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\def\ariadne{{\sc Ariadne}}
\def\lepto{{\sc Lepto}}
\def\jetset{{\sc Jetset}}
\def\herwig{{\sc HERWIG}}
\def\Pslash{{P\!\!\!\!/}}
\def\tPslash{$\Pslash$}
\def\as{\alpha_s}
\def\tas{$\as$}
\def\tordas{${\cal O}(\as)$}
\def\todo#1{{\bf $\rightarrow$ #1 $\leftarrow$}}
\def\zcut{{z_{\mbox{{\scriptsize cut}}}}}
\def\tzcut{$\zcut$}
\def\f2{F_2}
\def\tf2{$\f2$}
\def\q2{Q^2}
\def\tq2{$\q2$}
\def\et{E_\perp}
\def\tet{$\et$}
\def\kt{k_\perp}
\def\tkt{$\kt$}
\def\k2t{k_\perp^2}
\def\tk2t{$\k2t$}
\def\alb{\bar{\alpha}}
\def\talb{$\alb$}
\def\eq#1{eq.~(\ref{#1})}
\def\Eq#1{Eq.~(\ref{#1})}
\def\eqs#1{eqs.~(\ref{#1})}
\def\Eqs#1{Eqs.~(\ref{#1})}
\def\refc#1{ref.~\cite{#1}}
\def\refs#1{refs.~\cite{#1}}
\def\fig#1{fig.~\ref{#1}}
\def\figs#1{figs.~\ref{#1}}
\def\laeq{\,\lower3pt\hbox{$\buildrel < \over\sim$}\,}
\def\bigtimes{\scalebox{1.5}{$\times$}}

\newcommand{\ZETF}[3]{{\it Zh.~Eksp.~Teor.~Fiz.} {\bf #1} ({#3}) {#2}}
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\newcommand{\NPB}[3]{{\it Nucl.~Phys.} {\bf B#1} ({#3}) {#2}}
\newcommand{\PRD}[3]{{\it Phys.~Rev.} {\bf D#1} ({#3}) {#2}}
\newcommand{\PRL}[3]{{\it Phys.~Rev.~Lett.} {\bf #1} ({#3}) {#2}}
\newcommand{\ZPC}[3]{{\it Z.~Phys.} {\bf  C#1} ({#3}) {#2}}
\newcommand{\PR}[3]{{\it Phys.~Rep.} {\bf #1} ({#3}) {#2}}
\newcommand{\CPC}[3]{{\it Comput.~Phys.~Comm.} {\bf #1} ({#3}) {#2}}
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\newcommand{\preprint}{Report No.\ }
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\begin{document}

\begin{titlepage}
  \renewcommand{\thefootnote}{\fnsymbol{footnote}}
  \begin{flushright}
    LU-TP 97-21\\
    NORDITA-97/54 P\\
    \\
    September 1997\\
    Revised\footnote{The original publication was based
      on results from an implementation containing an error. In this
      revised version this error has been corrected, some of the
      beyond leading-log assumptions have been revised and so have some
      of the results.} December 1997

  \end{flushright}
  \begin{center}

    \vskip 10mm
    {\LARGE\bf The Linked Dipole Chain Monte Carlo}
    \vskip 15mm

    
    {\large Hamid Kharraziha}\\
    Dept.~of Theoretical Physics\\
    Slvegatan 14a\\
    S-223 62  Lund, Sweden\\
    hamid@thep.lu.se

    \vskip 10mm

    {\large Leif Lnnblad}\\
    NORDITA\\
    Blegdamsvej 17\\
    DK-2100 Kbenhavn , Denmark\\
    leif@nordita.dk

  \end{center}
  \vskip 30mm
  \begin{abstract}
    We present an implementation of the Linked Dipole Chain model for
    deeply inelastic $ep$ scattering into the framework of the
    \ariadne\ event generator. Using this implementation we obtain
    results both for the inclusive structure function as well as for
    exclusive properties of the hadronic final state.

  \end{abstract}

\end{titlepage}

\addtocounter{footnote}{-1}

\input LDCIntro.tex

\input LDCModel.tex

\input LDCMC.tex

\input LDCFit.tex

\input LDCResult.tex

\input LDCSum.tex

\section*{Acknowledgements}

We would like to thank Bo Andersson and Gsta Gustafson for important
contributions and discussions. We would also like to thank Hannes Jung
for providing us with a preliminary HZTOOL implementation of the H1
results on di-jet production \cite{R2H1}.

\begin{thebibliography}{10}
\itemsep -1mm

\bibitem{DGLAP} V.N.~Gribov, L.N.~Lipatov, \SJNP{15}{438}{1972} and 675;\\
  G.~Altarelli, G.~Parisi, \NPB{126}{298}{1977};\\
  Yu.L.~Dokshitzer, \JETP{46}{641}{1977}.

\bibitem{BFKL} E.A.~Kuraev, L.N.~Lipatov and V.S.~Fadin, \ZETF{72}{373}{1977},
  \JETP{45}{199}{1977}; \\
  Ya.Ya.~Balitsky and L.N.~Lipatov, \YF{28}{1597}{1978}, \SJNP{28}{822}{1978}.
  
\bibitem{HERAWS1} See eg.\ J.~Blmlein et al., Proceedings of the
  ``Future Physics at HERA'' workshop, Hamburg 1996, eds.\ 
  G.~Ingelman, A.~De~Roeck and R.~Klanner, vol.~1, p.~3,
   and references therein.
  
\bibitem{HERAWS2} See eg.\ M.~Erdmann et al., Proceedings of the
  ``Future Physics at HERA'' workshop, Hamburg 1996, eds.\ 
  G.~Ingelman, A.~De~Roeck and R.~Klanner, vol.~1, p.~500,
   and references therein.

\bibitem{CCFM} M.~Ciafaloni, \NPB{269}{49}{1988};\\
  S.~Catani, F.~Fiorani, G.~Machesini, \PLB{234}{339}{1990},
  \NPB{336}{18}{1990}.

\bibitem{HERWIG} G.~Marchesini et al., \CPC{67}{465}{1992}.
  
\bibitem{LEPTO} G.~Ingelman, A.~Edin and J.~Rathsman,
  \CPC{101}{108}{1997}, .
  
\bibitem{ForwardJets} H1 Collaboration, S.~Aid et al.,
  \PLB{356}{118}{1995}, .
  
\bibitem{ForwardParticles} H1 Collaboration, C.~Adloff et al.,
  \NPB{485}{3}{1997}, .

\bibitem{ARIADNE} L.~Lnnblad, \CPC{71}{15}{1992}.

\bibitem{BGLP} B.~Andersson et al., \ZPC{43}{621}{1989}.

\bibitem{SMALLX} G.~Marchesini, B.~Webber, \NPB{386}{215}{1992}.

\bibitem{LDC96} B.~Andersson, G.~Gustafson, J.~Samuelsson, \NPB{463}{217}{1996}.

\bibitem{kharrDIPOLE} G.~Gustafson, \PLB{175}{453}{1986};\\
  G.~Gustafson, U.~Petterson, \NPB{306}{746}{1988};\\ B.~Andersson,
  G.~Gustafson, L.~Lnnblad, \NPB{339}{393}{1990}.

\bibitem{kharrLDC2} B.~Andersson et al., \ZPC{71}{613}{1996}.

\bibitem{HamidDIS97} H.~Kharraziha, ``The LDCMonte Carlo'', talk
  presented at the DIS~'97 conference, Chicago, April 1997, to be
  published in the proceedings.
  
\bibitem{HamidNext} B.~Andersson, G.~Gustafson and H.~Kharraziha,\\
  LU-TP~97-29, aps1997nov19\_001, .

\bibitem{Kramer} R.D.~Peccei, R.~Rckl, \NPB{162}{125}{1980};\\
  Ch.~Rumpf, G.~Kramer, J.~Willrodt, \ZPC{7}{337}{1981}.

\bibitem{lambda} B.~Andersson, P.~Dahlqvist and G.~Gustafson,
  \PLB{214}{604}{1988}; \ZPC{44}{455}{1989}.

\bibitem{colrec} L.~Lnnblad, \ZPC{70}{107}{1996}.

\bibitem{JETSET} T.~Sjstrand, \CPC{82}{74}{1994}
  
\bibitem{H1F2} H1 collaboration, S.~Aid et al., \NPB{470}{3}{1996},
  .
  
\bibitem{ZEUSF2} ZEUS collaboration, M.~Derrick et al.,
  \ZPC{72}{399}{1996}, .
  
\bibitem{NMCF2} NMC collaboration, M.~Arneodo et al.,
  \PL{B364}{107}{1995}, .
  
\bibitem{E665F2} E665 collaboration, M.R.~Adams et al.,
  \PRD{54}{3006}{1996}
  
\bibitem{GRV} M.~Glck, E.~Reya and A.~Vogt, \ZPC{48}{471}{1990};
  \ZPC{53}{127}{1992}; \ZPC{67}{433}{1995}.

\bibitem{HZTOOL} J.~Bromley et al., Proceedings of the ``Future
  Physics at HERA'' workshop, Hamburg 1996, eds.\ G.~Ingelman,
  A.~De~Roeck and R.~Klanner, vol.~1, p.~611.
  
\bibitem{HZ95221} ZEUS Collaboration, M.~Derrick et al.,
  \ZPC{70}{1}{1996}, .

\bibitem{EMC} EMC Collaboration, M.~Arneodo et al., \ZPC{36}{527}{1987}.
  
\bibitem{R2H1} H1 Collaboration, T.~Ahmed et al., Contributed paper,
  Abstract 247, HEP97, Jerusalem, Israel, August 1997 (unpublished).
  
\bibitem{Kuhlen} M.~Kuhlen, \PLB{382}{441}{1996}

\bibitem{LeptoSCI} A.~Edin, G.~Ingelman, J.~Rathsman,
  \PLB{366}{371}{1996}, ; \ZPC{75}{57}{1997},
  .

\bibitem{HUB&co} H-U.~Bengtsson, \CPC{31}{323}{1984}.

\bibitem{Mike} M.H.~Seymour, \NPB{436}{443}{1995}

\end{thebibliography}

\input LDCApp.tex

\end{document}
\begin{appendix}
\section*{Appendix A: Colour connected matrix element corrections}
\label{clapp}
We will here describe the splitting functions that are used for the
local sub-collisions (see point~\ref{enum:splitfn} in
section~\ref{sec:MC}). These are derived from the corresponding {\em
  colour connected} $2\rightarrow 2$ QCD matrix elements presented in
\cite{HUB&co}.

In order to use matrix elements for deriving splitting functions, it
is necessary to take colour connections into account. We can
illustrate this statement with the matrix element for the
$g+g\rightarrow q+\bar{q}$ process. Suppose we are in a frame where
the incoming gluons are moving head on and parallel to the
longitudinal axis and let $\zeta_{+(-)}$ be the positive (negative)
light-cone momentum fraction for the quark and $\bar{\zeta}_{+(-)}$ be
the same for the anti-quark.  Because of symmetry between the quark
and the anti-quark it is clear that the matrix element must be
symmetric in these variables: $|{\cal
  M}|^2(\zeta_{+(-)},\bar{\zeta}_{+(-)})= |{\cal
  M}|^2(\bar{\zeta}_{+(-)},\zeta_{+(-)})$.  In the program, the
colours are connected randomly so that e.g. the quark has the same
probability to be connected with the gluon coming in from either the
proton or the photon side. This would mean that the two diagrams ${\rm
  a}')$ and ${\rm b}'\!\!\!\!\! \bigtimes \!\!)$ in \fig{colcon}
would be equally probable.

\begin{figure}[t]
\begin{center}
\epsfig{figure=colcon.eps,width=10cm, height=8cm}
\end{center}
\caption[dummy]{{\it Feynman vs.\ colour connection diagrams for the $g+g\rightarrow
q+q$ sub-process. If we do not average the colour states, the
diagrams a') and b') give separate contributions. The contribution
from diagram b') should be suppressed.}}
\label{colcon}
\end{figure}

This is because the matrix element is calculated with an averaging of
the colour states of the gluons. The corresponding colour connected
matrix element, $|{\cal M}_{cc}|^2$ is on the other hand calculated
with the assumption that the colour states of the incoming gluons are
known. This means that we get two separate contributions, one from the
case that the quark is colour connected with the gluon coming in from
one side, say the proton side, and one contribution from the case that
it is connected with the gluon from the photon side. Each of these
contributions is non-symmetric with respect to the quark and anti-quark
variables, while the sum of them still is symmetric.

A similar procedure is used in the ${\cal O}(\alpha\as)$ matrix
element correction (see point~\ref{enum:wend} in
section~\ref{sec:MC}), to separate the $\gamma g$ matrix element into
two contributions corresponding to quark and anti-quark scattering
respectively as described in \refc{Mike}.

For a colour connected matrix element 
$|{\cal M}_{cc}|^2(\hat{s},\hat{t},\hat{u})$, the
corresponding splitting function is given by the formula
\begin{eqnarray}
P(z)\propto z(1-z)\cdot |{\cal M}_{cc}|^2(1,z,1-z) \ \ \ \ 
{\rm with}\ \ z=z_+\approx z_-.
\end{eqnarray}
The splitting functions $\Pslash_{ijk}(z)$ where the colour factor 
is divided out and with the successive flavours $ijk$ for
the propagators, become
\begin{eqnarray}
\Pslash_{qqq}(z)&=& (1-z)^2\left[z^2+(1-z)^2\right]\nonumber \\
\Pslash_{qqg}(z)=\Pslash_{gqq}(z)&=&0 \\
\Pslash_{qgq}(z)&=&\frac{1-z}{z}\left[1+(1-z)^2\right]\nonumber \\
\Pslash_{gqg}(z)&=& (1-z)^2\left[z^2+(1-z)^2\right]\nonumber \\
\Pslash_{qgg}(z)=\Pslash_{ggq}(z)&=&\frac{1}{2z}
\left[1+(1-z)^2\right]^2\nonumber \\
\Pslash_{ggg}(z)&=&\frac{1}{3}\nonumber 
\left\{
\frac{\left[1-z(1-z)\right]^2}{z(1-z)}\cdot\theta(0.5-z)+2\frac{1-z}{z}
\left[1-z+z^2\right]^2
\right\}.
\end{eqnarray}
The $\theta$-function in the first term of $\Pslash_{ggg}$ is a
cut-off to prevent divergences. This is needed since the Sudakov
form factors that are used regularize only regions with ordered
virtualities. The splitting functions $\Pslash_{qqg}$ and
$\Pslash_{gqq}$ are set to zero since they correspond to the
same ($g+q\rightarrow g+q$) scattering as the splitting functions
$\Pslash_{qgg}$ and $\Pslash_{ggq}$ which have been chosen to take
the whole contribution.

Here we have summed the contributions from different colour 
connections for each flavour combination. The choice of colour
connections has an effect on the multiplicity and transverse
momentum distributions at the hadronic level. Therefore, we will
in the future use this information also to choose the colour
connections for the local sub-collisions.
\end{appendix}


% Local Variables: 
% mode: LaTeX
% TeX-master: "LDC"
% End: 
\section{Fitting the input parton densities}
\label{sec:pdf}

From \eq{eq:evolved} we can write the leading order expression for
\tf2\ as
\begin{eqnarray}
  \label{eq:F2}
  \f2=\sum_{i\ne 0} e_i^2 \sum_j\int_x^1\frac{dx_0}{x_0}
  \left[\right.\begin{array}[t]{l}G_{ij}(x,Q^2,x_0,k_{\perp 0}^2)+\\~\\
    S_j(Q^2,k_{\perp
    0}^2)\delta_{ij}(\ln{x}-\ln{x_0})\left.\right]
  x_0f_{0j}(x_0,k_{\perp 0}^2).\end{array}
\end{eqnarray}

For given values of $x$, $Q^2$, $x_0$, $k_{\perp 0}^2$, $i$ and $j$,
we can calculate $G_{ij}(x,Q^2,x_0,k_{\perp 0}^2)$ from steps
\ref{enum:wbegin} through \ref{enum:wend} in the previous section
using \eq{eq:Gexact}. For any given parametrization of the input
densities it is then possible to calculate \tf2\ and compare with
experimental data.

In principle one could also fit to other data, such as prompt photon
and jet production in hadron-hadron collisions. This is not possible
in our current implementation which only gives the evolved densities
for quarks and anti-quarks. This means that the input gluon
distribution is only constrained indirectly from the \tq2\ dependence
of \tf2, and in this paper we only make a very crude fit using only
four different parameters for $x_0f_{0j}(x_0,k_{\perp 0}^2)$.

The input densities are parametrized as
\begin{equation}
  \label{eq:f0param}
  xf_{0j}(x) = A_j x^{\alpha_j}(1-x)^{\beta_j}.
\end{equation}
For the valence distributions $u_v(x)$ and $d_v(x)$ we use the same
form, with $\beta_v=3$ leaving $\alpha_v$ free and using the normalization
\begin{equation}
  \label{eq:valnorm}
  \int_0^1u_v(x)dx=2,\mbox{~~~}\int_0^1d_v(x)dx=1
\end{equation}
to fix $A_{uv}$ and $A_{dv}$. For the gluon distribution, $\beta_g=4$
while $\alpha_g$ and $A_g$ are left free. All the sea-quarks
distributions have the same form with $\beta_S=4$, leaving $\alpha_S$
free and setting $2A_s=2A_{\bar{s}}=A_u=A_{\bar{u}}=A_d=A_{\bar{d}}$
so that the total momentum
\begin{equation}
  \label{eq:allnorm}
  \int_0^1dx\sum_jxf_{0j}(x)=1
\end{equation}
is conserved.

We only use data from proton \tf2\ measurements from H1\cite{H1F2},
ZEUS\cite{ZEUSF2}, NMC\cite{NMCF2} and E665\cite{E665F2} without
allowing for any normalization uncertainty factors. We use only data
for $\q2>1.5$ GeV and $x<0.5$ to ensure a reasonable length of the
evolution. We then make six sets of fits using different options in
the generation of the $G$ function:
\begin{Aenumerate}
\item LDC default: $k_{\perp 0}=0.6$ GeV,
   $\Lambda=0.22$ GeV.
\item \label{enum:DGLAP} DGLAP: As for A but only allow chains with
   monotonically increasing virtualities of the links from the proton
   side.
\item DGLAP': As for B, but chains where the virtuality of the link
   closest to the virtual photon is larger than $Q^2$ are
   permitted. We use this as a kind of higher-order corrected DGLAP
   evolution although, of course, not equivalent to NLO evolution.
\item As for A but $k_{\perp 0}=1$ GeV, to check the
  sensitivity to this cutoff.
\item As for A but without the Sudakov form factor. Instead
  $P_{q\rightarrow q}(z)$ is set to zero and $P_{g\rightarrow q}(z)$
  is nonzero only in the splitting closest to the photon.
\item As for A but $\beta_g = \beta_S = 5$ to check the
  sensitivity to the fit parameters.
\item As for A but only fitting to \tf2\ data with $x<0.1$, to reduce
  the sensitivity to the step size $\delta \ln{x_0/x}=0.2$ used when
  integrating \eq{eq:F2}, and to the high-$x$ form of the input
  parametrization.
\item As for A but allow the virtuality of some links to be below
  $k_{\perp 0}$ as long as the largest virtuality of two consecutive
  links always is above $k_{\perp 0}$.
\end{Aenumerate}

\begin{figure}[t]
  \vskip -1cm
  \hbox{
    \hskip -1.5cm
    \input figF2xa.tex
    \hskip -1.5cm
    \input figF2xb.tex
    }
  \begin{center}
    \input figF2xc.tex
  \end{center}
  \caption[dummy]{{\it The fitted \tf2\ as a
      function of $x$ for different strategies and for two different
      values of \tq2, compared with data from
      \cite{H1F2,ZEUSF2,NMCF2,E665F2}.}}
  \label{fig:F2x}
\end{figure}
\begin{figure}[t]
  \vskip -1cm
  \hbox{
    \hskip -1.5cm
    \input figgina.tex
    \hskip -1.5cm
    \input figginb.tex
    }
  \begin{center}
    \input figginc.tex
  \end{center}
  \caption[dummy]{{\it The fitted input gluon density as a function of
      $x$ for different strategies.}}
  \label{fig:gin}
\end{figure}

The results of the fits are presented in \figs{fig:F2x} and
\ref{fig:gin}. For the default case, the fit is quite acceptable. We
note in particular that the fitted input gluon density is slowly
decreasing with $1/x$, although we must keep in mind that the gluon
distribution is only indirectly constrained.

For the DGLAP case the fit is much worse. The number of allowed ISB
emissions is here strongly restricted, especially for small \tq2\ and
$x$. This results in much slower evolution which forces the input
densities to rise with $1/x$. In fit C, where the link closest to the
virtual photon is allowed to be above \tq2, the fit is on the other
hand again quite acceptable. The input gluon distribution is no longer
rising with $1/x$, on the other hand it is as strongly decreasing as
in fit A. It is known that \tf2\ can be fitted using conventional
DGLAP evolution with a valence-like flat input distribution at small
input scales, close to the one used here, as in the GRV
parametrizations \cite{GRV}.  Such fits can, however, not be directly
compared to this one, as we here have less parameters. But we can
conclude that the ISB chains with unordered \tkt\ do play a rle in
our case, although most of the effect can be obtained allowing only
for one `stepping down', closest to the photon in the chain.

Increasing the input scale to $1$ GeV in fit D makes a big difference
particularly at small $x$ as seen in \figs{fig:F2x}b and
\ref{fig:gin}b. Also here the number of allowed ISB emissions is
strongly restricted and again the input gluon is forced to increase
strongly with $1/x$. This should come as no surprise. We expect,
however, that when we below study the hadronic final states, the
result should be less sensitive to the input scale used.

The importance of the Sudakov form factor is apparent in
\figs{fig:F2x}b and \ref{fig:gin}b, comparing fits A and E, especially
for the input gluon distribution. Also in \figs{fig:F2x}b and
\ref{fig:gin}b we show the fit H, where some propagators below the
cutoff is allowed. The reproduction of \tf2\ does not change much, but
we see that the input gluon decreases slightly faster with $1/x$ than
for the default fit A.

The fits F and G in \figs{fig:F2x}c and \ref{fig:gin}c show how
sensitive the fit is to the input distribution is at large $x$.
Changing $\beta_g$ and $\beta_S$ from $4$ to $5$ does not influence
the fit very much and neither does the omission of the data points at
high $x$, although in both cases the input gluon is shifted somewhat
to higher $x$.

% Local Variables: 
% mode: LaTeX
% TeX-master: "LDC"
% End: 
\section{The Monte Carlo implementation}
\label{sec:MC}

%\subsubsection*{Implementation}

The LDC model has been implemented in a Monte Carlo program and some
results within the leading log approximation have already been
presented. Here we present a more complete implementation taking into
account some non-leading corrections and generating complete events
all the way down to the final state hadron level. What follows is a
step-by-step description of the procedure.

The basic formula for the evolution of the parton densities is
\begin{eqnarray}
  x f_i(x,Q^2) = \sum_j\int_x^1 \frac{dx_0}{x_0}
  \left[\right.\begin{array}[t]{l} G_{ij}(x,Q^2,x_0,k_{\perp 0}^2)+\\~\\
  \left. S_j(Q^2,k_{\perp
    0}^2)\delta_{ij}(\ln{x}-\ln{x_0})\right] x_0f_{0j}(x_0,k_{\perp 0}^2),\end{array}
  \label{eq:evolved}
\end{eqnarray}
Where $G_{ij}(x,Q^2,x_0,k_{\perp 0}^2)$ is the sum of the weights of
all chains starting with a parton $j$ at some low scale $k_{\perp
  0}^2$ carrying a momentum fraction $x_0$, and ending up with a
parton $i$ carrying a momentum fraction $x$ being hit by a photon with
virtuality $Q^2$. The delta function corresponds to the case of no
emissions. $G$ is positive definite, and to conserve the total
momentum, $\sum_j\int_0^1xf_j(\q2,x)dx = 1$, the delta function is
multiplied by a Sudakov form factor, $S_j(Q^2,k_{\perp 0}^2)$,
representing the probability that the parton $j$ with with momentum
fraction $x_0$ at $k_{\perp 0}^2$ has not split, and thus reduced its
momentum, when probed at a higher scale \tq2. This form factor will be
discussed in detail below.

The analytic approximate upper limiting function for $G$
\cite{kharrLDC2}, given by
\begin{eqnarray}
  \label{eq:Ganalytic}
  &G_{ij}(x,Q^2,x_0,k_{\perp 0}^2) \laeq G(Q^2/k_{\perp 0}^2,x/x_0) = 
  \sqrt{\frac{a}{b}}I_1(2\sqrt{ab})&\\
  &a=\sqrt{\alb}(\ln{Q^2/k_{\perp 0}^2}+\ln{x_0/x}),\mbox{~~~}
  b=\sqrt{\alb}\ln{x_0/x}&\nonumber
\end{eqnarray}
provides us with the starting point, and for each generated chain $c$
a number of multiplicative weights are calculated
$\omega_c=\Pi_l\omega^{(l)}_c$ so that the correct form of $G_{ij}$ is
obtained as
\begin{equation}
  G_{ij}(x,Q^2,x_0,k_{\perp 0}^2) =
  \bar{\omega}_{ij}(x,Q^2,x_0,k_{\perp 0}^2)G(Q^2/k_{\perp 0}^2,x_0/x),
  \label{eq:Gexact}
\end{equation}
with the average weight
\begin{equation}
  \bar{\omega}=\frac{1}{N}\sum_{c=1}^{N}\omega_c
\end{equation}

This is how it is done:
\begin{enumerate}
\item First the $x$, $Q^2$ and the flavour $i$ of the struck quark is
  chosen using evolved parton densities and the standard Born-level
  electro-weak matrix elements. This is currently done within the
  \lepto\ program \cite{LEPTO}.
\item Then the $x_0$ and flavour of the incoming parton is chosen
  according to \eqs{eq:evolved} and (\ref{eq:Ganalytic}) with
  $k_{\perp 0}$ as a given parameter.
\item The number of emissions is chosen from
  \label{enum:wbegin}
  \begin{equation}
    \sqrt{\frac{a}{b}}I_1(2\sqrt{ab}) =
    \sum_{n=1}^{\infty}\frac{a^{n}b^{n-1}}{n!(n-1)!}
  \end{equation}
\item The positive and negative light-cone momentum fractions,
  $z_{j+}$ and $z_{j-}$ which enters in each emission $j$ is generated
  according to the ordered integral
  \begin{equation}
    \frac{a^{n}b^{n-1}}{n!(n-1)!} =
    \int\alb^n\Pi_j\frac{dz_{j+}}{z_{j+}}\frac{dz_{j-}}{z_{j-}}
    \delta(\ln{x_0}+\sum_j \ln{z_{j+}}-\ln{x})
    \label{eq:zgen}
  \end{equation}
\item At this point we need to choose the
  flavours of each link. To do this we introduce the standard
  Altarelli-Parisi splitting functions and preliminary we use
their approximations, $\tilde{P}_{i\rightarrow j}(z)$, in the 
$z\rightarrow 0$ limit
  \begin{eqnarray}
    P_{q\rightarrow q}(z) &=& C_F\frac{1+z^2}{1-z}
\approx C_F,\nonumber\\
    P_{g\rightarrow g}(z) &=& 2N_C\frac{(1-z(1-z))^2}{z(1-z)}
\approx 2N_C\frac{1}{z},\nonumber\\
    P_{g\rightarrow q}(z) &=& T_R(z^2+(1-z)^2)
\approx T_R,\\
    P_{q\rightarrow g}(z) &=& C_F\frac{1+(1-z)^2}{z}
\approx C_F\frac{2}{z},\nonumber
  \end{eqnarray}
  We get the first weight factor as the approximated splitting
  functions summed over all possible flavour combinations,
  \begin{equation}
    \label{eq:omega1}
    \omega^{(0)} =
    \sum\Pi 
\frac{\tilde{P}_{i\rightarrow j}(z_+)}{\tilde{P}_{g\rightarrow g}(z_+)}.
  \end{equation}
\item We can then use \eq{eq:omega1} to generate a specific flavour
  combination according to their individual weights.
\item Next, we generate the azimuthal angles of each emission and
  construct the exact kinematics. The delta function in \eq{eq:zgen},
  which handles the conservation of positive light-cone momenta, does
  not take into account the transverse degrees of freedom. In
  particular it would give zero positive light-cone momentum for the
  struck quark, $q_{n+1}$, in the final-state. We therefore modify
  this delta function to exactly conserve the total energy and
  momentum, effectively setting $z_{n+}$ by hand to the value
  needed. However, for some values of the azimuth angles this is not
  possible and we get a weight factor corresponding to the allowed
  integration area $\Delta \phi_j$:
  \begin{equation}
    \label{eq:genzphi}
    \omega^{(1)}=\Pi_j\frac{1}{2\pi}\int_{\Delta \phi_j}d\phi_j.
  \end{equation}
\item Then we implement the condition that the transverse momenta
  (which is generalized to the transverse mass $m_\perp$ for massive
  partons) of an emitted parton must be larger than the smallest
  virtuality $v_{j\min}$ of the connecting links $j$ and $j-1$, and
  that all virtualities must be above $k_{\perp 0}^2$, giving us the
  second weight factor
  \begin{equation}
    \label{eq:omega2}
    \omega^{(2)}=\Pi_j\Theta(m_{\perp j}^2-v_{j\min})
    \Theta(v_{j\min}-k_{\perp 0}^2).
  \end{equation}
  We note that the weights are finite even if one of $v_j$ and
  $v_{j-1}$ goes to zero, and one could imagine replacing the second
  theta function in \eq{eq:omega2} with $\Theta(v_{j\max}-k_{\perp
    0}^2)$. This would reduce the dependency on $k_{\perp 0}$ and
  would allow for more unordered chains as discussed below.
  
\item Now we introduce the running of \tas, giving a fourth weight
  \begin{equation}
    \label{eq:omega4}
    \omega^{(3)} = \Pi_j \frac{1}{\ln(m_{\perp j}^2/\Lambda^2)}
  \end{equation}
\item 
\label{enum:splitfn} 
Having obtained the virtualities of the links, we can now
  correct the splitting functions in \eq{eq:omega1}. We get the
  following cases:
  \begin{itemize}
  \item $v_{j+1}>v_{j}>v_{j-1}$: Going
    upwards from the proton side we use $P_{f_{j-1}\rightarrow f_{j}}(z_{j+})$
  \item $v_{j+1}<v_{j}<v_{j-1}$: Going
    upwards from the photon side we use $P_{f_{j+1}\rightarrow f_{j}}(z_{j-})$
  \item $v_{j+1}<v_{j}>v_{j-1}$: Corresponds to a Rutherford
    scattering and $z_{j+}\approx z_{j-}\equiv z$. Here we use
    $2\rightarrow 2$ matrix elements taking into account colour
    connections as explained in Appendix A.
  \item $v_{j+1}>v_{j}<v_{j-1}$: Here we use the $z\rightarrow 0$
    limit, $\tilde{P}_{i\rightarrow j}(z)$ of the splitting functions,
    where $z=z_+$ if $v_{j+1}>v_{j-1}$ or else $z=z_-$.
  \end{itemize}
  Note that the colour factor is independent of the ordering of the
  virtualities. Therefor we have to correct for only the kinematical
  part of the splitting function by using reduced splitting functions
  \tPslash\ where the colour factor is divided out.  Here we also
  introduce the Sudakov form factor, to be discussed below and we can
  write the fourth weight factor
  \begin{equation}
    \label{eq:omega5}
    \omega^{(4)}=\Pi_j
    \frac{S_j(v_{j-1},v_{j},v_{j+1})\Pslash^{v_{j-1}v_{j}v_{j+1}}
                      _{f_{j-1}f_{j}f_{j+1}}(z_{j+},z_{j-})}
    {\tilde{\Pslash}_{f_{j-1}\rightarrow f_{j}}(z_{j+})},
  \end{equation}
  where $S_j(v_{j-1},v_{j},v_{j+1})$ is $S_{f_{j-1}}(v_j,v_{j-1})$ or
  $S_{f_{j+1}}(v_j,v_{j+1})$ depending on whether the virtuality is
  going up or down.

  $P_{q\rightarrow q}$ and $P_{g\rightarrow g}$ both have poles as
  $z\rightarrow 1$, corresponding to emission of low-energy
  gluons. Typically these should be counted as final-state emissions,
  but to be sure to avoid divergences we introduce a cutoff
  $z_{\mbox{\tiny cut}}=0.5$. See also the discussion of double
  counting below.
  
  Note also that we use mass less splitting functions, and the
  production of heavy quarks is only suppressed by the phase space.
  This should be improved in the future.

\item \label{enum:wend} The final weight factor is introduced to correct
  the emission closest to the photon, in the cases where $v_n>v_{n-1}$,
  to reproduce the exact ${\cal O}(\alpha\as)$ matrix element as given
  eg.\ in \refc{Kramer}:

  \begin{equation}
    \label{eq:omega6}
    \omega^{(5)}=\frac{{\cal M}(Q^2,x,z_{n+},z_{n-})}
    {P^{v_{n-1}v_{n}Q^2}_{f_{n-1}f_{n}\gamma}(z_{n+},z_{n-})}
  \end{equation}
\item The generated chain is now kept with a probability
  $\omega_c=\frac{1}{W}\Pi_l\omega^{(l)}_c$, where $W$ is a scale
  factor to avoid probabilities larger than one. There is in principle
  nothing preventing weight larger than one, but they turn out to be
  very rare. Nevertheless, it may happen, and it is important to check
  that $W$ is large enough so that the results are not influenced by
  this. Chains with $\omega>1$ may optionally be saved and retrieved
  again when an event with the same flavour and similar $x$ and \tq2\
  is requested. A chain will then be used on the average $\omega_c$
  times, each time with different final-state cascade and
  hadronization. Below we have used $W=1$ giving less than 0.1\%
  events with weight larger than one.
\item To prepare for the final-state dipole radiation the emitted
  partons must be connected together and form dipoles. In the case of
  quark links, this is straight forward, the incoming quark is simply
  connected to the first emitted gluon and so on until the struck
  quark. In the case of gluon links, there are two colour lines, and a
  radiated gluon can belong to either of these. This choice is done
  completely at random. The connection between the colour line of the
  incoming parton and the proton remnant is handled in the same way as
  in the default soft radiation model of \ariadne\cite{ARIADNE}.
  
  One could imagine using other methods for determining the
  colour-flow. One suggestion is to use the colour-flow which minimize
  the total string length\footnote{As defined eg.\ by the $\lambda$
  measure of \refc{lambda}}. One could also consider colour-{\bf
  re}connections, following eg.\ the model already implemented in
  \ariadne\ \cite{colrec}, possibly giving rise to large rapidity gaps
  among the final-state hadrons.
\item The constructed dipoles can then radiate more final-state gluons
  in the phase-space limited by the virtuality and the positive and
  negative light-cone momenta of the links in the chain as in
  \fig{LDCTRI}a. Note that also in the no-emission case corresponding
  to the delta function in \eq{eq:evolved}, some radiation is allowed
  within the triangular area defined by $q_+<-Q_+$ and $q_-<Q_-$. In
  the Breit frame, this is just the area of allowed FSB in a
  $e^+e^-\rightarrow q\bar{q}$ event with centre of mass energy $Q$.
\item Finally the final state dipole chains are hadronized according
  to the Lund string fragmentation model as implemented in \jetset\
  \cite{JETSET}.
  
\end{enumerate}

This concludes the description of the actual implementation. But
before we can start producing events we have to fix the parameters
involved. These are $\Lambda$, $k_{\perp 0}$, $W$, $z_{\mbox{\tiny
cut}}$ and the input parton densities $x_0f_{0j}(x_0,k_{\perp 0}^2)$.
$W$ is not really a physical parameter, and should be set large enough
so that the result no longer depend on it. It would be natural to take
$\Lambda$ and $k_{\perp 0}$ to be the values which have been tuned for
the final state dipole cascade to reproduce LEP data,
$\Lambda_{\mbox{\tiny LEP}}$ and $k_{\perp 0\mbox{\tiny LEP}}$. One
could, of course, use an increased cutoff $k_{\perp 0}>k_{\perp
0\mbox{\tiny LEP}}$ in the ISB. This would mean that more emissions
would be moved from the initial to the final state, which would
continue emitting down to $k_{\perp 0\mbox{\tiny LEP}}$. We therefore
expect the final result to be fairly stable w.r.t.\ such variations as
long as $k_{\perp 0}$ is not too large.

There is an additional complication with a large $k_{\perp 0}$ for the
cases where the virtuality drops below $k_{\perp 0}$ somewhere along
the chain, causing a zero weight in \eq{eq:omega2}. For the total
cross section, this does not matter, as such fluctuations are included
in the input parton densities at a lower $x_0$, corresponding to the
momentum fraction of the link closest to the photon which is below the
cutoff. For the final state, however, it means that we are excluding
some radiation close to the direction of the incoming hadron.

As discussed above, one could replace the second theta function in
\eq{eq:omega2} with $\Theta(v_{j\max}-k_{\perp 0}^2)$ since the
$m_\perp^2$ then would still be in the perturbative region. In this
way the result would be less sensitive to variations of $k_{\perp 0}$,
as the perturbative system on the proton side of the sub-cutoff link
would still be generated. This would not, of course, solve the problem
altogether as one can imagine chains where two or more consecutive
links are below the cutoff.

%\subsubsection*{The problem of double counting}
In step \ref{enum:splitfn} we have replaced the $N_c/z$ pole, which is
used in the emissions of the original leading log LDC model, with the
standard Altarelli-Parisi splitting functions. This should be a
sensible way of including some sub-leading effects, as long as the
splitting functions are regularized in a correct way. In each
emission, two particles are produced and one vanishes. For the parton
distributions, this means we must subtract and add partons
correspondingly. 

A simple way of treating this double counting problem is to add only
one of the produced partons assuming that the mother parton is not
affected by the emission.  The choice of which of the two partons to
add is not trivial. One way is to introduce a cut-off $z_{cut}=0.5$
allowing only $z<z_{cut}$. This is a good choice in the $g\rightarrow
g$ splitting since the gluon with $z>0.5$ is more similar to the
mother gluon. For the other emissions, which involve both quarks and
gluons, it is more important to make the choice which leads to a better
approximation of the quark distributions. This can be done by allowing
all $q \rightarrow g$ splittings, forbidding all $q \rightarrow q$
splittings and allowing $g \rightarrow q$ splittings only if the quark
(or anti-quark) interacts directly with the photon.

In a more sophisticated treatment, the subtraction of a parton is done
with Sudakov form factors corresponding to the probability for the
partons not to vanish before the splitting can occur. In this way, we
can take into account that e.g. the possibility for a gluon to split
into a quark anti-quark pair with low virtuality reduces its
contribution to emissions with higher virtualities. The suppression
factor becomes an exponential of an integral over the splitting
\begin{equation}
S_g=\exp\left[ -\int_0^1 P_{g\rightarrow q}(z)dz \int
\frac{\alpha_s(q_\perp^2)}{2\pi} \frac{dq_\perp^2}{q_\perp^2} \right].
\end{equation}
The region of integration corresponds to the region of allowed
emissions. Here we only use an approximate form. A more thorough
investigation will be presented in a future paper.

The lower limit on $q_\perp^2$ is given by the lowest virtuality of an
emission step multiplied by a fudge factor, $e^\delta$, to account
for the suppression of the emission probability for small
$q_\perp$-values (\eq{qtrest}). The value of $\delta=0.4$ that is used
has proven to be an effective cut-off in the leading log treatment of
the LDC model \cite{kharrLDC2,HamidNext}. The upper limit is set
to the highest virtuality of an emission step. One could imagine
having a higher, or a $z$ dependent limit, but for simplicity we only
use the virtuality in this publication.

The Sudakov factor due to the $g\rightarrow g$ splitting depends on
the choice of $z_{cut}$. For $z_{cut}=0.5$, there is no double
counting to correct for and for larger $z_{cut}$ the integration of
the splitting function is in the region $0.5<z<z_{cut}$. If $q
\rightarrow q$ splittings are allowed, quarks are suppressed with a
Sudakov factor where the integration region is given by $0<z<z_{cut}$.

The situation is quite different when we are interested in the final
state properties. Here, the Sudakov form factors are not as important
since they for most events roughly give an overall factor
$\prod_{j=1}^nS(v_{j-1},v_j,v_{j+1})\sim S(k_{\perp 0}^2,Q^2)$ which
does not have an influence on the relative contributions of different
final states. On the other hand, disallowing some of the initial state
emissions to reduce double counting has a large effect on the final
state properties since it reduces high $p_\perp$ emissions. This is a
problem except for gluon emissions with $z>0.5$ in regions with ordered
virtuality, since these gluons can be treated as final state emissions
as shown in \fig{fig:largez}.  Consequently, we would expect it to be
a good approximation for the final state generation to skip the
Sudakov form factors and to allow some of the splittings which lead to
double counting.

 
\begin{figure}[t]
  \begin{center}
    \hbox{
      \epsfig{figure=largezemm.eps,width=12cm,height=4cm}
      \hspace{0.5cm}
      \psfig{figure=koord.ps,width=2cm,height=4cm}
      }
    \caption[dummy]{{\it In the case where the virtuality is increasing,
        a splitting with $z$ larger than 0.5 corresponds to an
        emission of a gluon $q$ which is inside the phase space where
        subsequent FSB emissions is allowed. Such splittings should
        therefore be removed from the ISB chains to avoid double
        counting.}}
    \label{fig:largez}
  \end{center}
\end{figure}

It is clear that the result is very dependent on the non-perturbative
input parton densities, which are basically unknown. If eg.\ the gluon
density is very divergent at small $x$, the $x_0$ chosen from
\eq{eq:evolved} will tend to be small, limiting the total phase space
available for radiation $\Delta y\propto\ln{x_0/x}$. The input parton
densities can, however, be constrained somewhat from the total cross
section, and we can parametrize them and make a fit of the parameters
to eg.\ \tf2\ data at different $x$ and $Q^2$.

% Local Variables: 
% mode: LaTeX
% TeX-master: "LDC"
% End: 
\section{The Linked Dipole Chain model}
\label{sec:ldc}

It is well known that the cross section for DIS events is not
describable only by the lowest order perturbative terms. Still, it is
not necessary to consider all possible emissions. A large set of them
can be summed over and do, in principle, not affect the cross section.
In the LDC model \cite{LDC96}, the emissions that are considered to
contribute to the cross section are regarded as Initial State
Bremsstrahlung (ISB). The description of these emissions is based on
the CCFM \cite{CCFM} model which is a leading-log approximation 
of the structure function evolution in DIS. The CCFM model has been
modified by redefining which emissions should be counted as ISB,
resulting in a much simplified description.

To describe final state properties of DIS events, one must also
consider Final State Bremsstrahlung (FSB). In the LDC model this is
done within the framework of the Colour Dipole cascade Model (CDM)
\cite{kharrDIPOLE} which has previously proved to give a good
description of parton cascades in hadronic $e^+e^-$ events and DIS.
The general picture is that the initial parton ladder builds a chain
of linked colour dipoles and that the FSB is radiated from these
dipoles.

%\subsection{The Initial State Bremsstrahlung}
Let $\{q_i\}$ denote the momenta of emitted partons, $\{k_i\}$ denote
the momenta of the propagators (see \fig{fan})
\begin{figure}
  \begin{center}
    \epsfig{figure=fan.eps,width=10cm,height=4cm}
  \end{center}
  \caption[dummy]{{\it Lepton proton scattering with $n$ perturbative
      ISB emissions.  The emitted ISB partons are denoted $\{ q_i \}$
      and the propagators are denoted $\{ k_i \}$.}}
  \label{fan}
\end{figure}
and $z_{+i}$ be the positive light-cone momentum
fraction\footnote{Note that throughout this paper we work in the
$\gamma^*p$ centre of mass system with the proton along the positive
$z$-axis, except in section \ref{sec:res} when comparing with
experimental data, where the $\gamma^*$ is along the positive
$z$-axis.} of $k_i$ in each emission:
$k_{+i}=z_{+i}k_{+(i-1)}$. According to the CCFM model, the emissions
that contribute to the cross section (ISB) are ordered in rapidity and
energy. Furthermore, there is a restriction on the transverse momenta
of the continuing propagator in each emission:

\begin{equation}
k^2_{\perp i} > z_{+i}q^2_{\perp i}.
\end{equation}
The weight distribution, $dw$, of the initial chains 
factorizes, with the factors $dw_i$ given by the following expression
(${\bar \alpha}=3\alpha_s/\pi$):

\begin{eqnarray}
dw&=&dw_1dw_2\cdots dw_n, \nonumber \\
dw_i&=&{\bar \alpha} 
\frac{dz_{+i}}{z_{+i}} \frac{d^2q_{\perp i}}{\pi q^2_{\perp i}}
\Delta _{ne} \left(z_{+i},k_{\perp i},q_{\perp i}\right).
\end{eqnarray}
$\Delta _{ne}$ is the so called
non-eikonal form factor, given by the expression:
 
\begin{equation}
\Delta _{ne}\left(z,k_\perp,q_\perp\right)=
\exp \left[-{\bar \alpha} \log \left(\frac{1}{z}\right) 
\log \left(\frac{k^2_\perp}{zq^2_\perp}\right)\right].
\end{equation}

In the LDC model, the definition of the ISB is more restricted. Consequently,
more emissions are summed over and the expression for the weight distribution,
$dw_i$, is changed. The new restriction is that in each emission, the 
transverse momentum ($q_{\perp i}$) of the emitted parton must be larger
than the lower one of the transverse momenta of the surrounding propagators
\begin{equation}
q_{\perp i} > \min \left(k_{\perp i},k_{\perp (i-1)} \right).
\label{qtrest}
\end{equation}

In \fig{ccfmvsldc} we show an example of an emission which
belongs to the ISB according to the CCFM model but violates the
restriction in \eq{qtrest} and is regarded as FSB in the LDC model.

\begin{figure}[t]
  \vbox{
    \hbox{
      \epsfig{figure=ccfmvsldc1.eps,width=14cm,height=3cm}
      }
    \vspace{1.5cm}
    \hbox{
      \epsfig{figure=ccfmvsldc2.eps,width=11cm,height=4cm}
      \hspace{0.5cm}
      \psfig{figure=koord.ps,width=2cm,height=4cm}
      }
    }
  \caption[dummy]{{\it CCFM $\rightarrow$ LDC: The emission $q$ is
      regarded as ISB in the CCFM model but in LDC model it is
      regarded as FSB.  This is illustrated (above) with fan diagrams
      and (below) in a $\log p_\perp^2-$rapidity diagram.  Rapidity
      here is defined as $\log (p_+/p_\perp )$ and the dashed lines
      indicate equal $p_+$.}}
  \label{ccfmvsldc}
\end{figure}

Summing over these emissions, the weight distribution of each allowed
emission now becomes
\begin{equation}
  dw_i={\bar \alpha} \frac{dz_{+i}}{z_{+i}} \frac{d^2q_{\perp i}}{\pi
    q^2_{\perp i}}.
\end{equation}
This simplification of the expression for $dw_i$ is due to the fact
that one can interpret the non-eikonal form factor, $\Delta _{ne}$, as
a Sudakov form factor, that is, it is equal to the probability of not
violating the restriction in \eq{qtrest}.

By changing variables to the propagator momenta and integrating $dw_i$ 
over the azimuthal angle (in the transverse plane) it can be written
approximately as:

\begin{equation}
dw_i=\bar{\alpha} 
\frac{dk_{\perp i}^2}{k^2_{\perp i}}
\frac{dz_{+i}}{z_{+i}}
{\rm min} \left(1,\frac{k^2_{\perp i}}{k^2_{\perp (i-1)}} \right).
\end{equation}

It is instructive to look at three different possibilities for two
subsequent emissions, specified by different orderings of the
transverse momenta of the propagators, numbered $k_1$, $k_2$ and $k_3$
(with the same order as in \fig{fan}):

\begin{itemize}
\item $k_{\perp 1}<k_{\perp 2}<k_{\perp 3}$: This is included in DGLAP
  and gives the same weight as there: $dw_2dw_3 \propto
  \frac{1}{k^2_{\perp 2}} \frac{1}{k^2_{\perp 3}}$.
\item $k_{\perp 1}<k_{\perp 2}>k_{\perp 3}$: This gives the weight
  $dw_2dw_3 \propto \frac{1}{k^4_{\perp 2}}$ and resembles a hard
  sub-collision with a momentum transfer $\hat{t} \simeq k^2_{\perp
  2}$.
\item $k_{\perp 1}>k_{\perp 2}<k_{\perp 3}$: Here $dw_2dw_3 \propto
  \frac{1}{k^2_{\perp 1}} \frac{1}{k^2_{\perp 3}}$, it has no factor
  $\frac {1}{k^2_{\perp 2}}$ and is thus infrared safe!
\end{itemize} 

%\subsubsection{Leading Log Results}
The LDC model, without corrections to the leading log approximation,
has previously been studied and some qualitative results for the
structure functions and final state properties in DIS have been
presented in \refs{LDC96,kharrLDC2,HamidDIS97,HamidNext}. It is found
that the LDC model, just as the CCFM model, interpolates smoothly
between DGLAP and BFKL. The emissions along the rapidity axis can be
separated in two phases: For rapidities closest to the proton
(forward) direction, the ISB chain performs a BFKL like motion with a
constant mean $E_\perp$-flow. At a certain distance from the photon
end, the transverse momentum of the emissions begins to rise to the
photon virtuality, as expected by DGLAP. For the structure functions,
a DGLAP behaviour ($\exp(\mbox{const}\sqrt{-\log x})$) is shown for
moderate values of $x$, but for small $x$-values it has the BFKL
$x^{-\lambda}$ behaviour.

A prediction from the BFKL model is that $\log{p_\perp}$ of the
emitted partons along the chain would be described by a Gaussian
distribution with a growing width $\propto \log{1/x}$. From the result
of the LDC model, one can clearly see that this BFKL behaviour is
indeed present for a constant coupling but not for a running coupling.
This is illustrated in \fig{kharrET} where the parton density at a
certain rapidity is plotted as a function of $\log{p_\perp}$ for
different values of $x$-Bjorken, for constant and running coupling.
The Gaussian behaviour is observed for constant coupling, while for a
running coupling it appears to decay exponentially.  It seems though
that the exponential decay is significant only for events with very
small $x$-values and will probably not be visible in currently
available data.

\begin{figure}
  \begin{center}
    \input st3.tex
  \end{center}
  \caption[dummy]{{\it The decrease of log of the parton density with
      $\ln p_\perp^2$ at a certain rapidity, for different $x$-values,
      for running (solid line) and constant (dashed line) coupling.}}
  \label{kharrET}
\end{figure}

%\subsection{The Final State Bremsstrahlung}

The dipole model \cite{kharrDIPOLE} was originally developed for
final-state parton cascades from a quark-anti quark system.  The phase
space for gluon emission from a $q-{\bar q}$ pair is approximately
given by the triangular area in \fig{kharrTRI1}a. The gluons are
assumed to be radiated from a $q-{\bar q}$ colour dipole and after
each emission, the dipole is split into smaller dipoles
(\fig{kharrTRI1}b), which continue to radiate independently under a
$p_\perp $-ordering condition (shaded area). Also $g\rightarrow
q+{\bar q}$ splittings have been included in this model.  The size of
the dipole triangle is determined by the total $q-{\bar q}$ invariant
mass $W$.

\begin{figure}
    \hbox{
      \psfig{figure=tri0.ps,width=4.5cm,height=6cm}
      \hspace{0.2cm}
      \vbox{
        \hbox{
        \psfig{figure=tri1.ps,width=4.3cm,height=5.6cm}
        \hspace{0.2cm}
        \psfig{figure=tri2.ps,width=4.3cm,height=5.6cm}
        }
      \vspace{0.4cm}
      }
    }

  \caption[dummy]{{\it (a) The phase space of parton emission from a
      $q-{\bar q}$ pair.  (b) After the first emission, the phase
      space is split into two triangle which radiate gluons
      independently, but under a $p_\perp$-ordering condition.  (c) An
      event with four emissions.}}
  \label{kharrTRI1}
\end{figure}


The momenta $\{ q_i \}$ of the emitted ISB partons in DIS are plotted in
\fig{LDCTRI}a. Due to
the ordering in positive and negative light cone momenta, one can
insert the ISB into a dipole triangle with a size determined by the photon
negative light cone momentum (left edge) and the positive light cone
momentum of the incoming (non-perturbative) gluon (right edge). After
doing this, the FSB partons can be emitted in a similar way as for the
$q-{\bar q}$ parton shower (\fig{LDCTRI}b). The phase space of the FSB
is the shaded area in \fig{LDCTRI}a.

\begin{figure}

  \hbox{
    \epsfig{figure=triLDC1.eps,width=6cm,height=7cm}
    \hspace{2cm}
    \vbox{
      \epsfig{figure=triLDC2.eps,width=5.7cm,height=6.5cm}
      \vspace{0.5cm}
      }
    }

  \caption[dummy]{{\it (a) ISB emissions plotted in a dipole triangle.
      (b) The solid circles are ISB emissions and the dashed circles are
      FSB emissions.}}
  \label{LDCTRI}
\end{figure}

% Local Variables: 
% mode: LaTeX
% TeX-master: "LDC"
% End: 
\section{Results for the hadronic final states at HERA}
\label{sec:res}

The results for the hadronic final states will depend on the input
densities used. If eg.\ the input densities are increasing with
$1/x_0$, chains starting with low $x_0$ will be favoured and the
length of the chains will be shorter resulting in fewer emissions and
less activity in general due to the reduction of the available phase
space, in particular close to the direction of the incoming hadron.
Also a smaller input gluon density will result in fewer chains
initiated with a gluon, which has higher charge than an incoming
quark, and again the probability of emissions will become smaller. In
addition, the non-perturbative hadronization is smaller if only one
string is stretched between the perturbative system and the hadron
remnant in the case of an incoming quark.

\begin{figure}
  \vskip -1.8cm
  \hbox{
    \hskip -1.7cm
    \input figcompa.tex
    \hskip -1.2cm
    \input figcompb.tex
    }
  \vskip -0.2cm
  \hbox{
    \hskip -1.7cm
    \input figcompc.tex
    \hskip -1.2cm
    \input figcompd.tex
    }
  \vskip -0.2cm
  \hbox{
    \hskip -1.7cm
    \input figcompe.tex
    \hskip -1.2cm
    \input figcompf.tex
    }
  \vskip -3mm
  \caption[dummy]{{\it Comparison of the default LDC~A model with HERA
      data as given in \cite{HZTOOL} and other event generators. The
      distributions are as follows: (a) The transverse energy flow as
      a function of pseudo rapidity for events with $0.0001<x<0.0002$,
      $5<Q^2/\mbox{GeV}^2<10$ \cite{ForwardJets}.  (b) As (a) but for
      events with $0.003<x<0.01$, $20<Q^2/\mbox{GeV}^2<50$. (c) The
      average squared transverse momentum of charged particles as a
      function of $x_F$ \cite{HZ95221}. (d) The transverse momentum
      distribution of charged particles in the pseudo rapidity bin
      $0.5<\eta<1.5$ for events with $0.0002<x<0.0005$,
      $6<Q^2/\mbox{GeV}^2<10$ \cite{ForwardParticles}. (e) The
      pseudo rapidity distribution of charged particles with a
      transverse momentum larger than 1 GeV for the same kinematical
      bin as in (d). (f) The two-jet ratio $R_2(x)$ as a function of
      $x$ \cite{R2H1}. All measurements were made in the hadronic
      centre of mass system and only events without a large rapidity
      gap were included.  The full line is LDC~A, long-dashed is
      \ariadne\ 4.08 with default parameter settings, dotted is
      \lepto\ 6.4 with default parameter settings and short-dashed is
      the same but with SCI and the special sea-quark remnant
      treatment \cite{LeptoSCI} switched off.}}
  \label{fig:comp}
\end{figure}

\begin{figure}
  \vskip -1cm
  \hbox{
    \hskip -2.0cm
    \input fig3compa.tex
    \hskip -1.0cm
    \input fig3compb.tex
    }
  \vskip 0.5cm
  \hbox{
    \hskip -2.0cm
    \input fig3compc.tex
    \hskip -1.0cm
    \input fig3compd.tex
    }
  \caption[dummy]{{\it Comparison between different LDC strategies and
      data corresponding to \fig{fig:comp}e. In all cases the full
      line is the default LDC~A strategy. In (a) the dashed line is
      LDC~B using only DGLAP-like chains and dotted is LDC~C, i.e.\ 
      the same but allowing the virtuality of the link closest to the
      photon to be above \tq2. In (b) the long-dashed line is LDC~D
      with $k_{\perp 0}=1$ GeV, the short-dashed is LDC~A$_z$ with
      increased cutoff, $z_{\mbox{cut}}$, in the splitting functions
      and dotted is LDC~H, allowing some propagators below the cutoff.
      In (c) the dashed line is LDC~E without Sudakov form factors and
      disallowing all $q\rightarrow q$ and most $g\rightarrow q$
      splittings in the parton density fit and the dotted, LDC~A$_0$,
      is the same as LDC~A but using Sudakov only in the parton
      density fit. Finally in (d) the dashed line is LDC~F using a
      different form of the input densities at large $x$ and the
      dotted is LDC~G restricting the fit to \tf2\ data with
      $x<0.1$.}}
  \label{fig:comp3}
\end{figure}

To study the hadronic final states we use the HZTOOL package
\cite{HZTOOL} developed jointly by H1, ZEUS and theoreticians for
comparison between event generators and published experimental data.
We have selected six different distributions which have been shown to
be sensitive to details in the models used in Monte Carlo event
generators. The distributions presented in \fig{fig:comp}, which are
all measured in the hadronic centre of mass system, are as follows.
\begin{itemize}
\item[(a)] The \tet-flow as a function of the pseudo rapidity for two
  bins in $x$ and \tq2, one with low $x$ and low \tq2, and one with
  moderate $x$ and \tq2\ in (b) \cite{ForwardJets}.  The large amount
  of \tet\ in the forward direction was previously claimed to be a
  good signal of \tkt\ non-ordering in the ISB, but it has been shown
  that this effect can also be obtained by the introduction of
  additional non-perturbative effects.
\item[(c)] The so-called seagull plot with the average \tk2t\ as a function
  of Feynman-$x$ \cite{HZ95221}, which at eg.\ EMC
  \cite{EMC} was shown to be difficult to reproduce with event
  generators.
\item[(d)] The \tkt-distribution of charged particles in a forward
  pseudo rapidity bin. This was recently proposed
  \cite{ForwardParticles,Kuhlen} as a new signal for perturbative
  activity in the forward region indicating \tkt\ non-ordering: a
  high-\tkt\ tail would be difficult to reproduce by non-perturbative
  models, where such tails would be exponentially suppressed.
\item[(e)] In \cite{ForwardParticles} was also shown that the
  pseudo rapidity distribution of charged particles with $\kt>1$ GeV
  also could be a good signal for \tkt\ non-ordering.
\item[(f)] Result for the two-jet ratio $R_2$ was recently reported
  \cite{R2H1} and showed large differences between the currently used
  event generators.
\end{itemize}

In \fig{fig:comp} we see the results from LDC with default settings
and using fit A, labeled LDC~A, compared with data and with the
results for \ariadne\ using the default soft radiation model, with
\lepto\ with and without the additional non-perturbative (soft colour
interactions (SCI), and perturbative-like treatment of remnants in the
case of sea quarks) assumptions presented in \cite{LeptoSCI}. For the
\ariadne\ and \lepto\ models we confirm previous results where
\ariadne, which until now was the only model implementing \tkt\ 
non-ordering, reproduces the data very well while the DGLAP-based
\lepto\ has difficulties, especially without the additional
non-perturbative models. We find that the result for LDC is quite
acceptable, although not reproducing data as well as \ariadne.

To compare different LDC strategies, we show in \fig{fig:comp3} only
the number of charged particles with transverse momentum larger than 1
GeV as a function of pseudo rapidity for small $x$ and \tq2\ 
(\fig{fig:comp3}). The effects on the other distributions in
\fig{fig:comp} are very similar.

In \fig{fig:comp3}a we see the results for LDC when restricting to
DGLAP-like chains. In this case, corresponding to the lines marked
LDC~B, the result is very poor as expected.  Allowing the virtuality
of the link closest to the photon to be above \tq2\ as for LDC~C,
makes things much better and only slightly worse than the default
LDC~A. Navely one may expect this to give the same result as \lepto\ 
which uses the exact \tordas\ Matrix Element for the emission closest
to the photon, also allowing the first link to have a virtuality
larger than \tq2, and which adds on parton-showers  la DGLAP on such
configurations. But in LDC, even though no ISB emissions are allowed
between the highest virtuality link and the photon, there is still a
resummation of diagrams which are then replaced by FSB emissions.
\lepto, however, uses the 'bare' matrix element and does not include
any resummation.

For the line marked LDC~D in \fig{fig:comp3}b, the cutoff in \tkt\ for
the ISB is set to 1 GeV. In the previous section we saw that the
parton density functions in this case became dramatically different.
For the final state, however, the reduction of ISB is compensated by
final state dipole emissions, which are allowed in the whole rapidity
range below $\kt=1$ GeV down to the cutoff fitted to LEP data,
$k_{\perp 0\mbox{\tiny LEP}}=0.6$.  The effect of increasing \tzcut\ 
is also shown in \fig{fig:comp3}b for the line LDC~A$_z$. The
dependence on this cutoff is small, which is expected as most of the
emissions with $z>0.5$ are counted as FSB as explained in
\fig{fig:largez} above. Also in \fig{fig:comp3}b is shown the effects
of allowing the virtuality of some links below the cutoff. Again the
differences are small.

In \fig{fig:comp3}c we see the effect of different regularizations of
the splitting functions. The line LDC~E uses the fit E in the previous
section, where all $q\rightarrow q$ and most of the $g\rightarrow q$
splittings are disallowed and all Sudakov form factors set to 1. For
the final state generation, all splittings are again included, and we
see a clear enhancement w.r.t. the default LDC~A strategy which
includes Sudakov form factors. The line LDC~A$_0$ uses the same parton
densities as the default strategy, but excludes the Sudakov form
factors when generating the final states. The main effect of the form
factor is to scale down all weights, and since in the total number of
events is fixed, the effects on the final state is small. The fact
that LDC~E is as different is then mostly due to the difference in
the input parton densities.

Finally in \fig{fig:comp3}d, we show the effects of using different
fitting procedures for the input parton densities and we see that the
differences are small.

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3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
currentpoint stroke M
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 -1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
3 0 V
2 -1 V
2 0 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
currentpoint stroke M
3 0 V
2 0 V
LT1
1962 1121 M
180 0 V
600 1018 M
2 0 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
currentpoint stroke M
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
3 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
3 -2 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
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2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
currentpoint stroke M
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 -1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
3 0 V
2 -1 V
2 0 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
currentpoint stroke M
3 0 V
2 0 V
LT1
1962 1121 M
180 0 V
1056 1360 M
2 -2 V
2 -3 V
2 -3 V
2 -2 V
3 -3 V
2 -3 V
2 -3 V
2 -3 V
2 -3 V
2 -2 V
3 -3 V
2 -3 V
2 -3 V
2 -3 V
2 -2 V
2 -3 V
3 -3 V
2 -3 V
2 -2 V
2 -3 V
2 -3 V
2 -3 V
3 -2 V
2 -3 V
2 -3 V
2 -2 V
2 -3 V
2 -3 V
3 -3 V
2 -2 V
2 -3 V
2 -3 V
2 -2 V
2 -3 V
3 -2 V
2 -3 V
2 -3 V
2 -2 V
2 -3 V
3 -2 V
2 -3 V
2 -3 V
2 -2 V
2 -3 V
2 -2 V
3 -3 V
2 -2 V
2 -3 V
2 -2 V
2 -3 V
2 -3 V
3 -2 V
2 -3 V
2 -2 V
2 -3 V
2 -2 V
2 -2 V
3 -3 V
2 -2 V
2 -3 V
2 -2 V
2 -3 V
2 -2 V
3 -3 V
2 -2 V
2 -3 V
2 -2 V
2 -2 V
2 -3 V
3 -2 V
2 -3 V
2 -2 V
2 -2 V
2 -3 V
2 -2 V
3 -2 V
2 -3 V
2 -2 V
2 -2 V
2 -3 V
2 -2 V
3 -2 V
2 -3 V
2 -2 V
2 -2 V
2 -3 V
2 -2 V
3 -2 V
2 -2 V
2 -3 V
2 -2 V
2 -2 V
3 -3 V
2 -2 V
2 -2 V
2 -2 V
2 -3 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -3 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -3 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -3 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -3 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -3 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
currentpoint stroke M
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -2 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 0 V
2 -1 V
2 0 V
2 0 V
2 -1 V
2 0 V
3 0 V
2 -1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
LT2
1962 1021 M
180 0 V
600 393 M
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 1 V
2 0 V
2 1 V
currentpoint stroke M
3 0 V
2 0 V
2 1 V
2 0 V
2 1 V
3 0 V
2 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 1 V
2 0 V
2 1 V
2 0 V
2 1 V
2 0 V
3 1 V
2 0 V
2 1 V
2 0 V
2 1 V
2 0 V
3 1 V
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2 1 V
2 0 V
2 1 V
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3 1 V
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3 0 V
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2 -1 V
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3 -1 V
2 -1 V
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2 -1 V
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2 -1 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -3 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
3 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
2 -2 V
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2 0 V
2 0 V
2 0 V
2 0 V
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3 0 V
2 0 V
LT3
1962 921 M
180 0 V
600 499 M
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
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3 0 V
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2 0 V
2 0 V
3 0 V
2 1 V
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2 0 V
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2 1 V
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2 1 V
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3 0 V
2 0 V
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2 0 V
3 1 V
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2 0 V
2 1 V
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3 1 V
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2 0 V
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3 1 V
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3 0 V
2 1 V
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3 0 V
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3 1 V
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3 0 V
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2 1 V
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2 0 V
2 0 V
3 1 V
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2 0 V
2 0 V
2 1 V
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3 0 V
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3 0 V
2 0 V
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3 0 V
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2 1 V
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2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 1 V
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2 0 V
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3 0 V
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2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 1 V
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3 1 V
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2 0 V
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2 1 V
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3 0 V
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2 0 V
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2 -1 V
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3 0 V
2 0 V
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currentpoint stroke M
3 0 V
2 0 V
stroke
grestore
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2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
currentpoint stroke M
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 -1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
3 0 V
2 -1 V
2 0 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
currentpoint stroke M
3 0 V
2 0 V
LT1
1962 1121 M
180 0 V
600 477 M
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 1 V
2 0 V
2 0 V
currentpoint stroke M
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
2 0 V
3 1 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 1 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 1 V
2 0 V
2 0 V
2 0 V
3 1 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
3 0 V
2 -1 V
2 0 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 0 V
2 -1 V
2 0 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
3 -2 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -1 V
2 -2 V
2 -2 V
2 -1 V
2 -2 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -2 V
2 -1 V
2 -2 V
2 -1 V
3 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
3 -2 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -2 V
2 -1 V
2 -1 V
3 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
3 0 V
2 -1 V
2 -1 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 -1 V
2 0 V
2 -1 V
2 -1 V
2 0 V
3 -1 V
2 0 V
2 -1 V
2 0 V
2 -1 V
3 0 V
2 -1 V
2 0 V
2 -1 V
2 0 V
2 0 V
3 0 V
2 -1 V
2 0 V
2 0 V
2 -1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 -1 V
2 0 V
3 0 V
2 0 V
2 0 V
2 0 V
2 0 V
2 0 V
3 0 V
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