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le" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 24 0 0 0 1 2 1 2 2 2 2 1 1 1 
1 }2 1 0 0 12 12 1 0 1 0 2 2 19 1 }}
{SECT 0 {SECT 1 {PARA 256 "" 0 "" {TEXT -1 28 "Chapter 13\n\nAssume Fa
cility\n" }}{PARA 0 "" 0 "" {TEXT 261 31 "\251 Copyright 2003 by Andr
\351 Heck." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 0 "" 0 "
" {TEXT 260 1 "1" }{TEXT -1 14 ". Compute cos(" }{XPPEDIT 18 0 "n*Pi" 
"6#*&%\"nG\"\"\"%#PiGF%" }{TEXT -1 10 ") and sin(" }{XPPEDIT 18 0 "n*(
Pi/2)" "6#*&%\"nG\"\"\"*&%#PiGF%\"\"#!\"\"F%" }{TEXT -1 29 ") under th
e assumptions that " }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 79 " is\n-
 an integer.\n- an odd integer.\n- an odd and positive integer less th
an 3.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "c := cos(n*Pi);  s := sin(n*
Pi/2); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"cG-%$cosG6#*&%\"nG\"\"
\"%#PiGF*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG-%$sinG6#,$*(\"\"#!
\"\"%\"nG\"\"\"%#PiGF-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 
"c, s assuming n::integer;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$)!\"\"%
\"nG-%$sinG6#,$*(\"\"#F$F%\"\"\"%#PiGF,F," }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 21 "c, s assuming n::odd;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6$!\"\")F#,&*&\"\"#F#%\"nG\"\"\"F)#F)F'F#" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 39 "assume(n, odd): additionally(0<n, n<3):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "about(n);" }}{PARA 6 "" 1 "" 
{TEXT -1 25 "Originally n, renamed n~:" }}{PARA 6 "" 1 "" {TEXT -1 79 
"  is assumed to be: AndProp(LinearProp(2,integer,1),RealRange(Open(0)
,Open(3)))" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 5 "c, s;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$!\"\")F#,&*
&\"\"#F#%#n|irG\"\"\"F)#F)F'F#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "is(n=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "is(sin(n*Pi/2)=1);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "_EnvTry := hard:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "for
get(is):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "about(n);" }}
{PARA 6 "" 1 "" {TEXT -1 25 "Originally n, renamed n~:" }}{PARA 6 "" 
1 "" {TEXT -1 79 "  is assumed to be: AndProp(LinearProp(2,integer,1),
RealRange(Open(0),Open(3)))" }}{PARA 6 "" 1 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "is(n=1);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%FAILG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "c
oulditbe(n=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}}}{SECT 
1 {PARA 0 "" 0 "" {TEXT 257 2 "2." }{TEXT -1 67 " Add the interval (-1
,1) to the hierarchy of numerical properties.\n" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 57 "alias(absvaluelessthan1 = RealRange(Open(-1), Open(1)) ):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "addproperty(absvaluelessthan
1, \{real\}, \{\}): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "`pr
operty/ParentTable`[BottomProp];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<9
\"\"!\"\"\"%)operatorG%%typeG%)constantG%)fractionG%1ElementaryMatrixG
%+irrationalG%+NullMatrixG%*LinearMapG-%$NonG6#%*symmetricG%&primeG%(E
venMapG%'addmulG%'OddMapG%'scalarG%.GaussianPrimeG-%*RealRangeG6$,$%)i
nfinityG!\"\"-%%OpenG6#F$%2absvaluelessthan1G%/IdentityMatrixG%2Rectan
gularMatrixG%2MutuallyExclusiveG%+NullVectorG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 28 "`property/ChildTable`[real];" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#<'%)rationalG%+irrationalG-%*RealRangeG6$\"\"!%)infin
ityG-F'6$,$F*!\"\"-%%OpenG6#F)%2absvaluelessthan1G" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 28 "is(-1/2, absvaluelessthan1);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 25 "is(1, absvaluelessthan1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&f
alseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 0 "" 0 "" {TEXT 258 2 "3." }{TEXT -1 209 " A natural number is c
alled perfect when it is one more than the sum of its nontrivial divis
ors (e.g., 6=2+3+1 and 28=2+4+7+14+1 are perfect). Add the perfect num
bers to the hierarchy of numerical properties.\n" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 143 "`type/perfect` := proc(n)\n  if type(n, posint) then\n    eva
lb( 2*n = convert(numtheory[divisors](n), `+`) )\n  else\n    false\n \+
 end if\nend proc;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-type/perfectG
f*6#%\"nG6\"F(F(@%-%%typeG6$9$%'posintG-%&evalbG6#/,$*&\"\"#\"\"\"F-F6
F6-%(convertG6$-&%*numtheoryG6#%)divisorsG6#F-%\"+G%&falseGF(F(F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "addproperty(perfect, \{integ
er\}, \{\}): " }}}{PARA 0 "" 0 "" {TEXT -1 33 "Some examples to see if
 it works:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "is(6, perfect)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "is(28, perfect);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "is(11, per
fect);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 19 "assume(p, perfect);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "is(p, perfect);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "is(q, perf
ect);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT 262 2 "4.
" }{TEXT -1 4 " An " }{XPPEDIT 18 0 "n x n " "6#*(%\"nG\"\"\"%\"xGF%F$
F%" }{TEXT -1 37 " matrix A is called skew-diagonal if " }{XPPEDIT 18 
0 "A[ij]" "6#&%\"AG6#%#ijG" }{TEXT -1 10 "=0 unless " }{XPPEDIT 18 0 "
i" "6#%\"iG" }{TEXT -1 1 "+" }{XPPEDIT 18 0 "j" "6#%\"jG" }{TEXT -1 1 
"=" }{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 21 "+1. Add the property " 
}{TEXT 0 12 "skewdiagonal" }{TEXT -1 28 " to Maple's knowledge base.\n
" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart; " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 337 "`type/skewdiagonal` := proc(M)\n  \+
local i, j:\n  if type(M, 'matrix'(rational,square)) then\n    for i t
o linalg[rowdim](M) do\n      for j to linalg[coldim](M) do\n         \+
if i+j<>linalg[rowdim](M)+1 and M[i,j]<>0 then\n            return fal
se\n         end if\n      end do\n    end do:\n    return true \n  el
se\n    return false\n  fi\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "addproperty(skewdiagonal, \{SquareMatrix\}, \{NullMat
rix\}): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "`property/+`(sk
ewdiagonal, skewdiagonal) := skewdiagonal:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 48 "`property/^`(skewdiagonal, odd) := skewdiagonal:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "`property/^`(skewdiagonal,
 even) := diagonal:" }}}{PARA 0 "" 0 "" {TEXT -1 35 "A few examples to
 see how it works." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "M := m
atrix([[0,0,1], [0,2,0], [3,0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"MG-%'matrixG6#7%7%\"\"!F*\"\"\"7%F*\"\"#F*7%\"\"$F*F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "is(M, skewdiagonal);" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "is(M^3, skewdiagonal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&fals
eG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "is(evalm(M^3), skewdi
agonal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%FAILG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 17 "M3 := evalm(M^3);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#M3G-%'matrixG6#7%7%\"\"!F*\"\"$7%F*\"\")F*7%\"\"*F*F
*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "is(M3, skewdiagonal);
" }}}{EXCHG {PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 23 "is(M&^3, skewdiagonal);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#%&falseG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
30 "is(evalm(M&^3), skewdiagonal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
%%FAILG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "M3 := evalm(M&^3
);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#M3G-%'matrixG6#7%7%\"\"!F*\"
\"$7%F*\"\")F*7%\"\"*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "is(M3, skewdiagonal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "assume(A, skewdiagonal, n
, odd):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "is(A^n, skewdiag
onal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 24 "assume(B, skewdiagonal):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 22 "is(A+B, skewdiagonal);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}}}{MARK "0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 
2 33 1 1 }