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{SECT 0 {PARA 0 "" 0 "" {TEXT 271 26 "Section 1.1: Linear Spaces" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 3 "" 0 "" {TEXT 309 30 "
Maple Packages for Section 1.1" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 20 "with(LinearAlgebra):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
0 "" }{TEXT -1 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 497 "There is elementary idea in mathematics that is so imp
ortant that it crosses the curriculum. It is the notion of a linear sp
ace. Many examples come to mind. Consider the linear space of points i
n the plane, or the linear spaces 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 60 " is a collection of functions all \+
having the same domain and" }}{PARA 0 "" 0 "" {TEXT -1 16 "(a) if each
of " }{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 multiply 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 1 {PARA 3 "" 0 "" {TEXT 264
23 "The Answer Using Maple." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
39 "U:=Vector([1,2,3]); V:=Vector([4,5,6]);" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 37 "'U+V'=Add(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 rea
l numbers. " }}{PARA 0 "" 0 "" {TEXT 273 9 "Question:" }{TEXT -1 36 " \+
Is this collection a linear family?" }}{SECT 1 {PARA 3 "" 0 "" {TEXT
272 23 "The Answer Using Maple." }}{PARA 0 "" 0 "" {TEXT -1 78 "Just l
ook. The sum of two vectors in this collection is not in the collectio
n." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "U:=Vector([1,2,1]); V:
=Vector([4,5,1]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "'U+V'=
Add(U,V);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 280 67 "C([0,1]): the col
lection of continuous functions with domain [0,1]." }{TEXT -1 48 " An \+
example of a function in this collection is " }{XPPEDIT 18 0 "x^2;" "6
#*$%\"xG\"\"#" }{TEXT -1 256 ". A function not in this collection is 1
/x for this function does not have 0 in its domain. We know that the s
um of two continuous functions with domain [0, 1] is continuous and ha
s domain [0, 1]. This is true if the function is multiplied by a numbe
r too." }}{PARA 0 "" 0 "" {TEXT 279 9 "Question:" }{TEXT -1 81 " Give \+
an example of a function with domain [0, 1] that is not in this collec
tion." }}{SECT 1 {PARA 3 "" 0 "" {TEXT 278 23 "The Answer Using Maple.
" }}{PARA 0 "" 0 "" {TEXT -1 77 "We have only to describe a function w
ith domain [0,1] that is not continuous." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 20 "f:=x->signum(x-1/2);" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 49 "plot(f(x),x=0..1,view=[0..1,-1..1],discont=true);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1
264 "When considering a linear space, it is common to identify a finit
e collection of elements of the space having the property that every o
ther vector can be written as a finite linear combination of these. Th
at is, it is common to identify a collection of elements \{ " }
{XPPEDIT 18 0 "v[1],v[2];" "6$&%\"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 1 {PARA 3 "" 0 "" {TEXT 294
22 "The Answer Using Maple" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
99 "T1:=Multiply(r,Vector([1,-1,0]));\nT2:=Multiply(s,Vector([1,1,0]))
;\nT3:=Multiply(t,Vector([0,0,1]));" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 30 "rightside:=Add(Add(T1,T2),T3);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 62 "solve(\{a=rightside[1],b=rightside[2],c=rights
ide[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 co
llection 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 s
pace of cubic polynomials." }}{PARA 0 "" 0 "" {TEXT 305 9 "Question:"
}{TEXT -1 14 " Give numbers " }{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 1 {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 togeth
er all the powers of x." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "c
ollect(poly,x);" }}}{PARA 0 "" 0 "" {TEXT -1 69 "We equate coefficient
s for the two representations of the polynomial." }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 39 "solve(\{u=1,t=1,s-u=1,r-t=1\},\{r,s,t,u\});" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 ""
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 308 63 "The collection of vector
s \{ [1, -1, 1], [1, 1, 1], [1, 0, 1] \}." }{TEXT -1 36 " This collect
ion is not a basis for " }{XPPEDIT 18 0 "R^3;" "6#*$%\"RG\"\"$" }
{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 68 "Question: Show this col
lection is not a basis in two different ways." }}{SECT 1 {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 li
nearly independent. Most undergraduate linear algebra courses develop \+
many ways to answer this question. We simply use the definition and sh
ow the vectors are not linearly independent. In order to do this we as
k if there are non-zero numbers " }{TEXT 321 5 "a, b," }{TEXT -1 5 " a
nd " }{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 anything, a must have the same value, and " }{TEXT 327
1 "c" }{TEXT -1 7 " = - 2 " }{TEXT 328 1 "b" }{TEXT -1 20 ". For examp
le, 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." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{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." }}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 ""
{TEXT -1 712 "The world that we observe is most often non-linear. To u
nderstand the phenomena that we observe in the world, we need to have \+
an understanding of non-linear differential equations -- non-linear o
rdinary differential equations and non-linear partial differential equ
ations. Undergraduate studies in ordinary differential equations no do
ubt spent considerable time on the linear theory and some development \+
of the non-linear theory. A good understanding of the linear theory op
ens doors into understanding the nonlinear theory. Here, again, we beg
in to try to understand the structure of linear partial differential e
quations. First, be reminded what is a linear function, or a linear op
erator, on a linear space." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0
"" 0 "" {TEXT 313 11 "Definition:" }{TEXT -1 3 " A " }{TEXT 314 15 "li
near operator" }{TEXT -1 57 " L is a function with domain a vector spa
ce 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 spa
ce" }{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 " fo
r 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 determinant of a square matrix
is zero, then it has a non-trivial null space." }}{PARA 0 "" 0 ""
{TEXT 318 9 "Question:" }{TEXT -1 73 " Evaluate the determinant and fi
nd 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 1 {PARA 3 "" 0 "" {TEXT 317 22 "The
Answer Using Maple" }}{PARA 0 "" 0 "" {TEXT -1 29 "We find the determ
inant 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 response we get is that the vector [1
, 1, -2] is in the nullspace. We check this" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 29 "Multiply(M,Vector([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 "Lin
ear operators can be differential operators." }}{PARA 0 "" 0 "" {TEXT
319 9 "Question:" }{TEXT -1 100 " Find the null space for both these l
inear operators on the space of twice differentiable functions." }}
{PARA 0 "" 0 "" {TEXT -1 19 " " }{TEXT 341 25 "L(y) \+
= y '' - 2 y ' - 2 y" }{TEXT -1 21 " and " }{TEXT 342
26 "L(y) = y '' + 2 y ' - 2 y" }{TEXT -1 1 "." }}{SECT 1 {PARA 3 ""
0 "" {TEXT 320 22 "The Answer Using Maple" }}{PARA 0 "" 0 "" {TEXT -1
196 "To answer this question, we are asked to solve two second order, \+
constant coefficient differential equations such as we solved in under
graduate differential equations, likely even in the calculus." }}
{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));" }}}{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 has \+
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#&%\"CG
6#\"\"\"" }{TEXT -1 8 " f(x) + " }{XPPEDIT 18 0 "C[2];" "6#&%\"CG6#\"
\"#" }{TEXT -1 187 " g(x) form solutions. Humans might have said that \+
the null space for these differential operators is a two dimensional v
ector 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*dif
f(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),ou
tput=basis);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 501 "This section h
as served to remind you of those important ideas from linear algebra. \+
These ideas are core in this course. But these are chiefly algebraic i
deas. In an analysis course, we need to have a sense of closeness. Thu
s, we next turn to the geometry of vector spaces. Among other things, \+
we define a distance. We need to have a way to measure distance so tha
t if you are given two vectors -- or two functions -- in a linear spa
ce, 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 ge
ometric ideas are the subjects of the next lectures" }}{PARA 0 "" 0 "
" {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 51 "EMAIL: herod@math.gate
ch.edu or jherod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 38 "URL: htt
p://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 rights reserved" }}}{MARK "0 0
" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }