This is a post-script file divided in three pieces. Part 1: 159155 B Part 2: 145674 B Part 3: 137733 B LaTeX source available by e-mail request: jbarata@fma.if.usp.br . 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b(The)n(or)n(em)f(1.2)h(imme)n(diately)f(implies)i(absenc)n(e)g(of)e (metastability)i(for)-84 712 y(our)d(dilute)n(d)h(mo)n(del.)22 b(We)16 b(r)n(e)n(c)n(al)r(l)g(that)h(our)f(mo)n(dels)g(distinguish)h (the)g(c)n(ontribution)g(of)f(the)h(\014nite)g(clusters)h(fr)n(om)-84 772 y(the)h(c)n(ontribution)h(of)f(the)g(in\014nite)h(cluster.)28 b(When)19 b FC(p)d FB(=)h(1)i Fn(the)g(system)g(r)n(e)n(duc)n(es)f(to)h (the)g(deterministic)h(Ising)-84 832 y(ferr)n(omagnet)d(on)h(the)g (Cayley)g(tr)n(e)n(e)f(which)h(exhibits)g(metastability)h(as)e(in)h (the)g(me)n(an)f(\014eld)i(the)n(ory.)-11 968 y FB(Recen)o(tly)l(,)13 b(Dreifus,)i(Klein)f(and)i(P)o(erez)e([DKP])h(ha)o(v)o(e)f(established) h(in\014nite)f(di\013eren)o(tiabilit)o(y)f(of)j(the)f(quen-)-84 1028 y(c)o(hed)h(magnetization)f(for)i(general)f(disordered)g(spin)g (systems)g(using)g(a)h(mo)q(di\014ed)f(high)g(temp)q(erature)f(expan-) -84 1088 y(sion)h(whic)o(h)f(con)o(v)o(erges)f(in)i(the)f(presence)g (of)h(Gri\016ths')e(singularities.)21 b(W)l(e)15 b(here)g(are)h(able)f (to)h(study)g(the)f(righ)o(t)-84 1148 y(and)i(left)g(deriv)m(ativ)o(es) e(of)i Fz(m)g FB(at)g Fz(z)g FB(=)e(1)i(when)g(sp)q(on)o(taneous)i (magnetization)d(o)q(ccurs)h(and)h(the)e(high)h(temp)q(era-)-84 1208 y(ture)f(expansion)g(do)q(es)h(not)g(con)o(v)o(erge.)j(In)c(view)g (of)g(Theorems)f(1.1)i(and)g(1.2)f(it)g(su\016ces)g(to)h(examine)d (only)i(the)-84 1269 y(neigh)o(b)q(orho)q(o)q(d)i(of)f Fz(z)e FB(=)f(1)j(in)f Fz(z)i Fx(2)c Fs(R)593 1276 y Fy(+)636 1269 y FB(when)i Fz(\020)i(<)c(\020)875 1276 y Ft(c)892 1269 y FB(.)-84 1404 y FC(Theorem)i(1.6)j(\(Con)n(tin)n(uit) n(y)g(and)g(Di\013eren)n(tiabilit)n(y\))i Fn(F)l(or)f(any)h FB(0)f Fx(\024)g Fz(\020)k(<)19 b(\020)1504 1411 y Ft(c)1522 1404 y Fn(,)i Fz(z)h Fx(2)e Fs(R)1695 1411 y Fy(+)1742 1404 y Fn(and)h FC(p)f Fx(2)g Fz(\031)1972 1411 y Fy(0)1992 1404 y Fn(,)-84 1464 y(the)f(quenche)n(d)g(magnetization)h Fz(m)d Fn(given)j(by)e(\(1.7\))g(is)g(always)g(a)g(c)n(ontinuous)h (function)h(of)e Fz(z)i Fn(with)e Fz(m)p FB(\(1\))d(=)g(0)p Fn(.)-84 1525 y(Mor)n(e)n(over,)j Fz(F)26 b Fn(is)19 b(at)g(le)n(ast)g Fz(n)h Fn(-times)g(di\013er)n(entiable)g(in)f Fz(z)g Fx(2)e Fs(R)1114 1532 y Fy(+)1160 1525 y Fn(for)i FB(0)e Fz(<)g(\020)k(<)c(\020)1454 1532 y Ft(c)1490 1525 y Fn(and)j FC(p)d Fx(2)g Fz(\031)r Fn(,)i(pr)n(ovide)n(d)f(the)-84 1585 y(se)n(quenc)n(e)h Fx(f)p Fz(a)165 1592 y Ft(j)200 1585 y Fz(\026)229 1567 y Ft(nj)269 1585 y Fx(g)294 1558 y Fq(1)294 1601 y Ft(j)r Fy(=0)375 1585 y Fn(with)f Fz(\026)c FB(=)g Fz(\026)p FB(\()p Fz(\020)t(;)i(z)r FB(\))h Fn(given)i(by)505 1760 y Fz(\026)28 b FB(:=)f(4)p Fz(\020)t(H)t(z)g FB(+)e(2)879 1675 y Fr(2)880 1749 y(4)907 1687 y( )945 1726 y FB(4)p Fz(\020)t(H)t(z)14 b FB(+)d(1)p 945 1748 204 2 v 1035 1794 a(4)1153 1687 y Fr(!)1186 1699 y Fy(2)1217 1760 y FB(+)g Fz(\020)t(H)1335 1675 y Fr(3)1335 1749 y(5)1363 1686 y Fy(1)p Ft(=)p Fy(2)-84 1911 y Fn(is)17 b(summable,)i(i.e.,)691 1966 y FB(1)p 682 1988 43 2 v 682 2034 a Fz(n)p FB(!)738 1925 y 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b(that)h(\001)1081 1672 y Ft(j)1116 1665 y FB(dep)q(ends)g(on)g(the)f(generation)h(lev)o(el)d Fz(M)23 b FB(of)18 b(the)-84 1725 y(Ca)o(yley)c(tree)g(through)j(its)d (initial)g(condition.)21 b(The)15 b(distribution)g(of)g(\001)1254 1732 y Fy(0)1273 1725 y FB(,)g(when)g Fz(M)20 b Fx(!)13 b(1)p FB(,)i(is)g(w)o(orth)g(studying)-84 1785 y(b)o(y)i(its)g(o)o(wn)g (righ)o(t.)24 b(In)17 b(the)g(next)g(theorem)f(it)g(will)h(b)q(e)g (describ)q(ed)g(the)g(analytic)f(prop)q(erties)i(of)f(the)g(momen)o(ts) -84 1846 y Fx(M)-24 1853 y Ft(s)8 1846 y FB(=)d Fs(E)90 1853 y Ft(\030)120 1846 y FB([\001)175 1853 y Fy(0)194 1846 y FB(\()p Fz(\030)r FB(\))255 1827 y Ft(s)274 1846 y FB(],)h Fz(s)f Fx(2)g Fs(N)p FB(,)j(of)g(\001)564 1853 y Fy(0)599 1846 y FB(whic)o(h)f(can)g(b)q(e)h(written)e(as)775 1987 y Fx(M)835 1994 y Ft(s)881 1987 y FB(=)27 b Fx(F)982 1994 y Ft(s)1026 1987 y FB(+)d Fx(I)1115 1994 y Ft(s)1134 1987 y Fz(;)734 b FB(\(1.12\))-84 2073 y(where)16 b Fx(F)93 2080 y Ft(s)139 2073 y FB(:=)47 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Ft(s)1041 2537 y FB(:)28 b Fs(C)12 b Fx(\002)s FB([0)p Fz(;)c FB(1])t Fx(\002)s Fz(\031)18 b Fx(3)c FB(\()p Fz(z)r(;)8 b(\020)t(;)g FC(p)p FB(\))14 b Fx(7\000)-9 b(!)14 b(M)1729 2544 y Ft(s)1747 2537 y FB(\()p Fz(z)r(;)8 b(\020)t(;)g FC(p)p FB(\))13 b Fx(2)i Fs(C)-84 2597 y Fn(b)n(e)k(given)h(by)f(\(1.12\).)25 b(Then)19 b Fx(M)528 2604 y Ft(s)565 2597 y Fn(is)f(an)h(analytic)g(function)h(in)f Fz(z)h Fn(for)e Fx(j)p Fz(z)r Fx(j)e Fz(<)f FB(1)p Fn(.)26 b(In)19 b(addition,)g(for)f Fz(\020)i Fx(2)c FB([0)p Fz(;)24 b FB(1])p Fn(,)-84 2657 y Fx(M)-24 2664 y Ft(s)12 2657 y Fn(is)17 b(a)g(holomorphic)g(function)i(of)e Fz(z)j Fn(in)d(the)h(op)n(en)g(set)g Fx(f)p Fz(z)e Fx(2)e Fs(C)9 b FB(;)38 b Fx(j)p Fz(z)r Fx(j)14 b Fz(>)f FB(1)p Fx(g)f(n)p 1417 2618 41 2 v 11 w(Z)t Fn(,)p 1489 2618 V 17 w Fx(Z)22 b Fn(b)n(eing)c(the)g(closur)n(e)g(of)835 2764 y Fx(Z)g FB(:=)965 2722 y Fr([)954 2815 y Ft(a)p Fq(2)p Fj(N)1030 2764 y Fx(Z)1066 2771 y Ft(a)949 2930 y FB(8)p eop %%Page: 9 9 9 8 bop -84 79 a 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Fv(Concerning)g(this)g(restriction)h(see)g(the)f (Remark)f(7.11.)949 2930 y FB(9)p eop %%Page: 10 10 10 9 bop 1969 79 a Fm(2)-84 248 y FC(Remark)16 b(1.10)24 b Fn(A)g(simple)g(upp)n(er)f(b)n(ound)g(on)h(\(1.16\))f(for)g(the)h(p)n (ar)n(amagnetic)f(phase)g(c)n(an)h(b)n(e)f(obtaine)n(d)h(by)-84 309 y(using)c(Per)n(ez's)e(self-avoiding)j(r)n(andom)d(walk)i(estimate) g([P].)27 b(Only)20 b(one)f(term,)g(c)n(orr)n(esp)n(onding)f(to)i(the)f (single)-84 369 y(self-avoiding)f(p)n(ath)e(c)n(onne)n(cting)i FB(0)f Fn(to)f Fz(x)p Fn(,)g(c)n(ontributes)i(to)e(the)h(c)n(orr)n (elation)f(functions)i(in)e Fx(C)1620 376 y Ft(k)1642 369 y Fn(.)22 b(This)16 b(fact)g(shows)-84 429 y(that)j(the)h(Ising)g (mo)n(del)g(on)f(the)h(Cayley)f(tr)n(e)n(e)g(b)n(ehaves,)i(on)f(what)f (c)n(onc)n(erns)h(the)g(asymptotic)f(b)n(ehavior)h(of)f(the)-84 489 y(c)n(orr)n(elations,)c(as)f(a)h(one)g(dimensional)h(system.)21 b(As)15 b(a)f(c)n(onse)n(quenc)n(e,)j(the)e(c)n(orr)n(elation)f(length) j(is)d(always)h(\014nite,)-84 549 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Fz(\036)p FB(\()p Fz(n;)21 b FB(2)1463 229 y Ft(n)p Fy(+1)1543 249 y Fx(\000)11 b FB(1\))j Fx(\024)g FB(2)p Fz(\031)1757 201 y Fr(o)1793 249 y Fz(;)-84 359 y FB(where)711 420 y Fz(z)r FB(\()p Fz(n;)i(l)q FB(\))e Fz(\034)890 394 y Fy(\()p Ft(n)p Fy(\))884 434 y Ft(z)q Fy(\()p Ft(n;)6 b(l)p Fy(\))979 420 y FB(\(1\))28 b(=)g Fx(\000)p FB(1)p Fz(:)-84 507 y FB(Our)16 b(previous)g(remarks)f(sa)o(y)h(that)824 567 y Fz(\036)p FB(\()p Fz(n;)22 b FB(2)961 546 y Ft(n)984 567 y FB(\))14 b(=)g Fz(\031)-84 654 y FB(and)588 714 y Fz(\036)p FB(\()p Fz(n;)21 b(a)p FB(\))14 b(=)f(2)p Fz(\031)g Fx(\000)e Fz(\036)p FB(\()p Fz(n;)22 b FB(2)1062 694 y Ft(n)p Fy(+1)1142 714 y Fx(\000)11 b FB(1)g Fx(\000)g Fz(a)p FB(\))p Fz(;)-84 802 y FB(for)16 b(all)g Fz(a)22 b FB(=)g(1)p Fz(;)17 b(:)8 b(:)g(:)f(;)16 b FB(2)340 783 y Ft(n)364 802 y FB(.)-11 883 y(Let)g(us)h(denote)f(b)o(y)g Fu(R)405 890 y Ft(n)443 883 y Fx(\032)d FB([0)p Fz(;)22 b(\031)r FB(])15 b(the)h(set)545 993 y Fu(R)586 1000 y Ft(n)624 993 y FB(:=)d Fx(f)p Fz(\036)p FB(\()p Fz(n;)22 b(l)q FB(\))13 b Fx(2)h FB([0)p Fz(;)22 b(\031)r FB(])p Fz(;)56 b FB(1)14 b Fx(\024)f Fz(l)i Fx(\024)e FB(2)1306 973 y Ft(n)1330 993 y Fx(g)8 b Fz(:)-11 1125 y FB(In)16 b(the)g(next)g(t)o(w)o(o)g(subsections)g(w)o(e)g(will)f(establish)h (the)g(follo)o(wing)g(theorem:)-84 1260 y FC(Theorem)g(4.1)24 b Fn(The)18 b(set)g Fu(R)466 1267 y Ft(n)507 1260 y Fn(b)n(e)n(c)n (omes)g(a)f(dense)h(set)g(on)g FB([0)p Fz(;)k(\031)r FB(])16 b Fn(when)j Fz(n)14 b Fx(!)f(1)18 b Fn(for)e FB(0)f Fx(\024)e Fz(\020)18 b Fx(\024)c FB(1)p Fz(=)p FB(3)p Fn(.)1969 1396 y Fm(2)-11 1527 y FB(T)l(o)h(start)f(with)g(the)f (analysis,)h(let)f(us)i(lo)q(ok)f(in)f(more)g(detail)g(at)h(the)g (sequence)f Fz(z)r(\034)1488 1509 y Fy(\()p Ft(n)p Fy(\))1482 1540 y Ft(z)1538 1527 y FB(\(1\),)i Fz(n)f Fx(2)g Fs(N)p FB(,)g(for)h Fz(z)g Fx(2)f Fz(S)1973 1509 y Fy(1)1993 1527 y FB(.)-84 1587 y(Firstly)l(,)h(consider)g(the)h(function)h Fz(h)p FB(\()p Fz(w)q FB(\))f(for)h Fz(w)e Fx(2)f Fz(S)872 1569 y Fy(1)891 1587 y FB(.)21 b(Since)16 b(1)p Fz(=w)g FB(=)p 1204 1561 37 2 v 13 w Fz(w)i FB(w)o(e)e(can)g(write)714 1737 y Fz(h)p FB(\()p Fz(w)q FB(\))e(=)g Fz(w)918 1716 y Fy(2)946 1664 y Fr( )984 1703 y FB(1)d(+)g Fz(\020)p 1093 1677 V 4 w(w)p 984 1725 146 2 v 984 1771 a FB(1)g(+)g Fz(\020)t(w)1134 1664 y Fr(!)1167 1675 y Fy(2)1195 1737 y Fz(:)-84 1884 y FB(De\014ning)16 b Fz(w)f FB(=:)f Fz(e)249 1866 y Ft(i\022)296 1884 y FB(and)j Fz(h)p FB(\()p Fz(w)q FB(\))d(=:)f Fz(e)595 1866 y Ft(i\022)624 1855 y Fg(0)638 1884 y FB(,)i(it)h(is)g(easy)h(to)f(v)o(erify)f(the)h(relation)597 2025 y Fz(\022)621 2005 y Fq(0)661 2025 y FB(=)27 b(2)p Fz(\022)13 b Fx(\000)e FB(4)d(arctan)1012 1952 y Fr( )1095 1992 y Fz(\020)k FB(sin)c Fz(\022)p 1050 2014 216 2 v 1050 2060 a FB(1)j(+)g Fz(\020)i FB(cos)8 b Fz(\022)1270 1952 y Fr(!)1312 2025 y Fz(:)-11 2194 y FB(Hence,)20 b(de\014ning)g Fz(\036)h FB(through)g Fz(z)i FB(=:)d Fz(e)720 2176 y Ft(i\036)776 2194 y FB(and)h Fz(\036)904 2201 y Ft(j)942 2194 y FB(through)h Fz(z)10 b(\034)1191 2176 y Fy(\()p Ft(j)r Fy(\))1185 2206 y Ft(z)1236 2194 y FB(\(1\))22 b(=:)e Fz(e)1415 2176 y Ft(i\036)1448 2181 y Fl(j)1466 2194 y FB(,)h Fz(j)j Fx(2)d Fs(N)p FB(,)h(and)f(taking)g (in)o(to)-84 2254 y(accoun)o(t)16 b(that)h Fz(z)f(\034)268 2236 y Fy(\()p Ft(j)r Fy(+1\))262 2266 y Ft(z)358 2254 y FB(\(1\))e(=)g Fz(z)i(h)p FB(\()p Fz(z)10 b(\034)632 2236 y Fy(\()p Ft(j)r Fy(\))626 2266 y Ft(z)677 2254 y FB(\(1\)\),)16 b(w)o(e)g(ha)o(v)o(e)594 2362 y Fz(\036)623 2369 y Fy(0)724 2362 y FB(:=)41 b Fz(\036;)262 b(\036)14 b Fx(2)g FB([0)p Fz(;)8 b(\031)r FB(])p Fz(;)594 2483 y(\036)623 2490 y Ft(i)p Fy(+1)724 2483 y FB(:=)41 b Fz(\036)846 2490 y Fy(0)876 2483 y FB(+)11 b Fz(L)p FB(\()p Fz(\036)1006 2490 y Ft(i)1020 2483 y FB(\))83 b Fz(i)14 b Fx(2)g Fs(N)p Fz(;)1906 2423 y FB(\(4.3\))-84 2589 y(where)565 2668 y Fz(L)p FB(\()p Fz(x)p FB(\))g(:=)f(2)p Fz(x)e Fx(\000)g FB(4)d(arctan)1032 2595 y Fr( )1115 2635 y Fz(\020)13 b FB(sin)j Fz(x)p 1070 2657 228 2 v 1070 2702 a FB(1)c(+)f Fz(\020)h FB(cos)17 b Fz(x)1303 2595 y Fr(!)1344 2668 y Fz(;)548 b FB(\(4.4\))-84 2787 y Fz(x)13 b Fx(2)i Fs(R)e FB(and)k Fz(\020)h Fx(2)c FB([0)p Fz(;)21 b FB(1\).)h(These)16 b(equations)g(will)f(b)q(e)i(treated)f(as) h(a)f(discrete)g(dynamical)e(system)h(on)h Fs(R)p FB(.)937 2930 y(18)p eop %%Page: 19 19 19 18 bop -84 79 a FI(4.1)66 b(Prop)r(erties)23 b(of)f(the)g(Discrete)g (Dynamical)i(System)-84 193 y FC(Prop)r(osition)18 b(4.2)24 b Fn(F)l(or)19 b(al)r(l)h Fz(x)d Fx(2)h Fs(R)e Fn(and)k Fz(\020)h Fx(2)d FB([0)p Fz(;)25 b FB(1\))p Fn(,)20 b Fz(L)f Fn(is)h(a)f(c)n(ontinuous)h(monotone)g(incr)n(e)n(asing)g (function)-84 253 y(of)d Fz(x)g Fn(such)h(that)733 313 y Fz(L)766 293 y Fq(0)778 313 y FB(\()p Fz(x)p FB(\))c Fx(\025)f Fz(L)943 293 y Fq(0)955 313 y FB(\(0\))h Fx(\021)g Fz(\025)p FB(\()p Fz(\020)t FB(\))p Fz(;)717 b FB(\(4.5\))-84 401 y Fn(wher)n(e)742 479 y Fz(\025)p FB(\()p Fz(\020)t FB(\))28 b(:=)f(2)972 406 y Fr( )1011 446 y FB(1)11 b Fx(\000)g Fz(\020)p 1011 468 111 2 v 1011 514 a FB(1)g(+)g Fz(\020)1126 406 y Fr(!)1167 479 y Fz(:)725 b FB(\(4.6\))-84 595 y Fn(Mor)n(e)n(over,)16 b(the)i(function)h Fz(L)f Fn(has)f(the)h(fol)r(lowing)h(pr)n(op)n(erties:)-84 731 y FC(\(i\))k Fz(L)p FB(\()p Fz(x)11 b FB(+)g(2)p Fz(\031)r FB(\))j(=)g(4)p Fz(\031)f FB(+)e Fz(L)p FB(\()p Fz(x)p FB(\))p Fn(,)17 b Fz(x)d Fx(2)g Fs(R)p Fn(,)-84 832 y FC(\(ii\))23 b Fz(L)48 814 y Fq(00)69 832 y FB(\()p Fz(x)p FB(\))14 b Fx(\025)g FB(0)j Fn(,)h(for)f Fz(x)c Fx(2)h FB([0)p Fz(;)i(\031)r FB(])o Fn(,)-84 934 y FC(\(iii\))23 b Fz(L)p FB(\()p Fz(x)p FB(\))13 b(=)h(4)p Fz(\031)f Fx(\000)e Fz(L)p FB(\(2)p Fz(\031)i Fx(\000)e Fz(x)p FB(\))17 b Fn(,)g(for)g Fz(x)d Fx(2)g FB([)o(0)p Fz(;)j FB(2)p Fz(\031)r FB(])o Fn(,)-84 1095 y FC(\(iv\))23 b FB(lim)36 1125 y Ft(\020)r Fq(")p Fy(1)105 1095 y Fz(L)p FB(\()p Fz(x)p FB(\))14 b(=)270 995 y Fr(8)270 1033 y(>)270 1045 y(<)270 1120 y(>)270 1132 y(:)327 1034 y FB(0)72 b Fz(;)41 b FB(0)14 b Fx(\024)f Fz(x)h(<)g(\031)327 1094 y FB(2)p Fz(\031)44 b(;)d(x)13 b FB(=)h Fz(\031)327 1155 y FB(4)p Fz(\031)44 b(;)d(\031)15 b(<)f(x)f Fx(\024)h FB(2)p Fz(\031)764 1095 y Fn(.)1969 1328 y Fm(2)-84 1460 y FC(Pro)r(of.)21 b FB(By)16 b(de\014nition)559 1548 y Fz(L)592 1528 y Fq(0)604 1548 y FB(\()p Fz(x)p FB(\))d(=)h(2)767 1475 y Fr( )915 1515 y FB(1)d Fx(\000)g Fz(\020)1025 1497 y Fy(2)p 806 1537 349 2 v 806 1583 a FB(1)g(+)g(2)p Fz(\020)i FB(cos)c Fz(x)h FB(+)h Fz(\020)1134 1568 y Fy(2)1159 1475 y Fr(!)1206 1548 y Fx(\025)i Fz(\025)p FB(\()p Fz(\020)t FB(\))p Fz(:)543 b FB(\(4.7\))-84 1667 y(This)16 b(relation)g(sa)o(ys)h(that)f Fz(L)h FB(is)f(a)g(strictly)f (p)q(ositiv)o(e,)g(strictly)g(increasing)h(function)g(in)g Fs(R)1597 1674 y Fy(+)1635 1667 y Fx(n)11 b(f)p FB(0)p Fx(g)p FB(.)-11 1748 y(W)l(e)16 b(also)h(ha)o(v)o(e)615 1826 y Fz(L)648 1805 y Fq(00)669 1826 y FB(\()p Fz(x)p FB(\))d(=)848 1792 y(4)p Fz(\020)t FB(\(1)e Fx(\000)f Fz(\020)1027 1774 y Fy(2)1046 1792 y FB(\))d(sin)h Fz(x)p 806 1814 407 2 v 806 1865 a FB(\(1)i(+)g(2)p Fz(\020)i FB(cos)8 b Fz(x)j FB(+)g Fz(\020)1153 1850 y Fy(2)1173 1865 y FB(\))1192 1840 y Fy(2)1230 1826 y Fx(\025)j FB(0)-84 1943 y(for)j(0)e Fx(\024)g Fz(x)f(<)h(\031)r FB(.)22 b(Note)17 b(that,)f(for)i Fz(\020)g FB(=)d(1)i(and)h(0)d Fx(\024)f Fz(x)h(<)g(\031)r FB(,)g Fz(L)i FB(satis\014es)h(the)e (\014rst)h(order)g(di\013eren)o(tial)f(equation)-84 2003 y Fz(L)-51 1985 y Fq(0)-39 2003 y FB(\()p Fz(x)p FB(\))d(=)h(0)i(with)f (initial)f(data)i Fz(L)p FB(\(0\))f(=)e(0.)22 b(This)15 b(sho)o(ws)h(item)e(\()p FC(iv)q FB(\))g(since)g Fz(L)p FB(\()p Fz(\031)r FB(\))g(=)g(2)p Fz(\031)j FB(for)e(an)o(y)h Fz(\020)t FB(.)21 b(F)l(or)15 b(\()p FC(iii)p FB(\))o(,)g(w)o(e)-84 2063 y(use)375 2142 y(arctan)519 2069 y Fr( )602 2108 y Fz(\020)d FB(sin\(2)p Fz(\031)h Fx(\000)e Fz(x)p FB(\))p 557 2130 364 2 v 557 2176 a(1)g(+)g Fz(\020)h FB(cos)q(\(2)p Fz(\031)h Fx(\000)d Fz(x)p FB(\))925 2069 y Fr(!)972 2142 y FB(=)k Fx(\000)8 b FB(arctan)1214 2069 y Fr( )1297 2108 y Fz(\020)13 b FB(sin)8 b Fz(x)p 1252 2130 220 2 v 1252 2176 a FB(1)k(+)f Fz(\020)h FB(cos)d Fz(x)1476 2069 y Fr(!)1534 2142 y Fz(:)-84 2260 y FB(Item)14 b(\()p FC(i)p FB(\))i(is)g(clear.)p 1977 2342 30 30 v -84 2527 a FC(Corollary)i(4.3)24 b Fn(F)l(or)17 b Fz(x)c Fx(2)h Fs(R)479 2534 y Fy(+)523 2527 y Fn(and)k Fz(\020)g Fx(2)c FB([0)p Fz(;)22 b FB(1])17 b Fn(one)h(has)f(the)h(simple)g(b)n(ound)786 2637 y Fz(L)p FB(\()p Fz(x)p FB(\))31 b Fx(\025)g Fz(\025)p FB(\()p Fz(\020)t FB(\))18 b Fz(x:)769 b FB(\(4.8\))1969 2747 y Fm(2)937 2930 y FB(19)p eop %%Page: 20 20 20 19 bop -11 79 a FB(In)18 b(view)f(of)h(Prop)q(osition)i(4.2)e(and)h (since)e Fz(\025)p FB(\()p Fz(\020)t FB(\))g Fz(>)g FB(1)i(for)f(0)f Fx(\024)g Fz(\020)k(<)c FB(1)p Fz(=)p FB(3,)i(the)f(Liapuno)o(v)h(exp)q (onen)o(t)e(of)i(the)-84 139 y(discrete)c(dynamical)f(system)h(\(4.3\)) i(is)f(p)q(ositiv)o(e)f(in)h(this)g(range)h(of)g Fz(\020)t FB(.)k(W)l(e)16 b(ha)o(v)o(e)f(also)i(the)f(follo)o(wing)-84 275 y FC(Lemma)f(4.4)24 b Fn(If)18 b Fz(\036)291 282 y Fy(0)324 275 y Fx(2)c Fs(R)410 282 y Fy(+)454 275 y Fn(then)k Fz(\036)591 282 y Ft(i)619 275 y Fx(2)c Fs(R)705 282 y Fy(+)749 275 y Fn(for)j(al)r(l)i Fz(i)13 b Fx(2)h Fs(N)p Fn(,)19 b Fz(i)13 b Fx(\025)h FB(0)p Fn(.)1969 411 y Fm(2)-84 542 y FC(Pro)r(of.)21 b FB(Let)16 b(us)h(assume,)e(b)o (y)h(an)h(induction)e(argumen)o(t,)g(that)i Fz(\036)1140 549 y Ft(i)1168 542 y Fx(2)d Fs(R)1254 549 y Fy(+)1280 542 y FB(.)21 b(By)16 b(Corollary)g(4.3)h(one)f(has)702 652 y Fz(\036)731 659 y Ft(i)p Fy(+1)804 652 y Fx(\025)e Fz(\036)886 659 y Fy(0)916 652 y FB(+)d Fz(\025)p FB(\()p Fz(\020)t FB(\))17 b Fz(\036)1102 659 y Ft(i)1130 652 y Fx(\025)c FB(0)p Fz(;)686 b FB(\(4.9\))-84 762 y(b)o(y)16 b(the)g(h)o(yp)q(othesis.)21 b(Since)15 b(this)h(holds)h(for)g(all)e Fz(i)f Fx(2)g Fs(N)j FB(the)f(lemma)d(is)j(pro)o(v)o(en.)p 1977 844 30 30 v -11 975 a(F)l(rom)g(no)o(w)i(on)g(w)o(e)f(will)g (consider)g(only)g(starting)h(p)q(oin)o(ts)g(in)f Fs(R)1184 982 y Fy(+)1211 975 y FB(,)g(what)h(reduces)f(the)g(dynamical)f(system) -84 1035 y(to)h(a)f(dynamical)f(system)f(on)j Fs(R)519 1042 y Fy(+)546 1035 y FB(.)-84 1171 y FC(Lemma)e(4.5)24 b Fn(L)n(et)19 b Fz(\036)325 1153 y Fy(2)364 1171 y Fn(and)g Fz(\036)489 1153 y Fy(1)528 1171 y Fn(two)h(starting)f(p)n(oints)g (with)h Fz(\036)1084 1153 y Fy(2)1121 1171 y Fz(>)d(\036)1205 1153 y Fy(1)1241 1171 y Fx(\025)g FB(0)p Fn(.)28 b(Denote)20 b(by)f Fz(\036)1627 1153 y Ft(a)1627 1183 y(i)1648 1171 y Fn(,)h(with)f Fz(i)e Fx(2)g Fs(N)j Fn(and)-84 1231 y Fz(a)13 b FB(=)h(1)p Fn(,)j FB(2)g Fn(the)g(se)n(quenc)n(es)h(gener)n (ate)n(d)g(by)e(the)h(dynamic)n(al)g(system)g(having)g Fz(\036)1330 1213 y Ft(a)1367 1231 y Fn(as)g(starting)g(p)n(oint.)22 b(Then)17 b(for)f(al)r(l)-84 1291 y Fz(i)d Fx(2)h Fs(N)19 b Fn(one)f(has)f Fz(\036)255 1273 y Fy(2)255 1304 y Ft(i)289 1291 y Fz(>)c(\036)369 1273 y Fy(1)369 1304 y Ft(i)403 1291 y Fx(\025)g FB(0)p Fn(.)1969 1427 y Fm(2)-84 1558 y FC(Pro)r(of.)21 b FB(Assume,)14 b(b)o(y)i(an)h(induction)f(argumen)o (t,)e(that)j Fz(\036)1002 1540 y Fy(2)1002 1571 y Ft(i)1035 1558 y Fz(>)d(\036)1116 1540 y Fy(1)1116 1571 y Ft(i)1136 1558 y FB(.)21 b(Then)525 1668 y Fz(\036)554 1648 y Fy(2)554 1681 y Ft(i)p Fy(+1)624 1668 y Fx(\000)11 b Fz(\036)703 1648 y Fy(1)703 1681 y Ft(i)p Fy(+1)776 1668 y FB(=)j(\()p Fz(\036)876 1648 y Fy(2)906 1668 y Fx(\000)d Fz(\036)985 1648 y Fy(1)1005 1668 y FB(\))g(+)g Fz(L)p FB(\()p Fz(\036)1165 1648 y Fy(2)1165 1681 y Ft(i)1184 1668 y FB(\))g Fx(\000)g Fz(L)p FB(\()p Fz(\036)1345 1648 y Fy(1)1345 1681 y Ft(i)1365 1668 y FB(\))p Fz(:)484 b FB(\(4.10\))-11 1800 y(F)l(or)16 b Fz(y)g Fx(\025)d Fz(x)h Fx(\025)g FB(0)i(w)o(e)g(can)g(write)502 1923 y Fz(L)p FB(\()p Fz(y)r FB(\))11 b Fx(\000)f Fz(L)p FB(\()p Fz(x)p FB(\))k(=)824 1865 y Fr(Z)865 1878 y Ft(y)847 1959 y(x)911 1923 y Fz(L)944 1903 y Fq(0)955 1923 y FB(\()p Fz(s)p FB(\))j Fz(ds)d Fx(\025)f Fz(\025)p FB(\()p Fz(\020)t FB(\))k(\()p Fz(y)c Fx(\000)d Fz(x)p FB(\))p Fz(;)461 b FB(\(4.11\))-84 2048 y(b)o(y)16 b(Prop)q(osition)h(4.2.)22 b(Therefore)486 2158 y Fz(\036)515 2137 y Fy(2)515 2170 y Ft(i)p Fy(+1)585 2158 y Fx(\000)11 b Fz(\036)664 2137 y Fy(1)664 2170 y Ft(i)p Fy(+1)737 2158 y Fx(\025)i FB(\()p Fz(\036)837 2137 y Fy(2)868 2158 y Fx(\000)e Fz(\036)947 2137 y Fy(1)966 2158 y FB(\))g(+)g Fz(\025)p FB(\()p Fz(\020)t FB(\)\()p Fz(\036)1184 2137 y Fy(2)1184 2170 y Ft(i)1216 2158 y Fx(\000)f Fz(\036)1294 2137 y Fy(1)1294 2170 y Ft(i)1314 2158 y FB(\))k Fz(>)f FB(0)p Fz(;)446 b FB(\(4.12\))-84 2268 y(b)o(y)16 b(the)g(h)o(yp)q(othesis.)p 1977 2349 V -11 2481 a(Let)h(us)h(denote)f(b)o(y)f Fz(\036)396 2488 y Ft(i)410 2481 y FB(\()p Fz(\036)p FB(\))h(the)g Fz(i)p FB(-th)g(iterated)f(of)i(the)f(discrete)e(dynamical)h(system)f (ha)o(ving)i Fz(\036)g FB(as)h(starting)-84 2541 y(p)q(oin)o(t.)i(Note) 11 b(that)h Fz(\036)304 2548 y Ft(i)318 2541 y FB(\()p Fz(\036)p FB(\))g(is)f(a)i(con)o(tin)o(uous)e(function)h(of)g(the)g (starting)g(p)q(oin)o(t)g Fz(\036)p FB(,)g(since)f(it)g(is)h(a)g (\014nite)f(comp)q(osition)-84 2601 y(of)16 b(con)o(tin)o(uous)h (functions.)k(This)16 b(fact)g(and)h(the)f(previous)g(lemma)d (immediatel)o(y)g(imply:)-84 2737 y FC(Lemma)i(4.6)24 b Fn(F)l(or)17 b(al)r(l)i Fz(i)14 b Fx(2)g Fs(N)p Fn(,)k Fz(\036)546 2744 y Ft(i)560 2737 y FB(\()p Fz(\036)p FB(\))f Fn(is)h(a)f(strictly)h(incr)n(e)n(asing)f(c)n(ontinuous)i (function)g(on)e Fs(R)1678 2744 y Fy(+)1705 2737 y Fn(.)937 2930 y FB(20)p eop %%Page: 21 21 21 20 bop 1969 79 a Fm(2)-11 211 y FB(This)16 b(lemma)e(has)j(an)f(imp) q(ortan)o(t)g(consequence:)-84 346 y FC(Corollary)i(4.7)24 b Fn(The)17 b(sets)g Fu(R)496 353 y Ft(n)520 346 y Fn(,)g Fz(n)d Fx(2)g Fs(N)p Fn(,)k(ar)n(e)e(c)n(omp)n(ose)n(d)f(by)i FB(2)1092 328 y Ft(n)1133 346 y Fn(distinct)h(elements,)g(what)g(in)f (p)n(articular)f(says)-84 407 y(that)j(e)n(quation)i(\(4.1\))e(has,)g (for)g(e)n(ach)g Fz(n)f Fx(2)f Fs(N)p Fn(,)k(pr)n(e)n(cisely)e FB(2)1023 388 y Ft(n)p Fy(+1)1105 407 y Fx(\000)12 b FB(1)20 b Fn(distinct)g(solutions)g(in)g Fz(S)1677 388 y Fy(1)1696 407 y Fn(.)28 b(Each)20 b(set)g Fu(R)1983 414 y Ft(n)-84 467 y Fn(c)n(an)g(b)n(e)f(or)n(der)n(e)n(d)f(such)h (that)h FB(0)e Fx(\024)f Fz(\036)p FB(\()p Fz(n;)25 b(l)q FB(\))17 b Fz(<)g(\036)p FB(\()p Fz(n;)25 b(l)909 449 y Fq(0)921 467 y FB(\))17 b Fx(\024)g Fz(\031)k Fn(for)e Fz(l)f(<)f(l)1246 449 y Fq(0)1276 467 y Fn(and)j(the)g(elements)h Fz(\036)p FB(\()p Fz(n;)k(l)q FB(\))19 b Fn(ar)n(e)g(such)-84 527 y(that)667 587 y Fz(\036)696 594 y Ft(n)719 587 y FB(\()p Fz(\036)p FB(\()p Fz(n;)25 b(l)q FB(\)\))14 b(=)f Fz(\031)g FB(+)e(2)p Fz(\031)r FB(\()p Fz(l)h Fx(\000)e FB(1\))627 b(\(4.13\))-84 674 y Fn(for)17 b FB(1)d Fx(\024)g Fz(l)g Fx(\024)g FB(2)191 656 y Ft(n)214 674 y Fn(.)23 b(A)o(l)r(l)c(these)f(facts)g(hold)f(for)g(al)r(l)i Fz(\020)f Fx(2)c FB([0)p Fz(;)22 b FB(1\))p Fn(.)1969 810 y Fm(2)-84 941 y FC(Pro)r(of.)28 b FB(The)19 b(sets)f Fu(R)337 948 y Ft(n)361 941 y FB(,)h Fz(n)e Fx(2)h Fs(N)p FB(,)i(de\014ned)e(ab)q(o) o(v)o(e)h(can)g(also)g(b)q(e)f(c)o(haracterized)f(as)j(the)e(sets)h(of) f(all)g Fz(\036)p FB(\()p Fz(n;)26 b(l)q FB(\))17 b Fx(2)-84 1001 y FB([0)p Fz(;)22 b(\031)r FB(],)16 b(1)f Fx(\024)f Fz(l)h Fx(\024)g FB(2)264 983 y Ft(n)288 1001 y FB(,)h(for)h(whic)o(h)f Fz(\036)562 1008 y Ft(n)585 1001 y FB(\()p Fz(\036)p FB(\()p Fz(n;)23 b(l)q FB(\)\))14 b(=)h Fz(\031)d FB(+)g(2)p Fz(\031)r(n)1012 1008 y Ft(l)1041 1001 y FB(for)17 b(some)f Fz(n)1268 1008 y Ft(l)1296 1001 y Fx(2)f Fs(N)p FB(,)i(dep)q(ending)g (on)g Fz(l)q FB(.)22 b(T)l(o)c(see)e(that)-84 1062 y Fz(n)-55 1069 y Ft(l)-27 1062 y FB(=)e Fz(l)e Fx(\000)f FB(1)17 b(w)o(e)f(note)h(that,)g(due)g(to)g(Lemma)d(4.6,)j Fz(\036)893 1069 y Ft(n)916 1062 y FB(\()p Fz(\036)p FB(\))g(is)f(a)h(strictly)e(increasing)i(con)o(tin)o(uous)f(function)h (of)g Fz(\036)-84 1122 y FB(with)f Fz(\036)56 1129 y Ft(n)79 1122 y FB(\(0\))e(=)g(0)j(and)f Fz(\036)371 1129 y Ft(n)394 1122 y FB(\()p Fz(\031)r FB(\))e(=)f(\(2)570 1104 y Ft(n)p Fy(+1)650 1122 y Fx(\000)d FB(1\))p Fz(\031)r FB(.)21 b(Therefore,)16 b Fz(\036)1071 1129 y Ft(n)1108 1122 y FB(:)27 b([0)p Fz(;)22 b(\031)r FB(])12 b Fx(!)i FB([0)p Fz(;)22 b FB(\(2)1460 1104 y Ft(n)p Fy(+1)1539 1122 y Fx(\000)11 b FB(1\))p Fz(\031)r FB(])k(is)h(an)g(in)o(v)o (ertible)-84 1182 y(map)f(and)i(w)o(e)f(can)g(de\014ne)649 1242 y Fz(\036)p FB(\()p Fz(n;)24 b(l)q FB(\))13 b(:=)h Fz(\036)907 1222 y Fq(\000)p Fy(1)907 1255 y Ft(n)954 1242 y FB(\()p Fz(\031)e FB(+)f(2)p Fz(\031)r FB(\()p Fz(l)h Fx(\000)e FB(1\)\))609 b(\(4.14\))-84 1329 y(for)16 b(1)f Fx(\024)e Fz(l)i Fx(\024)e FB(2)187 1311 y Ft(n)211 1329 y FB(.)21 b(Clearly)16 b Fz(\036)p FB(\()p Fz(n;)21 b(l)q FB(\))14 b Fz(<)g(\036)p FB(\()p Fz(n;)21 b(l)757 1311 y Fq(0)769 1329 y FB(\))16 b(if)f Fz(l)g(<)f(l)946 1311 y Fq(0)957 1329 y FB(.)p 1977 1411 30 30 v -84 1627 a FI(4.2)66 b(The)22 b(Sets)g Fu(R)390 1636 y Fd(n)440 1627 y FI(in)i(the)e(F)-6 b(erromagnetic)24 b(Region)e(\()p FG(0)17 b Fe(\024)g Ff(\020)k Fe(\024)c FG(1)p Ff(=)p FG(3)p FI(\))-84 1741 y FB(In)k(this)f(subsection)h(w)o(e)g(will)e(sho) o(w)j(that,)g(in)e(the)h(ferromagnetic)e(region)i(\(0)h Fx(\024)g Fz(\020)j Fx(\024)d FB(1)p Fz(=)p FB(3\),)g(the)f(Lee-Y)l (ang)-84 1801 y(singularities)15 b(b)q(ecome)g(dense)h(in)g(the)g (whole)g(unit)g(circle)f Fz(S)1038 1783 y Fy(1)1074 1801 y FB(when)h(the)g(thermo)q(dynamic)d(limit)h(is)i(tak)o(en.)-11 1882 y(The)f(next)f(t)o(w)o(o)h(theorems)e(are)i(crucial)f(for)h (understanding)g(the)g(b)q(eha)o(vior)f(of)h(the)g(Lee-Y)l(ang)g (singularities)-84 1943 y(when)h Fz(n)h FB(go)q(es)g(to)g(in\014nit)o (y)l(.)-84 2078 y FC(Theorem)f(4.8)24 b Fn(With)18 b(the)g(ab)n(ove)g (de\014nitions)g(we)h(have)639 2188 y FB(lim)627 2213 y Ft(n)p Fq(!1)727 2188 y FB(\()p Fz(\036)p FB(\()p Fz(n;)j(l)12 b FB(+)f(1\))g Fx(\000)g Fz(\036)p FB(\()p Fz(n;)22 b(l)q FB(\)\))13 b(=)h(0)587 b(\(4.15\))-84 2309 y Fn(uniformly)17 b(in)h Fz(l)q Fn(,)f(for)g(al)r(l)i FB(0)14 b Fx(\024)f Fz(\020)18 b Fx(\024)c FB(1)p Fz(=)p FB(3)p Fn(.)1969 2444 y Fm(2)-84 2576 y FC(Pro)r(of.)21 b FB(Call)15 b Fz(\036)220 2583 y Ft(i)234 2576 y FB(\()p Fz(n;)22 b(l)q FB(\))14 b(:=)f Fz(\036)461 2583 y Ft(i)475 2576 y FB(\()p Fz(\036)p FB(\()p Fz(n;)21 b(l)q FB(\)\),)15 b(the)h Fz(i)f FB(-th)h(iteration)g(started)g(from)e(the)i(p)q(oin)o(t)g Fz(\036)p FB(\()p Fz(n;)21 b(l)q FB(\).)g(W)l(e)15 b(ha)o(v)o(e)g(for) -84 2636 y(0)f Fx(\024)g Fz(i)f Fx(\024)h Fz(n)d Fx(\000)g FB(1)41 2746 y Fz(\036)70 2753 y Ft(i)p Fy(+1)129 2746 y FB(\()p Fz(n;)22 b(l)11 b FB(+)g(1\))h Fx(\000)f Fz(\036)422 2753 y Ft(i)p Fy(+1)481 2746 y FB(\()p Fz(n;)21 b(l)q FB(\))42 b(=)f(\()p Fz(\036)p FB(\()p Fz(n;)22 b(l)11 b FB(+)g(1\))h Fx(\000)f Fz(\036)p FB(\()p Fz(n;)21 b(l)q FB(\)\))11 b(+)g Fz(L)p FB(\()p Fz(\036)1339 2753 y Ft(i)1353 2746 y FB(\()p Fz(n;)22 b(l)11 b FB(+)g(1\)\))h Fx(\000)f Fz(L)p FB(\()p Fz(\036)1717 2753 y Ft(i)1731 2746 y FB(\()p Fz(n;)21 b(l)q FB(\)\))p Fz(:)p FB(\(4.16\))937 2930 y(21)p eop %%Page: 22 22 22 21 bop -11 79 a FB(Since)15 b Fz(\036)145 86 y Ft(i)159 79 y FB(\()p Fz(n;)22 b(l)12 b FB(+)f(1\))j Fx(\025)g Fz(\036)458 86 y Ft(i)472 79 y FB(\()p Fz(n;)21 b(l)q FB(\),)16 b(w)o(e)g(can)g(write)317 189 y Fz(L)p FB(\()p Fz(\036)398 196 y Ft(i)413 189 y FB(\()p Fz(n;)21 b(l)12 b FB(+)f(1\)\))g Fx(\000)g Fz(L)p FB(\()p Fz(\036)776 196 y Ft(i)790 189 y FB(\()p Fz(n;)22 b(l)q FB(\)\))13 b Fx(\025)h Fz(\025)p FB(\()p Fz(\020)t FB(\)\()p Fz(\036)1133 196 y Ft(i)1147 189 y FB(\()p Fz(n;)22 b(l)12 b FB(+)f(1\))g Fx(\000)g Fz(\036)1440 196 y Ft(i)1454 189 y FB(\()p Fz(n;)22 b(l)q FB(\)\))p Fz(;)276 b FB(\(4.17\))-84 299 y(b)o(y)16 b(Lemma)e(4.5)i(and)h(\(4.11\).)22 b(Hence)4 409 y Fz(\036)33 416 y Ft(i)p Fy(+1)92 409 y FB(\()p Fz(n;)g(l)11 b FB(+)g(1\))h Fx(\000)f Fz(\036)385 416 y Ft(i)p Fy(+1)444 409 y FB(\()p Fz(n;)21 b(l)q FB(\))14 b Fx(\025)g FB(\()p Fz(\036)p FB(\()p Fz(n;)21 b(l)12 b FB(+)f(1\))g Fx(\000)g Fz(\036)p FB(\()p Fz(n;)22 b(l)q FB(\)\))10 b(+)h Fz(\025)p FB(\()p Fz(\020)t FB(\))d(\()q Fz(\036)1314 416 y Ft(i)1328 409 y FB(\()p Fz(n;)22 b(l)11 b FB(+)g(1\))h Fx(\000)f Fz(\036)1621 416 y Ft(i)1635 409 y FB(\()p Fz(n;)21 b(l)q FB(\)\))8 b Fz(:)88 b FB(\(4.18\))-11 541 y(Iterating)16 b(this)g(inequalit)o(y)e(w)o(e)i(obtain)274 684 y Fz(\036)303 691 y Ft(i)p Fy(+1)362 684 y FB(\()p Fz(n;)22 b(l)12 b FB(+)f(1\))g Fx(\000)g Fz(\036)655 691 y Ft(i)p Fy(+1)714 684 y FB(\()p Fz(n;)22 b(l)q FB(\))13 b Fx(\025)899 611 y Fr( )957 630 y Ft(i)933 643 y Fr(X)932 734 y Ft(a)p Fy(=0)1004 684 y Fz(\025)p FB(\()p Fz(\020)t FB(\))1095 664 y Ft(a)1116 611 y Fr(!)1157 684 y FB(\()p Fz(\036)p FB(\()p Fz(n;)22 b(l)11 b FB(+)g(1\))h Fx(\000)f Fz(\036)p FB(\()p Fz(n;)21 b(l)q FB(\)\))p Fz(:)233 b FB(\(4.19\))-11 850 y(Since)15 b Fz(\036)145 857 y Ft(n)169 850 y FB(\()p Fz(n;)22 b(l)11 b FB(+)g(1\))h Fx(\000)f Fz(\036)462 857 y Ft(n)485 850 y FB(\()p Fz(n;)22 b(l)q FB(\))13 b(=)h(2)p Fz(\031)r FB(,)h(w)o(e)h(get)603 983 y Fz(\036)p FB(\()p Fz(n;)21 b(l)12 b FB(+)f(1\))h Fx(\000)e Fz(\036)p FB(\()p Fz(n;)22 b(l)q FB(\))13 b Fx(\024)1181 949 y FB(2)p Fz(\031)p 1114 971 188 2 v 1114 1002 a Ft(n)p Fq(\000)p Fy(1)1117 1014 y Fr(X)1115 1105 y Ft(a)p Fy(=0)1189 1056 y Fz(\025)p FB(\()p Fz(\020)t FB(\))1280 1035 y Ft(a)1306 983 y Fz(:)562 b FB(\(4.20\))-11 1222 y(If)16 b Fz(\025)p FB(\()p Fz(\020)t FB(\))e Fx(\025)g FB(1,)i(what)h(happ)q (ens)g(for)g(0)d Fx(\024)f Fz(\020)18 b Fx(\024)c FB(1)p Fz(=)p FB(3,)j(w)o(e)f(ha)o(v)o(e)637 1332 y(lim)625 1356 y Ft(n)p Fq(!1)717 1332 y FB(\()p Fz(\036)p FB(\()p Fz(n;)21 b(l)12 b FB(+)f(1\))g Fx(\000)g Fz(\036)p FB(\()p Fz(n;)22 b(l)q FB(\)\))13 b(=)h(0)p Fz(;)584 b FB(\(4.21\))-84 1449 y(uniformly)14 b(in)i Fz(l)q FB(.)p 1977 1531 30 30 v -11 1662 a(W)l(e)g(can)h(also)f(pro)o(v)o(e)g(the)g(follo)o(wing)g (theorem:)-84 1798 y FC(Theorem)g(4.9)24 b Fn(With)18 b(the)g(ab)n(ove)g(de\014nitions)g(we)h(have)798 1908 y FB(lim)786 1933 y Ft(n)p Fq(!1)887 1908 y Fz(\036)p FB(\()p Fz(n;)25 b FB(1\))14 b(=)g(0)746 b(\(4.22\))-84 2028 y Fn(for)17 b(al)r(l)i FB(0)14 b Fx(\024)f Fz(\020)18 b Fx(\024)c FB(1)p Fz(=)p FB(3)p Fn(.)1969 2164 y Fm(2)-84 2295 y FC(Pro)r(of.)21 b FB(Since)718 2355 y Fz(\036)747 2362 y Ft(i)p Fy(+1)820 2355 y FB(=)14 b Fz(\036)p FB(\()p Fz(n;)24 b FB(1\))11 b(+)g Fz(L)p FB(\()p Fz(\036)1171 2362 y Ft(i)1185 2355 y FB(\))678 b(\(4.23\))-84 2443 y(with)16 b(0)e Fx(\024)g Fz(i)f Fx(\024)h Fz(n)d Fx(\000)g FB(1,)16 b(w)o(e)g(ha)o(v)o(e,)f(using)i(Corollary)f(4.3,)g(that)701 2553 y Fz(\036)730 2560 y Ft(i)p Fy(+1)803 2553 y Fx(\025)d Fz(\036)p FB(\()p Fz(n;)24 b FB(1\))12 b(+)f Fz(\025)p FB(\()p Fz(\020)t FB(\))p Fz(\036)1194 2560 y Ft(i)1208 2553 y Fz(;)660 b FB(\(4.24\))-84 2663 y(from)15 b(whic)o(h)h(w)o(e)f (conclude)h(that)669 2747 y Fz(\036)698 2754 y Ft(i)p Fy(+1)771 2747 y Fx(\025)823 2674 y Fr( )882 2693 y Ft(i)858 2705 y Fr(X)856 2796 y Ft(a)p Fy(=0)928 2747 y Fz(\025)p FB(\()p Fz(\020)t FB(\))1019 2726 y Ft(a)1041 2674 y Fr(!)1082 2747 y Fz(\036)p FB(\()p Fz(n;)24 b FB(1\))p Fz(:)628 b FB(\(4.25\))937 2930 y(22)p eop 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Fx(\024)c FB(1.)-84 1181 y FC(Remark)i(4.10)24 b Fn(Notic)n(e)18 b(that)g(b)n(oth)f(The)n(or)n(ems)g(4.8)g(and)h(4.9)f(hold)g(also)h(at) g(the)f(critic)n(al)h(p)n(oint)g Fz(\020)f FB(=)d(1)p Fz(=)p FB(3)p Fn(.)-84 1345 y FI(4.3)66 b(The)22 b(Sets)g Fu(R)390 1354 y Fd(n)440 1345 y FI(in)i(the)e(P)n(aramagnetic)h(Region) f(\()p FG(1)p Ff(=)p FG(3)c Ff(<)e(\020)21 b Fe(\024)c FG(1)p FI(\))-84 1459 y FB(In)i(Section)f(5)i(w)o(e)e(will)g(see)h (that,)h(for)f(eac)o(h)f(v)m(alue)h(of)h Fz(\020)j FB(in)c(the)f (paramagnetic)h(region)g(\(1)p Fz(=)p FB(3)g Fz(<)g(\020)k Fx(\024)18 b FB(1\),)i(the)-84 1519 y(sets)d Fu(S)53 1526 y Ft(n)94 1519 y FB(are)g(excluded)e(from)h(a)i(neigh)o(b)q(orho)q (o)q(d)g(of)g Fz(z)f FB(=)e(1.)23 b(Ho)o(w)o(ev)o(er,)15 b(one)i(should)h(exp)q(ect)e(that,)h(in)g(analogy)-84 1579 y(to)e(the)f(situation)h(in)g(the)f(ferromagnetic)f(region,)i(the) f(sets)h Fu(S)1078 1586 y Ft(n)1117 1579 y FB(b)q(ecome)e(dense)h(in)h (some)f(prop)q(er)h(subset)g(of)g Fz(S)1987 1561 y Fy(1)-84 1639 y FB(when)h(the)g(limit)e Fz(n)g Fx(!)f(1)j FB(is)h(tak)o(en.)-11 1721 y(In)h(this)f(subsection)h(w)o(e)f(will)g(pro)o(v)o(e)g(a)h(w)o (eak)o(er)f(statemen)o(t,)f(namely)g(that)i(there)f(exists)h(a)g(prop)q (er)g(subset)-84 1781 y Fz(A)-47 1788 y Ft(\024)-8 1781 y FB(of)f Fz(S)81 1763 y Fy(1)118 1781 y FB(suc)o(h)g(that)h Fu(S)377 1788 y Ft(n)409 1748 y Fr(T)451 1781 y Fz(A)488 1788 y Ft(\024)528 1781 y FB(b)q(ecomes)d(dense)i(on)h Fz(A)962 1788 y Ft(\024)1001 1781 y FB(when)f Fz(n)f Fx(!)f(1)p FB(.)23 b(Let)18 b(us)f(start)h(capturing)f(the)g(main)-84 1841 y(ingredien)o(ts)e(for)i(the)f(pro)q(of,)h(as)f(w)o(e)g(learned)g (from)f(the)h(last)h(subsection.)-11 1923 y(The)f(pro)q(ofs)i(presen)o (ted)d(in)h(the)g(previous)g(subsection)h(sho)o(w)g(clearly)d(that)j (the)f(phenomenon)g(of)g(the)g(Lee-)-84 1983 y(Y)l(ang)g(singularities) f(b)q(ecoming)g(dense)g(is)g(closely)g(related)g(to)h(the)f(p)q (ositivit)o(y)g(of)h(the)f(Liapuno)o(v)i(exp)q(onen)o(ts)e(of)-84 2043 y(our)20 b(discrete)e(dynamical)g(system.)29 b(In)19 b(order)h(to)g(obtain)g(analogous)h(results)e(for)h(the)f(region)h(1)p Fz(=)p FB(3)g Fz(<)g(\020)j Fx(\024)c FB(1)-84 2104 y(one)c(has)h(to)f (tak)o(e)f(in)o(to)h(accoun)o(t)g(that,)g(for)g(these)g(v)m(alues)g(of) g Fz(\020)t FB(,)g(the)g(deriv)m(ativ)o(e)e Fz(L)1434 2086 y Fq(0)1446 2104 y FB(\()p Fz(x)p FB(\))h(is)h(not)g(larger)g (than)h(1)f(for)-84 2164 y(all)h(p)q(oin)o(ts)g Fz(x)e Fx(2)g FB([0)p Fz(;)22 b FB(2)p Fz(\031)r FB(].)e(Ho)o(w)o(ev)o(er,)15 b(if)g(the)h(tra)s(jectory)g(of)h(the)f(dynamical)e(system)h(visits)g (frequen)o(tly)g(enough)-84 2224 y(regions)i(where)f Fz(L)256 2206 y Fq(0)285 2224 y FB(is)g(large,)h(a)g(p)q(ositiv)o(e)f (lo)o(w)o(er)g(b)q(ound)i(for)f(the)f(Liapuno)o(v)h(exp)q(onen)o(t)g (can)g(b)q(e)g(obtained)g(with)-84 2284 y(few)f(iterations.)22 b(This)16 b(can)h(b)q(e)f(the)h(case)f(if,)g(for)g(instance,)g(the)g(p) q(oin)o(ts)h(of)g(the)f(tra)s(jectory)g(successiv)o(ely)e(jump)-84 2344 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y(pro)o(v)o(en)j(that,)h(under)g(the)g(circumstances)e(that)i(pro) o(vide)f(\(4.31\),)i(the)f(set)f Fu(R)1380 2692 y Ft(n)1421 2651 y Fr(T)1464 2685 y FB([)p Fz(\024)1506 2667 y Fy(0)1525 2685 y Fz(;)25 b(\031)r FB(])17 b(b)q(ecomes)f(dense)i(on)-84 2745 y(itself)e(when)i Fz(n)e Fx(!)g(1)p FB(.)25 b(T)l(o)18 b(establish)g(that)g Fu(R)788 2752 y Ft(n)828 2712 y Fr(T)871 2745 y FB([)p Fz(\024)913 2727 y Fy(0)932 2745 y Fz(;)24 b(\031)r FB(])17 b(b)q(ecomes)f(indeed)h(dense)g(o)o(v)o(er)g (the)g(whole)h(in)o(terv)m(al)-84 2805 y([)o Fz(\024)-43 2787 y Fy(0)-23 2805 y Fz(;)k(\031)r FB(])15 b(w)o(e)h(will)f(also)i (need)f(the)g(follo)o(wing)g(result:)937 2930 y(24)p eop %%Page: 25 25 25 24 bop -84 79 a FC(Theorem)16 b(4.11)24 b Fn(L)n(et)18 b Fz(l)378 61 y Fy(0)412 79 y FB(=)c Fz(l)480 61 y Fy(0)499 79 y FB(\()p Fz(n)p FB(\))k Fn(b)n(e)g(the)h(lab)n(el)f(of)g(the)g (\014rst)g(singularity)g Fz(\036)p FB(\()p Fz(n;)f(l)1441 61 y Fy(0)1460 79 y FB(\))h Fn(in)g Fu(R)1598 86 y Ft(n)1639 79 y 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Fz(\024)769 1557 y Fy(0)788 1575 y Fz(;)22 b(\031)r FB(])15 b(b)q(ecomes)g(indeed)g(dense)h(o)o(v)o(er)g(the)f(whole)h(in)o(terv)m (al)f([)p Fz(\024)1908 1557 y Fy(0)1928 1575 y Fz(;)21 b(\031)r FB(])-84 1635 y(under)e(the)f(ab)q(o)o(v)o(e)h(assumptions)g (when)g Fz(n)f Fx(!)g(1)p FB(.)29 b(After)18 b(the)g(ab)q(o)o(v)o(e)h (results,)g(w)o(e)f(only)h(need)f(to)i(sho)o(w)f(that)-84 1695 y(the)e(sets)g Fu(R)138 1702 y Ft(n)178 1662 y Fr(T)221 1695 y FB([)p Fz(\024)263 1677 y Fy(0)283 1695 y Fz(;)23 b(\031)r FB(])16 b(con)o(tain)h(other)g(elemen)o(ts)d(than)k Fz(\031)g FB(for)f(the)g(considered)g(v)m(alues)g(of)g Fz(\020)k FB(and)d(for)f Fz(n)h FB(large)-84 1755 y(enough.)-11 1837 y(Since)e(w)o(e)g(already)h(kno)o(w)g(that,)g(for)g Fz(\020)i FB(=)c(1)p Fz(=)p FB(3,)j Fu(R)923 1844 y Ft(n)963 1837 y FB(b)q(ecomes)e(dense)g(on)i Fz(S)1393 1819 y Fy(1)1429 1837 y FB(when)f Fz(n)g FB(go)q(es)h(to)f(in\014nite,)f(w)o (e)-84 1897 y(can)f(argue)g(that,)f(for)h Fz(\020)j Fx(\021)c Fz(\020)438 1904 y Fy(1)472 1897 y FB(larger)h(but)f(close)g(enough)h (to)g(1)p Fz(=)p FB(3,)h(there)e(is)g(a)h(\014xed)f Fz(n)1491 1904 y Fy(0)1525 1897 y FB(large)h(enough)g(suc)o(h)f(that)-84 1957 y Fu(R)-43 1964 y Ft(n)-22 1969 y Fk(0)15 1924 y Fr(T)57 1957 y FB([)p Fz(\024)99 1939 y Fy(0)119 1957 y Fz(;)23 b(\031)r FB(])10 b Fx(n)i(f)p Fz(\031)r Fx(g)i(6)p FB(=)h Fx(;)p FB(.)24 b(This)17 b(is)g(b)q(ecause)g(the)g(elemen)o(ts)d (of)j Fu(R)1182 1964 y Ft(n)1203 1969 y Fk(0)1240 1957 y FB(dep)q(end)g(con)o(tin)o(uously)f(on)i Fz(\020)t FB(.)24 b(W)l(e)16 b(ha)o(v)o(e)-84 2018 y(seen)f(that)h Fu(R)167 2025 y Ft(n)208 1984 y Fr(T)251 2018 y FB([)p Fz(\024)293 2000 y Fy(0)312 2018 y Fz(;)22 b(\031)r FB(])14 b(b)q(ecomes)h(dense)g(on)h(itself)f(for)h Fz(n)e Fx(!)g(1)h FB(and)h(b)o(y)f(Theorem)g(4.11,)h(w)o(e)f(conclude)g(that)-84 2078 y(for)h Fz(\020)h Fx(\021)d Fz(\020)102 2085 y Fy(1)137 2078 y FB(the)h(set)h Fu(R)337 2085 y Ft(n)377 2045 y Fr(T)420 2078 y FB([)p Fz(\024)462 2060 y Fy(0)481 2078 y 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FB(\))1003 339 y Fr(\020)1029 387 y FB(1)e Fx(\000)f FB(3)p Fz(\020)1164 366 y Fy(2)1184 339 y Fr(\021)1217 387 y Fz(:)675 b FB(\(5.2\))-84 467 y Fn(Then,)18 b(for)f(al)r(l)h Fz(\036)234 474 y Fy(0)268 467 y Fz(<)c(\024)p Fn(,)j(the)h(\014xe)n(d)g(p)n(oint)f(e)n(quation) 810 560 y Fz(\036)c FB(=)h Fz(\036)933 567 y Fy(0)964 560 y FB(+)d Fz(L)p FB(\()p Fz(\036)p FB(\))793 b(\(5.3\))-84 653 y Fn(admits)23 b(a)h(unique)h(stable)g(solution)f Fz(\036)653 635 y Fq(\003)698 653 y FB(=)h Fz(\036)790 635 y Fq(\003)810 653 y FB(\()p Fz(\020)t(;)8 b(\036)905 660 y Fy(0)924 653 y FB(\))24 b Fn(to)f(which)h(the)g(dynamic)n(al)g (system)f(\(4.3\))h(c)n(onver)n(ges.)-84 713 y(Mor)n(e)n(over,)d(the)g (solution)h Fz(\036)451 695 y Fq(\003)492 713 y Fn(is)f(a)g(monotonic)n (al)r(ly)h(incr)n(e)n(asing)f(function)i(of)e(the)g(initial)h(c)n (ondition)f Fz(\036)1877 720 y Fy(0)1918 713 y Fn(with)-84 773 y FB(0)14 b Fz(<)g(\036)35 755 y Fq(\003)68 773 y Fz(<)p 120 734 V 14 w(\036)g Fx(\024)f Fz(\031)r 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Fz(x)e FB(at)i(least)f(once)g(in)g(this)g(domain.)21 b(The)16 b(constan)o(t)h Fz(\024)f FB(is)g(obtained)h(b)o(y)e(imp)q(osing)p 820 1396 V 820 1436 a Fz(\036)f FB(=)f Fz(\024)f FB(+)f Fz(L)p FB(\()p 1055 1396 V Fz(\036)p FB(\))-84 1529 y(and)17 b(the)f(tangency)g(condition)584 1651 y Fz(L)617 1630 y Fq(0)629 1651 y FB(\()p 648 1611 V Fz(\036)o FB(\))e(=)g(2)793 1578 y Fr( )941 1617 y FB(1)e Fx(\000)f Fz(\020)1052 1599 y Fy(2)p 832 1639 350 2 v 832 1687 a FB(1)g(+)g(2)p Fz(\020)i FB(cos)p 1047 1647 29 2 v 8 w Fz(\036)e FB(+)g Fz(\020)1161 1672 y Fy(2)1186 1578 y Fr(!)1233 1651 y FB(=)j(1)i Fz(:)-84 1780 y FB(These)h(equations)g(determine)d(uniquely) i Fz(\024)h FB(and)h(the)f(\014xed)f(p)q(oin)o(t)h(solution)p 1362 1741 V 18 w Fz(\036)g FB(to)g(\(5.3\))g(at)h Fz(\036)1675 1787 y Fy(0)1709 1780 y FB(=)e Fz(\024)p FB(,)g(in)h(accor-)-84 1841 y(dance)f(to)h(Theorem)e(5.1.)-11 1922 y(Next)g(w)o(e)h(will)f (sho)o(w)i(that,)f(the)g(iterates)g(of)h(the)f(discrete)f(map)g (\(4.3\))i(con)o(v)o(erges)e(to)i(the)f(\014xed)f(p)q(oin)o(t)i Fz(\036)1950 1904 y Fq(\003)1969 1922 y FB(.)-11 2004 y(Let)f Fz(\036)105 1986 y Fq(\003)139 2004 y FB(=)e Fz(\036)220 1986 y Fq(\003)239 2004 y FB(\()p Fz(\020)t(;)8 b(\036)334 2011 y Fy(0)354 2004 y FB(\))16 b(b)q(e)g(smallest)f (solution)h(to)h(\(5.3\).)k(It)16 b(follo)o(ws)g(from)g(Prop)q(osition) h(4.2)g(that)705 2146 y Fz(\036)d(<)g(\036)829 2153 y Fy(0)859 2146 y FB(+)d Fz(L)p FB(\()p Fz(\036)p FB(\))j Fz(<)g(\036)1103 2125 y Fq(\003)1136 2146 y Fz(<)p 1188 2106 V 14 w(\036)-84 2226 y FB(holds)j(for)h(all)e Fz(\036)g Fx(\024)f Fz(\036)316 2207 y Fq(\003)353 2226 y FB(and)i Fz(\036)477 2233 y Fy(0)512 2226 y Fz(<)e(\024)p FB(.)24 b(In)17 b(particular,)g(using)g(\(5.3\),)h Fz(\036)1222 2233 y Fy(0)1257 2226 y FB(=)d Fz(\036)1339 2207 y Fq(\003)1370 2226 y Fx(\000)c Fz(L)p FB(\()p Fz(\036)1501 2207 y Fq(\003)1521 2226 y FB(\))16 b Fz(<)f(\036)1638 2207 y Fq(\003)1657 2226 y FB(.)24 b(So,)18 b(\(4.3\))f(maps)-84 2286 y(the)f(in)o(terv)m (al)f([0)p Fz(;)8 b(\036)265 2268 y Fq(\003)284 2286 y FB(])16 b(in)o(to)g(itself)f(and)i(it)f(is)g(con)o(tractiv)o(e)f(in)h (this)g(domain:)508 2379 y Fx(j)p Fz(L)p FB(\()p Fz(\036)p FB(\))11 b Fx(\000)f Fz(L)p FB(\()p Fz(\033)r FB(\))p Fx(j)k(\024)f Fz(L)896 2358 y Fq(0)908 2379 y FB(\()p Fz(\036)956 2358 y Fq(\003)976 2379 y FB(\))8 b Fx(j)p Fz(\036)j Fx(\000)g Fz(\033)r Fx(j)i(\021)g Fz(q)1240 2358 y Fq(\003)1268 2379 y Fx(j)p Fz(\036)e Fx(\000)f Fz(\033)r Fx(j)-84 2472 y FB(for)20 b(all)g Fz(\036)p FB(,)h Fz(\033)h Fx(2)e FB([0)p Fz(;)8 b(\036)322 2454 y Fq(\003)342 2472 y FB(].)32 b(This)21 b(implies)c(con)o(v)o(ergence)i (of)h(the)g(sequence)f Fx(f)p Fz(\036)1368 2479 y Ft(j)1386 2472 y Fx(g)1411 2488 y Ft(j)r Fq(\025)p Fy(0)1495 2472 y FB(to)h(the)g(solution)g Fz(\036)1863 2454 y Fq(\003)1903 2472 y FB(since)-84 2541 y Fz(q)-60 2523 y Fq(\003)-27 2541 y Fz(<)14 b(L)58 2523 y Fq(0)70 2541 y FB(\()p 89 2502 V Fz(\036)o FB(\))g Fx(\021)g FB(1)i(and)h(concludes)f(the)g(pro)q (of)h(of)g(Theorem)e(5.1.)p 1977 2623 30 30 v -11 2745 a(W)l(e)d(are)h(no)o(w)g(in)g(p)q(osition)g(to)g(determine)d(the)i (domain)g(of)h(analyticit)o(y)e(of)i(the)f(magnetization)g Fz(m)g FB(according)-84 2805 y(to)17 b(Theorem)d(1.2.)22 b(W)l(e)16 b(b)q(egin)g(b)o(y)g(sho)o(wing)h(the)f(follo)o(wing)g (result.)937 2930 y(28)p eop %%Page: 29 29 29 28 bop -84 79 a FC(Lemma)15 b(5.3)24 b Fn(F)l(or)19 b FB(1)p Fz(=)p FB(3)g Fz(<)e(\020)k Fx(\024)c FB(1)i Fn(and)h FC(p)d Fx(2)g Fz(\031)r Fn(,)i(the)h(se)n(quenc)n(e)h Fx(f)p Fz(m)1200 86 y Ft(N)1250 79 y FB(=)c Fz(F)1337 86 y Ft(N)1383 79 y FB(+)12 b Fz(I)1455 86 y Ft(N)1488 79 y Fz(;)c(N)23 b Fx(2)17 b Fs(N)p Fx(g)j Fn(c)n(onver)n(ges)g(to)g(a) -84 139 y(c)n(ontinuous)e(function)354 135 y Fr(f)353 139 y Fz(m)c FB(=)473 123 y Fr(e)461 139 y Fz(F)k FB(+)566 123 y Fr(e)560 139 y Fz(I)j Fn(on)c Fz(S)705 121 y Fy(1)725 139 y Fz(=)m(A)783 146 y Ft(\024)823 139 y Fn(wher)n(e)h Fz(\024)c FB(=)f Fz(\024)p FB(\()p Fz(\020)t FB(\))18 b Fn(is)f(given)i(by)f(\(5.1\).)1969 271 y Fm(2)-84 400 y FC(Pro)r(of.)24 b FB(Let)17 b Fz(z)g FB(=)f Fz(e)297 382 y Ft(i\036)330 387 y Fk(0)366 400 y FB(with)h Fz(\036)507 407 y Fy(0)542 400 y Fx(2)f FB([0)p Fz(;)8 b(\031)r FB(\))16 b(and)i(recall)e(that)h Fz(w)1084 407 y Ft(j)1118 400 y FB(=)e Fz(e)1194 382 y Ft(i\036)1227 387 y Fl(j)1261 400 y Fx(\021)g Fz(z)r(\034)1367 382 y Fy(\()p Ft(j)r Fy(\))1361 412 y Ft(z)1412 400 y FB(\(1\),)j Fz(j)g Fx(2)e Fs(N)p FB(,)i(are)f(suc)o(h)g(that)g Fz(\036)1988 407 y Ft(j)-84 460 y FB(satisfy)f(the)g(discrete)f(map)h(\(4.3\).)21 b(By)16 b(Theorem)f(5.1,)h(w)o(e)g(ha)o(v)o(e)780 522 y Fr(\014)780 547 y(\014)780 572 y(\014)p FB(1)c(+)f Fz(z)r(\034)931 552 y Fy(\()p Ft(j)r Fy(\))925 584 y Ft(z)976 572 y FB(\(1\))1038 522 y Fr(\014)1038 547 y(\014)1038 572 y(\014)j Fx(\025)g Fz(\016)765 b FB(\(5.4\))-84 684 y(for)16 b(some)g Fz(\016)f FB(=)f Fz(\016)r FB(\()p Fz(\020)t(;)8 b(\036)321 691 y Fy(0)340 684 y FB(\))13 b Fz(>)h FB(0)j(pro)o(vided)e(1)p Fz(=)p FB(3)g Fz(<)f(\020)k Fx(\024)c FB(1)i(and)h(0)d Fx(\024)g Fz(\036)1151 691 y Fy(0)1184 684 y Fz(<)g(\024)p FB(.)21 b(As)c(a)f(consequence,)f(for)i (an)o(y)f Fz(")d(>)h FB(0,)362 828 y Fx(j)o Fz(F)407 835 y Ft(M)446 828 y FB(\()p Fz(z)r FB(\))d Fx(\000)g Fz(F)602 835 y Ft(N)635 828 y FB(\()p Fz(z)r FB(\))p Fx(j)j(\024)778 774 y Ft(M)t Fq(\000)p Fy(1)790 786 y Fr(X)779 878 y Ft(n)p Fy(=)p Ft(N)869 828 y Fz(a)895 835 y Ft(n)927 753 y Fr(\014)927 778 y(\014)927 803 y(\014)927 828 y(\014)927 853 y(\014)945 794 y FB(1)e Fx(\000)f Fz(z)r(\034)1083 776 y Fy(\()p Ft(j)r Fy(\))1077 806 y Ft(z)1128 794 y FB(\(1\))p 945 816 245 2 v 945 871 a(1)h(+)f Fz(z)r(\034)1082 846 y Fy(\()p Ft(j)r Fy(\))1076 875 y Ft(z)1128 871 y FB(\(1\))1195 753 y Fr(\014)1195 778 y(\014)1195 803 y(\014)1195 828 y(\014)1195 853 y(\014)1223 828 y Fx(\024)1281 794 y FB(2)p 1281 816 25 2 v 1281 862 a Fz(\016)1318 774 y Ft(M)t Fq(\000)p Fy(1)1329 786 y Fr(X)1319 878 y Ft(n)p Fy(=)p Ft(N)1409 828 y Fz(a)1435 835 y Ft(n)1472 828 y Fx(\024)j Fz(";)-84 971 y FB(for)g Fz(M)19 b(>)14 b(N)5 b FB(,)14 b Fz(N)19 b FB(large)14 b(enough,)h(since)e Fx(f)p Fz(a)701 978 y Ft(n)724 971 y Fx(g)749 978 y Ft(n)p Fq(2)p Fj(N)835 971 y FB(is)h(summable.)k(This) c(sa)o(ys)g(that)g(the)g(sequence)e(of)j(con)o(tin)o(uous)-84 1031 y(functions)21 b Fz(F)163 1038 y Ft(N)217 1031 y FB(con)o(v)o(erges)f(uniformly)f(to)i(a)g(con)o(tin)o(uous)g(function) 1233 1015 y Fr(e)1221 1031 y Fz(F)27 b FB(on)22 b Fz(S)1386 1013 y Fy(1)1419 1031 y Fx(n)15 b Fz(A)1496 1038 y Ft(\024)1539 1031 y FB(with)20 b Fz(\024)i FB(=)g Fz(\024)p FB(\()p Fz(\020)t FB(\))f(for)g(all)-84 1092 y(1)p Fz(=)p FB(3)15 b Fz(<)f(\020)k Fx(\024)13 b FB(1.)-11 1173 y(Analogously)l(,)j(under)g (the)g(same)g(conditions,)f(w)o(e)h(ha)o(v)o(e)305 1308 y Fx(j)p Fz(I)341 1315 y Ft(M)380 1308 y FB(\()p Fz(z)r FB(\))11 b Fx(\000)g Fz(I)526 1315 y Ft(N)559 1308 y FB(\()p Fz(z)r FB(\))p Fx(j)j(\024)f FB(2)734 1233 y Fr(\014)735 1258 y(\014)735 1283 y(\014)735 1308 y(\014)735 1333 y(\014)850 1274 y Fz(w)885 1281 y Ft(N)929 1274 y Fx(\000)e Fz(w)1014 1281 y Ft(M)p 754 1296 396 2 v 754 1342 a FB(\(1)g(+)g Fz(w)892 1349 y Ft(M)931 1342 y FB(\))e(\(1)i(+)g Fz(w)1097 1349 y Ft(N)1131 1342 y FB(\))1155 1233 y Fr(\014)1155 1258 y(\014)1155 1283 y(\014)1155 1308 y(\014)1155 1333 y(\014)1182 1308 y Fx(\024)1249 1274 y FB(2)p 1240 1296 44 2 v 1240 1342 a Fz(\016)1264 1327 y Fy(2)1296 1308 y Fx(j)p Fz(w)1345 1315 y Ft(N)1390 1308 y Fx(\000)g Fz(w)1475 1315 y Ft(M)1514 1308 y Fx(j)j(\024)f Fz(")-84 1442 y FB(for)19 b(an)o(y)f Fz(")f(>)g FB(0)i(pro)o(vided)f Fz(M)24 b FB(and)19 b Fz(N)k FB(are)c(large)f(enough,)h(as)g(a)g(consequence)e(of)i(Theorem)e (5.1.)28 b(Th)o(us,)19 b(the)-84 1502 y(sequence)f(of)i(con)o(tin)o (uous)f(functions)h Fz(I)663 1509 y Ft(N)696 1502 y FB(,)f Fz(N)25 b Fx(2)20 b Fs(N)p FB(,)g(con)o(v)o(erges)f(uniformly)e(to)j(a) g(con)o(tin)o(uous)f(function)1917 1486 y Fr(e)1910 1502 y Fz(I)k FB(on)-84 1562 y Fz(S)-51 1544 y Fy(1)-20 1562 y Fx(n)11 b Fz(A)53 1569 y Ft(\024)75 1562 y FB(.)-11 1644 y(This)16 b(concludes)g(the)g(pro)q(of)i(of)e(Theorem)f(5.3.)p 1977 1726 30 30 v -11 1855 a(By)g(using)g(the)g(\\Edge-of-the-W)l (edge")j(theorem,)13 b(the)i(analytic)g(function)g Fz(m)g FB(studied)g(in)g(Section)g(1.1)g(can)-84 1915 y(b)q(e)e(analytically)f (con)o(tin)o(ued)g(through)i(the)f(arc)g Fz(S)831 1897 y Fy(1)856 1915 y Fx(n)5 b Fz(A)923 1922 y Ft(\024)958 1915 y FB(pro)o(vided)1156 1911 y Fr(f)1155 1915 y Fz(m)13 b FB(is)g(the)g(limit)d(of)k Fz(m)e FB(when)i Fz(z)h FB(appro)o(ximates)-84 1975 y(the)h(arc)g Fz(S)114 1957 y Fy(1)145 1975 y Fx(n)11 b Fz(A)218 1982 y Ft(\024)240 1975 y FB(.)21 b(W)l(e)16 b(shall)g(establish)g(this)g(in)g(the)g (follo)o(wing)g(theorem.)-84 2106 y FC(Theorem)g(5.4)24 b Fn(L)n(et)d FB(1)p Fz(=)p FB(3)h Fz(<)e(\020)25 b Fx(\024)20 b FB(1)i Fn(and)f FC(p)g Fx(2)f Fz(\031)r Fn(.)33 b(L)n(et)21 b Fz(m)f FB(=)h Fz(m)p FB(\()p Fz(\020)t(;)8 b(z)r(;)g FC(p)p FB(\))20 b Fn(b)n(e)h(the)h(analytic)g(functions)g(on)g Fz(z)-84 2167 y Fn(state)n(d)17 b(in)h(The)n(or)n(em)f(1.1)g(for)g(the) h(domain)f Fx(fj)p Fz(z)r Fx(j)c Fz(<)h FB(1)p Fx(g)k([)f(fj)p Fz(z)r Fx(j)d Fz(>)f FB(1)p Fx(g)q Fn(.)22 b(If)17 b Fz(z)1297 2174 y Fy(0)1330 2167 y Fx(2)d Fz(S)1410 2148 y Fy(1)1441 2167 y Fx(n)d Fz(A)1514 2174 y Ft(\024)1536 2167 y Fn(,)17 b(we)i(have)720 2273 y FB(lim)710 2297 y Ft(z)q Fq(!)p Ft(z)778 2302 y Fk(0)805 2273 y Fx(j)p Fz(m)p FB(\()p Fz(z)r FB(\))11 b Fx(\000)986 2269 y Fr(f)985 2273 y Fz(m)p FB(\()p Fz(z)1070 2280 y Fy(0)1089 2273 y FB(\))p Fx(j)j FB(=)g(0)1969 2404 y Fm(2)-84 2533 y FC(Pro)r(of.)21 b FB(Giv)o(en)15 b Fz(z)h Fx(2)e Fs(C)28 b FB(de\014ne)16 b(a)h(sequence)e Fz(w)787 2540 y Ft(j)805 2533 y FB(,)h Fz(j)h Fx(2)d Fs(N)p FB(,)j(recursiv)o(ely)c(b)o(y)807 2639 y Fz(w)842 2646 y Ft(j)874 2639 y FB(=)h Fz(z)r(h)p FB(\()p Fz(w)1033 2646 y Ft(j)r Fq(\000)p Fy(1)1096 2639 y FB(\))791 b(\(5.5\))-84 2745 y(with)19 b Fz(w)65 2752 y Fy(0)103 2745 y FB(=)g Fz(z)r FB(.)29 b(Let)20 b(us)f(denote)g(b)o(y) g Fz(w)650 2752 y Fy(0)p Ft(;j)695 2745 y FB(,)h Fz(j)h Fx(2)e Fs(N)p FB(,)h(the)f(sequence)f(giv)o(en)g(b)o(y)h(\(5.5\))g (with)g Fz(z)i FB(=)e Fz(z)1741 2752 y Fy(0)1779 2745 y Fx(2)f Fz(S)1863 2727 y Fy(1)1896 2745 y Fx(n)13 b Fz(A)1971 2752 y Ft(\024)1993 2745 y FB(.)-84 2805 y(Note)j(that)g Fx(j)p Fz(w)188 2812 y Fy(0)p Ft(;j)234 2805 y Fx(j)e FB(=)f(1)k(and)g(if)f(one)g(writes)g Fz(w)762 2812 y Fy(0)p Ft(;j)821 2805 y FB(=)e Fz(e)896 2787 y Ft(i\036)929 2792 y Fl(j)947 2805 y FB(,)i(the)g(sequence)f Fz(\036)1292 2812 y Ft(j)1310 2805 y FB(,)h Fz(j)h Fx(2)d Fs(N)p FB(,)i(satis\014es) h(\(4.3\).)937 2930 y(29)p eop %%Page: 30 30 30 29 bop -11 79 a FB(W)l(e)16 b(ha)o(v)o(e)738 139 y Fx(j)p Fz(w)787 146 y Fy(0)817 139 y Fx(\000)11 b Fz(w)902 146 y Fy(0)p Ft(;)p Fy(0)949 139 y Fx(j)j FB(=)g Fx(j)p Fz(z)e Fx(\000)f Fz(z)1151 146 y Fy(0)1171 139 y Fx(j)721 b FB(\(5.6\))-84 227 y(and)17 b(for)f Fz(j)h FB(=)d(1)p Fz(;)8 b FB(2)p 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Ft(k)-84 2303 y FB(with)g(the)g(p)q(olynomial)f(\(1)c(+)g Fz(x)p FB(\))513 2278 y Ft(k)534 2303 y FB(.)-84 2438 y FC(Theorem)16 b(6.7)24 b Fn(Given)19 b Fz(k)e Fx(2)e Fs(N)517 2445 y Fy(+)547 2438 y Fn(,)j(let)h Fz(R)686 2445 y Ft(k)722 2438 y FB(:)14 b Fz(x)g Fx(2)h Fs(R)879 2445 y Fy(+)920 2438 y Fx(7\000)-9 b(!)15 b Fs(R)1054 2445 y Fy(+)1098 2438 y Fn(b)n(e)j(the)h(p)n(olynomial)e(de\014ne)n(d)i (by)f(\(6.24\).)23 b(Then,)-84 2498 y(for)17 b(any)g Fz(\016)e(>)f FB(1)p Fn(,)k(the)g(fol)r(lowing)i(holds)745 2559 y Fz(R)782 2566 y Ft(k)804 2559 y FB(\()p Fz(x)p FB(\))13 b Fx(\024)h Fz(C)971 2566 y Ft(\016)1007 2559 y FB(\()p Fz(\016)e FB(+)f Fz(x)p FB(\))1156 2538 y Ft(k)1882 2559 y FB(\(6.25\))-84 2646 y Fn(with)768 2706 y Fz(C)803 2713 y Ft(\016)836 2706 y FB(=)j(1)d(+)1018 2672 y Fz(x)p 977 2694 109 2 v 977 2740 a(\016)i Fx(\000)d FB(1)1141 2706 y Fz(:)937 2930 y FB(38)p eop %%Page: 39 39 39 38 bop 1969 79 a Fm(2)-11 199 y FB(Theorem)15 b(6.7)i(will)e(b)q(e)h (pro)o(v)o(en)g(in)g(the)g(next)f(subsection.)-11 281 y(Applying)i(Theorem)g(6.7)i(to)g(the)f(inequalit)o(y)e(\(6.23\),)j (giv)o(es)e(an)i(upp)q(er)f(b)q(ound)i(of)e(the)g(form)g(\(6.21\).)27 b(F)l(or)-84 341 y(an)o(y)16 b Fz(\016)f(>)f FB(1,)i(w)o(e)g(th)o(us)g (ha)o(v)o(e)684 351 y Fr(\014)684 376 y(\014)684 401 y(\014)o Fz(D)738 381 y Ft(k)769 401 y FB(\()p Fz(f)g Fx(\016)11 b Fz(g)r FB(\))908 351 y Fr(\014)908 376 y(\014)908 401 y(\014)j Fx(\024)g Fz(C)1024 408 y Ft(f)t Fq(\016)p Ft(g)1090 401 y Fz(k)r FB(!)8 b Fz(K)1184 381 y Ft(k)1180 414 y(f)t Fq(\016)p Ft(g)1882 401 y FB(\(6.26\))-84 480 y(where)689 552 y Fz(C)724 559 y Ft(f)t Fq(\016)p Ft(g)796 552 y FB(=)14 b Fz(C)883 559 y Ft(f)914 491 y Fr(\022)945 552 y FB(1)d(+)g Fz(C)1064 559 y Ft(g)1111 518 y Fz(K)1152 525 y Ft(f)p 1089 540 109 2 v 1089 586 a Fz(\016)h Fx(\000)f FB(1)1203 491 y Fr(\023)1882 552 y FB(\(6.27\))-84 647 y(and)688 707 y Fz(K)729 714 y Ft(f)t Fq(\016)p Ft(g)802 707 y FB(=)i(\()p Fz(\016)g FB(+)e Fz(C)991 714 y Ft(g)1019 707 y Fz(K)1060 714 y Ft(f)1083 707 y FB(\))e Fz(K)1152 714 y Ft(g)1221 707 y Fz(:)647 b FB(\(6.28\))-11 807 y(The)17 b(estimate)f(\(6.26\))h(can)h(b)q(e)f(used)g(to)h(con)o(trol)e (inductiv)o(ely)f(deriv)m(ativ)o(es)h(of)h Fz(w)1510 814 y Ft(n)1551 807 y FB(in)f(\(6.8\))i(pro)o(vided)e Fz(f)23 b FB(is)-84 867 y(replaced)15 b(b)o(y)h Fz(h)g FB(and)h Fz(g)i FB(replaced)c(b)o(y)h Fz(w)652 874 y Ft(n)p Fq(\000)p Fy(1)720 867 y FB(.)21 b(This)c(yields)e(the)h(follo)o (wing)g(theorem.)-84 980 y FC(Theorem)g(6.8)24 b Fn(F)l(or)17 b(\014xe)n(d)i Fz(n)p Fn(,)e Fz(k)f Fx(2)e Fs(N)p Fn(,)k Fz(k)e Fx(\025)e FB(1)p Fn(,)j(let)i Fz(W)942 962 y Ft(j)935 992 y(n)978 980 y Fn(b)n(e)e(given)i(by)f(\(6.9\))f(and)h(supp)n(ose) 587 1070 y Fz(W)640 1049 y Ft(j)633 1082 y(n)672 1070 y Fx(\024)c Fz(W)771 1077 y Ft(n)812 1070 y Fz(\026)841 1049 y Ft(j)841 1082 y(n)964 1070 y Fz(;)108 b(j)17 b FB(=)c(1)p Fz(;)8 b(:)g(:)g(:)g(;)g(k)549 b FB(\(6.29\))-84 1160 y Fn(holds)17 b(with)h Fz(W)192 1167 y Ft(n)233 1160 y Fn(and)g Fz(\026)357 1167 y Ft(n)398 1160 y Fn(p)n(ositive)g(c)n (onstants.)23 b(Then,)18 b(given)h Fz(\016)c(>)f FB(1)k Fn(and)f Fz(H)22 b Fn(as)17 b(in)h(L)n(emma)f(6.14,)756 1249 y Fz(W)809 1226 y Ft(j)802 1260 y(n)p Fy(+1)884 1249 y Fx(\024)d Fz(W)983 1256 y Ft(n)p Fy(+1)1069 1249 y Fz(\026)1098 1226 y Ft(j)1098 1260 y(n)p Fy(+1)-84 1339 y Fn(holds)j(for)g(al)r(l)i Fz(j)e FB(=)d(1)p Fz(;)8 b(:)g(:)g(:)f(;)h(k)r Fn(,)18 b(with)581 1457 y Fz(W)627 1464 y Ft(n)p Fy(+1)710 1457 y FB(=)13 b Fz(H)814 1397 y Fr(\022)845 1457 y Fz(z)g FB(+)935 1424 y(1)p 935 1446 25 2 v 935 1492 a Fz(\016)964 1397 y Fr(\023)1003 1384 y( )1036 1457 y FB(1)e(+)1155 1424 y(2)p Fz(\020)p 1125 1446 109 2 v 1125 1492 a(\016)i Fx(\000)d FB(1)1239 1457 y Fz(W)1285 1464 y Ft(n)1308 1384 y Fr(!)1882 1457 y FB(\(6.30\))-84 1576 y Fn(and)697 1636 y Fz(\026)726 1643 y Ft(n)p Fy(+1)809 1636 y FB(=)j(\()p Fz(\016)g FB(+)e(2)p Fz(\020)t(W)1058 1643 y Ft(n)1082 1636 y FB(\))d Fz(\026)1138 1643 y Ft(n)1212 1636 y Fz(:)656 b FB(\(6.31\))1969 1748 y Fm(2)-84 1868 y FC(Pro)r(of:)21 b FB(Di\013eren)o(tiating)16 b(\(6.8\))g Fz(j)s FB(-times)f(and)h(taking)h(absolute)g(v)m(alue,)e (giv)o(es)324 1947 y(1)p 318 1969 37 2 v 318 2014 a Fz(j)s FB(!)367 1930 y Fr(\014)367 1955 y(\014)367 1980 y(\014)p Fz(D)422 1960 y Ft(j)441 1980 y Fz(w)476 1987 y Ft(n)p Fy(+1)545 1930 y Fr(\014)545 1955 y(\014)545 1980 y(\014)e Fx(\024)h Fz(z)677 1947 y FB(1)p 671 1969 V 671 2014 a Fz(j)s FB(!)721 1930 y Fr(\014)721 1955 y(\014)721 1980 y(\014)p Fz(D)776 1960 y Ft(j)803 1980 y FB(\()p Fz(h)d Fx(\016)g Fz(w)932 1987 y Ft(n)956 1980 y FB(\))975 1930 y Fr(\014)975 1955 y(\014)975 1980 y(\014)g FB(+)1121 1947 y(1)p 1054 1969 160 2 v 1054 2014 a(\()p Fz(j)j Fx(\000)c FB(1\)!)1227 1930 y Fr(\014)1227 1955 y(\014)1227 1980 y(\014)o Fz(D)1281 1960 y Ft(j)r Fq(\000)p Fy(1)1354 1980 y FB(\()p Fz(h)h Fx(\016)g Fz(w)1483 1987 y Ft(n)1506 1980 y FB(\))1525 1930 y Fr(\014)1525 1955 y(\014)1525 1980 y(\014)57 b Fz(:)-11 2119 y 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2543 y FB(b)o(y)16 b(the)g(p)q(ositivit)o(y)f(of)h Fz(W)386 2550 y Ft(n)410 2543 y FB(,)g Fz(n)e Fx(2)g Fs(N)p FB(.)22 b(This)16 b(concludes)g(the)g(pro)q(of)h(of)g(Theorem)e (6.8.)p 1977 2625 30 30 v -11 2745 a(Theorem)k(6.5)h(and)h(equation)e (\(6.19\))i(are)f(particular)g(cases)g(of)g(the)g(follo)o(wing)g (result)f(obtained)h(b)o(y)g(the)-84 2805 y(induction)c(describ)q(ed)f (in)h(Theorem)f(6.8.)937 2930 y(39)p eop %%Page: 40 40 40 39 bop -84 79 a FC(Theorem)16 b(6.9)24 b Fn(Given)i FB(0)h Fz(<)g(\020)k(<)26 b FB(1)p Fz(=)p FB(3)p Fn(,)i Fz(z)h Fx(2)e Fs(R)897 86 y Fy(+)948 79 y Fn(and)d Fz(n)p Fn(,)i Fz(k)j Fx(2)e Fs(N)p Fn(,)h Fz(k)g Fx(\025)f FB(1)p Fn(.)44 b(Ther)n(e)24 b(exist)h(a)g(c)n(onstant)-84 139 y Fz(C)17 b FB(=)d Fz(C)t FB(\()p Fz(\020)t(;)8 b(z)r FB(\))13 b Fz(<)h Fx(1)j Fn(such)h(that)830 200 y Fz(W)883 179 y Ft(j)876 212 y(n)915 200 y Fx(\024)13 b Fz(C)21 b(\026)1052 179 y Ft(nj)1882 200 y FB(\(6.32\))-84 287 y Fn(holds)c(for)g(al)r(l)i Fz(j)e FB(=)d(1)p Fz(;)8 b(:)g(:)g(:)f(;)h(k)20 b Fn(with)473 449 y Fz(\026)p FB(\()p Fz(z)r FB(\))28 b(:=)f(4)p Fz(\020)t(H)t(z)h FB(+)c(2)910 364 y Fr(2)911 439 y(4)939 376 y( )977 416 y FB(4)p Fz(\020)t(H)t(z)13 b FB(+)e(1)p 977 438 204 2 v 1066 484 a(4)1185 376 y Fr(!)1218 388 y Fy(2)1248 449 y FB(+)g Fz(\020)t(H)1366 364 y Fr(3)1366 439 y(5)1394 375 y Fy(1)p Ft(=)p Fy(2)-84 600 y Fn(and)18 b Fz(H)j Fn(as)d(in)f(L)n(emma)g(6.14.)22 b(Mor)n(e)n(over,)16 b Fz(\026)i Fn(and)g Fz(C)i Fn(satis\014es)f(the)e(fol)r(lowing)j(r)n (elation)e Fz(\026)c FB(=)g(1)d(+)g(4)p Fz(\020)t(C)t Fn(.)1969 736 y Fm(2)-84 867 y FC(Pro)r(of:)21 b FB(Since)15 b Fz(w)251 874 y Fy(0)285 867 y FB(=)f Fz(z)r FB(,)h(w)o(e)h(ha)o(v)o (e)640 1034 y Fz(W)693 1010 y Ft(j)686 1045 y Fy(0)725 1034 y Fx(\024)778 934 y Fr(8)778 971 y(>)778 984 y(<)778 1059 y(>)778 1071 y(:)835 973 y FB(1)42 b(if)214 b Fz(j)16 b FB(=)e(1)835 1093 y(0)42 b(otherwise)-84 1203 y(whic)o(h)20 b(certainly)g(satis\014es)h(\(6.29\))h(with)f Fz(W)759 1210 y Fy(0)801 1203 y FB(=)h(1)f(and)h Fz(\026)1035 1210 y Fy(0)1077 1203 y FB(=)f(1.)36 b(This)22 b(will)d(b)q(e)j(the)e (initial)g(conditions)h(of)-84 1263 y(Theorem)15 b(6.8.)-11 1345 y(W)l(e)j(no)o(w)h(observ)o(e)f(that)h(the)f(sequence)f Fx(f)p Fz(W)824 1352 y Ft(n)848 1345 y Fx(g)873 1359 y Ft(n)p Fq(2)p Fj(N)945 1345 y FB(,)h(de\014ned)h(b)o(y)f(the)g (recurrence)f(\(6.30\))i(is)f(a)h(con)o(v)o(ergen)o(t)-84 1405 y(sequence)c(pro)o(vided)h Fz(\016)h FB(is)f(suc)o(h)g(that)h(2)p Fz(\020)t(H)c FB(\()p Fz(z)g FB(+)e(1)p Fz(=\016)r FB(\))e(/\()p Fz(\016)j Fx(\000)f FB(1\))j Fz(<)g FB(1)6 b(.)21 b(F)l(or)16 b(de\014niteness,)f(let)h Fz(\016)i FB(b)q(e)e(giv)o(en)f(b)o(y)763 1506 y(2)p Fz(\020)t(H)p 756 1528 109 2 v 756 1574 a(\016)d Fx(\000)f FB(1)878 1479 y Fr(\022)908 1540 y Fz(z)i FB(+)998 1506 y(1)p 998 1528 25 2 v 998 1574 a Fz(\016)1028 1479 y Fr(\023)1072 1540 y FB(=)1129 1506 y(1)p 1129 1528 V 1129 1574 a(2)1158 1540 y Fz(:)710 b FB(\(6.33\))-11 1691 y(Iterating)16 b(equation)g(\(6.30\),)h(yields)516 1832 y Fz(W)562 1839 y Ft(n)p Fy(+1)672 1832 y FB(=)757 1816 y Fr(f)753 1832 y Fz(H)805 1759 y Fr( )838 1832 y FB(1)12 b(+)928 1798 y(1)p 928 1820 V 928 1866 a(2)968 1832 y(+)f Fx(\001)d(\001)g(\001)j FB(+)1135 1771 y Fr(\022)1171 1798 y FB(1)p 1171 1820 V 1171 1866 a(2)1200 1771 y Fr(\023)1231 1779 y Ft(n)p Fy(+1)1308 1832 y Fz(W)1354 1839 y Fy(0)1373 1759 y Fr(!)672 1989 y Fx(\024)42 b FB(2)781 1973 y Fr(f)777 1989 y Fz(H)1882 1895 y FB(\(6.34\))-84 2108 y(where)61 2092 y Fr(f)57 2108 y Fz(H)18 b FB(=)c Fz(H)e FB(\()p Fz(z)h FB(+)e(1)p Fz(=\016)r FB(\))17 b(whic)o(h)e(can)i(b)q(e)f (written)g(as)354 2258 y Fr(f)350 2274 y Fz(H)i FB(=)465 2240 y Fz(\016)12 b Fx(\000)f FB(1)p 465 2262 109 2 v 494 2308 a(4)p Fz(\020)592 2274 y FB(=)649 2240 y(4)p Fz(\020)t(H)t(z)j Fx(\000)c FB(1)p 649 2262 204 2 v 726 2308 a(8)p Fz(\020)869 2274 y FB(+)935 2240 y(1)p 923 2262 50 2 v 923 2308 a(4)p Fz(\020)986 2188 y Fr(2)986 2263 y(4)1013 2201 y( )1051 2240 y FB(4)p Fz(\020)t(H)t(z)k FB(+)d(1)p 1051 2262 204 2 v 1140 2308 a(4)1259 2201 y Fr(!)1292 2212 y Fy(2)1323 2274 y FB(+)g Fz(\020)t(H)1441 2188 y Fr(3)1441 2263 y(5)1469 2200 y Fy(1)p Ft(=)p Fy(2)-84 2427 y FB(b)o(y)16 b(solving)g(equation)g(\(6.33\).)-11 2509 y(In)g(view)g(of)g(\(6.31\))h(and)g(\(6.34\),)f(w)o(e)g(ha)o(v)o (e)336 2648 y Fz(\026)365 2655 y Ft(n)p Fy(+1)448 2648 y FB(=)e Fz(\026)529 2655 y Fy(0)577 2594 y Ft(n)561 2606 y Fr(Y)557 2697 y Ft(j)r Fy(=0)627 2648 y FB(\()p Fz(\016)e FB(+)f(2)p Fz(\020)t(W)824 2655 y Ft(j)843 2648 y FB(\))j Fx(\024)929 2600 y Fr(\020)953 2648 y Fz(\016)f FB(+)e(4)p Fz(\020)1090 2632 y Fr(f)1086 2648 y Fz(H)1131 2600 y Fr(\021)1155 2611 y Ft(n)p Fy(+1)1238 2648 y FB(=)1290 2600 y Fr(\020)1314 2648 y FB(1)h(+)f(8)p Fz(\020)1452 2632 y Fr(f)1448 2648 y Fz(H)1493 2600 y Fr(\021)1517 2611 y Ft(n)p Fy(+1)1882 2648 y FB(\(6.35\))-84 2797 y(whic)o(h)k(concludes)h(the)g(pro)q(of)i(of)e(Theorem)f(6.9.)937 2930 y(40)p eop %%Page: 41 41 41 40 bop 1977 79 30 30 v -11 207 a FB(W)l(e)16 b(are)g(no)o(w)h(ready) f(to)h(\014nish)f(the)g(pro)q(of)h(of)g(Theorem)e(1.6.)-84 289 y FC(Pro)r(of)k(of)f(Theorem)f(1.6)f FB(\(Conclusion\))p FC(:)21 b FB(Di\013eren)o(tiating)16 b(\(6.2\))g Fz(k)r FB(-times,)e(giv)o(es)638 397 y(1)p 630 419 41 2 v 630 464 a Fz(k)r FB(!)675 430 y Fz(D)716 410 y Ft(k)739 430 y Fz(F)771 437 y Ft(N)818 430 y FB(=)869 376 y Ft(N)t Fq(\000)p Fy(1)878 389 y Fr(X)874 479 y Ft(n)p Fy(=1)954 430 y Fz(a)980 437 y Ft(n)1033 397 y FB(1)p 1025 419 V 1025 464 a Fz(k)r FB(!)1070 430 y Fz(D)1111 410 y Ft(k)1142 430 y FB([)o Fz(r)f Fx(\016)e Fz(w)1261 437 y Ft(n)1284 430 y FB(])584 b(\(6.36\))-84 575 y(whic)o(h)15 b(can)i(b)q(e)f (estimated)f(b)o(y)g(using)i(\(6.26\))g(pro)o(vided)e(w)o(e)h(compute)f Fz(r)1263 556 y Fy(\()p Ft(j)r Fy(\))1309 575 y FB(,)h Fz(j)h Fx(2)d Fs(N)p FB(.)22 b(By)15 b(an)i(explicitly)c(compu-)-84 635 y(tation,)j(w)o(e)g(ha)o(v)o(e)690 714 y Fz(r)713 694 y Fy(\()p Ft(j)r Fy(\))759 714 y FB(\()p Fz(x)p FB(\))d(=)h(\()p 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Ft(k)600 1237 y FB([)o Fz(r)d Fx(\016)e FB(\()p Fz(w)738 1244 y Ft(n)761 1237 y FB(\)])794 1187 y Fr(\014)794 1212 y(\014)794 1237 y(\014)i Fx(\024)h FB(2)906 1176 y Fr(\022)937 1237 y FB(1)e(+)1049 1203 y Fz(C)t(\021)p 1027 1225 109 2 v 1027 1271 a(\016)g Fx(\000)f FB(1)1140 1176 y Fr(\023)1179 1237 y FB(\()p Fz(\016)i FB(+)e Fz(C)t(\021)r FB(\))1365 1213 y Ft(k)1395 1237 y Fz(\026)1424 1217 y Ft(nk)1882 1237 y FB(\(6.37\))-84 1358 y(with)16 b Fz(\016)f(>)f FB(1.)-11 1440 y(In)i(view)g(of)g(\(6.36\))h(and)g(\(6.37\),)f(there)g (exist)f(\014nite)h(constan)o(ts,)g Fz(C)1237 1447 y Fy(1)1273 1440 y FB(and)h Fz(C)1403 1447 y Fy(2)1423 1440 y FB(,)f(suc)o(h)g(that)496 1547 y(1)p 488 1569 41 2 v 488 1615 a Fz(k)r FB(!)542 1531 y Fr(\014)542 1556 y(\014)542 1581 y(\014)p Fz(D)597 1560 y Ft(k)619 1581 y Fz(F)651 1588 y Ft(N)695 1581 y Fx(\000)11 b Fz(D)786 1560 y Ft(k)808 1581 y Fz(F)840 1588 y Ft(M)879 1531 y Fr(\014)879 1556 y(\014)879 1581 y(\014)j Fx(\024)f Fz(C)994 1588 y Fy(1)1014 1581 y Fz(C)1053 1560 y Ft(k)1049 1593 y Fy(2)1132 1527 y Ft(N)1118 1539 y Fr(X)1082 1631 y Ft(n)p Fy(=)p Ft(M)t Fy(+1)1230 1581 y Fz(a)1256 1588 y Ft(n)1279 1581 y Fz(\026)1308 1560 y Ft(k)q(n)1365 1581 y Fz(<)h(")-84 1737 y FB(for)22 b(an)o(y)g Fz(")h(>)g FB(0,)g(pro)o(vided)468 1689 y Fr(n)496 1737 y Fz(a)522 1744 y Ft(n)545 1737 y Fz(\026)574 1719 y Ft(k)q(n)617 1689 y Fr(o)645 1764 y Ft(n)p Fq(\025)p Fy(0)735 1737 y FB(is)f(a)g(summable)d(sequence)i(and)h Fz(M)5 b FB(,)23 b Fz(N)29 b(>)23 b(N)1637 1744 y Fy(0)1679 1737 y FB(for)f(some)e Fz(N)1925 1744 y Fy(0)1969 1737 y FB(=)-84 1823 y Fz(N)-45 1830 y Fy(0)-25 1823 y FB(\()p Fz(\020)12 b(;)c(z)i(;)16 b FC(p)p Fz(;)8 b(k)r FB(\))17 b Fz(<)g Fx(1)p FB(.)27 b(The)18 b(sequence)681 1775 y Fr(n)708 1823 y Fz(D)749 1805 y Ft(k)771 1823 y Fz(F)803 1830 y Ft(N)837 1775 y Fr(o)864 1850 y Ft(N)t Fq(\025)p Fy(0)943 1823 y FB(,)g(for)h(an)o(y) f Fz(k)h Fx(2)e Fs(N)p Fz(;)i FB(is)f(a)h(uniform)d(Cauc)o(h)o(y)i (sequence)f(of)-84 1895 y(con)o(tin)o(uous)f(function)g(of)h Fz(z)h FB(in)e Fs(R)544 1902 y Fy(+)570 1895 y FB(.)21 b(This)c(concludes)f(the)g(pro)q(of)h(of)f(Theorem)f(1.6.)p 1977 1976 30 30 v -84 2188 a FI(6.3)66 b(Pro)r(of)22 b(of)f(Theorem)h(6.7)-84 2302 y FB(Theorem)15 b(6.7)h(will)g(b)q(e)g (pro)o(v)o(en)f(\014rstly)h(assuming)g(that)h(the)f(follo)o(wing)g(upp) q(er)g(b)q(ound)751 2440 y Fz(R)788 2447 y Ft(k)810 2440 y FB(\()p Fz(x)p FB(\))d Fx(\024)h Fz(x)978 2386 y Ft(k)q Fq(\000)p Fy(1)980 2399 y Fr(X)979 2490 y Ft(j)r Fy(=0)1050 2440 y Fz(R)1087 2447 y Ft(j)1106 2440 y FB(\()p Fz(x)p FB(\))710 b(\(6.38\))-84 2586 y(holds)17 b(for)f(all)g Fz(x)d Fx(2)h Fs(R)313 2593 y Fy(+)340 2586 y Fz(;)i FB(with)g Fz(R)518 2593 y Fy(0)538 2586 y FB(\()p Fz(x)p FB(\))d Fx(\021)h FB(1.)21 b(The)c(inequalit)o(y)d(\(6.38\))j(will)e(b) q(e)h(established)g(afterw)o(ard.)-11 2668 y(W)l(e)g(pro)o(v)o(e)g (\(6.25\))g(b)o(y)g(induction.)21 b(F)l(or)16 b Fz(k)g FB(=)e(0,)713 2783 y Fz(R)750 2790 y Fy(0)770 2783 y FB(\()p Fz(x)p FB(\))f Fx(\021)h FB(1)g Fx(\024)g FB(1)d(+)1123 2750 y Fz(x)p 1082 2772 109 2 v 1082 2817 a(\016)i Fx(\000)e FB(1)1196 2783 y Fz(:)937 2930 y FB(41)p eop %%Page: 42 42 42 41 bop -11 79 a FB(No)o(w,)16 b(let)f(us)i(assume)e(\(6.25\))i(v)m (alid)f(for)h Fz(k)e FB(=)f(0)p Fz(;)8 b(:)g(:)g(:)g(;)g(n)p FB(,)16 b Fz(n)e Fx(2)g Fs(N)p FB(.)22 b(F)l(rom)15 b(\(6.38\),)i(w)o (e)e(ha)o(v)o(e)504 218 y Fz(R)541 225 y Ft(n)p Fy(+1)610 218 y FB(\()p Fz(x)p FB(\))41 b Fx(\024)g Fz(x)833 157 y Fr(\022)864 218 y FB(1)11 b(+)993 184 y Fz(x)p 953 206 109 2 v 953 252 a(\016)i Fx(\000)d FB(1)1067 157 y Fr(\023)1125 164 y Ft(n)1106 177 y Fr(X)1106 268 y Ft(j)r Fy(=0)1175 218 y FB(\()p Fz(\016)j FB(+)e Fz(x)p FB(\))1324 194 y Ft(j)717 418 y FB(=)42 b Fz(x)833 357 y Fr(\022)864 418 y FB(1)11 b(+)993 384 y Fz(x)p 953 406 V 953 452 a(\016)i Fx(\000)d FB(1)1067 357 y Fr(\023)1111 384 y FB(\()p Fz(\016)i FB(+)f Fz(x)p FB(\))1260 360 y Ft(n)p Fy(+1)1339 384 y Fx(\000)g FB(1)p 1111 406 304 2 v 1164 452 a Fz(x)g FB(+)g Fz(\016)h Fx(\000)f FB(1)717 579 y Fx(\024)797 518 y Fr(\022)828 579 y FB(1)g(+)957 545 y Fz(x)p 917 567 109 2 v 917 613 a(\016)i Fx(\000)d FB(1)1031 518 y Fr(\023)1070 579 y FB(\()p Fz(\016)i FB(+)f Fz(x)p FB(\))1219 554 y Ft(n)p Fy(+1)1882 579 y FB(\(6.39\))-84 708 y(since)k Fz(x)8 b FB(/)q(\()p Fz(x)j FB(+)g Fz(\016)h Fx(\000)f FB(1\))19 b Fz(<)14 b FB(1.)22 b(This)16 b(pro)o(v)o(es)g(Theorem)e(6.7)j(under)f(the)g (assumption)g(\(6.38\).)p 1977 790 30 30 v -84 921 a FC(Pro)r(of)j(of)f(\(6.38\):)j FB(De\014ne)615 1019 y Fz(y)639 1026 y Ft(`)669 1019 y FB(=)744 965 y Ft(`)721 978 y Fr(X)720 1069 y Ft(j)r Fy(=1)790 1019 y Fz(n)819 1026 y Ft(j)951 1019 y Fz(;)97 b(`)15 b FB(=)e(1)p Fz(;)8 b(:)g(:)g(:)g(;)g FB(k)-84 1145 y(and)17 b(notice)e(that)i(the)f(co)q (e\016cien)o(ts)f(of)h Fz(R)683 1152 y Ft(k)721 1145 y FB(can)h(b)q(e)f(written)g(as)420 1208 y Fr(\020)445 1223 y(P)489 1267 y Ft(j)515 1256 y Fz(n)544 1263 y Ft(j)563 1208 y Fr(\021)587 1256 y FB(!)p 410 1288 201 2 v 410 1334 a Fz(n)439 1341 y Fy(1)459 1334 y FB(!)8 b Fz(:)g(:)g(:)g(n)576 1341 y Ft(k)597 1334 y FB(!)629 1300 y(=)689 1266 y Fz(y)713 1273 y Fy(1)732 1266 y FB(!)p 686 1288 63 2 v 686 1334 a Fz(n)715 1341 y Fy(1)735 1334 y FB(!)790 1266 y Fz(y)814 1273 y Fy(2)833 1266 y FB(!)p 758 1288 120 2 v 758 1334 a Fz(n)787 1341 y Fy(2)807 1334 y FB(!)p Fz(y)845 1341 y Fy(1)864 1334 y FB(!)891 1300 y Fx(\001)g(\001)g(\001)1017 1266 y Fz(y)1041 1273 y Ft(k)1062 1266 y FB(!)p 963 1288 168 2 v 963 1334 a Fz(n)992 1341 y Ft(k)1013 1334 y FB(!)p Fz(y)1051 1341 y Ft(k)q Fq(\000)p Fy(1)1117 1334 y FB(!)1149 1300 y(=)1201 1239 y Fr(\022)1239 1266 y Fz(y)1263 1273 y Fy(2)1236 1334 y Fz(n)1265 1341 y Fy(2)1290 1239 y Fr(\023)1329 1300 y Fx(\001)g(\001)g(\001)1396 1239 y Fr(\022)1434 1266 y Fz(y)1458 1273 y Ft(k)1431 1334 y Fz(n)1460 1341 y Ft(k)1487 1239 y Fr(\023)-11 1457 y FB(With)16 b(this)g(notation,)h(\(6.24\))g(can)f(th)o(us)g(b)q(e)h (written)f(as)219 1605 y Fz(R)256 1612 y Ft(k)277 1605 y FB(\()p 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Fz(R)497 1793 y Fy([)p Ft(i;j)r Fy(])497 1831 y Ft(k)557 1818 y FB(\()p Fz(x)p FB(\))h(b)q(e)g(the)f(p)q(olynomial)g(of)h(order)g Fz(k)i FB(in)e Fz(x)g FB(obtained)g(from)f(\(6.40\))h(b)o(y)g(setting) -84 1878 y Fz(n)-55 1885 y Fy(1)-35 1878 y Fz(;)8 b(:)g(:)g(:)f(;)h(n) 103 1885 y Ft(i)p Fq(\000)p Fy(1)162 1878 y Fz(;)g(n)213 1885 y Ft(j)r Fy(+1)277 1878 y Fz(;)g(:)g(:)g(:)f(;)h(n)415 1885 y Ft(k)453 1878 y FB(all)15 b(equal)h(to)h(0,)f(i.e.,)69 2021 y Fz(R)106 1995 y Fy([)p Ft(i;j)r Fy(])106 2033 y Ft(k)166 2021 y FB(\()p Fz(x)p FB(\))e(=)297 1979 y Fr(X)318 2066 y Ft(n)340 2071 y Fl(i)366 2021 y Fz(x)394 2000 y Ft(n)415 2005 y Fl(i)453 1979 y Fr(X)455 2066 y Ft(n)476 2071 y Fl(i)p Fk(+1)536 1948 y Fr( )577 1987 y Fz(y)601 1994 y Ft(i)p Fy(+1)574 2055 y Fz(n)603 2062 y Ft(i)p Fy(+1)667 1948 y Fr(!)708 2021 y Fz(x)736 2000 y Ft(n)757 2005 y Fl(i)p Fk(+1)820 2021 y Fz(:)8 b(:)g(:)885 1979 y Fr(X)904 2066 y Ft(n)926 2071 y Fl(j)953 1948 y Fr( )994 1987 y Fz(y)1018 1994 y Ft(j)991 2055 y Fz(n)1020 2062 y Ft(j)1044 1948 y Fr(!)1085 2021 y Fz(x)1113 2000 y Ft(nj)1168 2021 y Fz(\016)1190 2028 y Ft(k)1220 2021 y FB(\()p Fz(in)1285 2028 y Ft(i)1309 2021 y FB(+)j(\()p Fz(i)g FB(+)g(1\))p Fz(n)1526 2028 y Ft(i)p Fy(+1)1597 2021 y FB(+)g Fx(\001)d(\001)g(\001)j FB(+)g Fz(j)s(n)1816 2028 y Ft(j)1834 2021 y FB(\))-11 2191 y(No)o(w,)j(note)g(that)g Fz(n)350 2198 y Ft(k)385 2191 y FB(=)g(0)p Fz(;)8 b FB(1)14 b(are)g(the)f(only)h(v)m(alues)g(whic)o(h)f(satisfy)g(the)h(equation)f Fz(n)1522 2198 y Fy(1)1548 2191 y FB(+)6 b(2)p Fz(n)1645 2198 y Fy(2)1671 2191 y FB(+)g Fx(\001)i(\001)g(\001)e FB(+)g Fz(k)r(n)1879 2198 y Ft(k)1914 2191 y FB(=)14 b Fz(k)r FB(.)-84 2251 y(In)f(the)g(case)g Fz(n)184 2258 y Ft(k)219 2251 y FB(=)h(1,)g(w)o(e)e(ha)o(v)o(e)h Fz(n)530 2258 y Fy(1)564 2251 y FB(=)g Fx(\001)8 b(\001)g(\001)14 b FB(=)g Fz(n)768 2258 y Ft(k)q Fq(\000)p Fy(1)848 2251 y FB(=)g(0)g(and)g Fz(y)1054 2258 y Ft(k)1088 2251 y FB(=)g(1)g(.)20 b(Equation)14 b(\(6.40\))f(can)h(th)o(us)f(b)q(e)g (rewritten)-84 2311 y(as)558 2365 y Fz(R)595 2372 y Ft(k)616 2365 y FB(\()p Fz(x)p FB(\))42 b(=)f Fz(R)840 2340 y Fy([1)p Ft(;k)q Fq(\000)p Fy(1])840 2378 y Ft(k)954 2365 y FB(\()p Fz(x)p FB(\))11 b(+)g Fz(x)724 2494 y FB(=)41 b Fz(R)840 2469 y Fy([2)p Ft(;k)q Fq(\000)p Fy(2])840 2507 y Ft(k)954 2494 y FB(\()p Fz(x)p FB(\))11 b(+)g Fz(xR)1145 2501 y Ft(k)q Fq(\000)p Fy(1)1211 2494 y FB(\()p Fz(x)p FB(\))g(+)g Fz(x)1882 2426 y FB(\(6.41\))-11 2615 y(In)16 b(the)g(second)g(line)g(of)g(\(6.41\),)h Fz(R)631 2590 y Fy([1)p Ft(;k)q Fq(\000)p Fy(1])631 2628 y Ft(k)745 2615 y FB(\()p Fz(x)p FB(\))f(has)h(b)q(een)f(written)f(as)i(a)g(sum)e (of)i(t)o(w)o(o)f(terms:)31 2756 y Fz(R)68 2731 y Fy([1)p Ft(;k)q Fq(\000)p Fy(1])68 2769 y Ft(k)182 2756 y FB(\()p Fz(x)p FB(\))e(=)f Fz(R)350 2731 y Fy([2)p Ft(;k)q Fq(\000)p Fy(1])350 2769 y Ft(k)464 2756 y FB(\()p Fz(x)p FB(\))e(+)610 2715 y Fr(X)606 2807 y Ft(n)628 2812 y Fk(1)645 2807 y Fq(6)p Fy(=0)698 2756 y Fz(x)726 2736 y Ft(n)747 2741 y Fk(1)775 2756 y Fz(:)d(:)g(:)859 2715 y Fr(X)857 2802 y Ft(n)879 2808 y Fl(k)q Fg(\000)p Fk(1)945 2683 y Fr( )986 2723 y Fz(y)1010 2730 y Ft(k)q Fq(\000)p Fy(1)983 2790 y Fz(n)1012 2797 y Ft(k)q Fq(\000)p Fy(1)1083 2683 y Fr(!)1125 2756 y Fz(x)1153 2736 y Ft(n)1174 2742 y Fl(k)q Fg(\000)p Fk(1)1250 2756 y Fz(\016)1272 2763 y Ft(k)1302 2756 y FB(\()p Fz(n)1350 2763 y Fy(1)1380 2756 y FB(+)j Fx(\001)d(\001)g(\001)k FB(+)f(\()p Fz(k)i Fx(\000)d FB(1\))f Fz(n)1735 2763 y Ft(k)q Fq(\000)p Fy(1)1802 2756 y FB(\))57 b Fz(:)937 2930 y FB(42)p eop %%Page: 43 43 43 42 bop -84 79 a FB(By)16 b(making)f(the)i(c)o(hange)f(of)h(v)m (ariables)g Fz(n)699 61 y Fq(0)699 92 y Fy(1)733 79 y FB(=)d Fz(n)814 86 y Fy(1)845 79 y Fx(\000)e FB(1)17 b(the)f(second)h(term)e(can)h(b)q(e)h(written)f(as)h Fz(xR)1748 86 y Ft(k)q Fq(\000)p Fy(1)1815 79 y FB(.)22 b(F)l(or)17 b(the)-84 139 y(\014rst)h(term,)d(w)o(e)i(note)g(that)h Fz(y)462 146 y Fy(2)497 139 y FB(=)e Fz(n)580 146 y 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530 y Fl(k)q Fg(\000)p Fk(1)542 405 y Fr( )582 445 y Fz(y)606 452 y Ft(k)q Fq(\000)p Fy(2)580 513 y Fz(n)609 520 y Ft(k)q Fq(\000)p Fy(2)680 405 y Fr(!)721 478 y Fz(x)749 458 y Ft(n)770 464 y Fl(k)q Fg(\000)p Fk(2)847 478 y Fz(\016)869 485 y Ft(k)q Fq(\000)p Fy(2)943 478 y FB(\(2)p Fz(n)1015 485 y Fy(2)1047 478 y Fx(\000)j FB(2)g(+)g Fx(\001)d(\001)g(\001)j FB(+)g(\()p Fz(k)i Fx(\000)e FB(2\))e Fz(n)1487 485 y Ft(k)q Fq(\000)p Fy(2)1553 478 y FB(\))14 b(=)g Fz(xR)1703 453 y Fy([2)p Ft(;k)q Fq(\000)p Fy(2])1703 491 y Ft(k)q Fq(\000)p Fy(2)-84 631 y FB(and)j(notice)e(that)i Fz(R)297 606 y Fy([2)p Ft(;k)q Fq(\000)p Fy(2])297 644 y Ft(k)q Fq(\000)p Fy(2)425 631 y Fx(\024)c Fz(R)514 638 y Ft(k)q Fq(\000)p Fy(2)597 631 y FB(b)o(y)j(the)g(p)q(ositivit)o(y)f(of)i Fz(x)p FB(.)-11 713 y(\(6.41\))g(can)f(th)o(us)h(b)q(e)f(written)g(as)482 819 y Fz(R)519 826 y Ft(k)540 819 y FB(\()p Fz(x)p FB(\))e Fx(\024)f Fz(R)709 794 y Fy([3)p Ft(;k)q Fq(\000)p Fy(3])709 832 y Ft(k)823 819 y FB(\()p Fz(x)p 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Ft(j)966 1204 y FB(\()p Fz(x)p FB(\))g Fx(\025)g FB(0)h(for)f Fz(x)g Fx(2)g Fs(R)1368 1211 y Fy(+)1395 1204 y FB(,)h(this)f (concludes)g(the)g(pro)q(of)h(of)-84 1264 y(\(6.38\).)p 1977 1346 30 30 v -84 1576 a FA(7)81 b(The)27 b(Analyticit)n(y)d (Domain)j(of)g Fb(M)1103 1587 y Fz(r)-84 1707 y FB(In)17 b(this)g(section)g(w)o(e)g(will)f(pro)o(v)o(e)g(Theorem)g(1.8)i (concerning)f(the)g(analyticit)o(y)e(domain)i(of)g Fx(M)1687 1714 y Ft(r)1706 1707 y FB(.)24 b(W)l(e)17 b(split)g(the)-84 1767 y(pro)q(of)g(in)f(t)o(w)o(o)g(subsections,)g(the)g(\014rst)h (dedicated)e(to)i(the)f(function)g Fx(I)1227 1774 y Ft(r)1262 1767 y FB(and)h(the)f(second)h(to)f(the)g(function)g Fx(F)1969 1774 y Ft(r)1988 1767 y FB(.)-84 1931 y FI(7.1)66 b(The)22 b(Analyticit)n(y)k(Domain)c(of)g Fe(I)919 1940 y Fd(r)-84 2045 y FB(In)16 b(this)g(subsection)g(w)o(e)g(will)f (establish)h(the)g(analyticit)o(y)f(results)h(on)h(the)f(functions)g Fx(I)1544 2052 y Ft(r)1563 2045 y FB(\()p Fz(z)r FB(\),)g Fz(r)f Fx(2)f Fs(N)p 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(subsections)f(according)h(to)f(whether)g Fz(z)g Fx(2)e Fz(D)1243 2812 y Ft(<)p Fy(1)1307 2805 y FB(or)i Fz(z)g Fx(2)e Fz(D)1492 2812 y Ft(>)p Fy(1)1540 2805 y FB(.)937 2930 y(43)p eop %%Page: 44 44 44 43 bop -84 79 a FC(7.2.1)55 b(Analyticit)n(y)17 b(in)i Fz(D)485 86 y Ft(<)p Fy(1)-84 193 y FB(W)l(e)d(no)o(w)h(pro)o(v)o(e)e (our)i(\014rst)f(result)g(on)h(the)f(analyticit)o(y)e(of)j Fx(F)1042 200 y Ft(r)1061 193 y FB(\()p Fz(z)r FB(\).)-84 326 y FC(Theorem)f(7.1)j(\(The)f(First)g(Analyticit)n(y)f(Theorem\))22 b Fn(The)g(function)i Fx(F)1410 333 y Ft(r)1428 326 y FB(\()p Fz(z)r FB(\))e Fn(de\014ne)n(d)h(in)f(\(1.13\))f(is)h(an)-84 387 y(analytic)c(function)h(of)e Fz(z)j Fn(in)d(the)h(op)n(en)g(set)g Fz(D)763 394 y Ft(<)p Fy(1)828 387 y Fn(for)f(al)r(l)i FC(p)14 b Fx(2)g Fz(\031)k Fn(and)g Fz(\020)g Fx(2)c FB([0)p Fz(;)i FB(1\))p Fn(.)1969 520 y Fm(2)-84 650 y FC(Pro)r(of.)21 b FB(The)16 b(partial)g(sums)483 793 y Fx(F)519 800 y Ft(r)o(;)10 b(N)587 793 y FB(\()p Fz(z)r FB(\))k(:=)747 739 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Fx(\024)e Fz(";)-84 1285 y FB(for)20 b(all)f Fz(N)25 b FB(and)20 b Fz(M)5 b FB(,)20 b Fz(N)25 b(<)19 b(M)5 b FB(,)21 b(large)e(enough,)i(since)e(the)g(sequence)f Fx(f)p Fz(a)1295 1292 y Ft(n)1318 1285 y Fz(;)28 b(n)19 b Fx(2)h Fs(N)p Fx(g)g FB(is)g(summable.)28 b(So,)21 b(the)-84 1345 y(partial)14 b(sums)f(form)f(a)i(uniform)f(Cauc)o(h)o(y) g(sequence)g(of)h(analytic)f(functions)h(and,)g(therefore,)f(their)g (limit)e(exists)-84 1405 y(and)17 b(is)f(analytic)f(in)h Fz(D)342 1412 y Ft(<)p Fy(1)390 1405 y FB(.)p 1977 1487 30 30 v -84 1687 a FC(7.2.2)55 b(Analyticit)n(y)17 b(in)i Fz(D)485 1694 y Ft(>)p Fy(1)532 1687 y Fx(n)p 557 1648 41 2 v(Z)-84 1800 y FB(W)l(e)i(no)o(w)i(pro)o(v)o(e)e(the)g(analyticit) o(y)f(results)i(on)g(the)g(function)f Fx(F)1143 1807 y Ft(r)1162 1800 y FB(\()p Fz(z)r FB(\))h(in)f Fz(D)1349 1807 y Ft(>)p Fy(1)1397 1800 y FB(.)38 b(W)l(e)21 b(start)h(b)o(y)g (studying)g(the)-84 1861 y(lo)q(cation)17 b(of)f(the)g(singularities)g (of)g Fx(F)604 1868 y Ft(r)623 1861 y FB(\()p Fz(z)r FB(\).)21 b(First)16 b(w)o(e)g(pro)o(v)o(e)f(the)h(follo)o(wing)g(lemm) o(a.)-84 1994 y FC(Lemma)f(7.2)24 b Fn(L)n(et)18 b Fx(B)d(\032)e Fs(C)30 b Fn(b)n(e)17 b(an)h(op)n(en)g(set)g(such)g(that)f Fz(\034)1006 1976 y Fy(\()p Ft(n)p Fy(\))1000 2006 y Ft(z)1057 1994 y FB(\(1\))p Fn(,)h Fz(n)c Fx(2)g Fs(N)p Fn(,)k(ar)n(e)f(analytic)h(on)g Fx(B)g Fn(and)639 2133 y FB(inf)636 2163 y Ft(z)q Fq(2B)729 2133 y FB(inf)723 2164 y Ft(n)p Fq(2)p Fj(N)816 2058 y Fr(\014)816 2083 y(\014)816 2108 y(\014)816 2133 y(\014)816 2158 y(\014)829 2133 y Fz(\034)856 2112 y Fy(\()p Ft(n)p Fy(\))850 2145 y Ft(z)907 2133 y FB(\(1\))12 b Fx(\000)1030 2060 y Fr( )1068 2099 y Fx(\000)p FB(1)p 1068 2121 64 2 v 1075 2167 a Fz(z)r(\020)1136 2060 y Fr(!)1169 2058 y(\014)1169 2083 y(\014)1169 2108 y(\014)1169 2133 y(\014)1169 2158 y(\014)1197 2133 y Fz(>)i FB(0)p Fz(:)619 b FB(\(7.3\))-84 2272 y Fn(Then)18 b Fx(F)79 2279 y Ft(r)98 2272 y FB(\()p Fz(z)r FB(\))f Fn(is)g(an)h(analytic)g(function)h(in)f Fx(B)r Fn(.)1969 2405 y Fm(2)-84 2587 y FC(Remark)e(7.3)24 b Fn(Note)h(that)k FB(inf)454 2618 y Ft(n)p Fq(2)p Fj(N)532 2537 y Fr(\014)532 2562 y(\014)532 2587 y(\014)p Fz(\034)573 2569 y Fy(\()p Ft(n)p Fy(\))567 2600 y Ft(z)624 2587 y FB(\(1\))11 b Fx(\000)g FB(\()p Fx(\000)p Fz(z)830 2569 y Fq(\000)p Fy(1)877 2587 y Fz(=\020)t FB(\))945 2537 y Fr(\014)945 2562 y(\014)945 2587 y(\014)25 b FB(=)h(0)e Fn(is)f(not)i(a)e(ne)n(c)n(essary)h(c)n(ondition)g(for)f Fx(F)1857 2594 y Ft(r)1899 2587 y Fn(to)h(b)n(e)-84 2678 y(singular)d(at)g Fz(z)r Fn(.)32 b(F)l(or)20 b(instanc)n(e,)j(for)d Fz(z)h FB(=)f Fx(\000)p FB(1)p Fn(,)h(one)h(has)1011 2629 y Fr(\014)1011 2654 y(\014)1011 2678 y(\014)p Fz(\034)1052 2660 y Fy(\()p Ft(n)p Fy(\))1046 2691 y Ft(z)1103 2678 y FB(\(1\))11 b Fx(\000)g FB(\()p Fx(\000)p Fz(z)1309 2660 y Fq(\000)p Fy(1)1356 2678 y Fz(=\020)t FB(\))1424 2629 y Fr(\014)1424 2654 y(\014)1424 2678 y(\014)20 b FB(=)f(1)p Fz(=\020)g Fx(\000)13 b FB(1)21 b Fn(for)f(al)r(l)i Fz(n)e Fx(2)g Fs(N)p Fn(.)-84 2745 y(However,)d(as)e(we)h(wil)r(l)h(se) n(e)e(b)n(elow,)i Fx(F)604 2752 y Ft(r)622 2745 y FB(\()p Fz(z)r FB(\))e Fn(is)g(not)h(analytic)g(at)f Fz(z)h FB(=)e Fx(\000)p FB(1)h Fn(b)n(e)n(c)n(ause)g Fx(\000)p FB(1)h Fn(is)f(an)g(ac)n(cumulation)i(p)n(oint)-84 2805 y(of)g(p)n(oles)h(of)f Fx(F)186 2812 y Ft(r)205 2805 y Fn(.)937 2930 y FB(44)p eop %%Page: 45 45 45 44 bop -84 79 a FC(Pro)r(of)53 b(of)27 b(Lemm)o(a)d(7.2.)42 b FB(If)23 b(condition)g(\(7.3\))g(holds)h(for)f(a)h(giv)o(en)e Fx(B)i FB(it)f(implies)d(that)k(there)f(exists)f(a)-84 139 y(neigh)o(b)q(orho)q(o)q(d)c Fx(O)f FB(of)g Fx(\000)p FB(1)p Fz(=\020)k FB(suc)o(h)16 b(that,)g(for)h(all)e Fz(z)h Fx(2)e(B)k FB(and)e(for)h(all)f Fz(n)e Fx(2)g Fs(N)p FB(,)780 228 y Fz(z)r(\034)832 207 y Fy(\()p Ft(n)p Fy(\))826 240 y Ft(z)882 228 y FB(\(1\))g Fx(2)g Fs(C)23 b Fx(n)11 b(O)q Fz(:)-84 316 y FB(Since)k Fz(\034)70 298 y Fy(\()p Ft(n)p Fy(+1\))64 328 y Ft(z)166 316 y FB(\(1\))f(=)g Fz(h)p FB(\()p Fz(z)r(\034)393 298 y Fy(\()p Ft(n)p Fy(\))387 328 y Ft(z)444 316 y FB(\(1\)\),)i(this)g(means)f (that)i(there)f(exists)f(a)i(p)q(ositiv)o(e)f(constan)o(t)g Fz(M)5 b FB(,)17 b(suc)o(h)f(that)g(for)h(all)-84 376 y Fz(z)f Fx(2)e(B)j FB(and)g(all)f Fz(n)e Fx(2)g Fs(N)p FB(,)813 436 y Fx(j)p Fz(\034)854 416 y Fy(\()p Ft(n)p Fy(\))848 449 y Ft(z)904 436 y FB(\(1\))p Fx(j)g(\024)g Fz(M)r(:)-84 514 y FB(Hence,)19 b(b)o(y)h(the)f(same)g(argumen)o(t)g (used)h(in)g(the)f(pro)q(of)i(of)g(Theorem)d(7.1,)j(the)f(partial)g (sums)f Fx(F)1757 521 y Ft(r)o(;)9 b(N)1825 514 y FB(\()p Fz(z)r FB(\))20 b(form)-84 574 y(a)f(uniform)f(Cauc)o(h)o(y)h(sequence) f(of)h(analytic)f(functions)h(in)g Fx(B)h FB(and)g(consequen)o(tly)l(,) e(their)g(limit)f(exists)h(and)i(is)-84 634 y(analytic)c(in)f Fx(B)r FB(.)p 1977 716 30 30 v -11 835 a(Our)h(task)h(no)o(w)g(is)f(to) g(lo)q(calize)f(the)h(p)q(oin)o(ts)h Fz(z)f Fx(2)e Fs(C)28 b FB(where)16 b(the)g(condition)699 952 y(inf)693 983 y Ft(n)p Fq(2)p Fj(N)772 877 y Fr(\014)772 902 y(\014)772 927 y(\014)772 952 y(\014)772 977 y(\014)786 952 y Fz(\034)813 931 y Fy(\()p Ft(n)p Fy(\))807 964 y Ft(z)863 952 y FB(\(1\))c Fx(\000)986 879 y Fr( )1024 918 y Fx(\000)p FB(1)p 1024 940 64 2 v 1031 986 a Fz(z)r(\020)1092 879 y Fr(!)1125 877 y(\014)1125 902 y(\014)1125 927 y(\014)1125 952 y(\014)1125 977 y(\014)1153 952 y FB(=)i(0)677 b(\(7.4\))-84 1068 y(holds.)23 b(The)17 b(next)g(prop)q(osition)h(sho)o(ws)f(that,)g(for)h (eac)o(h)e Fz(n)f Fx(2)g Fs(N)p FB(,)j(there)e(exist)g(at)h(least)g (one)g(p)q(oin)o(t)g Fz(z)g Fx(2)e Fs(C)29 b FB(suc)o(h)-84 1129 y(that)22 1079 y Fr(\014)22 1104 y(\014)22 1129 y(\014)o Fz(\034)62 1110 y Fy(\()p Ft(n)p Fy(\))56 1141 y Ft(z)113 1129 y FB(\(1\))12 b Fx(\000)e FB(\()p Fx(\000)p Fz(z)319 1110 y Fq(\000)p Fy(1)366 1129 y Fz(=\020)t FB(\))435 1079 y Fr(\014)435 1104 y(\014)435 1129 y(\014)j FB(=)h(0)j(holds.)-84 1248 y FC(Prop)r(osition)h(7.4)24 b Fn(F)l(or)18 b(e)n(ach)h Fz(n)d Fx(2)h Fs(N)i Fn(ther)n(e)g(is)f(at)h (le)n(ast)g(one)h(and)f(at)g(most)f FB(2)1408 1229 y Ft(n)p Fy(+1)1489 1248 y Fx(\000)12 b FB(1)19 b Fn(distinct)h (solutions)f(in)-84 1308 y Fs(C)29 b Fn(of)18 b(the)g(e)n(quation)740 1396 y Fz(\034)767 1376 y Fy(\()p Ft(n)p Fy(\))761 1409 y Ft(z)817 1396 y FB(\(1\))12 b Fx(\000)940 1323 y Fr( )978 1363 y Fx(\000)p FB(1)p 978 1385 V 985 1430 a Fz(z)r(\020)1046 1323 y Fr(!)1093 1396 y FB(=)i(0)p Fz(:)723 b FB(\(7.5\))1969 1523 y Fm(2)-84 1642 y FC(Pro)r(of.)21 b FB(As)16 b(observ)o(ed)g(in)g (Section)g(2,)g Fz(\034)671 1624 y Fy(\()p Ft(n)p Fy(\))665 1654 y Ft(z)721 1642 y FB(\(1\))h(is)f(the)g(ratio)h(of)f(t)o(w)o(o)g (p)q(olynomials)f(of)i(degree)f(2)1700 1624 y Ft(n)p Fy(+1)1780 1642 y Fx(\000)11 b FB(2:)784 1758 y Fz(\034)811 1737 y Fy(\()p Ft(n)p Fy(\))805 1770 y Ft(z)862 1758 y FB(\(1\))j(=)998 1724 y Fz(P)1029 1731 y Ft(n)1053 1724 y FB(\()p Fz(z)r FB(\))p 995 1746 125 2 v 995 1792 a Fz(Q)1034 1799 y Ft(n)1057 1792 y FB(\()p Fz(z)r FB(\))1125 1758 y Fz(:)-84 1874 y FB(Therefore,)h(\(7.5\))i(means)e(that)728 1956 y Fz(z)r(P)784 1963 y Ft(n)808 1956 y FB(\()p Fz(z)r FB(\))c(+)936 1922 y(1)p 936 1945 26 2 v 936 1990 a Fz(\020)966 1956 y(Q)1005 1963 y Ft(n)1028 1956 y FB(\()p Fz(z)r FB(\))j(=)f(0)p Fz(;)712 b FB(\(7.6\))-84 2061 y(whic)o(h)18 b(has)i(at)f(least)g(one)g(and)h(at)f(most)g(2)726 2043 y Ft(n)p Fy(+1)808 2061 y Fx(\000)12 b FB(1)20 b(p)q(ossible)f (distinct)f(solutions)i(in)e Fs(C)9 b FB(,)23 b(since)18 b(the)h(left)f(hand)-84 2122 y(side)g(is)h(a)h(p)q(olynomial)d(of)j (degree)e(2)603 2103 y Ft(n)p Fy(+1)685 2122 y Fx(\000)13 b FB(1.)30 b(Clearly)l(,)18 b(for)h Fz(n)g FB(=)f(0,)i(the)e(unique)h (solution)g(is)g(the)g(p)q(ole)g(of)g Fz(h)p FB(,)-84 2182 y Fz(z)d FB(=)d Fx(\000)p FB(1)p Fz(=\020)t FB(.)p 1977 2263 30 30 v -11 2382 a(Let)j(us)h(denote)f(b)o(y)g Fx(Z)400 2389 y Ft(n)440 2382 y FB(the)g(\014nite)f(set)i(of)f(all)g Fz(z)r FB('s)g(satisfying)g(\(7.5\))h(for)f Fz(n)e Fx(2)g Fs(N)p FB(.)22 b(De\014ne)821 2471 y Fx(Z)32 b FB(:=)979 2429 y Fr([)968 2522 y Ft(a)p Fq(2)p Fj(N)1044 2471 y Fx(Z)1080 2478 y Ft(a)-84 2603 y FB(and)17 b(let)p 82 2564 41 2 v 16 w Fx(Z)k FB(b)q(e)c(the)f(closure)h(of)g Fx(Z)t FB(.)22 b(Note)17 b(that)g(an)g(in\014nite)f(coun)o(table)g (union)h(of)g(closed)f(sets)h(as)h Fx(Z)t FB(,)e(is)h(not)g(in)-84 2663 y(general)f(a)h(closed)e(set.)22 b(The)16 b(set)g(di\013erence)f Fx(Z)798 2645 y Fy(0)832 2663 y FB(:=)p 897 2624 V 13 w Fx(Z)g(n)c(Z)20 b FB(is)d(the)f(set)g(of)g(accum)o(ulation)f(p)q(oin) o(ts)h(of)h Fx(Z)t FB(.)-11 2745 y(T)l(o)g(establish)e(that)i Fx(F)403 2752 y Ft(r)438 2745 y FB(is)e(holomorphic)g(in)h Fs(C)23 b Fx(n)p 904 2706 V 10 w(Z)d FB(w)o(e)c(need)f(to)i(describ)q (e)e(where)h(the)f(elemen)o(ts)f(of)i Fx(Z)k FB(and)-84 2805 y Fx(Z)-44 2787 y Fy(0)-8 2805 y FB(are)c(lo)q(cated.)937 2930 y(45)p eop %%Page: 46 46 46 45 bop -84 79 a FC(Theorem)16 b(7.5)24 b Fn(F)l(or)17 b(al)r(l)i Fz(\020)f Fx(2)c FB(\(0)p Fz(;)23 b FB(1\))17 b Fn(and)h Fz(k)e Fx(2)e Fs(N)19 b Fn(one)f(has)f Fx(Z)1088 86 y Ft(k)1123 79 y Fx(\032)d Fz(D)1216 86 y Ft(>)p Fy(1)1281 79 y Fn(with)k(the)g(ac)n(cumulation)h(p)n(oints)e(of)g Fx(Z)-84 139 y Fn(lying)h(in)g(the)g(unit)g(cir)n(cle,)g(i.)k(e.,)c Fx(Z)584 121 y Fy(0)618 139 y Fx(\032)13 b Fz(S)703 121 y Fy(1)723 139 y Fn(.)-11 221 y(In)18 b(addition,)775 281 y Fx(Z)811 288 y Ft(k)843 281 y Fx(\\)11 b FB([1)p Fz(;)22 b Fx(1)p FB(\))27 b(=)h Fx(;)-84 362 y Fn(holds)19 b(for)f(al)r(l)j Fz(k)e Fx(2)e Fs(N)p Fn(.)28 b(The)19 b(sets)g Fx(Z)599 369 y Ft(k)640 362 y Fn(ar)n(e)f(also)h(self-c)n (onjugate)n(d)i(in)f(the)f(sense)h(that)p 1529 336 25 2 v 19 w Fz(z)f Fx(2)e(Z)1657 369 y Ft(k)1697 362 y Fn(if)i Fz(z)g Fx(2)e(Z)1874 369 y Ft(k)1914 362 y Fn(and,)-84 422 y(for)g(al)r(l)i Fz(k)c Fx(2)f Fs(N)p Fn(,)19 b(one)f(has)726 483 y Fx(Z)762 490 y Ft(k)795 483 y Fx(\\)11 b FB(\()p Fx(\0001)p Fz(;)21 b Fx(\000)p FB(1\))28 b Fx(6)p FB(=)g Fx(;)p Fz(:)-84 563 y Fn(This)17 b(last)h(fact)g(to)n(gether)g(with)g Fx(Z)546 545 y Fy(0)579 563 y Fx(\032)c Fz(S)665 545 y Fy(1)702 563 y Fn(implies)k(that)f Fx(\000)p FB(1)d Fx(2)g(Z)1132 545 y Fy(0)1152 563 y Fn(.)1969 682 y Fm(2)-84 805 y FC(Pro)r(of)p FB(.)21 b(Firstly)l(,)15 b(observ)o(e)g(that)i(the) f(equation)810 920 y Fz(\034)837 899 y Fy(\()p Ft(k)q Fy(\))831 932 y Ft(z)886 920 y FB(\(1\))e(=)g Fx(\000)1070 886 y FB(1)p 1058 908 50 2 v 1058 954 a Fz(z)r(\020)1906 920 y FB(\(7.7\))-84 1046 y(has)h(no)h(solutions)f(for)g Fz(z)g Fx(2)f Fz(D)468 1053 y Fy(1)502 1046 y FB(:=)g Fz(D)608 1053 y Ft(<)p Fy(1)663 1046 y Fx([)8 b Fz(S)737 1028 y Fy(1)757 1046 y FB(,)14 b(since)g Fz(\034)930 1028 y Fy(\()p Ft(k)q Fy(\))924 1058 y Ft(z)979 1046 y FB(\(1\))h(maps)f Fz(D)1222 1053 y Fy(1)1257 1046 y FB(in)o(to)g(itself)g(and)h Fz(\020)j(<)c FB(1.)21 b(This)15 b(equation)-84 1106 y(has)20 b(no)f(solutions)h(for)g Fx(j)p Fz(z)r Fx(j)e(\025)g FB(1)p Fz(=\020)25 b FB(either,)18 b(since)g(in)h(this)g(region,)h(b)o(y)e(Corollary)i(2.2,)f(the)g(left)g (hand)g(side)g(of)-84 1166 y(\(7.7\))d(lies)g(in)g Fz(D)214 1173 y Ft(>)p Fy(1)261 1166 y FB(,)g(but)h(the)f(righ)o(t)f(hand)i (side)f(in)g Fz(D)899 1173 y Fy(1)919 1166 y FB(.)21 b(W)l(e)16 b(conclude)g(that)821 1262 y Fx(Z)32 b(\032)27 b Fz(D)995 1269 y Fy(1)p Ft(;)10 b Fy(1)p Ft(=\020)1088 1262 y Fz(:)-84 1357 y FB(Analogously)l(,)20 b(\(7.7\))f(has)h(no)g (solutions)f(in)g(\(1)p Fz(;)27 b Fx(1)p FB(\),)19 b(since)g(there)f (its)h(left)g(hand)g(side)g(is)g(p)q(ositiv)o(e)f(while)h(the)-84 1417 y(righ)o(t)d(hand)h(side)f(is)g(negativ)o(e.)-11 1499 y(T)l(o)k(pro)o(v)o(e)e(that)i Fx(Z)347 1481 y Fy(0)385 1499 y Fx(\032)f Fz(S)476 1481 y Fy(1)514 1499 y FB(w)o(e)g(pro)q(ceed) g(as)h(follo)o(ws.)30 b(The)19 b(elemen)o(ts)d(of)k Fx(Z)1423 1506 y Ft(n)1465 1499 y FB(are)g(the)f(p)q(oles)g(of)g Fz(\034)1848 1481 y Fy(\()p Ft(n)p Fy(+1\))1842 1511 y Ft(z)1944 1499 y FB(\(1\))-84 1559 y(in)j Fz(D)19 1566 y Ft(>)p Fy(1)66 1559 y FB(.)39 b(Therefore,)23 b(if)f Fz(w)j Fx(2)f(Z)565 1566 y Ft(n)589 1559 y FB(,)f Fz(w)662 1541 y Fq(\000)p Fy(1)732 1559 y FB(is)f(a)g(zero)g(of)h Fz(\034)1031 1541 y Fy(\()p Ft(n)p Fy(+1\))1025 1571 y Ft(z)1126 1559 y FB(\(1\))g(in)f Fz(D)1314 1566 y Ft(<)p Fy(1)1384 1559 y FB(.)39 b(The)22 b(sequence)f(of)h(functions)-84 1619 y Fz(\034)-57 1601 y Fy(\()p Ft(n)p Fy(\))-63 1631 y Ft(z)12 1619 y FB(con)o(v)o(erges)17 b(uniformly)f(in)i Fz(D)555 1626 y Ft(<)p Fy(1)621 1619 y FB(to)g(an)h(analytic)e (function)h Fz(\034)23 b FB(=)17 b Fz(\034)6 b FB(\()p Fz(z)r FB(\))17 b(\(Theorem)g(3.5\).)27 b(According)18 b(to)g(a)-84 1679 y(theorem)c(of)i(Hurwitz)f(\(see,)g(e.g.,)f([T]\))h (the)h(fact)f(that)h Fz(\034)22 b FB(has)16 b(no)h(zeros)e(on)h Fz(D)1367 1686 y Ft(<)p Fy(1)1431 1679 y FB(\(Theorem)e(3.6\))i (implies)d(that)-84 1739 y(the)k(set)h(of)g(zeros)g(of)g(the)f (functions)h Fz(\034)641 1721 y Fy(\()p Ft(n)p Fy(\))635 1752 y Ft(z)710 1739 y FB(has)g(no)g(limit)d(p)q(oin)o(ts)k(on)f Fz(D)1243 1746 y Ft(<)p Fy(1)1308 1739 y FB(and)h(therefore,)e(if)g (they)g(exist,)g(they)-84 1800 y(m)o(ust)e(lie)g(on)i Fz(S)203 1782 y Fy(1)222 1800 y FB(.)-11 1881 y(The)h(sets)g Fx(Z)224 1888 y Ft(k)263 1881 y FB(are)g(all)f(self-conjugated)h(b)o(y) f(the)h(de\014nition)f(of)h Fz(h)p FB(,)g(since)f(the)h(left)f(hand)h (side)f(of)h(\(7.6\))h(is)e(a)-84 1942 y(p)q(olynomial)g(with)h(real)g (co)q(e\016cien)o(ts.)25 b(This)18 b(p)q(olynomial)f(is)h(also)h(of)g (o)q(dd)g(degree,)e(namely)g(2)1688 1923 y Ft(k)q Fy(+1)1767 1942 y Fx(\000)12 b FB(1)19 b(.)26 b(Both)-84 2002 y(facts)20 b(together)h(imply)c(that)k(this)f(p)q(olynomial)f(has)i(at)g(least)f (one)g(real)g(ro)q(ot.)34 b(Ho)o(w)o(ev)o(er,)19 b(according)h(to)h (the)-84 2062 y(previous)16 b(remarks,)f(the)h(real)h(ro)q(ots)g (cannot)h(lie)d(in)h([)p Fx(\000)p FB(1)p Fz(;)22 b Fx(1)p FB(\).)g(So,)17 b(they)f(m)o(ust)f(b)q(e)h(negativ)o(e)g(and)h(lo)o(w)o (er)f(than)-84 2122 y Fx(\000)p FB(1.)29 b(F)l(rom)18 b(the)h(fact)g(that)g Fx(Z)487 2104 y Fy(0)525 2122 y Fx(\032)f Fz(S)615 2104 y Fy(1)654 2122 y FB(it)g(follo)o(ws)h(that)h (these)e(negativ)o(e)g(ro)q(ots)j(m)o(ust)c(con)o(v)o(erge)h(to)h Fx(\000)p FB(1)g(when)-84 2182 y Fz(k)d Fx(!)d(1)k FB(and)f(so)h Fx(\000)p FB(1)d Fx(2)g(Z)405 2164 y Fy(0)425 2182 y FB(.)21 b(This)c(sho)o(ws,)f(in)g(particular,)g(that)g Fx(Z)1166 2164 y Fy(0)1200 2182 y Fx(6)p FB(=)e Fx(;)p FB(.)p 1977 2264 30 30 v -84 2424 a FC(Remark)i(7.6)24 b Fn(Numeric)n(al)16 b(c)n(omputations)g(se)n(em)g(to)f(indic)n(ate)i (that)e Fx(Z)1246 2406 y Fy(0)1294 2424 y FB(=)27 b Fz(S)1392 2406 y Fy(1)1427 2424 y Fn(for)15 b(the)h(whole)h(ferr)n(omagnetic)-84 2484 y(r)n(e)n(gion)g FB(0)d Fx(\024)g Fz(\020)k(<)13 b FB(1)p Fz(=)p FB(3)p Fn(.)24 b(Unfortunately)19 b(we)f(wer)n(e)g(not) g(able)g(to)g(pr)n(ove)f(this)g(c)n(onje)n(ctur)n(e.)-11 2603 y FB(After)e(these)g(considerations)h(on)h(the)e(lo)q(cation)h(of) g(the)g(singularities)f(of)h Fz(\034)1385 2585 y Fy(\()p Ft(n)p Fy(\))1379 2615 y Ft(z)1435 2603 y FB(w)o(e)g(turn)f(to)h(the)g (pro)q(of)h(of)f(the)-84 2663 y(analyticit)o(y)e(statemen)o(t)h(on)i Fx(F)489 2670 y Ft(r)508 2663 y FB(.)-11 2745 y(As)d(ab)q(o)o(v)o(e,)h (let)f Fz(\034)6 b FB(\()p Fz(w)q FB(\),)14 b Fz(w)h Fx(2)f Fz(D)545 2752 y Ft(<)p Fy(1)593 2745 y FB(,)g(b)q(e)h(the)f (uniform)g(limit)d(function)k(of)g(the)f(sequence)f(of)i(functions)g Fz(\034)1880 2727 y Fy(\()p Ft(n)p Fy(\))1874 2757 y Ft(w)1931 2745 y FB(\(1\),)-84 2805 y Fz(n)f Fx(2)g Fs(N)p FB(.)937 2930 y(46)p eop %%Page: 47 47 47 46 bop -84 79 a FC(Lemma)15 b(7.7)24 b Fn(L)n(et)18 b Fx(B)i Fn(b)n(e)e(an)g(op)n(en)h(set)f(such)h(that)p 882 40 35 2 v 18 w Fx(B)d(\032)f Fz(D)1025 86 y Ft(>)p Fy(1)1084 79 y Fx(n)p 1121 40 41 2 v 12 w(Z)t Fn(,)j(wher)n(e)p 1332 40 35 2 v 18 w Fx(B)i Fn(is)e(the)g(closur)n(e)h(of)f Fx(B)r Fn(.)23 b(Then)c(the)-84 139 y(se)n(quenc)n(e)g Fz(\034)141 121 y Fy(\()p Ft(n)p Fy(\))135 152 y Ft(z)192 139 y FB(\(1\))p Fn(,)e Fz(n)d Fx(2)g Fs(N)p Fn(,)19 b(c)n(onver)n(ges)f(to)g(the)f(function)i FB(1)p Fz(=\034)6 b FB(\()p Fz(z)1112 121 y Fq(\000)p Fy(1)1160 139 y FB(\))17 b Fn(uniformly)g(on)h Fx(B)r Fn(.)-84 275 y FC(Pro)r(of)h(.)i FB(F)l(or)16 b Fz(z)g Fx(2)e(B)j FB(call)f Fz(w)f Fx(\021)e Fz(z)548 257 y Fq(\000)p Fy(1)612 275 y FB(the)j(\\image")f(of)i Fz(z)r FB(,)f(and)g(let)g(the)g(op)q(en)h(set)f Fx(B)1472 282 y Ft(i)1502 275 y FB(giv)o(en)f(b)o(y)559 385 y Fx(B)592 392 y Ft(i)634 385 y FB(:=)713 337 y Fr(n)741 385 y Fz(w)g Fx(2)f Fz(D)878 392 y Ft(<)p Fy(1)942 385 y FB(suc)o(h)i(that)g(w)1193 364 y Fq(\000)p Fy(1)1254 385 y Fx(2)e(B)1336 337 y Fr(o)-84 495 y FB(b)q(e)i(the)g(\\image)g(set")g(of)h Fx(B)r FB(.)-11 577 y(F)l(rom)c(Theorem)f(3.6)i(w)o(e)g(kno)o(w)g(that)g Fz(\034)6 b FB(\()p Fz(w)q FB(\))14 b(has)h(no)f(zeros)g(on)g Fz(D)1182 584 y Ft(<)p Fy(1)1230 577 y FB(.)20 b(Hence,)13 b(for)h Fz(w)h Fx(2)f(B)1623 584 y Ft(i)1637 577 y FB(,)g Fx(j)p Fz(\034)g FB(\()p Fz(w)q FB(\))8 b Fx(j)14 b Fz(>)f(\032)1900 584 y Fy(1)1920 577 y FB(,)h(for)-84 637 y(some)i(p)q(ositiv)o(e)g Fz(\032)245 644 y Fy(1)265 637 y FB(.)24 b(Next,)16 b(w)o(e)h(argue)h (that)f(there)g(is)g(also)h(a)f(p)q(ositiv)o(e)g(constan)o(t)g Fz(\032)1470 644 y Fy(2)1508 637 y FB(suc)o(h)f(that)i Fx(j)p Fz(\034)1766 619 y Fy(\()p Ft(n)p Fy(\))1760 649 y Ft(w)1817 637 y FB(\(1\))p Fx(j)d Fz(>)h(\032)1987 644 y Fy(2)-84 697 y FB(for)i(all)f Fz(n)f Fx(2)g Fs(N)i FB(and)g(all)f Fz(w)g Fx(2)f(B)507 704 y Ft(i)521 697 y FB(.)25 b(F)l(or,)18 b(notice)e(that)i(no)g(elemen)o(t)d(of)i(the)h (sequence)e(of)i(functions)f Fz(\034)1805 679 y Fy(\()p Ft(n)p Fy(\))1799 709 y Ft(w)1856 697 y FB(\(1\))h(has)-84 757 y(a)i(zero)g(in)g Fx(B)161 764 y Ft(i)175 757 y FB(.)33 b(Since)19 b(the)h(sequence)f(con)o(v)o(erges)g(uniformly)f(and)j(the)f (limit)d(function)j(has)h(no)f(zeros)g(in)g Fx(B)1979 764 y Ft(i)1993 757 y FB(,)-84 817 y(there)d(m)o(ust)g(b)q(e)h(a)g(p)q (ositiv)o(e)f(constan)o(t)h(suc)o(h)g(that)g Fx(j)p Fz(\034)915 799 y Fy(\()p Ft(n)p Fy(\))909 830 y Ft(w)965 817 y FB(\(1\))p Fx(j)f Fz(>)g(\032)1138 824 y Fy(2)1175 817 y FB(for)h(all)g Fz(w)f Fx(2)g(B)1456 824 y Ft(i)1488 817 y FB(and)h Fz(n)f Fx(2)f Fs(N)p FB(.)27 b(The)18 b(desired)-84 878 y(uniformit)o(y)c (follo)o(ws)i(no)o(w)h(from)299 943 y Fr(\014)299 968 y(\014)299 993 y(\014)299 1018 y(\014)299 1043 y(\014)313 1018 y Fz(\034)340 997 y Fy(\()p Ft(n)p Fy(\))334 1030 y Ft(z)402 1018 y Fx(\000)513 984 y FB(1)p 457 1006 137 2 v 457 1052 a Fz(\034)6 b FB(\()p Fz(z)528 1038 y Fq(\000)p Fy(1)574 1052 y FB(\))598 943 y Fr(\014)598 968 y(\014)598 993 y(\014)598 1018 y(\014)598 1043 y(\014)626 1018 y FB(=)678 943 y Fr(\014)678 968 y(\014)678 993 y(\014)678 1018 y(\014)678 1043 y(\014)697 984 y Fz(\034)g FB(\()p Fz(w)q FB(\))k Fx(\000)h Fz(\034)885 966 y Fy(\()p Ft(n)p Fy(\))879 997 y Ft(w)936 984 y FB(\(1\))p 697 1006 302 2 v 720 1062 a Fz(\034)6 b FB(\()p Fz(w)q FB(\))14 b Fz(\034)862 1036 y Fy(\()p Ft(n)p Fy(\))856 1065 y Ft(w)912 1062 y FB(\(1\))1003 943 y Fr(\014)1003 968 y(\014)1003 993 y(\014)1003 1018 y(\014)1003 1043 y(\014)1031 1018 y Fx(\024)1125 984 y FB(1)p 1089 1006 99 2 v 1089 1052 a Fz(\032)1114 1059 y Fy(1)1142 1052 y Fz(\032)1167 1059 y Fy(2)1205 1018 y Fx(j)p Fz(\034)6 b FB(\()p Fz(w)q FB(\))11 b Fx(\000)g Fz(\034)1408 997 y Fy(\()p Ft(n)p Fy(\))1402 1030 y Ft(w)1459 1018 y FB(\(1\))p Fx(j)j Fz(<)f(")283 b FB(\(7.8\))-84 1171 y(for)16 b(an)o(y)f(prescrib)q(ed)g Fz(")f(>)g FB(0,)h(uniformly)f(on)i Fx(B)779 1178 y Ft(i)808 1171 y FB(and)g(for)g Fz(n)f FB(large)h(enough)g(since,)f(on)h Fx(B)1539 1178 y Ft(i)1553 1171 y FB(,)f(the)g(sequence)f Fz(\034)1893 1153 y Fy(\()p Ft(n)p Fy(\))1887 1183 y Ft(w)1944 1171 y FB(\(1\))-84 1231 y(con)o(v)o(erges)h(uniformly)g(to)h Fz(\034)6 b FB(\()p Fz(w)q FB(\).)p 1977 1313 30 30 v -84 1498 a FC(Lemma)15 b(7.8)24 b Fn(L)n(et)18 b Fx(B)g Fn(b)n(e)g(as)f(in)h(L)n(emma)f(7.7.)22 b(Then)c(the)g(c)n(ondition)642 1637 y FB(inf)639 1675 y Ft(z)q Fq(2)p 680 1647 24 2 v(B)736 1637 y FB(inf)730 1668 y Ft(n)p Fq(2)p Fj(N)826 1562 y Fr(\014)826 1587 y(\014)826 1612 y(\014)826 1637 y(\014)826 1662 y(\014)840 1637 y Fz(\034)867 1616 y Fy(\()p Ft(n)p Fy(\))861 1649 y Ft(z)917 1637 y FB(\(1\))12 b Fx(\000)1041 1564 y Fr( )1079 1603 y Fx(\000)p FB(1)p 1079 1625 64 2 v 1086 1671 a Fz(z)r(\020)1147 1564 y Fr(!)1180 1562 y(\014)1180 1587 y(\014)1180 1612 y(\014)1180 1637 y(\014)1180 1662 y(\014)1207 1637 y FB(=)i(0)623 b(\(7.9\))-84 1778 y Fn(is)17 b(imp)n(ossible)h(for)f(any)g Fz(\020)h Fx(2)c FB(\(0)p Fz(;)22 b FB(1])p Fn(.)1969 1913 y Fm(2)-84 2045 y FC(Pro)r(of)e(.)k FB(By)16 b(the)h(de\014nition) g(of)g Fx(B)r FB(,)f(the)h(condition)g Fz(\034)940 2027 y Fy(\()p Ft(n)p Fy(\))934 2057 y Ft(z)991 2045 y FB(\(1\))12 b(+)g(1)p Fz(=)p FB(\()p Fz(z)r(\020)t FB(\))k(=)f(0)j(do)q(es)g(not)f (hold)h(for)f(an)o(y)g(\014nite)g Fz(n)p FB(.)-84 2105 y(Therefore,)e(condition)h(\(7.9\))h(sa)o(ys)f(that)654 2246 y(inf)643 2284 y Ft(z)q Fq(2)p 684 2256 24 2 v(B)747 2172 y Fr(\014)747 2196 y(\014)747 2221 y(\014)747 2246 y(\014)747 2271 y(\014)822 2213 y FB(1)p 765 2235 137 2 v 765 2280 a Fz(\034)6 b FB(\()p Fz(z)836 2266 y Fq(\000)p Fy(1)883 2280 y FB(\))932 2246 y Fx(\000)996 2173 y Fr( )1034 2213 y Fx(\000)p FB(1)p 1034 2235 64 2 v 1041 2280 a Fz(z)r(\020)1102 2173 y Fr(!)1134 2172 y(\014)1134 2196 y(\014)1134 2221 y(\014)1134 2246 y(\014)1134 2271 y(\014)1176 2246 y FB(=)28 b(0)p Fz(:)-84 2392 y FB(Since)p 48 2353 35 2 v 20 w Fx(B)23 b FB(is)e(closed)f(and)i(1)8 b(/)q Fz(\034)e FB(\()p Fz(z)534 2374 y Fq(\000)p Fy(1)581 2392 y FB(\))42 b(+)36 b(1)p Fz(=)p FB(\()p Fz(z)r(\020)t FB(\))22 b(is)f(a)h(con)o(tin)o(uous)f(function)g(of)g Fz(z)i FB(on)p 1598 2353 V 22 w Fx(B)q FB(,)f(this)f(implies)e(the)-84 2453 y(existence)e(of)j(a)f(p)q(oin)o(t)g Fz(u)f Fx(2)p 456 2414 V 19 w(B)i FB(suc)o(h)f(that)g Fz(\034)6 b FB(\()p Fz(u)804 2434 y Fq(\000)p Fy(1)851 2453 y FB(\))18 b(=)h Fx(\000)p Fz(u)8 b(\020)t FB(.)29 b(Since)18 b Fz(\034)6 b FB(\()p Fz(w)q FB(\))19 b(ful\014lls)f(the)h(\014xed)f(p)q(oin)o(t)i (equation)-82 2513 y(~)-26 b Fz(\034)20 b FB(=)13 b Fz(h)p FB(\()p Fz(w)f FB(~)-26 b Fz(\034)6 b FB(\))15 b(for)h Fz(w)f Fx(2)f Fz(D)372 2520 y Ft(<)p Fy(1)420 2513 y FB(,)h(w)o(e)h(conclude)f(that)h Fx(\000)p Fz(u)8 b(\020)17 b FB(=)d Fz(h)p FB(\()p Fx(\000)p Fz(\020)t FB(\))g Fx(\021)f FB(0.)22 b(But)15 b(this)h(is)f(imp)q(ossible)f(for)i Fz(\020)i(>)c FB(0)i(and)-84 2573 y Fz(u)e Fx(2)g Fz(D)45 2580 y Ft(>)p Fy(1)92 2573 y FB(.)p 1977 2655 30 30 v 937 2930 a(47)p eop %%Page: 48 48 48 47 bop -84 79 a FC(Theorem)16 b(7.9)j(\(The)f(Second)g(Analyticit)n (y)g(Theorem\))k Fn(The)c(function)h Fx(F)1456 86 y Ft(r)1475 79 y FB(\()p Fz(z)r FB(\))f Fn(de\014ne)n(d)h(in)f(\(1.13\))g(is)g(a) -84 139 y(holomorphic)f(function)i(of)e Fz(z)i Fn(in)f(the)g(op)n(en)g (set)g Fz(D)856 146 y Ft(>)p Fy(1)914 139 y Fx(n)p 950 100 41 2 v 11 w(Z)k Fn(for)17 b(al)r(l)h FC(p)c Fx(2)g Fz(\031)19 b Fn(and)f Fz(\020)g Fx(2)c FB(\(0)p Fz(;)i FB(1])p Fn(.)-11 221 y(If)26 b Fz(a)75 228 y Ft(n)128 221 y Fx(6)p FB(=)k(0)c Fn(the)h(singularities)g(of)f Fx(F)721 228 y Ft(r)765 221 y Fn(at)h(the)f(p)n(oints)g(of)g Fx(Z)1176 228 y Ft(n)1226 221 y Fn(ar)n(e)f(p)n(oles)h(of)g(or)n(der)f (not)i(gr)n(e)n(ater)e(than)-84 281 y FB(\(2)-41 263 y Ft(n)p Fy(+1)39 281 y Fx(\000)11 b FB(1\))d Fz(r)q Fn(.)1969 414 y Fm(2)-84 544 y FC(Pro)r(of)41 b(of)20 b(Theorem)e(7.9.)26 b FB(Pic)o(k)16 b(an)j(op)q(en)f(set)g Fx(B)h FB(suc)o(h)e(that)p 1158 505 35 2 v 18 w Fx(B)h(\032)e Fz(D)1304 551 y Ft(>)p Fy(1)1364 544 y Fx(n)p 1401 505 41 2 v 12 w(Z)t FB(,)h(where)p 1615 505 35 2 v 18 w Fx(B)i FB(is)e(the)h(closure)f(of)-84 604 y Fx(B)r FB(.)26 b(Then,)18 b(analyticit)o(y)e(of)j Fx(F)472 611 y Ft(r)508 604 y FB(on)g Fx(B)g FB(follo)o(ws)f(from)f(Lemma)f(7.2)i(and)h(Lemma)d(7.8.) 27 b(Since)17 b Fz(D)1710 611 y Ft(>)p Fy(1)1770 604 y Fx(n)p 1807 565 41 2 v 12 w(Z)22 b FB(can)c(b)q(e)-84 664 y(co)o(v)o(ered)d(b)o(y)g(suc)o(h)h(op)q(en)h(sets)g(the)f(theorem) e(is)i(pro)o(v)o(en.)p 1977 746 30 30 v -84 945 a FC(7.2.3)55 b(Estimates)17 b(on)h(the)h(Lo)r(cation)e(of)i(the)f(P)n(oles)h(of)g Fz(\034)1150 927 y Fy(\()p Ft(n)p Fy(\))1144 958 y Ft(z)1200 945 y FB(\(1\))-84 1059 y(T)l(o)24 b(giv)o(e)f(an)i(idea)f(of)g(ho)o(w) g(fast)h(the)e(sets)h Fx(Z)794 1066 y Ft(k)840 1059 y FB(accum)o(ulate)d(on)k Fz(S)1212 1041 y Fy(1)1255 1059 y FB(w)o(e)f(will)f(mak)o(e)f(use)i(of)g(the)g(follo)o(wing)-84 1119 y(Con)o(traction)17 b(Theorem.)-11 1201 y(First,)e(some)h (de\014nitions.)21 b(F)l(or)16 b Fz(w)f Fx(2)f Fs(C)9 b FB(,)19 b Fz(w)c Fx(6)p FB(=)f(1)p Fz(=\020)t FB(,)i(de\014ne)787 1333 y Fz(g)r FB(\()p Fz(w)q FB(\))e(:=)983 1299 y Fz(\020)h Fx(\000)c Fz(w)p 970 1322 147 2 v 970 1367 a(\020)t(w)i Fx(\000)e FB(1)1122 1333 y Fz(:)746 b FB(\(7.10\))-84 1471 y(and)17 b(let)e Fz(u)109 1453 y Fy(1)p Ft(=)p Fy(2)180 1471 y FB(b)q(e)i(some)e(branc)o(h)h(of)h(the)f(square)g(ro)q(ot)h (function)f(in)g Fs(C)9 b FB(.)24 b(De\014ne)482 1579 y Fz(h)510 1558 y Fq(\000)p Fy(1)510 1591 y(+)557 1579 y FB(\()p Fz(u)p FB(\))j(:=)h Fz(g)r FB(\()p Fz(u)802 1558 y Fy(1)p Ft(=)p Fy(2)856 1579 y FB(\))p Fz(;)106 b(h)1023 1558 y Fq(\000)p Fy(1)1023 1591 y Fq(\000)1070 1579 y FB(\()p Fz(u)p FB(\))27 b(:=)h Fz(g)r FB(\()p Fx(\000)p Fz(u)1354 1558 y Fy(1)p Ft(=)p Fy(2)1408 1579 y FB(\))p Fz(:)441 b FB(\(7.11\))-84 1686 y(Then)16 b(one)h(has)g Fz(h)11 b Fx(\016)g Fz(h)323 1668 y Fq(\000)p Fy(1)323 1699 y Fq(\006)370 1686 y FB(\()p Fz(z)r FB(\))j(=)f Fz(z)r FB(,)j Fx(8)p Fz(z)f Fx(2)f Fs(C)9 b FB(.)-84 1819 y FC(Theorem)16 b(7.10)24 b Fn(Ther)n(e)d(exists)g(a)g(numb)n(er)g Fz(\020)801 1826 y Fy(0)841 1819 y Fx(2)g FB([0)p Fz(;)28 b FB(1)p Fz(=)p FB(3\))p Fn(,)22 b(whose)g(appr)n(oximate)e(value)i(is) e Fz(\020)1733 1826 y Fy(0)1773 1819 y Fx(')g FB(0)p Fz(:)p FB(29559)p Fn(,)-84 1879 y(such)g(that)f(for)g Fz(\020)i Fx(2)c FB([0)p Fz(;)25 b(\020)400 1886 y Fy(0)420 1879 y FB(\))19 b Fn(ther)n(e)h(exists)g(a)f(strictly)g(p)n(ositive)h (function)g Fz(e)p FB(\()p Fz(\020)t FB(\))f Fn(such)h(that)f(for)g(al) r(l)h Fz(u)d Fx(2)h Fz(D)1903 1887 y Fy(1)p Ft(;)9 b(a)p Fy(\()p Ft(\020)r Fy(\))-84 1939 y Fn(with)787 2009 y Fz(a)p FB(\()p Fz(\020)t FB(\))k(:=)960 1976 y(1)p 960 1998 26 2 v 960 2044 a Fz(\020)1001 2009 y FB(+)e Fz(e)p FB(\()p Fz(\020)t FB(\))-84 2118 y Fn(one)18 b(has)792 2178 y Fx(j)p Fz(h)834 2157 y Fq(\000)p Fy(1)834 2190 y Fq(\006)881 2178 y FB(\()p Fz(u)p FB(\))p Fx(j)31 b Fz(<)g Fx(j)p Fz(u)p Fx(j)p Fz(:)751 b FB(\(7.12\))1969 2311 y Fm(2)-84 2492 y FC(Remark)16 b(7.11)24 b Fn(Numeric)n(al)19 b(c)n(omputations)f(indic)n(ate)h(to)f(b)n(e)h(imp)n(ossible)f(to)g (impr)n(ove)g(the)g(r)n(e)n(gion)g(of)g(validity)-84 2552 y(of)f(e)n(quation)h(\(7.12\))f(to)h Fz(\020)g Fx(\025)27 b Fz(\020)499 2559 y Fy(0)519 2552 y Fn(.)-84 2685 y FC(Remark)16 b(7.12)24 b Fn(As)c(alr)n(e)n(ady)e(observe)n(d,)i(the)g (ine)n(quality)h(\(7.12\))e(b)n(e)n(c)n(omes)g(an)h(e)n(quality)g(in)f Fz(S)1695 2667 y Fy(1)1715 2685 y Fn(,)h(which)g(is)f(the)-84 2745 y(internal)g(b)n(oundary)e(of)h Fz(D)408 2753 y Fy(1)p Ft(;)10 b(a)p Fy(\()p Ft(\020)r Fy(\))512 2745 y Fn(.)23 b(It)18 b(is)g(imp)n(ortant)f(to)h(note)h(also)f(that)g(the)g (set)h Fz(D)1447 2753 y Fy(1)p Ft(;)9 b(a)p Fy(\()p Ft(\020)r Fy(\))1568 2745 y Fn(c)n(ontains)19 b(the)f(the)g(p)n(ole)-84 2805 y Fz(z)e FB(=)d Fx(\000)p FB(1)p Fz(=\020)t Fn(,)18 b(of)g Fz(h)p Fn(.)937 2930 y FB(48)p eop %%Page: 49 49 49 48 bop -11 79 a FB(W)l(e)16 b(will)g(presen)o(t)g(the)g(pro)q(of)h (of)g(the)f(Con)o(traction)h(Theorem)e(in)i(Subsection)f(7.2.4.)22 b(Let)17 b(us)g(no)o(w)f(explore)-84 139 y(some)f(of)i(its)f (consequences.)-84 260 y FC(Theorem)g(7.13)24 b Fn(L)n(et)18 b Fz(\020)383 267 y Fy(0)420 260 y Fn(as)f(in)h(The)n(or)n(em)e(7.10.) 22 b(F)l(or)17 b(al)r(l)i Fz(\020)f Fx(2)c FB(\(0)p Fz(;)22 b(\020)1218 267 y Fy(0)1238 260 y FB(\))c Fn(and)f Fz(k)f Fx(2)e Fs(N)835 357 y Fx(Z)871 364 y Ft(k)906 357 y Fx(\032)g Fz(D)999 364 y Fy(1)p Ft(;)c(r)1053 370 y Fl(k)1074 357 y Fz(;)794 b FB(\(7.13\))-84 454 y Fn(holds)17 b(with)854 514 y Fz(r)876 521 y Ft(k)911 514 y FB(=)d Fz(\020)1001 478 y Fg(\000)p Fk(1)p 993 485 56 2 v 993 505 a Fl(k)q Fk(+1)1055 514 y Fz(:)-84 596 y Fn(Sinc)n(e)19 b(the)g Fx(Z)164 603 y Ft(k)204 596 y Fn(ar)n(e)f(\014nite)i(sets,)f(this)f (cle)n(arly)h(shows)f(explicitly)i(that)53 b(the)19 b(ac)n(cumulation)g (p)n(oints)f Fx(Z)1820 578 y Fy(0)1840 596 y Fn(of)g Fx(Z)23 b Fn(lie)-84 656 y(in)18 b(the)g(unit)g(cir)n(cle.)1969 777 y Fm(2)-84 901 y FC(Pro)r(of.)42 b FB(The)22 b(pro)q(of)g(of)g (Theorem)e(7.13)i(mak)o(es)e(use)i(of)f(the)h(Con)o(traction)f(Theorem) g(whic)o(h)f(requires)h(the)-84 961 y(follo)o(wing)g(tec)o(hnical)e (lemma.)33 b(W)l(e)21 b(note)h(that,)g(from)e(equation)h(\(7.7\),)i (there)d(exists)h(a)g(\014nite)g(sequence)f(of)-84 1021 y(signals)d Fx(f)p Fz(s)p FB(\()p Fz(l)q FB(\))c Fx(2)h(f\000)p Fz(;)8 b FB(+)p Fx(g)p Fz(;)21 b FB(1)14 b Fx(\024)g Fz(l)g Fx(\024)g Fz(k)r Fx(g)i FB(suc)o(h)g(that)453 1116 y(1)p 452 1138 25 2 v 452 1183 a Fz(z)482 1149 y(h)510 1129 y Fq(\000)p Fy(1)510 1164 y Ft(s)p Fy(\()p Ft(k)q Fy(\))584 1076 y Fr( )622 1116 y FB(1)p 621 1138 V 621 1183 a Fz(z)651 1149 y(h)679 1129 y Fq(\000)p Fy(1)679 1164 y Ft(s)p Fy(\()p Ft(k)q Fq(\000)p Fy(1\))798 1076 y Fr( )836 1116 y FB(1)p 836 1138 V 836 1183 a Fz(z)874 1149 y Fx(\001)8 b(\001)g(\001)946 1116 y FB(1)p 945 1138 V 945 1183 a Fz(z)975 1149 y(h)1003 1129 y Fq(\000)p Fy(1)1003 1164 y Ft(s)p Fy(\(1\))1075 1076 y Fr( )1108 1149 y Fx(\000)1164 1116 y FB(1)p 1152 1138 50 2 v 1152 1183 a Fz(z)r(\020)1207 1076 y Fr(!)1248 1149 y Fx(\001)g(\001)g(\001) 1306 1076 y Fr(!!)1385 1149 y FB(=)14 b(1)p Fz(:)407 b FB(\(7.14\))-84 1301 y FC(Lemma)15 b(7.14)25 b Fn(Given)13 b Fz(z)j Fx(2)e(Z)498 1308 y Ft(k)519 1301 y Fn(,)f Fz(k)j Fx(2)e Fs(N)p Fn(,)h Fz(k)h Fx(\025)d FB(1)p Fn(,)h(c)n(onsider)e(a)h (se)n(quenc)n(e)h(of)e(signals)h Fx(f)p Fz(s)p FB(\()p Fz(l)q FB(\))h Fx(2)g(f\000)p Fz(;)8 b FB(+)p Fx(g)p Fz(;)21 b FB(1)14 b Fx(\024)g Fz(l)g Fx(\024)g Fz(k)r Fx(g)-84 1361 y Fn(satisfying)k(\(7.14\))f(and)g(de\014ne)254 1489 y Fz(w)289 1496 y Ft(l)315 1489 y FB(:=)386 1456 y(1)p 386 1478 25 2 v 386 1524 a Fz(z)416 1489 y(h)444 1469 y Fq(\000)p Fy(1)444 1504 y Ft(s)p Fy(\()p Ft(l)p Fy(\))509 1416 y Fr( )547 1456 y FB(1)p 547 1478 V 547 1524 a Fz(z)576 1489 y(h)604 1469 y Fq(\000)p Fy(1)604 1504 y Ft(s)p Fy(\()p Ft(l)p Fq(\000)p Fy(1\))715 1416 y Fr( )753 1456 y FB(1)p 752 1478 V 752 1524 a Fz(z)791 1489 y Fx(\001)8 b(\001)g(\001)862 1456 y FB(1)p 862 1478 V 862 1524 a Fz(z)892 1489 y(h)920 1469 y Fq(\000)p Fy(1)920 1504 y Ft(s)p Fy(\(1\))992 1416 y Fr( )1025 1489 y Fx(\000)1081 1456 y FB(1)p 1069 1478 50 2 v 1069 1524 a Fz(z)r(\020)1123 1416 y Fr(!)1164 1489 y Fx(\001)g(\001)g(\001) 1223 1416 y Fr(!)o(!)1346 1489 y Fz(;)g(l)15 b Fx(2)f(f)p FB(1)p Fz(;)8 b(:)g(:)g(:)f(;)h(k)r Fx(g)p Fz(:)-84 1615 y Fn(Then)18 b(al)r(l)h Fz(w)148 1622 y Ft(l)161 1615 y Fn('s)e(b)n(elong)i(to)e Fz(D)458 1622 y Fy(1)p Ft(;)10 b(a)p Fy(\()p Ft(\020)r Fy(\))562 1615 y Fn(,)18 b(exc)n(ept,)g(of)f(c) n(ourse,)h Fz(w)1014 1622 y Ft(k)1052 1615 y Fn(which)g(is)g(e)n(qual)g (to)f FB(1)p Fn(.)1969 1735 y Fm(2)-84 1859 y FC(Pro)r(of.)39 b FB(T)l(o)22 b(pro)o(v)o(e)f(Lemma)f(7.14)j(w)o(e)f(tak)o(e,)g (without)h(loss)f(of)g(generalit)o(y)l(,)g Fz(k)k(>)d FB(1)g(and)g(note)f(that,)h(since)-84 1920 y Fz(z)16 b Fx(2)e Fz(D)42 1927 y Fy(1)p Ft(;)6 b Fy(1)p Ft(=\020)131 1920 y FB(,)15 b(one)i(has)g Fz(h)365 1899 y Fq(\000)p Fy(1)365 1934 y Ft(s)p Fy(\(1\))437 1920 y FB(\()p Fx(\000)p Fz(z)520 1902 y Fq(\000)p Fy(1)566 1920 y Fz(=\020)t FB(\))e Fx(2)f Fz(D)736 1927 y Fy(1)p Ft(;)6 b Fy(1)p Ft(=\020)825 1920 y FB(,)16 b(b)o(y)f(the)h(Con)o(traction)h(Theorem.)j (Hence,)14 b Fz(w)1694 1927 y Fy(1)1728 1920 y Fx(2)g Fz(D)1815 1927 y Ft(\020)r(;)c Fy(1)p Ft(=\020)1908 1920 y FB(.)-11 2007 y(On)i(the)f(other)g(hand)h Fz(w)419 2014 y Fy(1)450 2007 y FB(cannot)g(b)q(elong)g(to)g Fz(D)851 2014 y Ft(\020)r(;)e Fy(1)910 2007 y Fx([)q Fz(S)977 1989 y Fy(1)1008 2007 y FB(for)i(the)f(follo)o(wing)g(reason:)20 b Fz(h)1555 1989 y Fq(\000)p Fy(1)1555 2020 y Fq(\006)1614 2007 y FB(maps)11 b Fz(D)1777 2014 y Ft(<)p Fy(1)1825 2007 y Fx([)q Fz(S)1892 1989 y Fy(1)1924 2007 y FB(in)o(to)-84 2068 y(itself)16 b(and,)i(since)f(1)p Fz(=)p Fx(j)p Fz(z)r Fx(j)g Fz(<)f FB(1,)i(this)f(implies)e Fz(w)792 2075 y Fy(2)828 2068 y Fx(2)h Fz(D)917 2075 y Ft(<)p Fy(1)965 2068 y FB(,)h(what,)h(in)f(turn,)h(w)o(ould)g(imply)d(that)j(all)f (subsequen)o(t)-84 2128 y Fz(w)-49 2135 y Ft(l)-36 2128 y FB('s)f(w)o(ould)g(also)h(b)q(e)f(in)g Fz(D)416 2135 y Ft(<)p Fy(1)480 2128 y FB(and)h(so)g(they)f(could)g(not)g(reac)o(h)g (the)g(p)q(oin)o(t)g(1)h(after)f(a)h(\014nite)f(n)o(um)o(b)q(er)e(of)j (steps.)-11 2209 y(Therefore,)25 b Fz(w)268 2216 y Fy(1)315 2209 y Fx(2)j Fz(D)416 2217 y Fy(1)p Ft(;)10 b Fy(1)p Ft(=\020)533 2209 y FB(and)25 b(w)o(e)e(can)i(apply)f(again)h(the)f (Con)o(traction)g(Theorem)f(and)i(claim)d(that)-84 2270 y Fz(h)-56 2249 y Fq(\000)p Fy(1)-56 2284 y Ft(s)p Fy(\(2\))7 2270 y FB(\()p Fz(w)61 2277 y Fy(1)81 2270 y FB(\))e Fx(2)h Fz(D)214 2277 y Fy(1)p Ft(;)9 b Fy(1)p Ft(=\020)307 2270 y FB(.)32 b(So,)21 b(w)o(e)f(conclude)f(again)i(that)f Fz(w)997 2277 y Fy(2)1037 2270 y Fx(2)g Fz(D)1130 2277 y Ft(\020)r(;)10 b Fy(1)p Ft(=\020)1223 2270 y FB(.)33 b(If)19 b Fz(k)k FB(=)d(2)g(then,)g(b)o(y)g(the)g(h)o(yp)q(otheses,)-84 2330 y Fz(w)-49 2337 y Fy(2)-16 2330 y FB(=)14 b(1.)21 b(Otherwise,)12 b(if)g Fz(k)k(>)d FB(2,)h(rep)q(eating)e(the)h (previous)f(argumen)o(ts,)g(w)o(e)g(conclude)g(again)i(that)f Fz(w)1780 2337 y Fy(2)1813 2330 y Fx(2)h Fz(D)1900 2338 y Fy(1)p Ft(;)c Fy(1)p Ft(=\020)1993 2330 y FB(.)-84 2390 y(This)16 b(argumen)o(tation)g(can)g(b)q(e)h(rep)q(eated)f(a)g (\014nite)g(n)o(um)o(b)q(er)f(of)h(times)e(and)j(therefore)f(the)g (lemm)o(a)e(is)i(pro)o(v)o(en.)p 1977 2472 30 30 v -11 2596 a(Lemma)k(7.14)i(implies)d(that)j(all)f Fz(w)667 2603 y Ft(l)680 2596 y FB(,)h(1)i Fx(\024)e Fz(l)i(<)f(k)r FB(,)f(lie)f(in)g(the)g(region)h(of)g(v)m(alidit)o(y)e(of)i(the)f(Con)o (traction)-84 2656 y(Theorem.)f(Therefore,)15 b(applying)h(the)g (inequalit)o(y)f(\(7.12\))i(rep)q(eatedly)e(in)h(\(7.14\))h(w)o(e)f (get)807 2775 y(1)28 b Fz(<)977 2741 y FB(1)p 929 2763 119 2 v 929 2809 a Fx(j)p Fz(z)r Fx(j)982 2795 y Ft(k)q Fy(+1)1072 2741 y FB(1)p 1072 2763 26 2 v 1072 2809 a Fz(\020)1102 2775 y(;)937 2930 y FB(49)p eop %%Page: 50 50 50 49 bop -84 79 a FB(whic)o(h)14 b(pro)o(v)o(es)h(\(7.13\).)22 b(The)15 b(fact)g(that)h Fx(Z)703 61 y Fy(0)736 79 y Fx(\032)e Fz(S)822 61 y Fy(1)857 79 y FB(follo)o(ws)h(easily)g(from)f (\(7.13\).)21 b(This)16 b(completes)d(the)i(pro)q(of)h(of)-84 139 y(Theorem)f(7.13.)p 1977 221 30 30 v -11 353 a(The)h(Con)o (traction)h(Theorem)e(has)i(another)g(consequence.)-84 488 y FC(Theorem)f(7.15)24 b Fn(F)l(or)16 b(the)h(sets)g Fx(Z)578 495 y Ft(k)616 488 y Fn(de\014ne)n(d)h(ab)n(ove)f(we)g(have,)g (in)g(the)g(r)n(e)n(gion)f(of)g(validity)h(of)f(the)h(Contr)n(action) -84 548 y(The)n(or)n(em,)f(that)795 609 y Fx(Z)831 616 y Ft(l)869 609 y Fx(\\)25 b(Z)963 616 y Ft(m)1024 609 y FB(=)i Fx(;)p Fz(;)-84 696 y Fn(for)17 b(al)r(l)i Fz(l)q Fn(,)e Fz(m)c Fx(2)h Fs(N)p Fn(,)k Fz(l)d Fx(6)p FB(=)f Fz(m)p Fn(.)1969 831 y Fm(2)-84 963 y FC(Pro)r(of.)21 b FB(Let)16 b Fz(m)e(>)g(l)j FB(and)f(pic)o(k)f(a)i Fz(z)f Fx(2)e(Z)677 970 y Ft(l)715 963 y Fx(\\)25 b(Z)809 970 y Ft(m)842 963 y FB(.)d(Then)16 b(one)g(has)709 1101 y Fz(\034)736 1081 y Fy(\()p Ft(m)p Fq(\000)p Ft(l)p Fy(\))730 1113 y Ft(z)843 1028 y Fr( )876 1101 y Fx(\000)933 1067 y FB(1)p 920 1090 50 2 v 920 1135 a Fz(z)r(\020)975 1028 y Fr(!)1035 1101 y FB(=)28 b Fx(\000)1158 1067 y FB(1)p 1145 1090 V 1145 1135 a Fz(z)r(\020)1200 1101 y(:)-84 1242 y FB(This)16 b(means)g(also)g(that)h(there)f(exists)f(a)i (\014nite)f(sequence)f Fx(f)p Fz(s)p FB(\()p Fz(a)p FB(\))e Fx(2)h(f\000)p Fz(;)8 b FB(+)p Fx(g)p Fz(;)21 b FB(1)14 b Fx(\024)g Fz(a)f Fx(\024)h Fz(m)d Fx(\000)g Fz(l)q Fx(g)k FB(suc)o(h)h(that)356 1383 y Fx(\000)412 1350 y FB(1)p 400 1372 V 400 1418 a Fz(z)r(\020)468 1383 y FB(=)525 1350 y(1)p 525 1372 25 2 v 525 1418 a Fz(z)555 1383 y(h)583 1363 y Fq(\000)p Fy(1)583 1398 y Ft(s)p Fy(\()p Ft(m)p Fq(\000)p Ft(l)p Fy(\))707 1310 y Fr( )745 1350 y FB(1)p 744 1372 V 744 1418 a Fz(z)774 1383 y(h)802 1363 y Fq(\000)p Fy(1)802 1398 y Ft(s)p Fy(\()p Ft(m)p Fq(\000)p Ft(l)p Fq(\000)p Fy(1\))971 1310 y Fr( )1009 1350 y FB(1)p 1009 1372 V 1009 1418 a Fz(z)1047 1383 y Fx(\001)8 b(\001)g(\001)1119 1350 y FB(1)p 1119 1372 V 1119 1418 a Fz(z)1148 1383 y(h)1176 1363 y Fq(\000)p Fy(1)1176 1398 y Ft(s)p Fy(\(1\))1248 1310 y Fr( )1281 1383 y Fx(\000)1338 1350 y FB(1)p 1325 1372 50 2 v 1325 1418 a Fz(z)r(\020)1380 1310 y Fr(!)1421 1383 y Fx(\001)g(\001)g(\001) 1479 1310 y Fr(!!)1553 1383 y Fz(:)315 b FB(\(7.15\))-84 1522 y(Applying)15 b(the)h(Con)o(traction)h(Theorem)e(to)h(the)g(last)h (equalit)o(y)d(w)o(e)i(get)807 1618 y(1)p 781 1640 78 2 v 781 1685 a Fx(j)p Fz(z)r Fx(j)p Fz(\020)877 1651 y(<)1019 1618 y FB(1)p 934 1640 195 2 v 934 1685 a Fx(j)p Fz(z)r Fx(j)987 1671 y Ft(m)p Fq(\000)p Ft(l)p Fy(+1)1103 1685 y Fz(\020)1133 1651 y(;)-84 1793 y FB(what)h(means)e Fx(j)p Fz(z)r Fx(j)240 1775 y Ft(m)p Fq(\000)p Ft(l)325 1793 y Fz(<)f FB(1,)i(a)h(con)o(tradiction)f(since)f Fx(Z)924 1800 y Ft(l)948 1793 y Fx(\\)c(Z)1028 1800 y Ft(m)1076 1793 y Fx(\032)i Fz(D)1168 1801 y Fy(1)p Ft(;)d Fy(1)p Ft(=\020)1261 1793 y FB(.)p 1977 1875 30 30 v -84 2076 a FC(7.2.4)55 b(The)19 b(Con)n(traction)g(Theorem)-84 2190 y FB(This)14 b(subsection)h(is)f(dev)o(oted)g(to)g(the)h(pro)q(of) g(of)g(Theorem)e(7.10)i(whic)o(h)f(is)g(implied)d(b)o(y)j(the)g(follo)o (wing)h(theorem.)-84 2325 y FC(Theorem)h(7.16)24 b Fn(Ther)n(e)19 b(exists)g(a)g Fz(\020)618 2332 y Fy(0)654 2325 y Fx(2)d FB([0)p Fz(;)24 b FB(1)p Fz(=)p FB(3\))p Fn(,)c(whose)g(appr)n(oximate) e(value)i(is)e Fz(\020)1527 2332 y Fy(0)1564 2325 y Fx(')d FB(0)p Fz(:)p FB(29559)p Fn(,)21 b(such)e(that)-84 2385 y(ther)n(e)g(is)g(a)f(strictly)h(p)n(ositive)h(function)g Fz(f)5 b FB(\()p Fz(\020)t FB(\))19 b Fn(in)g(the)h(interval)g Fz(\020)g Fx(2)d FB([0)p Fz(;)24 b(\020)1295 2392 y Fy(0)1315 2385 y FB(\))19 b Fn(such)g(that,)h(for)e Fz(w)g Fx(2)e Fz(D)1803 2398 y Fy(1)p Ft(;)1861 2384 y Fk(1)p 1846 2390 46 2 v 1846 2395 a Fq(p)p 1875 2395 16 2 v 23 x Fl(\020)1896 2398 y Fy(+)p Ft(f)t Fy(\()p Ft(\020)r Fy(\))1992 2385 y Fn(,)-84 2465 y(i.e.,)i(for)730 2545 y FB(1)c Fz(<)g Fx(j)p Fz(w)q Fx(j)g Fz(<)976 2511 y FB(1)p 955 2533 67 2 v 955 2542 a Fx(p)p 996 2542 26 2 v 996 2580 a Fz(\020)1037 2545 y FB(+)d Fz(f)5 b FB(\()p Fz(\020)t FB(\))p Fz(;)-84 2656 y Fn(one)18 b(has)803 2716 y Fx(j)p Fz(g)r FB(\()p Fz(w)q FB(\))p Fx(j)27 b Fz(<)h Fx(j)p Fz(w)q Fx(j)1087 2696 y Fy(2)1106 2716 y Fz(:)937 2930 y FB(50)p eop %%Page: 51 51 51 50 bop 1969 79 a Fm(2)-11 207 y FB(In)16 b(view)g(of)g(\(7.10\),)h (\(7.11\))f(and)h(Theorem)e(7.16,)i(for)f Fz(u)e Fx(2)g Fs(C)28 b FB(suc)o(h)16 b(that)563 350 y(1)e Fz(<)g Fx(j)p Fz(u)p Fx(j)f Fz(<)774 277 y Fr( )833 316 y FB(1)p 812 339 67 2 v 812 348 a Fx(p)p 854 348 26 2 v 37 x Fz(\020)895 350 y FB(+)e Fz(f)5 b FB(\()p Fz(\020)t FB(\))1036 277 y Fr(!)1069 289 y Fy(2)1103 350 y FB(=:)1173 316 y(1)p 1173 339 V 1173 384 a Fz(\020)1214 350 y FB(+)11 b Fz(\017)p FB(\()p Fz(\020)t FB(\))p Fz(;)-84 482 y FB(one)16 b(has)671 542 y Fx(j)p Fz(h)713 522 y Fq(\000)p Fy(1)713 555 y Fq(\006)760 542 y FB(\()p Fz(u)p FB(\))p Fx(j)d FB(=)h Fx(j)p Fz(g)r FB(\()p Fx(\006)p Fz(u)1030 522 y Fy(1)p Ft(=)p Fy(2)1084 542 y FB(\))p Fx(j)g Fz(<)g Fx(j)p Fz(u)p Fx(j)p Fz(:)-11 648 y FB(This)i(concludes)g(the)g(pro)q(of)i(of)e (Theorem)f(7.10.)p 1977 730 30 30 v -84 858 a FC(Pro)r(of)38 b(of)19 b(Theorem)e(7.16.)25 b FB(W)l(riting)16 b Fz(w)g FB(=)e Fz(x)d FB(+)g Fz(iy)r FB(,)16 b(with)g Fz(x)p FB(,)g Fz(y)g Fx(2)e Fs(R)p FB(,)g(a)j(simple)d(computation)i(sho)o(ws) h(that)-84 918 y(w)o(e)f(ha)o(v)o(e)649 995 y Fx(j)p Fz(g)r FB(\()p Fz(w)q FB(\))p Fx(j)776 975 y Fy(2)809 995 y FB(=)878 962 y(\()p Fz(\020)922 944 y Fy(2)953 962 y FB(+)11 b Fx(j)p Fz(w)q Fx(j)1066 944 y Fy(2)1086 962 y FB(\))g Fx(\000)g FB(2)p Fz(\020)t(x)p 866 984 389 2 v 866 1030 a FB(\(1)h(+)f Fz(\020)995 1015 y Fy(2)1014 1030 y Fx(j)p Fz(w)q Fx(j)1078 1015 y Fy(2)1098 1030 y FB(\))g Fx(\000)g FB(2)p Fz(\020)t(x)1260 995 y(:)608 b FB(\(7.16\))-11 1126 y(T)l(o)17 b(analyze)f(the)g(righ)o(t)g(hand)g (side,)g(consider)g(the)g(follo)o(wing)g(lemm)o(a.)-84 1255 y FC(Lemma)f(7.17)25 b Fn(L)n(et)17 b Fz(t)c FB(:)g Fs(R)e Fx(7\000)-8 b(!)13 b Fs(R)p Fn(,)i(b)n(e)i(given)j(by)795 1380 y Fz(t)p FB(\()p Fz(a)p FB(\))13 b(:=)961 1347 y Fz(\013)e Fx(\000)g Fz(\014)s(a)p 961 1369 149 2 v 963 1415 a(\015)i Fx(\000)e Fz(\014)s(a)1114 1380 y(;)-84 1509 y Fn(for)k Fz(\013)p Fn(,)i Fz(\014)h Fn(and)e Fz(\015)g Fx(2)e Fs(R)f Fn(with)j Fz(\013)f(>)e(\015)k(>)d FB(0)i Fn(and)g Fz(\014)g Fx(\025)e FB(0)p Fn(.)22 b(Consider)16 b(the)g(op)n(en)g(interval)h Fz(J)h FB(:=)c(\()p Fx(\0001)p Fz(;)h(\015)s(=\014)s FB(\))p Fn(.)22 b(Then,)-84 1570 y Fz(t)17 b Fn(is)g(a)h(c)n(ontinuous,)g(strictly)g(p)n(ositive,)f (nowher)n(e)h(c)n(onstant)g(for)f Fz(\014)f(>)e FB(0)k Fn(and)g(incr)n(e)n(asing)f(function)i(in)f Fz(J)5 b Fn(.)1969 1698 y Fm(2)-84 1826 y FC(Pro)r(of.)21 b FB(Con)o(tin)o(uit)o (y)15 b(on)i(the)f(in)o(terv)m(al)f Fz(J)21 b FB(is)16 b(ob)o(vious.)21 b(A)16 b(computation)f(sho)o(ws)j(that)769 1954 y Fz(t)787 1934 y Fq(0)798 1954 y FB(\()p Fz(a)p FB(\))c(=)939 1920 y Fz(\014)s FB(\()p Fz(\013)d Fx(\000)g Fz(\015)s FB(\))p 932 1943 203 2 v 932 1988 a(\()p Fz(\015)j Fx(\000)d Fz(\014)s(a)p FB(\))1116 1974 y Fy(2)1140 1954 y Fz(:)-84 2083 y FB(All)k(claims)f(follo)o(w)i(immedi)o(ately)d(from)i (con)o(tin)o(uit)o(y)f(and)j(from)e(this)h(relation.)p 1977 2165 30 30 v -11 2292 a(Lo)q(oking)i(at)e(\(7.16\))h(w)o(e)f(can)g (mak)o(e)f(the)h(follo)o(wing)g(iden)o(ti\014cations:)419 2396 y Fz(\013)e Fx(\021)g Fz(\020)542 2376 y Fy(2)572 2396 y FB(+)d Fx(j)p Fz(w)q Fx(j)685 2376 y Fy(2)705 2396 y FB(;)57 b Fz(\014)16 b Fx(\021)d FB(2)p Fz(\020)t FB(;)57 b Fz(\015)17 b Fx(\021)d FB(1)d(+)g Fz(\020)1196 2376 y Fy(2)1216 2396 y Fx(j)p Fz(w)q Fx(j)1280 2376 y Fy(2)1300 2396 y FB(;)56 b Fz(a)14 b Fx(\021)f Fz(x:)378 b FB(\(7.17\))-84 2500 y(Note)16 b(that,)g(in)g(view)f(of)i(this)629 2560 y Fz(\013)12 b Fx(\000)e Fz(\015)17 b FB(=)d(\(1)d Fx(\000)g Fz(\020)944 2539 y Fy(2)964 2560 y FB(\)\()p Fx(j)p Fz(w)q Fx(j)1066 2539 y Fy(2)1097 2560 y Fx(\000)f FB(1\))15 b Fz(>)e FB(0)p Fz(;)589 b FB(\(7.18\))-84 2645 y(for)16 b(all)g Fz(w)f Fx(2)f Fz(D)195 2652 y Ft(>)p Fy(1)243 2645 y FB(.)21 b(Moreo)o(v)o(er,)15 b(the)h(condition)g Fz(a)d(<)h(\015)s(=\014)19 b FB(means)796 2778 y Fz(x)14 b(<)895 2744 y FB(1)d(+)g Fz(\020)1004 2726 y Fy(2)1024 2744 y Fx(j)p Fz(w)q Fx(j)1088 2726 y Fy(2)p 895 2766 213 2 v 976 2812 a FB(2)p Fz(\020)1113 2778 y(:)937 2930 y FB(51)p eop %%Page: 52 52 52 51 bop -84 79 a FB(This)16 b(condition)g(is)g(alw)o(a)o(ys)g (satis\014ed)h(if)f Fx(j)p Fz(w)q Fx(j)d Fz(<)h FB(1)p Fz(=\020)t FB(,)j(since)e Fz(x)f Fx(\024)f(j)p Fz(w)q Fx(j)k FB(and)f(since)g(the)g(stronger)h(condition)785 219 y Fx(j)p Fz(w)q Fx(j)d Fz(<)920 185 y FB(1)d(+)g Fz(\020)1029 167 y Fy(2)1049 185 y Fx(j)p Fz(w)q Fx(j)1113 167 y Fy(2)p 920 207 213 2 v 1001 253 a FB(2)p Fz(\020)-84 362 y FB(is)16 b(equiv)m(alen)o(t)f(to)h(the)g(condition)g(\(1)c Fx(\000)f Fz(\020)t Fx(j)p Fz(w)q Fx(j)p FB(\))766 338 y Fy(2)800 362 y Fz(>)j FB(0.)-11 444 y(In)i(view)g(of)g(Lemma)e(7.17)j (w)o(e)f(conclude)f(that,)i(for)f Fz(w)f Fx(2)f Fz(D)1098 452 y Fy(1)p Ft(;)c Fy(1)p Ft(=\020)1191 444 y FB(,)496 587 y Fx(j)p Fz(g)r FB(\()p Fz(w)q FB(\))p Fx(j)623 567 y Fy(2)656 587 y Fx(\024)726 553 y Fz(\020)751 535 y Fy(2)782 553 y FB(+)h Fx(j)p Fz(w)q Fx(j)895 535 y Fy(2)925 553 y Fx(\000)g FB(2)p Fz(\020)t Fx(j)p Fz(w)q Fx(j)p 714 576 388 2 v 714 621 a FB(1)g(+)g Fz(\020)823 607 y Fy(2)843 621 y Fx(j)p Fz(w)q Fx(j)907 607 y Fy(2)938 621 y Fx(\000)f FB(2)p Fz(\020)t Fx(j)p Fz(w)q Fx(j)1120 587 y FB(=)1189 553 y(\()p Fz(\020)15 b Fx(\000)10 b(j)p Fz(w)q Fx(j)p FB(\))1376 535 y Fy(2)p 1176 576 232 2 v 1176 621 a FB(\(1)i Fx(\000)f Fz(\020)t Fx(j)p Fz(w)q Fx(j)p 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Fz(s)618 2539 y Fy(3)647 2557 y FB(+)8 b Fz(s)716 2539 y Fy(2)745 2557 y FB(+)g Fz(s)h Fx(\000)f FB(1)16 b(has)f(one)h(real)e (ro)q(ot)i(at)g Fz(s)1362 2564 y Fy(0)1395 2557 y Fx(')e FB(0)p Fz(:)p FB(543689)j(and)f(t)o(w)o(o)f(complex)-84 2617 y(ro)q(ots)j(at)f Fz(s)123 2624 y Fq(\006)167 2617 y Fx(')d(\000)p FB(0)p Fz(:)p FB(77)e Fx(\006)f FB(1)p Fz(:)p FB(115)p Fz(i)p FB(.)23 b(W)l(e)17 b(call)f Fz(\020)767 2624 y Fy(0)801 2617 y FB(:=)e Fz(s)890 2599 y Fy(2)890 2629 y(0)910 2617 y FB(,)i(whic)o(h)g(giv)o(es)g Fz(\020)1221 2624 y Fy(0)1256 2617 y Fx(')e FB(0)p Fz(:)p FB(295597.)25 b(Th)o(us,)16 b(condition)h(\(7.22\))-84 2677 y(is)f(satis\014ed)g(for) h(0)d Fz(<)g(\020)k(<)c(\020)429 2684 y Fy(0)449 2677 y FB(.)21 b(With)16 b(this)g(the)g(pro)q(of)h(of)g(Lemma)d(7.19)j(is)f (complete.)p 1977 2759 30 30 v 937 2930 a(54)p eop %%Page: 55 55 55 54 bop -84 79 a FA(8)81 b(The)27 b(Tw)n(o-P)n(oin)n(t)g(F)-7 b(unction)-84 210 y FB(This)18 b(section)f(is)g(dedicated)g(to)h(the)f (pro)q(of)i(of)e(Theorems)g(1.9)h(and)g(1.11.)25 b(W)l(e)18 b(shall)f(consider)g(the)h(truncated)-84 270 y(t)o(w)o(o-p)q(oin)o(t)e (function)433 331 y Fx(h)p Fz(\033)480 338 y Fy(0)500 331 y FB(;)8 b Fz(\033)550 338 y Ft(x)571 331 y Fx(i)590 345 y Ft(M)638 331 y FB(\()p Fz(\030)r FB(\))15 b(=)e Fx(h)q Fz(\033)813 338 y Fy(0)832 331 y Fz(\033)860 338 y Ft(x)882 331 y Fx(i)901 345 y Ft(M)949 331 y FB(\()p Fz(\030)r FB(\))f Fx(\000)f(h)p Fz(\033)1119 338 y Fy(0)1138 331 y Fx(i)1158 345 y Ft(M)1206 331 y FB(\()p Fz(\030)r FB(\))25 b Fx(h)p Fz(\033)1339 338 y Ft(x)1361 331 y Fx(i)1380 345 y Ft(M)1428 331 y FB(\()p Fz(\030)r FB(\))417 b(\(8.1\))-84 418 y(where)535 488 y Fx(h)q Fz(\033)583 495 y Fy(0)602 488 y Fz(\033)630 495 y Ft(x)652 488 y Fx(i)671 502 y Ft(M)719 488 y FB(\()p Fz(\030)r FB(\))14 b(=)906 454 y(1)p 851 476 135 2 v 851 522 a Fz(Z)884 529 y Ft(M)924 522 y FB(\()p Fz(\030)r FB(\))999 446 y Fr(X)1018 533 y Ft(\033)1067 488 y Fz(\033)1095 495 y Fy(0)1131 488 y Fz(\033)1159 495 y Ft(x)1197 488 y Fz(e)1220 467 y Fq(\000)p Ft(\014)r(H)s Fy(\()p Ft(\033)q Fy(;)9 b Ft(\030)q Fy(\))1906 488 y FB(\(8.2\))-84 603 y(for)16 b(all)g Fz(x)e Fx(2)g(C)173 610 y Fy(2)p Ft(;)5 b(M)262 603 y FB(with)16 b Fz(M)22 b FB(large)16 b(enough.)-11 685 y(Let)h(us)g(\014rst)g(restrict)f(to)h(the)g(case)g Fz(\030)g Fx(\021)e FB(1.)23 b(T)l(o)18 b(compute)d(the)i(n)o(umerator) e(of)j(\(8.2\),)e(w)o(e)h(rep)q(eat)g(the)f(steps)-84 745 y(in)g(the)g(calculation)f(of)i(the)f(partition)g(function)g Fz(Z)870 752 y Ft(M)910 745 y FB(\()p Fz(\030)r FB(\))h(that)g(led)e (to)i(\(3.2\).)-11 833 y(Let)88 817 y Fr(e)77 833 y Fz(Z)110 840 y Ft(j)143 833 y FB(=)196 785 y Fr(\020)232 817 y(e)221 833 y Fz(Z)258 812 y Fy(+)254 844 y Ft(j)287 833 y Fz(;)320 817 y Fr(e)309 833 y Fz(Z)346 812 y Fq(\000)342 844 y Ft(j)375 785 y Fr(\021)400 833 y FB(,)g Fz(j)g FB(=)e(0)p Fz(;)8 b(:)g(:)g(:)f(;)h(M)d FB(,)17 b(b)q(e)g(a)g(sequence)f(of)h(t)o (w)o(o-comp)q(onen)o(t)e(v)o(ectors)h(de\014ned)h(recursiv)o(ely)-84 893 y(as)g(in)f(the)g(follo)o(wing.)-11 975 y(F)l(or)g Fz(j)h FB(=)d Fz(n)194 982 y Fy(0)225 975 y FB(+)d(1)p Fz(;)17 b(:)8 b(:)g(:)f(;)16 b(M)5 b FB(,)16 b(with)h Fz(n)647 982 y Fy(0)680 975 y FB(=dist\(0)p Fz(;)22 b(x)p FB(\),)964 959 y Fr(e)953 975 y Fz(Z)990 957 y Ft(\033)986 987 y(j)1027 975 y FB(=)1090 959 y Fr(b)1079 975 y Fz(Z)1116 957 y Ft(\033)1112 987 y(j)1155 975 y FB(satisfy)16 b(the)g(equation)h (\(3.1\),)f(i.e.,)495 1089 y Fr(b)484 1106 y Fz(Z)521 1085 y Ft(\033)517 1118 y(j)r Fq(\000)p Fy(1)595 1106 y FB(=)d(\()p Fz(\020)t(z)r FB(\))734 1081 y Fq(\000)p Fy(1)790 1057 y Fr(\020)815 1106 y Fz(\020)840 1085 y Fy(\(1)p Fq(\000)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)998 1089 y Fr(b)987 1106 y Fz(Z)1024 1085 y Fy(+)1020 1118 y Ft(j)1065 1106 y FB(+)e Fz(\020)1139 1085 y Fy(\(1+)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)1286 1106 y Fz(z)1338 1089 y Fr(b)1327 1106 y Fz(Z)1364 1085 y Fq(\000)1360 1118 y Ft(j)1394 1057 y Fr(\021)1419 1069 y Fy(2)1906 1106 y FB(\(8.3\))-84 1231 y(with)16 b(initial)f(conditions)412 1214 y Fr(b)401 1231 y Fz(Z)438 1210 y Fy(+)434 1243 y Ft(M)488 1231 y FB(=)551 1214 y Fr(b)540 1231 y Fz(Z)577 1210 y Fq(\000)573 1243 y Ft(M)626 1231 y FB(=)f(1.)-11 1312 y(F)l(or)i Fz(j)h Fx(\024)d Fz(n)195 1319 y Fy(0)215 1312 y FB(,)i(w)o(e)f(consider)h(a)h(linear)e(transformation)i(of)f (the)g(form)538 1442 y Fr(e)527 1458 y Fz(Z)564 1438 y Ft(\033)560 1471 y(j)r Fq(\000)p Fy(1)638 1458 y FB(=)701 1442 y Fr(b)690 1458 y Fz(Z)727 1438 y Ft(\033)723 1471 y(j)r Fq(\000)p Fy(1)791 1422 y Fz(\020)816 1404 y Fy(\(1)p Fq(\000)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)967 1406 y Fr(e)956 1422 y Fz(Z)993 1401 y Fy(+)989 1433 y Ft(j)1033 1422 y FB(+)11 b Fz(\020)1107 1404 y Fy(\(1+)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)1255 1422 y Fz(z)1307 1406 y Fr(e)1296 1422 y Fz(Z)1333 1401 y Fq(\000)1329 1433 y Ft(j)p 791 1447 572 2 v 791 1498 a Fz(\020)816 1483 y Fy(\(1)p Fq(\000)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)975 1482 y Fr(b)964 1498 y Fz(Z)1001 1477 y Fy(+)997 1509 y Ft(j)1041 1498 y FB(+)g Fz(\020)1115 1483 y Fy(\(1+)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)1255 1498 y Fz(z)1307 1482 y Fr(b)1296 1498 y Fz(Z)1333 1477 y Fq(\000)1329 1509 y Ft(j)1906 1458 y FB(\(8.4\))-84 1616 y(with)38 1600 y Fr(e)27 1616 y Fz(Z)64 1598 y Ft(\033)60 1628 y(n)81 1633 y Fk(0)115 1616 y FB(=)j Fz(\033)207 1600 y Fr(b)197 1616 y Fz(Z)234 1598 y Ft(\033)230 1628 y(n)251 1633 y Fk(0)270 1616 y FB(.)-11 1697 y(W)l(e)j(no)o(w)g(observ) o(e)f(that)h(the)f(sum)g(o)o(v)o(er)g(all)g(spin)h(con\014gurations)h (in)e(the)g(n)o(umerator)g(of)h(\(8.2\),)f(except)g(b)o(y)-84 1758 y(spin)c(at)g(the)f(origin,)h(is)g(determined)d(b)o(y)i(\(8.4\))h (and)g(the)f(sum)g(o)o(v)o(er)g(all)g(spin)h(con\014gurations)h(in)e (the)g(denominator)-84 1818 y(is)16 b(determined)e(b)o(y)h(\(8.3\).)22 b(The)16 b(t)o(w)o(o)g(p)q(oin)o(t)h(function)f(\(8.2\))g(can)h(th)o (us)f(b)q(e)g(written)g(in)g(the)g(follo)o(wing)g(form)668 1964 y Fx(h)p Fz(\033)715 1971 y Fy(0)735 1964 y Fz(\033)763 1971 y Ft(x)785 1964 y Fx(i)804 1979 y Ft(M)852 1964 y FB(\(1\))e(=)1012 1914 y Fr(e)1001 1930 y Fz(Z)1038 1910 y Fy(+)1034 1941 y(0)1078 1930 y Fx(\000)d Fz(z)1180 1914 y Fr(e)1169 1930 y Fz(Z)1206 1910 y Fq(\000)1202 1941 y Fy(0)p 985 1952 252 2 v 1012 1987 a Fr(b)1001 2004 y Fz(Z)1038 1983 y Fy(+)1034 2014 y(0)1079 2004 y FB(+)g Fz(z)1180 1987 y Fr(b)1169 2004 y Fz(Z)1206 1983 y Fq(\000)1202 2014 y Fy(0)1241 1964 y Fz(:)651 b FB(\(8.5\))-11 2136 y(In)17 b(the)g(follo)o(wing)g(lemm)o(a)e(the)i (equation)g(\(8.5\))h(will)e(b)q(e)h(reorganized)g(and)h(reexpressed)e (in)h(terms)f(of)1944 2120 y Fr(b)1933 2136 y Fz(Z)1970 2118 y Ft(\033)1966 2148 y(j)1993 2136 y FB(.)-84 2209 y(W)l(e)g(recall)f(the)h(de\014nition)440 2193 y Fr(b)431 2209 y FB(\001)472 2216 y Ft(j)504 2209 y FB(:=)580 2193 y Fr(b)569 2209 y Fz(Z)606 2188 y Fq(\000)602 2220 y Ft(j)644 2161 y Fr(.)688 2193 y(b)677 2209 y Fz(Z)714 2188 y Fy(+)710 2220 y Ft(j)749 2209 y FB(.)-84 2359 y FC(Lemma)f(8.1)24 b Fn(The)18 b(se)n(quenc)n(e)h(of)e(ve)n(ctors)740 2343 y Fr(e)729 2359 y Fz(Z)762 2366 y Ft(n)786 2359 y Fn(,)g Fz(n)d FB(=)g(1)p Fz(;)8 b(:)g(:)g(:)g(;)g(n)1076 2366 y Fy(0)1096 2359 y Fn(,)17 b(de\014ne)n(d)h(by)g(\(8.4\),)f(c)n (an)h(b)n(e)f(written)i(as)703 2453 y Fr(e)692 2469 y Fz(Z)729 2449 y Ft(\033)725 2482 y(n)p Fq(\000)p Fy(1)808 2469 y FB(=)14 b(\()p Fx(A)919 2476 y Ft(n)953 2469 y FB(+)d Fz(\033)r Fx(B)1065 2476 y Ft(n)1088 2469 y FB(\))1126 2453 y Fr(b)1115 2469 y Fz(Z)1152 2449 y Ft(\033)1148 2482 y(n)p Fq(\000)p Fy(1)1217 2469 y Fz(;)675 b FB(\(8.6\))-84 2579 y Fn(wher)n(e)561 2640 y Fx(A)601 2647 y Ft(n)638 2640 y FB(=)14 b Fz(A)727 2647 y Ft(n)750 2640 y Fx(B)783 2647 y Ft(n)p Fy(+1)862 2640 y FB(+)d Fz(:)d(:)g(:)i FB(+)h Fz(A)1065 2647 y Ft(n)1086 2652 y Fk(0)1104 2647 y Fq(\000)p Fy(1)1151 2640 y Fx(B)1184 2647 y Ft(n)1205 2652 y Fk(0)1235 2640 y FB(+)g Fz(A)1321 2647 y Ft(n)1342 2652 y Fk(0)1906 2640 y FB(\(8.7\))-84 2727 y Fn(and)501 2787 y Fx(B)534 2794 y Ft(j)566 2787 y FB(=)i Fz(B)654 2794 y Ft(j)673 2787 y Fz(B)710 2794 y Ft(j)r Fy(+1)781 2787 y Fz(:)8 b(:)g(:)g(B)884 2794 y Ft(n)905 2799 y Fk(0)1024 2787 y Fz(;)108 b(j)17 b FB(=)c Fz(n;)8 b(:)g(:)g(:)g(;)g(n) 1402 2794 y Fy(0)1906 2787 y FB(\(8.8\))937 2930 y(55)p eop %%Page: 56 56 56 55 bop -84 79 a Fn(Her)n(e)17 b Fz(A)69 86 y Ft(j)101 79 y FB(=)d Fz(A)190 86 y Ft(j)208 79 y FB(\()p Fz(\020)t(;)8 b(z)r FB(\))17 b Fn(and)h Fz(B)467 86 y Ft(j)499 79 y FB(=)13 b Fz(B)587 86 y Ft(j)606 79 y FB(\()p Fz(\020)t(;)8 b(z)r FB(\))p Fn(,)17 b Fz(j)g FB(=)c(1)p Fz(;)k(:)8 b(:)g(:)g(;)16 b(n)1016 86 y Fy(0)1036 79 y Fn(,)h(ar)n(e)g(given)i(by) 616 243 y Fz(A)653 250 y Ft(j)685 243 y FB(=)850 152 y Fr(\020)875 200 y Fz(z)909 184 y Fr(b)900 200 y FB(\001)941 207 y Ft(j)959 152 y Fr(\021)983 163 y Fq(\000)p Fy(1)1042 200 y Fx(\000)11 b Fz(z)1125 184 y Fr(b)1117 200 y FB(\001)1158 207 y Ft(j)p 742 232 542 2 v 742 246 a Fr(\020)767 295 y Fz(z)800 278 y Fr(b)792 295 y FB(\001)833 302 y Ft(j)850 246 y Fr(\021)875 258 y Fq(\000)p Fy(1)933 295 y FB(+)g Fz(z)1016 278 y Fr(b)1007 295 y FB(\001)1048 302 y Ft(j)1077 295 y FB(+)g Fz(\020)1151 280 y Fq(\000)p Fy(1)1210 295 y FB(+)g Fz(\020)1906 243 y FB(\(8.9\))-84 403 y Fn(and)618 480 y Fz(B)655 487 y Ft(j)687 480 y FB(=)936 447 y Fz(\020)961 428 y Fq(\000)p Fy(1)1019 447 y Fx(\000)g Fz(\020)p 744 469 V 744 484 a Fr(\020)769 532 y Fz(z)802 516 y Fr(b)794 532 y FB(\001)835 539 y Ft(j)853 484 y Fr(\021)877 495 y Fq(\000)p Fy(1)936 532 y FB(+)g Fz(z)1018 516 y Fr(b)1010 532 y FB(\001)1051 539 y Ft(j)1079 532 y FB(+)g Fz(\020)1153 517 y Fq(\000)p Fy(1)1212 532 y FB(+)g Fz(\020)1291 480 y(:)577 b FB(\(8.10\))1969 652 y Fm(2)-84 781 y FC(Pro)r(of.)24 b FB(W)l(e)17 b(shall)g(pro)o(v)o(e)g(Lemma)e(8.1)j(b)o(y)f(induction.) 24 b(W)l(e)17 b(let)f Fz(j)j FB(=)c Fz(n)1265 788 y Fy(0)1303 781 y FB(and)i(observ)o(e)g(that,)h(b)o(y)f(m)o(ultiply)o(ing)-84 857 y(the)h(n)o(umerator)g(and)h(the)f(denominator)g(of)h(\(8.4\))g(b)o (y)959 796 y Fr(\022)990 857 y Fz(\020)1015 839 y Fq(\000)p Fy(\(1)p Fq(\000)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)1182 809 y Fr(\020)1207 857 y Fz(z)1240 841 y Fr(b)1232 857 y FB(\001)1273 864 y Ft(n)1294 869 y Fk(0)1313 809 y Fr(\021)1338 820 y Fq(\000)p Fy(1)1396 857 y FB(+)11 b Fz(\020)1470 839 y Fq(\000)p Fy(\(1+)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)1629 796 y Fr(\023)1668 809 y(.)1712 841 y(b)1701 857 y Fz(Z)1738 839 y Fy(+)1734 869 y Ft(n)1755 874 y Fk(0)1781 857 y FB(,)18 b(it)h(can)f(b)q(e)-84 932 y(written)e(as)405 1067 y Fr(e)394 1083 y Fz(Z)431 1065 y Ft(\033)427 1096 y(n)448 1101 y Fk(0)466 1096 y Fq(\000)p Fy(1)p 394 1107 120 2 v 405 1142 a Fr(b)394 1158 y Fz(Z)431 1141 y Ft(\033)427 1169 y(n)448 1174 y Fk(0)466 1169 y Fq(\000)p Fy(1)532 1119 y FB(=)589 1027 y Fr(\020)614 1075 y Fz(z)647 1059 y Fr(b)639 1075 y FB(\001)680 1082 y Ft(n)701 1087 y Fk(0)720 1027 y Fr(\021)745 1039 y Fq(\000)p Fy(1)803 1075 y Fx(\000)11 b Fz(z)886 1059 y Fr(b)878 1075 y FB(\001)919 1082 y Ft(n)940 1087 y Fk(0)970 1075 y FB(+)g Fz(\020)1044 1057 y Fq(\000)p Ft(\033)1106 1075 y Fx(\000)g Fz(\020)1181 1057 y Ft(\033)p 589 1107 616 2 v 590 1122 a Fr(\020)614 1170 y Fz(z)648 1154 y Fr(b)639 1170 y FB(\001)680 1177 y Ft(n)701 1182 y Fk(0)721 1122 y Fr(\021)746 1133 y Fq(\000)p Fy(1)804 1170 y FB(+)g Fz(z)886 1154 y Fr(b)878 1170 y FB(\001)919 1177 y Ft(n)940 1182 y Fk(0)970 1170 y FB(+)g Fz(\020)1044 1156 y Fq(\000)p Ft(\033)1106 1170 y FB(+)g Fz(\020)1180 1156 y Ft(\033)1223 1119 y FB(=)j Fz(A)1312 1126 y Ft(n)1333 1131 y Fk(0)1363 1119 y FB(+)d Fz(\033)r(B)1479 1126 y Ft(n)1500 1131 y Fk(0)1520 1119 y Fz(:)348 b FB(\(8.11\))-11 1283 y(No)o(w,)16 b(let)f Fz(j)i FB(=)d Fz(n)d FB(+)g(1.)22 b(Assuming)15 b(\(8.6\))h(v)m(alid,)g (\(8.4\))g(can)h(b)q(e)f(written)g(as)484 1374 y Fr(e)473 1390 y Fz(Z)510 1372 y Ft(\033)506 1402 y(n)p 473 1412 61 2 v 484 1447 a Fr(b)473 1463 y Fz(Z)510 1449 y Ft(\033)506 1475 y(n)553 1424 y FB(=)d Fx(A)644 1431 y Ft(n)p Fy(+1)724 1424 y FB(+)778 1390 y Fz(\020)803 1372 y Fy(\(1)p Fq(\000)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)961 1390 y Fx(\000)e Fz(\020)1036 1372 y Fy(\(1+)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)1184 1390 y Fz(z)1234 1374 y Fr(b)1225 1390 y FB(\001)1266 1397 y Ft(n)p Fy(+1)p 778 1412 557 2 v 778 1463 a Fz(\020)803 1449 y Fy(\(1)p Fq(\000)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)962 1463 y FB(+)g Fz(\020)1036 1449 y Fy(\(1+)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)1183 1463 y Fz(z)1233 1447 y Fr(b)1225 1463 y FB(\001)1266 1470 y Ft(n)p Fy(+1)1339 1424 y Fx(B)1372 1431 y Ft(n)p Fy(+1)1440 1424 y Fz(:)428 b FB(\(8.12\))-84 1582 y(Multiplying)14 b(the)i(n)o(umerator)g(and)g(the)g(denominator)g (b)o(y)g Fz(\020)1059 1564 y Fq(\000)p Fy(\(1)p Fq(\000)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)1226 1534 y Fr(\020)1251 1582 y Fz(z)1284 1566 y Fr(b)1276 1582 y FB(\001)1317 1589 y Ft(n)p Fy(+1)1385 1534 y Fr(\021)1409 1546 y Fq(\000)p Fy(1)1468 1582 y FB(+)11 b Fz(\020)1542 1564 y Fq(\000)p Fy(\(1+)p Ft(\033)q Fy(\))p Ft(=)p Fy(2)1700 1582 y FB(,)16 b(giv)o(es)646 1686 y Fr(e)635 1702 y Fz(Z)672 1684 y Ft(\033)668 1714 y(n)p 635 1724 61 2 v 646 1759 a Fr(b)635 1775 y Fz(Z)672 1761 y Ft(\033)668 1788 y(n)714 1736 y FB(=)e Fx(A)806 1743 y Ft(n)p Fy(+1)886 1736 y FB(+)d(\()p Fz(A)991 1743 y Ft(n)1025 1736 y FB(+)g Fz(\033)r(B)1141 1743 y Ft(n)1164 1736 y FB(\))d Fx(B)1224 1743 y Ft(n)p Fy(+1)-84 1874 y FB(whic)o(h,)15 b(in)h(view)f(of)i(\(8.7\))f(and)h (\(8.8\),)f(concludes)g(the)g(pro)q(of)h(of)g(Lemma)d(8.1.)p 1977 1955 30 30 v -11 2084 a(No)o(w)i(w)o(e)g(pro)q(ceed)g(with)g(the)g (pro)q(of)i(of)e(Theorem)f(1.9.)22 b(Using)16 b(\(8.6\))g(to)h(compute) e(\(8.5\))h(giv)o(es)441 2225 y Fx(h)p Fz(\033)488 2232 y Fy(0)507 2225 y Fz(\033)535 2232 y Ft(x)557 2225 y Fx(i)576 2240 y Ft(M)624 2225 y FB(\(1\))42 b(=)813 2191 y(\()o Fx(A)871 2198 y Fy(1)902 2191 y FB(+)11 b Fx(B)984 2198 y Fy(1)1003 2191 y FB(\))1042 2175 y Fr(b)1031 2191 y Fz(Z)1068 2171 y Fy(+)1064 2202 y(0)1108 2191 y Fx(\000)g Fz(z)f FB(\()p Fx(A)1250 2198 y Fy(1)1281 2191 y Fx(\000)h(B)1364 2198 y Fy(1)1383 2191 y FB(\))1421 2175 y Fr(b)1410 2191 y Fz(Z)1447 2171 y Fq(\000)1443 2202 y Fy(0)p 813 2213 665 2 v 1047 2248 a Fr(b)1036 2265 y Fz(Z)1073 2244 y Fy(+)1069 2275 y(0)1113 2265 y FB(+)g Fz(z)1198 2248 y Fr(b)1187 2265 y Fz(Z)1224 2244 y Fq(\000)1220 2275 y Fy(0)728 2415 y FB(=)42 b Fx(A)848 2422 y Fy(1)872 2382 y FB(1)11 b Fx(\000)g Fz(z)991 2366 y Fr(b)982 2382 y FB(\001)1023 2389 y Fy(0)p 872 2404 171 2 v 872 2455 a FB(1)h(+)f Fz(z)991 2439 y Fr(b)982 2455 y FB(\001)1023 2462 y Fy(0)1059 2415 y FB(+)g Fx(B)1141 2422 y Fy(1)1160 2415 y Fz(:)1882 2316 y FB(\(8.13\))-11 2570 y(The)16 b(one-p)q(oin)o(t)g(function)f(at)h Fz(x)g FB(can)f(b)q(e)h (analogously)h(represen)o(ted)d(b)o(y)h(equations)h(of)g(the)f(form)g (\(8.5\))h(and)-84 2630 y(\(8.13\).)22 b(It)16 b(can)g(th)o(us)g(b)q(e) h(written)e(as)483 2771 y Fx(h)p Fz(\033)530 2778 y Ft(x)552 2771 y Fx(i)572 2786 y Ft(M)619 2771 y FB(\(1\))f(=)780 2721 y Fr(e)769 2738 y Fz(Z)806 2717 y Fy(+)802 2748 y(0)846 2738 y FB(+)d Fz(z)947 2721 y Fr(e)936 2738 y Fz(Z)973 2717 y Fq(\000)969 2748 y Fy(0)p 752 2760 251 2 v 780 2795 a Fr(b)769 2811 y Fz(Z)806 2790 y Fy(+)802 2822 y(0)846 2811 y FB(+)g Fz(z)947 2795 y Fr(b)936 2811 y Fz(Z)973 2790 y Fq(\000)969 2822 y Fy(0)1021 2771 y FB(=)j Fx(A)1113 2778 y Fy(1)1144 2771 y FB(+)d Fx(B)1226 2778 y Fy(1)1250 2738 y FB(1)h Fx(\000)f Fz(z)1369 2721 y Fr(b)1361 2738 y FB(\001)1402 2745 y Fy(0)p 1250 2760 171 2 v 1250 2811 a FB(1)h(+)f Fz(z)1369 2795 y Fr(b)1360 2811 y FB(\001)1401 2818 y Fy(0)1426 2771 y Fz(:)442 b FB(\(8.14\))937 2930 y(56)p eop %%Page: 57 57 57 56 bop -11 79 a FB(Inserting)16 b(\(8.13\))h(and)g(\(8.14\))g(in)o (to)e(\(8.1\))i(and)g(taking)f(in)o(to)g(accoun)o(t)g(\(3.4\),)g (yields)554 253 y Fx(h)p Fz(\033)601 260 y Fy(0)621 253 y FB(;)8 b Fz(\033)671 260 y Ft(x)692 253 y Fx(i)712 268 y Ft(M)760 253 y FB(\(1\))14 b(=)888 168 y Fr(2)888 242 y(4)915 253 y FB(1)e Fx(\000)1000 180 y Fr( )1038 219 y FB(1)g Fx(\000)f Fz(z)1157 203 y Fr(b)1149 219 y FB(\001)1190 226 y Fy(0)p 1038 242 171 2 v 1038 293 a FB(1)h(+)f Fz(z)1157 276 y Fr(b)1148 293 y FB(\001)1189 300 y Fy(0)1214 180 y Fr(!)1247 186 y Fy(2)1266 168 y Fr(3)1266 242 y(5)1302 253 y Fx(B)1335 260 y Fy(1)1355 253 y Fz(:)513 b FB(\(8.15\))-11 404 y(F)l(or)11 b(an)o(y)g(giv)o(en)g (v)m(alue)g(of)g Fz(\030)j FB(the)d(truncated)g(t)o(w)o(o-p)q(oin)o(t)g (function)g(is)g(still)f(giv)o(en)g(b)o(y)h(\(8.15\))h(pro)o(vided)e (that,)i(for)-84 464 y(eac)o(h)f Fz(B)58 471 y Ft(j)87 464 y FB(in)g Fx(B)172 471 y Fy(1)191 464 y FB(,)226 448 y Fr(b)217 464 y FB(\001)258 471 y Ft(j)287 464 y FB(is)g(replaced)g(b)o(y)f(the)h(random)g(v)m(ariables)g(\001)1072 471 y Ft(j)1101 464 y FB(and)h(the)f(v)m(ariable)g Fz(\020)k FB(is)c(replaced)g(b)o(y)f Fz(\020)1799 471 y Ft(j)1832 464 y FB(=)j Fz(e)1906 446 y Fq(\000)p Fy(2)p Ft(\014)r(\030)1989 451 y Fl(j)-84 524 y FB(\(see)19 b(remark)e(after)i(Prop)q(osition)h (3.1\).)30 b(In)18 b(the)h(thermo)q(dynamic)d(limit,)h Fz(M)24 b Fx(!)18 b(1)p FB(,)h(the)g(random)g(v)m(ariables)-84 584 y(\001)-43 591 y Ft(j)-25 584 y 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Fz(\020)1107 1535 y Fq(\000)p Fy(1)1165 1554 y Fx(\000)11 b Fz(\020)p 971 1576 381 2 v 971 1621 a(x)999 1607 y Fq(\000)p Fy(1)1057 1621 y FB(+)g Fz(x)g FB(+)g Fz(\020)1219 1607 y Fq(\000)p Fy(1)1277 1621 y FB(+)g Fz(\020)1356 1587 y(:)512 b FB(\(8.18\))-84 1698 y(Using)16 b(\(8.16\))h(in)f(the)g(exp)q(ected)f(v)m(alue)h(of)h (\(8.15\))g(giv)o(es)19 1847 y Fs(E)49 1854 y Ft(\030)79 1847 y Fx(h)q Fz(\033)127 1854 y Fy(0)146 1847 y FB(;)24 b Fz(\033)212 1854 y Ft(x)234 1847 y Fx(i)253 1862 y Ft(M)301 1847 y FB(\()p Fz(\030)r FB(\))42 b(=)f Fz(p)507 1854 y Fy(1)536 1847 y Fz(:)8 b(:)g(:)g(p)626 1854 y Ft(n)647 1859 y Fk(0)681 1847 y Fs(E)711 1854 y Ft(\030)728 1845 y Fg(0)744 1847 y Fz(s)p FB(\()p Fz(z)r FB(\001)852 1854 y Fy(0)871 1847 y FB(\))915 1762 y Fr(2)915 1836 y(4)954 1792 y Ft(n)975 1797 y Fk(0)946 1806 y Fr(Y)942 1897 y Ft(j)r Fy(=1)1012 1847 y Fz(t)p FB(\()p Fz(z)r FB(\001)1115 1854 y Ft(j)1132 1847 y FB(\))1151 1762 y Fr(3)1151 1836 y(5)404 2057 y FB(=)485 2003 y Ft(M)t Fq(\000)p Fy(1)496 2015 y Fr(X)483 2108 y Ft(k)q Fy(=)p Ft(n)550 2113 y Fk(0)577 2057 y Fz(a)603 2064 y Ft(k)640 2057 y Fz(s)p FB(\()p Fz(w)717 2064 y Ft(k)739 2057 y FB(\))782 1971 y Fr(2)782 2046 y(4)821 2002 y Ft(n)842 2007 y Fk(0)814 2015 y Fr(Y)810 2106 y Ft(j)r Fy(=1)879 2057 y Fz(t)p FB(\()p Fz(w)951 2064 y Ft(k)q Fq(\000)p Ft(j)1016 2057 y FB(\))1035 1971 y Fr(3)1035 2046 y(5)1073 2057 y FB(+)j Fz(p)1146 2064 y Fy(1)1175 2057 y Fz(:)d(:)g(:)f(p)1264 2064 y Ft(M)1321 2057 y Fz(s)p FB(\()p Fz(w)1398 2064 y Ft(M)1437 2057 y FB(\))1481 1971 y Fr(2)1481 2046 y(4)1520 2002 y Ft(n)1541 2007 y Fk(0)1512 2015 y Fr(Y)1508 2106 y Ft(j)r Fy(=1)1578 2057 y Fz(t)p FB(\()p Fz(w)1650 2064 y Ft(M)t Fq(\000)p Ft(j)1732 2057 y FB(\))1751 1971 y Fr(3)1751 2046 y(5)1882 1952 y FB(\(8.19\))-84 2212 y(where)20 b Fz(w)96 2219 y Ft(n)140 2212 y FB(=)h Fz(z)r(\034)251 2194 y Fy(\()p Ft(n)p Fy(\))245 2224 y Ft(z)302 2212 y FB(\(1\))p Fz(;)8 b(n)21 b Fx(2)g Fs(N)p FB(.)34 b(Here,)20 b(w)o(e)g(ha)o(v)o(e)g(\014rst)g(tak)o(en)g(partial)h(exp)q(ectations)f (with)g(resp)q(ect)g(to)h(the)-84 2272 y(v)m(ariables)d Fz(\030)141 2279 y Fy(1)161 2272 y Fz(;)8 b(:)g(:)g(:)f(\030)269 2279 y Ft(n)290 2284 y Fk(0)328 2272 y FB(and)19 b(use)f(the)f(fact)h (that)h(the)e(sequence)g(\001)1132 2279 y Ft(j)1150 2272 y FB(,)h Fz(j)i Fx(2)d Fs(N)p FB(,)h(satis\014es)h(the)e(recurrence)g (relation)-84 2332 y Fz(z)r FB(\001)-18 2339 y Ft(j)r Fq(\000)p Fy(1)66 2332 y FB(=)22 b Fz(z)r(h)179 2339 y Ft(j)197 2332 y FB(\()p Fz(z)r FB(\001)282 2339 y Ft(j)300 2332 y FB(\))e(with)h Fz(h)483 2339 y Ft(j)501 2332 y FB(\(1\))h(=)g(1)f(in)f(the)h(remaining)e(exp)q(ectations.)34 b(Equation)22 b(\(1.16\))f(then)g(follo)o(ws)-84 2392 y(since)15 b Fz(s)p FB(\()p Fz(x)p FB(\))g(and)h Fz(t)p FB(\()p Fz(x)p FB(\))f(are)g(con)o(tin)o(uous)g(in)g Fs(R)750 2399 y Fy(+)792 2392 y FB(and)h Fz(w)921 2399 y Ft(n)960 2392 y FB(con)o(v)o(erges)f(to)h(the)f(solution)h Fz(w)f FB(=)f Fz(w)q FB(\()p Fz(\020)t(;)8 b(z)r FB(\))15 b(of)h(the)f(\014xed)-84 2453 y(p)q(oin)o(t)h(equation)g Fz(w)g FB(=)d Fz(z)r(h)p FB(\()p Fz(w)q FB(\))k(in)f(this)g(domain)f (pro)o(vided)h Fz(z)f Fx(2)f Fs(R)1138 2460 y Fy(+)1165 2453 y FB(.)-11 2534 y(This)i(concludes)g(the)g(pro)q(of)i(of)e (Theorem)f(1.9.)p 1977 2616 30 30 v -11 2745 a(W)l(e)g(no)o(w)h(turn)g (to)f(the)h(pro)q(of)g(of)g(Theorem)e(1.11)i(on)g(the)f(quenc)o(hed)f (susceptibilit)o(y)g(at)h(origin)h Fz(\037)p FB(.)k(W)l(e)15 b(note)-84 2805 y(the)h(follo)o(wing)g(facts)g(on)h(the)f(function)g Fz(s)g FB(and)h Fz(t)f FB(\(the)g(pro)q(of)h(will)e(b)q(e)i(omitted\):) 937 2930 y(57)p eop %%Page: 58 58 58 57 bop -84 79 a FC(Prop)r(osition)18 b(8.2)24 b Fn(The)f(function)i Fz(s)f FB(:)g Fz(w)i Fx(2)f Fs(R)861 86 y Fy(+)912 79 y Fx(7\000)-9 b(!)25 b Fz(s)p FB(\()p Fz(w)q FB(\))f Fx(2)h Fs(R)1235 86 y Fy(+)1285 79 y Fn(and)e Fz(t)i FB(:)f(\()p Fz(\020)t(;)8 b(w)q FB(\))24 b Fx(2)h FB([0)p Fz(;)8 b FB(1])15 b Fx(\002)g Fs(R)1875 86 y Fy(+)1926 79 y Fx(7\000)-8 b(!)-84 139 y Fz(t)p FB(\()p Fz(\020)t(;)8 b(w)q FB(\))27 b Fx(2)g Fs(R)181 146 y Fy(+)232 139 y Fn(given)f(by)e(\(8.17\))h(and)f(\(8.18\),)i(r)n(esp)n(e)n(ctively,)g (have)g(a)e(maximum)h(value)g(at)g Fz(w)j FB(=)f(1)e Fn(with)-84 200 y Fz(s)p FB(\(1\))c(=)f(1)h Fn(and)h Fz(t)p FB(\()p Fz(\020)t(;)8 b FB(1\))20 b(=)g(\(1)14 b Fx(\000)g Fz(\020)t FB(\))p Fz(=)p FB(\(1)g(+)g Fz(\020)t FB(\))p Fn(,)22 b(ar)n(e)e(monotonic)n(al)r(ly)j(incr)n(e)n(asing)e (function)h(of)f Fz(w)h Fn(in)g FB([0)p Fz(;)8 b FB(1])20 b Fn(and)-84 260 y(satisfy)d Fz(s)p FB(\()p Fz(w)q FB(\))d(=)g Fz(s)p FB(\()p Fz(w)309 242 y Fq(\000)p Fy(1)356 260 y FB(\))j Fn(and)h Fz(t)p FB(\()p Fz(\020)t(;)8 b(w)q FB(\))13 b(=)h Fz(t)p FB(\()p Fz(\020)t(;)8 b(w)811 242 y Fq(\000)p Fy(1)858 260 y FB(\))p Fn(.)1969 391 y Fm(2)-11 520 y FB(F)l(or)15 b Fz(z)h Fx(2)e FB([0)p Fz(;)8 b FB(1\))15 b(w)o(e)g(recall)e(that)j Fz(w)618 527 y Ft(n)641 520 y Fz(;)8 b(n)14 b Fx(2)g Fs(N)p FB(,)i(is)e(a)i(monotonically)d 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y(4)918 1704 y Ft(n)902 1717 y Fr(Y)898 1808 y Ft(j)r Fy(=1)968 1758 y Fz(t)p FB(\()p Fz(w)1040 1765 y Ft(k)q Fq(\000)p Ft(j)1104 1758 y FB(\))1123 1673 y Fr(3)1123 1748 y(5)1162 1758 y FB(+)11 b Fz(a)16 b(s)p FB(\()p Fz(w)q FB(\))1374 1704 y Fq(1)1361 1717 y Fr(X)1358 1807 y Ft(n)p Fy(=0)1449 1758 y FB([2)p Fz(t)p FB(\()p Fz(w)q FB(\)])1592 1734 y Ft(n)379 1968 y FB(=)473 1914 y Fq(1)460 1926 y Fr(X)458 2019 y Ft(k)q Fy(=0)531 1968 y Fz(a)557 1975 y Ft(k)594 1968 y Fz(s)p FB(\()p Fz(w)671 1975 y Ft(k)692 1968 y FB(\))743 1914 y Ft(k)723 1926 y Fr(X)719 2017 y Ft(n)p Fy(=0)810 1968 y FB(2)834 1947 y Ft(n)867 1882 y Fr(2)867 1957 y(4)914 1914 y Ft(n)898 1926 y Fr(Y)894 2018 y Ft(j)r Fy(=1)964 1968 y Fz(t)p FB(\()p Fz(w)1036 1975 y Ft(k)q Fq(\000)p Ft(j)1100 1968 y FB(\))1119 1882 y Fr(3)1119 1957 y(5)1158 1968 y FB(+)1243 1934 y Fz(a)g(s)p FB(\()p Fz(w)q FB(\))p 1212 1956 202 2 v 1212 2002 a(1)11 b Fx(\000)g FB(2)p Fz(t)p FB(\()p Fz(w)q FB(\))1418 1968 y Fz(:)1882 1863 y FB(\(8.22\))-11 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FB(=)d(1)h(for)h(an)o(y)-84 2425 y Fz(\020)e Fx(2)c FB([0)p Fz(;)8 b FB(1])16 b(since,)f(b)o(y)h(the)g(dominated)f (con)o(v)o(ergence)f(theorem,)723 2562 y(lim)720 2592 y Ft(x)p Fq(\045)p Fy(1)816 2508 y Fq(1)803 2521 y Fr(X)801 2613 y Ft(k)q Fy(=0)879 2529 y FB(1)d Fx(\000)g Fz(x)992 2511 y Ft(k)p 879 2551 135 2 v 889 2596 a FB(1)h Fx(\000)f Fz(x)1026 2562 y(a)1052 2569 y Ft(k)1087 2562 y Fz(<)j Fx(1)p Fz(:)-11 2723 y FB(This)i(concludes)g(the)g(pro)q(of)i(of)e (Theorem)f(1.11.)p 1977 2805 30 30 v 937 2930 a(58)p eop %%Page: 59 59 59 58 bop -11 79 a FC(Ac)n(kno)n(wledgmen)n(ts.)20 b FB(W)l(e)15 b(are)g(indebted)f(to)i(C.)f(Bonato,)g(P)l(.)g(L.)g (Christiano,)g(H.)f(v)o(on)h(Dreifus,)g(S.)f(R.)h(de)-84 139 y(A.)g(Salinas)i(and)g(W.)f(F.)f(W)l(reszinski)g(for)i(suggestions) g(and)g(stim)o(ulating)e(discussions.)-84 327 y FA(References)-84 458 y FB([A)-5 b(W])50 b(M)23 b(Aizenman)e(and)j(J)f(W)l(ehr.)g (\\Rounding)h(E\013ects)g(of)g(Quenc)o(hed)e(Randomness)h(on)h (First-Order)76 518 y(Phase)17 b(T)l(ransitions".)f(Comm)o(un.)e(Math.) i(Ph)o(ys.)g FC(130)p FB(,)g(489-528)i(\(1990\).)-84 619 y([DKP])24 b(H)15 b(v)o(on)g(Dreifus,)f(A)h(Klein)f(and)i(J)f(F)g (P)o(erez.)f(\\T)l(aming)h(Gri\016ths)g(Singularities:)20 b(In\014nite)14 b(Di\013eren)o(tia-)76 680 y(bilit)o(y)g(of)j(the)f (Correlations".)g(Comm)o(un.)e(Math.)i(Ph)o(ys.)g FC(170)p FB(,)f(21-39)j(\(1995\).)-84 781 y([F])100 b(J)15 b(F)l(r\177)-24 b(ohlic)o(h.)13 b(\\Mathematical)h(Asp)q(ects)g(of)i(the)f(Ph)o(ysics)f (of)h(Disordered)g(Systems")f(in)h(\\Critical)f(Phe-)76 842 y(nomena,)24 b(Random)e(System)g(and)i(Gauge)g(Theories")f(edited)g (b)o(y)f(K)i(Osterw)o(alder)e(and)i(R)f(Stora.)76 902 y(Elsevier,)14 b(Amsterdam.)f(\(1986\).)-84 1003 y([F)l(e])82 b(William)13 b(F)l(eller.)g(\\An)i(In)o(tro)q(duction)g(to)h (probabilit)o(y)f(Theory)g(and)h(its)f(Applications",)g(V)l(ol.)f(2.)i (John)76 1064 y(Wiley)f(&)h(Sons.)g(Second)h(edition,)e(1971.)-84 1165 y([G])94 b(Rob)q(ert)22 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b(F)f(P)o(erez.)g(\\Con)o(trolling)h(the)g(E\013ect)f (of)i(Gri\016ths')e(Singularities)g(in)g(Random)h(F)l(erromagnets:)76 1873 y(Smo)q(othness)16 b(of)h(the)f(Magnetization".)g(Brazilian)f (Journal)i(of)f(Ph)o(ysics)g FC(23)p FB(,)g(356-362)i(\(1993\).)-84 1975 y([S])105 b(Andr\023)-24 b(as)16 b(S)q(\177)-26 b(ut})i(o.)17 b(\\W)l(eak)f(Singularit)o(y)f(and)i(Absence)e(of)h (Metastabilit)o(y)e(in)i(Random)f(Ising)h(F)l(erromag-)76 2035 y(nets".)g(J.)g(Ph)o(ys.)g(A:)f(Math.)h(Gen.)g FC(15)g FB(L749-L752)k(\(1982\).)-84 2137 y([T])97 b(E)22 b(C)h(Titc)o(hmarsh.) d(\\The)i(Theory)h(of)f(F)l(unctions".)g(Oxford)g(Univ)o(ersit)o(y)e (Press.)i(Second)g(Edition,)76 2197 y(1939.)937 2930 y(59)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF