%  First, use latex to compile the file. Second, use dvi2ps to become .ps file. You can use Gsview to watch it. But
%  I suggest you use ps2pdf to become .pdf file. Because .ps file is for making a final film.
%  There are nine types of backgrounds. You can revise the second parameter. Now I use the darkblue background. All
%  backgrounds(9) are the following:
%  alienglow, autumn, azure, contemporain, darkblue, frames, lignesbleues, nuancegris, troispoints.
% 14 new types of background in C:\CTeX\texmf\tex\latex\prosper
% gyom, rico, alcatel, blends, capsules, (corners,) fyma, (mancini,) prettybox, serpaggi, thomasd, whitecross, (winter, wj)
% Of course, you can make your background if you have the ability and you wish.

\documentclass[pdf,darkblue,slideColor,colorBG]{prosper}

\usepackage[latin1]{inputenc}
\usepackage{pstricks,pst-node,pst-text,pst-3d}
\usepackage{amsmath}
\usepackage{CJK}
\usepackage{graphicx}
% Definition of new colors
\newrgbcolor{LemonChiffon}{1. 0.98 0.8}
\newrgbcolor{LightBlue}{0.68 0.85 0.9}

% > BEGIN OF OVERLAPPED COLORS
% Code below devised by Pingji Deng (BNU/ - China, dengpj@eyou.com)
\newrgbcolor{LemonChiffon}{1. 0.98 0.8}
\newrgbcolor{LightBlue}{0.68 0.85 0.9}
\makeatletter
\newdimen\pst@dimz

% Draw two overlapped surfaces, with computation of the mixed color for
% the intersection of the surfaces
% #1=first  surface, #2=color of first  surface,
% #3=second surface, #4=color of second surface
\def\ColoredOverlappedSurfaces#1#2#3#4{%
\psset{fillstyle=solid}
% Decode the three components of the first RGB color
\DecodeRGBFirstColor{\csname color@#2\endcsname}%
\psset{fillcolor=#2}
% Draw first surface
#1
% Decode the three components of the second RGB color
\DecodeRGBSecondColor{\csname color@#4\endcsname}%
% Compute the mixed color
\BuildMixedColor
% Draw second surface
\psclip{\psset{fillcolor=#4}#3}
\psset{fillcolor=MixedColor}
% Redraw overlapped surface in the mixed color
#1
\endpsclip}

% Get the three components of the first color
\def\DecodeRGBFirstColor#1{%
\pst@expandafter\pst@getnumiii{#1} {} {} {} {}\@nil
\edef\pst@FirstColorR{\pst@tempg}%
\edef\pst@FirstColorG{\pst@temph}%
\edef\pst@FirstColorB{\pst@tempi}%
%\typeout{Color 1=\pst@tempg,\pst@temph,\pst@tempi}% Debug
}

% Get the three components of the second color
\def\DecodeRGBSecondColor#1{%
\pst@expandafter\pst@getnumiii{#1} {} {} {} {}\@nil
\edef\pst@SecondColorR{\pst@tempg}%
\edef\pst@SecondColorG{\pst@temph}%
\edef\pst@SecondColorB{\pst@tempi}%
%\typeout{Color 2=\pst@tempg,\pst@temph,\pst@tempi}% Debug
}

% Build the mixed RBG color (by means of each three components)
\def\BuildMixedColor{%
% Resulting R component
\pst@dimz=\pst@FirstColorR pt
\advance\pst@dimz\pst@SecondColorR pt
\divide\pst@dimz\tw@
\pst@dimtonum{\pst@dimz}{\pst@MixedColorR}%
% Resulting G component
\pst@dimz=\pst@FirstColorG pt
\advance\pst@dimz\pst@SecondColorG pt
\divide\pst@dimz\tw@
\pst@dimtonum{\pst@dimz}{\pst@MixedColorG}%
% Resulting B component
\pst@dimz=\pst@FirstColorB pt
\advance\pst@dimz\pst@SecondColorB pt
\divide\pst@dimz\tw@
\pst@dimtonum{\pst@dimz}{\pst@MixedColorB}%
% Definition of the mixed color MixedColor
\newrgbcolor{MixedColor}{%
\pst@MixedColorR\space \pst@MixedColorG\space \pst@MixedColorB}
%\typeout{Mixed color=\csname color@MixedColor\endcsname}% Debug
}
\makeatother
% < END OF OVERLAPPED COLORS

\begin{document}
\begin{CJK*}{GBK}{song}
\CJKtilde\sloppy
%---------------------------------------------------------------------- SLIDE 1-

\begin{slide}[Box]{\tiny\yellow Workshop on ``Probability and Applications'',\ Îäºº´óÑ§, 2004Äê6ÔÂ22ÈÕÖÁ26ÈÕ
\hskip2cm\hyperlink{S1}{$\triangleright$}
\hypertarget{S15}{}}
\vspace{1cm}

\centerline{\Large\green Brusselator Ä£ÐÍ}

\vskip1cm \centerline{\yellow Îâ²¨\ \ ÕÅÓà»Ô}

\vskip1cm \centerline{\yellow ±±¾©Ê¦·¶´óÑ§}

\end{slide}

%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 2-
\overlays{4}{
\begin{slide}[Blinds]{\small\yellow ÑÐ¾¿±³¾°}

·´Ó¦À©É¢¹ý³ÌÊÇ·Ç½ô×´Ì¬¿Õ¼äÉÏµÄÎÞÇîÁ£×ÓÂí¶û¿É·ò¹ý³Ì, ÓÃÀ´ÃèÊöÀýÈç»¯Ñ§·´Ó¦µÈµÄÓÐ¹ØÄ£ÐÍ.

\begin{itemstep}

\item ÑÏÊ¿½¡, ÀîÕ¼±ú(1980). ÎïÀíÑ§±¨, 29, 139-152.

\item H. Haken(1983). Synergetics.

\item G.Nicolis \& I. Prigogine(1977). Self-Organization in Non-Equilibrium Systems.

\item M.F. Chen(1992).
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 3-

\begin{slide}[Split]{\small\yellow ÑÐ¾¿±³¾°}

ÓÉÓÚ×´Ì¬¿Õ¼ä·Ç½ô, ÑÐ¾¿ÒªÀ§ÄÑµÃ¶à. ¶ÔÓÚµ¥ÎïÖÖÇéÐÎ, ÓÐÏµÍ³ÑÐ¾¿.
¶ø¶Ô¶àÎïÖÖÇéÐÎ, ËùµÃ½á¹û²»¹»·á¸», ÓÈÆäÔÚÎÞÇîÎ¬ÇéÐÎ.

\vskip0.2cm
Brusselator~Ä£ÐÍÊÇ¶àÎïÖÖµÄ»ù±¾Ä£ÐÍÖ®Ò», ÊÇÁ½ÎïÖÖµÄ.
×´Ì¬¿Õ¼äÎª~$(\mathbf Z_+^2)^S$, $S$ÊÇºÐ×Ó¼¯, $p_k(u,v)$~ÊÇ~$S$~ÉÏµÄ×ªÒÆ¸ÅÂÊ($k=1,2$)ÇÒ
$$e_{ui}(v,j)=\begin{cases} 1,\quad&\hbox{Èô}\ v=u,\ j=i;\\
0,&\hbox{ÆäËü}\ u,v\in S,\ i,j=1,2.\end{cases}$$
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 4-
\overlays{7}{
\begin{slide}[Wipe]{\small\yellow Brusselator~Ä£ÐÍ\hskip8cm\hyperlink{S4}{$\triangleright$}
\hypertarget{S11}{}}

ÉèºÐ×ÓÖÐÓÐ¹Ì¶¨ÊýÄ¿µÄ×÷ÓÃÏàµ±ÓÚ´ß»¯¼ÁµÄÁ½ÖÖÁ£×Ó~$A$~ºÍ~$B$.
²Î¼Ó·´Ó¦µÄÊÇÁ£×Ó~$X_1$~ºÍ~$X_2$. ºÐ×ÓÖÐµÄ»¯Ñ§·´Ó¦Ä£ÐÍÊÇ:

\vskip0.2cm
\small

\fromSlide*{2}{
\yellow »¯Ñ§·´Ó¦ \hskip75pt ËÙÂÊ\ $q(x,y)$}

\fromSlide*{3}{
$A\overset{\lambda_1}\longrightarrow X_1$ \hskip71pt
$\lambda_1 a(u)$,\hskip55pt $y=x+e_{u1}$}

\fromSlide*{4}{
$B+X_1\overset{\lambda_2}\longrightarrow X_2+D$\hskip19pt
$\lambda_2 b(u)x_1(u)$, \hskip27pt $y=x-e_{u1}+e_{u2}$}

\fromSlide*{5}{
$2X_1+X_2\overset{\lambda_3}\longrightarrow 3X_1$\hskip27pt
$\lambda_3 \binom{x_1(u)}2 x_2(u)$, \hskip15pt $y=x+e_{u1}-e_{u2}$}

\fromSlide*{6}{
$X_1\overset{\lambda_4}\longrightarrow C$\hskip73pt
$\lambda_4 x_1(u)$, \hskip48pt $y=x-e_{u1}$}

\onlySlide*{7}{
{\green À©É¢\ $k=1,2,v\ne u$\hskip22pt
$x_k(u)p_k(u,v)$,\hskip24pt $y=x-e_{uk}+e_{vk}$}}


\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 5-
\overlays{3}{
\begin{slide}[Box]{\small\yellow ÒÑÓÐ½á¹û}
\vspace{-0.3cm}
\begin{itemstep}
\item S.J. Yan \& M.F. Chen (1986). Chinese Ann. Math. 7B, 90-110.
\newline{\small ¶àÎ¬~$\rightarrow$~Ò»Î¬.  ´Ë·½·¨¶Ô´ó²¿·Ö»ù±¾Ä£ÐÍÓÐÐ§,
µ«¶Ô~Brusselator~Ä£ÐÍµÄ±éÀúÐÔÊ§Ð§. Î¨Ò»ÐÔ.}

\item º«¶«(1991). ÐÂ½®´óÑ§Ñ§±¨, 8, 37-40.
\newline {\small $|S|=1$, ±éÀú. ÎÞÀ©É¢.
$\sum_j q_{ij}y_j\le -1 (i\notin H),\ \sum_{i\in H}\sum_{j\ne i}q_{ij}y_j<\infty$
ÓÐÓÐÏÞ·Ç¸º½â, $H$ÊÇÒ»·Ç¿ÕÓÐÏÞ×Ó¼¯.}

\item ³Â½ðÎÄ(1995). ÊýÑ§ÎïÀíÑ§±¨, 15, 121-125.
\newline {\small $|S|<\infty$, Ö¸Êý±éÀú. ¹¹Ôì·Ç¸º½ô~$h$, $C\ge 0$, $c>0$:
$\Omega h\le C-ch$.}
\end{itemstep}

\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 6-
\overlays{3}{
\begin{slide}[Dissolve]{\small\yellow Ò»Ð©¸ÅÄî} Éè~$Q=(q_{ij})$~ ÊÇ
¿ÉÊý×´Ì¬¿Õ¼ä~$E$~ÉÏµÄÕýÔò~$Q$~¾ØÕó:

$q_{ij}\ge 0 (i\ne j)$, $q_i:=-q_{ii}=\sum_{j\ne i}q_{ij}<\infty
(i\in E)$, $Q$~Î¨Ò»µØÈ·¶¨~$P_t=(p_{ij}(t))$ ($Q$~¹ý³Ì»òÂíÊÏÁ´).

¼Ù¶¨~$Q$~²»¿ÉÔ¼ÓÐÆ½ÎÈ·Ö²¼~$\pi=(\pi_i)$.
\begin{itemstep}\small
\item  ±éÀú:
$\displaystyle\lim_{t\to\infty}|p_{ij}(t)-\pi_j|=0$.

\item Ö¸Êý±éÀú:
$\displaystyle\lim_{t\to\infty}{\yellow e^{\alpha t}}|p_{ij}(t)-\pi_j|=0$.

\item Ç¿±éÀú:
$\displaystyle\lim_{t\to\infty}\sup_i|p_{ij}(t)-\pi_j|=0$

\hskip3cm
$\Longleftrightarrow\displaystyle\lim_{t\to\infty}{\yellow e^{\beta t}}\sup_i|p_{ij}(t)-\pi_j|=0$.
\end{itemstep}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 7-
\overlays{2}{
\begin{slide}[Glitter]{\small\yellow Ç¿±éÀúµÄÒ»¸öÅÐ±ð×¼Ôò\hskip7cm\hyperlink{S7}{$\triangleright$}
\hypertarget{S13}{}}

Ç¿±éÀú~$\Longleftrightarrow$~ÏÂÁÐ·½³Ì×é
$$\begin{cases}
\sum_j q_{ij}y_j\le -1, \quad i\notin H,\\
\sum_{i\in H}\sum_{j\ne i}q_{ij}y_j<\infty
\end{cases}$$
ÓÐÓÐ½ç·Ç¸º½â, ÆäÖÐ$H$ÊÇÒ»·Ç¿ÕÓÐÏÞ×Ó¼¯.

\vskip0.5cm
\OnlySlide{2} Brusselator~Ä£ÐÍÒ»Ö±Ã»ÓÐ½øÕ¹£¡

%\OnlySlide{2}
% \rput(8,0.2){
%      \onlySlide{2}{
%        \psellipse[fillstyle=solid,fillcolor=LemonChiffon](0.3,-0.7)(1.5,1)}}
% \rput(8.5,-0.5){
%      \onlySlide{2}{\blue OH!NO!}}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 8-
\overlays{4}{
\begin{slide}[Blinds]{\small\yellow »ñµÃµÄ½á¹û}
µ±~$|S|=\infty$, Ö»¿¼ÂÇ~$E_*:=\{x\in E:|x|<\infty\}$~ÉÏµÄ~$Q$~¹ý³Ì, ÕâÀï~
$|x|:=\sum_{u\in S}(x_1(u)+x_2(u))$. ×¢Òâµ±~$|S|<\infty$~Ê±, $E=E_*$.

\vskip1cm
\FromSlide{2}{\vbox{\green ¶¨Àí \ {\yellow ¼Ù¶¨~$a=\sum_{u\in S}a(u)<\infty$. Ôò~$E_*$~ÉÏ
~Brusselator~Ä£ÐÍ¶ÔÓ¦µÄ~$Q$~¹ý³Ì·ÇÇ¿±éÀú.}}}

\vskip-4cm
\FromSlide{3}{\setlength{\unitlength}{2cm}
\begin{picture}(0,4)
    \thicklines
   \put(-0.1,1.85){\red\framebox(4.8,1.0){}}
\end{picture}
}

\vskip-3.3cm
\FromSlide{4}{
ÏÂÃæÖ»½²~$|S|=1$~µÄÇéÐÎ, ´ËÊ±~$E=\mathbf Z_+^2$.
}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 9-
\begin{slide}[Split]{\small\yellow $|S|=1$}
$$\aligned {\yellow (m,n)}&\to (m+1,n)\\
&\to (m-1,n+1)\\
&\to (m+1,n-1)\\
&\to (m-1,n)\endaligned
\quad
\aligned \hbox{ËÙÂÊ}\quad & \lambda_1 a\\
& \lambda_2 b m\\
& \lambda_3 m(m-1)n/2\\
& \lambda_4 m
\endaligned$$

\centerline{\yellow $(m,n)\in\mathbf Z_+^2$.}
\end{slide}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 10-
\overlays{3}{
\begin{slide}[Box]{\small\yellow Ë¼Â·ÓëYan-Chen·½·¨}
\setlength{\unitlength}{2cm}  %set the unit length for the figure below
  %\vspace{-.2in}
  %make a figure
  \begin{picture}(0,10)
    \thinlines
    \put(0.1,7.1){\vector(1,0){4.7}}
    \put(0.1,7.6){\line(1,0){4.7}}
    \put(0.1,8.1){\line(1,0){4.7}}
    \put(0.1,8.6){\line(1,0){4.7}}
    \put(0.1,9.1){\line(1,0){4.7}}
    \put(0.1,9.6){\line(1,0){4.7}}
    \put(0.1,10.1){\line(1,0){4.7}}

    \put(0.1,7.1){\vector(0,1){3.2}}
    \put(0.6,7.1){\line(0,1){3.2}}
    \put(1.1,7.1){\line(0,1){3.2}}
    \put(1.6,7.1){\line(0,1){3.2}}
    \put(2.1,7.1){\line(0,1){3.2}}
    \put(2.6,7.1){\line(0,1){3.2}}
    \put(3.1,7.1){\line(0,1){3.2}}
    \put(3.6,7.1){\line(0,1){3.2}}
    \put(4.1,7.1){\line(0,1){3.2}}
    \put(4.6,7.1){\line(0,1){3.2}}
    \put(5.1,9.5){\green$\mathbf Z_+^2$}

\FromSlide{2}{\yellow
    \put(0.6,7.1){\line(-1,1){0.5}}
    \put(1.1,7.1){\line(-1,1){1.0}}
    \put(1.6,7.1){\line(-1,1){1.5}}
    \put(2.1,7.1){\line(-1,1){2.0}}
    \put(2.6,7.1){\line(-1,1){2.5}}
    \put(3.1,7.1){\line(-1,1){3.0}}
    \put(3.6,7.1){\line(-1,1){3.2}}
    \put(4.1,7.1){\line(-1,1){3.2}}
    \put(4.6,7.1){\line(-1,1){3.2}}
    }
\FromSlide{3}{\green
    \put(0.1,6.8){$E_0$}
    \put(0.6,6.8){$E_1$}
    \put(1.1,6.8){$E_2$}
    \put(1.6,6.8){$E_3$}
    \put(2.1,6.8){$E_4$}
    \put(2.6,6.8){$E_5$}
    \put(3.1,6.8){$E_6$}
    \put(3.6,6.8){$E_7$}
    \put(4.1,6.8){$E_8$}
    \put(4.6,6.8){$\cdots$}
    }
\end{picture}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 11-
\overlays{3}{
\begin{slide}[Glitter]{\small\yellow ¶àÎ¬µ½Ò»Î¬\hskip9cm\hyperlink{S11}{$\triangleleft$}
\hypertarget{S4}{}}
»¯¶àÎ¬~$Q$~¹ý³ÌÎÊÌâÎªÒ»Î¬ÎÊÌâ. ±È½ÏµÄ·½·¨. ÓëÊìÖªµÄ(¿ØÖÆ)¹ý³Ì±È, ÈçÉúÃð¹ý³ÌºÍµ¥Éú¹ý³Ì, µÃ³ö
Î¨Ò»¡¢³£·µºÍ±éÀúÎÊÌâµÄÏÔÊ½³ä·ÖÌõ¼þ. ¶ÔÎ¨Ò»ÐÔºÜÓÐÐ§.

\vskip0.3cm
\FromSlide{2}{ÎªµÃµ½ÏÔÊ½±ØÒªÌõ¼þ, »¹ÊÇ¿ÉÒÔÓ¦ÓÃ±È½ÏµÄË¼Ïë. ´ËÊ±, ±»¿ØÖÆ¹ý³ÌÊÇÊìÖªµÄ¹ý³Ì. ¾Í~Brusselator~Ä£ÐÍ,
(±»)¿ØÖÆ¹ý³ÌÊÇ\yellow ÉúÃð¹ý³Ì.}  »÷ÖÐÊ±¾Ø. ·´Ö¤·¨($|S|<\infty$)ºÍñîºÏ·½·¨.

\vskip0.3cm
\FromSlide{3}{Î¨Ò», ³£·µ, ±éÀú, $\ell$ ±éÀú, Ö¸Êý±éÀúºÍÇ¿±éÀú .}

\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 12-
\overlays{3}{
\begin{slide}[Box]{\small\yellow $|S|=1$~µÄÄ£ÐÍ}

¶Ô~Brusselator~Ä£ÐÍ×÷ÈçÏÂÐÞÕý:
$$\aligned (m,0)&\to(m-1,0)\\
& \endaligned
\quad
\aligned \hbox{ËÙÂÊ}\quad &
\fromSlide*{2}{\lambda_4 k\rnode{NA}{\pscirclebox[linecolor=red]{\green m^2}}}
\onlySlide*{1}{\lambda_4 k\rnode{NA}{\pscirclebox[linecolor=red,linestyle=none]{\green m^2}}}
\\
&
\fromSlide*{3}{\lambda_4\rnode{NB}{\pscirclebox[linecolor=red]{\green m}}}
\onlySlide*{2}{\lambda_4\rnode{NB}{\pscirclebox[linecolor=red,linestyle=none]{\green m}}}
\endaligned$$
ÆäÖÐ³£Êý~$k>2\lambda_2b/\lambda_4$, ÆäËüµÄ²»±ä.

\fromSlide*{3}{\nccurve[linecolor=red,angleA=0,angleB=270]{->}{NB}{NA}}
\end{slide}
}
%------------------------------------------------------------------------------

%-------------------------------------------------------------------------SLIDE 13-
\overlays{3}{
\begin{slide}[Wipe]{\small\yellow $|S|=1$~µÄÄ£ÐÍ\hskip8.5cm\hyperlink{S13}{$\triangleleft$}
\hypertarget{S7}{}}

\FromSlide{2}{\green ½áÂÛ:}
\vskip0.3cm

\FromSlide{2}{
\setlength{\unitlength}{2cm}
\begin{picture}(0,2)
    \thicklines
   \put(-0.1,1.2){\red\framebox(5.2,0.8){\yellow
~Brusselator~ÐÞÕýÄ£ÐÍ¶ÔÓ¦µÄ~$Q$~¹ý³ÌÇ¿±éÀú.}}
\end{picture}}

\vspace{-1.3cm}
\OnlySlide{3}{\yellow
~Brusselator~Ä£ÐÍºÍÐÞÕýÄ£ÐÍ¶ÔÓ¦µÄ~$Q$~¹ý³Ì¾ù²»Âú×ã³¬~Poincar\'e~²»µÈÊ½.

\vspace{-3.8cm}
\setlength{\unitlength}{2cm}
\begin{picture}(0,4)
    \thicklines
   \put(-0.1,1.95){\red\framebox(5.5,1.0){}}
\end{picture}
}

\end{slide}
}
%------------------------------------------------------------------------------

%-------------------------------------------------------------------------SLIDE 14-
\overlays{2}{
\begin{slide}[Wipe]{\small\yellow ÎÊÌâ}

\FromSlide{1}{\yellow $|S|=\infty$, ±éÀú? Ö¸Êý±éÀú? µÈµÈ}

\vskip0.5cm
\OnlySlide{2}{\yellow $2\le |S|\le\infty$, ³¬~Poincar\'e~²»µÈÊ½?}
\end{slide}
}
%------------------------------------------------------------------------------

%---------------------------------------------------------------------- SLIDE 15-
\overlays{2}{
\begin{slide}[Dissolve]{\small\yellow ÖÂÐ»\hskip10cm\hyperlink{S15}{$\triangleleft$}
\hypertarget{S1}{}}
\FromSlide{1}
 \rput(5.6,-1.2){\psellipse[fillstyle=solid,fillcolor=LemonChiffon](0.3,-0.7)(1.5,1)}
 \rput(5.9,-1.9){\blue --Íê--}
\FromSlide{2}
 \rput(5.6,-3.2){\psellipse[fillstyle=solid,fillcolor=LightBlue](0.3,-0.7)(1.5,1)}
 \rput(5.9,-3.9){\red --Ð»Ð»--}

\end{slide}
}
%------------------------------------------------------------------------------
\clearpage
\end{CJK*}
\end{document}

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