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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 9 "Lecture 3" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 32 "Maple 6 versus Previous Versions" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Let's discuss once more some diffe
rences between  " }{TEXT 256 9 "Maple  6 " }{TEXT 259 23 "and previous
 versions. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 45 "In previous versions of Maple :\n\ninstead of  " }{TEXT 345 6 "
end if" }{TEXT -1 15 " one must use  " }{TEXT 346 2 "fi" }{TEXT -1 14 
",\ninstead of  " }{TEXT 347 3 "end" }{TEXT -1 1 " " }{TEXT 348 4 "pro
c" }{TEXT -1 15 " one must use  " }{TEXT 349 3 "end" }{TEXT -1 14 ",\n
instead of  " }{TEXT 350 6 "end do" }{TEXT -1 15 " one must use  " }
{TEXT 351 2 "od" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 12 "instea
d of  " }{TEXT 352 16 "return statement" }{TEXT -1 14 " one must use \+
" }{TEXT 353 17 "RETURN(statement)" }{TEXT -1 3 ".\n\n" }{TEXT 359 7 "
Maple 6" }{TEXT -1 46 " does however allow one to  use the old forms \+
" }{TEXT 354 2 "od" }{TEXT -1 2 ", " }{TEXT 355 2 "fi" }{TEXT -1 5 " a
nd " }{TEXT 356 3 "end" }{TEXT -1 10 ". In fact " }{TEXT 357 3 "end" }
{TEXT -1 65 " alone will work for any of these so-called delimiters.  \+
Thus in " }{TEXT 257 7 "Maple 6" }{TEXT -1 24 " we would usually write
:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 95 "f:=proc(x) \n  if x > 0 then \n   return sin(x); \n  \+
else \n   return cos(x); \n  end if; \nend proc:" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 7 "But in " }{TEXT 258 7 "Maple V" }{TEXT -1 46 " we M
UST write the same  procedure as follows:" }}}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 89 "f:=proc(x) \n  if x > 0 then \n   RETURN(sin(x)); \n \+
 else \n   RETURN(cos(x)); \n  fi; \nend:\n" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 29 "The latter will also work in " }{TEXT 358 7 "Maple 6" }
{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "The Concatenation Operator " }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Another difference is the " }
{TEXT 260 22 "concatenation operator" }{TEXT 268 2 ". " }{TEXT -1 13 "
 In Maple 6  " }{TEXT 360 6 "x || y" }{TEXT -1 42 " gives the concaten
ation of strings x and " }{TEXT 361 1 "y" }{TEXT -1 48 ". In previous \+
versions of Maple it was given by " }{TEXT 266 3 "x.y" }{TEXT -1 75 ".
  Concatenation is useful for  forming variables. Here's a simple exam
ple:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "for i from 1 to 5 d
o\n  x||i:=i^2;\nod;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "We have c
reated the four variables x1,x2,x3,x4 and assigned to xi the value " }
{XPPEDIT 18 0 "i^2;" "6#*$%\"iG\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x
1,x2,x3,x4;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 78 "The Maple help pa
ge for || states that the use of this operator is discourage." }{TEXT 
-1 89 "  In the next section we show how to use indexed variables for \+
the same purpose. Usually " }{TEXT 362 9 "cat(x,y) " }{TEXT -1 14 "is \+
the same as" }{TEXT 363 7 " x || y" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "The  func
tion " }{TEXT 267 3 "cat" }{TEXT -1 124 " may also be useful to get ri
d of extra commas when printing combinations of text and integers, as \+
in the following example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 
"restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "for i from 1 \+
to 4 do\n#print(`If i = `, i, ` then i^2 = `, i^2);\nprint(cat(`If i =
 `, i, ` then i^2 = `, i^2, `.`));\nend do;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 89 "Note that the
 cat operator will not work nicely with floating point numbers. For ex
ample:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "x||1.0:=3;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "cat(x,1.0):=3;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "x||1:=3; " }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 52 "But note that if we use cat we get an error message:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "cat(x,1):=3; " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 10 "Check out " }{TEXT 364 4 "?cat" }{TEXT 
-1 19 " for more examples." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Indexed Variables" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "An
 alternative way to create several variable is to use " }{TEXT 261 17 
"indexed variables" }{TEXT -1 34 ". These are variables of the form " 
}{TEXT 262 4 "x[i]" }{TEXT -1 25 " which Maple exhibits as " }
{XPPEDIT 263 0 "x[i];" "6#&%\"xG6#%\"iG" }{TEXT 264 2 " ." }{TEXT -1 
15 "  That is, the " }{TEXT 265 1 "i" }{TEXT -1 144 " becomes a subscr
ipt.  This is the same in Maple 6 as in previous versions of Maple.  [
Althought the notation is similar to that for arrays, an " }{TEXT 367 
16 "indexed variable" }{TEXT -1 11 " is not an " }{TEXT 366 5 "array" 
}{TEXT -1 80 ". We will discuss the distinction later. Indexed variabl
es are actually of type " }{TEXT 365 5 "table" }{TEXT -1 2 ".]" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 37 "for i from 1 to 5 do\n  x[i]:=i^2;\nod;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "One may also use doubly, triply, e
tc., subscribted variables as in the following examples:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "A[2,5]:=2;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "z[1,2,2,3]:=x^2 + sin(x);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 11 "Sequences, " }{TEXT 283 25 "(max, min, seq, ithprime)" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Pe
rhaps the most important " }{TEXT 368 5 "types" }{TEXT -1 4 " or " }
{TEXT 369 7 "classes" }{TEXT -1 25 " of objects in Maple are " }{TEXT 
274 16 "sequences, sets " }{TEXT -1 4 "and " }{TEXT 275 5 "lists" }
{TEXT -1 46 ". It is vital to understand these thoroughly. " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "We star
t with some examples of " }{TEXT 270 9 "sequences" }{TEXT -1 4 ": A " 
}{TEXT 276 8 "sequence" }{TEXT -1 3 "  (" }{TEXT 341 3 "aka" }{TEXT 
-1 4 " an " }{TEXT 339 10 "expression" }{TEXT -1 1 " " }{TEXT 340 8 "s
equence" }{TEXT -1 151 ") is just a bunch of expressions separated by \+
commas: The expressions may be numbers, variables, functions, etc...--
and they need not all be different." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 
"sequence1:=1,2,1,A,3,x,sin(t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 
"If " }{TEXT 277 9 "sequence1" }{TEXT -1 20 " is a sequence then " }
{TEXT 278 12 "sequence1[i]" }{TEXT -1 47 " is the i-th term of the seq
uence, for example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "sequ
ence1[5];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "sequence1[3];
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sequence2:=55,7,33,-1,4
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "The maximum and minimum of a
 sequence or real numbers is obtained by use of the functions " }
{TEXT 279 3 "max" }{TEXT -1 5 " and " }{TEXT 280 3 "min" }{TEXT -1 31 
", as in the following examples:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "max(sequence2);\nmin(sequence2);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "Note that \+
the solution to a " }{TEXT 370 5 "solve" }{TEXT -1 51 " command as in \+
the following example is given as a " }{TEXT 371 8 "sequence" }{TEXT 
-1 58 ". We name it so we can manipulate the different solutions:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p:=x^3-60*x^2 + 1100*x - 600
0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sequence3:=solve(p=0,
x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "sequence3[1]; " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "sequence3[2];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "sequence3[3];" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 13 "The function " }{TEXT 273 3 "seq" }{TEXT -1 280 " \+
is a very important way to form sequences. It can be used to do many t
hings that can be done with do loop and when that's possible should be
 done. Generally this is a faster way to go. For example, if we wish t
o apply some function f to the numbers -1, 2, 3,4,5, ..10 we can use \+
" }{TEXT 281 3 "seq" }{TEXT -1 12 " as follows:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "sequence4:=seq(f(i),i=-1..10);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 16 "For example, if " }{XPPEDIT 18 0 "f(x) = \+
x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }{TEXT -1 9 " we have:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "sequence4:=seq(i^2,i=-1..10);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "We could also form this sequence \+
with the following do loop.  We use s to indicate the sequence and we \+
start by setting s to be the NULL sequence:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 65 "s:=NULL:\nfor i from -1 to 10 do\n  s:=s,i^2:\nend
 do:\nsequence4:=s;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "In Maple i
thprime(i) is the i-th prime number. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "ithprime(1);\nithprime(2);\nithprime(3); " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 81 "We can get a sequence consisting of the f
irst  20 primes by the following method:" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "sequence5:=seq(ithpr
ime(k),k=1..15);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "One can " }
{TEXT 272 11 "concatenate" }{TEXT -1 23 " sequences as follows:\n" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "sequence4;\nsequence5;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sequence6:=sequence4,sequenc
e5;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "A simpler example would be
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "seqA:=a,b,c;\nseqB:=x,
y,z,w;\nseqC:=seqA,seqB,seqA;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "sequence7:=a,b,c,d,e,f,g,h;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
74 "One can select more than one element of a sequence at a time. For \+
example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "sequence7[2..4]
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "To get the first element on \+
the right use:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "sequence7
[-1];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "To get the second elemen
t from the right use:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "sequence7[-2];" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 38 "We can use this to reverse a sequence:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "S1:=seq(i,i=1..7);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "S2:=seq(S1[-i],i=1..7);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 31 "The empty sequence is given by " }{TEXT 
271 8 "NULL -- " }{TEXT 282 23 "as we aready say above." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "S3:=NULL;\n\nS4:=x,x,S3,y;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 5 "Sets " }{TEXT 284 39 "(union, intersect, nops, minus, me
mber)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "One can obtain a " }
{TEXT 289 3 "set" }{TEXT -1 8 " from a " }{TEXT 290 8 "sequence" }
{TEXT -1 109 " by placing braces around it. We illustrate this below. \+
We also illustrate some basic set operations such as " }{TEXT 287 40 "
union, intersect, minus, nops and member" }{TEXT -1 2 ". " }{TEXT 288 
79 "Note, however, that the user has no control over the order of elem
ents of a set" }{TEXT -1 89 ". To control the order of elements we mus
t use sequences or lists --as we will see below:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "set1:=\{4,3,3,2,1\};" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "set2:=\{4,4,5,5,6\};" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "set3:=set1 union set2;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "set4:=set1 intersect set2;" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 22 "set5:=set1 minus set2;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "Z[10]:=seq(i,i=0..9);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "set6:=\{Z[10]\};" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "seq(x^j-1,j=1..5);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "set7:=\{seq(x^j-1,j=1..5)\};" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 4 "The " }{TEXT 292 9 "empty set" }{TEXT -1 19 " is denote
d by \{\};\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "\{1,2,3\} i
ntersect \{4,5,6\};" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 285 4 "nops" }
{TEXT -1 49 " is used to find the number of elements in a set:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "nops(set7);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "nops(\{\});" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 33 "Note that nops will not work for " }{TEXT 291 9 "seque
nces" }{TEXT -1 1 "!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "nop
s(1,2,3,4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "But " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "nops(\{1,2,3,4\});" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 286 "If S is a set then S[i] is the i-th elem
ent of the set, but you have no way of knowing what order Maple assign
s to the elements of the set S. And the order may change in a single w
orksheet session. So do NOT make any assumptions about the order even \+
if you see the order on the monitor." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "set7;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "fo
r i from 1 to nops(set7) do i,set7[i]; end do;" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 286 10 "Membership" }{TEXT -1 31 " in a set is tested as fo
llows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "set7;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "member(x^2-1,set7);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "member(x,set7);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 61 "if member(x,set7) = false then set7:=set7 u
nion \{x\}; end if;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "Lists " }{TEXT 293 43 "(sort,  \+
nops,  op,  concatenation of lists)" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
297 5 "Lists" }{TEXT 300 1 " " }{TEXT -1 68 "may be formed by placing \+
a sequence in square brackets, as follows: " }{TEXT 299 36 "In this ca
se order is all important." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
23 "List1:=[4,0,2,2,3,1,4];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 301 5 "sort " }{TEXT -1 58 " is useful for rearranging a list in
 non-decreasing order." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s
ort(List1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 296 4 "nops" }{TEXT -1 
43 "(List1) gives the length of the list List1:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "N:=nops(List1);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 22 "Compare the following:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "nops([1,3,3,3,3]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "nops(\{1,3,3,3,3\});" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 20 "The i-th element in " }{TEXT 302 5 "List1" }{TEXT -1 13 "
 is given by " }{TEXT 303 9 "List1[i]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "for i from 1 to N do i,List1[i]; end do;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "List1[2..4];" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "List1;\nList1[-1];\nList1[-2];" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 3 "To " }{TEXT 298 11 "concatenate" }{TEXT 
-1 81 " lists we need to first recover the underlying sequence by appl
ying the function " }{TEXT 295 2 "op" }{TEXT -1 49 ". This also works \+
for sets, as we now illustrate:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "List1; \nop(List1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "set1;\nop(set1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Lis
t1;\nList2:=[x,y,z];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Lis
t3:=[op(List1),op(List2)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "List4:=[op(List3),`a new element`];" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 4 "The " }{TEXT 294 10 "empty list" }{TEXT -1 20 " is denoted
 by [ ] ;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 97 "Let's make a list of the primes numbers up to 100: We wil
l put them in a list we will call List5:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 89 "List5:=[]:\nfor i from 1 to 100 do\nif isprime(i) the
n List5:=[op(List5),i]; fi;\nod;\nList5;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 46 "Another way to create this list is as follows:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 76 "seq5:=NULL:\nfor i from 1 to 100 do\nif ispr
ime(i) then seq5:=seq5,i; fi;\nod;\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "seq5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Li
st5:=[seq5];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "As mentioned abo
ve one can also select elements of a list in reverse order. For exampl
e, the last element of the list List is found by List[-1]. The second \+
from last is given by List[-2], etc..." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "List:=[seq(i, i=1..10)];\na:=List[-1];\nb:=List[-2];
\nc:=List[-3];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "We can obtain t
he reverse of the list " }{TEXT 304 4 "List" }{TEXT -1 18 " above as f
ollows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "revList:=[seq(Li
st[-i],i=1..10)];" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 48 "Converting
 from list to set and from set to list" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 67 "One can convert lists to sets and vice versa by using the
 function " }{TEXT 305 7 "convert" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 35 "L:=[1,1,1,2,3];\nA:=convert(L, set);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "B:=\{1,2,3,4,5\};\nLL:=convert(B,li
st);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "We can convert from " }
{TEXT 306 3 "set" }{TEXT -1 5 " and " }{TEXT 307 4 "list" }{TEXT -1 4 
" to " }{TEXT 308 8 "sequence" }{TEXT -1 11 " by use of " }{TEXT 309 
2 "op" }{TEXT -1 16 " as shown above:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "op(L);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "op
(B);" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 " round,  floor, ceil" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 3 "" 0 "" {TEXT 372 5 "r
ound" }{TEXT 30 2 " (" }{TEXT 378 1 "x" }{TEXT 30 3 ") -" }{TEXT 377 
23 " the nearest integer to" }{TEXT 30 1 " " }{TEXT 380 1 "x" }}{PARA 
3 "" 0 "" {TEXT 373 9 "floor(x) " }{TEXT 30 3 " - " }{TEXT 375 20 "the
 largest integer " }{XPPEDIT 18 0 "n <= x;" "6#1%\"nG%\"xG" }{TEXT -1 
1 "." }}{PARA 3 "" 0 "" {TEXT 374 4 "ceil" }{TEXT 30 1 "(" }{TEXT 379 
1 "x" }{TEXT 30 4 ") - " }{TEXT 376 33 "the smallest integer n such th
at " }{XPPEDIT 18 0 "x <= n;" "6#1%\"xG%\"nG" }{TEXT -1 1 "." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Here are some e
xamples:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "L:=[-3.4,-3.6, 3.6,3.5, 3.4, 3.5, 3, -3]:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "for x in L do\nprint(`if x = `, x \+
, `   round(x) = `, round(x),  `   floor(x) = `, floor(x), `   ceil(x)
 = `, ceil(x));\nend do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Somet
imes one must apply evalf before invoking these functions. For example
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "floor(sin(10^8));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "floor(sin(10^10));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "floor(evalf(sin(10^10)));" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "floor(sqrt(10^9 + 1));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "floor(sqrt(10^10 + 1));" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "floor(evalf(sqrt(10^10+1)))
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "floor(evalf(sqrt(10^10
0 + 1)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 14 "The Digits of " }{XPPEDIT 18 0 "pi;" "6#%
#piG" }{TEXT -1 1 " " }}{EXCHG {PARA 256 "" 0 "" {TEXT 317 140 "Here w
e write a procedure to explore the question of whether or not the digi
ts 0 to 9 are uniformly distributed among the first N digits of " }
{XPPEDIT 319 0 "pi;" "6#%#piG" }{TEXT -1 0 "" }{TEXT 318 1 "." }{TEXT 
-1 1 "\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Use the command " }
{TEXT 310 13 "?convert/base" }{TEXT -1 37 " to find out more about the
 function " }{TEXT 311 7 "convert" }{TEXT -1 25 " used below.  Actuall
y,  " }{TEXT 312 7 "convert" }{TEXT -1 60 " has many uses in maple, as
 we will see later in the course." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT 313 6 "Digits" }{TEXT -1 129 " is an \"en
vironmental\" variable that tells how many digits are carried in float
ing point operations. The default is 10 as we see:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "Digits;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "
evalf(Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "n:=3: \nDigit
s:=n; " }{TEXT -1 66 "This sets the number of digits carried by floati
ng point numbers. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 13 "X:=evalf(Pi);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "10^(n-1)*X;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "X:=floor(10^(n-1)*X); " }{TEXT -1 51 "This converts X to an inte
ger with the same digits." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
24 "X:=convert(X,base, 10); " }{TEXT -1 178 "Note the digits appear in
 reversed order, but they are now in a list which makes them easier to
 handle. There are alternative ways to get at the digits that we may d
iscuss later." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "Let's do this ag
ain with n = 100 instead of just 3:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "n:=100:\nDigits:=n;\nX:=ev
alf(Pi,n);\nX:=floor(10^(n-1)*X);\nX:=convert(X,base, 10);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(X);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 85 "Now we shall count the number of times each digit appears
 in the first 100 digits of " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 
-1 142 ". We will let c[0] be the number of times 0 appears, c[1] be t
he number of times 1 appears, etc... First we initialize 10 variables \+
c[i] to 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "for i from 0 \+
to 9 do c[i]:=0; od;" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 54 "for k from 1 to n do\n  j:=X[k]; \n  c[j]:=c[j
] + 1;\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "[seq(c[j],j=
0..9)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "Another way to do the \+
countinng is as follows: First set the c[i]'s back to 0:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "for i from 0 to 9 do c[i]:=0; od:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "for k from 1 to n do \n  \+
c[X[k]]:=c[X[k]] + 1;\nod:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "[seq(c[j],j=0..9)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Now w
e construct a procedure to compute the number of times each digit appe
ars in the first n digits of " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT 
-1 113 ". Before constructing the procedure we do it step by step for \+
a small value of n so we can see what is happening." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 9 "restart:\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 85 "n:=40;\nDigits:=n;\nX:=evalf(Pi,n);\nX:=floor(10^(n-1
)*X);\nX:=convert(X,base, 10);\n  \n \n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "for i from 0 to 9 do c[i]:=0; end do:\n  " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "for k from 1 to n do\n   j:=X[k]; \+
\n   c[j]:=c[j] + 1;\nend do:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "[seq(c[j],j=0..9)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Now \+
we make this into a procedure:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 252 "numdigits:=proc(n)\nlocal X,c,i,k,j;\n  Digits:=n:\n  X:=eval
f(Pi,n):\n  X:=floor(10^(n-1)*X):\n  X:=convert(X,base, 10):\n  for i \+
from 0 to 9 do c[i]:=0; end do:\n  for k from 1 to n do\n   j:=X[k]; \+
\n   c[j]:=c[j] + 1;\n  end do:\n [seq(c[j],j=0..9)];\nend proc:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "numdigits(100);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "numdigits(1000);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "numdigits(10000);" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "At the webpage " }}{PARA 
0 "" 0 "" {TEXT -1 3 "   " }}{PARA 259 "" 0 "" {TEXT 315 59 "http://ww
w.cs.ruu.nl/wais/html/na-dir/sci-math-faq/pi.html " }{TEXT -1 0 "" }
{TEXT 314 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 93 "you can find out how mathematicians have been able to com
pute the first 51 billion digits of " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }
{TEXT -1 125 ". Maple will not be able to do this using the method we \+
have just described. More sophisticated techniques are necessary. At \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 316 42 "ht
tp://mathworld.wolfram.com/PiDigits.html" }{TEXT -1 1 " " }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "you will also see \+
some data on the distribution of the digits 0-9 in the known digits of
 " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 113 ". It appears to be tr
ue that the digits appear with approximately equal frequency--but this
 has not been proved. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 "Assignment 3 - Due Next Monday
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 328 
10 "Problem 1." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 62 "(a)  Prove that every prime p greater tha
n 2 satisfies either " }{TEXT 327 11 "p mod 8 = r" }{TEXT -1 127 " whe
re r is 1, 3, 5, or 7. Thus primes > 2 may be divided into four classe
s depending on their remainders when divided by  8.  " }{TEXT 344 99 "
No Maple calculations are necessary here, but write out your proof usi
ng Maple as a word processor." }{TEXT -1 2 "  " }{TEXT 321 6 "HINT: " 
}{TEXT -1 24 "Can p > 2 be a prime if " }{TEXT 320 12 "p mod 8  = r" }
{TEXT -1 65 "  when r = 0, 2, 4 or 6?  (Take each case separately.) No
te that " }{TEXT 330 12 "p mod 8 =  r" }{TEXT -1 12 " means that " }
{TEXT 331 10 "p = 8q + r" }{TEXT -1 7 " where " }{TEXT 332 30 "r = 0, \+
1, 2, 3, 4, 5, 6, or 7." }{TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT 
-1 9 "(b)  Set " }{TEXT 333 6 "N:=100" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 96 "     (i) Form four lists L[r] for r = 1,3,5,7 where \+
L[r] contains all primes 2< p < N such that " }{TEXT 329 11 "p mod 8 =
 r" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 63 "    (ii) Using Ma
ple count the number of primes in each list.\n\n" }{TEXT 322 10 "Probl
em 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 "
In the notation of Problem 1, make a procedure with input = arbitrary \+
 positive integer " }{TEXT 334 1 "N" }{TEXT -1 23 " and output = the l
ist " }{TEXT 326 48 "[nops(L[1]), nops(L[3]), nops(L[5]), nops(L[7])]
" }{TEXT -1 2 ". " }{TEXT 325 67 "Do some experiments to see for which
 (if any) r, L[r] is largest. ." }{TEXT -1 15 " [According to " }
{TEXT 323 56 "Dirichlet's Theorem on Primes in Arithmetic Progressions
" }{TEXT -1 13 " the numbers " }{TEXT 324 10 "nops(L[r])" }{TEXT -1 
38 " will be asymptotically the same.]  \n\n" }{TEXT 335 3 "(a)" }
{TEXT -1 29 " Test your procedure on some " }{TEXT 336 5 "small" }
{TEXT -1 13 " values of N " }{TEXT 337 52 "where you can check by hand
 if the answer is correct" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 338 3 "(b)" }{TEXT -1 64 " Then try it
 on some larger values of N. Try, for example,  N = " }{XPPEDIT 18 0 "
10^i;" "6#)\"#5%\"iG" }{TEXT -1 16 ",  i = 2,3,4, 5." }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 342 10 "Problem 3." }
{TEXT -1 3 " \n " }}{PARA 0 "" 0 "" {TEXT -1 82 "(a) Write a procedure
 whose input is a positive integer n and whose output is the " }{TEXT 
386 4 "set " }{TEXT -1 74 "of all squares of positive integers less th
an or equal to n.  Write it in " }{TEXT 387 18 "two different ways" }
{TEXT -1 12 ". (i) using " }{TEXT 384 3 "seq" }{TEXT -1 18 " and (ii) \+
using a " }{TEXT 385 7 "do loop" }{TEXT -1 54 ". Show output for n = 4
, 10, 20, 100 and 1000. [Hint: " }{XPPEDIT 18 0 "i^2 <= n;" "6#1*$%\"i
G\"\"#%\"nG" }{TEXT -1 18 " is equivalent to " }{XPPEDIT 18 0 "i <= sq
rt(n);" "6#1%\"iG-%%sqrtG6#%\"nG" }{TEXT -1 3 ".]\n" }}{PARA 0 "" 0 "
" {TEXT -1 22 "(b) Write a procedure " }{TEXT 390 6 "genseq" }{TEXT 
-1 55 " whose input is the integer n. and whose output is the " }
{TEXT 388 4 "list" }{TEXT -1 22 " of all integers  7*i " }{TEXT 381 4 
" mod" }{TEXT -1 112 " 11 for i from 1 to n.  Note that each of these \+
integers will be in the set \{0, 1, 2, . . .  , 10\}. Write it in " }
{TEXT 389 18 "two different ways" }{TEXT -1 11 " (i) using " }{TEXT 
382 3 "seq" }{TEXT -1 18 " and (ii) using a " }{TEXT 383 7 "do loop" }
{TEXT -1 42 ". Test each procedures for n = 11 and 110." }}{PARA 0 "" 
0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 343 10 "Problem 4." }{TEXT -1 3 "  \n" }}{PARA 0 "" 0 "" {TEXT 
-1 65 "(a) Write a procedure that will take as input a positive intege
r " }{TEXT 393 1 "n" }{TEXT -1 30 " and whose output will be the " }
{TEXT 399 4 "list" }{TEXT -1 1 " " }{TEXT 394 27 "[c[0], c[1], . . . ,
 c[10]]" }{TEXT -1 7 " where " }{TEXT 395 4 "c[i]" }{TEXT -1 51 " is t
he number of times that i appears in the list " }{TEXT 391 9 "genseq(n
)" }{TEXT -1 7 " where " }{TEXT 392 6 "genseq" }{TEXT -1 183 " is as d
escribed in Problem 3.  Test the procedure for n =  100, 1000 and 1000
0. Do the numbers 0, 1, . . ., 10 appear to be equally distributed?\n
\n(b) In the procedure genseq replace " }{TEXT 396 10 "7*i mod 11" }
{TEXT -1 4 " by " }{TEXT 397 10 "8*i mod 22" }{TEXT -1 47 ". The repea
t the instructions in part (a) with " }{TEXT 398 69 "[c[0], c[1], . . \+
. , c[10]] replaced by [c[0], c[1], . . . , c[21]]. " }{TEXT -1 65 "Do
 the numbers 0, 1, . . ., 21 appear to be equally distributed? " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 19 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "
" {TEXT -1 1 "\n" }}}}{MARK "10" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 1 1 2 33 1 1 }