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{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 271 26 "Section 1.1: Linear Spaces" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 309 30 "Maple Pac
kages for Section 1.1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "wit
h(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
{TEXT -1 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 499 "There is an elementary idea in mathematics that is so im
portant that it crosses the curriculum. It is the notion of a linear s
pace. Many examples come to mind. Consider the linear space of points \+
in the plane, or the linear space of vectors at which a particular mat
rix is zero, or the linear space of continuous functions on the interv
al [0,1]. This elementary idea of a linear space will be valuable enou
gh in this collections of lectures on partial differential equations t
o say again what it is." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT 257 11 "Definition:" }{TEXT -1 3 " A " }{TEXT 256 25 "linea
r space of functions" }{TEXT -1 66 " is a collection of functions all \+
having the same domain such that" }}{PARA 0 "" 0 "" {TEXT -1 7 "(a) if
 " }{TEXT 258 1 "f" }{TEXT -1 7 "  and  " }{TEXT 259 1 "g" }{TEXT -1 
35 "  belongs to the collection, then  " }{TEXT 260 5 "f + g" }{TEXT 
-1 16 "  does also, and" }}{PARA 0 "" 0 "" {TEXT -1 7 "(b) if " }
{TEXT 261 2 "f " }{TEXT -1 25 "is in the collection and " }{TEXT 262 
1 "r" }{TEXT -1 19 " is a number then  " }{TEXT 263 3 "r f" }{TEXT -1 
29 "  is also in the collection. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 213 "In these lectures, whenever I give a def
inition, I will try to give some examples of things that satisfy the d
efinition and some things that do not. Here are some collections: some
 are linear spaces, some are not." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 12 "Collecti
ons." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {XPPEDIT 266 
0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT 267 1 ":" }{TEXT -1 1 " " }{TEXT 265 
43 "the collection of three dimensional vectors" }{TEXT 277 111 ". Thi
s is a linear space. You know how to add objects in this collection an
d how to multiple them by numbers. \n" }{TEXT 269 10 "Question: " }
{TEXT 270 87 "Suppose u = [1, 2, 3] and v = [4, 5, 6]. Suppose r is a \+
number. What is u + v and r u? " }}{SECT 0 {PARA 3 "" 0 "" {TEXT 264 
23 "The Answer Using Maple." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "u:=<1,2,3>; v:=<4,5,6>;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
32 "'u+v'=u+v; \n'r*u'=Multiply(r,u);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 274 10 "Points in " }{XPPEDIT 275 0 "R^3;" "6#*$%\"RG\"\"$" }
{TEXT 276 23 " with last component 1:" }{TEXT -1 83 " A typical vector
 in this collection is [a, b, 1], where a and b are real numbers. " }}
{PARA 0 "" 0 "" {TEXT 273 9 "Question:" }{TEXT -1 36 " Is this collect
ion a linear family?" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 272 23 "The Answ
er Using Maple." }}{PARA 0 "" 0 "" {TEXT -1 78 "Just look. The sum of \+
two vectors in this collection is not in the collection." }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "u:=<1,2,1>; v:=<4,5,1>;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "'u+v'=u+v;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "unassign('u');" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT 280 67 "C([0,1]): the collection of continuo
us functions with domain [0,1]." }{TEXT -1 48 " An example of a functi
on in this collection is " }{XPPEDIT 18 0 "x^2;" "6#*$%\"xG\"\"#" }
{TEXT -1 258 ". A function not in this collection is 1/x, for this fun
ction does not have 0 in its domain. We know that the sum of two conti
nuous functions with domain [0, 1] is continuous and has domain [0, 1]
. This is true if the function is multiplied by a number, too." }}
{PARA 0 "" 0 "" {TEXT 279 9 "Question:" }{TEXT -1 81 " Give an example
 of a function with domain [0, 1] that is not in this collection." }}
{SECT 0 {PARA 3 "" 0 "" {TEXT 278 23 "The Answer Using Maple." }}
{PARA 0 "" 0 "" {TEXT -1 77 "We have only to describe a function with \+
domain [0,1] that is not continuous." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "f:=x->signum(x-1/2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "plot(f(x),x=0..1,view=[0..1,-1..1],discont=true);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
264 "When considering a linear space, it is common to identify a finit
e collection of elements of the space having the property that every o
ther vector can be written as a finite linear combination of these. Th
at is, it is common to identify a collection of elements \{ " }
{XPPEDIT 18 0 "v[1],v[2];" "6$&%\"vG6#\"\"\"&F$6#\"\"#" }{TEXT -1 7 ",
 ..., " }{XPPEDIT 18 0 "v[n];" "6#&%\"vG6#%\"nG" }{TEXT -1 31 " \} hav
ing the property that if " }{TEXT 281 1 "u" }{TEXT -1 91 " is any vect
or in the linear family, then u can be written as a linear  combinatio
n of the " }{TEXT 282 1 "v" }{TEXT -1 11 "'s:        " }}{PARA 0 "" 0 
"" {TEXT -1 38 "                                      " }{XPPEDIT 18 
0 "u = alpha[1]*v[1]+alpha[2]*v[2];" "6#/%\"uG,&*&&%&alphaG6#\"\"\"F*&
%\"vG6#F*F*F**&&F(6#\"\"#F*&F,6#F1F*F*" }{TEXT -1 1 " " }{TEXT 283 7 "
+ ... +" }{TEXT -1 1 " " }{XPPEDIT 18 0 "alpha[n]*v[n];" "6#*&&%&alpha
G6#%\"nG\"\"\"&%\"vG6#F'F(" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT 
-1 10 "where the " }{XPPEDIT 18 0 "alpha;" "6#%&alphaG" }{TEXT -1 45 "
's are numbers. In this case we say that the " }{TEXT 288 1 "v" }
{TEXT -1 3 "'s " }{TEXT 290 4 "span" }{TEXT -1 19 " the vector space. \+
" }{TEXT 289 1 " " }{TEXT -1 7 "If the " }{TEXT 286 1 "v" }{TEXT -1 
44 "'s are linearly independent, we call them a " }{TEXT 287 5 "basis
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 285 11 "Definition:" }{TEXT -1 3 " A " }{TEXT 284 5 "basis" }
{TEXT -1 70 " for a vector space is a collection of linearly independe
nt vectors \{ " }{XPPEDIT 18 0 "v[1],v[2];" "6$&%\"vG6#\"\"\"&F$6#\"\"
#" }{TEXT -1 7 ", ..., " }{XPPEDIT 18 0 "v[n];" "6#&%\"vG6#%\"nG" }
{TEXT -1 85 " \} such that any vector in the space can be written as a
 linear combination of these." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 292 13 "Basis Vectors" }}{PARA 0 "" 0 "" {TEXT 
293 63 "The collection of vectors \{ [1, -1, 0], [1, 1, 0], [0, 0, 1] \+
\}." }{TEXT -1 51 " This collection is linearly independent and spans \+
" }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT 295 10 "Question: " }{TEXT -1 13 "Give numbers " }{TEXT 
301 1 "r" }{TEXT -1 2 ", " }{TEXT 300 1 "s" }{TEXT -1 6 ", and " }
{TEXT 299 1 "t" }{TEXT -1 12 " such that [" }{TEXT 291 7 "a, b, c" }
{TEXT -1 4 "] = " }{XPPEDIT 18 0 "r*[1, -1, 0]+s*[1, 1, 0]+t*[0, 0, 1]
;" "6#,(*&%\"rG\"\"\"7%F&,$F&!\"\"\"\"!F&F&*&%\"sGF&7%F&F&F*F&F&*&%\"t
GF&7%F*F*F&F&F&" }{TEXT -1 1 "." }}{SECT 0 {PARA 3 "" 0 "" {TEXT 294 
22 "The Answer Using Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
78 "T1:=Multiply(r, <1,-1,0>);\nT2:=Multiply(s, <1,1,0>);\nT3:=Multipl
y(t, <0,0,1>);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "rightside
:=T1+T2+T3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "solve(\{a=ri
ghtside[1],b=rightside[2],c=rightside[3]\},\{r,s,t\});" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT 296 29 "The collection of functions \{" }
{XPPEDIT 297 0 "1,x,x^2-1,x^3-x;" "6&\"\"\"%\"xG,&*$F$\"\"#F#F#!\"\",&
*$F$\"\"$F#F$F(" }{TEXT 298 2 "\}." }{TEXT -1 89 " This collection is \+
linearly independent and spans the linear space of cubic polynomials.
" }}{PARA 0 "" 0 "" {TEXT 305 9 "Question:" }{TEXT -1 14 " Give number
s " }{TEXT 304 1 "r" }{TEXT -1 2 ", " }{TEXT 303 1 "s" }{TEXT -1 2 ", \+
" }{TEXT 302 3 "t, " }{TEXT -1 3 "and" }{TEXT 307 2 " u" }{TEXT -1 11 
" such that " }{XPPEDIT 18 0 "1+x+x^2+x^3;" "6#,*\"\"\"F$%\"xGF$*$F%\"
\"#F$*$F%\"\"$F$" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "r+s*x+t*(x^2-1)+u*
(x^3-x);" "6#,*%\"rG\"\"\"*&%\"sGF%%\"xGF%F%*&%\"tGF%,&*$F(\"\"#F%F%!
\"\"F%F%*&%\"uGF%,&*$F(\"\"$F%F(F.F%F%" }{TEXT -1 1 "." }}{SECT 0 
{PARA 3 "" 0 "" {TEXT 306 22 "The Answer Using Maple" }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 32 "poly:=r+s*x+t*(x^2-1)+u*(x^3-x);" }}}
{PARA 0 "" 0 "" {TEXT -1 40 "We collect together all the powers of x.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "collect(poly,x);" }}}
{PARA 0 "" 0 "" {TEXT -1 69 "We equate coefficients for the two repres
entations of the polynomial." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
46 "sol := solve(\{u=1,t=1,s-u=1,r-t=1\},\{r,s,t,u\});" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 20 "Verify the solution." }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "eval(poly, sol);" }}}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 308 63 "The collection of vectors \{ [
1, -1, 1], [1, 1, 1], [1, 0, 1] \}." }{TEXT -1 36 " This collection is
 not a basis for " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 2 
". " }}{PARA 0 "" 0 "" {TEXT -1 68 "Question: Show this collection is \+
not a basis in two different ways." }}{SECT 0 {PARA 3 "" 0 "" {TEXT 
312 22 "The Answer Using Maple" }}{PARA 0 "" 0 "" {TEXT 310 13 "First \+
method:" }{TEXT -1 278 " we show that the vectors are not linearly ind
ependent. Most undergraduate linear algebra courses develop many ways \+
to answer this question. We simply use the definition and show the vec
tors are not linearly independent. In order to do this we ask if there
 are non-zero numbers " }{TEXT 321 5 "a, b," }{TEXT -1 5 " and " }
{TEXT 322 1 "c" }{TEXT -1 11 " such that " }}{PARA 0 "" 0 "" {TEXT -1 
24 "                        " }{TEXT 323 1 "a" }{TEXT -1 14 " [1, -1, \+
1] + " }{TEXT 324 1 "b" }{TEXT -1 13 " [1, 1, 1] + " }{TEXT 325 1 "c" 
}{TEXT -1 15 " [1, 0, 1] = 0." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "solve(\{a+b+c=0,-a+b=0,a+b+c=0\},\{a,b,c\});" }}}{PARA 0 "" 0 "
" {TEXT -1 13 "We find that " }{TEXT 326 1 "b" }{TEXT -1 50 " can be a
nything, a must have the same value, and " }{TEXT 327 1 "c" }{TEXT -1 
7 " = - 2 " }{TEXT 328 1 "b" }{TEXT -1 20 ". For example, take " }
{TEXT 329 1 "a" }{TEXT -1 6 " = 1, " }{TEXT 330 1 "b" }{TEXT -1 10 " =
 1, and " }{TEXT 331 1 "c" }{TEXT -1 6 " = -2." }}{PARA 0 "" 0 "" 
{TEXT 311 15 "Second method: " }{TEXT -1 20 "we give a vector in " }
{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }{TEXT -1 469 " that can not be \+
written as a linear combination of these three. It is not in their spa
n. There are methods developed for doing this with matrices in most un
dergraduate linear algebra courses, but it would take us astray to dev
elop these. Here, we note that each of the vectors has the first and l
ast coordinates equal. Any vector in their span must have the same pro
perty. Check this by showing that no linear combination of the vectors
 will give the vector [1, 1, 2]." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 42 "solve(\{a+b+c=1, -a+b=1, a+b+c=2\},\{a,b,c\});" }}}{PARA 0 "" 
0 "" {TEXT -1 17 "We get no answer." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}{PARA 0 "" 0 "" {TEXT -1 712 "The world that we observe is most ofte
n non-linear. To understand the phenomena that we observe in the world
, we need to have an understanding of non-linear differential equation
s  -- non-linear ordinary differential equations and non-linear partia
l differential equations. Undergraduate studies in ordinary differenti
al equations no doubt spent considerable time on the linear theory and
 some development of the non-linear theory. A good understanding of th
e linear theory opens doors into understanding the nonlinear theory. H
ere, again, we begin to try to understand the structure of linear part
ial differential equations. First, be reminded what is a linear functi
on, or a linear operator, on a linear space." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 11 "Definition:" }{TEXT -1 3 " A \+
" }{TEXT 314 15 "linear operator" }{TEXT -1 57 " L is a function with \+
domain a vector space and for which" }}{PARA 0 "" 0 "" {TEXT -1 20 "  \+
                  " }{TEXT 332 1 "L" }{TEXT -1 2 "( " }{TEXT 333 7 "a \+
x + y" }{TEXT -1 5 " ) = " }{TEXT 334 13 "a L(x) + L(y)" }}{PARA 0 "" 
0 "" {TEXT -1 16 "for all vectors " }{TEXT 335 1 "x" }{TEXT -1 5 " and
 " }{TEXT 336 1 "y" }{TEXT -1 37 " in the vector space and all numbers
 " }{TEXT 337 1 "a" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT 315 11 "Definition:" }{TEXT -1 5 " The " }
{TEXT 316 10 "null space" }{TEXT -1 22 " of a linear operator " }
{TEXT 338 1 "L" }{TEXT -1 30 " is the collection of vectors " }{TEXT 
340 1 "x" }{TEXT -1 11 " for which " }}{PARA 0 "" 0 "" {TEXT -1 24 "  \+
                      " }{TEXT 339 5 "L(x) " }{TEXT -1 4 "= 0." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 84 "If the de
terminant of a square matrix is zero, then it has a non-trivial null s
pace." }}{PARA 0 "" 0 "" {TEXT 318 9 "Question:" }{TEXT -1 73 " Evalua
te the determinant and find the nullspace of the following matrix:" }}
{PARA 0 "" 0 "" {TEXT -1 26 "                      M:= " }{XPPEDIT 18 
0 "matrix([[1, 1, 1], [-1, 1, 0], [1, 1, 1]]);" "6#-%'matrixG6#7%7%\"
\"\"F(F(7%,$F(!\"\"F(\"\"!7%F(F(F(" }{TEXT -1 2 " ." }}{SECT 0 {PARA 
3 "" 0 "" {TEXT 317 22 "The Answer Using Maple" }}{PARA 0 "" 0 "" 
{TEXT -1 29 "We find the determinant of M." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 38 "M:=Matrix([[1,1,1],[-1,1,0],[1,1,1]]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Determinant(M);" }}}{PARA 0 "" 0 "
" {TEXT -1 23 "We find the null space." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "NullSpace(M);" }}}{PARA 0 "" 0 "" {TEXT -1 84 "The re
sponse we get is that the vector [1, 1, -2] is in the nullspace. We ch
eck this" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "M.<1,1,-2>;" }}}
{PARA 0 "" 0 "" {TEXT -1 12 "Bingo! 'Tis." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 47 "Linear operators can be differential operators." }}{PARA 
0 "" 0 "" {TEXT 319 9 "Question:" }{TEXT -1 100 " Find the null space \+
for both these linear operators on the space of twice differentiable f
unctions." }}{PARA 0 "" 0 "" {TEXT -1 19 "                   " }{TEXT 
341 23 "L(y) = y '' - y ' - 2 y" }{TEXT -1 21 "         and         " 
}{TEXT 342 24 "L(y) = y '' +  y ' - 2 y" }{TEXT -1 1 "." }}{SECT 0 
{PARA 3 "" 0 "" {TEXT 320 22 "The Answer Using Maple" }}{PARA 0 "" 0 "
" {TEXT -1 192 "To answer this question, we are asked to solve two sec
ond order, constant coefficient differential equations such as we solv
ed in undergraduate differential equations, likely even in calculus." 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "dsolve(diff(y(x),x,x)-2*di
ff(y(x),x)-2*y(x)=0,y(x));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
52 "dsolve(diff(y(x),x,x)+2*diff(y(x),x)+2*y(x)=0,y(x));" }}}{PARA 0 "
" 0 "" {TEXT -1 157 "It is important to understand the answer Maple ha
s presented. In both cases, Maple has provided two functions f(x) and \+
g(x) and stated that all combinations " }{XPPEDIT 18 0 "C[1];" "6#&%\"
CG6#\"\"\"" }{TEXT -1 8 " f(x) + " }{XPPEDIT 18 0 "C[2];" "6#&%\"CG6#
\"\"#" }{TEXT -1 187 " g(x) form solutions. Humans might have said tha
t the null space for these differential operators is a two dimensional
 vector space. In fact, here is the basis for the null space of each.
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "dsolve(diff(y(x),x,x)-2*
diff(y(x),x)-2*y(x)=0,y(x),output=basis);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 65 "dsolve(diff(y(x),x,x)+2*diff(y(x),x)+2*y(x)=0,y(x),
output=basis);" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 501 "This section has served to remind you of those important
 ideas from linear algebra. These ideas are core in this course. But t
hese are chiefly algebraic ideas. In an analysis course, we need to ha
ve a sense of closeness. Thus, we next turn to the geometry of vector \+
spaces. Among other things, we define a distance. We need to have a wa
y to measure distance so that  if you are given two vectors -- or two \+
functions -- in a linear space, you may expect to be able to determine
 how far apart they are. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 59 "These geometric ideas are the subjects of the next
 lectures" }}{PARA 0 "" 0 "" {TEXT -1 1 "." }}{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" }}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 1 1 2 33 1 1 }