{VERSION 6 0 "IBM INTEL LINUX" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple O utput" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }} {PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and tra ce have been redefined and unprotected\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 39 "Maximum value of k in all calculations." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "maxk:=18:" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 256 53 "Weakly semimagic homogeneous and affine 3x3 squares " }}{PARA 0 "" 0 "" {TEXT -1 108 "We calculate the number of weakly se mimagic squares: w[k] has upper bound x[i] " 0 "" {MPLTEXT 1 0 245 "for k from 1 to maxk do\n w[k]:=0:\n wa[k]:=0: \n for x[1] from 1 \+ to (k-1) do\n for x[2] from 1 to (k-1) do \n for x[3] from 1 to \+ (k-1) do \n rs:=x[1]+x[2]+x[3]: \n for x[4] from 1 to (k-1 ) do \n for x[5] from 1 to (k-1) do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 250 " x[6]:=rs-x[4]-x[5]:\n x[7]:=rs- x[1]-x[4]: \n x[8]:=rs-x[2]-x[5]: \n x[9]:=rs-x[3] -x[6]: \n if ( (x[6]>0) and (x[6]0) and (x[7]< k) and (x[8]>0) and (x[8]0) and (x[9] " 0 "" {MPLTEXT 1 0 74 " w[k]:=w[k]+1:\n \+ if (rs=k) then wa[k]:=wa[k]+1: fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 " fi:\n od:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 59 " od:\n od:\n od:\n od:\n print(k,w[k],w a[k]):\nod:" }}{PAGEBK }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"\"\"!F$ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$\"#9\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% \"\"%\"#()\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&\"$S$\"#@" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"'\"%,5\"#b" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"(\"%UC\"$?\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\" \")\"%:_\"$J#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"*\"&)35\"$1%" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"#5\"&\"3=\"$m'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#6\"&-0$\"%N5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#7 \"&$)*[\"%S:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#8\"&;b(\"%6A" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"#9\"'*[7\"\"%\"3$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6%\"#:\"'AF;\"%'=%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% \"#;\"'.&H#\"%lb" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#<\"'CmJ\"%gs" } }{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#=\"'<%G%\"%;$*" }}}{PAGEBK } {EXCHG {PARA 0 "" 0 "" {TEXT -1 214 "We expect a quasipolynomial for w (cubical) of degree 5 and a quasipolynomial for wa (affine) of degree 4. We don't know the period; the following calculations are set up t o do any desired period. The variables: " }}{PARA 0 "" 0 "" {TEXT -1 39 "p = assumed period of quasipolynomial, " }}{PARA 0 "" 0 "" {TEXT -1 35 "r (1<=r<=p) = constituent residue, " }}{PARA 0 "" 0 "" {TEXT -1 40 "deg = degree of polynomial, dp = deg+1." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "The first step sets up the period and degree." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "p:=2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "deg:=5 ; \ndp:=deg+1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$degG\"\"&" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dpG\"\"'" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 108 "Arrays to hold the coefficients of the cubical and aff ine polynomials. \"coef\" is a temporary working array." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "coef:=array(1..dp); \nwcoeff:=array(1..p,1..d p);\nwacoeff:=array(1..p,1..dp);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% %coefG-%&arrayG6$;\"\"\"\"\"'7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% 'wcoeffG-%&arrayG6%;\"\"\"\"\"#;F)\"\"'7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(wacoeffG-%&arrayG6%;\"\"\"\"\"#;F)\"\"'7\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 335 "The following proc edure will generate all the p different weak polynomials and p differe nt strong polynomials, factor them, and test by substituting the next \+ value of the argument, comparing to the raw data of the surplus perio d that was calculated in the first procedure). The polynomials will b e saved in \"wpoly[r]\" and \"spoly[r]\"." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "for r from 1 to p do" }}{PARA 0 "" 0 "" {TEXT -1 129 "The following procedure will generate the matrix of values for numbe rs mod r of the period for degree deg with any coefficients." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "V2:=array(1..dp,1..dp):\nV2a:=array(1..de g,1..deg):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "for n from 1 to dp do \n for k from 1 to dp do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 94 " V 2[k,n]:=(p*(k-1)+r)^(n-1):\n if ( (k " 0 "" {MPLTEXT 1 0 3 "od:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "print(V2);" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 27 "This part assumes degree 3." }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 57 "print([w[r],w[r+p],w[r+2*p],w[r+3*p],w[r+4*p], w[r+5*p]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "coef:=linsolve(V2,[w [r],w[r+p],w[r+2*p],w[r+3*p],w[r+4*p],w[r+5*p]]); " }{TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for j from 1 to dp do " }}{PARA 0 " > " 0 "" {MPLTEXT 1 0 91 " wcoeff[r,j]:=coef[j]:\nod;\nprint([wa[r],w a[r+p],wa[r+2*p],wa[r+3*p],wa[r+4*p],wa[r+5*p]]);" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 75 "coef:=linsolve(V2,[wa[r],wa[r+p],wa[r+2*p],wa[r+3*p ],wa[r+4*p],wa[r+5*p]]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "for j f rom 1 to dp do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 " wacoeff[r,j]:= coef[j]:\nod;\n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "wpoly[r]:=wcoef f[r,6]*x^5+wcoeff[r,5]*x^4+wcoeff[r,4]*x^3+wcoeff[r,3]*x^2+wcoeff[r,2] *x+wcoeff[r,1]; \nsubs(x=r+dp*p,wpoly[r]); \nfactor(wpoly[r]);\n" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 155 "wapoly[r]:=wacoeff[r,6]*x^5+wacoef f[r,5]*x^4+wacoeff[r,4]*x^3+wacoeff[r,3]*x^2+wacoeff[r,2]*x+wacoeff[r, 1]; \nsubs(x=r+dp*p,wapoly[r]); \nfactor(wapoly[r]);\n" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V2G-% &arrayG6%;\"\"\"\"\"'F(7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$V2aG- %&arrayG6%;\"\"\"\"\"&F(7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matr ixG6#7(7(\"\"\"F(F(F(F(F(7(F(\"\"$\"\"*\"#F\"#\")\"$V#7(F(\"\"&\"#D\"$ D\"\"$D'\"%DJ7(F(\"\"(\"#\\\"$V$\"%,C\"&2o\"7(F(F+F-\"$H(\"%hl\"&\\!f7 (F(\"#6\"$@\"\"%J8\"&TY\"\"'^5;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7( \"\"!\"#9\"$S$\"%UC\"&)35\"&-0$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%% coefG-%'vectorG6#7(!\"\"#\"#;\"\"&#!\"*\"\"##\"\"(F/#!\"$F/#\"\"$\"#5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!\"\"\"\"#@\"$?\"\"$1%\"%N5 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coefG-%'vectorG6#7(\"\"\"#!\"* \"\"%#\"#:\"\")#!\"$F,#F)F/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>& %&wpolyG6#\"\"\",.*&#\"\"$\"#5F'*$)%\"xG\"\"&F'F'F'*&#F+\"\"#F'*$)F/\" \"%F'F'!\"\"*&#\"\"(F3F'*$)F/F+F'F'F'*&#\"\"*F3F'*$)F/F3F'F'F7*&#\"#;F 0F'F/F'F'F'F7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"&;b(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,$**\"#5!\"\",&%\"xG\"\"\"F)F&F),(*&\"\"$F))F(\" \"#F)F)*&\"\"'F)F(F)F&\"\"&F)F),(*$F-F)F)*&F.F)F(F)F&F.F)F)F)" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%'wapolyG6#\"\"\",,F'F'*&#F'\"\")F'* $)%\"xG\"\"%F'F'F'*&#\"\"$F/F'*$)F.F2F'F'!\"\"*&#\"#:F+F'*$)F.\"\"#F'F 'F'*&#\"\"*F/F'F.F'F5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"%6A" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\")!\"\",&%\"xG\"\"\"F)F&F),&F(F )\"\"#F&F),(*$)F(F+F)F)*&\"\"$F)F(F)F&\"\"%F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#V2G-%&arrayG6%;\"\"\"\"\"'F(7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$V2aG-%&arrayG6%;\"\"\"\"\"&F(7\"" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#-%'matrixG6#7(7(\"\"\"\"\"#\"\"%\"\")\"#;\"#K7(F(F*F, \"#k\"$c#\"%C57(F(\"\"'\"#O\"$;#\"%'H\"\"%wx7(F(F+F/\"$7&\"%'4%\"&oF$7 (F(\"#5\"$+\"\"%+5\"&++\"\"'++57(F(\"#7\"$W\"\"%G<\"&O2#\"'K)[#" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"\"\"#()\"%,5\"%:_\"&\"3=\"&$)*[ " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coefG-%'vectorG6#7(!\"\"#\"#;\" \"&#!\"*\"\"##\"\"(F/#!\"$F/#\"\"$\"#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"!\"\"'\"#b\"$J#\"$m'\"%S:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%coefG-%'vectorG6#7(\"\"\"#!\"*\"\"%#\"#:\"\")#!\"$F,#F)F/\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%&wpolyG6#\"\"#,.*&#\"\"$\"#5\"\" \"*$)%\"xG\"\"&F-F-F-*&#F+F'F-*$)F0\"\"%F-F-!\"\"*&#\"\"(F'F-*$)F0F+F- F-F-*&#\"\"*F'F-*$)F0F'F-F-F7*&#\"#;F1F-F0F-F-F-F7" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#\"'*[7\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"#5! \"\",&%\"xG\"\"\"F)F&F),(*&\"\"$F))F(\"\"#F)F)*&\"\"'F)F(F)F&\"\"&F)F) ,(*$F-F)F)*&F.F)F(F)F&F.F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%' wapolyG6#\"\"#,,\"\"\"F)*&#F)\"\")F)*$)%\"xG\"\"%F)F)F)*&#\"\"$F0F)*$) F/F3F)F)!\"\"*&#\"#:F,F)*$)F/F'F)F)F)*&#\"\"*F0F)F/F)F6" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"%\"3$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$** \"\")!\"\",&%\"xG\"\"\"F)F&F),&F(F)\"\"#F&F),(*$)F(F+F)F)*&\"\"$F)F(F) F&\"\"%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "for r from 1 to p do: r: wapoly[r]: 8*wapoly[r]: factor(wapoly[r]): od;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"\"\"F$*&#F$\"\")F$*$)%\"xG\"\"%F$F$F$*&#\"\"$F+F$*$)F*F.F$F$ !\"\"*&#\"#:F'F$*$)F*\"\"#F$F$F$*&#\"\"*F+F$F*F$F1" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,,\"\")\"\"\"*$)%\"xG\"\"%F%F%*&\"\"'F%)F(\"\"$F%!\" \"*&\"#:F%)F(\"\"#F%F%*&\"#=F%F(F%F." }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$**\"\")!\"\",&%\"xG\"\"\"F)F&F),&F(F)\"\"#F&F),(*$)F(F+F)F)*&\"\"$ F)F(F)F&\"\"%F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"\"\"F$*&#F$\"\")F$*$)%\"xG\"\"%F$F$F$*&# \"\"$F+F$*$)F*F.F$F$!\"\"*&#\"#:F'F$*$)F*\"\"#F$F$F$*&#\"\"*F+F$F*F$F1 " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,\"\")\"\"\"*$)%\"xG\"\"%F%F%*&\" \"'F%)F(\"\"$F%!\"\"*&\"#:F%)F(\"\"#F%F%*&\"#=F%F(F%F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"\")!\"\",&%\"xG\"\"\"F)F&F),&F(F)\"\"#F&F),( *$)F(F+F)F)*&\"\"$F)F(F)F&\"\"%F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "r:=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "for k from 1 to maxk do " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " r:=r+1: " }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 " if (r>p) then r:=1: fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 " print(k,r,eval(wapoly[r],x=k)-wa[k]): \+ " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"F#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"#F #\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\" \"'\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"(\"\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"*\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"# 5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#6\"\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#8\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#9 \"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#:\"\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"#;\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#<\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#= \"\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "for r from 1 \+ to p do: r: wpoly[r]: 10*wpoly[r]: factor(wpoly[r]): od;" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&# \"\"$\"#5\"\"\"*$)%\"xG\"\"&F(F(F(*&#F&\"\"#F(*$)F+\"\"%F(F(!\"\"*&#\" \"(F/F(*$)F+F&F(F(F(*&#\"\"*F/F(*$)F+F/F(F(F3*&#\"#;F,F(F+F(F(F(F3" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&\"\"$\"\"\")%\"xG\"\"&F&F&*&\"#:F& )F(\"\"%F&!\"\"*&\"#NF&)F(F%F&F&*&\"#XF&)F(\"\"#F&F.*&\"#KF&F(F&F&\"#5 F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"#5!\"\",&%\"xG\"\"\"F)F&F) ,(*&\"\"$F))F(\"\"#F)F)*&\"\"'F)F(F)F&\"\"&F)F),(*$F-F)F)*&F.F)F(F)F&F .F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&#\"\"$\"#5\"\"\"*$)%\"xG\"\"&F(F(F(*&#F&\"\"#F(*$)F +\"\"%F(F(!\"\"*&#\"\"(F/F(*$)F+F&F(F(F(*&#\"\"*F/F(*$)F+F/F(F(F3*&#\" #;F,F(F+F(F(F(F3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*&\"\"$\"\"\")% \"xG\"\"&F&F&*&\"#:F&)F(\"\"%F&!\"\"*&\"#NF&)F(F%F&F&*&\"#XF&)F(\"\"#F &F.*&\"#KF&F(F&F&\"#5F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$**\"#5!\" \",&%\"xG\"\"\"F)F&F),(*&\"\"$F))F(\"\"#F)F)*&\"\"'F)F(F)F&\"\"&F)F),( *$F-F)F)*&F.F)F(F)F&F.F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 111 "r:=0:\nfor k from 1 to maxk do \n r:=r+1: \n if (r>p) then r: =1: fi:\n print(k,r,eval(wpoly[r],x=k)-w[k]): \nod:\n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"\"F#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6% \"\"#F#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"$\"\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"%\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"&\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\" \"'\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"(\"\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"\")\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"\"*\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"# 5\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#6\"\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"#7\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#8\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#9 \"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#:\"\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6%\"#;\"\"#\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#<\"\"\"\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%\"#= \"\"#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{MARK "2 1 0" 107 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }