\documentclass[12pt]{article}
\pagestyle{empty}

%%%%%%%%%%%%%%
% Packages
%%%%%%%%%%%%%%

\usepackage[english]{babel}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{vaucanson}
\usepackage{stmaryrd}
\usepackage{graphicx}

\def\ra{\rightarrow}
\def\al{\alpha}
\def\be{\beta}
\begin{document}



\begin{figure}
\begin{center}
\LargePicture\VCDraw{
\begin{VCPicture}{(0,0)(20,3)}
%\ChgEdgeLabelScale{0.8}
\ChgEdgeLabelScale{0.6}
\State[0]{(0,1)}{O}
\State[1]{(3,1)}{O1}

\State[n]{(6,1)}{A}
\StateVar[n+1]{(9,1)}{B}
\State[]{(12,1)}{C}
\StateVar[]{(15,1)}{D}
\ArcL{O}{O1}{\IOL{a^+}{1}}
\ArcL{O1}{O}{\IOL{a}{1}}
\ArcL{A}{B}{\IOL{a^+}{\sqrt{[n+1]_q}}}
\ArcL{B}{A}{\IOL{a}{\sqrt{[n+1]_q}}}
\ArcL{C}{D}{a^+}
\ArcL{D}{C}{a}
%\LoopN{A}{\StackTwoLabels{\IOL{a}{3}}{\IOL{b}{1}}}
%\LoopN[.75]{B}{\StackTwoLabels{\IOL{a}{1}}{\IOL{b}{4}}}
%\InitialR{w}{A}{3}
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\ChgEdgeLineStyle{dotted}
\InitialR{w}{A}{}
\FinalR{e}{D}{}
\FinalR{e}{B}{}
\FinalR{w}{O}{a|0}
\end{VCPicture}}
\end{center}
\caption{Classical Fock space}
\end{figure}

\begin{figure}
\begin{center}
\LargePicture\VCDraw{
\begin{VCPicture}{(0,0)(20,3)}
%\ChgEdgeLabelScale{0.8}
\ChgEdgeLabelScale{0.6}
\State[0]{(0,1)}{O}
\State[1]{(3,1)}{O1}

\State[n]{(6,1)}{A}
\StateVar[n+1]{(9,1)}{B}
\State[]{(12,1)}{C}
\StateVar[]{(15,1)}{D}
\ArcL{O}{O1}{\IOL{a^+}{1}}
\ArcL{O1}{O}{\IOL{a}{1}}
\ArcL{A}{B}{\IOL{a^+}{1}}
\ArcL{B}{A}{\IOL{a}{n+1}}
\ArcL{C}{D}{a^+}
\ArcL{D}{C}{a}
%\LoopN{A}{\StackTwoLabels{\IOL{a}{3}}{\IOL{b}{1}}}
%\LoopN[.75]{B}{\StackTwoLabels{\IOL{a}{1}}{\IOL{b}{4}}}
%\InitialR{w}{A}{3}
%\FinalR{e}{B}{1}
\ChgEdgeLineStyle{dotted}
\InitialR{w}{A}{}
\FinalR{e}{D}{}
\FinalR{e}{B}{}
\FinalR{w}{O}{a|0}
\end{VCPicture}}
\end{center}
\caption{Bargman-Fock representation (flat).\hfill \break
L'\'etat ``$n$'' est $z^n$ et $a^+\rightarrow z\ ;\ a\rightarrow D=\frac{d}{dz}$}
\end{figure}

\begin{figure}
\begin{center}
\LargePicture\VCDraw{
\begin{VCPicture}{(0,0)(20,3)}
%\ChgEdgeLabelScale{0.8}
\ChgEdgeLabelScale{0.6}
\State[0]{(0,1)}{O}
\State[1]{(3,1)}{O1}

\State[n]{(6,1)}{A}
\StateVar[n+1]{(9,1)}{B}
\State[]{(12,1)}{C}
\StateVar[]{(15,1)}{D}
\ArcL{O}{O1}{\IOL{a^+}{1}}
\ArcL{O1}{O}{\IOL{a}{1}}
\ArcL{A}{B}{\IOL{a^+}{n+1}}
\ArcL{B}{A}{\IOL{a}{1}}
\ArcL{C}{D}{a^+}
\ArcL{D}{C}{a}
%\LoopN{A}{\StackTwoLabels{\IOL{a}{3}}{\IOL{b}{1}}}
%\LoopN[.75]{B}{\StackTwoLabels{\IOL{a}{1}}{\IOL{b}{4}}}
%\InitialR{w}{A}{3}
%\FinalR{e}{B}{1}
\ChgEdgeLineStyle{dotted}
\InitialR{w}{A}{}
\FinalR{e}{D}{}
\FinalR{e}{B}{}
\FinalR{w}{O}{a|0}
\end{VCPicture}}
\end{center}
\caption{Repr\'esentation de Bargman-Fock avec \'etats normalis\'es.\hfill \break
L'état ``$n$'' is $\frac{z^n}{n!}$ and $a^+\rightarrow z\ ;\ a\rightarrow D=\frac{d}{dz}$}
\end{figure}

\begin{figure}
\begin{center}
\LargePicture\VCDraw{
\begin{VCPicture}{(0,0)(20,3)}
%\ChgEdgeLabelScale{0.8}
\ChgEdgeLabelScale{0.6}
\State[0]{(0,1)}{O}
\State[1]{(3,1)}{O1}

\State[n]{(6,1)}{A}
\StateVar[n+1]{(9,1)}{B}
\State[]{(12,1)}{C}
\StateVar[]{(15,1)}{D}
\ArcL{O}{O1}{\IOL{a^+}{1}}
\ArcL{O1}{O}{\IOL{a}{\sqrt{n+1}}}
\ArcL{A}{B}{\IOL{a^+}{\sqrt{n+1}}}
\ArcL{B}{A}{\IOL{a}{1}}
\ArcL{C}{D}{a^+}
\ArcL{D}{C}{a}
%\LoopN{A}{\StackTwoLabels{\IOL{a}{3}}{\IOL{b}{1}}}
%\LoopN[.75]{B}{\StackTwoLabels{\IOL{a}{1}}{\IOL{b}{4}}}
%\InitialR{w}{A}{3}
%\FinalR{e}{B}{1}
\ChgEdgeLineStyle{dotted}
\InitialR{w}{A}{}
\FinalR{e}{D}{}
\FinalR{e}{B}{}
\FinalR{w}{O}{a|0}
\end{VCPicture}}
\end{center}
\caption{Repr\'esentation de Bargman-Fock avec \'etats normalis\'es.\hfill \break
L'état ``$n$'' is $\frac{z^n}{n!}$ and $a^+\rightarrow z\ ;\ a\rightarrow D=\frac{d}{dz}$}
\end{figure}

\begin{figure}
\begin{center}
\LargePicture\VCDraw{
\begin{VCPicture}{(0,0)(20,3)}
%\ChgEdgeLabelScale{0.8}
\ChgEdgeLabelScale{0.6}
\State[0]{(0,1)}{O}
\State[1]{(3,1)}{O1}

\State[n]{(6,1)}{A}
\StateVar[n+1]{(9,1)}{B}
\State[]{(12,1)}{C}
\StateVar[]{(15,1)}{D}
\ArcL{O}{O1}{\IOL{a^+}{1}}
\ArcL{O1}{O}{\IOL{a}{1}}
\ArcL{A}{B}{\IOL{a^+}{1}}
\ArcL{B}{A}{\IOL{a}{[n+1]_q}}
\ArcL{C}{D}{a^+}
\ArcL{D}{C}{a}
%\LoopN{A}{\StackTwoLabels{\IOL{a}{3}}{\IOL{b}{1}}}
%\LoopN[.75]{B}{\StackTwoLabels{\IOL{a}{1}}{\IOL{b}{4}}}
%\InitialR{w}{A}{3}
%\FinalR{e}{B}{1}
\ChgEdgeLineStyle{dotted}
\InitialR{w}{A}{}
\FinalR{e}{D}{}
\FinalR{e}{B}{}
\FinalR{w}{O}{a|0}
\end{VCPicture}}
\end{center}
\caption{Bargman-Fock representation.\hfill \break
State ``$n$'' is $z^n$ and $a^+\rightarrow z\ ;\ a\rightarrow D_q$ with $D_q(f)=\frac{f(qz)-f(z)}{z(q-1)}$}
\end{figure}


\begin{figure}
\begin{center}
\LargePicture\VCDraw{
\begin{VCPicture}{(0,0)(20,3)}
%\ChgEdgeLabelScale{0.8}
\ChgEdgeLabelScale{0.6}
\State[0]{(0,1)}{O}
\State[1]{(3,1)}{O1}

\State[n]{(6,1)}{A}
\StateVar[n+1]{(9,1)}{B}
\State[]{(12,1)}{C}
\StateVar[]{(15,1)}{D}
\ArcL{O}{O1}{\IOL{a^+}{\beta_{1}}}
\ArcL{O1}{O}{\IOL{a}{\alpha_{1}}}
\ArcL{A}{B}{\IOL{a^+}{\beta_{n+1}}}
\ArcL{B}{A}{\IOL{a}{\alpha_{n+1}}}
\ArcL{C}{D}{a^+}
\ArcL{D}{C}{a}
%\LoopN{A}{\StackTwoLabels{\IOL{a}{3}}{\IOL{b}{1}}}
%\LoopN[.75]{B}{\StackTwoLabels{\IOL{a}{1}}{\IOL{b}{4}}}
%\InitialR{w}{A}{3}
%\FinalR{e}{B}{1}
\ChgEdgeLineStyle{dotted}
\InitialR{w}{A}{}
\FinalR{e}{D}{}
\FinalR{e}{B}{}
\FinalR{w}{O}{a|0}
\end{VCPicture}}
\end{center}
\caption{General setting: in order that the Fock space be bounded below, one must have $\alpha_0=0$.}
\end{figure}



\begin{figure}
\begin{center}
\LargePicture\VCDraw{
\begin{VCPicture}{(0,0)(20,3)}
%\ChgEdgeLabelScale{0.8}
\ChgEdgeLabelScale{0.6}
\State[0]{(0,1)}{O}
\State[1]{(3,1)}{O1}

\State[n]{(6,1)}{A}
\State[m]{(12,1)}{C}
\StateVar[]{(15,1)}{D}
\ArcL{O}{O1}{\IOL{a^+}{\beta_{1}}}
\ArcL{O1}{O}{\IOL{a}{\alpha_{1}}}
\ArcL{A}{C}{\IOL{w}{\omega_{w}}}
%\ArcL{B}{A}{\IOL{a}{\alpha_{n+1}}}
\ArcL{C}{D}{a^+}
\ArcL{D}{C}{a}
%\LoopN{A}{\StackTwoLabels{\IOL{a}{3}}{\IOL{b}{1}}}
%\LoopN[.75]{B}{\StackTwoLabels{\IOL{a}{1}}{\IOL{b}{4}}}
%\InitialR{w}{A}{3}
%\FinalR{e}{B}{1}
\ChgEdgeLineStyle{dotted}

\InitialR{w}{A}{}
\FinalR{e}{D}{}
%\FinalR{e}{B}{}
\FinalR{w}{O}{a|0}
\ChgStateLineStyle{dotted}
\State[]{(8,1)}{B}
\State[]{(10,1)}{B1}
\end{VCPicture}}
\end{center}
\caption{The transfer packet problem} 
\end{figure}

\begin{figure}
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture}(80,60)
%\thinlines 
%\linethickness{0.075mm} 
\linethickness{0.2mm} 
\multiput(0,0)(8,0){11}{\line(0,1){56}} \multiput(0,0)(0,8){8}{\line(1,0){80}}
\linethickness{1mm} 
\put(0,0){\line(1,1){8}} \put(8,8){\line(1,1){8}} \put(16,16){\line(1,-1){8}}
\put(24,8){\line(1,1){8}} \put(32,16){\line(1,-1){8}} \put(40,8){\line(1,1){8}} \put(48,16){\line(1,1){8}}
\put(56,24){\line(1,1){8}} \put(64,32){\line(1,1){8}} \put(72,40){\line(1,-1){8}} \put(0,0){\line(1,1){8}}
\put(2,-5){$a^{\dagger}$} \put(10,-5){$a^{\dagger}$} \put(19,-5){$a$} \put(26,-5){$a^{\dagger}$} \put(35,-5){$a$}
\put(42,-5){$a^{\dagger}$} \put(50,-5){$a^{\dagger}$} \put(58,-5){$a^{\dagger}$} \put(66,-5){$a^{\dagger}$}
\put(75,-5){$a$} 
%\multiput(16,16)(1,1){40}{\circle*{0.1}} \multiput(56,56)(1,-1){16}{\circle*{0.1}}
%\multiput(32,16)(1,1){32}{\circle*{0.1}} \multiput(24,24)(1,-1){8}{\circle*{0.1}}
%\multiput(32,32)(1,-1){16}{\circle*{0.1}} \multiput(40,40)(1,-1){16}{\circle*{0.1}}
%\multiput(48,48)(1,-1){16}{\circle*{0.1}}
\end{picture}
\label{figr1}
\end{center}
\caption{A word which transfers from $n$ to $n+4$.} 
\end{figure}

\newpage

\begin{equation}
	W_{k}^{(i)}=\Big\{w\in \{a,a^+\}^* \Big| \pi_e(w)=k \textrm{ and } |w|=i\Big\}
\end{equation}

\hspace{2cm}        with $k=m-n$ and $\pi_e(w)=|w|_{a^+}-|w|_a$.

\newpage
\begin{equation}
e_n.W_{m-n}^{(i)}=\omega_{n\ra m}^{(i)}\ e_m\ ;\ T_{n\rightarrow n+k}:=\sum_{i\geq 0}t^i\omega_{n\rightarrow n+k}^{(i)} 
\end{equation}

\vspace{2cm}

\begin{equation}
\Big(D_+ + D_-^{(n)}\Big)^* 
\end{equation}

\vspace{2cm}

\begin{equation}
a^+(D_+)^* 
\end{equation}

\vspace{2cm}

\begin{equation}
	D_+=a^+\Big(D_+\Big)^*a\ ;\ D_-^{(n)}=a\Big(D_-^{(n-1)}\Big)^*a^+\ ;\ D_-^{(0)}=\emptyset
\end{equation}

\vspace{2cm}

\begin{equation}
	T_{n\ra n}[t]=\frac{1}{1-\frac{t^2\al_{n+1}\be_{n+1}}{1-\frac{t^2\al_{n+2}\be_{n+2}}{1-\cdots}}-   
\frac{t^2\al_{n}\be_{n}}{1-\frac{t^2\al_{n-1}\be_{n-1}}{1-\cdots}	}}
\end{equation}

\vspace{2cm}


\begin{equation}
	T_{n\ra n}^+[t]=\frac{1}{1-\frac{t^2\al_{n+1}\be_{n+1}}{1-\frac{t^2\al_{n+2}\be_{n+2}}{1-\cdots}}}
\end{equation}


\end{document}