{VERSION 2 3 "SUN SPARC SOLARIS" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 256 "terminal" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 257 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE " Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 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 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "" 3 256 1 {CSTYLE " " -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R3 Font 0" -1 257 1 {CSTYLE "" -1 -1 "Helvetica" 1 14 0 0 0 0 2 1 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "R 3 Font 2" -1 258 1 {CSTYLE "" -1 -1 "Courier" 1 14 0 0 0 0 2 2 2 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT 256 47 "/home/m262f99/CHYZAK/work sheetsV.4/oredemo1.mws" }{MPLTEXT 1 0 0 "" }}{PARA 256 "" 0 "" {TEXT 257 105 "Math 262a, Fall 1999, Glenn Tesler\nOre Algebras, Non-Commuta tive Division and Euclidean Algorithm\n11/2/99" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 150 "This uses software packages by Frederick Chyzak. F ollow the installation instructions on the class homepage, or the foll owing commands will not work." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "with(Ore_algebra); with(Groebner); with(Holonomy); wi th(Mgfun);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#74%-Ore_to_DESolG%-Ore_t o_RESolG%,Ore_to_diffG%-Ore_to_shiftG%-annihilatorsG%)applyoprG%-diff_ algebraG%-poly_algebraG%/qshift_algebraG%/rand_skew_polyG%.shift_algeb raG%-skew_algebraG%*skew_elimG%+skew_gcdexG%*skew_pdivG%+skew_powerG%* skew_premG%-skew_productG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#75%*fglm_ algoG%'gbasisG%'gsolveG%+hilbertdimG%,hilbertpolyG%.hilbertseriesG%-in ter_reduceG%*is_finiteG%,is_solvableG%*leadcoeffG%(leadmonG%)leadtermG %(normalfG%/pretend_gbasisG%'reduceG%&spolyG%*termorderG%*testorderG%) univpolyG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7,%1algeq_to_dfiniteG%,df inite_addG%,dfinite_mulG%.holon_closureG%-holon_defintG%.holon_defqsum G%-holon_defsumG%/holon_diagonalG%5hypergeom_to_dfiniteG%.takayama_alg oG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7(%,diag_of_sysG%+int_of_sysG%+p ol_to_sysG%+sum_of_sysG%(sys*sysG%(sys+sysG" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 407 "Define a noncommutative algebra with a shift operator \n Sn f(n,k,x) = f(n+1,k,x)\nand a difference (Delta) operator\n Dk f(n,k,x) = f(n,k+1,x)-f(n,k,x)\nand a differential operator\n Dx f(n,k,x) = d/dx f(n,k,x)\n(There are only so many letters avai lable, so he used D to denote these two separate things.)\nThere are o ther predefined types of Ore algebras, as well as the ability to defin e your own." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "A:=skew_alge bra(shift=[Sn,n],delta=[Dk,k],diff=[Dx,x],polynom=k);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG%,Ore_algebraG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "We can multiply two operators together with skew_product( f,g,A):" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "showprod := proc(f,g,A)\n print(f &* g = skew_product(f,g,A)) \nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "showprod(x,Dx,A); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"xG%#DxG*&F'\"\"\"F(F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "showprod(Dx,x,A);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%#DxG%\"xG,&\"\"\"F**&F(F*F'F *F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "showprod(n,Sn,A); sh owprod(Sn,n,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%\"nG%#SnG *&F'\"\"\"F(F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%#SnG%\"nG* &,&F(\"\"\"F+F+F+F'F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "sh owprod(Dx,x^2,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$%#DxG*$% \"xG\"\"#,&*&F'\"\"\"F)F*F-F)F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 258 8 "Warning:" }{TEXT -1 380 " Maple understands commutative polynomials , but doesn't really understand noncommutative ones. Chyzak's softwar e recognizes this and works around it. Any monomial in x, Dx (or n, \+ Sn, etc.) that is properly of the form x^a Dx^b may be represented by \+ Maple in either order; that way, or Dx^b x^a, but Chyzak's software as sumes it's intended the x's be left and the Dx's be right." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "showprod(Sn*Dx,n *x^2,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%#&*G6$*&%#SnG\"\"\"%#Dx GF)*&%\"nGF)%\"xG\"\"#,&*&,&*&F-F)F,F)F.F-F.F)F(F)F)*(,&F+F)*$F-F.F)F) F(F)F*F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "We can also apply a n operator to a function:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "applyo pr(Dx,sin(x),A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$cosG6#%\"xG" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "applyopr(Dx,x,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "applyopr(Dx,x^2,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#,$%\"xG\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Random skew pol ynomials can be generated (for instance, to create random input for ro utines):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "rand_skew_poly(x,A);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$%\"xG\"\"&!#&)*$F%\"\"%!#b*$F%\" \"$!#P*$F%\"\"#!#NF%\"#(*\"#]\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "rand_skew_poly([x,Dx],A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$%\"xG\"\"&!\")*$F%\"\"%!#$**&,&F%\"#XF(\"#V\"\"\"%# DxGF/F/*&,&\"##*F/*$F%\"\"$!#iF/F0\"\"#F/" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 33 "rand_skew_poly([x,Dx],terms=5,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,!#h\"\"\"%#DxG!#]*$%\"xG\"\"$!#7*$F&F*!#=*&F)\"\"# F&F/\"#J" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 259 12 "Application." } {TEXT -1 61 " Find operators in this algebra that annihilate binomial( n,k)" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 " el := hypergeom_to_dfinite(binomial(n,k),A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#elG7%,**&%#DkG\"\"\",&%\"kGF)F)F)F)F)%\"nG!\"\"F+\" \"#F)F)%#DxG,(*&%#SnGF),(F,F)F)F)F+F-F)F)F,F-F-F)" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 72 "Verify that they do annihilate it. Apply the oper ators to the function." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "applyopr( el[1],binomial(n,k),A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,&*&,(\"\" \"F&%\"nG!\"\"%\"kG\"\"#F&-%)binomialG6$F'F)F&F&*&,&F)F&F&F&F&,&-F,6$F 'F/F&F+F(F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "sumtools[s impcomb](\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "map(applyopr,el,binomial(n,k),A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7%,&*&,(\"\"\"F'%\"nG!\"\"%\"kG\"\"# F'-%)binomialG6$F(F*F'F'*&,&F*F'F'F'F',&-F-6$F(F0F'F,F)F'F'\"\"!,&*&,& F(F)F)F'F'F,F'F'*&,(F(F'F'F'F*F)F'-F-6$,&F(F'F'F'F*F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "map(sumtools[simpcomb],\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"!F$F$" }}}{EXCHG {PARA 3 "" 0 " " {TEXT -1 35 "Noncommutative division in K(n)[Sn]" }{MPLTEXT 1 0 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "A := shift_algebra([Sn, n]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG%,Ore_algebraG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f1 := skew_power((n+1)*Sn,2, A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#f1G*&,(*$%\"nG\"\"#\"\"\"F( \"\"$F)F*F*%#SnGF)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "f2 := skew_product(Sn+5,f1,A) + Sn+9;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>% #f2G,**&,(*$%\"nG\"\"#\"\"&F)\"#:\"#5\"\"\"F.%#SnGF*F.*&,(F(F.F)F+\"\" 'F.F.F/\"\"$F.F/F.\"\"*F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "d1 := skew_pdiv(f2,(n+1)*Sn,Sn,A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%#d1G7%,&%\"nG\"\"\"F(F(,0*&%#SnG\"\"#F'F,F(*&F+F,F'F(\"\"$*&F+ F(F'F,\"\"&*&F+F(F'F(\"#5*$F+F,F,F+F0F(F(,&F'\"\"*F5F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "skew_product(d1[1],f2,A) - skew_pro duct(d1[2],(n+1)*Sn,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"nG\"\" *F%\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "d2 := skew_pdi v(f2,(n+1)*Sn,n,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#d2G7%\"\"\", **&%#SnG\"\"#%\"nGF&F&*$F)F*F**&F)F&F+F&\"\"&F)F.,&F)F&\"\"*F&" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "skew_product(d2[1],f2,A) - s kew_product(d2[2],(n+1)*Sn,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%# SnG\"\"\"\"\"*F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "skew_pdiv(p, q,x,A) ---> [ u, v, r ]\nwhere u*p - v*q = r of x-degree lower \+ than q.\nv and r are polynomials in x, while u is a coefficient." }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "skew_pdiv((n+3)^5,n^2+2,n,A);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\",**$%\"nG\"\"$F$*$F'\"\"#\"#:F '\"#))\"$S#F$,&F'\"$H#!$P#F$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "skew_pdiv((n+Sn)^3,Sn+n^2,n,A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7%\"\"\",&%\"nGF$%#SnG\"\"$,,*&F'F$F&F$!\"(*&F'\"\"#F&F$F(*$F'F(F$F '!\"$*$F'F-F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "skew_produ ct(\"[2],Sn+n^2,A) + \"[3];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,.*$%\" nG\"\"$\"\"\"*&,(F%\"\"(*$F%\"\"#F&F&F'F'%#SnGF'F'*&F-F'F%F'!\"(*&F-F, F%F'F&*$F-F&F'F-!\"$" }}}{EXCHG {PARA 3 "" 0 "" {TEXT -1 46 "Noncommut ative Euclidean Algorithm for K" }{MPLTEXT 1 0 0 "" }}{PARA 0 " " 0 "" {TEXT -1 19 "skew_gcdex(p,q,x,A)" }}{PARA 15 "" 0 "" {TEXT -1 368 "The function skew_gcdex performs an extended skew gcd algorithm o n the skew polynomials p and q viewed as polynomials in x with coeffic ients in their other indeterminates. It returns a list [g,a,b,u,v] suc h that up+vq=0 and ap+bq=g. Hence, g is a gcd of p and q (in an algebr a where all coefficient indeterminates are invertible), while up and v q are lcm's of p and q." }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 104 "P := skew_product(Sn^2+n*Sn+3,Sn+n,A);\nQ := skew_ product(Sn^3+n*Sn+3,Sn+n,A);\nG := skew_gcdex(P,Q,Sn,A);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%\"PG,*%\"nG\"\"$*&,(F'\"\"\"*$F&\"\"#F*F&F*F*% #SnGF*F**&,&F&F,F,F*F*F-F,F**$F-F'F*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#>%\"QG,,*$%#SnG\"\"%\"\"\"%\"nG\"\"$*&,(F+F)*$F*\"\"#F)F*F)F)F'F)F)* &F'F/F*F)F)*&,&F*F)F+F)F)F'F+F)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>% \"GG7',**$%\"nG\"\"#\"\"*%#SnG\"#a*&F+\"\"\"F(F.F*F(F,,,F'\"\"%*$F(\" \"$F.F*F.*&,&F(F0F2F.F.F+F.!\"\"*&,(F2F.F(!\"#F'F5F.F+F)F5,,F*F.F(F2F' !\"%F1F5*&,(!\"$F.F(F)F'F.F.F+F.F5,2*&F+F.F(F)F5F-!\"*F(F=*&F+F2F(F.F5 *$F+F2!\"'*$F+F)F.!#@F.F+!#<,0F?F.F-F**&F+F)F(F.F.F(F2FD\"\"'F+\"#<\"# @F." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "This says the GCD of (P,Q) is" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "G[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$%\"nG\"\"#\"\"*%#SnG\"#a*&F(\"\"\"F%F+F'F%F)" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,&\"\"'\"\"\"%\"nGF'F',&F(F'%#SnGF'F'\"\"*" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 163 "(Instead of working in K(n)[Sn], we use only polynomials, i.e., K, so this K(n)-multiple of what we expected (Sn+n) was produced to keep denominators clear.)" }} {PARA 0 "" 0 "" {TEXT -1 60 "This GCD can be expressed as a linear com bination g=a*P+b*Q:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "skew_product (G[2],P,A)+skew_product(G[3],Q,A);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6# ,:*&,*!#@\"\"\"*$%\"nG\"\"#\"#5F)F**$F)\"\"$F*F'%#SnG\"\"%F'F)\"#a*&,. F,\"\"(F)!#7F(\"\"**$F)F/\"\"&\"#=F'*$F)F7F'F'F.F'F'*&,,!#:F'F,F5F(F7F 6F*F)!\")F'F.F*F'*&,,F(\"#7F)!#>F,\"\")!#IF'F6F'F'F.F-F'*&,(!\"$F'F)F* F(F'F'F.F7F'*&,*\"#@F'F(!#5F)!\"#F,FKF'F.F/F'F(F5*&,.\"#OF'F,!\"(F(!\" *F9!\"\"F)FIF6!\"&F'F.F'F'*&,,F,FP\"#:F'F(FRF6FKF)FBF'F.F*F'*&,,F(F4F) \"#>F,F=\"#IF'F6FQF'F.F-F'*&,(F-F'F)FKF(FQF'F.F7F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "collect(\",Sn,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&,&\"#a\"\"\"%\"nG\"\"*F'%#SnGF'F'*&F(F',&\"\"'F'F(F 'F'F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "simplify(\"-G[1]); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Also, the LCM can be computed as a multiple of P or of Q: " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "skew_product(G[4],P,A):\ncollec t(\",Sn,factor);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,0*&,&%\"nG!\"\"! \"'\"\"\"F)%#SnG\"\"'F)*&,(*$F&\"\"#!\"#!#ZF)F&!#?F)F*\"\"&F)*&,*!$,\" F)*$F&\"\"$F'F.!#9F&!#kF)F*\"\"%F)*&,*!$M\"F)F.!#CF&!#&*F7F0F)F*F8F)*& ,,*$F&F;F'F&!$T\"!$@\"F)F.!#bF7!#7F)F*F/F)*&,*F7F(F&!$3\"F.!#a!$9\"F)F )F*F)F)*&F&F),&\"\"(F)F&F)F)!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "skew_product(G[5],Q,A):\ncollect(\",Sn,factor);" }} {PARA 12 "" 1 "" {XPPMATH 20 "6#,0*&,&\"\"'\"\"\"%\"nGF'F'%#SnGF&F'*&, (*$F(\"\"#F-\"#ZF'F(\"#?F'F)\"\"&F'*&,*\"$,\"F'*$F(\"\"$F'F,\"#9F(\"#k F'F)\"\"%F'*&,*\"$M\"F'F,\"#CF(\"#&*F4F-F'F)F5F'*&,,*$F(F8F'F(\"$T\"\" $@\"F'F,\"#bF4\"#7F'F)F-F'*&,*F4F&F(\"$3\"F,\"#a\"$9\"F'F'F)F'F'*&F(F' ,&\"\"(F'F(F'F'\"\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }} }}{MARK "34 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }