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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 28 "Musterloesung zur 7. Sitz
ung" }{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT 273 16 "Die A
ufgabe war:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 258 11 "Aufgabe 1. " }
{TEXT -1 84 "   In der Kombinatorik von Graphen wird bewiesen, dass mi
t den ganzzahligen Werten  " }{XPPEDIT 18 0 "a[p]" "&%\"aG6#%\"pG" }
{TEXT -1 47 "  , welche die kombinatorische Interpretation  " }}{PARA 
256 "" 0 "" {XPPEDIT 18 0 "a[p] " "&%\"aG6#%\"pG" }{TEXT -1 43 "  = ``
Anzahl aller gewurzelten Baeume mit  " }{XPPEDIT 18 0 "p" "I\"pG6\"" }
{TEXT -1 9 " Knoten''" }}{PARA 0 "" 0 "" {TEXT -1 51 "haben, die folge
nde formale Potenzreihenidentitaet " }}{PARA 0 "" 0 "" {TEXT -1 3 "   \+
" }{XPPEDIT 18 0 "sum(a[p+1]*x^p , p=0..infinity)*product( (1-x^p)^a[p
],p=1..infinity) = 1" "/*&-%$sumG6$*&&%\"aG6#,&%\"pG\"\"\"\"\"\"F-F-)%
\"xGF,F-/F,;\"\"!%)infinityGF--%(productG6$),&\"\"\"F-)F0F,!\"\"&F)6#F
,/F,;\"\"\"F4F-\"\"\"" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 22 
"in einer Unbestimmten " }{XPPEDIT 18 0 "x" "I\"xG6\"" }{TEXT -1 15 " \+
erfuellt ist. " }{TEXT 256 18 "Bestimme hieraus  " }{XPPEDIT 18 0 "a[1
], `` .. ``, a[51]" "6%&%\"aG6#\"\"\";%!GF(&F$6#\"#^" }{TEXT -1 2 "  \+
" }{TEXT 271 1 "!" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 274 18 "Die Anlei
tung war:" }}{EXCHG {PARA 0 "" 0 "" {TEXT 259 9 "Anleitung" }{TEXT 
260 2 ": " }{TEXT -1 6 " Sei  " }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }
{XPPEDIT 18 0 "A[N](x) = sum(a[p+1]*x^p , p=0..N)*` `,` `*B[N] (x)= pr
oduct( (1-x^p)^a[p],p=1..N)*`  ,`" "6$/-&%\"AG6#%\"NG6#%\"xG*&-%$sumG6
$*&&%\"aG6#,&%\"pG\"\"\"\"\"\"F5F5)F*F4F5/F4;\"\"!F(F5%\"~GF5/*&F;F5-&
%\"BG6#F(6#F*F5*&-%(productG6$),&\"\"\"F5)F*F4!\"\"&F16#F4/F4;\"\"\"F(
F5%$~~,GF5" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 4 "also" }}
{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "A[infinity](x)=A[N](x)
+O(x^(N+1))*` `,``" "6$/-&%\"AG6#%)infinityG6#%\"xG,&-&F&6#%\"NG6#F*\"
\"\"*&-%\"OG6#)F*,&F/F1\"\"\"F1F1%\"~GF1F1%!G" }}{PARA 0 "" 0 "" 
{TEXT -1 2 "  " }{XPPEDIT 18 0 "B[infinity](x)=B[N](x)*(1+x^(N+1))^(a[
N+1])*(`...`)*` = `*B[N](x)+O(x^(N+1))*` .`" "/-&%\"BG6#%)infinityG6#%
\"xG,&*,-&F%6#%\"NG6#F)\"\"\"),&\"\"\"F1)F),&F/F1\"\"\"F1F1&%\"aG6#,&F
/F1\"\"\"F1F1%$...GF1%$~=~GF1-&F%6#F/6#F)F1F1*&-%\"OG6#)F),&F/F1\"\"\"
F1F1%#~.GF1F1" }{MPLTEXT 1 0 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Mit   \+
" }{XPPEDIT 18 0 " A[infinity](x)*B[infinity](x) = 1 " "/*&-&%\"AG6#%)
infinityG6#%\"xG\"\"\"-&%\"BG6#F(6#F*F+\"\"\"" }{TEXT -1 10 "    folgt
 " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "t[N](x):= A[N](x)
*B[N](x)-1*` = `*A[infinity]*B[infinity]+O(x^(N+1)) -1 *` =  `*O(x^(N+
1))" ">-&%\"tG6#%\"NG6#%\"xG,**&-&%\"AG6#F'6#F)\"\"\"-&%\"BG6#F'6#F)F1
F1**\"\"\"F1%$~=~GF1&F.6#%)infinityGF1&F46#F<F1!\"\"-%\"OG6#)F),&F'F1
\"\"\"F1F1*(\"\"\"F1%%~=~~GF1-FA6#)F),&F'F1\"\"\"F1F1F?" }{TEXT -1 2 "
  " }}{PARA 0 "" 0 "" {TEXT -1 16 "fuer beliebiges " }{XPPEDIT 18 0 "N
=0,1,2" "6%/%\"NG\"\"!\"\"\"\"\"#" }{TEXT -1 29 " , ... . Die Taylor-R
eihe von" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "t[N] (x)= s
um(a[p+1]*x^p , p=0..N)*product( (1-x^p)^a[p],p=1..N) -1" "/-&%\"tG6#%
\"NG6#%\"xG,&*&-%$sumG6$*&&%\"aG6#,&%\"pG\"\"\"\"\"\"F5F5)F)F4F5/F4;\"
\"!F'F5-%(productG6$),&\"\"\"F5)F)F4!\"\"&F16#F4/F4;\"\"\"F'F5F5\"\"\"
FB" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 3 "um " }{XPPEDIT 18 0 
"x=0" "/%\"xG\"\"!" }{TEXT -1 28 " muss damit bis zur Ordnung " }
{XPPEDIT 18 0 "N" "I\"NG6\"" }{TEXT -1 58 "  verschwinden. Die Taylor-
Koeffizienten sind Polynome in " }{XPPEDIT 18 0 "a[1],a[2]" "6$&%\"aG6
#\"\"\"&F$6#\"\"#" }{TEXT -1 39 " , ... , so dass sich Gleichungen fue
r " }{XPPEDIT 18 0 "a[1]" "&%\"aG6#\"\"\"" }{TEXT -1 8 ", ... , " }
{XPPEDIT 18 0 "a[N+1]" "&%\"aG6#,&%\"NG\"\"\"\"\"\"F'" }{TEXT -1 65 " \+
ergeben, aus denen sich diese Werte rekursiv bestimmen lassen.  " }
{TEXT 276 30 "Waehle zunaechst ein kleines  " }{TEXT 272 1 "N" }{TEXT 
277 60 "  und betrachte die Struktur der resultierenden Gleichungen." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 9 "Achtung
: " }{TEXT -1 24 "es ist nicht sinnvoll,  " }{TEXT 261 2 "N=" }{TEXT 
-1 67 "50  zu waehlen und alle Gleichungen aus der Taylor-Entwicklung \+
von " }{XPPEDIT 18 0 "t[50](x)" "-&%\"tG6#\"#]6#%\"xG" }{TEXT -1 18 " \+
bis zur Ordnung  " }{XPPEDIT 18 0 "x^50" "*$%\"xG\"#]" }{TEXT -1 78 " \+
 berechnen zu wollen. Die Taylor-Koeffizienten (Polynome in den Unbeka
nnten  " }{XPPEDIT 18 0 "a[1],a[2],`` .. ``" "6%&%\"aG6#\"\"\"&F$6#\"
\"#;%!GF+" }{TEXT -1 7 ") sind " }{TEXT 263 6 "RIESIG" }{TEXT -1 35 " \+
!  Es ist sinnvoller, mit kleinem " }{TEXT 269 1 "N" }{TEXT -1 20 " zu
 starten, daraus " }{XPPEDIT 18 0 "a[1],`` .. ``,a[N+1] " "6%&%\"aG6#
\"\"\";%!GF(&F$6#,&%\"NG\"\"\"\"\"\"F-" }{TEXT -1 51 " zu bestimmen un
d mit diesen Werten zu groesseren  " }{TEXT 262 1 "N" }{TEXT -1 11 "  \+
zu gehen." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
43 "Eine Loesung ist mit den MAPLE-Funktionen  " }{TEXT 270 27 "sum, p
roduct, taylor, solve" }{TEXT -1 29 "  in wenigen Zeilen moeglich." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 15 "Zur Kont
rolle :" }{TEXT 267 1 " " }{TEXT -1 2 "  " }{XPPEDIT 18 0 "a[1]=1,a[2]
=1,a[3]=2,a[4]=4,a[5]=9" "6'/&%\"aG6#\"\"\"\"\"\"/&F%6#\"\"#\"\"\"/&F%
6#\"\"$\"\"#/&F%6#\"\"%\"\"%/&F%6#\"\"&\"\"*" }{TEXT -1 9 " , ... , " 
}{XPPEDIT 18 0 "a[10]=719" "/&%\"aG6#\"#5\"$>(" }{TEXT -1 9 " , ...  .
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
275 14 "Musterloesung:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res
tart:" }}{PARA 0 "" 0 "" {TEXT -1 40 "Die Struktur der Gleichungen, z.
B. fuer " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "N:=2:" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 119 "t[2]:=taylor (  sum  (  a[p+1]*x^p ,p=0..N) \n   \+
         * product( (1-x^p)^a[p],p=1..N) \n            - 1 , x=0 , N+1
);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Offensichtlich koennen hier
aus " }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "a[1]=1" "/&%\"
aG6#\"\"\"\"\"\"" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "a[2]=a[1]^2*``(``=1)" "/&%\"aG6#\"\"#*&&F$6#\"\"\"\"\"#
-%!G6#/F-\"\"\"\"\"\"" }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 2 
"  " }{XPPEDIT 18 0 "a[3]=2*a[1]*a[2]-a[1]^2*(a[1]-1)/2" "/&%\"aG6#\"
\"$,&*(\"\"#\"\"\"&F$6#\"\"\"F*&F$6#\"\"#F*F**(&F$6#\"\"\"\"\"#,&&F$6#
\"\"\"F*\"\"\"!\"\"F*\"\"#F;F;" }{TEXT -1 3 "   " }{XPPEDIT 18 0 "``(`
`=2)" "-%!G6#/F#\"\"#" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 17 
"bestimmt werden. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Allgemein g
ilt:  wenn " }{XPPEDIT 18 0 "a[1],``..``,a[N]" "6%&%\"aG6#\"\"\";%!GF(
&F$6#%\"NG" }{TEXT -1 34 " richtig bestimmt sind, dann folgt" }}{PARA 
0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "t[N-1](x)=O(x^N)" "/-&%\"tG6
#,&%\"NG\"\"\"\"\"\"!\"\"6#%\"xG-%\"OG6#)F-F(" }}{PARA 0 "" 0 "" 
{TEXT -1 9 "und damit" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 
0 "t[N](x)=t[N-1](x)+O(x^N)*` = `*O(x^N)*` .`" "/-&%\"tG6#%\"NG6#%\"xG
,&-&F%6#,&F'\"\"\"\"\"\"!\"\"6#F)F/**-%\"OG6#)F)F'F/%$~=~GF/-F56#)F)F'
F/%#~.GF/F/" }}{PARA 0 "" 0 "" {TEXT -1 37 "Genauer:  die Taylorentwic
klung von  " }{XPPEDIT 18 0 "t[N] (x)" "-&%\"tG6#%\"NG6#%\"xG" }{TEXT 
-1 18 "  beginnt dann mit" }}{PARA 0 "" 0 "" {TEXT -1 8 "        " }
{XPPEDIT 18 0 "t[N](x)  =(` `*a[N+1]-Polynom(a[1]..a[N])*` `)*x^N+ O(x
^(N+1))" "/-&%\"tG6#%\"NG6#%\"xG,&*&,&*&%\"~G\"\"\"&%\"aG6#,&F'F/\"\"
\"F/F/F/*&-%(PolynomG6#;&F16#\"\"\"&F16#F'F/F.F/!\"\"F/)F)F'F/F/-%\"OG
6#)F),&F'F/\"\"\"F/F/" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 22 
"wobei das Polynom in  " }{XPPEDIT 18 0 "a[1], ` ` .. ` `, a[N]" "6%&%
\"aG6#\"\"\";%\"~GF(&F$6#%\"NG" }{TEXT -1 58 "  zu einer Zahl evaluier
t. Der fuehrende Koeffizient von  " }{XPPEDIT 18 0 "t=t[N] " "/%\"tG&F
#6#%\"NG" }{TEXT -1 7 "  ist  " }{TEXT 266 7 "op(1,t)" }{TEXT -1 12 " \+
, bestimme " }{XPPEDIT 18 0 "a[N+1]" "&%\"aG6#,&%\"NG\"\"\"\"\"\"F'" }
{TEXT -1 6 " durch" }}{PARA 0 "" 0 "" {TEXT -1 6 "      " }{XPPEDIT 
18 0 "a[N+1] := solve( op(1,t)=0 , a[N=1] ); " ">&%\"aG6#,&%\"NG\"\"\"
\"\"\"F(-%&solveG6$/-%#opG6$\"\"\"%\"tG\"\"!&F$6#/F'\"\"\"" }{TEXT -1 
2 " ;" }}{PARA 0 "" 0 "" {TEXT -1 15 "Dann gehe von  " }{TEXT 264 1 "N
" }{TEXT -1 6 "  zu  " }{TEXT 265 3 "N+1" }{TEXT -1 12 " , bestimme " 
}{XPPEDIT 18 0 "a[N+2] " "&%\"aG6#,&%\"NG\"\"\"\"\"#F'" }{TEXT -1 11 "
 usw., bis " }{XPPEDIT 18 0 "a[1],``..``,a[101]" "6%&%\"aG6#\"\"\";%!G
F(&F$6#\"$,\"" }{TEXT -1 15 " bestimmt sind:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "for N \+
from 0 to 50 do\n  t:=taylor(    sum  ( a[k+1]*x^k ,k=0..N)\n         \+
    *product((1-x^k)^a[k],k=1..N)\n             - 1 , x=0,N+1);\n  a[N
+1]:=solve( op(1,t)=0 , a[N+1] );\nod;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}}{MARK "3" 0 }{VIEWOPTS 1 1 0 3 2 1804 }