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y(in)26 b(the)g(T)-8 b(aylor)n (\255Couette)26 b(problem.)39 b(These)26 b(fronts)f(ha)n(v)o(e)h(been)h (constructed)e(in)h([HS99].)40 b(The)26 b(stability)e(of)94 4555 y(the)h(spatially)f(periodic)g(T)-8 b(aylor)25 b(v)n(ortices)f (has)h(been)g(sho)n(wn)e(in)i([Schn98].)94 4733 y Fv(Notation.)d Fu(Throughout)e(this)g(paper)i(man)o(y)e(dif)n(ferent)h(constants)g (are)h(denoted)f(with)f(the)i(same)f(symbol)f Fr(C)7 b Fu(.)94 4911 y Fv(Ackno)o(wledgements.)22 b Fu(Guido)c(Schneider)i(w) o(ould)f(lik)o(e)g(to)g(thank)g(for)h(the)f(kind)g(hospitality)e(at)j (the)f(Physics)94 5031 y(Department)h(of)h(the)f(Uni)n(v)o(ersity)e(of) j(Gene)n(v)n(a.)34 b(This)19 b(w)o(ork)i(is)f(partially)f(supported)h (by)g(the)g(F)o(onds)g(National)94 5150 y(Suisse.)64 b(The)35 b(w)o(ork)f(of)g(Guido)f(Schneider)i(is)f(partially)f (supported)h(by)g(the)g(Deutsche)g(F)o(orschungsge\255)94 5270 y(meinschaft)25 b(DFG)g(under)f(the)h(grant)g(Mi459/2\2263.)p eop %%Page: 5 5 5 4 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2111 b Fv(5)94 194 y Fs(P)o(art)30 b(I.)f(A)h(simpli\002ed)i(pr)n(oblem)94 471 y(2.)e(The)g(model)g (equation)119 660 y Fu(Let)1514 780 y Fr(a)p Fq(\()p Fr(\030)5 b Fq(\))42 b(=)1870 740 y FE(1)p 1870 757 35 4 v 1870 814 a(2)1917 780 y Fq(\()p Fu(1)22 b Fq(+)g Fu(tanh)16 b Fr(\030)5 b Fq(\))24 b Fr(:)1214 b Fq(\()p Fu(2)p Fr(:)p Fu(1)p Fq(\))94 946 y Fu(W)-8 b(e)26 b(w)o(ant)e(to)h (study)f(the)g(equation)897 1132 y Fr(@)950 1159 y FD(t)985 1132 y Fr(u)p Fq(\()p Fr(x;)17 b(t)p Fq(\))43 b(=)i Fr(@)1482 1089 y FE(2)1476 1159 y FD(x)1526 1132 y Fr(u)p Fq(\()p Fr(x;)17 b(t)p Fq(\))k(+)h Fr(a)p Fq(\()p Fr(x)g Fp(\000)h Fr(ct)p Fq(\))p Fr(u)p Fq(\()p Fr(x;)17 b(t)p Fq(\))j(+)j Fr(u)p Fq(\()p Fr(x;)17 b(t)p Fq(\))2974 1089 y FD(p)3044 1132 y Fr(;)597 b Fq(\()p Fu(2)p Fr(:)p Fu(2)p Fq(\))94 1319 y Fu(with)25 b Fr(c)i(>)h Fu(0)d(and)g Fr(p)j(>)g Fu(3.)36 b(F)o(or)24 b(notational)g(simplicity)e(we)j(assume)g Fr(p)i Fp(2)h Fv(N)p Fu(.)316 1439 y(T)-8 b(o)21 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b(not)g(depend)h(on) f Fr(U)1599 5071 y FE(0)1639 5044 y Fg(.)94 5291 y Fv(Remark.)i Fu(Note)e(that)h(it)f(is)g(optimal)g(to)g(choose)j Fq(^)-52 b Fr(c)25 b Fu(arbitrarily)f(close)h(to)f Fr(c)p Fu(.)p eop %%Page: 9 9 9 8 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2111 b Fv(9)94 194 y(Pr)n(oof)o(.)24 b Fu(First)f(of)g(all,)h(we)f(re)n(write)h(the)f(equation)f(\(3.1\))i (for)g Fr(U)2216 221 y FD(t)2274 194 y Fu(in)f(terms)g(of)2741 169 y Fo(e)2727 194 y Fr(U)2795 221 y FD(t)2853 194 y Fu(and)3025 169 y Fo(f)3021 194 y Fr(W)3115 221 y FD(t)3150 194 y Fu(:)35 b(The)23 b(equation)g(for)94 314 y Fr(W)188 341 y FD(t)251 314 y Fq(=)28 b Fp(W)454 341 y FD(\014)t(;)r Fk(^)-42 b FD(ct)598 314 y Fr(U)666 341 y FD(t)726 314 y Fu(is)800 544 y Fr(@)853 571 y FD(t)888 544 y Fr(W)14 b Fq(\()p Fr(\030)5 b(;)17 b(t)p Fq(\))42 b(=)j Fr(@)1427 501 y FE(2)1421 571 y FD(\030)1466 544 y Fr(W)14 b Fq(\()p Fr(\030)5 b(;)17 b(t)p Fq(\))j(+)i(\()r(^)-52 b Fr(c)22 b Fp(\000)h Fu(2)p Fr(\014)5 b Fq(\))p Fr(@)2308 571 y FD(\030)2352 544 y Fr(W)14 b Fq(\()p Fr(\030)5 b(;)17 b(t)p Fq(\))1240 714 y(+)23 b Fr(a)p Fq(\()p Fr(\030)j Fp(\000)c Fq(\()p Fr(c)g Fp(\000)j Fq(^)-52 b Fr(c)p Fq(\))p Fr(t)p Fq(\))p Fr(W)14 b Fq(\()p Fr(\030)5 b(;)17 b(t)p Fq(\))i(+)k(\()p Fr(\014)2497 671 y FE(2)2559 714 y Fp(\000)g Fr(\014)7 b Fq(^)-52 b Fr(c)q Fq(\))p Fr(W)14 b Fq(\()p Fr(\030)5 b(;)17 b(t)p Fq(\))22 b Fr(:)3669 626 y Fq(\()p Fu(3)p Fr(:)p Fu(8)p Fq(\))94 918 y Fu(T)-8 b(aking)19 b(F)o(ourier)h(transforms,)f(we)h(then)f(\002nd,)i(omitting) c(the)j(ar)n(gument)f Fr(k)j Fu(and)e(using)e(the)i(notation)e(of)h (\(2.5\):)745 1128 y Fr(@)798 1155 y FD(t)848 1102 y Fo(e)833 1128 y Fr(U)901 1155 y FD(t)981 1128 y Fq(=)44 b Fp(\000)p Fr(k)1234 1085 y FE(2)1289 1102 y Fo(e)1275 1128 y Fr(U)1343 1155 y FD(t)1400 1128 y Fq(+)23 b(\()1555 1102 y Fo(e)1539 1128 y Fr(T)1597 1155 y FD(ct)1669 1128 y Fq(~)-52 b Fr(a)p Fq(\))22 b Fp(\003)1867 1102 y Fo(e)1853 1128 y Fr(U)1921 1155 y FD(t)1981 1128 y Fr(;)1660 b Fq(\()p Fu(3)p Fr(:)p Fu(9)p Fq(\))719 1294 y Fr(@)772 1321 y FD(t)811 1269 y Fo(f)807 1294 y Fr(W)901 1321 y FD(t)981 1294 y Fq(=)1102 1214 y Fo(\000)1148 1294 y Fr(\014)1209 1251 y FE(2)1272 1294 y Fp(\000)22 b Fr(\014)7 b Fq(^)-52 b Fr(c)23 b Fp(\000)g Fr(k)1653 1251 y FE(2)1715 1294 y Fq(+)f Fr(ik)s Fq(\()r(^)-52 b Fr(c)23 b Fp(\000)f Fu(2)p Fr(\014)5 b Fq(\))2257 1214 y Fo(\001)2307 1269 y(f)2303 1294 y Fr(W)2397 1321 y FD(t)2455 1294 y Fq(+)22 b(\()2610 1269 y Fo(e)2593 1294 y Fr(T)2651 1321 y Fk(\()p FD(c)p Fm(\000)r Fk(^)-42 b FD(c)p Fk(\))p FD(t)2883 1294 y Fq(~)-51 b Fr(a)p Fq(\))22 b Fp(\003)3072 1269 y Fo(f)3067 1294 y Fr(W)3161 1321 y FD(t)3222 1294 y Fr(:)369 b Fq(\()p Fu(3)p Fr(:)p Fu(10)p Fq(\))94 1504 y Fu(It)29 b(is)f(at)g(this)f(point)h(that)f(the)i(simultaneous)d (choice)i(of)h(tw)o(o)e(representations)h(for)g(the)h(solution)d(and)i (their)94 1623 y(associated)d(topologies)e(is)h(crucial.)316 1755 y(W)-8 b(e)25 b(\002rst)g(sho)n(w)f(that)1062 1730 y Fo(f)1058 1755 y Fr(W)1152 1782 y FD(t)1212 1755 y Fu(con)l(v)o(er)n(ges)h(to)f(0,)h FF(i.e)o(.)p Fu(,)f(we)h(sho)n(w)f (\(3.7\).)35 b(W)-8 b(e)26 b(\002nd)e(from)h(\(2.9\):)883 1979 y Fp(k)p Fq(\()989 1953 y Fo(e)972 1979 y Fr(T)1030 2005 y FD(\020)1077 1979 y Fq(~)-51 b Fr(a)p Fq(\))21 b Fp(\003)1283 1952 y Fq(~)1261 1979 y Fr(f)11 b Fp(k)49 b(\024)c(k)p Fr(a)p Fq(\()p Fp(\001)21 b(\000)h Fr(\020)7 b Fq(\))p Fp(k)1972 2005 y Fm(C)2018 1981 y Fj(2)2015 2032 y(b)p Fi(;\016)2129 1979 y Fp(\001)22 b(k)2250 1952 y Fq(~)2229 1979 y Fr(f)10 b Fp(k)49 b Fq(=)c Fp(k)p Fr(a)p Fp(k)2662 2005 y Fm(C)2708 1981 y Fj(2)2705 2032 y(b)p Fi(;\016)2819 1979 y Fp(\001)21 b(k)2940 1952 y Fq(~)2918 1979 y Fr(f)11 b Fp(k)29 b Fr(:)534 b Fq(\()p Fu(3)p Fr(:)p Fu(11)p Fq(\))94 2208 y Fu(Therefore,)26 b(\(3.4\))f(implies)1190 2430 y Fp(k)p Fq(\()1295 2404 y Fo(e)1279 2430 y Fr(T)1337 2456 y Fk(\()p FD(c)p Fm(\000)r Fk(^)-42 b FD(c)o Fk(\))p FD(t)1569 2430 y Fq(~)-51 b Fr(a)o Fq(\))22 b Fp(\003)1757 2404 y Fo(f)1753 2430 y Fr(W)1847 2456 y FD(t)1882 2430 y Fp(k)49 b(\024)c Fq(\()p Fu(1)22 b Fq(+)h Fr(K)7 b(\016)t Fq(\))p Fp(k)2547 2404 y Fo(f)2543 2430 y Fr(W)2637 2456 y FD(t)2671 2430 y Fp(k)30 b Fr(;)94 2639 y Fu(and)25 b(we)g(get)g(from)g(\(3.10\))f (the)h(bound)978 2821 y FE(1)p 978 2837 35 4 v 978 2895 a(2)1025 2860 y Fr(@)1078 2887 y FD(t)1113 2860 y Fp(k)1167 2835 y Fo(f)1163 2860 y Fr(W)1257 2887 y FD(t)1292 2860 y Fp(k)1342 2817 y FE(2)1426 2860 y Fp(\024)45 b Fq(\()p Fr(\014)1648 2817 y FE(2)1710 2860 y Fp(\000)23 b Fr(\014)7 b Fq(^)-52 b Fr(c)23 b Fq(+)f Fu(1)g Fq(+)h Fr(K)7 b(\016)26 b Fq(+)d Fr(K)2555 2887 y FE(1)2594 2860 y Fr(\016)t Fq(\))p Fp(k)2735 2835 y Fo(f)2731 2860 y Fr(W)2825 2887 y FD(t)2860 2860 y Fp(k)2910 2817 y FE(2)2974 2860 y Fr(;)94 3070 y Fu(for)30 b(a)g(constant)e Fr(K)760 3096 y FE(1)829 3070 y Fu(independent)h(of)g Fr(\016)38 b Fp(2)d Fq(\()p Fu(0)p Fr(;)17 b Fu(1].)48 b(The)29 b(term)g Fr(K)2412 3096 y FE(1)2452 3070 y Fr(\016)k Fu(comes)c(from)g(the)h (deri)n(v)n(ati)n(v)o(es)c(in)j(the)94 3189 y(norm)c Fp(k)d(\001)f(k)509 3212 y Fk(~)501 3229 y FE(H)557 3189 y Fj(2)p Fi(;\016)557 3256 y Fj(2)649 3189 y Fu(.)35 b(W)-8 b(e)26 b(choose)e Fr(\016)32 b(>)c Fu(0)d(so)f(small)g(that)1207 3445 y Fr(\014)1268 3402 y FE(2)1331 3445 y Fp(\000)e Fr(\014)7 b Fq(^)-52 b Fr(c)23 b Fq(+)g Fu(1)f Fq(+)g(\()p Fr(K)29 b Fq(+)23 b Fr(K)2166 3472 y FE(1)2205 3445 y Fq(\))p Fr(\016)32 b Fp(\024)c(\000)p Fu(3)p Fr(\015)6 b(=)p Fu(2)24 b Fr(:)94 3654 y Fu(Inte)o(grating)g(o)o(v)o(er)g Fr(t)h Fu(we)g(get)g(from)f(the)h(choice)g(of)g Fr(\014)5 b Fu(,)25 b Fr(\016)t Fu(,)g(and)i Fq(^)-52 b Fr(c)p Fu(:)1489 3876 y Fp(k)1543 3851 y Fo(f)1539 3876 y Fr(W)1633 3903 y FD(t)1668 3876 y Fp(k)49 b(\024)44 b Fr(e)1934 3833 y Fm(\000)p FE(3)p FD(\015)t(t=)p FE(2)2189 3876 y Fp(k)2243 3851 y Fo(f)2239 3876 y Fr(W)2333 3903 y FE(0)2373 3876 y Fp(k)29 b Fr(:)1139 b Fq(\()p Fu(3)p Fr(:)p Fu(12)p Fq(\))94 4085 y Fu(Thus,)25 b(we)g(ha)n(v)o(e)f(sho)n (wn)g(Eq.\(3.7\).)316 4205 y(Ne)o(xt,)g(we)h(study)954 4179 y Fo(e)940 4205 y Fr(U)11 b Fu(.)35 b(From)25 b(\(2.12\))g(and)g (deforming)f(the)g(contour)h(of)f(inte)o(gration,)f(we)i(get)276 4381 y Fo(\000\000)384 4436 y(e)368 4462 y Fr(T)426 4489 y FD(\020)473 4462 y Fq(~)-51 b Fr(a)524 4381 y Fo(\001)592 4462 y Fp(\003)686 4435 y Fq(~)664 4462 y Fr(f)724 4381 y Fo(\001)769 4462 y Fq(\()p Fr(k)s Fq(\))44 b(=)1068 4326 y Fo(Z)1164 4462 y Fu(d)p Fr(`)17 b(e)1319 4419 y Fm(\000)p FD(i\020)5 b Fk(\()p FD(k)r Fm(\000)p FD(`)p Fk(\))1658 4462 y Fq(~)-51 b Fr(a)o Fq(\()p Fr(k)25 b Fp(\000)e Fr(`)p Fq(\))2026 4435 y(~)2006 4462 y Fr(f)9 b Fq(\()p Fr(`)p Fq(\))946 4723 y(=)1068 4587 y Fo(Z)1164 4723 y Fu(d)p Fr(`)17 b(e)1319 4680 y Fm(\000)p FD(i\020)5 b Fk(\()p FD(k)r Fm(\000)p FD(`)p Fk(\))1658 4723 y Fq(~)-51 b Fr(a)o Fq(\()p Fr(k)25 b Fp(\000)e Fr(`)p Fq(\))2006 4642 y Fo(\000)2062 4698 y(f)2051 4723 y Fp(W)2149 4750 y FD(\014)t(;)r Fk(^)-42 b FD(c)o(t)2314 4697 y Fq(~)2292 4723 y Fr(f)2352 4642 y Fo(\001)2397 4723 y Fq(\()p Fr(`)22 b Fp(\000)g Fr(i\014)5 b Fq(\))p Fr(e)2779 4680 y Fm(\000)p FD(i`)r Fk(^)-42 b FD(c)r(t)946 4984 y Fq(=)1068 4849 y Fo(Z)1164 4984 y Fu(d)p Fr(`)17 b(e)1319 4941 y Fm(\000)p FD(i\020)5 b Fk(\()p FD(k)r Fm(\000)p FD(`)p Fm(\000)p FD(i\014)t Fk(\))1797 4984 y Fq(~)-51 b Fr(a)o Fq(\()p Fr(k)25 b Fp(\000)e Fr(`)f Fp(\000)g Fr(i\014)5 b Fq(\))2361 4903 y Fo(\000)2419 4959 y(f)2408 4984 y Fp(W)2506 5011 y FD(\014)t(;)r Fk(^)-42 b FD(c)o(t)2671 4958 y Fq(~)2649 4984 y Fr(f)2709 4903 y Fo(\001)2754 4984 y Fq(\()p Fr(`)p Fq(\))p Fr(e)2920 4941 y Fm(\000)p FD(i)p Fk(\()p FD(`)p Fk(+)p FD(i\014)t Fk(\))r(^)g FD(ct)946 5245 y Fq(=)45 b Fr(e)1114 5203 y Fm(\000)p FD(\014)t Fk(\()p FD(\020)5 b Fm(\000)r Fk(^)-42 b FD(ct)p Fk(\))1478 5110 y Fo(Z)1574 5245 y Fu(d)p Fr(`)17 b(e)1729 5203 y Fm(\000)p FD(i\020)5 b Fk(\()p FD(k)r Fm(\000)p FD(`)p Fk(\))2068 5245 y Fq(~)-51 b Fr(a)o Fq(\()p Fr(k)25 b Fp(\000)e Fr(`)e Fp(\000)i Fr(i\014)5 b Fq(\))2632 5165 y Fo(\000)2690 5220 y(f)2678 5245 y Fp(W)2776 5272 y FD(\014)t(;)r Fk(^)-42 b FD(ct)2941 5219 y Fq(~)2920 5245 y Fr(f)2979 5165 y Fo(\001)3025 5245 y Fq(\()p Fr(`)p Fq(\))p Fr(e)3191 5203 y Fm(\000)p FD(i`)r Fk(^)g FD(ct)3410 5245 y Fr(:)3619 4854 y Fq(\()p Fu(3)p Fr(:)p Fu(13)p Fq(\))p eop %%Page: 10 10 10 9 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(10)94 194 y Fu(Let)262 168 y Fq(~)261 194 y Fr(h)p Fq(\()p Fr(k)s Fq(\))39 b(=)g Fr(e)652 152 y Fm(\000)p FD(ictk)859 194 y Fq(~)-51 b Fr(a)p Fq(\()p Fr(k)29 b Fp(\000)e Fr(i\014)5 b Fq(\))34 b Fu(and)j Fq(~)-53 b Fr(g)s Fq(\()p Fr(k)s Fq(\))38 b(=)i Fr(e)1866 152 y Fm(\000)p FD(ik)t Fk(^)-42 b FD(c)q(t)2072 114 y Fo(\000)2129 169 y(f)2117 194 y Fp(W)2215 221 y FD(\014)t(;)r Fk(^)g FD(ct)2373 169 y Fo(e)2359 194 y Fr(U)2427 221 y FD(t)2462 114 y Fo(\001)2508 194 y Fq(\()p Fr(k)s Fq(\))38 b(=)i Fr(e)2842 152 y Fm(\000)p FD(ik)t Fk(^)-42 b FD(c)q(t)3052 169 y Fo(f)3048 194 y Fr(W)3142 221 y FD(t)3177 194 y Fq(\()p Fr(k)s Fq(\))p Fu(.)60 b(Then)33 b(\(3.13\))94 314 y(implies)1344 353 y Fo(\000)1407 408 y(e)1390 433 y Fr(T)1448 460 y FD(ct)1520 433 y Fq(~)-51 b Fr(a)1572 353 y Fo(\001)1640 433 y Fp(\003)1726 408 y Fo(e)1711 433 y Fr(U)1779 460 y FD(t)1859 433 y Fq(=)45 b Fr(e)2027 391 y Fm(\000)p FD(\014)t Fk(\()p FD(c)p Fm(\000)r Fk(^)-42 b FD(c)p Fk(\))p FD(t)2370 407 y Fq(~)2369 433 y Fr(h)22 b Fp(\003)k Fq(~)-54 b Fr(g)28 b(:)94 618 y Fu(From)d(this)f(we)h(conclude)g(that)869 835 y Fp(k)919 754 y Fo(\000)982 810 y(e)965 835 y Fr(T)1023 862 y FD(ct)1095 835 y Fq(~)-51 b Fr(a)1146 754 y Fo(\001)1214 835 y Fp(\003)1301 810 y Fo(e)1286 835 y Fr(U)1354 862 y FD(t)1389 835 y Fp(k)49 b Fq(=)c Fr(e)1656 792 y Fm(\000)p FD(\014)t Fk(\()p FD(c)p Fm(\000)r Fk(^)-42 b FD(c)p Fk(\))p FD(t)1998 835 y Fp(k)2049 809 y Fq(~)2048 835 y Fr(h)23 b Fp(\003)i Fq(~)-53 b Fr(g)s Fp(k)1488 1005 y(\024)45 b Fr(e)1656 962 y Fm(\000)p FD(\014)t Fk(\()p FD(c)p Fm(\000)r Fk(^)-42 b FD(c)p Fk(\))p FD(t)1998 1005 y Fp(k)2049 978 y Fq(~)2048 1005 y Fr(h)q Fp(k)38 b(k)s Fq(~)-53 b Fr(g)s Fp(k)1488 1179 y(\024)45 b Fr(C)7 b Fq(\()p Fu(1)22 b Fq(+)h Fr(tc)p Fq(\))2017 1137 y FE(2)2056 1179 y Fq(\()p Fu(1)f Fq(+)h Fr(t)r Fq(^)-52 b Fr(c)p Fq(\))2385 1137 y FE(2)2424 1179 y Fr(e)2470 1137 y Fm(\000)p FD(\014)t Fk(\()p FD(c)p Fm(\000)r Fk(^)-42 b FD(c)q Fk(\))p FD(t)2813 1179 y Fp(k)2867 1154 y Fo(f)2863 1179 y Fr(W)2957 1206 y FD(t)2992 1179 y Fp(k)29 b Fr(:)3619 1001 y Fq(\()p Fu(3)p Fr(:)p Fu(14)p Fq(\))94 1433 y Fu(On)f(the)g(other)g(hand,)h(from)f(\(3.7\))g (we)g(kno)n(w)f(that)h Fp(k)1939 1408 y Fo(f)1935 1433 y Fr(W)2029 1460 y FD(t)2064 1433 y Fp(k)33 b Fu(stays)27 b(bounded)g(\(it)h(actually)g(decays)g(e)o(xponen\255)94 1567 y(tially\),)c(and)h(thus)f(the)h(e)n(v)n(olution)e(equation)h(for) 1809 1542 y Fo(e)1795 1567 y Fr(U)1863 1594 y FD(t)1923 1567 y Fu(is)g(of)h(the)g(form)751 1809 y Fr(@)804 1836 y FD(t)854 1784 y Fo(e)839 1809 y Fr(U)907 1836 y FD(t)942 1809 y Fq(\()p Fr(k)s Fq(\))44 b(=)h Fp(\000)p Fr(k)1373 1766 y FE(2)1428 1784 y Fo(e)1413 1809 y Fr(U)1481 1836 y FD(t)1516 1809 y Fq(\()p Fr(k)s Fq(\))22 b(+)1771 1783 y(~)1770 1809 y Fr(h)q Fq(\()p Fr(k)s(;)17 b(t)p Fq(\)\()p Fu(1)j Fq(+)j Fr(tc)p Fq(\))2369 1766 y FE(2)2409 1809 y Fq(\()p Fu(1)e Fq(+)i Fr(t)r Fq(^)-52 b Fr(c)p Fq(\))2737 1766 y FE(2)2777 1809 y Fr(e)2823 1766 y Fm(\000)p FD(\014)t Fk(\()p FD(c)p Fm(\000)r Fk(^)-42 b FD(c)p Fk(\))p FD(t)3190 1809 y Fr(;)94 2048 y Fu(with)31 b Fp(k)354 2021 y Fq(~)353 2048 y Fr(h)p Fq(\()p Fp(\001)p Fr(;)17 b(t)p Fq(\))p Fp(k)34 b Fu(uniformly)c(bounded)g(in)h Fr(t)p Fu(.)55 b(Since,)33 b(by)e(construction,)i Fq(^)-52 b Fr(c)37 b(<)f(c)p Fu(,)d(we)f(conclude)e(that)h(\(3.6\))94 2167 y(holds,)g(using)e(well\255kno)n(wn)f(ar)n(guments)i(which)f(will)h(be) g(made)g(e)o(xplicit)e(in)i(the)g(proof)g(of)g(Theorem)g(4.1.)94 2287 y(The)25 b(proof)g(of)g(Proposition)e(3.1)i(is)f(complete.)p 3812 2233 63 2 v 3812 2291 2 59 v 3872 2291 V 3812 2293 63 2 v 94 2753 a Fs(4.)30 b(The)g(r)n(enormalization)g(appr)n(oach)h(f) m(or)f(the)g(simpli\002ed)h(pr)n(oblem)94 2957 y Fu(W)-8 b(e)38 b(consider)e(no)n(w)g(the)h(non\255linear)f(problem)g(\(2.2\))h (and)g(its)f(related)h(v)o(ersion)f(for)57 b Fq(~)-70 b Fr(w)3217 2984 y FD(t)3297 2957 y Fq(=)3430 2932 y Fo(f)3419 2957 y Fp(W)3517 2984 y FD(\014)t(;)r Fk(^)-42 b FD(c)o(t)3666 2957 y Fq(~)-56 b Fr(u)3717 2984 y FD(t)3797 2957 y Fq(=)94 3076 y Fp(F)10 b(W)274 3103 y FD(\014)t(;)r Fk(^)-42 b FD(c)o(t)417 3076 y Fr(u)474 3103 y FD(t)535 3076 y Fu(in)24 b(F)o(ourier)h(space.)36 b(It)25 b(tak)o(es)f(the)h (form)349 3333 y Fr(@)402 3359 y FD(t)443 3333 y Fq(~)-56 b Fr(u)494 3359 y FD(t)573 3333 y Fq(=)45 b Fp(\000)p Fr(k)827 3290 y FE(2)874 3333 y Fq(~)-56 b Fr(u)925 3359 y FD(t)982 3333 y Fq(+)1081 3252 y Fo(\000)1144 3307 y(e)1127 3333 y Fr(T)1185 3359 y FD(ct)1257 3333 y Fq(~)-51 b Fr(a)p Fq(\))22 b Fp(\003)28 b Fq(~)-57 b Fr(u)1498 3359 y FD(t)1556 3333 y Fq(+)29 b(~)-57 b Fr(u)1712 3285 y Fm(\003)p FD(p)1712 3357 y(t)1824 3333 y Fr(;)335 3511 y(@)388 3538 y FD(t)443 3511 y Fq(~)-70 b Fr(w)494 3538 y FD(t)573 3511 y Fq(=)695 3431 y Fo(\000)741 3511 y Fr(\014)802 3469 y FE(2)864 3511 y Fp(\000)23 b Fr(\014)7 b Fq(^)-52 b Fr(c)23 b Fp(\000)g Fr(k)1246 3469 y FE(2)1308 3511 y Fq(+)f Fr(ik)s Fq(\()r(^)-52 b Fr(c)23 b Fp(\000)f Fu(2)p Fr(\014)5 b Fq(\))1850 3431 y Fo(\001)1916 3511 y Fq(~)-70 b Fr(w)1967 3538 y FD(t)2025 3511 y Fq(+)2124 3431 y Fo(\000)2186 3486 y(e)2170 3511 y Fr(T)2228 3538 y Fk(\()p FD(c)p Fm(\000)r Fk(^)-42 b FD(c)p Fk(\))p FD(t)2460 3511 y Fq(~)-51 b Fr(a)p Fq(\))21 b Fp(\003)43 b Fq(~)-71 b Fr(w)2715 3538 y FD(t)2773 3511 y Fq(+)29 b(~)-56 b Fr(u)2930 3460 y Fm(\003)p Fk(\()p FD(p)p Fm(\000)p FE(1)p Fk(\))2930 3536 y FD(t)3198 3511 y Fp(\003)42 b Fq(~)-70 b Fr(w)3341 3538 y FD(t)3401 3511 y Fr(:)3669 3420 y Fq(\()p Fu(4)p Fr(:)p Fu(1)p Fq(\))316 3766 y Fu(Let)23 b Fr(M)569 3792 y FD(\014)645 3766 y Fu(be)h(the)f(operator)g (of)g(multiplication:)32 b Fq(\()p Fr(M)2127 3792 y FD(\014)2180 3766 y Fr(f)11 b Fq(\)\()p Fr(x)p Fq(\))26 b(=)i Fr(e)2591 3723 y FD(\014)t(x)2691 3766 y Fr(f)11 b Fq(\()p Fr(x)p Fq(\))p Fu(.)34 b(Choose)23 b(the)g(constants)h Fq(^)-52 b Fr(c)p Fu(,)94 3885 y(and)25 b Fr(\014)31 b Fu(such)24 b(that)h(the)o(y)f(satisfy)g(as)h(before)1425 4113 y(0)44 b Fr(>)h Fp(\000)p Fu(2)p Fr(\015)50 b Fq(=)44 b Fr(\014)2052 4070 y FE(2)2115 4113 y Fp(\000)23 b Fr(\014)7 b Fq(^)-52 b Fr(c)22 b Fq(+)h Fu(1)i Fr(;)94 4341 y Fu(and)g(\002x)g(them)g (henceforth.)35 b(Our)25 b(main)f(result)h(for)g(the)f(simpli\002ed)g (problem)g(is:)94 4533 y Fv(Theor)n(em)30 b(4.1.)44 b Fg(There)29 b(are)f(positi)n(v)o(e)e(constants)g Fr(R)q Fg(,)i Fr(C)35 b Fg(and)28 b Fr(\016)36 b Fp(2)c Fq(\()p Fu(0)p Fr(;)17 b Fu(1])27 b Fg(such)h(that)f(the)h(follo)n(wing)d (holds:)94 4652 y(Assume)37 b Fp(k)p Fr(u)560 4679 y FE(0)600 4652 y Fp(k)650 4679 y FE(H)705 4655 y Fj(2)705 4706 y(2)p Fi(;\016)826 4652 y Fq(+)30 b Fp(k)p Fr(M)1080 4679 y FD(\014)1133 4652 y Fr(u)1190 4679 y FE(0)1230 4652 y Fp(k)1280 4679 y FE(H)1335 4655 y Fj(2)1335 4706 y(2)p Fi(;\016)1473 4652 y Fp(\024)46 b Fr(R)q Fg(.)74 b(Then)37 b(the)h(solution)e Fr(u)2590 4679 y FD(t)2663 4652 y Fg(of)i(\(2.2\))f(with)g(initial)g(condition)94 4800 y Fr(u)151 4827 y FE(0)224 4800 y Fg(con)l(v)o(er)n(ges)31 b(to)h(a)g(Gaussian)g(in)g(the)g(sense)g(that)f(there)h(is)g(a)h (constant)e Fr(A)2775 4827 y Fm(\003)2858 4800 y Fq(=)38 b Fr(A)3048 4827 y Fm(\003)3094 4800 y Fq(\()p Fr(u)3190 4827 y FE(0)3230 4800 y Fq(\))31 b Fg(such)h(that)g(with)115 4931 y Fq(~)94 4957 y Fr( )t Fq(\()p Fr(k)s Fq(\))27 b(=)h Fr(e)474 4914 y Fm(\000)p FD(k)580 4872 y Fj(2)641 4957 y Fg(the)c(rescaled)i(solution)g Fq(~)-53 b Fr(v)t Fq(\()p Fr(k)s(;)17 b(t)p Fq(\))26 b(=)35 b(~)-57 b Fr(u)p Fq(\()p Fr(k)s(t)2069 4914 y Fm(\000)p FE(1)p FD(=)p FE(2)2247 4957 y Fr(;)17 b(t)p Fq(\))24 b Fg(satis\002es)1342 5236 y Fp(k)t Fq(~)-54 b Fr(v)1440 5263 y FD(t)1498 5236 y Fp(\000)22 b Fr(A)1672 5263 y Fm(\003)1738 5210 y Fq(~)1718 5236 y Fr( )s Fp(k)1844 5259 y Fk(~)1836 5276 y FE(H)1891 5236 y Fj(2)p Fi(;\016)1891 5303 y Fj(2)2027 5236 y Fp(\024)2284 5169 y Fr(C)7 b(R)p 2161 5213 401 4 v 2161 5306 a Fq(\()p Fr(t)22 b Fq(+)h Fu(1)p Fq(\))2447 5277 y FE(1)p FD(=)p FE(2)2599 5236 y Fr(:)1042 b Fq(\()p Fu(4)p Fr(:)p Fu(2)p Fq(\))p eop %%Page: 11 11 11 10 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(11)94 194 y Fg(Furthermore,)1112 319 y Fp(k)20 b Fq(~)-70 b Fr(w)1233 346 y FD(t)1268 319 y Fp(k)1326 342 y Fk(~)1318 359 y FE(H)1373 319 y Fj(2)p Fi(;\016)1373 386 y Fj(2)1510 319 y Fq(=)44 b Fp(kF)10 b(W)1861 346 y FD(\014)t(;)r Fk(^)-42 b FD(c)o(t)2004 319 y Fr(u)2061 346 y FD(t)2096 319 y Fp(k)2154 342 y Fk(~)2146 359 y FE(H)2201 319 y Fj(2)p Fi(;\016)2201 386 y Fj(2)2338 319 y Fp(\024)44 b Fr(C)7 b(R)q(e)2660 277 y Fm(\000)p FD(\015)t(t)2829 319 y Fr(:)316 602 y Fu(W)-8 b(e)27 b(shall)g(use)g(the)g(renormalization)f(technique)g(of)h ([BK92])h(to)f(sho)n(w)f(that)33 b Fq(~)-57 b Fr(u)3085 629 y FD(t)3147 602 y Fu(and)48 b Fq(~)-71 b Fr(w)3389 629 y FD(t)3452 602 y Fu(beha)n(v)o(e)27 b(\(as)94 739 y Fr(t)34 b Fp(!)f(1)p Fu(\))c(essentially)f(in)h(the)g(same)f(w)o(ay)i (as)f(their)f(linear)h(counterparts)2711 713 y Fo(e)2696 739 y Fr(U)2764 765 y FD(t)2828 739 y Fu(and)3005 713 y Fo(f)3001 739 y Fr(W)3095 765 y FD(t)3160 739 y Fu(from)f(the)h(pre)n (vious)94 858 y(section.)50 b(This)28 b(technique)h(consists,)g(see)h ([CEE92],)h(in)e(pushing)f(forw)o(ard)i(the)g(solution)d(for)j(some)f (time)94 978 y(and)22 b(then)g(rescaling)g(it.)34 b(This)21 b(process)h(mak)o(es)g(the)g(ef)n(fecti)n(v)o(e)f(non\255linearity)g (smaller)g(at)h(each)h(step,)f(so)g(that)94 1097 y(in)j(the)g(end)f (the)h(con)l(v)o(er)n(gence)g(properties)f(of)h(the)g(linearized)g (problem)f(are)h(obtained.)316 1217 y(W)-8 b(e)25 b(\002x)g(0)j Fr(<)g(\033)j Fp(\024)d Fu(1)d(and)g(introduce:)1556 1385 y Fo(\000)1612 1439 y Fq(~)1602 1466 y Fn(L)1694 1440 y Fq(~)1673 1466 y Fr(f)1732 1385 y Fo(\001)1778 1466 y Fq(\()p Ff({)t Fq(\))44 b(=)2113 1440 y(~)2092 1466 y Fr(f)11 b Fq(\()p Fr(\033)t Ff({)t Fq(\))24 b Fr(:)3669 1462 y Fq(\()p Fu(4)p Fr(:)p Fu(3)p Fq(\))94 1706 y Fu(This)h(is)f(again)g(a)h(linear)g(change)g(of)g(coordinates)f (in)h(function)e(space.)37 b(Note)24 b(that)735 1918 y Fo(\000)791 1972 y Fq(~)781 1999 y Fn(L)p Fq(\()912 1973 y(~)891 1999 y Fr(f)32 b Fp(\003)26 b Fq(~)-54 b Fr(g)s Fq(\))1134 1918 y Fo(\001)1180 1999 y Fq(\()p Ff({)t Fq(\))44 b(=)1494 1863 y Fo(Z)1590 1999 y Fu(d)p Ff({)1710 1956 y Fm(0)1776 1973 y Fq(~)1755 1999 y Fr(f)10 b Fq(\()p Fr(\033)t Ff({)26 b Fp(\000)d Ff({)2176 1956 y Fm(0)2204 1999 y Fq(\))s(~)-53 b Fr(g)s Fq(\()p Ff({)2403 1956 y Fm(0)2431 1999 y Fq(\))1372 2260 y(=)45 b Fr(\033)1571 2125 y Fo(Z)1667 2260 y Fu(d)p Fq(\()p Fr(\033)1817 2217 y Fm(\000)p FE(1)1918 2260 y Ff({)1988 2217 y Fm(0)2017 2260 y Fq(\))2094 2234 y(~)2073 2260 y Fr(f)10 b Fq(\()p Fr(\033)t Ff({)25 b Fp(\000)e Fr(\033)t(\033)2545 2217 y Fm(\000)p FE(1)2646 2260 y Ff({)2716 2217 y Fm(0)2745 2260 y Fq(\))s(~)-53 b Fr(g)r Fq(\()p Fr(\033)t(\033)2995 2217 y Fm(\000)p FE(1)3096 2260 y Ff({)3166 2217 y Fm(0)3195 2260 y Fq(\))1372 2479 y(=)45 b Fr(\033)1555 2399 y Fo(\000)1600 2479 y Fq(\()1650 2452 y(~)1639 2479 y Fn(L)1731 2453 y Fq(~)1710 2479 y Fr(f)11 b Fq(\))21 b Fp(\003)h Fq(\()1952 2452 y(~)1941 2479 y Fn(L)t Fq(~)-54 b Fr(g)s Fq(\))2102 2399 y Fo(\001)2148 2479 y Fq(\()p Ff({)t Fq(\))25 b Fr(:)3669 2214 y Fq(\()p Fu(4)p Fr(:)p Fu(4)p Fq(\))94 2717 y Fu(Furthermore,)979 2761 y Fo(\000)1036 2815 y Fq(~)1025 2842 y Fn(L)p Fq(\()1152 2817 y Fo(e)1135 2842 y Fr(T)1193 2869 y FD(\020)1240 2842 y Fq(~)-51 b Fr(a)p Fq(\))1331 2761 y Fo(\001)1376 2842 y Fq(\()p Ff({)t Fq(\))44 b(=)h Fr(e)1736 2799 y FD(i\020)5 b(\033)s Fe({)1918 2842 y Fq(~)-52 b Fr(a)p Fq(\()p Fr(\033)t Ff({)t Fq(\))43 b(=)2343 2761 y Fo(\000)2406 2817 y(e)2389 2842 y Fr(T)2447 2869 y FD(\033)s(\020)2542 2842 y Fq(\()2592 2815 y(~)2581 2842 y Fn(L)q Fq(~)-51 b Fr(a)o Fq(\))2743 2761 y Fo(\001)2789 2842 y Fq(\()p Ff({)t Fq(\))24 b Fr(;)94 3037 y Fu(and)h(therefore)h (we)f(ha)n(v)o(e)1252 3135 y Fq(~)1241 3162 y Fn(L)1313 3081 y Fo(\000)1358 3162 y Fq(\()1414 3137 y Fo(e)1397 3162 y Fr(T)1455 3189 y FD(\020)1502 3162 y Fq(~)-51 b Fr(a)p Fq(\))22 b Fp(\003)1708 3136 y Fq(~)1686 3162 y Fr(f)1746 3081 y Fo(\001)1836 3162 y Fq(=)45 b Fr(\033)t Fq(\()2074 3137 y Fo(e)2058 3162 y Fr(T)2116 3189 y FD(\033)s(\020)2220 3135 y Fq(~)2210 3162 y Fn(L)q Fq(~)-51 b Fr(a)p Fq(\))21 b Fp(\003)h Fq(\()2516 3135 y(~)2505 3162 y Fn(L)2598 3136 y Fq(~)2576 3162 y Fr(f)11 b Fq(\))25 b Fr(:)941 b Fq(\()p Fu(4)p Fr(:)p Fu(5)p Fq(\))316 3357 y Fu(W)-8 b(e)25 b(ne)o(xt)f(de\002ne)562 3587 y Fq(~)-56 b Fr(u)613 3614 y FD(n;\034)735 3587 y Fq(\()p Ff({)t Fq(\))45 b(=)1049 3507 y Fo(\000)1106 3560 y Fq(~)1095 3587 y Fn(L)1166 3544 y FD(n)1227 3587 y Fq(~)-57 b Fr(u)1277 3507 y Fo(\001)1323 3587 y Fq(\()p Ff({)t Fr(;)17 b(\033)1538 3544 y Fm(\000)p FE(2)p FD(n)1688 3587 y 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11 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(12)94 194 y Fu(W)-8 b(e)30 b(see)f(that)g(under)g(these)g(rescalings)g(the)g(coef)n(\002cients)g (of)g(the)g(non\255linear)f(terms)h(go)g(to)g(0)g(as)g Fr(n)k Fp(!)g(1)p Fu(.)94 314 y(W)-8 b(e)26 b(will)e(no)n(w)g(put)g (this)g(observ)n(ation)f(into)h(more)h(mathematical)e(form.)316 433 y(The)29 b(equation)f(\(4.1\))g(is)h(of)g(the)f(form)h Fr(@)1725 460 y FD(t)1760 433 y Fr(X)1843 460 y FD(t)1911 433 y Fq(=)k Fr(L)2089 353 y Fo(\000)2135 433 y Fr(X)2218 460 y FD(t)2253 353 y Fo(\001)2323 433 y Fq(+)24 b Fn(N)2503 353 y Fo(\000)2549 433 y Fr(X)2632 460 y FD(t)2666 353 y Fo(\001)2712 433 y Fu(,)30 b(where)f Fr(L)g Fu(contains)f(the)g (linear)94 553 y(parts)c(with)f(the)g(e)o(xception)f(of)i(those)f (depending)g(on)h Fq(~)-51 b Fr(a)2023 580 y FD(n)2100 553 y Fu(and)24 b Fn(N)g Fu(denotes)f(the)g(other)h(terms.)34 b(W)-8 b(e)24 b(can)g(write)94 672 y(the)h(solution)e(as)1109 837 y Fr(X)1192 864 y 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b(these)94 4511 y(a)g(priori)e(bounds)g(we)h(sho)n(w)f(Proposition)f(4.5.)54 b(From)31 b(these)f(results,)i(Theorem)f(4.1)f(will)g(follo)n(w)g (rather)94 4630 y(simply)24 b(by)g(a)h(contraction)f(ar)n(gument.)p eop %%Page: 13 13 13 12 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(13)94 194 y(4.1.)25 b(The)h(scaled)f (linear)g(pr)n(oblem)94 383 y Fu(Here,)d(we)d(deri)n(v)o(e)g(the)g (essential)g(bounds)f(on)i(the)f(in\003uence)h(of)f(the)h(term)f Fr(a)p Fq(\()p Fr(x)10 b Fp(\000)g Fr(ct)p Fq(\))p Fr(u)p Fq(\()p Fr(x;)17 b(t)p Fq(\))h Fu(in)i(the)f(equation)94 503 y(for)49 b Fq(~)-71 b Fr(w)309 530 y FD(t)373 503 y Fu(under)27 b(the)h(scalings)f(introduced)g(abo)o(v)o(e.)43 b(Note)28 b(\002rst)g(that,)g(from)f(de\002nition)g(\(2.6\))h(and)f (\(4.3\),)i(we)94 623 y(ha)n(v)o(e)717 815 y Fp(k)778 788 y Fq(~)767 815 y Fn(L)859 789 y Fq(~)838 815 y Fr(f)10 b Fp(k)947 772 y FE(2)1031 815 y Fq(=)45 b Fr(\033)1214 772 y Fm(\000)p FE(1)1332 680 y Fo(Z)1428 815 y Fu(d)p Fq(\()p Fr(\033)t Ff({)t Fq(\))1795 691 y FE(2)1741 720 y Fo(X)1720 935 y FD(j;`)p Fk(=)p FE(0)1922 815 y Fr(\016)1970 772 y FE(2)p FD(`)2043 815 y Fr(\033)2104 772 y Fm(\000)p FE(2)p FD(`)2239 815 y Fr(\033)2300 772 y FE(2)p FD(j)2377 815 y Fp(j)p Fq(\()p Fr(@)2503 772 y FD(j)2565 789 y Fq(~)2544 815 y Fr(f)10 b Fq(\)\()p Fr(\033)t Ff({)t Fq(\))p Fp(j)2879 772 y FE(2)2918 815 y Fq(\()p Fr(\033)t Ff({)t Fq(\))3127 772 y FE(2)p FD(`)3224 815 y Fr(:)94 1101 y Fu(From)25 b(this)f(we)h(conclude)g(immediately)e(that)h(for)h (0)j Fr(<)g(\033)j(<)d Fu(1:)956 1331 y Fp(k)1017 1304 y Fq(~)1006 1331 y Fn(L)1099 1305 y Fq(~)1077 1331 y Fr(f)11 b Fp(k)49 b(\024)c Fr(\033)1419 1288 y Fm(\000)p FE(5)p FD(=)p FE(2)1596 1331 y Fp(k)1667 1305 y Fq(~)1646 1331 y Fr(f)10 b Fp(k)80 b Fu(and)75 b Fp(k)2115 1304 y Fq(~)2104 1331 y Fn(L)2175 1288 y Fm(\000)p FE(1)2298 1305 y Fq(~)2277 1331 y Fr(f)10 b Fp(k)49 b(\024)c Fr(\033)2618 1288 y Fm(\000)p FE(3)p FD(=)p FE(2)2795 1331 y Fp(k)2867 1305 y Fq(~)2845 1331 y Fr(f)11 b Fp(k)29 b Fr(:)607 b Fq(\()p Fu(4)p Fr(:)p Fu(10)p Fq(\))94 1561 y Fu(W)-8 b(e)26 b(ne)o(xt)e(bound)735 1536 y Fq(~)719 1561 y Fr(S)780 1588 y FD(n;\034)t(;\034)963 1568 y Fh(0)994 1561 y Fu(.)35 b(Recall)26 b(that)e(we)h(are)h(assuming)d Fr(\014)2258 1518 y FE(2)2321 1561 y Fp(\000)f Fr(\014)7 b Fq(^)-52 b Fr(c)23 b Fq(+)g Fu(1)k Fq(=)h Fp(\000)p Fu(2)p Fr(\015)34 b(<)28 b Fu(0.)94 1765 y Fv(Lemma)e(4.2.)35 b Fg(F)o(or)25 b(all)g Fr(")964 1722 y Fm(0)1020 1765 y Fp(2)j Fq(\()p Fu(0)p Fr(;)17 b Fu(1)p Fq(\))23 b Fg(there)i(e)o(xists)e(a)j Fr(C)1975 1792 y FD(")2013 1772 y Fh(0)2072 1765 y Fr(>)i Fu(0)d Fg(such)f(that)h(for)g Fu(1)i Fr(>)i(\034)38 b(>)28 b(\034)3200 1722 y Fm(0)3256 1765 y Fp(\025)g Fu(0)d Fg(one)f(has)1056 2031 y Fp(k)1122 2006 y Fq(~)1106 2031 y Fr(S)1167 2058 y FD(n;\034)t(;\034)1350 2038 y Fh(0)1402 2005 y Fq(~)1380 2031 y Fr(f)11 b Fp(k)49 b(\024)c Fr(C)1732 2058 y FD(")1770 2038 y Fh(0)1802 2031 y Fr(\033)1863 1989 y Fm(\000)p FD(")1963 1946 y Fh(0)1989 1989 y FD(n)2043 2031 y Fr(e)2089 1989 y Fm(\000)p FD(\015)t(\033)2246 1946 y Fh(\000)p Fj(2)p Fi(n)2374 1989 y Fk(\()p FD(\034)8 b Fm(\000)p FD(\034)2557 1946 y Fh(0)2584 1989 y Fk(\))p FD(=)p FE(2)2696 2031 y Fp(k)2767 2005 y Fq(~)2746 2031 y Fr(f)i Fp(k)30 b Fr(;)3619 2017 y Fq(\()p Fu(4)p Fr(:)p Fu(11)p Fq(\))94 2260 y Fg(for)c(all)e Fr(n)k Fp(2)g Fv(N)p Fg(.)94 2495 y Fv(Pr)n(oof)o(.)d Fu(W)-8 b(e)25 b(consider)g(the)f(equation)g Fr(@)1464 2522 y FD(\034)1535 2468 y Fq(~)1514 2495 y Fr(f)1563 2522 y FD(\034)1640 2495 y Fq(=)1768 2470 y(~)1745 2495 y Fr(G)1823 2522 y FD(n;\034)1968 2468 y Fq(~)1946 2495 y Fr(f)1995 2522 y FD(\034)2045 2495 y Fu(,)h(whose)f(solution)f(is)2833 2468 y Fq(~)2812 2495 y Fr(f)2861 2522 y FD(\034)2938 2495 y Fq(=)3060 2470 y(~)3043 2495 y Fr(S)3104 2522 y FD(n;\034)t(;\034)3287 2502 y Fh(0)3339 2468 y Fq(~)3318 2495 y Fr(f)3367 2522 y FD(\034)3412 2502 y Fh(0)3443 2495 y Fu(:)1103 2753 y Fr(@)1156 2780 y FD(\034)1227 2726 y Fq(~)1206 2753 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FE(2)2671 4860 y Fr(:)94 5145 y Fu(W)-8 b(e)26 b(use)1276 5264 y Fp(k)p Fr(e)1372 5221 y FD(\025)1419 5239 y Fi(n)1468 5221 y FD(\034)1539 5238 y Fq(~)1518 5264 y Fr(f)10 b Fp(k)1635 5287 y Fk(~)1627 5304 y FE(H)1682 5264 y Fj(2)p Fi(;\016)1682 5331 y Fj(0)1802 5264 y Fp(\024)28 b(k)p Fr(e)2003 5221 y FD(\025)2050 5239 y Fi(n)2099 5221 y FD(\034)2149 5264 y Fp(k)2199 5291 y Fm(C)2245 5267 y Fj(2)2242 5318 y(b)p Fi(;\016)2334 5264 y Fp(k)2405 5238 y Fq(~)2384 5264 y Fr(f)10 b Fp(k)2501 5287 y Fk(~)2493 5304 y FE(H)2549 5264 y Fj(2)p Fi(;\016)2549 5331 y Fj(0)2665 5264 y Fr(;)p eop %%Page: 14 14 14 13 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(14)94 194 y Fu(and)353 423 y Fp(k)p Fr(e)449 380 y FD(\025)496 398 y Fi(n)546 380 y FD(\034)595 423 y Fp(k)645 450 y Fm(C)691 426 y Fj(2)688 476 y(b)p Fi(;\016)808 423 y Fp(\024)28 b(k)p Fr(e)1009 380 y FD(\025)1056 398 y Fi(n)1105 380 y FD(\034)1155 423 y Fp(k)1205 450 y Fm(C)1251 425 y Fj(0)1248 475 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Fq(\()p Fr(\034)32 b Fp(\000)22 b Fr(\034)3232 5221 y Fm(0)3260 5264 y Fq(\))3299 5221 y FE(1)p FD(=)p FE(2)3414 5184 y Fo(\001)3485 5264 y Fr(:)106 b Fq(\()p Fu(4)p Fr(:)p Fu(14)p Fq(\))p eop %%Page: 15 15 15 14 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(15)94 194 y Fu(T)-8 b(o)25 b(bound)f(the)h(second)f(po)n(wer)h(of)32 b Fq(~)-57 b Fr(q)t Fu(,)24 b(choose)h Fr(")1769 151 y Fm(0)1825 194 y Fp(2)j Fq(\()p Fu(0)p Fr(;)17 b Fu(1)p Fq(\))p Fu(.)34 b(Then)618 449 y Fp(k)7 b Fq(~)-57 b Fr(q)716 407 y FE(2)778 423 y Fq(~)756 449 y Fr(f)805 476 y FD(\034)855 449 y Fp(k)913 472 y Fk(~)905 489 y FE(H)960 449 y Fj(2)p Fi(;\016)960 516 y Fj(0)1113 449 y Fp(\024)61 b Fr(e)1297 407 y Fk(\()p FD(\014)1377 364 y Fj(2)1407 407 y Fm(\000)p FD(\014)6 b Fk(^)-42 b FD(c)p Fk(+)p FE(3)p FD(\015)t(=)p FE(2)p Fk(\))p FD(\033)1852 364 y Fh(\000)p Fj(2)p Fi(n)1979 407 y Fk(\()p FD(\034)8 b Fm(\000)p FD(\034)2162 364 y Fh(0)2189 407 y Fk(\))2225 449 y Fp(k)f Fq(~)-57 b Fr(q)2323 407 y FE(2)2384 423 y Fq(~)2363 449 y Fr(f)2412 476 y FD(\034)2457 456 y Fh(0)2488 449 y Fp(k)2546 472 y Fk(~)2538 489 y FE(H)2594 449 y Fj(2)p Fi(;\016)2594 516 y Fj(0)1090 692 y Fq(+)1190 556 y Fo(Z)1290 599 y FD(\034)1245 783 y(\034)1290 763 y Fh(0)1336 692 y Fu(d)p Fr(s)17 b(e)1496 649 y Fk(\()p FD(\014)1576 606 y Fj(2)1605 649 y Fm(\000)p FD(\014)6 b Fk(^)-42 b FD(c)p Fk(+)p FE(3)p FD(\015)t(=)p FE(2)p Fk(\))p FD(\033)2050 606 y Fh(\000)p Fj(2)p Fi(n)2178 649 y Fk(\()p FD(\034)8 b Fm(\000)p FD(s)p Fk(\))2417 692 y Fu(sup)2406 777 y Fe({)s Fm(2)p Fl(R)2584 577 y Fo(\014)2584 637 y(\014)2584 697 y(\014)2617 692 y Fr(e)2663 649 y Fm(\000)p Fe({)2782 606 y Fj(2)2812 649 y Fk(\()p FD(\034)g Fm(\000)p FD(s)p Fk(\))3024 692 y Fp(j)p Ff({)t Fp(j)3150 649 y FE(2)p Fm(\000)p FD(")3285 606 y Fh(0)3317 577 y Fo(\014)3317 637 y(\014)3317 697 y(\014)1090 946 y Fp(\002)23 b Fr(\033)1251 903 y Fm(\000)p FE(2)p FD(n)1402 946 y Fr(\033)1463 903 y Fm(\000)p FD(")1563 860 y Fh(0)1589 903 y FD(n)1644 946 y Fp(k)p Fr(a)1747 973 y FD(n)1800 946 y Fp(k)1850 988 y Fm(C)1896 954 y Fj(0)p Fi(;")1978 939 y Fh(0)1893 1013 y Fj(b)2032 946 y Fp(k)2103 920 y Fq(~)2082 946 y Fr(f)2131 973 y FD(s)2173 946 y Fp(k)2231 968 y Fk(~)2223 985 y FE(H)2278 946 y Fj(2)p Fi(;\016)2278 1012 y Fj(0)2395 946 y Fr(;)94 1238 y Fu(where)28 b Fp(k)p Fr(g)t Fp(k)517 1280 y Fm(C)563 1246 y Fj(0)p Fi(;")645 1231 y Fh(0)560 1305 y Fj(b)712 1238 y Fq(=)k Fu(sup)959 1265 y FD(x)p Fm(2)p Fl(R)1130 1238 y Fp(j)p Fr(g)t Fq(\()p Fr(x)p Fq(\))p Fp(j)21 b Fq(+)j Fu(sup)1634 1265 y FD(x)p Fm(2)p Fl(R)1804 1238 y Fp(j)p Fq(\()p Fp(F)1953 1195 y Fm(\000)p FE(1)2054 1238 y Fq(\()19 b(~)-69 b Fr(m)t Fq(~)-53 b Fr(g)s Fq(\)\()p Fr(x)p Fq(\))p Fp(j)26 b Fu(with)45 b Fq(~)-68 b Fr(m)p Fq(\()p Fr(k)s Fq(\))31 b(=)g Fp(j)p Fu(1)23 b Fq(+)h Fr(k)3281 1195 y FE(2)3321 1238 y Fp(j)3349 1195 y FD(")3387 1152 y Fh(0)3413 1195 y FD(=)p FE(2)3494 1238 y Fu(.)43 b(Clearly)-6 b(,)94 1390 y Fp(k)p Fr(a)197 1417 y FD(n)251 1390 y Fp(k)301 1432 y Fm(C)347 1398 y Fj(0)p Fi(;")429 1383 y Fh(0)344 1457 y Fj(b)491 1390 y Fu(is)24 b(\002nite)h(and)g(using)f(the)g(estimate)g(\(4.13\))h(to)f (bound)g Fp(k)p Fr(f)2466 1417 y FD(s)2508 1390 y Fp(k)2566 1413 y Fk(~)2558 1430 y FE(H)2613 1390 y Fj(2)p Fi(;\016)2613 1457 y Fj(0)2705 1390 y Fu(,)h(we)g(get)108 1688 y Fp(k)7 b Fq(~)-57 b Fr(q)206 1645 y FE(2)267 1662 y Fq(~)246 1688 y Fr(f)295 1715 y FD(\034)344 1688 y Fp(k)402 1711 y Fk(~)394 1728 y FE(H)449 1688 y Fj(2)p Fi(;\016)449 1755 y Fj(0)585 1688 y Fp(\024)45 b Fr(C)7 b Fp(k)p Fq(\()p Fu(1)22 b Fq(+)30 b(~)-57 b Fr(q)1094 1645 y FE(2)1134 1688 y Fq(\))1194 1662 y(~)1173 1688 y Fr(f)1222 1715 y FD(\034)1267 1695 y Fh(0)1298 1688 y Fp(k)1356 1711 y Fk(~)1348 1728 y FE(H)1403 1688 y Fj(2)p Fi(;\016)1403 1755 y Fj(0)1495 1688 y Fr(e)1541 1645 y Fk(\()p FD(\014)1621 1603 y Fj(2)1651 1645 y Fm(\000)p FD(\014)6 b Fk(^)-42 b FD(c)p Fk(+)p FE(1)p Fk(+)p FE(3)p FD(\015)t(=)p FE(2)p Fk(\))p FD(\033)2192 1603 y Fh(\000)p Fj(2)p 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(this)f(term)g(by)h(the)f(assumptions)f(on)3582 3692 y Fq(~)3560 3718 y Fr(f)11 b Fu(.)74 b(The)94 3838 y(deri)n(v)n(ati)n (v)o(es)24 b(are)i(handled)g(similarly)-6 b(,)24 b(e)o(xcept)h(that)g (there)i(is)e(no)h(need)g(to)f(di)n(vide)g(and)g(multiply)f(by)i(po)n (wers)94 3957 y(of)f Fr(\033)t Ff({)k Fu(since)c(each)g(deri)n(v)n(ati) n(v)o(e)e(produces)h(a)h(f)o(actor)h Fr(\033)t Fu(.)316 4077 y(W)-8 b(e)23 b(no)n(w)f(bound)g(the)g(other)h(terms)f(in)h (\(4.25\).)35 b(The)22 b(\002rst)h(term)g(is)f(bounded)g(using)g (\(4.29\))g(and)h(yields)94 4196 y(a)j(bound)e(\(in)587 4172 y Fq(~)573 4196 y Fu(H)650 4148 y FE(2)p FD(;\016)650 4225 y FE(2)752 4196 y Fu(\))h(of)1807 4318 y Fr(\032)1859 4275 y FD(u)1859 4345 y(n)p Fm(\000)p FE(1)2010 4318 y Fr(=)p Fu(2)f Fr(:)1457 b Fq(\()p Fu(4)p Fr(:)p Fu(30)p Fq(\))94 4509 y Fu(The)25 b(second)g(and)g(third)f(terms)g(ha)n(v)o(e)h (been)g(bounded)f(in)g(\(4.21\))h(and)f(\(4.22\):)1300 4778 y Fr(C)7 b(e)1424 4735 y Fm(\000)p FD(C)e(\033)1597 4692 y 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5206 y Fj(2)3244 5291 y Fr(:)p eop %%Page: 19 19 19 18 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(19)94 194 y Fu(The)25 b(\002rst)g(e)o(xpression)f(v)n(anishes)f(and)i(we)g(get)g(a)g(bound)f (\(in)2218 170 y Fq(~)2204 194 y Fu(H)2281 146 y FE(2)p FD(;\016)2281 223 y FE(2)2384 194 y Fu(\):)1097 471 y Fp(k)1175 446 y Fq(~)1147 471 y Fr(X)1230 498 y FD(n)1283 471 y Fp(k)49 b(\024)c Fr(C)7 b(e)1628 428 y Fm(\000)p FD(C)e(\033)1801 385 y Fh(\000)p Fi(n)1910 471 y Fr(R)1987 428 y FD(w)1986 498 y(n)2073 471 y Fq(+)22 b Fr(C)7 b(\033)2311 428 y FD(n)p Fk(\()p FD(p)p Fm(\000)p FE(3)p Fk(\))2566 471 y Fq(\()p Fr(R)2682 428 y FD(u)2681 498 y(n)2734 471 y Fq(\))2773 428 y FD(p)2844 471 y Fr(:)747 b Fq(\()p Fu(4)p Fr(:)p Fu(32)p Fq(\))94 710 y Fu(Collecting)25 b(the)f(bounds)g(\(4.30\)\226\(4.32\),)g(the)h(assertion)f(\(4.27\))h (for)k Fq(~)-54 b Fr(r)2553 737 y FD(n)2632 710 y Fu(follo)n(ws.)34 b(Finally)-6 b(,)24 b(the)h(bounds)e(on)94 830 y Fr(\032)146 787 y FD(w)146 857 y(n)235 830 y Fu(follo)n(w)h(as)h(those)f(in)g (Lemma)h(4.4.)35 b(The)25 b(proof)g(of)g(Proposition)e(4.5)h(is)h (complete.)p 3812 776 63 2 v 3812 834 2 59 v 3872 834 V 3812 836 63 2 v 94 1129 a Fv(Pr)n(oof)17 b(of)g(Theor)n(em)h(4.1.)e Fu(The)h(proof)f(is)h(an)f(induction)f(ar)n(gument,)j(using)e (repeatedly)g(the)h(abo)o(v)o(e)e(estimates.)94 1248 y(Again)20 b(we)g(write)g Fr(C)28 b Fu(for)21 b(\(positi)n(v)o(e\))d (constants)h(which)h(can)h(be)f(chosen)g(independent)f(of)i Fr(\033)j Fu(and)c Fr(n)p Fu(.)34 b(Assume)94 1368 y(that)25 b Fr(R)j Fq(=)g Fu(sup)616 1395 y FD(n)p Fm(2)p Fl(N)791 1368 y Fr(R)868 1325 y FD(u)867 1395 y(n)949 1368 y Fr(<)g Fp(1)d Fu(e)o(xists.)34 b(From)25 b(Lemma)f(4.4)g(we)h(observ)o(e)g (for)g Fr(\033)31 b(>)d Fu(0)d(suf)n(\002ciently)f(small)881 1717 y Fr(R)958 1674 y FD(w)957 1744 y(n)1083 1717 y Fp(\024)1341 1643 y Fr(C)7 b(\033)1480 1600 y Fm(\000)p FE(5)p FD(=)p FE(2)p Fm(\000)p FD(n")1802 1557 y Fh(0)1834 1643 y Fr(\032)1886 1600 y FD(w)1886 1670 y(n)p Fm(\000)p FE(1)p 1233 1694 912 4 v 1233 1787 a Fu(1)22 b Fp(\000)h Fr(C)7 b(\033)1544 1758 y FD(n)p Fk(\()p FD(p)p Fm(\000)p FE(1)p Fm(\000)p FD(")1862 1738 y Fh(0)1889 1758 y Fk(\))1925 1787 y Fr(R)2002 1758 y FD(p)p Fm(\000)p FE(1)2201 1717 y Fp(\024)45 b Fr(C)24 b(\033)2479 1674 y Fm(\000)p FE(5)p FD(=)p FE(2)p Fm(\000)p FD(n")2801 1631 y Fh(0)2832 1717 y Fr(\032)2884 1674 y FD(w)2884 1744 y(n)p Fm(\000)p FE(1)3060 1717 y Fr(;)892 2028 y(R)969 1985 y FD(u)968 2055 y(n)1083 2028 y Fp(\024)1233 1954 y Fr(C)7 b(\033)1372 1911 y Fm(\000)p FE(5)p FD(=)p FE(2)1550 1954 y Fr(\032)1602 1911 y FD(u)1602 1981 y(n)p Fm(\000)p FE(1)1775 1954 y Fq(+)22 b Fr(C)7 b(e)1998 1911 y Fm(\000)p FD(C)e(\033)2171 1868 y Fh(\000)p Fi(n)2280 1954 y Fr(R)2357 1911 y FD(w)2356 1981 y(n)p 1233 2005 1188 4 v 1435 2098 a Fu(1)22 b Fp(\000)g Fr(C)7 b(\033)1745 2069 y FD(n)p Fk(\()p FD(p)p Fm(\000)p FE(3)p Fk(\))2000 2098 y Fr(R)2077 2069 y FD(p)p Fm(\000)p FE(1)1083 2265 y Fp(\024)61 b Fr(C)7 b(\033)1360 2222 y Fm(\000)p FE(5)p FD(=)p FE(2)1538 2265 y Fr(\032)1590 2222 y FD(u)1590 2292 y(n)p Fm(\000)p FE(1)1763 2265 y Fq(+)22 b Fr(C)7 b(e)1986 2222 y Fm(\000)p FD(C)e(\033)2159 2180 y Fh(\000)p Fi(n)2268 2265 y Fr(\032)2320 2222 y FD(w)2320 2292 y(n)p Fm(\000)p FE(1)2496 2265 y Fr(;)3619 1940 y Fq(\()p Fu(4)p Fr(:)p Fu(33)p Fq(\))94 2500 y Fu(with)25 b(a)g(constant)f Fr(C)32 b Fu(which)24 b(can)h(be)g(chosen)g (independent)f(of)h Fr(R)q Fu(.)35 b(Using)24 b(Proposition)f(4.5)i(we) g(\002nd)614 2770 y Fp(j)p Fr(A)717 2797 y FD(n)793 2770 y Fp(\000)e Fr(A)968 2797 y FD(n)p Fm(\000)p FE(1)1119 2770 y Fp(j)44 b(\024)h Fr(C)7 b(e)1437 2727 y Fm(\000)p FD(C)e(\033)1610 2684 y Fh(\000)p Fi(n)1718 2770 y Fr(\032)1770 2727 y FD(w)1770 2797 y(n)p Fm(\000)p FE(1)1943 2770 y Fq(+)23 b Fr(C)7 b(\033)2182 2727 y FD(n)p Fk(\()p FD(p)p Fm(\000)p FE(3)p Fk(\))2436 2770 y Fr(\033)2497 2727 y Fm(\000)p FE(5)p FD(=)p FE(2)2674 2770 y Fr(\032)2726 2727 y FD(u)2726 2797 y(n)p Fm(\000)p FE(1)2902 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Fu(implies)28 b(the)h(e)o(xponential)f (decay)i(of)115 4928 y Fq(~)-71 b Fr(w)165 4955 y FD(t)201 4928 y Fu(.)36 b(The)25 b(proof)f(of)h(Theorem)g(4.1)f(is)h(complete.)p 3812 4874 63 2 v 3812 4933 2 59 v 3872 4933 V 3812 4935 63 2 v eop %%Page: 20 20 20 19 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(20)94 194 y Fs(P)o(art)30 b(II.)f(The)h(Swift\255Hohenber)o(g)h(equation)94 524 y(5.)f(Bloch)g(wa)m(v)o(es)94 732 y Fu(Since)c(the)e(problem)g(we)h (consider)g(tak)o(es)f(place)h(in)g(a)g(setting)f(with)g(a)h FF(periodic)f Fu(background)g(pro)o(vided)f(by)94 851 y(the)17 b(stationary)g(solution)e(of)i(the)g(Swift\255Hohenber)n(g,)i (it)e(is)f(natural)h(to)g(w)o(ork)g(with)f(the)h(Bloch)h (representation)94 971 y(of)25 b(the)g(functions.)35 b(F)o(or)25 b(additional)e(informations)g(see)i([RS72].)316 1090 y(The)k(starting)g(point)f(of)i(Bloch)f(w)o(a)n(v)o(e)h(analysis)e (in)h(case)i(of)e(a)h(2)p Fr(\031)t Fu(\226periodic)e(underlying)g (pattern)i(is)94 1210 y(the)25 b(follo)n(wing)e(relation)705 1524 y Fr(u)p Fq(\()p Fr(x)p Fq(\))44 b(=)1063 1389 y Fo(Z)1159 1524 y Fu(d)p Fr(k)19 b(e)1326 1481 y FD(ik)r(x)1456 1524 y Fq(~)-57 b Fr(u)p Fq(\()p Fr(k)s Fq(\))44 b(=)1808 1430 y Fo(X)1805 1643 y FD(n)p Fm(2)p Fl(Z)1971 1389 y Fo(Z)2070 1432 y FE(1)p FD(=)p FE(2)2026 1615 y Fm(\000)p FE(1)p FD(=)p FE(2)2201 1524 y Fu(d)p Fr(`)17 b(e)2356 1481 y FD(i)p Fk(\()p FD(n)p Fk(+)p FD(`)p Fk(\))p FD(x)2646 1524 y Fq(~)-57 b Fr(u)p Fq(\()p Fr(n)22 b Fq(+)h Fr(`)p Fq(\))941 1843 y(=)1063 1708 y Fo(Z)1162 1750 y FE(1)p FD(=)p FE(2)1118 1934 y Fm(\000)p FE(1)p FD(=)p FE(2)1293 1843 y Fu(d)p Fr(`)1420 1749 y Fo(X)1417 1962 y FD(n)p Fm(2)p Fl(Z)1583 1843 y Fr(e)1629 1800 y FD(i)p Fk(\()p FD(n)p Fk(+)p FD(`)p Fk(\))p FD(x)1920 1843 y Fq(~)-56 b Fr(u)p Fq(\()p Fr(n)22 b Fq(+)g Fr(`)p Fq(\))44 b(=)2438 1708 y Fo(Z)2537 1750 y FE(1)p FD(=)p FE(2)2493 1934 y Fm(\000)p FE(1)p FD(=)p FE(2)2667 1843 y Fu(d)p Fr(`)17 b(e)2822 1800 y FD(i`x)2940 1843 y Fq(^)-57 b Fr(u)p Fq(\()p Fr(`;)17 b(x)p Fq(\))24 b Fr(;)3669 1703 y Fq(\()p Fu(5)p Fr(:)p Fu(1)p Fq(\))94 2192 y Fu(where)i(we)f(de\002ne)1080 2249 y Fo(\000)1126 2330 y Fp(T)h Fr(u)1263 2249 y Fo(\001)1308 2330 y Fq(\()p Fr(`;)17 b(x)p Fq(\))43 b Fp(\021)51 b Fq(^)-56 b Fr(u)p Fq(\()p Fr(`;)17 b(x)p Fq(\))42 b(=)2141 2235 y Fo(X)2138 2449 y FD(n)p Fm(2)p Fl(Z)2304 2330 y Fr(e)2350 2287 y FD(inx)2484 2330 y Fq(~)-56 b Fr(u)p Fq(\()p Fr(n)22 b Fq(+)g Fr(`)p Fq(\))i Fr(:)781 b Fq(\()p Fu(5)p Fr(:)p Fu(2)p Fq(\))94 2630 y Fu(The)34 b(operator)g Fp(T)59 b Fu(will)32 b(play)h(a)h(r)1274 2629 y(\210)1266 2630 y(ole)g(analogous)e(to)h(that)g(played)h(by)f(the)g(F)o(ourier)h (transform)e Fp(F)44 b Fu(for)33 b(the)94 2749 y(simpli\002ed)24 b(problem)g(of)h(P)o(art)g(I.)g(W)-8 b(e)25 b(will)f(use)h(analogous)e (notation:)94 2941 y Fv(Notation.)h Fu(If)h Fr(f)34 b Fu(denotes)23 b(a)i(function,)e(then)1683 2915 y Fq(^)1661 2941 y Fr(f)35 b Fu(is)23 b(de\002ned)i(by)2298 2915 y Fq(^)2276 2941 y Fr(f)39 b 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Fq(=)45 b Fu(2)p Fr(\031)1863 3797 y Fo(Z)1918 4023 y Fl(R)1970 3933 y Fu(d)p Fr(k)s Fp(j)6 b Fq(~)-56 b Fr(u)p Fq(\()p Fr(k)s Fq(\))p Fp(j)2321 3890 y FE(2)1614 4208 y Fq(=)45 b Fu(2)p Fr(\031)1865 4114 y Fo(X)1863 4327 y FD(n)p Fm(2)p Fl(Z)2029 4073 y Fo(Z)2128 4116 y FE(1)p FD(=)p FE(2)2084 4299 y Fm(\000)p FE(1)p FD(=)p FE(2)2259 4208 y Fu(d)p Fr(`)17 b Fp(j)6 b Fq(~)-56 b Fr(u)n Fq(\()p Fr(n)22 b Fq(+)g Fr(`)p Fq(\))p Fp(j)2780 4165 y FE(2)1614 4527 y Fq(=)45 b Fu(2)p Fr(\031)1863 4392 y Fo(Z)1962 4434 y FE(1)p FD(=)p FE(2)1918 4618 y Fm(\000)p FE(1)p FD(=)p FE(2)2092 4527 y Fu(d)p Fr(`)2203 4433 y Fo(X)2201 4646 y FD(n)p Fm(2)p Fl(Z)2366 4527 y Fp(j)6 b Fq(~)-56 b Fr(u)p Fq(\()p Fr(n)22 b Fq(+)g Fr(`)p Fq(\))p Fp(j)2780 4484 y FE(2)1614 4846 y Fq(=)45 b Fu(2)p Fr(\031)1863 4711 y Fo(Z)1962 4753 y FE(1)p FD(=)p FE(2)1918 4937 y(1)p FD(=)p FE(2)2074 4846 y Fu(d)p Fr(`)2183 4711 y Fo(Z)2282 4753 y FE(2)p FD(\031)2238 4937 y FE(0)2368 4846 y Fu(d)p Fr(x)17 b Fp(j)6 b Fq(^)-56 b Fr(u)o Fq(\()p Fr(`;)17 b(x)p Fq(\))p Fp(j)2826 4803 y FE(2)2888 4846 y Fr(:)94 5172 y Fu(The)26 b(sum)f(and)g(the)h(inte)o(gral)e(can)i(be)g(interchanged)f(in)g (\(5.1\))h(due)f(to)g(Fubini')-5 b(s)25 b(theorem)g(when)g Fr(u)h Fu(is)f(in)g(the)94 5291 y(Schw)o(artz)h(space)f Fp(S)7 b Fu(.)p eop %%Page: 21 21 21 20 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(21)316 194 y Fu(W)-8 b(e)26 b(shall)e(use)i(frequently)f(the)g(follo)n(wing)e(fundamental)i (properties)g(\(which)g(follo)n(w)f(at)i(once)f(from)94 314 y(\(5.2\)\):)1198 430 y Fq(^)-57 b Fr(u)p Fq(\()p Fr(`;)17 b(x)p Fq(\))43 b(=)i Fr(e)1681 388 y FD(ix)1766 430 y Fq(^)-57 b Fr(u)p Fq(\()p Fr(`)22 b Fq(+)g Fu(1)p Fr(;)17 b(x)p Fq(\))24 b Fr(;)1198 580 y Fq(^)-57 b Fr(u)p Fq(\()p Fr(`;)17 b(x)p Fq(\))43 b(=)51 b(^)-56 b Fr(u)p Fq(\()p Fr(`;)17 b(x)j Fq(+)j Fu(2)p Fr(\031)t Fq(\))h Fr(;)1198 744 y Fq(^)-57 b Fr(u)p Fq(\()p Fr(`;)17 b(x)p Fq(\))43 b(=)p 1635 663 58 4 v 51 w(^)-56 b Fr(u)p Fq(\()p 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Fp(\001)22 b(\000)g Fr(i\014)5 b(;)17 b Fp(\001)p Fq(\))p Fp(k)2959 4959 y Fk(^)2951 4976 y FE(H)3006 4935 y Fj(2)p Fi(;\016)3006 5003 y Fj(2)3123 4935 y Fr(:)518 b Fq(\()p Fu(5)p Fr(:)p Fu(8)p Fq(\))94 5150 y Fu(Thus,)28 b(apart)g(from)g(notational)e(dif)n(ferences,)i(we)g (can)h(w)o(ork)e(in)g(the)h(Bloch)g(spaces)g(with)f(much)g(the)g(same) 94 5270 y(bounds)d(as)h(in)g(the)f(spaces)h(used)g(for)g(the)f(model)g (problem)g(of)h(the)g(pre)n(vious)e(sections.)p eop %%Page: 22 22 22 21 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(22)94 194 y Fs(6.)30 b(The)g(linearized) i(pr)n(oblem)94 389 y Fu(W)-8 b(e)31 b(discuss)e(here)h(again)f(the)h (beha)n(vior)f(of)h(the)g(linearized)g(problem)f(as)h(in)f(Section)h (3,)h(b)n(ut)e(no)n(w)g(for)h(the)94 508 y(Swift\255Hohenber)n(g)c (equation.)35 b(The)26 b(discussion)d(will)h(again)h(be)g(split)f(in)h (an)g(aspect)g(behind)f(the)h(front)g(and)94 628 y(one)d(ahead)g(of)f (the)g(front.)35 b(In)21 b(Section)g(3,)h(the)f(beha)n(vior)h(of)f(the) g(problem)g(in)g(the)g(b)n(ulk)f(behind)h(the)g(tra)n(v)o(eling)94 748 y(front)34 b(w)o(as)g(dif)n(fusi)n(v)o(e)e(by)h(construction,)i (and)f(the)f(only)h(dif)n(\002culty)e(w)o(as)i(to)g(understand)f(the)g (r)3487 747 y(\210)3479 748 y(ole)h(of)g(the)94 867 y(decay)f(of)f Fr(a)g Fu(to)g(0)g(\(as)g Fr(e)945 824 y Fm(\000)p FD(\014)t Fm(j)p FD(x)p Fm(j)1154 867 y Fu(\))h(as)f Fr(x)38 b Fp(!)f(\0001)p Fu(.)58 b(F)o(or)32 b(the)g(problem)g(of)g(the)g (Swift\255Hohenber)n(g)g(equation,)94 987 y(the)i(situation)e(is)h (similar)l(,)i(leading)e(again)f(to)i(dif)n(fusi)n(v)o(e)d(beha)n(vior) -5 b(.)61 b(Ho)n(we)n(v)o(er)l(,)35 b(this)d(observ)n(ation)h(is)g(not) 94 1106 y(ob)o(vious.)58 b(Therefore,)35 b(the)d(\002rst)h(problem)f (consists)f(in)h(sho)n(wing)f(the)h(dif)n(fusi)n(v)o(e)f(beha)n(vior)-5 b(.)58 b(In)33 b(order)g(to)94 1226 y(obtain)d(optimal)f(results)g(for) h(the)g(analysis)g(ahead)g(of)g(the)g(front,)i FF(i.e)o(.)p 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(considered)f(in)g(the)h(sequel)f(together)g(with)g(the)h (non\255linear)94 2746 y(terms.)g(As)24 b(a)i(consequence,)e(the)h (linear)g(equation)f(dominating)f(the)h(beha)n(vior)h(behind)f(the)h (front)f(is)h(gi)n(v)o(en)94 2866 y(by)1730 2991 y Fr(@)1783 3018 y FD(t)1818 2991 y Fr(v)48 b Fq(=)d Fn(M)p Fr(v)29 b(:)1430 b Fq(\()p Fu(6)p Fr(:)p Fu(2)p Fq(\))94 3186 y Fu(W)-8 b(e)19 b(recall)g(those)f(features)g(of)h(the)f(proof)g(of)g (dif)n(fusi)n(v)o(e)e(stability)h(of)h([Schn96,)i(Schn98])e(which)g (are)h(rele)n(v)n(ant)94 3305 y(to)25 b(the)g(study)e(of)i(\(6.2\).)316 3425 y(In)e(order)h(to)f(do)g(this,)f(we)i(need)f(to)g(localize)g(the)g (spectrum)g(of)g Fn(M)p Fu(.)36 b(Since)23 b(this)g(is)f (well\255documented,)94 3544 y(we)j(just)e(summarize)g(the)g(results.) 35 b(As)24 b(the)f(linearized)h(problem)f(has)h(periodic)f(coef)n (\002cients,)h(the)g(operator)119 3637 y Fq(^)94 3664 y Fn(M)k Fq(=)g Fp(T)e Fn(M)p Fp(T)582 3621 y Fm(\000)p FE(1)705 3664 y Fu(equals)20 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5291 y Fq(\))29 b Fu(the)g(eigenfunction)p eop %%Page: 23 23 23 22 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(23)94 194 y Fr(')159 221 y FD(`)198 194 y Fq(\()p Fr(x)p Fq(\))27 b(=)h Fr(w)536 221 y FD(`;)p FE(1)633 194 y Fq(\()p Fr(x)p Fq(\))c Fu(of)h(the)f(main) g(branch)h Fr(\026)1626 221 y FE(1)1666 194 y Fq(\()p Fr(`)p Fq(\))f Fu(is)g(well)g(de\002ned)h(\(and)g(a)f(continuation)f (of)i Fr(@)3369 221 y FD(x)3419 194 y Fr(U)3472 221 y Fm(\003)3533 194 y Fq(\()p Fr(x)p Fq(\))p Fu(\))f(as)h Fr(`)94 339 y Fu(is)g(v)n(aried)f(a)o(w)o(ay)h(from)f(0.)36 b(Corresponding)24 b(to)h(this)e(we)j(de\002ne)f(the)g(central)g (projections)3250 314 y Fq(^)3227 339 y Fr(P)3291 366 y FE(c)3327 339 y Fq(\()p Fr(`)p Fq(\))f Fu(by)1564 563 y Fq(^)1542 589 y Fr(P)1606 615 y FE(c)1642 589 y Fq(\()p Fr(`)p Fq(\))p Fr(f)53 b Fq(=)45 b Fp(h)16 b Fq(\026)-66 b Fr(')2090 615 y FD(`)2128 589 y Fr(;)17 b(f)11 b Fp(i)p Fr(')2337 615 y FD(`)2399 589 y Fr(;)94 826 y Fu(where)42 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1240 y FE(c)2021 1213 y Fu(and)2213 1188 y Fq(^)2190 1213 y Fr(E)2264 1240 y FE(s)2321 1213 y Fu(by:)1044 1438 y Fq(^)1021 1463 y Fr(E)1095 1490 y FE(c)1131 1463 y Fq(\()p Fr(`)p Fq(\))43 b(=)i Fr(\037)p Fq(\()p Fr(`)p Fq(\))1619 1438 y(^)1598 1463 y Fr(P)1662 1490 y FE(c)1697 1463 y Fq(\()p Fr(`)p Fq(\))24 b Fr(;)2008 1438 y Fq(^)1985 1463 y Fr(E)2059 1490 y FE(s)2091 1463 y Fq(\()p Fr(`)p Fq(\))43 b(=)i Fv(1)p Fq(\()p Fr(`)p Fq(\))21 b Fp(\000)2689 1438 y Fq(^)2666 1463 y Fr(E)2740 1490 y FE(c)2776 1463 y Fq(\()p Fr(`)p Fq(\))j Fr(:)94 1713 y Fu(It)h(will)f(be)h(useful)g(to)f (de\002ne)h(auxiliary)f(\223mode)h(\002lters\224)2107 1688 y Fq(^)2084 1713 y Fr(E)2164 1670 y FE(h)2158 1740 y(c)2228 1713 y Fu(and)2420 1688 y Fq(^)2397 1713 y Fr(E)2477 1670 y FE(h)2471 1740 y(s)2541 1713 y Fu(by)872 1938 y Fq(^)849 1963 y Fr(E)929 1920 y FE(h)923 1990 y(c)968 1963 y Fq(\()p Fr(`)p Fq(\))44 b(=)g Fr(\037)p Fq(\()p Fr(`=)p Fu(2)p Fq(\))1557 1938 y(^)1535 1963 y Fr(P)1599 1990 y FE(c)1634 1963 y Fq(\()p Fr(`)p Fq(\))24 b Fr(;)1945 1938 y Fq(^)1922 1963 y Fr(E)2002 1920 y FE(h)1996 1990 y(s)2041 1963 y Fq(\()p Fr(`)p Fq(\))43 b(=)i Fv(1)p Fq(\()p Fr(`)p Fq(\))21 b Fp(\000)i Fr(\037)p Fq(\()p Fu(2)p Fr(`)p Fq(\))2870 1938 y(^)2849 1963 y Fr(P)2913 1990 y FE(c)2948 1963 y Fq(\()p Fr(`)p Fq(\))h Fr(:)94 2200 y Fu(These)h(de\002nitions)f(are)h(made)g(in)g(such)f(a)h(w)o(ay)g (that)1396 2412 y Fq(^)1373 2438 y Fr(E)1453 2395 y FE(h)1447 2464 y(c)1515 2412 y Fq(^)1492 2438 y Fr(E)1566 2464 y FE(c)1646 2438 y Fq(=)1791 2412 y(^)1768 2438 y Fr(E)1842 2464 y FE(c)1902 2438 y Fr(;)2069 2412 y Fq(^)2046 2438 y Fr(E)2126 2395 y FE(h)2120 2464 y(s)2188 2412 y Fq(^)2165 2438 y Fr(E)2239 2464 y FE(s)2315 2438 y Fq(=)2460 2412 y(^)2437 2438 y Fr(E)2511 2464 y FE(s)2568 2438 y Fr(;)94 2687 y Fu(which)g(will)f(be)h(used)f(to)h(replace)g(the)g(\(missing\))e (projection)h(property)h(of)2754 2662 y Fq(^)2731 2687 y Fr(E)2805 2714 y FE(c)2866 2687 y Fu(and)3058 2662 y Fq(^)3035 2687 y Fr(E)3109 2714 y FE(s)3140 2687 y Fu(.)316 2807 y(W)-8 b(e)18 b(ne)o(xt)e(e)o(xtend)h(the)g (de\002nitions)f(\(4.3\))h(of)h(Section)f(4)g(to)g(the)g(Bloch)h (spaces.)33 b(T)-8 b(o)17 b(a)n(v)n(oid)g(cumbersome)94 2926 y(notation,)25 b(we)i(shall)e(use)h(mostly)f(the)h(same)g(symbols) e(as)i(in)g(that)g(section.)39 b(Thus,)25 b(with)h Fr(\033)32 b(<)e Fu(1)c(as)h(before,)94 3046 y(we)f(let)e(no)n(w)1457 3085 y Fo(\000)1511 3138 y(b)1503 3165 y Fn(L)7 b Fq(^)-57 b Fr(u)1631 3085 y Fo(\001)1677 3165 y Fq(\()p Ff({)t Fr(;)17 b(x)p Fq(\))43 b(=)51 b(^)-56 b Fr(u)p Fq(\()p Fr(\033)t Ff({)t Fr(;)17 b(x)p Fq(\))24 b Fr(:)94 3354 y Fu(Note)34 b(that)f(here,)k(and)c(else)n(where,)j(the)e(scaling)f (does)g(not)g(act)h(on)g(the)f Fr(x)h Fu(v)n(ariable,)i(only)d(on)g (the)h(quasi\255)94 3474 y(momentum)25 b Ff({)t Fu(.)41 b(The)26 b(no)o(v)o(elty)f(of)h(renormalization)f(in)h(Bloch)h(space)g (here)g(is)e(that)h(since)h(the)f(inte)o(gration)94 3593 y(re)o(gion)g(o)o(v)o(er)g(the)g Fr(`)g 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Fq(\))p Fu(.)36 b(The)25 b(function)1480 5291 y Fq(^)-54 b Fr(v)1524 5318 y FD(t)1560 5291 y Fq(\()p Fr(`;)17 b(x)p Fq(\))42 b(=)j Fr(e)1992 5248 y FD(\026)2040 5266 y Fj(1)2071 5248 y Fk(\()p FD(`)p Fk(\))p FD(t)2202 5291 y Fr(')2267 5318 y FD(`)2305 5291 y Fq(\()p Fr(x)p Fq(\))24 b Fr(;)p eop %%Page: 24 24 24 23 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(24)94 194 y Fu(solv)o(es)24 b(the)h(equation)1450 314 y Fr(@)1503 341 y FD(t)1542 314 y Fq(^)-54 b Fr(v)1586 341 y FD(t)1621 314 y Fq(\()p Fr(`;)17 b Fp(\001)p Fq(\))42 b(=)j Fn(M)2076 341 y FD(`)2114 314 y Fq(\()t(^)-54 b Fr(v)2201 341 y FD(t)2236 314 y Fq(\()p Fr(`;)17 b Fp(\001)p Fq(\)\))23 b Fr(:)94 503 y Fu(Because)j(of)f(the)g(nature)g(of)g(the)f(spectrum)g Fr(\026)1684 530 y FE(1)1724 503 y Fq(\()p Fr(`)p Fq(\))p Fu(,)g(this)g(solution)f(satis\002es)1099 777 y Fq(^)-54 b Fr(v)1143 804 y FD(t)1178 777 y Fq(\()p Fr(`t)1295 735 y Fm(\000)p FE(1)p FD(=)p FE(2)1472 777 y Fr(;)17 b(x)p 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b(^)-52 b Fr(c)p Fq(\()p Fr(i)p Fq(\()p Fr(`)21 b Fq(+)i Fr(i\014)5 b Fq(\))23 b(+)f Fr(@)3114 5318 y FD(x)3165 5291 y Fq(\))3208 5266 y Fo(c)3204 5291 y Fr(W)38 b(:)305 b Fq(\()p Fu(6)p Fr(:)p Fu(7)p Fq(\))p eop %%Page: 26 26 26 25 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(26)94 194 y Fu(In)37 b(order)f(to)g (control)f(these)h(solutions)e(we)i(use)g(that)g(the)g(linearized)g (system)f(\(6.6\))h(e)n(v)n(olv)o(es)e(in)i(such)g(a)94 314 y(w)o(ay)g(that)g(during)f(times)g(of)g(order)i Fn(O)p Fq(\()p Fu(1)p Fr(=")1649 271 y FE(2)1689 314 y Fq(\))e Fu(it)h(can)g(be)g(approximated)f(by)g(the)h(associated)f(linearized)94 433 y(Ginzb)n(ur)n(g\255Landau)25 b(equation)699 672 y Fr(@)752 699 y FD(\034)802 672 y Fr(A)i Fq(=)h Fu(4)p Fq(\()p Fr(@)1151 699 y FD(X)1248 672 y Fp(\000)23 b Fr(\014)1389 699 y FD(A)1469 672 y Fq(\))1508 630 y FE(2)1548 672 y Fr(A)f Fq(+)g Fr(c)1787 699 y FD(B)1856 672 y Fq(\()p Fr(@)1948 699 y FD(X)2046 672 y Fp(\000)g 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b(the)f(follo)n(wing)f(we)i(consider)g(a)g(modulated)e(front)i(with)f (v)o(elocity)f Fr(c)i Fu(and)g(a)g(gi)n(v)o(en)e(\(suf)n(\002ciently)94 314 y(small\))24 b(bifurcation)h(parameter)g Fr(")j(>)g Fu(0)d(for)g(which)f(there)h(are)h(a)f Fr(\014)30 b Fu(and)25 b(a)j Fq(^)-53 b Fr(c)28 b Fp(2)g Fq(\()p Fu(0)p Fr(;)17 b(c)p Fq(\))24 b Fu(which)g(satisfy:)1524 534 y Fr(\045)p Fq(\()r(^)-52 b Fr(c)o(;)17 b(\014)5 b(;)17 b(")p Fq(\))27 b(=)h Fp(\000)p Fu(2)p Fr(\015)33 b(<)28 b Fu(0)d Fr(:)1174 b Fq(\()p Fu(6)p Fr(:)p Fu(11)p Fq(\))94 826 y Fv(Pr)n(oposition)28 b(6.3.)45 b Fg(Suppose)27 b(that)h(the)f(abo)o(v)o(e)g(stability)f (condition)g(\(6.11\))i(is)f(satis\002ed.)45 b(Then)27 b(there)h(is)g(a)94 958 y Fr(\016)k Fp(2)c Fq(\()p Fu(0)p Fr(;)17 b Fu(1])22 b Fg(such)g(that:)34 b(There)23 b(is)f(a)h Fr(C)35 b(<)28 b Fp(1)23 b Fg(for)f(which)h(the)f(functions)2612 932 y Fo(c)2608 958 y Fr(W)2702 985 y FD(t)2764 958 y Fq(=)2881 932 y Fo(c)2870 958 y Fp(W)2968 985 y FD(\014)t(;)r Fk(^)-42 b 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Fj(2)1262 3007 y(2)p Fi(;\016)1381 2954 y Fp(\024)29 b Fr(R)20 b Fg(and)f(denote)g(by)h Fr(v)2199 2981 y FD(t)2254 2954 y Fg(the)f(solution)f(of)h(\(1.4\))h(with)e (initial)h(condition)94 3120 y Fr(v)142 3147 y FE(0)183 3120 y Fg(.)48 b(Let)438 3093 y Fq(~)418 3120 y Fr( )s Fq(\()p Fr(`)p Fq(\))33 b(=)h Fu(e)o(xp)o Fq(\()p Fp(\000)p Fr(c)1051 3147 y FE(1)1091 3120 y Fr(`)1133 3077 y FE(2)1173 3120 y Fq(\))p Fg(.)48 b(There)29 b(is)g(a)g(constant)f Fr(A)2151 3147 y Fm(\003)2230 3120 y Fq(=)34 b Fr(A)2416 3147 y Fm(\003)2462 3120 y Fq(\()p Fr(v)2549 3147 y FE(0)2589 3120 y Fq(\))28 b Fg(such)h(that)g(the)f(rescaled)i(solution)98 3253 y Fq(^)-54 b Fr(v)146 3210 y FE(r)142 3280 y FD(t)178 3253 y Fq(\()p Fr(`;)17 b(x)p Fq(\))26 b(=)32 b(^)-54 b Fr(v)579 3280 y FD(t)614 3253 y Fq(\()p Fr(`t)731 3210 y Fm(\000)p FE(1)p FD(=)p FE(2)908 3253 y Fr(;)17 b(x)p Fq(\))24 b Fg(satis\002es)1207 3526 y Fp(k)t Fq(^)-54 b Fr(v)1309 3483 y FE(r)1305 3552 y FD(t)1363 3526 y Fp(\000)22 b Fr(A)1537 3552 y Fm(\003)1603 3499 y Fq(~)1583 3526 y Fr( )s(@)1704 3552 y FD(x)1754 3526 y Fr(U)1807 3552 y Fm(\003)1868 3526 y Fp(k)1918 3552 y Fm(K)1979 3575 y Fj(1)p Fi(=)2026 3534 y Fh(p)p 2080 3534 29 3 v 2080 3575 a Fi(t)2162 3526 y Fp(\024)2419 3458 y Fr(C)7 b(R)p 2296 3503 401 4 v 2296 3596 a Fq(\()p Fr(t)22 b Fq(+)h Fu(1)p Fq(\))2582 3567 y FE(1)p FD(=)p FE(4)2734 3526 y Fr(:)907 b Fq(\()p Fu(7)p Fr(:)p Fu(1)p Fq(\))94 3798 y Fg(Furthermore,)1104 3918 y Fp(k)18 b Fo(b)-73 b Fr(w)1225 3945 y FD(t)1260 3918 y Fp(k)1310 3945 y Fm(K)1371 3968 y Fj(1)p Fi(=)1418 3926 y Fh(p)p 1472 3926 29 3 v 1472 3968 a Fi(t)1555 3918 y Fq(=)44 b Fp(k)1738 3893 y Fo(c)1726 3918 y Fp(W)1824 3945 y FD(\014)t(;)r Fk(^)-42 b FD(ct)1972 3918 y Fq(^)-54 b Fr(v)2016 3945 y FD(t)2051 3918 y Fp(k)2101 3945 y Fm(K)2162 3968 y Fj(1)p Fi(=)2209 3926 y Fh(p)p 2263 3926 V 2263 3968 a Fi(t)2345 3918 y Fp(\024)45 b Fr(C)7 b(R)q(e)2668 3875 y Fm(\000)p FD(\015)t(t)2837 3918 y Fr(:)804 b Fq(\()p Fu(7)p Fr(:)p Fu(2)p Fq(\))94 4230 y Fv(Remarks.)144 4350 y Fp(\017)50 b Fu(The)25 b(inequality)e(\(7.1\))i(really)g(says)f (that)h(the)f(dif)n(ference)1386 4601 y Fq(^)-54 b Fr(v)1430 4628 y FD(t)1466 4601 y Fq(\()p Fr(`t)1583 4558 y Fm(\000)p FE(1)p FD(=)p FE(2)1760 4601 y Fr(;)17 b(x)p Fq(\))k Fp(\000)h Fr(A)2096 4628 y Fm(\003)2142 4601 y Fr(e)2188 4558 y Fm(\000)p FD(c)2286 4576 y Fj(1)2316 4558 y FD(`)2349 4516 y Fj(2)2384 4601 y Fr(@)2437 4628 y FD(x)2487 4601 y Fr(U)2540 4628 y Fm(\003)2601 4601 y Fq(\()p Fr(x)p Fq(\))244 4821 y Fu(is)k(small,)g(where)h Fr(U)928 4848 y Fm(\003)1016 4821 y Fu(is)f(the)g(periodic)g(solution)f(\(see)i (Eq.\(1.3\)\))f(of)h(the)f(Swift\255Hohenber)n(g)h(equation.)244 4941 y(Expressed)d(in)g(the)g(laboratory)g(frame,)g(this)g(means)g (that)g FF(an)g(initial)e(perturbation)g Fr(v)3225 4968 y FE(0)3265 4941 y Fq(\()p Fr(x)p Fq(\))i FF(will)g(go)g(to)g(0)244 5060 y(lik)o(e)1175 5229 y Fr(v)1223 5255 y FD(t)1259 5229 y Fq(\()p Fr(x)p Fq(\))43 b Fp(\031)i Fr(A)1634 5255 y Fm(\003)1680 5229 y Fq(\()p Fr(v)1767 5255 y FE(0)1806 5229 y Fq(\))1845 5096 y Fo(r)p 1945 5096 143 4 v 1986 5161 a Fr(\031)p 1957 5206 119 4 v 1957 5297 a(c)2000 5324 y FE(1)2040 5297 y Fr(t)2104 5229 y Fu(e)o(xp)p Fq(\()2298 5161 y Fp(\000)p Fr(x)2432 5118 y FE(2)p 2297 5206 175 4 v 2300 5297 a Fu(4)p Fr(c)2393 5324 y FE(1)2433 5297 y Fr(t)2484 5229 y Fq(\))17 b Fr(@)2593 5255 y FD(x)2642 5229 y Fr(U)2710 5255 y Fm(\003)2756 5229 y Fq(\()p Fr(x)p Fq(\))24 b Fr(;)p eop %%Page: 28 28 28 27 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(28)244 194 y Fu(when)29 b Fr(t)34 b Fp(!)f(1)p Fu(,)d(uniformly)e(for)i Fr(x)j Fp(2)i Fv(R)p Fu(.)49 b(See)30 b([Schn96].)49 b(In)29 b(particular)l(,)i(this)d(means)h(that)f(near)i(the)244 314 y(e)o(xtrema)24 b(of)h Fr(U)764 341 y Fm(\003)835 314 y Fu(the)g(con)l(v)o(er)n(gence)g(is)f(f)o(aster)h(than)g Fn(O)p Fq(\()p Fr(t)2181 271 y Fm(\000)p FE(1)p FD(=)p FE(2)2359 314 y Fq(\))f Fu(since)h(at)g(those)f(points)f Fr(@)3305 341 y FD(x)3355 314 y Fr(U)3423 341 y Fm(\003)3494 314 y Fu(v)n(anishes.)144 433 y Fp(\017)50 b Fu(The)20 b(inequality)f(\(7.2\))h(gi)n(v)o(es)e(some)i(more)g(precise)g(bound)f (on)h(the)g(gro)n(wth)f(of)h(a)g(perturbation)g(ahead)g(of)244 553 y(the)27 b(front,)g(because)h(it)f(says)g(that)f(this)h (perturbation)f(decays)h(e)o(xponentially)e(in)i(the)g(weighted)g (norm.)244 672 y(More)e(e)o(xplicitly)-6 b(,)22 b(we)j(ha)n(v)o(e)f(at) h(least)g(a)g(bound)1518 885 y Fp(j)p Fr(v)1594 912 y FD(t)1629 885 y Fq(\()p Fr(x)d Fq(+)h Fr(ct)p Fq(\))p Fp(j)43 b(\024)i Fr(C)7 b(e)2282 842 y FD(\014)t(x)p Fm(\000)p FD(\015)2484 799 y Fh(0)2512 842 y FD(t)2572 885 y Fr(;)244 1072 y Fu(with)24 b Fr(\015)504 1029 y Fm(0)556 1072 y Fu(slightly)f(smaller)h(than)g Fr(\015)144 1192 y Fp(\017)50 b Fu(The)26 b(decay)g Fq(\()p Fr(t)d Fq(+)g Fu(1)p Fq(\))970 1149 y Fm(\000)p FE(1)p FD(=)p FE(4)1174 1192 y Fu(in)j(\(7.1\))g(can)g(be)g(impro)o(v)o(ed)e(easily)i (to)f Fq(\()p Fr(t)e Fq(+)g Fu(1)p Fq(\))2833 1149 y Fm(\000)p FE(1)p FD(=)p FE(2)p Fk(+)p FD(")3136 1192 y Fu(for)j(an)o(y)g Fr(")j(>)h Fu(0.)40 b(W)-8 b(e)244 1311 y(ha)n(v)o(e)24 b(chosen)h Fr(")j Fq(=)g Fu(1)p Fr(=)p Fu(4)d(to)f(k)o(eep)h(the)g(notation)e(at)i(a)g(reasonable)g(le) n(v)o(el.)94 1479 y Fv(Pr)n(oof)o(.)i Fu(As)f(we)h(e)o(xplained)e (before,)i(the)g(proof)f(is)g(similar)g(to)g(the)g(one)g(in)h(Section)f (3)g(e)o(xcept)h(that)f(no)n(w)f(the)94 1599 y(function)i(behind)f(the) h(front)f(is)h(split)e(into)h(a)i(dif)n(fusi)n(v)o(e)c(part)31 b Fq(^)-54 b Fr(v)2271 1626 y FE(c)2335 1599 y Fu(and)27 b(into)f(an)h(e)o(xponentially)e(damped)h(part)98 1718 y Fq(^)-54 b Fr(v)142 1745 y FE(s)175 1718 y Fu(,)25 b(and)f(correspondingly)g(there)h(will)f(be)h(a)g(fe)n(w)g(more)f (equations.)316 1838 y(In)36 b(Bloch)g(space)g(the)g(initial)f (conditions)f(satisfy)h Fp(k)t Fq(^)-54 b Fr(v)2238 1865 y FE(0)2278 1838 y Fp(k)2336 1862 y Fk(^)2328 1879 y FE(H)2383 1838 y Fj(2)p Fi(;\016)2383 1906 y Fj(2)2503 1838 y Fq(+)29 b Fp(k)t Fq(^)-54 b Fr(v)2707 1865 y FE(0)2747 1838 y Fq(\()p Fp(\001)28 b(\000)h Fr(i\014)5 b(;)17 b Fp(\001)p Fq(\))p Fp(k)3213 1862 y Fk(^)3205 1879 y FE(H)3259 1838 y Fj(2)p Fi(;\016)3259 1906 y Fj(2)3411 1838 y Fp(\024)60 b Fr(R)q Fu(.)69 b(The)94 2010 y(system)28 b(for)h(the)g(v)n(ariables)j Fq(^)-54 b Fr(v)1130 2037 y FE(c)1195 2010 y Fu(and)32 b Fq(^)-53 b Fr(v)1416 2037 y FE(s)1477 2010 y Fu(with)28 b(initial)f(conditions)k Fq(^)-54 b Fr(v)2436 2037 y FE(c)2472 2010 y Fp(j)2500 2037 y FD(t)p Fk(=)p FE(0)2664 2010 y Fq(=)2798 1985 y(^)2775 2010 y Fr(E)2849 2037 y FE(c)2888 2010 y Fq(^)g Fr(v)t Fp(j)2964 2037 y FD(t)p Fk(=)p FE(0)3095 2010 y Fu(,)34 b Fq(^)-54 b Fr(v)3198 2037 y FE(s)3230 2010 y Fp(j)3258 2037 y FD(t)p Fk(=)p FE(0)3422 2010 y Fq(=)3556 1985 y(^)3533 2010 y Fr(E)3607 2037 y FE(s)3642 2010 y Fq(^)g Fr(v)t Fp(j)3718 2037 y FD(t)p Fk(=)p FE(0)3849 2010 y Fu(,)94 2148 y(and)29 b(for)g(the)g(v)n(ariable)46 b Fo(b)-73 b Fr(w)36 b Fq(=)1140 2122 y Fo(c)1128 2148 y Fp(W)1226 2175 y FD(\014)t(;)r Fk(^)-42 b FD(c)o(t)1373 2148 y Fq(^)-54 b Fr(v)33 b Fu(with)28 b(initial)f(conditions)45 b Fo(b)-73 b Fr(w)s Fp(j)2463 2175 y FD(t)p Fk(=)p FE(0)2627 2148 y Fq(=)2750 2122 y Fo(c)2738 2148 y Fp(W)2836 2175 y FD(\014)t(;)p FE(0)2952 2148 y Fq(^)-53 b Fr(v)s Fp(j)3028 2175 y FD(t)p Fk(=)p FE(0)3188 2148 y Fu(is)28 b(gi)n(v)o(en)g(in)g (Bloch)94 2267 y(space)e(by)1098 2398 y Fr(@)1151 2425 y FD(t)1190 2398 y Fq(^)-54 b Fr(v)1234 2425 y FE(c)1315 2398 y Fq(=)1458 2371 y Fo(b)1437 2398 y Fn(M)s Fq(^)g Fr(v)1582 2425 y FE(c)1641 2398 y Fq(+)1763 2373 y(^)1740 2398 y Fr(E)1814 2425 y FE(c)1879 2373 y Fq(^)1850 2398 y Fp(H)q Fq(\()t(^)g Fr(v)2022 2425 y FE(c)2058 2398 y Fr(;)20 b Fq(^)-53 b Fr(v)2151 2425 y FE(s)2183 2398 y Fq(\))22 b(+)2366 2373 y(^)2343 2398 y Fr(E)2417 2425 y FE(c)2464 2371 y Fo(b)2453 2398 y Fn(N)p Fq(\()t(^)-54 b Fr(v)2618 2425 y FE(c)2654 2398 y Fr(;)20 b Fq(^)-53 b Fr(v)2747 2425 y FE(s)2779 2398 y Fq(\))24 b Fr(;)1102 2572 y(@)1155 2599 y FD(t)1194 2572 y Fq(^)-54 b Fr(v)1238 2599 y FE(s)1315 2572 y Fq(=)1458 2545 y Fo(b)1437 2572 y Fn(M)s Fq(^)g Fr(v)1582 2599 y FE(s)1637 2572 y Fq(+)1760 2546 y(^)1737 2572 y Fr(E)1811 2599 y FE(s)1871 2546 y Fq(^)1842 2572 y Fp(H)q Fq(\()t(^)g Fr(v)2014 2599 y FE(c)2050 2572 y Fr(;)20 b Fq(^)-53 b Fr(v)2143 2599 y FE(s)2175 2572 y Fq(\))22 b(+)2359 2546 y(^)2336 2572 y Fr(E)2410 2599 y FE(s)2453 2545 y Fo(b)2441 2572 y Fn(N)q Fq(\()t(^)-54 b Fr(v)2607 2599 y FE(c)2643 2572 y Fr(;)20 b Fq(^)-53 b Fr(v)2736 2599 y FE(s)2767 2572 y Fq(\))25 b Fr(;)1109 2745 y(@)1162 2772 y FD(t)1214 2745 y Fo(b)-72 b Fr(w)47 b Fq(=)1458 2718 y Fo(b)1437 2745 y Fn(M)1535 2772 y FE(w)1607 2745 y Fo(b)-72 b Fr(w)25 b Fq(+)1797 2718 y Fo(b)1786 2745 y Fn(N)1864 2772 y FE(w)1918 2745 y Fq(\()t(^)-54 b Fr(v)2005 2772 y FE(c)2041 2745 y Fr(;)20 b Fq(^)-53 b Fr(v)2134 2772 y FE(s)2166 2745 y Fr(;)34 b Fo(b)-72 b Fr(w)r Fq(\))25 b Fr(;)3669 2564 y Fq(\()p Fu(7)p Fr(:)p Fu(3)p Fq(\))94 2906 y Fu(where,)h(see)f 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3600 y Fo(b)1426 3627 y Fn(M)1524 3654 y FD(\014)t(;)r Fk(^)-42 b FD(c)n(t)1689 3627 y Fq(+)1810 3600 y Fo(b)1788 3627 y Fn(M)1886 3654 y Fl(i)p FD(;\014)t(;)r Fk(^)g FD(c)o(t)2097 3627 y Fr(;)734 3779 y Fo(b)722 3806 y Fn(N)800 3833 y FE(w)855 3806 y Fq(\()t(^)-54 b Fr(v)942 3833 y FE(c)978 3806 y Fr(;)20 b Fq(^)-53 b Fr(v)1071 3833 y FE(s)1103 3806 y Fr(;)33 b Fo(b)-71 b Fr(w)r Fq(\))44 b(=)h Fp(\000)p Fu(3)p Fr(U)1606 3833 y Fm(\003)1689 3806 y Fp(\001)25 b Fq(^)-53 b Fr(v)42 b Fp(\003)1819 3798 y Fc(\015)1924 3806 y Fo(b)-73 b Fr(w)25 b Fp(\000)e Fu(3)2175 3781 y Fo(b)2152 3806 y Fr(K)2237 3833 y FD(ct)2345 3806 y Fp(\003)2335 3798 y Fc(\015)2426 3806 y Fq(^)-54 b Fr(v)43 b Fp(\003)2503 3798 y Fc(\015)2607 3806 y Fo(b)-73 b Fr(w)25 b Fp(\000)i Fq(^)-54 b Fr(v)43 b Fp(\003)2866 3798 y Fc(\015)2956 3806 y Fq(^)-54 b Fr(v)43 b Fp(\003)3033 3798 y Fc(\015)3137 3806 y Fo(b)-73 b Fr(w)28 b(:)94 3993 y Fu(It)d(is)g(useful)f(to)g 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y(Under)28 b(the)g(scaling)g(used)f(belo)n(w)g(the)h(ne)n(w)g(v)n(ariable)33 b Fq(^)-56 b Fr(u)2056 4531 y FD(s)2127 4504 y Fu(con)l(v)o(er)n(ges)27 b(to)h(zero,)h(while)f(the)f(old)h(v)n(ariable)j Fq(^)-54 b Fr(v)3831 4531 y FD(s)94 4624 y Fu(con)l(v)o(er)n(ges)25 b(to)f(a)i(nontri)n(vial)c(e)o(xpression.)316 4743 y(Under)j(this)f (transform)g(\(7.3\))h(becomes)1141 4938 y Fr(@)1194 4965 y FD(t)1236 4938 y Fq(^)-57 b Fr(u)1286 4965 y FE(c)1366 4938 y Fq(=)1510 4911 y Fo(b)1488 4938 y Fn(M)6 b Fq(^)-56 b Fr(u)1643 4965 y FE(c)1701 4938 y Fq(+)1812 4911 y Fo(b)1801 4938 y Fn(N)1879 4965 y FE(c)p FD(;)p Fl(i)1957 4938 y Fq(\()6 b(^)-56 b Fr(u)2053 4965 y FE(c)2089 4938 y Fr(;)22 b Fq(^)-55 b Fr(u)2191 4965 y FE(s)2222 4938 y Fq(\))22 b(+)2394 4911 y Fo(b)2383 4938 y Fn(N)2461 4965 y FE(c)2496 4938 y Fq(\()6 b(^)-56 b Fr(u)2592 4965 y FE(c)2627 4938 y Fr(;)23 b Fq(^)-56 b Fr(u)2729 4965 y FE(s)2761 4938 y Fq(\))p Fr(;)1145 5117 y(@)1198 5144 y FD(t)1239 5117 y Fq(^)g Fr(u)1290 5144 y FE(s)1366 5117 y Fq(=)1510 5090 y Fo(b)1488 5117 y Fn(M)6 b Fq(^)-56 b Fr(u)1643 5144 y FE(s)1697 5117 y Fq(+)1809 5090 y Fo(b)1797 5117 y Fn(N)1875 5144 y FE(s)p FD(;)p Fl(i)1949 5117 y Fq(\()6 b(^)-56 b Fr(u)2045 5144 y FE(c)2081 5117 y Fr(;)23 b Fq(^)-56 b Fr(u)2183 5144 y FE(s)2214 5117 y Fq(\))22 b(+)2386 5090 y Fo(b)2375 5117 y Fn(N)2453 5144 y FE(s)2484 5117 y Fq(\()6 b(^)-56 b Fr(u)2580 5144 y FE(c)2616 5117 y Fr(;)23 b Fq(^)-56 b Fr(u)2718 5144 y FE(s)2749 5117 y Fq(\))p Fr(;)1160 5296 y(@)1213 5323 y FD(t)1266 5296 y Fo(b)-73 b Fr(w)47 b Fq(=)1510 5269 y Fo(b)1488 5296 y Fn(M)1586 5323 y FE(w)1659 5296 y Fo(b)-73 b Fr(w)26 b Fq(+)1849 5269 y Fo(b)1837 5296 y Fn(N)1915 5323 y FE(w)1970 5296 y Fq(\()t(^)-54 b Fr(v)2057 5323 y FE(c)2093 5296 y Fr(;)20 b Fq(^)-53 b Fr(v)2186 5323 y FE(s)2217 5296 y Fr(;)34 b Fo(b)-72 b Fr(w)s Fq(\))24 b Fr(;)3669 5110 y Fq(\()p Fu(7)p Fr(:)p Fu(5)p Fq(\))p eop %%Page: 29 29 29 28 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(29)94 194 y Fu(where)294 586 y Fo(b)282 613 y Fn(N)360 640 y FE(c)p FD(;)p Fl(i)438 613 y Fq(\()6 b(^)-56 b Fr(u)534 640 y FE(c)570 613 y Fr(;)22 b Fq(^)-55 b Fr(u)672 640 y FE(s)703 613 y Fq(\))44 b(=)931 588 y(^)908 613 y Fr(E)982 640 y FE(c)1046 588 y Fq(^)1018 613 y Fp(H)q Fq(\()6 b(^)-56 b Fr(u)1199 640 y FE(c)1234 613 y Fr(;)1300 586 y Fo(b)1279 613 y Fn(M)1377 571 y Fm(\000)p FE(1)1502 588 y Fq(^)1479 613 y Fr(E)1553 640 y FE(s)1584 613 y Fq(\()p Fu(3)p Fr(U)1726 640 y Fm(\003)1809 613 y Fp(\001)28 b Fq(^)-56 b Fr(u)1916 640 y FE(c)1990 613 y Fp(\003)1980 605 y Fc(\015)2073 613 y Fq(^)g Fr(u)2124 640 y FE(c)2160 613 y Fq(\))22 b(+)29 b(^)-57 b Fr(u)2377 640 y FE(s)2410 613 y Fq(\))24 b Fr(;)297 766 y Fo(b)286 793 y Fn(N)364 820 y FE(s)p FD(;)p Fl(i)438 793 y Fq(\()6 b(^)-56 b Fr(u)534 820 y FE(c)570 793 y Fr(;)22 b Fq(^)-55 b Fr(u)672 820 y FE(s)703 793 y Fq(\))44 b(=)931 768 y(^)908 793 y Fr(E)982 820 y FE(s)1042 768 y Fq(^)1014 793 y Fp(H)q Fq(\()6 b(^)-56 b Fr(u)1195 820 y FE(c)1231 793 y Fr(;)1296 766 y Fo(b)1276 793 y Fn(M)1374 750 y Fm(\000)p FE(1)1498 768 y Fq(^)1475 793 y Fr(E)1549 820 y FE(s)1580 793 y Fq(\()p Fu(3)p Fr(U)1722 820 y Fm(\003)1805 793 y Fp(\001)28 b Fq(^)-56 b Fr(u)1912 820 y FE(c)1986 793 y Fp(\003)1976 785 y Fc(\015)2069 793 y Fq(^)g Fr(u)2120 820 y FE(c)2156 793 y Fq(\))22 b(+)29 b(^)-56 b Fr(u)2374 820 y FE(s)2406 793 y Fq(\))24 b Fr(;)337 945 y Fo(b)325 972 y Fn(N)403 999 y FE(c)438 972 y Fq(\()6 b(^)-56 b Fr(u)534 999 y FE(c)570 972 y Fr(;)22 b Fq(^)-55 b Fr(u)672 999 y FE(s)703 972 y Fq(\))44 b(=)931 947 y(^)908 972 y Fr(E)982 999 y FE(c)1029 945 y Fo(b)1018 972 y Fn(N)p Fq(\()6 b(^)-56 b Fr(u)1192 999 y FE(c)1228 972 y Fr(;)1293 945 y Fo(b)1273 972 y Fn(M)1371 929 y Fm(\000)p FE(1)1495 947 y Fq(^)1472 972 y Fr(E)1546 999 y FE(s)1578 972 y Fq(\()p Fu(3)p Fr(U)1720 999 y Fm(\003)1802 972 y Fp(\001)28 b Fq(^)-56 b Fr(u)1909 999 y FE(c)1984 972 y Fp(\003)1974 964 y Fc(\015)2067 972 y Fq(^)f Fr(u)2117 999 y FE(c)2153 972 y Fq(\))22 b(+)29 b(^)-56 b Fr(u)2371 999 y FE(s)2403 972 y Fq(\))25 b Fr(;)340 1119 y Fo(b)329 1146 y Fn(N)407 1173 y FE(s)438 1146 y Fq(\()6 b(^)-56 b Fr(u)534 1173 y FE(c)570 1146 y Fr(;)22 b Fq(^)-55 b Fr(u)672 1173 y FE(s)703 1146 y Fq(\))44 b(=)931 1121 y(^)908 1146 y Fr(E)982 1173 y FE(s)1025 1119 y Fo(b)1014 1146 y Fn(N)p Fq(\()6 b(^)-56 b Fr(u)1188 1173 y FE(c)1224 1146 y Fr(;)1289 1119 y Fo(b)1269 1146 y Fn(M)1367 1103 y Fm(\000)p FE(1)1491 1121 y Fq(^)1468 1146 y Fr(E)1542 1173 y FE(s)1574 1146 y Fq(\()p Fu(3)p Fr(U)1716 1173 y Fm(\003)1798 1146 y Fp(\001)29 b Fq(^)-57 b Fr(u)1905 1173 y FE(c)1980 1146 y Fp(\003)1970 1138 y Fc(\015)2063 1146 y Fq(^)g Fr(u)2113 1173 y FE(c)2149 1146 y Fq(\))22 b(+)29 b(^)-56 b Fr(u)2367 1173 y FE(s)2399 1146 y Fq(\))22 b Fp(\000)h Fr(@)2613 1173 y FD(t)2648 1146 y Fu([)2702 1119 y Fo(b)2681 1146 y Fn(M)2779 1103 y Fm(\000)p FE(1)2904 1121 y Fq(^)2881 1146 y Fr(E)2955 1173 y FE(s)2986 1146 y Fq(\()p Fu(3)p Fr(U)3128 1173 y 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Fn(M)3455 4965 y FE(s)p FD(;n)3560 4938 y Fq(\()p Fr(`)p Fq(\))g Fu(as)h(a)94 5057 y(function)24 b(of)h Fr(`)g Fu(can)g(be)g(bounded)f (from)h(abo)o(v)o(e)e(by)i(a)g(strictly)f(ne)o(gati)n(v)o(e)e (parabola.)p 3812 5003 63 2 v 3812 5062 2 59 v 3872 5062 V 3812 5064 63 2 v 316 5291 a(Ne)o(xt,)i(we)h(bound)987 5266 y Fo(b)973 5291 y Fr(S)1034 5318 y FD(n)1088 5291 y Fq(\()p Fr(\034)6 b(;)17 b(\034)1277 5248 y Fm(0)1303 5291 y Fq(\))25 b Fu(as)g(de\002ned)g(through)f(\(7.9\))h(and)f(state)h (the)g(analog)f(of)h(Lemma)f(4.2.)p eop %%Page: 31 31 31 30 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(31)94 194 y(Lemma)29 b(7.3.)45 b Fg(Suppose)28 b(that)g(the)g(stability)e(condition)g(\(6.11\))i(is)g (satis\002ed.)45 b(Then)28 b(there)g(is)g(a)g Fr(\016)37 b Fp(2)32 b Fq(\()p Fu(0)p Fr(;)17 b Fu(1])94 314 y Fg(such)j(that)f (for)h(all)f Fr(")767 271 y Fm(0)823 314 y Fp(2)29 b Fq(\()p Fu(0)p Fr(;)17 b Fu(1)p Fq(\))h Fg(there)i(e)o(xists)e(a)i Fr(C)1758 341 y FD(")1796 321 y Fh(0)1855 314 y Fr(>)28 b Fu(0)20 b Fg(such)f(that)h(for)f Fu(1)28 b Fr(>)g(\034)39 b(>)28 b(\034)2963 271 y Fm(0)3018 314 y Fp(\025)g Fu(0)20 b Fg(and)g(all)f Fr(\033)31 b Fp(2)d Fq(\()p Fu(0)p Fr(;)17 b Fu(1])94 433 y Fg(one)25 b(has)416 674 y Fp(k)480 649 y Fo(b)466 674 y Fr(S)527 701 y FD(n)581 674 y Fq(\()p Fr(\034)6 b(;)17 b(\034)770 631 y Fm(0)797 674 y Fq(\))g Fo(b)-72 b Fr(w)r Fp(k)959 701 y Fm(K)1020 719 y Fi(\033)1063 704 y(n)1165 674 y Fp(\024)45 b Fr(C)1358 701 y FD(")1396 681 y Fh(0)1428 674 y Fr(\033)1489 631 y Fm(\000)p FD(")1589 588 y Fh(0)1615 631 y FD(n)1669 674 y Fr(e)1715 631 y Fm(\000)p FD(\015)t(\033)1872 588 y Fh(\000)p Fj(2)p Fi(n)2001 631 y Fk(\()p FD(\034)8 b Fm(\000)p FD(\034)2184 588 y Fh(0)2211 631 y Fk(\))p FD(=)p FE(2)2322 674 y Fq(\()p Fr(\034)33 b Fp(\000)23 b Fr(\034)2593 631 y Fm(0)2621 674 y Fq(\))2660 631 y FD(")2698 588 y Fh(0)2724 631 y Fm(\000)p FE(1)2826 674 y Fp(k)17 b Fo(b)-72 b Fr(w)s Fp(k)3000 701 y Fm(K)3061 719 y Fi(\033)3104 704 y(n)3152 719 y(;")3209 704 y Fh(0)3270 674 y Fr(;)3619 667 y Fq(\()p Fu(7)p Fr(:)p Fu(10)p Fq(\))94 921 y Fg(for)26 b(all)e Fr(n)k Fp(2)g Fv(N)p Fg(.)316 1099 y Fu(The)c(proof)g(of)g (Lemma)g(7.3)g(follo)n(ws)f(closely)g(the)h(one)g(of)h(Lemma)e(4.2)h (in)g(Section)g(4.1.)35 b(Therefore,)94 1218 y(it)25 b(will)f(be)h(omitted)e(here.)36 b(W)-8 b(e)26 b(only)e(remark)h(that)f (the)h(estimate)f(for)h(the)f(solution)f(of)i(\(7.9\))1209 1483 y Fp(k)1280 1457 y Fq(^)1259 1483 y Fr(f)1308 1510 y FD(\034)1358 1483 y Fp(k)1427 1499 y Fk(^)1408 1517 y FD(H)1478 1483 y Fj(2)p Fi(;\016)1473 1543 y Fj(0)1598 1483 y Fp(\024)j Fr(C)7 b(e)1827 1440 y FD(\015)t(\033)1922 1397 y Fh(\000)p Fj(2)p Fi(n)2050 1440 y Fk(\()p FD(\034)h Fm(\000)p FD(\034)2233 1397 y Fh(0)2260 1440 y Fk(\))p FD(=)p FE(2)2372 1483 y Fp(k)2443 1457 y Fq(^)2422 1483 y Fr(f)2471 1510 y FD(\034)2516 1490 y Fh(0)2547 1483 y Fp(k)2617 1499 y Fk(^)2597 1517 y FD(H)2667 1483 y Fj(2)p Fi(;\016)2662 1543 y Fj(0)94 1734 y Fu(associated)30 b(to)g(\(4.13\))g(can)h(be)g(obtained)e(e)o(xactly)h(in)g(the)g(same)g (w)o(ay)h(as)f(\(6.12\).)52 b(The)30 b(estimates)g(for)g(the)94 1854 y(weights)24 b(in)h Fr(`)f Fu(and)h(the)g(deri)n(v)n(ati)n(v)o(es) d(with)i(respect)h(to)f Fr(x)h Fu(follo)n(w)f(again)g(as)h(in)f(the)h (proof)f(of)h(Lemma)g(4.2.)94 2209 y Fv(7.2.)g(The)h(scaled)f (non\255linear)h(terms)94 2398 y Fu(Ne)o(xt)f(we)g(estimate)f(the)g (scaled)h(non\255linear)f(terms)h(in)f Fn(N)2091 2425 y FE(c)p FD(;n)2199 2398 y Fu(,)h Fn(N)2327 2425 y FE(s)p FD(;n)2431 2398 y Fu(,)g(and)g Fn(N)2728 2425 y FE(w)p FD(;n)2855 2398 y Fu(.)94 2602 y Fv(Lemma)35 b(7.4.)61 b Fg(Suppose)33 b Fu(max)p Fp(fk)t Fq(^)-54 b Fr(v)1357 2629 y FE(c)p FD(;n)1465 2602 y Fp(k)1515 2629 y Fm(K)1576 2647 y Fi(\033)1619 2632 y(n)1677 2602 y Fr(;)17 b Fp(k)t Fq(^)-54 b Fr(v)1820 2629 y FE(s)p FD(;n)1925 2602 y Fp(k)1975 2629 y Fm(K)2036 2647 y Fi(\033)2079 2632 y(n)2136 2602 y Fr(;)17 b Fp(k)g Fo(b)-72 b Fr(w)2302 2629 y FD(n)2356 2602 y Fp(k)2406 2629 y Fm(K)2467 2647 y Fi(\033)2510 2632 y(n)2568 2602 y Fp(g)39 b(\024)i Fu(1)p Fg(.)61 b(Then)34 b(for)f(all)g Fr(")3478 2559 y Fm(0)3547 2602 y Fp(2)40 b Fq(\()p Fu(0)p Fr(;)17 b Fu(1)p Fq(\))94 2722 y Fg(there)26 b(e)o(xist)d Fr(C)601 2748 y FE(1)641 2722 y Fr(;)17 b(C)757 2748 y FD(")795 2728 y Fh(0)854 2722 y Fr(>)28 b Fu(0)d Fg(such)f(that)h(for)g(all)f Fr(\033)31 b Fp(2)e Fq(\()p Fu(0)p Fr(;)17 b Fu(1])23 b Fg(one)i(has)666 2971 y Fp(k)728 2944 y Fo(b)716 2971 y Fn(N)794 2998 y FE(c)p FD(;n)901 2971 y Fp(k)951 2998 y Fm(K)1012 3016 y Fi(\033)1055 3001 y(n)1103 3016 y(;)p Fj(1)p Fi(=)p Fj(4)1267 2971 y Fp(\024)17 b Fr(C)1432 2998 y FE(1)1472 2971 y Fr(\033)1533 2929 y FE(5)p FD(n=)p FE(2)1697 2971 y Fq(\()p Fp(k)t Fq(^)-54 b Fr(v)1834 2998 y FE(c)p FD(;n)1943 2971 y Fp(k)1993 2998 y Fm(K)2054 3016 y Fi(\033)2097 3001 y(n)2177 2971 y Fq(+)22 b Fp(k)t Fq(^)-54 b Fr(v)2374 2998 y FE(s)p FD(;n)2480 2971 y Fp(k)2530 2998 y Fm(K)2591 3016 y Fi(\033)2634 3001 y(n)2691 2971 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Fc(\015)490 4769 y Fq(^)-54 b Fr(v)34 b Fp(\003)550 4761 y Fc(\015)654 4769 y Fo(b)-73 b Fr(w)s Fu(.)35 b(From)21 b(\(7.11\))g(we)g(obtain)f(a)i Fr(\033)1803 4726 y FD(n)1877 4769 y Fu(for)g(the)f(scaled)g(v)o(ersion)f(of)25 b Fq(^)-54 b Fr(v)34 b Fp(\003)2909 4761 y Fc(\015)3013 4769 y Fo(b)-73 b Fr(w)s Fu(.)35 b(W)-8 b(e)21 b(loose)g Fr(\033)3573 4726 y Fm(\000)p FD(")3673 4683 y Fh(0)3699 4726 y FD(n)3774 4769 y Fu(by)94 4903 y(taking)h(the)g(norm)g(in)g Fp(K)919 4930 y FD(\033)968 4910 y Fi(n)1016 4930 y FD(;")1078 4910 y Fh(0)1132 4903 y Fu(due)g(to)g(the)g(f)o(act)g(that)1908 4878 y Fo(b)1884 4903 y Fr(K)1969 4930 y FD(ct)2062 4903 y Fu(is)g(\002x)o(ed)g(and)g(does)g(not)g(scale)g(when)g(time)g(e)n(v)n (olv)o(es.)316 5052 y(ii\))g(W)-8 b(e)23 b(use)f(again)f(\(7.11\))h(to) g(obtain)g(the)g(estimates)f(for)2294 5025 y Fo(b)2283 5052 y Fn(N)2361 5079 y FE(s)p FD(;n)2465 5052 y Fu(.)35 b(The)22 b(only)g(dif)n(\002culty)f(stems)g(from)h(the)94 5172 y(term)877 5291 y Fr(@)930 5318 y FD(t)965 5291 y Fu([)1019 5264 y Fo(b)998 5291 y Fn(M)1096 5248 y Fm(\000)p FE(1)1221 5266 y Fq(^)1198 5291 y Fr(E)1272 5318 y FE(s)1303 5291 y Fq(\()p Fu(3)p Fr(U)1445 5318 y Fm(\003)1528 5291 y Fp(\001)28 b Fq(^)-56 b Fr(u)1635 5318 y FE(c)1710 5291 y Fp(\003)1700 5283 y Fc(\015)1792 5291 y Fq(^)g Fr(u)1843 5318 y FE(c)1879 5291 y Fq(\))p Fu(])28 b Fq(=)2105 5264 y Fo(b)2084 5291 y Fn(M)2182 5248 y Fm(\000)p FE(1)2307 5266 y Fq(^)2284 5291 y Fr(E)2358 5318 y FE(s)2389 5291 y Fq(\()p Fu(6)p Fr(U)2531 5318 y Fm(\003)2614 5291 y Fp(\001)g Fq(^)-56 b Fr(u)2721 5318 y FE(c)2796 5291 y Fp(\003)2786 5283 y Fc(\015)2872 5291 y Fr(@)2925 5318 y FD(t)2966 5291 y Fq(^)g Fr(u)3017 5318 y FE(c)3053 5291 y Fq(\))p eop %%Page: 32 32 32 31 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(32)94 194 y Fu(coming)33 b(from)h(the)g(change)g(of)g(coordinates)f(\(7.4\).)63 b(This)33 b(can)h(be)g(estimated)f(in)g(the)h(required)f(w)o(ay)h(by)94 314 y(e)o(xpressing)27 b Fr(@)600 341 y FD(t)641 314 y Fq(^)-56 b Fr(u)692 341 y FE(c)755 314 y Fu(by)27 b(the)h(right)e (hand)h(side)h(of)f(\(7.5\),)h(by)f(using)g(then)g(the)g(points)f (ii.1\)\226ii.3\))g(and)h(the)h(f)o(act)94 433 y(we)e(already)f(ha)n(v) o(e)f(a)h(f)o(actor)h Fr(\033)1150 390 y FD(n)1228 433 y Fu(by)31 b Fq(^)-56 b Fr(u)1410 460 y FE(c)1484 433 y Fp(\003)1474 425 y Fc(\015)1561 433 y Fr(@)1614 460 y FD(t)1655 433 y Fq(^)g Fr(u)1706 460 y FE(c)1767 433 y Fu(using)24 b(again)g(\(7.11\).)316 553 y(ii.1\))g(The)h(\002rst)g (bound)f(for)h(the)g(terms)f(on)g(the)h(right)f(hand)h(side)f(of)h (\(7.5\))g(is)1123 818 y Fp(k)1194 791 y Fo(b)1173 818 y Fn(M)1271 845 y FD(c;n)1388 818 y Fq(^)-54 b Fr(v)1432 845 y FE(c)p FD(;n)1542 818 y Fp(k)1592 845 y Fm(K)1653 863 y Fi(\033)1696 848 y(n)1743 863 y(;\032)1840 818 y Fp(\024)28 b Fr(C)7 b(\033)2084 775 y FE(2)p Fk(\()p FE(1)p Fm(\000)p FD(\032)p Fk(\))p FD(n)2374 818 y Fp(k)t Fq(^)-54 b Fr(v)2472 845 y FE(c)p FD(;n)2581 818 y Fp(k)2631 845 y Fm(K)2692 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b(there)g(e)o(xists)e(a)i Fr(C)35 b(>)28 b Fu(0)d Fg(such)g(that)f(for)h(all)g Fr(\033)31 b Fp(2)d Fq(\()p Fu(0)p Fr(;)17 b Fu(1])24 b Fg(we)h(ha)n(v)o(e)702 1791 y Fp(k)p Fq(\()799 1764 y Fo(b)791 1791 y Fn(L)862 1818 y FD(\033)916 1791 y Fr(\026)p Fq(\))6 b(^)-56 b Fr(u)p Fp(k)1122 1818 y Fm(K)1183 1836 y Fi(\033)n(;\032)1318 1791 y Fp(\024)28 b Fr(C)7 b(\033)1562 1748 y FE(2)p Fk(\()p FE(1)p Fm(\000)p FD(\032)p Fk(\))1802 1791 y Fp(k)p Fr(\026)p Fp(k)1962 1818 y Fm(C)2008 1798 y Fj(2)2005 1845 y(per)2074 1818 y Fk(\()p FE([)p Fm(\000)p FE(1)p FD(=)p FE(2)p FD(;)p FE(1)p FD(=)p FE(2)p Fk(\))p FD(;)p Fm(C)2537 1798 y Fj(2)2567 1818 y Fk(\(\()p FE(0)p FD(;)p FE(2)p FD(\031)s Fk(\))p FD(;)p Fl(C)p Fk(\)\))2944 1791 y Fp(k)f Fq(^)-56 b Fr(u)p Fp(k)3101 1818 y Fm(K)3162 1836 y Fi(\033)3239 1791 y Fr(:)352 b Fq(\()p Fu(7)p Fr(:)p Fu(12)p Fq(\))94 2207 y Fv(Pr)n(oof)o(.)25 b Fu(This)f(follo)n(ws)f(since)1293 2516 y(sup)1294 2603 y FD(`)p Fm(2)p Fl(R)1448 2516 y Fp(j)1488 2449 y Fr(`)1530 2406 y FE(2)p Fk(\()p FE(1)p Fm(\000)p FD(\032)p Fk(\))1770 2449 y Fr(\033)1831 2406 y FE(2)p Fk(\()p FE(1)p Fm(\000)p FD(\032)p Fk(\))p 1488 2493 584 4 v 1511 2586 a Fq(\()p Fu(1)f Fq(+)h Fr(`)1764 2557 y FE(2)1803 2586 y Fq(\))1842 2557 y Fk(\()p FE(1)p Fm(\000)p FD(\032)p Fk(\))2083 2516 y Fp(j)k Fr(<)h(C)7 b(\033)2382 2473 y FE(2)p Fk(\()p FE(1)p Fm(\000)p FD(\032)p Fk(\))2648 2516 y Fr(:)p 3812 2765 63 2 v 3812 2823 2 59 v 3872 2823 V 3812 2825 63 2 v 316 3078 a Fu(ii.2\))24 b(By)h(Lemma)g(7.8)f(belo)n(w)g(the)g(term) 1756 3051 y Fo(b)1744 3078 y Fn(N)1822 3105 y FE(c)p FD(;)p Fl(i)p FD(;n)1998 3078 y Fu(is)g(e)o(xponentially)f(small)h(in)g (terms)h(of)f Fr(\033)t Fu(.)316 3197 y(ii.3\))g(From)h(\(7.11\))f(we)i (easily)e(obtain)1074 3463 y Fp(k)1136 3436 y Fo(b)1124 3463 y Fn(N)1202 3489 y FE(c)p FD(;n)1310 3463 y Fp(k)1360 3489 y Fm(K)1421 3507 y Fi(\033)1464 3492 y(n)1549 3463 y Fp(\024)k Fr(\033)1715 3420 y FD(n)1769 3463 y Fq(\()p Fp(k)t Fq(^)-54 b Fr(v)1906 3489 y FE(c)p 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b(The)23 b(detailed)g(e)o(xpression)e(for)j Fr(K)3505 1522 y FE(1)3567 1495 y Fu(is)f(gi)n(v)o(en)94 1615 y(in)i(\(7.13\))g (belo)n(w)-6 b(.)316 1734 y(The)26 b(case)h Fr(n)j Fq(=)g Fr(m)h Fq(=)f Fr(k)j Fq(=)d Fr(`)g Fq(=)g Fu(0)c(corresponds)g(to)g (the)g(spatially)g(periodic)g(case.)41 b(In)26 b(the)g(spatially)94 1854 y(periodic)f(case)g(there)g(e)o(xists)f(a)h(center)g(manifold)1466 2038 y Fq(\000)j(=)g Fp(f)p Fr(u)g Fq(=)g Fr(U)1969 2065 y FE(0)p FD(;a)2101 2038 y Fp(j)c Fr(a)k Fp(2)g Fv(R)p Fp(g)c Fr(;)94 2222 y Fu(consisting)18 b(of)i(the)g(spatially)f (periodic)g(\002x)o(ed)h(points)e(related)i(to)g(each)g(other)g(by)f (the)h(translation)f(in)l(v)n(ariance)94 2341 y(of)28 b(the)g(original)e(Swift\255Hohenber)n(g)i(equation.)44 b(By)28 b(a)g(formal)f(calculation)g(it)g(turns)g(out)g(that)g(the)h (\003o)n(w)f(of)94 2461 y(the)e(one\255dimensional)e(center)i(manifold) f Fq(\000)h Fu(is)g(determined)f(by)g(the)h(ordinary)f(dif)n(ferential) g(equation)870 2626 y Fr(d)p 852 2670 88 4 v 852 2762 a(dt)951 2693 y(a)k Fq(=)g Fu(0)22 b Fp(\001)g Fr(a)g Fq(+)g Fr(K)1518 2720 y FE(1)1558 2693 y Fq(\()p Fu(0)p Fr(;)17 b Fu(0)p Fr(;)g Fu(0)p Fq(\))p Fr(a)1929 2650 y FE(2)1988 2693 y Fq(+)22 b Fr(K)2172 2720 y FE(2)2212 2693 y Fq(\()p Fu(0)p Fr(;)17 b Fu(0)p Fr(;)g Fu(0)p Fr(;)g Fu(0)p Fq(\))p Fr(a)2678 2650 y FE(3)2736 2693 y Fq(+)23 b Fn(O)p Fq(\()p Fr(a)2998 2650 y FE(4)3037 2693 y Fq(\))i Fr(:)94 2924 y Fu(Since)34 b(the)f(center)h(manifold)e (consists)g(of)i(\002x)o(ed)f(points)f(the)h(\003o)n(w)g Fr(a)40 b Fq(=)g Fr(a)p Fq(\()p Fr(t)p Fq(\))32 b Fu(is)h(tri)n(vial,)h FF(i.e)o(.)p Fu(,)3522 2884 y FD(d)p 3507 2901 72 4 v 3507 2958 a(dt)3590 2924 y Fr(a)39 b Fq(=)h Fu(0.)94 3043 y(Consequently)-6 b(,)24 b(we)h(obtain)f Fr(K)1186 3070 y FE(1)1225 3043 y Fq(\()p Fu(0)p Fr(;)17 b Fu(0)p Fr(;)g Fu(0)p Fq(\))25 b(=)k Fr(K)1759 3070 y FE(2)1798 3043 y Fq(\()p Fu(0)p Fr(;)17 b Fu(0)p Fr(;)g Fu(0)p Fr(;)g Fu(0)p Fq(\))24 b(=)k Fu(0.)36 b(Therefore,)697 3227 y Fp(j)p Fr(K)810 3254 y FE(2)849 3227 y Fq(\()p Fr(`;)17 b(`)j Fp(\000)j Fr(m;)17 b(m)22 b Fp(\000)h Fr(k)s(;)17 b(k)s Fq(\))p Fp(j)43 b(\024)i Fr(C)7 b Fq(\()p Fp(j)p Fr(`)p Fp(j)21 b Fq(+)h Fp(j)p Fr(`)g Fp(\000)g Fr(m)p Fp(j)h Fq(+)f Fp(j)p Fr(m)h Fp(\000)f Fr(k)s Fp(j)g Fq(+)h Fp(j)p Fr(k)s Fp(j)p Fq(\))h Fr(;)94 3411 y Fu(and)h(so)g (\(7.11\))f(and)h(\(7.12\))g(imply)996 3595 y Fp(k)t Fq(^)-54 b Fr(s)1093 3621 y FE(2)1132 3595 y Fp(k)1182 3621 y Fm(K)1243 3639 y Fi(\033)1286 3624 y(n)1334 3639 y(;)p Fj(1)p Fi(=)p Fj(2)1498 3595 y Fp(\024)44 b Fr(C)7 b(\033)1758 3552 y FE(3)p FD(n)1847 3595 y Fq(\()p Fp(k)t Fq(^)-54 b Fr(v)1984 3621 y FE(c)p FD(;n)2093 3595 y Fp(k)2143 3621 y Fm(K)2204 3639 y Fi(\033)2247 3624 y(n)2327 3595 y Fq(+)23 b Fp(k)t Fq(^)-54 b Fr(v)2525 3621 y FE(s)p FD(;n)2630 3595 y Fp(k)2680 3621 y Fm(K)2741 3639 y Fi(\033)2784 3624 y(n)2842 3595 y Fq(\))2881 3552 y FE(2)2945 3595 y Fr(:)94 3794 y Fu(Interestingly)29 b(it)h(turned)f(out)g(that)h(the)g (\002rst)g(deri)n(v)n(ati)n(v)o(es)d(of)j Fr(K)2328 3821 y FE(1)2397 3794 y Fu(v)n(anish)f(as)h(well.)51 b(Since)30 b(the)g(eigen)l(v)n(alue)94 3925 y(problem)24 b Fn(M)549 3952 y FD(`)587 3925 y Fr(')652 3952 y FD(`)718 3925 y Fq(=)29 b Fr(\026)884 3952 y FE(1)923 3925 y Fq(\()p Fr(`)p Fq(\))p Fr(')1108 3952 y FD(`)1170 3925 y Fu(is)24 b(self\255adjoint,)g(the)g(projection)2362 3900 y Fq(^)2340 3925 y Fr(P)2404 3952 y FE(c)2439 3925 y Fq(\()p Fr(`)p Fq(\))g Fu(is)g(orthogonal)f(in)h(L)3293 3883 y FE(2)3333 3925 y Fq(\()p Fu(0)p Fr(;)17 b Fu(2)p Fr(\031)t Fq(\))22 b Fu(and)j(is)94 4057 y(gi)n(v)o(en)f(by)484 4032 y Fq(^)461 4057 y Fr(P)525 4084 y FE(c)561 4057 y Fq(\()p Fr(`)p Fq(\))p Fr(u)j Fq(=)h(\()909 3977 y Fo(R)p 992 3971 238 4 v 992 4057 a Fr(')1057 4084 y FD(`)1095 4057 y Fq(\()p Fr(x)p Fq(\))p Fr(u)p Fq(\()p Fr(`;)17 b(x)p Fq(\))p Fr(dx)p Fq(\))p Fr(')1722 4084 y FD(`)1758 4057 y Fq(\()p Fp(\001)p Fq(\))p Fu(.)35 b(Thus)865 4298 y Fr(K)950 4325 y FE(1)989 4298 y Fq(\()p Fr(`;)17 b(`)j Fp(\000)j Fr(m;)17 b(m)p Fq(\))28 b(=)g Fu(3)1735 4162 y Fo(Z)1830 4298 y Fu(d)p Fr(x)p 1954 4211 V 17 w(')2019 4325 y FD(`)2057 4298 y Fq(\()p Fr(x)p Fq(\))p Fr(')2257 4325 y FD(`)p Fm(\000)p FD(m)2428 4298 y Fq(\()p Fr(x)p Fq(\))p Fr(')2628 4325 y FD(m)2704 4298 y Fq(\()p Fr(x)p Fq(\))p Fr(U)11 b Fq(\()p Fr(x)p Fq(\))23 b Fr(:)515 b Fq(\()p Fu(7)p Fr(:)p Fu(13)p Fq(\))94 4548 y Fu(Expanding)24 b Fr(')616 4574 y FD(`)655 4548 y Fq(\()p Fr(x)p Fq(\))j(=)h Fr(@)975 4574 y FD(x)1025 4548 y Fr(U)11 b Fq(\()p Fr(x)p Fq(\))21 b(+)i Fr(i`g)t Fq(\()p Fr(x)p Fq(\))d(+)j Fn(O)p Fq(\()p Fr(`)1894 4505 y FE(2)1934 4548 y Fq(\))p Fr(;)h Fu(with)g Fr(g)t Fq(\()p Fr(x)p Fq(\))i Fp(2)i Fv(R)d Fu(yields)423 4793 y Fr(K)508 4820 y FE(1)548 4793 y Fq(\()p Fr(`;)17 b(`)j Fp(\000)i Fr(m;)17 b(m)p Fq(\))28 b(=)p Fu(3)1265 4657 y Fo(Z)1361 4793 y Fu(d)p Fr(x)1485 4653 y Fo(\022)1558 4793 y Fq(\()p Fr(@)1650 4820 y FD(x)1700 4793 y Fr(U)11 b Fq(\()p Fr(x)p Fq(\)\))1953 4750 y FE(3)1992 4793 y Fr(U)g Fq(\()p Fr(x)p Fq(\))1221 5017 y Fp(\000)22 b Fr(i`g)t Fq(\()p Fr(x)p Fq(\)\()p Fr(@)1675 5044 y FD(x)1723 5017 y Fr(U)11 b Fq(\()p Fr(x)p Fq(\)\))1976 4975 y FE(2)2015 5017 y Fr(U)g Fq(\()p Fr(x)p Fq(\))22 b(+)g Fr(i)p Fq(\()p Fr(`)g Fp(\000)h Fr(m)p Fq(\))p Fr(g)t Fq(\()p Fr(x)p Fq(\)\()p Fr(@)2992 5044 y FD(x)3040 5017 y Fr(U)11 b Fq(\()p Fr(x)p Fq(\)\))3293 4975 y FE(2)3332 5017 y Fr(U)g Fq(\()p Fr(x)p Fq(\))1221 5237 y(+)22 b(\()p Fr(@)1412 5263 y FD(x)1462 5237 y Fr(U)11 b Fq(\()p Fr(x)p Fq(\)\))1715 5194 y FE(2)1754 5237 y Fr(img)t Fq(\()p Fr(x)p Fq(\))p Fr(U)g Fq(\()p Fr(x)p Fq(\))21 b(+)h Fn(O)p Fq(\()p Fr(`)2547 5194 y FE(2)2609 5237 y Fq(+)h(\()p Fr(`)e Fp(\000)i Fr(m)p Fq(\))3037 5194 y FE(2)3099 5237 y Fq(+)g Fr(m)3286 5194 y FE(2)3326 5237 y Fq(\))3365 5096 y Fo(\023)3463 5237 y Fr(:)p eop %%Page: 34 34 34 33 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(34)94 194 y Fu(Note)26 b(that)e Fr(U)11 b Fq(\()p Fr(x)p Fq(\))25 b Fu(is)g(an)h(e)n(v)o(en)e (function,)h(so)g Fr(@)1698 221 y FD(x)1748 194 y Fr(U)36 b Fu(is)25 b(odd,)g(which)g(pro)o(v)o(es)f(again)g Fr(K)3026 221 y FE(1)3066 194 y Fq(\()p Fu(0)p Fr(;)17 b Fu(0)p Fr(;)g Fu(0)p Fq(\))26 b(=)j Fu(0.)37 b(Since,)94 314 y(in)25 b(addition,)e(the)i(\002rst)g(order)g(terms)f(cancel)i(we)f(ha) n(v)o(e)1035 566 y Fp(j)p Fr(K)1148 593 y FE(1)1187 566 y Fq(\()p Fr(`;)17 b(`)j Fp(\000)j Fr(m;)17 b(m)p Fq(\))p Fp(j)44 b(\024)h Fr(C)7 b Fp(j)p Fr(`)2075 523 y FE(2)2136 566 y Fq(+)23 b(\()p Fr(`)e Fp(\000)i Fr(m)p Fq(\))2564 523 y FE(2)2626 566 y Fq(+)g Fr(m)2813 523 y FE(2)2853 566 y Fp(j)i Fr(;)94 818 y Fu(and)g(so)g(from)f(\(7.11\))h(and)g (\(7.12\))958 1070 y Fp(k)t Fq(^)-54 b Fr(s)1055 1097 y FE(1)1094 1070 y Fp(k)1144 1097 y Fm(K)1205 1115 y Fi(\033)1248 1100 y(n)1296 1115 y(;)p Fj(1)p Fi(=)p Fj(4)1460 1070 y Fp(\024)45 b Fr(C)7 b(\033)1721 1027 y FE(5)p FD(n=)p FE(2)1885 1070 y Fq(\()p Fp(k)t Fq(^)-54 b Fr(v)2022 1097 y FE(c)p FD(;n)2131 1070 y Fp(k)2181 1097 y Fm(K)2242 1115 y Fi(\033)2285 1100 y(n)2365 1070 y Fq(+)22 b Fp(k)t Fq(^)-54 b Fr(v)2562 1097 y FE(s)p FD(;n)2668 1070 y Fp(k)2718 1097 y Fm(K)2779 1115 y Fi(\033)2822 1100 y(n)2879 1070 y Fq(\))2918 1027 y FE(2)2983 1070 y Fr(:)94 1337 y Fu(Summing)24 b(the)h(estimates)e(sho)n(ws)h(the)g(assertion.)p 3812 1283 63 2 v 3812 1341 2 59 v 3872 1341 V 3812 1343 63 2 v 94 1832 a Fv(7.3.)h(Bounds)h(on)f(the)h(integrals)94 2034 y Fu(Here)35 b(we)e(estimate)g(the)h(inte)o(grals)e(in)h(the)g(v)n (ariation)g(of)g(constant)g(formula)g(in)g(terms)g(of)h(the)f(follo)n (wing)94 2154 y(quantities.)94 2345 y Fv(De\002nition)26 b(7.6.)36 b Fg(F)o(or)24 b(all)h Fr(n)p Fg(,)f(we)i(de\002ne)94 2597 y Fr(R)171 2555 y FD(u)170 2624 y FE(cs)p FD(;n)350 2597 y Fq(=)117 b Fu(sup)472 2689 y FD(\034)8 b Fm(2)p FE([)p FD(\033)643 2669 y Fj(2)672 2689 y FD(;)p FE(1])771 2597 y Fp(k)t Fq(^)-54 b Fr(v)869 2624 y FE(c)p FD(;n)978 2597 y Fq(\()p Fr(\034)11 b Fq(\))p Fp(k)1161 2624 y Fm(K)1222 2642 y Fi(\033)1265 2627 y(n)1342 2597 y Fq(+)93 b Fu(sup)1440 2689 y FD(\034)8 b Fm(2)p FE([)p FD(\033)1611 2669 y Fj(2)1640 2689 y FD(;)p FE(1])1738 2597 y Fp(k)t Fq(^)-54 b Fr(v)1836 2624 y FE(s)p FD(;n)1942 2597 y Fq(\()p Fr(\034)11 b Fq(\))p Fp(k)2125 2624 y Fm(K)2186 2642 y Fi(\033)2229 2627 y(n)2310 2597 y Fr(;)116 b Fg(and)99 b Fr(R)2774 2555 y FD(w)2773 2624 y(n)2882 2597 y Fq(=)117 b Fu(sup)3004 2689 y FD(\034)8 b Fm(2)p FE([)p FD(\033)3175 2669 y Fj(2)3204 2689 y FD(;)p FE(1])3303 2597 y Fp(k)17 b Fo(b)-72 b Fr(w)3424 2624 y FD(n)3478 2597 y Fq(\()p Fr(\034)11 b Fq(\))p Fp(k)3661 2624 y Fm(K)3722 2642 y Fi(\033)3765 2627 y(n)3846 2597 y Fr(:)316 2990 y Fu(In)25 b(the)f(follo)n(wing)f(tw)o(o)i(lemmas)e(we)j(estimate)e(the)g(inte)o (grals)g(appearing)g(in)h(\(7.6\)\226\(7.8\).)94 3182 y Fv(Lemma)h(7.7.)35 b Fg(Assume)24 b Fr(R)1054 3139 y FD(u)1053 3209 y FE(cs)p FD(;n)1211 3182 y Fq(+)f Fr(R)1388 3139 y FD(w)1387 3209 y(n)1479 3182 y Fp(\024)28 b Fu(1)p Fg(.)36 b(Then)25 b(for)g(all)f Fu(1)k Fp(\025)g Fr(\034)38 b Fp(\025)29 b Fr(\033)2622 3139 y FE(2)2686 3182 y Fg(and)c(all)f Fr(\033)31 b Fp(2)d Fq(\()p Fu(0)p Fr(;)17 b Fu(1])24 b Fg(one)h(has)380 3500 y Fp(k)p Fr(\033)491 3458 y Fm(\000)p FE(2)p FD(n)658 3365 y Fo(Z)758 3408 y FD(\034)714 3591 y(\033)763 3571 y Fj(2)804 3500 y Fu(d)p Fr(\034)909 3458 y Fm(0)954 3500 y Fr(e)1000 3458 y FD(\033)1049 3415 y Fh(\000)p Fj(2)p Fi(n)1190 3452 y Fo(b)1176 3458 y Fb(M)1258 3476 y Fj(c)p Fi(;n)1352 3458 y Fk(\()p FD(\034)8 b Fm(\000)p FD(\034)1535 3415 y Fh(0)1562 3458 y Fk(\))1598 3420 y Fo(\000)1655 3473 y(b)1644 3500 y Fn(N)1722 3527 y FE(c)p FD(;n)1830 3500 y Fq(\()t(^)-54 b Fr(v)1917 3527 y FE(c)1953 3500 y Fr(;)20 b Fq(^)-53 b Fr(v)2046 3527 y FE(s)2077 3500 y Fq(\))2116 3420 y Fo(\001)2162 3500 y Fq(\()p Fp(\001)p Fr(;)17 b Fp(\001)p Fr(;)g(\034)2402 3458 y Fm(0)2427 3500 y Fq(\))p Fp(k)2516 3527 y Fm(K)2577 3545 y Fi(\033)2620 3530 y(n)2705 3500 y Fp(\024)28 b Fr(C)7 b(\033)2949 3458 y FD(n=)p FE(2)3079 3500 y Fq(\()p 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3767 y Fq(\))3369 3724 y FE(2)3433 3767 y Fr(;)743 4033 y Fp(k)810 3897 y Fo(Z)909 3940 y FD(\034)865 4124 y(\033)914 4104 y Fj(2)955 4033 y Fu(d)p Fr(\034)1060 3990 y Fm(0)1118 4008 y Fo(b)1104 4033 y Fr(S)1165 4060 y FD(n)1220 4033 y Fq(\()p Fr(t;)17 b(\034)1395 3990 y Fm(0)1422 4033 y Fq(\))1461 3952 y Fo(\000)1517 4006 y(b)1506 4033 y Fn(N)1584 4060 y FE(w)p FD(;n)1711 4033 y Fq(\()t(^)-54 b Fr(v)1798 4060 y FE(c)1834 4033 y Fr(;)20 b Fq(^)-53 b Fr(v)1927 4060 y FE(s)1959 4033 y Fr(;)34 b Fo(b)-72 b Fr(w)r Fq(\))2116 3952 y Fo(\001)2162 4033 y Fq(\()p Fp(\001)p Fr(;)17 b Fp(\001)p Fr(;)g(\034)2402 3990 y Fm(0)2427 4033 y Fq(\))p Fp(k)2516 4060 y Fm(K)2577 4078 y Fi(\033)2620 4063 y(n)2705 4033 y Fp(\024)53 b Fr(C)7 b(\033)2974 3990 y FD(n)p Fk(\()p FE(1)p Fm(\000)p FD(")3189 3947 y Fh(0)3216 3990 y Fk(\))3252 4033 y Fr(R)3329 3990 y FD(u)3328 4060 y FE(cs)p FD(;n)3464 4033 y Fr(R)3541 3990 y FD(w)3540 4060 y(n)3630 4033 y Fr(:)94 4471 y Fv(Pr)n(oof)o(.)24 b Fu(W)-8 b(e)24 b(\002rst)f(use)h(Lemma)f(7.2)g (and)g(Lemma)g(7.4.)35 b(F)o(or)24 b(the)f(second)h(inte)o(gral)e(in)h (\(7.6\))h(we)g(get)f(a)h(bound)775 4770 y(sup)703 4862 y FD(\034)8 b Fm(2)p FE([)p FD(\033)874 4842 y Fj(2)903 4862 y FD(;)p FE(1])1001 4770 y Fp(k)p Fr(\033)1112 4727 y Fm(\000)p FE(2)p FD(n)1280 4635 y Fo(Z)1379 4678 y FD(\034)1335 4861 y(\033)1384 4841 y Fj(2)1426 4770 y Fu(d)p Fr(\034)1531 4727 y Fm(0)1558 4770 y Fr(e)1604 4727 y FD(\033)1653 4685 y Fh(\000)p Fj(2)p Fi(n)1794 4721 y Fo(b)1781 4727 y Fb(M)1863 4745 y Fj(c)p Fi(;n)1957 4727 y Fk(\()p FD(\034)g Fm(\000)p FD(\034)2140 4685 y Fh(0)2167 4727 y Fk(\))2203 4690 y Fo(\000)2260 4743 y(b)2248 4770 y Fn(N)2326 4797 y FE(c)p FD(;n)2434 4770 y Fq(\()t(^)-54 b Fr(v)2521 4797 y FE(c)2557 4770 y Fr(;)20 b Fq(^)-53 b Fr(v)2650 4797 y FE(s)2682 4770 y Fq(\))2721 4690 y Fo(\001)2766 4770 y Fq(\()p Fp(\001)p Fr(;)17 b Fp(\001)p Fr(;)g(\034)3006 4727 y Fm(0)3032 4770 y Fq(\))p Fp(k)3121 4797 y Fm(K)3182 4815 y Fi(\033)3225 4800 y(n)730 5065 y Fp(\024)45 b Fr(C)7 b(\033)991 5022 y Fm(\000)p FE(2)p FD(n)1142 5065 y Fq(\()p Fr(R)1258 5022 y FD(u)1257 5092 y FE(cs)p FD(;n)1393 5065 y Fq(\))1432 5022 y FE(2)1471 5065 y Fr(\033)1532 5022 y FE(5)p FD(n=)p FE(2)1713 4929 y Fo(Z)1813 4972 y FE(1)1769 5156 y FD(\033)1818 5136 y Fj(2)1849 5065 y Fu(d)p Fr(\034)1954 5022 y Fm(0)1982 5065 y Fq(\()p Fu(1)22 b Fp(\000)g Fr(\034)2247 5022 y Fm(0)2275 5065 y Fq(\))2314 5022 y Fm(\000)p FE(3)p FD(=)p FE(4)730 5291 y Fp(\024)45 b Fr(C)7 b(\033)991 5248 y FD(n=)p FE(2)1121 5291 y Fq(\()p Fr(R)1237 5248 y FD(u)1236 5318 y FE(cs)p FD(;n)1371 5291 y Fq(\))1410 5248 y FE(2)1475 5291 y Fr(:)p eop %%Page: 35 35 35 34 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(35)94 194 y Fu(F)o(or)25 b(the)g(second)f(inte)o(gral)g(in)h(\(7.7\))f(we)i(\002nd)e(similarly) 732 510 y(sup)660 601 y FD(\034)8 b Fm(2)p FE([)p FD(\033)831 581 y Fj(2)860 601 y FD(;)p FE(1])942 510 y Fp(k)p Fr(\033)1053 467 y Fm(\000)p FE(7)p FD(n=)p FE(2)1296 374 y Fo(Z)1396 417 y FD(\034)1351 600 y(\033)1400 580 y Fj(2)1442 510 y Fu(d)p Fr(\034)1547 467 y Fm(0)1591 510 y Fr(e)1637 467 y FD(\033)1686 424 y Fh(\000)p Fj(2)p Fi(n)1827 461 y Fo(b)1814 467 y Fb(M)1896 485 y Fj(s)p Fi(;n)1987 467 y Fk(\()p FD(\034)g Fm(\000)p FD(\034)2170 424 y Fh(0)2197 467 y Fk(\))2233 429 y Fo(\000)2290 483 y(b)2278 510 y Fn(N)2356 537 y FE(s)p FD(;n)2461 510 y Fq(\()t(^)-54 b Fr(v)2548 537 y FE(c)2584 510 y Fr(;)20 b Fq(^)-53 b Fr(v)2677 537 y FE(s)2708 510 y Fq(\))2747 429 y Fo(\001)2793 510 y Fq(\()p Fp(\001)p Fr(;)17 b Fp(\001)p Fr(;)g(\034)3033 467 y Fm(0)3058 510 y Fq(\))p Fp(k)3147 537 y Fm(K)3208 554 y Fi(\033)3251 539 y(n)970 804 y Fp(\024)44 b Fr(C)7 b Fq(\()p Fr(R)1285 761 y FD(u)1284 831 y FE(cs)p FD(;n)1420 804 y Fq(\))1459 761 y FE(2)1499 804 y Fr(\033)1560 761 y Fm(\000)p FE(3)p FD(n=)p FE(2)1803 669 y Fo(Z)1902 712 y FE(1)1858 895 y FD(\033)1907 875 y Fj(2)1939 804 y Fu(d)p Fr(\034)2044 761 y Fm(0)2088 804 y Fr(e)2134 761 y Fm(\000)p FD(C)e(\033)2307 719 y Fh(\000)p Fj(2)p Fi(n)2435 761 y Fk(\()p FE(1)p Fm(\000)p FD(\034)2608 719 y Fh(0)2635 761 y Fk(\))2671 804 y Fq(\()p Fu(1)22 b Fp(\000)h Fr(\034)2937 761 y Fm(0)2964 804 y Fq(\))3003 761 y Fm(\000)p FE(1)p FD(=)p FE(2)970 1030 y Fp(\024)44 b Fr(C)7 b(\033)1230 987 y FD(n=)p FE(2)1360 1030 y Fq(\()p Fr(R)1476 987 y FD(u)1475 1057 y FE(cs)p FD(;n)1610 1030 y Fq(\))1649 987 y FE(2)1714 1030 y Fr(:)94 1296 y Fu(F)o(or)25 b(the)g(inte)o(gral)f(in)g(\(7.8\))h(we)g(\002nd,)g(using)f(no)n(w)g (Lemma)g(7.3)g(and)h(Lemma)f(7.4,)h(a)g(bound)456 1611 y Fr(C)7 b(\033)595 1568 y Fm(\000)p FE(2)p FD(n)763 1476 y Fo(Z)862 1519 y FD(\034)818 1702 y(\033)867 1682 y Fj(2)892 1611 y Fu(d)p Fr(\034)997 1568 y Fm(0)1041 1611 y Fq(\()p Fr(\033)1141 1568 y Fm(\000)p FD(")1241 1526 y Fh(0)1267 1568 y FD(n=)p FE(2)1397 1611 y Fr(e)1443 1568 y Fm(\000)p FD(\015)t(\033)1600 1526 y Fh(\000)p Fj(2)p Fi(n)1729 1568 y Fk(\()p FD(\034)h Fm(\000)p FD(\034)1912 1526 y Fh(0)1938 1568 y Fk(\))p FD(=)p FE(2)2050 1611 y Fq(\()p Fr(\034)33 b Fp(\000)23 b Fr(\034)2321 1568 y Fm(0)2348 1611 y Fq(\))2387 1568 y FD(")2425 1526 y Fh(0)2452 1568 y FD(=)p FE(2)p Fm(\000)p FE(1)2630 1611 y Fq(\)\()p Fr(\033)2769 1568 y Fk(\()p FE(1)p Fm(\000)p FD(")2935 1526 y Fh(0)2960 1568 y FD(=)p FE(2)p Fk(\))p FD(n)3121 1611 y Fr(R)3198 1568 y FD(u)3197 1638 y FE(cs)p FD(;n)3333 1611 y Fr(R)3410 1568 y FD(w)3409 1638 y(n)3474 1611 y Fq(\))940 1855 y Fp(\024)44 b Fr(C)7 b(\033)1200 1812 y FD(n)p Fk(\()p Fm(\000)p FE(1)p Fm(\000)p FD(")1477 1770 y Fh(0)1505 1812 y Fk(\))1541 1855 y Fr(\033)1602 1812 y FE(2)p FD(n)1690 1855 y Fr(R)1767 1812 y FD(u)1766 1882 y FE(cs)p FD(;n)1902 1855 y Fr(R)1979 1812 y FD(w)1978 1882 y(n)2087 1855 y Fp(\024)45 b Fr(C)7 b(\033)2348 1812 y FD(n)p Fk(\()p FE(1)p Fm(\000)p FD(")2563 1770 y Fh(0)2590 1812 y Fk(\))2626 1855 y Fr(R)2703 1812 y FD(u)2702 1882 y FE(cs)p FD(;n)2838 1855 y Fr(R)2915 1812 y FD(w)2914 1882 y(n)3003 1855 y Fr(:)p 3812 2056 63 2 v 3812 2114 2 59 v 3872 2114 V 3812 2116 63 2 v 94 2459 a Fv(Lemma)26 b(7.8.)35 b Fg(Assume)24 b Fr(R)1054 2417 y FD(u)1053 2486 y FE(cs)p FD(;n)1211 2459 y Fq(+)f Fr(R)1388 2417 y FD(w)1387 2486 y(n)1479 2459 y Fp(\024)28 b Fu(1)p Fg(.)36 b(Then)25 b(for)g(all)f Fu(1)k Fp(\025)g Fr(\034)38 b Fp(\025)29 b Fr(\033)2622 2417 y FE(2)2686 2459 y Fg(and)c(all)f Fr(\033)31 b Fp(2)d Fq(\()p Fu(0)p Fr(;)17 b Fu(1)p Fq(\))24 b Fg(one)g(has)290 2794 y Fp(k)p Fr(\033)401 2751 y Fm(\000)p FE(2)p FD(n)568 2658 y Fo(Z)668 2701 y FD(\034)624 2884 y(\033)673 2864 y Fj(2)714 2794 y Fu(d)p Fr(\034)819 2751 y Fm(0)864 2794 y Fr(e)910 2751 y FD(\033)959 2708 y Fh(\000)p Fj(2)p Fi(n)1099 2745 y Fo(b)1086 2751 y Fb(M)1168 2769 y Fj(c)p Fi(;n)1262 2751 y Fk(\()p FD(\034)8 b Fm(\000)p FD(\034)1445 2708 y Fh(0)1472 2751 y Fk(\))1508 2713 y Fo(\000)1565 2767 y(b)1554 2794 y Fn(N)1632 2821 y FE(c)p FD(;)p Fl(i)p FD(;n)1783 2794 y Fq(\()t(^)-54 b Fr(v)1870 2821 y FE(c)1906 2794 y Fr(;)20 b Fq(^)-53 b Fr(v)1999 2821 y FE(s)2030 2794 y Fq(\))2069 2713 y Fo(\001)2115 2794 y Fq(\()p Fp(\001)p Fr(;)17 b Fp(\001)p Fr(;)g(\034)2355 2751 y Fm(0)2380 2794 y Fq(\))p Fp(k)2469 2821 y Fm(K)2530 2838 y Fi(\033)2573 2823 y(n)2658 2794 y Fp(\024)62 b Fr(C)7 b(e)2921 2751 y Fm(\000)p Fk(\()p FD(\014)t Fk(\()p FD(c)p Fm(\000)r Fk(^)-42 b FD(c)p Fk(\)+)p FD(\015)t Fk(\))p FD(\033)3446 2708 y Fh(\000)p Fi(n)3554 2794 y Fr(R)3631 2751 y FD(w)3630 2821 y(n)3720 2794 y Fr(;)221 3060 y Fp(k)p Fr(\033)332 3017 y Fm(\000)p FE(7)p FD(n=)p FE(2)575 2924 y Fo(Z)674 2967 y FD(\034)630 3151 y(\033)679 3131 y Fj(2)721 3060 y Fu(d)p Fr(\034)826 3017 y Fm(0)870 3060 y Fr(e)916 3017 y FD(\033)965 2974 y Fh(\000)p Fj(2)p Fi(n)1106 3011 y Fo(b)1093 3017 y Fb(M)1175 3035 y Fj(s)p Fi(;n)1266 3017 y Fk(\()p FD(\034)8 b Fm(\000)p FD(\034)1449 2974 y Fh(0)1476 3017 y Fk(\))1512 2979 y Fo(\000)1569 3033 y(b)1557 3060 y Fn(N)1635 3087 y FE(s)p FD(;)p Fl(i)p FD(;n)1783 3060 y Fq(\()t(^)-54 b Fr(v)1870 3087 y FE(c)1906 3060 y Fr(;)20 b Fq(^)-53 b Fr(v)1999 3087 y FE(s)2030 3060 y Fq(\))2069 2979 y Fo(\001)2115 3060 y Fq(\()p Fp(\001)p Fr(;)17 b Fp(\001)p Fr(;)g(\034)2355 3017 y Fm(0)2380 3060 y Fq(\))p Fp(k)2469 3087 y Fm(K)2530 3105 y Fi(\033)2573 3090 y(n)2658 3060 y Fp(\024)62 b Fr(C)7 b(e)2921 3017 y Fm(\000)p Fk(\()p FD(\014)t Fk(\()p FD(c)p Fm(\000)r Fk(^)-42 b FD(c)p Fk(\)+)p FD(\015)t Fk(\))p FD(\033)3446 2974 y Fh(\000)p Fi(n)3554 3060 y Fr(R)3631 3017 y FD(w)3630 3087 y(n)3720 3060 y Fr(:)94 3537 y Fv(Pr)n(oof)o(.)37 b Fu(W)-8 b(e)38 b(restrict)f(ourselv)o(es)f(to)h (the)h(linear)f(part)g Fn(M)2114 3563 y Fl(i)2138 3537 y Fu(.)74 b(A)37 b(typical)g(term)g(of)g(\(7.6\)\227the)g(\002rst)g(in) g(the)94 3656 y(de\002nition)24 b(of)h Fn(M)707 3683 y Fl(i)756 3656 y Fu(in)g(\(1.8\)\227can)g(be)f(re)n(written)h(as)469 3986 y Fr(\033)530 3943 y Fm(\000)p FE(2)p FD(n)681 3845 y Fo(\022)754 3850 y(Z)854 3893 y FD(\034)810 4076 y(\033)859 4057 y Fj(2)900 3986 y Fu(d)p Fr(\034)1005 3943 y Fm(0)1050 3986 y Fr(e)1109 3937 y Fo(b)1096 3943 y Fb(M)1178 3961 y Fj(c)p Fi(;n)1271 3943 y Fk(\()p FD(\034)8 b Fm(\000)p FD(\034)1454 3900 y Fh(0)1481 3943 y Fk(\))1525 3959 y Fo(b)1517 3986 y Fn(L)1588 3943 y FD(n)1643 3905 y Fo(\000)1712 3961 y(b)1689 3986 y Fr(K)1774 4013 y FD(c\033)1859 3993 y Fh(\000)p Fj(2)p Fi(n)1985 4013 y FD(\034)2030 3993 y Fh(0)2101 3986 y Fp(\003)2091 3978 y Fc(\015)2177 3986 y Fq(\()2224 3959 y Fo(b)2216 3986 y Fn(L)2287 3943 y Fm(\000)p FD(n)2410 3986 y Fq(^)-56 b Fr(u)2461 4013 y FD(n;\034)2579 3993 y Fh(0)2611 3986 y Fq(\))2650 3905 y Fo(\001)2695 3845 y(\023)2768 3986 y Fq(\()p Ff({)t Fr(;)17 b(x)p Fq(\))g Fr(U)11 b Fq(\()p Fr(x)p Fq(\))709 4265 y(=)44 b Fr(\033)891 4222 y Fm(\000)p FE(2)p FD(n)1042 4124 y Fo(\022)1116 4129 y(Z)1215 4172 y FD(\034)1171 4355 y(\033)1220 4336 y Fj(2)1262 4265 y Fu(d)p Fr(\034)1367 4222 y Fm(0)1411 4265 y Fr(e)1470 4216 y Fo(b)1457 4222 y Fb(M)1539 4240 y Fj(c)p Fi(;n)1633 4222 y Fk(\()p FD(\034)8 b Fm(\000)p FD(\034)1816 4179 y Fh(0)1843 4222 y Fk(\))1887 4238 y Fo(b)1879 4265 y Fn(L)1950 4222 y FD(n)2004 4184 y Fo(\000)2074 4239 y(b)2050 4265 y Fr(K)2135 4292 y FD(c\033)2220 4272 y Fh(\000)p Fj(2)p Fi(n)2347 4292 y FD(\034)2392 4272 y Fh(0)2462 4265 y Fp(\003)2452 4257 y Fc(\015)2545 4265 y Fq(^)-56 b Fr(u)2596 4292 y FD(\033)2645 4272 y Fh(\000)p Fj(2)p Fi(n)2772 4292 y FD(\034)2817 4272 y Fh(0)2849 4184 y Fo(\001)2894 4124 y(\023)2968 4265 y Fq(\()p Ff({)t Fr(;)17 b(x)p Fq(\))g Fr(U)11 b Fq(\()p Fr(x)p Fq(\))23 b Fr(:)94 4585 y Fu(Note)i(ne)o(xt)f(that)338 4760 y Fo(\000)407 4816 y(b)384 4841 y Fr(K)469 4868 y FD(c\033)554 4848 y Fh(\000)p Fj(2)p Fi(n)680 4868 y FD(\034)725 4848 y Fh(0)796 4841 y Fp(\003)786 4833 y Fc(\015)879 4841 y Fq(^)-57 b Fr(u)929 4868 y FD(\033)978 4848 y Fh(\000)p Fj(2)p Fi(n)1106 4868 y FD(\034)1151 4848 y Fh(0)1182 4760 y Fo(\001)1228 4841 y Fq(\()p Ff({)t Fr(;)17 b(x)p Fq(\))785 5051 y(=)906 4916 y Fo(Z)1003 5051 y Fu(d)p Fr(`)1134 5026 y Fo(b)1112 5051 y Fr(K)1197 5078 y FD(c\033)1282 5058 y Fh(\000)p Fj(2)p Fi(n)1407 5078 y FD(\034)1452 5058 y Fh(0)1484 5051 y Fq(\()p Ff({)26 b Fp(\000)d Fr(`)f Fp(\000)g Fr(i\014)5 b(;)17 b(x)p Fq(\))34 b Fo(b)-72 b Fr(w)s Fq(\()p Fr(`;)17 b(x;)g(\033)2494 5009 y Fm(\000)p FE(2)p FD(n)2642 5051 y Fr(\034)2697 5009 y Fm(0)2725 5051 y Fq(\))p Fr(e)2810 5009 y Fm(\000)p FD(i`)r Fk(^)-42 b FD(c)q(\033)3019 4966 y Fh(\000)p Fj(2)p Fi(n)3146 5009 y FD(\034)3191 4966 y Fh(0)3223 5051 y Fr(e)3269 5009 y Fm(\000)p FD(\015)t(\033)3426 4966 y Fh(\000)p Fj(2)p Fi(n)3554 5009 y FD(\034)3599 4966 y Fh(0)779 5296 y Fp(\002)23 b Fr(e)925 5253 y Fm(\000)p FD(\014)t Fk(\()p FD(c)p Fm(\000)r Fk(^)-42 b FD(c)p Fk(\))p FD(\033)1281 5211 y Fh(\000)p Fj(2)p Fi(n)1408 5253 y FD(\034)1453 5211 y Fh(0)1485 5296 y Fr(e)1531 5253 y FD(i)p Fk(\()p Fe({)s Fm(\000)p FD(`)p Fk(\))p FD(c\033)1858 5211 y Fh(\000)p Fj(2)p Fi(n)1986 5253 y FD(\034)2031 5211 y Fh(0)2088 5296 y Fr(:)p eop %%Page: 36 36 36 35 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(36)94 194 y Fu(Using)24 b(this)g(identity)-6 b(,)23 b(we)i(get)g(\(because)g(e)o(xp)p Fq(\()1738 167 y Fo(b)1717 194 y Fn(M)1815 221 y FE(c)p FD(;n)1923 194 y Fq(\()p Fr(\034)33 b Fp(\000)22 b Fr(\034)2193 151 y Fm(0)2221 194 y Fq(\)\))j Fu(is)f(bounded\):)213 420 y Fr(\033)274 378 y Fm(\000)p FE(2)p FD(n)442 420 y Fp(k)509 285 y Fo(Z)608 328 y FD(\034)564 511 y(\033)613 491 y Fj(2)654 420 y Fu(d)p Fr(\034)759 378 y Fm(0)804 420 y Fr(e)863 371 y Fo(b)850 377 y Fb(M)932 396 y Fj(c)p Fi(;n)1025 378 y Fk(\()p FD(\034)8 b Fm(\000)p FD(\034)1208 335 y Fh(0)1235 378 y Fk(\))1280 393 y Fo(b)1272 420 y Fn(L)1343 378 y FD(n)1397 340 y Fo(\000)1466 395 y(b)1443 420 y Fr(K)1528 447 y FD(c\033)1613 427 y Fh(\000)p Fj(2)p Fi(n)1739 447 y FD(\034)1784 427 y Fh(0)1855 420 y Fp(\003)1845 412 y Fc(\015)1938 420 y Fq(^)-57 b Fr(u)1988 447 y FD(n;\034)2106 427 y Fh(0)2138 340 y Fo(\001)2184 420 y Fp(k)2234 447 y Fm(K)2295 465 y Fi(\033)2338 450 y(n)453 687 y Fp(\024)45 b Fr(C)7 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FD(\033)1220 5194 y Fh(\000)p Fj(2)p Fi(n)1361 5231 y Fo(b)1347 5237 y Fb(M)1429 5255 y Fj(c)p Fi(;n)1523 5237 y Fk(\()p FE(1)p Fm(\000)p FD(\033)1700 5194 y Fj(2)1730 5237 y Fk(\))1770 5280 y Fq(^)-54 b Fr(r)1811 5307 y FD(n)p Fm(\000)p FE(1)1962 5280 y Fq(\()p Fr(\033)t Ff({)t Fr(;)17 b(x)p Fq(\))p Fp(k)2323 5307 y Fm(K)2384 5325 y Fi(\033)2427 5310 y(n)2528 5280 y Fp(\024)2662 5241 y FE(1)p 2662 5257 35 4 v 2662 5314 a(2)2708 5280 y Fp(k)t Fq(^)-54 b Fr(r)2803 5307 y FD(n)p Fm(\000)p FE(1)2955 5280 y Fp(k)3005 5307 y Fm(K)3066 5331 y Fi(\033)3109 5316 y(n)p Fh(\000)p Fj(1)3270 5280 y Fr(;)321 b Fq(\()p Fu(7)p Fr(:)p Fu(27)p Fq(\))p eop %%Page: 39 39 39 38 bop 94 -45 a Ft(St)-7 b(ability)38 b(of)h(Modula)-7 b(ted)38 b(Fr)n(onts)2061 b Fv(39)94 194 y Fu(as)25 b(in)g(the)f(proof) h(of)g(Proposition)e(4.5.)36 b(This)24 b(leads)h(for)g(the)f(\002rst)h (term)g(in)f(\(7.22\))h(to)f(a)h(bound)f(\(in)h Fp(K)3602 221 y FD(\033)3651 201 y Fi(n)3704 194 y Fu(\))1807 440 y Fr(\032)1859 397 y FD(r)1859 467 y(n)p Fm(\000)p FE(1)2010 440 y Fr(=)p Fu(2)f Fr(:)1457 b Fq(\()p Fu(7)p Fr(:)p Fu(28)p Fq(\))94 686 y Fu(The)25 b(second)g(and)g(third)f(term)g(ha)n (v)o(e)h(been)g(bounded)f(in)g(the)h(proof)g(of)f(Lemma)h(7.7)f(and)h (Lemma)f(7.8)h(by)1371 963 y Fr(C)7 b(e)1495 920 y Fm(\000)p FD(C)e(\033)1668 877 y Fh(\000)p Fi(n)1777 963 y Fr(R)1854 920 y FD(u)1853 990 y(n)1929 963 y Fq(+)23 b Fr(C)7 b(\033)2168 920 y FD(n=)p FE(2)2297 963 y Fq(\()p Fr(R)2413 920 y FD(u)2412 990 y(n)2466 963 y Fq(\))2505 920 y FE(2)2569 963 y Fr(:)1022 b Fq(\()p Fu(7)p Fr(:)p Fu(29)p Fq(\))94 1209 y Fu(Finally)-6 b(,)24 b(the)h(last)f(term)489 1462 y Fo(b)463 1487 y Fr(X)546 1514 y FD(n)600 1487 y Fq(\()p Ff({)t Fr(;)17 b(x)p Fq(\))43 b Fp(\021)i Fr(e)1061 1445 y FD(\033)1110 1402 y Fh(\000)p Fj(2)p Fi(n)1251 1439 y Fo(b)1238 1445 y Fb(M)1320 1463 y Fj(c)p Fi(;n)1413 1445 y Fk(\()p FE(1)p Fm(\000)p FD(\033)1590 1402 y Fj(2)1620 1445 y Fk(\))1656 1487 y Fr(A)1731 1514 y FD(n)p Fm(\000)p FE(1)1903 1461 y Fq(~)1882 1487 y Fr( )t Fq(\()p Fr(\033)t Ff({)t Fq(\))p Fr(')2225 1514 y FD(\033)2274 1494 y Fh(\000)p Fi(n)2376 1514 y Fe({)2438 1487 y Fq(\()p Fr(x)p Fq(\))22 b Fp(\000)g Fr(A)2769 1514 y FD(n)2844 1461 y Fq(~)2823 1487 y Fr( )t Fq(\()p Ff({)t Fq(\))p Fr(')3105 1514 y FD(\033)3154 1494 y Fh(\000)p Fi(n)3256 1514 y Fe({)3318 1487 y Fq(\()p Fr(x)p Fq(\))j Fr(;)94 1733 y Fu(in)g(\(7.22\))g(leads)f (to)h(a)g(bound)f(\(in)g Fp(K)1349 1760 y FD(\033)1398 1740 y Fi(n)1452 1733 y Fu(\):)815 2015 y Fp(k)891 1990 y Fo(b)865 2015 y Fr(X)948 2042 y FD(n)1002 2015 y Fp(k)49 b(\024)44 b Fr(C)7 b(e)1346 1973 y Fm(\000)p FD(C)e(\033)1519 1930 y Fh(\000)p Fi(n)1628 2015 y Fr(R)1705 1973 y FD(w)1704 2042 y(n)p Fm(\000)p FE(1)1877 2015 y Fq(+)23 b Fr(C)7 b(\033)2116 1973 y FD(n=)p FE(2)2245 2015 y Fq(\()p Fr(R)2361 1973 y FD(u)2360 2042 y FE(cs)p FD(;n)2496 2015 y Fq(\))2535 1973 y FE(2)2597 2015 y Fq(+)22 b Fr(C)7 b(\033)2835 1973 y FD(n)2889 2015 y Fr(R)2966 1973 y FD(u)2965 2042 y FE(cs)p FD(;n)3126 2015 y Fr(;)465 b Fq(\()p Fu(7)p Fr(:)p Fu(30)p Fq(\))94 2282 y Fu(where)29 b(the)e(last)g(term)h(is)f (due)h(to)f Fr(\026)1323 2309 y FE(1)1363 2282 y Fq(\()p Fr(`)p Fq(\))j(=)j Fp(\000)p Fr(c)1743 2309 y FE(1)1783 2282 y Fr(`)1825 2239 y FE(2)1888 2282 y Fq(+)24 b Fn(O)p Fq(\()p Fr(`)2140 2239 y FE(3)2180 2282 y Fq(\))j Fu(not)h(being)f(e)o (xactly)g(a)h(parabola.)44 b(F)o(or)28 b(details)94 2401 y(see)k([Schn96].)53 b(Collecting)30 b(the)h(bounds,)g(the)f(assertion) g(\(7.24\))h(for)k Fq(^)-54 b Fr(r)2641 2428 y FD(n)2726 2401 y Fu(follo)n(ws.)52 b(Finally)-6 b(,)31 b(the)f(bounds)94 2521 y(on)h Fr(\032)277 2478 y FD(w)277 2548 y(n)373 2521 y Fu(follo)n(w)f(the)h(in)f(the)i(same)f(w)o(ay)g(as)g(those)g(in) g(Lemma)f(7.11.)55 b(The)31 b(proof)g(of)g(Proposition)f(7.12)h(is)94 2641 y(complete.)p 3812 2586 63 2 v 3812 2645 2 59 v 3872 2645 V 3812 2647 63 2 v 94 2953 a Fv(Pr)n(oof)c(of)f(Theor)n(em)h (7.1.)e Fu(As)h(before)h(the)e(proof)h(is)g(just)e(an)i(induction)f(ar) n(gument,)g(using)g(repeatedly)h(the)94 3072 y(abo)o(v)o(e)31 b(estimates.)53 b(Again)31 b(we)g(write)g Fr(C)39 b Fu(for)31 b(constants)f(which)h(can)h(be)f(chosen)g(independent)f(of)h Fr(\033)k Fu(and)94 3192 y Fr(n)p Fu(.)66 b(Assume)34 b(that)g Fr(R)42 b Fq(=)g Fu(sup)1161 3219 y FD(n)p Fm(2)p Fl(N)1336 3192 y Fr(R)1413 3149 y FD(u)1412 3219 y FE(cs)p FD(;n)1589 3192 y Fr(<)g Fp(1)34 b Fu(e)o(xists.)65 b(From)34 b(Lemma)h(7.11)f(we)h(observ)o(e)f(for)h Fr(\033)45 b(>)d Fu(0)94 3312 y(suf)n(\002ciently)24 b(small,)1033 3657 y Fr(R)1110 3614 y FD(w)1109 3684 y(n)1234 3657 y Fp(\024)1385 3583 y Fr(C)7 b(\033)1524 3540 y Fm(\000)p FE(5)p FD(=)p FE(2)p Fm(\000)p FD(n")1846 3497 y Fh(0)1878 3583 y Fr(\032)1930 3540 y FD(w)1930 3610 y(n)p Fm(\000)p FE(1)p 1385 3634 696 4 v 1400 3727 a Fu(1)22 b Fp(\000)h Fr(C)7 b(\033)1711 3698 y FD(n)p Fk(\()p FE(1)p Fm(\000)p FD(")1926 3678 y Fh(0)1953 3698 y Fk(\))1989 3727 y Fr(R)2137 3657 y Fp(\024)44 b Fr(C)7 b(\033)2397 3614 y Fm(\000)p FE(5)p FD(=)p FE(2)p Fm(\000)p FD(n")2719 3571 y Fh(0)2751 3657 y Fr(\032)2803 3614 y FD(w)2803 3684 y(n)p Fm(\000)p FE(1)2979 3657 y Fr(;)962 3973 y(R)1039 3931 y FD(u)1038 4000 y FE(cs)p FD(;n)1234 3973 y Fp(\024)1385 3894 y Fr(C)g(\033)1524 3851 y Fm(\000)p FE(4)1626 3894 y Fr(\032)1678 3851 y FD(u)1678 3921 y FE(cs)p FD(;n)p Fm(\000)p FE(1)1932 3894 y Fq(+)23 b Fr(C)7 b(e)2156 3851 y Fm(\000)p FD(C)e(\033)2329 3808 y Fh(\000)p Fi(n)2438 3894 y Fr(R)2515 3851 y FD(w)2514 3921 y(n)p 1385 3951 1194 4 v 1723 4044 a Fu(1)22 b Fp(\000)h Fr(C)7 b(\033)2034 4015 y FD(n=)p FE(2)2164 4044 y Fr(R)1234 4211 y Fp(\024)62 b Fr(C)7 b(\033)1512 4168 y Fm(\000)p FE(4)1614 4211 y Fr(\032)1666 4168 y FD(u)1666 4238 y FE(cs)p FD(;n)p Fm(\000)p FE(1)1921 4211 y Fq(+)22 b Fr(C)7 b(e)2144 4168 y Fm(\000)p FD(C)e(\033)2317 4125 y Fh(\000)p Fi(n)2426 4211 y Fr(\032)2478 4168 y FD(w)2478 4238 y(n)p Fm(\000)p FE(1)2653 4211 y Fr(;)3619 3885 y Fq(\()p Fu(7)p Fr(:)p Fu(31)p Fq(\))94 4455 y Fu(with)25 b(a)g(constant)f Fr(C)32 b Fu(which)24 b(can)h(be)g(chosen)g (independent)f(of)h Fr(R)q Fu(.)35 b(Using)24 b(Proposition)f(7.12)i (we)g(\002nd)674 4732 y Fp(j)p Fr(A)777 4759 y FD(n)852 4732 y Fp(\000)e Fr(A)1027 4759 y FD(n)p Fm(\000)p FE(1)1178 4732 y Fp(j)44 b(\024)h Fr(C)7 b(e)1496 4689 y Fm(\000)p FD(C)e(\033)1669 4647 y Fh(\000)p Fi(n)1777 4732 y Fr(\032)1829 4689 y FD(w)1829 4759 y(n)p Fm(\000)p FE(1)2002 4732 y Fq(+)23 b Fr(C)7 b(\033)2241 4689 y FD(n=)p FE(2)2371 4732 y Fr(\033)2432 4689 y Fm(\000)p FE(4)2533 4732 y Fr(\032)2585 4689 y FD(u)2585 4759 y FE(cs)p FD(;n)p Fm(\000)p FE(1)2843 4732 y Fr(;)1100 4931 y(\032)1152 4889 y FD(r)1152 4958 y(n)1250 4931 y Fp(\024)45 b Fr(\032)1424 4889 y FD(r)1424 4958 y(n)p Fm(\000)p FE(1)1575 4931 y Fr(=)p Fu(2)21 b Fq(+)i Fr(C)7 b(e)1920 4889 y Fm(\000)p FD(C)e(\033)2093 4846 y Fh(\000)p Fi(n)2202 4931 y Fr(\032)2254 4889 y FD(w)2254 4958 y(n)p Fm(\000)p FE(1)2427 4931 y Fq(+)22 b Fr(C)7 b(\033)2665 4889 y FD(n=)p FE(2)2795 4931 y Fr(\033)2856 4889 y Fm(\000)p FE(4)2958 4931 y Fr(\032)3010 4889 y FD(u)3010 4958 y FE(cs)p FD(;n)p Fm(\000)p FE(1)3267 4931 y Fr(;)1035 5096 y(\032)1087 5054 y FD(u)1087 5123 y FE(cs)p FD(;n)1267 5096 y Fp(\024)44 b Fr(C)7 b Fq(\()p Fp(j)p Fr(A)1608 5123 y FD(n)1662 5096 y Fp(j)22 b Fq(+)h Fr(\032)1864 5054 y FD(r)1864 5123 y(n)1917 5096 y Fq(\))i Fr(;)1090 5296 y(\032)1142 5253 y FD(w)1142 5323 y(n)1250 5296 y Fp(\024)45 b Fr(C)7 b(e)1496 5253 y Fm(\000)p FD(C)e(\033)1669 5210 y Fh(\000)p Fj(2)p Fi(n)1802 5296 y Fr(\032)1854 5253 y FD(w)1854 5323 y(n)p Fm(\000)p FE(1)2027 5296 y Fq(+)23 b Fr(C)7 b(\033)2266 5253 y FD(n)p Fk(\()p FE(1)p Fm(\000)p FD(")2481 5210 y Fh(0)2508 5253 y Fk(\))2544 5296 y Fr(\033)2605 5253 y Fm(\000)p FE(5)p FD(=)p FE(2)p Fm(\000)p FD(n")2927 5210 y Fh(0)2958 5296 y Fr(\032)3010 5253 y FD(w)3010 5323 y(n)p Fm(\000)p FE(1)3186 5296 y Fr(:)p eop %%Page: 40 40 40 39 bop 94 -45 a Ft(St)-7 b(ability)34 b(of)g(Modula)-7 b(ted)34 b(Fr)n(onts)2074 b Fv(40)94 194 y Fu(Therefore,)26 b(we)f(can)g(choose)g Fr(\033)31 b(>)d Fu(0)d(so)f(small)g(that)h(for)g Fr(n)i(>)h Fu(9:)949 433 y Fp(j)p Fr(A)1052 460 y FD(n)1128 433 y Fp(\000)23 b Fr(A)1303 460 y FD(n)p Fm(\000)p FE(1)1454 433 y Fp(j)60 b(\024)28 b Fr(\032)1699 390 y FD(w)1699 460 y(n)p Fm(\000)p FE(1)1850 433 y Fr(=)p Fu(10)22 b Fq(+)g Fr(\033)2182 390 y FD(n)p Fm(\000)p FE(9)2333 433 y Fq(\()p Fp(j)p Fr(A)2475 460 y FD(n)p Fm(\000)p FE(1)2626 433 y Fp(j)g Fq(+)g Fr(\032)2827 390 y FD(r)2827 460 y(n)2881 433 y Fq(\))p Fr(;)1376 598 y(\032)1428 555 y FD(r)1428 625 y(n)1542 598 y Fp(\024)28 b Fu(3)p Fr(\032)1749 555 y FD(r)1749 625 y(n)p Fm(\000)p FE(1)1900 598 y Fr(=)p Fu(4)22 b Fq(+)g Fr(\032)2173 555 y FD(w)2173 625 y(n)p Fm(\000)p FE(1)2324 598 y Fr(=)p Fu(10)g Fq(+)h Fr(\033)2657 555 y FD(n)p Fm(\000)p FE(9)2807 598 y Fp(j)p Fr(A)2910 625 y FD(n)2964 598 y Fp(j)p Fr(;)1365 747 y(\032)1417 705 y FD(w)1417 774 y(n)1542 747 y Fp(\024)28 b Fr(\032)1699 705 y FD(w)1699 774 y(n)p Fm(\000)p FE(1)1850 747 y Fr(=)p Fu(10)d Fr(:)94 987 y Fu(Thus,)32 b(the)f(sequence)g(of)f Fr(A)1088 1013 y FD(n)1173 987 y Fu(con)l(v)o(er)n(ges)h(geometrically) e(to)i(a)g(\002nite)f(limit)g Fr(A)2893 1013 y Fm(\003)2938 987 y Fu(.)54 b(Furthermore,)32 b(we)f(\002nd)94 1106 y(that)25 b(lim)402 1133 y FD(n)p Fm(!1)632 1106 y Fr(\032)684 1063 y FD(r)684 1133 y(n)766 1106 y Fq(=)k Fu(0,)c(and)g(lim)1274 1133 y FD(n)p Fm(!1)1503 1106 y Fr(\032)1555 1063 y FD(w)1555 1133 y(n)1648 1106 y Fq(=)j Fu(0.)37 b(Since)26 b(the)f(quantities)f Fp(j)p Fr(A)2775 1133 y FD(n)2828 1106 y Fp(j)p Fu(,)h Fr(\032)2958 1063 y FD(r)2958 1133 y(n)3012 1106 y Fu(,)g Fr(\032)3114 1063 y FD(w)3114 1133 y(n)3204 1106 y Fu(increase)g(only)g (for)94 1226 y(at)i(most)f(9)h(steps)f(the)h(term)f Fr(C)7 b(R)28 b Fu(in)e(\(7.31\))h(stays)f(less)g(than)h(1)p Fr(=)p Fu(2)f(if)h(we)g(choose)f Fp(j)p Fr(A)3043 1253 y FE(1)3083 1226 y Fp(j)p Fu(,)h Fr(\032)3215 1183 y FD(r)3215 1253 y FE(1)3258 1226 y Fu(,)h Fr(\032)3363 1183 y FD(w)3363 1253 y FE(1)3457 1226 y Fq(=)j Fn(O)p Fq(\()p Fr(\033)3735 1183 y FD(m)3810 1226 y Fq(\))p Fu(,)94 1345 y(for)26 b(a)f(suf)n(\002ciently)e(lar)n(ge)j Fr(m)i(>)g Fu(0.)36 b(From)25 b(\(7.31\))f(the)h(e)o(xistence)f(of)h(a) g(\002nite)g(constant)f Fr(R)k Fq(=)g Fu(sup)3488 1372 y FD(n)p Fm(2)p Fl(N)3662 1345 y Fr(R)3739 1302 y FD(u)3738 1372 y FE(cs)p FD(;n)94 1465 y Fu(follo)n(ws)c(.)38 b(Finally)-6 b(,)25 b(the)g(scaling)g(of)h Fr(w)1441 1492 y FD(n)1495 1465 y Fq(\()p Fp(\001)p Fr(;)17 b Fp(\001)p Fr(;)g(\034)11 b Fq(\))23 b Fu(implies)h(the)h(e)o(xponential)f(decay)i(of)g Fr(w)s Fq(\()p Fr(t)p Fq(\))p Fu(.)37 b(The)26 b(proof)f(of)94 1584 y(Theorem)g(7.1)f(is)h(complete.)p 3812 1530 63 2 v 3812 1589 2 59 v 3872 1589 V 3812 1591 63 2 v 94 2063 a Fs(Refer)n(ences)94 2312 y FB([A)-7 b(W78])112 b(D.G.)32 b(Aronson,)g(H.)h(W)-6 b(einber)o(ger)l(.:)16 b(Multidimensional)34 b(nonlinear)f(dif)n(fusion)f(arising)i(in)f (population)g(genetics.)466 2401 y Fa(Adv)-6 b(.)33 b(Math.)h FA(30)f FB(\(1978\),)e(33\22676.)94 2491 y([BK92])127 b(J.)33 b(Bricmont,)h(A.)e(K)o(upiainen.:)18 b(Renormalization)34 b(group)e(and)h(the)h(Ginzb)n(ur)o(g\226Landau)e(equation.)i Fa(Comm.)e(Math.)466 2581 y(Phys.)h FA(150)g FB(\(1992\),)f (193\226208.)94 2670 y([BK94])127 b(J.)33 b(Bricmont,)h(A.)e(K)o (upiainen.:)18 b(Stability)34 b(of)f(mo)o(ving)f(fronts)g(in)h(the)h (Ginzb)n(ur)o(g\226Landau)e(equation.)i Fa(Comm.)466 2760 y(Math.)g(Phys.)f FA(159)f FB(\(1994\),)g(287\226318.)94 2850 y([CE86])136 b(P)-9 b(.)34 b(Collet,)g(J.\226P)-9 b(.)33 b(Eckmann.:)18 b(The)33 b(e)o(xistence)h(of)f(dendritic)h (fronts.)e Fa(Comm.)g(Math.)h(Phys.)g FA(107)g FB(\(1986\),)f (39\22692.)94 2939 y([CE87])136 b(P)-9 b(.)34 b(Collet,)g(J.\226P)-9 b(.)33 b(Eckmann.:)18 b(The)33 b(stability)h(of)e(modulated)i(fronts.)e Fa(Helv)-6 b(.)34 b(Phys.)f(Acta)g FA(60)g FB(\(1987\),)f(969\226991.) 94 3029 y([CE90a])101 b(P)-9 b(.)34 b(Collet,)g(J.\226P)-9 b(.)33 b(Eckmann.:)18 b Fa(Instabilities)33 b(and)g(fr)l(onts)h(in)f(e) n(xtended)i(systems.)e FB(1990.)g(Princeton,)h(Princeton)466 3119 y(Uni)n(v)o(ersity)f(Press.)94 3208 y([CE90b])96 b(P)-9 b(.)34 b(Collet,)g(J.\226P)-9 b(.)33 b(Eckmann.:)18 b(The)33 b(time)h(dependent)f(amplitude)h(equation)g(for)e(the)i (Swift\226Hohenber)o(g)d(problem.)466 3298 y Fa(Comm.)i(Math.)g(Phys.)g FA(132)g FB(\(1990\),)e(139\226153.)94 3388 y([CEE92])87 b(P)-9 b(.)34 b(Collet,)g(J.\226P)-9 b(.)33 b(Eckmann,)g(H.)g (Epstein.:)18 b(Dif)n(fusi)n(v)o(e)32 b(repair)h(for)f(the)h(Ginsb)n (ur)o(g\226Landau)f(equation.)i Fa(Helv)-6 b(.)33 b(Phys.)466 3477 y(Acta)h FA(65)e FB(\(1992\),)g(56\22692.)94 3567 y([DL83])131 b(G.)33 b(Dee,)g(J.)h(S.)f(Langer)l(.:)17 b(Propagating)33 b(pattern)g(selection.)i Fa(Phys.)e(Re)o(v)-6 b(.)33 b(Lett.)h FA(50)f FB(\(1983\),)f(383\226386.)94 3657 y([Eck65])114 b(W)-7 b(.)33 b(Eckhaus.:)18 b Fa(Studies)33 b(in)g(nonlinear)h(stability)g(theory)l(.)f FB(Springer)g(tracts)h(in)f (Nat.)g(Phil.)h(V)-10 b(ol.)32 b(6,)h(1965.)94 3746 y([EW91])114 b(J.\226P)-9 b(.)34 b(Eckmann,)f(C.E.)g(W)-6 b(ayne.:)17 b(Propagating)33 b(fronts)g(and)g(the)g(center)h(manifold)f(theorem.)h Fa(Comm.)e(Math.)h(Phys)466 3836 y FA(136)g FB(\(1991\),)f(285\226307.) 94 3926 y([EW94])114 b(J.\226P)-9 b(.)34 b(Eckmann,)f(C.E.)g(W)-6 b(ayne.:)17 b(The)34 b(non\226linear)f(stability)h(of)e(front)g (solutions)i(for)e(parabolic)h(partial)h(dif)n(feren\255)466 4015 y(tial)g(equations.)g Fa(Comm.)e(Math.)i(Phys.)f FA(161)f FB(\(1994\),)g(323\226334.)94 4105 y([EWW97])39 b(J.\226P)-9 b(.)34 b(Eckmann,)f(C.E.)g(W)-6 b(ayne,)33 b(P)-9 b(.)34 b(W)m(ittwer)l(.:)17 b(Geometric)33 b(stability)h (analysis)g(of)f(periodic)g(solutions)h(of)e(the)466 4195 y(Swift\226Hohenber)o(g)g(equation.)h Fa(Comm.)g(Math.)g(Phys.)g FA(190)g FB(\(1997\),)e(173\226211.)94 4284 y([Ga94])145 b(T)-6 b(.)33 b(Gallay)-5 b(.:)18 b(Local)34 b(stability)g(of)e (critical)j(fronts)d(in)h(nonlinear)g(parabolic)h(partial)g(dif)n (ferential)f(equations.)h Fa(Non\255)466 4374 y(linearity)h FA(7)e FB(\(1994\),)e(741\226764.)94 4464 y([HS99])136 b(M.)33 b(Haragus,)g(G.)g(Schneider)l(.:)17 b(Bifurcating)34 b(fronts)e(for)g(the)h(T)-6 b(aylor)n(\226Couette)34 b(problem)f(in)g(in\002nite)h(c)o(ylinders.)466 4553 y Fa(Zeitsc)o(hrift)h(f)855 4552 y(\250)848 4553 y(ur)e(Ang)o(e)o (wandte)h(Mathematik)f(und)g(Physik)h(\(ZAMP\))e FA(50)h FB(\(1999\),)f(120\226151.)94 4643 y([KSM92])65 b(P)-9 b(.)34 b(Kirrmann,)e(G.)h(Schneider)m(,)g(A.)g(Mielk)o(e:)18 b(The)33 b(v)n(alidity)h(of)f(modulation)g(equations)h(for)e(e)o (xtended)i(systems)466 4733 y(with)f(cubic)h(nonlinearities.)g Fa(Pr)l(oceedings)g(of)f(the)h(Royal)f(Society)h(of)g(Edinb)n(ur)m(gh)e FA(122A)h FB(\(1992\),)e(85\22691.)94 4822 y([RS72])141 b(M.)33 b(Reed,)h(B.)f(Simon.:)18 b Fa(Methods)34 b(of)f(Modern)g (Mathematical)h(Physics)g(I\226IV)-10 b(.)32 b FB(Ne)n(w)g(Y)-9 b(ork,)32 b(Academic)j(Press,)466 4912 y(1972.)94 5002 y([Sa77])159 b(D.H.)32 b(Sattinger)l(.:)51 b(W)-6 b(eighted)34 b(norms)f(for)f(the)h(stability)h(of)f(tra)n(v)o(elling)h(w)o(a)n(v)o (es.)g Fa(J)n(.)f(Dif)o(f)o(.)g(Eqns.)f FA(25)h FB(\(1977\),)466 5091 y(130\226144.)94 5181 y([Schn94])79 b(G.)33 b(Schneider)l(.:)18 b(Error)31 b(estimates)k(for)d(the)i(Ginzb)n(ur)o(g\226Landau)e (approximation.)h Fa(J)n(.)g(Appl.)g(Math.)g(Physics)h FA(45)466 5271 y FB(\(1994\),)e(433\226457.)p eop %%Page: 41 41 41 40 bop 94 -45 a Ft(St)-7 b(ability)34 b(of)g(Modula)-7 b(ted)34 b(Fr)n(onts)2074 b Fv(41)94 194 y FB([Schn96])79 b(G.)33 b(Schneider)l(.:)18 b(Dif)n(fusi)n(v)o(e)32 b(stability)i(of)e (spatial)j(periodic)e(solutions)g(of)g(the)g(Swift\226Hohenber)o(g)f (equation.)466 284 y Fa(Comm.)h(Math.)g(Phys.)g FA(178)g FB(\(1996\),)e(679\226702.)94 373 y([Schn98])79 b(G.)33 b(Schneider)l(.:)18 b(Nonlinear)33 b(stability)h(of)e(T)-6 b(aylor)n(\226v)n(ortices)34 b(in)f(in\002nite)h(c)o(ylinders.)f Fa(Ar)m(c)o(hive)h(for)f(Rational)g(Me\255)466 463 y(c)o(hanics)h(and)f (Analysis)g FA(144)g FB(\(1998\),)e(121\226200.)94 553 y([T)-6 b(a97])160 b(M.)33 b(E.)g(T)-6 b(aylor)l(.:)16 b(P)o(artial)34 b(Dif)n(ferential)f(Equations)g(I:)g(Basic)i(Theory)-5 b(.Appl.)31 b(Math.)i(Sciences)i FA(115)p FB(,)e(Springer)466 642 y(1997.)94 732 y([vH91])140 b(A.)33 b(v)n(an)g(Harten.:)17 b(On)33 b(the)h(v)n(alidity)f(of)g(Ginzb)n(ur)o(g\226Landau')l(s)e (equation.)j Fa(J)n(.)f(Nonlinear)h(Science)h FA(1)e FB(\(1991\),)466 822 y(397\226422.)94 911 y([W)-6 b(a97])134 b(C.E.)33 b(W)-6 b(ayne.:)18 b(In)m(v)n(ariant)33 b(manifolds)g(for)f (parabolic)i(partial)f(dif)n(ferential)h(equations)f(on)g(unbounded)g (domains.)466 1001 y Fa(Ar)m(c)o(h.)g(Rat.)g(Mec)o(h.)h(Anal.)f FA(138)f FB(\(1997\),)g(279\226306.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0003160635705--