This is a multi-part message in MIME format. ---------------0004201223115 Content-Type: text/plain; name="00-193.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="00-193.keywords" Fronts, Traveling Waves, Stability, Extended Fisher-Kolmogorov Equation ---------------0004201223115 Content-Type: application/postscript; name="jan18.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="jan18.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: jan18.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -o jan18.ps jan18.dvi %DVIPSParameters: dpi=300, compressed, comments removed %DVIPSSource: TeX output 2000.04.13:1024 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 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Fm(dB)1322 2013 y Fo(\()p Fn(B)s Fo(\))1400 2021 y Fr(j)p Fm(B)r Fj(=1)1505 2013 y Fn(<)f Fl(\000)p Fo(1.)33 b(Also,)20 b(since)15 2073 y Fl(N)29 b Fo(is)23 b(a)f(nonlinear)h(term) d(w)o(e)i(kno)o(w)h(that)859 2054 y Fm(d)p Fr(N)p 859 2062 V 862 2091 a Fm(dB)916 2073 y Fo(\()p Fn(B)s Fo(\))994 2081 y Fr(j)p Fm(B)r Fj(=0)1103 2073 y Fo(=)h(0)f(holds.)40 b(The)23 b(c)o(hoice)e(of)h Fn(D)27 b(>)d Fo(0)f(is)15 2134 y(dictated)17 b(b)o(y)g(the)h(ph)o(ysical)f(requiremen)n(t)e(that) j(the)g(mo)q(del)e(is)i(stable)f(at)i(short)f(w)o(a)o(v)o(elengths.)25 b(W)l(e)15 2194 y(will)18 b(study)i(the)f(regime)f(in)h(whic)o(h)g Fn(D)i Fo(is)f(v)o(ery)e(large.)31 b(The)20 b(amplitude)e Fn(B)k Fo(is)d(c)o(hosen)h(to)g(b)q(e)f(real.)15 2254 y(W)l(e)11 b(are)h(in)o(terested)e(in)h(the)h(existence)e(and)i (stabilit)o(y)e(of)i(tra)o(v)o(eling)f(fron)o(t)g(solutions)i(to)f (this)f(equation.)15 2374 y(Equation)i(\(1.1\))f(is)g(found,)i(with)e (a)h(quadratic)f(\()p Fl(N)7 b Fo(\()p Fn(B)s Fo(\))14 b(=)f Fl(\000)p Fn(B)1174 2356 y Fj(2)1193 2374 y Fo(\))g(or)g(a)f (cubic)g(nonlinearit)o(y)f(\()p Fl(N)c Fo(\()p Fn(B)s Fo(\))14 b(=)15 2434 y Fl(\000)p Fn(B)94 2416 y Fj(3)113 2434 y Fo(\),)e(to)h(describ)q(e)f(the)g(b)q(eha)o(viour)g(of)g(small)f (solutions)i(of)f(a)h(reaction-di\013usion)f(system)f(near)i(a)f(co-)15 2495 y(dimension)e(2)i(p)q(oin)o(t,)h(see)e([28)q(,)g(30)q(,)g(29)q(].) 19 b(There,)12 b(a)h(general)e(reaction-di\013usion)i(system)d(is)i (studied)f(for)15 2555 y(small)17 b(p)q(erturbations)i(of)g(the)f (zero-solution)h(where)f(an)h(amplitude)d(equation)i(approac)o(h)i(is)e (used.)15 2615 y(Near)j(criticalit)o(y)l(,)e(the)i(b)q(eha)o(viour)g (of)h(these)f(solutions)h(is)f(describ)q(ed)g(b)o(y)g(the)g (Ginzburg-Landau)15 2675 y(\(GL\))h(equation.)38 b(Ho)o(w)o(ev)o(er,)21 b(mo)o(ving)f(to)o(w)o(ards)i(a)g(co-dimension)f(2)h(p)q(oin)o(t)f(in)h (parameter-space,)p eop %%Page: 2 2 2 1 bop 15 68 a Fo(this)15 b(GL-equation)i(cannot)g(b)q(e)f(used)g(an)o 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Fn(x)13 b Fo(=)h Fn("x)i Fo(in)g(equation)g(\(1.1\))g(w)o(e)g(obtain) 559 1080 y Fn(B)596 1087 y Fm(t)624 1080 y Fo(=)e Fn(B)g Fo(+)d Fn(")799 1060 y Fj(2)818 1080 y Fn(D)q(B)898 1087 y Fj(~)-20 b Fm(x)s Fj(~)g Fm(x)950 1080 y Fl(\000)11 b Fn(")1023 1060 y Fj(4)1043 1080 y Fn(B)1082 1087 y Fj(~)-20 b Fm(x)r Fj(~)g Fm(x)q Fj(~)g Fm(x)r Fj(~)g Fm(x)1172 1080 y Fo(+)11 b Fl(N)c Fo(\()p Fn(B)s Fo(\))p Fn(:)443 b Fo(\(1.2\))15 1190 y(Therefore,)14 b(w)o(e)g(c)o(ho)q(ose)i Fn(D)f Fo(=)592 1171 y Fj(1)p 584 1179 34 2 v 584 1208 a Fm(")600 1198 y Fi(2)638 1190 y Fo(where)f Fn(")h Fo(is)g(in)o(tro)q (duced)f(to)h(b)q(e)g(a)g(small)f(parameter.)19 b(This)c(implies)15 1251 y(that)20 b(w)o(e)e(tak)o(e)h Fn(D)j Fo(large.)30 b(T)l(aking)20 b Fn(D)h Fo(large)e(is)h(in)o(teresting)e(b)q(ecause)h (this)h(c)o(hoice)e(corresp)q(onds)i(to)15 1311 y(studying)13 b(a)g(reaction-di\013usion)h(system)d(where)i(the)g(rates)g(of)g (di\013usion)h(are)f(v)o(ery)f(di\013eren)o(t,)g(see)h([30])15 1371 y(for)18 b(a)g(expression)f(of)h Fn(D)i Fo(in)d(terms)f(of)i (these)g(di\013usion)g(co)q(e\016cien)o(ts.)24 b(The)18 b(c)o(hoice)e Fn(D)j Fo(=)1707 1351 y Fj(1)p 1699 1359 V 1699 1388 a Fm(")1715 1379 y Fi(2)1755 1371 y Fo(implies)15 1431 y(that)h(equation)g(\(1.1\))g(is)g(a)g(singular)h(p)q(erturbation) f(of)g(the)g(so-called)g(Fisher-Kolmogoro)o(v)f(\(FK\))15 1491 y(equation)766 1552 y Fn(u)794 1559 y Fm(t)822 1552 y Fo(=)14 b Fn(u)902 1559 y Fm(xx)955 1552 y Fo(+)d Fn(u)g Fl(\000)f Fn(u)1120 1531 y Fj(2)1140 1552 y Fn(;)15 1639 y Fo(asso)q(ciated)18 b(with)f(the)g(names)f(of)i(Fisher)f([13])g(and)h (Kolmogoro)o(v,)e(P)o(etro)o(vsky)g(and)i(Piskuno)o(v)f([23].)15 1699 y(Equation)i(\(1.2\))g(is)g(an)h(in)o(teresting)e(p)q(erturbation) h(of)g(the)g(FK-equation)g(since)f(the)h(fourth)g(order)15 1759 y(term)c(w)o(ould)i(come)f(in)o(to)h(the)g(equation)g(as)g(one)h (of)f(the)g(higher)g(order)g(terms)f(when)h(doing)h(an)f(am-)15 1819 y(plitude)k(equation)h(approac)o(h.)39 b(The)22 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y(studies)22 b(of)h(fron)o(ts)f(in)g (the)g(eFK-equation,)h([27,)f(25)q(,)g(1])g(and)h([3])f(fo)q(cussed)h (on)g(the)f(existence)e(of)15 1278 y(fron)o(ts)e(connecting)f Fn(B)i Fo(=)d(+1)i(to)f Fn(B)i Fo(=)d Fl(\000)p Fo(1)i(whic)o(h)f(are)h (stable-stable)f(fron)o(ts.)26 b(As)17 b(far)h(as)g(w)o(e)f(kno)o(w)15 1338 y(the)g(only)h(other)g(rigorous)h(w)o(ork)e(on)i(unstable-stable)f (fron)o(ts)g(in)f(the)h(eFK-equation)f(is)h([3],)f(whic)o(h)15 1398 y(pro)o(v)o(es)i(that)i(suc)o(h)f(fron)o(ts)h(exist)e(if)h(one)g (c)o(ho)q(oses)i(a)e(nonlinear)g(term)f Fl(N)7 b Fo(\()p Fn(B)s Fo(\))20 b(=)h Fl(\000)p Fn(B)1643 1380 y Fj(2)1682 1398 y Fo(and)g(whic)o(h)15 1458 y(pro)o(v)o(es)15 b(the)g(existence)f (of)h(fron)o(ts)h(connecting)f Fn(B)i Fo(=)c Fl(\000)p Fn(a)i Fo(to)h Fn(B)g Fo(=)e Fl(\006)p Fo(1)i(for)f Fl(N)7 b Fo(\()p Fn(B)s Fo(\))14 b(=)g Fn(a)p Fo(\(1)9 b Fl(\000)g Fn(B)1749 1440 y Fj(2)1769 1458 y Fo(\))g Fl(\000)g Fn(B)1885 1440 y Fj(3)15 1518 y Fo(where)17 b(0)f Fn(<)f(a)g(<)h 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(equations)f(has)i(b)q(een)e(studied)g(extensiv)o(ely)l(,)e(where)i(v)m (arious)h(metho)q(ds)f(w)o(ere)g(used.)15 609 y(F)l(or)d(example,)e ([23)q(,)i(13)q(,)g(2])g(used)h(comparison)e(theorems)h(based)g(on)h (the)g(Maxim)o(um)10 b(Principle.)19 b(Us-)15 669 y(ing)d(functional)g (analytic)f(tec)o(hniques,)f(renormalisation)h(group)i(metho)q(ds)f(or) g(energy)g(functionals,)15 730 y(a)j(lo)q(cal)f(stabilit)o(y)f (analysis)h(in)g(suitable)g(w)o(eigh)o(ted)g(spaces)g(has)h(b)q(een)g (studied)f(b)o(y)g(Sattinger)g([31],)15 790 y(Kirc)o(hg\177)-24 b(assner)17 b([22],)g(Kapitula)h([21],)f(Bricmon)o(t)e(and)j(Kupiainen) f([4],)g(Ec)o(kmann)f(and)i(W)l(a)o(yne)f([10])15 850 y(and)g(Galla)o(y)e(and)i(Raugel)f([15)q(,)g(16,)g(17)q(].)15 970 y(The)f(fron)o(t)h(solutions)f(w)o(e)g(construct)h(are)f(only)g (exp)q(ected)g(to)h(b)q(e)f(stable)g(if)g(w)o(e)g(restrict)g(ourselv)o (es)f(to)15 1031 y(p)q(erturbations)19 b(whic)o(h)e(deca)o(y)g(to)i(0)f (su\016cien)o(tly)e(fast)j(as)g Fn(x)d Fl(!)h(1)p 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Fh(0)1025 1486 y Fo(\(1\))h Fn(<)e Fl(\000)p Fo(1)h(as)h(stated)f(ab)q (o)o(v)o(e,)g(and)g(in)f(addition)15 1546 y(that)22 b(the)f(co)q (e\016cien)o(t)f(of)i(either)f(the)h(quadratic)f(or)h(cubic)f(term)f (in)i(the)f(p)q(o)o(w)o(er)h(series)f(for)h Fl(N)29 b Fo(is)15 1606 y(non-zero.)15 1705 y Fk(Remark)16 b(1.1)24 b Ff(In)17 b(p)n(articular,)f(the)g(ab)n(ove)h(assumptions)f(hold)h (for)e(either)i(of)f(the)h(c)n(ommon)f(choic)n(es)15 1765 y(of)h Fl(N)7 b Fo(\()p Fn(u)p Fo(\))14 b(=)g Fl(\000)p Fn(u)319 1747 y Fj(2)355 1765 y Ff(or)j Fl(N)7 b Fo(\()p Fn(u)p Fo(\))14 b(=)g Fl(\000)p Fn(u)665 1747 y Fj(3)684 1765 y Ff(.)15 1864 y Fo(W)l(e)d(substitute)g(the)g(ab)q(o)o(v)o(e)g (expression)f(\(1.6\))i(in)o(to)f(equation)g(\(1.1\))g(and)h(use)f(the) g(fact)g(that)h Fn(\036)p Fo(\()p Fn("x)q Fl(\000)q Fn(ct)p Fo(\))15 1925 y(satis\014es)k(equation)h(\(1.1\).)k(Then,)16 b(using)h(that)f Fn(D)g Fo(=)1024 1905 y Fj(1)p 1016 1913 34 2 v 1016 1942 a Fm(")1032 1932 y Fi(2)1054 1925 y Fo(,)g(leads)g(to)94 2027 y Fn(@)s(w)p 94 2050 65 2 v 104 2095 a(@)s(t)206 2061 y Fo(=)41 b(\(1)11 b Fl(\000)g Fo(6)p Fn(")436 2041 y Fj(4)456 2061 y Fn(\015)484 2041 y Fj(2)504 2061 y Fo(\))p Fn(w)558 2068 y Fm(\020)r(\020)606 2061 y Fo(+)g(\()p Fn(c)g Fl(\000)g Fo(2)p Fn(\015)k Fo(+)c(4)p Fn(")916 2041 y Fj(4)936 2061 y Fn(\015)964 2041 y Fj(3)983 2061 y Fo(\))p Fn(w)1037 2068 y Fm(\020)1068 2061 y Fo(+)g(\(1)h Fl(\000)e Fn(c\015)15 b Fo(+)c Fn(\015)1359 2041 y Fj(2)1389 2061 y Fl(\000)g Fn(")1462 2041 y Fj(4)1482 2061 y Fn(\015)1510 2041 y Fj(4)1529 2061 y Fo(\))p Fn(w)i Fo(+)e(4)p Fn(")1692 2041 y Fj(4)1712 2061 y Fn(\015)s(w)1775 2068 y Fm(\020)r(\020)r(\020)285 2154 y Fl(\000)p Fn(")347 2133 y Fj(4)366 2154 y Fn(w)401 2161 y Fm(\020)r(\020)r(\020)r(\020)485 2154 y Fo(+)g Fn(w)q(g)r(;)1195 b Fo(\(1.7\))15 2261 y(where)646 2331 y Fn(w)q(g)16 b Fo(=)779 2297 y(1)p 778 2319 26 2 v 778 2365 a Fn(a)809 2331 y Fo(\()p Fl(N)7 b Fo(\()p Fn(\036)k Fo(+)g Fn(aw)q Fo(\))g Fl(\000)g(N)c 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y(in)f(damp)q(ed)h(h)o(yp)q(erb)q(olic)f(equations.)27 b(The)18 b(c)o(hoice)e(of)j Fn(\015)i Fo(implies)15 b(that)j(w)o(e)g(m) o(ust)e(tak)o(e)i Fn(c)f Fl(\025)f Fn(c)1788 2657 y Fr(\003)1826 2675 y Fo(and)947 2800 y(4)p eop %%Page: 5 5 5 4 bop 15 68 a Fo(that)16 b Fn(\015)k Fo(equals)c(the)g(exp)q(onen)o (tial)f(deca)o(y)h(rate)g(of)g Fn(\036)h Fo(as)g Fn(\020)g Fl(!)d(1)p Fo(.)21 b(With)16 b(this)g(c)o(hoice)f(of)i Fn(\015)s Fo(,)f(equation)15 128 y(\(1.7\))g(reduces)g(to)162 226 y Fn(@)s(w)p 162 248 65 2 v 172 294 a(@)s(t)246 260 y Fo(=)e(\(1)d Fl(\000)g Fo(6)p Fn(")449 239 y Fj(4)469 260 y Fn(\015)497 239 y Fj(2)517 260 y Fo(\))p Fn(w)571 267 y Fm(\020)r(\020)619 260 y Fo(+)g(\()p Fn(c)g Fl(\000)g Fo(2)p Fn(\015)j Fo(+)d(4)p Fn(")928 239 y Fj(4)948 260 y Fn(\015)976 239 y Fj(3)996 260 y Fo(\))p Fn(w)1050 267 y Fm(\020)1081 260 y Fo(+)g(4)p Fn(")1177 239 y Fj(4)1197 260 y Fn(\015)s(w)1260 267 y Fm(\020)r(\020)r(\020)1326 260 y Fl(\000)g Fn(")1399 239 y Fj(4)1419 260 y Fn(w)1454 267 y Fm(\020)r(\020)r(\020)r(\020)1538 260 y Fo(+)g Fn(w)q(g)r(:)142 b Fo(\(1.9\))15 380 y(Using)16 b(a)h(T)l(a)o(ylor)f (expansion,)g(the)g(term)e Fn(w)q(g)19 b Fo(can)d(b)q(e)h(rewritten)e (as)718 487 y Fn(w)q(g)h Fo(=)e Fl(N)893 455 y Fh(0)906 487 y Fo(\()p Fn(\036)p Fo(\))p Fn(w)e Fo(+)f Fn(aw)1131 466 y Fj(2)1151 487 y Fn(R;)15 594 y Fo(where)19 b(w)o(e)g(kno)o(w)h (that)g(there)f(exists)f(a)i(0)g Fn(<)g(\021)h(<)e(aw)i Fo(suc)o(h)e(that)h Fn(R)g Fo(=)1410 574 y Fj(1)p 1410 582 18 2 v 1410 611 a(2)1433 594 y Fl(N)1481 564 y Fh(00)1503 594 y Fo(\()p Fn(\036)13 b Fo(+)h Fn(\021)r Fo(\).)30 b(F)l(rom)19 b(the)15 654 y(fact)11 b(that)h(the)f(nonlinearit)o(y)g Fl(N)18 b Fo(is)12 b(analytic,)f(it)g(follo)o(ws)g(that)h Fn(R)g Fo(is)f(b)q(ounded)i(b)o(y)e(a)h(constan)o(t)g Fn(K)1772 661 y Fm(R)1815 654 y Fn(>)i Fo(0.)15 775 y(The)i(main)f (results)h(of)h(section)e(5)i(can)f(b)q(e)h(summarised)d(in)i(the)g (follo)o(wing)f(theorem)15 874 y Fk(Theorem)h(1.2)24 b Ff(L)n(et)16 b Fl(N)23 b Ff(satisfy)15 b(the)h(ab)n(ove)h (assumptions.)22 b(Ther)n(e)15 b(exists)i(an)f Fn(")1512 881 y Fj(0)1545 874 y Fn(>)e Fo(0)i Ff(such)g(that)g(for)15 934 y Fn(")d(<)h(")126 941 y Fj(0)163 934 y Ff(and)k Fn(c)c Fl(\025)f Fn(c)366 916 y Fr(\003)386 934 y Ff(,)18 b(ther)n(e)f(exists)h(a)g Fn(\016)d(>)f Fo(0)k Ff(such)f(that)h(if)f (we)i(cho)n(ose)e Fn( )f Fl(2)e Fn(X)1470 916 y Fj(5)1490 934 y Ff(,)j(the)h(function)h(sp)n(ac)n(e)15 994 y(de\014ne)n(d)f(b)n (elow,)f(with)h Fl(jj)p Fn( )r Fl(jj)518 1003 y Fm(X)550 993 y Fi(3)581 994 y Fl(\024)c Fn(\016)k Ff(and,)f(let)h Fn(w)g Ff(b)n(e)f(the)g(c)n(orr)n(esp)n(onding)f(solution)i(of)f (\(1.9\))f(such)h(that)15 1055 y Fn(w)q Fo(\(0\))d(=)g Fn( )19 b Ff(and)e(let)i Fn(B)h Ff(b)n(e)e(as)f(in)h(\(1.6\),)f(then) 622 1162 y Fo(lim)615 1189 y Fm(t)p Fr(!1)706 1162 y Fl(jj)p Fn(B)s Fo(\()p Fl(\001)p Fn(;)8 b(t)p Fo(\))i Fl(\000)h Fn(\036)p Fo(\()p Fn(")f Fl(\001)h(\000)p Fn(ct)p Fo(\))p Fl(jj)1157 1169 y Fm(L)1181 1160 y Fh(1)1229 1162 y Fo(=)i(0)15 1279 y Ff(i.e.)22 b(the)c(solution)h Fn(\036)e Ff(is)g(stable.)15 1378 y Fo(W)l(e)k(de\014ne)h(the)f(follo)o (wing)g(function)h(spaces.)38 b(By)21 b Fn(H)1059 1360 y Fm(j)1100 1378 y Fo(=)j Fn(H)1206 1360 y Fm(j)1224 1378 y Fo(\()p Fk(R)1285 1360 y Fm(n)1308 1378 y Fo(\))e(for)g Fn(j)k Fl(2)d Fk(N)f Fo(w)o(e)f(denote)h(the)15 1439 y(usual)e(\(real\))f(Sob)q(olev)i(space)f(of)g(order)g Fn(j)j Fo(o)o(v)o(er)c Fk(R)985 1420 y Fm(n)1008 1439 y Fo(,)h(with)g Fn(H)1201 1420 y Fj(0)1221 1439 y Fo(\()p Fk(R)1282 1420 y Fm(n)1305 1439 y Fo(\))g(=)g Fn(L)1435 1420 y Fj(2)1455 1439 y Fo(\()p Fk(R)1516 1420 y Fm(n)1539 1439 y Fo(\).)33 b(W)l(e)19 b(denote)h(b)o(y)15 1499 y Fn(H)59 1481 y Fm(j)55 1511 y(a)96 1499 y Fo(=)f Fn(H)197 1481 y Fm(j)193 1511 y(a)216 1499 y Fo(\()p Fk(R)277 1481 y Fm(n)300 1499 y Fo(\))g(the)g(w)o(eigh)o(ted)f(Sob)q(olev)i (space)f(de\014ned)g(b)o(y)g(the)g(norm)f Fl(jj)p Fn(w)q Fl(jj)1500 1514 y Fm(H)1532 1498 y Fe(j)1529 1520 y(a)1569 1499 y Fo(=)h Fl(jj)p Fn(aw)q Fl(jj)1744 1508 y Fm(H)1776 1498 y Fe(j)1793 1499 y Fo(.)30 b(W)l(e)15 1564 y(also)18 b(set)g Fn(H)236 1546 y Fj(0)232 1577 y Fm(a)256 1564 y Fo(\()p Fk(R)317 1546 y Fm(n)340 1564 y Fo(\))f(=)g Fn(L)464 1546 y Fj(2)464 1577 y Fm(a)485 1564 y Fo(\()p Fk(R)546 1546 y Fm(n)569 1564 y Fo(\).)26 b(Finally)l(,)17 b(w)o(e)g(write)h Fn(X)1050 1546 y Fm(j)1086 1564 y Fo(for)g(the)g(in)o (tersection)f Fn(H)1556 1546 y Fm(j)1586 1564 y Fl(\\)c Fn(H)1676 1546 y Fm(j)1672 1577 y(a)1712 1564 y Fo(equipp)q(ed)15 1625 y(with)j(the)g(norm)f Fl(jj)p Fn(w)q Fl(jj)429 1607 y Fj(2)429 1639 y Fm(X)461 1630 y Fe(j)493 1625 y Fo(=)e Fl(jj)p Fn(w)q Fl(jj)636 1607 y Fj(2)636 1639 y Fm(H)668 1630 y Fe(j)697 1625 y Fo(+)e Fl(jj)p Fn(w)q Fl(jj)838 1607 y Fj(2)838 1646 y Fm(H)870 1630 y Fe(j)867 1652 y(a)15 1745 y Fo(T)l(o)17 b(pro)o(v)o(e)e(Theorem)g(1.2)i(w)o(e)e(in)o (tro)q(duce)h(the)g(follo)o(wing)g(functionals)184 1872 y Fn(E)s Fo(\()p Fn(t)p Fo(\))41 b(=)400 1813 y Fg(Z)441 1826 y Fr(1)423 1907 y(\0001)501 1838 y Fo(1)p 501 1860 25 2 v 501 1906 a(2)530 1872 y Fn(Aw)603 1851 y Fj(2)634 1872 y Fo(+)688 1838 y(1)p 688 1860 V 688 1906 a(2)717 1872 y Fn(B)s(w)793 1851 y Fj(2)792 1884 y Fm(\020)824 1872 y Fo(+)878 1838 y(1)p 878 1860 V 878 1906 a(2)907 1872 y Fn(C)t(w)982 1851 y Fj(2)981 1884 y Fm(\020)r(\020)1029 1872 y Fo(+)1083 1838 y(1)p 1083 1860 V 1083 1906 a(2)1112 1872 y Fn(")1135 1851 y Fj(6)1155 1872 y Fn(w)1191 1851 y Fj(2)1190 1884 y Fm(\020)r(\020)r(\020)1245 1872 y Fn(d\020)t(;)471 b Fo(\(1.10\))185 1993 y Fn(F)7 b Fo(\()p Fn(t)p Fo(\))40 b(=)400 1935 y Fg(Z)441 1948 y Fr(1)423 2029 y(\0001)501 1960 y Fo(1)p 501 1982 V 501 2027 a(2)543 1981 y(~)530 1993 y Fn(A)p Fo(\()p Fn(aw)q Fo(\))667 1973 y Fj(2)697 1993 y Fo(+)751 1960 y(1)p 751 1982 V 751 2027 a(2)792 1981 y(~)781 1993 y Fn(B)r Fo(\(\()p Fn(aw)q Fo(\))939 2000 y Fm(\020)959 1993 y Fo(\))978 1973 y Fj(2)1009 1993 y Fo(+)1062 1960 y(1)p 1062 1982 V 1062 2027 a(2)1103 1981 y(~)1092 1993 y Fn(C)s Fo(\(\()p Fn(aw)q Fo(\))1249 2000 y Fm(\020)r(\020)1287 1993 y Fo(\))1306 1973 y Fj(2)1336 1993 y Fo(+)1390 1960 y(1)p 1390 1982 V 1390 2027 a(2)1420 1993 y Fn(")1443 1973 y Fj(6)1462 1993 y Fo(\(\()p Fn(aw)q Fo(\))1581 2000 y Fm(\020)r(\020)r(\020)1637 1993 y Fo(\))1656 1973 y Fj(2)1675 1993 y Fn(d\020)t(;)h Fo(\(1.11\))172 2086 y Fn(F)204 2093 y Fj(1)223 2086 y Fo(\()p Fn(t)p Fo(\))g(=)h Fn(\014)s(E)s Fo(\()p Fn(t)p Fo(\))10 b(+)h Fn(F)c Fo(\()p Fn(t)p Fo(\))p Fn(;)1086 b Fo(\(1.12\))15 2194 y(where)14 b Fn(A;)8 b(B)s(;)g(C)q(;)344 2181 y Fo(~)333 2194 y Fn(A)n(;)401 2181 y Fo(~)390 2194 y Fn(B)r(;)462 2181 y Fo(~)451 2194 y Fn(C)18 b Fo(and)c Fn(\014)j Fo(are)d(p)q(ositiv)o(e) f(constan)o(ts)i(that)g(are)f(c)o(hosen)g(suc)o(h)g(that)h(they)e (satisfy)15 2254 y(Lemma)h(5.7)i(and)h(Lemma)d(5.9)j(\(these)f (functionals)g(are)g(mo)q(delled)f(on)h(those)h(of)g([16]\).)15 2374 y(W)l(e)f(will)g(sho)o(w)i(in)f(section)f(5)i(that)f(when)g Fn(w)h Fo(is)f(su\016cien)o(tly)e(small)g(-this)i(is)g(satis\014ed)g(b) q(ecause)h Fn(w)g Fo(is)15 2434 y(a)h(p)q(erturbation)g(of)g(the)f(tra) o(v)o(eling)f(w)o(a)o(v)o(e)h(solution-)h Fn(E)j Fo(and)d Fn(F)1206 2441 y Fj(1)1244 2434 y Fo(are)g(non-increasing)g(functions)f (of)15 2495 y(time.)g(F)l(rom)12 b(this)h(the)f(con)o(v)o(ergence)g(of) h Fn(w)i Fo(to)e(zero)g(follo)o(ws)g(and)h(therefore)e(the)h(solution)g Fn(\036)g Fo(is)g(stable.)15 2615 y(Our)f(results)g(also)g(shed)g(some) f(ligh)o(t)g(on)i(the)f(v)o(ery)e(in)o(teresting)h(ideas)h(of)g(Eb)q (ert)h(and)f(v)m(an)h(Saarlo)q(os)g([9].)15 2675 y(Although)h(their)f (argumen)o(ts)g(are)g(formal)g(these)g(authors)i(predict)e(the)h (nonlinear)f(stabilit)o(y)g(of)h(what)947 2800 y(5)p eop %%Page: 6 6 6 5 bop 15 68 a Fo(they)15 b(refer)g(to)h(as)h(`pulled)d(fron)o(ts')i (and)g(also)g(predict)f(the)h(rate)g(and)g(form)f(of)h(con)o(v)o (ergence)e(to)o(w)o(ards)15 128 y(suc)o(h)19 b(fron)o(ts.)32 b(In)19 b(the)g(con)o(text)g(of)h(our)g(mo)q(del,)f(pulled)g(fron)o(ts) h(are)f(precisely)f(those)i(with)g Fn(c)f Fo(=)h Fn(c)p Fl(\003)15 188 y Fo(and)d(in)g(accordance)g(with)g(the)g(predictions)f (of)i([9],)e(w)o(e)h(sho)o(w)g(that)h(these)e(fron)o(ts)i(are)f (stable.)23 b(F)l(ur-)15 248 y(thermore,)13 b([9])i(predict)f(that)i (if)f(the)g(fron)o(t)g(solution)h Fn(\036)p Fo(\()p Fn(\020)t Fo(\))d Fl(\031)h Fn(\013e)1213 230 y Fr(\000)p Fm(\015)r(\020)1296 248 y Fo(as)i Fn(\020)h Fl(!)d(1)h Fo(\(or)h Fn(\036)p Fo(\()p Fn(\020)t Fo(\))d Fl(\031)h Fn(\013\020)t(e)1837 230 y Fr(\000)p Fm(\015)r(\020)15 308 y Fo(for)19 b Fn(c)e Fo(=)h Fn(c)207 290 y Fr(\003)226 308 y Fo(\),)h(then)f(the)h(fron)o(t) f(should)h(b)q(e)g(stable)f(with)g(resp)q(ect)g(to)h(p)q(erturbations)h (whic)o(h)d(ob)q(ey)i(a)15 369 y(steepness)c(condition)h(whic)o(h)f (implies)e(that)j(the)g(p)q(erturbation)g(go)q(es)h(to)f(zero)g(as)g Fn(\020)k Fo(go)q(es)d(to)f(in\014nit)o(y)15 429 y(faster)g(than)g Fn(e)285 411 y Fr(\000)p Fm(\015)r(\020)352 429 y Fo(.)22 b(This)16 b(should)g(b)q(e)g(compared)f(with)h(our)g(result)g(whic)o(h) 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(itself,)e(again)i(as)g(predicted)e(b)o(y)h([9].)20 b(Indeed,)14 b(if)g(the)g(p)q(erturbation)h(deca)o(ys)f(with)15 790 y(a)i(rate)h Fn(e)179 772 y Fr(\000)p Fm(\016)q(\020)242 790 y Fo(,)f(with)g Fn(\016)g(<)e(\015)s Fo(,)h(then)i(w)o(e)e(see)h (from)f(the)h(discussion)h(just)f(prior)h(to)f(\(1.9\))h(that)g(the)f (fron)o(t)15 850 y(will)f(b)q(e)h(linearly)f(unstable.)15 952 y Fk(Remark)h(1.3)24 b Ff(The)d(outline)i(of)d(this)h(p)n(ap)n(er)f (is)g(as)h(fol)r(lows:)31 b(in)21 b(se)n(ctions)g(2)g(and)g(3)g(we)g (pr)n(ove)g(the)15 1012 y(existenc)n(e)d(of)f(tr)n(aveling)h(fr)n(onts) e(and)g(in)h(se)n(ction)h(4)e(we)i(determine)f(the)g(critic)n(al)g(sp)n (e)n(e)n(d.)22 b(In)16 b(se)n(ction)i(5)15 1072 y(the)g(stability)g(of) f(these)h(tr)n(aveling)h(fr)n(onts)e(is)g(studie)n(d.)15 1239 y Fp(2)81 b(The)26 b(existence)g(of)h(tra)n(v)n(eling)e(fron)n(ts) 15 1363 y Fd(2.1)66 b(The)22 b(equation)i(for)e(tra)n(v)n(eling)i(fron) n(ts.)15 1455 y Fo(In)12 b(this)g(section)f(w)o(e)h(will)f(deriv)o(e)g (the)h(equation)g(whic)o(h)f(describ)q(es)h(the)g(b)q(eha)o(viour)g(of) h(tra)o(v)o(eling)d(fron)o(ts)15 1515 y(of)16 b(equation)g(\(1.1\).)22 b(T)l(ra)o(v)o(eling)15 b(fron)o(ts)h(are)h(solutions)g(of)f(the)g (form)751 1625 y Fn(B)s Fo(\()p Fn(x;)8 b(t)p Fo(\))k(=)i Fn(\036)p Fo(\()p Fn(x)d Fl(\000)g Fo(~)-25 b Fn(ct)p Fo(\))p Fn(;)635 b Fo(\(2.1\))15 1735 y(where)22 b(~)-25 b Fn(c)21 b Fo(is)h(the)f(w)o(a)o(v)o(esp)q(eed.)36 b(W)l(e)21 b(are)g(in)o(terested)g(in)g(solutions)h(where)f(lim)1508 1742 y Fm(\030)q Fr(!1)1606 1735 y Fn(\036)p Fo(\()p Fn(\030)r Fo(\))i(=)g(0)f(and)15 1795 y(lim)83 1802 y Fm(\030)q Fr(!\0001)208 1795 y Fn(\036)p Fo(\()p Fn(\030)r Fo(\))14 b(=)g(1.)15 1916 y(Substituting)i(\(2.1\))h(in)o(to)f (equation)g(\(1.1\))g(giv)o(es)602 2026 y Fl(\000)q Fo(~)-25 b Fn(c)o(\036)690 2033 y Fm(\030)723 2026 y Fo(=)14 b Fn(\036)d Fo(+)g Fn(D)q(\036)934 2033 y Fm(\030)q(\030)981 2026 y Fl(\000)g Fn(\036)1060 2033 y 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userdict /Mathdict 100 dict dup begin put % The externally visible functions are: % MathPictureStart- start page. % MathPictureEnd - finish off page. % MathSubStart - start a sub-page. % MathSubEnd - finish off a sub-page. % Mdot - draw a dot. % Mtetra - draw a filled tetragon. % Metetra - draw a filled tetragon with black edges. % Mistroke - intermediate stroke of multi-stroke line/curve. % Mfstroke - final stroke of multi-stroke line/curve. % Msboxa - compute coordinates of text bounding box. % Mshowa - plot characters. % MathScale - compute scaling info to contain array of points. %start of ISOLatin1 stuff /ISOLatin1Encoding dup where { pop pop } { [ /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /minus /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /dotlessi /grave /acute /circumflex /tilde /macron /breve /dotaccent /dieresis /.notdef /ring /cedilla /.notdef /hungarumlaut /ogonek /caron /space /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def } ifelse /MFontDict 50 dict def /MStrCat { exch dup length 2 index length add string dup 3 1 roll copy length exch dup 4 2 roll exch putinterval } bind def /MCreateEncoding { 1 index 255 string cvs (-) MStrCat 1 index MStrCat cvn exch (Encoding) MStrCat cvn dup where { exch get } { pop StandardEncoding } ifelse 3 1 roll dup MFontDict exch known not { 1 index findfont dup length dict begin {1 index /FID ne {def} {pop pop} ifelse} forall /Encoding 3 index def currentdict end 1 index exch definefont pop MFontDict 1 index null put } if exch pop exch pop } bind def /ISOLatin1 { (ISOLatin1) MCreateEncoding } bind def %end of ISOLatin1 stuff % Set up for the start of a page. /MathPictureStart { /Msaveobj save def /findresource where { pop /WRI-Mathematica-prolog /ProcSet findresource begin } { Mathdict begin } ifelse userdict begin /MathPictureStartHook where { /MathPictureStartHook get exec } if Mtransform Mlmarg Mbmarg translate Mwidth Mlmarg Mrmarg add sub /Mwidth exch def Mheight Mbmarg Mtmarg add sub /Mheight exch def /Mtmatrix matrix currentmatrix def /Mgmatrix matrix currentmatrix def /MathDPSFlag where { pop clientsync } if } def % Finish off a page. /MathPictureEnd { end end Msaveobj restore /MathDPSFlag where { pop } { showpage } ifelse } def %MFill fills the drawing area with the current color. /MFill { 0 0 moveto Mwidth 0 lineto Mwidth Mheight lineto 0 Mheight lineto fill } def %New device-specific optimizing operators /Mfill /.devicefill where { pop /.devicefill load def } { /fill load def } ifelse /Mstroke /.devicestroke where { pop /.devicestroke load def } { /stroke load def } ifelse % xmin xmax ymin ymax MPlotRegion alters the origin, Mwidth and Mheight % so that the picture fills the altered region /MPlotRegion { 3 index Mwidth mul 2 index Mheight mul translate exch sub Mheight mul /Mheight exch def exch sub Mwidth mul /Mwidth exch def } def % Given a rectangle, set it up as a sub-picture. /MathSubStart { Momatrix Mgmatrix Mtmatrix Mwidth Mheight 7 -2 roll moveto Mtmatrix setmatrix currentpoint Mgmatrix setmatrix 9 -2 roll moveto Mtmatrix setmatrix currentpoint 2 copy translate /Mtmatrix matrix currentmatrix def 3 -1 roll exch sub /Mheight exch def sub /Mwidth exch def } def % Restore the saved state left by the matching MathSubStart. % Note, we also leave with the new Mgmatrix as the current matrix. /MathSubEnd { /Mheight exch def /Mwidth exch def /Mtmatrix exch def dup setmatrix /Mgmatrix exch def /Momatrix exch def } def % Given a point, draw a dot. /Mdot { newpath currentlinewidth 2 div 0 360 arc fill } bind def % Given 4 points, draw the corresponding filled tetragon. /Mtetra { moveto lineto lineto lineto Mfill } bind def % Given 4 points, draw the corresponding filled tetragon with black edges. % Note, this leaves the gray level at 0 (for compatibility with the old % C code. /Metetra { moveto lineto lineto lineto closepath gsave Mfill grestore 0 setgray Mstroke } bind def % Mistroke is called to stroke intermediate parts of a path. It makes % sure to resynchronize the dashing pattern and to leave the current point % as the final point of the path. /Mistroke { flattenpath 0 0 0 { 4 2 roll pop pop } { 4 -1 roll 2 index sub dup mul 4 -1 roll 2 index sub dup mul add sqrt 4 -1 roll add 3 1 roll } { stop } { stop } pathforall pop pop currentpoint Mstroke moveto currentdash 3 -1 roll add setdash } bind def % Mfstroke is called to stroke the final parts of a path. It resets % the dashing pattern to compensate for any adjustments made by Mistroke. /Mfstroke { Mstroke currentdash pop 0 setdash } bind def % Mrotsboxa is the same as Msboxa except that it takes an angle % from the stack and the box is calculated for text rendered at this angle. % It gsaves in case we are starting a MathSubStart % save Mrot so that Msboxa can convert bouding box back to the non-rotated % coordinate system % call Mrotcheck to alter the offsets into the rotated system % converts Mtmatrix to render in the rotated system % the calls Msboxa which does all the work % at the end Mtmatrix is restored and Mrot is reset to 0 /Mrotsboxa { gsave dup /Mrot exch def Mrotcheck Mtmatrix dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def grestore Msboxa 3 -1 roll /Mtmatrix exch def /Mrot 0 def } bind def % Given an array of strings ([str...]), which represent consecutive lines % of text, a position in graphics coordinates (gx,gy) and a position in % the bounding box coordinates (sx,sy), compute the low and high coordinates % of the resulting text, in the [gx gy tx ty] form, which corresponds to % the point (gx,gy) (in graphics coordinates) plus the offset (tx,ty) (in % text coordinates). % Note, Msboxa assumes that the current matrix is the text matrix. % Mboxout is called in case we are in Mouter to make the box bigger % Mboxrot adjusts the box to account for a rotation to convert the box % into a nonrotated coordinate system % /Msboxa { newpath 5 -1 roll Mvboxa pop Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Msboxa1 5 -3 roll Msboxa1 Mboxrot [ 7 -2 roll 2 copy [ 3 1 roll 10 -1 roll 9 -1 roll ] 6 1 roll 5 -2 roll ] } bind def % Msboxa1 is an internal function which, given a bounding box coordinate % (sz), and the bounding box limits (blz and bhz), computes the actual % offsets (tlz = (blz-bhz)(sz+1)/2, thz = (blz-bhz)(sz-1)/2). /Msboxa1 { sub 2 div dup 2 index 1 add mul 3 -1 roll -1 add 3 -1 roll mul } bind def % Given a (non-empty) array of strings ([str...]) which represent consecutive % lines of text, compute the total bounding box assuming that we start at % (0,0) and the array of y offsets to place the lines correctly. % Note, Mvboxa assumes that the current matrix is the text matrix. % Note, Mvboxa does not alter the current path. % The vertical spacing is set so that the bounding boxes of adjacent lines % are .3 times the width of an `m' apart. /Mvboxa { gsave newpath [ true 3 -1 roll { Mbbox 5 -1 roll { 0 5 1 roll } { 7 -1 roll exch sub (m) stringwidth pop .3 mul sub 7 1 roll 6 -1 roll 4 -1 roll Mmin 3 -1 roll 5 index add 5 -1 roll 4 -1 roll Mmax 4 -1 roll } ifelse false } forall { stop } if counttomark 1 add 4 roll ] grestore } bind def % Given a string, compute the bounding box assuming that we start at (0,0). % Note, the path is assumed to be empty, and we are assumed to be in text % coordinates. Allows for long strings. /Mbbox { 1 dict begin 0 0 moveto /temp (T) def { gsave currentpoint newpath moveto temp 0 3 -1 roll put temp false charpath flattenpath currentpoint pathbbox grestore moveto lineto moveto} forall pathbbox newpath end } bind def % Compute the minimum of two numbers. /Mmin { 2 copy gt { exch } if pop } bind def % Compute the maximum of two numbers. /Mmax { 2 copy lt { exch } if pop } bind def % % Mrotwork saves Mrot (not really needed but saved anyway) % calls Mrotcheck to adjust the offsets into the rotated coordinate system % converts Mtmatrix to render in the rotated system % /Mrotwork { dup /Mrot exch def Mrotcheck Mtmatrix dup setmatrix 7 1 roll 4 index 4 index translate rotate 3 index -1 mul 3 index -1 mul translate /Mtmatrix matrix currentmatrix def Mgmatrix setmatrix } def % % Mrotshowa is the same as Mshowa except that it takes an angle % from the stack and the text is rendered at this angle. % It call Mrotwork to do all the rotation work. % Then it calls Mshowa which does all the drawing work. % At the end Mtmatrix is restored and Mrot is reset to zero. % /Mrotshowa { Mrotwork Mshowa /Mtmatrix exch def /Mrot 0 def } def % % Draws rotated text with a background. % /Mrotback { 7 1 roll dup 8 1 roll Mrotwork 8 -2 roll Mshowback /Mtmatrix exch def /Mrot 0 def } def % % Given an array of strings ([str...]), which represent consecutive lines % of text, a position in graphics coordinates (gx,gy) and a position in % the bounding box coordinates (sx,sy), display the strings. % Mboxout is called in case we are in Mouter % /Mshowa { 4 -2 roll moveto 2 index Mtmatrix setmatrix Mvboxa 7 1 roll Mboxout 6 -1 roll 5 -1 roll 4 -1 roll Mshowa1 4 1 roll Mshowa1 rmoveto currentpoint Mshowax pop pop pop pop Mgmatrix setmatrix } def % % The background box of text is increased by the % size of an "m" % /Madjust { (m) stringwidth pop mul dup 4 -1 roll add 3 1 roll add } def % % % /Mbuildcoord { 7 index 7 index 4 -1 roll 4 -1 roll index exch index Mabsadd } def % % If the text is rotated then translate to the drawing point. % Rotate then translate back. Do not do this if the angle % of rotation is 0. % /Mrotate { 8 index 0 ne { 13 copy 4 -2 roll pop pop 6 -2 roll pop pop 4 -1 roll 3 -1 roll sub 6 -1 roll mul 3 1 roll sub 4 -1 roll mul 3 -1 roll 5 1 roll Mabsadd 2 copy translate 3 -1 roll rotate exch -1 mul exch -1 mul translate } if 9 -1 roll pop } def % % Draw text but into a box with a certain color. % Uses Msboxa to build the box. % /Mshowback { gsave exec 6 copy pop Msboxa aload pop 0.5 Madjust 5 -1 roll aload pop Mrotate -0.5 Madjust 4 -1 roll pop 3 -1 roll pop 3 4 Mbuildcoord moveto 3 4 Mbuildcoord lineto 3 6 Mbuildcoord lineto 5 6 Mbuildcoord lineto 5 4 Mbuildcoord lineto pop pop pop pop pop pop fill grestore Mshowa } def % This is used for fixedwidth fonts % It simply shows each string and advances the y direction by the offset % /Mshowax { 0 1 4 index length -1 add { 2 index 4 index 2 index get 3 index add moveto 4 index exch get Mfixdash { Mfixdashp } if show } for } def % Fix if all dashes and length > 1 /Mfixdashp { dup length 1 gt 1 index true exch { 45 eq and } forall and { gsave (--) stringwidth pop (-) stringwidth pop sub 2 div 0 rmoveto dup length 1 sub { (-) show } repeat grestore } if } bind def % Mshowa1 is an internal routine which, given a bounding box coordinate % (sz), and the bounding box limits if we started drawing at 0 (tlz and thz), % computes the offset at which to start drawing % (relz = (sz-1)tlz/2 - (sz+1)thz/2 = sz(tlz-thz)/2-(tlz+thz)/2). /Mshowa1 { 2 copy add 4 1 roll sub mul sub -2 div } bind def % Given the x and y scaling to user coordinates and an array of points to % fit (xbias xscale ybias yscale [pnts]), set up the scaling. The array % must contain atleast two points, and the last two must be of the form % [gxlow gylow 0 0] and [gxhigh gyhigh 0 0]. % Note, MathScale assumes that we are already scaled so that the active area % is the rectangle [0,Mwidth-Mlmarg-Mrmarg]x[0,Mheight-Mbmarg-Mtmarg]. % also keep bias and scale info for PostScript commands /MathScale { Mwidth Mheight Mlp /MathScaleHook where { /MathScaleHook get exec } if translate scale pop pop pop pop pop pop pop pop /yscale exch def /ybias exch def /xscale exch def /xbias exch def /Momatrix xscale yscale matrix scale xbias ybias matrix translate matrix concatmatrix def /Mgmatrix matrix currentmatrix def } def % Given a non-empty array of points to fit ([p]) and a maximum width (sx) % and height (sy) find the largest scale (Ax and Ay) and offsets (Bx and By) % such that the transformation % [gx gy tx ty] -> (Ax gx + tx + bx, Ay gy + ty + By) % maps the points into the rectangle [0,sx]x[0,sy] /Mlp { 3 copy Mlpfirst { Mnodistort { Mmin dup } if 4 index 2 index 2 index Mlprun 16 -8 roll 21 8 roll 11 index 11 -1 roll 10 -4 roll Mlp1 8 index 9 -5 roll Mlp1 4 -1 roll and { exit } if 15 -8 roll pop pop pop pop pop pop pop pop 3 -1 roll pop pop } loop exch 3 1 roll 7 -3 roll pop pop pop } def % Given an array of points in the [gx gy tx ty] form, with the last two % being [gxlow gylow 0 0] and [gxhigh gyhigh 0 0], and the width and height % (sx and sy) in which to fit them, compute the maximum scaling (Ax and Ay). /Mlpfirst { 3 -1 roll dup length 2 copy -2 add get aload pop pop pop 4 -2 roll -1 add get aload pop pop pop 6 -1 roll 3 -1 roll 5 -1 roll sub div 4 1 roll exch sub div } bind def % Given a non-empty array of points to fit ([pnts]) and scale factors % for graphics->text (Ax and Ay), compute the limiting points. /Mlprun { 2 copy 4 index 0 get dup 4 1 roll Mlprun1 3 copy 8 -2 roll 9 -1 roll { 3 copy Mlprun1 3 copy 11 -3 roll /gt Mlpminmax 8 3 roll 11 -3 roll /lt Mlpminmax 8 3 roll } forall pop pop pop pop 3 1 roll pop pop aload pop 5 -1 roll aload pop 8 copy exch 6 -1 roll Mlprun2 8 2 roll 4 -1 roll Mlprun2 6 2 roll 3 -1 roll Mlprun2 4 2 roll exch Mlprun2 6 2 roll } bind def % Given scale factors for graphics->text (Ax and Ay) and a point in the % [gx gy tx ty] form, return the text x and y coordinate that results. /Mlprun1 { aload pop exch 6 -1 roll 5 -1 roll mul add 4 -2 roll mul 3 -1 roll add } bind def % Given a low and high coordinate, compute the center and width. /Mlprun2 { 2 copy add 2 div 3 1 roll exch sub } bind def % Given two points stored as [gx gy tx ty] followed by the scaled % result (rx, ry), and a comparison function (lt or gt) leave the % point which is the minimum (or maximum) in each dimension. /Mlpminmax { cvx 2 index 6 index 2 index exec { 7 -3 roll 4 -1 roll } if 1 index 5 index 3 -1 roll exec { 4 1 roll pop 5 -1 roll aload pop pop 4 -1 roll aload pop [ 8 -2 roll pop 5 -2 roll pop 6 -2 roll pop 5 -1 roll ] 4 1 roll pop } { pop pop pop } ifelse } bind def % Given a size (s), graphics->text scale (A), text center (ct), text % width (wt), graphics center (cg) and graphics width (wg), compute % a new graphics->text scale (Anew) and offset (B) and whether or not % we are done. % Note, the mysterious .99999 is magic juju which is supposed to ward % off the possibility that floating point errors would cause this % routine to return the old A and yet claim not-done. /Mlp1 { 5 index 3 index sub 5 index 2 index mul 1 index le 1 index 0 le or dup not { 1 index 3 index div .99999 mul 8 -1 roll pop 7 1 roll } if 8 -1 roll 2 div 7 -2 roll pop sub 5 index 6 -3 roll pop pop mul sub exch } bind def % The following are the workings of the tick, axes and plot labels. % NOTE a possible source of confusion is that for xticks ie tickmarks on % the x axis we keep y information and vice versa for yticks % % When Minner is found then % % assumes that box starts at zero on the left lower side % % 0) if outflag = 1 then intop = 0, inrht = 0 outflag = 0 % 1) Save intop largest top of box % 2) Save inrht largest rht of box % 3) set inflag notifies that inner marks are present % % When Mouter is found then % % if inflag is set then % 1) get vecx and vecy off the stack (points in direcn to move) % 2) vecx < 1 xadrht = inrht*abs(vecx) % 3) vecx > 1 xadlft = inrht*abs(vecx) % 4) vecy < 1 yadtop = intop*abs(vecy) % 5) vecy > 1 yadbot = intop*abs(vecy) % 6) set outflag = 1 % 7) clear inflag, inrht, intop % guaranteed to be zero if no inner is present?? % % % These all have effects in Mrotsboxa and Mrotshowa % check inflag and if set % % 1) increase top of bbox by yadtop % 2) decrease bot of bbox by yadbot % 3) increase rht of bbox by xadrht % 4) increase lft of bbox by xadlft % 5) clear outflag, yadtop, yadbot, % xadrht, xadlft % % % This saves the top right corner of the bounding box as a side effect % This is to allow the adjustment of text placed with Mouter so that % it misses the Minner text. It is assumed that the ang is 0 % in the same way it is assumed that the text of Mouter is 0 or 90 % /Minner { outflag 1 eq { /outflag 0 def /intop 0 def /inrht 0 def } if 5 index gsave Mtmatrix setmatrix Mvboxa pop grestore 3 -1 roll pop dup intop gt { /intop exch def } { pop } ifelse dup inrht gt { /inrht exch def } { pop } ifelse pop /inflag 1 def } def % This takes two number off the stack and uses them as a vector in graphics % coordinates which points in the direction in which the Mouter text is to move % it calculates the bouding box adjustments yadtop yadbot xadrht and xadlft % these are in Mboxout to adjust the bounding box to compensate. /Mouter { /xadrht 0 def /xadlft 0 def /yadtop 0 def /yadbot 0 def inflag 1 eq { dup 0 lt { dup intop mul neg /yadtop exch def } if dup 0 gt { dup intop mul /yadbot exch def } if pop dup 0 lt { dup inrht mul neg /xadrht exch def } if dup 0 gt { dup inrht mul /xadlft exch def } if pop /outflag 1 def } { pop pop} ifelse /inflag 0 def /inrht 0 def /intop 0 def } def % % This adjusts the bounding box to account for adjacent text % This allows the two text strings to avoid each other % current matrix is the text matrix /Mboxout { outflag 1 eq { 4 -1 roll xadlft leadjust add sub 4 1 roll 3 -1 roll yadbot leadjust add sub 3 1 roll exch xadrht leadjust add add exch yadtop leadjust add add /outflag 0 def /xadlft 0 def /yadbot 0 def /xadrht 0 def /yadtop 0 def } if } def /leadjust { (m) stringwidth pop .5 mul } bind def % The offsets sx and sy refer to the graphics coordinate system % thus they must be altered if a rotation has taken place. % We must also change the bounding box computations for Minner % /Mrotcheck { dup 90 eq { % % Mouter only applies to strings which are either at 0 or 90 % sort out the box adjust factors % % xadrht -> yadbot % xadlft -> yadtop % yadtop -> xadrht % yadbot -> xadlft yadbot /yadbot xadrht def /xadrht yadtop def /yadtop xadlft def /xadlft exch def } if dup cos 1 index sin Checkaux dup cos 1 index sin neg exch Checkaux 3 1 roll pop pop } def % % Checkaux is an auxilliary function for Mrotcheck it multiplies a % row vector by a column vector % /Checkaux { 4 index exch 4 index mul 3 1 roll mul add 4 1 roll } bind def % % Mboxrot converts the bounding box back from the rotated coordinate % system to the Mgmatrix system to compensate for a rotation % It has the opposite functionality of Mrotcheck % This is not the most neatest or most efficient implementation but it works % /Mboxrot { Mrot 90 eq { % old tlx thx tly thy % new -thy -tly tlx thx % brotaux 4 2 roll } if Mrot 180 eq { % old tlx thx tly thy % new -thx -tlx -thx -tly % 4 2 roll brotaux 4 2 roll brotaux } if Mrot 270 eq { % old tlx thx tly thy % new tly thy -thx -tlx % 4 2 roll brotaux } if } def % % auxilliary function negate and reverse /brotaux { neg exch neg } bind def % % Mabsproc takes a measurement in the default user units and converts % it to the present units. This allows absolute thickness and dashing % to work. It works by using a {0, x} vector and using the RMS of the result. /Mabsproc { 0 matrix defaultmatrix dtransform idtransform dup mul exch dup mul add sqrt } bind def % % Mabswid allows the linewidth to be specified in absolute coordinates % It does this by recording the graphics transformation matrix at the % begining of the plot. % This will break if the scaling in the x and y directions is % different. This is the case if Mnodistort is false % /Mabswid { Mabsproc setlinewidth } bind def % Mabsdash allows the dashing pattern to be specified in absolute coordinates % It does this by recording the graphics transformation matrix at the % begining of the plot. % This will break if the scaling in the x and y directions is % different. This is the case if Mnodistort is false % /Mabsdash { exch [ exch { Mabsproc } forall ] exch setdash } bind def % % Mabs takes a pair of coordinates in points % and converts them into a pair of coordinates in % the current drawing scheme % /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def % % Mabs takes a pair of numbers in drawing coordinates % and a pair in points. It converts the points to % drawing coordinates and adds them to the first pair. % /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def % % These two procedures are for using Offset coordinates % with MathScale computations % /Mabsfix1 { 3 -1 roll /ar exch def ar 2 get 3 -1 roll add exch ar 3 get add ar 3 3 -1 roll put ar 2 3 -1 roll put ar } bind def /Mabsfix { 2 copy 6 -1 roll 3 1 roll Mabsfix1 4 1 roll Mabsfix1 } bind def %MBeginOrig start coordinates in user coordinates /MBeginOrig { Momatrix concat} bind def %MEndOrig start coordinates in user coordinates /MEndOrig { Mgmatrix setmatrix} bind def /sampledsound where { pop } { /MathDPSFlag where { pop /sampledsound { clientsync } def } { /sampledsound { exch pop exch 5 1 roll mul 4 idiv mul 2 idiv exch pop exch /Mtempproc exch def { Mtempproc pop } repeat } def } ifelse } ifelse % Define setrgbcolor for those ancient PostScript devices lacking it /setrgbcolor dup where { pop pop } { { .114 mul exch .587 mul add exch .299 mul add setgray } bind def } ifelse % % now simple conversion of cmykcolor to rgbcolor % subtract k and then take complements /setcmykcolor where { pop} { /setcmykcolor { 4 1 roll [ 4 1 roll ] { 1 index sub 1 sub neg dup 0 lt { pop 0 } if dup 1 gt { pop 1 } if exch } forall pop setrgbcolor } bind def } ifelse % Here are the short operators /g /setgray load def /k /setcmykcolor load def /m /moveto load def /p /gsave load def /r /setrgbcolor load def /w /setlinewidth load def /C /curveto load def /F /Mfill load def /L /lineto load def /P /grestore load def /s /Mstroke load def /Mcharproc { currentfile (x) readhexstring pop 0 get exch div } bind def /Mshadeproc { dup 3 1 roll { dup Mcharproc 3 1 roll } repeat 1 eq { setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /Mrectproc { 3 index 2 index moveto 2 index 3 -1 roll lineto dup 3 1 roll lineto lineto Mfill } bind def /Mcolorimage { 7 1 roll pop pop matrix invertmatrix concat 2 exch exp 1 sub 3 1 roll 1 1 2 index { 1 1 4 index { dup 1 sub exch 2 index dup 1 sub exch 7 index 9 index Mshadeproc Mrectproc } for pop } for pop pop pop pop } bind def /Mimage { pop matrix invertmatrix concat 2 exch exp 1 sub 3 1 roll 1 1 2 index { 1 1 4 index { dup 1 sub exch 2 index dup 1 sub exch 7 index Mcharproc setgray Mrectproc } for pop } for pop pop pop } bind def % Default values for variables /Mlmarg 1.0 72 mul def /Mrmarg 1.0 72 mul def /Mbmarg 1.0 72 mul def /Mtmarg 1.0 72 mul def /Mwidth 8.5 72 mul def /Mheight 11 72 mul def /Mtransform { } bind def /Mnodistort true def /Mfixdash false def /Mrot 0 def /intop 0 def /inrht 0 def /inflag 0 def /outflag 0 def /xadrht 0 def /xadlft 0 def /yadtop 0 def /yadbot 0 def /defineresource where { pop /WRI-Mathematica-prolog currentdict /ProcSet defineresource pop } if end %KLUDGE until Mathematica output uses resources directly userdict /MathPictureStart { /findresource where { pop /WRI-Mathematica-prolog /ProcSet findresource } { Mathdict } ifelse /MathPictureStart get exec } put MathPictureStart /Mabs { Mgmatrix idtransform Mtmatrix dtransform } bind def /Mabsadd { Mabs 3 -1 roll add 3 1 roll add exch } bind def /Courier findfont 10 scalefont setfont % Scaling calculations 0.0993953 0.377929 0.213492 0.634921 [ [ 0 0 0 0 ] [ 1 .3 0 0 ] ] MathScale % Start of Graphics 1 setlinecap 1 setlinejoin newpath 0 g .5 Mabswid [ ] 0 setdash .47729 .21343 m .47644 .21239 L .47309 .20823 L .45422 .18546 L .42892 .15677 L .37499 .10301 L .33422 .06963 L .31021 .05318 L .28581 .03908 L .26199 .02807 L .2493 .02339 L .23533 .01925 L .2211 .01619 L .2076 .01444 L .19145 .01391 L .1767 .01505 L .16085 .01818 L .14622 .02302 L .13346 .02896 L .12017 .0371 L .10641 .04796 L .09291 .06153 L .08223 .07485 L .07192 .09052 L .06344 .10625 L .05673 .12125 L .05097 .13693 L .04659 .15159 L .04267 .16859 L .0398 .18666 L .03845 .20195 L .03817 .21727 L .03908 .23225 L .04091 .24461 L .04389 .25648 L .04574 .26173 L .04798 .26693 L .05226 .27437 L .05737 .28045 L .06257 .28446 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% % Polyline gs clippath 6003 3645 m 6123 3675 l 6003 3705 l 6150 3705 l 6150 3645 l cp clip n 6000 3675 m 6150 3675 l gr % Polyline gs clippath 9078 3645 m 9198 3675 l 9078 3705 l 9225 3705 l 9225 3645 l cp clip n 9075 3675 m 9225 3675 l gr % Polyline gs clippath 7245 3447 m 7275 3327 l 7305 3447 l 7305 3300 l 7245 3300 l cp clip n 7275 3525 m 7275 3300 l gr % Polyline gs clippath 4095 3372 m 4125 3252 l 4155 3372 l 4155 3225 l 4095 3225 l cp clip n 4125 3375 m 4125 3225 l gr 7.500 slw % Polyline gs clippath 6003 3645 m 6123 3675 l 6003 3705 l 6165 3705 l 6165 3645 l cp clip n 5925 3675 m 6150 3675 l gs col-1 s gr gr % arrowhead n 6003 3645 m 6123 3675 l 6003 3705 l col-1 s % Polyline gs clippath 9078 3645 m 9198 3675 l 9078 3705 l 9240 3705 l 9240 3645 l cp clip n 9000 3675 m 9225 3675 l gs col-1 s gr gr % arrowhead n 9078 3645 m 9198 3675 l 9078 3705 l col-1 s % Polyline gs clippath 4140 3297 m 4170 3177 l 4200 3297 l 4200 3135 l 4140 3135 l cp clip n 4170 3375 m 4170 3150 l gs col-1 s gr gr % arrowhead n 4140 3297 m 4170 3177 l 4200 3297 l col-1 s % Polyline gs clippath 7210 3372 m 7240 3252 l 7270 3372 l 7270 3210 l 7210 3210 l cp clip n 7240 3450 m 7240 3225 l gs col-1 s gr gr % arrowhead n 7210 3372 m 7240 3252 l 7270 3372 l col-1 s $F2psEnd rs %%EndDocument @endspecial 196 524 a Fo(a.)857 184 y Fn(u)1027 524 y Fo(b.)1631 184 y Fn(u)1130 70 y(v)357 51 y(v)202 633 y Fo(Figure)16 b(1:)21 b(The)16 b(solutions)h(in)f(the)g(\()p Fn(u;)8 b(v)r Fo(\)-plane)15 b(for)i(a.)k Fn(c)14 b(<)g(c)1361 615 y Fr(\003)1397 633 y Fo(and)j(b.)k Fn(c)14 b Fl(\025)g Fn(c)1663 615 y Fr(\003)1682 633 y Fo(.)15 828 y(attracted)i(to)h(the)f (origin)g(b)o(y)g(the)g(\015o)o(w)g(on)h Fn(M)876 835 y Fm(\017)893 828 y Fo(.)15 948 y(Next)g(consider)h(the)g(p)q(oin)o(t)g (\(1)p Fn(;)8 b Fo(0)p Fn(;)g Fo(0)p Fn(;)g Fo(0\).)28 b(Note)18 b(that)g(b)q(ecause)h(of)f(the)g(fact)g(that)h(w)o(e)f (assumed)f(that)20 989 y Fm(d)p Fr(N)p 20 997 52 2 v 26 1026 a Fm(du)77 1009 y Fo(\()p Fn(u)p Fo(\))143 1016 y Fr(j)p Fm(u)p Fj(=1)241 1009 y Fn(<)22 b Fl(\000)p Fo(1,)g(this)e(p)q(oin)o(t)h(is)g(a)g(saddle)g(for)g(all)f Fn(c)i(>)g Fo(0.)35 b(Since)20 b(the)g(\015o)o(w)i(on)f Fn(M)1657 1016 y Fm(\017)1694 1009 y Fo(is)g(a)g(small)15 1069 y(p)q(erturbation)h(of)h(the)e(\015o)o(w)i(de\014ned)f(b)o(y)f (\(2.8\),)i(an)o(y)f(\014nite)g(piece)e(of)j(the)e(unstable)i(manifold) d(of)15 1129 y(\(1)p Fn(;)8 b Fo(0)p Fn(;)g Fo(0)p Fn(;)g Fo(0\),)20 b(restricted)e(to)h Fn(M)578 1136 y Fm(\017)614 1129 y Fo(will)f(v)m(ary)h(smo)q(othly)f(with)h Fn(\017)p Fo(.)29 b(In)19 b(particular,)f(since)h(the)f(unstable)15 1189 y(manifold)g(of)i(the)f(\014xed)g(p)q(oin)o(t)h(\(1)p Fn(;)8 b Fo(0\))20 b(of)g(\(2.8\))f(is)h(asymptotic)e(to)i(the)f (origin)h(for)f Fn(c)h(>)f Fo(0,)h(w)o(e)f(can)15 1249 y(c)o(ho)q(ose)f Fn(\017)191 1256 y Fj(0)226 1249 y Fn(>)e Fo(0)h(\(p)q(ossibly)h(smaller)d(than)j(the)f Fn(\017)918 1256 y Fj(0)955 1249 y Fo(c)o(hosen)g(in)h(the)f(previous)g (paragraph\))i(suc)o(h)e(that)15 1310 y(for)i Fn(c)113 1317 y Fj(0)152 1310 y Fo(and)h Fn(s)f Fo(as)h(ab)q(o)o(v)o(e,)g(the)f (unstable)g(manifold)f(of)i(\(1)p Fn(;)8 b Fo(0)p Fn(;)g Fo(0)p Fn(;)g Fo(0\))20 b(for)f(the)g(\015o)o(w)h(de\014ned)f(b)o(y)g (\(2.5\),)15 1370 y(restricted)d(to)h Fn(M)340 1377 y Fm(\017)357 1370 y Fo(,)g(passes)h(within)e(a)i(distance)e Fn(s)h Fo(of)h(the)f(origin.)23 b(But)17 b(then)g(b)o(y)g(the)g (results)f(of)i(the)15 1430 y(previous)i(paragraph,)k(it)c(m)o(ust)g(b) q(e)h(attracted)g(to)g(the)f(origin)h({)g(i.e.)34 b(the)21 b(unstable)g(manifold)e(of)15 1490 y(\(1)p Fn(;)8 b Fo(0)p Fn(;)g Fo(0)p Fn(;)g Fo(0\))17 b(is)f(asymptotic)f(to)i(\(0)p Fn(;)8 b Fo(0)p Fn(;)g Fo(0)p Fn(;)g Fo(0\))17 b(as)g Fn(\030)f Fl(!)e(1)p Fo(,)i(and)g(w)o(e)g(ha)o(v)o(e)g(pro)o(v)o(en:)15 1586 y Fk(Prop)r(osition)i(3.1)24 b Ff(Fix)d Fn(c)525 1593 y Fj(0)566 1586 y Fn(>)g Fo(0)g Ff(\(arbitr)n(arily)f(smal)r(l\).) 35 b(Ther)n(e)21 b(exists)h(an)g Fn(\017)1481 1593 y Fj(0)1521 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Fm(@)r Fr(N)p 276 1077 55 2 v 283 1106 a Fm(@)r(u)336 1108 y Fr(j)p Fm(u)p Fj(=0)428 1089 y Fo(=)g(0\).)24 b(Th)o(us,)17 b(the)g(eigen)o(v)m(alues)f(of)i(the)e(linearisation)h (of)g(the)g(v)o(ector)f(\014eld)h(on)15 1155 y Fn(M)62 1162 y Fm(\017)95 1155 y Fo(at)f(the)g(origin)g(are)220 1284 y Fn(\025)248 1291 y Fr(\006)291 1284 y Fo(=)348 1251 y(1)p 348 1273 25 2 v 348 1318 a(2)377 1284 y Fl(f\000)p Fn(c)11 b Fo(+)g Fn(@)548 1291 y Fm(v)568 1284 y Fn(H)612 1264 y Fm(\017)608 1297 y(w)637 1284 y Fo(\(0)p Fn(;)d Fo(0\))k Fl(\006)807 1232 y Fg(q)p 848 1232 797 2 v 848 1284 a Fo(\()p Fn(c)f Fl(\000)g Fn(@)975 1291 y Fm(v)995 1284 y Fn(H)1039 1270 y Fm(\017)1035 1297 y(w)1064 1284 y Fo(\(0)p Fn(;)d Fo(0\)\))1191 1270 y Fj(2)1222 1284 y Fl(\000)j Fo(4\(1)h Fl(\000)f Fn(@)1427 1291 y Fm(u)1449 1284 y Fn(H)1493 1270 y Fm(\017)1489 1297 y(w)1517 1284 y Fo(\(0)p Fn(;)d Fo(0\)\))q Fl(g)16 b Fn(:)15 1407 y Fo(F)l(rom)h(this)h(expression,)h(w)o(e)f(see)g(that)g(b)q(oth)i(of)f 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Fo(\(0)p Fn(;)8 b Fo(0\))28 b(=)15 1937 y Fn(\017)35 1919 y Fj(4)54 1937 y Fo(\(1)13 b Fl(\000)e Fn(c)181 1919 y Fj(2)201 1937 y Fo(\))h(+)g(hot)q(,)17 b(while)g Fn(@)539 1944 y Fm(u)561 1937 y Fn(H)605 1919 y Fm(\017)601 1950 y(w)630 1937 y Fo(\(0)p Fn(;)8 b Fo(0\))17 b(=)f Fn(\017)829 1919 y Fj(4)849 1937 y Fo(\(2)p Fn(c)c Fl(\000)g Fn(c)997 1919 y Fj(3)1017 1937 y Fo(\))g(+)g(hot.)26 b(Inserting)17 b(these)h(in)o(to)f(the)h(previous)15 1998 y(relation)e(w)o(e)f(\014nd:)15 2099 y Fk(Prop)r(osition)j(4.1)24 b Ff(The)f(fr)n(ont)f(solutions)i(of)f(the)h(eFK-e)n(quation)h(\(1.1\)) e(c)n(onstructe)n(d)g(ab)n(ove)h(ar)n(e)15 2160 y(monotonic)18 b(if)f(their)g(sp)n(e)n(e)n(d)f Fn(c)h Ff(satis\014es)g(\(4.3\).)22 b(F)l(or)17 b(smal)r(l)h Fn(\017)p Ff(,)f(this)g(implies)g(that)g(the)h (sp)n(e)n(e)n(d)e(satis\014es)15 2220 y Fn(c)e Fl(\025)f Fo(2)f Fl(\000)e Fn(\017)207 2202 y Fj(4)238 2220 y Fo(+)h Fn(:)d(:)g(:)o Ff(.)15 2334 y Fk(Remark)16 b(4.2)24 b Ff(The)d(critic)n(al)h(sp)n(e)n(e)n(d)e 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(the)f(tra)o(v)o(eling)g(w)o(a)o(v)o(e)g(solutions)h(that)h(w)o(e)e (found)i(in)15 237 y(the)g(previous)g(sections.)21 b(W)l(e)16 b(will)f(pro)o(v)o(e)g(that)i(for)g Fn(c)c Fl(\025)h Fn(c)1090 219 y Fr(\003)1126 237 y Fo(the)i(solutions)h(are)f(stable.) 15 358 y(First,)k(w)o(e)g(substitute)g(all)f(the)h(rescalings)g(bac)o (k)g(in)g(the)g(tra)o(v)o(eling)e(w)o(a)o(v)o(e)h(solution)i Fn(\036)p Fo(\()p Fn(y)r Fo(\))e(that)i(w)o(e)15 418 y(found.)g(In)13 b(the)h(previous)g(section)f(w)o(e)h(in)o(tro)q(duced) f(the)h(rescalings)g Fn(\020)k Fo(=)13 b Fn("\030)k Fo(=)d Fn(")p Fo(\()p Fn(x)6 b Fl(\000)h Fo(~)-25 b Fn(ct)p Fo(\))13 b(and)i Fn(c)e Fo(=)h Fn(")q Fo(~)-25 b Fn(c)p Fo(.)15 478 y(So,)605 538 y Fn(\036)p Fo(\()p Fn(y)r Fo(\))13 b(=)h Fn(\036)p Fo(\()p Fn(")p Fo(\()p Fn(x)d Fl(\000)g Fo(~)-25 b Fn(ct)p Fo(\)\))14 b(=)g Fn(\036)p Fo(\()p Fn("x)c Fl(\000)h Fn(ct)p Fo(\))p Fn(:)489 b Fo(\(5.1\))15 625 y(Recall)16 b(that)i(w)o(e)g(also)g(ha)o(v)o(e)f (that)h(lim)732 632 y Fm(\020)r 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Fo(\(0\))g(=)g(0,)15 987 y Fl(N)7 b Fo(\(1\))17 b(=)g Fl(\000)p Fo(1,)h Fl(N)340 957 y Fh(0)354 987 y Fo(\()p Fn(\036)p Fo(\))e Fn(<)h Fo(0)h(for)h Fn(\036)d Fl(2)h Fo(\(0)p Fn(;)8 b Fo(1\))19 b(and)g Fl(N)978 957 y Fh(0)991 987 y Fo(\(1\))e Fn(<)g Fl(\000)p Fo(1)h(and)h(that)f(the)g (co)q(e\016cien)o(t)e(of)j(either)15 1047 y(the)h(quadratic)h(or)h (cubic)e(term)f(in)h(the)h(p)q(o)o(w)o(er)g(series)f(for)i Fl(N)28 b Fo(is)21 b(non-zero.)35 b(Since)20 b Fl(N)29 b Fo(is)20 b(a)i(non-)15 1107 y(linear)c(function)i(where)f Fl(N)540 1077 y Fh(0)553 1107 y Fo(\()p Fn(\036)p Fo(\))g Fn(<)g Fo(0)h(for)g(all)e(0)i Fn(<)f(\036)g(<)g Fo(1)h(w)o(e)f(can)h (\014nd)f(a)h(function)1655 1094 y(~)1644 1107 y Fn(F)26 b Fo(suc)o(h)19 b(that)15 1167 y Fl(N)63 1137 y Fh(0)76 1167 y Fo(\()p Fn(\036)p Fo(\))14 b(:=)f Fl(F)5 b Fo(\()p Fn(\036)p Fo(\))14 b(=)f Fl(\000)p Fn(\036)474 1155 y Fo(~)463 1167 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))16 b(and)690 1155 y(~)679 1167 y Fn(F)7 b Fo(\()p Fn(\036)p Fo(\))13 b Fn(>)h Fo(0)j(for)f(all)g(0)e Fn(<)g(\036)g(<)f Fo(1.)15 1288 y(As)21 b(explained)g(in)g(the)g(in)o(tro)q(duction)h(w)o(e)f (will)f(study)i(a)g(p)q(erturbation)g(of)g(these)f(tra)o(v)o(eling)g(w) o(a)o(v)o(e)15 1348 y(solutions:)658 1408 y Fn(B)s Fo(\()p Fn(x;)8 b(t)p Fo(\))13 b(=)g Fn(\036)p Fo(\()p Fn(\020)t Fo(\))e(+)g Fn(a)p Fo(\()p Fn(\020)t Fo(\))p Fn(w)q Fo(\()p Fn(\020)t(;)d(t)p Fo(\))p Fn(;)542 b Fo(\(5.2\))15 1495 y(where)16 b Fn(a)p Fo(\()p Fn(\020)t Fo(\))e(=)h Fn(e)335 1477 y Fr(\000)p Fm(\015)r(\020)419 1495 y Fo(and)i Fn(\015)g(>)e Fo(0)i(is)g(the)f(smallest)f(p)q(ositiv)o(e)h(ro)q(ot)i(of)f(1)11 b Fl(\000)h Fn(c\015)i Fo(+)d Fn(\015)1537 1477 y Fj(2)1568 1495 y Fl(\000)g Fn(")1641 1477 y Fj(4)1661 1495 y Fn(\015)1689 1477 y Fj(4)1723 1495 y Fo(=)k(0.)23 b(W)l(e)15 1555 y(will)17 b(assume)h(that)i Fn(w)g Fo(is)e(a)h(function)g(that)g(is)f (small.)28 b(Also,)18 b(w)o(e)g(kno)o(w)h(that)h Fn(w)f Fo(satis\014es)h(equation)15 1615 y(\(1.9\).)h(W)l(e)16 b(will)f(\014rst)i(sho)o(w)g(the)f(lo)q(cal)g(existence)e(of)j (solutions)g(to)f(\(1.9\).)15 1736 y(Recall)g(that)i(the)g(function)f (spaces)h(w)o(ere)f(de\014ned)h(as)g(follo)o(ws.)26 b(By)17 b Fn(H)1350 1718 y Fm(j)1385 1736 y Fo(=)f Fn(H)1483 1718 y Fm(j)1502 1736 y Fo(\()p Fk(R)1563 1718 y Fm(n)1586 1736 y Fo(\))i(for)g Fn(j)h Fl(2)d Fk(N)i Fo(w)o(e)15 1796 y(denote)c(the)f(usual)i(\(real\))e(Sob)q(olev)i(space)f(of)g (order)g Fn(j)j Fo(o)o(v)o(er)c Fk(R)1174 1778 y Fm(n)1197 1796 y Fo(,)h(with)g Fn(H)1378 1778 y Fj(0)1398 1796 y Fo(\()p Fk(R)1459 1778 y Fm(n)1483 1796 y Fo(\))f(=)h Fn(L)1600 1778 y Fj(2)1620 1796 y Fo(\()p Fk(R)1681 1778 y Fm(n)1704 1796 y Fo(\).)21 b(W)l(e)13 b(de-)15 1856 y(note)g(b)o(y)g Fn(H)229 1838 y Fm(j)225 1869 y(a)262 1856 y Fo(=)g Fn(H)357 1838 y Fm(j)353 1869 y(a)376 1856 y Fo(\()p Fk(R)437 1838 y Fm(n)460 1856 y Fo(\))h(the)f(w)o(eigh)o(ted) f(Sob)q(olev)i(space)g(de\014ned)f(b)o(y)g(the)g(norm)g Fl(jj)p Fn(w)q Fl(jj)1609 1872 y Fm(H)1641 1856 y Fe(j)1638 1878 y(a)1672 1856 y Fo(=)h Fl(jj)p Fn(aw)q Fl(jj)1842 1865 y Fm(H)1874 1856 y Fe(j)t Fo(.)15 1922 y(W)l(e)e(also)i(set)e Fn(H)306 1904 y Fj(0)302 1934 y Fm(a)326 1922 y Fo(\()p Fk(R)387 1904 y Fm(n)410 1922 y Fo(\))i(=)g Fn(L)528 1904 y Fj(2)528 1934 y Fm(a)549 1922 y Fo(\()p Fk(R)610 1904 y Fm(n)633 1922 y Fo(\).)20 b(Finally)l(,)12 b(w)o(e)g(write)g Fn(X)1092 1904 y Fm(j)1124 1922 y Fo(for)h(the)f(in)o(tersection)f Fn(H)1577 1904 y Fm(j)1600 1922 y Fl(\\)t Fn(H)1681 1904 y Fm(j)1677 1934 y(a)1712 1922 y Fo(equipp)q(ed)15 1982 y(with)16 b(the)g(norm)f Fl(jj)p Fn(w)q Fl(jj)429 1964 y Fj(2)429 1997 y Fm(X)461 1987 y Fe(j)493 1982 y Fo(=)e Fl(jj)p Fn(w)q Fl(jj)636 1964 y Fj(2)636 1997 y Fm(H)668 1987 y Fe(j)697 1982 y Fo(+)e Fl(jj)p Fn(w)q Fl(jj)838 1964 y Fj(2)838 2003 y Fm(H)870 1987 y Fe(j)867 2009 y(a)887 1982 y Fo(.)15 2103 y(The)16 b(main)f(results)h(of)h(this)f (section)f(can)i(b)q(e)f(summarised)e(in)i(the)g(follo)o(wing)g (theorem)15 2204 y Fk(Theorem)g(5.1)24 b Ff(L)n(et)16 b Fl(N)23 b Ff(satisfy)15 b(the)h(ab)n(ove)h(assumptions.)22 b(Ther)n(e)15 b(exists)i(an)f Fn(")1512 2211 y Fj(0)1545 2204 y Fn(>)e Fo(0)i Ff(such)g(that)g(for)15 2264 y Fn(")d(<)h(")126 2271 y Fj(0)161 2264 y Ff(and)i Fn(c)d Fl(\025)h Fn(c)362 2246 y Fr(\003)382 2264 y Ff(,)h(ther)n(e)h(exists)g(a)f Fn(\016)g(>)f Fo(0)i Ff(such)f(that)h(if)f(we)h(cho)n(ose)g Fn( )f Fl(2)f Fn(X)1444 2246 y Fj(5)1479 2264 y Ff(with)i Fl(jj)p Fn( )r Fl(jj)1673 2273 y Fm(X)1705 2263 y Fi(3)1737 2264 y Fn(<)e(\016)i Ff(and)15 2325 y(let)f Fn(w)f Ff(b)n(e)g(the)h(c)n (orr)n(esp)n(onding)e(solution)h Fn(w)h Fl(2)f Fn(C)883 2307 y Fj(0)902 2325 y Fo(\([0)p Fn(;)8 b(T)f Fo(])p Fn(;)h(X)1097 2307 y Fj(5)1116 2325 y Fo(\))s Fl(\\)s Fn(C)1213 2307 y Fj(1)1232 2325 y Fo(\(\(0)p Fn(;)g(T)f Fo(\))p Fn(;)h(X)1437 2307 y Fj(5)1456 2325 y Fo(\))14 b Ff(such)g(that)g Fn(w)q Fo(\(0\))g(=)g Fn( )r Ff(,)15 2385 y(then)652 2445 y Fo(lim)644 2472 y Fm(t)p Fr(!1)735 2445 y Fl(jj)p Fn(B)s Fo(\()p Fl(\001)p Fn(;)8 b(t)p Fo(\))i Fl(\000)h Fn(\036)p Fo(\()p Fn(")g Fl(\001)f(\000)p Fn(ct)p Fo(\))p Fl(jj)j Fo(=)h(0)15 2542 y Ff(i.e.)22 b(the)c(solution)h Fn(\036)e Ff(is)g(stable.)935 2800 y Fo(12)p eop %%Page: 13 13 13 12 bop 15 68 a Fd(5.1)66 b(Existence)23 b(of)f(solutions)g(to)h (equation)g(\(1.9\).)15 160 y Fo(In)c(this)g(section,)g(w)o(e)g(pro)o (v)o(e)f(lo)q(cal)h(existence)f(and)i(uniqueness)e(of)i(solutions)g(to) f(\(1.9\).)31 b(This)19 b(is)g(a)15 220 y(standard)e(result)f(and)h(w)o (e)f(include)f(it)g(only)h(for)h(completeness.)15 310 y Fk(Prop)r(osition)h(5.2)24 b Ff(Ther)n(e)18 b(exists)h(an)g Fn(")783 317 y Fj(0)819 310 y Fn(>)d Fo(0)k Ff(such)f(that)g(if)f(we)i (\014x)f Fn(")d(<)g(")1436 317 y Fj(0)1475 310 y Ff(and)j(cho)n(ose)f Fn( )g Fl(2)e Fn(X)1869 292 y Fj(5)1890 310 y Ff(,)15 370 y(then)24 b(ther)n(e)f(exists)h(a)f(time)h Fn(T)30 b Fo(=)25 b Fn(T)7 b Fo(\()p Fn(";)h( )r Fo(\))23 b Fn(>)h Fo(0)f Ff(such)h(that)f(\(1.9\))g(has)g(a)g(unique)i(solution)f Fn(w)h Fl(2)15 430 y Fn(C)54 412 y Fj(0)73 430 y Fo(\([0)p Fn(;)8 b(T)f Fo(])p Fn(;)h(X)268 412 y Fj(5)287 430 y Fo(\))j Fl(\\)g Fn(C)400 412 y Fj(1)419 430 y Fo(\(\(0)p Fn(;)d(T)f Fo(\))p Fn(;)h(X)624 412 y Fj(5)644 430 y Fo(\))p Ff(.)15 520 y Fk(Pro)r(of:)15 b Fo(W)l(e)h(rewrite)g(\(1.9\))g (as)774 580 y Fn(w)809 587 y Fm(t)838 580 y Fo(=)d Fn(Lw)g Fo(+)e Fn(M)5 b Fo(\()p Fn(w)q Fo(\))15 662 y(where)16 b Fn(L)g Fo(is)g(the)g(linear)g(op)q(erator)263 758 y Fn(L)e Fo(=)f(\(1)f Fl(\000)f Fo(6)p Fn(")513 737 y Fj(4)533 758 y Fn(\015)561 737 y Fj(2)580 758 y Fo(\))p Fn(@)628 737 y Fj(2)625 770 y Fm(\020)658 758 y Fo(+)g(\()p Fn(c)h Fl(\000)e Fo(2)p Fn(\015)15 b Fo(+)c(4)p Fn(")968 737 y Fj(4)988 758 y Fn(\015)1016 737 y Fj(3)1035 758 y Fo(\))p Fn(@)1080 765 y Fm(\020)1111 758 y Fo(+)g(4)p Fn(")1207 737 y Fj(4)1227 758 y Fn(\015)s(@)1284 737 y Fj(3)1281 770 y Fm(\020)1314 758 y Fl(\000)g Fn(")1387 737 y Fj(4)1407 758 y Fn(@)1436 737 y Fj(4)1433 770 y Fm(\020)1466 758 y Fo(+)g Fl(N)1563 725 y Fh(0)1576 758 y Fo(\()p Fn(\036)p Fo(\))p Fn(;)15 854 y Fo(and)k Fn(M)5 b Fo(\()p Fn(w)q Fo(\))14 b(=)g Fn(aw)362 836 y Fj(2)382 854 y Fn(R)p Fo(.)21 b(The)14 b(op)q(erator)i Fn(L)f Fo(is)f(a)h(b)q(ounded)g(p)q (erturbation)g(of)g(the)f(constan)o(t)h(co)q(e\016cien)o(t)15 914 y(op)q(erator)347 974 y Fn(L)380 981 y Fj(0)414 974 y Fo(=)e(\(1)f Fl(\000)f Fo(6)p Fn(")617 954 y Fj(4)637 974 y Fn(\015)665 954 y Fj(2)684 974 y Fo(\))p Fn(@)732 954 y Fj(2)729 986 y Fm(\020)763 974 y Fo(+)g(\()p Fn(c)g Fl(\000)f Fo(2)p Fn(\015)15 b Fo(+)c(4)p Fn(")1072 954 y Fj(4)1092 974 y Fn(\015)1120 954 y Fj(3)1139 974 y Fo(\))p Fn(@)1184 981 y Fm(\020)1215 974 y Fo(+)g(4)p Fn(")1311 954 y Fj(4)1331 974 y Fn(\015)s(@)1388 954 y Fj(3)1385 986 y Fm(\020)1418 974 y Fl(\000)g Fn(")1491 954 y Fj(4)1511 974 y Fn(@)1540 954 y Fj(4)1537 986 y Fm(\020)1559 974 y Fn(:)15 1055 y(L)48 1062 y Fj(0)81 1055 y Fo(is)i(the)f(generator)i(of)f(an)h(analytic)e(semigroup)g(of)i (op)q(erators)g(on)g Fn(X)1324 1037 y Fj(5)1357 1055 y Fo(as)f(one)h(can)f(v)o(erify)e(b)o(y)i(direct)15 1116 y(computation,)g(and)h(hence)e(b)o(y)h(Theorem)f(3.2.1)i(of)g([24],)f Fn(L)h Fo(also)g(generates)f(an)h(analytic)f(semigroup.)15 1236 y(It)22 b(is)g(also)h(easy)g(to)g(c)o(hec)o(k)d(that)j Fn(M)30 b Fo(:)24 b Fn(X)821 1218 y Fj(5)866 1236 y Fl(!)g Fn(X)984 1218 y Fj(5)1027 1236 y Fo(is)e(Lipsc)o(hitz-con)o(tin)o (uous.)39 b(Th)o(us,)24 b(standard)15 1296 y(results)16 b(in)g(semigroup)f(theory)h(imply)e(that)j(there)e(exists)h(a)h Fn(T)22 b Fo(suc)o(h)16 b(that,)575 1414 y Fn(w)q Fo(\()p Fn(t)p Fo(\))e(=)g Fn(e)756 1393 y Fm(Lt)794 1414 y Fn( )f Fo(+)888 1355 y Fg(Z)929 1369 y Fm(T)911 1450 y Fj(0)965 1414 y Fn(e)988 1393 y Fm(L)p Fj(\()p Fm(t)p Fr(\000)p Fm(\034)t Fj(\))1101 1414 y Fn(M)5 b Fo(\()p Fn(w)q Fo(\()p Fn(\034)h Fo(\)\))p Fn(d\034)15 1529 y Fo(has)17 b(an)f(unique)g (solution)h Fn(w)e Fl(2)f Fn(C)649 1511 y Fj(0)668 1529 y Fo(\([0)p Fn(;)8 b(T)f Fo(])p Fn(;)h(X)863 1511 y Fj(5)881 1529 y Fo(\))j Fl(\\)h Fn(C)995 1511 y Fj(1)1014 1529 y Fo(\(\(0)p Fn(;)c(T)f Fo(\))p Fn(;)h(X)1219 1511 y Fj(5)1238 1529 y Fo(\).)596 b Fc(2)15 1732 y Fd(5.2)66 b(The)22 b(energy)h(functionals.)15 1824 y Fo(T)l(o)h(pro)o(v)o(e)e (stabilit)o(y)g(w)o(e)h(use)h(the)f(three)g(functionals)g Fn(E)s(;)8 b(F)30 b Fo(and)24 b Fn(F)1335 1831 y Fj(1)1377 1824 y Fo(de\014ned)f(in)g(\(1.10\),)j(\(1.11\))15 1884 y(and)c(\(1.12\).)39 b(W)l(e)21 b(will)g(sho)o(w)i(that)f Fn(E)j Fo(and)d Fn(F)910 1891 y Fj(1)951 1884 y Fo(are)g (non-increasing)h(functions)f(of)g(time)d(and)k(as)15 1945 y(explained)17 b(in)g(the)h(in)o(tro)q(duction,)f(this)h(implies)d (that)k(the)e(solution)h(is)g(stable.)26 b(These)18 b(results)f(can)15 2005 y(b)q(e)f(summarised)e(in)i(the)g(follo)o(wing)g(theorems)15 2095 y Fk(Theorem)g(5.3)24 b Ff(L)n(et)16 b Fl(N)23 b Ff(satisfy)15 b(the)h(assumptions)g(ab)n(ove.)23 b(Ther)n(e)15 b(exists)i(an)f Fn(")1512 2102 y Fj(0)1545 2095 y Fn(>)e Fo(0)i Ff(such)g(that)g(for)15 2155 y Fn(")d(<)h(")126 2162 y Fj(0)163 2155 y Ff(and)k Fn(c)c Fl(\025)f Fn(c)366 2137 y Fr(\003)386 2155 y Ff(,)k(ther)n(e)h(exists)g(a)f Fn(\016)f(>)d Fo(0)18 b Ff(such)g(that)g(if)f Fn(w)e Fl(2)f Fn(X)1245 2137 y Fj(5)1283 2155 y Ff(and)843 2251 y Fl(jj)p Fn(w)q Fl(jj)935 2259 y Fm(X)967 2250 y Fi(3)1000 2251 y Fl(\024)f Fn(\016)730 b Fo(\(5.3\))15 2347 y Ff(for)17 b Fn(t)c Fo(=)h(0)p Ff(,)k(then)g(for)f(al)r(l)i Fn(t)13 b Fl(\025)h Fo(0)881 2395 y Fn(@)s(E)p 881 2417 68 2 v 891 2463 a(@)s(t)967 2429 y Fl(\024)f Fo(0)761 b(\(5.4\))15 2523 y Ff(and)17 b(ther)n(e)h(exists)g(a)f(p)n(ositive)h(c)n(onstant)g Fn(d)g Ff(such)g(that)808 2607 y Fn(@)s(F)869 2614 y Fj(1)p 808 2629 80 2 v 824 2675 a Fn(@)s(t)906 2641 y Fl(\024)c(\000)p Fn(dF)7 b Fo(\()p Fn(t)p Fo(\))686 b(\(5.5\))935 2800 y(13)p eop %%Page: 14 14 14 13 bop 15 68 a Fo(Because)12 b(of)h(the)f(fact)h(that)g(the)f (inequalities)f(\(5.4\))i(and)h(\(5.5\))f(imply)d(that)j Fn(E)7 b Fo(+)t Fn(F)1528 75 y Fj(1)1560 68 y Fo(is)12 b(monotonically)15 128 y(decreasing)20 b(as)i(a)f(function)f(of)h Fn(t)p Fo(,)g(the)f(b)q(ound)i(\(5.3\))f(holds)g(for)g(all)f(times)f Fn(t)i(>)h Fo(0.)34 b(The)21 b(theorem)15 188 y(is)g(pro)o(v)o(ed)h(b)o (y)f(explicitly)e(calculating)781 168 y Fm(@)r(E)p 781 176 49 2 v 789 205 a(@)r(t)856 188 y Fo(and)962 168 y Fm(@)r(F)p 962 176 48 2 v 969 205 a(@)r(t)1036 188 y Fo(and)k(using)f(the)g(fact)g(that)1578 168 y Fm(@)r(w)p 1578 176 47 2 v 1585 205 a(@)r(t)1651 188 y Fo(satis\014es)h(the)15 248 y(equation)17 b(\(1.9\).)23 b(Then)17 b(all)g(the)g(di\013eren)o(t) f(in)o(tegrals)g(can)h(b)q(e)g(separately)g(appro)o(ximated.)22 b(W)l(e)17 b(will)15 308 y(giv)o(e)d(the)g(pro)q(of)i(of)f(this)g (theorem)e(in)i(the)f(next)h(subsection.)20 b(Note)15 b(that)g(from)f(\(5.3\))h(it)f(follo)o(ws)h(that)15 369 y Fl(jj)p Fn(w)q Fl(jj)107 376 y Fm(L)131 367 y Fh(1)180 369 y Fl(\024)g Fn(\016)r Fo(,)i Fl(jj)p Fn(aw)q Fl(jj)407 376 y Fm(L)431 367 y Fh(1)480 369 y Fl(\024)e Fn(\016)k Fo(and)f Fl(jj)p Fo(\()p Fn(aw)q Fo(\))799 376 y Fm(\020)818 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Fm(\020)r(\020)r(\020)r(\020)454 2579 y Fo(+)p Fn(w)q Fl(F)5 b Fo(\()p Fn(\036)p Fo(\))11 b(+)g Fn(aw)758 2558 y Fj(2)778 2579 y Fn(R)p Fo(\))375 2666 y(=)41 b Fl(\000)p Fo(\(1)12 b Fl(\000)e Fo(6)p Fn(\015)649 2646 y Fj(2)670 2666 y Fn(")693 2646 y Fj(4)712 2666 y Fo(\))739 2608 y Fg(Z)789 2666 y Fn(w)825 2646 y Fj(2)824 2679 y Fm(\020)856 2666 y Fl(\000)h Fn(")929 2646 y Fj(4)957 2608 y Fg(Z)1007 2666 y Fn(w)1043 2646 y Fj(2)1042 2679 y Fm(\020)r(\020)1090 2666 y Fl(\000)1140 2608 y Fg(Z)1190 2666 y Fn(\036)1230 2654 y Fo(~)1219 2666 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))p Fn(w)1360 2646 y Fj(2)1391 2666 y Fo(+)1440 2608 y Fg(Z)1490 2666 y Fn(aw)1552 2646 y Fj(3)1571 2666 y Fn(R;)935 2800 y Fo(14)p eop %%Page: 15 15 15 14 bop 15 68 a Fo(where)15 b(w)o(e)g(used)g(\(1.9\).)21 b(W)l(e)15 b(in)o(tegrated)g(b)o(y)g(parts)h(and)g(used)f(that)h Fn(w)h Fo(is)e(a)h(smo)q(oth)f(function.)21 b(Here)15 128 y(and)g(from)f(no)o(w)h(on)h(w)o(e)e(denote)651 92 y Fg(R)679 105 y Fr(1)670 140 y(\0001)756 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Fn(\036)p Fo(\()p Fn(\020)t Fo(\))f Fl(\025)g Fn(\013a)p Fo(\()p Fn(\020)t Fo(\))h(for)h(some)f Fn(\013)f(>)g Fo(0)i(and)g Fl(jj)p Fn(w)q Fl(jj)846 1220 y Fm(L)870 1211 y Fh(1)920 1213 y Fl(\024)e Fn(\016)r Fo(.)25 b(Therefore)17 b(\(1)c Fl(\000)1374 1193 y Fm(\016)p 1371 1201 23 2 v 1371 1230 a(\013)1399 1213 y Fo(\))p Fn(\036)j Fl(\024)g Fn(\036)c Fo(+)f Fn(\021)18 b Fl(\024)e Fo(\(1)d(+)1818 1193 y Fm(\016)p 1815 1201 V 1815 1230 a(\013)1843 1213 y Fo(\))p Fn(\036)p Fo(.)15 1273 y(No)o(w)j(consider) 687 1354 y Fl(j)739 1321 y Fn(R)p 706 1343 106 2 v 716 1381 a Fo(~)706 1393 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))816 1354 y Fl(j)13 b Fo(=)900 1321 y(1)p 900 1343 25 2 v 900 1389 a(2)929 1354 y Fl(j)948 1321 y Fn(\036)p Fl(N)1025 1291 y Fh(00)1048 1321 y Fo(\()p Fn(\036)e Fo(+)g Fn(\021)r Fo(\))p 948 1343 253 2 v 1010 1389 a Fl(N)1058 1365 y Fh(0)1072 1389 y Fo(\()p Fn(\036)p Fo(\))1205 1354 y Fl(j)p Fn(:)15 1481 y Fo(Since)17 b Fl(N)7 b Fo(\()p Fn(\036)p Fo(\))17 b(is)h(analytic)f(and)h Fl(N)658 1451 y Fh(0)671 1481 y Fo(\()p Fn(\036)p Fo(\))e Fn(<)h Fo(0)h(for)g(0)e Fn(<)g(\036)h(<)f Fo(1,)i(there)f(exists)g(a)h(p)q(ositiv)o(e)f (constan)o(t)27 b(~)-33 b Fn(c)1882 1488 y Fm(f)15 1541 y Fo(suc)o(h)16 b(that)g Fl(j)274 1521 y Fm(R)p 249 1529 76 2 v 257 1553 a Fj(~)249 1562 y Fm(F)6 b Fj(\()p Fm(\036)p Fj(\))330 1541 y Fl(j)14 b(\024)23 b Fo(~)-34 b Fn(c)431 1548 y Fm(f)470 1541 y Fo(for)17 b Fn(\020)h Fl(\025)13 b Fo(0.)22 b(This)16 b(giv)o(es)g(that)598 1633 y Fg(Z)622 1728 y Fm(\020)r Fr(\025)p Fj(0)695 1692 y Fl(j)p Fn(Raw)808 1672 y Fj(3)828 1692 y Fl(j)d(\024)913 1658 y Fn(\016)e Fo(~)-33 b Fn(c)958 1665 y Fm(f)p 913 1680 68 2 v 931 1726 a Fn(\013)993 1633 y Fg(Z)1016 1728 y Fm(\020)r Fr(\025)p Fj(0)1090 1692 y Fl(j)1115 1679 y Fo(~)1104 1692 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fn(\036w)1288 1672 y Fj(2)1307 1692 y Fn(;)15 1826 y Fo(where)21 b(w)o(e)h(again)g (use)g(that)g Fn(\036)p Fo(\()p Fn(\020)t Fo(\))i Fl(\025)f Fn(\013a)p Fo(\()p Fn(\020)t Fo(\))e(for)h Fn(\020)28 b Fl(\025)23 b Fo(0)f(and)g Fl(jj)p Fn(w)q Fl(jj)1323 1833 y Fm(L)1347 1824 y Fh(1)1405 1826 y Fl(\024)h Fn(\016)r Fo(.)37 b(Com)o(bining)21 b(these)15 1886 y(estimates)14 b(giv)o(es)i(us)g(the)g(b)q(ound)i(as)f(in)f(the)g(Lemma.)846 b Fc(2)15 2006 y Fo(No)o(w)16 b(w)o(e)g(di\013eren)o(tiate)474 1987 y Fj(1)p 474 1995 18 2 v 474 2023 a(2)505 1971 y Fg(R)541 2006 y Fn(w)577 1988 y Fj(2)576 2018 y Fm(\020)613 2006 y Fo(with)g(resp)q(ect)g(to)g Fn(t)259 2122 y Fo(1)p 259 2144 25 2 v 259 2190 a(2)293 2118 y Fn(@)s Fo(\()341 2083 y Fg(R)376 2118 y Fn(w)412 2100 y Fj(2)411 2131 y Fm(\020)432 2118 y Fo(\))p 293 2144 159 2 v 349 2190 a Fn(@)s(t)470 2155 y Fo(=)e Fl(\000)569 2097 y Fg(Z)619 2155 y Fn(w)654 2162 y Fm(\020)r(\020)691 2155 y Fn(w)726 2162 y Fm(t)755 2155 y Fo(=)g Fl(\000)p Fo(\(1)d Fl(\000)g Fo(6)p Fn(\015)1002 2135 y Fj(2)1022 2155 y Fn(")1045 2135 y Fj(4)1064 2155 y Fo(\))1091 2097 y Fg(Z)1141 2155 y Fn(w)1177 2135 y Fj(2)1176 2168 y Fm(\020)r(\020)1225 2155 y Fl(\000)g Fn(")1298 2135 y Fj(4)1317 2155 y Fn(w)1353 2135 y Fj(2)1352 2168 y Fm(\020)r(\020)r(\020)1419 2155 y Fl(\000)1468 2097 y Fg(Z)1518 2155 y Fn(w)1553 2162 y Fm(\020)r(\020)1591 2155 y Fn(w)q(g)r(:)15 2281 y Fo(W)l(e)i(study) 228 2245 y Fg(R)264 2281 y Fn(w)299 2288 y Fm(\020)r(\020)337 2281 y Fn(w)q(g)j Fo(separately)l(.)k(Using)14 b(the)f(remark)f(follo)o (wing)i(equation)g(\(1.9\))g(it)f(can)h(b)q(e)g(written)15 2341 y(as)319 2393 y Fg(Z)368 2451 y Fn(w)403 2458 y Fm(\020)r(\020)441 2451 y Fn(w)q(g)44 b Fo(=)623 2393 y Fg(Z)673 2451 y Fl(F)5 b Fo(\()p Fn(\036)p Fo(\))p Fn(w)q(w)852 2458 y Fm(\020)r(\020)900 2451 y Fo(+)949 2393 y Fg(Z)999 2451 y Fn(aw)1061 2431 y Fj(2)1081 2451 y Fn(w)1116 2458 y Fm(\020)r(\020)1153 2451 y Fn(R)544 2567 y Fo(=)41 b Fl(\000)670 2509 y Fg(Z)720 2567 y Fl(F)5 b Fo(\()p Fn(\036)p Fo(\))p Fn(w)864 2547 y Fj(2)863 2580 y Fm(\020)895 2567 y Fo(+)949 2533 y(1)p 949 2556 25 2 v 949 2601 a(2)986 2509 y Fg(Z)1036 2567 y Fn(@)1065 2547 y Fj(2)1062 2580 y Fm(\020)1084 2567 y Fo(\()p Fl(F)g Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)1266 2547 y Fj(2)1297 2567 y Fo(+)1346 2509 y Fg(Z)1396 2567 y Fn(aw)1458 2547 y Fj(2)1477 2567 y Fn(w)1512 2574 y Fm(\020)r(\020)1550 2567 y Fn(R:)935 2800 y Fo(15)p eop %%Page: 16 16 16 15 bop 15 68 a Fo(Applying)16 b(the)h(Cauc)o(h)o(y-Sc)o(h)o(w)o(arz) f(inequalit)o(y)l(,)f(the)i(inequalit)o(y)f Fl(j)p Fn(ab)p Fl(j)e(\024)1375 48 y Fj(1)p 1375 56 18 2 v 1375 85 a(2)1397 68 y Fn(a)1423 50 y Fj(2)1454 68 y Fo(+)1509 48 y Fj(1)p 1509 56 V 1509 85 a(2)1532 68 y Fn(b)1553 50 y Fj(2)1589 68 y Fo(and)k(the)f(b)q(ound)15 128 y(for)f Fn(R)h Fo(w)o(e)f(obtain) 192 248 y Fl(j)214 189 y Fg(Z)264 248 y Fn(aw)326 227 y Fj(2)346 248 y Fn(w)381 255 y Fm(\020)r(\020)418 248 y Fn(R)p Fl(j)e(\024)541 214 y Fo(1)p 541 236 25 2 v 541 282 a(2)570 248 y(\()589 189 y Fg(Z)639 248 y Fl(j)p Fn(R)p Fl(j)p Fn(a)730 227 y Fj(2)750 248 y Fn(w)786 227 y Fj(4)817 248 y Fo(+)d Fn(K)907 255 y Fm(R)944 189 y Fg(Z)994 248 y Fn(w)1030 227 y Fj(2)1029 260 y Fm(\020)r(\020)1067 248 y Fo(\))i Fl(\024)1157 214 y Fo(1)p 1157 236 V 1157 282 a(2)1186 248 y(\()p Fn(\016)1237 189 y Fg(Z)1287 248 y Fl(j)p Fn(R)p Fl(j)p Fn(aw)1414 227 y Fj(3)1445 248 y Fo(+)e Fn(K)1535 255 y Fm(R)1572 189 y Fg(Z)1622 248 y Fn(w)1658 227 y Fj(2)1657 260 y Fm(\020)r(\020)1695 248 y Fo(\))p Fn(:)15 370 y Fo(The)16 b(\014rst)h(in)o(tegral)e(can)h (b)q(e)h(estimated)d(b)o(y)i(Lemma)e(5.6)j(and)g(th)o(us)110 441 y Fg(Z)160 500 y Fn(w)195 507 y Fm(\020)r(\020)232 500 y Fn(w)q(g)f Fl(\024)e(\000)407 441 y Fg(Z)457 500 y Fl(F)5 b Fo(\()p Fn(\036)p Fo(\))p Fn(w)601 479 y Fj(2)600 512 y Fm(\020)631 500 y Fo(+)685 466 y(1)p 685 488 V 685 534 a(2)723 441 y Fg(Z)773 500 y Fn(@)802 479 y Fj(2)799 512 y Fm(\020)821 500 y Fo(\()p Fl(F)g Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)1003 479 y Fj(2)1033 500 y Fo(+)1087 466 y(1)p 1087 488 V 1087 534 a(2)1117 500 y(\()p Fn(\016)1160 479 y Fj(2)1179 500 y Fn(K)1220 507 y Fj(1)1248 441 y Fg(Z)1298 500 y Fl(j)1323 487 y Fo(~)1312 500 y Fn(F)h Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fn(\036w)1496 479 y Fj(2)1527 500 y Fo(+)11 b Fn(K)1617 507 y Fm(R)1655 441 y Fg(Z)1704 500 y Fn(w)1740 479 y Fj(2)1739 512 y Fm(\020)r(\020)1777 500 y Fo(\))p Fn(:)15 730 y Fo(In)16 b(lik)o(e)e(fashion)228 827 y(1)p 228 849 V 228 895 a(2)262 824 y Fn(@)s Fo(\()310 788 y Fg(R)345 824 y Fn(w)381 805 y Fj(2)380 836 y Fm(\020)r(\020)418 824 y Fo(\))p 262 849 175 2 v 326 895 a Fn(@)s(t)456 861 y Fo(=)508 802 y Fg(Z)557 861 y Fn(w)592 868 y Fm(\020)r(\020)r (\020)r(\020)665 861 y Fn(w)700 868 y Fm(t)729 861 y Fo(=)g Fl(\000)p Fo(\(1)d Fl(\000)g Fo(6)p Fn(\015)976 840 y Fj(2)996 861 y Fn(")1019 840 y Fj(4)1038 861 y Fo(\))1065 802 y Fg(Z)1115 861 y Fn(w)1151 840 y Fj(2)1150 873 y Fm(\020)r(\020)r(\020)1217 861 y Fl(\000)g Fn(")1290 840 y Fj(4)1309 861 y Fn(w)1345 840 y Fj(2)1344 873 y Fm(\020)r(\020)r(\020)r(\020)1428 861 y Fo(+)1477 802 y Fg(Z)1527 861 y Fn(w)1562 868 y Fm(\020)r(\020)r(\020)r(\020)1635 861 y Fn(w)q(g)15 983 y Fo(and)17 b(w)o(e)e(\014nd)i(that)129 1044 y Fg(Z)179 1103 y Fn(w)214 1110 y Fm(\020)r(\020)r(\020)r(\020)287 1103 y Fn(w)q(g)g Fo(=)414 1044 y Fg(Z)464 1103 y Fl(F)5 b Fo(\()p Fn(\036)p Fo(\))p Fn(w)608 1082 y Fj(2)607 1115 y Fm(\020)r(\020)655 1103 y Fl(\000)11 b Fo(2)737 1044 y Fg(Z)788 1103 y Fn(@)817 1082 y Fj(2)814 1115 y Fm(\020)836 1103 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)1018 1082 y Fj(2)1017 1115 y Fm(\020)1048 1103 y Fo(+)1102 1069 y(1)p 1102 1091 25 2 v 1102 1137 a(2)1140 1044 y Fg(Z)1190 1103 y Fn(@)1219 1082 y Fj(4)1216 1115 y Fm(\020)1238 1103 y Fo(\()p Fl(F)g Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)1420 1082 y Fj(2)1450 1103 y Fo(+)1499 1044 y Fg(Z)1549 1103 y Fn(w)1584 1110 y Fm(\020)r(\020)r(\020)r(\020) 1657 1103 y Fn(aw)1719 1082 y Fj(2)1739 1103 y Fn(R;)15 1226 y Fo(where)16 b(in)g(a)g(similar)e(w)o(a)o(y)i(as)h(ab)q(o)o(v)o (e)365 1348 y Fl(j)387 1289 y Fg(Z)436 1348 y Fn(w)471 1355 y Fm(\020)r(\020)r(\020)r(\020)545 1348 y Fn(aw)607 1327 y Fj(2)626 1348 y Fn(R)p Fl(j)d(\024)744 1289 y Fg(Z)794 1348 y Fl(j)p Fn(w)843 1355 y Fm(\020)r(\020)r(\020)898 1348 y Fl(j)p Fo(\()p Fl(j)p Fn(R)982 1355 y Fm(\020)1013 1348 y Fl(\000)c Fn(\015)s(R)p Fl(j)p Fn(aw)1203 1327 y Fj(2)1234 1348 y Fo(+)h(2)p Fn(aK)1374 1355 y Fm(R)1404 1348 y Fl(j)p Fn(w)q(w)1489 1355 y Fm(\020)1508 1348 y Fl(j)p Fo(\))p Fn(:)15 1473 y Fo(These)17 b(t)o(w)o(o)g(in)o(tegrals) g(can)h(b)q(e)f(b)q(ounded)i(b)o(y)e(using)g(the)g(Cauc)o(h)o(y-Sc)o(h) o(w)o(arz-inequalit)o(y)e(and)j Fl(j)p Fn(ab)p Fl(j)d(\024)20 1514 y Fj(1)p 20 1522 18 2 v 20 1550 a(2)42 1533 y Fo(\()p Fn(a)87 1515 y Fj(2)118 1533 y Fo(+)c Fn(b)188 1515 y Fj(2)207 1533 y Fo(\))233 1612 y Fg(Z)282 1670 y Fl(j)p Fn(R)333 1677 y Fm(\020)364 1670 y Fl(\000)g Fn(\015)s(R)p Fl(j)p Fn(aw)555 1650 y Fj(2)575 1670 y Fl(j)p Fn(w)624 1677 y Fm(\020)r(\020)r(\020)679 1670 y Fl(j)41 b(\024)820 1636 y Fo(1)p 820 1659 25 2 v 820 1704 a(2)857 1612 y Fg(Z)907 1670 y Fl(j)p Fn(R)958 1677 y Fm(\020)989 1670 y Fl(\000)11 b Fn(\015)s(R)p Fl(jj)p Fn(aw)1194 1650 y Fj(3)1214 1670 y Fl(j)g Fo(+)1293 1636 y(1)p 1293 1659 V 1293 1704 a(2)1335 1658 y(~)1322 1670 y Fn(K)1363 1677 y Fm(R)1401 1612 y Fg(Z)1450 1670 y Fn(a)p Fl(j)p Fn(w)q Fl(j)p Fn(w)1576 1650 y Fj(2)1575 1682 y Fm(\020)r(\020)r(\020)815 1787 y Fl(\024)872 1753 y Fo(1)p 872 1775 V 872 1821 a(2)910 1728 y Fg(Z)951 1787 y Fo(\()p Fl(j)p Fn(R)1021 1794 y Fm(\020)1041 1787 y Fl(j)g Fo(+)g Fn(\015)s Fl(j)p Fn(R)p Fl(j)p Fo(\))p Fn(a)p Fl(j)p Fn(w)1303 1766 y Fj(3)1323 1787 y Fl(j)g Fo(+)1402 1753 y(1)p 1402 1775 V 1402 1821 a(2)1431 1787 y Fn(\016)1467 1774 y Fo(~)1455 1787 y Fn(K)1496 1794 y Fm(R)1533 1728 y Fg(Z)1583 1787 y Fn(w)1619 1766 y Fj(2)1618 1799 y Fm(\020)r(\020)r(\020)1673 1787 y Fn(:)15 1909 y Fo(W)l(e)16 b(b)q(ound)h(the)f(\014rst)h(in)o (tegral)e(b)o(y)510 2005 y(1)p 510 2027 V 510 2073 a(2)547 1980 y Fg(Z)589 2039 y Fo(\()p Fl(j)p Fn(R)659 2046 y Fm(\020)679 2039 y Fl(j)c Fo(+)g Fn(\015)s Fl(j)p Fn(R)p Fl(j)p Fo(\))p Fn(a)p Fl(j)p Fn(w)941 2018 y Fj(3)960 2039 y Fl(j)j(\024)1054 2026 y Fo(~)1040 2039 y Fn(K)1081 2046 y Fj(1)1102 2039 y Fn(\016)1133 1980 y Fg(Z)1183 2039 y Fl(j)1208 2026 y Fo(~)1197 2039 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fn(\036w)1381 2018 y Fj(2)1401 2039 y Fn(;)389 b Fo(\(5.6\))15 2162 y(using)16 b(a)h(similar)d (argumen)o(t)h(as)i(in)f(obtaining)h(the)f(inequalit)o(y)e(in)i(Lemma)e (5.6.)22 b(And)258 2233 y Fg(Z)308 2291 y Fn(a)p Fl(j)p Fn(w)q(w)419 2298 y Fm(\020)439 2291 y Fn(w)474 2298 y Fm(\020)r(\020)r(\020)529 2291 y Fl(j)14 b(\024)614 2257 y Fo(1)p 614 2279 V 614 2325 a(2)652 2233 y Fg(Z)702 2291 y Fn(a)p Fl(j)p Fn(w)q Fl(j)p Fn(w)828 2271 y Fj(2)827 2303 y Fm(\020)858 2291 y Fo(+)912 2257 y(1)p 912 2279 V 912 2325 a(2)950 2233 y Fg(Z)1000 2291 y Fn(a)p Fl(j)p Fn(w)q Fl(j)p Fn(w)1126 2271 y Fj(2)1125 2303 y Fm(\020)r(\020)r(\020) 1193 2291 y Fl(\024)1251 2257 y Fo(1)p 1251 2279 V 1251 2325 a(2)1280 2291 y Fn(\016)r Fo(\()1323 2233 y Fg(Z)1372 2291 y Fn(w)1408 2271 y Fj(2)1407 2303 y Fm(\020)1439 2291 y Fo(+)1488 2233 y Fg(Z)1538 2291 y Fn(w)1574 2271 y Fj(2)1573 2303 y Fm(\020)r(\020)r(\020)1628 2291 y Fo(\))p Fn(:)15 2414 y Fo(This)i(leads)g(to)201 2475 y Fg(Z)251 2534 y Fn(w)286 2541 y Fm(\020)r(\020)r(\020)r(\020)359 2534 y Fn(w)q(g)44 b Fl(\024)542 2475 y Fg(Z)592 2534 y Fl(F)5 b Fo(\()p Fn(\036)p Fo(\))p Fn(w)736 2513 y Fj(2)735 2546 y Fm(\020)r(\020)783 2534 y Fl(\000)11 b Fo(2)865 2475 y Fg(Z)915 2534 y Fn(@)944 2513 y Fj(2)941 2546 y Fm(\020)964 2534 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)1146 2513 y Fj(2)1145 2546 y Fm(\020)1176 2534 y Fo(+)1230 2500 y(1)p 1230 2522 V 1230 2568 a(2)1268 2475 y Fg(Z)1318 2534 y Fn(@)1347 2513 y Fj(4)1344 2546 y Fm(\020)1366 2534 y Fo(\()p Fl(F)g Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)1548 2513 y Fj(2)542 2650 y Fo(+)p Fn(\016)r Fo(\()636 2638 y(~)623 2650 y Fn(K)664 2657 y Fj(1)692 2592 y Fg(Z)742 2650 y Fl(j)767 2638 y Fo(~)756 2650 y Fn(F)h Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fn(\036w)940 2630 y Fj(2)971 2650 y Fo(+)11 b(\()p Fn(K)1080 2657 y Fm(R)1120 2650 y Fo(+)1174 2616 y(1)p 1174 2639 V 1174 2684 a(2)1216 2638 y(~)1203 2650 y Fn(K)1244 2657 y Fm(R)1274 2650 y Fo(\))1301 2592 y Fg(Z)1351 2650 y Fn(w)1387 2630 y Fj(2)1386 2662 y Fm(\020)r(\020)r(\020)1452 2650 y Fo(+)g Fn(K)1542 2657 y Fm(R)1580 2592 y Fg(Z)1630 2650 y Fn(w)1666 2630 y Fj(2)1665 2662 y Fm(\020)1686 2650 y Fo(\))p Fn(:)935 2800 y Fo(16)p eop %%Page: 17 17 17 16 bop 15 68 a Fo(Finally)210 161 y(1)p 210 183 25 2 v 210 228 a(2)244 157 y Fn(@)281 122 y Fg(R)317 157 y Fn(w)353 139 y Fj(2)352 170 y Fm(\020)r(\020)r(\020)p 244 183 164 2 v 303 228 a Fn(@)s(t)426 194 y Fo(=)14 b Fl(\000)525 136 y Fg(Z)575 194 y Fn(w)611 174 y Fj(\(6\))658 194 y Fn(w)693 201 y Fm(t)722 194 y Fo(=)f Fl(\000)p Fo(\(1)f Fl(\000)e Fo(6)p Fn(\015)968 174 y Fj(2)988 194 y Fn(")1011 174 y Fj(4)1031 194 y Fo(\))1058 136 y Fg(Z)1108 194 y Fn(w)1144 174 y Fj(2)1143 207 y Fm(\020)r(\020)r (\020)r(\020)1227 194 y Fl(\000)h Fn(")1300 174 y Fj(4)1320 194 y Fn(w)1356 174 y Fj(2)1355 207 y Fm(\020)r(\020)r(\020)r(\020)r (\020)1457 194 y Fl(\000)1506 136 y Fg(Z)1556 194 y Fn(w)1592 174 y Fj(\(6\))1640 194 y Fn(w)q(g)r(;)15 304 y Fo(where)204 351 y Fg(Z)254 410 y Fn(w)290 389 y Fj(\(6\))337 410 y Fn(w)q(g)44 b Fo(=)e Fl(\000)567 351 y Fg(Z)616 410 y Fl(F)5 b Fo(\()p Fn(\036)p Fo(\))p Fn(w)760 389 y Fj(2)759 422 y Fm(\020)r(\020)r(\020)825 410 y Fo(+)879 376 y(3)p 879 398 25 2 v 879 444 a(2)917 351 y Fg(Z)967 410 y Fn(@)996 389 y Fj(2)993 422 y Fm(\020)1015 410 y Fo(\()p Fl(F)g Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)1197 389 y Fj(2)1196 422 y Fm(\020)r(\020)1244 410 y Fl(\000)11 b Fo(3)1326 351 y Fg(Z)1376 410 y Fn(@)1405 389 y Fj(2)1402 422 y Fm(\020)1425 410 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)1606 417 y Fm(\020)1625 410 y Fn(w)1660 417 y Fm(\020)r(\020)r(\020)520 519 y Fl(\000)567 461 y Fg(Z)616 519 y Fn(@)645 499 y Fj(3)642 531 y Fm(\020)665 519 y Fo(\()p Fl(F)g Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)q(w)882 526 y Fm(\020)r(\020)r(\020)948 519 y Fo(+)997 461 y Fg(Z)1047 519 y Fn(aw)1109 499 y Fj(2)1128 519 y Fn(w)1164 499 y Fj(\(6\))1211 519 y Fn(R:)15 628 y Fo(This)15 b(last)g(term)f (still)g(had)h(to)h(b)q(e)f(b)q(ounded,)h(w)o(e)f(will)f(do)h(this)g (later)g(on.)21 b(Com)o(bining)14 b(the)h(estimates)15 688 y(ab)q(o)o(v)o(e)h(leads)g(to)20 763 y Fn(@)s(E)p 20 785 68 2 v 30 831 a(@)s(t)134 797 y Fo(=)42 b Fl(\000)p Fn(A)298 738 y Fg(Z)347 797 y Fn(\036)387 784 y Fo(~)376 797 y Fn(F)7 b Fo(\()p Fn(\036)p Fo(\))p Fn(w)518 776 y Fj(2)548 797 y Fl(\000)k Fn(A)p Fo(\(1)g Fl(\000)g Fo(6)p Fn(\015)791 776 y Fj(2)811 797 y Fn(")834 776 y Fj(4)854 797 y Fo(\))881 738 y Fg(Z)931 797 y Fn(w)967 776 y Fj(2)966 809 y Fm(\020)998 797 y Fo(+)g(\()p Fl(\000)p Fn(A")1165 776 y Fj(4)1195 797 y Fl(\000)f Fn(B)s Fo(\(1)h Fl(\000)g Fo(6)p Fn(\015)1440 776 y Fj(2)1460 797 y Fn(")1483 776 y Fj(4)1503 797 y Fo(\)\))1549 738 y Fg(Z)1599 797 y Fn(w)1635 776 y Fj(2)1634 809 y Fm(\020)r(\020)214 906 y Fo(+\()p Fl(\000)p Fn(B)s(")373 886 y Fj(4)403 906 y Fl(\000)g Fn(C)t Fo(\(1)g Fl(\000)f Fo(6)p Fn(\015)647 886 y Fj(2)667 906 y Fn(")690 886 y Fj(4)710 906 y Fo(\)\))756 848 y Fg(Z)806 906 y Fn(w)842 886 y Fj(2)841 919 y Fm(\020)r(\020)r (\020)907 906 y Fo(+)h(\()p Fl(\000)p Fn(C)t(")1076 886 y Fj(4)1106 906 y Fl(\000)g Fn(")1179 886 y Fj(6)1199 906 y Fo(\(1)g Fl(\000)g Fo(6)p Fn(\015)1355 886 y Fj(2)1375 906 y Fn(")1398 886 y Fj(4)1417 906 y Fo(\)\))1463 848 y Fg(Z)1514 906 y Fn(w)1550 886 y Fj(2)1549 919 y Fm(\020)r(\020)r (\020)r(\020)1633 906 y Fl(\000)f Fn(")1705 886 y Fj(10)1751 848 y Fg(Z)1801 906 y Fn(w)1837 886 y Fj(2)1836 919 y Fm(\020)r(\020)r(\020)r(\020)r(\020)214 1022 y Fo(+)p Fn(A)297 964 y Fg(Z)347 1022 y Fn(w)383 1002 y Fj(2)403 1022 y Fn(g)j Fl(\000)e Fn(B)536 964 y Fg(Z)586 1022 y Fn(w)621 1029 y Fm(\020)r(\020)659 1022 y Fn(w)q(g)i Fo(+)e Fn(C)827 964 y Fg(Z)877 1022 y Fn(w)912 1029 y Fm(\020)r(\020)r(\020)r(\020)985 1022 y Fn(w)q(g)i Fl(\000)1112 989 y Fo(1)p 1112 1011 25 2 v 1112 1056 a(2)1141 1022 y Fn(")1164 1002 y Fj(6)1192 964 y Fg(Z)1242 1022 y Fn(w)1278 1002 y Fj(\(6\))1325 1022 y Fn(w)q(g)134 1150 y Fl(\024)214 1091 y Fg(Z)255 1150 y Fo(\()p Fl(\000)p Fn(A)p Fo(\()380 1137 y(~)369 1150 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))11 b Fl(\000)g Fn(\016)r(K)600 1157 y Fj(1)619 1150 y Fl(j)644 1137 y Fo(~)633 1150 y Fn(F)c Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fo(\))j Fl(\000)h Fn(B)s Fo(\()896 1113 y Fn(@)925 1094 y Fj(2)922 1125 y Fm(\020)944 1113 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p 896 1138 194 2 v 966 1184 a(2)p Fn(\036)1105 1150 y Fo(+)1159 1116 y(1)p 1159 1138 25 2 v 1159 1184 a(2)1189 1150 y Fn(\016)1213 1129 y Fj(2)1232 1150 y Fn(K)1273 1157 y Fj(1)1293 1150 y Fl(j)1318 1137 y Fo(~)1307 1150 y Fn(F)h Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fo(\))11 b(+)g Fn(C)t Fo(\()1568 1113 y Fn(@)1597 1094 y Fj(4)1594 1125 y Fm(\020)1616 1113 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p 1568 1138 194 2 v 1638 1184 a(2)p Fn(\036)214 1269 y Fo(+)p Fn(\016)288 1256 y Fo(~)276 1269 y Fn(K)317 1276 y Fj(1)337 1269 y Fl(j)361 1256 y Fo(~)351 1269 y Fn(F)h Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fo(\)\))p Fn(\036w)573 1248 y Fj(2)603 1269 y Fo(+)652 1210 y Fg(Z)694 1269 y Fo(\()p Fl(\000)p Fn(A)p Fo(\(1)k Fl(\000)h Fo(6)p Fn(\015)944 1248 y Fj(2)964 1269 y Fn(")987 1248 y Fj(4)1007 1269 y Fo(\))g Fl(\000)g Fn(B)s(\036)1166 1256 y Fo(~)1156 1269 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))k(+)h Fn(C)t Fo(\()p Fl(\000)p Fo(2)p Fn(@)1470 1248 y Fj(2)1467 1281 y Fm(\020)1489 1269 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))214 1385 y(+)257 1351 y(1)p 257 1373 25 2 v 257 1419 a(2)286 1385 y Fn(\016)r(K)351 1392 y Fm(R)380 1385 y Fo(\)\))p Fn(w)454 1364 y Fj(2)453 1397 y Fm(\020)485 1385 y Fo(+)534 1326 y Fg(Z)576 1385 y Fo(\()p Fl(\000)p Fn(A")694 1364 y Fj(4)723 1385 y Fl(\000)11 b Fn(B)s Fo(\(1)g Fl(\000)g Fo(6)p Fn(\015)969 1364 y Fj(2)989 1385 y Fn(")1012 1364 y Fj(4)1043 1385 y Fo(+)1097 1351 y(1)p 1097 1373 V 1097 1419 a(2)1126 1385 y Fn(K)1167 1392 y Fm(R)1196 1385 y Fo(\))h Fl(\000)e Fn(C)t(\036)1355 1372 y Fo(~)1344 1385 y Fn(F)c Fo(\()p Fn(\036)p Fo(\))11 b Fl(\000)1515 1351 y Fo(3)p 1515 1373 V 1515 1419 a(4)1544 1385 y Fn(")1567 1364 y Fj(6)1587 1385 y Fn(@)1616 1364 y Fj(2)1613 1397 y Fm(\020)1635 1385 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\)\))p Fn(w)1836 1364 y Fj(2)1835 1397 y Fm(\020)r(\020)214 1443 y Fg(Z)255 1501 y Fo(\()p Fl(\000)p Fn(B)s(")376 1481 y Fj(4)406 1501 y Fl(\000)11 b Fn(C)t Fo(\(1)g Fl(\000)g Fo(6)p Fn(\015)651 1481 y Fj(2)671 1501 y Fn(")694 1481 y Fj(4)725 1501 y Fl(\000)f Fn(\016)r Fo(\()822 1468 y(1)p 822 1490 V 822 1535 a(2)864 1489 y(~)851 1501 y Fn(K)892 1508 y Fm(R)933 1501 y Fo(+)h Fn(K)1023 1508 y Fm(R)1052 1501 y Fo(\)\))g(+)1155 1468 y(1)p 1155 1490 V 1155 1535 a(2)1184 1501 y Fn(")1207 1481 y Fj(6)1227 1501 y Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)1390 1481 y Fj(2)1389 1514 y Fm(\020)r(\020)r(\020)1455 1501 y Fl(\000)11 b Fn(")1528 1481 y Fj(10)1573 1443 y Fg(Z)1623 1501 y Fn(w)1659 1481 y Fj(2)1658 1514 y Fm(\020)r(\020)r(\020)r(\020)r (\020)214 1618 y Fo(+)260 1559 y Fg(Z)302 1618 y Fo(\()p Fl(\000)p Fn(C)t(")422 1597 y Fj(4)452 1618 y Fl(\000)f Fn(")524 1597 y Fj(6)544 1618 y Fo(\(1)h Fl(\000)g Fo(6)p Fn(\015)700 1597 y Fj(2)720 1618 y Fn(")743 1597 y Fj(4)763 1618 y Fo(\)\))p Fn(w)837 1597 y Fj(2)836 1630 y Fm(\020)r(\020)r(\020) r(\020)920 1618 y Fo(+)974 1584 y(3)p 974 1606 V 974 1652 a(2)1003 1618 y Fn(")1026 1597 y Fj(6)1054 1559 y Fg(Z)1104 1618 y Fl(j)p Fn(@)1147 1597 y Fj(2)1144 1630 y Fm(\020)1166 1618 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)1347 1625 y Fm(\020)1366 1618 y Fn(w)1401 1625 y Fm(\020)r(\020)r(\020)1456 1618 y Fl(j)214 1734 y Fo(+)257 1701 y(1)p 257 1723 V 257 1769 a(2)286 1734 y Fn(")309 1714 y Fj(6)337 1676 y Fg(Z)387 1734 y Fl(j)p Fn(@)430 1714 y Fj(3)427 1747 y Fm(\020)449 1734 y Fo(\()p Fl(F)g Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)q(w)666 1741 y Fm(\020)r(\020)r(\020)721 1734 y Fl(j)11 b Fo(+)800 1701 y(1)p 800 1723 V 800 1769 a(2)829 1734 y Fn(")852 1714 y Fj(6)880 1676 y Fg(Z)930 1734 y Fl(j)p Fn(aw)1006 1714 y Fj(2)1025 1734 y Fn(w)1061 1714 y Fj(\(6\))1109 1734 y Fn(R)p Fl(j)p Fn(:)652 b Fo(\(5.7\))15 1844 y(The)16 b(last)g(three)g(in)o(tegrals)g(can)g(all)g(b)q(e)g(b)q(ounded.)22 b(This)17 b(is)f(done)g(as)h(follo)o(ws)430 1960 y Fl(j)p Fn(")467 1939 y Fj(6)495 1901 y Fg(Z)545 1960 y Fn(@)574 1939 y Fj(2)571 1972 y Fm(\020)593 1960 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)774 1967 y Fm(\020)793 1960 y Fn(w)828 1967 y Fm(\020)r(\020)r(\020)884 1960 y Fl(j)13 b(\024)969 1926 y Fo(1)p 969 1948 V 969 1994 a(2)998 1960 y Fn(K)t Fo(\()p Fn(")1085 1939 y Fj(5)1113 1901 y Fg(Z)1163 1960 y Fn(w)1199 1939 y Fj(2)1198 1972 y Fm(\020)1230 1960 y Fo(+)e Fn(")1302 1939 y Fj(7)1330 1901 y Fg(Z)1380 1960 y Fn(w)1416 1939 y Fj(2)1415 1972 y Fm(\020)r(\020)r(\020)1470 1960 y Fo(\))15 2069 y(and)317 2150 y Fl(j)p Fn(")354 2129 y Fj(6)381 2091 y Fg(Z)431 2150 y Fn(@)460 2129 y Fj(3)457 2162 y Fm(\020)479 2150 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p Fn(w)q(w)696 2157 y Fm(\020)r(\020)r(\020)751 2150 y Fl(j)14 b(\024)837 2116 y Fo(1)p 837 2138 V 837 2184 a(2)866 2150 y(\()p Fn(")908 2129 y Fj(5)936 2091 y Fg(Z)991 2112 y Fn(@)1020 2094 y Fj(3)1017 2125 y Fm(\020)1039 2112 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p 991 2138 194 2 v 1073 2184 a Fn(\036)1189 2150 y(\036w)1254 2129 y Fj(2)1285 2150 y Fo(+)11 b Fn(")1357 2129 y Fj(7)1377 2150 y Fn(K)1430 2091 y Fg(Z)1480 2150 y Fn(w)1516 2129 y Fj(2)1515 2162 y Fm(\020)r(\020)r(\020)1570 2150 y Fo(\))p Fn(:)15 2258 y Fo(where)19 b(w)o(e)g(use)h(that)f(b)q(oth)i Fn(@)578 2240 y Fj(2)575 2271 y Fm(\020)597 2258 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))19 b(and)h Fn(@)889 2240 y Fj(3)886 2271 y Fm(\020)908 2258 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))19 b(are)h(b)q(ounded)g(b)o(y)f(a)h(constan)o (t.)32 b(The)19 b(term)15 2324 y Fn(")38 2306 y Fj(6)66 2289 y Fg(R)102 2324 y Fn(aw)164 2306 y Fj(2)183 2324 y Fn(w)219 2306 y Fj(\(6\))266 2324 y Fn(R)f Fo(m)o(ust)d(\014rst)h(b)q (e)h(in)o(tegrated)f(partially)l(,)f(then)i(it)f(can)g(b)q(e)h(b)q (ounded)g(in)g(a)f(similar)f(w)o(a)o(y)15 2384 y(\(see)h(app)q(endix)g (B\))g(as)292 2503 y Fn(")315 2482 y Fj(6)335 2503 y Fl(j)357 2444 y Fg(Z)407 2503 y Fn(aw)469 2482 y Fj(2)488 2503 y Fn(w)524 2482 y Fj(\(6\))572 2503 y Fn(R)p Fl(j)e(\024)g Fn(K)t Fo(\()754 2444 y Fg(Z)803 2503 y Fl(j)828 2490 y Fo(~)817 2503 y Fn(F)7 b Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fn(\036w)1002 2482 y Fj(2)1032 2503 y Fo(+)k Fn(\016)1105 2482 y Fj(2)1124 2503 y Fn(")1147 2482 y Fj(2)1175 2444 y Fg(Z)1225 2503 y Fn(w)1261 2482 y Fj(2)1260 2515 y Fm(\020)1281 2503 y Fo(\))g(+)1365 2469 y(3)p 1365 2491 25 2 v 1365 2537 a(4)1394 2503 y Fn(")1417 2482 y Fj(10)1463 2444 y Fg(Z)1513 2503 y Fn(w)1549 2482 y Fj(\(5\))1595 2471 y Fi(2)1613 2503 y Fn(;)15 2615 y Fo(for)16 b(some)f(constan)o(t)h Fn(K)i(>)c Fo(0.)22 b(Th)o(us,)16 b(the)f(last)h(three)g(terms)e(in)i (\(5.7\))g(can)g(b)q(e)h(tak)o(en)e(in)o(to)h(the)f(other)15 2675 y(terms)g(of)h(\(5.7\).)22 b(W)l(e)16 b(use)g(the)g(follo)o(wing)g (Lemma)e(to)i(b)q(ound)1189 2656 y Fm(dE)p 1189 2664 46 2 v 1196 2692 a(dt)1256 2675 y Fo(further.)935 2800 y(17)p eop %%Page: 18 18 18 17 bop 15 68 a Fk(Lemm)o(a)16 b(5.7)24 b Ff(F)l(r)n(om)e(the)i (assumptions)f(on)h(the)f(nonline)n(arity)h Fl(N)31 b Ff(it)23 b(fol)r(lows)i(that)e(for)g Fn(")g Ff(and)g Fn(\016)15 128 y Ff(su\016ciently)18 b(smal)r(l,)h(the)e(c)n(onstants)h Fn(A)p Ff(,)f Fn(B)i Ff(and)e Fn(C)k Ff(c)n(an)c(b)n(e)h(chosen)g(such) f(that)h(the)f(\014rst)g(term)g(in)1854 108 y Fm(dE)p 1854 116 46 2 v 1861 145 a(dt)15 195 y Ff(is)g(smal)r(ler)i(than)e Fl(\000)394 175 y Fj(1)p 394 183 18 2 v 394 212 a(2)425 159 y Fg(R)461 195 y Fn(A\036)537 182 y Fo(~)527 195 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))p Fn(w)668 177 y Fj(2)705 195 y Ff(and)18 b(al)r(l)g(the)g(other)g(terms)f(ar)n(e)g(ne)n(gative.) 15 294 y Fk(Pro)r(of:)c Fo(F)l(or)h(the)g(\014rst)h(part)f(of)g(the)g (Lemma,)e(w)o(e)i(m)o(ust)e(pro)o(v)o(e)h(that)i(there)e(exist)h(p)q (ositiv)o(e)f(constan)o(ts)15 354 y Fn(A)p Fo(,)i Fn(B)k Fo(and)e Fn(C)j Fo(suc)o(h)c(that)98 491 y Fn(A)p Fo(\()p Fl(\000)198 457 y Fo(1)p 198 479 25 2 v 198 525 a(2)237 478 y(~)226 491 y Fn(F)7 b Fo(\()p Fn(\036)p Fo(\))k(+)g Fn(\016)r(K)457 498 y Fj(1)476 491 y Fl(j)501 478 y Fo(~)490 491 y Fn(F)c Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fo(\))j Fl(\000)h Fn(B)s Fo(\()753 454 y Fn(@)782 436 y Fj(2)779 466 y Fm(\020)801 454 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p 753 479 194 2 v 823 525 a(2)p Fn(\036)962 491 y Fo(+)1016 457 y(1)p 1016 479 25 2 v 1016 525 a(2)1046 491 y Fn(\016)1070 470 y Fj(2)1089 491 y Fn(K)1130 498 y Fj(1)1150 491 y Fl(j)1175 478 y Fo(~)1164 491 y Fn(F)h Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fo(\))11 b(+)g Fn(C)t Fo(\()1425 454 y Fn(@)1454 436 y Fj(4)1451 466 y Fm(\020)1472 454 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p 1425 479 194 2 v 1494 525 a(2)p Fn(\036)1634 491 y Fo(+)11 b Fn(\016)1720 478 y Fo(~)1707 491 y Fn(K)1748 498 y Fj(1)1768 491 y Fl(j)1793 478 y Fo(~)1782 491 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fo(\))98 628 y(+)p Fn(")159 607 y Fj(5)183 591 y Fn(@)212 573 y Fj(3)209 603 y Fm(\020)231 591 y Fo(\()p Fl(F)f Fo(\()p Fn(\036)p Fo(\)\))p 183 616 V 253 662 a(2)p Fn(\036)393 628 y Fo(+)11 b Fn(")465 607 y Fj(2)485 628 y Fn(K)t Fl(j)554 615 y Fo(~)544 628 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))p Fl(j)1156 b Fo(\(5.8\))15 757 y(is)18 b(negativ)o(e.)27 b(W)l(e)18 b(pro)o(v)o(e)g(the)g(ab)q(o)o(v)o(e)h(in)f(t)o(w)o(o)g(parts,)i(for)e Fn(\036)g Fl(\025)f Fn(\036)1234 764 y Fj(0)1273 757 y Fo(and)i Fn(\036)e Fl(\024)h Fn(\036)1502 764 y Fj(0)1540 757 y Fo(where)g Fn(\036)1712 764 y Fj(0)1750 757 y Fo(is)g(su\016-)15 817 y(cien)o(tly)c(small.)20 b(F)l(or)d Fn(\036)d Fl(\025)f Fn(\036)531 824 y Fj(0)567 817 y Fo(all)j(the)g(terms)f(are)i(b)q (ounded)g(a)o(w)o(a)o(y)f(from)f(0,)i(th)o(us)f(when)h Fn(A)f Fo(is)g(c)o(hosen)15 877 y(su\016cien)o(tly)f(large,)i(the)g (expression)f(is)h(alw)o(a)o(ys)g(negativ)o(e.)23 b(When)17 b Fn(\036)e Fl(!)h Fo(0,)h(the)g(\014rst)g(term)e(in)i(\(5.8\))15 937 y(also)d(go)q(es)h(in)e(the)g(limit)e(to)j(0)g(and)h(the)e(ab)q(o)o (v)o(e)h(argumen)o(t)e(do)q(es)j(not)f(hold)g(an)o(ymore.)k(In)c(this)f (case)h(w)o(e)15 998 y(use)g(that)g(from)e(the)i(fact)g(that)g Fl(N)21 b Fo(is)13 b(analytic)g(w)o(e)h(kno)o(w)f(that)i(w)o(e)e(can)h (write)f(it)g(as)i Fl(N)21 b Fo(=)1675 964 y Fg(P)1719 978 y Fr(1)1719 1010 y Fm(k)q Fj(=)p Fm(k)1783 1015 y Fi(0)1812 998 y Fn(c)1833 1005 y Fm(k)1854 998 y Fn(\036)1883 979 y Fm(k)15 1064 y Fo(where)c Fn(k)182 1071 y Fj(0)219 1064 y Fl(\025)f Fo(2.)26 b(Then)478 1052 y(~)467 1064 y Fn(F)d Fo(=)16 b Fl(\000)623 1031 y Fg(P)667 1044 y Fr(1)667 1077 y Fm(k)q Fj(=)p Fm(k)731 1082 y Fi(0)760 1064 y Fn(k)r(c)808 1071 y Fm(k)829 1064 y Fn(\036)858 1046 y Fm(k)q Fr(\000)p Fj(2)942 1064 y Fo(and)j Fl(F)5 b Fo(\()p Fn(\036)p Fo(\))16 b(=)1217 1031 y Fg(P)1261 1044 y Fr(1)1261 1077 y Fm(k)q Fj(=)p Fm(k)1325 1082 y Fi(0)1354 1064 y Fn(k)r(c)1402 1071 y Fm(k)1423 1064 y Fn(\036)1452 1046 y Fm(k)q Fr(\000)p Fj(1)1518 1064 y Fo(.)26 b(Also,)18 b(recall)e(that)15 1124 y Fn(\036)g Fl(\031)h Fn(c)137 1131 y Fj(1)156 1124 y Fn(e)179 1106 y Fr(\000)p Fm(\015)r(\020)264 1124 y Fo(for)i Fn(\020)h Fl(!)d(1)h Fo(if)f Fn(c)g(>)f(c)676 1106 y Fr(\003)714 1124 y Fo(\(or)i Fn(\036)f Fl(\031)f Fn(c)916 1131 y Fj(1)936 1124 y Fn(\020)t(e)984 1106 y Fr(\000)p Fm(\015)r(\020)1069 1124 y Fo(if)h Fn(c)g Fo(=)g Fn(c)1229 1106 y Fr(\003)1249 1124 y Fo(\),)g(see)h(remark)e(3.3.)27 b(The)18 b(leading)15 1185 y(order)e(term)e(in)i(\(5.8\))g(b)q(ecomes)f Fl(\031)f(\000)722 1165 y Fj(1)p 722 1173 18 2 v 722 1202 a(2)744 1185 y Fn(A)q Fo(~)-25 b Fn(ce)825 1167 y Fr(\000)p Fm(\015)r(\020)r Fj(\()p Fm(k)922 1172 y Fi(0)939 1167 y Fr(\000)p Fj(2\))1016 1185 y Fo(\(here)15 b(w)o(e)h(assume)f Fn(c)f(>)g(c)1488 1167 y Fr(\003)1508 1185 y Fo(,)h(the)h(mo)q(di\014cations)15 1245 y(when)f Fn(c)f Fo(=)g Fn(c)249 1227 y Fr(\003)284 1245 y Fo(are)h(ob)o(vious\).)21 b(Th)o(us,)16 b(b)o(y)e(c)o(ho)q (osing)j Fn(A)e Fo(large)g(enough,)h(\(5.8\))f(can)h(b)q(e)f(made)f (negativ)o(e)15 1305 y(for)h Fn(")f Fo(and)i Fn(\016)g Fo(su\016cien)o(tly)c(small.)19 b(Using)c(a)g(similar)e(argumen)o(t)g (w)o(e)h(can)h(sho)o(w)h(that)f(the)f(other)h(terms)15 1365 y(in)h(\(5.7\))g(are)g(negativ)o(e)g(for)g(appropriate)h(c)o (hoices)e(of)i Fn(A)p Fo(,)e Fn(B)k Fo(and)e Fn(C)j Fo(.)562 b Fc(2)15 1486 y Fo(F)l(rom)15 b(the)h(ab)q(o)o(v)o(e)g(Lemma,)e(w)o(e) h(can)i(conclude)e(that)i(for)g(su\016cien)o(tly)d(small)g Fn(")i Fo(and)h Fn(\016)r Fo(,)664 1581 y Fn(dE)p 664 1603 65 2 v 674 1648 a(dt)747 1614 y Fl(\024)c(\000)843 1581 y Fo(1)p 843 1603 25 2 v 843 1648 a(2)881 1556 y Fg(Z)931 1614 y Fn(A\036)1007 1602 y Fo(~)997 1614 y Fn(F)5 b Fo(\()p Fn(\036)p Fo(\))p Fn(w)1137 1594 y Fj(2)1171 1614 y Fn(<)14 b Fo(0)p Fn(:)15 1734 y Fk(Remark)i(5.8)24 b Ff(Note)e(that)g(for)f Fn(c)g(<)g(c)752 1716 y Fr(\003)793 1734 y Ff(the)h(pr)n(e)n(c)n(e)n(ding)f(ar)n(gument)h(fails)g(sinc)n(e) g Fn(\036)f Ff(p)n(asses)g(thr)n(ough)15 1794 y(zer)n(o)c(and)g(we)h(c) n(an)g(no)g(longer)g(c)n(ontr)n(ol)758 1774 y Fm(@)r(E)p 758 1782 49 2 v 765 1811 a(@)r(t)811 1794 y Ff(.)15 1893 y Fo(No)o(w,)d(w)o(e)h(will)f(compute)g(the)h(deriv)m(ativ)o(e)f(of)h Fn(F)7 b Fo(\()p Fn(t)p Fo(\))15 b(with)h(resp)q(ect)g(to)h Fn(t)p Fo(.)104 1995 y(1)p 104 2018 25 2 v 104 2063 a(2)139 1995 y Fn(@)s Fo(\()187 1960 y Fg(R)214 1995 y Fo(\()p Fn(aw)q Fo(\))314 1977 y Fj(2)333 1995 y Fo(\))p 139 2018 214 2 v 222 2063 a Fn(@)s(t)371 2029 y Fo(=)423 1971 y Fg(Z)464 2029 y Fo(\()p Fn(aw)q Fo(\))564 2009 y Fj(2)595 2029 y Fl(\000)645 1971 y Fg(Z)686 2029 y Fo(\(\()p Fn(aw)q Fo(\))805 2036 y Fm(\020)825 2029 y Fo(\))844 2009 y Fj(2)875 2029 y Fl(\000)10 b Fn(")947 2009 y Fj(4)975 1971 y Fg(Z)1017 2029 y Fo(\(\()p Fn(aw)q Fo(\))1136 2036 y Fm(\020)r(\020)1173 2029 y Fo(\))1192 2009 y Fj(2)1223 2029 y Fl(\000)1273 1971 y Fg(Z)1322 2029 y Fn(a)1348 2009 y Fj(2)1368 2029 y Fn(w)1404 2009 y Fj(2)1424 2029 y Fn(\036)1464 2017 y Fo(~)1453 2029 y Fn(F)c Fo(\()p Fn(\036)p Fo(\))11 b(+)1618 1971 y Fg(Z)1668 2029 y Fn(a)1694 2009 y Fj(3)1713 2029 y Fn(w)1749 2009 y Fj(3)1769 2029 y Fn(R:)15 2149 y Fo(Here)k(w)o(e)h(b)q(ound)676 2209 y Fl(j)698 2150 y Fg(Z)748 2209 y Fn(a)774 2188 y Fj(3)793 2209 y Fn(w)829 2188 y Fj(3)849 2209 y Fn(R)p Fl(j)e(\024)g Fn(K)1008 2216 y Fm(R)1037 2209 y Fn(\016)1069 2150 y Fg(Z)1110 2209 y Fo(\()p Fn(aw)q Fo(\))1210 2188 y Fj(2)1230 2209 y Fn(:)15 2307 y Fo(Also)163 2397 y(1)p 163 2419 25 2 v 163 2465 a(2)197 2397 y Fn(@)s Fo(\()245 2361 y Fg(R)272 2397 y Fo(\(\()p Fn(aw)q Fo(\))391 2404 y Fm(\020)411 2397 y Fo(\))430 2379 y Fj(2)449 2397 y Fo(\))p 197 2419 272 2 v 310 2465 a Fn(@)s(t)487 2431 y Fo(=)539 2372 y Fg(Z)580 2431 y Fo(\(\()p Fn(aw)q Fo(\))699 2438 y Fm(\020)719 2431 y Fo(\))738 2410 y Fj(2)769 2431 y Fl(\000)819 2372 y Fg(Z)860 2431 y Fo(\(\()p Fn(aw)q Fo(\))979 2438 y Fm(\020)r(\020)1017 2431 y Fo(\))1036 2410 y Fj(2)1066 2431 y Fl(\000)d Fn(")1139 2410 y Fj(4)1167 2372 y Fg(Z)1208 2431 y Fo(\(\()p Fn(aw)q Fo(\))1327 2438 y Fm(\020)r(\020)r(\020)1383 2431 y Fo(\))1402 2410 y Fj(2)1432 2431 y Fl(\000)1482 2372 y Fg(Z)1524 2431 y Fo(\()p Fn(aw)q Fo(\))1624 2438 y Fm(\020)r(\020)1661 2431 y Fn(aw)q(g)r(:)15 2550 y Fo(Here)184 2607 y Fg(Z)226 2666 y Fo(\()p Fn(aw)q Fo(\))326 2673 y Fm(\020)r(\020)363 2666 y Fn(aw)q(g)16 b Fo(=)521 2632 y(1)p 521 2654 25 2 v 521 2700 a(2)559 2607 y Fg(Z)608 2666 y Fn(@)637 2645 y Fj(2)634 2678 y Fm(\020)657 2666 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\)\()p Fn(aw)q Fo(\))903 2645 y Fj(2)933 2666 y Fl(\000)982 2607 y Fg(Z)1032 2666 y Fl(F)g Fo(\()p Fn(\036)p Fo(\)\(\()p Fn(aw)q Fo(\))1259 2673 y Fm(\020)1278 2666 y Fo(\))1297 2645 y Fj(2)1328 2666 y Fo(+)1377 2607 y Fg(Z)1427 2666 y Fn(R)p Fo(\()p Fn(aw)q Fo(\))1564 2645 y Fj(2)1584 2666 y Fo(\()p Fn(aw)q Fo(\))1684 2673 y Fm(\020)r(\020)1721 2666 y Fn(:)935 2800 y Fo(18)p eop %%Page: 19 19 19 18 bop 15 68 a Fo(This)16 b(last)h(in)o(tegral)e(can)h(b)q(e)h(b)q (ounded)g(as)195 174 y Fl(j)217 116 y Fg(Z)267 174 y Fn(R)p Fo(\()p Fn(aw)q Fo(\))404 154 y Fj(2)424 174 y Fo(\()p Fn(aw)q Fo(\))524 181 y Fm(\020)r(\020)561 174 y Fl(j)d(\024)g Fn(K)683 181 y Fm(R)712 174 y Fn(\016)744 116 y Fg(Z)794 174 y Fl(j)p Fn(aw)q Fo(\()p Fn(aw)q Fo(\))970 181 y Fm(\020)r(\020)1007 174 y Fl(j)f(\024)1092 141 y Fo(1)p 1092 163 25 2 v 1092 209 a(2)1121 174 y Fn(K)1162 181 y Fm(R)1192 174 y Fn(\016)r Fo(\()1235 116 y Fg(Z)1276 174 y Fo(\()p Fn(aw)q Fo(\))1376 154 y Fj(2)1406 174 y Fo(+)1455 116 y Fg(Z)1497 174 y Fo(\(\()p Fn(aw)q Fo(\))1616 181 y Fm(\020)r(\020)1653 174 y Fo(\))1672 154 y Fj(2)1692 174 y Fo(\))p Fn(:)15 274 y Fo(W)l(e)j(obtain)g(that)110 345 y(1)p 110 367 V 110 413 a(2)144 345 y Fn(@)s Fo(\()192 310 y Fg(R)219 345 y Fo(\(\()p Fn(aw)q Fo(\))338 352 y Fm(\020)r(\020)376 345 y Fo(\))395 327 y Fj(2)414 345 y Fo(\))p 144 367 290 2 v 266 413 a Fn(@)s(t)452 379 y Fo(=)504 320 y Fg(Z)545 379 y Fo(\(\()p Fn(aw)q Fo(\))664 386 y Fm(\020)r(\020)702 379 y Fo(\))721 358 y Fj(2)751 379 y Fl(\000)801 320 y Fg(Z)843 379 y Fo(\(\()p Fn(aw)q Fo(\))962 386 y Fm(\020)r(\020)r(\020)1017 379 y Fo(\))1036 358 y Fj(2)1067 379 y Fl(\000)10 b Fn(")1139 358 y Fj(4)1167 320 y Fg(Z)1209 379 y Fo(\(\()p Fn(aw)q Fo(\))1328 386 y Fm(\020)r(\020)r(\020)r(\020)1401 379 y Fo(\))1420 358 y Fj(2)1451 379 y Fo(+)1500 320 y Fg(Z)1541 379 y Fo(\()p Fn(aw)q Fo(\))1641 386 y Fm(\020)r(\020)r(\020)r(\020)1714 379 y Fn(aw)q(g)r(;)15 479 y Fo(where)82 518 y Fg(Z)124 576 y Fo(\()p Fn(aw)q Fo(\))224 583 y Fm(\020)r(\020)r(\020)r(\020)297 576 y Fn(aw)q(g)43 b Fo(=)505 518 y Fg(Z)554 576 y Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\(\()p Fn(aw)q Fo(\))781 583 y Fm(\020)r(\020)818 576 y Fo(\))837 556 y Fj(2)868 576 y Fl(\000)11 b Fo(2)950 518 y Fg(Z)1000 576 y Fn(@)1029 556 y Fj(2)1026 589 y Fm(\020)1049 576 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\)\(\()p Fn(aw)q Fo(\))1314 583 y Fm(\020)1333 576 y Fo(\))1352 556 y Fj(2)1382 576 y Fo(+)1436 543 y(1)p 1436 565 25 2 v 1436 610 a(2)1474 518 y Fg(Z)1524 576 y Fn(@)1553 556 y Fj(4)1550 589 y Fm(\020)1572 576 y Fo(\()p Fl(F)g Fo(\()p Fn(\036)p Fo(\)\)\()p Fn(aw)q Fo(\))1818 556 y Fj(2)505 686 y Fo(+)551 627 y Fg(Z)592 686 y Fo(\()p Fn(aw)q Fo(\))692 693 y Fm(\020)r(\020)r(\020) r(\020)765 686 y Fo(\()p Fn(aw)q Fo(\))865 665 y Fj(2)885 686 y Fn(R)15 785 y Fo(and)15 873 y Fl(j)37 814 y Fg(Z)78 873 y Fo(\()p Fn(aw)q Fo(\))178 880 y Fm(\020)r(\020)r(\020)r(\020)251 873 y Fo(\()p Fn(aw)q Fo(\))351 852 y Fj(2)371 873 y Fn(R)p Fl(j)14 b Fo(=)g Fl(j)510 814 y Fg(Z)551 873 y Fo(\()p Fn(aw)q Fo(\))651 880 y Fm(\020)r(\020)r(\020)707 873 y Fo(\()p Fn(R)763 880 y Fm(\020)783 873 y Fo(\()p Fn(aw)q Fo(\))883 852 y Fj(2)902 873 y Fo(+2)p Fn(Raw)q Fo(\()p Fn(aw)q Fo(\))1163 880 y Fm(\020)1183 873 y Fo(\))p Fl(j)g(\024)g Fn(K)1324 880 y Fj(1)1344 873 y Fn(\016)r Fo(\()1387 814 y Fg(Z)1428 873 y Fo(\()p Fn(aw)q Fo(\))1528 852 y Fj(2)1547 873 y Fo(+)1585 814 y Fg(Z)1627 873 y Fo(\()p Fn(aw)q Fo(\))1727 852 y Fj(2)1727 885 y Fm(\020)1746 873 y Fo(+)1784 814 y Fg(Z)1826 873 y Fo(\()p Fn(aw)q Fo(\))1926 852 y Fj(2)1926 885 y Fm(\020)r(\020)r(\020)1981 873 y Fo(\))p Fn(;)15 972 y Fo(for)i(some)f(p)q(ositiv)o(e)h(constan)o (t)h Fn(K)630 979 y Fj(1)650 972 y Fo(.)k(Finally)l(,)109 1053 y(1)p 109 1075 V 109 1121 a(2)143 1053 y Fn(@)s Fo(\()191 1017 y Fg(R)218 1053 y Fo(\(\()p Fn(aw)q Fo(\))337 1060 y Fm(\020)r(\020)r(\020)392 1053 y Fo(\))411 1035 y Fj(2)431 1053 y Fo(\))p 143 1075 307 2 v 273 1121 a Fn(@)s(t)469 1086 y Fo(=)521 1028 y Fg(Z)562 1086 y Fo(\(\()p Fn(aw)q Fo(\))681 1093 y Fm(\020)r(\020)r(\020)736 1086 y Fo(\))755 1066 y Fj(2)786 1086 y Fl(\000)836 1028 y Fg(Z)877 1086 y Fo(\(\()p Fn(aw)q Fo(\))996 1093 y Fm(\020)r(\020)r (\020)r(\020)1069 1086 y Fo(\))1088 1066 y Fj(2)1119 1086 y Fl(\000)11 b Fn(")1192 1066 y Fj(4)1220 1028 y Fg(Z)1261 1086 y Fo(\(\()p Fn(aw)q Fo(\))1380 1066 y Fj(\(5\))1427 1086 y Fo(\))1446 1066 y Fj(2)1477 1086 y Fl(\000)1527 1028 y Fg(Z)1568 1086 y Fo(\()p Fn(aw)q Fo(\))1668 1066 y Fj(\(6\))1715 1086 y Fn(aw)q(g)r(;)15 1186 y Fo(where)22 1225 y Fg(Z)63 1284 y Fo(\()p Fn(aw)q Fo(\))163 1263 y Fj(\(6\))210 1284 y Fn(aw)q(g)43 b Fo(=)f Fl(\000)465 1225 y Fg(Z)515 1284 y Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\(\()p Fn(aw)q Fo(\))742 1291 y Fm(\020)r(\020)r(\020)796 1284 y Fo(\))815 1263 y Fj(2)846 1284 y Fo(+)900 1250 y(3)p 900 1272 25 2 v 900 1318 a(2)938 1225 y Fg(Z)988 1284 y Fn(@)1017 1263 y Fj(2)1014 1296 y Fm(\020)1036 1284 y Fo(\()p Fl(F)g Fo(\()p Fn(\036)p Fo(\)\)\(\()p Fn(aw)q Fo(\))1301 1291 y Fm(\020)r(\020)1338 1284 y Fo(\))1357 1263 y Fj(2)1387 1284 y Fl(\000)1437 1225 y Fg(Z)1487 1284 y Fn(@)1516 1263 y Fj(3)1513 1296 y Fm(\020)1535 1284 y Fo(\()p Fl(F)g Fo(\()p Fn(\036)p Fo(\)\))p Fn(aw)q Fo(\()p Fn(aw)q Fo(\))1843 1291 y Fm(\020)r(\020)r (\020)418 1393 y Fl(\000)p Fo(3)489 1334 y Fg(Z)539 1393 y Fn(@)568 1372 y Fj(2)565 1405 y Fm(\020)587 1393 y Fo(\()p Fl(F)g Fo(\()p Fn(\036)p Fo(\)\)\()p Fn(aw)q Fo(\))833 1400 y Fm(\020)852 1393 y Fo(\()p Fn(aw)q Fo(\))952 1400 y Fm(\020)r(\020)r(\020)1019 1393 y Fo(+)1068 1334 y Fg(Z)1109 1393 y Fo(\()p Fn(aw)q Fo(\))1209 1372 y Fj(\(6\))1256 1393 y Fo(\()p Fn(aw)q Fo(\))1356 1372 y Fj(2)1376 1393 y Fn(R:)15 1499 y Fo(Note)18 b(that)h(in)f(the)g (functional)g Fn(F)7 b Fo(\()p Fn(t)p Fo(\),)17 b(the)h(quan)o(tit)o(y) 1033 1479 y Fj(1)p 1033 1487 18 2 v 1033 1516 a(2)1064 1463 y Fg(R)1092 1499 y Fo(\(\()p Fn(aw)q Fo(\))1211 1506 y Fm(\020)r(\020)r(\020)1266 1499 y Fo(\))1285 1481 y Fj(2)1323 1499 y Fo(has)h(a)g(prefactor)f(of)h Fn(\017)1742 1481 y Fj(6)1761 1499 y Fo(.)28 b(Mul-)15 1559 y(tiplying)20 b(the)g(previous)h(equation)g(b)o(y)g Fn(\017)784 1541 y Fj(6)803 1559 y Fo(,)h(the)f(last)g(term)e(can)i(b)q(e)g(b)q(ounded)h (as)g(in)f(App)q(endix)f(B)15 1619 y(b)o(y)208 1692 y Fn(")231 1671 y Fj(6)251 1692 y Fl(j)273 1633 y Fg(Z)315 1692 y Fo(\()p Fn(aw)q Fo(\))415 1671 y Fj(2)434 1692 y Fn(w)470 1671 y Fj(\(6\))517 1692 y Fn(R)p Fl(j)15 b(\024)e Fn(c)656 1699 y Fj(6)676 1692 y Fn(\016)r Fo(\()p Fn(")742 1671 y Fj(2)769 1633 y Fg(Z)811 1692 y Fo(\()p Fn(aw)q Fo(\))911 1671 y Fj(2)941 1692 y Fo(+)e Fn(")1013 1671 y Fj(2)1041 1633 y Fg(Z)1083 1692 y Fo(\(\()p Fn(aw)q Fo(\))1202 1699 y Fm(\020)1221 1692 y Fo(\))1240 1671 y Fj(2)1271 1692 y Fo(+)g Fn(")1343 1671 y Fj(10)1388 1633 y Fg(Z)1430 1692 y Fo(\(\()p Fn(aw)q Fo(\))1549 1699 y Fm(\020)r(\020)r(\020)r(\020)r(\020)1640 1692 y Fo(\))1659 1671 y Fj(2)1678 1692 y Fo(\))p Fn(;)15 1781 y Fo(for)16 b(some)f(constan)o(t)i Fn(c)429 1788 y Fj(6)463 1781 y Fn(>)d Fo(0.)21 b(This)c(leads)f(to)20 1855 y Fn(@)s(F)p 20 1877 67 2 v 30 1923 a(@)s(t)133 1889 y Fo(=)213 1830 y Fg(Z)276 1876 y Fo(~)263 1889 y Fn(A)p Fo(\(1)11 b Fl(\000)g Fn(\036)444 1876 y Fo(~)433 1889 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\)\)\()p Fn(aw)q Fo(\))657 1868 y Fj(2)687 1889 y Fo(+)11 b(\()p Fl(\000)807 1876 y Fo(~)794 1889 y Fn(A)g Fo(+)902 1876 y(~)891 1889 y Fn(B)r Fo(\))957 1830 y Fg(Z)999 1889 y Fo(\(\()p Fn(aw)q Fo(\))1118 1896 y Fm(\020)1138 1889 y Fo(\))1157 1868 y Fj(2)1187 1889 y Fo(+)g(\()p Fl(\000)p Fn(")1317 1868 y Fj(4)1350 1876 y Fo(~)1337 1889 y Fn(A)f Fl(\000)1446 1876 y Fo(~)1434 1889 y Fn(B)k Fo(+)1545 1876 y(~)1534 1889 y Fn(C)s Fo(\))1599 1830 y Fg(Z)1641 1889 y Fo(\(\()p Fn(aw)q Fo(\))1760 1896 y Fm(\020)r(\020)1797 1889 y Fo(\))1816 1868 y Fj(2)213 1998 y Fo(+\()p Fl(\000)p Fn(")332 1977 y Fj(4)363 1985 y Fo(~)351 1998 y Fn(B)g Fl(\000)463 1985 y Fo(~)452 1998 y Fn(C)g Fo(+)d Fn(")573 1977 y Fj(6)593 1998 y Fo(\))620 1939 y Fg(Z)662 1998 y Fo(\(\()p Fn(aw)q Fo(\))781 2005 y Fm(\020)r(\020)r(\020)836 1998 y Fo(\))855 1977 y Fj(2)886 1998 y Fo(+)g(\()p Fl(\000)p Fn(")1016 1977 y Fj(4)1046 1985 y Fo(~)1035 1998 y Fn(C)k Fl(\000)10 b Fn(")1157 1977 y Fj(6)1177 1998 y Fo(\))1204 1939 y Fg(Z)1246 1998 y Fo(\(\()p Fn(aw)q Fo(\))1365 2005 y Fm(\020)r(\020)r(\020)r(\020)1438 1998 y Fo(\))1457 1977 y Fj(2)1487 1998 y Fl(\000)h Fn(")1560 1977 y Fj(10)1606 1939 y Fg(Z)1647 1998 y Fo(\(\()p Fn(aw)q Fo(\))1766 1977 y Fj(\(5\))1813 1998 y Fo(\))1832 1977 y Fj(2)213 2107 y Fo(+)264 2094 y(~)251 2107 y Fn(A)296 2048 y Fg(Z)346 2107 y Fn(Ra)409 2086 y Fj(3)429 2107 y Fn(w)465 2086 y Fj(3)496 2107 y Fl(\000)557 2094 y Fo(~)546 2107 y Fn(B)593 2048 y Fg(Z)635 2107 y Fo(\()p Fn(aw)q Fo(\))735 2114 y Fm(\020)r(\020)772 2107 y Fn(aw)q(g)i Fo(+)930 2094 y(~)919 2107 y Fn(C)966 2048 y Fg(Z)1008 2107 y Fo(\()p Fn(aw)q Fo(\))1108 2114 y Fm(\020)r(\020)r(\020)r(\020)1181 2107 y Fn(aw)q(g)g Fl(\000)d Fn(")1351 2086 y Fj(6)1379 2048 y Fg(Z)1421 2107 y Fo(\()p Fn(aw)q Fo(\))1521 2086 y Fj(\(6\))1568 2107 y Fn(aw)q(g)133 2223 y Fl(\024)213 2164 y Fg(Z)255 2223 y Fo(\()287 2210 y(~)274 2223 y Fn(A)o Fo(\(1)i Fl(\000)f Fn(\036)454 2210 y Fo(~)444 2223 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))11 b(+)g Fn(K)650 2230 y Fm(R)679 2223 y Fn(\016)r Fo(\))g Fl(\000)787 2189 y Fo(1)p 787 2211 25 2 v 787 2257 a(2)828 2210 y(~)817 2223 y Fn(B)r Fo(\()p Fn(@)904 2202 y Fj(2)901 2235 y Fm(\020)923 2223 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))11 b(+)g Fn(\016)r(K)1194 2230 y Fm(R)1223 2223 y Fo(\))g(+)1313 2210 y(~)1302 2223 y Fn(C)s Fo(\()1364 2189 y(1)p 1364 2211 V 1364 2257 a(2)1394 2223 y Fn(@)1423 2202 y Fj(4)1420 2235 y Fm(\020)1442 2223 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))10 b(+)h Fn(K)1688 2230 y Fj(1)1709 2223 y Fn(\016)r Fo(\))213 2332 y Fl(\000)p Fn(c)273 2339 y Fj(6)293 2332 y Fn(")316 2312 y Fj(2)335 2332 y Fn(\016)r Fo(\)\()p Fn(aw)q Fo(\))478 2312 y Fj(2)508 2332 y Fo(+)557 2274 y Fg(Z)599 2332 y Fo(\()p Fl(\000)669 2320 y Fo(~)657 2332 y Fn(A)f Fo(+)765 2320 y(~)753 2332 y Fn(B)s Fo(\(1)h(+)g Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))11 b Fl(\000)1095 2320 y Fo(~)1084 2332 y Fn(C)s Fo(\(2)p Fn(@)1194 2312 y Fj(2)1191 2345 y Fm(\020)1214 2332 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))10 b Fl(\000)h Fn(K)1461 2339 y Fj(1)1481 2332 y Fn(\016)r Fo(\))g Fl(\000)f Fn(c)1605 2339 y Fj(6)1625 2332 y Fn(")1648 2312 y Fj(2)1668 2332 y Fn(\016)r Fo(\)\(\()p Fn(aw)q Fo(\))1830 2339 y Fm(\020)1849 2332 y Fo(\))1868 2312 y Fj(2)213 2390 y Fg(Z)255 2448 y Fo(\()p Fl(\000)p Fn(")336 2428 y Fj(4)368 2436 y Fo(~)355 2448 y Fn(A)h Fl(\000)464 2436 y Fo(~)453 2448 y Fn(B)r Fo(\(1)h(+)600 2415 y(1)p 600 2437 V 600 2482 a(2)630 2448 y Fn(K)671 2455 y Fm(R)700 2448 y Fn(\016)r Fo(\))f(+)814 2436 y(~)803 2448 y Fn(C)s Fo(\(1)h(+)f Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))10 b Fl(\000)1137 2415 y Fo(3)p 1137 2437 V 1137 2482 a(2)1166 2448 y Fn(")1189 2428 y Fj(6)1209 2448 y Fn(@)1238 2428 y Fj(2)1235 2461 y Fm(\020)1257 2448 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\)\)\(\()p Fn(aw)q Fo(\))1541 2455 y Fm(\020)r(\020)1578 2448 y Fo(\))1597 2428 y Fj(2)213 2558 y Fo(+)259 2499 y Fg(Z)301 2558 y Fo(\()p Fl(\000)p Fn(")382 2537 y Fj(4)413 2545 y Fo(~)401 2558 y Fn(B)14 b Fl(\000)513 2545 y Fo(~)502 2558 y Fn(C)s Fo(\(1)e Fl(\000)e Fn(\016)r(K)709 2565 y Fj(1)729 2558 y Fo(\))h(+)g Fn(")831 2537 y Fj(6)851 2558 y Fo(\(1)g(+)g Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\)\)\(\()p Fn(aw)q Fo(\))1219 2565 y Fm(\020)r(\020)r(\020)1274 2558 y Fo(\))1293 2537 y Fj(2)1324 2558 y Fl(\000)10 b Fn(")1396 2537 y Fj(10)1434 2558 y Fo(\(1)h(+)g Fn(\016)r(c)1582 2565 y Fj(6)1601 2558 y Fo(\))1628 2499 y Fg(Z)1670 2558 y Fo(\(\()p Fn(aw)q Fo(\))1789 2565 y Fm(\020)r(\020)r(\020)r(\020)r (\020)1880 2558 y Fo(\))1899 2537 y Fj(2)213 2666 y Fo(+3)p Fn(")298 2646 y Fj(6)326 2608 y Fg(Z)376 2666 y Fl(j)p Fn(@)419 2646 y Fj(2)416 2679 y Fm(\020)438 2666 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\)\()p Fn(aw)q Fo(\))684 2673 y Fm(\020)703 2666 y Fo(\()p Fn(aw)q Fo(\))803 2673 y Fm(\020)r(\020)r(\020)859 2666 y Fl(j)10 b Fo(+)h Fn(")955 2646 y Fj(6)983 2608 y Fg(Z)1033 2666 y Fl(j)p Fn(@)1076 2646 y Fj(3)1073 2679 y Fm(\020)1095 2666 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))p Fn(aw)q Fo(\()p Fn(aw)q Fo(\))1403 2673 y Fm(\020)r(\020)r(\020)1458 2666 y Fl(j)p Fn(:)332 b Fo(\(5.9\))935 2800 y(19)p eop %%Page: 20 20 20 19 bop 15 68 a Fo(In)16 b(a)h(similar)d(w)o(a)o(y)i(as)i(b)q(efore,) e(the)g(last)h(t)o(w)o(o)f(terms)f(can)i(b)q(e)g(b)q(ounded)g(b)o(y)f Fn(")1449 50 y Fj(6)1468 68 y Fn(c)1489 75 y Fj(2)1509 68 y Fo(\()1528 32 y Fg(R)1556 68 y Fo(\()p Fn(aw)q Fo(\))1656 50 y Fj(2)1687 68 y Fo(+)11 b(\()p Fn(aw)q Fo(\))1836 50 y Fj(2)1836 80 y Fm(\020)1867 68 y Fo(+)15 128 y(\()p Fn(aw)q Fo(\))115 110 y Fj(2)115 140 y Fm(\020)r(\020)r(\020)170 128 y Fo(\))19 b(with)g Fn(c)343 135 y Fj(2)382 128 y Fo(a)h(p)q(ositiv)o(e)e(constan)o(t)i(and)g(these)f(can)g(again)i(b)q (e)e(tak)o(en)g(in)o(to)g(the)g(other)g(terms)15 188 y(in)25 b(\(5.9\).)50 b(Then)26 b(all)f(the)g(terms)f(in)i(\(5.9\),)h (except)e(for)h(the)f(\014rst)h(one,)i(can)e(b)q(e)f(b)q(ounded)i(b)o (y)15 255 y Fl(\000)p Fn(d)p Fo(\()111 227 y Fj(~)103 235 y Fm(B)p 103 243 29 2 v 108 272 a Fj(2)144 219 y Fg(R)172 255 y Fo(\()p Fn(aw)q Fo(\))272 237 y Fj(2)272 267 y Fm(\020)305 255 y Fo(+)368 227 y Fj(~)361 235 y Fm(C)p 361 243 28 2 v 366 272 a Fj(2)401 219 y Fg(R)429 255 y Fo(\()p Fn(aw)q Fo(\))529 237 y Fj(2)529 267 y Fm(\020)r(\020)579 255 y Fo(+)635 235 y Fm(")651 224 y Fi(6)p 635 243 34 2 v 643 272 a Fj(2)682 219 y Fg(R)710 255 y Fo(\()p Fn(aw)q Fo(\))810 237 y Fj(2)810 267 y Fm(\020)r(\020)r(\020)865 255 y Fo(\))19 b(for)g(some)f(p)q(ositiv)o(e) h(constan)o(t)g Fn(d)h Fo(su\016cien)o(tly)d(small.)15 315 y(This)f(is)h(done)f(is)h(a)g(similar)d(w)o(a)o(y)i(as)h(in)f (Lemma)f(5.7)i(b)o(y)f(using)h(the)f(assumptions)g(on)h Fl(N)7 b Fo(.)22 b(Th)o(us)17 b(for)15 375 y Fn(")f Fo(and)h Fn(\016)g Fo(su\016cien)o(tly)e(small)f(w)o(e)i(\014nd)h(that)98 460 y Fn(dF)p 98 482 64 2 v 108 528 a(dt)208 494 y Fl(\024)288 435 y Fg(Z)330 494 y Fo(\()362 481 y(~)349 494 y Fn(A)p Fo(\(1)11 b Fl(\000)g Fn(\036)530 481 y Fo(~)519 494 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))11 b(+)g Fn(K)725 501 y Fm(R)754 494 y Fn(\016)r Fo(\))g Fl(\000)863 460 y Fo(1)p 863 482 25 2 v 863 528 a(2)904 481 y(~)892 494 y Fn(B)r Fo(\()p Fn(@)979 473 y Fj(2)976 506 y Fm(\020)999 494 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))10 b(+)h Fn(\016)r(K)1269 501 y Fm(R)1298 494 y Fo(\))g(+)1388 481 y(~)1377 494 y Fn(C)t Fo(\()1440 460 y(1)p 1440 482 V 1440 528 a(2)1469 494 y Fn(@)1498 473 y Fj(4)1495 506 y Fm(\020)1517 494 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))11 b(+)g Fn(K)1764 501 y Fj(1)1784 494 y Fn(\016)r Fo(\))288 624 y Fl(\000)p Fn(c)348 631 y Fj(6)368 624 y Fn(")391 603 y Fj(2)410 624 y Fn(\016)i Fo(+)e Fn(c)515 631 y Fj(2)535 624 y Fn(")558 603 y Fj(6)588 624 y Fo(+)g Fn(d)681 577 y Fo(~)667 590 y Fn(A)p 667 612 37 2 v 673 658 a Fo(2)709 624 y(\)\()p Fn(aw)q Fo(\))828 603 y Fj(2)859 624 y Fl(\000)g Fn(dF)c Fo(\()p Fn(t)p Fo(\))p Fn(:)15 840 y Fo(No)o(w)16 b(w)o(e)g(study)g(the)g(deriv)m(ativ)o(e)f(of)h Fn(F)727 847 y Fj(1)763 840 y Fo(with)g(resp)q(ect)g(to)g Fn(t)20 925 y(dF)77 932 y Fj(1)p 20 947 77 2 v 37 993 a Fn(dt)112 959 y Fo(+)11 b Fn(dF)c Fo(\()p Fn(t)p Fo(\))42 b(=)f Fn(\014)438 925 y(dE)p 438 947 65 2 v 448 993 a(dt)518 959 y Fo(+)572 925 y Fn(dF)p 572 947 64 2 v 582 993 a(dt)652 959 y Fo(+)11 b Fn(dF)c Fo(\()p Fn(t)p Fo(\))322 1078 y Fl(\024)402 1019 y Fg(Z)444 1078 y Fo(\()p Fl(\000)507 1044 y Fo(1)p 507 1066 25 2 v 507 1112 a(2)536 1078 y Fn(\014)s(A\036)643 1065 y Fo(~)633 1078 y Fn(F)e Fo(\()p Fn(\036)p Fo(\))11 b(+)g Fn(a)823 1057 y Fj(2)842 1078 y Fo(\()874 1065 y(~)861 1078 y Fn(A)p Fo(\(1)g Fl(\000)g Fn(\036)1042 1065 y Fo(~)1031 1078 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))11 b(+)1201 1044 y Fn(d)p 1201 1066 26 2 v 1201 1112 a Fo(2)1243 1078 y(+)g Fn(K)1333 1085 y Fm(R)1362 1078 y Fn(\016)r Fo(\))f Fl(\000)1470 1044 y Fo(1)p 1470 1066 25 2 v 1470 1112 a(2)1511 1065 y(~)1500 1078 y Fn(B)r Fo(\()p Fn(@)1587 1057 y Fj(2)1584 1090 y Fm(\020)1606 1078 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))11 b(+)g Fn(\016)r(K)1877 1085 y Fm(R)1906 1078 y Fo(\))402 1194 y(+)451 1182 y(~)440 1194 y Fn(C)t Fo(\()503 1161 y(1)p 503 1183 V 503 1229 a(2)532 1194 y Fn(@)561 1174 y Fj(4)558 1207 y Fm(\020)580 1194 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))11 b(+)g Fn(K)827 1201 y Fj(1)847 1194 y Fn(\016)r Fo(\))g Fl(\000)f Fn(c)971 1201 y Fj(6)991 1194 y Fn(")1014 1174 y Fj(2)1034 1194 y Fn(\016)i Fo(+)f Fn(")1140 1174 y Fj(5)1160 1194 y Fn(c)1181 1201 y Fj(2)1200 1194 y Fo(\)\))p Fn(w)1274 1174 y Fj(2)1294 1194 y Fn(:)15 1304 y Fo(The)16 b(follo)o(wing)g (Lemma)e(leads)i(to)h(the)f(inequalit)o(y)e(as)j(stated)g(in)f(Theorem) e(5.3.)15 1394 y Fk(Lemm)o(a)i(5.9)24 b Ff(F)l(r)n(om)e(the)i (assumptions)f(on)h(the)f(nonline)n(arity)h Fl(N)31 b Ff(it)23 b(fol)r(lows)i(that)e(for)g Fn(")g Ff(and)g Fn(\016)15 1454 y Ff(su\016ciently)c(smal)r(l,)f(the)g(c)n(onstant)g Fn(\014)i Ff(c)n(an)e(b)n(e)g(chosen)g(such)g(that)270 1572 y Fl(\000)314 1538 y Fo(1)p 314 1560 V 314 1606 a(2)343 1572 y Fn(\014)s(A\036)450 1560 y Fo(~)440 1572 y Fn(F)5 b Fo(\()p Fn(\036)p Fo(\))11 b(+)g Fn(a)630 1552 y Fj(2)650 1572 y Fo(\()682 1560 y(~)669 1572 y Fn(A)o Fo(\(1)h Fl(\000)e Fn(\036)849 1560 y Fo(~)838 1572 y Fn(F)d Fo(\()p Fn(\036)p Fo(\))k(+)1009 1538 y Fn(d)p 1009 1560 26 2 v 1009 1606 a Fo(2)1050 1572 y(+)g Fn(K)1140 1579 y Fm(R)1169 1572 y Fn(\016)r Fo(\))g Fl(\000)1278 1538 y Fo(1)p 1278 1560 25 2 v 1278 1606 a(2)1319 1560 y(~)1307 1572 y Fn(B)s Fo(\()p Fn(@)1395 1552 y Fj(2)1392 1584 y Fm(\020)1414 1572 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))10 b(+)h Fn(\016)r(K)1684 1579 y Fm(R)1713 1572 y Fo(\))270 1689 y(+)319 1676 y(~)308 1689 y Fn(C)t Fo(\()371 1655 y(1)p 371 1677 V 371 1723 a(2)400 1689 y Fn(@)429 1668 y Fj(4)426 1701 y Fm(\020)448 1689 y Fo(\()p Fl(F)5 b Fo(\()p Fn(\036)p Fo(\)\))11 b(+)g Fn(K)695 1696 y Fj(1)715 1689 y Fn(\016)r Fo(\))f Fl(\000)h Fn(c)839 1696 y Fj(6)859 1689 y Fn(")882 1668 y Fj(2)902 1689 y Fn(\016)h Fo(+)f Fn(")1008 1668 y Fj(5)1028 1689 y Fn(c)1049 1696 y Fj(2)1068 1689 y Fo(\))693 b(\(5.10\))15 1796 y Ff(is)17 b(ne)n(gative.)15 1886 y Fk(Pro)r(of:)f Fo(The)h(pro)q(of)h(is)e(similar)f(to)i(that)g(of)g(Lemma)d(5.7)k(and)f (w)o(e)f(also)h(use)g(that)g Fn(a)p Fo(\()p Fn(\020)t Fo(\))e(=)f Fn(e)1733 1867 y Fr(\000)p Fm(\015)r(\020)1800 1886 y Fo(.)23 b(W)l(e)15 1946 y(consider)14 b(t)o(w)o(o)f(cases;)i Fn(\036)f Fl(\025)f Fn(\036)551 1953 y Fj(0)585 1946 y Fo(and)i Fn(\036)e(<)h(\036)801 1953 y Fj(0)835 1946 y Fo(where)f Fn(\036)1002 1953 y Fj(0)1036 1946 y Fo(is)h(su\016cien)o (tly)e(small.)19 b(If)13 b Fn(\036)h Fl(\025)g Fn(\036)1642 1953 y Fj(0)1661 1946 y Fo(,)1700 1933 y(~)1689 1946 y Fn(F)7 b Fo(\()p Fn(\036)p Fo(\))13 b Fl(\025)h Fn(f)1885 1953 y Fj(0)15 2006 y Fo(and)23 b(the)f(result)f(follo)o(ws)i(imm)o (ediatel)o(y)c(b)o(y)j(taking)h Fn(\014)h Fo(su\016cien)o(tly)c(large.) 40 b(T)l(o)22 b(handle)h(the)f(case)15 2066 y Fn(\036)e(<)h(\036)152 2073 y Fj(0)172 2066 y Fo(,)f(supp)q(ose)i Fl(N)7 b Fo(\()p Fn(\036)p Fo(\))21 b(=)588 2033 y Fg(P)632 2046 y Fr(1)632 2078 y Fm(k)q Fj(=)p Fm(k)696 2083 y Fi(0)725 2066 y Fn(c)746 2073 y Fm(k)767 2066 y Fn(\036)796 2048 y Fm(k)817 2066 y Fo(.)34 b(Then)20 b Fn(\036)1036 2054 y Fo(~)1025 2066 y Fn(F)6 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y(remaining)d(ones)j(and)g(the)f (result)g(follo)o(ws.)1043 b Fc(2)15 2548 y Fo(This)16 b(Lemma)e(implies)g(that)i(for)h Fn(")f Fo(and)h Fn(\016)h Fo(su\016cien)o(tly)c(small)803 2632 y Fn(dF)860 2639 y Fj(1)p 803 2654 77 2 v 820 2700 a Fn(dt)898 2666 y(<)g Fl(\000)p Fn(dF)7 b Fo(\()p Fn(t)p Fo(\))p Fn(;)935 2800 y Fo(20)p eop %%Page: 21 21 21 20 bop 15 68 a Fo(for)16 b(some)f(su\016cien)o(tly)g(small)f (constan)o(t)j Fn(d)p Fo(.)22 b(This)16 b(concludes)g(the)g(pro)q(of)h (of)g(theorem)d(5.3.)15 188 y Fk(Ac)n(kno)n(wledgemen)n(ts)h Fo(The)h(w)o(ork)f(of)i(CEW)f(w)o(as)h(supp)q(orted)f(in)g(part)g(b)o (y)g(the)f(NSF)h(gran)o(t)g(DMS-)15 248 y(9803164)21 b(and)e(the)f(w)o(ork)h(of)g(VR)f(w)o(as)h(supp)q(orted)g(in)f(part)h (b)o(y)f(the)g(Dutc)o(h)h(Science)e(Organisation)15 308 y(and)g(in)f(part)g(b)o(y)g(the)g(NSF)g(gran)o(t)g(DMS-9631755.)15 535 y Fp(A)80 b(The)27 b(function)f Fb(H)701 509 y Fn(\017)695 553 y(w)734 535 y Fa(\()p Fb(u;)12 b(v)s Fa(\))p Fp(.)15 644 y Fo(In)18 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b(this)g(in)o(to)g(the)g(condition)g(\(4.3\))h(leads)f(to)h Fn(c)c Fl(\025)h Fo(2)d Fl(\000)g Fn(")1280 2600 y Fj(4)1311 2618 y Fo(+)g Fn(hot)p Fo(.)935 2800 y(21)p eop %%Page: 22 22 22 21 bop 15 68 a Fp(B)81 b(App)r(endix)25 b(B)15 177 y Fo(In)11 b(this)g(app)q(endix,)h(w)o(e)f(b)q(ound)h(the)f(term)f Fn(")808 159 y Fj(6)827 177 y Fl(j)849 142 y Fg(R)885 177 y Fn(Raw)984 159 y Fj(2)1004 177 y Fn(w)1040 159 y Fj(\(6\))1088 177 y Fl(j)k(\024)f Fn(")1191 159 y Fj(6)1211 177 y Fo(\()p Fl(j)1252 142 y Fg(R)1288 177 y Fn(R)1325 184 y Fm(\020)1345 177 y Fn(aw)1407 159 y Fj(2)1426 177 y Fn(w)1462 159 y Fj(\(5\))1510 177 y Fl(j)q Fo(+)q Fl(j)1586 142 y Fg(R)1621 177 y Fn(\015)s(Raw)1748 159 y Fj(2)1768 177 y Fn(w)1804 159 y Fj(\(5\))1852 177 y Fl(j)q Fo(+)15 237 y Fl(j)p Fo(2)61 202 y Fg(R)97 237 y Fn(Raw)q(w)231 244 y Fm(\020)251 237 y Fn(w)287 219 y Fj(\(5\))335 237 y Fl(j)p Fo(\).)36 b(W)l(e)21 b(study)g(all)g(these)g(terms)f (separately)l(.)36 b(Applying)21 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Fo(+)1617 398 y(1)p 1617 420 V 1617 465 a(4)1646 431 y Fn(")1669 411 y Fj(10)1715 373 y Fg(Z)1765 431 y Fn(w)1801 411 y Fj(\(5\))1847 399 y Fi(2)15 554 y Fo(where)16 b(w)o(e)f(also)i(use)g(inequalit)o(y)d (\(5.6\).)21 b(Also,)15 683 y Fn(")38 662 y Fj(6)57 683 y Fl(j)79 624 y Fg(Z)129 683 y Fn(Raw)228 662 y Fj(2)248 683 y Fn(w)284 662 y Fj(\(5\))332 683 y Fl(j)13 b(\024)h Fn(")435 662 y Fj(2)463 624 y Fg(Z)513 683 y Fn(R)550 662 y Fj(2)570 683 y Fn(a)596 662 y Fj(2)615 683 y Fn(w)651 662 y Fj(4)671 683 y Fo(+)714 649 y(1)p 714 671 V 714 717 a(4)743 683 y Fn(")766 662 y Fj(10)812 624 y Fg(Z)862 683 y Fn(w)898 662 y Fj(\(5\))944 650 y Fi(2)976 683 y Fl(\024)1034 649 y Fo(1)p 1034 671 V 1034 717 a(2)1063 683 y Fn(")1086 662 y Fj(2)1106 683 y Fn(\016)1130 662 y Fj(2)1149 683 y Fo(\(2+)1239 649 y(1)p 1235 671 32 2 v 1235 717 a Fn(\013)1271 683 y Fo(\))p Fn(K)1335 662 y Fj(2)1331 695 y Fm(R)1369 624 y Fg(Z)1419 683 y Fl(j)1444 670 y Fo(~)1433 683 y Fn(F)6 b Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fn(\036w)1617 662 y Fj(2)1637 683 y Fo(+)1680 649 y(1)p 1680 671 25 2 v 1680 717 a(4)1709 683 y Fn(")1732 662 y Fj(10)1777 624 y Fg(Z)1827 683 y Fn(w)1863 662 y Fj(\(5\))1909 650 y Fi(2)15 802 y Fo(and)17 b(\014nally)82 928 y Fl(j)p Fo(2)128 870 y Fg(Z)179 928 y Fn(Raw)q(w)313 935 y Fm(\020)333 928 y Fn(w)369 908 y Fj(\(5\))416 928 y Fl(j)d(\024)f Fo(4)p Fn(")543 908 y Fj(2)572 870 y Fg(Z)621 928 y Fn(R)658 908 y Fj(2)679 928 y Fn(a)705 908 y Fj(2)724 928 y Fn(w)760 908 y Fj(2)780 928 y Fn(w)816 908 y Fj(2)815 941 y Fm(\020)847 928 y Fo(+)901 895 y(1)p 901 917 V 901 963 a(4)930 928 y Fn(")953 908 y Fj(10)999 870 y Fg(Z)1048 928 y Fn(w)1084 908 y Fj(\(5\))1130 896 y Fi(2)1163 928 y Fl(\024)h Fo(4)p Fn(")1263 908 y Fj(2)1283 928 y Fn(K)1328 908 y Fj(2)1324 941 y Fm(R)1353 928 y Fn(\016)1377 908 y Fj(2)1404 870 y Fg(Z)1454 928 y Fn(w)1490 908 y Fj(2)1489 941 y Fm(\020)1521 928 y Fo(+)1575 895 y(1)p 1575 917 V 1575 963 a(4)1605 928 y Fn(")1628 908 y Fj(10)1673 870 y Fg(Z)1723 928 y Fn(w)1759 908 y Fj(\(5\))1805 896 y Fi(2)1823 928 y Fn(:)15 1048 y Fo(Therefore,)h(taking)i(together) f(all)g(these)g(terms)e(w)o(e)i(\014nd)h(that)292 1174 y Fn(")315 1154 y Fj(6)335 1174 y Fl(j)357 1116 y Fg(Z)407 1174 y Fn(Raw)506 1154 y Fj(2)526 1174 y Fn(w)562 1154 y Fj(\(6\))609 1174 y Fl(j)d(\024)g Fn(K)t Fo(\()754 1116 y Fg(Z)803 1174 y Fl(j)828 1162 y Fo(~)817 1174 y Fn(F)7 b Fo(\()p Fn(\036)p Fo(\))p Fl(j)p Fn(\036w)1002 1154 y Fj(2)1032 1174 y Fo(+)k Fn(\016)1105 1154 y Fj(2)1124 1174 y Fn(")1147 1154 y Fj(2)1175 1116 y Fg(Z)1225 1174 y Fn(w)1261 1154 y Fj(2)1260 1187 y Fm(\020)1281 1174 y Fo(\))g(+)1365 1141 y(3)p 1365 1163 V 1365 1209 a(4)1394 1174 y Fn(")1417 1154 y Fj(10)1463 1116 y Fg(Z)1513 1174 y Fn(w)1549 1154 y Fj(\(5\))1595 1142 y Fi(2)1613 1174 y Fn(;)15 1294 y Fo(for)16 b(some)f(constan)o(t)i Fn(K)h(>)c Fo(0.)15 1460 y Fp(References)39 1569 y Fo([1])24 b(M.E.)11 b(Akv)o(eld,)g(J.)g(Hulshof)h(\(1998\))i(T)l(ra)o(v)o(elling)c(w)o(a)o (v)o(e)i(solutions)g(of)h(a)f(fourth)h(order)f(semi-linear)115 1630 y(di\013usion)17 b(equation,)e Ff(Appl.)j(Math.)f(L)n(ett.)f Fk(11)p Fo(,)g(115-120.)39 1730 y([2])24 b(D.G.)c(Aronson,)i(H.F.)d(W)l (ein)o(b)q(erger)g(\(1975\))j(Nonlinear)e(di\013usion)g(in)g(p)q (opulation)i(genetics,)115 1791 y(com)o(bustion,)c(and)h(nerv)o(e)e (pulse)h(propagation,)j Ff(L)n(e)n(ctur)n(e)e(Notes)h(in)g(Mathematics) g(446,)g(Par-)115 1851 y(tial)g(Di\013er)n(ential)g(Equations)h(and)f (R)n(elate)n(d)g(T)l(opics)p Fo(,)f(ed.)f(J.A.)g(Goldstein,)g (Springer-V)l(erlag,)115 1911 y(Berlin,)c(5-49.)39 2012 y([3])24 b(J.B.)16 b(v)m(an)i(den)f(Berg,)g(J.)g(Hulshof,)g(R.)g(v)m (an)h(der)f(V)l(orst)g(\(1999\))i(T)l(ra)o(v)o(elling)d(w)o(a)o(v)o(es) g(for)i(fourth-)115 2072 y(order)e(semilinear)e(parab)q(olic)j (equations,)f(preprin)o(t)f(Leiden)h(Univ)o(ersit)o(y)l(.)39 2173 y([4])24 b(J.)13 b(Bricmon)o(t,)f(A.)h(Kupiainen)g(\(1994\))j (Stabilit)o(y)c(of)i(mo)o(ving)e(fron)o(ts)i(in)g(the)g (Ginzburg-Landau)115 2233 y(equation,)i Ff(Comm.)h(Math.)g(Phys.)e Fk(159)p Fo(,)h(287-318.)39 2334 y([5])24 b(E.)15 b(Bo)q(densc)o(hatz,) f(M.)g(Kaiser,)h(L.)f(Kramer,)g(W.)g(P)o(esc)o(h,)g(A.)g(W)l(eb)q(er,)g (W.)h(Zimm)o(erm)o(an)d(\(1990\))115 2394 y(P)o(atterns)h(and)g (defects)f(in)g(liquid)f(crystals,)i Ff(New)i(tr)n(ends)f(in)g(nonline) n(ar)h(dynamics)f(and)g(p)n(attern)115 2454 y(forming)23 b(phenomena:)35 b(The)23 b(ge)n(ometry)g(of)g(none)n(quilibrium)h Fo(Eds.)e(P)l(.)g(Coullet,)h(P)l(.)f(Huerre,)115 2514 y(NA)l(TO)16 b(ASI)f(Series,)g(Plen)o(um)f(Press,)i(New)g(Y)l(ork,)f (111.)39 2615 y([6])24 b(B.)19 b(Bu\013oni,)i(A.R.)e(Champneys,)h(J.F.) f(T)l(oland)i(\(1996\))h(Bifurcation)d(and)i(coalescence)e(of)h(a)115 2675 y(plethora)14 b(of)h(homo)q(clinic)d(orbits)i(for)h(a)f (Hamiltonian)f(system,)f Ff(J.)j(Dyn.)h(Di\013.)f(Eq.)f Fk(8)p Fo(,)g(221-279.)935 2800 y(22)p eop %%Page: 23 23 23 22 bop 39 68 a Fo([7])24 b(P)l(.)18 b(Coullet,)g(C.)g(Elphic)o(k,)f (D.)i(Repaux)f(\(1987\))i(Nature)e(of)h(spatial)g(c)o(haos,)g Ff(Phys.)f(R)n(ev.)i(L)n(ett.)115 128 y Fk(58)p Fo(,)c(431-434.)39 229 y([8])24 b(G.T.)16 b(Dee,)g(W.)g(v)m(an)h(Saarlo)q(os)i(\(1988\))f (Bistable)d(systems)g(with)i(propagating)h(fron)o(ts)f(leading)115 290 y(to)g(pattern)f(formation,)f Ff(Phys.)i(R)n(ev.)g(L)n(ett.)f Fk(60)p Fo(,)g(2641-2644)q(.)39 391 y([9])24 b(U.)16 b(Eb)q(ert,)h(W.)g(v)m(an)h(Saarlo)q(os)h(\(1999\))f(F)l(ron)o(t)f (propagation)i(in)o(to)e(unstable)g(states:)24 b(univ)o(ersal)115 451 y(algebraic)c(con)o(v)o(ergence)f(to)o(w)o(ards)i(uniformly)d (translating)k(pulled)d(fron)o(ts,)i(preprin)o(t)f(Leiden)115 512 y(Univ)o(ersit)o(y)l(.)15 613 y([10])k(J.-P)l(.)e(Ec)o(kmann,)f (C.E.)h(W)l(a)o(yne)g(\(1994\))i(The)e(non-linear)g(stabilit)o(y)f(of)h (fron)o(t)h(solutions)f(for)115 673 y(parab)q(olic)17 b(partial)f(di\013eren)o(tial)f(equations,)h Ff(Commun.)h(Math.)g (Phys.)f Fk(161\(2\))p Fo(,)f(323-334.)15 775 y([11])24 b(N.)18 b(F)l(enic)o(hel)f(\(1979\))j(Geometric)d(singular)i(p)q (erturbation)h(theory)f(for)g(ordinary)g(di\013eren)o(tial)115 835 y(equations,)d Ff(J.)h(Di\013.)g(Eq.)g Fk(31)p Fo(,)e(53-98.)15 937 y([12])24 b(P)l(.C.)f(Fife)f(\(1979\))i(Mathematical)d(asp)q(ects)j (of)f(reacting)g(and)h(di\013using)f(systems,)g Ff(L)n(e)n(ctur)n(e)115 997 y(Notes)18 b(in)g(Biomathematics)f Fk(28)p Fo(,)f(Springer-V)l (erlag,)f(New)h(Y)l(ork.)15 1099 y([13])24 b(R.A.)12 b(Fisher)h(\(1937\))i(The)e(adv)m(ance)h(of)g(adv)m(an)o(tageous)h (genes,)f Ff(A)o(nn.)i(of)f(Eugenics)g Fk(7)p Fo(,)e(355-369.)15 1200 y([14])24 b(S.)d(F)l(o)q(can)o(t,)i(Th.)e(Galla)o(y)g(\(1998\))j (Existence)c(and)i(stabilit)o(y)e(of)i(propagating)h(fron)o(ts)f(for)g (an)115 1260 y(auto)q(catalytic)16 b(reaction-di\013usion)h(system,)d Ff(Phys.)j(D)f Fk(120)p Fo(,)g(346-368.)15 1362 y([15])24 b(Th.)g(Galla)o(y)g(\(1994\))i(Lo)q(cal)f(stabilit)o(y)e(of)h(critical) f(fron)o(ts)h(in)g(nonlinear)g(parab)q(olic)h(partial)115 1422 y(di\013eren)o(tial)15 b(equations,)h Ff(Nonline)n(arity)h Fk(7)p Fo(,)e(741-764.)15 1524 y([16])24 b(Th.)18 b(Galla)o(y)l(,)f(G.) h(Raugel)g(\(1997\))h(Stabilit)o(y)d(of)j(tra)o(v)o(elling)d(w)o(a)o(v) o(es)h(for)h(a)g(damp)q(ed)f(h)o(yp)q(erb)q(olic)115 1584 y(equation,)f Ff(Z.)h(angew.)i(Math.)e(Phys.)f Fk(48)p Fo(,)g(451-479.)15 1686 y([17])24 b(Th.)13 b(Galla)o(y)l(,)f(G.)h (Raugel)f(\(1998\))j(Scaling)d(v)m(ariables)h(and)g(asymptotic)f (expansions)h(in)f(damp)q(ed)115 1746 y(w)o(a)o(v)o(e)j(equations,)h Ff(J.)h(Di\013.)g(Eq.)g Fk(150)p Fo(,)f(42-97.)15 1847 y([18])24 b(C.K.R.T.)10 b(Jones)h(\(1995\))i(Geometric)c(singular)i(p)q (erturbation)h(theory)e(in)h Ff(Dynamic)n(al)i(systems,)115 1908 y(Monte)n(c)n(atibi)21 b(T)l(erme,)h(1994)p Fo(,)e(Lecture)g (Notes)g(in)g(Mathematics,)e Fk(1609)p Fo(,)j(R.)e(Johnson)j(\(ed.\),) 115 1968 y(Springer-V)l(erlag.)15 2069 y([19])i(W.D.)13 b(Kalies,)f(J.)h(Kw)o(apisz,)g(R.C.A.M.)e(v)m(an)j(der)f(V)l(orst)g (\(1998\))i(Homotop)o(y)d(classes)h(for)h(stable)115 2130 y(connections)j(b)q(et)o(w)o(een)g(Hamiltonian)f(saddle-fo)q(cus)i (equilibria,)e Ff(Comm.)i(Math.)g(Phys.)f Fk(193)p Fo(,)115 2190 y(337-371.)15 2291 y([20])24 b(W.D.)15 b(Kalies,)f(R.C.A.M.)f(v)m (an)i(der)g(V)l(orst)h(\(1996\))g(Multitransition)f(homo)q(clinic)e (and)i(hetero-)115 2352 y(clinic)f(solutions)j(of)g(the)f(eFK)f (equation,)h Ff(J.)h(Di\013.)g(Eq.)g Fk(131)p Fo(,)f(209-228.)15 2453 y([21])24 b(T.)c(Kapitula)g(\(1994\))i(On)f(the)f(stabilit)o(y)f (of)h(tra)o(v)o(elling)f(w)o(a)o(v)o(es)h(in)g(w)o(eigh)o(ted)f Fn(L)1638 2435 y Fr(1)1696 2453 y Fo(spaces,)i Ff(J.)115 2513 y(Di\013.)c(Eq.)g Fk(112)p Fo(,)f(179-215.)15 2615 y([22])24 b(K.)17 b(Kirc)o(hg\177)-24 b(assner)18 b(\(1992\))i(On)e (the)f(nonlinear)h(dynamics)f(of)h(tra)o(v)o(elling)e(fron)o(ts,)i Ff(J.)h(Di\013.)f(Eq.)115 2675 y Fk(96)p Fo(,)e(256-278.)935 2800 y(23)p eop %%Page: 24 24 24 23 bop 15 68 a Fo([23])24 b(A.N.)17 b(Kolmogoro)o(v,)g(I.G.)h(P)o (etro)o(vskii,)e(N.S.)h(Piskuno)o(v)h(\(1937\))i(Etude)f(de)f(la)g (di\013usion)h(a)o(v)o(ec)115 128 y(croissance)i(de)g(la)h(quan)o(tit)o (\023)-23 b(e)20 b(de)h(mati)o(\022)-23 b(ere)20 b(et)h(son)h (application)f(\022)-24 b(a)22 b(un)f(probl)o(\022)-23 b(eme)20 b(biologique,)115 188 y Ff(Mosc)n(ow)d(Univ.)i(Math.)e(Bul)r (l.)h Fk(1)p Fo(,)d(1-25.)15 290 y([24])24 b(A.)13 b(P)o(azy)h (\(1983\))h(Semigroups)e(of)h(linear)f(op)q(erators)j(and)e (applications)g(to)g(partial)g(di\013eren)o(tial)115 350 y(equations,)i Ff(Appl.)i(Math.)f(Sci.)g Fk(44)p Fo(,)f(Springer-V)l(erlag,)f(New)h(Y)l(ork.)15 452 y([25])24 b(L.A.)16 b(P)o(eletier,)f(W.C.)i(T)l(ro)o(y)g(\(1995\))i(Spatial)e (patterns)g(describ)q(ed)g(b)o(y)g(the)g(extended)f(Fisher-)115 512 y(Kolmogoro)o(v)g(\(eFK\))f(equation:)21 b(Kinks,)16 b Ff(Di\013.)h(Int.)h(Eq.)e Fk(8)p Fo(,)g(1279-1304.)15 613 y([26])24 b(L.A.)16 b(P)o(eletier,)f(W.C.)i(T)l(ro)o(y)g(\(1997\))i (Spatial)e(patterns)g(describ)q(ed)g(b)o(y)g(the)g(extended)f(Fisher-) 115 674 y(Kolmogoro)o(v)11 b(\(eFK\))g(equation:)19 b(P)o(erio)q(dic)11 b(solutions,)h Ff(SIAM)i(J.)e(Math.)h(A)o(nal.)f Fk(28)p Fo(,)h(1318-1353.)15 775 y([27])24 b(L.A.)18 b(P)o(eletier,)f(W.C.)h(T) l(ro)o(y)l(,)h(R.C.A.M.)d(v)m(an)j(der)g(V)l(orst)f(\(1995\))j (Stationary)e(solutions)g(of)g(a)115 836 y(fourth-order)e(nonlinear)f (di\013usion)h(equation,)f Ff(J.)h(Di\013.)g(Eq.)f Fk(31)p Fo(,)g(301-314.)15 937 y([28])24 b(V.)11 b(Rottsc)o(h\177)-24 b(afer)11 b(\(1998\))i(Co-dimension)d(2)i(phenomena)f(in)g(pattern)g (formation,)h(Ph.)f(D.)g(thesis,)115 997 y(Utrec)o(h)o(t)j(Univ)o (ersit)o(y)l(,)g(the)i(Netherlands.)15 1099 y([29])24 b(V.)17 b(Rottsc)o(h\177)-24 b(afer)17 b(\(1999\))j(Multi-bump)15 b(patterns)j(b)o(y)g(a)g(normal)e(form)h(approac)o(h,)h(Cen)o(ter)f (for)115 1159 y(BioDynamics)d(preprin)o(t)i(nr.)g(16,)g(Boston)h(Univ)o (ersit)o(y)l(.)15 1261 y([30])24 b(V.)d(Rottsc)o(h\177)-24 b(afer,)24 b(A.)d(Do)q(elman)h(\(1998\))h(On)g(the)f(transition)g(from) g(the)g(Ginzburg-Landau)115 1321 y(equation)16 b(to)h(the)f(extended)f (Fisher-Kolmogoro)o(v)g(equation,)h Ff(Physic)n(a)h(D)f Fk(118)p Fo(,)g(261-292.)15 1423 y([31])24 b(D.H.)14 b(Sattinger)h(\(1976\))i(On)e(the)g(stabilit)o(y)f(of)h(w)o(a)o(v)o(es) g(of)g(nonlinear)g(parab)q(olic)h(systems,)e Ff(A)n(dv.)115 1483 y(Math.)i Fk(22)p Fo(,)g(312-355.)15 1585 y([32])24 b(A.I.)11 b(V)l(olp)q(ert,)i(V.A.)f(V)l(olp)q(ert,)g(V.A.)g(V)l(olp)q (ert)g(\(1994\))j(T)l(ra)o(v)o(eling)d(w)o(a)o(v)o(e)g(solutions)i(of)f (parab)q(olic)115 1645 y(systems,)i Ff(T)l(r)n(anslations)i(of)g(Math.) h(Mono)n(gr)n(aphs)c Fo(V)l(ol.)i Fk(140)p Fo(,)f(AMS.)15 1747 y([33])24 b(W.)14 b(Zimm)o(erm)o(an)d(\(1991\))16 b(Propagating)g(fron)o(ts)e(near)g(a)h(Lifshitz)f(p)q(oin)o(t,)g Ff(Phys.)h(R)n(ev.)g(L)n(ett.)f Fk(66)p Fo(,)115 1807 y(1546.)935 2800 y(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0004201223115--