Sharedwww / tables / parity / images.plOpen in CoCalc
Author: William A. Stein
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# LaTeX2HTML 98.1p1 release (March 2nd, 1998)
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# Associate images original text with physical files.
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$key = q/{inline}sigmaR_S,m{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img50.gif"
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ALT="$\sigma R_{S,m}$">|;
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$key = q/{inline}sigma{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="17" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img23.gif"
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ALT="$\sigma$">|;
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$key = q/{inline}x_imapstoepsilon_i{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="63" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img45.gif"
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ALT="$x_{i} \mapsto \epsilon_{i}$">|;
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$key = q/{inline}a_1,a_2,ldots,a_k{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="107" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img52.gif"
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ALT="$a_{1}, a_{2}, \ldots, a_{k}$">|;
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$key = q/{inline}t=prod(x_s+epsilon_s){inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="121" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img39.gif"
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ALT="$t = \prod (x_{s} + \epsilon_{s})$">|;
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$key = q/{inline}2^n-1times2^n-1{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="101" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img20.gif"
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ALT="$2^{n-1} \times 2^{n-1}$">|;
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$key = q/{inline}pi_b{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="23" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img17.gif"
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ALT="$\pi_{b}$">|;
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$key = q/{inline}ngeq1{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img28.gif"
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ALT="$n \geq 1$">|;
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$key = q/{inline}ngeq2{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img32.gif"
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ALT="$n \geq 2$">|;
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$key = q/{inline}2^@mathcalP(S){inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="98" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
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SRC="img2.gif"
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ALT="$2^{{\@mathcal P}(S)}$">|;
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$key = q/{displaymath}R_S:=fracmboxbfF_2[x_s,:,sinS](x_s^2-x_s,:,sinS).{displaymath}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="198" HEIGHT="52"
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SRC="img6.gif"
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ALT="\begin{displaymath}R_{S}
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:= \frac{\mbox{\bf F}_2[x_{s}\, :\, s \in S]}{(x_{s}^{2}- x_{s}\, : \, s \in S)}.
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\end{displaymath}">|;
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$key = q/{inline}Tvarphi{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="31" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img11.gif"
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ALT="$T\varphi$">|;
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$key = q/{inline}sigma:=1+sum_sinSx_s{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="142" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img10.gif"
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ALT="$\sigma := 1 + \sum_{s \in S} x_{s}$">|;
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$key = q/{inline}sigma=1{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img14.gif"
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ALT="$\sigma = 1$">|;
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$key = q/{inline}epsilon_i{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img44.gif"
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ALT="$\epsilon_{i}$">|;
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$key = q/{displaymath}Tvarphi=sum_binT,prod_sinbx_s.{displaymath}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="127" HEIGHT="53"
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SRC="img9.gif"
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ALT="\begin{displaymath}T\varphi= \sum_{b \in T} \,\prod_{s \in b} x_{s}.
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\end{displaymath}">|;
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$key = q/{inline}(Tvarphi)(b){inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="67" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img16.gif"
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ALT="$(T \varphi)(b)$">|;
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$key = q/{inline}sigmaR{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="31" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img38.gif"
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ALT="$\sigma R$">|;
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$key = q/{inline}|b|,|mathcalP(b)capT|{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img46.gif"
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ALT="$\vert b\vert \, \vert{\mathcal P}(b) \cap T\vert$">|;
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$key = q/{inline}|T|_a_1,a_2,ldots,a_k{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="97" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img53.gif"
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ALT="$\vert T\vert _{a_{1}, a_{2}, \ldots, a_{k}}$">|;
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$key = q/{inline}sigma=1+sum_sinSx_s{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="137" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img22.gif"
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ALT="$\sigma = 1 + \sum_{s\in S} x_s$">|;
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$key = q/{inline}mathcalP(mathcalP(S)){inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img8.gif"
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ALT="${\mathcal P}({\mathcal P}(S))$">|;
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$key = q/{inline}varphi{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="18" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img7.gif"
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ALT="$\varphi$">|;
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$key = q/{inline}2^3cdot2^n-2{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="58" HEIGHT="24" ALIGN="BOTTOM" BORDER="0"
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SRC="img33.gif"
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ALT="$2^{3 \cdot 2^{n-2}}$">|;
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$key = q/{inline}b_1,b_2,ldots,b_2^n-1{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="125" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img19.gif"
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ALT="$b_{1}, b_{2}, \ldots, b_{2^{n-1}}$">|;
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$key = q/{inline}epsilon_s{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img41.gif"
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ALT="$\epsilon_{s}$">|;
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$key = q/{inline}cvarphi{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img15.gif"
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ALT="$c \varphi$">|;
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$key = q/{inline}T=(sigmat)varphi^-1{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="106" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img31.gif"
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ALT="$T = (\sigma t) \varphi^{-1}$">|;
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$key = q/{displaymath}R_S,m:=fracbfZslashmbfZ[x_s,:,sinS](x_s^2-x_s,:,sinS).{displaymath}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="219" HEIGHT="52"
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SRC="img47.gif"
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ALT="\begin{displaymath}R_{S,m}
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:= \frac{{\bf Z}/m{\bf Z}[x_{s}\, :\, s \in S]}{(x_{s}^{2}- x_{s}\, :
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\, s \in S)}.
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\end{displaymath}">|;
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$key = q/{inline}mathcalP(S){inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="46" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img4.gif"
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ALT="${\mathcal P}(S)$">|;
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$key = q/{inline}T^prime{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="23" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
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SRC="img29.gif"
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ALT="$T^{\prime}$">|;
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$key = q/{inline}(1+sigma)R{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="76" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img25.gif"
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ALT="$(1 + \sigma) R$">|;
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$key = q/{inline}sigmat=t+sum_sinSx_st{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="147" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img40.gif"
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ALT="$\sigma t = t + \sum_{s \in S} x_{s} t$">|;
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$key = q/{inline}m^sum_i=0^lfloornslashmrfloornchoosemi{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="109" HEIGHT="28" ALIGN="BOTTOM" BORDER="0"
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SRC="img51.gif"
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ALT="$m^{\sum_{i=0}^{\lfloor n/m \rfloor} {n \choose mi}}$">|;
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$key = q/{inline}tpi_b=t(b){inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="82" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img18.gif"
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ALT="$t \pi_{b} = t(b)$">|;
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$key = q/{inline}sigmaR_S{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img1.gif"
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ALT="$\sigma R_{S}$">|;
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$key = q/{inline}mboxbfF_2{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img3.gif"
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ALT="$\mbox{\bf F}_2$">|;
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$key = q/{inline}prod_i=1^n(x_i+epsilon_i){inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="110" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img43.gif"
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ALT="$\prod_{i=1}^{n} (x_{i} + \epsilon_{i})$">|;
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$key = q/{inline}sigmainP{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="53" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img24.gif"
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ALT="$\sigma \in P$">|;
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$key = q/{inline}varphi^-1(sigmat){inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="68" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img34.gif"
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ALT="$\varphi^{-1}(\sigma t)$">|;
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$key = q/{inline}(sigmat)varphi^-1{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="68" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
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SRC="img30.gif"
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ALT="$(\sigma t) \varphi^{-1}$">|;
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$key = q/{inline}sum_|a|<mprod_sina(-x_s){inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="152" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img49.gif"
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ALT="$\sum_{\vert a\vert < m} \prod_{s \in a} (-x_{s})$">|;
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$key = q/{inline}2^mathcalP(S){inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="47" HEIGHT="22" ALIGN="BOTTOM" BORDER="0"
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SRC="img35.gif"
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ALT="$2^{{\mathcal P}(S)}$">|;
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$key = q/{inline}mboxbullet{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="15" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img13.gif"
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ALT="$\mbox{$\bullet$}$">|;
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$key = q/{inline}|b|,|mathcalP(b)capT|_k,k+1,ldots,k+m-1{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="213" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img54.gif"
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ALT="$\vert b\vert \, \vert{\mathcal P}(b) \cap T\vert _{k, k+1, \ldots, k+m-1}$">|;
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$key = q/{inline}mboxbfF_2^n{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="28" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
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SRC="img36.gif"
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ALT="$\mbox{\bf F}_2^{n}$">|;
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$key = q/{inline}mathcalP(b)capT{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="75" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img5.gif"
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ALT="${\mathcal P}(b) \cap T$">|;
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$key = q/{inline}sigmat{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="23" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
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SRC="img12.gif"
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ALT="$\sigma t$">|;
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$key = q/{inline}{prod_sinS(x_s+epsilon_s),|,epsilon_sinmboxbfF_2}{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="204" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img37.gif"
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ALT="$\{\prod_{s \in S} (x_{s} + \epsilon_{s}) \,
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\vert \, \epsilon_{s} \in \mbox{\bf F}_2\}$">|;
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$key = q/{inline}prod(x_s+epsilon_s){inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="90" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img42.gif"
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ALT="$\prod (x_{s} + \epsilon_{s})$">|;
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$key = q/{inline}(b_ivarphi)pi_b_j{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="69" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img21.gif"
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ALT="$(b_{i} \varphi) \pi_{b_{j}}$">|;
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$key = q/{inline}(m,k),|mathcalP(b)capT|_k{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="145" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
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SRC="img55.gif"
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ALT="$(m,k) \, \vert{\mathcal P}(b) \cap T\vert _{k}$">|;
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$key = q/{inline}Box{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="19" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.gif"
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ALT="$\Box$">|;
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$key = q/{inline}prod_sinax_s{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="66" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
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SRC="img48.gif"
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ALT="$\prod_{s \in a} x_{s}$">|;
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$key = q/{inline}1+sigma{inline}MSF=2.5;AAT;/;
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$cached_env_img{$key} = q|<IMG
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WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
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SRC="img26.gif"
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ALT="$1 + \sigma$">|;
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1;
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