{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Script MT \+ Bold" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Script MT Bo ld" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 283 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 284 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 285 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "Norma l" -1 256 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 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 41 "Section \+ 1.6: A Maximal Orthonormal Family" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 3 "" 0 "" {TEXT 258 30 "Maple Packages for Section 1.6" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 246 "In the previous sections, we have develo ped the notions of an orthogonal family, or of an orthonormal family. \+ We emphasized the importance of these in Fourier Analysis. There is on e final notion that we introduce in this section. It is that of a " } {TEXT 274 7 "maximal" }{TEXT -1 102 " orthonormal family. It should be come apparent what is the importance of this notion in this section. \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 139 "We r eview that important Fourier Inequality. Suppose that \{ E, \} \+ is a linear space on which there is defined an inner product. If \{ " }{XPPEDIT 18 0 "phi[1];" "6#&%$phiG6#\"\"\"" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "phi[2];" "6#&%$phiG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[3];" "6#&%$phiG6#\"\"$" }{TEXT -1 14 ", ... \} is an " } {TEXT 285 11 "orthonormal" }{TEXT -1 12 " family and " }{TEXT 275 1 "f " }{TEXT -1 22 " is in the space, then" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 8 " 0 " }{XPPEDIT 18 0 "` ` <= ` `; " "6#1%\"~GF$" }{TEXT -1 2 " " }{XPPEDIT 18 0 "abs(f-sum(a[p]*phi[p], p))^2" "6#*$-%$absG6#,&%\"fG\"\"\"-%$sumG6$*&&%\"aG6#%\"pGF)&%$phiG6#F 1F)F1!\"\"\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "abs(f)^2" "6#*$-%$a bsG6#%\"fG\"\"#" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(abs(- a[p])^2,p);" "6#-%$sumG6$*$-%$absG6#,&-%$<,>G6$%\"fG6#&%$phiG6#%\"pG\" \"\"&%\"aG6#F3!\"\"\"\"#F3" }{TEXT -1 4 " - " }{XPPEDIT 18 0 "sum(abs ()^2,p);" "6#-%$sumG6$*$-%$absG6#-%$<,>G6$%\"fG6#&%$phiG6#% \"pG\"\"#F2" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "Several important facts follow from this inequa lity." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 7 " Fact 1:" }{TEXT -1 44 " Suppose that n is a positive integer, that " } {TEXT 259 1 "S" }{TEXT -1 18 " is the span of \{ " }{XPPEDIT 18 0 "phi [1];" "6#&%$phiG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[2];" "6 #&%$phiG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[3];" "6#&%$phiG6 #\"\"$" }{TEXT -1 8 ", ... , " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#% \"nG" }{TEXT -1 13 " \}, and that " }{TEXT 276 1 "f" }{TEXT -1 33 " is in E. The closest element in " }{TEXT 260 1 "S" }{TEXT -1 34 " is giv en by the Fourier Expansion" }}{PARA 0 "" 0 "" {TEXT -1 71 " \+ approximation for f = " } {XPPEDIT 18 0 "~;" "6#%\"|irG" }{TEXT -1 2 " " }{XPPEDIT 18 0 "sum(`< `*f*`,`*phi[p]*`>`*phi[p],p = 1 .. n);" "6#-%$sumG6$*.%\"GF(&F,6#F.F(/F.;F(%\"nG" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 48 "We have argued that this is true in S ection 1.3." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 7 "Fact 2:" }{TEXT -1 6 " If \{ " }{XPPEDIT 18 0 "phi[1];" "6#&%$p hiG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[2];" "6#&%$phiG6#\" \"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[3];" "6#&%$phiG6#\"\"$" } {TEXT -1 57 ", ... \} is an infinite sequence, then the number series \+ " }{XPPEDIT 18 0 "sum(abs(`<`*f*`,`*phi[p]*`>`)^2,p);" "6#-%$sumG6$*$ -%$absG6#*,%\"GF,\"\"#F2" } {TEXT -1 11 " converges." }}{PARA 0 "" 0 "" {TEXT -1 38 "We argue this as follows: By choosing " }{XPPEDIT 18 0 "a[p];" "6#&%\"aG6#%\"pG" } {TEXT -1 5 " = < " }{TEXT 262 1 "f" }{TEXT -1 3 " , " }{XPPEDIT 18 0 " phi[p];" "6#&%$phiG6#%\"pG" }{TEXT -1 196 " > we get Bessel's inequali ty. From this we see that the number series is increasing and bounded \+ above. That is, it is a series of non-negative numbers and bounded abo ve. Thus, it has to converge." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 264 7 "Fact 3:" }{TEXT -1 6 " If \{ " }{XPPEDIT 18 0 "phi[1];" "6#&%$phiG6#\"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "ph i[2];" "6#&%$phiG6#\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[3];" "6 #&%$phiG6#\"\"$" }{TEXT -1 62 ", ... \} is an infinite sequence, then \+ the series in E given by" }}{PARA 0 "" 0 "" {TEXT -1 51 " \+ " }{XPPEDIT 18 0 "sum(`<`*f*`,`* phi[p]*`>`*phi[p],p = 1 .. infinity);" "6#-%$sumG6$*.%\"GF(&F,6#F.F(/F.;F(%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "converges in E." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "This follows from Fact \+ 2 through a series of steps: Suppose m > n." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "sum(`<`*f *`,`*phi[p]*`>`*phi[p],p = 1 .. m);" "6#-%$sumG6$*.%\"GF(&F,6#F.F(/F.;F(%\"mG" }{TEXT -1 3 " " } {TEXT 265 1 "-" }{TEXT -1 3 " " }{XPPEDIT 18 0 "sum(`<`*f*`,`*phi[p] *`>`*phi[p],p = 1 .. n);" "6#-%$sumG6$*.%\"GF(&F,6#F.F(/F.;F(%\"nG" }{TEXT -1 3 " " }{TEXT 266 1 "=" }{TEXT -1 2 " " }{XPPEDIT 18 0 "sum(`<`*f*`,`*phi[p]*`>`*phi[p] ,p = n .. m);" "6#-%$sumG6$*.%\"GF(&F,6#F.F(/F.;%\"nG%\"mG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Also, " }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 18 0 "abs(sum(`<`*f*`,`*phi[p]*`>`*phi[p] ,p = n .. m))^2;" "6#*$-%$absG6#-%$sumG6$*.%\"GF,&F06#F2F,/F2;%\"nG%\"mG\"\"#" }{TEXT -1 5 " = \+ " }{XPPEDIT 18 0 "sum(` `,p = n .. m);" "6#-%$sumG6$%\"~G/%\"pG;%\"nG% \"mG" }{TEXT -1 8 "| < f , " }{XPPEDIT 18 0 "phi[p];" "6#&%$phiG6#%\"p G" }{TEXT -1 4 " > |" }{XPPEDIT 18 0 "` `^2;" "6#*$%\"~G\"\"#" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "sum(` `,p = 1 .. m);" "6#-%$sumG6$%\"~G/% \"pG;\"\"\"%\"mG" }{TEXT -1 8 "| < f , " }{XPPEDIT 18 0 "phi[p];" "6#& %$phiG6#%\"pG" }{TEXT -1 4 " > |" }{XPPEDIT 18 0 "` `^2;" "6#*$%\"~G\" \"#" }{TEXT -1 1 " " }{TEXT 267 1 "-" }{TEXT -1 2 " " }{XPPEDIT 18 0 "sum(` `,p = 1 .. n);" "6#-%$sumG6$%\"~G/%\"pG;\"\"\"%\"nG" }{TEXT -1 8 "| < f , " }{XPPEDIT 18 0 "phi[p];" "6#&%$phiG6#%\"pG" }{TEXT -1 4 " > |" }{XPPEDIT 18 0 "` `^2;" "6#*$%\"~G\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 59 "Using Fact 2, we have the convergence of the vector series." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 268 8 "Remark: " }{TEXT -1 148 "The naive novice might think that having an infinite orthonormal family and having tha t the series of Fact 3 converges implies that it converges to " } {TEXT 277 1 "f" }{TEXT -1 188 ". Only a minute or two of consideration , however, will yield the folly of this mistake. Here is a specific ex ample. Take the orthonormal sequence to be the infinite sequence consi sting of " }}{PARA 0 "" 0 "" {TEXT -1 36 " \+ \{2 sin(2 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 13 " x), 2 sin(4 \+ " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 13 " x), 2 sin(6 " } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 12 " x), ... \}. " }}{PARA 0 " " 0 "" {TEXT -1 5 "Take " }{TEXT 278 1 "f" }{TEXT -1 6 " to be" }} {PARA 0 "" 0 "" {TEXT -1 30 " " }{TEXT 279 1 "f" }{TEXT -1 1 "(" }{TEXT 280 1 "x" }{TEXT -1 4 ") = " } {XPPEDIT 18 0 "x^2*(1-x);" "6#*&%\"xG\"\"#,&\"\"\"F'F$!\"\"F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 43 "Look at how badly the approxim ation misses." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=x->x^2*(1-x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "for n from 1 to 10 do\n a[n]:=int(f(x)*sqrt(2)*sin( 2*n*Pi*x),x=0..1);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 " approx:=x->sum(a[p]*sqrt(2)*sin(2*p*Pi*x),p=1..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "plot([f(x),approx(x)],x=0..1,color=[black ,red]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "We missed all the od d terms. There were not enough orthonormal vectors. One needs a " } {TEXT 269 7 "maximal" }{TEXT -1 44 " family to be able to guarantee co nvergence." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 271 11 "Definition:" }{TEXT -1 27 " An orthogonal sequence is " } {TEXT 273 7 "maximal" }{TEXT -1 5 ", or " }{TEXT 270 8 "complete" } {TEXT -1 105 ", if the only vector in the space that is orthogonal to \+ every element in the sequence is the zero vector." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 272 6 "Fact 4" } {TEXT -1 17 ". Suppose that \{ " }{XPPEDIT 18 0 "phi[1];" "6#&%$phiG6# \"\"\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "phi[2];" "6#&%$phiG6#\"\"#" } {TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" } {TEXT -1 49 " , ...\} is a maximal orthonormal sequence. Then " }} {PARA 0 "" 0 "" {TEXT -1 34 " " } {XPPEDIT 18 0 "sum(*phi[p],p = 1 .. infinity);" "6#-%$sumG6$ *&-%$<,>G6$%\"fG6#&%$phiG6#%\"pG\"\"\"&F-6#F/F0/F/;F0%)infinityG" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "converges to " }{TEXT 281 2 "f " }{TEXT -1 8 "in norm." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 61 "We argue this by showing that the element g defined as f " }{TEXT 282 1 "-" }{TEXT -1 2 " " }{XPPEDIT 18 0 "sum(*phi[p],p = 1 .. infinity);" "6#-%$sumG6$*&-%$<,>G6$% \"fG6#&%$phiG6#%\"pG\"\"\"&F-6#F/F0/F/;F0%)infinityG" }{TEXT -1 24 " \+ is orthogonal to each " }{XPPEDIT 18 0 "phi[p];" "6#&%$phiG6#%\"pG" } {TEXT -1 27 ". Just do the computations:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 " < g , " }{XPPEDIT 18 0 "phi[n]; " "6#&%$phiG6#%\"nG" }{TEXT -1 11 " > = < f , " }{XPPEDIT 18 0 "phi[n] ;" "6#&%$phiG6#%\"nG" }{TEXT -1 3 " > " }{TEXT 283 1 "-" }{TEXT -1 7 " < f , " }{XPPEDIT 18 0 "phi[n];" "6#&%$phiG6#%\"nG" }{TEXT -1 7 " > = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 52 "He re, we have used the fact that the dot product of " }{XPPEDIT 18 0 "ph i[p];" "6#&%$phiG6#%\"pG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi[n]; " "6#&%$phiG6#%\"nG" }{TEXT -1 134 " is zero, unless p = n. In that ca se, the dot product is 1. Because the orthonormal sequence was maximal and g is orthogonal to every " }{XPPEDIT 18 0 "phi[p];" "6#&%$phiG6#% \"pG" }{TEXT -1 57 ", it must be that g is zero, and the series conver ged to " }{TEXT 284 1 "f" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "We need to repair the previous exa mple by taking all the sine terms:" }}{PARA 0 "" 0 "" {TEXT -1 35 " \+ \{2 sin(" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" } {TEXT -1 13 " x), 2 sin(2 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 13 " x), 2 sin(3 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 11 " x), . .. \}." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "f:=x->x^2*(1-x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "for n from 1 to 10 do\n \+ a[n]:=int(f(x)*sqrt(2)*sin(n*Pi*x),x=0..1);\nod;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 49 "approx:=x->sum(a[p]*sqrt(2)*sin(p*Pi*x),p=1. .10);" }}}{PARA 0 "" 0 "" {TEXT -1 74 "We off set the graph of f just \+ a little so that we can see the difference." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 54 "plot([f(x)+0.003,approx(x)],x=0..1,color=[black,red ]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 390 "This was the culmination of the review of concepts from Linear Algebra that are needed to unde rstand some of the methods for solving partial differential equations \+ in this set of lecture notes. The new and important idea of this Secti on is the notion of a maximal orthonormal family. With such a family, \+ we can approximate in norm all vectors in the space having the associa ted dot product." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 51 "EMAIL: herod@math.gatech.edu or jherod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 38 "URL: http://www.math.gatech.edu/~herod" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 36 "Copyright \+ \251 2003 by James V. Herod" }}{PARA 256 "" 0 "" {TEXT -1 19 "All ri ghts reserved" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0 0" 1 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }