{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 1 0 0 0 1 }{CSTYLE "Text" -1 200 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Maple Out put" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 2 0 2 0 2 2 0 1 }{PSTYLE "Left Justified Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 2 0 2 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "NOTE: This is not reliable for non-nested sets." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res tart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "s:=0:c:=7;a:=28;b: =37;d:=9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG\"\"(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"aG\"#G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% \"bG\"#P" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"dG\"\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 233 "a and b are the seeds. The sets G[k] a re generated by taking all elements of G[k-1] times c, and then also a dding d to the largest. Thus, G[1]=\{a,b\}, G[2]=\{ac, bc, bc+d\}, et c. The numbers above correspond to what we did Wednesday." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "s:=\{\}:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 85 "The set W below is the set of postage amounts we can mak e with denominations a and b." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "W:=\{\}:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for i from 0 to a- 1 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "for j from 0 to floor((a*b- a-b-i*b)/a) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "W:=W union \{i*b+ j*a\}" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "od:od:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "s:=[seq(i,i=1..a*b-a-b)]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "s:=convert(s,set):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "S[1]:=s minus W;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>&%\"SG6#\"\" \"\"#?\"#@\"#A\"#B\"#C\"#D\"#E\"#F\"#H\"#I\"#J\"#K\"#L \"#M\"#N\"#O\"#Q\"#R\"#S\"#T\"#U\"#V\"#W\"#X\"#Y\"#Z\"#[\"#\\\"#]\"#^ \"#_\"#`\"#a\"#b\"#d\"#e\"#f\"#g\"#h\"#i\"#j\"#k\"#m\"#n\"#o\"#p\"#q\" #r\"#s\"#t\"#v\"#w\"#x\"#y\"#z\"#!)\"#\")\"##)\"#$)\"#&)\"#')\"#()\"#) )\"#*)\"#!*\"#\"*\"##*\"#%*\"#&*\"#'*\"#(*\"#)*\"#**\"$+\"\"$,\"\"$.\" \"$/\"\"$0\"\"$1\"\"$2\"\"$3\"\"$4\"\"$5\"\"$8\"\"$9\"\"$:\"\"$;\"\"$< \"\"$=\"\"$>\"\"$?\"\"$A\"\"$B\"\"$C\"\"$D\"\"$E\"\"$F\"\"$G\"\"$H\"\" $J\"\"$K\"\"$L\"\"$M\"\"$N\"\"$O\"\"$P\"\"$Q\"\"$T\"\"$U\"\"$V\"\"$W\" \"$X\"\"$Y\"\"$Z\"\"$]\"\"$^\"\"$_\"\"$`\"\"$a\"\"$b\"\"$c\"\"$d\"\"$f \"\"$g\"\"$h\"\"$i\"\"$j\"\"$k\"\"$l\"\"$m\"\"$p\"\"$q\"\"$r\"\"$s\"\" $t\"\"$u\"\"$v\"\"$y\"\"$z\"\"$!=\"$\"=\"$#=\"$$=\"$%=\"$(=\"$)=\"$*= \"$!>\"$\">\"$#>\"$$>\"$%>\"$(>\"$)>\"$*>\"$+#\"$,#\"$-#\"$.#\"$1#\"$2 #\"$3#\"$4#\"$5#\"$6#\"$7#\"$:#\"$;#\"$<#\"$=#\"$>#\"$?#\"$@#\"$D#\"$E #\"$F#\"$G#\"$H#\"$I#\"$J#\"$M#\"$N#\"$O#\"$P#\"$Q#\"$R#\"$S#\"$V#\"$W #\"$X#\"$Y#\"$Z#\"$[#\"$\\#\"$`#\"$a#\"$b#\"$c#\"$d#\"$e#\"$i#\"$j#\"$ k#\"$l#\"$m#\"$n#\"$o#\"$r#\"$s#\"$t#\"$u#\"$v#\"$w#\"$x#\"$\"G\"$#G\" $$G\"$%G\"$&G\"$'G\"$!H\"$\"H\"$#H\"$$H\"$%H\"$&H\"$*H\"$+$\"$,$\"$-$ \"$.$\"$/$\"$0$\"$4$\"$5$\"$6$\"$7$\"$8$\"$9$\"$=$\"$>$\"$?$\"$@$\"$A$ \"$B$\"$F$\"$G$\"$H$\"$I$\"$J$\"$K$\"$P$\"$Q$\"$R$\"$S$\"$T$\"$U$\"$Y$ \"$Z$\"$[$\"$\\$\"$]$\"$^$\"$b$\"$c$\"$d$\"$e$\"$f$\"$g$\"$l$\"$m$\"$n $\"$o$\"$p$\"$u$\"$v$\"$w$\"$x$\"$y$\"$z$\"$$Q\"$%Q\"$&Q\"$'Q\"$(Q\"$) Q\"$$R\"$%R\"$&R\"$'R\"$(R\"$-%\"$.%\"$/%\"$0%\"$1%\"$6%\"$7%\"$8%\"$9 %\"$:%\"$;%\"$@%\"$A%\"$B%\"$C%\"$D%\"$I%\"$J%\"$K%\"$L%\"$M%\"$R%\"$S %\"$T%\"$U%\"$V%\"$\\%\"$]%\"$^%\"$_%\"$`%\"$e%\"$f%\"$g%\"$h%\"$i%\"$ n%\"$o%\"$p%\"$q%\"$r%\"$x%\"$y%\"$z%\"$![\"$'[\"$([\"$)[\"$*[\"$!\\\" $&\\\"$'\\\"$(\\\"$)\\\"$*\\\"$0&\"$1&\"$2&\"$3&\"$9&\"$:&\"$;&\"$<&\" $B&\"$C&\"$D&\"$E&\"$F&\"$L&\"$M&\"$N&\"$O&\"$U&\"$V&\"$W&\"$X&\"$^&\" $_&\"$`&\"$a&\"$h&\"$i&\"$j&\"$k&\"$q&\"$r&\"$s&\"$t&\"$z&\"$!e\"$\"e \"$#e\"$*e\"$!f\"$\"f\"$)f\"$*f\"$+'\"$,'\"$2'\"$3'\"$4'\"$5'\"$<'\"$= '\"$>'\"$E'\"$F'\"$G'\"$N'\"$O'\"$P'\"$Q'\"$X'\"$Y'\"$Z'\"$a'\"$b'\"$c '\"$j'\"$k'\"$l'\"$t'\"$u'\"$v'\"$#o\"$$o\"$%o\"$\"p\"$#p\"$$p\"$,(\"$ -(\"$5(\"$6(\"$7(\"$>(\"$?(\"$@(\"$H(\"$I(\"$Q(\"$R(\"$Z(\"$[(\"$\\(\" $d(\"$e(\"$m(\"$n(\"$v(\"$w(\"$&y\"$'y\"$%z\"$&z\"$.)\"$/)\"$8)\"$A)\" $B)\"$J)\"$K)\"$T)\"$])\"$f)\"$g)\"$p)\"$y)\"$())\"$(*)\"$1*\"$:*\"$M* \"$V*\"$r*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 200 168 "The S[k] and G[k] sets are as in the paper. (G[k] gives the available stamp denominati ons, and S[k] gives the postage amounts that *cannot* be made with tho se stamps.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "G[1]:=[a,b]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%\"GG6#\"\"\"7$\"#G\"#P" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "for i from 2 to 20 do G[i]:= vector(i+1,0):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for j from 1 to i do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "G[i][j]:=c*G[i-1][j]" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "G[i][i+1]:=G[i][i]+d:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "for i from 2 to 20 do S[i ]:=\{\}:od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for k from 2 to 3 do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "upperlimit:=(a-1)*G[k][2]+(c-1) *sum(G[k][q],q=3..k+1)-G[k][1]+40000:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "for n from 1 to upperlimit do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "R:=(d^(-1)*n mod c)*(b*c^(k-1)+d*(c^(k-1)-1)/(c-1));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "if (n-R)/c in S[k-1] or n " 0 "" {MPLTEXT 1 0 28 "od;print(S[k-1] m inus S[k]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+upperlimitG\"&0%[" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<(\"$o#\"$F&\"$O&\"$'y\"$&z\"$/)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%+upperlimitG\"'X,6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<2\"%w= \"%&)=\"%*o$\"%)p$\"%_P\"%hP\"%qP\"%-b\"%6b\"%lb\"%ub\"%$e&\"%Gc\"%Pc \"%Yc\"%bc" }}}{EXCHG }{EXCHG {PARA 0 "" 0 "" {TEXT 200 102 "The upper limits are questionable with the new formulas. 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