%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %% 513 K, Postscript, 20 pages, 2 figures BODY %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: bg1.dvi %%Pages: 20 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentPaperSizes: A4 %%EndComments %DVIPSCommandLine: dvips bg1 %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1998.05.14:0940 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N 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b Fh(Consider)h(the)h(standar)-5 b(d)48 b(map)53 b Fq(\(1.1\))47 b Fh(with)g Fo(!)52 b Fq(=)c Fo(p=q)33 b Fq(+)d Fo(i\021)s Fh(,)49 b Fo(p;)15 b(q)52 b Fl(2)d Fm(Z)p Fh(,)270 4345 y Fo(p)20 b Fl(^)g Fo(q)28 b Fq(=)d(1)33 b Fh(and)g Fo(\021)c Fl(2)c Fm(R)s Fh(.)47 b(Then)33 b(the)g(fol)5 b(lowing)34 b(r)-5 b(esults)33 b(hold.)370 4494 y Fq(1.)42 b Fh(F)-7 b(or)44 b(\014xe)-5 b(d)43 b Fo(\021)k Fl(6)p Fq(=)c(0)g Fh(the)g(function)f Fo(u)p Fq(\()p Fo(\013;)15 b(")p Fq(\))p Fh(,)47 b(de\014ne)-5 b(d)44 b(by)50 b Fq(\(1.2\))r Fh(,)45 b(is)e(divisible)f(by)h Fo(")482 4623 y Fh(and)32 b(jointly)g(analytic)h(in)e Fq(\()p Fo(\013;)15 b(")p Fq(\))33 b Fh(in)f(the)f(pr)-5 b(o)g(duct)34 b(of)d(a)h(strip)h(ar)-5 b(ound)33 b(the)e(r)-5 b(e)g(al)33 b(axis)482 4753 y(in)38 b(the)g(c)-5 b(omplex)40 b Fo(\013)e Fh(plane)h(and)h(a)e(neighb)-5 b(orho)g(o)g(d)42 b Fl(j)p Fo(")p Fl(j)36 b Fo(<)e(")2536 4767 y Fi(0)2614 4753 y Fh(of)39 b(the)f(origin)h(in)f(the)482 4882 y(c)-5 b(omplex)34 b Fo(")f Fh(plane,)g(with)h Fo(")1410 4896 y Fi(0)1475 4882 y Fq(=)25 b Fo(O)s Fq(\()p Fo(\021)1726 4849 y Fi(2)p Fn(=q)1835 4882 y Fq(\))p Fh(.)p eop %%Page: 6 6 6 5 bop 390 46 a Fi(6)739 b(ALBER)-6 b(TO)30 b(BERRETTI)h(AND)f(GUIDO)h (GENTILE)490 241 y Fq(2.)42 b Fh(The)33 b(function)h Fo(v)s Fq(\()p Fo(\013;)15 b(")p Fq(\))29 b(=)d Fo(u)p Fq(\()p Fo(\013;)15 b Fq(\(2)p Fo(\031)s(\021)s Fq(\))1933 208 y Fi(2)p Fn(=q)2045 241 y Fo(")p Fq(\))34 b Fh(is)f(wel)5 b(l)34 b(de\014ne)-5 b(d)34 b(for)g Fo(\021)c Fl(!)d Fq(0)33 b Fh(and)i(c)-5 b(on-)602 370 y(ver)g(ges)33 b(to)h(a)g(function)39 b Fq(\026)-51 b Fo(u)p Fq(\()p Fo(\013;)15 b(")p Fq(\))p Fh(,)36 b(divisible)d(by)g Fo(")2257 337 y Fn(q)2329 370 y Fh(and)h(analytic)h(in)e Fo(")3005 337 y Fn(q)3076 370 y Fh(in)h(a)f(neigh-)602 500 y(b)-5 b(orho)g(o)g(d)44 b(of)e(the)f(origin,)j(which)e(is)f Fq(2)p Fo(\031)s(=q)k Fh(p)-5 b(erio)g(dic)42 b(and)h(solves)e(the)h (di\013er)-5 b(ential)602 629 y(e)g(quation:)1282 776 y(d)1329 743 y Fi(2)1374 776 y Fq(\026)-51 b Fo(u)p Fq(\()p Fo(\013)p Fq(\))p 1282 817 268 4 v 1343 900 a Fh(d)q Fo(\013)1448 874 y Fi(2)1584 838 y Fq(=)25 b Fo(C)1745 856 y Fn(p=q)1854 838 y Fo(")1896 800 y Fn(q)1950 838 y Fq(sin)14 b(\()p Fo(q)s Fq(\()p Fo(\013)21 b Fq(+)26 b(\026)-51 b Fo(u)p Fq(\()p Fo(\013)p Fq(\)\)\))667 b(\(1.11\))602 1022 y Fh(with)33 b(b)-5 b(oundary)35 b(c)-5 b(onditions)40 b Fq(\026)-51 b Fo(u)p Fq(\(0\))27 b(=)j(\026)-50 b Fo(u)p Fq(\(2)p Fo(\031)s Fq(\))26 b(=)f(0)p Fh(,)33 b(for)g(some)h(c)-5 b(onstant)34 b Fo(C)3170 1041 y Fn(p=q)3279 1022 y Fh(.)490 1216 y Fq(This)28 b(theorem)j(will)d(b)s(e)h(pro)m(v)m(ed)i(through)f (a)h(series)e(of)i(lemmata.)925 1658 y(2.)51 b Fp(Estima)-6 b(tes)33 b(on)h(the)f(Radius)h(of)g(Conver)n(gence)490 1841 y Fq(The)29 b(small)g(divisor)g Fo(\015)5 b Fq(\()p Fo(\027)h Fq(\))30 b(de\014ned)g(b)m(y)h(\(1.6\))h(satis\014es)e(the)h (b)s(ound:)1226 2104 y Fl(j)p Fo(\015)5 b Fq(\()p Fo(\027)h Fq(\))p Fl(j)26 b(\025)1571 1917 y Fk(8)1571 1999 y(<)1571 2163 y(:)1652 2034 y Fo(c)p Fl(j)p Fo(\027)6 b(\021)s Fl(j)1840 2001 y Fi(2)1880 2034 y Fo(;)91 b Fq(for)30 b Fo(\027)h Fl(2)25 b Fo(q)s Fm(Z)16 b Fl(n)21 b(f)p Fq(0)p Fl(g)p Fo(;)1652 2190 y(cq)1735 2157 y Fj(\000)p Fi(2)1829 2190 y Fo(;)142 b Fq(otherwise,)3323 2104 y(\(2.1\))390 2362 y(for)30 b(some)h(p)s(ositiv)m(e)e(constan)m(t)j Fo(c)p Fq(,)f(as)f(it)g(can)h(b)s(e)f(easily)f(c)m(hec)m(k)m(ed)k(from) d(the)g(inequalities:)1123 2570 y Fl(j)15 b Fq(cos)h Fo(z)25 b Fl(\000)20 b Fq(1)p Fl(j)26 b(\025)1660 2509 y(j)p Fo(z)t Fl(j)1756 2476 y Fi(2)p 1660 2549 137 4 v 1705 2632 a Fq(4)1806 2570 y Fo(;)106 b Fq(for)30 b Fl(j)15 b Fq(Re)h Fo(z)t Fl(j)26 b(\024)2442 2509 y Fo(\031)p 2442 2549 56 4 v 2447 2632 a Fq(4)2507 2570 y Fo(;)1123 2810 y Fl(j)15 b Fq(cos)h Fo(z)25 b Fl(\000)20 b Fq(1)p Fl(j)26 b(\025)1660 2749 y(j)15 b Fq(Im)g Fo(z)t Fl(j)1895 2716 y Fi(2)p 1660 2790 276 4 v 1775 2873 a Fq(2)1945 2810 y Fo(;)1123 3034 y Fl(j)g Fq(cos)h Fo(z)25 b Fl(\000)20 b Fq(1)p Fl(j)26 b(\025)1660 2972 y Fq(1)p 1660 3013 46 4 v 1660 3096 a(2)1715 3034 y Fo(;)106 b Fq(for)31 b Fl(j)15 b Fq(Re)g Fo(z)35 b Fq(mo)s(d)30 b(2)p Fo(\031)s Fl(j)c(\025)2687 2972 y Fo(\031)p 2687 3013 56 4 v 2692 3096 a Fq(4)2752 3034 y Fo(;)3323 2787 y Fq(\(2.2\))390 3225 y(holding)c(for)h(an)m(y)h(complex)f Fo(z)t Fq(;)k(for)c(example,) i(one)f(can)g(tak)m(e)i Fo(c)f Fq(=)g(min)o Fl(f)p Fo(q)2871 3192 y Fi(2)2910 3225 y Fo(=)p Fq(2)p Fo(;)15 b Fq(2)p Fo(\031)3140 3192 y Fi(2)3182 3225 y Fl(g)25 b(\025)g Fq(1)p Fo(=)p Fq(2.)490 3355 y(Then,)36 b(giv)m(en)h(a)f(tree)h Fo(#)p Fq(,)h(w)m(e)e(can)h(asso)s(ciate)g(to)g(eac)m(h)g(line)e Fo(`)h Fq(of)g Fo(#)g Fq(a)g(scale)h(lab)s(el)d Fo(n)3452 3370 y Fn(`)3485 3355 y Fq(,)390 3484 y(setting)k Fo(n)751 3499 y Fn(`)822 3484 y Fq(=)g(0)g(if)f(its)h(momen)m(tum)g Fo(\027)1770 3499 y Fn(`)1841 3484 y Fq(is)f(a)h(m)m(ultiple)e(of)j Fo(q)s Fq(,)h(and)d Fo(n)2847 3499 y Fn(`)2918 3484 y Fq(=)h(1)g(otherwise.)390 3614 y(Giv)m(en)e(a)h(tree)f Fo(#)p Fq(,)i(a)e Fh(cluster)47 b Fo(T)h Fq(of)37 b Fo(#)f Fq(is)f(a)h(maximal)f(connected)i(set)g(of)f(lines)f(on)h(scale)390 3743 y Fo(n)42 b Fq(=)g(1;)47 b(w)m(e)41 b(shall)e(sa)m(y)i(that)h(suc) m(h)e(lines)f(are)i Fh(internal)51 b Fq(to)42 b Fo(T)13 b Fq(,)43 b(and)d(write)g(sometimes)390 3873 y Fo(`)33 b Fl(2)f Fo(T)13 b Fq(.)55 b(The)34 b(lines)g(outside)g(the)h(clusters) f(are)i(all)e(on)h(scale)g Fo(n)d Fq(=)h(0,)k(and)d(eac)m(h)j(cluster) 390 4002 y(has)c(an)f(arbitrary)g(n)m(um)m(b)s(er)f Fo(p)1452 4016 y Fn(T)1536 4002 y Fl(\025)e Fq(0)34 b(of)f(en)m(tering)f(lines)f (but)h(only)g(one)h(exiting)f(line.)47 b(A)390 4132 y(no)s(de)35 b Fo(u)g Fq(will)d(b)s(e)j(considered)f(in)m(ternal)g(to)i Fo(T)13 b Fq(,)36 b(and)f(w)m(e)h(shall)d(write)i Fo(u)e Fl(2)g Fo(T)13 b Fq(,)36 b(if)f(at)h(least)390 4261 y(the)31 b(exiting)e(line)g(or)h(one)h(of)f(its)g(en)m(tering)g(lines)f(is)g(in) g Fo(T)13 b Fq(.)490 4391 y(A)30 b(cluster)g Fo(V)50 b Fq(will)28 b(b)s(e)i(called)f(a)i Fh(r)-5 b(esonanc)g(e)39 b Fq(if:)1728 4479 y Fk(X)1722 4676 y Fn(u)p Fj(2)p Fn(V)1881 4566 y Fo(\027)1926 4580 y Fn(u)1997 4566 y Fq(=)25 b(0)p Fo(;)1160 b Fq(\(2.3\))390 4806 y(and,)35 b(in)d(suc)m(h)i(a)g(case,)j (the)d(exiting)f(line)g(of)h(the)g(cluster)f Fo(V)55 b Fq(will)31 b(b)s(e)i(called)h(a)g Fh(r)-5 b(esonant)390 4936 y(line)7 b Fq(;)30 b(w)m(e)h(also)g(denote)g(with)e Fo(k)1460 4950 y Fn(V)1551 4936 y Fq(the)i(n)m(um)m(b)s(er)e(of)h(no)s (des)g(in)m(ternal)f(to)i Fo(V)20 b Fq(.)p eop %%Page: 7 7 7 6 bop 955 46 a Fi(SCALING)31 b(PR)n(OPER)-6 b(TIES)31 b(OF)f(THE)h(ST)-6 b(AND)n(ARD)30 b(MAP)649 b(7)370 241 y Fq(Giv)m(en)22 b(a)g(resonance)h Fo(V)d Fq(,)k(with)d(resonan)m(t)i (line)d Fo(`)1980 255 y Fn(V)2041 241 y Fq(,)k(w)m(e)f(can)f(de\014ne)f (its)h Fh(r)-5 b(esonanc)g(e)27 b(factor)270 370 y Fl(V)326 384 y Fn(V)387 370 y Fq(\()p Fo(#)p Fq(\))j(as:)535 606 y Fl(V)591 620 y Fn(V)652 606 y Fq(\()p Fo(#)p Fq(\))25 b(=)g Fo(\015)5 b Fq(\()p Fo(\027)1029 621 y Fn(`)1058 632 y Fe(V)1116 606 y Fq(\))1181 519 y Fk(Y)1166 716 y Fn(u)p Fj(2)p Fn(V)1372 544 y Fq(1)p 1336 585 117 4 v 1336 668 a Fo(p)1382 682 y Fn(u)1427 668 y Fq(!)1482 544 y Fo(\027)1533 500 y Fn(p)1569 508 y Fe(u)1609 500 y Fi(+1)1527 556 y Fn(u)p 1472 585 242 4 v 1472 668 a Fo(\015)g Fq(\()p Fo(\027)1604 683 y Fn(`)1633 691 y Fe(u)1678 668 y Fq(\))1749 606 y(=)1845 450 y Fk(")1912 519 y(Y)1898 716 y Fn(u)p Fj(2)p Fn(V)2068 544 y Fo(\027)2119 500 y Fn(p)2155 508 y Fe(u)2195 500 y Fi(+1)2113 556 y Fn(u)p 2068 585 222 4 v 2120 668 a Fo(p)2166 682 y Fn(u)2211 668 y Fq(!)2299 450 y Fk(#)15 b(")2429 519 y(Y)2420 717 y Fn(`)p Fj(2)p Fn(V)2676 544 y Fq(1)p 2578 585 242 4 v 2578 668 a Fo(\015)5 b Fq(\()p Fo(\027)2710 683 y Fn(`)2739 691 y Fe(u)2784 668 y Fq(\))2830 450 y Fk(#)2898 606 y Fq(;)280 b(\(2.4\))270 847 y(of)34 b(course)f(the)h(resonance)g(factor)g(will)d(dep)s(end)g(on)i Fo(#)h Fq(only)e(through)h(the)g(momen)m(ta)i(of)270 977 y(the)j(incoming)f(lines)g(of)h Fo(V)59 b Fq(and)37 b(on)h(the)h(mo)s(de)f(lab)s(els)e Fo(\027)2314 991 y Fn(u)2359 977 y Fq('s)i(of)h(the)f(no)s(des)g(inside)e(the)270 1106 y(resonance)f Fo(V)20 b Fq(.)53 b(The)34 b(factor)i Fo(\015)5 b Fq(\()p Fo(\027)1428 1121 y Fn(`)1457 1132 y Fe(V)1515 1106 y Fq(\))34 b(in)g(\(2.4\))i(simply)c(cancels)j(the)f (propagator)i(in)d(the)270 1236 y(pro)s(duct)c(corresp)s(onding)g(to)i (the)f(exiting)g(line)e(\(whic)m(h)i(is)f(the)i(resonan)m(t)g(line)d Fo(`)3055 1250 y Fn(V)3116 1236 y Fq(\).)370 1365 y(Note)j(that,)h(b)m (y)f(\(2.1\))r(,)f(w)m(e)h(ha)m(v)m(e)h(the)e(b)s(ound:)1422 1540 y Fl(j)q(V)1504 1554 y Fn(V)1564 1540 y Fq(\()p Fo(#)p Fq(\))p Fl(j)c(\024)f Fo(c)1874 1502 y Fj(\000)p Fn(k)1966 1513 y Fe(V)2024 1540 y Fo(q)2068 1502 y Fi(2)p Fn(k)2140 1513 y Fe(V)2197 1540 y Fo(;)981 b Fq(\(2.5\))270 1715 y(as)31 b(all)e(lines)f(inside)h Fo(V)50 b Fq(are)31 b(on)f(scale)h Fo(n)25 b Fq(=)g(1.)370 1844 y(If)37 b Fo(N)541 1858 y Fn(n)588 1844 y Fq(\()p Fo(#)p Fq(\))i(,)i Fo(n)d Fq(=)g(0)p Fo(;)15 b Fq(1,)42 b(denotes)c(the)h(n)m(um)m(b)s(er) e(of)h(lines)f(in)g Fo(#)h Fq(on)g(scale)h Fo(n)p Fq(,)h(w)m(e)f(ha)m (v)m(e)270 1974 y(trivially)28 b(using)h(again)h(\(2.1\))i(that)f(for)f (a)h(\014xed)f(tree)h Fo(#)p Fq(:)1306 2149 y Fl(j)q Fq(V)-8 b(al\()p Fo(#)p Fq(\))p Fl(j)26 b(\024)f Fo(c)1773 2111 y Fj(\000)p Fn(k)1871 2149 y Fo(q)1915 2111 y Fi(2)p Fn(N)2006 2120 y Ff(1)2045 2149 y Fl(j)p Fo(\021)s Fl(j)2143 2111 y Fj(\000)p Fi(2)p Fn(N)2289 2120 y Ff(0)2328 2149 y Fo(:)850 b Fq(\(2.6\))270 2324 y(Ho)m(w)m(ev)m(er,)50 b(ev)m(en)45 b(if)e(the)i(b)s(ound)d(\(2.6\))k(cannot)f(b)s(e)e(impro)m (v)m(ed)h Fh(for)i(a)f(single)h(tr)-5 b(e)g(e)p Fq(,)48 b(w)m(e)270 2453 y(shall)33 b(see)i(that)h(b)m(y)e(p)s(erforming)f(a)i (partial)f(resummation)f(a)i(cancellation)g(mec)m(hanism)270 2583 y(app)s(ears,)29 b(allo)m(wing)e(us)i(to)g(obtain,)g(for)g(the)g (co)s(e\016cien)m(ts)h(\(1.8\))h(of)e(the)g(Lindstedt)e(series,)270 2712 y(a)k(b)s(etter)f(estimate,)i(and)d(th)m(us)h(leading)g(to)h(the)f (pro)s(of)g(of)g(the)h(theorem.)270 2906 y Fv(Lemma)i(1.)45 b Fh(L)-5 b(et)42 b Fo(N)1016 2873 y Fj(\003)1006 2931 y Fi(0)1056 2906 y Fq(\()p Fo(#)p Fq(\))g Fh(b)-5 b(e)41 b(the)h(numb)-5 b(er)42 b(of)g(lines)f(on)h(sc)-5 b(ale)43 b Fo(n)d Fq(=)i(0)f Fh(which)i(ar)-5 b(e)42 b(not)270 3036 y(r)-5 b(esonant,)34 b(in)f(a)g(given)f(tr)-5 b(e)g(e)33 b Fo(#)p Fh(.)42 b(We)32 b(then)h(have)g(the)g(b)-5 b(ound:)1530 3239 y Fo(N)1613 3201 y Fj(\003)1603 3261 y Fi(0)1652 3239 y Fq(\()p Fo(#)p Fq(\))26 b Fl(\024)1898 3111 y Fk(\026)1961 3177 y Fo(k)p 1961 3218 51 4 v 1964 3301 a(q)2021 3111 y Fk(\027)2089 3239 y Fo(;)1089 b Fq(\(2.7\))270 3447 y Fh(wher)-5 b(e)35 b Fo(k)j Fh(is)c(the)h(or)-5 b(der)36 b(of)e(the)h(tr)-5 b(e)g(e)35 b(and)g Fl(b)p Fo(x)p Fl(c)g Fh(is)g(the)f(highest)h(inte)-5 b(ger)35 b(smal)5 b(ler)35 b(or)h(e)-5 b(qual)270 3577 y(to)33 b Fo(x)p Fh(.)270 3771 y(Pr)-5 b(o)g(of.)43 b Fq(F)-8 b(or)26 b Fo(k)j(<)c(q)s Fq(,)h Fo(N)1059 3738 y Fj(\003)1049 3795 y Fi(0)1098 3771 y Fq(\()p Fo(#)p Fq(\))g(is)f(clearly)g(0,)i(and) e(for)g Fo(k)j Fq(=)d Fo(q)s Fq(,)i Fo(N)2374 3738 y Fj(\003)2364 3795 y Fi(0)2413 3771 y Fq(\()p Fo(#)p Fq(\))f Fl(\024)f Fq(1,)i(so)f(in)e(these)i(cases)270 3900 y(the)31 b(b)s(ound)d(\(2.7\))k(is)d(trivially)f(satis\014ed.)40 b(So)30 b(consider)g(the)g(case)i Fo(k)c(>)d(q)s Fq(.)370 4030 y(If)i(the)g(tree)i Fo(#)e Fq(has)g(the)h(ro)s(ot)g(line)e(on)h (scale)h Fo(n)d Fq(=)g(1,)k(or)e(on)h(scale)f Fo(n)e Fq(=)g(0)j(and)f(resonan)m(t,)270 4159 y(then)43 b(the)h(b)s(ound)e (\(2.7\))j(follo)m(ws)e(inductiv)m(ely)-8 b(.)79 b(In)43 b(fact,)48 b(calling)43 b Fo(#)2735 4173 y Fi(1)2774 4159 y Fq(,)h Fo(:)15 b(:)g(:)31 b Fq(,)47 b Fo(#)3105 4173 y Fn(p)3141 4181 y Fe(u)3178 4196 y Ff(0)3264 4159 y Fq(the)270 4289 y(subtrees)31 b(of)h Fo(#)g Fq(whic)m(h)f(ha)m(v)m(e) i(as)f(ro)s(ot)g(the)h(last)e(no)s(de)h Fo(u)2205 4303 y Fi(0)2276 4289 y Fq(of)g Fo(#)p Fq(,)g(and)g Fo(k)2718 4303 y Fi(1)2758 4289 y Fq(,)g Fo(:)15 b(:)g(:)31 b Fq(,)i Fo(k)3056 4303 y Fn(p)3092 4311 y Fe(u)3129 4326 y Ff(0)3203 4289 y Fq(their)270 4418 y(orders,)d(one)h(has:)1149 4611 y Fo(N)1232 4573 y Fj(\003)1222 4633 y Fi(0)1271 4611 y Fq(\()p Fo(#)p Fq(\))26 b(=)1527 4482 y Fn(p)1563 4490 y Fe(u)1600 4505 y Ff(0)1517 4525 y Fk(X)1521 4720 y Fn(j)t Fi(=1)1663 4611 y Fo(N)1746 4573 y Fj(\003)1736 4633 y Fi(0)1786 4611 y Fq(\()p Fo(#)1875 4625 y Fn(j)1911 4611 y Fq(\))g Fl(\024)2078 4549 y Fo(k)d Fl(\000)d Fq(1)p 2078 4590 207 4 v 2159 4673 a Fo(q)2320 4611 y(<)2426 4549 y(k)p 2426 4590 51 4 v 2429 4673 a(q)2486 4611 y(:)370 4836 y Fq(On)29 b(the)i(other)g(hand,)f(if)g(the)h(ro)s(ot)g(line)e(is) h(on)g(scale)h Fo(n)26 b Fq(=)f(0)31 b(and)g(non)f(resonan)m(t,)h(then) 270 4966 y(the)i(lines)d(en)m(tering)j Fo(u)1044 4980 y Fi(0)1115 4966 y Fq(cannot)g(b)s(e)f(all)g(on)g(scale)h Fo(n)28 b Fq(=)g(0,)34 b(otherwise)e Fo(\027)2756 4980 y Fn(u)2797 4989 y Ff(0)2864 4966 y Fq(=)c(0,)34 b(whic)m(h)d(is)270 5095 y(not)i(allo)m(w)m(ed.)46 b(Then)32 b(at)h(least)g(one)f(line)f (will)f(ha)m(v)m(e)j(scale)g Fo(n)28 b Fq(=)h(1:)45 b(let)32 b Fo(T)45 b Fq(b)s(e)32 b(the)h(cluster)p eop %%Page: 8 8 8 7 bop 390 46 a Fi(8)739 b(ALBER)-6 b(TO)30 b(BERRETTI)h(AND)f(GUIDO)h (GENTILE)390 241 y Fq(con)m(taining)37 b(it.)60 b(The)37 b(cluster)f Fo(T)50 b Fq(will)34 b(ha)m(v)m(e)39 b Fo(p)2023 255 y Fn(T)2114 241 y Fq(en)m(tering)e(lines,)h(with)d Fo(p)2973 255 y Fn(T)3065 241 y Fl(\025)h Fq(0,)j(and)e({)390 370 y(as)e(w)m(e)h(just)e(assumed)g({)h(it)g(is)f(not)h(a)g(resonance;) j(then)2375 302 y Fk(P)2471 397 y Fn(u)p Fj(2)p Fn(T)2629 370 y Fo(\027)2674 384 y Fn(u)2752 370 y Fl(6)p Fq(=)32 b(0,)37 b(so)e(there)g(m)m(ust)390 500 y(b)s(e)f(at)h(least)f Fo(q)j Fq(no)s(des,)e(hence)f Fo(q)26 b Fl(\000)c Fq(1)35 b(lines,)f(inside)d Fo(T)13 b Fq(.)52 b(The)34 b(subtrees)g(en)m (tering)g(in)m(to)g Fo(T)390 629 y Fq(will)27 b(ha)m(v)m(e,)k(resp)s (ectiv)m(ely)-8 b(,)30 b Fo(k)1355 643 y Fi(1)1394 629 y Fq(,)g Fo(:)15 b(:)g(:)31 b Fq(,)f Fo(k)1687 643 y Fn(p)1723 654 y Fe(T)1805 629 y Fq(no)s(des,)g(with)e Fo(q)21 b Fq(+)2443 561 y Fk(P)2539 585 y Fn(p)2575 596 y Fe(T)2539 656 y Fn(j)t Fi(=1)2681 629 y Fo(k)2728 643 y Fn(j)2790 629 y Fl(\024)k Fo(k)33 b Fq(so)c(that,)i(again)390 759 y(inductiv)m(ely:)1113 941 y Fo(N)1196 903 y Fj(\003)1186 963 y Fi(0)1235 941 y Fq(\()p Fo(#)p Fq(\))26 b(=)f(1)c(+)1661 822 y Fn(p)1697 833 y Fe(T)1638 855 y Fk(X)1642 1050 y Fn(j)t Fi(=1)1784 941 y Fo(N)1867 903 y Fj(\003)1857 963 y Fi(0)1907 941 y Fq(\()p Fo(#)1996 955 y Fn(j)2032 941 y Fq(\))26 b Fl(\024)f Fq(1)20 b(+)2355 879 y Fo(k)k Fl(\000)c Fo(q)p 2355 920 206 4 v 2436 1003 a(q)2596 941 y Fl(\024)2702 879 y Fo(k)p 2702 920 51 4 v 2705 1003 a(q)2762 941 y(:)390 1171 y Fq(Then)37 b Fo(N)718 1138 y Fj(\003)708 1196 y Fi(0)796 1171 y Fq(will)f(b)s(e)i(b)s(ounded) e(b)m(y)i Fo(k)s(=q)s Fq(.)65 b(As)39 b Fo(N)2076 1138 y Fj(\003)2066 1196 y Fi(0)2153 1171 y Fq(has)f(to)i(b)s(e)d(an)i(in)m (teger)f(n)m(um)m(b)s(er,)i(the)390 1301 y(assertion)30 b(follo)m(ws.)p 3444 1301 4 62 v 3448 1243 55 4 v 3448 1301 V 3502 1301 4 62 v 490 1480 a(W)-8 b(e)33 b(no)m(w)f(sho)m(w)g(ho) m(w)g(to)h(construct)g(a)f(suitable)f(partial)g(resummation)g(of)h(the) h(Lind-)390 1610 y(stedt)c(series,)f(that)g(is)g(w)m(e)g(shall)f (de\014ne)g(families)f(of)i(trees)h(to)g(b)s(e)f(group)s(ed)f(and)g(b)s (ounded)390 1739 y(together)32 b(to)f(obtain)f(extra)h Fo(\021)j Fq(factors.)490 1869 y(Giv)m(en)26 b(a)g(tree)h Fo(#)f Fq(and)f(a)i(resonance)f Fo(V)47 b Fq(with)24 b Fo(p)2080 1883 y Fn(V)2167 1869 y Fq(incoming)h(lines)f Fo(`)2793 1883 y Fi(1)2832 1869 y Fq(,)j Fo(:)15 b(:)g(:)31 b Fq(,)c Fo(`)3110 1883 y Fn(p)3146 1894 y Fe(V)3229 1869 y Fq(and)f Fo(k)3449 1883 y Fn(V)390 1998 y Fq(no)s(des,)i(w)m(e)g (de\014ne)f(the)h(family)e Fl(F)1554 2012 y Fn(V)1615 1998 y Fq(\()p Fo(#)p Fq(\))j(of)e Fo(V)48 b Fq(in)27 b Fo(#)g Fq(as)h(the)g(set)h(of)e(trees)i(obtained)e(from)g Fo(#)390 2128 y Fq(b)m(y)h(the)g(action)g(of)f(a)i(group)e(of)h (transformation)f Fl(P)2148 2142 y Fn(V)2236 2128 y Fq(on)h Fo(#)p Fq(,)g(generated)h(b)m(y)f(the)f(follo)m(wing)390 2257 y(op)s(erations.)490 2405 y(1.)42 b(Detac)m(h)29 b(the)e(line)e Fo(`)1267 2419 y Fi(1)1307 2405 y Fq(,)j(and)e(reattac)m (h)j(it)e(to)g(all)f(of)h(the)h(no)s(des)e(of)h(the)g(resonance;)i(for) 602 2535 y(eac)m(h)c(of)f(the)g(so)g(obtained)f(trees,)j(detac)m(h)f (the)f(line)f Fo(`)2395 2549 y Fi(2)2458 2535 y Fq(and)g(reattac)m(h)j (it)d(to)i(all)e(of)h(the)602 2665 y(no)s(des)29 b(of)i(the)g (resonance;)g(and)f(so)h(on)f(for)g(eac)m(h)i(en)m(tering)e(line)f(of)h (the)h(resonance.)602 2794 y(In)e(this)h(w)m(a)m(y)h(w)m(e)g(obtain)f Fo(k)1544 2750 y Fn(p)1580 2761 y Fe(V)1541 2823 y Fn(V)1667 2794 y Fq(di\013eren)m(t)g(trees.)490 2924 y(2.)42 b(In)32 b(a)h(giv)m(en)g(tree,)i(eac)m(h)f(no)s(de)e Fo(u)e Fl(2)f Fo(V)53 b Fq(will)30 b(ha)m(v)m(e)k Fo(p)2386 2938 y Fn(u)2464 2924 y Fq(en)m(tering)f(lines,)f(of)h(whic)m(h)e Fo(s)3465 2938 y Fn(u)602 3053 y Fq(are)39 b(inside)e Fo(V)59 b Fq(and)38 b Fo(r)1367 3067 y Fn(u)1452 3053 y Fq(=)h Fo(p)1608 3067 y Fn(u)1679 3053 y Fl(\000)25 b Fo(s)1818 3067 y Fn(u)1902 3053 y Fq(are)39 b(outside)g Fo(V)59 b Fq(\()p Fh(i.e.)40 b Fq(are)f(en)m(tering)g(lines)e(in)602 3183 y Fo(V)20 b Fq(\).)45 b(Then)31 b(w)m(e)h(can)g(apply)e(to)i(the)g (set)g(of)g(lines)e(en)m(tering)i Fo(u)f Fq(a)h(transformation)f(in)602 3312 y(the)d(group)f(obtained)g(as)h(the)g(quotien)m(t)f(of)h(the)g (group)f(of)h(p)s(erm)m(utations)f(of)g(the)h Fo(p)3465 3326 y Fn(u)602 3442 y Fq(lines)k(b)m(y)i(the)g(groups)g(of)g(p)s(erm)m (utations)f(of)h(the)h Fo(s)2379 3456 y Fn(u)2457 3442 y Fq(in)m(ternal)e(en)m(tering)h(lines)f(and)602 3571 y(of)j(p)s(erm)m(utations)e(of)i(the)g Fo(p)1589 3585 y Fn(u)1670 3571 y Fq(en)m(tering)f(lines)f(outside)h Fo(V)20 b Fq(:)52 b(in)34 b(this)h(w)m(a)m(y)h(for)g(eac)m(h)602 3701 y(no)s(de)30 b Fo(u)25 b Fl(2)g Fo(V)20 b Fq(,)30 b(a)h(n)m(um)m(b)s(er)e(of)i(trees)g(equal)f(to:)1815 3842 y(1)p 1780 3883 117 4 v 1780 3966 a Fo(p)1826 3980 y Fn(u)1870 3966 y Fq(!)1906 3776 y Fk(\022)1973 3842 y Fo(p)2019 3856 y Fn(u)1974 3966 y Fo(s)2017 3980 y Fn(u)2063 3776 y Fk(\023)3323 3904 y Fq(\(2.8\))602 4107 y(is)f(obtained.)490 4236 y(3.)42 b(Flip)28 b(all)i(the)g(mo)s(de)g (lab)s(els)f(inside)f Fo(V)50 b Fq(sim)m(ultaneously)-8 b(.)490 4385 y(W)g(e)32 b(shall)d(call)i(transformations)f(of)h(t)m(yp) s(e)g(1,)h(2,)g(3)f(the)h(op)s(erations)e(describ)s(ed)f(in)h(the)390 4514 y(tree)h(items)f(ab)s(o)m(v)m(e.)490 4644 y(Of)39 b(course,)k(the)d(group)g Fl(P)1441 4658 y Fn(V)1542 4644 y Fq(acts)h(on)f(all)f(trees)i(of)f(a)g(giv)m(en)h(order)e(con)m (taining)h(the)390 4773 y(resonance)28 b Fo(V)20 b Fq(,)28 b(and)f(its)f(action)i(on)f(the)h(set)f(of)h(trees)g(of)f(a)h(giv)m(en) f(order)g Fo(k)j Fq(con)m(taining)d(the)390 4903 y(resonance)g Fo(V)47 b Fq(de\014nes)25 b(an)h(equiv)-5 b(alence)26 b(relation,)h(and)f(the)h(equiv)-5 b(alence)25 b(classes)i(are)g(the) 390 5032 y Fh(r)-5 b(esonant)35 b(families)j Fl(F)1170 5046 y Fn(V)1231 5032 y Fq(\()p Fo(#)p Fq(\).)p eop %%Page: 9 9 9 8 bop 955 46 a Fi(SCALING)31 b(PR)n(OPER)-6 b(TIES)31 b(OF)f(THE)h(ST)-6 b(AND)n(ARD)30 b(MAP)649 b(9)370 241 y Fq(The)28 b(follo)m(wing)f(lemma)h(is)f(the)i(crucial)e(one,)j(where) e(cancellations)g(b)s(et)m(w)m(een)h(trees)g(in)270 370 y(the)35 b(same)g(family)f(are)h(exploited.)53 b(W)-8 b(e)37 b(state)f(it)e(here)h(and)f(use)h(it)g(to)g(pro)m(v)m(e)h(the)f (main)270 500 y(estimate)30 b(on)g(the)g(radius)d(of)j(con)m(v)m (ergence)i(of)e(the)g(Lindstedt)e(series,)h(and)g(p)s(ostp)s(one)g(its) 270 629 y(pro)s(of)h(to)h(the)f(next)h(section.)270 815 y Fv(Lemma)i(2.)45 b Fh(Given)25 b(a)f(tr)-5 b(e)g(e)25 b Fo(#)p Fh(,)h(with)f(a)g(r)-5 b(esonanc)g(e)26 b Fo(V)20 b Fh(,)26 b(if)e Fo(\027)2293 829 y Fi(1)2332 815 y Fh(,)g Fo(:)15 b(:)g(:)31 b Fh(,)26 b Fo(\027)2619 829 y Fn(p)2655 840 y Fe(V)2736 815 y Fh(ar)-5 b(e)26 b(the)e(momenta)270 944 y(\015owing)34 b(thr)-5 b(ough)34 b(the)f(entering)g(lines)f Fo(`)1669 958 y Fi(1)1709 944 y Fh(,)g Fo(:)15 b(:)g(:)31 b Fh(,)i Fo(`)2004 958 y Fn(p)2040 969 y Fe(V)2129 944 y Fh(of)g Fo(V)20 b Fh(,)33 b(we)g(have)g(the)g(b)-5 b(ound:)806 1149 y Fq(1)p 704 1190 251 4 v 704 1273 a Fl(F)769 1287 y Fn(V)830 1273 y Fq(\()p Fo(#)p Fq(\))980 1025 y Fk(\014)980 1079 y(\014)980 1134 y(\014)980 1188 y(\014)980 1243 y(\014)980 1297 y(\014)1100 1125 y(X)1010 1326 y Fn(#)1052 1307 y Fd(0)1074 1326 y Fj(2F)1171 1337 y Fe(V)1225 1326 y Fi(\()p Fn(#)p Fi(\))1337 1211 y Fl(V)1393 1225 y Fn(V)1454 1211 y Fq(\()p Fo(#)1543 1173 y Fj(0)1566 1211 y Fq(\))1601 1025 y Fk(\014)1601 1079 y(\014)1601 1134 y(\014)1601 1188 y(\014)1601 1243 y(\014)1601 1297 y(\014)1657 1211 y Fl(\024)25 b Fo(c)1792 1173 y Fj(\000)p Fn(k)1884 1184 y Fe(V)1942 1211 y Fo(q)1986 1173 y Fi(2)p Fn(k)2058 1184 y Fe(V)2115 1211 y Fo(D)2190 1225 y Fi(0)2329 1092 y Fn(p)2365 1103 y Fe(V)2308 1125 y Fk(X)2245 1323 y Fn(m;m)2389 1304 y Fd(0)2412 1323 y Fi(=1)2517 1134 y Fk(\014)2517 1188 y(\014)2547 1211 y Fo(\027)2592 1225 y Fn(m)2659 1211 y Fo(\027)2704 1226 y Fn(m)2766 1207 y Fd(0)2793 1211 y Fo(\021)2841 1173 y Fi(2)2880 1134 y Fk(\014)2880 1188 y(\014)2926 1211 y Fo(;)252 b Fq(\(2.9\))270 1469 y Fh(for)33 b(some)h(c)-5 b(onstant)34 b Fo(D)1085 1483 y Fi(0)1125 1469 y Fh(.)370 1654 y Fq(As)g(the)h(t)m(ypical)f (tree)h(ma)m(y)g(con)m(tain)g(more)f(than)g(one)h(resonance,)h(w)m(e)f (need)g(the)f(fol-)270 1784 y(lo)m(wing)29 b(easy)i(corollary)f(of)h (the)f(ab)s(o)m(v)m(e)i(lemma.)270 1969 y Fv(Corollary)-9 b(.)46 b Fh(Given)c(a)h(tr)-5 b(e)g(e)43 b Fo(#)e Fh(with)j(r)-5 b(esonanc)g(es)44 b Fo(V)2137 1983 y Fi(1)2176 1969 y Fh(,)e Fo(:)15 b(:)g(:)31 b Fh(,)45 b Fo(V)2508 1983 y Fn(s)2545 1969 y Fh(,)f(c)-5 b(onsider)43 b(the)g(family)270 2099 y Fl(F)9 b Fq(\()p Fo(#)p Fq(\))36 b Fh(obtaine)-5 b(d)36 b(by)f(the)g(action)h(on)f Fo(#)f Fh(of)i(al)5 b(l)35 b(the)g(gr)-5 b(oups)36 b Fl(P)2386 2113 y Fn(V)2427 2122 y Ff(1)2466 2099 y Fh(,)f Fo(:)15 b(:)g(:)31 b Fh(,)k Fl(P)2791 2113 y Fn(V)2832 2124 y Fe(S)2881 2099 y Fh(;)h(the)f(numb)-5 b(er)270 2228 y(of)33 b(tr)-5 b(e)g(es)33 b(in)g(this)g(family)h(is)e (given)g(by)h Fo(k)1652 2247 y Fj(F)7 b Fi(\()p Fn(#)p Fi(\))1835 2228 y Fq(=)1931 2160 y Fk(Q)2017 2186 y Fn(s)2017 2255 y(i)p Fi(=1)2151 2228 y Fl(jF)2241 2242 y Fn(V)2282 2252 y Fe(i)2313 2228 y Fq(\()p Fo(#)p Fq(\))p Fl(j)q Fh(,)32 b(and:)924 2442 y Fq(1)p 844 2482 206 4 v 844 2566 a Fo(k)891 2584 y Fj(F)7 b Fi(\()p Fn(#)p Fi(\))1074 2317 y Fk(\014)1074 2372 y(\014)1074 2426 y(\014)1074 2481 y(\014)1074 2535 y(\014)1074 2590 y(\014)1172 2417 y(X)1105 2618 y Fn(#)1147 2599 y Fd(0)1169 2618 y Fj(2F)g Fi(\()p Fn(#)p Fi(\))1385 2503 y Fq(V)-8 b(al\()p Fo(#)1605 2466 y Fj(0)1629 2503 y Fq(\))1664 2317 y Fk(\014)1664 2372 y(\014)1664 2426 y(\014)1664 2481 y(\014)1664 2535 y(\014)1664 2590 y(\014)1720 2503 y Fl(\024)25 b Fo(c)1855 2466 y Fj(\000)p Fn(k)1953 2503 y Fo(q)1997 2466 y Fi(2)p Fn(N)2088 2475 y Ff(1)2127 2503 y Fl(j)p Fo(\021)s Fl(j)2225 2466 y Fj(\000)p Fi(2)p Fn(N)2371 2475 y Ff(0)2410 2503 y Fo(D)2488 2466 y Fn(s)2485 2526 y Fi(0)2525 2503 y Fo(k)2575 2466 y Fi(2)2615 2503 y Fl(j)p Fo(\021)s Fl(j)2713 2466 y Fi(2)p Fn(s)2786 2503 y Fo(;)347 b Fq(\(2.10\))270 2761 y Fh(with)33 b(the)g(same)h(c)-5 b(onstant)34 b Fo(D)1288 2775 y Fi(0)1360 2761 y Fh(of)f(lemma)h(2.)270 2947 y(Pr)-5 b(o)g(of.)43 b Fq(Simply)c(use)i(that)h(for)g(eac)m(h)g (resonance)h Fo(V)61 b Fq(a)42 b(factor)2510 2869 y Fk(\014)2510 2924 y(\014)2540 2947 y Fo(\027)2585 2961 y Fn(m)2652 2947 y Fo(\027)2697 2962 y Fn(m)2759 2943 y Fd(0)2786 2947 y Fo(\021)2834 2914 y Fi(2)2874 2869 y Fk(\014)2874 2924 y(\014)2945 2947 y Fq(arises)f(from)270 3076 y(lemma)i(2,)48 b(while)41 b(the)j(propagators)g(corresp)s(onding)e(to)i(the)g(t)m(w)m (o)h(lines)c Fo(`)2963 3090 y Fn(m)3073 3076 y Fq(and)i Fo(`)3301 3091 y Fn(m)3363 3072 y Fd(0)270 3206 y Fq(can)32 b(b)s(e)f(b)s(ounded,)f(resp)s(ectiv)m(ely)-8 b(,)31 b(b)m(y)h Fo(c)1642 3173 y Fj(\000)p Fi(1)1752 3206 y Fl(j)p Fo(\027)1822 3220 y Fn(m)1888 3206 y Fo(\021)s Fl(j)1962 3164 y Fj(\000)p Fi(2)2088 3206 y Fq(and)f Fo(c)2305 3173 y Fj(\000)p Fi(1)2414 3206 y Fl(j)q Fo(\027)2485 3221 y Fn(m)2547 3202 y Fd(0)2573 3206 y Fo(\021)s Fl(j)2647 3164 y Fj(\000)p Fi(2)2741 3206 y Fq(,)h(b)m(y)h(\(2.1\))r(.)44 b(Then)270 3335 y(just)26 b(note)h(that)g(the)g(cancellation)f(mec)m (hanisms)g(op)s(erating)g(for)g(eac)m(h)i(resonance)f(do)g(not)270 3465 y(in)m(terfere)c(with)f(eac)m(h)i(other)f(\()p Fh(i.e.)j Fq(there)d(is)f(no)h Fh(c)-5 b(anc)g(el)5 b(lation)28 b(overlap)6 b Fq(\):)38 b(this)22 b(follo)m(ws)g(from)270 3594 y(the)31 b(fact)g(that,)g(for)f(an)m(y)h(tree)g Fo(#)1375 3561 y Fj(0)1423 3594 y Fl(2)25 b(F)9 b Fq(\()p Fo(#)p Fq(\),)32 b(w)m(e)e(can)h(write:)1305 3822 y(V)-8 b(al)q(\()p Fo(#)1526 3784 y Fj(0)1549 3822 y Fq(\))26 b(=)f Fl(B)s Fq(\()p Fo(#)p Fq(\))1950 3708 y Fn(s)1908 3735 y Fk(Y)1909 3931 y Fn(i)p Fi(=1)2039 3822 y Fl(V)2095 3836 y Fn(V)2136 3846 y Fe(i)2166 3822 y Fq(\()p Fo(#)2255 3784 y Fj(0)2279 3822 y Fq(\))p Fo(;)819 b Fq(\(2.11\))270 4064 y(where)33 b(the)h(factor)h Fl(B)s Fq(\()p Fo(#)p Fq(\))f(assumes)f(the)h(same)h(v)-5 b(alue)33 b(indep)s(enden)m(tly)e (from)i Fo(#)3028 4031 y Fj(0)3051 4064 y Fq(.)51 b(There-)270 4193 y(fore)31 b(w)m(e)h(can)f(use)g(the)g(trivial)e(inequalit)m(y)1760 4116 y Fk(\014)1760 4170 y(\014)1791 4193 y Fo(\027)1836 4207 y Fn(m)1902 4193 y Fo(\027)1947 4208 y Fn(m)2009 4189 y Fd(0)2036 4193 y Fo(\021)2084 4160 y Fi(2)2124 4116 y Fk(\014)2124 4170 y(\014)2169 4193 y Fl(j)p Fo(\027)2239 4208 y Fn(m)2301 4189 y Fd(0)2328 4193 y Fo(\021)s Fl(j)2402 4151 y Fj(\000)p Fi(2)2511 4193 y Fl(j)q Fo(\027)2582 4207 y Fn(m)2648 4193 y Fo(\021)s Fl(j)2722 4151 y Fj(\000)p Fi(2)2842 4193 y Fl(\024)e(j)p Fo(\021)s Fl(j)3038 4160 y Fi(2)3109 4193 y Fq(for)k(an)m(y)270 4323 y(resonance)j Fo(V)741 4337 y Fn(i)769 4323 y Fq(.)50 b(Moreo)m(v)m(er)36 b(the)e(n)m(um)m(b)s(er)e(of)h(addends)f(app)s(earing)h(in)f(the)h(sum) g(in)f(\(2.9\))270 4452 y(ma)m(y)f(b)s(e)f(b)s(ounded)e(b)m(y)i Fo(k)1138 4419 y Fi(2)1135 4479 y Fn(V)1176 4489 y Fe(i)1237 4452 y Fq(for)h(eac)m(h)g(resonance,)g(and)2198 4384 y Fk(Q)2284 4479 y Fn(i)2327 4452 y Fo(k)2377 4419 y Fi(2)2374 4479 y Fn(V)2415 4489 y Fe(i)2471 4452 y Fl(\024)25 b Fo(k)2617 4419 y Fi(2)2657 4452 y Fq(.)p 3324 4452 4 62 v 3328 4394 55 4 v 3328 4452 V 3382 4452 4 62 v 370 4631 a(Finally)i(w)m(e)j(ha)m(v)m(e)h(the)f(main)f(lemma)g(on)g (the)h(con)m(v)m(ergence)i(radius)c(of)i(the)f(Lindstedt)270 4761 y(series.)270 4946 y Fv(Lemma)k(3.)45 b Fh(L)-5 b(et)45 b Fo(\021)50 b Fl(6)p Fq(=)d(0)p Fh(,)g(and)f Fo(\032)p Fq(\()p Fo(\021)s Fq(\))f Fh(the)g(r)-5 b(adius)46 b(of)f(c)-5 b(onver)g(genc)g(e)45 b(of)f(the)h(Lindste)-5 b(dt)270 5076 y(series)36 b Fq(\(1.9\))30 b Fh(as)f(a)g(function)g(of)g Fo(\021)s Fh(.)40 b(Then)29 b(ther)-5 b(e)30 b(exists)e(a)h(p)-5 b(ositive)30 b(c)-5 b(onstant)30 b Fo(r)h Fh(such)e(that)270 5205 y Fo(\032)p Fq(\()p Fo(\021)s Fq(\))d Fl(\025)f Fo(r)s Fl(j)p Fo(\021)s Fl(j)699 5172 y Fi(2)p Fn(=q)808 5205 y Fh(.)p eop %%Page: 10 10 10 9 bop 390 46 a Fi(10)704 b(ALBER)-6 b(TO)30 b(BERRETTI)h(AND)f (GUIDO)h(GENTILE)390 241 y Fh(Pr)-5 b(o)g(of.)43 b Fq(Let)35 b Fo(N)921 208 y Fn(R)911 265 y Fi(0)1013 241 y Fq(b)s(e)f(the)h(n)m (um)m(b)s(er)f(of)g(resonan)m(t)i(lines)c(on)j(scale)g Fo(n)d Fq(=)g(0,)k(so)f(that)g Fo(N)3367 255 y Fi(0)3439 241 y Fq(=)390 370 y Fo(N)473 337 y Fj(\003)463 395 y Fi(0)539 370 y Fq(+)27 b Fo(N)720 337 y Fn(R)710 395 y Fi(0)778 370 y Fq(.)71 b(F)-8 b(or)41 b(the)f(resonances)h(an)g(o)m (v)m(erall)f(gain)h Fo(D)2397 337 y Fn(s)2394 395 y Fi(0)2433 370 y Fo(k)2483 337 y Fi(2)2523 370 y Fl(j)p Fo(\021)s Fl(j)2621 337 y Fi(2)p Fn(s)2734 370 y Fq(is)f(obtained,)i(b)m(y)f(the) 390 500 y(corollary)30 b(to)h(lemma)f(2.)41 b(So)30 b(w)m(e)h(ha)m(v)m (e:)721 696 y(1)p 641 737 206 4 v 641 820 a Fo(k)688 838 y Fj(F)7 b Fi(\()p Fn(#)p Fi(\))872 571 y Fk(\014)872 626 y(\014)872 680 y(\014)872 735 y(\014)872 789 y(\014)872 844 y(\014)969 671 y(X)902 872 y Fn(#)944 854 y Fd(0)967 872 y Fj(2F)g Fi(\()p Fn(#)p Fi(\))1183 757 y Fq(V)-8 b(al\()p Fo(#)p Fq(\))1438 571 y Fk(\014)1438 626 y(\014)1438 680 y(\014)1438 735 y(\014)1438 789 y(\014)1438 844 y(\014)1494 757 y Fl(\024)25 b Fo(c)1629 720 y Fj(\000)p Fn(k)1727 757 y Fo(q)1771 720 y Fi(2)p Fn(N)1862 729 y Ff(1)1901 757 y Fl(j)p Fo(\021)s Fl(j)1999 720 y Fj(\000)p Fi(2)p Fn(N)2152 696 y Fd(\003)2145 741 y Ff(0)2193 757 y Fl(j)p Fo(\021)s Fl(j)2291 720 y Fj(\000)p Fi(2)p Fn(N)2444 696 y Fe(R)2437 741 y Ff(0)2500 757 y Fo(D)2578 706 y Fn(N)2641 682 y Fe(R)2634 726 y Ff(0)2575 783 y Fi(0)2695 757 y Fo(k)2745 720 y Fi(2)2785 757 y Fl(j)p Fo(\021)s Fl(j)2883 720 y Fi(2)p Fn(N)2981 696 y Fe(R)2974 741 y Ff(0)1494 1039 y Fl(\024)1590 966 y Fk(\000)1632 1039 y Fo(c)1671 1002 y Fj(\000)p Fi(1)1765 1039 y Fo(e)1807 1002 y Fi(2)1847 1039 y Fo(D)1922 1053 y Fi(0)1962 1039 y Fo(q)2006 1002 y Fi(2)2045 966 y Fk(\001)2087 985 y Fn(k)2145 1039 y Fl(j)p Fo(\021)s Fl(j)2243 1002 y Fj(\000)p Fi(2)p Fn(k)r(=q)2445 1039 y Fo(;)3278 849 y Fq(\(2.12\))390 1215 y(as)34 b Fo(k)555 1182 y Fi(2)626 1215 y Fl(\024)d Fo(e)770 1182 y Fi(2)p Fn(k)882 1215 y Fq(and)i(the)h(n)m(um)m(b)s(er)f (of)h(resonances)g(is)f(equal)h(to)h(the)f(n)m(um)m(b)s(er)e(of)i (resonan)m(t)390 1344 y(lines,)29 b Fh(i.e.)j Fo(s)25 b Fq(=)g Fo(N)1028 1312 y Fn(R)1018 1369 y Fi(0)1085 1344 y Fq(.)41 b(Therefore,)30 b(writing)f(the)h(sum)g(o)m(v)m(er)h (all)f(trees)h(as:)1152 1456 y Fk(X)1110 1654 y Fn(#)p Fj(2T)1238 1666 y Fe(\027;k)1341 1543 y Fq(V)-8 b(al)q(\()p Fo(#)p Fq(\))25 b(=)1761 1456 y Fk(X)1718 1654 y Fn(#)p Fj(2T)1846 1666 y Fe(\027;k)2040 1481 y Fq(1)p 1960 1522 V 1960 1605 a Fo(k)2007 1623 y Fj(F)7 b Fi(\()p Fn(#)p Fi(\))2257 1456 y Fk(X)2190 1658 y Fn(#)2232 1639 y Fd(0)2254 1658 y Fj(2F)g Fi(\()p Fn(#)p Fi(\))2470 1543 y Fq(V)-8 b(al)q(\()p Fo(#)2691 1505 y Fj(0)2714 1543 y Fq(\))p Fo(;)504 b Fq(\(2.13\))390 1801 y(w)m(e)31 b(can)g(conclude)e(that:)935 1909 y Fk(\014)935 1964 y(\014)935 2018 y(\014)966 2014 y Fo(u)1018 1976 y Fi(\()p Fn(k)r Fi(\))1018 2036 y Fn(\027)1115 1909 y Fk(\014)1115 1964 y(\014)1115 2018 y(\014)1171 2014 y Fl(\024)c Fo(D)1345 1976 y Fn(k)1342 2036 y Fi(1)1387 2014 y Fl(j)p Fo(\021)s Fl(j)1485 1976 y Fj(\000)p Fi(2)p Fn(k)r(=q)1688 2014 y Fo(;)1910 1855 y Fk(\014)1910 1909 y(\014)1910 1964 y(\014)1910 2018 y(\014)1910 2073 y(\014)1941 1927 y(X)1941 2125 y Fn(\027)t Fj(2)p Fg(Z)2087 2014 y Fo(e)2129 1976 y Fn(i\027)t(\013)2242 2014 y Fo(u)2294 1976 y Fi(\()p Fn(k)r Fi(\))2294 2036 y Fn(\027)2392 1855 y Fk(\014)2392 1909 y(\014)2392 1964 y(\014)2392 2018 y(\014)2392 2073 y(\014)2447 2014 y Fl(\024)g Fo(D)2621 1976 y Fn(k)2618 2036 y Fi(1)2664 2014 y Fl(j)p Fo(\021)s Fl(j)2762 1976 y Fj(\000)p Fi(2)p Fn(k)r(=q)3278 2014 y Fq(\(2.14\))390 2260 y(for)f(some)h(constan)m(ts)g Fo(D)1216 2274 y Fi(1)1280 2260 y Fq(\(see)g(commen)m(ts)h(after)e (\(1.8\))j(on)d(the)g(n)m(um)m(b)s(er)f(of)i(trees)g(in)d Fl(T)3352 2275 y Fn(\027;k)3449 2260 y Fq(\).)390 2389 y(A)30 b(comparison)f(b)s(et)m(w)m(een)h(\(1.8\))r(,)g(\(2.12\))i(and)d (\(2.13\))i(giv)m(es)f Fo(D)2556 2403 y Fi(1)2621 2389 y Fq(=)25 b(4)p Fo(cD)2876 2403 y Fi(0)2917 2389 y Fo(q)2961 2356 y Fi(2)3000 2389 y Fq(;)30 b(this)f(implies)390 2519 y(that)i Fo(\032)p Fq(\()p Fo(\021)s Fq(\))26 b Fl(\025)f Fo(r)s Fl(j)p Fo(\021)s Fl(j)1016 2486 y Fi(2)p Fn(=q)1125 2519 y Fq(,)31 b(for)f(some)h({)g(explicitly)d(computable)h ({)i(constan)m(t)h Fo(r)s Fq(.)p 3444 2519 4 62 v 3448 2461 55 4 v 3448 2519 V 3502 2519 4 62 v 1556 3010 a(3.)51 b Fp(Cancella)-6 b(tions)490 3193 y Fq(In)26 b(this)h(section)h(w)m(e)g (pro)m(v)m(e)g(lemma)f(2,)i(b)m(y)f(exhibiting)d(the)j(cancellation)f (mec)m(hanisms)390 3323 y(b)s(et)m(w)m(een)k(trees)g(b)s(elonging)e(to) i(the)f(same)h(family)-8 b(.)390 3502 y Fh(Pr)j(o)g(of)34 b(of)f(lemma)h(2.)41 b Fq(By)27 b(neglecting)e(the)h(trees)h(obtained)e (\015ipping)e(all)h(the)i(mo)s(de)g(lab)s(els)390 3632 y Fo(\027)435 3646 y Fn(u)480 3632 y Fq(,)f Fo(u)h Fl(2)f Fo(V)20 b Fq(,)25 b(the)g(resonance)f(factors)h(\(2.4\))h(asso)s (ciated)f(to)f(the)h(trees)f(in)f Fl(F)2935 3646 y Fn(V)2996 3632 y Fq(\()p Fo(#)p Fq(\))i(can)f(di\013er)390 3761 y(from)30 b(eac)m(h)h(other)g(b)s(ecause)f(of)h(the)g(follo)m(wing)d (three)j(reasons.)490 3910 y(1.)42 b(Some)30 b(of)h(the)f(propagators)h (are)g(di\013eren)m(t,)f(as)g(the)h(momen)m(tum)f(\015o)m(wing)g (through)602 4039 y(a)43 b(line)e Fo(`)i Fq(in)m(ternal)e(to)j Fo(V)63 b Fq(dep)s(ends)41 b(on)i(the)g(momen)m(tum)g(\015o)m(wing)f (through)g(the)602 4169 y(en)m(tering)26 b(line)f Fo(`)1155 4183 y Fn(m)1248 4169 y Fq(only)h(if)g Fo(`)g Fq(is)g(on)h(the)f(path)h (leading)e(from)h Fo(`)2718 4183 y Fn(m)2812 4169 y Fq(to)h(the)g (exiting)f(line)602 4298 y(of)k Fo(V)21 b Fq(.)490 4428 y(2.)42 b(The)d(n)m(umerator)g(is)g(giv)m(en)g(b)m(y)g(a)h(common)g (factor)g(\()p Fh(i.e.)h Fq(a)f(factor)g(whic)m(h)e(is)h(the)602 4557 y(same)27 b(for)f(all)g(trees\))h(times)g Fo(\027)1618 4571 y Fn(w)1669 4580 y Ff(1)1722 4557 y Fo(:)15 b(:)g(:)h(\027)1888 4571 y Fn(w)1939 4579 y Fe(p)1971 4596 y(V)2032 4557 y Fq(,)28 b(if)e Fo(w)2230 4571 y Fn(p)2266 4580 y Ff(1)2304 4557 y Fq(,)h Fo(:)15 b(:)g(:)31 b Fq(,)d Fo(w)2610 4571 y Fn(p)2646 4582 y Fe(V)2730 4557 y Fq(are)f(the)f(no)s(des)g(inside) 602 4687 y Fo(V)50 b Fq(whic)m(h)29 b(the)i(en)m(tering)f(lines)f Fo(`)1719 4701 y Fi(1)1758 4687 y Fq(,)i Fo(:)15 b(:)g(:)31 b Fq(,)g Fo(`)2044 4701 y Fn(p)2080 4712 y Fe(V)2167 4687 y Fq(are)g(attac)m(hed)h(to.)490 4816 y(3.)42 b(Some)c(of)g(the)h (com)m(binatorial)e(factors)i Fo(p)2059 4830 y Fn(u)2103 4816 y Fq(!)2128 4783 y Fj(\000)p Fi(1)2261 4816 y Fq(are)g(di\013eren) m(t,)g(as)g(the)f(n)m(um)m(b)s(er)f(of)602 4946 y(lines)31 b(en)m(tering)h(the)h(no)s(des)f(can)g(c)m(hange,)j(and)d(the)h(p)s (erm)m(utations)e(of)i(the)g(exiting)602 5075 y(lines)18 b(in)h(the)i(de\014nition)d(of)j(the)f(group)g Fl(P)1987 5089 y Fn(V)2068 5075 y Fq(pro)s(duces,)i(for)e(eac)m(h)i(no)s(de)d Fo(u)p Fq(,)k(a)e(binomial)602 5205 y(factor)31 b(as)g(in)e(\(2.8\))r (.)p eop %%Page: 11 11 11 10 bop 955 46 a Fi(SCALING)31 b(PR)n(OPER)-6 b(TIES)31 b(OF)f(THE)h(ST)-6 b(AND)n(ARD)30 b(MAP)613 b(11)270 241 y Fq(If)38 b(w)m(e)i(p)s(erform)d(also)i(the)g(mo)s(de)f (\015ipping)e(of)j(the)g(lab)s(els)e Fo(\027)2396 255 y Fn(u)2441 241 y Fq(,)k Fo(u)f Fl(2)e Fo(V)21 b Fq(,)41 b(other)e(c)m(hanges)270 370 y(are)28 b(in)m(tro)s(duced:)38 b(the)28 b(n)m(umerator)g(will)d(c)m(hange)k(b)m(y)f(a)g(m)m (ultiplicativ)m(e)e(factor)j(\()p Fl(\000)p Fq(1\))3181 337 y Fn(p)3217 348 y Fe(V)3270 337 y Fi(+1)3365 370 y Fq(,)270 500 y(and)43 b(the)g(argumen)m(ts)g(of)h(the)f(propagators)h (will)c(b)s(e)j(mo)s(di\014ed)e(in)g(the)j(ob)m(vious)e(w)m(a)m(y)-8 b(.)270 629 y(An)m(yw)m(a)m(y)25 b(w)m(e)g(shall)e(not)h(consider)f (cancellations)h(b)s(et)m(w)m(een)h(the)g(trees)g(obtained)e(b)m(y)h (mo)s(de)270 759 y(lab)s(els)40 b(\015ipping,)i(except)g(for)g(the)g (case)h(in)d(whic)m(h)h(there)h(is)e(only)h(one)h(incoming)f(line)270 888 y(\()p Fh(i.e.)34 b Fo(p)511 902 y Fn(V)601 888 y Fq(=)29 b(1\),)34 b(and)e(w)m(e)h(shall)e(see)i(that)g(in)e(suc)m(h)h (a)h(case)h(the)f(resonance)g(factor)h(enjo)m(ys)270 1018 y(a)d(strong)f(parit)m(y)g(prop)s(ert)m(y)-8 b(.)370 1147 y(The)30 b(momen)m(tum)h(\015o)m(wing)g(through)f(a)i(line)d Fo(`)1978 1161 y Fn(u)2054 1147 y Fq(inside)g(a)i(resonance)h(will)d (dep)s(end)g(on)270 1277 y(the)37 b(mo)s(des)f(of)g(the)h(no)s(des)e Fo(w)j Fl(2)d Fo(V)57 b Fq(suc)m(h)36 b(that)h Fo(w)h Fl(\024)d Fo(u)i Fq(and)f(on)g(the)h(momen)m(ta)g(of)g(the)270 1406 y(en)m(tering)e(lines)f(of)h(the)g(resonance)h Fh(only)i(if)f(the) g(latter)i(end)e(into)h(no)-5 b(des)38 b(which)g(pr)-5 b(e)g(c)g(e)g(de)270 1536 y Fo(u)p Fq(;)31 b(w)m(e)f(call)g Fo(L)740 1550 y Fn(u)815 1536 y Fq(the)h(set)g(of)f(suc)m(h)g(lines.) 370 1665 y(F)-8 b(or)31 b(an)m(y)f Fo(`)742 1679 y Fn(u)818 1665 y Fq(in)m(ternal)f(to)i(a)g(resonance)g Fo(V)50 b Fq(w)m(e)31 b(ha)m(v)m(e:)1347 1840 y Fo(\027)1392 1855 y Fn(`)1421 1863 y Fe(u)1491 1840 y Fq(=)1599 1754 y Fk(X)1587 1951 y Fn(w)r Fj(2)p Fn(V)1591 2008 y(w)r Fj(\024)p Fn(u)1758 1840 y Fo(\027)1803 1854 y Fn(w)1880 1840 y Fq(+)1998 1754 y Fk(X)1971 1952 y Fn(`)2000 1933 y Fd(0)2022 1952 y Fj(2)p Fn(L)2117 1960 y Fe(u)2173 1840 y Fo(\027)2218 1856 y Fn(`)2247 1837 y Fd(0)2273 1840 y Fo(;)905 b Fq(\(3.1\))270 2145 y(and)31 b(in)f(the)h(corresp)s (onding)e(propagator,)k(w)m(e)f(can)f(write)g(the)g(argumen)m(t)h(of)g (the)f(cosine)270 2275 y(as:)1011 2481 y(2)p Fo(\031)s(p)p 1011 2521 147 4 v 1063 2605 a(q)1195 2456 y Fk(X)1183 2653 y Fn(w)r Fj(2)p Fn(V)1187 2710 y(w)r Fj(\024)p Fn(u)1354 2542 y Fo(\027)1399 2556 y Fn(w)1476 2542 y Fq(+)20 b(2)p Fo(\031)s(i)1713 2332 y Fk(2)1713 2492 y(6)1713 2550 y(4)1787 2456 y(X)1774 2653 y Fn(w)r Fj(2)p Fn(V)1778 2710 y(w)r Fj(\024)p Fn(u)1946 2542 y Fo(\021)s(\027)2039 2556 y Fn(w)2116 2542 y Fq(+)2235 2456 y Fk(X)2207 2654 y Fn(`)2236 2635 y Fd(0)2258 2654 y Fj(2)p Fn(L)2353 2662 y Fe(u)2409 2542 y Fo(\021)s(\027)2502 2557 y Fn(`)2531 2539 y Fd(0)2558 2332 y Fk(3)2558 2492 y(7)2558 2550 y(5)2633 2542 y Fo(:)545 b Fq(\(3.2\))370 2854 y(W)-8 b(e)29 b(shall)e(consider)g(the)i(resonance)g(factor)h Fl(V)1970 2868 y Fn(V)2030 2854 y Fq(\()p Fo(#)2119 2821 y Fj(0)2143 2854 y Fq(\))e(as)h(a)g(function)e(of)i(the)g(quan)m (tities)270 2984 y Fo(\026)325 2998 y Fi(1)397 2984 y Fl(\021)k Fo(\021)s(\027)594 2999 y Fn(`)623 3008 y Ff(1)662 2984 y Fq(,)i Fo(:)15 b(:)g(:)32 b Fq(,)k Fo(\026)975 2998 y Fn(p)1011 3009 y Fe(V)1101 2984 y Fl(\021)d Fo(\021)s(\027)1298 2999 y Fn(`)1327 3007 y Fe(p)1359 3024 y(V)1456 2984 y Fq(of)i(the)h(incoming)d(lines)h Fo(`)2370 2998 y Fi(1)2409 2984 y Fq(,)h Fo(:)15 b(:)g(:)31 b Fq(,)37 b Fo(`)2705 2998 y Fn(p)2741 3009 y Fe(V)2798 2984 y Fq(,)f Fh(i.e.)h Fl(V)3078 2998 y Fn(V)3138 2984 y Fq(\()p Fo(#)3227 2951 y Fj(0)3251 2984 y Fq(\))c Fl(\021)270 3113 y(V)326 3127 y Fn(V)387 3113 y Fq(\()p Fo(#)476 3080 y Fj(0)499 3113 y Fq(;)15 b Fo(\021)s(\027)632 3128 y Fn(`)661 3137 y Ff(1)700 3113 y Fo(;)g(:)g(:)g(:)32 b(;)15 b(\021)s(\027)1010 3128 y Fn(`)1039 3136 y Fe(p)1071 3153 y(V)1133 3113 y Fq(\).)41 b(F)-8 b(or)31 b(an)m(y)g Fo(u)26 b Fl(2)e Fo(V)d Fq(,)30 b(w)m(e)h(ha)m(v)m(e)h Fo(L)2267 3127 y Fn(u)2337 3113 y Fl(\022)25 b(f)p Fo(`)2516 3127 y Fi(1)2555 3113 y Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(`)2810 3127 y Fn(p)2846 3138 y Fe(V)2904 3113 y Fl(g)p Fq(.)370 3243 y(W)-8 b(e)31 b(can)g(write:)353 3485 y Fl(V)409 3499 y Fn(V)470 3485 y Fq(\()p Fo(#)559 3447 y Fj(0)582 3485 y Fq(;)15 b Fo(\021)s(\027)715 3500 y Fn(`)744 3509 y Ff(1)783 3485 y Fo(;)g(:)g(:)g(:)32 b(;)15 b(\021)s(\027)1093 3500 y Fn(`)1122 3508 y Fe(p)1154 3525 y(V)1216 3485 y Fq(\))26 b(=)f Fl(V)1429 3499 y Fn(V)1490 3485 y Fq(\()p Fo(#)1579 3447 y Fj(0)1602 3485 y Fq(;)15 b(0)p Fo(;)g(:)g(:)g(:)33 b(;)15 b Fq(0\))21 b(+)2129 3366 y Fn(p)2165 3377 y Fe(V)2108 3398 y Fk(X)2097 3593 y Fn(m)p Fi(=1)2265 3485 y Fo(\021)s(\027)2358 3500 y Fn(`)2387 3508 y Fe(m)2520 3423 y Fo(@)p 2460 3464 175 4 v 2460 3547 a(@)5 b(\026)2568 3561 y Fn(m)2644 3485 y Fl(V)2700 3499 y Fn(V)2761 3485 y Fq(\()p Fo(#)2850 3447 y Fj(0)2873 3485 y Fq(;)15 b(0)p Fo(;)g(:)g(:)g(:)33 b(;)15 b Fq(0\))458 3799 y(+)633 3680 y Fn(p)669 3691 y Fe(V)612 3712 y Fk(X)549 3910 y Fn(m;m)693 3892 y Fd(0)716 3910 y Fi(=1)821 3799 y Fo(\021)869 3761 y Fi(2)909 3799 y Fo(\027)954 3814 y Fn(`)983 3822 y Fe(m)1046 3799 y Fo(\027)1091 3814 y Fn(`)1120 3833 y Fe(m)1174 3819 y Fd(0)1220 3675 y Fk(Z)1311 3701 y Fi(1)1270 3881 y(0)1365 3799 y Fq(d)p Fo(t)g Fq(\(1)21 b Fl(\000)f Fo(t)p Fq(\))1873 3737 y Fo(@)1926 3704 y Fi(2)p 1734 3778 372 4 v 1734 3861 a Fo(@)5 b(\026)1842 3875 y Fn(m)1908 3861 y Fo(@)g(\026)2016 3876 y Fn(m)2078 3857 y Fd(0)2115 3799 y Fl(V)2171 3813 y Fn(V)2232 3799 y Fq(\()p Fo(#)2321 3761 y Fj(0)2344 3799 y Fq(;)15 b Fo(t\021)s(\027)2510 3814 y Fn(`)2539 3823 y Ff(1)2578 3799 y Fo(;)g(:)g(:)g(:)32 b(;)15 b(t\021)s(\027)2921 3814 y Fn(`)2950 3822 y Fe(p)2982 3839 y(V)3044 3799 y Fq(\))p Fo(;)99 b Fq(\(3.3\))270 4049 y(where:)1497 4121 y Fo(@)p 1436 4161 175 4 v 1436 4245 a(@)5 b(\026)1544 4259 y Fn(m)1621 4182 y Fl(V)1677 4196 y Fn(V)1737 4182 y Fq(\()p Fo(#)1826 4145 y Fj(0)1850 4182 y Fq(;)15 b(0)p Fo(;)g(:)g(:)g(:)33 b(;)15 b Fq(0\))270 4363 y(denotes)28 b(the)f(\014rst)g(deriv)-5 b(ativ)m(e)27 b(of)g Fl(V)1504 4377 y Fn(V)1565 4363 y Fq(\()p Fo(#)1654 4330 y Fj(0)1677 4363 y Fq(;)15 b Fo(\026)1772 4377 y Fi(1)1812 4363 y Fo(;)g(:)g(:)g(:)32 b(;)15 b(\026)2084 4377 y Fn(p)2120 4388 y Fe(V)2177 4363 y Fq(\))28 b(with)e(resp)s(ect)h(to)h(the)f (argumen)m(t)270 4493 y Fo(\026)325 4507 y Fn(m)391 4493 y Fq(,)34 b(computed)e(in)f Fo(\026)1034 4507 y Fi(1)1102 4493 y Fq(=)e Fo(:)15 b(:)g(:)30 b Fq(=)e Fo(\026)1491 4507 y Fn(p)1527 4518 y Fe(V)1613 4493 y Fq(=)g(0,)34 b(while)d(the)h(term)h(in)e(the)i(second)f(line)f(is)h(the)270 4622 y(in)m(tegral)e(in)m(terp)s(olation)f(form)m(ula)g(for)h(the)h (second)f(order)g(remainder.)370 4752 y(Of)c(course)h(w)m(e)h(use)f (here)g(the)g(fact)h(that)g(the)f(resonance)g(factor)i(is)d(a)h (function)f(of)h(class)270 4881 y Fo(C)342 4848 y Fi(2)408 4881 y Fq({)g(actually)g(analytic)f({)i(in)e(the)h(parameters)g Fo(\026)2016 4895 y Fi(1)2055 4881 y Fq(,)g Fo(:)15 b(:)g(:)32 b Fq(,)c Fo(\026)2352 4895 y Fn(p)2388 4906 y Fe(V)2445 4881 y Fq(.)39 b(As)27 b(it)g(can)g(b)s(e)f(seen)h(from)270 5011 y(\(3.2\))r(,)34 b(the)g(resonance)f(v)-5 b(alue)33 b(dep)s(ends)f(on)h Fo(\026)1870 5025 y Fi(1)1909 5011 y Fq(,)h Fo(:)15 b(:)g(:)31 b Fq(,)j Fo(\026)2218 5025 y Fn(p)2254 5036 y Fe(V)2344 5011 y Fq(only)f(through)f(the)i(last)f (sum)270 5140 y(app)s(earing)c(in)g(the)i(argumen)m(t)f(of)h(the)g (propagators.)p eop %%Page: 12 12 12 11 bop 390 46 a Fi(12)704 b(ALBER)-6 b(TO)30 b(BERRETTI)h(AND)f (GUIDO)h(GENTILE)490 241 y Fq(Note)e(that,)g(as)g(the)f(incoming)f (lines)f(are)j(on)f(scale)g Fo(n)d Fq(=)g(0,)k Fo(\027)2592 256 y Fn(`)2621 264 y Fe(m)2712 241 y Fq(is)e(a)h(m)m(ultiple)e(of)i Fo(q)j Fq(for)390 370 y(eac)m(h)g Fo(m)25 b Fq(=)g(1)p Fo(;)15 b(:)g(:)g(:)32 b(;)15 b(p)1103 384 y Fn(V)1164 370 y Fq(.)41 b(Note)31 b(also)f(that)h(the)f(tree)h(v)-5 b(alues)29 b(V)-8 b(al\()p Fo(#)2656 337 y Fj(0)2680 370 y Fq(\))30 b(di\013er)f(only)g(as)h(far)g(as)390 500 y(the)i(resonance)f(factors)i(are)e(concerned,)h(b)s(ecause)g(all)e (the)i(other)f(factors)h(con)m(tributing)390 629 y(to)j(the)f(tree)g(v) -5 b(alue)34 b(are)g(equal,)h Fh(i.e.)g Fq(V)-8 b(al\()p Fo(#)1895 596 y Fj(0)1919 629 y Fq(\))31 b(=)g Fl(A)p Fq(\()p Fo(#)p Fq(\))p Fl(V)2340 643 y Fn(V)2401 629 y Fq(\()p Fo(#)2490 596 y Fj(0)2513 629 y Fq(\),)k(where)f Fl(A)p Fq(\()p Fo(#)p Fq(\))g(is)f(a)h(factor)390 759 y(whic)m(h)29 b(has)h(the)h(same)g(v)-5 b(alue)29 b(for)i(all)e Fo(#)1751 726 y Fj(0)1799 759 y Fl(2)c(F)1950 773 y Fn(V)2011 759 y Fq(\()p Fo(#)p Fq(\).)490 888 y(The)k(\014rst)f(term)i(in)e (\(3.3\))k(is)c(the)i(term)f(whic)m(h)g(arises)g(from)g(the)g (resonance)i(factor)f(b)m(y)390 1018 y(neglecting)g(the)h(c)m(hange)g (in)e(the)i(momen)m(ta.)490 1147 y(De\014ne)800 1124 y(~)776 1147 y Fl(F)841 1161 y Fn(V)902 1147 y Fq(\()p Fo(#)p Fq(\))36 b(as)f(the)g(equiv)-5 b(alence)35 b(class)g(of)g(the)g (trees)h(in)e Fl(F)2710 1161 y Fn(V)2771 1147 y Fq(\()p Fo(#)p Fq(\),)j(suc)m(h)e(that)h(t)m(w)m(o)390 1277 y(trees)d(in)e Fl(F)783 1291 y Fn(V)844 1277 y Fq(\()p Fo(#)p Fq(\))i(are)g (considered)e(equiv)-5 b(alen)m(t)32 b(if)f(eac)m(h)j(one)f(can)f(b)s (e)g(obtained)g(from)g(the)390 1406 y(other)f(b)m(y)f(the)g(action)h (of)g(a)f(transformation)g(of)h(the)f(group)g Fl(P)2563 1420 y Fn(V)2655 1406 y Fq(of)g(t)m(yp)s(e)h(2.)490 1536 y(Then,)40 b(summing)d(the)i(v)-5 b(alues)38 b(of)h(the)g(trees)g(b)s (elonging)f(to)h Fl(F)2713 1550 y Fn(V)2774 1536 y Fq(\()p Fo(#)p Fq(\),)j(w)m(e)d(can)g(group)390 1665 y(them)30 b(in)m(to)f(subfamilies)e(of)i(inequiv)-5 b(alen)m(t)28 b(trees)i(whose)g(con)m(tributions)e(are)i(di\013eren)m(t)f(as)390 1795 y(for)h(eac)m(h)i(no)s(de)d(there)i(is)e(a)i(factor:)1612 1919 y(1)p 1577 1959 117 4 v 1577 2043 a Fo(p)1623 2057 y Fn(u)1668 2043 y Fq(!)1703 1852 y Fk(\022)1770 1919 y Fo(p)1816 1933 y Fn(u)1771 2043 y Fo(s)1814 2057 y Fn(u)1860 1852 y Fk(\023)1953 1980 y Fq(=)2148 1919 y(1)p 2059 1959 225 4 v 2059 2043 a Fo(s)2102 2057 y Fn(u)2146 2043 y Fq(!)p Fo(r)2212 2057 y Fn(u)2257 2043 y Fq(!)2293 1980 y Fo(;)1005 b Fq(\(3.4\))390 2189 y(as)40 b(all)g(terms)g(whic)m (h)f(are)h(obtained)g(b)m(y)g(p)s(erm)m(utations)f(are)i(summed)e (together)j(\(this)390 2318 y(giv)m(es)27 b(the)f(binomial)e(co)s (e\016cien)m(t)j(in)f(the)g(left)g(hand)g(side)f(of)i(the)f(ab)s(o)m(v) m(e)i(equation\),)f(times)390 2448 y(a)k(factor:)1558 2622 y Fo(\027)1609 2585 y Fn(p)1645 2593 y Fe(u)1685 2585 y Fi(+1)1603 2645 y Fn(u)1805 2622 y Fq(=)25 b Fo(\027)1952 2585 y Fi(\()p Fn(s)2012 2593 y Fe(u)2052 2585 y Fi(+1\)+)p Fn(r)2256 2593 y Fe(u)1946 2645 y Fn(u)2302 2622 y Fo(;)996 b Fq(\(3.5\))390 2797 y(times)30 b(a)h(propagator)g(1)p Fo(=\015)5 b Fq(\()p Fo(\027)1396 2812 y Fn(`)1425 2820 y Fe(u)1471 2797 y Fq(\).)490 2927 y(Then)29 b(for)h Fo(\026)25 b Fq(=)g(0)31 b(w)m(e)g(can)f(write:)732 3007 y Fk(X)642 3209 y Fn(#)684 3190 y Fd(0)706 3209 y Fj(2F)803 3220 y Fe(V)857 3209 y Fi(\()p Fn(#)p Fi(\))969 3094 y Fq(V)-8 b(al\()p Fo(#)1189 3056 y Fj(0)1213 3094 y Fq(\))25 b(=)g Fo(A)p Fq(\()p Fo(#)p Fq(\))1667 3007 y Fk(X)1576 3209 y Fn(#)1618 3190 y Fd(0)1641 3209 y Fj(2F)1738 3220 y Fe(V)1792 3209 y Fi(\()p Fn(#)p Fi(\))1904 3094 y Fl(V)1960 3108 y Fn(V)2021 3094 y Fq(\()p Fo(#)2110 3056 y Fj(0)2133 3094 y Fq(\))1273 3434 y(=)g Fo(A)p Fq(\()p Fo(#)p Fq(\))1667 3348 y Fk(X)1576 3549 y Fn(#)1618 3530 y Fd(0)1641 3549 y Fj(2F)1738 3560 y Fe(V)1792 3549 y Fi(\()p Fn(#)p Fi(\))1904 3279 y Fk(")1972 3348 y(Y)1957 3545 y Fn(u)p Fj(2)p Fn(V)2161 3372 y Fq(1)p 2127 3413 113 4 v 2127 3496 a Fo(s)2170 3510 y Fn(u)2215 3496 y Fq(!)2271 3372 y Fo(\027)2322 3339 y Fn(s)2355 3347 y Fe(u)2396 3339 y Fi(+1)2316 3395 y Fn(u)p 2260 3413 242 4 v 2260 3496 a Fo(\015)5 b Fq(\()p Fo(\027)2392 3511 y Fn(`)2421 3519 y Fe(u)2466 3496 y Fq(\))2511 3279 y Fk(#)16 b( )2666 3348 y(Y)2652 3545 y Fn(u)p Fj(2)p Fn(V)2821 3372 y Fo(\027)2872 3339 y Fn(r)2904 3347 y Fe(u)2866 3395 y Fn(u)p 2821 3413 128 4 v 2829 3496 a Fo(r)2870 3510 y Fn(u)2916 3496 y Fq(!)2959 3279 y Fk(!)1273 3774 y Fq(=)25 b Fo(A)p Fq(\()p Fo(#)p Fq(\))1576 3619 y Fk(")1644 3688 y(Y)1630 3885 y Fn(u)p Fj(2)p Fn(V)1834 3713 y Fq(1)p 1800 3754 113 4 v 1800 3837 a Fo(s)1843 3851 y Fn(u)1888 3837 y Fq(!)1944 3713 y Fo(\027)1995 3680 y Fn(s)2028 3688 y Fe(u)2068 3680 y Fi(+1)1989 3735 y Fn(u)p 1933 3754 242 4 v 1933 3837 a Fo(\015)5 b Fq(\()p Fo(\027)2065 3852 y Fn(`)2094 3860 y Fe(u)2139 3837 y Fq(\))2184 3619 y Fk(#)2343 3688 y(X)2252 3901 y Fn(#)2294 3882 y Fd(0)2317 3901 y Fj(2)2383 3884 y Fi(~)2364 3901 y Fj(F)2414 3912 y Fe(V)2468 3901 y Fi(\()p Fn(#)p Fi(\))2580 3619 y Fk( )2666 3688 y(Y)2652 3885 y Fn(u)p Fj(2)p Fn(V)2821 3713 y Fo(\027)2872 3680 y Fn(r)2904 3688 y Fe(u)2866 3735 y Fn(u)p 2821 3754 128 4 v 2829 3837 a Fo(r)2870 3851 y Fn(u)2916 3837 y Fq(!)2959 3619 y Fk(!)3046 3774 y Fo(;)3323 3485 y Fq(\(3.6\))390 4043 y(where)27 b(w)m(e)g(ha)m(v)m(e)h(used)e(that)i (for)f Fo(\026)e Fq(=)g(0)i(the)g(factors)h(in)e(square)h(brac)m(k)m (ets)h(ha)m(v)m(e)g(the)g(same)390 4173 y(v)-5 b(alue)30 b(for)g(all)f Fo(#)944 4140 y Fj(0)993 4173 y Fl(2)1103 4150 y Fq(~)1078 4173 y Fl(F)1143 4187 y Fn(V)1205 4173 y Fq(\()p Fo(#)p Fq(\).)41 b(The)30 b(last)g(sum)f(in)g(\(3.6\))k(can)d (b)s(e)g(rewritten)f(as:)722 4324 y Fk(X)632 4537 y Fn(#)674 4518 y Fd(0)696 4537 y Fj(2)762 4520 y Fi(~)743 4537 y Fj(F)793 4548 y Fe(V)847 4537 y Fi(\()p Fn(#)p Fi(\))959 4255 y Fk(")1026 4324 y(Y)1012 4521 y Fn(u)p Fj(2)p Fn(V)1182 4349 y Fo(\027)1233 4316 y Fn(r)1265 4324 y Fe(u)1227 4372 y Fn(u)p 1182 4390 V 1190 4473 a Fo(r)1231 4487 y Fn(u)1276 4473 y Fq(!)1319 4255 y Fk(#)1398 4411 y Fq(=)1645 4324 y Fk(X)1594 4526 y Fj(f)p Fn(r)1661 4534 y Fe(u)1702 4526 y Fj(\025)p Fi(0)p Fj(g)1494 4551 y Fc(P)1568 4622 y Fe(u)p Fd(2)p Fe(V)1712 4601 y Fn(r)1744 4609 y Fe(u)1784 4601 y Fi(=)p Fn(p)1875 4612 y Fe(V)1958 4324 y Fk(Y)1943 4521 y Fn(u)p Fj(2)p Fn(V)2113 4349 y Fo(\027)2164 4316 y Fn(r)2196 4324 y Fe(u)2158 4372 y Fn(u)p 2113 4390 V 2121 4473 a Fo(r)2162 4487 y Fn(u)2207 4473 y Fq(!)2276 4411 y(=)2425 4349 y(1)p 2382 4390 132 4 v 2382 4473 a Fo(p)2428 4487 y Fn(V)2488 4473 y Fq(!)2539 4255 y Fk( )2618 4324 y(X)2611 4521 y Fn(u)p Fj(2)p Fn(V)2771 4411 y Fo(\027)2816 4425 y Fn(u)2861 4255 y Fk(!)2933 4278 y Fn(p)2969 4289 y Fe(V)3041 4411 y Fo(;)257 b Fq(\(3.7\))390 4757 y(so)25 b(that)g(a)g(quan)m(tit)m(y)g(prop)s(ortional)d(to)j(the)g Fo(p)1934 4771 y Fn(V)1995 4757 y Fq(-th)f(p)s(o)m(w)m(er)h(of)2493 4689 y Fk(P)2589 4784 y Fn(u)p Fj(2)p Fn(V)2753 4757 y Fo(\027)2798 4771 y Fn(u)2867 4757 y Fq(is)f(obtained.)38 b(But)390 4887 y(suc)m(h)30 b(a)h(sum)e(is)h(zero)h(b)m(y)f (de\014nition)e(of)j(resonance)g({)f(see)h(\(2.3\))r(.)490 5016 y(Also)37 b(the)h(second)h(term)f(in)e(\(3.3\))k(v)-5 b(anishes,)39 b(after)f(summing)e(o)m(v)m(er)k(the)e(trees)g Fo(#)3388 4983 y Fj(0)3449 5016 y Fl(2)390 5146 y(F)455 5160 y Fn(V)516 5146 y Fq(\()p Fo(#)p Fq(\).)i(T)-8 b(o)26 b(pro)m(v)m(e)h(this)e(w)m(e)h(shall)f(consider)g(separately)h(the)g (cases)h Fo(p)2786 5160 y Fn(V)2872 5146 y Fl(\025)e Fq(2)h(and)f Fo(p)3257 5160 y Fn(V)3343 5146 y Fq(=)g(1.)p eop %%Page: 13 13 13 12 bop 955 46 a Fi(SCALING)31 b(PR)n(OPER)-6 b(TIES)31 b(OF)f(THE)h(ST)-6 b(AND)n(ARD)30 b(MAP)613 b(13)370 241 y Fq(In)32 b(the)h(\014rst)f(case,)j(when)d(the)h(deriv)-5 b(ativ)m(e)32 b(\()q Fo(@)5 b(=@)g(\026)2116 255 y Fn(m)2183 241 y Fq(\))16 b Fl(V)2290 255 y Fn(V)2350 241 y Fq(\()p Fo(#)p Fq(;)f(0)p Fo(;)g(:)g(:)g(:)33 b(;)15 b Fq(0\))34 b(is)e(considered,)270 370 y(let)h(us)f(compare)h(all)f(the)h(trees)g Fo(#)1451 337 y Fj(0)1507 370 y Fq(in)f(the)h(subfamily)d(of)2322 347 y(~)2298 370 y Fl(F)2363 384 y Fn(V)2424 370 y Fq(\()p Fo(#)p Fq(\))j(in)f(whic)m(h)f(the)i(line)f Fo(`)3324 384 y Fn(m)270 500 y Fq(is)c(k)m(ept)h(\014xed)g(\(call)34 b(\026)-51 b Fo(u)29 b Fq(the)g(no)s(de)f(whic)m(h)g(suc)m(h)h(a)g (line)e(en)m(ters\),)k(while)26 b(all)i(other)h(lines)f(are)270 629 y(shifted)35 b(\()p Fh(i.e.)j Fq(detac)m(hed)f(and)e(reattac)m(hed) j(to)f(all)e(no)s(des)h(inside)e(the)i(resonance\).)59 b(The)270 759 y(di\013erence)36 b(with)g(resp)s(ect)h(to)h(the)f (previous)f(case,)k(discussed)35 b(ab)s(o)m(v)m(e,)40 b(is)c(that)h(the)h(line)270 888 y(with)29 b(momen)m(tum)g Fo(\027)998 903 y Fn(`)1027 911 y Fe(m)1120 888 y Fq(can)h(b)s(e)f(c)m (hosen)i(in)e Fo(r)1852 902 y Fi(\026)-40 b Fn(u)1921 888 y Fq(w)m(a)m(ys)31 b(among)g(the)f Fo(r)2636 902 y Fi(\026)-40 b Fn(u)2706 888 y Fq(lines)28 b(en)m(tering)i(the)270 1018 y(no)s(de)36 b(\026)-51 b Fo(u)25 b Fl(2)g Fo(V)50 b Fq(and)30 b(outside)g Fo(V)20 b Fq(.)41 b(This)28 b(means)j(that)g(w) m(e)f(can)h(write:)1336 1173 y Fo(\027)1387 1129 y Fn(p)1423 1137 y Fe(u)1463 1129 y Fi(+1)1381 1185 y Fn(u)p 1336 1213 222 4 v 1389 1297 a Fo(p)1435 1311 y Fn(u)1479 1297 y Fq(!)1567 1106 y Fk(\022)1634 1173 y Fo(p)1680 1187 y Fn(u)1636 1297 y Fo(s)1679 1311 y Fn(u)1725 1106 y Fk(\023)1817 1234 y Fq(=)1923 1173 y Fo(\027)1974 1125 y Fi(\()p Fn(s)2034 1133 y Fe(u)2075 1125 y Fi(+1\)+)p Fn(r)2279 1133 y Fe(u)1968 1185 y Fn(u)p 1923 1213 402 4 v 2011 1297 a Fo(s)2054 1311 y Fn(u)2099 1297 y Fq(!)p Fo(r)2165 1311 y Fn(u)2210 1297 y Fq(!)3203 1234 y(\(3.8\))270 1437 y(for)f(all)f(no)s(des)h Fo(u)25 b Fl(6)p Fq(=)31 b(\026)-51 b Fo(u)p Fq(,)31 b(and:)1155 1606 y Fo(\027)1206 1562 y Fn(p)1247 1570 y Ff(\026)-35 b Fe(u)1282 1562 y Fi(+1)1205 1628 y(\026)-40 b Fn(u)p 1155 1647 222 4 v 1208 1730 a Fo(p)1259 1744 y Fi(\026)g Fn(u)1299 1730 y Fq(!)1387 1540 y Fk(\022)1454 1606 y Fo(p)1505 1620 y Fi(\026)g Fn(u)1455 1730 y Fo(s)1503 1744 y Fi(\026)g Fn(u)1544 1540 y Fk(\023)1611 1668 y Fo(r)1657 1682 y Fi(\026)g Fn(u)1722 1668 y Fq(=)1828 1606 y Fo(\027)1879 1559 y Fi(\()p Fn(s)1944 1567 y Ff(\026)-35 b Fe(u)1980 1559 y Fi(+1\)+\()p Fn(r)2216 1567 y Ff(\026)g Fe(u)2253 1559 y Fj(\000)p Fi(1\))1878 1628 y(\026)-40 b Fn(u)p 1828 1647 547 4 v 1876 1730 a Fo(s)1924 1744 y Fi(\026)g Fn(u)1963 1730 y Fq(!\()p Fo(r)2069 1744 y Fi(\026)g Fn(u)2130 1730 y Fl(\000)20 b Fq(1\)!)2384 1668 y Fo(\027)2434 1682 y Fi(\026)-40 b Fn(u)2474 1668 y Fo(;)704 b Fq(\(3.9\))270 1876 y(for)36 b(\026)-51 b Fo(u)q Fq(.)41 b(Then)30 b(w)m(e)h(ha)m(v)m (e)h(an)f(expression)e(analogous)i(to)h(\(3.6\))q(,)f(with)f(the)h (only)e(di\013erence)270 2006 y(that)22 b(the)g(lab)s(els)d Fl(f)p Fo(r)938 2020 y Fn(u)984 2006 y Fl(g)j Fq(ha)m(v)m(e)g(to)g(b)s (e)f(replaced)g(with)f(lab)s(els)g Fl(f)p Fo(r)2352 1973 y Fj(0)2349 2028 y Fn(u)2394 2006 y Fl(g)p Fq(,)k(de\014ned)c(as)i Fo(r)2939 1973 y Fj(0)2936 2028 y Fn(u)3006 2006 y Fq(=)j Fo(r)3143 2020 y Fn(u)3191 2006 y Fl(\000)r Fo(\016)3304 2020 y Fn(u)5 b Fi(\026)-40 b Fn(u)270 2135 y Fl(8)p Fo(u)25 b Fl(2)g Fo(V)20 b Fq(,)30 b(suc)m(h)f(that)1012 2067 y Fk(P)1108 2162 y Fn(u)p Fj(2)p Fn(V)1272 2135 y Fo(r)1316 2102 y Fj(0)1313 2158 y Fn(u)1384 2135 y Fq(=)24 b Fo(p)1525 2149 y Fn(V)1605 2135 y Fl(\000)18 b Fq(1:)41 b(so)30 b(the)f(last)h(sum)f(in)f(the)i(third)d(line)h(of)37 b(\(3.6\))270 2265 y(has)30 b(to)h(b)s(e)f(replaced)g(b)m(y:)894 2496 y Fo(\027)944 2510 y Fi(\026)-40 b Fn(u)1089 2410 y Fk(X)999 2623 y Fn(#)1041 2604 y Fd(0)1063 2623 y Fj(2)1129 2606 y Fi(~)1110 2623 y Fj(F)1160 2634 y Fe(V)1214 2623 y Fi(\()p Fn(#)p Fi(\))1340 2410 y Fk(Y)1326 2607 y Fn(u)p Fj(2)p Fn(V)1496 2435 y Fo(\027)1547 2387 y Fn(r)1581 2364 y Fd(0)1579 2404 y Fe(u)1541 2447 y Fn(u)p 1496 2476 128 4 v 1504 2559 a Fo(r)1548 2533 y Fj(0)1545 2581 y Fn(u)1590 2559 y Fq(!)1659 2496 y(=)25 b Fo(\027)1805 2510 y Fi(\026)-40 b Fn(u)2056 2410 y Fk(X)2005 2613 y Fj(f)p Fn(r)2074 2589 y Fd(0)2072 2629 y Fe(u)2113 2613 y Fj(g\025)p Fi(0)1860 2640 y Fc(P)1934 2710 y Fe(u)p Fd(2)p Fe(V)2078 2689 y Fn(r)2112 2666 y Fd(0)2110 2706 y Fe(u)2150 2689 y Fi(=)p Fn(p)2241 2700 y Fe(V)2294 2689 y Fj(\000)p Fi(1)2414 2410 y Fk(Y)2399 2607 y Fn(u)p Fj(2)p Fn(V)2569 2435 y Fo(\027)2620 2387 y Fn(r)2654 2364 y Fd(0)2652 2404 y Fe(u)2614 2447 y Fn(u)p 2569 2476 V 2577 2559 a Fo(r)2621 2533 y Fj(0)2618 2581 y Fn(u)2663 2559 y Fq(!)1659 2940 y(=)1899 2878 y Fo(\027)1949 2892 y Fi(\026)g Fn(u)p 1765 2919 360 4 v 1765 3002 a Fq(\()p Fo(p)1846 3016 y Fn(V)1927 3002 y Fl(\000)20 b Fq(1\)!)2149 2785 y Fk( )2228 2854 y(X)2221 3050 y Fn(u)p Fj(2)p Fn(V)2381 2940 y Fo(\027)2426 2954 y Fn(u)2471 2785 y Fk(!)2543 2807 y Fn(p)2579 2818 y Fe(V)2631 2807 y Fj(\000)p Fi(1)2741 2940 y Fo(;)3158 2723 y Fq(\(3.10\))270 3181 y(so)31 b(that)g(w)m(e)f(ha)m(v)m(e)i(again)e(v)-5 b(anishing)28 b(con)m(tributions.)370 3310 y(On)38 b(the)h(con)m(trary) -8 b(,)42 b(if)c Fo(p)1225 3324 y Fn(V)1325 3310 y Fq(=)h(1,)i(the)e (ab)s(o)m(v)m(e)h(reasoning)e(do)s(es)h(not)g(apply)e(\(as)j(there)270 3440 y(is)e(only)g(one)i(incoming)e(line\).)65 b(An)m(yw)m(a)m(y)40 b(the)g(function)e(\()p Fo(@)5 b(=@)g(\026)2549 3454 y Fi(1)2589 3440 y Fq(\))16 b Fl(V)2696 3454 y Fn(V)2756 3440 y Fq(\()p Fo(#)p Fq(;)f(0\))41 b(is)d(an)h(o)s(dd)270 3569 y(function,)25 b(as)g(all)f(the)h(propagators)g(are)h(ev)m(en)f (in)f(their)g(argumen)m(ts)h(\(so)g(that)h(the)f(deriv)m(ed)270 3699 y(one)34 b(b)s(ecomes)h(o)s(dd\))e(and)h(the)g(n)m(umerator)g(con) m(tains)h(an)f(ev)m(en)h(n)m(um)m(b)s(er)e(of)h Fo(\027)3000 3713 y Fn(u)3045 3699 y Fq('s.)52 b(Then)270 3828 y(b)m(y)35 b(rev)m(ersing)f(the)h(signs)e(of)i(the)g(lab)s(els)e Fo(\027)1752 3842 y Fn(u)1797 3828 y Fq(,)j Fo(u)d Fl(2)f Fo(V)20 b Fq(,)36 b(the)f(n)m(umerator)g(will)d(not)j(c)m(hange)270 3958 y(\(as)c Fo(p)463 3972 y Fn(V)543 3958 y Fq(+)19 b(1)25 b(=)g(2\),)31 b(while)d(the)i(o)m(v)m(erall)g(sign)f(of)i(the)f (denominator)f(c)m(hanges,)i(so)f(that)h(the)270 4087 y(sum)e(o)m(v)m(er)j(the)f(t)m(w)m(o)g(considered)f(tree)h(v)-5 b(alues)29 b(giv)m(es)i(zero.)370 4217 y(The)e(in)m(tegral)h(app)s (earing)e(in)g(the)j(third)d(term)h(in)g(\(3.3\))j(giv)m(es,)e(for)g (eac)m(h)h Fo(#)2980 4184 y Fj(0)3028 4217 y Fl(2)25 b(F)3179 4231 y Fn(V)3240 4217 y Fq(\()p Fo(#)p Fq(\),)270 4346 y(a)36 b(con)m(tribution)e(b)s(ounded)g(b)m(y)h Fo(d)1429 4360 y Fi(0)1469 4346 y Fo(q)1513 4313 y Fi(2)1588 4346 y Fq(times)g(the)h(original)e(b)s(ound)f(on)j Fl(jV)2830 4360 y Fn(V)2891 4346 y Fq(\()p Fo(#)p Fq(\))p Fl(j)q Fq(.)56 b(In)35 b(fact)270 4476 y(the)d(propagators)f(of)h(all)e(lines) g(inside)f Fo(V)51 b Fq(are)32 b(b)s(ounded)d(b)m(y)i Fo(c)2424 4443 y Fj(\000)p Fi(1)2519 4476 y Fo(q)2563 4443 y Fi(2)2602 4476 y Fq(,)h(and)f(their)f(\014rst)h(and)270 4605 y(second)e(deriv)-5 b(ativ)m(es,)29 b(resp)s(ectiv)m(ely)-8 b(,)29 b(b)m(y)f Fo(d)1722 4619 y Fi(1)1762 4605 y Fo(q)1806 4572 y Fi(3)1874 4605 y Fq(and)g Fo(d)2096 4619 y Fi(2)2136 4605 y Fo(q)2180 4572 y Fi(4)2248 4605 y Fq(for)h(some)g(constan)m(ts)h Fo(d)3060 4619 y Fi(1)3128 4605 y Fq(and)e Fo(d)3350 4619 y Fi(2)270 4735 y Fq(\(suc)m(h)d(lines)f(remain)g(on)h(scale)h Fo(n)f Fq(=)g(1:)38 b(the)26 b(incoming)e(lines)f(con)m(tribute)i(a)h (quan)m(tit)m(y)f(that)270 4864 y(mo)s(di\014es)k(only)g(the)i (imaginary)e(part)h(of)g(the)h(momen)m(ta)g(of)g(lines)e(inside)f Fo(V)20 b Fq(\).)370 4994 y(Then)29 b(the)i(lemma)e(follo)m(ws,)h(with) f Fo(D)1662 5008 y Fi(0)1727 4994 y Fq(=)c(max)p Fl(f)p Fo(cd)2123 4961 y Fi(2)2123 5018 y(1)2164 4994 y Fo(;)15 b(d)2251 5008 y Fi(2)2291 4994 y Fl(g)p Fo(q)2380 4961 y Fi(2)2420 4994 y Fq(.)p 3324 4994 4 62 v 3328 4936 55 4 v 3328 4994 V 3382 4994 4 62 v eop %%Page: 14 14 14 13 bop 390 46 a Fi(14)704 b(ALBER)-6 b(TO)30 b(BERRETTI)h(AND)f (GUIDO)h(GENTILE)988 241 y Fq(4.)50 b Fp(Comput)-6 b(a)g(tion)34 b(of)f(the)h(critical)g(exponent)490 424 y Fq(Let)29 b(us)f(consider)f(the)i(co)s(e\016cien)m(t)g Fo(u)1747 438 y Fn(\027)1791 424 y Fq(\()p Fo(")p Fq(\))g(in)e(\(1.4\))r(.)40 b(By)29 b(de\014nition)d(of)j(momen)m(tum,)g(in)390 554 y(\(1.8\))37 b(only)e(trees)g(of)h(order)f Fo(k)h Fl(\025)e(j)p Fo(\027)6 b Fl(j)35 b Fq(can)h(con)m(tribute)f(to)h Fo(u)2497 568 y Fn(\027)2540 554 y Fq(\()p Fo(")p Fq(\),)i(so)d(that)h Fo(u)3085 506 y Fi(\()p Fn(k)r Fi(\))3085 566 y Fn(\027)3216 554 y Fq(=)d(0)j(for)390 683 y Fo(k)28 b(<)d(\027)6 b Fq(.)41 b(Therefore)30 b(w)m(e)h(can)f(write:)1025 892 y Fo(u)1077 906 y Fn(\027)1121 892 y Fq(\()p Fo(")p Fq(\))c(=)1375 805 y Fk(X)1355 1007 y Fn(k)r Fj(\025j)p Fn(\027)t Fj(j)1542 892 y Fo(")1584 854 y Fn(k)1627 892 y Fo(u)1679 854 y Fi(\()p Fn(k)r Fi(\))1679 914 y Fn(\027)1802 892 y Fq(=)e Fo(")1939 854 y Fj(j)p Fn(\027)t Fj(j)2022 892 y Fo(u)2074 854 y Fi(\()p Fj(j)p Fn(\027)t Fj(j)p Fi(\))2074 914 y Fn(\027)2232 892 y Fq(+)2418 778 y Fj(1)2388 805 y Fk(X)2323 1007 y Fn(k)r Fi(=)p Fj(j)p Fn(\027)t Fj(j)p Fi(+1)2600 892 y Fo(")2642 854 y Fn(k)2685 892 y Fo(u)2737 854 y Fi(\()p Fn(k)r Fi(\))2737 914 y Fn(\027)2834 892 y Fo(;)464 b Fq(\(4.1\))390 1184 y(and)34 b(use)h(the)g(\014rst)f(b)s (ound)f(in)h(\(2.14\))j(for)e Fo(u)1945 1136 y Fi(\()p Fn(k)r Fi(\))1945 1196 y Fn(\027)2043 1184 y Fq(,)h Fo(k)g(>)d Fl(j)p Fo(\027)6 b Fl(j)p Fq(,)36 b(in)e(order)g(to)i(b)s(ound)d(the)i (last)390 1313 y(sum)29 b(in)g(\(4.1\))k(b)m(y:)1195 1390 y Fk(\014)1195 1444 y(\014)1195 1499 y(\014)1195 1553 y(\014)1195 1608 y(\014)1195 1662 y(\014)1321 1462 y Fj(1)1290 1490 y Fk(X)1225 1691 y Fn(k)r Fi(=)p Fj(j)p Fn(\027)t Fj(j)p Fi(+1)1502 1576 y Fo(")1544 1539 y Fn(k)1587 1576 y Fo(u)1639 1539 y Fi(\()p Fn(k)r Fi(\))1639 1599 y Fn(\027)1737 1390 y Fk(\014)1737 1444 y(\014)1737 1499 y(\014)1737 1553 y(\014)1737 1608 y(\014)1737 1662 y(\014)1792 1576 y Fl(\024)25 b Fq(2)1948 1475 y Fk(\020)2003 1576 y Fo(D)2078 1590 y Fi(1)2118 1576 y Fl(j)p Fo(\021)s Fl(j)2216 1539 y Fj(\000)p Fi(2)p Fn(=q)2380 1576 y Fo(")2422 1475 y Fk(\021)2477 1498 y Fj(j)p Fn(\027)t Fj(j)p Fi(+1)2665 1576 y Fo(;)633 b Fq(\(4.2\))390 1868 y(pro)m(vided)27 b(that)h Fl(j)p Fo(")p Fl(j)e Fo(<)f(D)1248 1830 y Fj(\000)p Fi(1)1245 1894 y(1)1342 1868 y Fl(j)p Fo(\021)s Fl(j)1440 1835 y Fi(2)p Fn(=q)1550 1868 y Fo(=)p Fq(2.)40 b(The)28 b(co)s(e\016cien)m(t)h Fo(u)2364 1820 y Fi(\()p Fj(j)p Fn(\027)t Fj(j)p Fi(\))2364 1880 y Fn(\027)2529 1868 y Fq(in)d(\(4.1\))k(can)e(b)s(e)f(expressed)390 1998 y(in)37 b(terms)h(of)h(trees)g(ha)m(ving)f(all)f(the)i(mo)s(des)f Fo(\027)2037 2012 y Fn(u)2120 1998 y Fq(=)g Fo(\033)s Fq(,)j(where)d Fo(\033)k Fq(=)c(sign)14 b Fo(\027)6 b Fq(.)65 b(F)-8 b(rom)39 b(the)390 2127 y(de\014nition)31 b(of)j(tree)g(v)-5 b(alue)33 b(in)f(\(1.8\))j(it)e(is)f(easy)i(to)g (see)g(that)g(suc)m(h)f(trees)h(ha)m(v)m(e)h(the)f(same)390 2257 y(pro)s(duct)728 2189 y Fk(Q)814 2284 y Fn(u)p Fj(2)p Fn(#)963 2257 y Fo(\027)1014 2213 y Fn(p)1050 2221 y Fe(u)1090 2213 y Fi(+1)1008 2269 y Fn(u)1212 2257 y Fq(app)s(earing)25 b(in)h(\(1.8\))r(,)i(whic)m(h)e(is)h(giv)m(en)g(b)m(y)g Fo(\033)2731 2224 y Fi(2)p Fj(j)p Fn(\027)t Fj(j)p Fi(+1)2964 2257 y Fq(=)e Fo(\033)s Fq(.)40 b(F)-8 b(urther-)390 2386 y(more)29 b(among)f(suc)m(h)g(trees)h(there)g(will)d(also)i(b)s(e) g(trees)h(ha)m(ving)f Fl(bj)p Fo(\027)6 b Fl(j)p Fo(=q)s Fl(c)29 b Fq(propagators)g(with)390 2516 y(momen)m(tum)f Fo(q)s Fq(,)g(2)p Fo(q)s Fq(,)h Fo(:)15 b(:)g(:)31 b Fq(,)e Fl(bj)p Fo(\027)6 b Fl(j)p Fo(=q)s Fl(c)p Fo(q)s Fq(:)40 b(for)28 b(instance)g(the)g(linear)e(tree)j(\()p Fh(i.e.)h Fq(the)e(tree)h(whose)390 2645 y(no)s(des)c(ha)m(v)m(e)h(all) e(only)h(one)g(en)m(tering)h(line\).)37 b(Therefore)25 b(there)h(will)d(b)s(e)h(trees)i(whose)f(v)-5 b(alue)390 2775 y(will)26 b(b)s(e)j(b)s(ounded)e Fh(fr)-5 b(om)32 b(b)-5 b(elow)40 b Fq(b)m(y)29 b Fo(O)1726 2701 y Fk(\000)1768 2775 y Fl(j)p Fo(\021)s Fl(j)1866 2742 y Fj(\000)p Fi(2)p Fj(bj)p Fn(\027)t Fj(j)p Fn(=q)r Fj(c)2171 2701 y Fk(\001)2213 2775 y Fq(.)40 b(When)29 b(the)g(limit)e Fo(\021)i Fl(!)c Fq(0)k(is)g(tak)m(en,)390 2904 y(as)34 b(the)f(quan)m(tities)g Fo(\015)5 b Fq(\()p Fo(\027)h Fq(\))34 b(in)e(\(1.5\))k(for)d Fo(\021)h Fq(=)c(0)k(are)g(alw)m(a)m(ys)g(non)e(p)s(ositiv)m(e,)i(then) f(eac)m(h)i(of)390 3034 y(suc)m(h)30 b(trees)h Fo(#)f Fq(will)e(ha)m(v)m(e)j(a)g(v)-5 b(alue)30 b(whic)m(h)f(is)g(formally:) 821 3193 y(V)-8 b(al\()p Fo(#)p Fq(\))26 b(=)f Fl(\000)p Fo(i\033)s(B)5 b Fq(\()p Fo(#)p Fq(\))p Fo(\021)1601 3155 y Fj(\000)p Fi(2)p Fj(bj)p Fn(\027)t Fj(j)p Fn(=q)r Fj(c)1906 3193 y Fo(;)879 3490 y(B)g Fq(\()p Fo(#)p Fq(\))25 b(=)g(\()p Fl(\000)p Fq(1\))1384 3452 y Fn(k)1443 3334 y Fk(")1503 3404 y(Y)1496 3601 y Fn(u)p Fj(2)p Fn(#)1686 3428 y Fq(1)p 1651 3469 117 4 v 1651 3552 a Fo(p)1697 3566 y Fn(u)1742 3552 y Fq(!)1777 3334 y Fk(#)1845 3280 y(2)1845 3440 y(6)1845 3498 y(4)1929 3404 y(Y)1928 3601 y Fn(`)p Fj(2)p Fn(#)1906 3658 y(n)1949 3670 y Fe(`)1978 3658 y Fi(=0)2215 3428 y Fq(1)p 2094 3469 289 4 v 2094 3552 a(\(2)p Fo(\031)s(\027)2274 3567 y Fn(`)2307 3552 y Fq(\))2342 3526 y Fi(2)2392 3280 y Fk(3)2392 3440 y(7)2392 3498 y(5)2468 3280 y(2)2468 3440 y(6)2468 3498 y(4)2552 3404 y(Y)2551 3601 y Fn(`)p Fj(2)p Fn(#)2529 3658 y(n)2572 3670 y Fe(`)2601 3658 y Fi(=1)2820 3428 y Fq(1)p 2717 3469 252 4 v 2717 3552 a Fl(j)p Fo(\015)5 b Fq(\()p Fo(\027)2874 3567 y Fn(`)2907 3552 y Fq(\))p Fl(j)2978 3490 y Fo(;)3018 3280 y Fk(3)3018 3440 y(7)3018 3498 y(5)3323 3415 y Fq(\(4.3\))390 3808 y(as)31 b(the)f(propagators)h(\(1.6\))h(div)m(erge)f(as)f(\(2)p Fo(\031)s(i\027)6 b(\021)s Fq(\))2101 3775 y Fj(\000)p Fi(2)2228 3808 y Fq(for)30 b Fo(\021)e Fl(!)e Fq(0.)490 3937 y(This)i(means)i(that)h(no)g(cancellation)f(is)f(p)s(ossible)f(b)s (et)m(w)m(een)j(suc)m(h)f(trees,)h(so)g(that:)1295 4112 y Fo(u)1347 4075 y Fi(\()p Fj(j)p Fn(\027)t Fj(j)p Fi(\))1347 4135 y Fn(\027)1509 4112 y Fq(=)25 b Fo(A)1673 4126 y Fn(\027)1717 4112 y Fl(j)p Fo(\021)s Fl(j)1815 4075 y Fj(\000)p Fi(2)p Fj(bj)p Fn(\027)t Fj(j)p Fn(=q)r Fj(c)2120 4112 y Fo(;)106 b Fl(j)p Fo(A)2344 4126 y Fn(\027)2388 4112 y Fl(j)26 b Fo(>)e Fq(0)p Fo(:)719 b Fq(\(4.4\))490 4287 y(If)32 b(w)m(e)i(w)m(an)m(t)g(that,)g(in)e(the)h(limit)e Fo(\021)j Fl(!)29 b Fq(0,)35 b(the)e(co)s(e\016cien)m(t)h Fo(u)2600 4301 y Fn(\027)2643 4287 y Fq(\()p Fo(")p Fq(\))g(non)f(only) f(do)s(es)h(not)390 4417 y(div)m(erge)d(but)g(also)g(do)s(es)g(not)h(v) -5 b(anish,)29 b(w)m(e)i(ha)m(v)m(e)h(to)f(imp)s(ose)e(that:)1504 4591 y(0)d Fo(<)33 b Fq(lim)1671 4649 y Fn(\021)r Fj(!)p Fi(0)1829 4591 y Fo(")1871 4554 y Fj(j)p Fn(\027)t Fj(j)1954 4591 y Fo(u)2006 4554 y Fi(\()p Fj(j)p Fn(\027)t Fj(j)p Fi(\))2006 4614 y Fn(\027)2169 4591 y Fo(<)25 b Fl(1)p Fo(;)942 b Fq(\(4.5\))390 4808 y(so)27 b(that)g(\(4.4\))i(implies)24 b(that)j Fo(")g Fq(has)f(to)i(b)s(e)e(tak)m(en)h(of)g(order)g Fo(O)s Fq(\()p Fl(j)p Fo(\021)s Fl(j)2641 4775 y Fi(2)p Fn(=q)2750 4808 y Fq(\).)40 b(In)26 b(suc)m(h)g(a)h(w)m(a)m(y)h(all)390 4937 y(the)k(limits)d(\(4.5\))34 b(exist,)e(for)f(an)m(y)h Fo(\027)6 b Fq(,)32 b(and)f(they)h(are)g(v)-5 b(anishing)29 b(except)k(for)e Fl(j)p Fo(\027)6 b Fl(j)32 b Fq(m)m(ultiple)390 5067 y(of)f Fo(q)s Fq(.)p eop %%Page: 15 15 15 14 bop 955 46 a Fi(SCALING)31 b(PR)n(OPER)-6 b(TIES)31 b(OF)f(THE)h(ST)-6 b(AND)n(ARD)30 b(MAP)613 b(15)370 241 y Fq(Moreo)m(v)m(er,)29 b(when)d(w)m(e)h(compute)h Fo(u)1573 255 y Fn(\027)1616 241 y Fq(\()p Fo(")p Fq(\),)h(with)c Fo(\027)32 b Fq(m)m(ultiple)25 b(of)i Fo(q)s Fq(,)g(only)f(the)h(co)s (e\016cien)m(ts)270 370 y Fo(u)322 322 y Fi(\()p Fn(k)r Fi(\))322 382 y Fn(\027)463 370 y Fq(with)42 b Fo(k)47 b Fq(m)m(ultiple)41 b(of)j Fl(j)p Fo(\027)6 b Fl(j)44 b Fq(will)d(con)m(tribute)i(to)h(the)g(limit)d Fo(\021)51 b Fl(!)c Fq(0,)g(as)d(all)f(other)270 500 y(con)m(tributions)38 b(arise)i(from)f(trees)i(con)m(taining)e(resonances,)k(and)c(the)h (analysis)f(of)h(the)270 629 y(sect.)47 b(2)33 b(sho)m(ws)f(that)h(the) f(propagators)h(corresp)s(onding)e(to)i(the)f(resonan)m(t)h(lines)e(do) h(not)270 759 y(in)m(tro)s(duce)h(new)g(denominators)g(small)f(in)g Fo(")i Fq({)g(as)g(a)g(consequence)h(of)e(the)h(cancellation)270 888 y(mec)m(hanism)41 b(discussed)e(in)h(sect.)75 b(3)42 b({)g(while)d(eac)m(h)k(new)d(no)s(de)h(con)m(tributes)g(a)h(factor)270 1018 y Fl(j)p Fo(\021)s Fl(j)368 985 y Fi(2)p Fn(=q)478 1018 y Fq(,)30 b(b)m(y)i(\(4.4\))g(and)e(\(4.5\))q(.)1481 1459 y(5.)50 b Fp(Asymptotics)370 1643 y Fq(Lemma)27 b(3)g(implies)d(that)k(the)f(function)f Fo(v)s Fq(\()p Fo(\013;)15 b(")p Fq(\))27 b(=)e Fo(u)p Fq(\()p Fo(\013;)15 b Fq(\(2)p Fo(\031)s(\021)s Fq(\))2551 1610 y Fi(2)p Fn(=q)2663 1643 y Fo(")p Fq(\))27 b(admits)f(a)i(T)-8 b(a)m(ylor)270 1772 y(series)43 b(in)g Fo(")i Fq(con)m(v)m(ergen)m(t)h (for)e Fl(j)p Fo(")p Fl(j)50 b Fo(<)d(")1657 1786 y Fi(0)1746 1772 y Fq(=)g Fo(O)s Fq(\(1\))f(\(in)d Fo(\021)s Fq(\).)83 b(The)43 b Fo(k)s Fq(-th)i(order)e(F)-8 b(ourier)270 1902 y(co)s(e\016cien)m(ts)31 b(of)g Fo(v)i Fq(are)e(de\014ned)e(b)m (y:)1419 2076 y Fo(v)1466 2039 y Fi(\()p Fn(k)r Fi(\))1463 2099 y Fn(\027)1589 2076 y Fq(=)c Fo(u)1737 2039 y Fi(\()p Fn(k)r Fi(\))1737 2099 y Fn(\027)1834 2076 y Fq(\(2)p Fo(\031)s(\021)s Fq(\))2052 2039 y Fi(2)p Fn(k)r(=q)2201 2076 y Fo(;)977 b Fq(\(5.1\))270 2251 y(so)31 b(that,)g(if)e(w)m(e)i (in)m(tro)s(duce)e(the)i(function:)1414 2426 y(\026)-51 b Fo(u)p Fq(\()p Fo(\013;)15 b(")p Fq(\))27 b(=)34 b(lim)1793 2484 y Fn(\021)r Fj(!)p Fi(0)1952 2426 y Fo(v)s Fq(\()p Fo(\013;)15 b(")p Fq(\))p Fo(;)969 b Fq(\(5.2\))270 2623 y(w)m(e)31 b(ha)m(v)m(e)g(that:)1282 2828 y(\026)-51 b Fo(u)p Fq(\()p Fo(\013;)15 b(")p Fq(\))27 b(=)1692 2714 y Fj(1)1661 2741 y Fk(X)1663 2939 y Fn(k)r Fi(=1)1808 2741 y Fk(X)1808 2939 y Fn(\027)t Fj(2)p Fg(Z)1954 2828 y Fo(")1996 2790 y Fn(k)2039 2828 y Fo(e)2081 2790 y Fn(i\027)t(\013)2200 2828 y Fq(\026)-51 b Fo(u)2246 2790 y Fi(\()p Fn(k)r Fi(\))2246 2850 y Fn(\027)2344 2828 y Fo(;)789 b Fq(\(5.3a\))1396 3080 y(\026)-51 b Fo(u)1442 3043 y Fi(\()p Fn(k)r Fi(\))1442 3103 y Fn(\027)1565 3080 y Fq(=)34 b(lim)1661 3138 y Fn(\021)r Fj(!)p Fi(0)1820 3080 y Fo(u)1872 3043 y Fi(\()p Fn(k)r Fi(\))1872 3103 y Fn(\027)1970 3080 y Fq(\(2)p Fo(\031)s(\021)s Fq(\))2188 3043 y Fi(2)p Fn(k)r(=q)2336 3080 y Fo(;)792 b Fq(\(5.3b\))270 3282 y(and)40 b(the)g(series)g(in)f(\(5.3a\))k(con)m(v)m(erges)f (absolutely)e(b)m(y)h(\(2.14\))r(:)61 b(this)39 b(means)i(that)g(the) 270 3411 y(co)s(e\016cien)m(ts)e(\026)-51 b Fo(u)784 3363 y Fi(\()p Fn(k)r Fi(\))784 3423 y Fn(\027)914 3411 y Fq(are)32 b(w)m(ell)g(de\014ned,)f(and,)i(from)e(the)i(analysis)d(of) j(the)f(last)g(section,)h(w)m(e)270 3541 y(kno)m(w)c(that)g(only)f(the) g(F)-8 b(ourier)28 b(co)s(e\016cien)m(ts)i(with)d(mo)s(des)h(m)m (ultiples)e(of)j Fo(q)i Fq(surviv)m(e)d(when)270 3670 y(the)h(limit)e Fo(\021)i Fl(!)c Fq(0)30 b(is)e(tak)m(en;)j (furthermore,)d(the)i(function)j(\026)-51 b Fo(u)p Fq(\()p Fo(\013;)15 b(")p Fq(\))32 b(is)c(analytic)g(in)g Fo(")i Fq(for)f Fo(")270 3800 y Fq(small)g(enough)h(and)g Fo(\021)j Fq(indep)s(enden)m(t,)c(and)h(p)s(erio)s(dic)d(in)i Fo(\013)i Fq(with)e(p)s(erio)s(d)f(2)p Fo(\031)s(=q)s Fq(.)370 3929 y(Summarizing,)g(w)m(e)i(ha)m(v)m(e:)420 4199 y(\026)-51 b Fo(u)466 4162 y Fi(\()p Fn(k)r Fi(\))466 4222 y Fn(\027)589 4199 y Fq(=)685 4013 y Fk(8)685 4095 y(<)685 4258 y(:)766 4121 y Fl(\000)847 4076 y Fi(\(2)p Fn(\031)r Fi(\))979 4053 y Fe(k)q(=q)p 846 4100 234 4 v 926 4157 a Fi(2)961 4138 y Fe(k)1105 4035 y Fk(X)1236 4028 y Fd(0)1263 4175 y Fn(#)p Fj(2T)1391 4187 y Fe(\027;k)1498 4121 y Fq(V)-8 b(al)1629 4084 y Fj(0)1653 4121 y Fq(\()p Fo(#)p Fq(\))p Fo(;)91 b Fq(if)30 b Fo(\027)h Fl(2)24 b Fo(q)s Fm(Z)16 b Fl(n)21 b(f)p Fq(0)p Fl(g)p Fo(;)32 b(k)c Fl(2)d Fo(q)s Fm(Z)16 b Fl(n)k(f)p Fq(0)p Fl(g)p Fo(;)766 4305 y Fq(0)p Fo(;)1057 b Fq(otherwise,)3203 4199 y(\(5.4\))270 4526 y(where)548 4440 y Fk(X)680 4433 y Fd(0)737 4526 y Fq(means)30 b(that)i(only)e(trees)h(without)f(resonances)h(ha)m(v)m(e)h(to)f(b)s(e) f(summed)g(o)m(v)m(er)270 4658 y(and)44 b(V)-8 b(al)592 4621 y Fj(0)616 4658 y Fq(\()p Fo(#)p Fq(\))45 b(di\013ers)e(from)i(V) -8 b(al\()p Fo(#)p Fq(\))45 b(as)g(it)g(is)e(de\014ned)h(in)f(\(1.8\))k (inasm)m(uc)m(h)d(as,)49 b(for)44 b Fo(\027)270 4787 y Fq(m)m(ultiple)28 b(of)j Fo(q)s Fq(,)f(the)h(denominator)e Fo(\015)5 b Fq(\()p Fo(\027)h Fq(\))31 b(has)f(to)h(b)s(e)f(replaced)g (with)f(\(2)p Fo(\031)s(i\027)6 b Fq(\))2935 4754 y Fi(2)2976 4787 y Fq(.)370 4917 y(Let)37 b(us)g(no)m(w)g(consider)f(all)g(trees)i (of)f(order)g Fo(q)j Fq(con)m(tributing)35 b(to)j Fo(\027)k Fq(=)37 b Fo(\033)s(q)s Fq(,)i(with)d Fo(\033)j Fq(=)270 5046 y(sign)14 b Fo(\027)6 b Fq(:)50 b(the)36 b(v)-5 b(alues)34 b(of)h(suc)m(h)g(trees)h(can)f(b)s(e)g(read)g(from)g (\(1.8\))r(,)h(and)f(one)g(sees)h(that)g(the)p eop %%Page: 16 16 16 15 bop 390 46 a Fi(16)704 b(ALBER)-6 b(TO)30 b(BERRETTI)h(AND)f (GUIDO)h(GENTILE)390 241 y Fq(n)m(umerator)f(is)g(iden)m(tically)e Fl(\000)p Fo(i\033)s Fq(,)j(so)g(that:)570 346 y Fk(X)511 543 y Fn(#)p Fj(2T)639 551 y Fe(\033)r(q)r(;q)776 432 y Fq(V)-8 b(al)907 395 y Fj(0)930 432 y Fq(\()p Fo(#)p Fq(\))26 b(=)f(\(2)p Fo(\031)s Fq(\))1346 395 y Fi(2)1460 346 y Fk(X)1402 543 y Fn(#)p Fj(2T)1530 551 y Fe(\033)r(q)r(;q)1651 432 y Fq(\()p Fl(\000)p Fo(i\033)s Fq(\))1901 346 y Fk(Y)1893 543 y Fn(u)p Fj(2)p Fn(#)2084 371 y Fq(1)p 2049 411 117 4 v 2049 495 a Fo(p)2095 509 y Fn(u)2139 495 y Fq(!)2283 371 y(1)p 2185 411 242 4 v 2185 495 a Fo(\015)5 b Fq(\()p Fo(\027)2317 510 y Fn(`)2346 518 y Fe(u)2391 495 y Fq(\))2461 432 y(=)2641 371 y(1)p 2567 411 193 4 v 2567 495 a(\()p Fo(i\027)h Fq(\))2719 468 y Fi(2)2769 432 y Fq(\()p Fl(\000)p Fo(i\033)s Fq(\))p Fl(S)3051 451 y Fn(p=q)3161 432 y Fo(;)137 b Fq(\(5.5\))390 700 y(so)26 b(de\014ning)d(the)i(expression)f Fl(S)1474 718 y Fn(p=q)1608 700 y Fq(\(whic)m(h)h(do)s(es)g(not)g(dep)s (ends)f(in)g Fo(\033)s Fq(\):)38 b(the)26 b(factor)g(\()p Fo(i\027)6 b Fq(\))3415 667 y Fj(\000)p Fi(2)390 829 y Fq(arises)28 b(from)g(\(2)p Fo(\031)s Fq(\))1020 796 y Fi(2)1089 829 y Fq(times)g(the)g(propagator)i Fo(\015)5 b Fq(\()p Fo(\027)h Fq(\),)29 b(whic)m(h)e(app)s(ears)h(in)f(an)m(y)i (tree)g(in)e Fl(T)3376 843 y Fn(\033)r(q)r(;q)390 959 y Fq(so)43 b(it)e(can)i(b)s(e)f(put)g(in)f(evidence.)76 b(F)-8 b(or)43 b(instance,)i(for)e Fo(p=q)48 b Fq(=)d(0)p Fo(=)p Fq(1,)i Fo(p=q)h Fq(=)d(1)p Fo(=)p Fq(2)f(and)390 1088 y Fo(p=q)28 b Fq(=)d(1)p Fo(=)p Fq(3,)32 b(b)m(y)e(explicit)f (computation,)i(w)m(e)g(\014nd:)460 1243 y Fl(S)515 1262 y Fi(0)p Fn(=)p Fi(1)650 1243 y Fq(=)25 b(1)p Fo(;)460 1447 y Fl(S)515 1465 y Fi(1)p Fn(=)p Fi(2)650 1447 y Fq(=)965 1385 y(1)p 756 1426 465 4 v 756 1509 a(2\(cos)16 b Fo(\031)24 b Fl(\000)c Fq(1\))1256 1447 y(=)25 b Fl(\000)1433 1385 y Fq(1)p 1433 1426 46 4 v 1433 1509 a(4)1488 1447 y Fo(;)460 1693 y Fl(S)515 1711 y Fi(1)p Fn(=)p Fi(3)650 1693 y Fq(=)1061 1631 y(1)p 756 1672 657 4 v 756 1755 a(2\(cos)q(\(4)p Fo(\031)s(=)p Fq(3\))e Fl(\000)d Fq(1\))1738 1631 y(1)p 1432 1672 V 1432 1755 a(2\(cos)r(\(2)p Fo(\031)s(=)p Fq(3\))i Fl(\000)e Fq(1\))2119 1693 y(+)2220 1631 y(1)p 2220 1672 46 4 v 2220 1755 a(2)2636 1631 y(1)p 2285 1672 747 4 v 2285 1755 a([2\(cos)r(\(4)p Fo(\031)s(=)p Fq(3\))i Fl(\000)e Fq(1\)])2991 1729 y Fi(2)3067 1693 y Fq(=)3173 1631 y(1)p 3173 1672 46 4 v 3173 1755 a(6)3228 1693 y Fo(;)3323 1494 y Fq(\(5.6\))390 1900 y(as)31 b(it)f(can)g(b)s(e)g (easily)f(computed)i(from)f(the)g(de\014nition)e(\(5.5\))r(.)490 2029 y(W)-8 b(e)31 b(ha)m(v)m(e)h(the)e(follo)m(wing)f(lemma.)390 2223 y Fv(Lemma)k(4.)45 b Fh(The)33 b(c)-5 b(o)g(e\016cients)41 b Fq(\(5.3b\))34 b Fh(satisfy)g(the)f(r)-5 b(e)g(cursion)34 b(r)-5 b(elation:)636 2456 y Fq(\026)-51 b Fo(u)682 2418 y Fi(\()p Fn(k)r Fi(\))682 2478 y Fn(\027)805 2456 y Fq(=)975 2394 y(1)p 911 2435 174 4 v 911 2518 a(2)956 2492 y Fn(q)r Fj(\000)p Fi(1)1178 2394 y Fq(1)p 1105 2435 193 4 v 1105 2518 a(\()p Fo(i\027)6 b Fq(\))1257 2492 y Fi(2)1352 2342 y Fj(1)1322 2370 y Fk(X)1325 2564 y Fn(p)p Fi(=1)1491 2394 y Fq(1)p 1478 2435 71 4 v 1478 2518 a Fo(p)p Fq(!)1569 2385 y Fl(S)1624 2404 y Fn(p=q)p 1569 2435 164 4 v 1628 2518 a Fq(2)1743 2456 y(\()p Fl(\000)p Fo(i\033)s Fq(\)\()p Fo(i\033)s(q)s Fq(\))2170 2418 y Fn(p)2409 2370 y Fk(X)2254 2561 y Fn(\033)r(q)r Fi(+)p Fn(\027)2420 2570 y Ff(1)2455 2561 y Fi(+)p Fn(:::)o Fi(+)p Fn(\027)2659 2569 y Fe(p)2227 2634 y Fn(k)2264 2643 y Ff(1)2298 2634 y Fi(+)p Fn(:::)o Fi(+)p Fn(k)2504 2642 y Fe(p)2540 2634 y Fi(=)p Fn(k)r Fj(\000)p Fn(q)2777 2337 y(p)2737 2370 y Fk(Y)2738 2565 y Fn(i)p Fi(=1)2874 2456 y Fq(\026)-51 b Fo(u)2920 2418 y Fi(\()p Fn(k)2984 2428 y Fe(i)3011 2418 y Fi(\))2920 2478 y Fn(\027)2955 2488 y Fe(i)3043 2456 y Fo(;)255 b Fq(\(5.7\))390 2783 y Fh(wher)-5 b(e)34 b(the)f(c)-5 b(onstant)34 b Fl(S)1215 2802 y Fn(p=q)1357 2783 y Fh(is)e(de\014ne)-5 b(d)34 b(in)39 b Fq(\(5.5\))r Fh(.)390 2977 y(Pr)-5 b(o)g(of.)43 b Fq(A)22 b(generic)h(tree)g(of)f(order)g Fo(k)28 b Fq(=)d Fo(\024q)s Fq(,)g Fo(\024)g Fl(\025)g Fq(1,)g(and)d(momen)m(tum)g Fo(\027)31 b Fq(=)25 b Fo(nq)s Fq(,)e Fo(n)i Fl(\025)g Fq(1,)g(can)390 3107 y(b)s(e)32 b(obtained)g(starting)g(from)g(a)h (tree)g Fo(#)1766 3121 y Fi(0)1837 3107 y Fq(of)g(order)f Fo(q)j Fq(b)m(y)d(attac)m(hing)i(to)f(its)f(no)s(des)g Fo(p)c Fl(\025)h Fq(0)390 3236 y(trees)j Fo(#)663 3250 y Fi(1)702 3236 y Fq(,)f Fo(:)15 b(:)g(:)32 b Fq(,)f Fo(#)1005 3250 y Fn(p)1076 3236 y Fq(of)g(orders)g Fo(k)1502 3250 y Fi(1)1568 3236 y Fq(=)26 b Fo(\024)1717 3250 y Fi(1)1757 3236 y Fo(q)s Fq(,)31 b Fo(:)15 b(:)g(:)32 b Fq(,)f Fo(k)2097 3250 y Fn(p)2164 3236 y Fq(=)26 b Fo(\024)2313 3250 y Fn(p)2353 3236 y Fo(q)s Fq(,)32 b(with)e Fo(\024)2714 3250 y Fi(1)2774 3236 y Fq(+)21 b Fo(:)15 b(:)g(:)22 b Fq(+)e Fo(\024)3136 3250 y Fn(p)3203 3236 y Fq(=)26 b Fo(\024)21 b Fl(\000)g Fq(1)390 3366 y(and)j(total)i(momen) m(ta)g Fo(\027)1209 3380 y Fi(1)1274 3366 y Fq(=)f Fo(n)1425 3380 y Fi(1)1464 3366 y Fo(q)s Fq(,)g Fo(:)15 b(:)g(:)31 b Fq(,)p Fo(\027)1764 3380 y Fn(p)1829 3366 y Fq(=)25 b Fo(n)1980 3380 y Fn(p)2019 3366 y Fo(q)j Fq(with)c Fo(n)2345 3380 y Fi(1)2393 3366 y Fq(+)9 b Fo(:)15 b(:)g(:)c Fq(+)e Fo(n)2724 3380 y Fn(p)2788 3366 y Fq(=)25 b Fo(n)9 b Fl(\000)g Fq(1.)40 b(Eac)m(h)25 b(tree)390 3495 y(can)33 b(b)s(e)f(attac)m(hed)i(to)f(an)m(y)g(no)s(de)f(of)g Fo(#)1726 3509 y Fi(0)1798 3495 y Fq(so)h(that)g(the)f(com)m (binatorial)g(factor)h(asso)s(ciated)390 3625 y(to)40 b(an)m(y)f(no)s(de)g Fo(u)g Fl(2)h Fo(#)1165 3639 y Fi(0)1204 3625 y Fq(will)c(b)s(e)j Fo(p)1563 3639 y Fn(u)1608 3625 y Fq(!)1633 3592 y Fj(\000)p Fi(1)1727 3625 y Fq(,)j(with)37 b Fo(p)2055 3639 y Fn(u)2140 3625 y Fq(=)i Fo(s)2293 3639 y Fn(u)2364 3625 y Fq(+)26 b Fo(r)2502 3639 y Fn(u)2547 3625 y Fq(,)41 b(if)d Fo(s)2748 3639 y Fn(u)2832 3625 y Fq(is)g(the)h(n)m(um)m(b)s(er)f(of)390 3754 y(lines)29 b(connecting)i Fo(u)g Fq(to)h(other)f(no)s(des)g(of)g Fo(#)1901 3768 y Fi(0)1971 3754 y Fq(and)f Fo(r)2189 3768 y Fn(u)2265 3754 y Fq(is)g(the)i(n)m(um)m(b)s(er)d(of)j(trees)f (attac)m(hed)390 3884 y(to)36 b Fo(u)p Fq(.)56 b(Then)35 b(b)m(y)g(construction)1539 3816 y Fk(P)1635 3911 y Fn(u)p Fj(2)p Fn(#)1765 3920 y Ff(0)1819 3884 y Fo(r)1860 3898 y Fn(u)1939 3884 y Fq(=)e Fo(p)p Fq(.)56 b(Note)37 b(that)f(t)m(w)m(o)h (trees)f(in)e(whic)m(h)h(one)390 4014 y(of)g(the)g(\014rst)g Fo(s)895 4028 y Fn(u)974 4014 y Fq(subtrees)g(\(with)f(ro)s(ot)h(line)e (b)s(elonging)h(to)h Fo(#)2542 4028 y Fi(0)2582 4014 y Fq(\))g(is)f(p)s(erm)m(uted)g(with)g(one)390 4143 y(of)k(the)h (remaining)d Fo(r)1139 4157 y Fn(u)1222 4143 y Fq(can)j(not)f(b)s(e)f (iden)m(tical,)j(as)e(they)g(ha)m(v)m(e)i(a)e(di\013eren)m(t)g(n)m(um)m (b)s(er)f(of)390 4273 y(no)s(des:)i(only)27 b(the)h(latter)g(will)d(ha) m(v)m(e)k(a)g(n)m(um)m(b)s(er)d(of)i(no)s(des)f(whic)m(h)g(is)g(m)m (ultiple)e(of)j Fo(q)s Fq(.)40 b(If)27 b(w)m(e)390 4402 y(sum)i(together)j(all)d(trees)h(whic)m(h)f(can)i(b)s(e)e(obtained)h (from)f(eac)m(h)i(other)g(b)m(y)f(c)m(ho)s(osing)g(in)f(a)390 4532 y(di\013eren)m(t)h(w)m(a)m(y)h(the)g Fo(s)1133 4546 y Fn(u)1208 4532 y Fq(subtrees)f(with)f(ro)s(ot)h(line)f(b)s(elonging)g (to)i Fo(#)2713 4546 y Fi(0)2782 4532 y Fq(and)f(the)h(remaining)390 4661 y Fo(r)431 4675 y Fn(u)501 4661 y Fq(subtrees,)25 b(w)m(e)h(ha)m(v)m(e)1209 4593 y Fk(Q)1295 4688 y Fn(u)p Fj(2)p Fn(#)1425 4697 y Ff(0)1478 4661 y Fo(p)1524 4675 y Fn(u)1569 4661 y Fq(!)p Fo(s)1637 4675 y Fn(u)1682 4661 y Fq(!)1707 4628 y Fj(\000)p Fi(1)1802 4661 y Fo(r)1843 4675 y Fn(u)1888 4661 y Fq(!)1913 4628 y Fj(\000)p Fi(1)2032 4661 y Fq(terms,)g(so)f(that,)i(b)m(y)e(taking)g(in)m(to)f(accoun)m(t) 390 4791 y(that)31 b(\(see)g(\(5.5\))r(\):)1196 4987 y(V)-8 b(al)1327 4949 y Fj(0)1351 4987 y Fq(\()p Fo(#)1440 5001 y Fi(0)1479 4987 y Fq(\))26 b(=)f(\(2)p Fo(\031)s Fq(\))1806 4949 y Fi(2)1846 4987 y Fq(\()p Fl(\000)p Fo(i\033)s Fq(\))2114 4901 y Fk(Y)2088 5098 y Fn(u)p Fj(2)p Fn(#)2218 5107 y Ff(0)2313 4926 y Fq(1)p 2279 4966 113 4 v 2279 5049 a Fo(s)2322 5063 y Fn(u)2367 5049 y Fq(!)2510 4926 y(1)p 2412 4966 242 4 v 2412 5049 a Fo(\015)5 b Fq(\()p Fo(\027)2544 5064 y Fn(`)2573 5072 y Fe(u)2618 5049 y Fq(\))2664 4987 y Fo(;)634 b Fq(\(5.8\))p eop %%Page: 17 17 17 16 bop 955 46 a Fi(SCALING)31 b(PR)n(OPER)-6 b(TIES)31 b(OF)f(THE)h(ST)-6 b(AND)n(ARD)30 b(MAP)613 b(17)270 241 y Fq(and)30 b(that,)i(b)m(y)f(shifting)e(the)i(subtrees)f(attac)m (hed)j(to)e(the)g(no)s(des)g(of)g Fo(#)2691 255 y Fi(0)2730 241 y Fq(,)g(the)g(momen)m(tum)270 370 y(\015o)m(wing)23 b(through)g(an)m(y)h(line)e(of)i Fo(#)1392 384 y Fi(0)1455 370 y Fq(can)g(v)-5 b(ary)24 b(b)m(y)f(an)h(amoun)m(t)g(prop)s (ortional)e(to)i(a)g(m)m(ultiple)270 500 y(of)31 b Fo(q)s Fq(,)f(so)h(that)g(the)f(corresp)s(onding)e(propagator)k(do)s(es)e(not) g(c)m(hange,)i(w)m(e)f(can)g(write:)322 727 y(V)-8 b(al)453 690 y Fj(0)476 727 y Fq(\()p Fo(#)p Fq(\))26 b(=)798 641 y Fk(X)722 839 y Fn(#)764 848 y Ff(0)799 839 y Fj(2T)885 847 y Fe(\033)r(q)r(;q)1021 727 y Fq(V)-8 b(al\()p Fo(#)1241 741 y Fi(0)1281 727 y Fq(\))1467 641 y Fk(X)1416 842 y Fj(f)p Fn(r)1483 850 y Fe(u)1524 842 y Fj(\025)p Fi(0)p Fj(g)1331 868 y Fc(P)1406 938 y Fe(u)p Fd(2)p Fe(#)1522 953 y Ff(0)1572 918 y Fn(r)1604 926 y Fe(u)1644 918 y Fi(=)p Fn(p)1774 641 y Fk(Y)1750 839 y Fn(u)p Fj(2)p Fn(#)1880 848 y Ff(0)1972 666 y Fq(1)p 1940 706 112 4 v 1940 790 a Fo(r)1981 804 y Fn(u)2026 790 y Fq(!)2106 614 y Fj(1)2076 641 y Fk(X)2079 835 y Fn(p)p Fi(=0)2207 727 y Fq(\()p Fo(i\033)s Fq(\))2363 690 y Fn(p)2459 641 y Fk(X)2591 635 y Fd(0)2393 839 y Fn(#)2435 848 y Ff(1)2469 839 y Fn(;:::)11 b(;#)2622 847 y Fe(p)2713 609 y Fn(p)2672 641 y Fk(Y)2673 836 y Fn(i)p Fi(=1)2804 727 y Fq(V)-8 b(al)2935 690 y Fj(0)2958 727 y Fq(\()p Fo(#)3047 741 y Fn(i)3076 727 y Fq(\))p Fo(;)67 b Fq(\(5.9\))270 1086 y(where)538 1053 y Fj(0)596 1086 y Fq(recalls)34 b(the)i(constrain)m(t) f(on)g(the)h(trees)f Fo(#)2042 1100 y Fi(1)2082 1086 y Fq(,)g Fo(:)15 b(:)g(:)31 b Fq(,)37 b Fo(#)2394 1100 y Fn(p)2468 1086 y Fq(describ)s(ed)d(ab)s(o)m(v)m(e.)56 b(Then)270 1215 y(just)30 b(note)h(that:)1566 1323 y Fk(Y)1508 1525 y Fj(f)p Fn(r)1575 1533 y Fe(u)1616 1525 y Fj(\025)p Fi(0)p Fj(g)1422 1550 y Fc(P)1497 1621 y Fe(u)p Fd(2)p Fe(#)1613 1636 y Ff(0)1663 1600 y Fn(r)1695 1608 y Fe(u)1736 1600 y Fi(=)p Fn(p)1884 1348 y Fq(1)p 1851 1389 V 1851 1472 a Fo(r)1892 1486 y Fn(u)1937 1472 y Fq(!)1998 1410 y(=)2104 1348 y Fo(q)2148 1315 y Fn(p)p 2104 1389 84 4 v 2110 1472 a Fo(p)p Fq(!)2197 1410 y Fo(;)936 b Fq(\(5.10\))270 1767 y(to)31 b(deduce)f(from)g(\(5.9\))i (that:)492 1999 y(V)-8 b(al)623 1962 y Fj(0)646 1999 y Fq(\()p Fo(#)p Fq(\))26 b(=)968 1913 y Fk(X)892 2111 y Fn(#)934 2120 y Ff(0)969 2111 y Fj(2T)1055 2119 y Fe(\033)r(q)r(;q) 1191 1999 y Fq(V)-8 b(al)1322 1962 y Fj(0)1345 1999 y Fq(\()p Fo(#)1434 2013 y Fi(0)1474 1999 y Fq(\))1555 1886 y Fj(1)1525 1913 y Fk(X)1527 2107 y Fn(p)p Fi(=1)1694 1938 y Fq(1)p 1681 1978 71 4 v 1681 2062 a Fo(p)p Fq(!)1762 1999 y(\()p Fo(i\033)s(q)s Fq(\))1962 1962 y Fn(p)2058 1913 y Fk(X)2189 1907 y Fd(0)1991 2111 y Fn(#)2033 2120 y Ff(1)2068 2111 y Fn(;:::)10 b(;#)2220 2119 y Fe(p)2311 1881 y Fn(p)2271 1913 y Fk(Y)2272 2108 y Fn(i)p Fi(=1)2387 1999 y Fq(\()p Fo(i\033)s(q)s Fq(\))15 b(V)-8 b(al)2735 1962 y Fj(0)2758 1999 y Fq(\()p Fo(#)2847 2013 y Fn(i)2875 1999 y Fq(\))p Fo(:)223 b Fq(\(5.11\))270 2259 y(Then)29 b(from)h(\(5.4\))i(and)e(\(5.11\))j(w)m(e)e(read)f(that:)585 2492 y(\026)-51 b Fo(u)631 2454 y Fi(\()p Fn(k)r Fi(\))631 2514 y Fn(\027)754 2492 y Fq(=)879 2430 y(1)p 860 2471 84 4 v 860 2554 a(2)905 2528 y Fn(q)1026 2406 y Fk(X)969 2603 y Fn(#)1011 2612 y Ff(0)1045 2603 y Fj(2T)1131 2611 y Fe(q)r(;q)1230 2492 y Fq(V)-8 b(al)1361 2454 y Fj(0)1385 2492 y Fq(\()p Fo(#)1474 2506 y Fi(0)1513 2492 y Fq(\))1594 2378 y Fj(1)1564 2406 y Fk(X)1566 2600 y Fn(p)p Fi(=0)1733 2430 y Fq(1)p 1720 2471 71 4 v 1720 2554 a Fo(p)p Fq(!)2001 2406 y Fk(X)1819 2603 y Fn(k)1856 2612 y Ff(1)1890 2603 y Fi(+)p Fn(:::)o Fi(+)p Fn(k)2096 2611 y Fe(p)2132 2603 y Fi(=)p Fn(k)r Fj(\000)p Fn(q)1816 2670 y(\033)r Fi(+)p Fn(\027)1948 2679 y Ff(1)1983 2670 y Fi(+)p Fn(:::)o Fi(+)p Fn(\027)2187 2678 y Fe(p)2223 2670 y Fi(=)p Fn(\027)2372 2374 y(p)2332 2406 y Fk(Y)2333 2601 y Fn(i)p Fi(=1)2448 2492 y Fq(\()p Fo(i\033)s(q)s Fq(\))6 b(\026)-51 b Fo(u)2700 2454 y Fi(\()p Fn(k)2764 2464 y Fe(i)2791 2454 y Fi(\))2700 2514 y Fn(\027)2735 2524 y Fe(i)2823 2492 y Fo(:)310 b Fq(\(5.12\))270 2819 y(and)30 b(using)g(\(5.5\))i(w)m(e)f(obtain)f (\(5.7\))r(.)p 3324 2819 4 62 v 3328 2761 55 4 v 3328 2819 V 3382 2819 4 62 v 270 2999 a Fv(Lemma)j(5.)45 b Fh(The)31 b(function)37 b Fq(\026)-51 b Fo(u)p Fq(\()p Fo(\013)p Fq(\))27 b Fl(\021)j Fq(\026)-50 b Fo(u)p Fq(\()p Fo(\013;)15 b(")p Fq(\))32 b Fh(in)38 b Fq(\(5.2\))32 b Fh(satis\014es)f(the)g(di\013er)-5 b(ential)32 b(e)-5 b(qua-)270 3129 y(tion:)1157 3258 y(d)1204 3225 y Fi(2)1249 3258 y Fq(\026)-51 b Fo(u)p Fq(\()p Fo(\013)p Fq(\))p 1157 3299 268 4 v 1218 3382 a Fh(d)q Fo(\013)1323 3356 y Fi(2)1459 3320 y Fq(=)25 b Fo(C)1620 3338 y Fn(p=q)1729 3320 y Fo(")1771 3282 y Fn(q)1825 3320 y Fq(sin)n(\()p Fo(q)s Fq(\()p Fo(\013)d Fq(+)j(\026)-50 b Fo(u)p Fq(\()p Fo(\013)p Fq(\)\)\))p Fo(;)662 b Fq(\(5.13\))270 3504 y Fh(with)33 b(b)-5 b(oundary)35 b(c)-5 b(onditions)40 b Fq(\026)-51 b Fo(u)p Fq(\(0\))27 b(=)k(\026)-51 b Fo(u)p Fq(\(2)p Fo(\031)s Fq(\))26 b(=)f(0)p Fh(,)33 b(and:)582 3701 y Fo(C)647 3719 y Fn(p=q)781 3701 y Fq(=)25 b(2)922 3663 y Fj(\000)p Fi(\()p Fn(q)r Fj(\000)p Fi(1\))1160 3701 y Fl(S)1215 3719 y Fn(p=q)1349 3701 y Fq(=)g(2)1490 3663 y Fj(\000)p Fi(\()p Fn(q)r Fj(\000)p Fi(1\))1784 3614 y Fk(X)1744 3812 y Fn(#)p Fj(2T)1872 3820 y Fe(q)r(;q)1955 3701 y Fq(\(2)p Fo(\031)s(i\027)6 b Fq(\))2207 3663 y Fi(2)2270 3614 y Fk(Y)2263 3812 y Fn(u)p Fj(2)p Fn(#)2453 3639 y Fq(1)p 2418 3680 117 4 v 2418 3763 a Fo(p)2464 3777 y Fn(u)2509 3763 y Fq(!)2652 3639 y(1)p 2554 3680 242 4 v 2554 3763 a Fo(\015)f Fq(\()p Fo(\027)2686 3778 y Fn(`)2715 3786 y Fe(u)2760 3763 y Fq(\))2806 3701 y Fo(;)327 b Fq(\(5.14\))270 3968 y Fh(wher)-5 b(e)34 b(the)f(factor)g Fq(\(2)p Fo(\031)s(i\027)6 b Fq(\))1193 3935 y Fi(2)1267 3968 y Fh(simply)33 b(c)-5 b(anc)g(els)34 b(the)f(pr)-5 b(op)g(agator)37 b(of)c(the)g(r)-5 b(o)g(ot)34 b(line)f(of)g Fo(#)p Fh(.)270 4148 y(Pr)-5 b(o)g(of.)43 b Fq(Simply)33 b(write)i(\(5.13\))j(in)c(F)-8 b(ourier)35 b(space,)j(and)d(write)g (the)h(recursion)e(relations)270 4278 y(de\014ning)28 b(the)i(co)s(e\016cien)m(ts,)h(b)m(y)f(taking)g(in)m(to)f(accoun)m(t)j (that)e(only)f(orders)h(and)f(momen)m(ta)270 4408 y(m)m(ultiples)g(of)i Fo(q)j Fq(can)d(o)s(ccur,)h(as)f(it)g(can)g(b)s(e)g(immediately)e(v)m (eri\014ed)h(b)m(y)h(induction.)41 b(Then)270 4537 y(w)m(e)31 b(obtain:)566 4747 y(\026)-51 b Fo(u)612 4709 y Fi(\()p Fn(k)r Fi(\))612 4769 y Fn(\027)735 4747 y Fq(=)914 4685 y(1)p 841 4726 193 4 v 841 4809 a(\()p Fo(i\027)6 b Fq(\))993 4783 y Fi(2)1088 4633 y Fj(1)1058 4660 y Fk(X)1061 4855 y Fn(p)p Fi(=1)1227 4685 y Fq(1)p 1214 4726 71 4 v 1214 4809 a Fo(p)p Fq(!)1305 4676 y Fo(C)1370 4695 y Fn(p=q)p 1305 4726 174 4 v 1369 4809 a Fq(2)1489 4747 y(\()p Fl(\000)p Fo(i\033)s Fq(\)\()p Fo(i\033)s(q)s Fq(\))1916 4709 y Fn(p)2174 4660 y Fk(X)1973 4852 y Fn(\033)r(q)r Fi(+)p Fn(\027)2139 4861 y Ff(1)2174 4852 y Fi(+)p Fn(:::)n Fi(+)p Fn(\027)2377 4860 y Fe(p)2413 4852 y Fi(=)p Fn(\027)1992 4925 y(k)2029 4934 y Ff(1)2064 4925 y Fi(+)p Fn(:::)n Fi(+)p Fn(k)2269 4933 y Fe(p)2305 4925 y Fi(=)p Fn(k)r Fj(\000)p Fn(q)2562 4628 y(p)2522 4660 y Fk(Y)2523 4856 y Fn(i)p Fi(=1)2659 4747 y Fq(\026)-51 b Fo(u)2705 4709 y Fi(\()p Fn(k)2769 4719 y Fe(i)2796 4709 y Fi(\))2705 4769 y Fn(\027)2740 4779 y Fe(i)2828 4747 y Fo(;)305 b Fq(\(5.15\))270 5074 y(with)38 b Fo(\033)44 b Fq(=)c Fl(\006)p Fq(1.)68 b(Therefore)39 b(w)m(e)h(ha)m(v)m(e)g(the)g(same)g (expression)e(as)h(\(5.7\))j(pro)m(vided)c(the)270 5204 y(constan)m(t)32 b Fl(C)685 5222 y Fn(p=q)793 5204 y Fq(is)e(c)m(hosen)h(as)f(in)f(\(5.14\))s(.)p 3324 5204 4 62 v 3328 5146 55 4 v 3328 5204 V 3382 5204 4 62 v eop %%Page: 18 18 18 17 bop 390 46 a Fi(18)704 b(ALBER)-6 b(TO)30 b(BERRETTI)h(AND)f (GUIDO)h(GENTILE)490 241 y Fq(W)-8 b(e)35 b(note)f(that)h(lemma)e(5)h (giv)m(es)g(the)g(v)-5 b(alues)34 b Fo(C)2157 259 y Fi(0)p Fn(=)p Fi(1)2297 241 y Fq(=)d(1,)36 b Fo(C)2570 259 y Fi(1)p Fn(=)p Fi(2)2711 241 y Fq(=)30 b Fl(\000)p Fq(1)p Fo(=)p Fq(8)35 b(and)e Fo(C)3298 259 y Fi(1)p Fn(=)p Fi(3)3439 241 y Fq(=)390 370 y(1)p Fo(=)p Fq(24,)27 b(whic)m(h)22 b(are)i(consisten)m(t)f(with)f(the)i(results)e(of)h([5)q(].)39 b(W)-8 b(e)24 b(therefore)g(pro)m(v)m(e)g(the)g(conjec-)390 500 y(ture)d(prop)s(osed)f(in)f([5)q(],)k(and)e(giv)m(e)g(an)g (explicit)f(form)m(ula)g(for)h(the)g(constan)m(t)i Fo(C)2989 518 y Fn(p=q)3118 500 y Fq(app)s(earing)390 629 y(in)29 b(the)i(di\013eren)m(tial)d(equation.)1607 1071 y(6.)51 b Fp(Conclusions)490 1254 y Fq(The)29 b(ab)s(o)m(v)m(e)j(lemmata)f(pro) m(v)m(e)g(all)e(the)i(claims)e(in)g(the)i(main)e(theorem)i(in)e(sect.) 42 b(1.)390 1433 y Fh(Pr)-5 b(o)g(of)34 b(of)f(the)g(the)-5 b(or)g(em.)43 b Fq(Statemen)m(t)32 b(\(1\))f(is)e(ob)m(vious,)g(as)i (for)e Fo(\021)g Fl(6)p Fq(=)c(0)30 b(there)g(are)h(no)e(small)390 1563 y(divisors)i(and)h(the)h(the)g(con)m(v)m(ergence)i(of)e(the)g (Lindstedt)e(series)h(can)i(b)s(e)e(pro)m(v)m(ed)h(b)m(y)g(ele-)390 1692 y(men)m(tary)e(means;)f(the)h(only)f(non)f(ob)m(vious)h(statemen)m (t)j(is)c(the)i(b)s(eha)m(viour)e(of)h(the)h(radius)390 1822 y(of)g(con)m(v)m(ergence)h(as)f Fo(\021)e Fl(!)c Fq(0,)31 b(and)f(the)g(result)f(is)h(pro)m(v)m(ed)g(b)m(y)h(lemma)f(3.) 490 1951 y(Statemen)m(t)h(\(2\))g(is)e(pro)m(v)m(ed)i(b)m(y)f(lemma)f (3,)i(b)m(y)f(the)g(com)m(binatorics)g(in)f(sect.)41 b(4)31 b(and)e(b)m(y)390 2081 y(lemma)h(5.)p 3444 2081 4 62 v 3448 2023 55 4 v 3448 2081 V 3502 2081 4 62 v 490 2260 a(W)-8 b(e)30 b(also)g(note)g(that)g(the)g(considerations)e (of)h([5)q(])h(ab)s(out)f(the)h(analyticit)m(y)f(of)g(the)h(limit)390 2390 y(function)44 b(\026)-51 b Fo(u)40 b Fq(in)e(b)s(oth)h Fo(\013)g Fq(and)g Fo(")h Fq(apply)-8 b(,)41 b(th)m(us)e(giving)g(a)g (rigorous)g(explanation)g(to)h(the)390 2519 y(n)m(umerical)29 b(\014ndings)f(of)i([10)r(],)g([5)q(].)490 2649 y(W)-8 b(e)40 b(observ)m(e)f(that)g(the)g(restriction)f(on)h(the)g(w)m(a)m(y)g (w)m(e)g(tak)m(e)i(the)e(limit)d Fo(!)42 b Fl(!)e Fo(p=q)h Fq(in)390 2778 y(the)31 b(complex)f(plane)g(is)g(tak)m(en)i(only)e(for) g(the)h(sak)m(e)h(of)f(simplicit)m(y:)38 b(in)30 b(fact,)i(it)e(is)g (easy)h(to)390 2908 y(mo)s(dify)f(the)i(pro)s(ofs)f(in)f(suc)m(h)i(a)g (w)m(a)m(y)g(that)h(an)m(y)f(path)f(in)g(the)h(complex)f(plane,)h Fh(pr)-5 b(ovide)g(d)390 3037 y(it)33 b(is)f(not)h(tangent)h(to)f(the)g (r)-5 b(e)g(al)34 b(axis)p Fq(,)d(can)g(b)s(e)f(tak)m(en.)490 3167 y(More)h(precisely)-8 b(,)29 b(let:)1632 3342 y Fo(!)f Fq(=)1823 3280 y Fo(p)p 1823 3321 46 4 v 1824 3404 a(q)1899 3342 y Fq(+)20 b Fo(\020)27 b Fq(+)20 b Fo(i\021)s(;)1071 b Fq(\(6.1\))390 3539 y(with)29 b Fo(p;)15 b(q)28 b Fl(2)d Fm(Z)p Fq(,)h Fo(p)20 b Fl(^)g Fo(q)28 b Fq(=)d(1,)31 b(and)f Fo(\020)7 b(;)15 b(\021)28 b Fl(2)d Fm(R)s Fq(,)37 b(with:)1529 3713 y Fl(j)p Fo(\021)s Fl(j)26 b(\025)f Fo(a)p Fl(j)p Fo(\020)7 b Fl(j)p Fo(;)197 b(a)25 b(>)g Fq(0)p Fo(;)968 b Fq(\(6.2\))390 3888 y(in)26 b(the)i(limit)d Fo(\021)k Fl(!)c Fq(0.)40 b(Condition)26 b(\(6.2\))j(de\014nes)e(a)h (cone)g(in)e(the)i(complex)f Fo(!)k Fq(plane,)c(with)390 4018 y(its)f(v)m(ertex)i(in)e Fo(p=q)k Fq(and)c(its)h(slop)s(e)e(equal) i(to)g Fo(a)p Fq(;)i(an)m(y)e(path)g(inside)d(this)i(cone)i(tends)e(to) i Fo(p=q)390 4147 y Fq(non-tangen)m(tially)-8 b(.)490 4277 y(First)28 b(w)m(e)i(sho)m(w)g(that)g(inequalities)d(lik)m(e)h (\(2.1\))k(can)d(b)s(e)g(deriv)m(ed)g(under)f(the)h(condition)390 4406 y(\(6.2\))r(.)43 b(The)30 b(\014rst)g(inequalit)m(y)g(holds)f(in)h (fact)i(trivially)d(also)i(in)e(this)h(more)i(general)f(case,)390 4536 y(as,)g(for)f Fo(\027)36 b Fq(m)m(ultiple)28 b(of)j Fo(q)s Fq(,)f(w)m(e)h(can)g(write:)1115 4711 y(cos\(2)p Fo(\031)s Fq([)p Fo(p=q)25 b Fq(+)20 b Fo(\020)26 b Fq(+)20 b Fo(i\021)s Fq(])p Fo(\027)6 b Fq(\))27 b(=)d(cos)q(\(2)p Fo(\031)s Fq([)p Fo(\020)k Fq(+)20 b Fo(i\021)s Fq(])p Fo(\027)6 b Fq(\))p Fo(;)554 b Fq(\(6.3\))390 4885 y(so)31 b(that:)1132 5060 y(2)p Fl(j)15 b Fq(cos)r(\(2)p Fo(\031)s Fq([)p Fo(p=q)24 b Fq(+)c Fo(\020)27 b Fq(+)19 b Fo(i\021)s Fq(])p Fo(\027)6 b Fq(\))22 b Fl(\000)e Fq(1)p Fl(j)26 b(\025)f Fq(4)p Fo(\031)2499 5023 y Fi(2)2539 5060 y Fl(j)p Fo(\027)6 b(\021)s Fl(j)2688 5023 y Fi(2)2728 5060 y Fo(;)570 b Fq(\(6.4\))p eop %%Page: 19 19 19 18 bop 955 46 a Fi(SCALING)31 b(PR)n(OPER)-6 b(TIES)31 b(OF)f(THE)h(ST)-6 b(AND)n(ARD)30 b(MAP)613 b(19)270 241 y Fq(b)m(y)30 b(the)h(second)f(inequalit)m(y)f(in)g(\(2.2\))r(.)370 370 y(If)f Fo(\027)j Fl(6)p Fq(=)25 b(0)41 b(\(mo)s(d)29 b Fo(q)s Fq(\),)h(w)m(e)f(can)g(use)f(the)h(third)e(inequalit)m(y)g(in) g(\(2.2\))k(to)e(deduce)f(that,)i(b)m(y)270 500 y(denoting)g Fo(x)25 b Fq(=)g([)p Fo(p=q)e Fq(+)d Fo(\020)7 b Fq(])p Fo(\027)f Fq(,)30 b Fl(j)p Fo(\015)5 b Fq(\()p Fo(\027)h Fq(\))p Fl(j)27 b(\025)e Fq(1)p Fo(=)p Fq(2)31 b(for)f(2)p Fo(\031)s Fl(j)p Fo(x)p Fl(j)d(\025)e Fo(\031)s(=)p Fq(4.)370 629 y(If)30 b(2)p Fo(\031)s Fl(j)p Fo(x)p Fl(j)c(\024)f Fo(\031)s(=)p Fq(4)31 b(w)m(e)g(can)g(use)f(the)g(\014rst)g(inequalit)m (y)f(in)g(\(2.2\))j(and)e(write:)1318 804 y Fl(j)p Fo(\015)5 b Fq(\()p Fo(\027)h Fq(\))p Fl(j)27 b(\025)e Fq(2)p Fo(\031)1764 767 y Fi(2)1804 804 y Fq(\()p Fl(j)p Fo(x)p Fl(j)1941 767 y Fi(2)2001 804 y Fq(+)20 b Fl(j)p Fo(\021)s(\027)6 b Fl(j)2241 767 y Fi(2)2281 804 y Fq(\))p Fo(:)862 b Fq(\(6.5\))270 979 y(Then,)43 b(if)c Fl(j)p Fo(x)p Fl(j)k(\024)f Fq(\(2)p Fo(q)s Fq(\))1056 946 y Fj(\000)p Fi(1)1152 979 y Fq(,)h(one)e(has)g Fl(j)p Fo(\020)7 b(\027)f Fl(j)42 b(\025)g Fq(\(2)p Fo(q)s Fq(\))2032 946 y Fj(\000)p Fi(1)2168 979 y Fq(as)f Fl(j)p Fq([)p Fo(p=q)s Fq(])p Fo(\027)6 b Fl(j)43 b(\025)f Fq(1,)i(so)d(that)h Fl(j)p Fo(\027)6 b Fl(j)42 b(\025)270 1109 y Fq(\(2)p Fo(q)s Fl(j)p Fo(\020)7 b Fl(j)p Fq(\))526 1076 y Fj(\000)p Fi(1)621 1109 y Fq(,)48 b Fh(i.e.)d Fl(j)p Fo(\021)s(\027)6 b Fl(j)48 b(\025)g Fq(\(2)p Fo(q)s Fq(\))1340 1076 y Fj(\000)p Fi(1)1435 1109 y Fl(j)p Fo(\021)s(=\020)7 b Fl(j)p Fq(:)69 b(this)43 b(means)h(that)h(in)e(suc)m(h)h(a)g(case)h Fl(j)p Fo(\015)5 b Fq(\()p Fo(\027)h Fq(\))p Fl(j)49 b(\025)270 1238 y Fq(4)p Fo(\031)370 1205 y Fi(2)410 1238 y Fq(\(2)p Fo(q)s Fq(\))569 1205 y Fj(\000)p Fi(1)664 1238 y Fo(a)p Fq(.)41 b(If)30 b Fl(j)p Fo(x)p Fl(j)c(\025)f Fq(\(2)p Fo(q)s Fq(\))1252 1205 y Fj(\000)p Fi(1)1347 1238 y Fq(,)30 b(then)h Fl(j)p Fo(\015)5 b Fq(\()p Fo(\027)h Fq(\))p Fl(j)26 b(\025)f Fo(\031)2010 1205 y Fi(2)2049 1238 y Fo(q)2093 1205 y Fj(\000)p Fi(2)2187 1238 y Fo(=)p Fq(2.)370 1368 y(Then)41 b(w)m(e)h(can)g(write)f(again)h(the)g(same)g (inequalities)e(as)i(in)e(\(2.1\))r(,)45 b(with)c(the)h(only)270 1497 y(di\013erence)30 b(that)h(no)m(w)f Fo(c)c Fq(=)f(min)n Fl(f)p Fo(a\031)1523 1464 y Fi(2)1563 1497 y Fo(=)p Fq(2)p Fo(;)15 b(q)1737 1464 y Fi(2)1778 1497 y Fo(=)p Fq(2)p Fl(g)26 b(\025)f Fo(a=)p Fq(2.)370 1627 y(As)39 b(our)h(analysis)e(is)h (based)h(on)f(the)h(inequalities)e(\(2.1\))j(and)e(on)h(the)g (de\014nition)e(of)270 1756 y(resonance)29 b(\(2.3\))r(,)g(it)f(can)h (b)s(e)f(rep)s(eated)g(essen)m(tially)g(unc)m(hanged)g(in)f(the)i(case) h(\(6.1\))q(,)g(and)270 1886 y(the)37 b(same)g(results)f(hold.)58 b(In)37 b(fact,)i(the)e(pro)s(of)f(of)h(lemma)g(2)g(can)g(b)s(e)f (carried)g(out)h(in)f(a)270 2015 y(similar)20 b(manner,)j(b)m(y)f (expressing)f(the)h(resonance)h(v)-5 b(alue)22 b(as)g(a)h(function)e (of)h(the)h(quan)m(tities)270 2145 y Fo(\030)t(\027)359 2160 y Fn(`)388 2169 y Ff(1)426 2145 y Fq(,)36 b Fo(:)15 b(:)g(:)31 b Fq(,)37 b Fo(\030)t(\027)774 2160 y Fn(`)803 2168 y Fe(p)835 2185 y(V)896 2145 y Fq(,)g(with)d Fo(\030)j Fq(=)d Fo(\020)c Fq(+)23 b Fo(i\021)s Fq(;)38 b(then)d(b)m(y)h(taking)f (in)m(to)g(accoun)m(t)i(that)f(for)f(an)m(y)g Fo(\027)270 2274 y Fq(suc)m(h)30 b(that)h Fo(\027)g Fq(=)25 b(0)41 b(\(mo)s(d)30 b Fo(q)s Fq(\):)1492 2449 y Fl(j)p Fo(\015)5 b Fq(\()p Fo(\027)h Fq(\))p Fl(j)27 b(\025)e Fq(4)p Fo(\031)s Fl(j)p Fo(\027)6 b(\021)s Fl(j)2087 2411 y Fi(2)2127 2449 y Fo(;)1051 b Fq(\(6.6\))270 2634 y(and)38 b Fl(j)p Fo(\027)6 b(\020)h Fl(j)38 b(\024)750 2553 y Fk(p)p 840 2553 285 4 v 840 2634 a Fo(\020)887 2607 y Fi(2)947 2634 y Fq(+)20 b Fo(\021)1086 2607 y Fi(2)1164 2634 y Fl(\024)38 b(j)p Fo(\021)s Fl(j)p Fq(\(1)27 b(+)e Fo(a)1622 2601 y Fj(\000)p Fi(1)1716 2634 y Fq(\),)41 b(one)e(sees)f(that)h(the)f (cancellation)g(mec)m(han-)270 2763 y(isms)f(op)s(erate)i(exactly)g(in) e(the)i(same)f(w)m(a)m(y)i(as)e(b)s(efore,)i(and)e(the)h(second)f (order)g(terms)270 2893 y(can)47 b(b)s(e)f(dealt)h(with)e(as)i(in)e (sect.)91 b(3,)51 b(with)45 b(the)i(only)f(di\013erence)g(that)h(no)m (w)g Fo(D)3227 2907 y Fi(0)3319 2893 y Fq(=)270 3022 y(max)p Fl(f)p Fo(cd)570 2989 y Fi(2)570 3046 y(1)611 3022 y Fo(;)15 b(d)698 3036 y Fi(2)738 3022 y Fl(g)p Fo(q)827 2989 y Fi(2)867 3022 y Fq(\(1)21 b(+)f Fo(a)1107 2989 y Fj(\000)p Fi(1)1201 3022 y Fq(\).)370 3152 y(Once)g(the)g(p)s (erturbation)f(parameter)i Fo(")f Fq(has)g(b)s(een)g(scaled)g(to)h(\(2) p Fo(\031)s(\030)t Fq(\))2679 3119 y Fi(2)p Fn(=q)2789 3152 y Fo(")p Fq(,)i(the)e(surviving)270 3281 y(terms)28 b(are)h(exactly)g(the)g(same)g(as)g(b)s(efore,)f(so)h(that)g(all)f(of)g (the)h(ab)s(o)m(v)m(e)h(discussions)25 b(apply)270 3411 y Fh(verb)-5 b(atim)7 b Fq(;)32 b(in)d(fact:)1313 3586 y(lim)1304 3643 y Fn(\021)r Fj(!)p Fi(0)1463 3586 y Fo(\015)5 b Fq(\()p Fo(\027)h Fq(\)[2)p Fo(\031)s Fq(\()p Fo(\020)28 b Fq(+)20 b Fo(i\021)s Fq(\)])2094 3548 y Fj(\000)p Fi(2)2215 3586 y Fq(=)25 b(1)847 b(\(6.7\))270 3787 y(for)30 b Fo(\027)36 b Fq(m)m(ultiple)28 b(of)j Fo(q)i Fq(and)d Fo(\020)7 b Fq(,)30 b Fo(\021)j Fq(satisfying)e(\(6.2\))r(.)370 3917 y(Our)37 b(main)g(results)g(therefore)h(still)e(apply)h(pro)m (vided)g(the)i(path)f(tak)m(en)h(b)m(y)f Fo(!)j Fq(while)270 4046 y(tending)30 b(to)i Fo(p=q)j Fq(is)30 b(not)i(tangen)m(tial)f(to)h (the)g(real)f(axis,)g(so)h(that)g(\(6.2\))h(applies)c(for)j(some)270 4176 y Fo(a)p Fq(.)1565 4617 y Fp(References)308 4781 y Fu([1])40 b(MacKa)n(y)-6 b(,)31 b(R.)f(S.,)i Fr(T)-6 b(r)l(ansition)32 b(to)g(Chaos)h(for)e(A)n(r)l(e)l(a-pr)l(eserving)k (Maps)p Fu(,)d(Springer)e(Lect.)h(Notes)f(in)428 4890 y(Ph)n(ysics,)c(v)n(ol.)g(247,)h(p.)f(390)h(\(1985\))308 5000 y([2])40 b(Gen)n(tile,)26 b(G.)h(and)e(Mastropietro,)j(V.,)e Fr(KAM)h(The)l(or)l(em)h(R)l(evisite)l(d)p Fu(,)f(Ph)n(ys.)f Fb(D)j(90)p Fu(,)d(225)h(\(1996\))308 5109 y([3])40 b(Galla)n(v)n (otti,)27 b(G.,)g Fr(Twistless)h(KAM)f(tori)p Fu(,)g(Comm)n(un.)d (Math.)i(Ph)n(ys.)f Fb(164)p Fu(,)i(145)g(\(1994\))p eop %%Page: 20 20 20 19 bop 390 46 a Fi(20)704 b(ALBER)-6 b(TO)30 b(BERRETTI)h(AND)f (GUIDO)h(GENTILE)428 241 y Fu([4])40 b(Gen)n(tile,)34 b(G.)e(and)f(Mastropietro,)k(V.,)e Fr(Metho)l(ds)h(for)f(the)g(A)n (nalysis)h(of)e(the)i(Lindste)l(dt)g(Series)g(for)548 350 y(KAM)d(tori)g(and)g(R)l(enormalizability)h(in)e(Classic)l(al)h(Me) l(chanics.)h(A)f(R)l(eview)g(with)g(Some)g(Applic)l(a-)548 460 y(tions.)26 b Fu(Rev.)f(Math.)h(Ph)n(ys.)g Fb(8)p Fu(,)g(393)h(\(1996\))428 570 y([5])40 b(Berretti,)25 b(A.)f(and)f(Marmi,)i(S.,)f Fr(Sc)l(aling)i(ne)l(ar)h(R)l(esonanc)l(es) h(and)e(Complex)g(R)l(otation)h(Numb)l(ers)g(for)548 679 y(the)h(Standar)l(d)i(Map)p Fu(,)c(Nonlinearit)n(y)h Fb(7)p Fu(,)f(603)h(\(1994\))428 789 y([6])40 b(Berretti,)33 b(A.,)f Fr(A)n(nalyticity,)h(Sc)l(aling)f(and)h(R)l(enormalization)f (for)g(Some)h(Complex)f(A)n(nalytic)g(Dy-)548 898 y(namic)l(al)27 b(Systems)p Fu(,)h(1997)f(\(to)e(app)r(ear)i(in)e(Planetary)h(and)g (Space)f(Sciences\))428 1008 y([7])40 b(Berretti,)31 b(A.,)f(Marmi,)g(S.)f(and)f(Sauzin,)i(D.,)g Fr(Limit)g(at)h(R)l (esonanc)l(es)i(of)d(Line)l(arizations)i(of)f(Some)548 1117 y(Complex)d(A)n(nalytic)g(Dynamic)l(al)f(Systems)p Fu(,)g(1998)h(\(in)d(preparation\))428 1227 y([8])40 b(Marmi,)34 b(S.)e(and)g(Stark,)i(J.,)h Fr(On)e(the)i(Standar)l(d)g (Map)f(Critic)l(al)g(F)-6 b(unction)p Fu(,)35 b(Nonlinearit)n(y)e Fb(5)p Fu(,)i(743)548 1337 y(\(1992\))428 1446 y([9])40 b(Marmi,)28 b(S.,)g(Moussa,)i(P)-6 b(.,)28 b(Y)-6 b(o)r(ccoz,)29 b(J.)f(C.,)h Fr(The)g(Brjuno)g(F)-6 b(unctions)31 b(and)f(their)f(R)l (e)l(gularity)i(Pr)l(op-)548 1556 y(erties)p Fu(,)c(preprin)n(t)e (Sacla)n(y)h(95/028,)i(1995)390 1665 y([10])40 b(Berretti,)33 b(A.)e(and)g(Marmi,)i(S.,)f Fr(Standar)l(d)j(Map)d(at)h(Complex)g(R)l (otation)h(Numb)l(ers:)46 b(Cr)l(e)l(ation)33 b(of)548 1775 y(Natur)l(al)28 b(Boundaries)p Fu(,)g(Ph)n(ys.)d(Rev.)g(Lett.)h Fb(68)p Fu(,)h(1443)g(\(1992\))490 1980 y Ft(Alber)-5 b(to)39 b(Berretti,)h(Dip)-5 b(ar)g(timento)37 b(di)h(Ma)-5 b(tema)g(tica,)40 b(I)r(I)f(Universit)2892 1974 y(\022)2890 1980 y(a)e(di)h(R)n(oma)g(\(Tor)390 2090 y(Ver)n(ga)-5 b(t)g(a\),)31 b(Via)f(della)i(Ricer)n(ca)e(Scientifica,)g(00133)j(R)n (oma,)f(It)-5 b(al)g(y)31 b(and)g(INFN,)g(Sez.)45 b(Tor)390 2200 y(Ver)n(ga)-5 b(t)g(a)490 2309 y Fr(E-mail)27 b(addr)l(ess)6 b Fu(:)36 b Fa(berretti@roma2.infn.it)490 2496 y Ft(Guido)e(Gentile,)j (Dip)-5 b(ar)g(timento)34 b(di)h(Ma)-5 b(tema)g(tica,)36 b(Universit)2641 2490 y(\022)2639 2496 y(a)e(di)h(R)n(oma)h(3,)h(Lar)n (go)f(S.)390 2605 y(Leonardo)28 b(Murialdo)g(1,)g(00146)i(R)n(oma,)e (It)-5 b(al)g(y)490 2715 y Fr(E-mail)27 b(addr)l(ess)6 b Fu(:)36 b Fa(gentile@matrm3.mat.uniroma3.)q(it)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ENDBODY