{VERSION 4 0 "IBM INTEL NT" "4.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 0 0 128 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 "" 0 21 "" 0 1 0 0 0 1 0 0 0 0 2 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 128 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 128 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 3 0 3 0 2 2 0 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 1 3 1 } 1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "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 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Map le Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 128 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 3 0 3 0 2 2 0 1 }{PSTYLE "R3 Font 0" -1 256 1 {CSTYLE "" -1 -1 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "R3 Font 2" -1 257 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 128 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 16 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 36 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 3 0 3 0 2 2 19 1 }{PSTYLE "Maple Output" -1 260 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Title" -1 261 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 3 0 3 0 2 2 19 1 }} {SECT 0 {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 261 "" 0 "" {TEXT 256 19 "Fractions continues" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Incantations initiales" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "libname; with(pldx);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q2C:/MapleV6/update6\"Q/C:\\MAPLEV6/libF$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%\\ppldx~v6-21~~;~~author~:~ ~~;~~last~modif~(V6)~:~Jeu~30/10/2003G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7@%&XtensG%+convert/D2G%,ever_loadedG%'isdiagG%'sinsinG%*xHospit alG%'xcloneG%%xcolG%&xcol3G%)xcollectG%'xcolorG%$xcsG%(xdchvarG%%xeigG %&xeiggG%)xfenetreG%(xfubiniG%&xjoliG%.xmanip_sommesG%(xmk_pivG%&xplif G%'xpurgeG%%xputG%+xreg_falsiG%)xres_iniG%)xreverseG%&xtensG%&xtreeG%& xtrigG%0zmanip_productsG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "_EnvExplicit:= true; interface(labelling=false);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%-_EnvExplicitG%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "with(plots, display):kernelopts(ASSERT=true);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "`eval/Mult`:= proc() op(args[1]); Multiply(%); end;; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*eval/MultGR6\"F&F&F&C$-%#opG6#& 9\"6#\"\"\"-%)MultiplyG6#%\"%GF&F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "sayval:= proc(zz::uneval) local tmp; eval(eval(zz), \+ ploc=1); \ntmp:= eval(%); print(zz=tmp); tmp; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 172 "with(LinearAlgebra) ; macro(Det=Determinant, Tr=Transpose, CP=CharacteristicPolynomial, Mu l=Multiply); Xpand:= proc() map(expand, args) end;\nmacro(divi=numtheo ry[divisors]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7fq%$AddG%(AdjointG% 3BackwardSubstituteG%+BandMatrixG%&BasisG%-BezoutMatrixG%/BidiagonalFo rmG%-BilinearFormG%5CharacteristicMatrixG%9CharacteristicPolynomialG%' ColumnG%0ColumnDimensionG%0ColumnOperationG%,ColumnSpaceG%0CompanionMa trixG%0ConditionNumberG%/ConstantMatrixG%/ConstantVectorG%2CreatePermu tationG%-CrossProductG%-DeleteColumnG%*DeleteRowG%,DeterminantG%/Diago nalMatrixG%*DimensionG%+DimensionsG%+DotProductG%,EigenvaluesG%-Eigenv ectorsG%&EqualG%2ForwardSubstituteG%.FrobeniusFormG%2GenerateEquations G%/GenerateMatrixG%2GetResultDataTypeG%/GetResultShapeG%5GivensRotatio nMatrixG%,GramSchmidtG%-HankelMatrixG%,HermiteFormG%3HermitianTranspos eG%/HessenbergFormG%.HilbertMatrixG%2HouseholderMatrixG%/IdentityMatri xG%2IntersectionBasisG%+IsDefiniteG%-IsOrthogonalG%*IsSimilarG%*IsUnit aryG%2JordanBlockMatrixG%+JordanFormG%(LA_MainG%0LUDecompositionG%-Lea stSquaresG%,LinearSolveG%$MapG%%Map2G%*MatrixAddG%.MatrixInverseG%5Mat rixMatrixMultiplyG%+MatrixNormG%5MatrixScalarMultiplyG%5MatrixVectorMu ltiplyG%2MinimalPolynomialG%&MinorG%)MultiplyG%,NoUserValueG%%NormG%*N ormalizeG%*NullSpaceG%3OuterProductMatrixG%*PermanentG%&PivotG%0QRDeco mpositionG%-RandomMatrixG%-RandomVectorG%%RankG%$RowG%-RowDimensionG%- RowOperationG%)RowSpaceG%-ScalarMatrixG%/ScalarMultiplyG%-ScalarVector G%*SchurFormG%/SingularValuesG%*SmithFormG%*SubMatrixG%*SubVectorG%)Su mBasisG%0SylvesterMatrixG%/ToeplitzMatrixG%&TraceG%*TransposeG%0Tridia gonalFormG%+UnitVectorG%2VandermondeMatrixG%*VectorAddG%,VectorAngleG% 5VectorMatrixMultiplyG%+VectorNormG%5VectorScalarMultiplyG%+ZeroMatrix G%+ZeroVectorG%$ZipG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6&%$DetG%#TrG%#C PG%$MulG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&XpandGR6\"F&F&F&-%$mapG 6$%'expandG9\"F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6'%$DetG%#TrG%#C PG%$MulG%%diviG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 158 "aff:= p roc(); if type(args[1],list) then return procname(op(args[1]))\nelif n args=1 then return args[1]\nelse return args[1]+1/``(procname(args[2.. -1])) fi end;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%$affGR6\"F&F&F&@'-% %typeG6$&9\"6#\"\"\"%%listGO-9!6#-%#opG6#F+/9#F.OF+O,&F+F.*&F.F.-%!G6# -F26#&F,6#;\"\"#!\"\"FFF.F&F&F&" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Quotients pa rtiels (*)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "cf:= proc(z); \+ Digits:=100; convert(evalf(z), confrac); %[1..min(30,nops(%)-1)]; end: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sayval(cf(Pi)): " }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#cfG6#%#PiG7@\"\"$\"\"(\"#:\"\"\"\" $#HF,F,F,\"\"#F,F)F,\"#9F.F,F,F.F.F.F.F,\"#%)F.F,F,F+F)\"#8F,\"\"%" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "sayval(cf(5842/2379)): sayv al(cf(2+sqrt(7))): sayval(cf((exp(2)+1)/(exp(2)-1))): " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#cfG6##\"%Ue\"%zB7+\"\"#F+\"\"&\"\"(\"\"$\"\"\" F/F/F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#cfG6#,&\"\"#\"\"\"-%%sqr tG6#\"\"(F)7@\"\"%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)F) F)F/F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#cfG6#*&,&-%$expG6#\"\"# \"\"\"F-F-F-,&F)F-F-!\"\"F/7@F-\"\"$\"\"&\"\"(\"\"*\"#6\"#8\"#:\"#<\"# >\"#@\"#B\"#D\"#F\"#H\"#J\"#L\"#N\"#P\"#R\"#T\"#V\"#X\"#Z\"#\\\"#^\"#` \"#b\"#d\"#f" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 "Convergents (*)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "les_quotients:= cf(5842/2379); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%.les_quotientsG7+\"\"#F&\"\"&\"\"(\"\"$\"\"\"F*F *F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "les_quotients: aff(% )=yjoli(aff(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&\"\"#\"\"\"*&F& F&-%!G6#,&F%F&*&F&F&-F)6#,&\"\"&F&*&F&F&-F)6#,&\"\"(F&*&F&F&-F)6#,&\" \"$F&*&F&F&-F)6#,&F&F&*&F&F&-F)6#,&F&F&*&F&F&-F)6#,&F&F&*&F&F&-F)6#F%! \"\"F&FJF&FJF&FJF&FJF&FJF&FJF&FJF&#\"%Ue\"%zB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "seq( (aff= yjoli@aff)(les_quotients[1..k]), k=1. .nops(les_quotients));" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+/\"\"#F$/,&F $\"\"\"*&F'F'-%!G6#F$!\"\"F'#\"\"&F$/,&F$F'*&F'F'-F*6#,&F$F'*&F'F'-F*6 #F.F,F'F,F'#\"#F\"#6/,&F$F'*&F'F'-F*6#,&F$F'*&F'F'-F*6#,&F.F'*&F'F'-F* 6#\"\"(F,F'F,F'F,F'#\"$%>\"#z/,&F$F'*&F'F'-F*6#,&F$F'*&F'F'-F*6#,&F.F' *&F'F'-F*6#,&FHF'*&F'F'-F*6#\"\"$F,F'F,F'F,F'F,F'#\"$4'\"$[#/,&F$F'*&F 'F'-F*6#,&F$F'*&F'F'-F*6#,&F.F'*&F'F'-F*6#,&FHF'*&F'F'-F*6#,&FgnF'*&F' F'-F*6#F'F,F'F,F'F,F'F,F'F,F'#\"$.)\"$F$/,&F$F'*&F'F'-F*6#,&F$F'*&F'F' -F*6#,&F.F'*&F'F'-F*6#,&FHF'*&F'F'-F*6#,&FgnF'*&F'F'-F*6#,&F'F'F]pF'F, F'F,F'F,F'F,F'F,F'#\"%79\"$v&/,&F$F'*&F'F'-F*6#,&F$F'*&F'F'-F*6#,&F.F' *&F'F'-F*6#,&FHF'*&F'F'-F*6#,&FgnF'*&F'F'-F*6#,&F'F'FeqF'F,F'F,F'F,F'F ,F'F,F'#\"%:A\"$-*/,&F$F'*&F'F'-F*6#,&F$F'*&F'F'-F*6#,&F.F'*&F'F'-F*6# ,&FHF'*&F'F'-F*6#,&FgnF'*&F'F'-F*6#,&F'F'*&F'F'-F*6#,&F'F'*&F'F'-F*6#, &F'F'F(F'F,F'F,F'F,F'F,F'F,F'F,F'F,F'#\"%Ue\"%zB" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "convert(5842/2379, confrac, 'cvg');\n[``(0)/` `(1),1/``(0), ``(cvg[1])/``(1), op(cvg[2..-2])];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7+\"\"#F$\"\"&\"\"(\"\"$\"\"\"F(F(F$" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#7,*&-%!G6#\"\"!\"\"\"-F&6#F)!\"\"*&F)F)F%F,*&-F&6#\" \"#F)F*F,#\"\"&F1#\"#F\"#6#\"$%>\"#z#\"$4'\"$[##\"$.)\"$F$#\"%79\"$v&# \"%:A\"$-*" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 22 "Les matrices sp\351ciales" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "mat2hom:= proc(m) global z; (m[1,1] *z+m[1,2])/ (m[2,1]*z+m[2,2]);\nunapply(%,z); end;" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%(mat2homGR6#%\"mG6\"F(F(C$*&,&*&&9$6$\"\"\"F0F0%\"z GF0F0&F.6$F0\"\"#F0F0,&*&&F.6$F4F0F0F1F0F0&F.6$F4F4F0!\"\"-%(unapplyG6 $%\"%GF1F(6#F1F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 21 "tt is T^n, hh is ST^n" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 149 "tt:= t-> <<1|-t>,<0|1> >: %, mat2hom(tt(a))(z); \nhh:= a-> <<0|1>,<1|-a>>: %, mat2hom(hh(a))( z);\nmh:= a-> <,<1|0>>: %, expand(mat2hom(mh(a))(z));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$R6#%\"tG6\"6$%)operatorG%&arrowGF&-%7rtable/ ConstructColumnG6$-%4rtable/ConstructRowG6$\"\"\",$9$!\"\"-F.6$\"\"!F0 F&F&F&,&%\"zGF0%\"aGF3" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$R6#%\"aG6\"6 $%)operatorG%&arrowGF&-%7rtable/ConstructColumnG6$-%4rtable/ConstructR owG6$\"\"!\"\"\"-F.6$F1,$9$!\"\"F&F&F&*&F1F1,&%\"zGF1F%F6F6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$R6#%\"aG6\"6$%)operatorG%&arrowGF&-%7rtable/ ConstructColumnG6$-%4rtable/ConstructRowG6$9$\"\"\"-F.6$F1\"\"!F&F&F&, &F%F1*&F1F1%\"zG!\"\"F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 " hh(a).tt(t); tt(-t).mh(a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABL EG6$\"(GSn(-%'MATRIXG6#7$7$\"\"!\"\"\"7$F-,&%\"tG!\"\"%\"aGF1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(%=Uy-%'MATRIXG6#7$7$,&% \"aG\"\"\"%\"tGF.F.7$F.\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 267 "HH:= proc(); if nargs=0 then Matrix([[1, 0], [0, 1]]) \n elif \+ nargs=1 then hh(args) else procname(args[2..nargs]).hh(args[1]) fi end :\nmH:= proc(); if nargs=0 then Matrix([[1, 0], [0, 1]]) \n elif narg s=1 then mh(args) else mh(args[1]).procname(args[2..nargs]) fi end:" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "HH(1,2)- hh(2).hh(1), \nmH (1,2,3)-mh(1).mh(2).mh(3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6$-%'RTAB LEG6$\"(O@#z-%'MATRIXG6#7$7$\"\"!F,F+-F$6$\")W(G8#F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "ms:= <,>; ma:= <,>; m A:= <,>;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#msG-%'RTABLE G6$\")'f!H@-%'MATRIXG6#7$7$%\"uG%\"vG7$F/%\"wG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#maG-%'RTABLEG6$\")[*z8#-%'MATRIXG6#7$7$%\"aG%\"bG7$% \"cG%\"dG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mAG-%'RTABLEG6$\")WrX@ -%'MATRIXG6#7$7$%\"AG%\"BG7$%\"CG%#DDG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "abc:= proc(mz); a=mz[1,1], b=mz[1,2], c=mz[2,1], d=m z[2,2]; end;\nuvw:= proc(mz); u=mz[1,1], v=mz[1,2], w=mz[2,2]; end;" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$abcGR6#%#mzG6\"F(F(6&/%\"aG&9$6$\" \"\"F//%\"bG&F-6$F/\"\"#/%\"cG&F-6$F4F//%\"dG&F-6$F4F4F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$uvwGR6#%#mzG6\"F(F(6%/%\"uG&9$6$\"\"\"F// %\"vG&F-6$F/\"\"#/%\"wG&F-6$F4F4F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 231 "pf:= proc(ma) numer(z-mat2hom(ma)(z)); collect(%/coe ff(%,z,2),z,normal); end;\nnupf:= proc(ma) global z; local z_; if narg s=2 then z_:= args[2] else z_:= z; fi; \nma[2,1]*z_^2+(ma[2,2]-ma[1,1] )*z_-ma [1,2]; collect(%,z_,normal); end:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#pfGR6#%#maG6\"F(F(C$-%&numerG6#,&%\"zG\"\"\"--%(mat2 homG6#9$6#F.!\"\"-%(collectG6%*&%\"%GF/-%&coeffG6%F;F.\"\"#F6F.%'norma lGF(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "pf(ma); nupf(ma );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"zG\"\"#\"\"\"F(*&*&,&%\" dGF(%\"aG!\"\"F(F&F(F(%\"cGF.F(*&%\"bGF(F/F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"cG\"\"\")%\"zG\"\"#F&F&*&,&%\"dGF&%\"aG!\"\"F&F( F&F&%\"bGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 23 "Calcul des p\351riodes (*)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 521 "perper:=proc(d) \nlocal a,b,c, n,j,t,bb,per, li;\nn :=floor(sqrt(d)); li:=table();\nif n*n=d then return [ ] fi;\nif modp( d,4)=0 \nthen a:= 1; b:= (n-modp(n,2)); c:= (d-b^2)/4; \nelse ASSERT( \+ modp(d,4)=1); \n a:=1; b:= (n-1+modp(n,2)); c:= (d-b^2)/4; \nfi ;\n# a*z^ 2-b*z-c: print(%, solve(%));\nfor j from 0 do \nt:= floor((b+n)/ 2/a); li[j]:=t; \nc:=a; b:=2*a*t-b; a:=(d-b*b)/4/c;\nif a=c then per:= 2*j+1; break; fi;\nif b=bb then per:= 2*j; break ; fi;\nbb:=b; \nod; return [seq(li[i], i=0..j), seq(li[per-i], i=j+1..per-1)];\nend: " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 59 "cf:= proc(n); Digits:= 40: convert(evalf(n), confra c); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "cfb:= proc(n); \+ Digits:= 40: convert(sqrt(n*1.), confrac); end:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 66 "cfb100:= proc(n); Digits:= 100: convert(sqrt(n *1.), confrac); end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "cfa := proc(n); Digits:= 40: convert((-1+sqrt(n*1.))/2, confrac);\nend:" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 36 "Calculs pour les thm de la section 3" }}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 128 "pol:= z-> a*z^2 -b*z-c; qol:= A*zeta**2-B *zeta-C;\ncollect(-zeta^2*pol(q+1/zeta), zeta);\n\{coeffs(%-qol, zeta) \};\nsolve(%,\{A,B,C\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$polGR6# %\"zG6\"6$%)operatorG%&arrowGF(,(*&%\"aG\"\"\")9$\"\"#F/F/*&%\"bGF/F1F /!\"\"%\"cGF5F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$qolG,(*&%\"A G\"\"\")%%zetaG\"\"#F(F(*&%\"BGF(F*F(!\"\"%\"CGF." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,(*&%\"aG\"\"\")%\"qG\"\"#F(!\"\"*&%\"bGF(F*F(F(%\" cGF(F()%%zetaGF+F(F(*&,&*&F'F(F*F(!\"#F.F(F(F1F(F(F'F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#<%,*%\"AG!\"\"*&%\"aG\"\"\")%\"qG\"\"#F)F&*&%\"bGF )F+F)F)%\"cGF),&%\"CGF)F(F&,(%\"BGF)*(F,F)F(F)F+F)F&F.F)" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#<%/%\"BG,&*&%\"aG\"\"\"%\"qGF)\"\"#%\"bG!\"\"/% \"AG,(*&F(F))F*F+F)F-*&F,F)F*F)F)%\"cGF)/%\"CGF(" }}}{EXCHG {PARA 0 " " 0 "" {MPLTEXT 0 21 31 "xxx:= (55+sqrt(3))/87; cf(xxx);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%$xxxG,&#\"#b\"#()\"\"\"*&#F)F(F)-%%sqrtG6#\"\" $F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7K\"\"!\"\"\"F%F%\"\"'F%\"#A \"\"#F%\"\"$F(F(\"#L\"#8F+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 0 21 273 "(numer@expan d)((z-xxx)*(z-subs(sqrt(3)=-sqrt(3),xxx)));\n(sort@expand)(z^2*subs(z= 1/z,%));\n(sort@expand)(z^2*subs(z=1+1/z,%));\n(sort@expand)(z^2*subs( z=1+1/z,%));\n(sort@expand)(z^2*subs(z=1+1/z,%)); # map(subs,[z=-1, z= 0, z=1],%);\npol:= (sort@expand)(-z^2*subs(z=6+1/z,%)); " }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,(*$)%\"zG\"\"#\"\"\"\"%pv*&\"%q&*F(F&F(!\"\"\"% AIF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"zG\"\"#\"\"\"\"%AI*&\" %q&*F(F&F(!\"\"\"%pvF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"zG\" \"#\"\"\"\"%@5*&\"%ENF(F&F(!\"\"\"%AIF(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"zG\"\"#\"\"\"\"$<&*&\"%%[\"F(F&F(!\"\"\"%@5F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"zG\"\"#\"\"\"\"#a*&\"$]%F(F&F(!\"\" \"$<&F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$polG,(*$)%\"zG\"\"#\"\" \"\"$R#*&\"$)>F*F(F*!\"\"\"#aF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "mS:= mh(0); mT:= tt(-1); mU:= Matrix([[-1,0],[0,1]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mSG-%'RTABLEG6$\")#z*y@-%'MATRIXG6# 7$7$\"\"!\"\"\"7$F/F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mTG-%'RTAB LEG6$\"([a)y-%'MATRIXG6#7$7$\"\"\"F.7$\"\"!F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#mUG-%'RTABLEG6$\"(gMh(-%'MATRIXG6#7$7$!\"\"\"\"!7$F/ \"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "1/mT=mU.mT.mU; 1/ mS=-mU.mS.mU;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'RTABLEG6$\")#*G`@ -%'MATRIXG6#7$7$\"\"\"!\"\"7$\"\"!F--F%6$\"(!G%3)F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%'RTABLEG6$\"(K8%z-%'MATRIXG6#7$7$\"\"!\"\"\"7$F.F- -F%6$\"(KZn(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "pol2ma:= proc(pol); Matrix([[-coeff(pol,z,1)/2, -coeff(pol,z,0)],[coeff(pol,z, 2), coeff(pol,z,1)/2]]); end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'po l2maGR6#%$polG6\"F(F(-%'MatrixG6#7$7$,$-%&coeffG6%9$%\"zG\"\"\"#!\"\" \"\"#,$-F06%F2F3\"\"!F67$-F06%F2F3F7,$F/#F4F7F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "pol_:= alpha*z*z-beta*z-gamma; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%pol_G,(*&%&alphaG\"\"\")%\"zG\"\"#F(F(*&% %betaGF(F*F(!\"\"%&gammaGF." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "pol2ma(pol_); -mU.%.mU; nupf(%%), nupf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(ozh(-%'MATRIXG6#7$7$,$%%betaG#\"\"\"\"\" #%&gammaG7$%&alphaG,$F-#!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' RTABLEG6$\"(%ymy-%'MATRIXG6#7$7$,$%%betaG#!\"\"\"\"#%&gammaG7$%&alphaG ,$F-#\"\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,(*&%&alphaG\"\"\")% \"zG\"\"#F&F&*&%%betaGF&F(F&!\"\"%&gammaGF,,(F$F&F*F&F-F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "pol2ma(pol_); (nupf)(%);\n(subs)(z =(a*z+b)/(c*z+d), pol_ ): pol__:= collect(normal(%*(c*z+d)^2/Det(ma)), z);\nmap(normal, ((1/ma)).pol2ma(pol_).ma); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "(ASSERT@expand)( (nupf)(%)= (pol__));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(3[%y-%'MATRIXG6#7$7$,$%%betaG#\"\" \"\"\"#%&gammaG7$%&alphaG,$F-#!\"\"F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%&alphaG\"\"\")%\"zG\"\"#F&F&*&%%betaGF&F(F&!\"\"%&gammaGF," }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%&pol__G,(*&*&,(*(%%betaG\"\"\"%\"aGF +%\"cGF+!\"\"*&%&gammaGF+)F-\"\"#F+F.*&%&alphaGF+)F,F2F+F+F+)%\"zGF2F+ F+,&*&F,F+%\"dGF+F.*&%\"bGF+F-F+F+F.F.*&*&,**(F*F+F,F+F:F+F.**F2F+F0F+ F-F+F:F+F.**F2F+F4F+F,F+F " 0 "" {MPLTEXT 1 0 36 "1/mh(q).pol2ma(pol_).mh(q); nupf(%);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")c#fD#-%'MATRIXG6#7$7$,& *&%\"qG\"\"\"%&alphaGF/F/*&#F/\"\"#F/%%betaGF/!\"\"F07$,(*&,&F4#F/F3F- F5F/F.F/F/%&gammaGF/*(F:F/F.F/F4F/F/F9" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,(*&%\"qG\"\"\"%%betaGF(F(*&)F'\"\"#F(%&alphaGF(!\"\"%&gamma GF(F()%\"zGF,F(F(*&,&F)F(*(F,F(F'F(F-F(F.F(F1F(F(F-F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "-4*Det(pol2ma(pol_));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*$)%%betaG\"\"#\"\"\"F(*(\"\"%F(%&gammaGF(%&al phaGF(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "mpp2:= Matrix([ [b/2,c],[a,-b/2]]); numer(pf(%)); #1/mS.mpp2.mS, 1/mT.mpp2.mT;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mpp2G-%'RTABLEG6$\")/@%=#-%'MATRIXG 6#7$7$,$%\"bG#\"\"\"\"\"#%\"cG7$%\"aG,$F/#!\"\"F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"aG\"\"\")%\"zG\"\"#F&F&*&%\"bGF&F(F&!\"\"%\"cGF, " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 138 "mpp1:= Matrix([[(b+1)/ 2,c],[a,-(b-1)/2]]); numer(pf(%)); # 1/mS.mpp1.mS, 1/mT.mpp1.mT;\nsubs (a=1, b=1, [mpp2, 1/mT.mpp2.mT]): %, map(pf,%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%mpp1G-%'RTABLEG6$\"(o0L(-%'MATRIXG6#7$7$,&%\"bG#\"\" \"\"\"#F0F1%\"cG7$%\"aG,&F/#!\"\"F2F0F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"aG\"\"\")%\"zG\"\"#F&F&*&%\"bGF&F(F&!\"\"%\"cGF," }} {PARA 11 "" 1 "" {XPPMATH 20 "6$7$-%'RTABLEG6$\")[aZ@-%'MATRIXG6#7$7$# \"\"\"\"\"#%\"cG7$F.#!\"\"F/-F%6$\")'\\v9#-F)6#7$7$F2F07$F.F-7$,(*$)% \"zGF/F.F.F@F3F0F3,(F>F.F@F.F0F3" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "nxtpol:= proc(pol);\nsolve(\{pol, z>0\}); (eval@subs )(%, floor(z));\n(sort@expand)(-z^2*subs(z=%+1/z, pol));\nend: " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 17 "mH(a,b); nupf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#-%'RTABLEG6$\"(3%\\\")-%'MATRIXG6#7$7$,&*&%\"aG\"\"\"%\"bGF/F/F/F/F. 7$F0F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"bG\"\"\")%\"zG\"\"#F& F&*(%\"aGF&F%F&F(F&!\"\"F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "(a+b*sqrt(d))/2; (z-%)*(z-subs(sqrt(d)=-sqrt(d),%)); discrim(%,z );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"aG#\"\"\"\"\"#*(F%F&%\"bGF& -%%sqrtG6#%\"dGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(%\"zG\"\"\" *&#F&\"\"#F&%\"aGF&!\"\"*&#F&F)F&*&%\"bGF&-%%sqrtG6#%\"dGF&F&F+F&,(F%F &*&#F&F)F&F*F&F+*(#F&F)F&F/F&F0F&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&)%\"bG\"\"#\"\"\"%\"dGF'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "(n+sqrt(d)); (z-%)*(z-subs(sqrt(d)=-sqrt(d),%)): collect(%,z, e xpand);\nsubs(z=0,%), xcs(subs(z=-1,%), n); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"nG\"\"\"*$-%%sqrtG6#%\"dGF%F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"zG\"\"#\"\"\"F(*(F'F(%\"nGF(F&F(!\"\"*$)F*F'F(F (%\"dGF+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*$)%\"nG\"\"#\"\"\"F(%\" dG!\"\",&*$),&F&F(F(F(F'F(F(F)F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "(n+delta+sqrt(d))/2; 4*(z-%)*(z-subs(sqrt(d)=-sqrt(d ),%)): collect(%,z, expand);\nxcs(subs(z=0,%),n), xcs(subs(z=-1,%), n) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(%\"nG#\"\"\"\"\"#*&F%F&%&deltaG F&F&*&F%F&-%%sqrtG6#%\"dGF&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$) %\"zG\"\"#\"\"\"\"\"%*&,&%\"nG!\"%*&F)F(%&deltaGF(!\"\"F(F&F(F(*$)F,F' F(F(*(F'F(F,F(F/F(F(*$)F/F'F(F(%\"dGF0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&*$),&%\"nG\"\"\"%&deltaGF(\"\"#F(F(%\"dG!\"\",&*$),(F'F(F*F(F) F(F*F(F(F+F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "1/hh(m).ma; collect(pf(%), [z,a]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6 $\")'\\i6#-%'MATRIXG6#7$7$,&*&%\"mG\"\"\"%\"aGF/F/%\"cGF/,&*&F.F/%\"bG F/F/%\"dGF/7$F0F4" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"zG\"\"#\" \"\"F(*&,&%\"mG!\"\"*&,&%\"cGF,%\"bGF(F(%\"aGF,F(F(F&F(F(*&,&*&F+F(F0F (F,%\"dGF,F(F1F,F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "The omega case (*)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "dis:= 4*33; perper(%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ disG\"$K\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7&\"#5\"\"\"\"\"#F%" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "floor(sqrt(dis)): (%-modp(%, 2))/2; pol:= expand((z-%-sqrt(dis/4))*(z-%+sqrt(dis/4))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$ polG,(*$)%\"zG\"\"#\"\"\"F**&\"#5F*F(F*!\"\"\"\")F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "mXb:= Matrix([[solve(pol)],[1,1]]); mP0: = Matrix([[-coeff(pol,z)/2, -coeff(pol,z,0)],[1,+coeff(pol,z)/2]]); AS SERT(pf(%)=pol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mXbG-%'RTABLEG6 $\"(Kjo(-%'MATRIXG6#7$7$,&\"\"&\"\"\"*$-%%sqrtG6#\"#LF0F0,&F/F0F1!\"\" 7$F0F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mP0G-%'RTABLEG6$\")SfV@-% 'MATRIXG6#7$7$\"\"&\"\")7$\"\"\"!\"&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "motifb:= perper(dis); ff:= nops(%)-1; undices:= [0,1 /4, seq(op([k-1/2,k]),k=1..ff+1), ff+1+1/4, ff+1+1/2,ff+2]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'motifbG7&\"#5\"\"\"\"\"#F'" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#ffG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %(undicesG7/\"\"!#\"\"\"\"\"%#F(\"\"#F(#\"\"$F+F+#\"\"&F+F-#\"\"(F+F)# \"# " 0 "" {MPLTEXT 1 0 116 "Act:= \+ table(): Act[0]:=(1/mT).mT: for k from 1 to ff+1 do Act[k]:= 1/hh(moti fb[k]): od: Act[k]:= 1/hh(motifb[1]): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 203 "act:= table(): for k from 1 to ff+2 do act[k]:= mS; \+ od:\nact[0]:=(1/mT).mT: act[1/4], act[1/2], act[ff+1+1/4], act[ff+1+1/ 2]:= 1/tt(motifb[1]/2)$4: for k from 2 to 1+ff do act[k-1/2]:= mT^(mot ifb[k]); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "PP:='PP': P_[0]:= pol; for k to ff+ 1 do P_[k]:= nxtpol(%); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#P_G 6#\"\"!,(*$)%\"zG\"\"#\"\"\"F-*&\"#5F-F+F-!\"\"\"\")F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#P_G6#\"\"\",(*$)%\"zG\"\"#F'\"\")*&\"#5F'F+F'! \"\"F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#P_G6#\"\"#,(*$)%\"zGF' \"\"\"\"\"$*&\"\"'F,F+F,!\"\"\"\")F0" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>&%#P_G6#\"\"$,(*$)%\"zG\"\"#\"\"\"\"\")*&\"\"'F-F+F-!\"\"F'F1" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#P_G6#\"\"%,(*$)%\"zG\"\"#\"\"\"F-* &\"#5F-F+F-!\"\"\"\")F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "mHu:= table(): mHu[0]:= 1/mS.mS: for k from 1 to ff+1 do mHu[k]:=%.A ct[k]; od: mHu[k]:=%.(1/hh(motifb[1])): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "mHH:= table(): mHH[0]:= mHu[0]: for k in undices do m HH[k]:=%.act[k]; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 156 "mP_ini:= Matrix([[-coeff(p ol,z)/2, -coeff(pol,z,0)],[1,+coeff(pol,z)/2]]):\nmP:= table(): for i \+ from -1 to ff+1 do mP[i]:= (1/mHu[i]).mP_ini.(mHu[i]); od: " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "mPP:= table(): for i in undi ces do mPP[i]:= (1/mHH[i]).mP_ini.(mHH[i]); od:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "seq([i, Act[i], act[i], (nupf)(mP[i]),(nupf)(mPP[i]), mPP[i], mU.mPP[ i].mU],i= undices):\nmatrix([%]): map(proc(z) if has(z, \{mHu, mP, mHH , mPP, Act\}) then `` else z fi end, %);\nfeuille:=%:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7/7)\"\"!-%'RTABLEG6$\"(s#>z-%'MATRIXG 6#7$7$\"\"\"F(7$F(F2-F*6$\");&o6#F-,(*$)%\"zG\"\"#F2F2*&\"#5F2F:F2!\" \"\"\")F>F7-F*6$\")733A-F.6#7$7$\"\"&F?7$F2!\"&-F*6$\")#\\`7#-F.6#7$7$ FG!\")7$F>FI7)#F2\"\"%%!G-F*6$\")G([9#-F.6#7$7$F2FGF3FV,&F8F2\"#LF>-F* 6$\")WL!=#-F.6#7$7$F(FinF1-F*6$\")cQp@-F.6#7$7$F(!#L7$F>F(7)#F2F;FVFWF V,(F8F2*&F=F2F:F2F2F?F>-F*6$\"(3v0)-F.6#7$7$FIF?Fgn-F*6$\")O*oC#-F.6#7 $7$FIFQ7$F>FG7)F2-F*6$\"(cv!y-F.6#7$7$F=F2F1-F*6$\")#z*y@-F.6#7$F3F1,( F8F?*&F=F2F:F2F>F2F>F[r-F*6$\");L@@-F.6#7$7$FGF27$F?FI-F*6$\"(+r%y-F.6 #7$7$FGF>7$FQFI7)#\"\"$F;FV-F*6$\"([a)y-F.6#7$7$F2F2F3FV,(F8F?*&\"\"'F 2F:F2F2F_sF>-F*6$\")s!4<#-F.6#7$7$!\"$F_s7$F?F_s-F*6$\")k7G@-F.6#7$7$F atFat7$FQF_s7)F;-F*6$\")Sn]@-F.6#7$FfsF1Feq,(F8F_s*&FisF2F:F2F>F?F>Fbu -F*6$\"(g>$z-F.6#7$7$F_sF?7$F_sFat-F*6$\")3\"G<#-F.6#7$7$F_sFQFit7)#FG F;FV-F*6$\")w9n@-F.6#7$7$F2F;F3FV,(F8F_s*&FisF2F:F2F2F?F>-F*6$\")/f5@- F.6#7$7$FatF?7$F_sF_s-F*6$\")OI#>#-F.6#7$7$FatFQF`t7)F_s-F*6$\")g)[E#- F.6#7$7$F;F2F1Feq,(F8F?*&FisF2F:F2F>F_sF>Fex-F*6$\")s9X@-F.6#7$Few7$F? Fat-F*6$\")K$H?#-F.6#7$F[v7$FQFat7)#\"\"(F;FVF`sFV,(F8F?*&F=F2F:F2F2F2 F>-F*6$\")!)f#=#-F.6#7$7$FIF27$F?FG-F*6$\");h;A-F.6#7$7$FIF>7$FQFG7)FU -F*6$\"))G&z@F_uFeqF7F7-F*6$\")Oo/AFC-F*6$\"([r)yFM7)#\"#)F`r-F*6$\")%[m:#Fhr " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "seq( [mU.mPP[i].mU + mP P[ff+1+1/2-i]], i= undices[1..-2]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 .7#-%'RTABLEG6$\")3YL@-%'MATRIXG6#7$7$\"\"!F-F,7#-F%6$\")+2x@F(7#-F%6$ \")?%o?#F(7#-F%6$\")SD9AF(7#-F%6$\")cYL@F(7#-F%6$\"(!Q9wF(7#-F%6$\")g \\X@F(7#-F%6$\")GRCAF(7#-F%6$\")CJ4@F(7#-F%6$\")Kki@F(7#-F%6$\")+j*=#F (7#-F%6$\")!='3AF(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "seq( \+ [subs(z=-z, nupf(mPP[i])) - nupf(mPP[ff+1+1/2-i])], i= undices[1..-2]) ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6.7#\"\"!F#F#F#F#F#F#F#F#F#F#F#" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 132 "mil:= (ff+1)/2; ava:= mPP[ mil]; apr:= mPP[mil+1/2]; \n-mU.ava.mU=apr; mPP[ff+1-mil]=mS.apr.mS; \nmPP[ff+1-mil]= -mS.mU.mPP[mil].mU.mS;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$milG\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$avaG-%'RTABLE G6$\")g\"QM\"-%'MATRIXG6#7$7$\"\"'\"\"(7$\"\"$!\"'" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%$aprG-%'RTABLEG6$\")#>ZM\"-%'MATRIXG6#7$7$!\"'\"\"( 7$\"\"$\"\"'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'RTABLEG6$\")gea8-% 'MATRIXG6#7$7$!\"'\"\"(7$\"\"$\"\"'-F%6$\")#>ZM\"F(" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%'RTABLEG6$\")g\"QM\"-%'MATRIXG6#7$7$\"\"'\"\"(7$\" \"$!\"'-F%6$\")sza8-F)6#7$7$F-F07$F.F1" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'RTABLEG6$\")g\"QM\"-%'MATRIXG6#7$7$\"\"'\"\"(7$\"\"$!\"'-F%6 $\")w)p>\"-F)6#7$7$F-F07$F.F1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 " 2n does domines all other coefficients" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "a*z^2-b*x-c;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&%\"aG\"\" \")%\"zG\"\"#F&F&*&%\"bGF&%\"xGF&!\"\"%\"cGF-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 18 "The theta case (*)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "dis:= 33; dis:= 57; perper(%); " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "# dis:= 4*41; dis:= 872; perper(%); nops(%)-1;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%$disG\"#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$disG\"#d" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(\"\"( \"\"$\"\"\"F&F&F%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "floor( sqrt(dis)): (%+modp(%,2)-1)/2; pol:= expand((z-%-sqrt(dis/4))*(z-%+sqr t(dis/4))); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"\"(\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$polG,(*$)%\"zG\"\"#\"\"\"F**&\"\"(F*F(F*! \"\"F)F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 209 "mXa:= Matrix([ [solve(pol)],[1,1]]); \nmP0:= Matrix([[-coeff(pol,z)/2, -coeff(pol,z,0 )],[1,+coeff(pol,z)/2]]); \nmP0:= Matrix([[-coeff(pol,z)/2-1/2, -coeff (pol,z,0)],[1,+coeff(pol,z)/2-1/2]]); \nASSERT(pf(%)=pol);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mXaG-%'RTABLEG6$\")CHb8-%'MATRIXG6#7$7$,& #\"\"(\"\"#\"\"\"*&#F2F1F2-%%sqrtG6#\"#dF2F2,&F/F2*&#F2F1F2*$F5F2F2!\" \"7$F2F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mP0G-%'RTABLEG6$\")k\\b 8-%'MATRIXG6#7$7$#\"\"(\"\"#F07$\"\"\"#!\"(F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$mP0G-%'RTABLEG6$\"(OR(z-%'MATRIXG6#7$7$\"\"$\"\"#7$ \"\"\"!\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "motifa:= per per(dis); ff:= nops(%)-1; undices:= [0,1/6,2/6, seq(op([k-1/2,k]),k=1. .ff+1), ff+1+1/6, ff+1+2/6, ff+1+1/2,ff+2]; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'motifaG7(\"\"(\"\"$\"\"\"F(F(F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ffG\"\"&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(undic esG75\"\"!#\"\"\"\"\"'#F(\"\"$#F(\"\"#F(#F+F-F-#\"\"&F-F+#\"\"(F-\"\"% #\"\"*F-F0#\"#6F-F)#\"#PF)#\"#>F+#\"#8F-F2" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 116 "Act:= table(): Act[0]:=(1/mT).mT: for k from 1 to \+ ff+1 do Act[k]:= 1/hh(motifa[k]): od: Act[k]:= 1/hh(motifa[1]): " }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 241 "act:= table(): for k from \+ 1 to ff+2 do act[k]:= mS; od:\nact[0]:=(1/mT).mT: \nact[1/6], act[1/2] , act[ff+1+1/6], act[ff+1+1/2]:= 1/tt((motifa[1]-1)/2)$4:\nact[1/3], a ct[ff+1+1/3]:= mT$2: \nfor k from 2 to 1+ff do act[k-1/2]:= mT^(motifa [k]); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "seq(act[k],k=u ndices);" }}{PARA 11 "" 1 "" {XPPMATH 20 "65-%'RTABLEG6$\")#zhN\"-%'MA TRIXG6#7$7$\"\"\"\"\"!7$F-F,-F$6$\")wRc8-F(6#7$7$F,\"\"$F.-F$6$\")o/#[ \"-F(6#7$7$F,F,F.F/-F$6$\")[\"=[\"-F(6#7$F.F+-F$6$\"))3mN\"F2F>F7F>F7F >F7F>-F$6$\")_S+7F2F>F/F7F/F>" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "PP:='PP': P_[0]:= pol; for k to ff+1 do P_[k]:= nxtpol(%); od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#P_G6#\"\"!,(*$)%\"zG\"\"#\"\"\"F -*&\"\"(F-F+F-!\"\"F,F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#P_G6#\" \"\",(*$)%\"zG\"\"#F'F,*&\"\"(F'F+F'!\"\"F'F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#P_G6#\"\"#,(*$)%\"zGF'\"\"\"\"\"%*&\"\"&F,F+F,!\"\" F'F0" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#P_G6#\"\"$,(*$)%\"zG\"\"# \"\"\"F'*&F'F-F+F-!\"\"\"\"%F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%# P_G6#\"\"%,(*$)%\"zG\"\"#\"\"\"F'*&\"\"$F-F+F-!\"\"F/F0" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>&%#P_G6#\"\"&,(*$)%\"zG\"\"#\"\"\"F,*&F'F-F+F-! \"\"\"\"%F/" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%#P_G6#\"\"',(*$)%\"z G\"\"#\"\"\"F-*&\"\"(F-F+F-!\"\"F,F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "mHu:= table(): mHu[0]:= 1/mS.mS: for k from 1 to ff +1 do mHu[k]:=%.Act[k]; od: mHu[k]:=%.(1/hh(motifa[1])): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "seq(mHu[k], k=0..ff+2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6*-%'RTABLEG6$\")78d8-%'MATRIXG6#7$7$\"\"\"\" \"!7$F-F,-F$6$\")GPd8-F(6#7$7$\"\"(F,F+-F$6$\")Sed8-F(6#7$7$\"#AF67$\" \"$F,-F$6$\")O\"yN\"-F(6#7$7$\"#HF>7$\"\"%F@-F$6$\")+Ce8-F(6#7$7$\"#^F H7$F6FJ-F$6$\")sYe8-F(6#7$7$\"#!)FR7$\"#6F6-F$6$\")g&*e8-F(6#7$7$\"$\" HFen7$\"#SFgn-F$6$\"(oH(y-F(6#7$7$\"%<@F_o7$F_oFao" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "mHH:= table(): mHH[0]:= mHu[0]: for k in un dices do mHH[k]:=%.act[k]; od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "mP:= table(): fo r i from 0 to ff+2 do mP[i]:= (1/mHu[i]).mP0.(mHu[i]); od: " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "mPP:= table(): for i in undi ces do mPP[i]:= (1/mHH[i]).mP0.(mHH[i]); od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "(dis-1)/4;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\" #9" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 187 "seq([i, Act[i], act[ i], (nupf)(mP[i]),(nupf)(mPP[i]), mPP[i], mU.mPP[i].mU],i= undices):\n matrix([%]): map(proc(z) if has(z, \{mH, mP, mHH, mPP, Act\}) then `` \+ else z fi end, %);\nfeuille:=%:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' matrixG6#757)\"\"!-%'RTABLEG6$\")wVf8-%'MATRIXG6#7$7$\"\"\"F(7$F(F2-F* 6$\")#zhN\"F-,(*$)%\"zG\"\"#F2F2*&\"\"(F2F:F2!\"\"F;F>F7-F*6$\")Cmf8-F .6#7$7$\"\"$F;7$F2!\"%-F*6$\"))yK?\"-F.6#7$7$FF!\"#7$F>FH7)#F2\"\"'%!G -F*6$\")wRc8-F.6#7$7$F2FFF3FU,(F8F2F:F>\"#9F>-F*6$\")+K$=\"-F.6#7$7$F( Fhn7$F2F>-F*6$\")G`g8-F.6#7$7$F(!#97$F>F>7)#F2FFFU-F*6$\")o/#[\"-F.6#7 $7$F2F2F3FU,(F8F2F:F2FhnF>-F*6$\")wvg8-F.6#7$7$F>FhnF1-F*6$\")s)4O\"-F .6#7$7$F>Fho7$F>F(7)#F2F;FUFVFU,(F8F2*&F=F2F:F2F2F;F>-F*6$\")wTh8-F.6# 7$7$FHF;Ffn-F*6$\"))G;O\"-F.6#7$7$FHFP7$F>FF7)F2-F*6$\"(wk\\(-F.6#7$7$ F=F2F1-F*6$\")[\"=[\"-F.6#7$F3F1,(F8F;*&F=F2F:F2F>F2F>Fds-F*6$\")O\">O \"-F.6#7$7$FFF27$F;FH-F*6$\")S127-F.6#7$7$FFF>7$FPFH7)#FFF;FU-F*6$\")) 3mN\"FYFU,(F8F;*&\"\"&F2F:F2F2\"\"%F>-F*6$\")!Qd[\"-F.6#7$7$!\"$F^u7$F ;F;-F*6$\")#\\f[\"-F.6#7$7$FfuFH7$FPF;7)F;-F*6$\");Z'[\"-F.6#7$F\\tF1F ^s,(F8F^u*&F]uF2F:F2F>F;F>Fgv-F*6$\")#Hp[\"-F.6#7$Fgu7$F^uFfu-F*6$\"(/ P:)-F.6#7$7$F;FP7$FHFfu7)#F]uF;FUF\\pFU,(F8F^u*&FFF2F:F2F2FFF>-F*6$\") sP([\"-F.6#7$7$FPFF7$F^uF2-F*6$\")K\")p6-F.6#7$7$FPFfu7$FHF27)FF-F*6$ \")g'y[\"-F.6#7$FbpF1F^s,(F8FF*&FFF2F:F2F>F^uF>Fcy-F*6$\");4)[\"-F.6#7 $7$F2F^uFO-F*6$\")cH)[\"-F.6#7$FG7$FfuFP7)#F=F;FUF\\pFU,(F8FF*&FFF2F:F 2F2F^uF>-F*6$\")#H()[\"-F.6#7$7$FPF^uF\\t-F*6$\")g'*)[\"-F.6#7$Fet7$Ff uF27)F^u-F*6$\")W=*[\"F`yF^s,(F8F^u*&FFF2F:F2F>FFF>Fi[l-F*6$\")[T*[\"- F.6#7$Ffn7$F^uFP-F*6$\")7%)*[\"-F.6#7$7$F2FfuFdr7)#\"\"*F;FUF\\pFU,(F8 F^u*&F]uF2F:F2F2F;F>-F*6$\")o^x6-F.6#7$7$FfuF;7$F^uF;-F*6$\")%G,\\\"-F .6#7$FbzF]r7)F]u-F*6$\")cN!\\\"F`yF^s,(F8F;*&F]uF2F:F2F>F^uF>F`^l-F*6$ \"(kz&z-F.6#7$7$F;F^u7$F;Ffu-F*6$\")g+\"\\\"-F.6#7$F]tFjx7)#\"#6F;FU-F *6$\")_S+7FYFU,(F8F;*&F=F2F:F2F2F2F>-F*6$\")3B\"\\\"-F.6#7$F[y7$F;FF-F *6$\")?W\"\\\"-F.6#7$7$FHF>Fbx7)FT-F*6$\")7'=\\\"FdvF^sF7F7-F*6$\"()Gc \")FB-F*6$\")'*G#\\\"FL7)#\"#PFTFUFVFUFgn-F*6$\")Or#\\\"F\\o-F*6$\")?$ H\\\"Fdo7)#\"#>FFFUF\\pFUFcp-F*6$\")Sm#>\"Fgp-F*6$\")3q&=\"F^q7)#\"#8F ;FUFVFUFeq-F*6$\")'pO\\\"Fjq-F*6$\")!))Q\\\"Far7)F=-F*6$\")76%\\\"FjrF ^sFdsFds-F*6$\")7N%\\\"Fis-F*6$\")Kw%\\\"Fat" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "seq( [i, ff+1+1/2-i], i= undices[1..-2]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "647$\"\"!#\"#8\"\"#7$#\"\"\"\"\"'#\"#>\" \"$7$#F*F.#\"#PF+7$#F*F'F+7$F*#\"#6F'7$#F.F'\"\"&7$F'#\"\"*F'7$#F:F'\" \"%7$F.#\"\"(F'7$FBF.7$F@F?7$F " 0 "" {MPLTEXT 1 0 58 "seq( [mU.mPP[i].mU + mP P[ff+1+1/2-i]], i= undices[1..-2]);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6 47#-%'RTABLEG6$\")W(\\\\\"-%'MATRIXG6#7$7$!\"\"\"\"!7$F.F-7#-F%6$\");? &\\\"F(7#-F%6$\")?Z&\\\"F(7#-F%6$\")[q&\\\"F(7#-F%6$\")/$f\\\"F(7#-F%6 $\")3;'\\\"F(7#-F%6$\")KY(\\\"F(7#-F%6$\");o(\\\"F(7#-F%6$\")G*y\\\"F( 7#-F%6$\")'H\")\\\"F(7#-F%6$\")gb)\\\"F(7#-F%6$\")[j)\\\"F(7#-F%6$\")) Q))\\\"F(7#-F%6$\")c2*\\\"F(7#-F%6$\")K^*\\\"F(7#-F%6$\")Kt*\\\"F(7#-F %6$\")O'**\\\"F(7#-F%6$\")+R+:F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "seq( [subs(z=-z, nupf(mPP[i])) - nupf(mPP[ff+1+1/2-i] )], i= undices[1..-2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "647#\"\"!F#F# F#F#F#F#F#F#F#F#F#F#F#F#F#F#F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 "mil:= (ff+1)/2; ff+1-mil; ava:= mPP[mil]; apr:= mPP[mil+1/2]; \n-mU.ava.mU=apr; mPP[mil+1]=mS.apr.mS;\nmPP[mil+1]= -mS.mU.mPP[mil] .mU.mS;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$milG\"\"$" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$avaG -%'RTABLEG6$\");4)[\"-%'MATRIXG6#7$7$\"\"\"\"\"%7$\"\"$!\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$aprG-%'RTABLEG6$\")#H()[\"-%'MATRIXG6#7$7 $!\"#\"\"%7$\"\"$\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'RTABLEG 6$\")Kh+:-%'MATRIXG6#7$7$!\"\"\"\"%7$\"\"$\"\"#-F%6$\")#H()[\"-F)6#7$7 $!\"#F.7$F0\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'RTABLEG6$\")[ T*[\"-%'MATRIXG6#7$7$\"\"\"\"\"$7$\"\"%!\"#-F%6$\")s\"3]\"F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'RTABLEG6$\")[T*[\"-%'MATRIXG6#7$7$\"\"\" \"\"$7$\"\"%!\"#-F%6$\")K:u6-F)6#7$7$\"\"#F.7$F0!\"\"" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Algorithme de factorisation" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 243 "it:= proc(ma) local qq; if nargs=0 then return NULL \nelif type(args[1], numeric) then return args\nelif ma[1,2]=0 then re turn args[2..-1] fi;\nqq:=min(iquo(ma[1,1],ma[2,1]), iquo(ma[1,2],ma[2 ,2]));\nMatrix([[0,1],[1,-%]]).ma, args[2..-1], qq;\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Une matrice unitaire" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 113 "nit:= proc() global aa,bb,cc,dd,mma; igcdex(aa,bb, 'dd','cc'); mma:= Matrix([[aa/%,bb/%],[abs(cc),abs(dd)]]); end:" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 5 "essai" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "aa, bb := 4085, 1674: tmp:= nit(): sqq:= (it@@12)(mma ); tmp=mH(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$sqqG6,\"\"#\"\"\"F &F'\"\"&F&F'\"\"$F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%'RTABLEG6$ \")CL,:-%'MATRIXG6#7$7$\"%&3%\"%u;7$\"%\"\\\"\"$6'-F%6$\")s^t6F(" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "seq(mh(j),j= sqq); `.`(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6,-%'RTABLEG6$\")w!=]\"-%'MATRIXG6#7$7 $\"\"#\"\"\"7$F-\"\"!-F$6$\");,-:-F(6#7$7$F-F-F.-F$6$\")_W-:F'-F$6$\") #\\E]\"F3-F$6$\"))yG]\"-F(6#7$7$\"\"&F-F.-F$6$\")%3J]\"F'-F$6$\"(g(4zF 3-F$6$\")+f.:-F(6#7$7$\"\"$F-F.-F$6$\")Sz.:F'-F$6$\")_A/:F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\")+X/:-%'MATRIXG6#7$7$\"%&3%\"% u;7$\"%\"\\\"\"$6'" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 21 " Pour un motif donn\351 " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "mh(a).ms.mh(a): map(expand, \+ %); Det(%)= Det(ms);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\" )kn/:-%'MATRIXG6#7$7$,(*&)%\"aG\"\"#\"\"\"%\"uGF1F1*(F0F1F/F1%\"vGF1F1 %\"wGF1,&*&F/F1F2F1F1F4F17$F6F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,& *&%\"uG\"\"\"%\"wGF'F'*$)%\"vG\"\"#F'!\"\"F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "ms; mh(2*n).ms: map(expand, %); poly:= collect(p f(%),[z,n]); Det(%%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$ \"))e2[\"-%'MATRIXG6#7$7$%\"uG%\"vG7$F-%\"wG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'RTABLEG6$\"(?wx(-%'MATRIXG6#7$7$,&*&%\"nG\"\"\"%\"uG F/\"\"#%\"vGF/,&*&F.F/F2F/F1%\"wGF/7$F0F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%polyG,**$)%\"zG\"\"#\"\"\"F**(F)F*%\"nGF*F(F*!\"\"*& *(F)F*%\"vGF*F,F*F*%\"uGF-F-*&%\"wGF*F1F-F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"vG\"\"#\"\"\"F(*&%\"uGF(%\"wGF(!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 61 "; < <,> .% | % >; collect(-Determinant(%), z);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Pour tout choix de n, le point fixe est situ\351 ent re 2n et 2n+1...\na condition que u > v et u > w " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "(normal@subs)(z=2*n, poly*u); (normal@subs)(z=2*n+ 1, poly*u): \ncollect(% , n, normal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"nG\"\"\"%\"vGF&!\"#%\"wG!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*&,&%\"uG\"\"#*&F'\"\"\"%\"vGF)!\"\"F)%\"nGF)F)F&F)%\"wGF+" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 334 "mkvaln:= proc(poly) local \+ A0,B0,C0,A,B,C,dd, eqn; \neqn:= coeff(poly,z,0); \nA0, B0, C0:=coeff(n umer(%),n,0), coeff(numer(%),n,1), denom(%);\nif B0=0 then return n=mu fi;\ndd:= igcd(-B0,C0); A,B,C:= A0/dd,B0/dd,C0/dd;\nif type(A,numeric ) and not type(A, integer) then error sprintf(\"poly= %a, A= %a\", pol y,a) fi;\nn=modp(-A/B, C)+mu*C; end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "mkvalm:= subs(n=m, eval(mkvaln)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "valn:=mkvaln(poly);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%valnG/% \"nG,&-%%modpG6$,$*&%\"wG\"\"\"%\"vG!\"\"#F0\"\"#*&%\"uGF.-%%igcdG6$,$ F/F2F4F0F.*&*&%#muGF.F4F.F.F5F0F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "(mh(2*n)).ms: pf(%); mkvaln(%):\ncas_u_impair=subs(i gcd(2*v,u)=1, %);\ncas_u_pair=subs(u=2*U, w=2*W, igcd(2*v,2*U)=2, %%); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"zG\"\"#\"\"\"F(*(F'F(%\"nG F(F&F(!\"\"*&,&*&F*F(%\"vGF(F'%\"wGF(F(%\"uGF+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%-cas_u_impairG/%\"nG,&-%%modpG6$,$*&%\"wG\"\"\"%\"vG! \"\"#F0\"\"#%\"uGF.*&%#muGF.F3F.F." }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%+cas_u_pairG/%\"nG,&-%%modpG6$,$*&%\"WG\"\"\"%\"vG!\"\"F0%\"UGF.*&%# muGF.F1F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ra:= 1+rand( 5): " }}}{EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 321 "to 55 do motif:= 'r a()' $4; motif:= motif,seq(motif[5-k],k=1..4); \ntmp:= mH(motif); \n# \+ vapa:= evalb([modp(tmp[1,1],2), modp(tmp[1,2],2), modp(tmp[2,2],2)]=[0 ,1,1]);\nvapa:= evalb([modp(tmp[1,1],2), modp(tmp[2,2],2)]=[0,1]);\nmh (2*n).tmp; pf(%); \ntry mkvaln(%);\nASSERT(vapa=false); \ncatch : \nAS SERT(vapa=true);\nend try; \nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "(mh(2*n+1)).ms: pf(%); mkvaln(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,(*$)%\"zG\"\"#\"\"\"F(*&,&!\"\"F(*&F'F(%\"nGF(F+F(F& F(F(*&,(%\"vGF(%\"wGF(*(F'F(F-F(F0F(F(F(%\"uGF+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"nG,&-%%modpG6$,$*&,&%\"vG!\"\"%\"wGF-\"\"\"F,F-#F/ \"\"#*&%\"uGF/-%%igcdG6$,$F,F1F3F-F/*&*&%#muGF/F3F/F/F4F-F/" }}} {EXCHG {PARA 0 "" 0 "" {MPLTEXT 0 21 264 "to 20 do motif:= 'ra()' $3; \+ motif:= motif,seq(motif[4-k],k=1..3); \ntmp:= mH(motif); \nvapa:= eval b([modp(tmp[1,1],2), modp(tmp[2,2]+tmp[2,1],2)]=[0,1]);\nmh(2*n+1).tmp ; pf(%); \ntry mkvaln(%);\nASSERT(vapa=false, faux); \ncatch : \nASSER T(vapa=true, vrai);\nend try; \nod:" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 44 "Th\351or\350 me de triplement, \351criture matricielle" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display(figure, lignes, view=[0..jusqua,0..jusqua]); " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6+-%'POINTSG6f a`l7$$\"$>'\"\"!$\"%Z=F)7$$\"$F\"F)$\"$B%F)7$$\"$>\"F)$\"$2%F)7$$\"$r \"F)$\"$v&F)7$$\"$R#F)$\"$r(F)7$$\"$&=F)$\"$L&F)7$$\"#zF)$\"$j#F)7$$\" $4\"F)$\"$\"GF)7$$\"#LF)$\"#(*F)7$$\"#BF)$\"#\"*F)7$$\"$V#F)$\"$&pF)7$ $\"#JF)$\"#$)F)7$$\"$r%F)$\"%(Q\"F)7$$\"#\")F)$\"$x#F)7$$\"$H\"F)$\"$p $F)7$F[o$\"$n#F)7$$\"#^F)$\"$(=F)7$$\"#&*F)$\"$v#F)7$$\"#XF)$\"$d\"F)7 $F`pF27$$\"#$*F)$\"$4$F)7$$\"$J$F)$\"%F5F)7$$\"$f#F)$\"$F(F)7$$\"$H$F) $\"$d*F)7$$\"$\\&F)$\"%X;F)7$$\"#TF)$\"$`\"F)7$F[o$\"$N#F)7$$\"$.&F)$ \"%*\\\"F)7$$\"#bF)$\"$R\"F)7$$\"$$\\F)$\"%l9F)7$$\"$\\\"F)$\"$`%F)7$$ \"$f\"F)$\"$Z&F)7$$\"$.\"F)$\"$.$F)7$$\"$*fF)$\"%>=F)7$$\"$d#F)$\"$8(F )7$$\"#jF)F<7$FFF<7$$\"$`#F)$\"$X(F)7$F4$\"%Z6F)7$$\"$h$F)$\"%(4\"F)7$ F`q$\"$l#F)7$$\"$:#F)$\"$:'F)7$$\"$@$F)Far7$$\"$j\"F)$\"$\"\\F)7$FP$\" $@\"F)7$$\"$^\"F)$\"$>%F)7$$\"$&[F)F]t7$$\"$>#F)$\"$Z'F)7$$\"$*=F)Fdr7 $$\"$V(F)$\"%>AF)7$$\"$:&F)$\"%v:F)7$FR$\"$(HF)7$$\"$h%F)$\"%`8F)7$$\" $N\"F)F^o7$$\"$J%F)$\"%.8F)7$Fdu$\"$l(F)7$$\"$*>F)$\"$V&F)7$$\"$t\"F)$ \"$x%F)7$$\"$2$F)$\"$**)F)7$$\"$*\\F)$\"%N:F)7$$\"$X%F)$\"%08F)7$$\"$x \"F)$\"$x&F)7$$\"$,\"F)$\"$0$F)7$FRFjv7$$\"$D\"F)$\"$8%F)7$$\"$L#F)$\" $`'F)7$Fho$\"$H%F)7$$\"$0#F)Fdr7$$\"$t$F)Fgv7$$\"$Z(F)$\"%JAF)7$Fc\\l$ \"$`&F)7$F`t$\"$D%F)7$$\"$d$F)Fgv7$F<$\"$2(F)7$$\"$\"RF)$\"%(=\"F)7$F` q$\"$*GF)7$$\"$0%F)$\"%47F)7$$\"\"*F)$\"#PF)7$F_[l$\"$T&F)7$F\\u$\"$^* F)7$Fby$\"%.;F)7$$\"$:$F)$\"$\"**F)7$F]w$\"$f'F)7$Fh\\lFgy7$$\"$F#F)$ \"$N'F)7$$\"$$HF)$\"$p)F)7$$\"$^%F)$\"%R8F)7$Fbx$\"%t9F)7$$\"$J#F)Fhal 7$$\"$X\"F)$\"$P%F)7$$\"$h\"F)$\"$<&F)7$FirF`t7$FUFep7$$\"$*RF)$\"%R7F )7$Fex$\"$$oF)7$F_[l$\"$t%F)7$$\"$h#F)$\"$*yF)7$F]x$\"$j%F)7$$\"#hF)$ \"$4#F)7$$\"$\"=F)Ff^l7$F^]l$\"$*QF)7$$\"$<#F)$\"$0(F)7$$\"$2\"F)$\"$n $F)7$F\\_l$\"%`6F)7$Fh\\l$\"$X$F)7$F\\el$\"$8'F)7$F[o$\"$z#F)7$Fi[l$\" %^:F)7$$\"$(GF)$\"$V)F)7$$\"#()F)$\"$r#F)7$F[oFZ7$$\"$@#F)$\"$\\'F)7$F ZFa^l7$Figl$\"$X'F)7$Fbdl$\"$T)F)7$Fb_l$\"%j6F)7$$\"#xF)Feel7$$\"$0&F) $\"%L:F)7$F\\u$\"$j)F)7$F[s$\"$\\%F)7$$\"#rF)$\"$b#F)7$$\"#nF)F]w7$$\" $B&F)$\"%$e\"F)7$$\"$J&F)$\"%b:F)7$F[cl$\"$l%F)7$$\"$b%F)$\"%b8F)7$$\" $8\"F)$\"$L$F)7$Fjv$\"$D)F)7$$\"$P#F)$\"$P(F)7$F[oFhil7$$\"$&HF)$\"$N* F)7$$\"$<\"F)$\"$,$F)7$$\"#ZF)F77$F^]lFb[m7$Fin$\"$B\"F)7$$\"$0\"F)Feo 7$FjqF>7$FRF_r7$$\"$&eF)$\"%&o\"F)7$F_z$\"$6%F)7$$\"$,%F)F\\`l7$$\"$&G F)$\"$X)F)7$F^jl$\"%6;F)7$Fc\\lFjy7$$\"$n&F)$\"%&p\"F)7$$\"$(oF)$\"%v? 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