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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 22 "Working with Maple V.4" }
}}{EXCHG {PARA 19 "" 0 "" {TEXT -1 61 "Version: 9 Sep 96\nŠ Esben Slot
h Andersen, esa@business.auc.dk" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 
17 "Table of contents" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 136 "Here is \+
a table of contents that - if the worksheet is copied correctly - will
 function as hyperlinks to the individual sections.\n\n     " }
{HYPERLNK 17 "Introduction to Maple" 1 "" "Introduction" }{TEXT -1 11 
"     \n     " }{HYPERLNK 17 "Maple as a calculator" 1 "" "Calculator
" }{TEXT -1 8 "  \n     " }{HYPERLNK 17 "Maple as a calculator with na
med memory cells" 1 "" "NamedCells" }{TEXT -1 8 "  \n     " }
{HYPERLNK 17 "Maple as an algebra machine" 1 "" "Algebra" }{TEXT -1 8 
"  \n     " }{HYPERLNK 17 "Maple as a graphical machine" 1 "" "Graphic
al" }{TEXT -1 8 "  \n     " }{HYPERLNK 17 "Maple as an equation solver
" 1 "" "EquationSolver" }{TEXT -1 8 "  \n     " }{HYPERLNK 17 "Some da
ta structures in Maple" 1 "" "DataStructures" }{TEXT -1 8 "  \n     " 
}{HYPERLNK 17 "Saving and reading of files" 1 "" "Files" }{TEXT -1 8 "
  \n     " }{HYPERLNK 17 "Creating and using procedures" 1 "" "Procedu
res" }{TEXT -1 8 "  \n     " }{HYPERLNK 17 "Differential equations in \+
Maple" 1 "" "DifferentialEquations" }{TEXT -1 8 "  \n     " }
{HYPERLNK 17 "Graph theory with Maple" 1 "" "GraphTheory" }{TEXT -1 8 
"  \n     " }{HYPERLNK 17 "Creating, saving and reading packages of pr
ocedures" 1 "" "Packages" }{TEXT -1 8 "  \n     " }{HYPERLNK 17 "Creat
ing and using Maple worksheets" 1 "" "Worksheets" }{TEXT -1 8 "  \n   \+
  " }{HYPERLNK 17 "References" 1 "" "References" }{TEXT -1 8 "  \n    \+
 " }{HYPERLNK 17 "Endnotes" 1 "" "Endnotes" }{TEXT -1 3 "  \n" }}}}
{SECT 0 {PARA 3 "" 0 "Introduction" {TEXT -1 21 "Introduction to Maple
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 323 "Maple is primarily a very str
ong system for doing computer algebra (or symbolic mathematics), but i
t also supports numerical analysis and graphical presentations. Furthe
rmore, Maple V.4's worksheet interface can be used for producing techn
ical documents that integrates text, mathematical symbols, graphics an
d hyperlinks. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 631 "These differen
t functions of the Maple package means that it can be compared with se
veral other products. It has some of the facilities of an equation edi
tor, but it can also solve surprisingly many types of equations. It ca
n draw a graph of a set of results, but it can also recalculate the un
derlying function or procedure with new parameters. It can be accesses
 through individual commands, but you can also create, save and reload
 procedures that gradually expands the facilities of the system. In ot
her words, Maple is a versalite system for solving problems that to so
me extent relies on mathematics and computer programming:" }}}{EXCHG 
{PARA 15 "" 0 "" {TEXT -1 106 "Maple is an interpreted (immediate-mode
) language like BASIC or LISP. You write a command, press <enter> [" }
{HYPERLNK 17 "Endnote 1" 1 "" "1." }{TEXT -1 76 "], and then Maple per
forms the required task (if it is specified correctly)." }}}{EXCHG 
{PARA 15 "" 0 "" {TEXT -1 401 "Maple is an extensible language. You ar
e, so to say, adding a new word to the vocabulary of Maple each time y
ou define a procedure. This procedure can be used in your next command
. The procedure can even be used for defining other procedures; these \+
second-level procedures can be used for defining a third layer of proc
edures, and so on. You can even define a procedure in terms of itself \+
(recursion)." }}}{EXCHG {PARA 15 "" 0 "" {TEXT -1 303 "Maple is an evo
lving program to which many mathematicians and scientists have contrib
uted. The background for this is Maple's extensibility. New packages a
re gradually being included, and better algorithms substitute old ones
. This means that newer versions of Maple is not exactly the same as o
ld one [" }{HYPERLNK 17 "Endnote 2" 1 "" "2." }{TEXT -1 2 "]." }}}
{EXCHG {PARA 15 "" 0 "" {TEXT -1 286 "Maple provides a worksheet envir
onment that integrates commands to Maple's mathematical engine with th
e results (text or graphics) and with your comments to both the comman
ds and the output. The present document is an example of how to use Ma
ple worksheets (read more in the section on " }{HYPERLNK 17 "Worksheet
s" 1 "" "Worksheets" }{TEXT -1 2 ")." }}}{EXCHG {PARA 15 "" 0 "" 
{TEXT -1 369 "Maple is allows symbolic mathematics. It is built to wor
k with algebraic expressions, and this is not only an advantage for de
aling with mathematical expressions. Even when you program, you often \+
want to postpone the precise specification of a part of the program. M
aple helps by treating many undefined elements of the program as symbo
ls that may later obtain a value." }}}{EXCHG {PARA 15 "" 0 "" {TEXT 
-1 150 "Maple provides detailed facilities for manipulating graphics (
although you should turn to e.g. Excel if you want to produce piechart
s and histograms)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 639 "There is a
nother mathematics package that works more or less like Maple (Wolfram
's Mathematica). However, when you work in an environment (like Aalbor
g University) where a Maple V.4 site licence allows everyone to use th
e program freely, there is little choice. Furthermore, mathematicians \+
seem to prefer Maple because they can inspect and modify nearly all of
 Maple's procedures. This means that the Maple package is not only the
 result of its originator: the Symbolic Computation Group at the Unive
rsity of Waterloo, Canada. Especially Maple's very large library of pr
ocedures and functions is also the outcome of a collective endeavour.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 566 "The choice of Maple is in no
 way based in its simple computational efficiency. Maple simulation mo
dels with many numerical calculations runs at least 50 times slower th
an similar models programmed in C. However, it should be noted that ma
ny more problems and much more time relate to the understanding and re
vision of programs than to running them on a computer. Furthermore, wh
en a prototype is running without bugs and in an well-defined manner, \+
then experts can translate it to e.g. C.  The transformation to the ob
ject-oriented paradigm of C++ is more difficult." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 398 "With respect to economics teaching and the develo
pment of complex model prototypes Maple appears to be a good choice. B
ut there are significant barrier to enter the Maple community. The pre
sent worksheets try to demonstrate the increased power obtainable thro
ugh experimental mathematics and computer modelling. Thus they indicat
e that the effort needed to overcome the obstacles is well justified.
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 226 " The present worksheet gives
 the necessary acquaintance with Maple needeed for using and (slightly
) modifying a set of worksheets on evolutionary modelling for economis
ts. The topics covers are listed in the table of contents." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 229 "NOTE: As always in Maple worksheets, you
 should make sure that the cursor is placed in the (red) Maple input (
in the next section). When you press <enter>, Maple will perform the r
equired task (if it setup is made correctly, see " }{HYPERLNK 17 "Endn
ote 1" 1 "" "1." }{TEXT -1 223 "). Then there will normally be printed
 (blue) Maple output, and the cursor will jump to the next Maple input
. Thus you can again press <enter> when you have tried to figure out w
hat has happened and what is about to happen." }}}}{SECT 0 {PARA 3 "" 
0 "Calculator" {TEXT -1 21 "Maple as a calculator" }}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 34 "We begin our Maple session with a " }{XPPEDIT 18 
0 "restart" "I(restartG6\"" }{TEXT -1 157 " which means that all exist
ing results in Maple's memory are deleted (this is important to avoid \+
potential confusion if you have already made some commands)." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 12 "If we write " }{XPPEDIT 18 0 "5+5" ",&\"\"&\"\"\"F
#F$" }{TEXT -1 307 " in an input field (e.g. marked with red text) and
 press <enter>, Maple consider it as an incomplete command (to add 5 t
o 5 and display the result). The command is incomplete because Maple e
xecutes statements or commands that must end with a semicolon. The sta
ment can run over several lines (see, however, " }{HYPERLNK 17 "Endnot
e 1" 1 "" "1." }{TEXT -1 4 ").  " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "5 + 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "5 \+
- 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "5*5;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "5^5;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "3125/5" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 201 "
Oops! We forgot a semicolon. Put the cursor after 5, then type semicol
on and press <enter>. Now the cursor should have jumped to the next in
put field, and you can continue reading and pressing <enter>." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 320 "Maple is designed by mathematicia
ns who do not like to be imprecise. A rational number is kept like tha
t unless the user asks the program to treat it (evaluate it) as an (im
precise) floating-point number with decimals. The user can specify the
 number of actual digits of the floating-point number (default is 10 d
igits)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "5/10;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "5/8;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 12 "evalf(5/10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "evalf(5/10,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "eva
lf(5/8,2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(25/8,2)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "evalf(1111111/10,2); #
 very imprecise!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "27^(1/2)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(27^(1/2),4);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "5!;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 4 "25!;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 
"125!;" }}}}{SECT 0 {PARA 3 "" 0 "NamedCells" {TEXT -1 45 "Maple as a \+
calculator with named memory cells" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
135 "To simplify the treatment of complicated numerical expressions we
 create short names (abstractions) for their compoments. For instance:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "a1 := 5;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a2 := 5^5;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 10 "a3 := 10!;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "a4 := a3/(a1*a2);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "a5 := ln(a4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "evalf(a5,3);" }}}}{SECT 0 {PARA 3 "" 0 "Algebra" {TEXT -1 27 "
Maple as an algebra machine" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 122 "Ma
ple is not primarily designed for numerical calculations. Instead it h
as the ability of treating symbols systematically." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 6 "a + b;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "2*x^2 - 24*x - 5;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "x^2 - 2*x + 5;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
112 "In Maple we can eaily create short names (abstractions) for such \+
functions. This can be done in arrow notation. " }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 33 "c_arrow := x -> 2*x^2 - 24*x - 5;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "c_arrow(5);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 139 "We shall not use the arrow notation much in these w
orksheets (it is only for simple cases). Instead we shall give a funct
ion a simple name:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "c := \+
2*x^2 - 24*x - 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "d := x
^2 - 4*x + 3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "e := (x*d)
^(1/2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "When we know the numer
ical value of " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 109 ", we also
 know the numerical value of the expressions. For instance, we look fo
r the values of the functions " }{XPPEDIT 18 0 "c" "I\"cG6\"" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "d" "I\"dG6\"" }{TEXT -1 5 " for " }
{XPPEDIT 18 0 "x=5" "/%\"xG\"\"&" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 7 "x := 5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 2 "c;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "d;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 61 "The problem with Maple is that the value \+
of a variable (e.g. " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 151 ") i
s remembered. So if we move the cursor backwards in the worksheet, the
n the output is different compared with the situation before the assig
nmenmt (" }{XPPEDIT 18 0 "x := 5" ">%\"xG\"\"&" }{TEXT -1 21 "). We ha
ve to make a " }{XPPEDIT 18 0 "restart" "I(restartG6\"" }{TEXT -1 89 "
 (as we did in the beginning of this worksheet) or we can make an expl
icit unassignment (" }{XPPEDIT 18 0 "x := 'x'" ">%\"xG.F#" }{TEXT -1 
2 ")." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "c;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "x := 'x';" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 2 "c;" }}}}{SECT 0 {PARA 3 "" 0 "Graphical" {TEXT -1 
28 "Maple as a graphical machine" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
61 "We can also get a plot of the functions we have just defined." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "plot(c, x=-10..30, title = `
Plot of c(x)`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "plot(d,x
=-3..10, title = `Plot of d(x) = x^2 - 4*x + 3`);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 55 "plot([c,d], x=-5..30, title = `Plot of c(x) \+
and d(x)`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 105 "We can also plot \+
series of related functions. Let us consider a simple Cobb-Douglas pro
duction function (" }{XPPEDIT 18 0 "Q=A*K^a*L^(1-a)" "/%\"QG*(%\"AG\"
\"\")%\"KG%\"aGF&)%\"LG,&F&F&F)!\"\"F&" }{TEXT -1 250 "). First we mak
e a three-dimentional plot of for a given A and a given a. The specifi
cation of the Cobb-Douglas function is easy. However, the plot definit
ion is rather complex, so do not think too much about it. Instead you \+
can consider the outcome." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "Q := A*K^a*L^(1-a);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 136 
"A :=1; a:=.5; plot3d(Q,K=0..10,L=0..10, axes=BOXED, orientation=[-120
,40], style=PATCHCONTOUR, colour=Q, title=`Cobb-Douglas plot # 1`);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "If we project the contour lines
 of the \"production mountain\" into the K-L plane, we have the normal
 way of depicting a Cobb-Douglas function. By changing the values for \+
" }{XPPEDIT 18 0 "A" "I\"AG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "a
" "I\"aG6\"" }{TEXT -1 122 " in the above command (red text) and press
ing <enter>, we can consider the consequences for the form of the func
tion plot." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 406 "It is possible to \+
look at the production mountain from different angles. To do so, selec
t the plot, keep pressing the mouse while over one of the corners of t
he box, and rotate the box with the mouse pressed. Finally, you have t
o select the button: \"redraw the plot\". This reorientation of the pl
ot can be done in a way so that we obtain a normal two-dimentional plo
t of isoquants, i.e. curves defined by  " }{XPPEDIT 18 0 "Q(K,L)=Q[0]
" "/-%\"QG6$%\"KG%\"LG&F$6#\"\"!" }{TEXT -1 21 ". Let us have a look:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 133 "A :=1; a:=.5; plot3d(Q
,K=0..10,L=0..10, axes=BOX, orientation=[-90,0], style=PATCHCONTOUR, c
olour=Q, \ntitle=`Cobb-Douglas plot # 2`);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 208 "To check that we still have a three-dimentional picture,
 you can try to move the box around the figure. But we may also look f
or a more standard presentation of the isoquants. To create some varia
tion we set " }{XPPEDIT 18 0 "a=.3" "/%\"aG$\"\"$!\"\"" }{TEXT -1 1 ".
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 134 "A :=1; a:=.3; plot3d(Q
,K=0..10,L=0..10, axes=NORMAL, orientation=[-90,0], style=CONTOUR, col
our=black, title=`Cobb-Douglas plot # 3`);" }}}}{SECT 0 {PARA 3 "" 0 "
EquationSolver" {TEXT -1 27 "Maple as an equation solver" }}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 65 "If we look a little back and consider the
 plots of the functions " }{XPPEDIT 18 0 "c(x)" "-%\"cG6#%\"xG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "d(x)" "-%\"dG6#%\"xG" }{TEXT -1 
187 ", we can approximately see the solutions to the functions. Howeve
r, we can also solve them explicitly. The solutions are listed after e
ach other. In the first case (with two solutions to " }{XPPEDIT 18 0 "
c" "I\"cG6\"" }{TEXT -1 162 ") the solutions are difficult to interpre
t in relation to the graph (because Maple upholds the procese solution
s). Therefore, we ask for a floating point version." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "c1 := solve(c = 0, x);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 12 "evalf(c1,4);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "d1 := solve(d = 0, x);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 75 "We can also find precisely where the two functions are eq
ual to each other:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "cd1 :
= solve(c = d, x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "evalf
(cd1,4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 238 "Maple's equation-sol
ving facilities are also relevant to the analysis of Cobb-Douglas func
tions. We shall only consider the standard issue of deriving marginal \+
products of capital and labour from the already defined Cobb-Douglas f
unction " }{XPPEDIT 18 0 "Q" "I\"QG6\"" }{TEXT -1 63 ". To obtain the \+
general solutions we have to remember to reset " }{XPPEDIT 18 0 "A" "I
\"AG6\"" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 19 "
, and the function " }{XPPEDIT 18 0 "Q" "I\"QG6\"" }{TEXT -1 1 "." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "A := 'A': a := 'a': Q := 'Q'
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Q := A*K^a*L^(1-a);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "MP[K] := diff(Q,K);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "MP[L] := diff(Q,L);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Before we can consider other facil
ities of equation solving (like " }{HYPERLNK 17 "differential equation
s" 1 "" "DifferentialEquations" }{TEXT -1 49 "), we shall consider som
e other aspects of Maple." }}}}{SECT 0 {PARA 3 "" 0 "DataStructures" 
{TEXT -1 29 "Some data structures in Maple" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 333 "To get Maple to perform more complex tasks, we need data
 structures. Let us start with ordered sequences and lists and unorder
ed sets. Remark that Maple distinguishes between small letters and cap
ital letters. Remark also that variables do not have to be declared be
fore they are used (in contrast to e.g. Pascal and C++). Variables " }
{TEXT 261 3 "can" }{TEXT -1 108 " be declared explicitly,but they can \+
also be implicitly declared by their first use (like in BASIC or LISP)
." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "Sequence := a,b,1,2;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Sequence[1];" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "Sequence[1..3];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "list := [a,b,c,1,2,3,a,2];" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 117 "Oops! We used a name that is reserved for Maple expres
sions. However, Maple will accept the name with a small change." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "List := [a,b,c,1,2,3,a,2];" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "List[1];" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "List[2..5];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "List[8];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 
"nops(List);   # Number of list elements" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "Set1 := \{a,b,d,1,2,3,a,2\};" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 20 "Set2 := \{a,b,c,d,e\};" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 39 "Set1 union Set2;  # Set theoretic union" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Set1 minus Set2;  # Set theo
retic difference" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 248 "We now turn \+
to the potentially very complex tables. We start by considering the in
clusion of simple elements and retrival of their values. To make thing
s somewhat clearer we start by declaring the table (even though this i
s not necessary in Maple)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
41 "Table := table();   # We create the table" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 35 "Table[1] := 1;   # We store a value" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Table[2] := 4;   # We store another
 value" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Table[2];   # We \+
retrieve a value" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Now we fill o
ut the table. It is done by a loop that runs from 3 to 10. Remember th
e " }{XPPEDIT 18 0 "do" "?(6\"\"\"\"F$F#%%trueGF#" }{TEXT -1 37 " pare
nthesis around the statement(s)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "for i from 3 to 10 do\nTable[i] := i^2;\nod;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "Table[9];" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 42 "for i from 1 to 10 do\nprint(Table[i]);\nod
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 100 "The table we have under con
struction can also include quite different kinds of indexes and elemen
ts:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Table[1,1] := Yawn;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Table[1,1];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "Table[Nelwin,2,8] := Interesting;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Table[1,1] <> Table[Nelwi
n,2,8];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "print(Table);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 111 "If we try to access an element \+
of the table that has not yet been created, we receive an echo of our \+
statement:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Table[Nelwin,
1,1];" }}}}{SECT 0 {PARA 3 "" 0 "Files" {TEXT -1 24 "Saving and readin
g files" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 309 "File names depend on t
he individual operation systems as does the directory in which user fi
les should be placed. Since the present worksheet is intended to be us
ed on different platforms, we shall only say little about files. Howev
er, here are a few ideas (others are mentioned in the section on proce
dures)." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "One important task is
 to save your results. Maybe a complex computation have created a huge
 number of elements in " }{XPPEDIT 18 0 "Table" "I&TableG6\"" }{TEXT 
-1 270 ". We have found out what is the pathname and filename. Now we \+
want to save the table. (To avoid the wrong statements to be performed
, all statements are written as \"comments\" that are ignored by Maple
. To perform some of them, we have to remove the comment markers (#).)
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "To save the file on the hardd
isk of a Power Macintosh we e.g. write:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "# save Table,\n# `MacintoshHD:MapleVR4:Modelling:Tabl
e.m`;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Later we can retrieve th
e " }{XPPEDIT 18 0 "Table" "I&TableG6\"" }{TEXT -1 21 " and its elemen
ts by:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "# read\n# `Macint
oshHD:MapleVR4:Modelling:Table.m`;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "# Table[9];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "A
lternatively we can use a Windows system and save on a discette by:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "# save Table, `xxxxxxx:Tab
le.m`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "# read `xxxxxxx:T
able.m`;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "# Table[9];" }}
}}{SECT 0 {PARA 3 "" 0 "Procedures" {TEXT -1 29 "Creating and using pr
ocedures" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 180 "Maple statements can \+
be collected into procedures. Such procedures are first defined. Later
 they can be called whenever it is relevant. We shall start with a ver
y simple procedure." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "hell
o := proc()\nprint(`Important information on Maple procedures.`);\nend
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "hello();" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 451 "We see that after the procedure has been
 defined, it can be called by its name followed by a pair of parenthes
es. We also see that Maple echo our definition of the procedure with t
he formatted procedure text. To avoid this echo we shall in the follow
ing finish definitions with a colon instead of a semicolon. The proced
ure can easily be extended, especially if we give it an input (an argu
ment). We also specify the data type of acceptable inputs (a " }
{XPPEDIT 18 0 "name" "I%nameG%*protectedG" }{TEXT -1 49 " as it is def
ined in Maple, e.g. with no blanks)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 159 "hello := proc(onename::name)\n\n# This procedure req
uires the input of a\n# single name\n\nprint(onename,`here is importan
t information on Maple procedures.`);\nend:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 13 "hello(Esben);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "hello(117);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 275 "
Let us now make a couple of somewhat more complex procedures. First, w
e make a procedure that takes the two functions we have earlier plotte
d. For any argument the procedure selects the value of the function th
at is largest. The procedure introduces the selection statement (" }
{XPPEDIT 18 0 "if c>d then  else" "@%2%\"dG%\"cG6\"F&" }{TEXT -1 2 ").
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "biggest := proc(x)\nlo
cal c,d;\nc := 2*x^2 - 24*x - 5;\nd := x^2 - 4*x + 3;\nif d < c then c
;\nelse d;\nfi;\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 393 "Oops! W
e forgot that Maple distinguishes between global variables and variabl
es that are local to a procedure. No variable in the procedure is supp
osed to be globally available, so we can just accept Maple's warning. \+
However, it is useful to be explicit about the availability of variabl
es (we might avoid much confusion later). Therefore, we rewrite the pr
ocedure (including a little layout)." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 109 "biggest := proc(x)\nlocal c,d;\n\nc := 2*x^2 - 24*x \+
- 5;\nd := x^2 - 4*x + 3;\n\nif d < c then c;\nelse d;\nfi;\n\nend:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "biggest(0);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "biggest(5);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 259 "We now write a plotting procedure. It takes as inpu
t a function of one variable and a range for plotting. It prints a plo
t of the function in the given range, together with a title of the plo
t. In the procedure we include type checking of the second argument." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 409 "biggestplot := proc(inte
rval::range)\n\nlocal header,start,finish,plotpoints, x;\n\n# This fun
ction takes two inputs:\n# a name of a function with one argument\n# a
nd a plot interval (like 25..100).\n# Outputs is a graph of the functi
on.\n\nheader := cat(`Plot of \"biggest\"`);\nstart := lhs(interval);
\nfinish := rhs(interval);\nplotpoints := seq([x,biggest(x)],x=start..
finish);\nplot([plotpoints], title = header);\n\nend:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "biggestplot(-3..24);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 23 "We shall also define a " }{XPPEDIT 18 0 "
CoDo" "I%CoDoG6\"" }{TEXT -1 58 " procedure for plotting Cobb-Douglas \+
production functions:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 289 "C
oDo := proc(A,a)\nlocal Q;\n\n# This procedure takes two arguments:\n#
 A and a from the function A*K^a*L^(1-a)\n\nQ := A*K^a*L^(1-a);\n\npri
nt(`A is`,A, `  a is`, a);\nplot3d(Q,K=0..10,L=0..10, axes=BOXED, orie
ntation=[-120,40], style=PATCHCONTOUR, colour=Q, title=cat(`Cobb-Dougl
as plot`));\n\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "CoDo
(1,.3);" }}}}{SECT 0 {PARA 3 "" 0 "DifferentialEquations" {TEXT -1 33 
"Differential equations with Maple" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
53 "Let us take a simple differential equation like\n     " }{XPPEDIT 
18 0 "diff(y,t) = -2+t+y" "/-%%diffG6$%\"yG%\"tG,(\"\"#!\"\"F'\"\"\"F&
F+" }{TEXT -1 35 "\nWe start by writing this equation:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "deq := diff(y(t),t) = - 2 + t - y(t
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 144 "We consider some graphical
 and analytical tools for treating exploring this equation. The graphi
cal tools are provided by the following command." }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 14 "with(DEtools):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 99 "Given these tools we can make a plot indicating the direc
tion field for the differential equation. " }{XPPEDIT 18 0 "t" "I\"tG6
\"" }{TEXT -1 40 " is measured at the horizontal axis and " }{XPPEDIT 
18 0 "y(t)" "-%\"yG6#%\"tG" }{TEXT -1 22 " at the vertical axis." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "DEplot(deq,[y(t)], t=0..4, y
=-1..1, \ntitle = `Direction field for deq`);" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 284 "Let us now consider how a trajectory in the direction
 field will look like. Let us start at the origin (0,0). The arrows in
dicates that the movement will first be towards south-east, then east \+
and finally north-east. That this interpretation is correct can be see
n from the next plot." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "DE
plot(deq,[y(t)], t=0..4, [[0,0]], y=-1..1,\ntitle=`Trajectory of deq, \+
init=(0,0)`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "We can also cons
ider a family of trajectories by giving Maple a list of initial points
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 135 "DEplot(deq,[y(t)], t=
0..4,\n[[0,-1],[0,-.5],[0,0],[0,.5],[0,1]],\ny=-1.5..1, linecolour=t, \+
dirgrid=[15,15],\ntitle=`Trajectories of deq`);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 108 "We now turn to the analytical solution of our diffe
rential equation. Here we have a standard Maple command (" }{XPPEDIT 
18 0 "dsolve" "I'dsolveG%*protectedG" }{TEXT -1 79 "). First we find t
he general solution (with the arbitrary initial value called " }
{XPPEDIT 18 0 "_Cl" "I$_ClG6\"" }{TEXT -1 58 ") which is placed after \+
the expression that it multiplies:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "dsolve(deq, y(t));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 59 "Given an initial value, we can find a specific solution to " }
{XPPEDIT 18 0 "deq" "I$deqG6\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 39 "spec_sol := dsolve([deq,y(0)=0], y(t));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "This specific solution is itself a
n equation that gives the value of " }{XPPEDIT 18 0 "y(t)" "-%\"yG6#%
\"tG" }{TEXT -1 18 " for any value of " }{XPPEDIT 18 0 "t" "I\"tG6\"" 
}{TEXT -1 201 ". We transform the solution to a Maple procedure and ck
ecks it with a couple of arguments. The procedure is a bit tricky, so \+
we better forget its details at the moment. The result is, however, co
rrect." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "sdeq := proc(tim
e)\nlocal f,result,floatresult;\nf := rhs(spec_sol);\nresult := subs(t
=time,f);\nfloatresult := evalf(result,2);\nend:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 8 "sdeq(0);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "sdeq(1); sdeq(2); sdeq(3); sdeq(4);" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 104 "We end by plotting the solution function for t
he initial value (0,0) without the direction field arrows:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot(sdeq(t), t=0..5, y=-1..2,\ntit
le=`Solution curve, init=(0,0)`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
504 "This figure is based on the analycal solution of our differential
 equation. Thus it is precise (except for the floating-point evaluatio
n which can howver, be improved tas far as our comput resources permit
). On the other hand, the same curve in the earlier graphs was produce
d by means of a numerical calculations applied directly to the differe
ntial equation before we knew the analytical solution. However, many d
ifferential equations have no analytical solutions. For them we need c
omputer simulation." }}}}{SECT 0 {PARA 3 "" 0 "GraphTheory" {TEXT -1 
23 "Graph theory with Maple" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 225 "We
 now move from continuous to discrete mathematics. The example is grap
h theory. For instance, we want to explore tools for a formal represen
tation of the structure and evolution of production trees and industri
al networks. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "When dealing wit
h graphs, we start from a set of " }{XPPEDIT 18 0 "N" "I\"NG6\"" }
{TEXT -1 48 " different elements called nodes (or vertices), " }
{XPPEDIT 18 0 "V" "I\"VG6\"" }{TEXT -1 42 ". To define a graph we also
 need a set of " }{XPPEDIT 18 0 "K" "I\"KG6\"" }{TEXT -1 48 " differen
t elements called branches (or edges), " }{XPPEDIT 18 0 "E" "I\"EG6\"
" }{TEXT -1 117 ". A branch is a connection between pairs of nodes. An
 undirected branch is characterised by a set of two nodes, e.g. " }
{XPPEDIT 18 0 "\{a,b\}" "<$%\"aG%\"bG" }{TEXT -1 77 ". A directed bran
ch (or arrow) is characterised by a list of two nodes, e.g. " }
{XPPEDIT 18 0 "[a,b]" "7$%\"aG%\"bG" }{TEXT -1 83 "; the directed bran
ch goes from the first to the second node of the list. A graph, " }
{XPPEDIT 18 0 "G" "I\"GG6\"" }{TEXT -1 69 ", is a list consisting of a
 set of nodes and a set of branches, i.e. " }{XPPEDIT 18 0 "G=[V,E]" "
/%\"GG7$%\"VG%\"EG" }{TEXT -1 312 ". A graph can be created by two typ
es of assignments: one type (new) gives an empty structure; another ty
pe (graph) gives a graph with initial nodes and branches. When a graph
 has been created, it can be modified by adding or deleting nodes and \+
branches. Furthermore, weights can be given to nodes and branches. " }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "To handle graphs we start by loa
ding links to the package " }{XPPEDIT 18 0 "networks" "I)networksG%*pr
otectedG" }{TEXT -1 203 ". We start by creating a tree with directed b
ranches. (When ordering the networks facilities and creating graphs, r
emember to end the commands with a colon - otherwise you will see too \+
much information.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(
networks):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "V1 := \{a,b,c
,d\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "E1 := \{[a,b], [a,
c]\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "G1 := graph(V1, E1
):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "draw(Linear([d,a],[c,
b]), G1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 257 "The drawing shows a
 representation of the graph with 4 nodes and 2 branches. The graph ca
n also be described in other ways. One method is the adjacency table t
hat shows the possibility of coming from one node through another by m
eans of a (directed) branch." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "adjacency(G1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 300 "What we \+
have drawn is an example of a tree. A tree is a graph that is connecte
d and has no circuit. Trees are created, modified and inspected as gra
phs. The drawing also shows an oriented tree. An orientation of a tree
 is obtained by making one of the nodes the root node of the tree (in \+
the crawing " }{XPPEDIT 18 0 "root=a" "/%%rootG%\"aG" }{TEXT -1 102 ")
. Thus the rooted tree is defined by nodes, branches and the root. Let
 us look at another tree (with " }{XPPEDIT 18 0 "root=0" "/%%rootG\"\"
!" }{TEXT -1 2 ")." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "V2 :=
 \{0,1,2,3,4,5,6,7,8\};\011" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
81 "E2 := \{[0,1],[1,2],[1,3],[2,4], \013                       [2,5],
[3,6],[4,7],[4,8]\};" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "G2 \+
:= graph(V2, E2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "draw(L
inear([0],[1],[3,2],[6,5,4],[8,7]), G2);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 335 "Another type of graph is created by gradually adding ext
ra branches between randomly chosen nodes. Thereby we create an evolvi
ng grahp with interesting properties that e.g. have been studied by Ar
tificial Life research. In the following we shall study random graphs \+
with 15 nodes. We start by constructing a procedure for this purpose.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 383 "RandGraph :=proc(noden
um,branchprob)\nlocal Rgraph,Edges,Components;\n\n# This procedure req
uires the number of\n# edges and the probability of a branch.\n# It de
livers a plot and informative text.\n\nRgraph := random(nodenum,prob=b
ranchprob);\nEdges := nops(edges(Rgraph));\nComponents := nops(compone
nts(Rgraph));\n\nprint(`K/N `,Edges/nodenum,` Components `,Components)
; \ndraw(Rgraph);\n\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The \+
procedure will give information about the " }{XPPEDIT 18 0 "K/N" "*&%
\"KG\"\"\"%\"NG!\"\"" }{TEXT -1 405 " relationship as well as the numb
er of individual components (single nodes or interconnected nodes) in \+
the graph. Unless you specify the \"seed\" of the ransom-number genera
tor, the a call to the procedure will normally give very different res
ults. It is the statistical properties of the results that is of inter
est. But at present we can only experiment with repeated calls. This r
equires a lot of plots [" }{HYPERLNK 17 "Endnote 3" 1 "" "3." }{TEXT 
-1 2 "]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RandGraph(15,0.
1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RandGraph(15,0.1);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RandGraph(15,0.2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RandGraph(15,0.3);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RandGraph(15,0.4);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 371 "One of the issues of the theory o
f random graphs is at wheich parameter value there is a very rapid inc
rease on the probability of obtaining a single integrated component. I
n other words, we want to know how few branches are needed before we c
an be almost sure that there is only a single component. The other cas
e where all possible connections are drawn is much easier:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "RandGraph(15,1);" }}}}{SECT 0 
{PARA 3 "" 0 "Packages" {TEXT -1 51 "Creating, saving and reading pack
ages of procedures" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 169 "When we cre
ate complex models with a large number of procedures, we may easily fo
rget to load one of the procedures into Maple. This problem can be sol
ved by creating a " }{XPPEDIT 18 0 "helper" "I'helperG6\"" }{TEXT -1 
154 " package of procedures that can be saved as a Maple-format file (
an *.m file). This file can then be loaded as a whole when we start ou
r simulation work. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 214 "hel
per[calculator] := proc()\nprint(`Maple can be used as a calculator.`)
;\nprint(`Maple does not by itself transform to decimal numbers.`);\np
rint(`Write e.g. evalf(5/8, 4); - where 4 is the number of digits.`);
\nend:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 262 "helper[variables
] := proc()\nprint(`A variable has a name that starts with a letter.`)
;\nprint(`The rest of the name can be either letters or numbers.`);\np
rint(`No spaces and special characters are allowed.`);\nprint(`Undersc
ores are allowed, e.g. Table_27.`);\nend:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 215 "To define a place for the package at the harddisk, we us
e a hack that is a little complex and will not necessarily function (i
t depends on the setup of Maple). [Remome the #'es if you dare to try \+
out the procedure.]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "# fi
lepath := cat(substring(\n#     libname[nops([libname])],1..-4),\n#   \+
  `Helper.m`);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "# save he
lper, filepath;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 142 "We have now s
aved the package. To be sure that it has disappeared, we restart. Then
 we look for the package, load it, and (hopefully) find it." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "# restart;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 14 "# variables();" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 86 "# filepath := cat(substring(\n#     libname[nops([
libname])],1..-4),\n#     `Helper.m`);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "# read filepath;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "# with(helper);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "# calculator();" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "# variables();" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
305 "These helper texts are very limited. However, Maple has a very ex
tensive help system. Either we can browse through the help pages or we
 can go for a specific topic. Concerning the naming of variables we wr
ite \"?name\", press <enter>, read the text, close the help window, an
d return in our main workshweet." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "?name;" }}}}{SECT 0 {PARA 3 "" 0 "Worksheets" {TEXT 
-1 35 "Creating and using Maple worksheets" }}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 21 "The worksheet concept" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
341 "Through the experiences relating to the present worksheet much kn
owledge about Maple worksheets has already been obtained. Here is a sl
ightly more formal exposition of Maple worksheets. This exposition may
 help you to make your own worksheets (but the exposition does not cov
er all the necessary topics (see e.g. Heal et al., 1996, 12-20).  " }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 373 "A new worksheet looks like a fil
e for a word processor program (like Word or WordPerfect). However, wh
en you look at the present worksheet, it is obvious that text processi
ng is only one aspect of a worksheet. Text processing takes place in T
ext Cells while the writing of input intended for Maple takes place in
 Input Cells. Finally Maple answers by creating Output Cells." }}}}
{SECT 0 {PARA 4 "" 0 "" {TEXT -1 10 "Text Cells" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 377 "Text Cells are not the default for Maple. That the \+
worksheet has not started with a Text Cell is indicated by the Maple i
nput prompt in the beginning of the cell (> ). You should not try to d
elete this input prompt. Instead you select the menu <intert/text inpu
t> or type <command+t>. Now you can write ordinary text. Be aware that
 Maple can be a bit sluggish for good typists." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 312 "If you are using the same setup as the present auth
or, then a <return> will only mean <newline>. Therefore, you have to a
sk explicitly for a new Cell. This is done by <command+j> which insert
s a new Cell after the Cell in which the cursor is placed. The new cel
l is an Input Cell so you change it by <command+t>." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 325 "You can format individual words or the whole c
ell (by Maple's styles). You can also insert mathematical expressions \+
into the text. The securest way seems to continue a couple of characte
rs beyond the place where the mathematical expression should be insert
ed, then place the cursor at the precise place and finally click the \+
" }{XPPEDIT 18 0 "Sigma" "I&SigmaG6\"" }{TEXT -1 220 " in the tools li
ne of Maple. Now a questionmark is placed in the text and the cursor i
s ready in a special Excel-like entry line. Here you write a valid Map
le expression and the formatted result is inserted in your text." }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 230 "You may rather easily change the \+
mathematics. Place the cursor in the math text; then place the cursor \+
at the relevant place in the entry line; finally, do the necessary rev
isions. Let us have a try. In an equation we want to use " }{XPPEDIT 
18 0 "alpha" "I&alphaG6\"" }{TEXT -1 21 " (alpha) instead of  " }
{XPPEDIT 18 0 "a" "I\"aG6\"" }{TEXT -1 24 ". Here is the equation: " }
{XPPEDIT 18 0 "Q=A*K^a*L^(1-a)" "/%\"QG*(%\"AG\"\"\")%\"KG%\"aGF&)%\"L
G,&F&F&F)!\"\"F&" }{TEXT -1 280 ". To to change it, mark the expressio
n by clicking the cursor in the middle of it [do it now]. Then place t
he cursor just after the \"a\" in the entry line. Write \"alpha\" inst
ead. Now a Greek alpha is shown in the text. Do similarly with the sec
ond \"a\". Now the equation should be: " }{XPPEDIT 18 0 "Q=A*K^alpha*L
^(1-alpha)" "/%\"QG*(%\"AG\"\"\")%\"KG%&alphaGF&)%\"LG,&F&F&F)!\"\"F&
" }{TEXT -1 1 "." }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 11 "Input Cells
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "As mentioned, and Input Cell i
s marked with an input prompt. This is an empty Input Cell:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 255 "Input Cells are used for writing \"living\" Maple expres
sions. When we press <enter> the input is given to Maple's engine. How
ever, Maple can only interpret correct statements. If the statement is
 not correct, we get often get an error message. Here is one:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "a^(1/2;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 232 "Oops! We forgot the left parenthesis. Sometimes t
hings are more difficult. We might even commit a semantic error that i
s not recognised by Maple. It will probably be revealed as we go along
 with our computations. Here is an example:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 21 "restart; Q := A(K*L);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 125 "This looks nice, but it is probably incorrect. Maple con
siders the right-hand expression as a call to the undefined function \+
" }{XPPEDIT 18 0 "A" "I\"AG6\"" }{TEXT -1 19 " with the argument " }
{XPPEDIT 18 0 "K" "I\"KG6\"" }{TEXT -1 7 " times " }{XPPEDIT 18 0 "L" 
"I\"LG6\"" }{TEXT -1 41 ". Probably the intension was to multiply " }
{XPPEDIT 18 0 "A" "I\"AG6\"" }{TEXT -1 19 " by the product of " }
{XPPEDIT 18 0 "K" "I\"KG6\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "L" "I
\"LG6\"" }{TEXT -1 71 ". However the latter expression should be writt
en in the following way:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "
A*(K*L);" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 "Output Cells" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 239 "Output Cells are not produced by \+
the user by by Maple inself. It includes text formatted as you require
 (e.g. simple text or formatted text). You can copy this text and past
e it into your word processor. The result may be unformatted text." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "Output Cells may also contain Ma
ple plots (if you ask Maple to put plots within the worksheet instead \+
of in separate windows). Even plots can be selected, copied and pasted
 into your word processor." }}}}}{SECT 0 {PARA 3 "" 0 "References" 
{TEXT -1 10 "References" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Anderse
n, E.S. (1996), " }{TEXT 256 56 "Evolutionary Economics: Post-Schumpet
erian Contributions" }{TEXT -1 33 ", paperback edn., Pinter, London." 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Friedman, F.L., and Koffman, E.
B. (1995), " }{TEXT 257 50 "Problem Solving, Abstraction, and Design U
sing C++" }{TEXT -1 32 ", Addison-Wesley, Reading, Mass." }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 52 "Heal, K.M., Hansen, M.L., and Rickard, K.
M. (1996), " }{TEXT 258 22 "Maple V Learning Guide" }{TEXT -1 21 ", Sp
ringer, New Your." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 82 "Monagan, M.B
., Geddes, K.O., Heal, K.M., Labahn, G., and Vorkoetter, S.M. (1996), \+
" }{TEXT 259 25 "Maple V Programming Guide" }{TEXT -1 21 ", Springer, \+
New York." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Redfern, D. (1996), \+
" }{TEXT 260 37 "The Maple Handbook: Maple V Release 4" }{TEXT -1 21 "
, Springer, New York." }}}}{SECT 0 {PARA 3 "" 0 "Endnotes" {TEXT -1 8 
"Endnotes" }}{EXCHG {PARA 0 "" 0 "1." {TEXT -1 567 "1. There are two s
et-ups of Maple. In one setup <return> means \"insert newline characte
r\", and <enter> means a command to the Maple engine: \"if this is a M
aple input, then perform the required task\". In another setup <shift \+
return> means \"insert a new line\", and <return> means the same as <e
nter>, namely that a command is given to the Maple engine: \"if this i
s a Maple input, then perform the required task\". In order to read th
e present notes easily, you should secure that your version of Maple h
as the first type of setup. Otherwise, you should change the setup." }
}}{EXCHG {PARA 0 "" 0 "2." {TEXT -1 500 "2. The differences between th
e revisions of Maple are both revolutionary and evolutionary. The work
sheet interface is , so to say, a revolutionary change: it will not wo
rk at all in Maple V Revision 2. Most of the changes of mathematical e
xpressions and programming language found in Maple V Revision 4 are of
 an evolutionary character. Although they will not work in Maple V Rev
ision 3 or Maple V Revision 2, there are still hope: with a little mod
ification most of the commands can be made to work." }}}{EXCHG {PARA 
0 "" 0 "3." {TEXT -1 154 "3. Maple's plots can easily be resized (e.g.
 made smaller) by catching the lower right corner of the plot window a
nd then drag the window with the mouse. " }}}}}{MARK "2 0 0" 17 }
{VIEWOPTS 1 1 0 1 1 1803 }