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Sitzung" }{MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 29 "Erinnerung an die 17. S itzung" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 61 "Die Musterloesung17.mws \+ wird nach dem 4.7.97 lesbar gemacht." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}}{EXCHG {PARA 264 "" 0 "" {TEXT -1 90 " ----- schneide das Worksh eet hier ab und reiche die Loesung des unteren Teils ein ! -----" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 451 "Die folgende Aufgabe soll geloest und zur Korrektur abgeliefert werden (ihre erfolgreiche Bearbeitung z aehlt als Teilkriterium fuer die Vergabe des Praktikumsscheins zu dies em Kurs). Ergaenze dazu den Rest dieses Worksheets mit den MAPLE-Befeh len, die die Loesung der Aufgabe liefern. Trage Namen und (falls vorha nden) die flcaXX-Nummer ein. Raeume auf: entferne eventuelle Ausgabe n aus dem Worksheet. Speichere das Worksheet, etwa unter dem Namen " }{TEXT 312 11 "Loesung.mws" }{TEXT -1 53 " , dann reiche diese Datei \+ per electronic mail ein: " }}{PARA 263 "" 0 "" {TEXT -1 1 " " }{TEXT 314 48 "elm flca00@rz.uni-frankfurt.de < Loesung.mws " }}{PARA 0 " " 0 "" {TEXT 319 34 "Abgabe: bis Freitag, den 18.7.97" }{TEXT -1 29 ". Nach diesem Datum wird in " }{TEXT 315 24 "/home/fb12/kurse/flca00 " }{TEXT -1 15 "das Worksheet " }{TEXT 313 19 "Musterloesung18.mws" }{TEXT -1 130 " lesbar gemacht. Korrigierte Versionen der eingesandte n Loesungen werden individuell per electronic mail zurueckgeschickt we rden." }}{PARA 0 "" 0 "" {TEXT 316 10 "Trage ein:" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT 321 18 "Name und Vorname :" }{TEXT -1 6 " \+ " }{TEXT 320 19 "Mustermann, Hermann" }}{PARA 0 "" 0 "" {TEXT 322 18 " flcaXX-Nummer :" }{TEXT -1 7 " " }{TEXT 317 6 "flca??" }} {PARA 0 "" 0 "" {TEXT 323 18 "oder email :" }{TEXT -1 7 " \+ " }{TEXT 318 26 "flca??@rz.uni-frankfurt.de" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 528 "Die folgende (letzte) Aufgabe 6 diese Kurses ist das ``G esellenstueck'' im Umgang mit MAPLE: es werden vom System kaum Hilfsmi ttel zur nichtkommutativen Algebra zur Verfuegung gestellt, so dass ge eignete Strukturen selbst programmiert werden muessen. Der aufwendigst e Teil ist Aufgabe 6.a), Teil 6.b) ist trivial, ebenso Teil 6.c). T eil 6.d) kann, falls das nichtkommutative Produkt und die darauf aufba uende Exponentialfunktion richtig implementiert sind, sehr schnell gel oest werden (erzeuge Gleichungen, loese sie mittels " }{TEXT 361 5 "so lve" }{TEXT -1 3 " )." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 256 "" 0 "" {TEXT -1 68 "Eine abstrakte mathematische Struktur: \+ Lie -Algebren und Lie-Gruppen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 8 "Es sei " }{TEXT 330 1 "g" }{TEXT -1 42 " \+ eine Algebra ueber dem Skalarenkoerper " }{TEXT 333 1 "K" }{TEXT -1 3 " (=" }{TEXT 336 1 "R" }{TEXT -1 8 " bzw. = " }{TEXT 337 1 "C" } {TEXT -1 56 "), d.h., es existiert ein Produkt (Multiplikation) \+ " }}{PARA 0 "" 0 "" {TEXT 331 24 " &m : g x g -> g " }{TEXT -1 3 ", " }}{PARA 0 "" 0 "" {TEXT -1 25 "das dem Distibutivgesetz" }} {PARA 0 "" 0 "" {TEXT -1 8 " i) " }{TEXT 327 59 "(x+y) &m (a+b) = x &m a + x &m b + y &m a + y &m b" }{TEXT -1 9 " , " } {TEXT 328 8 "x,y,a,b " }{TEXT -1 5 " in " }{TEXT 329 1 "g" }}{PARA 0 "" 0 "" {TEXT -1 8 " ii) " }{TEXT 332 64 "(alpha x ) &m (beta y ) \+ = alpha beta (x &m y ) , x,y " }{TEXT -1 2 "in" }{TEXT 334 21 " g , alpha, beta " }{TEXT -1 2 "in" }{TEXT 335 3 " K" }} {PARA 0 "" 0 "" {TEXT -1 37 "genuege. Das Produkt sei assoziativ" }} {PARA 0 "" 0 "" {TEXT -1 7 "iii) " }{TEXT 338 36 "( x &m y ) &m z = x &m ( y &m z)" }}{PARA 0 "" 0 "" {TEXT -1 45 "aber nicht notwendi gerweise kommmutativ: " }{TEXT 339 20 "x &m y <> y &m x" } {TEXT -1 42 " . Es existiere ein Eins-Element namens " }{TEXT 256 6 "Eins " }{TEXT -1 2 "in" }{TEXT 258 3 " g" }{TEXT -1 21 " mit der E igenschaft" }}{PARA 0 "" 0 "" {TEXT -1 7 "iv) " }{TEXT 340 34 "x &m Eins = Eins &m x = x " }{TEXT -1 15 " fuer jedes " }{TEXT 341 2 "x " }{TEXT -1 3 "in " }{TEXT 342 1 "g" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 266 "" 0 "" {TEXT -1 0 "" }} {PARA 257 "" 0 "" {TEXT -1 15 "Aufgabe 6.a): " }{TEXT 257 30 "impleme ntiere eine Funktion " }{TEXT 302 1 "`" }{TEXT 259 24 "&m`:=proc(a,b ) ... end ," }{TEXT 303 98 " die dieses Produkt repraesentiert. Sie soll die Eigenschaften i) ... iv) realisieren, d.h., " }}{PARA 265 "" 0 "" {TEXT 343 108 "i)+ ii) Summen in den Argumenten sollen expand iert werden, skalare Faktoren sollen vorgezogen werden, z.B.:" }} {PARA 0 "" 0 "" {TEXT -1 15 " " }{TEXT 344 56 "(3*x + y \+ ) &m (2* b ) --> 6 (x &m b) + 2 (y &m b)" }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{TEXT 345 37 "(3*x + y ) &m Eins - -> 3 x + y" }}{PARA 0 "" 0 "" {TEXT -1 101 "iii) Die Assoziativitaet kann dadurch implementiert werden, dass man in vernesteten Aufruf en von " }{TEXT 346 2 "&m" }{TEXT -1 52 " Klammerungen systematisch ` `nach rechts schiebt'':" }}{PARA 0 "" 0 "" {TEXT -1 15 " \+ " }{TEXT 347 40 "(x &m y) &m z --> x &m ( y &m z )" }}{PARA 0 "" 0 "" {TEXT -1 15 "iv) Der Name " }{TEXT 348 5 "Eins " }{TEXT -1 12 " soll von " }{TEXT 349 2 "&m" }{TEXT -1 43 " als neutrales Elem ent verarbeitet werden." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 324 10 "Weiterhin:" }}{PARA 0 "" 0 "" {TEXT -1 131 "Es soll die Moeglichkeit vorgesehen sein, beliebige Namen als Skalare zu dek larieren, die dann mit ii) aus dem formalen Produkt " }{TEXT 304 2 " &m" }{TEXT -1 53 " herausgezogen werden. Lege dazu eine globale Menge \+ " }{TEXT 260 9 "scalars " }{TEXT -1 48 "an, welche diejenigen Namen enthaelt, die von " }{TEXT 305 2 "&m" }{TEXT -1 93 " als Skalare auf gefasst werden sollen (alle anderen symbolischen Namen sind als Elemen te von " }{TEXT 350 1 "g" }{TEXT -1 69 " aufzufassen). Die Skalare sin d damit die MAPLE-Konstanten (vom Typ " }{TEXT 261 12 "constant ), " }{TEXT -1 82 "sowie algebraische Ausdruecke, welche aus Konstanten un d (Potenzen von) Namen in " }{TEXT 262 8 " scalars" }{TEXT -1 17 " au fgebaut sind:" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 277 25 "scal ars:= \{ eps , alpha\};" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 306 87 "(3*x + eps* y) &m (alpha^2*beta) --> 3 alpha^2 (x &m x) + \+ eps alpha^2 (y &m beta)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 325 10 "Weiterhin:" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 46 "fuer die spaeter folgenden Aufgabenteile soll " }{TEXT 307 2 "&m" }{TEXT -1 40 " so implementiert werden, dass Produkte " } {TEXT 280 16 " x &m y &m z ..." }{TEXT -1 2 " " }}{PARA 0 "" 0 "" {TEXT -1 14 "mit mehr als " }{TEXT 279 5 "Tiefe" }{TEXT -1 23 " Fakt oren als Symbol " }{TEXT 281 14 "O(g^(Tiefe+1))" }{TEXT -1 39 " zuru eckgeliefert werden. Hierbei sei " }{TEXT 282 5 "Tiefe" }{TEXT -1 44 " ein global vorzugebender ganzzahliger Wert:" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 283 43 "Tiefe:=3; ( 2*x+ y&m z) &m a &m (Eins+b) " }}{PARA 258 "" 0 "" {TEXT -1 79 " --> 2 ( x &m a ) + 2 (x &m a &m b) + y &m z &m a + O(g^4)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 351 9 "Anleitung" }{TEXT -1 27 ": beachte Kapitel 8.10 (``" }{TEXT 326 20 "Operatorschreibweise" }{TEXT -1 32 "'') der Vorlesung/des Skriptes. " }{MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 284 7 "Achtung" }{TEXT -1 67 " beim Auste sten: es bietet sich an, die benoetigten Prozeduren mit " }{TEXT 287 15 "option remember" }{TEXT -1 46 " zu implementieren. Aendert man den Wert von " }{TEXT 286 5 "Tiefe" }{TEXT -1 39 " , so ist vor weitere n Rechnungen die " }{TEXT 285 14 "remember table" }{TEXT -1 13 " zu lo eschen!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 13 "Aufgabe 6.b):" }{TEXT -1 20 " Implementiere auf " } {TEXT 352 1 "g" }{TEXT -1 7 " den " }{TEXT 288 14 "``Kommutator``" } {TEXT -1 24 " , dies ist das Produkt " }}{PARA 0 "" 0 "" {TEXT -1 12 " " }{TEXT 353 55 "&c: g x g -> g , x &c y := x &m y - y &m x ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Dieses ``Lie-Produkt`` macht g zu einer sogenannt en Lie-Algebra. Der Kommutator is offensichtlich antisymmetrisch " }}{PARA 0 "" 0 "" {TEXT -1 12 " " }{TEXT 354 22 "x &c y = - y &c x " }{TEXT -1 3 " , " }}{PARA 0 "" 0 "" {TEXT -1 65 " aber n icht assoziativ. Es gilt die sogenannte Jacobi-Identitaet" }}{PARA 0 "" 0 "" {TEXT -1 6 " " }{TEXT 355 65 "x &c ( y &c z ) + y &c \+ ( z &c x ) + z &c ( x &c y ) = 0" }{TEXT -1 18 " fuer alle \+ " }{TEXT 356 5 "x,y,z" }{TEXT -1 4 " in " }{TEXT 357 1 "g" }{TEXT -1 1 "." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "Platz;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Test der Jacobi-Identitaet:" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 56 "x &c ( y &c z ) + y &c ( z &c x ) + z &c ( x &c \+ y ) ;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "Die " }{TEXT 265 19 "Expo nentialfunktion" }{TEXT -1 7 " auf " }{TEXT 358 1 "g" }{TEXT -1 27 " ist die (formale) Reihe " }}{PARA 0 "" 0 "" {TEXT -1 4 " " } {TEXT 270 70 "Exp(x) = Eins + x + (x &m x) / 2! + (x &m x &m \+ x) / 3! + ..." }}{PARA 0 "" 0 "" {TEXT -1 24 "mit der Eigenschaft \+ " }{TEXT 271 54 "Exp(x) &m Exp (-x) = Exp (-x) &m Exp (x) = Ei ns" }{TEXT -1 3 " . " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 13 "Aufgabe 6.c):" }{TEXT -1 43 " Implementiere \+ die Exponentialfunktion " }{TEXT 264 24 "Exp:=proc(x) ... end !" } {TEXT -1 5 " Da " }{TEXT 308 2 "&m" }{TEXT -1 16 " nicht mehr als " } {TEXT 278 5 "Tiefe" }{TEXT -1 101 " Multiplikationen zulaesst, brauch t die Reihe nur bis zur entsprechenden Potenz berechnet zu werden." }} {PARA 0 "" 0 "" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "T este " }{TEXT 289 3 "Exp" }{TEXT -1 22 ": es muss z.B. gelten:" }} {PARA 0 "" 0 "" {TEXT -1 4 " " }{TEXT 359 47 "Exp(x) &m Exp(-x) = \+ Eins + O(g^(Tiefe+1))" }}{PARA 0 "" 0 "" {TEXT -1 4 " " } {TEXT 360 46 "Exp(x) &m Exp( x) = Exp(2x) + O(g^(Tiefe+1))" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Die Theorie der Lie-Algebren und G ruppen liefert das Resultat, dass " }{TEXT 274 16 "Exp(x) &m Exp(y) " }{TEXT -1 7 " mit " }{TEXT 273 8 "Exp(x+y)" }{TEXT -1 23 " uebere instimmt, wenn " }{TEXT 276 1 "x" }{TEXT -1 5 " und " }{TEXT 275 1 "y " }{TEXT -1 24 " vertauschen, d.h., wenn" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 309 30 "x &c y = x &m y - y &m x = 0" }}{PARA 0 "" 0 " " {TEXT -1 139 "gilt. Allgemein ist der Unterschied zwischen den Expon entialtermen nur durch Kommutatoren bestimmt. So gilt z.B. die folgend e Entwicklung (" }{TEXT 290 31 "Baker-Campbell-Hausdorff-Formel" } {TEXT -1 2 "):" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 310 62 "Exp(-x) &m Exp(-y) &m Exp(x+y) \n = \+ Exp( alpha1 (x &c y)" }}{PARA 259 "" 0 "" {TEXT -1 42 " \+ + alpha2 (x &c ( x &c y) )" }}{PARA 260 "" 0 "" {TEXT -1 93 " \+ + alpha3 (y &c ( x &c y) )\n + alpha4 (x &c \+ ( x &c (x &c y) ) )" }}{PARA 261 "" 0 "" {TEXT -1 473 " \+ + alpha5 (x &c ( y &c (x &c y) ) )\n + alpha6 (y &c ( \+ y &c (y &c x) ) )\n + alpha7 (x &c ( x &c (x &c (x &c y )) ) )\n + alpha8 (x &c ( x &c (y &c (x &c y)) ) )\n \+ + alpha9 (y &c ( x &c (x &c (x &c y)) ) )\n \+ + alpha10 (x &c ( y &c (y &c (x &c y)) ) )\n + alpha11 \+ (y &c ( y &c (x &c (x &c y)) ) )\n + alpha12 (y &c ( y & c (y &c (x &c y)) ) )\n + O(g^6)" }}{PARA 262 "" 0 "" {TEXT -1 14 " )" }}{PARA 0 "" 0 "" {TEXT -1 13 "mit gewiss en " }{TEXT 291 12 "universellen" }{TEXT -1 53 " (von der Algebra unab haengigen) skalaren Konstanten " }{TEXT 269 14 "alphal, alpha2" } {TEXT -1 7 ", ... " }}{PARA 0 "" 0 "" {TEXT -1 74 "Anmerkung: Diese E ntwicklung ist folgendermassen zu verstehen: setzt man " }{TEXT 272 17 "x=eps*a , y=eps*b" }{TEXT -1 31 " mit einem skalaren Paremeter " }{TEXT 268 3 "eps" }{TEXT -1 39 " , so gilt die Formel bis zur Ordnun g " }{TEXT 311 8 "O(eps^6)" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 13 "Aufgabe 6.d):" }{TEXT -1 97 " Bestimme die universellen Konstanten alpha1,...,alpha12 der Baker-Campbell-Hausdorff-Formel !" }}{PARA 0 "" 0 "" {TEXT -1 4 " \+ " }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "tmp1 :=Exp(-x) &m Exp(-y) &m Exp(x+y);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 525 "tmp2:=Exp( \n alpha1 *(x &c y)\n + alpha2 *(x &c ( x &c y ) )\n + alpha3 *(y &c ( x &c y) )\n + alpha4 *(x &c ( x &c (x &c y) ) )\n + alpha5 *(x &c ( y &c (x &c y) ) )\n + alpha6 *(y &c \+ ( y &c (y &c x) ) )\n + alpha7 *(x &c ( x &c (x &c (x &c y)) ) )\n \+ + alpha8 *(x &c ( x &c (y &c (x &c y)) ) )\n + alpha9 *(y &c ( x &c (x &c (x &c y)) ) )\n + alpha10*(x &c ( y &c (y &c (x &c y)) ) \+ )\n + alpha11*(y &c ( y &c (x &c (x &c y)) ) )\n + alpha12*(y &c ( y &c (y &c (x &c y)) ) )\n + O(g^6)\n );" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "DiesSollVerschwinden:=tmp1-tmp2;" }}}}{MARK "4 1 0" 251 }{VIEWOPTS 1 1 0 3 2 1804 }