# LaTeX2HTML 2002 (1.62)
# Associate images original text with physical files.


$key = q/y^0=0,y^1=a_1^1x_1,y^2=a_1^2x_1,y^3=a_2^1x_2,y^4=a_2^2x_2,y^{2n-1}=a_n^1x_n,y^{2n}=a_n^2x_n,y^{2n+1}=a_1^1x_1+a_2^1x_2,y^{2n+2}=a_1^2x_1+a_2^2x_2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="876" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="$y^0=0, y^1=a_1^1 x_1, y^2=a_1^2 x_1,\
y^3=a_2^1 x_2, y^4=a_2^2 x_2,\\\\
 y^{...
...{2n}= a_n^2 x_n,
 y^{2n+1}=a_1^1 x_1+a_2^1 x_2,
 y^{2n+2}=a_1^2 x_1+a_2^2 x_2$">|; 

$key = q/y_j=d(x_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="$y_j= d (x_j)$">|; 

$key = q/i=1,...,m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$i=1,...,m$">|; 

$key = q/displaystylep(y^n);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$\displaystyle p(y^n)$">|; 

$key = q/U(x_j,a_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="$U(x_j,a_j)$">|; 

$key = q/p(0.5);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img228.png"
 ALT="$p(0.5)$">|; 

$key = q/n=m=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$n=m=1$">|; 

$key = q/w^n=(x^n,a^n)inB,n=0,1,2,...;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="$w^n=(x^n,a^n) \in B ,\
n=0,1,2,...$">|; 

$key = q/y^1=c_1(x_1)+sum_{i=2}^mc_i(x_i);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="$y^1=c_1( x_1)+\sum_{i=2}^m c_i(x_i)$">|; 

$key = q/x_i=z_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="$x_i=z_i$">|; 

$key = q/f[0]=0.0,f[1]=0.125,f[2]=0.25,f[3]=0.325,f[4]=0.5,f[5]=0.625,f[6]=0.75,f[7]=0.875,f[8]=1.0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="791" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img243.png"
 ALT="$f[0]=0.0, f[1]=0.125, f[2]=0.25, f[3]=0.325,\
f[4]=0.5, f[5]=0.625, f[6]=0.75, f[7]=0.875, f[8]=1.0$">|; 

$key = q/displaystylep_2p_2^1p_1,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$\displaystyle p_2 p_2^1 p_1,$">|; 

$key = q/displaystyleV(x,a)=sum_{y}v(y)p(y),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="$\displaystyle V(x,a)=\sum_{y} v(y) p(y),$">|; 

$key = q/r_{prudent}<r_{average}<r_{rich}<r_{risky};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="245" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img237.png"
 ALT="$r_{prudent} &lt; r_{average} &lt; r_{rich} &lt;r_{risky}$">|; 

$key = q/displaystylep(y^{n+2});MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$\displaystyle p(y^{n+2})$">|; 

$key = q/y_j=s_ja_j-(1-a_j)(n_j-s_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img188.png"
 ALT="$y_j=s_j a_j-(1-a_j)(n_j-s_j)$">|; 

$key = q/u(2)=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img223.png"
 ALT="$u(2)=1$">|; 

$key = q/(x^1,a^1);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="$(x^1,a^1)$">|; 

$key = q/y^j,j=1,2,....;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$y^j, j=1,2,....$">|; 

$key = q/C;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img210.png"
 ALT="$C$">|; 

$key = q/displaystyleL(2.0,1.0,1.5,p(1.5))=;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img232.png"
 ALT="$\displaystyle L(2.0,1.0,1.5,p(1.5))=$">|; 

$key = q/displaystylep_1prod_{ine1}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$\displaystyle p_1 \prod_{i \ne 1} q_i,$">|; 

$key = q/M;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$M$">|; 

$key = q/y=c(x);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$y= c (x)$">|; 

$key = q/(a_1,a_2);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img206.png"
 ALT="$(a_1,a_2)$">|; 

$key = q/displaystylep(y^{2n+1});MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$\displaystyle p(y^{2n+1})$">|; 

$key = q/displaystylei=0,1,...,4.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img236.png"
 ALT="$\displaystyle i=0,1,...,4.$">|; 

$key = q/{figure}{centerline{epsfig{file=flexible.crop2.in.eps,width=12.0cm}}{{{{figure};FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="543" HEIGHT="591" BORDER="0"
 SRC="|."$dir".q|img238.png"
 ALT="\begin{figure}\centerline{
\epsfig{file=flexible.crop2.in.eps,width=12.0cm}
}\end{figure}">|; 

$key = q/q_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$q_i$">|; 

$key = q/x_i=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="$x_i=1$">|; 

$key = q/p(1.5);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img229.png"
 ALT="$p(1.5)$">|; 

$key = q/a;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$a$">|; 

$key = q/displaystylep(y^4);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="$\displaystyle p(y^4)$">|; 

$key = q/y=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img217.png"
 ALT="$y=1$">|; 

$key = q/k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="$k$">|; 

$key = q/L(A,B,p)={pA+(1-p)B};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img207.png"
 ALT="$L(A,B,p)=\{pA+(1-p)B\}$">|; 

$key = q/displaystylep_2p_2^1q_1,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$\displaystyle p_2 p_2^1 q_1,$">|; 

$key = q/p;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="$p$">|; 

$key = q/y=(1-a)n+l;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="116" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="$y=(1-a)n+l$">|; 

$key = q/z;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="$z$">|; 

$key = q/x=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="$x=0$">|; 

$key = q/displaystyleV(x,a)=sum_yv(y)p(y),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="$\displaystyle V(x,a)=\sum_y v(y) p(y),$">|; 

$key = q/displaystyleU_K(x)=1slashKsum_{k=1}^Ku(y^k).;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$\displaystyle U_K(x)= 1/K \sum_{k=1}^K u(y^k).$">|; 

$key = q/u(0)=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img222.png"
 ALT="$u(0)=0$">|; 

$key = q/y=d(x);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="$y= d (x)$">|; 

$key = q/y^i,i=1,...,n+m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$y^i, i=1,...,n+m$">|; 

$key = q/displaystylep(k,l,q)=e^{-lambda}frac{lambda^l}{l!};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="$\displaystyle p(k,l,q)=e^{-\lambda} \frac {\lambda^l}{l!}$">|; 

$key = q/displaystyle[(y=0.5)approx{p(0.5)(y=1)+(1-p(0.5))(y=0)}]nonumber;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="353" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img231.png"
 ALT="$\displaystyle [(y=0.5) \approx \{p(0.5)(y=1) +(1-p(0.5))(y= 0) \}] \nonumber$">|; 

$key = q/a_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$a_i$">|; 

$key = q/p_i=1-q_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$p_i=1-q_i$">|; 

$key = q/displaystylea_j^1=argmax_{a_j}V_j(x_j^0,a_j,a_l^0,lnej),j=1,2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="300" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img201.png"
 ALT="$\displaystyle a_j^1 = arg \max_{a_j}
V_j(x_j^0, a_j,a_l^0, l \ne j), j=1,2$">|; 

$key = q/displaystylep_2p_2^2q_1,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$\displaystyle p_2 p_2^2 q_1,$">|; 

$key = q/u_i(y_i);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="$u_i(y_i)$">|; 

$key = q/x_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="$x_j$">|; 

$key = q/p(y^j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$p(y^j)$">|; 

$key = q/ax;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="$a x$">|; 

$key = q/V_j(x_j,a_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img190.png"
 ALT="$V_j(x_j,a_j)$">|; 

$key = q/displaystylep(y^{2n-1});MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$\displaystyle p(y^{2n-1})$">|; 

$key = q/displaystyleU(x,a)=sum_{y}u(y)p(y),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="$\displaystyle U(x,a)=\sum_{y} u(y) p(y),$">|; 

$key = q/displaystylex^1=argmax_{x}U(x,a^0),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="$\displaystyle x^1 = arg \max_{x}
U(x, a^0),$">|; 

$key = q/(x_j^0,a_j^0,j=1,2);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="$(x_j^0,a_j^0, j=1,2)$">|; 

$key = q/a_ix_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="$a_i x_i$">|; 

$key = q/j=1,2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="$j=1,2$">|; 

$key = q/displaystylep(y^2);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$\displaystyle p(y^2)$">|; 

$key = q/eta_i^k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$\eta_i^k $">|; 

$key = q/u(B)=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img215.png"
 ALT="$u(B)=0$">|; 

$key = q/s_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img187.png"
 ALT="$s_j$">|; 

$key = q/f[0]=0.0,f[1]=0.1,f[2]=0.2,f[3]=0.25,f[4]=0.3,f[5]=0.4,f[6]=0.7,f[7]=0.9,f[8]=1.0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="720" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img242.png"
 ALT="$f[0]=0.0, f[1]=0.1, f[2]=0.2, f[3]=0.25,\
f[4]=0.3, f[5]=0.4, f[6]=0.7, f[7]=0.9, f[8]=1.0$">|; 

$key = q/displaystylep(y^{2n+2});MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$\displaystyle p(y^{2n+2})$">|; 

$key = q/B;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="$B$">|; 

$key = q/n=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$n=0$">|; 

$key = q/L;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$L$">|; 

$key = q/displaystyleu(y)=left{array{{ll}0<col_mark><mbox_mark>{if{0ley<a{}<row_mark>1<col_mark><mbox_mark>{if{aleylec{}array{.right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="195" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="$\displaystyle u(y)=
\left\{
\begin{array}{ll}
0 &amp; \mbox{if $0 \le y &lt; a$} \\\\
1 &amp; \mbox{if $a \le y \le c$}
\end{array}.
\right.$">|; 

$key = q/displaystylea^1=argmax_{a}V(x^0,a_),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="161" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="$\displaystyle a^1 = arg \max_{a}
V(x^0, a_),$">|; 

$key = q/displaystylew^{n}=T(w^n).;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="$\displaystyle w^{n} = T(w^n).$">|; 

$key = q/z=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="$z=1$">|; 

$key = q/u(0)=0,u(0.5)=p(0.5),u(1)=p(1),u(1.5)=u(1)+p(1.5)(u(2)-u(1))u(2)=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="587" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img234.png"
 ALT="$u(0)=0, u(0.5)=p(0.5),
 u(1)=p(1), u(1.5)=u(1) + p(1.5) (u(2)-u(1)) u(2)=1$">|; 

$key = q/x_i=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="$x_i=0$">|; 

$key = q/y=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img219.png"
 ALT="$y=0$">|; 

$key = q/j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img186.png"
 ALT="$j$">|; 

$key = q/p_ic_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$p_ic_i$">|; 

$key = q/displaystyled(x_j)=a_js_j-(1-a_j)(n_j-s_j),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="232" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="$\displaystyle d(x_j)=
a_j s_j-(1-a_j)(n_j-s_j),$">|; 

$key = q/y;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="$y$">|; 

$key = q/y=0.5;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img226.png"
 ALT="$y=0.5$">|; 

$key = q/y^k,k=0,...,5;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$y^k, k=0,...,5$">|; 

$key = q/y=0,y=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img225.png"
 ALT="$y=0, y=1$">|; 

$key = q/n_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="$n_j$">|; 

$key = q/y^0=0,y_1=a_1x_1y^2=a_2^1x_2,y^3=a_2^2x_2,y^4=a_1x_1+a_2^1x_2,y^5=a_1x_1+a_2^2x_2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="543" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$y^0=0, y_1= a_1 x_1 y^2=a_2^1 x_2, y^3=a_2^2 x_2, \\\\
y^4=a_1 x_1+ a_2^1 x_2, y^5=a_1 x_1+ a_2^2 x_2$">|; 

$key = q/displaystyleU(x)={{bf{E}u(y)=int_0^{infty}u(y)p(y)dy.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="52" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$\displaystyle U(x)={\bf E} u(y) = \int_0^{\infty} u(y) p(y) dy.$">|; 

$key = q/displaystylec(x)=left{array{{ll}z-ax<col_mark><mbox_mark>{if{delta=1{}<row_mark>(1-a)x<col_mark><mbox_mark>{if{delta=0{}array{.right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="$\displaystyle c(x)=
\left\{
\begin{array}{ll}
z- a x &amp; \mbox{if $\delta=1$}\\\\
(1-a) x &amp; \mbox{if $\delta=0$}
\end{array}.
\right.$">|; 

$key = q/y_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="$y_j$">|; 

$key = q/a_j,j=1,2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="$a_j,  j=1,2$">|; 

$key = q/{figure}{centerline{epsfig{file=flexible.crop.out.eps,width=12.0cm}}{{{{figure};FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="543" HEIGHT="481" BORDER="0"
 SRC="|."$dir".q|img239.png"
 ALT="\begin{figure}\centerline{
\epsfig{file=flexible.crop.out.eps,width=12.0cm}
}\end{figure}">|; 

$key = q/w^n=(x_j^n,a_j^n,j=1,2)inB,n=0,1,2,...;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img202.png"
 ALT="$w^n=(x_j^n,a_j^n, j=1,2) \in B ,\
n=0,1,2,...$">|; 

$key = q/displaystylep(y^0);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$\displaystyle p(y^0)$">|; 

$key = q/x_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$x_i$">|; 

$key = q/(1-p);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img209.png"
 ALT="$(1-p)$">|; 

$key = q/p(y_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="$p(y_j)$">|; 

$key = q/(x_1,x_2,a_1,a_2);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img204.png"
 ALT="$(x_1,x_2,a_1,a_2)$">|; 

$key = q/c_i=max_jc_j=c;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="$c_i=\max_j c_j=c$">|; 

$key = q/w^{n+1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="$w^{n+1}$">|; 

$key = q/x_i^*=1slashm,i=1,...,m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$x_i^*=1/m, i=1,...,m$">|; 

$key = q/y^k=sum_{i=1}^mc_i(x_i);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="$y^k=\sum_{i=1}^m c_i (x_i)$">|; 

$key = q/p(C);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img211.png"
 ALT="$p(C)$">|; 

$key = q/displaystyleprod_{i}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$\displaystyle \prod_{i} q_i,$">|; 

$key = q/displaystyled(x)=an-(1-a)(n-s),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="$\displaystyle d(x)=
a n-(1-a)(n-s),$">|; 

$key = q/displaystyle..................................................;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$\displaystyle ..................................................$">|; 

$key = q/displaystyle...................................;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="159" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$\displaystyle ...................................$">|; 

$key = q/displaystylemin_{winB}parallelw-T(w)parallel^2.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="$\displaystyle \min_{w \in B} \parallel w - T(w) \parallel^2.$">|; 

$key = q/(x^0,a^0);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="$(x^0,a^0)$">|; 

$key = q/(x^0,a^0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="$(x^0,a^0$">|; 

$key = q/displaystylemax_xU(x),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$\displaystyle \max_x U(x),$">|; 

$key = q/displaystylep_2p_2^1prod_{ine2}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$\displaystyle p_2 p_2^1 \prod_{i \ne 2} q_i,$">|; 

$key = q/displaystylep_1q_2,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="$\displaystyle p_1 q_2,$">|; 

$key = q/displaystylep_np_n^1prod_{inen}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$\displaystyle p_n p_n^1 \prod_{i \ne n} q_i ,$">|; 

$key = q/y^0=0,y^1=a_1x_1,y^2=a_2x_2,y^n=a_nx_n,y^{n+1}=a_1x_1+a_2x_2,y^{n+2}=a_1x_1+a_3x_3;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="587" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$y^0=0, y^1=a_1 x_1, y^2=a_2 x_2, y^n= a_n x_n,\ y^{n+1}=a_1 x_1+a_2 x_2, y^{n+2}=a_1 x_1 + a_3 x_3$">|; 

$key = q/A;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img208.png"
 ALT="$A$">|; 

$key = q/K;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$K$">|; 

$key = q/displaystyleU(x,a)=sum_{k=1}^Mu(y^k)p(y^k),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="$\displaystyle U(x,a)=\sum_{k=1}^M u(y^k) p(y^k),$">|; 

$key = q/U(x);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$U(x)$">|; 

$key = q/y=1.5;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img227.png"
 ALT="$y=1.5$">|; 

$key = q/i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$i$">|; 

$key = q/p_i=P{delta_i=1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$p_i=P\{\delta_i=1\}$">|; 

$key = q/n;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="$n$">|; 

$key = q/s;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="$s$">|; 

$key = q/yk[i]=0.25*i,i=0,...,8;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img240.png"
 ALT="$yk[i]=0.25*i,\
i=0,...,8$">|; 

$key = q/displaystylec(x)=(1-a)n+l,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="$\displaystyle c(x)=
(1-a)n+l,$">|; 

$key = q/alpha_i>0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$\alpha_i&gt;0$">|; 

$key = q/x;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$x$">|; 

$key = q/p_i^l,sum_{l=1}^Lp_i^l=1,p_i^lge0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$p_i^l, \sum_{l=1}^L p_i^l=1, p_i^l \ge 0$">|; 

$key = q/displaystylep(y^5);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="$\displaystyle p(y^5)$">|; 

$key = q/displaystyleu(y)=left{array{{ll}0<col_mark><mbox_mark>{if{0ley<c{}<row_mark>1<col_mark><mbox_mark>{if{y=c{}array{.right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="193" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="$\displaystyle u(y)=
\left\{
\begin{array}{ll}
0 &amp; \mbox{if $0 \le y &lt; c$} \\\\
1 &amp; \mbox{if $ y = c$}
\end{array}.
\right.$">|; 

$key = q/y=an-(1-a)(n-s);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="$y=an-(1-a)(n-s)$">|; 

$key = q/displaystyle.............;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$\displaystyle .............$">|; 

$key = q/p_i,i=1,...,n;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$p_i, i=1,...,n$">|; 

$key = q/displaystyleu(y)=cy.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$\displaystyle u(y)= c y.$">|; 

$key = q/(x_1,x_2);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img205.png"
 ALT="$(x_1,x_2)$">|; 

$key = q/u(C)=p(C);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img216.png"
 ALT="$u(C)=p(C)$">|; 

$key = q/displaystyleu(y)>y,ifr<ylec.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="$\displaystyle u(y) &gt; y, if r &lt; y \le c.$">|; 

$key = q/displaystyleL(A,B,C,p(C))=[Capprox{p(C)A+(1-p(C))B}].;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img212.png"
 ALT="$\displaystyle L(A,B,C,p(C))=[C \approx \{p(C) A + (1-p(C)) B\}].$">|; 

$key = q/approx;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img213.png"
 ALT="$\approx$">|; 

$key = q/(x_j^0,a_j^0,j=1.2);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="117" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="$(x_j^0,a_j^0, j=1.2)$">|; 

$key = q/displaystyley_i^k=left{array{{ll}c_ix_i,<col_mark><mbox_mark>{if{eta_i^kin[0,p_i]{}<row_mark>0,<col_mark><mbox_mark>{otherwise}array{right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="206" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$\displaystyle y_i^k=
\left\{
\begin{array}{ll}
c_i x_i, &amp; \mbox{if $\eta_i^k \in [0,p_i]$} \\\\
0, &amp; \mbox{otherwise}
\end{array}\right.$">|; 

$key = q/displaystylep(y_j)=p(n_j,s_j,q)=P{y=s_ja_j-(1-a_j)(n_j-s_j)}.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="382" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img189.png"
 ALT="$\displaystyle p(y_j)=p(n_j,s_j,q)=P\{y=s_j a_j-(1-a_j)(n_j-s_j)\}.$">|; 

$key = q/u(y^k);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$u(y^k)$">|; 

$key = q/p(y);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$p(y)$">|; 

$key = q/c_ix_igea,i=1,...,m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="$c_i x_i \ge
a, i=1,...,m$">|; 

$key = q/displaystylep_2prod_{ine2}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$\displaystyle p_2
\prod_{i \ne 2} q_i,$">|; 

$key = q/delta_i=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="$\delta_i=1$">|; 

$key = q/a=(a_i,i=1,...,m);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$a=(a_i, i=1,...,m)$">|; 

$key = q/p(y^k);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$p(y^k)$">|; 

$key = q/L=2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$L=2$">|; 

$key = q/v_j(y_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="$v_j(y_j)$">|; 

$key = q/displaystylep_1p_1^1prod_{ine1}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$\displaystyle p_1 p_1^1 \prod_{i \ne 1} q_i,$">|; 

$key = q/p_x(y^k);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$p_x(y^k)$">|; 

$key = q/x=(x_i,...,x_n),sum_i=1,x_ige0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$x=(x_i,...,x_n),  \sum_i=1, x_i \ge 0$">|; 

$key = q/displaystyleU(x,a)=sum_yu(y)p(y),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="$\displaystyle U(x,a)=\sum_y u(y) p(y),$">|; 

$key = q/x=1slashm;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$x=1/m$">|; 

$key = q/displaystylep_1p_3prod_{ine1,ine3}q_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$\displaystyle p_1 p_3 \prod_{i
\ne 1, i \ne 3} q_i$">|; 

$key = q/p(1);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img220.png"
 ALT="$p(1)$">|; 

$key = q/displaystylep(y^3);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$\displaystyle p(y^3)$">|; 

$key = q/T;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="$T$">|; 

$key = q/V_j(x,a_j);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="$V_j(x,a_j)$">|; 

$key = q/displaystyleu(y)=u(y_i)+p(y_i)(u(y_{i+1})-u(y_i)),y_iley<y_{i+1},;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="375" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img235.png"
 ALT="$\displaystyle u(y)=u(y_i) + p(y_i) (u(y_{i+1}) - u(y_i)),  y_i \le y
&lt;y_{i+1},$">|; 

$key = q/displaystylep(y)=p(k,l,q)=P{y=(1-a)n+l}.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="276" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="$\displaystyle p(y)=p(k,l,q)=P\{y=(1-a)n+l\}.$">|; 

$key = q/displaystylep(y)=p(n,s,q)=P{y=sa-(1-a)(n-s)}.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="$\displaystyle p(y)=p(n,s,q)=P\{y=sa-(1-a)(n-s)\}.$">|; 

$key = q/c;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="$c$">|; 

$key = q/x=sum_{i=1}^nx_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="$x=\sum_{i=1}^n x_i$">|; 

$key = q/x_j=n_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img185.png"
 ALT="$x_j=n_j$">|; 

$key = q/r;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="$r$">|; 

$key = q/displaystyle[(y=1.0)approx{p(1.5)(y=2.0)+(1-p(1.5))(y=1)}].nonumber;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img233.png"
 ALT="$\displaystyle [(y=1.0) \approx \{p(1.5)(y=2.0) +(1-p(1.5))(y=1) \}]. \nonumber$">|; 

$key = q/displaystylep_2p_2^2p_1.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$\displaystyle p_2 p_2^2 p_1.$">|; 

$key = q/displaystyleV_j(x,a)=sum_{y_j}v_j(y_j)p(y_j),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="$\displaystyle V_j(x,a)=\sum_{y_j} v_j(y_j) p(y_j),$">|; 

$key = q/displaystyle(x_1^1,x_2^1)=argmax_{x_1,x_2}U(x_1,x_2,a_1^0,a_2^0),;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="251" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img200.png"
 ALT="$\displaystyle (x_1^1,x_2^1) = arg \max_{x_1,x_2}
U(x_1,x_2, a_1^0,a_2^0),$">|; 

$key = q/y^k;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$y^k$">|; 

$key = q/z_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="$z_i$">|; 

$key = q/u(1)=p(1);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img224.png"
 ALT="$u(1)=p(1)$">|; 

$key = q/displaystyley^k=sum_{i=1}^ny_i^k,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$\displaystyle y^k= \sum_{i=1}^n y_i^k ,$">|; 

$key = q/p=P{delta=1};MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$p=P\{\delta=1\}$">|; 

$key = q/displaystyleu(y)=y,ify=r,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="$\displaystyle u(y)=y, if y= r,$">|; 

$key = q/p(y^i);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$p(y^i)$">|; 

$key = q/w^n;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="$w^n$">|; 

$key = q/x=n;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="$x=n$">|; 

$key = q/u(1)=u(0)+p(1)(u(2)-u(0));MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="223" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img221.png"
 ALT="$u(1)= u(0)+p(1)(u(2)-u(0))$">|; 

$key = q/(x_j^1,a_j^1,j=1,2);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img197.png"
 ALT="$(x_j^1,a_j^1, j=1,2)$">|; 

$key = q/q=1-p;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="$q=1-p$">|; 

$key = q/displaystyleL(1.0,0.0,0.5,p(0.5))=;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="167" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img230.png"
 ALT="$\displaystyle L(1.0,0.0,0.5,p(0.5))=$">|; 

$key = q/U(x,a);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="$U(x,a)$">|; 

$key = q/delta=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$\delta=1$">|; 

$key = q/displaystylep(y^1);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$\displaystyle p(y^1)$">|; 

$key = q/alpha_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="$\alpha_i$">|; 

$key = q/displaystylep_2p_2^2prod_{ine2}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$\displaystyle p_2 p_2^2 \prod_{i \ne 2} q_i,$">|; 

$key = q/k=m-n;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="$k=m-n$">|; 

$key = q/displaystyleU(x)=sum_{k=1}^Mu(y^k)p(y^k).;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="164" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$\displaystyle U(x)=\sum_{k=1}^M u(y^k) p(y^k).$">|; 

$key = q/displaystylep_np_n^2prod_{inen}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$\displaystyle p_n p_n^2 \prod_{i \ne n} q_i ,$">|; 

$key = q/textstyle=;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$\textstyle =$">|; 

$key = q/delta_i=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$\delta_i=0$">|; 

$key = q/p_i,i=n+j,j=1,...,m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="$p_i, i=n+j,  j=1,...,m$">|; 

$key = q/displaystylep_1p_1^2p_2^p2^2prod_{ine1,ine2}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$\displaystyle p_1 p_1^2 p_2^p2^2 \prod_{i
\ne 1, i \ne 2} q_i,$">|; 

$key = q/a_i^l,l=1,...,L;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$a_i^l, l=1,...,L$">|; 

$key = q/y_i=c_ix_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$y_i=c_i x_i$">|; 

$key = q/nlem;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="$n \le m$">|; 

$key = q/a=1slashmmin_ic_ix_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$a= 1/m min_i c_i x_i$">|; 

$key = q/x_ilez_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$x_i \le z_i$">|; 

$key = q/displaystylep(y^{n+1});MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$\displaystyle p(y^{n+1})$">|; 

$key = q/N;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img203.png"
 ALT="$N$">|; 

$key = q/displaystylep(y^{2n});MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$\displaystyle p(y^{2n})$">|; 

$key = q/xlez;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="$x \le z$">|; 

$key = q/displaystylec_i(x_i)=left{array{{ll}z_i-a_ix_i<col_mark><mbox_mark>{if{delta_i=1row_mark>(1-a_i)x_i<col_mark><mbox_mark>{if{delta_i=0{}array{.right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="238" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="$\displaystyle c_i(x_i)=
\left\{
\begin{array}{ll}
z_i- a_i x_i &amp; \mbox{if $\delta_i=1$}\\\\
(1-a_i) x_i &amp; \mbox{if $\delta_i=0$}
\end{array}.
\right.$">|; 

$key = q/y=2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img218.png"
 ALT="$y=2$">|; 

$key = q/c=max_ic_ix_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="$c=\max_i c_i x_i$">|; 

$key = q/displaystyleq_1q_2,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$\displaystyle q_1 q_2,$">|; 

$key = q/l;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="9" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="$l$">|; 

$key = q/k-l;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="$k-l$">|; 

$key = q/displaystyleu(y)<y,if0ley<r,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="$\displaystyle u(y) &lt; y, if 0 \le y &lt; r,$">|; 

$key = q/p_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$p_i$">|; 

$key = q/x=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="$x=1$">|; 

$key = q/displaystylep_1p_2prod_{ine1,ine2}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$\displaystyle p_1 p_2 \prod_{i
\ne 1, i \ne 2} q_i ,$">|; 

$key = q/U(y);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$U(y)$">|; 

$key = q/displaystylep_nprod_{inen}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$\displaystyle p_n
\prod_{i \ne n} q_i ,$">|; 

$key = q/u(y);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="$u(y)$">|; 

$key = q/lambda=qk;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="$\lambda =qk$">|; 

$key = q/displaystyleU(x)=lim_{Ktoinfty}U_K(x).;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$\displaystyle U(x)= \lim_{K \to \infty} U_K(x).$">|; 

$key = q/a_j;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="$a_j$">|; 

$key = q/displaystylesum_{i=1}^nx_i=1,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="60" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$\displaystyle \sum_{i=1}^n x_i = 1,$">|; 

$key = q/f[0]=0.0,f[1]=0.1,f[2]=0.2,f[3]=0.25,f[4]=0.5,f[5]=0.7,f[6]=0.85,f[7]=0.95,f[8]=1.0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="736" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img244.png"
 ALT="$f[0]=0.0, f[1]=0.1, f[2]=0.2, f[3]=0.25,\
f[4]=0.5, f[5]=0.7, f[6]=0.85, f[7]=0.95, f[8]=1.0$">|; 

$key = q/p(y^1)=p_1prod_{ineq1}(1-p_i);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="168" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="$p(y^1)=p_1 \prod_{i \neq 1} (1-p_i)$">|; 

$key = q/c_1(x_1)=z_1-a_1x_1,c_i(x_i)=(1-a_i)x_i,i=2,...,m;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$c_1(x_1)= z_1-a_1 x_1,\\\\
c_i(x_i)=(1-a_i)x_i,  i=2,...,m$">|; 

$key = q/y=y^k,k=1,...,M;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$y=y^k, k=1,...,M$">|; 

$key = q/displaystylex_i=z_i=1.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="$\displaystyle x_i=z_i=1.$">|; 

$key = q/f[0]=0.0,f[1]=0.3,f[2]=0.5,f[3]=0.7,f[4]=0.8,f[5]=0.85,f[6]=0.9,f[7]=0.95,f[8]=1.0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="728" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img241.png"
 ALT="$f[0]=0.0, f[1]=0.3, f[2]=0.5, f[3]=0.7,\
f[4]=0.8, f[5]=0.85, f[6]=0.9, f[7]=0.95, f[8]=1.0$">|; 

$key = q/v(y);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="$v(y)$">|; 

$key = q/displaystylew^{n+1}=T(w^n),n=0,1,2,...;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="$\displaystyle w^{n+1} = T(w^n), n=0,1,2,...$">|; 

$key = q/displaystylex_ige0.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$\displaystyle x_i \ge 0.$">|; 

$key = q/c_i=1+alpha_i;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$c_i=1+\alpha_i$">|; 

$key = q/displaystylep_1p_1^2prod_{ine1}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$\displaystyle p_1 p_1^2 \prod_{i \ne 1} q_i,$">|; 

$key = q/p_i^l;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$p_i^l$">|; 

$key = q/a_i^l;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$a_i^l$">|; 

$key = q/u(A)=1;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img214.png"
 ALT="$u(A)=1$">|; 

$key = q/V(x,a);MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="$V(x,a)$">|; 

$key = q/displaystyled(x)=left{array{{ll}ax,<col_mark><mbox_mark>{if{delta=1{}<row_mark>-(1-a)x<col_mark><mbox_mark>{if{delta=0{}array{.right.;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="226" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="$\displaystyle d(x)=
\left\{
\begin{array}{ll}
a x, &amp; \mbox{if $\delta=1$} \\\\
-(1-a) x &amp; \mbox{if $\delta=0$}
\end{array}.
\right.$">|; 

$key = q/delta=0;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="15" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="$\delta=0$">|; 

$key = q/displaystylep_1p_1^1p_2p_2^1prod_{ine1,ine2}q_i,;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="132" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$\displaystyle p_1 p_1^1 p_2 p_2^1 \prod_{i
\ne 1, i \ne 2} q_i ,$">|; 

$key = q/x_j,j=1,2;MSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="$x_j,  j=1,2$">|; 

1;

