# LaTeX2HTML 98.1p1 release (March 2nd, 1998)
# Associate images original text with physical files.


$key = q/{inline}sigmaR_S,m{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img50.gif"
 ALT="$\sigma R_{S,m}$">|; 

$key = q/{inline}sigma{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img23.gif"
 ALT="$\sigma$">|; 

$key = q/{inline}x_imapstoepsilon_i{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img45.gif"
 ALT="$x_{i} \mapsto \epsilon_{i}$">|; 

$key = q/{inline}a_1,a_2,ldots,a_k{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img52.gif"
 ALT="$a_{1}, a_{2}, \ldots, a_{k}$">|; 

$key = q/{inline}t=prod(x_s+epsilon_s){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img39.gif"
 ALT="$t = \prod (x_{s} + \epsilon_{s})$">|; 

$key = q/{inline}2^n-1times2^n-1{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img20.gif"
 ALT="$2^{n-1} \times 2^{n-1}$">|; 

$key = q/{inline}pi_b{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img17.gif"
 ALT="$\pi_{b}$">|; 

$key = q/{inline}ngeq1{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img28.gif"
 ALT="$n \geq 1$">|; 

$key = q/{inline}ngeq2{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img32.gif"
 ALT="$n \geq 2$">|; 

$key = q/{inline}2^@mathcalP(S){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="img2.gif"
 ALT="$2^{{\@mathcal P}(S)}$">|; 

$key = q/{displaymath}R_S:=fracmboxbfF_2[x_s,:,sinS](x_s^2-x_s,:,sinS).{displaymath}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="198" HEIGHT="52"
 SRC="img6.gif"
 ALT="\begin{displaymath}R_{S}
:= \frac{\mbox{\bf F}_2[x_{s}\, :\, s \in S]}{(x_{s}^{2}- x_{s}\, : \, s \in S)}.
\end{displaymath}">|; 

$key = q/{inline}Tvarphi{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img11.gif"
 ALT="$T\varphi$">|; 

$key = q/{inline}sigma:=1+sum_sinSx_s{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img10.gif"
 ALT="$\sigma := 1 + \sum_{s \in S} x_{s}$">|; 

$key = q/{inline}sigma=1{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img14.gif"
 ALT="$\sigma = 1$">|; 

$key = q/{inline}epsilon_i{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img44.gif"
 ALT="$\epsilon_{i}$">|; 

$key = q/{displaymath}Tvarphi=sum_binT,prod_sinbx_s.{displaymath}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="53"
 SRC="img9.gif"
 ALT="\begin{displaymath}T\varphi= \sum_{b \in T} \,\prod_{s \in b} x_{s}.
\end{displaymath}">|; 

$key = q/{inline}(Tvarphi)(b){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img16.gif"
 ALT="$(T \varphi)(b)$">|; 

$key = q/{inline}sigmaR{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img38.gif"
 ALT="$\sigma R$">|; 

$key = q/{inline}|b|,|mathcalP(b)capT|{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img46.gif"
 ALT="$\vert b\vert \, \vert{\mathcal P}(b) \cap T\vert$">|; 

$key = q/{inline}|T|_a_1,a_2,ldots,a_k{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img53.gif"
 ALT="$\vert T\vert _{a_{1}, a_{2}, \ldots, a_{k}}$">|; 

$key = q/{inline}sigma=1+sum_sinSx_s{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img22.gif"
 ALT="$\sigma = 1 + \sum_{s\in S} x_s$">|; 

$key = q/{inline}mathcalP(mathcalP(S)){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img8.gif"
 ALT="${\mathcal P}({\mathcal P}(S))$">|; 

$key = q/{inline}varphi{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img7.gif"
 ALT="$\varphi$">|; 

$key = q/{inline}2^3cdot2^n-2{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
 SRC="img33.gif"
 ALT="$2^{3 \cdot 2^{n-2}}$">|; 

$key = q/{inline}b_1,b_2,ldots,b_2^n-1{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img19.gif"
 ALT="$b_{1}, b_{2}, \ldots, b_{2^{n-1}}$">|; 

$key = q/{inline}epsilon_s{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img41.gif"
 ALT="$\epsilon_{s}$">|; 

$key = q/{inline}cvarphi{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img15.gif"
 ALT="$c \varphi$">|; 

$key = q/{inline}T=(sigmat)varphi^-1{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img31.gif"
 ALT="$T = (\sigma t) \varphi^{-1}$">|; 

$key = q/{displaymath}R_S,m:=fracbfZslashmbfZ[x_s,:,sinS](x_s^2-x_s,:,sinS).{displaymath}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="219" HEIGHT="52"
 SRC="img47.gif"
 ALT="\begin{displaymath}R_{S,m}
:= \frac{{\bf Z}/m{\bf Z}[x_{s}\, :\, s \in S]}{(x_{s}^{2}- x_{s}\, :
\, s \in S)}.
\end{displaymath}">|; 

$key = q/{inline}mathcalP(S){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img4.gif"
 ALT="${\mathcal P}(S)$">|; 

$key = q/{inline}T^prime{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="img29.gif"
 ALT="$T^{\prime}$">|; 

$key = q/{inline}(1+sigma)R{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img25.gif"
 ALT="$(1 + \sigma) R$">|; 

$key = q/{inline}sigmat=t+sum_sinSx_st{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img40.gif"
 ALT="$\sigma t = t + \sum_{s \in S} x_{s} t$">|; 

$key = q/{inline}m^sum_i=0^lfloornslashmrfloornchoosemi{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
 SRC="img51.gif"
 ALT="$m^{\sum_{i=0}^{\lfloor n/m \rfloor} {n \choose mi}}$">|; 

$key = q/{inline}tpi_b=t(b){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img18.gif"
 ALT="$t \pi_{b} = t(b)$">|; 

$key = q/{inline}sigmaR_S{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img1.gif"
 ALT="$\sigma R_{S}$">|; 

$key = q/{inline}mboxbfF_2{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img3.gif"
 ALT="$\mbox{\bf F}_2$">|; 

$key = q/{inline}prod_i=1^n(x_i+epsilon_i){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img43.gif"
 ALT="$\prod_{i=1}^{n} (x_{i} + \epsilon_{i})$">|; 

$key = q/{inline}sigmainP{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img24.gif"
 ALT="$\sigma \in P$">|; 

$key = q/{inline}varphi^-1(sigmat){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img34.gif"
 ALT="$\varphi^{-1}(\sigma t)$">|; 

$key = q/{inline}(sigmat)varphi^-1{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="img30.gif"
 ALT="$(\sigma t) \varphi^{-1}$">|; 

$key = q/{inline}sum_|a|<mprod_sina(-x_s){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img49.gif"
 ALT="$\sum_{\vert a\vert < m} \prod_{s \in a} (-x_{s})$">|; 

$key = q/{inline}2^mathcalP(S){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
 SRC="img35.gif"
 ALT="$2^{{\mathcal P}(S)}$">|; 

$key = q/{inline}mboxbullet{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img13.gif"
 ALT="$\mbox{$\bullet$}$">|; 

$key = q/{inline}|b|,|mathcalP(b)capT|_k,k+1,ldots,k+m-1{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img54.gif"
 ALT="$\vert b\vert \, \vert{\mathcal P}(b) \cap T\vert _{k, k+1, \ldots, k+m-1}$">|; 

$key = q/{inline}mboxbfF_2^n{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="img36.gif"
 ALT="$\mbox{\bf F}_2^{n}$">|; 

$key = q/{inline}mathcalP(b)capT{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img5.gif"
 ALT="${\mathcal P}(b) \cap T$">|; 

$key = q/{inline}sigmat{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="img12.gif"
 ALT="$\sigma t$">|; 

$key = q/{inline}{prod_sinS(x_s+epsilon_s),|,epsilon_sinmboxbfF_2}{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img37.gif"
 ALT="$\{\prod_{s \in S} (x_{s} + \epsilon_{s}) \,
\vert \, \epsilon_{s} \in \mbox{\bf F}_2\}$">|; 

$key = q/{inline}prod(x_s+epsilon_s){inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img42.gif"
 ALT="$\prod (x_{s} + \epsilon_{s})$">|; 

$key = q/{inline}(b_ivarphi)pi_b_j{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img21.gif"
 ALT="$(b_{i} \varphi) \pi_{b_{j}}$">|; 

$key = q/{inline}(m,k),|mathcalP(b)capT|_k{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="img55.gif"
 ALT="$(m,k) \, \vert{\mathcal P}(b) \cap T\vert _{k}$">|; 

$key = q/{inline}Box{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="img27.gif"
 ALT="$\Box$">|; 

$key = q/{inline}prod_sinax_s{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="img48.gif"
 ALT="$\prod_{s \in a} x_{s}$">|; 

$key = q/{inline}1+sigma{inline}MSF=2.5;AAT;/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="img26.gif"
 ALT="$1 + \sigma$">|; 

1;

