{VERSION 3 0 "SUN SPARC SOLARIS" "3.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 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 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 1 14 0 0 0 0 0 0 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 -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 8 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 "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 75 "Here are some sample MA PLE calculations for polynomials over finite fields." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "Three polynomials:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "a:= x^3+4*x^2+9*x+10; \nb:= x^4+2*x-2;\nc:= 2*x^3-3* x+4*x^2-6;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,**$)%\"xG\"\"$\"\"\"\"\"\"*$)F(\"\"#F*\"\"%F(\" \"*\"#5F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"bG,(*$)%\"xG\"\"%\"\" \"\"\"\"F(\"\"#!\"#F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG,**$)% \"xG\"\"$\"\"\"\"\"#F(!\"$*$)F(F+F*\"\"%!\"'\"\"\"" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 258 62 "Here are examples of calculations on a,b,c ove r the rationals." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "By default, al l most \"lower case\" commands operate over the rationals." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "a*b;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,**$)%\"xG\"\"$\"\"\"\"\"\"*$)F'\"\"#F)\"\"%F'\"\"*\"#5F*F*,(* $)F'F.F)F*F'F-!\"#F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "e xpand(a*b);expand(a^3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,2*$)%\"xG \"\"(\"\"\"\"\"\"*$)F&\"\"%F(\"#7*$)F&\"\"$F(\"\"'*$)F&F1F(F,*$)F&\"\" #F(\"#5*$)F&\"\"&F(\"\"*F&F6!#?F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#, 6%\"xG\"%+F\"%+5\"\"\"*$)F$\"\"$\"\"\"\"%*=$*$)F$\"\"#F+\"%IO*$)F$\"\" %F+\"%#*>*$)F$\"\"(F+\"#v*$)F$\"\"'F+\"$5$*$)F$\"\"&F+\"$:**$)F$\"\"*F +F'*$)F$\"\")F+\"#7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "factor(a); factor(b); factor(c);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\"\"\"\"\"#F&F&,(*$)F%F'\"\" \"F&F%F'\"\"&F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"%\" \"\"\"\"\"F&\"\"#!\"#F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,&%\"xG\" \"\"\"\"#F&F&,&*$)F%F'\"\"\"F'!\"$F&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "irreduc(a); irreduc(b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gcd(a,b); gcd(a,c);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG \"\"\"\"\"#F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "Here is the ext ended Euclidian algorithm for finding polynomials s, t such that s(x) \+ a(x) + t(x) b(x) = 1" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "gcdex(a,b,x ,'s','t');" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "s;t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,* #\"$.\"\"$0*\"\"\"%\"xG#!$,\"F&*$)F(\"\"#\"\"\"#\"#\\F&*$)F(\"\"$F.#! \"(\"$i$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(#\"#D\"$i$\"\"\"*$)%\"xG \"\"#\"\"\"#\"\"(F&F*#\"#@\"$0*" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 257 47 "Here we repeat the above calculations modulo 3." }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "Redefine a,b,c by reducing them mod 3." } }{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a:=a mod 3;\nb:=b mod 3;\nc:=c mod 3;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG,(*$)%\"xG \"\"$\"\"\"\"\"\"*$)F(\"\"#F*F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%\"bG,(*$)%\"xG\"\"%\"\"\"\"\"\"F(\"\"#F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG,&*$)%\"xG\"\"$\"\"\"\"\"#*$)F(F+F*\"\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "For calculations in GF(3) capitali ze most commands and postfix \" mod 3\"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "Expand(a*b) mod 3; Expand(a^3) mod 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,,*$)%\"xG\"\"(\"\"\"\"\"\"*$)F&\"\"'F(F)*$)F&\"\" #F(F)F&F/F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%\"xG\"\"*\"\"\" \"\"\"*$)F&\"\"'F(F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Factor(a) mod 3; \nFactor(b) mod 3; \nFactor(c) mod 3;" }}{PARA 0 "" 0 "" {TEXT -1 55 "Note that b factor s in GF(3), but not in the rationals!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,(*$)%\"xG\"\"#\"\"\"\"\"\"F'F(F(F*F*,&F'F*F(F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&,**$)%\"xG\"\"$\"\"\"\"\"\"*$)F'\"\"#F)F-F'F*F*F *F*,&F'F*F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&)%\"xG\"\"#\"\" \",&F&\"\"\"F'F*F*F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Irr educ(a) mod 3; Irreduc(b) mod 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%& falseG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Gcd(a,b) mod 3; Gcd(a,c) mod 3;" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&%\"xG\"\"\"\"\"#F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Gcdex(a,b,x,'s','t') mod 3 ;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "s;" }{TEXT -1 0 "" }{MPLTEXT 1 0 2 "t;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"$\"\"\"\"\"\"*$)F&\"\"#F(F,F&F,F, F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%\"xG\"\"#\"\"\"F'F'\"\"\" " }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 262 36 "Constructing and computing in GF(81)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "To construct GF(81) \+ = " }{XPPEDIT 18 0 "Z[3];" "6#&%\"ZG6#\"\"$" }{TEXT -1 72 "[x]/f(x), w e first find an irreducible polynomial f(x) of degree 4 over " } {XPPEDIT 18 0 "Z[3];" "6#&%\"ZG6#\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 106 "We try some random ones until we get one. Each exe cution of the following should produce a new candidate." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "f:=x^4 + randpoly(x,degree=3,coeffs =rand(3),dense);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Irreduc(f) mod \+ 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,(*$)%\"xG\"\"%\"\"\"\"\" \"\"\"#F+*$)F(F,F*F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Actually, let us just proceed with the following, which I found ahead of time." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "f:=x^4 + 2*x^3 + x + 1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Irreduc(f) mod 3;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# >%\"fG,**$)%\"xG\"\"%\"\"\"\"\"\"*$)F(\"\"$F*\"\"#F(F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "We give a short name for the root of f(x)." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "alias( alpha=RootOf(f) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%\"IG%&alphaG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "W e can verrify that MAPLE now understands " }{XPPEDIT 18 0 "alpha;" "6# %&alphaG" }{TEXT -1 19 " to be a root of f." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify( eval(f,x=alpha) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Now each \+ field element can be represented by a polynomial in " }{XPPEDIT 18 0 " alpha;" "6#%&alphaG" }{TEXT -1 25 " having degree at most 3," }}{PARA 0 "" 0 "" {TEXT -1 25 "and with coefficients in " }{XPPEDIT 18 0 "Z[3] ;" "6#&%\"ZG6#\"\"$" }{TEXT -1 53 ". There are exactly 81 such polynom ials, including 0." }}{PARA 0 "" 0 "" {TEXT -1 63 "This is analogous t o how complex numbers can be represented as " }{XPPEDIT 18 0 "a+bI;" " 6#,&%\"aG\"\"\"%#bIGF%" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "I = Root Of(x^2+1);" "6#/%\"IG-%'RootOfG6#,&*$)%\"xG\"\"#\"\"\"F-\"\"\"F-" } {TEXT -1 1 "," }}{PARA 0 "" 0 "" {TEXT -1 22 "and with coefficients " }{XPPEDIT 18 0 "a,b;" "6$%\"aG%\"bG" }{TEXT -1 20 " being real numbers ." }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 259 26 "Basic Arithmetic in GF(81) " }}{PARA 0 "" 0 "" {TEXT -1 99 "We use the command \"Normal( ... ) mo d 3\" command as a \"wrapper for basic arithmetic: For example: " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "Normal ( alpha^5 ) mod 3;" } {TEXT -1 71 " \+ Simplifying [" }{XPPEDIT 18 0 "alpha^5;" "6#*$%&alphaG\"\"&" }{TEXT -1 15 "] in the field." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%&alpha G\"\"$\"\"\"\"\"\"*$)F&\"\"#F(F,F&F)F,F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Normal( (1+alpha^3)*(2+alpha^2+2*alpha^3) ) mod 3;" } {TEXT -1 26 " Multiplying in the field." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%&alphaG\"\"$\"\"\"\"\"#*$)F&F)F(F)F)\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "N" }{MPLTEXT 1 0 49 "ormal( (1+alpha ^3)/(2+alpha^2+2*alpha^3) ) mod 3;" }{TEXT -1 23 " Dividing in the fie ld." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*$)%&alphaG\"\"$\"\"\"\"\"\" \"\"#F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "Normal((alpha^2 \+ + 2*alpha + 2)^(-1)) mod 3;" }{TEXT -1 38 " Calculatin g inverses." }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,(*$)%&alphaG\"\"$\"\" \"\"\"#*$)F&F)F(F)F&F)" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 260 37 "Find ing a primitive element in GF(81)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "Is " }{XPPEDIT 18 0 "beta := alpha^2+2;" "6#>%%betaG,&*$%&alphaG\" \"#\"\"\"\"\"#F)" }{TEXT -1 41 " a primitive element? We need only te st " }{XPPEDIT 18 0 "beta^i;" "6#)%%betaG%\"iG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "i;" "6#%\"iG" }{TEXT -1 22 " is a divisor of 81-1." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "div:=numtheory[divisors](80 );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$divG<,\"\"\"\"\"#\"\"%\"\"&\" \")\"#5\"#;\"#?\"#S\"#!)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "beta:=alpha^2+2;\nfor i in div do\n if Normal( beta^i ) mod 3 = 1 t hen print(i) fi\nod;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG,&*$)% &alphaG\"\"#\"\"\"\"\"\"F)F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#S" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#!)" }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "beta = alpha^2+2;" "6#/%%betaG,&*$%&alphaG\"\"#\"\"\"\" \"#F)" }{TEXT -1 24 " is not primitive since " }{XPPEDIT 18 0 "beta^40 = 1;" "6#/*$%%betaG\"#S\"\"\"" }{TEXT -1 24 ". Try again with, say, \+ " }{XPPEDIT 18 0 "beta := alpha^3+alpha;" "6#>%%betaG,&*$%&alphaG\"\"$ \"\"\"F'F)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "beta:=alpha^3+alpha;\nfor i in numtheory[divisors](80) do\n if \+ Normal( beta^i ) mod 3 = 1 then print(i) fi\nod;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG,&*$)%&alphaG \"\"$\"\"\"\"\"\"F(F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#!)" }}} {PARA 0 "" 0 "" {TEXT -1 5 "Yes, " }{XPPEDIT 18 0 "beta = alpha^3+alph a;" "6#/%%betaG,&*$%&alphaG\"\"$\"\"\"F'F)" }{TEXT -1 26 " is primitiv e in GF(81) = " }{XPPEDIT 18 0 "Z[3];" "6#&%\"ZG6#\"\"$" }{TEXT -1 1 " [" }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 4 "]/f(" }{XPPEDIT 18 0 "x; " "6#%\"xG" }{TEXT -1 1 ")" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 261 40 "C onstructing Zech's Log Table for GF(81)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "To make Zech's Log Table, we can write a function which w hen given field element " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 0 " " }{TEXT -1 10 ", returns " }{XPPEDIT 18 0 "i;" "6#%\"iG" }{TEXT -1 1 " " }{TEXT -1 6 "where " }{XPPEDIT 18 0 "x = beta^i;" "6#/%\"xG)%%beta G%\"iG" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 162 "exponent ofbeta:=proc(x) local i; global beta; option remember;\n for i from 0 \+ to 79 do\n if x = Normal( beta^i ) mod 3 then RETURN(i) fi od;\n RETU RN(infinity) end:" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "We build the matrix with columns " }{XPPEDIT 18 0 "beta^i,i,z(i); " "6%)%%betaG%\"iGF%-%\"zG6#F%" }{TEXT -1 68 ". This may take a while , as the exponentofbeta function is so slow." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 221 "M:=matrix(81,3):\nM[1,1]:=[0,0,0,0]: M[1,2]:=infinit y: M[1,3]:=0:\nfor i from 2 to 81 do\n y:= Normal( beta^(i-1) ) mod 3 ; \n M[i,1]:=[seq(coeff(y,alpha,j),j=0..3)];\n M[i,2]:=i-1;\n M[i,3 ]:=exponentofbeta( y+1 mod 3);\n od:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "print(M);" }{TEXT -1 21 " Here is the table!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'matrixG6#7]p7%7&\"\"!F)F)F)%)infini tyGF)7%7&F)\"\"\"F)F-F-\"#:7%7&F)\"\"#F1F1F1\"#P7%7&F-F)F1F1\"\"$\"#X7 %7&F)F)F)F-\"\"%\"#t7%7&F-F)F-F-\"\"&\"#V7%7&F)F1F)F-\"\"'\"#J7%7&F1F- F)F)\"\"(\"#G7%7&F1F-F-F)\"\")\"#z7%7&F-F1F)F1\"\"*\"#b7%7&F)F1F-F1\"# 5\"#?7%7&F1F1F)F)\"#6\"#o7%7&F-F)F1F-\"#7\"#f7%7&F1F)F1F)\"#8\"#;7%7&F -F-F-F)\"#9FI7%7&F-F-F)F-F.\"#\\7%7&F)F)F1F)Fhn\"#=7%7&F-F1F-F-\"#<\"# H7%7&F-F)F1F)FaoFgn7%7&F-F)F-F1\"#>\"#_7%7&F-F1F-F1FR\"#q7%7&F1F)F)F- \"#@F97%7&F-F1F-F)\"#A\"#y7%7&F)F)F-F1\"#BFjo7%7&F-F-F-F-\"#C\"#x7%7&F 1F-F-F1\"#D\"#m7%7&F)F1F1F-\"#E\"#l7%7&F)F)F-F-\"#FF=7%7&F)F-F)F)FF\"# ^7%7&F1F1F-F-Feo\"#O7%7&F-F-F1F-\"#I\"#g7%7&F-F1F)F-FB\"#K7%7&F1F1F)F- FdrFA7%7&F1F)F)F1\"#L\"#W7%7&F1F1F1F-\"#MFcq7%7&F)F1F-F)\"#NFdp7%7&F)F 1F-F-F]rFdo7%7&F-F1F1F1F2\"#k7%7&F-F-F1F)\"#Q\"#i7%7&F)F1F1F)\"#R\"#[7 %7&F1F)F)F)\"#SF*7%7&F)F1F)F1\"#TFM7%7&F)F-F-F-\"#UF[q7%7&F1F)F-F-F>Fg q7%7&F)F)F)F1Fjr\"#h7%7&F1F)F1F1F6\"#n7%7&F)F-F)F1\"#Y\"#s7%7&F-F1F)F) \"#ZFU7%7&F-F1F1F)F]t\"#a7%7&F1F-F)F-F^oF-7%7&F)F-F1F-\"#]F`r7%7&F-F-F )F)FjqFE7%7&F1F)F-F1F[pFhp7%7&F-F)F-F)\"#`\"#e7%7&F1F1F1F)FhuF\\t7%7&F 1F1F)F1FNFct7%7&F)F)F-F)\"#cFdv7%7&F1F-F1F1\"#d\"#w7%7&F1F)F-F)FevF\\w 7%7&F1F)F1F-FZ\"#j7%7&F1F-F1F-FarF]v7%7&F-F)F)F1F[uFir7%7&F1F-F1F)Fis \"#v7%7&F)F)F1F-FewFY7%7&F1F1F1F1FesF17%7&F-F1F1F-FdqF]s7%7&F)F-F-F1F` q\"#u7%7&F)F)F1F1F^uF57%7&F)F1F)F)FVFeu7%7&F-F-F1F1\"#pF_w7%7&F1F1F-F1 F^pFQ7%7&F1F-F)F1\"#rFau7%7&F-F-F)F1FbuFay7%7&F-F)F)F-F:Fap7%7&F-F-F-F 1FexF_q7%7&F)F-F1F)F\\xFhs7%7&F)F-F1F1F`wF\\y7%7&F1F-F-F-F\\qFft7%7&F1 F1F-F)FepF`s7%7&F)F-F-F)FJF[o7%7&F-F)F)F)\"#!)F`t" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 0 "" }}}}}}{MARK "2" 0 }{VIEWOPTS 1 1 0 1 1 1803 }