# LaTeX2HTML 96.1 (Feb 5, 1996)
# Associate images original text with physical files.

$key = q/{_inline}$a_l${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=11 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline14115" SRC="img19.gif"  > '; 
$key = q/{figure}[btp]bfSOURCE:small<verbatim_mark>verbatim357normalsizebfOUTPUT:small<verbatim_mark>verbatim358normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=285 HEIGHT=649 ALIGN=BOTTOM ALT="figure2958" SRC="img67.gif"  > '; 
$key = q/{_inline}$x_0,ldots,x_mboxttdim+mboxttshiftbmodN,ldots,x_R-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=275 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline14149" SRC="img29.gif"  > '; 
$key = q/{figure}[p]scriptsize<verbatim_mark>verbatim354normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=454 HEIGHT=353 ALIGN=BOTTOM ALT="figure2942" SRC="img64.gif"  > '; 
$key = q/{figure}[p]scriptsize<verbatim_mark>verbatim356normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=419 HEIGHT=327 ALIGN=BOTTOM ALT="figure2952" SRC="img66.gif"  > '; 
$key = q/{_inline}$x_0,ldots,x_mboxttdim,ldots,x_R-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=156 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline14147" SRC="img28.gif"  > '; 
$key = q/{_inline}$r_l${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=10 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline14111" SRC="img17.gif"  > '; 
$key = q/{_inline}$A_1,ldots,A_n${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=74 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline14089" SRC="img9.gif"  > '; 
$key = q/{_inline}$s_l${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=10 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline14113" SRC="img18.gif"  > '; 
$key = q/{_inline}$x_0,ldots,x_r-1,x_r+1,ldots,x_R-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=192 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline14367" SRC="img48.gif"  > '; 
$key = q/{figure}[btp]centerlinepsfigfigure=cyclicsubker.eps,height=2in{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=483 HEIGHT=227 ALIGN=BOTTOM ALT="figure10313" SRC="img69.gif"  > '; 
$key = q/{displaymath}mboxtta.grp()subseteqmboxttapg{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=301 HEIGHT=15 ALIGN=BOTTOM ALT="displaymath7925" SRC="img27.gif"  > '; 
$key = q/{_inline}${0,1}${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=36 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline14079" SRC="img6.gif"  > '; 
$key = q/{_inline}$x_r${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=14 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline14337" SRC="img46.gif"  > '; 
$key = q/{_inline}$A_i${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=15 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline14101" SRC="img12.gif"  > '; 
$key = q/{_inline}$x_0,ldots,x_R-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=85 HEIGHT=13 ALIGN=MIDDLE ALT="tex2html_wrap_inline14365" SRC="img47.gif"  > '; 
$key = q/{_inline}$mboxttsubs_r${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=38 HEIGHT=20 ALIGN=MIDDLE ALT="tex2html_wrap_inline14243" SRC="img42.gif"  > '; 
$key = q/{_inline}$x_0,ldots,x_S-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=84 HEIGHT=13 ALIGN=MIDDLE ALT="tex2html_wrap_inline14231" SRC="img39.gif"  > '; 
$key = q/{_inline}$0,ldots,${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=39 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline14121" SRC="img20.gif"  > '; 
$key = q/{_inline}$x_0,ldots,x_mboxttdim-1,x_mboxttdim+1,ldots,x_R-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=230 HEIGHT=15 ALIGN=MIDDLE ALT="tex2html_wrap_inline14163" SRC="img32.gif"  > '; 
$key = q/{_inline}$x_mboxttdim${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=34 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline14585" SRC="img55.gif"  > '; 
$key = q/{figure}[p]bfSOURCE:small<verbatim_mark>verbatim363normalsizebfOUTPUT:small<verbatim_mark>verbatim364normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=390 HEIGHT=473 ALIGN=BOTTOM ALT="figure3005" SRC="img71.gif"  > '; 
$key = q/{figure}[p]scriptsize<verbatim_mark>verbatim346normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=325 HEIGHT=368 ALIGN=BOTTOM ALT="figure2879" SRC="img60.gif"  > '; 
$key = q/{_inline}$mboxttwhi_mboxact${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=45 HEIGHT=20 ALIGN=MIDDLE ALT="tex2html_wrap_inline14199" SRC="img34.gif"  > '; 
$key = q/{displaymath}prod_arraycx_mboxscriptsizettdim=0mboxscriptsizettmsk[x_0,ldots,x_mboxscriptsizettdim-1,x_mboxscriptsizettdim,quadquadquadx_mboxscriptsizettdim+1,ldots,x_R-1mboxscriptsizett]neq0array^N-1mboxttsrc[x_0,ldots,x_mboxttdim-1,x_mboxttdim,x_mboxttdim+1,ldots,x_R-1mboxtt]{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=509 HEIGHT=95 ALIGN=BOTTOM ALT="displaymath9348" SRC="img53.gif"  > '; 
$key = q/{_inline}$(x_0,ldots,x_R-1)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=98 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline14459" SRC="img49.gif"  > '; 
$key = q/{_inline}$mboxttwlo_mboxact${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=45 HEIGHT=20 ALIGN=MIDDLE ALT="tex2html_wrap_inline14197" SRC="img33.gif"  > '; 
$key = q/{figure}[btp]bfSOURCE:small<verbatim_mark>verbatim350normalsizebfOUTPUT:small<verbatim_mark>verbatim351normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=418 HEIGHT=594 ALIGN=BOTTOM ALT="figure2906" SRC="img62.gif"  > '; 
$key = q/{_inline}$|A_i|${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=23 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline14107" SRC="img15.gif"  > '; 
$key = q/{figure}[btp]bfSOURCE:small<verbatim_mark>verbatim352normalsizebfOUTPUT:small<verbatim_mark>verbatim353normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=185 HEIGHT=579 ALIGN=BOTTOM ALT="figure2924" SRC="img63.gif"  > '; 
$key = q/{figure}[bp]centerlinepsfigfigure=blocksubker.eps,height=2in{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=535 HEIGHT=229 ALIGN=BOTTOM ALT="figure10308" SRC="img68.gif"  > '; 
$key = q/{_inline}${0,1,2}${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=51 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline14081" SRC="img7.gif"  > '; 
$key = q/{_inline}$x_R-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=33 HEIGHT=12 ALIGN=MIDDLE ALT="tex2html_wrap_inline14327" SRC="img45.gif"  > '; 
$key = q/{figure}[p]small<verbatim_mark>verbatim82normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=479 HEIGHT=712 ALIGN=BOTTOM ALT="figure293" SRC="img23.gif"  > '; 
$key = q/{_inline}$a_i${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=11 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline14097" SRC="img11.gif"  > '; 
$key = q/{figure}[btp]centerlinepsfigfigure=blocksubs.eps,height=4in{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=505 HEIGHT=458 ALIGN=BOTTOM ALT="figure7609" SRC="img21.gif"  > '; 
$key = q/{displaymath}prod_x_mboxscriptsizettdim=0^N-1mboxttsrc[x_0,ldots,x_mboxttdim-1,x_mboxttdim,x_mboxttdim+1,ldots,x_R-1mboxtt]{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=433 HEIGHT=49 ALIGN=BOTTOM ALT="displaymath9290" SRC="img52.gif"  > '; 
$key = q/{_inline}$(mboxtta.rng(0).size(),ldots,mboxtta.rng(R-1).size())${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=313 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline14139" SRC="img25.gif"  > '; 
$key = q/{_inline}$0,ldotsP-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=75 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline14069" SRC="img4.gif"  > '; 
$key = q/{figure}[p]scriptsize<verbatim_mark>verbatim355normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=325 HEIGHT=495 ALIGN=BOTTOM ALT="figure2947" SRC="img65.gif"  > '; 
$key = q/{figure}[btp]centerlinepsfigfigure=cyclicsubs.eps,height=4in{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=463 HEIGHT=459 ALIGN=BOTTOM ALT="figure7613" SRC="img22.gif"  > '; 
$key = q/{displaymath}sum_arraycx_mboxscriptsizettdim=0mboxscriptsizettmsk[x_0,ldots,x_mboxscriptsizettdim-1,x_mboxscriptsizettdim,quadquadquadx_mboxscriptsizettdim+1,ldots,x_R-1mboxscriptsizett]neq0array^N-1mboxttsrc[x_0,ldots,x_mboxttdim-1,x_mboxttdim,x_mboxttdim+1,ldots,x_R-1mboxtt]{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=509 HEIGHT=95 ALIGN=BOTTOM ALT="displaymath9223" SRC="img51.gif"  > '; 
$key = q/{_inline}$r_i${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=10 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline14103" SRC="img13.gif"  > '; 
$key = q/{_inline}$s_i${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=10 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline14105" SRC="img14.gif"  > '; 
$key = q/{_inline}$1,ldots,n${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=52 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline14095" SRC="img10.gif"  > '; 
$key = q/{displaymath}arraylllmboxtta.grp();SPMamp;approx;SPMamp;mboxttb.grp()mboxtta.rng(0);SPMamp;approx;SPMamp;mboxttb.rng(0);SPMamp;vdotsmboxtta.rng(R-1);SPMamp;approx;SPMamp;mboxttb.rng(R-1)array{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=354 HEIGHT=102 ALIGN=BOTTOM ALT="displaymath7889" SRC="img26.gif"  > '; 
$key = q/{_inline}$(x_0,ldots,x_S-1)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=97 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline14275" SRC="img43.gif"  > '; 
$key = q/{_inline}$x_0,ldots,x_mboxttdim+mboxttshift,ldots,x_R-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=216 HEIGHT=20 ALIGN=MIDDLE ALT="tex2html_wrap_inline14153" SRC="img30.gif"  > '; 
$key = q/{_inline}$0ldotsP-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=70 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline14049" SRC="img1.gif"  > '; 
$key = q/{_inline}$mboxttwhi_mboxacttt)${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=53 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline14203" SRC="img35.gif"  > '; 
$key = q/{figure}[p]bfSOURCE:small<verbatim_mark>verbatim340normalsizebfOUTPUT:small<verbatim_mark>verbatim341normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=390 HEIGHT=385 ALIGN=BOTTOM ALT="figure2851" SRC="img58.gif"  > '; 
$key = q/{_inline}$x_0${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=15 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline14323" SRC="img44.gif"  > '; 
$key = q/{figure}[btp]bfSOURCE:small<verbatim_mark>verbatim359normalsizebfOUTPUT:small<verbatim_mark>verbatim360normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=418 HEIGHT=594 ALIGN=BOTTOM ALT="figure2987" SRC="img70.gif"  > '; 
$key = q/{_inline}$mboxttsubs_0${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=39 HEIGHT=20 ALIGN=MIDDLE ALT="tex2html_wrap_inline14233" SRC="img40.gif"  > '; 
$key = q/{figure}[p]bfSOURCE:small<verbatim_mark>verbatim342normalsizebfOUTPUT:small<verbatim_mark>verbatim343normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=390 HEIGHT=385 ALIGN=BOTTOM ALT="figure2856" SRC="img59.gif"  > '; 
$key = q/{displaymath}sum_x_mboxscriptsizettdim=0^N-1mboxttsrc[x_0,ldots,x_mboxttdim-1,x_mboxttdim,x_mboxttdim+1,ldots,x_R-1mboxtt]{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=433 HEIGHT=49 ALIGN=BOTTOM ALT="displaymath9165" SRC="img50.gif"  > '; 
$key = q/{_inline}$A_l${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=15 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline14109" SRC="img16.gif"  > '; 
$key = q/{_inline}$0,ldots,A-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=81 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline14055" SRC="img2.gif"  > '; 
$key = q/{figure}[btp]centerlinepsfigfigure=groupsEg.eps,height=2.5in{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=287 HEIGHT=294 ALIGN=BOTTOM ALT="figure7518" SRC="img8.gif"  > '; 
$key = q/{figure}[btp]centerlinepsfigfigure=WriteHalo.eps,height=3in{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=355 HEIGHT=346 ALIGN=BOTTOM ALT="figure8239" SRC="img38.gif"  > '; 
$key = q/{displaymath}sum_i=0^mboxttdim.size()mboxttblocks[imboxtt]{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=331 HEIGHT=50 ALIGN=BOTTOM ALT="displaymath7782" SRC="img24.gif"  > '; 
$key = q/{figure}[p]bfSOURCE:small<verbatim_mark>verbatim338normalsizebfOUTPUT:small<verbatim_mark>verbatim339normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=390 HEIGHT=473 ALIGN=BOTTOM ALT="figure2846" SRC="img57.gif"  > '; 
$key = q/{_inline}$x_mboxttdim+mboxttshift${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=95 HEIGHT=20 ALIGN=MIDDLE ALT="tex2html_wrap_inline14155" SRC="img31.gif"  > '; 
$key = q/{figure}[p]bfSOURCE:small<verbatim_mark>verbatim348normalsizebfOUTPUT:small<verbatim_mark>verbatim349normalsize{figure}/;
$cached_env_img{$key} = ' <IMG WIDTH=390 HEIGHT=613 ALIGN=BOTTOM ALT="figure2886" SRC="img61.gif"  > '; 
$key = q/{displaymath}mboxttmsk[x_0,ldots,x_mboxttdim-1,x_mboxttdim,x_mboxttdim+1,ldots,x_R-1mboxtt]neq0{displaymath}/;
$cached_env_img{$key} = ' <IMG WIDTH=427 HEIGHT=16 ALIGN=BOTTOM ALT="displaymath9481" SRC="img56.gif"  > '; 
$key = q/{_inline}$0,ldots,N-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=84 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline14075" SRC="img5.gif"  > '; 
$key = q/{_inline}$0,ldots,R-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=82 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline14063" SRC="img3.gif"  > '; 
$key = q/{_inline}$x_0,ldots,x_mboxttdim-1,x_mboxttdim,x_mboxttdim+1,ldots,x_R-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=273 HEIGHT=14 ALIGN=MIDDLE ALT="tex2html_wrap_inline14583" SRC="img54.gif"  > '; 
$key = q/{_inline}$mboxttsubs_R-1${_inline}/;
$cached_env_img{$key} = ' <IMG WIDTH=57 HEIGHT=18 ALIGN=MIDDLE ALT="tex2html_wrap_inline14237" SRC="img41.gif"  > '; 

1;

