16 eigenvalues for
the spin $1/2$ kink Hamiltonian for the field $(B,0,A/6)$ on a
chain of 13 sites and $\D=2.25$; the $z$-component, $B_3=A/6=
\sqrt{1-\D^{-1}}/6$ is chosen small compared to the gap
$1-\D^{-1}$ of $H_0^{+-}$. Notice the ground state energy, the gap
above it, and the branches descending from $\pm B_3/2$, and
$1-\D^{-1}-B_3/2$. }
\end{figure}

\begin{figure}[t]
\resizebox{3.5in}{3in}{\includegraphics{kink3.eps}}
\caption{\label{fig3} Energy of the lowest 20 eigenvalues for the
spin $1/2$ kink Hamiltonian with the field $(B,0,3)$ on a chain of
13 sites and $\D=2.25$; the field in the $z$-direction is chosen
large compared to $1-\D^{-1}$. Notice that the branches stemming
from $B_3/2$ are too high in energy to be plotted here. We see the
ground state energy, the gap above it, and the energy of $\psi_e$
descending from $1-\D^{-1}-B_3/2$. In between there are $b-y+1=6$
states bending downwards from $-B_3/2=-1.5$. }
\end{figure}

Next, we consider the kink sector and magnetic fields of the form
$B_1=B_2=0,B=B_3\not=0$. Let $B>0$. Then, as $b$, the size of the
system increases, the ground state tends to the all spin-down
state, $|\!\downarrow\rangle$. This vector is no longer in the
infinite volume kink-sector very much as $e^{ikx}$ is not a
genuine (i.e., normalizable) eigenvector of the Laplacian on the
real line. In other words, $|\!\downarrow\rangle$ is part of the
continuous spectrum. So let us consider the orthogonal sequence of
kink states, $\psi_n$; $n$, as usual, is the total $z$-component.
Then, the sequence $\langle\psi_n,H^{+-}(0,0,B),\psi_n\rangle$
converges to $-jB$ as $n\to-\infty$. Since the spectrum is closed
and $-jB$ is the least possible eigenvalue it has to be the ground
state energy. Therefore, in the infinite chain limit, $-jB$ is
contained in the continuous spectrum, and is hence non-isolated.
We conjecture that there is no other continuous spectrum close to
$-j|B|$, and thus $-j|B|$ is purely an accumulation point of
eigenvectors. We do not give a proof of this here. Similarily, if
$B<0$, then the bottom of the spectrum is $jB$ and there is no gap
above the ground state, which is obviously the all spin-up state.
We illustrate the low energy spectrum in Figure \ref{fig4}.

\begin{figure}[t]
\resizebox{3.5in}{3in}{\includegraphics{kink4.eps}}
\caption{\label{fig4} Low energy spectrum in the spin $1/2$ kink
sector for the field $(0,0,1.5)$ on a chain of 11 sites with
$\D=2.25$. The high energy spectrum is similar to Fig.~\ref{fig6}
and will not be reprinted. $n$ is the total $z$-component. }
\end{figure}

Let us collect our results in the following proposition:
\begin{prop}[Kink sector, $B_1^2+B_2^2=0$]\label{prop:kink-2}
The bottom of the spectrum of $H^{+-}(0,0,B)$ is equal to $-j|B|$,
which is part of the continuous spectrum. Excitations above the
ground state are gapless.
\end{prop}
Now we consider the Hamiltonian $H^{++}(\vec{B})$. It is useful to
decompose this as a sum of a kink and anti-kink Hamiltonian
($b\ge3$):
 \beax H^{++}_{[1,b]} +  jA\left(-j{\bf 1} + 2S_y^3\right)
 = H^{+-}_{[1,y]} +
    H^{-+}_{[y,b]}  ,
 \eeax
and thus
 \bea \label{decomp} H_{[1,b]}^{++}(\vec{B}) - j^2 A {\bf 1}
&=&H^{+-}_{[1,y]}(\mfr{B_1}{2},\mfr{B_2}{2},\mfr{B_3}{2}-jA) \nonumber
\\
&&+ H^{-+}_{[y,b]}(\mfr{B_1}{2},\mfr{B_2}{2},\mfr{B_3}{2}-jA).
 \eea
We start with the case $B_1^2+B_2^2>0$. As in the kink sector we
will find a unique ground state. Let
 $$
 z=-j\frac{\|(B_1,B_2,B_3-2jA\| + B_3 -2jA}{B_1-iB_2}q^y,
 $$
then according to Proposition \ref{prop:kink-1}, $\psi_{[1,y]}(z)$
is the unique ground state of
$H^{+-}_{[1,y]}(B_1/2,B_2/2,B_2/2-jA)$, while the anti-kink state
$\tilde{\psi}_{[y,b]}(z)$ is the corresponding ground state of
$H^{-+}_{[y,b]}(B_1/2,B_2/2,B_3/2-jA)$. They are both product
states which happen to have the same component
$\chi_y(z_1)=\tilde{\chi}_y(z_2)$, because
 \beax\lefteqn{\mfr{1}{2}(B_1 S_y^1 + B_2 S_y^2 + (B_3 - 2jA)
S_y^3)\chi^\sharp_y(z) }
\\
&=&\big[-\frac{j}{2}\|(B_1, B_2, B_3 -2j A\|\big]\chi^\sharp_y(z),
 \eeax
where $\chi^\sharp$ stands for either $\chi$ or $\tilde{\chi}$. Thus
 $$
 \psi(\vec{B})=\bigotimes_{x=1}^y \chi_x(z) \otimes \bigotimes_{x=y+1}^b
 \tilde{\chi}_x(z)
 $$
is the unique ground state of $H^{++}_0(\vec{B})$ with energy
$-j\|(B_1,B_2,B_3-2j A\| + j^2 A$.

Similar to the kink-sector, the vector
 $$
 \bigotimes_{x=1}^y \chi_x(-z) \otimes \bigotimes_{x=y+1}^b
 \tilde{\chi}_x(-z)
 $$
is another eigenstate with energy ${j}\|(B_1,B_2,B_3-2j A\| + j^2
A$.

Finally, we come to the case $B_1=B_2=0,B=B_3$. When $B=0$, it was
proven in \cite{NS} that for spin $1/2$, and in the infinite
volume limit $1-\D^{-1}$ is the gap above the ground state. It can
also be shown that there exists a gap, $\d$, for higher spins
although no precise estimates are known. This implies that
uniformly in the size of the lattice, that the all spin-up vector
is the unique ground state for $B < B_c = \d/(2j)$, where $\d$ is
the (strictly positive) gap of $H^{++}_0$.
%The first excited state is certainly bounded from below in energy
%by $\d -j|B|$.
As we mentioned in the Introduction, the value $B=A/(2j)$ is very
particular and interesting. It has been analyzed \cite{St2} in the
context of droplet states for spin $1/2$ but again the method
extends to general $j$. In fact, the set of ground states is
infinitely degenerate (in the infinite volume) and consists of
pairs of symmetric kink-antikink states (i.e.~droplets), all of
which have the same energy $j A$. The magnetization profile in the
$z$-direction is symmetric with respect to the center of the field
at $y$. Excitations are gapless because large droplets which are
antisymmetric with respect to $y$ come arbitrarily close in energy
to $j A$.

Since the ground state energy is concave, and since for $B=0$ and
$B=A/2j$, the all spin-up vector is a ground state, we conclude
that for all $B < A/(2j)$, the all spin-up vector is the unique
ground state. Numerical experiments for spin $1/2$ indicate that
in the region $B < A$ the eigenvalues are ordered by their total
$S^3$-value such that the second-lowest eigenstate is in the
sector with one overturned spin, and has energy ${\cal E}_-(B)=1-
\sqrt{\D^{-2}+B^2} + \frac{1}{2}|B|$, see (\ref{excit}); its
(infinite volume) derivation is given in Appendix C. The
third-lowest eigenvector is in the sector with two overturned
spins, and so on. The lowest eigenvalues with respect to the total
$S^3$-component accumulate at the line $A - B/2$. They all meet at
the critical value $A$, where the ground state becomes infinitely
degenerate. Assuming that this ordering holds true, we conjecture
that the gap for spin $1/2$ and $B\le A$ equals $1-\sqrt{\D^{-2} +
B^2} +\frac{1}{2}(|B|-B)$, which converges to $1-\frac{1}{2\D}$
for large $-B$, and vanishes at $B=A$.

For $B > A/(2j)$, the all spin-down state is the unique ground
state with energy $A/(2j) - jB $, which is part of the continuous
spectrum; in fact, $A/(2j) - jB $ is purely an accumulation point
of eigenvectors. It seems that for $B>A/2j$ the eigenvalues are
also ordered according to their total $z$-component, $n$, but this
time in the opposite way. I.e.~lower $n$ means lower energy, and
clearly the lowest is the all spin-down state. Similar to the kink
sector, we will not prove here that the rest of the continuous
spectrum is separated from the ground state.

The situation is illustrated in Fig.~\ref{fig5}-\ref{fig7} for
$j=1/2$, cf.~also Figure \ref{fig1}. Let us summarize our results
in the following proposition:

% use the file droplet1.m to generate the figure, save under droplet1.fig
% AND droplet1.eps (color)
% by using the print command
%\begin{center}
%\resizebox{15truecm}{10truecm}{\includegraphics{droplet1.eps}} \\
%\end{center}
\begin{figure}[t]
\resizebox{3.5in}{3in}{\includegraphics{droplet2.eps}}
\caption{\label{fig5} The five lowest eigenvalues in the spin
$1/2$ droplet sector for a chain of 13 sites with magnetic field
$(0,0,B)$ and $\D=2.25$. The eigenvalues are numbered by $n$, the
number of overturned spins; e.g.~$n=0$ means the all spin-up
state. }
\end{figure}

%\bigskip\noindent{\bf Comparison to XY-model:} It is amusing to compare our
%results with the results on perturbation of the spin $1/2$
%XY-model by magnetic fields as was done by Hume and Robinson
%\cite{Hume}. For the XY-model only magnetic fields in the
%$z$-direction can be handled directly by using the Jordan-Wigner
%transformation; in our opinion a field in any other direction
%poses an interesting and open problem.

% use droplet6&7

\begin{figure}[t]
\resizebox{3.5in}{3in}{\includegraphics{droplet7.eps}}
\caption{\label{fig6}Here, we plot the full spectrum of the spin
1/2 droplet Hamiltonian $H^{++}(0,0,1.5A)$ for 13 sites, and
$\D=2.25$. The index $n$ is the number of overturned spins. The
ground state is the all spin-down state, i.e.~$n=13$.}
\end{figure}

\begin{figure}[t]
\resizebox{3.5in}{3in}{\includegraphics{droplet6.eps}}
\caption{\label{fig7} Here, we amplify the low energy spectrum for
the same Hamiltonian as in Figure \ref{fig6}. At $n=13$ we have
indicated the ground state. Clearly, one sees monotonicity of
energy vs $n$.}
\end{figure}

\begin{prop}[Droplet sector] \label{droplet sector}
The ground state of the droplet Hamiltonian,
$H^{++}(\vec{B})$, on a chain of length $b\ge3$ depends on the
magnetic field $\vec{B}$ in the following way:
\begin{enumerate}

\item If $B_1^2+B_2^2>0$, then the ground state is unique. The ground
state energy is $-{j}\|(B_1,B_2,B_3 - 2jA)\| + j^2 A$.

\item If $B_1=B_2=0,B=B_3$, and $B_c = A/(2j)$, then for $B < B_c$ the unique
ground state of $H^{++}(0,0,B)$ is the all spin-up vector with
energy $jB$. For $B = B_c$ the ground states are droplet states
which are (in the thermodynamic limit) infinitely degenerate with
energy $jA$. In infinite volume, excitations above these ground
states are gapless. For $B > B_c$, the all spin-down state is
the unique ground state, which is an accumulation point of
eigenvectors. Hence, excitations are gapless.
\end{enumerate}
\end{prop}

\section{Estimate for the spectral gap in the case $j=1/2$}

Here we prove a uniform lower bound on the difference between the
ground state energy and the energy of the first excited state for
the spin $1/2$ Hamiltonians $H^{+-}_{[1,b]}(\vec{B})$ and
$H^{++}_{[1,b]}(\vec{B})$ on a finite chain $[1,b]$ with the
impurity field at $y$.

Before we prove these gap inequalities we introduce the methods
which were invented in \cite{Na1} and \cite{Na2}. Let $\C_i,
i=0,\ldots,N$ be a sequence of connected intervals with
$\bigcup_{i=0}^N \C_i = [1,b]$, and such that two intervals have
at most one lattice point in common. Let $h_{\C_i}\ge 0$ be some
(local) Hamiltonians acting on $\mathbb C^{2|\C_i|}$, and define
\be H_{[1,b]} = \sum_{i=0}^N h_{\C_i}. \ee $H_{[1,b]}$ acts on
${\cal H}_b = \bigotimes_{i=1}^b \mathbb C^2$. We assume that
$\ker{H_{[1,b]}}\not=\{0\}$. Let $\g_i$ denote the gap of
$h_{\C_i}$, i.e.~the smallest non-zero eigenvalue of $h_{\C_i}$.
It is clear that \be \ker H_{[1,b]} = \bigcap_{i=0}^N \ker
h_{\C_i} . \ee Let $\L\subset[1,b]$, then we define $G_\L$ to be
the orthogonal projection onto the \be \ker \sum_{i:\C_i\subset
\L} h_{\C_i}. \ee We use the convention that if
$\C_i\not\subset\L$ for any $i$, then we set $G_\L=\bf 1$. From
these definitions we derive the following properties:
\begin{enumerate}
\item $G_{\L} G_{\L'} = G_{\L'} G_{\L} = G_{\L'}$ if $\L\subset\L'$.
\item $G_{\L} G_{\L'} = G_{\L'} G_{\L}$ if $\L\cap\L'=\emptyset$.
\item $h_{\C_i}\ge \g_i ({\bf 1} - G_{\C_i})$.
\end{enumerate}
Next we define the intervals $\L_i=\bigcup_{j\le i}\C_j$, and operators
$E_i, i\ge0$, on ${\cal H}_b$ by
 \be E_i = \left\{\begin{array}{lcl}{\bf 1} - G_{\L_0}& \mbox{for} &i=0
                   \\G_{\L_i} - G_{\L_{i+1}}&\mbox{for} &1\le i<N
           \\G_{[1,b]}&\mbox{for}&i=N
         \end{array} \right. .
 \ee
These operators are mutually commuting projections adding up to ${\bf 1}$, i.e.
 \be
 E_i^* = E_i,\quad E_iE_j=\d_{ij} E_i,\quad \sum_{i=0}^N E_i ={\bf 1}.
 \ee
The key assumption in order to deduce a gap for $H_{[1,b]}$ from
the gaps of $h_{\C_i}$ is the following assumption:
\begin{assumption} \label{assumption} There exists a positive constant
$\e$ such
that $0\le \e < 1/\sqrt{2}$ and
 \be \label{ass1}
 E_i G_{\C_{i+1}} E_i \le \e^2 E_i, \quad 0\le i\le N-1 ,
 \ee
or equivalently,
 \be \label{key est}
 \|G_{\C_{i+1}}E_i\|\le \e, \quad 0\le i\le N-1.
 \ee
Further, we assume that the gaps, $\g_i$, for the local
Hamiltonians are bounded from below, i.e.~$\g_i\ge\g>0$.
\end{assumption}

The conditions (\ref{ass1}) and (\ref{key est}) are equivalent due
to $G_{\C_{i+1}}E_i =G_{\C_{i+1}}G_{\L_i}$ $-G_{\L_{i+1}}$.

Now we are ready to state the main theorem which we apply in all
three case below.

\begin{thm}[Nachtergaele \cite{Na1}] With the above definitions
and under the assumptions in (\ref{key est}) let $\psi$ be orthogonal to 
the ground states of $H_{[1,b]}$. Then
 \be\label{general gap}
 (\psi, H_{[1,b]}\psi) \ge \g (1-\sqrt{2}\e)^2\|\psi\|^2.
 \ee
I.e.~the gap in the spectrum of $H_{[1,b]}$ above 0 is at least
$\g (1-\sqrt{2}\e)^2$.
\end{thm}

\begin{proof}Let $\psi$ be orthogonal to the ground state, i.e.
$G_{[1,b]}\psi=0$. Then,
$\|\psi\|^2 = \sum_{0\le n< N}\|E_n \psi\|^2$.

We estimate $\|E_n \psi\|^2$ in terms of $(\psi,H_{\C_n}(\vec{B})\psi)$
as follows. First notice, that for $m\le n-2$, or $m\ge n+1$,
$E_m G_{\C_n} = G_{\C_n} E_m$.
Now we insert $G_{\C_n}$ and the resolution of identity, $\{E_n\}$, and we get
 \bea
 \lefteqn{\|E_n \psi\|^2}
 \label{3.11}\nonumber\\
 &=& (\psi,({\bf 1}- G_{\C_n}) E_n\psi) + (\psi,\sum_{0\le m< N}
 E_m G_{\C_n} E_n
                         \psi)
 \nonumber\\
 &=&(\psi,({\bf 1}- G_{\C_n}) E_n\psi) + ((E_{n-1} +E_{n})
 \psi,G_{\C_n} E_n\psi).\hspace{.7cm}
 \eea
Let $c_1,c_2>0$, then
 \bea \lefteqn{\|E_n \psi\|^2}
 \nonumber
 \\ &\le&\frac{1}{2c_1} (\psi, ({\bf 1} - G_{\C_n})\psi)
      + \frac{c_1}{2} (\psi,E_n\psi)
 \label{3.12}
 \\
 &&+\frac{1}{2c_2} (\psi,E_n G_{\C_n}E_n\psi) + \frac{c_2}{2}
 (\psi,(E_{n-1} + E_{n})^2\psi), \nonumber
 \eea
where we used the inequality
 $$
 |(\phi_1,\phi_2)| \le \frac{1}{2c} \|\phi_1\|^2 + \frac{c}{2} \|\phi_2\|^2
 $$
in both terms of (\ref{3.11}). Let $\gamma = \min{\{\g_{\C_i}\}}$ be the
minimum
of the gaps of the Hamiltonians $h_{\C_i}$.

The first term on the rhs of (\ref{3.12}) is less than
$\frac{1}{2c_1 \gamma} (\psi, h_{\C_n} \psi)$. Now, assuming the key
estimate,
$\|G_{\C_n} E_n\| < \e$, we see that
 \beax\lefteqn{\left(2-c_1 - \frac{\e^2}{c_2}\right) \|E_n\psi\|^2 - c_2
 \|(E_{n-1}+E_{n})\psi\|^2}
 \\
 &\le&\frac{1}{c_1\gamma} (\psi,h_{\C_n} \psi) .\hspace{4cm}
 \eeax
We sum over $n$ using $\|\psi\|^2 = \sum_{0\le n<N}\|E_n \psi\|^2$ from
above and get
 $$
 \left(2-c_1 - \frac{\e^2}{c_2} - 2c_2\right) \|\psi\|^2 - c_2 \|\psi\|^2 \le
 \frac{1}{c_1\gamma} (\psi, H_{[1,b]}\psi) .
 $$
Finally, we optimize the constants $c_1,c_2$ yielding
$c_1=1-\e\sqrt{2}, c_2=\e/\sqrt{2}$. This proves the gap
inequality.
\end{proof}

In all the upcoming proofs on the various gaps we use the same
definition of subsets $\C_n,\L_n$ of $[1,b]$ and projections $G_n,
E_n$. As usual $y\in[1,b]$ denotes the spot of the magnetic field.
Let $n_l,n_r$ be some non-negative integers such that
$n_r+n_l\ge1$, and assume that $n_r>0$; the choice of $n_l,n_r$ in
general will depend on $\D$ and $\vec{B}$. The idea behind the
definition of $\C_n$ is that we cover the chain $[1,b]$ by adding
points to an initially chosen interval $\C_0=[y-n_l,y+n_r]$ in an
alternating manner. First we add a point to the right of $\C_0$,
then to the left until we reach the point 1. Then we add points
only to the right of $\C_{2(y-n_l-1)}$ until we finish at $b$.
More precisely, we define the sets $C_n$ in the following way:

\begin{defin} \label{def}
Let $\C_0=[y-n_l,y+n_r]$, where we may assume that $y-n_l-1\le b-y-n_r$
such that
$\C_0\subset [1,b]$. The intervals for $n>0$ are then
$\C_1 = [y+n_r,y+n_r+1],\C_2 =
[y-n_l-1,y-n_l],\ldots, \C_{2(y-n_l-1)} = [1,2], \C_{2(y-n_l)-1} =
[2y+n_r-n_l-1,2y+n_r-n_l],\ldots,\C_{b-(n_l+n_r)-1} = [b-1,b]$.
\end{defin}

We start with the kink case.

\begin{prop}[Kink sector, $B_1^2+B_2^2>0$] \label{prop3.1}
Let $\psi$ be orthogonal to the ground state of the kink Hamiltonian,
$H^{+-}_{[1,b]}(\vec{B})$, on a chain of length $b$. Then, there exists a
strictly positive
function $g^{+-}(\vec{B},\D)$ and a function $0\le\e(\vec{B},\D)<1/\sqrt{2}$,
which are both independent of $b$, such that the following gap inequality is
satisfied
 \bea \label{gap alpha}
 (\psi,H^{+-}_{[1,b]}(\vec{B})\psi)
    \ge g^{+-}(\vec{B},\D) (1-\sqrt{2}\e(\vec{B},\D))^2\|\psi\|^2. \hspace{.7cm}
 \eea
\end{prop}
\begin{proof}
First we shift the ground state energy to be 0, and define the new Hamiltonian
 $$
 H_{[1,b]}(\vec{B}) = H^{+-}_{[1,b]}(\vec{B}) - \mfr{\|\vec{B}\|}{2} {\bf 1} .
 $$
With the set-up from Definition (\ref{def}) we
can write the Hamiltonian, $H_{[1,b]}(\vec{B})$, in the following form:
 \beax
 H_{[1,b]}(\vec{B})  &=& \sum_{i=0}^{b-(n_l+n_r)-1} h_{\C_i}
 \eeax
using
 \beax
  h_{\C_i} &=&\left\{\begin{array}{lcr}H_{\C_0}(\vec{B})&\mbox{ for }& i=0\\
  h_{x x+1}^{+-}&\mbox{ for }& i>0
               \end{array},\right.
 \eeax
where $\C_i = [x, x+1]$ for some $1\le x<y-n_l$ or $y+n_r \le x < b$, and with
$h_{x x+1}^{+-}$ from (\ref{h^{+-}}).

We can express the gap conditions as
\begin{enumerate}
\item $H_{\L_0}(\vec{B}) \ge g_{\L_0}^{+-}(\vec{B},\D) ({\bf 1} - G_{0})$,
where
$g_{\L_0}^{+-}(\vec{B},\D)$ is the gap for the finite chain Hamiltonian,
$H^{+-}_{\L_0}(\vec{B})$.
\item $h_{x x+1}^{+-} = {\bf 1} - G_{[x x+1]}$ for $1\le x<y-n_l$
and $y+n_r \le x< b$.
\end{enumerate}
Let $\g = \min{\{g_{\L_0}^{+-}(\vec{B,\D}),1\}}$ which is strictly positive.
Finally, we need to verify the second condition in Assumption
\ref{assumption}, and define
 \be
 C_n:= \sup_{0\not=\psi\in{\cal H}_{\Lambda_{n+1}} : E_n\psi = \psi}
          \frac{\|G_{\C_n}\psi\|}{\|\psi\|}.
 \ee
So let $\psi$ satisfy
 \be
 G_n\psi = \psi\quad\mbox{and}\quad G_{n+1}\psi = 0 .
 \ee
First, let $n=2m, 0\le m\le y-n_l-1$; the case $n\ge 2(y-n_l)$ is similar, and
the case of odd
$1\le n < 2(y-n_l)$ will be considered later.

Let $\psi$ be a ground state of $\L_n$, i.e.~$G_n\psi=\psi$
such that $G_{n+1}\psi=0$. Then
with the definition from (\ref{chi})
 $$
 \psi = \bigotimes_{i={y-n_l-m}}^{y+n_r+m} \chi_i(z) \otimes
 \chi^\perp_{y-n_r+m+1}(z),
 $$
where $\chi^\perp_{x}(z)$ is perpendicular to $\chi_x(z)$.
Let us make some definitions and call
$f:=-\mfr{1}{2}\frac{\|\vec{B}\| +B_3}{B_1-iB_2}$,
$\chi:=\chi_{y+n_r+m}(z) = |\!\!\uparrow\rangle +
f q^{-n_r-m}|\!\!\downarrow\rangle$,
$\chi^\perp:=\chi_{y+n_r+m}(z) = f q^{-n_r-m-1}$ $|\!\!\uparrow\rangle -
|\!\!\downarrow\rangle$,
and $(1+q^2)^{1/2}|\xi\rangle =  q|\!\!\uparrow\!\downarrow\rangle -
|\!\!\downarrow\!
\uparrow\rangle$.
Then,
 \beax \frac{\|G_{\C_n}\psi\|^2}{\|\psi\|^2}
 &=& 1- \frac{\||\xi\rangle\langle\xi|\chi\otimes
      \chi^\perp\rangle\|^2}{\|\chi\otimes\chi^\perp\|^2}
 \\
 &=&1 - \frac{q^2}{1+q^2}
 \frac{1+|f|^2 q^{-2(n_r+m+1)}}{1+|f|^2 q^{-2(n_r+m)}}.
 \eeax
Let $m=0$, then we choose $n_r$ such that rhs is less than $1/2$.
The condition for $C_0<1/\sqrt{2}$ is thus $|f| q^{-n_r} > 1$. By
monotonicity it is clear that the condition,
$C_{n=2m}<1/\sqrt{2}$, holds true for $m\ge0$.

Now we come to odd integers, $n=2m+1$ with $0\le m\le y-n_l-1$.
Let $\psi$ satisfy $G_n \psi =\psi$, and $G_{n+1}\psi=0$, then
$\psi$ is of the form $\psi = \chi^\perp_{y-n_l-m-1}(z) \otimes
\bigotimes_{i=y-n_l-m}^{y+n_r+m} \chi_i(z)$. We have thus
 \beax
 \frac{\|G_{\C_n}\psi\|^2}{\|\psi\|^2} &=& 1- \frac{1}{1+q^2}
      \frac{1+|f|^2 q^{2(n_l+m+1)}}{1+|f|^2 q^{2(n_l+m)}} .
 \eeax
Let $m=0$, then we choose $n_l$ such that rhs is less than $1/2$.
This is accomplished if $1>|f| q^{n_l}$. By monotonicity,
$C_{n=2m+1}<1/\sqrt{2}$ for $m\ge0$. Our condition for the choice
of $n_l,n_r$ is thus
 $$
 q^{n_r} < \frac{1}{2}\left|\frac{\|\vec{B}\|+B_3}{B_1-iB_2}\right|
<q^{-n_l}.
$$
\end{proof}

\begin{remark}
It is clear that there always exist integers $n_l$ and $n_r$ such
that $1> |f| q^{n_l}$ and $1<|f| q^{-n_r}$ are satisfied. Now
suppose that
$q<\frac{1}{2}\left|\frac{\|\vec{B}\|+B_3}{B_1-iB_2}\right| < 1$,
then we may choose $n_l=0$ and $n_r=1$; for example, if the
$\vec{B}$-field has no $B_3$-component, then $q$ has to be less
than $\frac{1}{2}$. In this case, $g^{+-}(\vec{B},\D) =
g_2^{+-}(\vec{B},\D)$ is found explicitly in Appendix B, see
(\ref{kinkgap}).
\end{remark}


\begin{prop}[Droplet sector] $B_1^2+B_2^2>0$.
Let $\psi$ be orthogonal to the ground state of the droplet Hamiltonian,
$H^{++}_{[1,b]}(\vec{B})$, on a chain of length $b$. Then, there exists a
strictly positive
function $g^{++}(\vec{B},\D)$ and a
positive function $0\le\e(\vec{B},\D)<1/\sqrt{2}$,
which are both independent of $b$, such that the following gap inequality is
satisfied
 \be
 (\psi,H^{++}_{[1,b]}(\vec{B})\psi) \ge g^{++}(\vec{B},\D)
    (1-\sqrt{2}\e(\vec{B},\D))^2\|\psi\|^2.
 \ee
\end{prop}

\begin{proof}
First, we need to shift the ground state energy, and define a new Hamiltonian
 $$
 H_{[1,b]}(\vec{B})
 = H^{++}_{[1,b]}(\vec{B}) + \mfr{1}{4}\|(B_1,B_2,B_3 - A)\| -
                       \mfr{1}{4} A.
 $$
Using the sets from Definition (\ref{def}) we have the decomposition
 \beax
 H_{[1,b]}(\vec{B})&=&\sum_{i=0}^{b-(n_l+n_r)-1} h_{\C_i},
 \eeax
with
 \beax
 h_{\C_i}=\left\{\begin{array}{ll}H_{\C_0}(\vec{B})&\mbox{ for } i=0 \\
 h_{x x+1}^{+-}&\mbox{ for some $x$: } 1\le x<y-n_l\\
 h_{x x+1}^{-+}&\mbox{ for some $x$: } y+n_r \le x < b ,
 \end{array}\right.
 \eeax
depending on whether $\C_i$ is to left(right) of $y$. $h_{x
x+1}^{+-}$ and $h_{x x+1}^{++}$ are taken from equations
(\ref{h^{+-}}), respectively (\ref{h^{++}}). We have the following
gap properties:
\begin{enumerate}
\item $H_{\L_0}(B) \ge g^{++}_{\C_0}(\vec{B},\D) ({\bf 1} - G_0)$,
where $g^{++}_{\C_0}(B)$
is the gap for the Hamiltonian, $H_{\C_0}(\vec{B})$.
\item $h_{x x+1}^{+-} = {\bf 1} - G_{[x x+1]}$ for $1\le x<y-n_l$.
\item $h_{x x+1}^{-+} = {\bf 1} - G_{[x x+1]}$ for $y+n_r \le x< b$.
\end{enumerate}
We are left with verifying the key estimate (\ref{key est}). So
let $0\not=\psi$ satisfy $G_n\psi=\psi$ such that $G_{n+1}\psi=0$.
If the interval $C_{i+1}$ is to the left of $y$ then we have the
same situation as in the previous proof with the condition
$1>|f|q^{n_l}$ with the slightly modified ground state energy $f =
\frac{1}{2}\frac{\|(B_1,B_2,B_3-A)\| +B_3-A}{B_1-iB_2}$.

If the interval $C_{i+1}$ is to the right of $y$, then we will
arrive at the same condition for $n_r$. This is true by symmetry
but one can easily derive this in the very same way we did in the
other case.
\end{proof}

\begin{remark} The same remarks are in order here for the droplet
Hamiltonian.
So let us suppose that
$q<\frac{1}{2}\left|\frac{\|\vec{B}\|+B_3}{B_1-iB_2}\right| < 1$,
then we choose $n_l=0$ and $n_r=1$. In this case,
$g^{++}(\vec{B},\D) = g_2^{++}(\vec{B},\D)$ is explicitly
calculated in Appendix B, see (\ref{dropletgap1}).
\end{remark}

\begin{prop}[Droplet sector] Let $B_1=B_2=0$, and $B<A$.
Let $\psi$ be orthogonal to the all spin-up ground state of the droplet
Hamiltonian,
$H^{++}_{[1,b]}(0,0,{B})$, on a chain of length $b\ge3$, and let
$g^{+}_3(B,\D)$ be the gap for the
three-site Hamiltonian from eq (\ref{dropletgap2}). Then,
 \be
 (\psi,H^{++}_{[1,b]}({B})\psi) \ge 2 g^{+}_3(B,\D) \left(\frac{1}{\sqrt{2}}-
 \sqrt{\frac{q}{q+q^{-1}}}\right)^2\|\psi\|^2.
 \ee
\end{prop}

\begin{proof}
Again, we need to shift the ground state energy, and define a new Hamiltonian
$$H_{[1,b]}(B) = H^{++}_{[1,b]}(0,0,B) - \mfr{B}{2} {\bf 1}.
$$
As before, we use the same decomposition of $H_{[1,b]}(B)$ into local
Hamiltonians,
$h_{\C_i}$.
The first gap condition of $\C_0$ has to be changed into
 $$
 H_{\L_0}(B) \ge g^{+}_{\C_0}(B,\D) ({\bf 1} - G_0).
 $$
We only need to compute
 $$
 C_n:= \sup_{0\not=\psi\in{\cal H}_{\L_{n+1}} : E_n\psi = \psi}
        \frac{\|G_{\C_n}\psi\|}{\|\psi\|}.
 $$
So let us take a (non-zero) vector $\psi$ such that $G_n\psi=\psi$
and $G_{n+1}\psi=0$. If $\C_{n+1}$ is to the left of $y$, then
$\psi = |\!\downarrow\rangle\otimes|\!\uparrow
\cdots\uparrow\rangle$, and $G_{\C_n} = {\bf 1} -
|\xi\rangle\langle\xi|$. Then
$$\frac{\|G_{\C_n} \psi\|^2}{\|\psi\|^2} = \frac{1}{1+q^2},
$$
which is less than $1/2$, and $(1-\sqrt{2}\e)^2= 2\left(\frac{1}{\sqrt{2}}-
\sqrt{\frac{q}{q+q^{-1}}}\right)^2$.

By symmetry this is also the condition if $\C_{n+1}$ is to the right of $y$.
More precisely,
$\psi=|\!\uparrow\cdots\uparrow\rangle \otimes |\!\downarrow\rangle$, and
$G_{\C_n} = {\bf 1} - |\zeta\rangle\langle\zeta|$
with $(1+q^2)^{{1}/{2}}|\zeta\rangle  =
q |\!\downarrow\uparrow\rangle - |\!\uparrow\downarrow\rangle$.

By choosing $\C_0=[y-1,y+1]$, we have verified the statement. The
three-site gap, $g^{+}_{3}(B,\D)$ is calculated in Appendix B.
\end{proof}

\begin{appendix}

\section{Proof of Proposition \ref{prop:kink-1}}

\begin{proof} First, it is clear that the bounded perturbation
$V=\vec{B}\cdot\vec{S}_y$ can shift the ground state energy of
$H_0^{+-}$ by no more than its norm, $j\|\vec{B}\|$. We claim, and
show below, that the product state
$\psi(z=-j\frac{\|\vec{B}\|+B_3}{B_1-iB_2}\,q^{y})$, which is a
ground state of $H_0^{+-}$, is a also a ground state of $V$.
Therefore, we have found a ground state of $H_0^{+-}(\vec{B})$.
That it is the unique ground state follows by combining two facts:
1) $\psi(z)$ is the unique kink state with this property, which we
will show, and 2) the vectors $\psi(z)$, for arbitrary complex
$z$, span the full ground state space of $H_0^{+-}$ \cite{GW,KN3}
and there is gap to the rest of the spectrum \cite{KN1,KNS}. So,
it only remains to prove that among all vectors $\psi(z)$, there
is a unique one that is a ground state and that the corresponding
value of $z$ is as stated in the proposition.

Since $\psi(z)$ is of product form, and $V$ acts non-trivially
only at site $y$, we are left to show that
 \be \label{7}
 \vec{B}\cdot\vec{S}_y \chi_y(z) = -j\|\vec{B}\|\chi_y(z) .
 \ee
Without loss of generality, we may assume that $\|\vec{B}\|=1$.
Then, $zq^{-y}=-j\frac{1+B_3}{B_1-iB_2}$. Now, checking all $2j+1$
vector components in (\ref{7}) we obtain
 \beax
 \lefteqn{\frac{1}{2} \r_n(B_1+iB_2) w_{n+1} (z q^{-y})^{j-n-1} + n
 B_3 w_n (zq^{-y})^{j-n}}
 %\hspace{5cm}
 \\ &&+\frac{1}{2} \r_{n-1}(B_1-iB_2) w_{n-1} (z
 q^{-y})^{j-n+1}\hspace{2cm}
 \\
 &=&j w_n (zq^{-y})^{j-n},
 \eeax
with $\r_n= \sqrt{j(j+1)-n-n^2}$, and for $|n|\le j$. This leads
to the following equation:
 \beax\lefteqn{\frac{1}{2} \r_n  w_{n+1}
 (1-B_3^2) + \frac{1}{2} \r_{n-1} w_{n-1} (1+B_3)^2 }
 \\
 &&-nB_3 w_n (1+B_3)
 \\
 &=&j w_n (1+B_3).\hspace{3.5cm}
 \eeax
By a straightforward calculations one verifies that
 \beax \frac{1}{2} \r_n w_{n+1} + \frac{1}{2} \r_{n-1} w_{n-1}  &=& j w_n,\\
      \r_{n-1}  w_{n-1} &= &(j+n) w_n,\\
      -\frac{1}{2} \r_n w_{n+1} + \frac{1}{2} \r_{n-1} w_{n-1} &=&n w_n.
 \eeax
This proves (\ref{7}).
\end{proof}


\section{Explicit diagonalizations of small-site spin $1/2$ Hamiltonians}

\subsection{$H^{+-}_{12}(\vec{B})$}

Here we diagonalize the two-site Hamiltonian,
$H^{+-}_{12}(\vec{B})$, with magnetic field not parallel to
$z$-axis at $y=1$. By the {\sf XX} symmetry we may assume $B_2=0$.
Since we already know two eigenvalues, namely,
$\pm\frac{1}{2}\sqrt{B_1^2+B_3^2}$, it is best to factor them out
from the characteristic equation. Another way is to diagonalize
the Hamiltonian restricted to the orthogonal complement of the two
known eigenvectors. The Hamiltonian is of the form
 \beax \lefteqn{2 H^{+-}_{[1,2]}(B_1,B_3)}
 \\
 &=&\left(\begin{array}{cccc} B_3&0          &B_1        &0\\
                             0  &1-A+B_3&-\D^{-1}   &B_1\\
                 B_1&-\D^{-1}   &1+A-B_3&0\\
                 0  &B_1        &0          &-B_3
     \end{array}
   \right) .
 \eeax
The characteristic polynomial, $p$, is equal to
 \beax p(t)
 &=&(t^2-B_3^2)(t^2-2t+2AB_3-2B_1^2-B_3^2)
 \\
 &+&2B_1^2(t-AB_3) + B_1^4.
 \eeax
We divide this polynomial by $t^2-B_1^2-B_3^2$ (Note that we have
multiplied the Hamiltonian by two) obtaining
$$p(t)= (t^2 -2t -B_1^2 - B_3^2 +2B_3 A)(t^2-B_1^2-B_3^2).$$
The two eigenvalues we are looking for are thus
$$t_\pm = 1\pm \sqrt{1 + B_1^2 + B_3^2 - 2 B_3 A}.
$$
One can easily verify that
 $$
 \sqrt{B_1^2+B_3^2}\ge1-\sqrt{1+B_1^2+B_3^2-2B_3A}.
 $$
Hence, the gap between the lowest eigenvalues of
$H^{+-}_{[1,2]}(B_1,B_2,B_3)$ is
equal to
 \be\label{kinkgap}
   g_2^{+-}(\vec{B}) = \mfr{1}{2}  - \mfr{1}{2} \sqrt{1 + \|\vec{B}\|^2 - 2B_3A}
                  +\mfr{1}{2} \|\vec{B}\|,
 \ee
which is a positive function.

\subsection{$H^{++}_{[12]}(\vec{B}), B_1^2+B_2^2>0$}

The diagonalization of the two-site droplet Hamiltonian,
$H^{++}_{[12]}(\vec{B})$,
with the field at $y=1$ is very similar to the two-site kink Hamiltonian.
We have
 \beax
 \lefteqn{2 H^{++}_{[12]}(B_1,B_3) -A{\bf 1}}
 \\
 &=&\left(\begin{array}{cccc} B_3-A&0       &B_1        &0\\
                             0        &1+B_3   &-\D^{-1}   &B_1\\
                 B_1      &-\D^{-1}&1-B_3      &0\\
                 0        &B_1     &0          &A-B_3
     \end{array}
   \right) .
 \eeax
The characteristic polynomial, $q$, of the rhs is equal to
 \beax
 q(t)
 &=&(t^2-(B_3-A)^2)(t^2 - 2t - 2B_1^2 -B_3^2 + A^2)
 \\
 &-&2B_1^2(t-A^2) + B_1^4.
 \eeax
$t=\pm \sqrt{B_1^2+(B_3-A)^2}$ are two roots and we factor them out
from $q(t)$, and obtain
 \beax
 q(t)&=&(t^2 - 2t - B_1^2 - B_3^2 + A^2) (t^2-B_1^2-(B_3-A^2).
 \eeax
The two new eigenvalues of $H^{++}_{[12]}(\vec{B})$ are
 $$
 t_\pm =  \mfr{1}{2} \left(1 \pm \sqrt{1 + \|\vec{B}\|^2 -A^2} +A\right) .
 $$
The gap above the ground state is therefore
 \bea \label{dropletgap1}
 g^{++}_2(\vec{B},\D) &= &
     \mfr{1}{2}\left(1 - \sqrt{1 + \|\vec{B}\|^2 -A^2} \right.
 \nonumber
 \\
 &&+\left.\sqrt{B_1^2 +B_2^2 +(B_3-A)^2}\right).
 \eea

\subsection{$H^{++}_{[13]}(0,0,B), B<A$}

Since in this case there is only a magnetic field in the
$z$-direction we can easily diagonalize the three-site droplet
Hamiltonian, $H=H^{++}_{[13]}(0,0,B)$, with the field in the
middle at $y=2$. Then $H$ commutes with the symmetry $S:S(u_{\s_1}
\otimes u_{\s_2} \otimes u_{\s_3}) = u_{\s_3}\otimes
u_{\s_2}\otimes u_{\s_1}$, where $\sigma_i=\pm$. We choose the
following eigenbasis of $S$:
 \beax v_1&=&(1,0)\otimes(1,0)\otimes(1,0),
 \\
 v_2&=&(0,1)\otimes(0,1)\otimes(0,1),
 \\
 v_3&=&\mfr{1}{\sqrt{2}}(1,0)\otimes(1,0)\otimes(0,1) - \mfr{1}{\sqrt{2}}
       (0,1)\otimes(1,0)\otimes(1,0),
 \\
 v_4&=&\mfr{1}{\sqrt{2}}(1,0)\otimes(0,1)\otimes(0,1) -
       \mfr{1}{\sqrt{2}}(0,1)\otimes(0,1)\otimes(1,0),
 \\
 v_5&=&\mfr{1}{\sqrt{2}}(1,0)\otimes(1,0)\otimes(0,1) +\mfr{1}{\sqrt{2}}
       (0,1)\otimes(1,0)\otimes(1,0),
 \\
 v_6&=&(1,0)\otimes(0,1)\otimes(1,0),
 \\
 v_7&=&(0,1)\otimes(1,0)\otimes(0,1) ,
 \\
 v_8&=&\mfr{1}{\sqrt{2}}(0,1)\otimes(0,1)\otimes(1,0)  + \mfr{1}{\sqrt{2}}
       (1,0)\otimes(0,1)\otimes(0,1).
 \eeax
$v_1,v_2,v_3,v_4$ are eigenvectors of $H$ with eigenvalues
$e_1=\frac{B}{2}$, $e_2=A-\frac{B}{2},e_3=\frac{1}{2}(A+1+B)$ and
$e_4=\frac{1}{2}(A+1-B)$, respectively. What remains
are two copies of the two-dimensional matrix (due to the symmetry $S$)
 \beax N(B)&=&\mfr{1}{2}\left(\begin{array}{cc}A+1+B&-(\sqrt{2}\D)^{-1}
             \\-(\sqrt{2}\D)^{-1}&A+1-B\end{array}\right).
 \eeax
The matrix $N$ is equal to $H^{++}(B)$ reduced to the span$\{v_5,v_6\}$,
as well as
to span$\{v_7,v_8\}$. The eigenvalues are equal to
 \bea
 e_5 = e_7&=& \mfr{1}{2}\left(A+1 - \sqrt{\mfr{1}{2}\D^{-2} + B^2}\right),
 \\
 e_6 = e_8&=& \mfr{1}{2}\left(A+1 + \sqrt{\mfr{1}{2}\D^{-2} + B^2}\right).
 \eea
Notice that for $B<\bar{B}=\frac{3A^2-4A+1}{4(1-A)}$,
 \be
 e_1 < e_5 = e_7 < e_2.
 \ee
This says that the lowest energies in the total $S^3$-sectors are
ordered (though not strictly) by their energy. The gap for $B\le
A$ is equal to
 \bea \label{dropletgap2}
 \lefteqn{g^{+}_3(B,\D)}
 \\
 && = \left\{\begin{array}{lcl}\mfr{1}{2}
 \left(1+A-\sqrt{\mfr{1}{2}\D^{-2} + B^2} -
 B \right)&\mbox{ for }&B\le \bar{B}\\
             A-B&\mbox{ for }& \bar{B}\le B\le A
             \end{array}\right..
 \nonumber
 \eea

\section{Excitation ${\cal E}_-(B)$}

Here we calculate the lowest eigenvalue of $\tilde{H}(B)=H_0^{++}
+ B (S_0^3-\frac{1}{2})$ in the sector with one overturned spin.
Since we want to avoid complications from finite chain boundary
effects, we prefer to treat the infinite volume case with the
magnetic field at 0, say. We first take $B\ge0$.

Let $\psi=\sum_x a_x S_x^-|\!\!\uparrow\rangle$. Then, for $\psi$
being an eigenvector of $\tilde{H}(B)$ with energy $\tilde{\cal
E}$, we have $\langle\uparrow\!| S_x^+ \tilde{H}(B)|\psi\rangle
=\tilde{{\cal E}} a_x$, and thus we get the equations
 \bea
 a_{x+1}&=&2\D (1-{\cal E}) a_x -a_{x-1}, \quad\mbox{for}\quad
 |x|>1
 \label{first}\\
 a_1&=&2\D(1-\tilde{{\cal E}} + B)a_0 -a_{-1} .
 \label{second}
 \eea
It turns out that in addition to the pure absolutely continuous
spectrum of the discrete Laplacian (in the units here, it is the
interval $[1-\D^{-1},1+\D^{-1}]$) there are two (a highest and a
lowest) eigenvalue generated by the perturbation $B S_0^3$. Let
 $$
 r_\pm=\D(1-\tilde{{\cal E}}) \pm\sqrt{\D^2(1-\tilde{{\cal E}})^2-1}
 $$
be the solutions to the characteristic polynomial. Then, all
solutions of (\ref{first}) are of the form $a_x = \a_1 r^x + \a_2
r^{-x}$ for $|x|>1$. Notice that $r_+=r_-^{-1}$. We now look for
the solution $a_x=r^{|x|}$ with $r=r_-$ which produces an
eigenvector. With this choice, we have $|r_-|<1$, we insert this
into (\ref{second}). Then we get
 $$
 \D B=\sqrt{\D^2(1-\tilde{{\cal E}})^2-1},
 $$
from which we conclude $\tilde{{\cal E}}_\pm(B) =
1\pm\sqrt{B^2+\D^{-2}}$. From the gap at $B=0$ we know \cite{KN1}
that $\tilde{{\cal E}}(0) = 1-\D^{-1}$. Thus, the correct solution
is $\tilde{{\cal E}}_-(B)$ which, for the original Hamiltonian of
interest, namely $H_0^{++} + B S_0^3$, has to be shifted back by
$B/2$.

Similarly, if $B\le 0$, then we study $\tilde{H}(B) = H_0^{++} + B
(S_0^3 + \frac{1}{2})$ which amounts to replacing $B$ by $-B$ in
(\ref{first}-\ref{second}).

The lowest energy state of $H_0^{++} + B S_0^3$ in the sector with one
overturned spin is thus
 \be \label{excit}
 {\cal E}_-(B) = 1-\sqrt{B^2+\D^{-2}} +\frac{1}{2}|B|.
 \ee
\end{appendix}

\begin{acknowledgments}
This material is based upon work supported by the National Science
Foundation under Grant \# DMS-0070774. B.N. would like to thank
the Dipartimento de Matematica of the Universit\`a de Bologna,
where this work was initiated, for warm hospitality. We are
indepted to Daniel Ueltschi for a very thorough reading of the
manuscript and the figure in the introduction. W.S. wants to thank
Shannon Starr for providing the matlab programs \cite{St2} with
which many of the propositions in this article where tested, and
dedicates this paper to the memory of his uncle Ernst.
\end{acknowledgments}

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  \bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Tatara}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{60}},
  \bibinfo{pages}{15970} (\bibinfo{year}{1999}).

\bibitem[{\citenamefont{Koma and Yamanaka}()}]{KY1}
\bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Koma}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Yamanaka}},
  \bibinfo{note}{preprint}.

\bibitem[{\citenamefont{Koma and Yamanaka}(1999)}]{KY2}
\bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Koma}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Yamanaka}},
  \bibinfo{journal}{J. Magn. Soc. Japan} \textbf{\bibinfo{volume}{23}},
  \bibinfo{pages}{141} (\bibinfo{year}{1999}).

\bibitem[{\citenamefont{Koma and Yamanaka}(2000)}]{KY3}
\bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Koma}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{M.}~\bibnamefont{Yamanaka}}
  (\bibinfo{year}{2000}), \eprint{cond-mat/0007099}.

\bibitem[{\citenamefont{Mibu et~al.}(1998)\citenamefont{Mibu, Nagahama, Shinjo,
  and Ono}}]{MNSO}
\bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Mibu}},
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Nagahama}},
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Shinjo}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Ono}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{58}},
  \bibinfo{pages}{6442} (\bibinfo{year}{1998}).

\bibitem[{\citenamefont{Nakanishi and Y.O.~Nakamura}(2000)}]{NN}
\bibinfo{author}{\bibfnamefont{K.}~\bibnamefont{Nakanishi}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{Y.}~\bibnamefont{Y.O.~Nakamura}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{61}},
  \bibinfo{pages}{11279} (\bibinfo{year}{2000}).

\bibitem[{\citenamefont{Tantara and Fukuyama}(1997)}]{TF}
\bibinfo{author}{\bibfnamefont{G.}~\bibnamefont{Tantara}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{H.}~\bibnamefont{Fukuyama}},
  \bibinfo{journal}{Phys. Rev. Lett.} \textbf{\bibinfo{volume}{78}},
  \bibinfo{pages}{3773} (\bibinfo{year}{1997}).

\bibitem[{\citenamefont{Wegrowe et~al.}(2000)\citenamefont{Wegrowe, Comment,
  Jaccard, J.-Ph., Dempsey, and Nozieres}}]{WCJADN}
\bibinfo{author}{\bibfnamefont{J.-E.} \bibnamefont{Wegrowe}},
  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{Comment}},
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  \bibinfo{author}{\bibfnamefont{A.}~\bibnamefont{J.-Ph.}},
  \bibinfo{author}{\bibfnamefont{N.}~\bibnamefont{Dempsey}}, \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{J.-P.} \bibnamefont{Nozieres}},
  \bibinfo{journal}{Phys. Rev. B} \textbf{\bibinfo{volume}{61}},
  \bibinfo{pages}{12216} (\bibinfo{year}{2000}).

\bibitem[{\citenamefont{Koma and Nachtergaele}(1996)}]{KN2}
\bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Koma}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Nachtergaele}},
  \bibinfo{journal}{Abstracts of the AMS} \textbf{\bibinfo{volume}{17}},
  \bibinfo{pages}{146} (\bibinfo{year}{1996}), \bibinfo{note}{and unpublished
  notes}.

\bibitem[{\citenamefont{Bolina et~al.}(2000)\citenamefont{Bolina, Contucci,
  Nachtergaele, and Starr}}]{BCNS}
\bibinfo{author}{\bibfnamefont{O.}~\bibnamefont{Bolina}},
  \bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Contucci}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Nachtergaele}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Starr}},
  \bibinfo{journal}{Comm. Math. Phys.} \textbf{\bibinfo{volume}{212}},
  \bibinfo{pages}{63} (\bibinfo{year}{2000}), \eprint{math-ph/9908018}.

\bibitem[{\citenamefont{Caputo and Martinelli}(2001)}]{CapMart1}
\bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Caputo}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Martinelli}},
  \bibinfo{journal}{Commun. Math. Phys.} \textbf{\bibinfo{volume}{226}},
  \bibinfo{pages}{323} (\bibinfo{year}{2001}), \eprint{cond-mat/0106523}.

\bibitem[{\citenamefont{Caputo and Martinelli}(2002)}]{CapMart2}
\bibinfo{author}{\bibfnamefont{P.}~\bibnamefont{Caputo}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{F.}~\bibnamefont{Martinelli}}
  (\bibinfo{year}{2002}), \bibinfo{note}{preprint}, \eprint{math.PR/0202025}.

\bibitem[{\citenamefont{Matsui}(1996)}]{Mat}
\bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Matsui}},
  \bibinfo{journal}{Lett. Math. Phys.} \textbf{\bibinfo{volume}{37}},
  \bibinfo{pages}{397} (\bibinfo{year}{1996}).

\bibitem[{\citenamefont{Koma and Nachtergaele}(1998)}]{KN3}
\bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Koma}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Nachtergaele}},
  \bibinfo{journal}{Adv. Theor. Math. Phys.} \textbf{\bibinfo{volume}{2}},
  \bibinfo{pages}{533} (\bibinfo{year}{1998}), \eprint{cond-mat/9709208}.

\bibitem[{\citenamefont{Koma and Nachtergaele}(1997)}]{KN1}
\bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Koma}} \bibnamefont{and}
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Nachtergaele}},
  \bibinfo{journal}{Lett. Math. Phys.} \textbf{\bibinfo{volume}{40}},
  \bibinfo{pages}{1} (\bibinfo{year}{1997}).

\bibitem[{\citenamefont{Nachtergaele}(1996)}]{Na1}
\bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Nachtergaele}},
  \bibinfo{journal}{Comm. Math. Phys.} \textbf{\bibinfo{volume}{175}},
  \bibinfo{pages}{565} (\bibinfo{year}{1996}).

\bibitem[{\citenamefont{Nachtergaele and Starr}(2001)}]{NS}
\bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Nachtergaele}}
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Starr}},
  \bibinfo{journal}{Comm. Math. Phys.} \textbf{\bibinfo{volume}{218}},
  \bibinfo{pages}{569} (\bibinfo{year}{2001}), \eprint{math-ph/0009002}.

\bibitem[{\citenamefont{Starr}(2001)}]{St2}
\bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Starr}}, Ph.D. thesis,
  \bibinfo{school}{U. C. Davis}, \bibinfo{address}{Davis, CA 95616}
  (\bibinfo{year}{2001}), \eprint{math-ph/0106024}.

\bibitem[{\citenamefont{Nachtergaele}()}]{Na2}
\bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Nachtergaele}},
  \bibinfo{note}{preprint}, \eprint{cond-mat/9501098}.

\bibitem[{\citenamefont{Koma et~al.}()\citenamefont{Koma, Nachtergaele, and
  Starr}}]{KNS}
\bibinfo{author}{\bibfnamefont{T.}~\bibnamefont{Koma}},
  \bibinfo{author}{\bibfnamefont{B.}~\bibnamefont{Nachtergaele}},
  \bibnamefont{and} \bibinfo{author}{\bibfnamefont{S.}~\bibnamefont{Starr}},
  \bibinfo{note}{submitted}, \eprint{math-ph/0110017}.

\end{thebibliography}

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/PP {closepath eofill} bdef
/DP {closepath stroke} bdef
/MR {4 -2 roll moveto dup  0 exch rlineto exch 0 rlineto
  neg 0 exch rlineto closepath} bdef
/FR {MR stroke} bdef
/PR {MR fill} bdef
/L1i {{currentfile picstr readhexstring pop} image} bdef
/tMatrix matrix def
/MakeOval {newpath tMatrix currentmatrix pop translate scale
0 0 1 0 360 arc tMatrix setmatrix} bdef
/FO {MakeOval stroke} bdef
/PO {MakeOval fill} bdef
/PD {currentlinecap 1 cap 3 1 roll 2 copy mt lineto stroke
   cap PDlw -1 eq not {PDlw w /PDlw -1 def} if} def
/FA {newpath tMatrix currentmatrix pop translate scale
  0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef
/PA {newpath tMatrix currentmatrix pop	translate 0 0 moveto scale
  0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef
/FAn {newpath tMatrix currentmatrix pop translate scale
  0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef
/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale
  0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef
/vradius 0 def /hradius 0 def /lry 0 def
/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def
/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef
  /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly
  vradius add translate hradius vradius scale 0 0 1 180 270 arc 
  tMatrix setmatrix lrx hradius sub uly vradius add translate
  hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix
  lrx hradius sub lry vradius sub translate hradius vradius scale
  0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub
  translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix
  closepath} bdef
/FRR {MRR stroke } bdef
/PRR {MRR fill } bdef
/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def
  newpath tMatrix currentmatrix pop ulx rad add uly rad add translate
  rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad
  sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix
  closepath} bdef
/FlrRR {MlrRR stroke } bdef
/PlrRR {MlrRR fill } bdef
/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def
  newpath tMatrix currentmatrix pop ulx rad add uly rad add translate
  rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad
  sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix
  closepath} bdef
/FtbRR {MtbRR stroke } bdef
/PtbRR {MtbRR fill } bdef
/stri 6 array def /dtri 6 array def
/smat 6 array def /dmat 6 array def
/tmat1 6 array def /tmat2 6 array def /dif 3 array def
/asub {/ind2 exch def /ind1 exch def dup dup
  ind1 get exch ind2 get sub exch } bdef
/tri_to_matrix {
  2 0 asub 3 1 asub 4 0 asub 5 1 asub
  dup 0 get exch 1 get 7 -1 roll astore } bdef
/compute_transform {
  dmat dtri tri_to_matrix tmat1 invertmatrix 
  smat stri tri_to_matrix tmat2 concatmatrix } bdef
/ds {stri astore pop} bdef
/dt {dtri astore pop} bdef
/db {2 copy /cols xdef /rows xdef mul dup 3 mul string
  currentfile exch readhexstring pop
  dup 0 3 index getinterval /rbmap xdef
  dup 2 index dup getinterval /gbmap xdef
  1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef
/it {gs np dtri aload pop moveto lineto lineto cp c
  cols rows 8 compute_transform 
  rbmap gbmap bbmap true 3 colorimage gr}bdef
/il {newpath moveto lineto stroke}bdef
currentdict end def
%%EndProlog

%%BeginSetup
MathWorks begin

0 cap

end
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%%Page: 1 1
%%BeginPageSetup
%%PageBoundingBox:    55   200   549   610
MathWorks begin
bpage
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%%BeginObject: obj1
bplot

/dpi2point 12 def
portraitMode 0216 7344 csm

  447    21  5925  4912 MR c np
92 dict begin %Colortable dictionary
/c0 { 0 0 0 sr} bdef
/c1 { 1 1 1 sr} bdef
/c2 { 1 0 0 sr} bdef
/c3 { 0 1 0 sr} bdef
/c4 { 0 0 1 sr} bdef
/c5 { 1 1 0 sr} bdef
/c6 { 1 0 1 sr} bdef
/c7 { 0 1 1 sr} bdef
c0
1 j
1 sg
   0    0 6914 5188 PR
6 w
0 4227 5357 0 0 -4227 899 4616 4 MP
PP
-5357 0 0 4227 5357 0 0 -4227 899 4616 5 MP stroke
4 w
DO
SO
6 w
0 sg
 899 4616 mt 6256 4616 L
 899 4616 mt  899  389 L
1083 4616 mt 1083 4562 L
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 120 FMSR

1050 4761 mt 
(0) s
1453 4616 mt 1453 4562 L
1420 4761 mt 
(1) s
1822 4616 mt 1822 4562 L
1789 4761 mt 
(2) s
2192 4616 mt 2192 4562 L
2159 4761 mt 
(3) s
2561 4616 mt 2561 4562 L
2528 4761 mt 
(4) s
2930 4616 mt 2930 4562 L
2897 4761 mt 
(5) s
3300 4616 mt 3300 4562 L
3267 4761 mt 
(6) s
3669 4616 mt 3669 4562 L
3636 4761 mt 
(7) s
4039 4616 mt 4039 4562 L
4006 4761 mt 
(8) s
4408 4616 mt 4408 4562 L
4375 4761 mt 
(9) s
4778 4616 mt 4778 4562 L
4712 4761 mt 
(10) s
5147 4616 mt 5147 4562 L
5081 4761 mt 
(11) s
5517 4616 mt 5517 4562 L
5451 4761 mt 
(12) s
5886 4616 mt 5886 4562 L
5820 4761 mt 
(13) s
6256 4616 mt 6256 4562 L
6190 4761 mt 
(14) s
 899 4263 mt  952 4263 L
 628 4307 mt 
(-0.5) s
 899 3559 mt  952 3559 L
 798 3603 mt 
(0) s
 899 2854 mt  952 2854 L
 698 2898 mt 
(0.5) s
 899 2150 mt  952 2150 L
 798 2194 mt 
(1) s
 899 1445 mt  952 1445 L
 698 1489 mt 
(1.5) s
 899  741 mt  952  741 L
 798  785 mt 
(2) s
gs 899 389 5358 4228 MR c np
185 0 1360 3813 2 MP stroke
185 0 1360 1619 2 MP stroke
185 0 1360 1545 2 MP stroke
185 0 1360 707 2 MP stroke
185 0 1360 460 2 MP stroke
184 0 1730 4050 2 MP stroke
184 0 1730 3420 2 MP stroke
184 0 1730 2431 2 MP stroke
184 0 1730 2369 2 MP stroke
184 0 1730 1251 2 MP stroke
184 0 1730 1179 2 MP stroke
184 0 1730 758 2 MP stroke
184 0 1730 748 2 MP stroke
185 0 2099 4164 2 MP stroke
185 0 2099 3775 2 MP stroke
185 0 2099 3629 2 MP stroke
185 0 2099 2583 2 MP stroke
185 0 2099 2553 2 MP stroke
185 0 2099 2171 2 MP stroke
185 0 2099 2124 2 MP stroke
185 0 2099 1444 2 MP stroke
185 0 2099 1439 2 MP stroke
185 0 2099 1329 2 MP stroke
185 0 2099 1265 2 MP stroke
185 0 2099 1004 2 MP stroke
185 0 2099 911 2 MP stroke
185 0 2099 860 2 MP stroke
185 0 2099 556 2 MP stroke
185 0 2099 554 2 MP stroke
184 0 2469 4214 2 MP stroke
184 0 2469 4075 2 MP stroke
184 0 2469 3757 2 MP stroke
184 0 2469 3748 2 MP stroke
184 0 2469 2665 2 MP stroke
184 0 2469 2630 2 MP stroke
184 0 2469 2442 2 MP stroke
184 0 2469 2398 2 MP stroke
184 0 2469 2314 2 MP stroke
184 0 2469 2202 2 MP stroke
184 0 2469 1539 2 MP stroke
184 0 2469 1538 2 MP stroke
184 0 2469 1356 2 MP stroke
184 0 2469 1354 2 MP stroke
184 0 2469 1330 2 MP stroke
184 0 2469 1282 2 MP stroke
184 0 2469 1228 2 MP stroke
184 0 2469 1116 2 MP stroke
184 0 2469 1115 2 MP stroke
184 0 2469 984 2 MP stroke
184 0 2469 894 2 MP stroke
184 0 2469 778 2 MP stroke
184 0 2469 774 2 MP stroke
184 0 2469 767 2 MP stroke
184 0 2469 609 2 MP stroke
184 0 2469 603 2 MP stroke
185 0 2838 4239 2 MP stroke
185 0 2838 4152 2 MP stroke
185 0 2838 4121 2 MP stroke
185 0 2838 3768 2 MP stroke
185 0 2838 3768 2 MP stroke
185 0 2838 2726 2 MP stroke
185 0 2838 2685 2 MP stroke
185 0 2838 2600 2 MP stroke
185 0 2838 2579 2 MP stroke
185 0 2838 2426 2 MP stroke
185 0 2838 2425 2 MP stroke
185 0 2838 2359 2 MP stroke
185 0 2838 2294 2 MP stroke
185 0 2838 1579 2 MP stroke
185 0 2838 1578 2 MP stroke
185 0 2838 1509 2 MP stroke
185 0 2838 1509 2 MP stroke
185 0 2838 1406 2 MP stroke
185 0 2838 1396 2 MP stroke
185 0 2838 1395 2 MP stroke
185 0 2838 1315 2 MP stroke
185 0 2838 1266 2 MP stroke
185 0 2838 1246 2 MP stroke
185 0 2838 1184 2 MP stroke
185 0 2838 1175 2 MP stroke
185 0 2838 1135 2 MP stroke
185 0 2838 983 2 MP stroke
185 0 2838 962 2 MP stroke
185 0 2838 937 2 MP stroke
185 0 2838 838 2 MP stroke
185 0 2838 790 2 MP stroke
185 0 2838 790 2 MP stroke
185 0 2838 605 2 MP stroke
185 0 2838 605 2 MP stroke
185 0 2838 449 2 MP stroke
185 0 2838 445 2 MP stroke
184 0 3208 4251 2 MP stroke
184 0 3208 4217 2 MP stroke
184 0 3208 4150 2 MP stroke
184 0 3208 4148 2 MP stroke
184 0 3208 3772 2 MP stroke
184 0 3208 3772 2 MP stroke
184 0 3208 2766 2 MP stroke
184 0 3208 2729 2 MP stroke
184 0 3208 2701 2 MP stroke
184 0 3208 2622 2 MP stroke
184 0 3208 2620 2 MP stroke
184 0 3208 2585 2 MP stroke
184 0 3208 2437 2 MP stroke
184 0 3208 2437 2 MP stroke
184 0 3208 2342 2 MP stroke
184 0 3208 2337 2 MP stroke
184 0 3208 1631 2 MP stroke
184 0 3208 1629 2 MP stroke
184 0 3208 1555 2 MP stroke
184 0 3208 1555 2 MP stroke
184 0 3208 1471 2 MP stroke
184 0 3208 1453 2 MP stroke
184 0 3208 1447 2 MP stroke
184 0 3208 1374 2 MP stroke
184 0 3208 1358 2 MP stroke
184 0 3208 1271 2 MP stroke
184 0 3208 1271 2 MP stroke
184 0 3208 1199 2 MP stroke
184 0 3208 1199 2 MP stroke
184 0 3208 1150 2 MP stroke
184 0 3208 1112 2 MP stroke
184 0 3208 1088 2 MP stroke
184 0 3208 993 2 MP stroke
184 0 3208 991 2 MP stroke
184 0 3208 929 2 MP stroke
184 0 3208 911 2 MP stroke
184 0 3208 767 2 MP stroke
184 0 3208 767 2 MP stroke
184 0 3208 612 2 MP stroke
184 0 3208 612 2 MP stroke
184 0 3208 568 2 MP stroke
184 0 3208 567 2 MP stroke
184 0 3208 469 2 MP stroke
184 0 3208 456 2 MP stroke
185 0 3577 4257 2 MP stroke
185 0 3577 4236 2 MP stroke
185 0 3577 4229 2 MP stroke
185 0 3577 4153 2 MP stroke
185 0 3577 4153 2 MP stroke
185 0 3577 3772 2 MP stroke
185 0 3577 3772 2 MP stroke
185 0 3577 2791 2 MP stroke
185 0 3577 2768 2 MP stroke
185 0 3577 2732 2 MP stroke
185 0 3577 2730 2 MP stroke
185 0 3577 2628 2 MP stroke
185 0 3577 2628 2 MP stroke
185 0 3577 2625 2 MP stroke
185 0 3577 2600 2 MP stroke
185 0 3577 2344 2 MP stroke
185 0 3577 2343 2 MP stroke
185 0 3577 1657 2 MP stroke
185 0 3577 1656 2 MP stroke
185 0 3577 1633 2 MP stroke
185 0 3577 1632 2 MP stroke
185 0 3577 1508 2 MP stroke
185 0 3577 1482 2 MP stroke
185 0 3577 1466 2 MP stroke
185 0 3577 1418 2 MP stroke
185 0 3577 1381 2 MP stroke
185 0 3577 1380 2 MP stroke
185 0 3577 1271 2 MP stroke
185 0 3577 1271 2 MP stroke
185 0 3577 1176 2 MP stroke
185 0 3577 1170 2 MP stroke
185 0 3577 1130 2 MP stroke
185 0 3577 1123 2 MP stroke
185 0 3577 1082 2 MP stroke
185 0 3577 998 2 MP stroke
185 0 3577 998 2 MP stroke
185 0 3577 933 2 MP stroke
185 0 3577 932 2 MP stroke
185 0 3577 795 2 MP stroke
185 0 3577 795 2 MP stroke
185 0 3577 629 2 MP stroke
185 0 3577 628 2 MP stroke
185 0 3577 596 2 MP stroke
185 0 3577 594 2 MP stroke
185 0 3577 484 2 MP stroke
185 0 3577 479 2 MP stroke
185 0 3946 4260 2 MP stroke
185 0 3946 4252 2 MP stroke
185 0 3946 4236 2 MP stroke
185 0 3946 4235 2 MP stroke
185 0 3946 4153 2 MP stroke
185 0 3946 4153 2 MP stroke
185 0 3946 2806 2 MP stroke
185 0 3946 2786 2 MP stroke
185 0 3946 2783 2 MP stroke
185 0 3946 2736 2 MP stroke
185 0 3946 2736 2 MP stroke
185 0 3946 2653 2 MP stroke
185 0 3946 2618 2 MP stroke
185 0 3946 2616 2 MP stroke
185 0 3946 2345 2 MP stroke
185 0 3946 2345 2 MP stroke
185 0 3946 1675 2 MP stroke
185 0 3946 1674 2 MP stroke
185 0 3946 1655 2 MP stroke
185 0 3946 1648 2 MP stroke
185 0 3946 1499 2 MP stroke
185 0 3946 1486 2 MP stroke
185 0 3946 1484 2 MP stroke
185 0 3946 1463 2 MP stroke
185 0 3946 1455 2 MP stroke
185 0 3946 1384 2 MP stroke
185 0 3946 1384 2 MP stroke
185 0 3946 1179 2 MP stroke
185 0 3946 1175 2 MP stroke
185 0 3946 1169 2 MP stroke
185 0 3946 1163 2 MP stroke
185 0 3946 1140 2 MP stroke
185 0 3946 1113 2 MP stroke
185 0 3946 1105 2 MP stroke
185 0 3946 935 2 MP stroke
185 0 3946 935 2 MP stroke
185 0 3946 802 2 MP stroke
185 0 3946 802 2 MP stroke
185 0 3946 647 2 MP stroke
185 0 3946 647 2 MP stroke
185 0 3946 618 2 MP stroke
185 0 3946 615 2 MP stroke
185 0 3946 498 2 MP stroke
185 0 3946 476 2 MP stroke
185 0 4316 4262 2 MP stroke
185 0 4316 4257 2 MP stroke
185 0 4316 4255 2 MP stroke
185 0 4316 4236 2 MP stroke
185 0 4316 4236 2 MP stroke
185 0 4316 2814 2 MP stroke
185 0 4316 2808 2 MP stroke
185 0 4316 2788 2 MP stroke
185 0 4316 2788 2 MP stroke
185 0 4316 2662 2 MP stroke
185 0 4316 2656 2 MP stroke
185 0 4316 2619 2 MP stroke
185 0 4316 2619 2 MP stroke
185 0 4316 1683 2 MP stroke
185 0 4316 1682 2 MP stroke
185 0 4316 1671 2 MP stroke
185 0 4316 1630 2 MP stroke
185 0 4316 1506 2 MP stroke
185 0 4316 1499 2 MP stroke
185 0 4316 1487 2 MP stroke
185 0 4316 1487 2 MP stroke
185 0 4316 1463 2 MP stroke
185 0 4316 1463 2 MP stroke
185 0 4316 1215 2 MP stroke
185 0 4316 1172 2 MP stroke
185 0 4316 1172 2 MP stroke
185 0 4316 1149 2 MP stroke
185 0 4316 1140 2 MP stroke
185 0 4316 1113 2 MP stroke
185 0 4316 1113 2 MP stroke
185 0 4316 807 2 MP stroke
185 0 4316 801 2 MP stroke
185 0 4316 655 2 MP stroke
185 0 4316 655 2 MP stroke
185 0 4316 624 2 MP stroke
185 0 4316 616 2 MP stroke
185 0 4316 488 2 MP stroke
185 0 4316 488 2 MP stroke
185 0 4685 4263 2 MP stroke
185 0 4685 4261 2 MP stroke
185 0 4685 4257 2 MP stroke
185 0 4685 4256 2 MP stroke
185 0 4685 2818 2 MP stroke
185 0 4685 2812 2 MP stroke
185 0 4685 2812 2 MP stroke
185 0 4685 2668 2 MP stroke
185 0 4685 2661 2 MP stroke
185 0 4685 2660 2 MP stroke
185 0 4685 1693 2 MP stroke
185 0 4685 1690 2 MP stroke
185 0 4685 1642 2 MP stroke
185 0 4685 1640 2 MP stroke
185 0 4685 1511 2 MP stroke
185 0 4685 1504 2 MP stroke
185 0 4685 1504 2 MP stroke
185 0 4685 1219 2 MP stroke
185 0 4685 1217 2 MP stroke
185 0 4685 1153 2 MP stroke
185 0 4685 1146 2 MP stroke
185 0 4685 1145 2 MP stroke
185 0 4685 808 2 MP stroke
185 0 4685 802 2 MP stroke
185 0 4685 658 2 MP stroke
185 0 4685 658 2 MP stroke
185 0 4685 623 2 MP stroke
185 0 4685 618 2 MP stroke
185 0 5055 4263 2 MP stroke
185 0 5055 4262 2 MP stroke
185 0 5055 4261 2 MP stroke
185 0 5055 2820 2 MP stroke
185 0 5055 2819 2 MP stroke
185 0 5055 2670 2 MP stroke
185 0 5055 2669 2 MP stroke
185 0 5055 1701 2 MP stroke
185 0 5055 1662 2 MP stroke
185 0 5055 1512 2 MP stroke
185 0 5055 1511 2 MP stroke
185 0 5055 1222 2 MP stroke
185 0 5055 1155 2 MP stroke
185 0 5055 1153 2 MP stroke
185 0 5055 665 2 MP stroke
185 0 5055 651 2 MP stroke
185 0 5424 4263 2 MP stroke
185 0 5424 4263 2 MP stroke
185 0 5424 2821 2 MP stroke
185 0 5424 2672 2 MP stroke
185 0 5424 1513 2 MP stroke
185 0 5424 1156 2 MP stroke
185 0 1360 3813 2 MP stroke
184 0 1730 4050 2 MP stroke
184 0 1730 3420 2 MP stroke
185 0 2099 4164 2 MP stroke
185 0 2099 3775 2 MP stroke
185 0 2099 3629 2 MP stroke
184 0 2469 4214 2 MP stroke
184 0 2469 4075 2 MP stroke
184 0 2469 3757 2 MP stroke
184 0 2469 3748 2 MP stroke
185 0 2838 4239 2 MP stroke
185 0 2838 4152 2 MP stroke
185 0 2838 4121 2 MP stroke
185 0 2838 3768 2 MP stroke
185 0 2838 3768 2 MP stroke
184 0 3208 4251 2 MP stroke
184 0 3208 4217 2 MP stroke
184 0 3208 4150 2 MP stroke
184 0 3208 4148 2 MP stroke
184 0 3208 3772 2 MP stroke
184 0 3208 3772 2 MP stroke
185 0 3577 4257 2 MP stroke
185 0 3577 4236 2 MP stroke
185 0 3577 4229 2 MP stroke
185 0 3577 4153 2 MP stroke
185 0 3577 4153 2 MP stroke
185 0 3577 3772 2 MP stroke
185 0 3577 3772 2 MP stroke
185 0 3946 4260 2 MP stroke
185 0 3946 4252 2 MP stroke
185 0 3946 4236 2 MP stroke
185 0 3946 4235 2 MP stroke
185 0 3946 4153 2 MP stroke
185 0 3946 4153 2 MP stroke
185 0 4316 4262 2 MP stroke
185 0 4316 4257 2 MP stroke
185 0 4316 4255 2 MP stroke
185 0 4316 4236 2 MP stroke
185 0 4316 4236 2 MP stroke
185 0 4685 4263 2 MP stroke
185 0 4685 4261 2 MP stroke
185 0 4685 4257 2 MP stroke
185 0 4685 4256 2 MP stroke
185 0 5055 4263 2 MP stroke
185 0 5055 4262 2 MP stroke
185 0 5055 4261 2 MP stroke
185 0 5424 4263 2 MP stroke
185 0 5424 4263 2 MP stroke
/c8 { 0.000000 0.000000 1.000000 sr} bdef
c8
184 0 5794 4263 2 MP stroke
0 sg
185 0 991 1445 2 MP stroke
gr

 573 2741 mt  -90 rotate
(2E / A\() s
90 rotate
%%IncludeResource: font Symbol
/Symbol /ISOLatin1Encoding 120 FMSR

 573 2375 mt  -90 rotate
(D) s
90 rotate
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 120 FMSR

 573 2302 mt  -90 rotate
(\)) s
90 rotate
3543 4904 mt 
(n) s
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 132 FMSR

2253  224 mt 
(Amplified low energy spectrum of H) s
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 96 FMSR

4328  158 mt 
(++) s
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 132 FMSR

4440  224 mt 
(\(0,0,B\)) s

end

eplot
%%EndObject

epage
end

showpage

%%Trailer
%%EOF

---------------0204052034356
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%%BeginProlog
% MathWorks dictionary
/MathWorks 160 dict begin
% definition operators
/bdef {bind def} bind def
/ldef {load def} bind def
/xdef {exch def} bdef
/xstore {exch store} bdef
% operator abbreviations
/c  /clip ldef
/cc /concat ldef
/cp /closepath ldef
/gr /grestore ldef
/gs /gsave ldef
/mt /moveto ldef
/np /newpath ldef
/cm /currentmatrix ldef
/sm /setmatrix ldef
/rm /rmoveto ldef
/rl /rlineto ldef
/s /show ldef
/sc {setcmykcolor} bdef
/sr /setrgbcolor ldef
/sg /setgray ldef
/w /setlinewidth ldef
/j /setlinejoin ldef
/cap /setlinecap ldef
/rc {rectclip} bdef
/rf {rectfill} bdef
% page state control
/pgsv () def
/bpage {/pgsv save def} bdef
/epage {pgsv restore} bdef
/bplot /gsave ldef
/eplot {stroke grestore} bdef
% orientation switch
/portraitMode 0 def /landscapeMode 1 def /rotateMode 2 def
% coordinate system mappings
/dpi2point 0 def
% font control
/FontSize 0 def
/FMS {/FontSize xstore findfont [FontSize 0 0 FontSize neg 0 0]
  makefont setfont} bdef
/reencode {exch dup where {pop load} {pop StandardEncoding} ifelse
  exch dup 3 1 roll findfont dup length dict begin
  { 1 index /FID ne {def}{pop pop} ifelse } forall
  /Encoding exch def currentdict end definefont pop} bdef
/isroman {findfont /CharStrings get /Agrave known} bdef
/FMSR {3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse
  exch FMS} bdef
/csm {1 dpi2point div -1 dpi2point div scale neg translate
 dup landscapeMode eq {pop -90 rotate}
  {rotateMode eq {90 rotate} if} ifelse} bdef
% line types: solid, dotted, dashed, dotdash
/SO { [] 0 setdash } bdef
/DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef
/DA { [6 dpi2point mul] 0 setdash } bdef
/DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4
  dpi2point mul] 0 setdash } bdef
% macros for lines and objects
/L {lineto stroke} bdef
/MP {3 1 roll moveto 1 sub {rlineto} repeat} bdef
/AP {{rlineto} repeat} bdef
/PDlw -1 def
/W {/PDlw currentlinewidth def setlinewidth} def
/PP {closepath eofill} bdef
/DP {closepath stroke} bdef
/MR {4 -2 roll moveto dup  0 exch rlineto exch 0 rlineto
  neg 0 exch rlineto closepath} bdef
/FR {MR stroke} bdef
/PR {MR fill} bdef
/L1i {{currentfile picstr readhexstring pop} image} bdef
/tMatrix matrix def
/MakeOval {newpath tMatrix currentmatrix pop translate scale
0 0 1 0 360 arc tMatrix setmatrix} bdef
/FO {MakeOval stroke} bdef
/PO {MakeOval fill} bdef
/PD {currentlinecap 1 cap 3 1 roll 2 copy mt lineto stroke
   cap PDlw -1 eq not {PDlw w /PDlw -1 def} if} def
/FA {newpath tMatrix currentmatrix pop translate scale
  0 0 1 5 -2 roll arc tMatrix setmatrix stroke} bdef
/PA {newpath tMatrix currentmatrix pop	translate 0 0 moveto scale
  0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill} bdef
/FAn {newpath tMatrix currentmatrix pop translate scale
  0 0 1 5 -2 roll arcn tMatrix setmatrix stroke} bdef
/PAn {newpath tMatrix currentmatrix pop translate 0 0 moveto scale
  0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill} bdef
/vradius 0 def /hradius 0 def /lry 0 def
/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def
/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef
  /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly
  vradius add translate hradius vradius scale 0 0 1 180 270 arc 
  tMatrix setmatrix lrx hradius sub uly vradius add translate
  hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix
  lrx hradius sub lry vradius sub translate hradius vradius scale
  0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub
  translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix
  closepath} bdef
/FRR {MRR stroke } bdef
/PRR {MRR fill } bdef
/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def
  newpath tMatrix currentmatrix pop ulx rad add uly rad add translate
  rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad
  sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix
  closepath} bdef
/FlrRR {MlrRR stroke } bdef
/PlrRR {MlrRR fill } bdef
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 899 4616 mt  899  389 L
6256 4616 mt 6256  389 L
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3655 4761 mt 
(4 ) s
4725 4616 mt 4725 4562 L
4725  389 mt 4725  442 L
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(6 ) s
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6000  389 mt 6000  442 L
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(8.5 ) s
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(-5) s
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 899 3207 mt  952 3207 L
6256 3207 mt 6202 3207 L
 728 3251 mt 
(-3) s
 899 2502 mt  952 2502 L
6256 2502 mt 6202 2502 L
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(0) s
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 899 4616 mt  899  389 L
6256 4616 mt 6256  389 L
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51 36 51 35 51 35 51 35 51 36 51 35 51 35 51 35 
51 35 51 36 51 35 51 35 51 35 51 35 51 36 51 35 
51 35 51 35 51 36 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 35 51 36 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 35 51 36 51 35 51 35 
51 35 51 35 51 36 51 35 51 35 51 35 51 35 52 36 
51 35 51 35 51 35 51 35 51 36 51 35 51 35 51 35 
51 35 51 35 51 36 51 35 51 35 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 34 51 35 51 34 51 34 
51 34 51 32 51 30 51 25 51 10 51 -10 51 -25 51 -30 
51 -32 51 -34 51 -34 51 -34 51 -35 51 -34 51 -35 51 -35 
51 -35 51 -35 51 -35 950 1589 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 34 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 34 51 35 51 35 
51 35 51 35 51 34 51 35 51 35 51 34 51 35 51 35 
51 34 51 35 51 35 51 34 51 35 51 34 51 35 51 34 
51 35 51 34 51 35 51 34 51 34 51 34 51 35 52 34 
51 34 51 34 51 34 51 33 51 34 51 34 51 33 51 34 
51 33 51 34 51 33 51 33 51 32 51 33 51 32 51 32 
51 32 51 32 51 31 51 31 51 30 51 30 51 29 51 29 
51 28 51 27 51 26 51 25 51 23 51 22 51 20 51 18 
51 15 51 12 51 10 51 6 51 2 51 -2 51 -6 51 -10 
51 -12 51 -15 51 -18 51 -20 51 -22 51 -23 51 -25 51 -26 
51 -27 51 -28 51 -29 950 1409 100 MP stroke
51 35 51 36 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 36 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 34 51 35 51 35 51 35 51 35 
51 35 51 34 51 35 51 35 51 35 51 34 51 35 52 35 
51 34 51 35 51 34 51 34 51 33 51 34 51 33 51 33 
51 33 51 33 51 33 51 33 51 32 51 32 51 32 51 32 
51 32 51 31 51 30 51 31 51 29 51 30 51 28 51 28 
51 27 51 26 51 25 51 24 51 22 51 21 51 19 51 16 
51 14 51 11 51 9 51 5 51 1 51 -1 51 -5 51 -9 
51 -11 51 -14 51 -16 51 -19 51 -21 51 -22 51 -24 51 -25 
51 -26 51 -27 51 -28 950 1394 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 34 51 35 51 35 51 35 51 35 51 35 
51 35 51 34 51 35 51 35 51 35 51 34 51 35 51 35 
51 35 51 34 51 35 51 35 51 34 51 35 51 35 51 34 
51 35 51 34 51 35 51 34 51 35 51 34 51 34 51 35 
51 34 51 34 51 35 51 34 51 34 51 34 51 34 52 34 
51 34 51 34 51 34 51 34 51 34 51 33 51 33 51 34 
51 33 51 33 51 32 51 33 51 32 51 33 51 32 51 31 
51 32 51 31 51 30 51 31 51 29 51 30 51 28 51 28 
51 27 51 26 51 25 51 24 51 22 51 20 51 19 51 16 
51 14 51 12 51 8 51 5 51 1 51 -1 51 -5 51 -8 
51 -12 51 -14 51 -16 51 -19 51 -20 51 -22 51 -24 51 -25 
51 -26 51 -27 51 -28 950 1393 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 34 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 34 51 35 
51 35 51 35 51 35 51 34 51 35 51 35 51 35 51 34 
51 35 51 35 51 35 51 34 51 35 51 34 51 35 51 35 
51 34 51 35 51 34 51 35 51 34 51 34 51 35 51 34 
51 34 51 35 51 34 51 34 51 34 51 34 51 34 52 34 
51 34 51 34 51 34 51 33 51 34 51 33 51 34 51 33 
51 33 51 33 51 33 51 32 51 33 51 32 51 32 51 31 
51 32 51 31 51 30 51 31 51 29 51 30 51 28 51 28 
51 27 51 26 51 25 51 24 51 22 51 20 51 19 51 16 
51 14 51 12 51 8 51 5 51 1 51 -1 51 -5 51 -8 
51 -12 51 -14 51 -16 51 -19 51 -20 51 -22 51 -24 51 -25 
51 -26 51 -27 51 -28 950 1393 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 34 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 34 
51 35 51 35 51 35 51 35 51 34 51 35 51 35 51 35 
51 34 51 35 51 35 51 34 51 35 51 35 51 34 51 35 
51 34 51 35 51 34 51 35 51 34 51 35 51 34 51 34 
51 35 51 34 51 34 51 34 51 34 51 34 51 34 52 34 
51 34 51 34 51 34 51 33 51 34 51 34 51 34 51 33 
51 33 51 33 51 33 51 32 51 33 51 32 51 32 51 31 
51 32 51 31 51 31 51 30 51 29 51 30 51 28 51 28 
51 27 51 26 51 25 51 24 51 22 51 20 51 19 51 16 
51 14 51 11 51 8 51 5 51 2 51 -2 51 -5 51 -8 
51 -11 51 -14 51 -16 51 -19 51 -20 51 -22 51 -24 51 -25 
51 -26 51 -27 51 -28 950 1392 100 MP stroke
51 35 51 35 51 34 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 34 51 35 51 35 51 35 
51 35 51 35 51 34 51 35 51 35 51 35 51 34 51 35 
51 35 51 35 51 34 51 35 51 35 51 34 51 35 51 34 
51 35 51 34 51 35 51 34 51 35 51 34 51 35 51 34 
51 34 51 34 51 35 51 34 51 34 51 34 51 34 52 34 
51 34 51 34 51 33 51 34 51 34 51 34 51 34 51 34 
51 34 51 34 51 34 51 34 51 34 51 33 51 34 51 33 
51 33 51 32 51 31 51 31 51 30 51 29 51 28 51 28 
51 26 51 25 51 25 51 22 51 22 51 19 51 18 51 15 
51 12 51 10 51 7 51 4 51 1 51 -1 51 -4 51 -7 
51 -10 51 -12 51 -15 51 -18 51 -19 51 -22 51 -22 51 -25 
51 -25 51 -26 51 -28 950 1375 100 MP stroke
51 35 51 35 51 35 51 35 51 36 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 34 
51 35 51 35 51 35 51 35 51 35 51 35 51 34 51 35 
51 35 51 35 51 34 51 35 51 35 51 34 51 35 52 34 
51 35 51 34 51 35 51 34 51 34 51 34 51 33 51 33 
51 33 51 33 51 33 51 32 51 32 51 32 51 32 51 32 
51 31 51 32 51 32 51 31 51 32 51 32 51 32 51 31 
51 30 51 31 51 29 51 28 51 28 51 26 51 24 51 22 
51 20 51 16 51 13 51 8 51 3 51 -3 51 -8 51 -13 
51 -16 51 -20 51 -22 51 -24 51 -26 51 -28 51 -28 51 -29 
51 -31 51 -30 51 -31 950 1356 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 36 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 34 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 34 51 35 51 35 51 35 
51 34 51 35 51 35 51 34 51 35 51 35 51 34 52 35 
51 34 51 35 51 34 51 35 51 34 51 33 51 34 51 34 
51 34 51 34 51 34 51 34 51 34 51 33 51 34 51 33 
51 33 51 33 51 33 51 32 51 32 51 32 51 31 51 31 
51 31 51 29 51 29 51 28 51 26 51 25 51 23 51 21 
51 19 51 15 51 11 51 7 51 2 51 -2 51 -7 51 -11 
51 -15 51 -19 51 -21 51 -23 51 -25 51 -26 51 -28 51 -29 
51 -29 51 -31 51 -31 950 1338 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 36 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 34 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 34 51 35 51 35 
51 35 51 34 51 35 51 35 51 34 51 35 51 34 52 35 
51 35 51 34 51 34 51 35 51 34 51 34 51 35 51 34 
51 34 51 34 51 34 51 33 51 34 51 34 51 33 51 33 
51 34 51 32 51 33 51 32 51 33 51 31 51 32 51 30 
51 31 51 29 51 29 51 28 51 26 51 25 51 23 51 21 
51 18 51 15 51 11 51 7 51 3 51 -3 51 -7 51 -11 
51 -15 51 -18 51 -21 51 -23 51 -25 51 -26 51 -28 51 -29 
51 -29 51 -31 51 -30 950 1336 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 35 51 35 51 34 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 34 51 35 51 35 51 35 51 35 51 34 
51 35 51 35 51 34 51 35 51 35 51 34 51 35 52 34 
51 35 51 34 51 34 51 34 51 34 51 34 51 35 51 34 
51 34 51 34 51 34 51 34 51 33 51 34 51 33 51 34 
51 33 51 33 51 32 51 33 51 32 51 31 51 32 51 31 
51 30 51 29 51 29 51 28 51 26 51 25 51 23 51 21 
51 18 51 15 51 11 51 7 51 3 51 -3 51 -7 51 -11 
51 -15 51 -18 51 -21 51 -23 51 -25 51 -26 51 -28 51 -29 
51 -29 51 -30 51 -31 950 1336 100 MP stroke
51 34 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 34 51 35 51 35 51 35 51 35 51 34 51 35 
51 35 51 35 51 34 51 35 51 35 51 35 51 34 51 35 
51 34 51 35 51 35 51 34 51 35 51 34 51 35 51 34 
51 35 51 34 51 35 51 34 51 34 51 35 51 34 51 34 
51 34 51 34 51 34 51 34 51 34 51 34 51 34 52 34 
51 34 51 34 51 34 51 34 51 34 51 34 51 34 51 34 
51 34 51 34 51 34 51 34 51 34 51 33 51 34 51 33 
51 33 51 33 51 32 51 33 51 32 51 31 51 32 51 31 
51 30 51 29 51 29 51 27 51 27 51 25 51 23 51 21 
51 18 51 14 51 11 51 7 51 3 51 -3 51 -7 51 -11 
51 -14 51 -18 51 -21 51 -23 51 -25 51 -27 51 -27 51 -29 
51 -29 51 -30 51 -31 950 1335 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 34 51 35 51 35 51 35 51 35 51 35 51 35 51 34 
51 35 51 35 51 35 51 34 51 35 51 35 51 34 51 35 
51 35 51 34 51 35 51 34 51 35 51 34 51 35 52 34 
51 34 51 35 51 34 51 34 51 34 51 34 51 34 51 34 
51 34 51 34 51 33 51 34 51 33 51 33 51 33 51 33 
51 33 51 32 51 32 51 32 51 31 51 31 51 31 51 30 
51 29 51 29 51 27 51 27 51 25 51 23 51 21 51 19 
51 17 51 12 51 10 51 5 51 2 51 -2 51 -5 51 -10 
51 -12 51 -17 51 -19 51 -21 51 -23 51 -25 51 -27 51 -27 
51 -29 51 -29 51 -30 950 1316 100 MP stroke
51 35 51 35 51 36 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 36 51 35 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 35 51 35 51 36 51 35 
51 35 51 35 51 35 51 36 51 35 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 35 51 35 51 36 51 35 
51 35 51 35 51 35 51 35 51 36 51 35 51 35 52 35 
51 35 51 35 51 35 51 36 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 34 51 35 51 35 
51 34 51 34 51 35 51 34 51 34 51 34 51 33 51 29 
51 31 51 29 51 26 51 21 51 7 51 -7 51 -21 51 -26 
51 -29 51 -31 51 -29 51 -33 51 -34 51 -34 51 -34 51 -35 
51 -34 51 -34 51 -35 950 1151 100 MP stroke
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 679 2560 mt  -90 rotate
(E) s
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gs 899 389 5358 4228 MR c np
c8
51 35 51 35 51 36 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 35 51 35 51 36 51 35 51 35 
51 35 51 35 51 36 51 35 51 35 51 35 51 35 51 35 
51 35 51 36 51 35 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
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51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 34 51 35 51 35 51 34 51 34 
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51 23 51 18 51 11 51 7 51 2 51 -2 51 -7 51 -11 
51 -18 51 -23 51 -26 51 -19 51 -20 51 -28 51 -29 51 -30 
51 -31 51 -32 51 -32 950 1043 100 MP stroke
51 36 51 35 51 35 51 35 51 35 51 36 51 35 51 35 
51 35 51 35 51 36 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 35 51 36 51 35 51 35 51 35 
51 35 51 36 51 35 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 35 51 36 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 52 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 34 51 35 
51 35 51 35 51 34 51 35 51 34 51 35 51 35 51 34 
51 35 51 35 51 34 51 34 51 34 51 34 51 34 51 33 
51 31 51 30 51 30 51 28 51 27 51 22 51 24 51 22 
51 19 51 15 51 11 51 7 51 2 51 -2 51 -7 51 -11 
51 -15 51 -19 51 -22 51 -24 51 -22 51 -27 51 -28 51 -30 
51 -30 51 -31 51 -33 950 1036 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 36 51 35 51 35 
51 35 51 35 51 35 51 36 51 35 51 35 51 35 51 35 
51 35 51 36 51 35 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 36 51 35 51 35 51 35 51 35 52 35 
51 35 51 35 51 36 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 34 51 35 51 32 51 33 51 32 51 32 
51 31 51 30 51 29 51 29 51 27 51 25 51 23 51 21 
51 18 51 15 51 10 51 7 51 3 51 -3 51 -7 51 -10 
51 -15 51 -18 51 -21 51 -23 51 -25 51 -27 51 -29 51 -29 
51 -30 51 -31 51 -32 950 1034 100 MP stroke
51 35 51 35 51 35 51 36 51 35 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 35 51 36 51 35 51 35 
51 35 51 35 51 36 51 35 51 35 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 35 51 35 51 36 51 35 
51 35 51 35 51 35 51 35 51 36 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 36 51 35 51 35 52 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 34 51 35 51 35 
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51 18 51 16 51 11 51 6 51 5 51 -5 51 -6 51 -11 
51 -16 51 -18 51 -21 51 -23 51 -22 51 -27 51 -28 51 -30 
51 -32 51 -32 51 -32 950 1034 100 MP stroke
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51 -34 51 -34 51 -34 51 -32 51 -30 51 -25 51 -10 51 10 
51 25 51 30 51 32 51 34 51 34 51 34 51 35 51 34 
51 35 51 35 51 35 51 35 51 35 950 597 30 MP stroke
gr

c8
DA
0 sg
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-776 0 1679 726 2 MP stroke
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 459  698 mt 
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 718  818 mt 
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 771  758 mt 
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 474 1239 mt 
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 624 1299 mt 
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/dpi2point 12 def
portraitMode 0216 7344 csm

  577   274  5729  4624 MR c np
92 dict begin %Colortable dictionary
/c0 { 0 0 0 sr} bdef
/c1 { 1 1 1 sr} bdef
/c2 { 1 0 0 sr} bdef
/c3 { 0 1 0 sr} bdef
/c4 { 0 0 1 sr} bdef
/c5 { 1 1 0 sr} bdef
/c6 { 1 0 1 sr} bdef
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c0
1 j
1 sg
   0    0 6914 5188 PR
6 w
0 4227 5357 0 0 -4227 899 4616 4 MP
PP
-5357 0 0 4227 5357 0 0 -4227 899 4616 5 MP stroke
4 w
DO
SO
6 w
0 sg
 899 4616 mt 6256 4616 L
 899  389 mt 6256  389 L
 899 4616 mt  899  389 L
6256 4616 mt 6256  389 L
 899 4616 mt 6256 4616 L
 899 4616 mt  899  389 L
 899 4616 mt  899 4562 L
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 713 4761 mt 
(-1.5  ) s
1664 4616 mt 1664 4562 L
1664  389 mt 1664  442 L
1614 4761 mt 
(0 ) s
2684 4616 mt 2684 4562 L
2684  389 mt 2684  442 L
2651 4761 mt 
(2) s
3705 4616 mt 3705 4562 L
3705  389 mt 3705  442 L
3655 4761 mt 
(4 ) s
4725 4616 mt 4725 4562 L
4725  389 mt 4725  442 L
4675 4761 mt 
(6 ) s
6000 4616 mt 6000 4562 L
6000  389 mt 6000  442 L
5900 4761 mt 
(8.5 ) s
 899 4616 mt  952 4616 L
6256 4616 mt 6202 4616 L
 728 4660 mt 
(-5) s
 899 3911 mt  952 3911 L
6256 3911 mt 6202 3911 L
 728 3955 mt 
(-4) s
 899 3207 mt  952 3207 L
6256 3207 mt 6202 3207 L
 728 3251 mt 
(-3) s
 899 2502 mt  952 2502 L
6256 2502 mt 6202 2502 L
 728 2546 mt 
(-2) s
 899 1798 mt  952 1798 L
6256 1798 mt 6202 1798 L
 728 1842 mt 
(-1) s
 899 1093 mt  952 1093 L
6256 1093 mt 6202 1093 L
 798 1137 mt 
(0) s
 899  389 mt  952  389 L
6256  389 mt 6202  389 L
 798  433 mt 
(1) s
 899 4616 mt 6256 4616 L
 899  389 mt 6256  389 L
 899 4616 mt  899  389 L
6256 4616 mt 6256  389 L
gs 899 389 5358 4228 MR c np
/c8 { 0.000000 0.000000 1.000000 sr} bdef
c8
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 34 
51 35 51 35 51 35 51 35 51 35 51 35 51 34 51 35 
51 35 51 35 51 34 51 35 51 35 51 35 51 34 51 35 
51 34 51 35 51 35 51 34 51 35 51 34 51 34 52 35 
51 34 51 34 51 34 51 35 51 34 51 34 51 34 51 34 
51 33 51 34 51 34 51 33 51 33 51 33 51 33 51 33 
51 33 51 32 51 32 51 32 51 31 51 31 51 31 51 30 
51 29 51 29 51 27 51 27 51 25 51 24 51 23 51 20 
51 18 51 15 51 12 51 9 51 5 51 -5 51 -9 51 -12 
51 -15 51 -18 51 -20 51 -23 51 -24 51 -25 51 -27 51 -27 
51 -29 51 -29 51 -30 950 1386 100 MP stroke
51 35 51 35 51 36 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 36 51 35 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 36 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 35 51 36 51 35 51 35 
51 35 51 36 51 35 51 35 51 35 51 35 51 36 51 35 
51 35 51 35 51 36 51 35 51 35 51 35 51 35 52 36 
51 35 51 35 51 35 51 36 51 35 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 36 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 35 51 36 51 35 51 35 
51 35 51 36 51 35 51 35 51 35 51 35 51 36 51 35 
51 35 51 35 51 36 51 35 51 35 51 -35 51 -35 51 -36 
51 -35 51 -35 51 -35 51 -36 51 -35 51 -35 51 -35 51 -35 
51 -36 51 -35 51 -35 950 1586 100 MP stroke
51 35 51 36 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 34 51 35 51 35 51 35 
51 35 51 35 51 34 51 35 51 35 51 35 51 35 51 34 
51 35 51 35 51 34 51 35 51 34 51 35 51 34 52 35 
51 34 51 35 51 34 51 34 51 35 51 34 51 34 51 34 
51 34 51 33 51 34 51 34 51 33 51 34 51 33 51 33 
51 32 51 33 51 32 51 32 51 32 51 31 51 31 51 30 
51 29 51 29 51 27 51 27 51 25 51 24 51 22 51 19 
51 17 51 14 51 11 51 7 51 3 51 -3 51 -7 51 -11 
51 -14 51 -17 51 -19 51 -22 51 -24 51 -25 51 -27 51 -27 
51 -29 51 -29 51 -30 950 1377 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 34 51 35 51 35 51 35 
51 35 51 35 51 35 51 34 51 35 51 35 51 35 51 34 
51 35 51 35 51 35 51 34 51 35 51 34 51 35 51 35 
51 34 51 35 51 34 51 35 51 34 51 34 51 35 52 34 
51 34 51 34 51 34 51 34 51 34 51 34 51 34 51 34 
51 33 51 34 51 33 51 34 51 33 51 33 51 32 51 33 
51 32 51 32 51 32 51 31 51 31 51 31 51 30 51 29 
51 29 51 28 51 26 51 26 51 24 51 23 51 21 51 18 
51 16 51 13 51 10 51 6 51 2 51 -2 51 -6 51 -10 
51 -13 51 -16 51 -18 51 -21 51 -23 51 -24 51 -26 51 -26 
51 -28 51 -29 51 -29 950 1364 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 34 51 35 51 35 
51 35 51 35 51 35 51 35 51 34 51 35 51 35 51 35 
51 34 51 35 51 35 51 34 51 35 51 35 51 34 51 35 
51 34 51 35 51 34 51 35 51 34 51 35 51 34 52 34 
51 34 51 35 51 34 51 34 51 34 51 34 51 33 51 34 
51 34 51 33 51 34 51 33 51 33 51 33 51 33 51 32 
51 33 51 32 51 31 51 32 51 31 51 30 51 30 51 30 
51 28 51 28 51 27 51 26 51 24 51 23 51 20 51 19 
51 16 51 13 51 9 51 6 51 2 51 -2 51 -6 51 -9 
51 -13 51 -16 51 -19 51 -20 51 -23 51 -24 51 -26 51 -27 
51 -28 51 -28 51 -30 950 1364 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 34 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 34 51 35 51 35 51 35 51 35 51 34 51 35 
51 35 51 34 51 35 51 35 51 34 51 35 51 35 51 34 
51 35 51 34 51 35 51 34 51 35 51 34 51 34 52 34 
51 35 51 34 51 34 51 34 51 34 51 34 51 34 51 33 
51 34 51 33 51 34 51 33 51 33 51 33 51 33 51 32 
51 33 51 32 51 31 51 32 51 31 51 30 51 30 51 29 
51 29 51 28 51 27 51 25 51 24 51 23 51 21 51 18 
51 16 51 13 51 9 51 6 51 2 51 -2 51 -6 51 -9 
51 -13 51 -16 51 -18 51 -21 51 -23 51 -24 51 -25 51 -27 
51 -28 51 -29 51 -29 950 1363 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 34 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 34 51 35 51 35 51 35 51 35 51 34 51 35 
51 35 51 34 51 35 51 35 51 34 51 35 51 35 51 34 
51 35 51 34 51 35 51 34 51 35 51 34 51 34 52 34 
51 35 51 34 51 34 51 34 51 34 51 34 51 34 51 33 
51 34 51 33 51 34 51 33 51 33 51 33 51 33 51 32 
51 33 51 32 51 31 51 32 51 31 51 30 51 30 51 29 
51 29 51 28 51 27 51 25 51 24 51 23 51 21 51 18 
51 16 51 13 51 9 51 6 51 2 51 -2 51 -6 51 -9 
51 -13 51 -16 51 -18 51 -21 51 -23 51 -24 51 -25 51 -27 
51 -28 51 -29 51 -29 950 1363 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 34 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 34 51 35 51 35 51 35 51 34 51 35 
51 35 51 35 51 34 51 35 51 35 51 34 51 35 51 34 
51 35 51 34 51 35 51 34 51 35 51 34 51 34 52 35 
51 34 51 34 51 34 51 34 51 34 51 34 51 34 51 33 
51 34 51 34 51 33 51 33 51 33 51 33 51 33 51 32 
51 33 51 32 51 31 51 32 51 31 51 30 51 30 51 29 
51 29 51 28 51 27 51 25 51 25 51 22 51 21 51 18 
51 16 51 13 51 9 51 6 51 2 51 -2 51 -6 51 -9 
51 -13 51 -16 51 -18 51 -21 51 -22 51 -25 51 -25 51 -27 
51 -28 51 -29 51 -29 950 1363 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 34 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 34 51 35 51 35 51 35 51 34 51 35 
51 35 51 35 51 34 51 35 51 35 51 34 51 35 51 34 
51 35 51 34 51 35 51 34 51 35 51 34 51 34 52 35 
51 34 51 34 51 34 51 34 51 34 51 34 51 34 51 33 
51 34 51 34 51 33 51 33 51 33 51 33 51 33 51 32 
51 33 51 32 51 31 51 32 51 31 51 30 51 30 51 30 
51 28 51 28 51 27 51 25 51 25 51 22 51 21 51 18 
51 16 51 13 51 9 51 6 51 2 51 -2 51 -6 51 -9 
51 -13 51 -16 51 -18 51 -21 51 -22 51 -25 51 -25 51 -27 
51 -28 51 -28 51 -30 950 1363 100 MP stroke
51 36 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 34 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 34 51 35 
51 35 51 35 51 35 51 35 51 34 51 35 51 35 51 35 
51 34 51 35 51 35 51 34 51 35 51 35 51 34 51 35 
51 34 51 35 51 34 51 35 51 34 51 34 51 35 52 34 
51 34 51 34 51 34 51 34 51 34 51 34 51 34 51 34 
51 33 51 34 51 33 51 33 51 33 51 33 51 33 51 32 
51 33 51 32 51 31 51 32 51 31 51 30 51 30 51 29 
51 29 51 28 51 26 51 26 51 24 51 23 51 20 51 19 
51 15 51 13 51 9 51 6 51 2 51 -2 51 -6 51 -9 
51 -13 51 -15 51 -19 51 -20 51 -23 51 -24 51 -26 51 -26 
51 -28 51 -29 51 -29 950 1362 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 34 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 34 51 35 51 35 51 35 51 35 51 34 51 35 51 35 
51 35 51 34 51 35 51 35 51 34 51 35 51 34 51 35 
51 35 51 34 51 35 51 34 51 34 51 35 51 34 52 34 
51 34 51 35 51 34 51 34 51 34 51 34 51 33 51 34 
51 34 51 33 51 33 51 34 51 33 51 33 51 32 51 33 
51 32 51 32 51 32 51 31 51 31 51 30 51 30 51 30 
51 28 51 28 51 27 51 25 51 25 51 22 51 21 51 18 
51 16 51 12 51 10 51 5 51 2 51 -2 51 -5 51 -10 
51 -12 51 -16 51 -18 51 -21 51 -22 51 -25 51 -25 51 -27 
51 -28 51 -28 51 -30 950 1362 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 34 51 35 51 35 51 35 51 35 
51 35 51 35 51 34 51 35 51 35 51 35 51 35 51 34 
51 35 51 35 51 34 51 35 51 35 51 34 51 35 51 35 
51 34 51 35 51 34 51 35 51 34 51 35 51 34 51 35 
51 34 51 34 51 35 51 34 51 34 51 34 51 34 52 34 
51 34 51 34 51 34 51 34 51 33 51 34 51 33 51 34 
51 33 51 33 51 33 51 33 51 33 51 32 51 32 51 32 
51 32 51 32 51 31 51 30 51 31 51 29 51 30 51 28 
51 28 51 27 51 26 51 24 51 23 51 22 51 19 51 18 
51 14 51 12 51 8 51 4 51 1 51 -1 51 -4 51 -8 
51 -12 51 -14 51 -18 51 -19 51 -22 51 -23 51 -24 51 -26 
51 -27 51 -28 51 -28 950 1347 100 MP stroke
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 34 51 35 51 35 51 35 51 35 51 35 51 35 51 34 
51 35 51 35 51 35 51 35 51 34 51 35 51 35 51 34 
51 35 51 35 51 34 51 35 51 34 51 35 51 34 51 35 
51 34 51 35 51 34 51 35 51 34 51 34 51 34 52 34 
51 35 51 34 51 34 51 33 51 34 51 34 51 34 51 33 
51 34 51 33 51 33 51 33 51 33 51 32 51 33 51 32 
51 32 51 32 51 31 51 31 51 30 51 30 51 30 51 28 
51 28 51 27 51 26 51 24 51 23 51 21 51 19 51 16 
51 14 51 10 51 6 51 3 51 -1 51 1 51 -3 51 -6 
51 -10 51 -14 51 -16 51 -19 51 -21 51 -23 51 -24 51 -26 
51 -27 51 -28 51 -28 950 1337 100 MP stroke
51 36 51 35 51 35 51 35 51 35 51 36 51 35 51 35 
51 35 51 35 51 36 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 36 51 35 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 35 51 36 51 35 51 35 
51 35 51 35 51 35 51 36 51 35 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 35 51 35 51 36 52 35 
51 35 51 35 51 35 51 35 51 35 51 36 51 35 51 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 34 51 35 51 35 51 35 51 35 
51 35 51 35 51 34 51 35 51 35 51 34 51 32 51 33 
51 33 51 31 51 31 51 29 51 24 51 -24 51 -29 51 -31 
51 -31 51 -33 51 -33 51 -32 51 -34 51 -35 51 -35 51 -34 
51 -35 51 -35 51 -35 950 1149 100 MP stroke
gr

c8
0 sg
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3524 4867 mt 
(B) s
 704 2558 mt  -90 rotate
(E) s
90 rotate
gs 899 389 5358 4228 MR c np
c8
51 35 51 35 51 35 51 35 51 36 51 35 51 35 51 35 
51 35 51 36 51 35 51 35 51 35 51 35 51 36 51 35 
51 35 51 35 51 35 51 35 51 36 51 35 51 35 51 35 
51 35 51 35 51 36 51 35 51 35 51 35 51 35 51 35 
51 36 51 35 51 35 51 35 51 35 51 35 51 35 51 36 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 52 35 
51 35 51 35 51 35 51 35 51 35 51 35 51 35 51 35 
51 35 51 35 51 35 51 35 51 34 51 35 51 35 51 34 
51 35 51 34 51 34 51 35 51 34 51 33 51 34 51 34 
51 33 51 34 51 32 51 32 51 31 51 23 51 27 51 25 
51 22 51 20 51 15 51 9 51 2 51 -2 51 -9 51 -15 
51 -20 51 -22 51 -25 51 -27 51 -23 51 -31 51 -32 51 -32 
51 -34 51 -33 51 -34 950 1032 100 MP stroke
51 -35 51 -35 51 -35 51 -36 51 -35 51 -35 51 -35 51 -35 
51 -36 51 -35 51 -35 51 -35 51 -36 51 -35 51 -35 51 -35 
51 -35 51 -36 51 -35 51 -35 51 -35 51 -36 51 -35 51 -35 
51 -35 51 -35 51 -36 51 -35 51 -35 950 1586 30 MP stroke
gr

c8
DA
0 sg
0 -4205 1656 4604 2 MP stroke
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 609  759 mt 
(1-) s
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 745  759 mt 
(D) s
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 818  699 mt 
(-1) s
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 927  759 mt 
( ) s
SO
c8
-686 -498 1656 1080 2 MP stroke
DO
0 sg
-759 0 1665 692 2 MP stroke
-750 0 1656 1088 2 MP stroke
SO

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% operator abbreviations
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(4 ) s
4725 4616 mt 4725 4562 L
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(6 ) s
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51 -16 51 -18 51 -20 950 1474 100 MP stroke
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51 41 51 41 51 41 51 40 51 41 51 40 51 40 51 40 
51 40 51 39 51 40 51 39 51 39 51 38 51 39 51 38 
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51 -5 51 -5 51 -7 51 -8 51 -9 51 -10 51 -11 51 -13 
51 -13 51 -15 51 -15 950 1444 100 MP stroke
51 43 51 42 51 43 51 42 51 43 51 42 51 42 51 42 
51 42 51 41 51 42 51 42 51 41 51 41 51 41 51 41 
51 41 51 41 51 40 51 41 51 40 51 40 51 40 51 40 
51 39 51 39 51 40 51 38 51 39 51 39 51 38 51 38 
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51 -4 51 -6 51 -6 51 -8 51 -9 51 -10 51 -11 51 -12 
51 -14 51 -14 51 -15 950 1442 100 MP stroke
51 43 51 43 51 42 51 42 51 43 51 42 51 42 51 42 
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51 40 51 39 51 39 51 39 51 39 51 38 51 39 51 37 
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51 -14 51 -14 51 -15 950 1442 100 MP stroke
51 42 51 43 51 42 51 43 51 42 51 42 51 42 51 42 
51 42 51 42 51 41 51 42 51 41 51 42 51 41 51 41 
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51 28 51 27 51 26 51 26 51 25 51 24 51 24 51 22 
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51 6 51 4 51 3 51 2 51 0 51 0 51 -2 51 -3 
51 -4 51 -6 51 -6 51 -8 51 -9 51 -10 51 -11 51 -12 
51 -14 51 -14 51 -15 950 1440 100 MP stroke
51 42 51 43 51 42 51 42 51 43 51 42 51 42 51 42 
51 41 51 42 51 42 51 41 51 41 51 42 51 41 51 41 
51 40 51 41 51 40 51 41 51 40 51 40 51 40 51 39 
51 40 51 39 51 39 51 39 51 38 51 39 51 38 51 38 
51 37 51 38 51 37 51 37 51 36 51 36 51 36 51 36 
51 35 51 35 51 34 51 35 51 33 51 34 51 33 52 32 
51 32 51 32 51 31 51 30 51 30 51 29 51 29 51 29 
51 27 51 27 51 26 51 26 51 25 51 24 51 23 51 23 
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51 14 51 13 51 12 51 11 51 10 51 9 51 8 51 6 
51 6 51 4 51 3 51 2 51 0 51 0 51 -2 51 -3 
51 -4 51 -6 51 -6 51 -8 51 -9 51 -10 51 -11 51 -12 
51 -13 51 -14 51 -15 950 1404 100 MP stroke
51 45 51 45 51 45 51 45 51 45 51 45 51 44 51 45 
51 45 51 45 51 44 51 45 51 44 51 44 51 45 51 44 
51 44 51 44 51 44 51 44 51 44 51 43 51 44 51 44 
51 43 51 43 51 43 51 43 51 43 51 43 51 42 51 43 
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51 40 51 41 51 40 51 39 51 40 51 39 51 38 52 38 
51 38 51 38 51 37 51 36 51 36 51 36 51 35 51 34 
51 34 51 33 51 32 51 32 51 31 51 30 51 29 51 29 
51 27 51 27 51 25 51 25 51 23 51 23 51 21 51 20 
51 18 51 18 51 16 51 15 51 13 51 12 51 10 51 8 
51 8 51 5 51 4 51 3 51 0 51 0 51 -3 51 -4 
51 -5 51 -8 51 -8 51 -10 51 -12 51 -13 51 -15 51 -16 
51 -18 51 -18 51 -20 950 919 100 MP stroke
51 45 51 46 51 45 51 46 51 45 51 45 51 46 51 45 
51 45 51 45 51 46 51 45 51 45 51 45 51 45 51 45 
51 45 51 44 51 45 51 45 51 44 51 45 51 44 51 45 
51 44 51 44 51 44 51 43 51 41 51 41 51 41 51 41 
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51 39 51 39 51 38 51 38 51 38 51 37 51 37 52 36 
51 37 51 36 51 35 51 35 51 34 51 34 51 34 51 32 
51 33 51 31 51 31 51 31 51 29 51 29 51 28 51 28 
51 26 51 26 51 24 51 24 51 22 51 22 51 20 51 19 
51 16 51 14 51 13 51 11 51 11 51 9 51 8 51 7 
51 5 51 5 51 3 51 2 51 0 51 0 51 -2 51 -3 
51 -5 51 -5 51 -7 51 -8 51 -9 51 -11 51 -11 51 -13 
51 -14 51 -16 51 -19 950 892 100 MP stroke
51 46 51 45 51 46 51 45 51 46 51 45 51 45 51 46 
51 45 51 45 51 45 51 45 51 46 51 45 51 45 51 45 
51 44 51 45 51 45 51 45 51 44 51 45 51 45 51 44 
51 44 51 42 51 42 51 43 51 44 51 44 51 44 51 43 
51 44 51 43 51 43 51 43 51 43 51 41 51 40 51 40 
51 39 51 40 51 39 51 39 51 38 51 39 51 38 52 37 
51 38 51 37 51 36 51 37 51 35 51 36 51 34 51 35 
51 33 51 33 51 29 51 27 51 26 51 26 51 24 51 24 
51 22 51 22 51 21 51 20 51 19 51 19 51 17 51 16 
51 16 51 13 51 13 51 11 51 10 51 9 51 8 51 7 
51 6 51 4 51 3 51 2 51 0 51 0 51 -2 51 -3 
51 -4 51 -6 51 -7 51 -8 51 -9 51 -10 51 -11 51 -13 
51 -13 51 -16 51 -16 950 887 100 MP stroke
51 46 51 45 51 46 51 45 51 46 51 45 51 46 51 45 
51 45 51 45 51 46 51 45 51 45 51 45 51 45 51 45 
51 45 51 45 51 44 51 45 51 45 51 44 51 45 51 44 
51 44 51 45 51 44 51 44 51 44 51 44 51 44 51 43 
51 44 51 43 51 42 51 41 51 40 51 43 51 43 51 42 
51 43 51 42 51 42 51 41 51 42 51 41 51 41 52 41 
51 40 51 40 51 34 51 32 51 32 51 30 51 31 51 29 
51 29 51 29 51 31 51 27 51 26 51 25 51 24 51 23 
51 23 51 21 51 21 51 20 51 19 51 17 51 17 51 16 
51 15 51 13 51 13 51 11 51 10 51 10 51 7 51 7 
51 6 51 4 51 3 51 2 51 0 51 0 51 -2 51 -3 
51 -4 51 -6 51 -7 51 -7 51 -10 51 -10 51 -11 51 -13 
51 -13 51 -15 51 -16 950 886 100 MP stroke
51 45 51 46 51 45 51 46 51 45 51 45 51 46 51 45 
51 45 51 46 51 45 51 45 51 45 51 45 51 45 51 45 
51 45 51 45 51 45 51 44 51 45 51 45 51 44 51 44 
51 43 51 44 51 44 51 44 51 44 51 44 51 44 51 44 
51 43 51 44 51 43 51 43 51 43 51 43 51 43 51 42 
51 43 51 42 51 42 51 42 51 42 51 41 51 41 52 41 
51 41 51 33 51 38 51 34 51 31 51 30 51 30 51 29 
51 29 51 28 51 27 51 27 51 25 51 25 51 24 51 24 
51 22 51 22 51 20 51 20 51 19 51 17 51 17 51 16 
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51 6 51 4 51 3 51 2 51 1 51 -1 51 -2 51 -3 
51 -4 51 -6 51 -6 51 -8 51 -9 51 -11 51 -11 51 -13 
51 -13 51 -15 51 -16 950 886 100 MP stroke
51 45 51 44 51 44 51 44 51 44 51 44 51 44 51 44 
51 43 51 44 51 43 51 44 51 43 51 43 51 44 51 43 
51 43 51 43 51 42 51 43 51 43 51 42 51 42 51 43 
51 44 51 44 51 44 51 44 51 44 51 44 51 44 51 44 
51 43 51 42 51 43 51 43 51 43 51 43 51 43 51 43 
51 42 51 43 51 42 51 42 51 41 51 42 51 41 52 41 
51 40 51 40 51 33 51 32 51 31 51 31 51 29 51 30 
51 28 51 28 51 27 51 27 51 25 51 25 51 24 51 24 
51 22 51 22 51 20 51 20 51 19 51 18 51 16 51 16 
51 16 51 14 51 13 51 11 51 10 51 9 51 8 51 7 
51 6 51 4 51 3 51 2 51 0 51 0 51 -2 51 -3 
51 -4 51 -6 51 -7 51 -8 51 -9 51 -10 51 -11 51 -13 
51 -14 51 -16 51 -16 950 886 100 MP stroke
51 45 51 45 51 46 51 45 51 46 51 45 51 45 51 45 
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51 7 51 5 51 4 51 2 51 1 51 -1 51 -2 51 -4 
51 -5 51 -7 51 -9 51 -10 51 -11 51 -13 51 -14 51 -15 
51 -16 51 -15 51 -16 950 884 100 MP stroke
51 45 51 44 51 44 51 45 51 44 51 44 51 44 51 44 
51 44 51 44 51 43 51 45 51 45 51 45 51 45 51 45 
51 45 51 45 51 44 51 45 51 44 51 45 51 44 51 45 
51 44 51 44 51 44 51 44 51 44 51 44 51 44 51 43 
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51 28 51 28 51 27 51 29 51 31 51 29 51 24 51 24 
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51 6 51 4 51 3 51 2 51 1 51 -1 51 -2 51 -3 
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51 -15 51 -15 51 -16 950 867 100 MP stroke
51 44 51 43 51 44 51 43 51 44 51 43 51 44 51 43 
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51 29 51 29 51 28 51 27 51 26 51 28 51 26 51 24 
51 22 51 22 51 21 51 20 51 19 51 19 51 17 51 16 
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51 6 51 4 51 4 51 1 51 1 51 -1 51 -1 51 -4 
51 -4 51 -6 51 -6 51 -8 51 -10 51 -10 51 -12 51 -12 
51 -14 51 -15 51 -16 950 865 100 MP stroke
51 44 51 44 51 44 51 43 51 44 51 43 51 43 51 43 
51 44 51 43 51 42 51 43 51 43 51 43 51 42 51 42 
51 43 51 42 51 42 51 41 51 42 51 42 51 41 51 41 
51 41 51 41 51 41 51 40 51 40 51 40 51 40 51 40 
51 39 51 39 51 39 51 38 51 39 51 38 51 37 51 37 
51 37 51 37 51 36 51 35 51 37 51 34 51 34 52 34 
51 33 51 33 51 38 51 32 51 37 51 31 51 31 51 29 
51 29 51 28 51 28 51 27 51 26 51 25 51 27 51 23 
51 23 51 22 51 21 51 20 51 19 51 18 51 17 51 16 
51 15 51 14 51 13 51 11 51 11 51 9 51 8 51 7 
51 6 51 4 51 3 51 2 51 1 51 -1 51 -2 51 -3 
51 -4 51 -6 51 -7 51 -8 51 -9 51 -11 51 -11 51 -13 
51 -14 51 -15 51 -16 950 864 100 MP stroke
51 43 51 44 51 43 51 44 51 43 51 43 51 43 51 43 
51 43 51 43 51 42 51 43 51 42 51 43 51 42 51 42 
51 42 51 42 51 41 51 42 51 41 51 41 51 41 51 41 
51 41 51 40 51 41 51 40 51 40 51 39 51 40 51 39 
51 39 51 38 51 39 51 38 51 38 51 37 51 37 51 37 
51 37 51 36 51 35 51 36 51 35 51 34 51 34 52 34 
51 33 51 34 51 37 51 38 51 32 51 32 51 31 51 29 
51 29 51 28 51 28 51 27 51 26 51 25 51 27 51 28 
51 28 51 25 51 22 51 20 51 19 51 18 51 16 51 16 
51 15 51 13 51 13 51 11 51 10 51 9 51 8 51 7 
51 5 51 5 51 3 51 2 51 0 51 0 51 -2 51 -3 
51 -5 51 -5 51 -7 51 -8 51 -9 51 -10 51 -11 51 -13 
51 -13 51 -15 51 -16 950 849 100 MP stroke
51 43 51 44 51 43 51 44 51 43 51 43 51 43 51 43 
51 43 51 43 51 43 51 42 51 43 51 42 51 42 51 42 
51 42 51 42 51 42 51 41 51 41 51 42 51 41 51 40 
51 41 51 41 51 40 51 40 51 40 51 39 51 40 51 39 
51 39 51 39 51 38 51 38 51 38 51 37 51 37 51 37 
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51 31 51 28 51 27 51 26 51 26 51 25 51 24 51 24 
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51 -14 51 -15 51 -16 950 832 100 MP stroke
51 44 51 44 51 43 51 43 51 44 51 43 51 43 51 43 
51 43 51 42 51 43 51 43 51 42 51 42 51 42 51 43 
51 41 51 42 51 42 51 41 51 42 51 41 51 41 51 41 
51 40 51 41 51 40 51 40 51 40 51 40 51 39 51 39 
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 522 2557 mt  -90 rotate
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 515 1349 mt 
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/lrx 0 def /uly 0 def /ulx 0 def /rad 0 def
/MRR {/vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef
  /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly
  vradius add translate hradius vradius scale 0 0 1 180 270 arc 
  tMatrix setmatrix lrx hradius sub uly vradius add translate
  hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix
  lrx hradius sub lry vradius sub translate hradius vradius scale
  0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub
  translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix
  closepath} bdef
/FRR {MRR stroke } bdef
/PRR {MRR fill } bdef
/MlrRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def
  newpath tMatrix currentmatrix pop ulx rad add uly rad add translate
  rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad
  sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix
  closepath} bdef
/FlrRR {MlrRR stroke } bdef
/PlrRR {MlrRR fill } bdef
/MtbRR {/lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def
  newpath tMatrix currentmatrix pop ulx rad add uly rad add translate
  rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad
  sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix
  closepath} bdef
/FtbRR {MtbRR stroke } bdef
/PtbRR {MtbRR fill } bdef
/stri 6 array def /dtri 6 array def
/smat 6 array def /dmat 6 array def
/tmat1 6 array def /tmat2 6 array def /dif 3 array def
/asub {/ind2 exch def /ind1 exch def dup dup
  ind1 get exch ind2 get sub exch } bdef
/tri_to_matrix {
  2 0 asub 3 1 asub 4 0 asub 5 1 asub
  dup 0 get exch 1 get 7 -1 roll astore } bdef
/compute_transform {
  dmat dtri tri_to_matrix tmat1 invertmatrix 
  smat stri tri_to_matrix tmat2 concatmatrix } bdef
/ds {stri astore pop} bdef
/dt {dtri astore pop} bdef
/db {2 copy /cols xdef /rows xdef mul dup 3 mul string
  currentfile exch readhexstring pop
  dup 0 3 index getinterval /rbmap xdef
  dup 2 index dup getinterval /gbmap xdef
  1 index dup 2 mul exch getinterval /bbmap xdef pop pop}bdef
/it {gs np dtri aload pop moveto lineto lineto cp c
  cols rows 8 compute_transform 
  rbmap gbmap bbmap true 3 colorimage gr}bdef
/il {newpath moveto lineto stroke}bdef
currentdict end def
%%EndProlog

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MathWorks begin

0 cap

end
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bplot

/dpi2point 12 def
portraitMode 0216 7344 csm

  446    23  5860  4921 MR c np
92 dict begin %Colortable dictionary
/c0 { 0 0 0 sr} bdef
/c1 { 1 1 1 sr} bdef
/c2 { 1 0 0 sr} bdef
/c3 { 0 1 0 sr} bdef
/c4 { 0 0 1 sr} bdef
/c5 { 1 1 0 sr} bdef
/c6 { 1 0 1 sr} bdef
/c7 { 0 1 1 sr} bdef
c0
1 j
1 sg
   0    0 6914 5188 PR
6 w
0 4227 5357 0 0 -4227 899 4616 4 MP
PP
-5357 0 0 4227 5357 0 0 -4227 899 4616 5 MP stroke
4 w
DO
SO
6 w
0 sg
 899 4616 mt 6256 4616 L
 899 4616 mt  899  389 L
1122 4616 mt 1122 4562 L
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 120 FMSR

1039 4761 mt 
(11 ) s
1568 4616 mt 1568 4562 L
1535 4761 mt 
(9) s
2015 4616 mt 2015 4562 L
1965 4761 mt 
(7 ) s
2461 4616 mt 2461 4562 L
2411 4761 mt 
(5 ) s
2907 4616 mt 2907 4562 L
2857 4761 mt 
(3 ) s
3354 4616 mt 3354 4562 L
3304 4761 mt 
(1 ) s
3800 4616 mt 3800 4562 L
3697 4761 mt 
(-1) s
4247 4616 mt 4247 4562 L
4144 4761 mt 
(-3) s
4693 4616 mt 4693 4562 L
4590 4761 mt 
(-5) s
5139 4616 mt 5139 4562 L
5036 4761 mt 
(-7) s
5586 4616 mt 5586 4562 L
5483 4761 mt 
(-9) s
6032 4616 mt 6032 4562 L
5896 4761 mt 
(-11) s
 899 4616 mt  952 4616 L
 728 4660 mt 
(-1) s
 899 3702 mt  952 3702 L
 628 3746 mt 
(-0.6) s
 899 2788 mt  952 2788 L
 628 2832 mt 
(-0.2) s
 899 1874 mt  952 1874 L
 698 1918 mt 
(0.2) s
 899  960 mt  952  960 L
 698 1004 mt 
(0.6) s
gs 899 389 5358 4228 MR c np
223 0 1457 1907 2 MP stroke
223 0 1457 617 2 MP stroke
223 0 1903 1966 2 MP stroke
223 0 1903 1907 2 MP stroke
223 0 1903 1878 2 MP stroke
223 0 1903 617 2 MP stroke
223 0 1903 453 2 MP stroke
224 0 2349 1995 2 MP stroke
224 0 2349 1975 2 MP stroke
224 0 2349 1923 2 MP stroke
224 0 2349 1921 2 MP stroke
224 0 2349 1910 2 MP stroke
224 0 2349 1863 2 MP stroke
224 0 2349 616 2 MP stroke
224 0 2349 499 2 MP stroke
224 0 2349 454 2 MP stroke
224 0 2349 430 2 MP stroke
223 0 2796 2095 2 MP stroke
223 0 2796 2019 2 MP stroke
223 0 2796 1994 2 MP stroke
223 0 2796 1980 2 MP stroke
223 0 2796 1970 2 MP stroke
223 0 2796 1925 2 MP stroke
223 0 2796 1923 2 MP stroke
223 0 2796 1910 2 MP stroke
223 0 2796 1838 2 MP stroke
223 0 2796 1726 2 MP stroke
223 0 2796 602 2 MP stroke
223 0 2796 565 2 MP stroke
223 0 2796 522 2 MP stroke
223 0 2796 509 2 MP stroke
223 0 2796 469 2 MP stroke
223 0 2796 465 2 MP stroke
223 0 2796 456 2 MP stroke
223 0 2796 412 2 MP stroke
223 0 3242 2592 2 MP stroke
223 0 3242 2273 2 MP stroke
223 0 3242 2152 2 MP stroke
223 0 3242 2026 2 MP stroke
223 0 3242 2023 2 MP stroke
223 0 3242 1994 2 MP stroke
223 0 3242 1975 2 MP stroke
223 0 3242 1925 2 MP stroke
223 0 3242 1924 2 MP stroke
223 0 3242 1915 2 MP stroke
223 0 3242 1887 2 MP stroke
223 0 3242 1799 2 MP stroke
223 0 3242 1786 2 MP stroke
223 0 3242 1277 2 MP stroke
223 0 3242 849 2 MP stroke
223 0 3242 634 2 MP stroke
223 0 3242 567 2 MP stroke
223 0 3242 547 2 MP stroke
223 0 3242 522 2 MP stroke
223 0 3242 520 2 MP stroke
223 0 3242 500 2 MP stroke
223 0 3242 470 2 MP stroke
223 0 3242 469 2 MP stroke
223 0 3242 453 2 MP stroke
223 0 3689 3971 2 MP stroke
223 0 3689 2652 2 MP stroke
223 0 3689 2312 2 MP stroke
223 0 3689 2156 2 MP stroke
223 0 3689 2030 2 MP stroke
223 0 3689 2023 2 MP stroke
223 0 3689 1994 2 MP stroke
223 0 3689 1975 2 MP stroke
223 0 3689 1925 2 MP stroke
223 0 3689 1924 2 MP stroke
223 0 3689 1915 2 MP stroke
223 0 3689 1890 2 MP stroke
223 0 3689 1821 2 MP stroke
223 0 3689 1790 2 MP stroke
223 0 3689 1298 2 MP stroke
223 0 3689 1132 2 MP stroke
223 0 3689 893 2 MP stroke
223 0 3689 815 2 MP stroke
223 0 3689 681 2 MP stroke
223 0 3689 638 2 MP stroke
223 0 3689 566 2 MP stroke
223 0 3689 558 2 MP stroke
223 0 3689 550 2 MP stroke
223 0 3689 520 2 MP stroke
223 0 3689 504 2 MP stroke
223 0 3689 469 2 MP stroke
223 0 3689 457 2 MP stroke
223 0 3689 422 2 MP stroke
223 0 4135 4040 2 MP stroke
223 0 4135 2656 2 MP stroke
223 0 4135 2512 2 MP stroke
223 0 4135 2316 2 MP stroke
223 0 4135 2156 2 MP stroke
223 0 4135 2030 2 MP stroke
223 0 4135 2023 2 MP stroke
223 0 4135 2013 2 MP stroke
223 0 4135 1994 2 MP stroke
223 0 4135 1975 2 MP stroke
223 0 4135 1924 2 MP stroke
223 0 4135 1915 2 MP stroke
223 0 4135 1891 2 MP stroke
223 0 4135 1820 2 MP stroke
223 0 4135 1790 2 MP stroke
223 0 4135 1390 2 MP stroke
223 0 4135 1294 2 MP stroke
223 0 4135 1180 2 MP stroke
223 0 4135 898 2 MP stroke
223 0 4135 870 2 MP stroke
223 0 4135 845 2 MP stroke
223 0 4135 685 2 MP stroke
223 0 4135 672 2 MP stroke
223 0 4135 632 2 MP stroke
223 0 4135 568 2 MP stroke
223 0 4135 559 2 MP stroke
223 0 4135 550 2 MP stroke
223 0 4135 504 2 MP stroke
223 0 4135 457 2 MP stroke
223 0 4135 429 2 MP stroke
223 0 4135 412 2 MP stroke
224 0 4581 4044 2 MP stroke
224 0 4581 2656 2 MP stroke
224 0 4581 2567 2 MP stroke
224 0 4581 2317 2 MP stroke
224 0 4581 2156 2 MP stroke
224 0 4581 2053 2 MP stroke
224 0 4581 2030 2 MP stroke
224 0 4581 2023 2 MP stroke
224 0 4581 2011 2 MP stroke
224 0 4581 1975 2 MP stroke
224 0 4581 1915 2 MP stroke
224 0 4581 1891 2 MP stroke
224 0 4581 1822 2 MP stroke
224 0 4581 1797 2 MP stroke
224 0 4581 1790 2 MP stroke
224 0 4581 1433 2 MP stroke
224 0 4581 1296 2 MP stroke
224 0 4581 1182 2 MP stroke
224 0 4581 933 2 MP stroke
224 0 4581 894 2 MP stroke
224 0 4581 848 2 MP stroke
224 0 4581 686 2 MP stroke
224 0 4581 674 2 MP stroke
224 0 4581 658 2 MP stroke
224 0 4581 568 2 MP stroke
224 0 4581 561 2 MP stroke
224 0 4581 449 2 MP stroke
224 0 4581 421 2 MP stroke
223 0 5028 4044 2 MP stroke
223 0 5028 2656 2 MP stroke
223 0 5028 2570 2 MP stroke
223 0 5028 2317 2 MP stroke
223 0 5028 2156 2 MP stroke
223 0 5028 2055 2 MP stroke
223 0 5028 2044 2 MP stroke
223 0 5028 2030 2 MP stroke
223 0 5028 1891 2 MP stroke
223 0 5028 1829 2 MP stroke
223 0 5028 1822 2 MP stroke
223 0 5028 1790 2 MP stroke
223 0 5028 1436 2 MP stroke
223 0 5028 1296 2 MP stroke
223 0 5028 1182 2 MP stroke
223 0 5028 936 2 MP stroke
223 0 5028 894 2 MP stroke
223 0 5028 848 2 MP stroke
223 0 5028 674 2 MP stroke
223 0 5028 568 2 MP stroke
223 0 5474 4044 2 MP stroke
223 0 5474 2656 2 MP stroke
223 0 5474 2570 2 MP stroke
223 0 5474 2317 2 MP stroke
223 0 5474 2055 2 MP stroke
223 0 5474 1822 2 MP stroke
223 0 5474 1436 2 MP stroke
223 0 5474 1296 2 MP stroke
223 0 5474 936 2 MP stroke
223 0 5474 894 2 MP stroke
/c8 { 0.000000 0.000000 1.000000 sr} bdef
c8
223 0 5921 4044 2 MP stroke
0 sg
223 0 1010 617 2 MP stroke
gr

%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 132 FMSR

 573 2547 mt  -90 rotate
(E) s
90 rotate
3503 4913 mt 
(2n) s
2257  226 mt 
(Amplified low energy spectrum of H) s
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 96 FMSR

4332  160 mt 
(+-) s
%%IncludeResource: font Helvetica
/Helvetica /ISOLatin1Encoding 132 FMSR

4444  226 mt 
(\(0,0,B\) ) s

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%%Trailer
%%EOF

---------------0204052034356--
