Content-Type: multipart/mixed; boundary="-------------9812300141887" This is a multi-part message in MIME format. ---------------9812300141887 Content-Type: text/plain; name="98-772.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="98-772.keywords" Dirichlet forms, von Neumann algebras, standard forms, Markovian semigroups, quantum spin systems ---------------9812300141887 Content-Type: application/postscript; name="df.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="df.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: df.dvi %%Pages: 15 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips -f df.dvi %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1998.12.30:1532 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 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5119 y(\(b\))48 b Fm(E)c Ff(is)34 b(a)h(Dirichlet)f(form.)1738 5943 y Fn(4)p eop %%Page: 5 5 5 4 bop -246 279 a Fi(3)161 b(Construction)52 b(of)i(Diric)l(hlet)g(F) -13 b(orms)-246 498 y Fn(In)41 b(this)f(section)h(w)m(e)g(construct)g (Diric)m(hlet)e(forms)g(on)i(the)f(natural)g(standard)g(forms)g(of)g(v) m(on)h(Neumann)-246 619 y(algebras)23 b(asso)s(ciated)g(with)g(the)g(T) -8 b(omita-T)g(ak)m(esaki)22 b(theory[BR,T)-8 b(ak].)42 b(The)24 b(forms)f(w)m(e)h(construct)g(generate)-246 739 y(Mark)m(o)m(vian)33 b(semigroups.)-99 859 y(Let)i Fm(M)f Fn(b)s(e)h(a)f Fl(\033)t Fn(-\014nite)g(v)m(on)i(Neumann)f (algebra)e(acting)h(on)h(a)f(Hilb)s(ert)f(space)j Fm(H)g Fn(and)f(let)f Fl(\030)3382 874 y Fk(0)3452 859 y Fm(2)e(H)k Fn(b)s(e)-246 980 y(a)k(cyclic)f(and)h(sep)s(erating)g(v)m(ector)h(for) e Fm(M)p Fn(.)65 b(W)-8 b(e)40 b(use)h(\001)f(and)g Fl(J)49 b Fn(to)40 b(denote)g(resp)s(ectiv)m(ely)-8 b(,)43 b(the)d(mo)s(dular) -246 1100 y(op)s(erator)e(and)g(the)h(mo)s(dular)d(conjugation)g(asso)s 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b(\014rst)h(sho)m(w)f(that)j(~)-52 b Fl(g)43 b Fn(is)c(a)g(non-negativ)m(e)h(function.)64 b(Using)40 b(the)g(residue)g(in)m(tegration)e(metho)s(d,)j(it)e(is)-246 5143 y(easy)34 b(to)e(c)m(hec)m(k)j(that)764 5257 y Fd(Z)863 5283 y Fe(1)819 5483 y(\0001)949 5393 y Fn(\(cosh)17 b Fl(k)s Fn(\))1280 5352 y Fe(\000)p Fk(1)1374 5393 y Fl(e)1419 5352 y Fh(ik)r(t)1528 5393 y Fl(dk)86 b Fn(=)d(2)p Fl(\031)t(e)2028 5352 y Fe(\000)2093 5325 y Fc(\031)p 2093 5337 38 3 v 2097 5378 a Fb(2)2141 5352 y Fe(j)p Fh(t)p Fe(j)2210 5393 y Fn(\(1)21 b(+)i Fl(e)2462 5352 y Fe(\000)p Fh(\031)r Fe(j)p Fh(t)p Fe(j)2628 5393 y Fn(\))2666 5352 y Fe(\000)p Fk(1)1716 5613 y Fn(=)83 b(2)p Fl(\031)t Fn(\()p Fl(e)2076 5544 y Fc(\031)p 2076 5556 V 2080 5598 a Fb(2)2124 5571 y Fh(t)2176 5613 y Fn(+)22 b Fl(e)2319 5571 y Fe(\000)2384 5544 y Fc(\031)p 2384 5556 V 2388 5598 a Fb(2)2432 5571 y Fh(t)2461 5613 y Fn(\))2499 5571 y Fe(\000)p Fk(1)2594 5613 y Fl(:)949 b Fn(\(3.2\))1738 5943 y(5)p eop %%Page: 6 6 6 5 bop -246 279 a 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b(all)e(requiremen)m(ts.)44 b(Since)31 b Fl(f)42 b Fn(is)31 b(the)h(con)m(v)m(olution)f(of)g Fl(g)j Fn(and)e(exp)q(\()p Fm(\000)p Fl(t)2673 1018 y Fk(2)2713 1054 y Fl(=)p Fn(2\),)f Fl(f)42 b Fn(is)31 b(non-negativ)m(e)g(on)-246 1175 y Fg(R)5 b Fn(.)50 b(Notice)32 b(that)g(for)g(an)m(y)i Fl(t)27 b Fm(2)i Fg(R)725 1436 y Fl(f)11 b Fn(\()p Fl(t)22 b Fn(+)g Fl(i=)p Fn(4\))g(+)g Fl(f)11 b Fn(\()p Fl(t)22 b Fm(\000)g Fl(i=)p Fn(4\))84 b(=)2012 1368 y(2)p 1941 1413 191 4 v 1941 1433 a Fm(p)p 2025 1433 108 4 v 2025 1515 a Fn(2)p Fl(\031)2159 1300 y Fd(Z)2275 1436 y Fl(e)2320 1395 y Fe(\000)2385 1368 y Fb(1)p 2385 1380 31 3 v 2385 1421 a(2)2425 1395 y Fh(k)2464 1371 y Fb(2)2503 1436 y Fl(e)2548 1395 y Fe(\000)p Fh(ik)r(t)2695 1436 y Fl(dk)1771 1626 y Fm(\025)g Fn(0)p Fl(:)-246 1844 y Fn(It)33 b(is)f(easy)i(to)e(c) m(hec)m(k)j(that)d(for)g(an)m(y)h Fl(n)28 b Fm(2)g Fg(N)48 b Fn(there)33 b(is)f(a)h(p)s(ositiv)m(e)e(constan)m(t)j Fl(M)10 b Fn(\()p Fl(n)p Fn(\))33 b(suc)m(h)h(that)f(the)g(b)s(ound) 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y(3.1\(c\))34 b(holds)g(with)h Fl(p)g Ff(gr)-5 b(e)g(ater)35 b(than)f Fl(\013)23 b Fn(+)f(1)p Ff(.)45 b(Se)-5 b(e)34 b(the)h(pr)-5 b(o)g(of)34 b(of)h(the)g(p)-5 b(art)35 b(\(c\))f(of)h(The)-5 b(or)g(em)33 b(3.3.)-99 2146 y Fn(The)g(follo)m(wing)d(is)i(a)g(consequence)k(of)c(Theorem)h(3.3:)-246 2349 y Fj(Theorem)k(3.5.)50 b Ff(L)-5 b(et)45 b Fm(f)p Fl(T)738 2364 y Fh(t)768 2349 y Fm(g)818 2364 y Fh(t)p Fe(\025)p Fk(0)982 2349 y Ff(b)-5 b(e)45 b(the)h(semigr)-5 b(oup)44 b(gener)-5 b(ate)g(d)44 b(by)i(the)f(form)f Fn(\()p Fm(E)9 b Fl(;)17 b Fm(H)q Fn(\))45 b Ff(in)g(The)-5 b(or)g(em)44 b(3.3.)-246 2470 y(Then)34 b Fm(f)p Fl(T)115 2485 y Fh(t)145 2470 y Fm(g)195 2485 y Fh(t)p Fe(\025)p Fk(0)349 2470 y Ff(is)h(a)g Fl(J)9 b Ff(-r)-5 b(e)g(al,)34 b(str)-5 b(ongly)35 b(c)-5 b(ontinuous,)35 b(symmetric)f(Markovian)g (semigr)-5 b(oup.)-246 2781 y Fj(Pro)s(of.)49 b Fn(It)25 b(follo)m(ws)f(from)g(Theorem)i(3.3\(a\))e(that)h Fl(T)1680 2796 y Fh(t)1710 2781 y Fn(\()p Fl(\030)1791 2796 y Fk(0)1830 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Fn(].)-99 5271 y(\(c\))32 b(By)h(the)g(expression)h(of)e Fm(E)9 b Fn([)p Fl(\030)c Fn(])32 b(in)g(\(3.5\),)g Fm(E)9 b Fn(\()p Fl(\030)1667 5286 y Fk(+)1725 5271 y Fl(;)17 b(\030)1812 5286 y Fe(\000)1870 5271 y Fn(\))33 b(can)g(b)s(e)f(written)h(as)1352 5491 y Fm(E)9 b Fn(\()p Fl(\030)1495 5506 y Fk(+)1553 5491 y Fl(;)17 b(\030)1640 5506 y Fe(\000)1699 5491 y Fn(\))27 b(=)h(I)22 b(+)55 b(I)s(I)p Fl(;)1414 b Fn(\(3.6\))1738 5943 y(7)p eop %%Page: 8 8 8 7 bop -246 279 a Fn(where)97 523 y(I)83 b(=)374 387 y Fd(Z)490 442 y(\000)536 523 y Fn(\()p Fl(\033)629 538 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)808 523 y Fn(\()p Fl(x)p Fn(\))p Fl(\030)982 538 y Fk(+)1041 523 y Fl(;)17 b(\033)1140 538 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)1320 523 y Fn(\()p Fl(x)p Fn(\))p Fl(\030)1494 538 y Fe(\000)1553 523 y Fn(\))22 b(+)g(\()p Fl(\033)1804 538 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)1983 523 y Fn(\()p Fl(x)p Fn(\))p Fl(\030)2157 538 y Fe(\000)2216 523 y Fl(;)17 b(\033)2315 538 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)2494 523 y Fn(\()p Fl(x)p Fn(\))p Fl(\030)2668 538 y Fk(+)2727 523 y Fn(\))2765 442 y Fd(\001)2828 523 y Fl(f)11 b Fn(\()p Fl(t)p Fn(\))17 b Fl(dt;)442 b Fn(\(3.7\))59 777 y(I)s(I)83 b(=)g Fm(\000)468 642 y Fd(Z)584 697 y(\000)630 777 y Fn(\()p Fl(\033)723 793 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)902 777 y Fn(\()p Fl(x)p Fn(\))p Fl(\030)1076 792 y Fk(+)1135 777 y Fl(;)17 b(j)23 b Fn(\()p Fl(\033)1335 793 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)1514 777 y Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)1726 792 y Fe(\000)1785 777 y Fn(\))f(+)g(\()p Fl(j)h Fn(\()p Fl(\033)2137 793 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)2316 777 y Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)2528 792 y Fk(+)2587 777 y Fl(;)17 b(\033)2686 793 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)2865 777 y Fn(\()p Fl(x)p Fn(\))p Fl(\030)3039 792 y Fe(\000)3098 777 y Fn(\))3136 697 y Fd(\001)3198 777 y Fl(f)11 b Fn(\()p Fl(t)p Fn(\))17 b Fl(dt:)-246 1046 y Fn(By)47 b(Theorem)g(4)f(of)g([Ara],)j Fm(M)p Fl(\030)1020 1061 y Fk(+)1130 1046 y Fm(?)i(M)p Fl(\030)1421 1061 y Fe(\000)1480 1046 y Fn(,)e(whic)m(h)e(implies)d(I)51 b(=)g(0.)85 b(Next,)50 b(w)m(e)e(consider)e(I)s(I.)h(Since)-246 1166 y Fl(\033)-191 1182 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)-11 1166 y Fn(\()p Fl(x)p Fn(\))120 1130 y Fe(\003)187 1166 y Fn(=)28 b Fl(\033)346 1182 y Fh(t)p Fk(+)p Fh(i=)p Fk(4)525 1166 y Fn(\()p Fl(x)p Fn(\))33 b(and)g Fl(j)22 b Fn(\()p Fl(\033)1034 1182 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)1214 1166 y Fn(\()p Fl(x)p Fn(\)\))28 b(=)f Fl(\033)1569 1182 y Fh(t)p Fk(+)p Fh(i=)p Fk(4)1749 1166 y Fn(\()p Fl(j)6 b Fn(\()p Fl(x)p Fn(\)\),)385 1386 y(\()p Fl(\033)478 1402 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)658 1386 y Fn(\()p Fl(x)p Fn(\))p Fl(\030)832 1401 y Fk(+)907 1386 y Fl(;)34 b(j)22 b Fn(\()p Fl(\033)1123 1402 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)1303 1386 y Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)1515 1401 y Fe(\000)1574 1386 y Fn(\))83 b(=)g(\()p Fl(\033)1947 1402 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)2126 1386 y Fn(\()p Fl(x)p Fn(\))p Fl(\030)2300 1401 y Fk(+)2376 1386 y Fl(;)33 b Fn(\()p Fl(\033)2529 1402 y Fh(t)p Fk(+)p Fh(i=)p Fk(4)2708 1386 y Fn(\()p Fl(j)6 b Fn(\()p Fl(x)p Fn(\)\)\))p Fl(\030)3042 1401 y Fe(\000)3101 1386 y Fn(\))1695 1531 y(=)83 b(\()17 b Fl(\030)1952 1546 y Fk(+)2026 1531 y Fl(;)34 b(\033)2142 1547 y Fh(t)p Fk(+)p Fh(i=)p Fk(4)2321 1531 y Fn(\()p Fl(xj)6 b Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)2672 1546 y Fe(\000)2731 1531 y Fn(\))801 b(\(3.8\))-246 1751 y(and)527 1971 y(\()p Fl(j)23 b Fn(\()p Fl(\033)721 1987 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)900 1971 y Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)1112 1986 y Fk(+)1188 1971 y Fl(;)33 b Fn(\()p Fl(\033)1341 1987 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)1520 1971 y Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)1732 1986 y Fe(\000)1791 1971 y Fn(\))83 b(=)g(\()p Fl(\033)2164 1987 y Fh(t)p Fk(+)p Fh(i=)p Fk(4)2344 1971 y Fn(\()p Fl(xj)6 b Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)2695 1986 y Fk(+)2770 1971 y Fl(;)34 b(\030)2874 1986 y Fe(\000)2932 1971 y Fn(\))1912 2117 y(=)83 b(\()p Fl(\030)2152 2132 y Fk(+)2227 2117 y Fl(;)34 b(\033)2343 2132 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)2522 2117 y Fn(\()p Fl(xj)6 b Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)2873 2132 y Fe(\000)2932 2117 y Fn(\))p Fl(:)573 b Fn(\(3.9\))-246 2337 y(It)33 b(follo)m(ws)e(from)h(\(3.7\)-\(3.9\))f (that)439 2605 y(I)s(I)c(=)h Fm(\000)737 2470 y Fd(Z)853 2524 y(\000)899 2605 y Fn(\()p Fl(\030)980 2620 y Fk(+)1039 2605 y Fl(;)17 b(\033)1138 2621 y Fh(t)p Fk(+)p Fh(i=)p Fk(4)1317 2605 y Fn(\()p Fl(xj)6 b Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)1668 2620 y Fe(\000)1727 2605 y Fn(\))22 b(+)g(\()p Fl(\030)1966 2620 y Fk(+)2025 2605 y Fl(;)17 b(\033)2124 2621 y Fh(t)p Fe(\000)p Fh(i=)p Fk(4)2303 2605 y Fn(\()p Fl(xj)6 b Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)2654 2620 y Fe(\000)2713 2605 y Fn(\))2751 2524 y Fd(\001)2813 2605 y Fl(f)11 b Fn(\()p Fl(t)p Fn(\))17 b Fl(dt)-246 2874 y Fn(Using)30 b(the)g(Cauc)m(h)m(y)i(in)m(tegral)c(theorem,)j(the) f(assumption)f(in)g(the)i(theorem)e(and)h(the)h(prop)s(ert)m(y)f(\(c\)) g(in)g(the)-246 2994 y(De\014nition)h(3.1,)h(w)m(e)i(obtain)e(that)599 3257 y(I)s(I)c(=)f Fm(\000)897 3121 y Fd(Z)997 3257 y Fn(\()p Fl(\030)1078 3272 y Fk(+)1137 3257 y Fl(;)17 b(\033)1236 3272 y Fh(t)1265 3257 y Fn(\()p Fl(xj)6 b Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)1616 3272 y Fe(\000)1676 3257 y Fn(\))1714 3146 y Fd(\020)1773 3257 y Fl(f)11 b Fn(\()p Fl(t)22 b Fm(\000)h Fl(i=)p Fn(4\))f(+)g Fl(f)11 b Fn(\()p Fl(t)22 b Fn(+)g Fl(i=)p Fn(4\))2737 3146 y Fd(\021)2812 3257 y Fl(dt:)-246 3525 y Fn(Since)35 b Fl(\033)66 3540 y Fh(t)96 3525 y Fn(\()p Fl(xj)6 b Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)447 3540 y Fe(\000)537 3525 y Fm(2)31 b(P)8 b Fn(,)35 b(\()p Fl(\030)854 3540 y Fk(+)913 3525 y Fl(;)17 b(\033)1012 3540 y Fh(t)1042 3525 y Fn(\()p Fl(xj)6 b Fn(\()p Fl(x)p Fn(\)\))p Fl(\030)1393 3540 y Fe(\000)1452 3525 y Fn(\))31 b Fm(\025)g Fn(0)j(for)g Fl(t)d Fm(2)g Fg(R)5 b Fn(.)55 b(By)35 b(the)g(prop)s(ert)m(y)g(\(b\))f (in)g(the)h(De\014nition)-246 3646 y(3.1,)e(w)m(e)g(conclude)g(that)g (I)s(I)f Fm(\024)c Fn(0.)44 b(This)32 b(pro)m(v)m(ed)i(the)f(part)g 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Fl(e)2722 4705 y Fe(\000)p Fh(tH)2904 4741 y Fn(on)j Fm(H)i Fn(constructed)g(in)-246 4862 y(Section)d(3,)f(and)h(in)m(v)m(estigate)f(detailed)g(prop)s (erties)g(of)h Fl(L)p Fn(.)-99 4982 y(Let)f(us)i(consider)f(the)g (symmetric)e(em)m(b)s(edding:)1370 5202 y Fl(i)1403 5217 y Fk(0)1550 5202 y Fn(:)107 b Fm(M)27 b(!)h(H)1221 5347 y Fl(i)1254 5362 y Fk(0)1293 5347 y Fn(\()p Fl(A)p Fn(\))83 b(=)g(\001)1765 5306 y Fk(1)p Fh(=)p Fk(4)1875 5347 y Fl(A)p Fn(\001)2029 5306 y Fe(\000)p Fk(1)p Fh(=)p Fk(4)2195 5347 y Fl(\030)2238 5362 y Fk(0)2277 5347 y Fl(:)1738 5943 y Fn(8)p eop %%Page: 9 9 9 8 bop -246 279 a Fn(Then)41 b Fl(L)f Fn(and)g Fl(H)47 b Fn(are)40 b(related)g(b)m(y)g Fl(i)1125 294 y Fk(0)1192 279 y Fm(\016)27 b Fl(L)40 b Fn(=)g Fl(H)34 b Fm(\016)27 b Fl(i)1716 294 y Fk(0)1796 279 y Fn([Cip2].)64 b(Note)40 b(that)g Fl(j)6 b Fn(\()p Fl(A)p Fn(\))p Fl(\030)2842 294 y Fk(0)2921 279 y Fn(=)40 b(\001)3118 243 y Fk(1)p Fh(=)p Fk(2)3228 279 y Fl(A)3301 243 y Fe(\003)3340 279 y Fl(\030)3383 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Fh(t)1013 1894 y Fn(\()p Fl(x)p Fn(\))p Fl(;)34 b(A)1278 1853 y Fe(\003)1334 1894 y Fn(]\001)1442 1853 y Fk(1)p Fh(=)p Fk(2)1552 1894 y Fn([)p Fl(\033)1634 1909 y Fh(t)1664 1894 y Fn(\()p Fl(x)p Fn(\))17 b Fl(;)g Fn(\001)1937 1853 y Fe(\000)p Fk(1)p Fh(=)p Fk(2)2102 1894 y Fl(A)p Fn(\001)2256 1853 y Fk(1)p Fh(=)p Fk(2)2367 1894 y Fn(]\001)2475 1853 y Fe(\000)p Fk(1)p Fh(=)p Fk(2)808 2039 y Fn(+\001)965 1998 y Fe(\000)p Fk(1)p Fh(=)p Fk(2)1130 2039 y Fn([)p Fl(\033)1212 2054 y Fh(t)1243 2039 y Fn(\()p Fl(x)p Fn(\))g Fl(;)g Fn(\001)1516 1998 y Fk(1)p Fh(=)p Fk(2)1625 2039 y Fl(A)1698 1998 y Fe(\003)1738 2039 y Fn(\001)1819 1998 y Fe(\000)p Fk(1)p Fh(=)p Fk(2)1984 2039 y Fn(]\001)2092 1998 y Fk(1)p Fh(=)p Fk(2)2203 2039 y Fn([)g Fl(\033)2302 2054 y Fh(t)2331 2039 y Fn(\()p Fl(x)p Fn(\))p Fl(;)34 b(A)p Fn(])808 2185 y(+)p Fl(A)957 2144 y Fe(\003)997 2185 y Fn(\001)1078 2144 y Fe(\000)p Fk(1)p Fh(=)p Fk(2)1243 2185 y Fn([)17 b Fl(\033)1342 2200 y Fh(t)1372 2185 y Fn(\()p Fl(x)p Fn(\))g 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Fm(\001)g(\001)g(\001)31 b Fl(;)17 b(D)s Fm(g)31 b Fn(w)m(e)j(represen)m(t)g(the)f(sum)g(on)f (the)h(righ)m(t)f(hand)h(side)g(of)f(\(5.8\))g(as)h(follo)m(ws:)682 5370 y Fd(X)675 5582 y Fh(N)7 b Fe(2)p Fa(N)1069 5370 y Fd(X)883 5586 y Fh(N)g Fe(\000)p Fk(1)p Fe(\024)p Fh(d)p Fk(\()p Fh(j;)p Fk(\003\))p Fh()1868 883 y Fc(N)p 1862 895 V 1862 936 a Fb(2)p Fc(\027)1958 820 y Fm(jj)17 b Fn([)p Fl(\033)2113 835 y Fh(t)2142 820 y Fn(\()p Fl(x)2235 779 y Fh(a)2235 845 y(j)2277 820 y Fn(\))p Fl(;)33 b(A)17 b Fn(])g Fm(jj)2565 779 y Fk(2)2603 820 y Fl(f)11 b Fn(\()p Fl(t)p Fn(\))17 b Fl(dt:)618 b Fn(\(5.11\))-246 1132 y(No)m(w,)34 b(using)e(the)h(\014nite)f(sp)s(eed)i(of)e(propagation)f (\(5.9\),)h(one)h(has)g(that)935 1381 y(I)970 1396 y Fk(1)1037 1381 y Fm(\024)1144 1286 y Fd(X)1142 1498 y Fk(N)p Fe(2)p Fa(N)1513 1286 y Fd(X)1340 1502 y Fk(N)p Fe(\000)p Fk(1)p Fe(\024)p Fk(d\(j)p Fh(;)p Fk(\003\))p Fh(<)p Fk(N)1856 1314 y Fn(N)p 1856 1358 74 4 v 1865 1449 a Fl(\027)1939 1381 y Fn(D\(A\)e)2206 1340 y Fe(\000)p Fh(\024)p Fk(N)2386 1381 y Fl(<)28 b Fm(1)931 b Fn(\(5.12\))-246 1687 y(On)33 b(the)g(other)g(hand,)g(w)m(e)g(use)h(the)f(deca)m(y)h (prop)s(ert)m(y)f(of)f Fl(f)43 b Fn(to)33 b(conclude)g(that)586 1931 y(I)621 1946 y Fk(2)743 1931 y Fm(\024)911 1836 y Fd(X)904 2048 y Fh(N)7 b Fe(2)p Fa(N)1297 1836 y Fd(X)1112 2052 y Fh(N)g Fe(\000)p Fk(1)p Fe(\024)p Fh(d)p Fk(\()p Fh(j;)p Fk(\003\))p Fh()2104 1994 y Fc(N)p 2098 2006 65 3 v 2098 2047 a Fb(2)p Fc(\027)2177 1931 y Fn(\(1)22 b(+)g Fm(j)p Fl(t)p Fm(j)p Fn(\))2513 1890 y Fe(\000)p Fk(\()p Fh(d)p Fk(+1+)p Fh(\017)p Fk(\))2853 1931 y Fl(dt)743 2216 y Fm(\024)911 2122 y Fd(X)904 2334 y Fh(N)7 b Fe(2)p Fa(N)1297 2122 y Fd(X)1112 2338 y Fh(N)g Fe(\000)p Fk(1)p Fe(\024)p Fh(d)p Fk(\()p Fh(j;)p Fk(\003\))p Fh(