Content-Type: multipart/mixed; boundary="-------------0501280249931" This is a multi-part message in MIME format. ---------------0501280249931 Content-Type: text/plain; name="05-36.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-36.keywords" Dissipative systems; Periodically forced systems; Quasi-periodically forced systems; Lindstedt series; Renormalization group; Divergent series; Borel summability. ---------------0501280249931 Content-Type: application/postscript; name="gbd.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="gbd.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: fish5.dvi %%Pages: 24 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o gbd.ps fish5 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.01.27:1058 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}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{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/id 0 N/rw 0 N/rc 0 N/gp 0 N/cp 0 N/G 0 N/CharBuilder{save 3 1 roll S A/base get 2 index get S/BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]/id Ci N/rw Cw 7 add 8 idiv string N/rc 0 N/gp 0 N/cp 0 N{ rc 0 ne{rc 1 sub/rc X rw}{G}ifelse}imagemask restore}B/G{{id gp get/gp gp 1 add N A 18 mod S 18 idiv pl S get exec}loop}B/adv{cp add/cp X}B /chg{rw cp id gp 4 index getinterval putinterval A gp add/gp X adv}B/nd{ /cp 0 N rw exit}B/lsh{rw cp 2 copy get A 0 eq{pop 1}{A 255 eq{pop 254}{ A A add 255 and S 1 and or}ifelse}ifelse put 1 adv}B/rsh{rw cp 2 copy get A 0 eq{pop 128}{A 255 eq{pop 127}{A 2 idiv S 128 and or}ifelse} ifelse put 1 adv}B/clr{rw cp 2 index string putinterval adv}B/set{rw cp fillstr 0 4 index getinterval putinterval adv}B/fillstr 18 string 0 1 17 {2 copy 255 put pop}for N/pl[{adv 1 chg}{adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{adv rsh nd}{1 add adv}{/rc X nd}{ 1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]A{bind pop} forall N/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: special.pro %! 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Fv(f)29 b Fy(b)r(oth)22 b(analytic)e(in)i(their)f(argumen)n (ts,)g(can)g(b)r(e)g(dealt)g(with)h(essen)n(tially)e(in)i(the)f(same)g (w)n(a)n(y)-7 b(.)33 b(Simply)-7 b(,)118 2417 y(w)n(e)27 b(ha)n(v)n(e)g(to)g(imp)r(ose)h(a)f(non-degeneracy)e(condition)j(on)f (the)h(function)g Fv(g)s Fy(,)f(whic)n(h)h(reads)e(as)1171 2614 y Fs(9)p Fv(x)1264 2626 y Fu(0)1329 2614 y Fy(suc)n(h)h(that)h Fv(g)s Fy(\()p Fv(x)1818 2626 y Fu(0)1856 2614 y Fy(\))23 b(=)g Fv(f)2040 2626 y Fu(0)2104 2614 y Fy(and)28 b Fv(g)2309 2580 y FA(0)2332 2614 y Fy(\()p Fv(x)2411 2626 y Fu(0)2449 2614 y Fy(\))23 b Fs(6)p Fy(=)g(0)p Fv(:)881 b Fy(\(1)p Fv(:)p Fy(6\))118 2811 y(In)30 b(the)g(particular)e(case)h(of)h (homogeneous)e Fv(g)s Fy(\()p Fv(x)p Fy(\),)i(that)g(is)g Fv(g)s Fy(\()p Fv(x)p Fy(\))d(=)f Fv(\033)s(x)2383 2781 y Fz(p)2422 2811 y Fy(,)31 b(with)f Fv(p)c Fs(\025)g Fy(2)j(an)h(in)n(teger)e(and)i Fv(\033)g Fs(2)d Fr(R)11 b Fy(,)118 2917 y(the)27 b(condition)g(is)g(automatically)e (satis\014ed)i(if)g Fv(p)g Fy(is)f(o)r(dd)h(\(for)g(an)n(y)f(v)-5 b(alue)27 b(of)f Fv(\033)s Fy(\),)i(while)f(it)g(requires)f Fv(\033)s(f)3393 2929 y Fu(0)3453 2917 y Fv(>)d Fy(0)j(for)118 3024 y Fv(p)i Fy(ev)n(en,)f({)g(as)g(assumed)g(in)h(\(1.1\).)189 3130 y(The)36 b(pap)r(er)f(is)g(organized)f(as)h(follo)n(ws.)60 b(F)-7 b(or)35 b(exp)r(ository)g(clearness)f(w)n(e)h(start)g(with)h (the)g(case)f(of)h(p)r(erio)r(dic)118 3236 y(forcing)25 b(terms.)36 b(In)26 b(Sections)f(2)g(and)h(3)f(w)n(e)h(sho)n(w)e(that)i (a)g(p)r(erio)r(dic)f(solution)g(with)i(frequency)e Fv(!)j Fy(in)e(the)g(form)g(of)118 3343 y(a)g(formal)g(p)r(o)n(w)n(er)f (series)h(in)h Fv(")f Fy(\(p)r(erturbation)g(series\))g(is)g(w)n(ell)h (de\014ned)g(to)f(all)g(orders,)g(and)g(it)h(admits)f(a)h(natural)118 3449 y(graphical)22 b(represen)n(tation.)34 b(In)24 b(Section)g(4)f(w)n (e)h(study)g(further)g(suc)n(h)f(a)g(series,)h(and)g(w)n(e)f(see)h (that)g(there)f(is)h(strong)118 3555 y(evidence)h(to)h(exp)r(ect)f (that)h(it)g(div)n(erges.)35 b(The)25 b(b)r(est)h(b)r(ounds)f(that)h(w) n(e)f(are)g(able)g(to)g(pro)n(vide)f(for)h(the)h(co)r(e\016cien)n(ts) 118 3661 y(gro)n(w)k(as)h(factorials.)48 b(T)-7 b(o)32 b(obtain)f(b)r(ounds)h(whic)n(h)g(allo)n(w)f(summabilit)n(y)g(of)h(the) g(p)r(erturbation)g(series)e(w)n(e)i(ha)n(v)n(e)118 3768 y(to)e(p)r(erform)h(a)f(suitable)g(summation)g(in)h(order)e(to)i(giv)n (e)e(the)i(series)f(a)g(meaning.)45 b(This)30 b(is)h(done)f(in)h (Section)f(5,)118 3874 y(and)25 b(the)h(resummed)f(series)g(is)g(found) h(to)f(represen)n(t)f(a)h(2)p Fv(\031)s(=!)s Fy(-p)r(erio)r(dic)f (solution)h(whic)n(h)g(is)h(Borel-summable)d(in)118 3980 y Fv(")p Fy(.)38 b(T)-7 b(o)28 b(pro)n(v)n(e)e(the)j(latter)e(prop)r (ert)n(y)g(w)n(e)h(rely)f(on)h(Nev)-5 b(anlinna's)28 b(impro)n(v)n(emen)n(t)f(of)h(W)-7 b(atson's)27 b(theorem)h([12].)38 b(In)118 4087 y(Section)27 b(6)f(w)n(e)g(consider)g(the)h(case)f(of)g (quasi-p)r(erio)r(dic)g(forcing)f(terms.)37 b(W)-7 b(e)27 b(\014nd)g(that)g(the)g(p)r(erturbation)f(series)118 4193 y(is)35 b(w)n(ell)f(de\014ned)h(if)g(the)g(frequency)f(v)n(ector)f (of)i(the)g(forcing)f(term)g(satis\014es)g(a)g(Diophan)n(tine)h (condition,)h(and,)118 4299 y(b)n(y)d(using)h(Renormalization)d(Group)i (tec)n(hniques)h(in)g(order)e(to)h(deal)g(with)h(the)g(small)f (divisors)g(problem,)h(w)n(e)118 4406 y(\014nd)c(that)f(the)h(resummed) e(series)g(still)i(con)n(v)n(erges)c(to)j(a)g(quasi-p)r(erio)r(dic)f (solution,)h(and)g(it)g(de\014nes)g(a)g(function)118 4512 y(analytic)i(in)h(a)f(domain)g(con)n(taining)g(the)h(origin)e(in)i (its)g(b)r(oundary)-7 b(.)48 b(W)-7 b(e)32 b(shall)f(see)g(that)h(the)g (b)r(ounds)g(w)n(e)f(\014nd)118 4618 y(do)26 b(not)g(allo)n(w)g(us)g (an)n(y)g(more)f(to)h(obtain)g(Borel)f(summabilit)n(y)-7 b(,)27 b(unlik)n(e)f(the)h(case)e(of)i(p)r(erio)r(dic)f(forcing)f (terms.)37 b(In)118 4724 y(Section)c(7)f(w)n(e)g(discuss)g(ho)n(w)g(to) h(extend)g(the)g(analysis)e(to)h(more)g(general)f(nonlinearities)h Fv(g)s Fy(\()p Fv(x)p Fy(\),)j(b)n(y)d(requiring)118 4831 y(the)c(condition)f(\(1.6\))h(to)f(b)r(e)h(satis\014ed.)189 4937 y(The)f(in)n(terest)g(of)g(the)g(approac)n(h)f(w)n(e)g(prop)r(ose) g(is)h(that)h(it)f(allo)n(ws)f(using)h(p)r(erturbation)f(theory)h(whic) n(h)g(can)f(b)r(e)118 5043 y(v)n(ery)g(natural)h(in)h(problems)e(in)i (whic)n(h)g(a)f(small)g(parameter)f(app)r(ears.)35 b(In)28 b(fact)f(analyticit)n(y)g(in)h Fv(")f Fy(for)g Fv(")g Fy(close)g(to)118 5150 y(0)k(\(that)i(is)e(in)h Fv(\015)k Fy(for)31 b Fv(\015)37 b Fy(large)30 b(enough\))h(could)h(b)r(e)g(pro)n (v)n(ed)e(v)n(ery)g(lik)n(ely)h(with)i(other)e(tec)n(hniques,)h(but)g (a)g(naiv)n(e)1893 5349 y(2)p eop %%Page: 3 3 3 2 bop 118 319 a Fy(expansion)31 b(in)h(p)r(o)n(w)n(ers)e(of)i Fv(")f Fy(is)h(prev)n(en)n(ted)e(b)n(y)i(the)g(lac)n(k)f(of)g (analyticit)n(y)g(in)h(a)f(neigh)n(b)r(ourho)r(o)r(d)g(of)h(the)g (origin.)118 426 y(On)25 b(the)g(other)g(hand)g(the)g(p)r(erturbation)g (series)e(giv)n(es)h(a)h(v)n(ery)f(accurate)f(description)i(of)g(the)g (solution,)g(hence)g(it)118 532 y(is)h(imp)r(ortan)n(t)f(to)h(kno)n(w)f (that)h(suc)n(h)g(a)f(series)g(is)g(an)h(asymptotic)f(series,)g(and)h (its)g(use)g(is)f(fully)i(justi\014ed.)37 b(Finally)118 638 y(w)n(e)26 b(can)f(men)n(tion)h(that)g(the)g(quasi-p)r(erio)r(dic)f (solution)g(w)n(e)h(in)n(v)n(estigate)e(is)i(of)g(ph)n(ysical)f(relev) -5 b(ance,)25 b(hence)h(it)g(can)118 744 y(b)r(e)31 b(useful)f(to)g (study)g(its)g(prop)r(erties.)44 b(F)-7 b(or)29 b(instance)h(in)g(the)h (case)e(of)h(the)h(aforemen)n(tioned)d(resistor-inductor-)118 851 y(v)-5 b(aractor)24 b(circuit)h(in)h(Ref.)f([5])h(the)f(2)p Fv(\031)s(=!)s Fy(-p)r(erio)r(dic)f(solution)h(is)h(n)n(umerically)e (found)i(to)f(attract)g(an)n(y)f(tra)5 b(jectory)118 957 y(whic)n(h)28 b(remains)e(b)r(ounded)i(in)g(phase)f(space.)1465 1205 y FB(2.)50 b(F)-9 b(ormal)36 b(analysis)118 1382 y Fy(Consider)27 b(\014rst)g(\(1.1\))g(for)g Fv(d)d Fy(=)e(1,)28 b(that)g(is)1498 1563 y Fv(")5 b Fy(\177)-47 b Fv(x)18 b Fy(+)33 b(_)-38 b Fv(x)19 b Fy(+)f Fv("x)1920 1528 y Fu(2)1981 1563 y Fy(=)k Fv("f)9 b Fy(\()p Fv(!)s(t)p Fy(\))p Fv(;)1209 b Fy(\(2)p Fv(:)p Fy(1\))118 1743 y(with)28 b Fv(f)9 b Fy(\()p Fv( )s Fy(\))28 b(giv)n(en)f(b)n(y)g(\(1.4\).)36 b(W)-7 b(e)28 b(lo)r(ok)f(for)g(b)r(ounded)h(solutions)f(\(if)h(an)n (y\))f(whic)n(h)g(are)g(analytic)g(in)g Fv(")p Fy(,)h(that)f(is)h(of) 118 1849 y(the)g(form)1551 1995 y Fv(x)p Fy(\()p Fv(t)p Fy(\))c(=)1831 1891 y FA(1)1804 1916 y Fq(X)1804 2095 y Fz(k)q Fu(=0)1938 1995 y Fv(")1977 1961 y Fz(k)2018 1995 y Fv(x)2065 1961 y Fu(\()p Fz(k)q Fu(\))2158 1995 y Fy(\()p Fv(t)p Fy(\))p Fv(:)1263 b Fy(\(2)p Fv(:)p Fy(2\))118 2220 y(Inserting)30 b(\(2.2\))g(in)n(to)g(\(2.1\))g(and)g (equating)g(terms)g(of)g(the)h(same)f(T)-7 b(a)n(ylor)28 b(order)h(w)n(e)h(\014nd)h(the)g(set)f(of)g(recursiv)n(e)118 2327 y(equations)1041 2429 y(_)-38 b Fv(x)1073 2395 y Fu(\(0\))1186 2429 y Fy(=)23 b(0)p Fv(;)1041 2569 y Fy(_)-38 b Fv(x)1073 2535 y Fu(\(1\))1186 2569 y Fy(=)23 b Fs(\000)5 b Fy(\177)-47 b Fv(x)1386 2535 y Fu(\(0\))1493 2569 y Fs(\000)18 b Fv(x)1623 2535 y Fu(\(0\)2)1764 2569 y Fy(+)g Fv(f)t(;)1037 2718 y Fy(_)-37 b Fv(x)1070 2684 y Fu(\()p Fz(k)q Fu(\))1186 2718 y Fy(=)23 b Fs(\000)5 b Fy(\177)-47 b Fv(x)1386 2684 y Fu(\()p Fz(k)q FA(\000)p Fu(1\))1582 2718 y Fs(\000)1784 2640 y Fq(X)1665 2818 y Fz(k)1700 2826 y Fm(1)1732 2818 y Fu(+)p Fz(k)1818 2826 y Fm(2)1851 2818 y Fu(=)p Fz(k)q FA(\000)p Fu(1)2038 2718 y Fv(x)2085 2684 y Fu(\()p Fz(k)2146 2692 y Fm(1)2179 2684 y Fu(\))2209 2718 y Fv(x)2256 2684 y Fu(\()p Fz(k)2317 2692 y Fm(2)2350 2684 y Fu(\))2380 2718 y Fv(;)180 b(k)26 b Fs(\025)c Fy(2)p Fv(:)3538 2614 y Fy(\(2)p Fv(:)p Fy(3\))118 2964 y(F)-7 b(rom)26 b(the)h(\014rst)g(equation)f(\(zeroth)g(order\))g(w)n (e)g(obtain)g(that)h Fv(x)2124 2934 y Fu(\(0\))2241 2964 y Fy(has)f(to)g(b)r(e)h(constan)n(t,)g(sa)n(y)e Fv(x)3147 2934 y Fu(\(0\))3260 2964 y Fy(=)d Fv(c)3383 2976 y Fu(0)3447 2964 y Fy(with)27 b Fv(c)3671 2976 y Fu(0)118 3070 y Fy(to)c(b)r(e)g(determined.)35 b(The)23 b(second)f(equation)f (\(\014rst)i(order\))f(can)g(giv)n(e)f(a)i(b)r(ounded)g(solution)f (only)g(if)h Fs(\000)p Fv(c)3362 3040 y Fu(2)3362 3091 y(0)3407 3070 y Fy(+)8 b Fv(\013)23 b Fy(=)g(0,)118 3176 y(whic)n(h)i(\014xes)f Fv(c)573 3188 y Fu(0)634 3176 y Fy(=)721 3117 y Fs(p)p 790 3117 54 4 v 59 x Fv(\013)g Fy(=)f Fv(a)h Fy(and)h(giv)n(es)f Fv(x)1430 3146 y Fu(\(1\))1519 3176 y Fy(\()p Fv(t)p Fy(\))i(as)e(a)g(p)r(erio)r(dic)h(function)g (with)h(the)f(same)f(p)r(erio)r(d)h(of)f(the)i(forcing)118 3283 y(term:)1212 3419 y Fv(x)1259 3385 y Fu(\(1\))1348 3419 y Fy(\()p Fv(t)p Fy(\))e(=)e Fv(x)1600 3385 y Fu(\(1\))1690 3419 y Fy(\(0\))c(+)1897 3306 y Fq(Z)1981 3327 y Fz(t)1944 3495 y Fu(0)2024 3419 y Fy(d)p Fv(t)2100 3385 y FA(0)2137 3419 y Fy(\()p Fv(f)9 b Fy(\()p Fv(!)s(t)2336 3385 y FA(0)2359 3419 y Fy(\))19 b Fs(\000)f Fv(\013)p Fy(\))c Fv(:)923 b Fy(\(2)p Fv(:)p Fy(4\))118 3628 y(As)25 b(eac)n(h)e Fv(x)468 3598 y Fu(\()p Fz(k)q Fu(\))561 3628 y Fy(\()p Fv(t)p Fy(\))i(dep)r(ends)g(on)f(the)h(functions)g Fv(x)1652 3598 y Fu(\()p Fz(k)1714 3573 y Fl(0)1737 3598 y Fu(\))1768 3628 y Fy(\()p Fv(t)p Fy(\))g(with)g Fv(k)2119 3598 y FA(0)2165 3628 y Fv(<)d(k)s Fy(,)j(w)n(e)f(exp)r(ect)h(that)g(if)g (there)f(is)g(an)n(y)g(p)r(erio)r(dic)118 3734 y(solution)j(then)h(it)g (m)n(ust)g(ha)n(v)n(e)e(the)i(same)f(p)r(erio)r(d)h(as)f(the)h(forcing) f(term.)189 3841 y(T)-7 b(o)28 b(con)n(tin)n(ue)g(the)h(analysis)e(to)h (all)g(orders)f(it)i(is)f(more)f(con)n(v)n(enien)n(t)h(to)g(write)g (the)h(recursiv)n(e)d(equations)i(\(2.3\))118 3947 y(in)k(F)-7 b(ourier)30 b(space.)47 b(The)31 b(analysis)f(to)h(\014rst)g(order)f (and)h(the)h(considerations)d(ab)r(o)n(v)n(e)h(motiv)-5 b(ate)31 b(us)g(to)g(write)g(in)118 4053 y(\(2.2\))1154 4220 y Fv(x)p Fy(\()p Fv(t)p Fy(\))24 b(=)1434 4116 y FA(1)1407 4141 y Fq(X)1407 4319 y Fz(k)q Fu(=0)1542 4220 y Fv(")1581 4185 y Fz(k)1621 4220 y Fv(x)1668 4185 y Fu(\()p Fz(k)q Fu(\))1761 4220 y Fy(\()p Fv(t)p Fy(\))g(=)1994 4116 y FA(1)1967 4141 y Fq(X)1967 4319 y Fz(k)q Fu(=0)2101 4220 y Fv(")2140 4185 y Fz(k)2199 4141 y Fq(X)2195 4327 y Fz(\027)t FA(2)p Fo(Z)2337 4220 y Fv(e)2376 4185 y Fz(i\027)t(!)r(t)2509 4220 y Fv(x)2556 4185 y Fu(\()p Fz(k)q Fu(\))2556 4240 y Fz(\027)2649 4220 y Fv(;)866 b Fy(\(2)p Fv(:)p Fy(5\))118 4457 y(whic)n(h)28 b(inserted)f(in)n(to)g (\(2.3\))h(giv)n(es)e(for)h Fv(\027)h Fs(6)p Fy(=)23 b(0)689 4638 y Fv(x)736 4604 y Fu(\(0\))736 4659 y Fz(\027)848 4638 y Fy(=)g(0)p Fv(;)689 4814 y(x)736 4780 y Fu(\(1\))736 4835 y Fz(\027)848 4814 y Fy(=)970 4758 y Fv(f)1011 4770 y Fz(\027)p 946 4795 130 4 v 946 4871 a Fv(i!)s(\027)1085 4814 y(;)685 5014 y(x)732 4980 y Fu(\()p Fz(k)q Fu(\))732 5035 y Fz(\027)848 5014 y Fy(=)g Fs(\000)p Fy(\()p Fv(i!)s(\027)5 b Fy(\))14 b Fv(x)1256 4980 y Fu(\()p Fz(k)q FA(\000)p Fu(1\))1256 5035 y Fz(\027)1452 5014 y Fs(\000)1589 4958 y Fy(1)p 1545 4995 V 1545 5071 a Fv(i!)s(\027)1828 4935 y Fq(X)1708 5114 y Fz(k)1743 5122 y Fm(1)1776 5114 y Fu(+)p Fz(k)1862 5122 y Fm(2)1895 5114 y Fu(=)p Fz(k)q FA(\000)p Fu(1)1778 5162 y Fn(k)1809 5174 y Fm(1)1842 5162 y Fn(;k)1892 5174 y Fm(2)1925 5162 y Fl(\025)p Fm(0)2166 4935 y Fq(X)2091 5110 y Fz(\027)2124 5118 y Fm(1)2157 5110 y Fu(+)p Fz(\027)2241 5118 y Fm(2)2273 5110 y Fu(=)p Fz(\027)2375 5014 y Fv(x)2422 4980 y Fu(\()p Fz(k)2483 4988 y Fm(1)2516 4980 y Fu(\))2422 5035 y Fz(\027)2455 5043 y Fm(1)2547 5014 y Fv(x)2594 4980 y Fu(\()p Fz(k)2655 4988 y Fm(2)2688 4980 y Fu(\))2594 5035 y Fz(\027)2627 5043 y Fm(2)2718 5014 y Fv(;)180 b(k)26 b Fs(\025)c Fy(2)p Fv(;)3538 4892 y Fy(\(2)p Fv(:)p Fy(6\))1893 5349 y(3)p eop %%Page: 4 4 4 3 bop 118 319 a Fy(pro)n(vided)27 b(that)h(one)f(has)g(for)g Fv(\027)h Fy(=)23 b(0)1160 526 y(0)g(=)f Fs(\000)p Fv(x)1424 483 y Fu(\(0\)2)1424 548 y(0)1565 526 y Fy(+)c Fv(f)1689 538 y Fu(0)1726 526 y Fv(;)1160 675 y Fy(0)23 b(=)1399 596 y Fq(X)1322 775 y Fz(k)1357 783 y Fm(1)1390 775 y Fu(+)p Fz(k)1476 783 y Fm(2)1509 775 y Fu(=)p Fz(k)1349 823 y Fn(k)1380 835 y Fm(1)1413 823 y Fn(;k)1463 835 y Fm(2)1496 823 y Fl(\025)p Fm(0)1693 596 y Fq(X)1620 772 y Fz(\027)1653 780 y Fm(1)1686 772 y Fu(+)p Fz(\027)1770 780 y Fm(2)1802 772 y Fu(=0)1900 675 y Fv(x)1947 641 y Fu(\()p Fz(k)2008 649 y Fm(1)2041 641 y Fu(\))1947 696 y Fz(\027)1980 704 y Fm(1)2071 675 y Fv(x)2118 641 y Fu(\()p Fz(k)2179 649 y Fm(2)2213 641 y Fu(\))2118 696 y Fz(\027)2151 704 y Fm(2)2243 675 y Fv(;)180 b(k)26 b Fs(\025)c Fy(1)p Fv(:)3538 662 y Fy(\(2)p Fv(:)p Fy(7\))118 1034 y(If)28 b(w)n(e)f(set)h Fv(x)500 991 y Fu(\()p Fz(k)q Fu(\))500 1057 y(0)616 1034 y Fy(=)23 b Fv(c)740 1046 y Fz(k)808 1034 y Fy(then)28 b(the)g(\014rst)g(of)f(\(2.7\))h(\014xes,) f(as)g(already)f(noted,)1671 1236 y Fv(c)1707 1248 y Fu(0)1768 1236 y Fy(=)c Fv(a)h Fy(=)2010 1172 y Fs(p)p 2079 1172 54 4 v 64 x Fv(\013;)1383 b Fy(\(2)p Fv(:)p Fy(8\))118 1438 y(b)r(ecause)27 b(one)g(has)h Fv(f)767 1450 y Fu(0)827 1438 y Fy(=)22 b Fv(\013)i(>)e Fy(0,)27 b(while)h(the)g(second)f(of)h(\(2.7\))f(giv)n(es)1505 1611 y Fz(k)1463 1636 y Fq(X)1452 1814 y Fz(k)1488 1798 y Fl(0)1511 1814 y Fu(=0)1627 1636 y Fq(X)1609 1822 y Fz(\027)1642 1830 y Fm(1)1674 1822 y FA(2)p Fo(Z)1779 1715 y Fv(x)1826 1680 y Fu(\()p Fz(k)q FA(\000)p Fz(k)1976 1655 y Fl(0)2001 1680 y Fu(\))1826 1735 y Fz(\027)1859 1743 y Fm(1)2031 1715 y Fv(x)2078 1671 y Fu(\()p Fz(k)2140 1646 y Fl(0)2163 1671 y Fu(\))2078 1735 y FA(\000)p Fz(\027)2163 1743 y Fm(1)2223 1715 y Fy(=)c(0)p Fv(:)1162 b Fy(\(2)p Fv(:)p Fy(9\))118 2005 y(The)32 b(latter)g(equation,)g(b)n(y)g(taking)f (in)n(to)h(accoun)n(t)f(\(2.8\))h(and)g(the)g(\014rst)g(of)g(\(2.6\),)h (can)e(b)r(e)i(more)e(con)n(v)n(enien)n(tly)118 2111 y(written)d(as)830 2272 y Fv(c)866 2284 y Fu(1)926 2272 y Fy(=)23 b(0)p Fv(;)345 b(c)1460 2284 y Fz(k)1524 2272 y Fy(=)23 b Fs(\000)1723 2216 y Fy(1)p 1687 2253 115 4 v 1687 2329 a(2)p Fv(c)1765 2341 y Fu(0)1835 2168 y Fz(k)q FA(\000)p Fu(1)1836 2193 y Fq(X)1825 2372 y Fz(k)1861 2355 y Fl(0)1884 2372 y Fu(=1)2000 2193 y Fq(X)1982 2379 y Fz(\027)2015 2387 y Fm(1)2047 2379 y FA(2)p Fo(Z)2152 2272 y Fv(x)2199 2237 y Fu(\()p Fz(k)q FA(\000)p Fz(k)2349 2212 y Fl(0)2374 2237 y Fu(\))2199 2292 y Fz(\027)2232 2300 y Fm(1)2404 2272 y Fv(x)2451 2229 y Fu(\()p Fz(k)2513 2204 y Fl(0)2536 2229 y Fu(\))2451 2292 y FA(\000)p Fz(\027)2536 2300 y Fm(1)2573 2272 y Fv(;)180 b(k)26 b Fs(\025)c Fy(2)p Fv(;)499 b Fy(\(2)p Fv(:)p Fy(10\))118 2518 y(whic)n(h)30 b(pro)n(vides)f(an)h(iterativ)n(e)f(de\014nition)h(of)g(the)h(co)r (e\016cien)n(ts)f Fv(c)2210 2530 y Fz(k)2280 2518 y Fy(as)g(the)g(righ) n(t)g(hand)g(side)g(dep)r(ends)g(only)g(on)118 2624 y(the)j(co)r (e\016cien)n(ts)e Fv(c)726 2636 y Fz(k)762 2620 y Fl(0)822 2624 y Fy(with)i Fv(k)1062 2594 y FA(0)1116 2624 y Fv(<)d(k)s Fy(.)51 b(T)-7 b(o)31 b(deduce)i Fv(c)1775 2636 y Fu(1)1843 2624 y Fy(=)d(0)i(w)n(e)g(used)g(the)g(\014rst)g(of)h(\(2.6\),)g(whic)n (h,)g(inserted)f(in)n(to)118 2731 y(\(2.9\))27 b(for)h Fv(k)d Fy(=)e(1,)k(giv)n(es)g(2)p Fv(c)974 2743 y Fu(0)1010 2731 y Fv(c)1046 2743 y Fu(1)1107 2731 y Fy(=)22 b(0,)27 b(hence)h Fv(c)1553 2743 y Fu(1)1613 2731 y Fy(=)23 b(0)k(as)g Fv(c)1908 2743 y Fu(0)1968 2731 y Fs(6)p Fy(=)c(0.)189 2837 y(The)28 b(follo)n(wing)e(result)h(holds.)118 3014 y Fk(Lemma)33 b(1.)43 b Fw(Consider)34 b(\(2.1\))f(with)g Fv(f)40 b Fw(given)33 b(by)g(\(1.4\))g(Then)g(ther)l(e)f(exists)g(a)g (formal)i(p)l(ower)f(series)f(solution)118 3120 y(\(2.2\))41 b(whose)g(c)l(o)l(e\016cients)f Fv(x)1068 3090 y Fu(\()p Fz(k)q Fu(\))1161 3120 y Fy(\()p Fv(t)p Fy(\))g Fw(ar)l(e)g(analytic)h (in)f Fv(t)p Fw(.)68 b(If)40 b Fv(f)48 b Fw(is)40 b(a)g(trigonometric)h (p)l(olynomial,)k(that)40 b(is)g(in)118 3227 y(\(1.4\))33 b(one)g(has)g Fs(j)p Fv(\027)5 b Fs(j)27 b(\024)g Fv(N)41 b Fw(for)33 b(some)g Fv(N)j Fs(2)27 b Fr(N)15 b Fw(,)33 b(then)f(for)h(al)t(l)g Fv(k)e Fs(\025)26 b Fy(0)32 b Fw(the)g(functions)g Fv(x)2855 3197 y Fu(\()p Fz(k)q Fu(\))2949 3227 y Fy(\()p Fv(t)p Fy(\))g Fw(ar)l(e)h(trigonometric)118 3333 y(p)l(olynomials)41 b(of)f(or)l(der)g Fy([\()p Fv(k)29 b Fy(+)c(1\))p 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y Fg(1)691 1682 y Fi(+)c Fh(k)798 1691 y Fg(2)847 1682 y Fi(+)g Fh(k)954 1691 y Fg(3)1003 1682 y Fi(+)f(10)24 b(if)e Fh(v)1264 1691 y Fg(0)1326 1682 y Fh(=)-42 b Fe(2)19 b Fh(V)1426 1691 y Fg(0)1461 1682 y Fi(\()p Fh(\022)r Fi(\))24 b(and)g Fh(k)d Fi(=)e Fh(k)1880 1691 y Fg(1)1929 1682 y Fi(+)c Fh(k)2036 1691 y Fg(2)2085 1682 y Fi(+)g Fh(k)2192 1691 y Fg(3)2241 1682 y Fi(+)f(9)24 b(if)e Fh(v)2467 1691 y Fg(0)2521 1682 y Fe(2)e Fh(V)2629 1691 y Fg(0)2664 1682 y Fi(\()p Fh(\022)r Fi(\).)31 b(In)24 b(the)g(latter)g(case)472 1788 y(one)h(m)n(ust)e(ha)n(v)n(e)h Fh(k)976 1797 y Fg(1)1030 1788 y Fh(>)c Fi(0)k(b)r(ecause)h(of)e(the)i (constrain)n(t)f(in)g(the)g(sum)f(app)r(earing)h(in)f(\(3.6\).)118 2111 y Fy(can)k(con)n(tain)g(more)g(than)h(one)f(or)g(none)g(elemen)n (t.)189 2230 y(Note)33 b(that)g(only)f(lines)h(coming)f(out)h(either)f (from)h(no)r(des)f(in)h Fv(V)2236 2242 y Fu(0)2274 2230 y Fy(\()p Fv(\022)r Fy(\))g Fs(\032)e Fv(V)2556 2242 y Fu(2)2594 2230 y Fy(\()p Fv(\022)r Fy(\))i(or)f(from)h(white)g (bullets)g(ha)n(v)n(e)118 2337 y(v)-5 b(anishing)27 b(momen)n(tum.)189 2456 y(The)f(order)f(of)h(the)h(tree)f(is)g(then)h(giv)n(en)e(b)n(y)h (the)h(n)n(um)n(b)r(er)f(of)g(elemen)n(ts)h(of)f Fv(V)19 b Fy(\()p Fv(\022)r Fy(\))d Fs([)g Fv(E)2861 2468 y Fz(B)2919 2456 y Fy(\()p Fv(\022)r Fy(\))27 b(min)n(us)f(the)h(n)n(um)n(b)r(er) 118 2563 y(of)i(elemen)n(ts)f(of)h Fv(V)698 2575 y Fu(0)736 2563 y Fy(\()p Fv(\022)r Fy(\),)h(that)f(is)f Fv(k)g Fy(=)d Fs(j)p Fv(V)19 b Fy(\()p Fv(\022)r Fy(\))p Fs(j)h Fy(+)e Fs(j)p Fv(E)1725 2575 y Fz(B)1783 2563 y Fy(\()p Fv(\022)r Fy(\))p Fs(j)i(\000)f(j)p Fv(V)2086 2575 y Fu(0)2124 2563 y Fy(\()p Fv(\022)r Fy(\))p Fs(j)p Fy(.)41 b(Of)29 b(course)e Fv(v)2730 2575 y Fu(0)2792 2563 y Fs(2)f Fv(V)2921 2575 y Fu(0)2958 2563 y Fy(\()p Fv(\022)r Fy(\))k(if)f(and)g(only)f(if)h(the)118 2669 y(momen)n(tum)i(of)f(the)h (ro)r(ot)f(line)h(is)f(v)-5 b(anishing,)31 b(that)g(is)f Fv(\022)h Fs(2)d(T)2088 2681 y Fz(k)q(;)p Fu(0)2212 2669 y Fy(for)i(some)g Fv(k)h Fs(\025)d Fy(2.)45 b(It)31 b(is)f(imp)r(ortan) n(t)g(to)h(stress)118 2775 y(that)23 b(no)g(line)g(en)n(tering)e(a)i(v) n(ertex)f Fv(v)k Fs(2)d Fv(V)1375 2787 y Fu(0)1413 2775 y Fy(\()p Fv(\022)r Fy(\))h(can)e(come)g(out)h(from)f(a)h(white)g (bullet)g(\(whic)n(h)g(no)n(w)f(has)g(necessarily)118 2881 y(an)k(order)e(lab)r(el)i(0\),)g(b)r(ecause)g(this)g(w)n(ould)f(b) r(e)i(against)d(the)j(constrain)n(t)d(in)i(the)h(sum)f(\(3.6\).)36 b(This)26 b(means)f(that)h(if)118 2988 y(t)n(w)n(o)j(lines)h(carrying)e (zero)h(momen)n(tum)i(en)n(ter)e(the)h(same)g(v)n(ertex)f Fv(v)k Fy(\(so)d(that)g Fv(v)g Fs(2)e Fv(V)2829 3000 y Fu(0)2867 2988 y Fy(\()p Fv(\022)r Fy(\))j(according)d(to)i (\(3.2\)\),)118 3094 y(then)e(none)g(of)f(them)h(can)f(exit)h(a)f (white)h(bullet.)189 3214 y(But)35 b(up)h(to)f(these)g(minor)g (di\013erences)g(a)g(tree)f(represen)n(tation)g(lik)n(e)h(in)g(\(3.5\)) g(and)g(\(3.6\))g(still)g(holds.)60 b(The)118 3320 y(adv)-5 b(an)n(tage)31 b(of)i(these)g(mo)r(di\014ed)g(rules)g(is)g(that)g(no)n (w)f(the)h(tree)g(v)-5 b(alues)32 b(are)g(expressed)g(no)g(longer)g(in) h(terms)g(of)118 3426 y(constan)n(ts)g Fv(c)528 3438 y Fz(k)603 3426 y Fy(to)g(b)r(e)h(determined,)i(but)e(only)g(in)g (terms)f(of)h Fv(c)2118 3438 y Fu(0)2189 3426 y Fy(whic)n(h)g(is)f(kno) n(wn.)55 b(A)34 b(tree)g(dra)n(wn)f(according)118 3532 y(these)e(new)g(rules)f(is)g(represen)n(ted)g(as)g(in)h(Figure)f(3.8)g (with)h Fv(k)2083 3544 y Fu(1)2149 3532 y Fy(=)d Fv(k)2285 3544 y Fu(2)2350 3532 y Fy(=)g Fv(k)2486 3544 y Fu(3)2552 3532 y Fy(=)g(0)i(\(and)h(in)g(particular)e(a)h(tree)h(of)118 3648 y(this)24 b(kind)g(can)f(con)n(tribute)g(only)g(to)h Fv(x)1325 3605 y Fu(\()p Fz(k)q Fu(\))1325 3658 y Fz(\027)1441 3648 y Fy(with)g Fv(\027)29 b Fs(6)p Fy(=)22 b(0\).)36 b(Note)23 b(that)h(w)n(e)f(could)h(a)n(v)n(oid)e(dra)n(wing)g(the)i (order)e(lab)r(els)118 3754 y(asso)r(ciated)28 b(with)j(the)f(endp)r (oin)n(ts,)g(as)f(they)h(are)e(uniquely)i(determined)g(as)f Fv(k)g Fy(=)d(0)k(for)f(the)h(white)g(bullets)g(and)118 3861 y Fv(k)c Fy(=)d(1)h(for)g(the)h(blac)n(k)f(bullets.)36 b(Of)25 b(course,)f(with)h(resp)r(ect)g(to)f(the)h(caption)g(of)f(that) h(Figure,)g(no)n(w)f(the)h(order)e Fv(k)28 b Fy(is)118 3967 y(giv)n(en)g(b)n(y)g(the)h(n)n(um)n(b)r(er)g(of)f(elemen)n(ts)h (in)g Fv(V)19 b Fy(\()p Fv(\022)r Fy(\))29 b(plus)g(the)g(n)n(um)n(b)r (er)f(of)h(elemen)n(ts)f(in)h Fv(E)2854 3979 y Fz(B)2912 3967 y Fy(\()p Fv(\022)r Fy(\))g(min)n(us)g(the)g(n)n(um)n(b)r(er)118 4073 y(of)f(elemen)n(ts)f(in)h Fv(V)697 4085 y Fu(0)735 4073 y Fy(\()p Fv(\022)r Fy(\).)1440 4321 y FB(4.)50 b(F)-9 b(ormal)36 b(solutions)118 4512 y Fy(The)25 b(sum)h(o)n(v)n(er)d (the)j(trees)f(in)g(\(3.5\))g(and)g(\(3.6\),)h(with)g(the)f(new)h (de\014nition)f(of)h(the)f(set)h Fs(T)2910 4524 y Fz(k)q(;\027)3033 4512 y Fy(giv)n(en)e(at)i(the)f(end)h(of)118 4618 y(Section)e(3,)h(can) e(b)r(e)i(p)r(erformed)e(b)n(y)h(summing)g(o)n(v)n(er)e(all)i(p)r (ossible)g(\\tree)f(shap)r(es")g(\(that)i(is)f(trees)f(without)i(lab)r (els)118 4724 y(or)g Fw(unlab)l(el)t(le)l(d)30 b(tr)l(e)l(es)p Fy(\),)c(and,)h(for)f(a)g(\014xed)g(shap)r(e,)g(o)n(v)n(er)f(all)h(p)r (ossible)g(assignmen)n(ts)f(of)h(mo)r(de)g(lab)r(els.)36 b(In)27 b(the)g(case)118 4831 y(of)k(a)g(trigonometric)f(p)r(olynomial) g(of)i(degree)e Fv(N)40 b Fy(the)32 b(latter)f(can)f(b)r(e)i(b)r (ounded)g(b)n(y)f(\(2)p Fv(N)9 b Fy(\))3003 4801 y FA(j)p Fz(E)s Fu(\()p Fz(\022)r Fu(\))p FA(j)3183 4831 y Fy(,)33 b(b)r(ecause)e(eac)n(h)118 4937 y(endp)r(oin)n(t)c Fv(v)j Fy(can)c(ha)n(v)n(e)f(either)h(a)g(mo)r(de)h(lab)r(el)f Fv(\027)1639 4949 y Fz(v)1701 4937 y Fs(6)p Fy(=)d(0,)j(with)h Fs(j)p Fv(\027)2132 4949 y Fz(v)2172 4937 y Fs(j)c(\024)g Fv(N)9 b Fy(,)26 b(or)g(the)g(mo)r(de)h(lab)r(el)f Fv(\027)3136 4949 y Fz(v)3199 4937 y Fy(=)c(0,)27 b(while)f(the)118 5043 y(case)g(of)h(analytic)g(functions)g(\(or)f(ev)n(en)h(to)g(obtain) f(b)r(ounds)i(whic)n(h)e(are)g(uniform)h(in)h Fv(N)9 b Fy(\))27 b(has)f(to)h(b)r(e)h(discussed)e(a)118 5150 y(little)i(more)f(carefully)-7 b(.)36 b(The)28 b(n)n(um)n(b)r(er)f(of)h (unlab)r(elled)g(trees)e(with)j Fv(P)39 b Fy(no)r(des)27 b(\(v)n(ertices)g(and)g(endp)r(oin)n(ts\))h(can)f(b)r(e)1893 5349 y(9)p eop %%Page: 10 10 10 9 bop 118 319 a Fy(b)r(ounded)28 b(b)n(y)f(2)614 289 y Fu(2)p Fz(P)702 319 y Fy(.)189 426 y(Recall)j(that)g Fv(V)672 438 y Fz(s)708 426 y Fy(\()p Fv(\022)r Fy(\))h(denotes)f(the)g (set)h(of)f(v)n(ertices)f Fv(v)k Fy(suc)n(h)d(that)g Fv(s)2309 438 y Fz(v)2376 426 y Fy(=)d Fv(s)p Fy(;)k(of)f(course)f Fv(V)2963 438 y Fu(1)3001 426 y Fy(\()p Fv(\022)r Fy(\))21 b Fs([)f Fv(V)3250 438 y Fu(2)3288 426 y Fy(\()p Fv(\022)r Fy(\))28 b(=)f Fv(V)19 b Fy(\()p Fv(\022)r Fy(\),)118 532 y(and)28 b Fv(V)328 544 y Fu(0)365 532 y Fy(\()p Fv(\022)r Fy(\))c Fs(\032)f Fv(V)630 544 y Fu(2)667 532 y Fy(\()p Fv(\022)r Fy(\).)38 b(Analogously)26 b(w)n(e)h(can)h(set)1218 728 y Fv(L)1275 740 y Fu(0)1312 728 y Fy(\()p Fv(\022)r Fy(\))c(=)f Fs(f)o Fv(`)g Fs(2)g Fv(L)p Fy(\()p Fv(\022)r Fy(\))h(:)f Fv(n)1988 740 y Fz(`)2042 728 y Fy(=)g(0)p Fv(;)14 b Fs(g)f Fv(;)1218 859 y(L)1275 871 y Fu(1)1312 859 y Fy(\()p Fv(\022)r Fy(\))24 b(=)f Fs(f)o Fv(`)g Fs(2)g Fv(L)p Fy(\()p Fv(\022)r Fy(\))h(:)f Fv(`)f Fy(=)h Fv(`)2118 871 y Fz(v)2157 859 y Fv(;)14 b(v)26 b Fs(2)e Fv(V)2387 871 y Fu(1)2424 859 y Fy(\()p Fv(\022)r Fy(\))p Fs(g)14 b Fv(;)1218 991 y(L)1275 1003 y Fu(2)1312 991 y Fy(\()p Fv(\022)r Fy(\))24 b(=)f Fv(L)p Fy(\()p Fv(\022)r Fy(\))18 b Fs(n)1769 923 y Fq(\000)1807 991 y Fv(L)1864 1003 y Fu(0)1901 991 y Fy(\()p Fv(\022)r Fy(\))h Fs([)g Fv(L)2156 1003 y Fu(1)2193 991 y Fy(\()p Fv(\022)r Fy(\))2298 923 y Fq(\001)2337 991 y Fv(;)3538 859 y Fy(\(4)p Fv(:)p Fy(1\))118 1182 y(with)28 b(the)g(splitting)g(made)g(in)f(suc)n(h)h(a)f (w)n(a)n(y)f(that)i(one)f(has)646 1318 y Fq(\014)646 1368 y(\014)646 1417 y(\014)753 1334 y(Y)687 1516 y Fz(v)r FA(2)p Fz(V)806 1524 y Fm(1)839 1516 y Fu(\()p Fz(\022)r Fu(\))938 1413 y Fv(F)991 1425 y Fz(v)1031 1318 y Fq(\014)1031 1368 y(\014)1031 1417 y(\014)1072 1318 y(\014)1072 1368 y(\014)1072 1417 y(\014)1179 1334 y(Y)1114 1516 y Fz(`)p FA(2)p Fz(L)1233 1524 y Fm(1)1264 1516 y Fu(\()p Fz(\022)r Fu(\))1364 1413 y Fv(g)1404 1425 y Fz(`)1435 1318 y Fq(\014)1435 1368 y(\014)1435 1417 y(\014)1486 1413 y Fs(\024)1638 1334 y Fq(Y)1573 1516 y Fz(`)p FA(2)p Fz(L)1692 1524 y Fm(1)1724 1516 y Fu(\()p Fz(\022)r Fu(\))1823 1413 y Fs(j)p Fv(!)s(\027)1942 1425 y Fz(`)1974 1413 y Fs(j)14 b Fv(;)2214 1318 y Fq(\014)2214 1368 y(\014)2214 1417 y(\014)2320 1334 y(Y)2255 1516 y Fz(`)p FA(2)p Fz(L)2374 1524 y Fm(2)2406 1516 y Fu(\()p Fz(\022)r Fu(\))2505 1413 y Fv(g)2545 1425 y Fz(`)2576 1318 y Fq(\014)2576 1368 y(\014)2576 1417 y(\014)2627 1413 y Fs(\024)2779 1334 y Fq(Y)2715 1516 y Fz(`)p FA(2)p Fz(L)2834 1524 y Fm(2)2865 1516 y Fu(\()p Fz(\022)r Fu(\))3040 1357 y Fy(1)p 2974 1394 174 4 v 2974 1470 a Fs(j)p Fv(!)s(\027)3093 1482 y Fz(`)3125 1470 y Fs(j)3158 1413 y Fv(;)646 1620 y Fq(\014)646 1670 y(\014)646 1720 y(\014)753 1637 y(Y)687 1819 y Fz(v)r FA(2)p Fz(V)806 1827 y Fm(0)839 1819 y Fu(\()p Fz(\022)r Fu(\))938 1716 y Fv(F)991 1728 y Fz(v)1031 1620 y Fq(\014)1031 1670 y(\014)1031 1720 y(\014)1082 1716 y Fs(\024)1169 1599 y Fq(\022)1277 1660 y Fy(1)p 1240 1697 115 4 v 1240 1773 a(2)p Fv(c)1318 1785 y Fu(0)1365 1599 y Fq(\023)1426 1616 y FA(j)p Fz(V)1485 1624 y Fm(0)1517 1616 y Fu(\()p Fz(\022)r Fu(\))p FA(j)1640 1716 y Fv(;)1843 1620 y Fq(\014)1843 1670 y(\014)1843 1720 y(\014)1971 1637 y(Y)1885 1819 y Fz(v)r FA(2)p Fz(E)2014 1827 y Fn(W)2078 1819 y Fu(\()p Fz(\022)r Fu(\))2177 1716 y Fv(F)2230 1728 y Fz(v)2270 1620 y Fq(\014)2270 1670 y(\014)2270 1720 y(\014)2321 1716 y Fs(\024)22 b Fv(c)2444 1673 y FA(j)p Fz(E)2513 1681 y Fn(W)2577 1673 y Fu(\()p Fz(\022)r Fu(\))p FA(j)2444 1738 y Fu(0)2687 1716 y Fv(;)646 1875 y Fq(\014)646 1925 y(\014)646 1975 y(\014)766 1892 y(Y)687 2074 y Fz(v)r FA(2)p Fz(E)816 2082 y Fn(B)865 2074 y Fu(\()p Fz(\022)r Fu(\))964 1971 y Fv(F)1017 1983 y Fz(v)1057 1875 y Fq(\014)1057 1925 y(\014)1057 1975 y(\014)1108 1971 y Fs(\024)h Fv(F)1261 1936 y FA(j)p Fz(E)1330 1944 y Fn(B)1378 1936 y Fu(\()p Fz(\022)r Fu(\))p FA(j)1580 1892 y Fq(Y)1501 2074 y Fz(v)r FA(2)p Fz(E)1630 2082 y Fn(B)1679 2074 y Fu(\()p Fz(\022)r Fu(\))1778 1971 y Fy(e)1815 1936 y FA(\000)p Fz(\030)r FA(j)p Fz(\027)1952 1944 y Fn(v)1988 1936 y FA(j)2012 1971 y Fv(;)3538 1721 y Fy(\(4)p Fv(:)p Fy(2\))118 2264 y(where)k(for)g(eac)n(h)g(line)h Fv(`)f Fy(one)g(has)g Fs(j)p Fv(\027)1255 2276 y Fz(`)1287 2264 y Fs(j)d(\024)1421 2202 y Fq(P)1509 2289 y Fz(v)r FA(2)p Fz(E)1638 2297 y Fn(W)1702 2289 y Fu(\()p Fz(\022)r Fu(\))1805 2264 y Fs(j)p Fv(\027)1869 2276 y Fz(v)1909 2264 y Fs(j)p Fy(.)189 2370 y(The)k(follo)n(wing)e(result)h(is)h (useful)g(when)g(lo)r(oking)e(for)h(b)r(ounds)h(on)f(the)h(tree)g(v)-5 b(alues.)118 2547 y Fk(Lemma)26 b(3.)35 b Fw(Given)28 b(a)f(tr)l(e)l(e)f Fv(\022)j Fw(with)e(br)l(anching)h(numb)l(er)e Fv(s)h Fw(one)g(has)g Fs(j)p Fv(E)5 b Fy(\()p 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FA(\000)p Fz(\030)r FA(j)p Fz(\027)2278 4034 y Fn(v)2313 4026 y FA(j)p Fz(=)p Fu(2)2404 3968 y Fq(\021)2468 3943 y(\022)2539 4004 y Fy(2)p Fv(k)s Fs(j)p Fv(!)s Fs(j)p 2539 4041 189 4 v 2613 4117 a Fv(\030)2737 3943 y Fq(\023)2798 3960 y Fz(k)2853 4060 y Fv(;)3538 3946 y Fy(\(4)p Fv(:)p Fy(3\))118 4349 y(and)d(in)h(the)g(second)e (line)i(the)f(pro)r(duct)h(can)f(b)r(e)g(used)h(to)f(p)r(erform)g(the)g (sum)h(o)n(v)n(er)d(the)j(F)-7 b(ourier)22 b(lab)r(els)h({)g(and)g (this)118 4455 y(giv)n(es)i(a)i(factor)e Fv(F)690 4425 y Fz(k)731 4455 y Fv(B)798 4425 y Fz(k)794 4476 y Fu(2)839 4455 y Fy(,)i(with)g Fv(B)1140 4467 y Fu(2)1200 4455 y Fy(=)c(2e)1367 4425 y FA(\000)p Fz(\030)r(=)p Fu(2)1522 4455 y Fy(\(1)16 b Fs(\000)g Fy(e)1730 4425 y FA(\000)p Fz(\030)r(=)p Fu(2)1885 4455 y Fy(\))1917 4425 y FA(\000)p Fu(1)2006 4455 y Fy(,)27 b({)f(while)h(the)g(last)f(factor)g(is)g(b)r (ounded)h(b)n(y)f Fv(A)3471 4467 y Fu(1)3509 4455 y Fv(B)3576 4425 y Fz(k)3572 4476 y Fu(1)3617 4455 y Fv(k)s Fy(!,)118 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b(+)f Fc(R)1747 5087 y Fz(N)1810 5075 y Fy(\()p Fv(")p Fy(\))p Fv(;)184 b Fs(j)p Fc(R)2226 5087 y Fz(N)2289 5075 y Fy(\()p Fv(")p Fy(\))p Fs(j)23 b(\024)g Fv(AB)2655 5040 y Fz(N)2718 5075 y Fv(N)9 b Fy(!)p Fs(j)p Fv(")p Fs(j)2902 5040 y Fz(N)2965 5075 y Fv(;)508 b Fy(\(5)p Fv(:)p Fy(16\))1872 5349 y(15)p eop %%Page: 16 16 16 15 bop 118 319 a Fw(wher)l(e)31 b(the)e(c)l(onstants)g Fv(A)h Fw(and)h Fv(B)i Fw(ar)l(e)e(uniform)f(in)g Fv(N)38 b Fw(and)30 b(in)g Fv(")p Fw(.)118 499 y(Pr)l(o)l(of.)37 b Fy(W)-7 b(rite)24 b Fv(x)p Fy(\()p Fv(t)p Fy(\))g(as)e Fv(x)p Fy(\()p Fv(t)p Fy(\))i(=)f Fv(x)1161 511 y Fz(N)1225 499 y Fy(\()p Fv(t)p Fy(\))10 b(+)g Fc(R)1485 511 y Fz(N)1548 499 y Fy(\()p Fv(t)p Fy(\),)25 b(where)e Fv(x)1973 511 y Fz(N)2036 499 y Fy(\()p Fv(t)p Fy(\))h(is)f(giv)n(en)g(b)n(y)f(the)i (sum)f(of)g(the)h(\014rst)f Fv(N)18 b Fs(\000)10 b Fy(1)22 b(orders)118 605 y(of)27 b(the)h(formal)f(p)r(o)n(w)n(er)f(series)g (expansion)g(of)i(the)f(solution)g Fv(x)p Fy(\()p Fv(t)p 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b(Quasi-p)s(erio)s(dic)37 b(forcing)g(terms)118 853 y Fy(In)25 b(the)g(case)f(of)g(analytic)g (quasi-p)r(erio)r(dic)g(forcing)f(terms,)i(w)n(e)g(shall)f(assume)g(a)g (Diophan)n(tine)h(condition)f(on)g(the)118 959 y(rotation)j(v)n(ector)f Ft(!)s Fy(,)h(that)h(is)1250 1072 y Fs(j)p Ft(!)21 b Fs(\001)d Ft(\027)6 b Fs(j)23 b(\025)g Fv(C)1642 1084 y Fu(0)1679 1072 y Fs(j)p Ft(\027)6 b Fs(j)1779 1038 y FA(\000)p Fz(\034)2039 1072 y Fs(8)p Ft(\027)27 b Fs(2)d Fr(Z)2306 1024 y Fz(d)2363 1072 y Fs(n)18 b(f)p Fk(0)p Fs(g)p Fv(;)960 b Fy(\(6)p Fv(:)p Fy(1\))118 1243 y(where)24 b Fs(j)p Ft(\027)6 b Fs(j)23 b Fy(=)g Fs(j)p Ft(\027)6 b Fs(j)666 1255 y Fu(1)726 1243 y Fs(\021)22 b(j)p Fv(\027)877 1255 y Fu(1)915 1243 y Fs(j)13 b Fy(+)g Fv(:)h(:)g(:)e Fy(+)h Fs(j)p Fv(\027)1280 1255 y Fz(d)1318 1243 y Fs(j)p Fy(,)25 b(and)g Fv(C)1607 1255 y Fu(0)1670 1243 y Fy(and)f Fv(\034)35 b Fy(are)23 b(p)r(ositiv)n(e)i(constan)n(ts.)35 b(W)-7 b(e)25 b(need)g Fv(\034)33 b Fs(\025)22 b Fv(d)13 b Fs(\000)g Fy(1)24 b(in)h(order)118 1349 y(to)h(ha)n(v)n(e)e(a)h (non-v)n(oid)g(set)g(of)h(v)n(ectors)e(satisfying)h(the)h(condition)f (\(6.1\),)h(and)f Fv(\034)33 b(>)23 b(d)15 b Fs(\000)f Fy(1)25 b(in)h(order)e(to)h(ha)n(v)n(e)g(a)g(full)118 1455 y(measure)c(set)h(of)g(suc)n(h)g(v)n(ectors.)34 b(F)-7 b(or)21 b(simplicit)n(y)h(\(and)g(without)h(loss)e(of)h (generalit)n(y\))f(w)n(e)h(can)f(assume)h Fv(C)3407 1467 y Fu(0)3467 1455 y Fv(<)h(\015)5 b(=)p Fy(2,)118 1562 y(with)28 b Fv(\015)g Fy(=)23 b(min)p Fs(f)p Fy(1)p Fv(;)14 b Fs(j)p Fv(c)p Fs(jg)p Fy(,)26 b(where)h Fv(c)h Fy(is)f(a)g(suitable)h (constan)n(t)f(to)g(b)r(e)h(\014xed)g(as)f Fv(c)c Fy(=)f Fs(\000)p Fy(2)p Fv(c)2805 1574 y Fu(0)2842 1562 y Fy(,)27 b(with)i Fv(c)3118 1574 y Fu(0)3178 1562 y Fy(=)3265 1502 y Fs(p)p 3335 1502 54 4 v 3335 1562 a Fv(\013)p Fy(.)189 1670 y(The)f(equation)f(of)g(motion)h(can)f(b)r(e)h(written)g (in)f(F)-7 b(ourier)27 b(space)g(as)1045 1882 y Fv(i)p Ft(!)21 b Fs(\001)d Ft(\027)i Fy(\(1)e(+)g Fv(i")p Ft(!)i Fs(\001)f Ft(\027)6 b Fy(\))14 b Fv(x)1776 1894 y Fp(\027)1841 1882 y Fy(+)k Fv(")2060 1803 y Fq(X)1977 1978 y Fp(\027)2015 1986 y Fm(1)2048 1978 y Fu(+)p Fp(\027)2137 1986 y Fm(2)2169 1978 y Fu(=)p Fp(\027)2277 1882 y Fv(x)2324 1894 y Fp(\027)2362 1902 y Fm(1)2399 1882 y Fv(x)2446 1894 y Fp(\027)2484 1902 y Fm(2)2545 1882 y Fy(=)k Fv("f)2712 1894 y Fp(\027)2759 1882 y Fv(;)756 b Fy(\(6)p Fv(:)p Fy(2\))118 2166 y(and)28 b(the)g(formal)e(expansion)h(for)g(a)g(quasi-p)r(erio)r(dic)g(solution) g(with)h(frequency)f(v)n(ector)f Ft(!)31 b Fy(reads)26 b(as)1119 2434 y Fv(x)p Fy(\()p Fv(t)p Fy(\))e(=)1399 2330 y FA(1)1372 2355 y Fq(X)1372 2534 y Fz(k)q Fu(=0)1506 2434 y Fv(")1545 2400 y Fz(k)1586 2434 y Fv(x)1633 2400 y Fu(\()p Fz(k)q Fu(\))1726 2434 y Fy(\()p Fv(t)p Fy(\))g(=)1959 2330 y FA(1)1932 2355 y Fq(X)1931 2534 y Fz(k)q Fu(=0)2066 2434 y Fv(")2105 2400 y Fz(k)2184 2355 y Fq(X)2159 2550 y Fp(\027)5 b FA(2)p Fo(Z)2293 2522 y Fn(d)2342 2434 y Fy(e)2379 2400 y Fz(i)p Fp(\027)g FA(\001)p Fp(!)r Fz(t)2544 2434 y Fv(x)2591 2391 y Fu(\()p Fz(k)q Fu(\))2591 2448 y Fp(\027)2685 2434 y Fv(;)830 b Fy(\(6)p Fv(:)p Fy(3\))118 2763 y(and)38 b(to)f(see)g(that)h(the)g(co)r(e\016cien)n(ts) g Fv(x)1366 2720 y Fu(\()p Fz(k)q Fu(\))1366 2778 y Fp(\027)1497 2763 y Fy(are)e(w)n(ell)i(de\014ned)g(to)f(all)h(orders)e Fv(k)42 b Fs(\025)e Fy(0)d(one)g(can)g(pro)r(ceed)g(as)g(in)118 2870 y(Section)g(2,)h(with)f(no)g(extra)f(di\016cult)n(y)-7 b(.)64 b(In)37 b(particular)e(the)i(Diophan)n(tine)g(condition)f (\(6.1\))h(is)f(su\016cien)n(t)h(to)118 2976 y(assure)26 b(analyticit)n(y)h(in)h Fv(t)g Fy(of)f(the)h(co)r(e\016cien)n(ts)f Fv(x)1640 2946 y Fu(\()p Fz(k)q Fu(\))1734 2976 y Fy(\()p Fv(t)p Fy(\).)189 3084 y(Also)i(the)i(graphical)d(represen)n(tation)g (can)h(b)r(e)h(w)n(ork)n(ed)e(out)i(as)f(in)h(Section)g(3.)43 b(The)30 b(only)f(di\013erence)h(is)f(that)118 3191 y(no)n(w)i(the)g (propagators)d(of)k(the)f(lines)g(with)h(non-v)-5 b(anishing)30 b(momen)n(tum)i Ft(\027)2553 3203 y Fz(`)2584 3191 y Fy(,)h(whic)n(h)e(is)g(de\014ned)g(according)f(to)118 3297 y(\(3.2\),)k(with)f(the)g(v)n(ectors)f(replacing)f(the)i(scalars,) f(are)g(giv)n(en)g(b)n(y)h(1)p Fv(=)p Fy(\()p Fv(i)p Ft(!)23 b Fs(\001)f Ft(\027)2590 3309 y Fz(`)2622 3297 y Fy(\),)34 b(the)g(no)r(de)e(factors)g(asso)r(ciated)118 3403 y(with)i(the)h(v)n(ertices)d Fv(v)37 b Fy(with)e Fv(s)1081 3415 y Fz(v)1154 3403 y Fy(=)e(1)g(are)g(giv)n(en)g(b)n(y)g Fv(F)1869 3415 y Fz(v)1943 3403 y Fy(=)g Fs(\000)p Fy(\()p Fv(i)p Ft(!)24 b Fs(\001)f Ft(\027)2345 3415 y Fz(`)2373 3423 y Fn(v)2413 3403 y Fy(\))2445 3373 y Fu(2)2482 3403 y Fy(,)36 b(and)d(the)h(no)r(de)g(factors)f(asso)r(ciated)118 3521 y(to)e(the)g(blac)n(k)f(bullets)i Fv(v)i Fy(are)c(giv)n(en)g(b)n (y)h Fv(F)1469 3533 y Fz(v)1537 3521 y Fy(=)d Fv(f)1671 3533 y Fp(\027)1709 3541 y Fn(v)1749 3521 y Fy(,)k(with)f Ft(\027)2044 3533 y Fz(v)2112 3521 y Fs(2)e Fr(Z)2263 3473 y Fz(d)2322 3521 y Fs(n)20 b(f)p Fk(0)p Fs(g)p Fy(.)46 b(All)32 b(the)f(other)f(notations)g(remain)118 3627 y(unc)n(hanged.)189 3736 y(This)d(yields)g(that)h(the)g(propagators)c (and)j(the)h(no)r(de)f(factors)f(can)h(b)r(e)h(b)r(ounded)g(as)e(in)i (\(4.2\))f(and)g(\(4.3\),)g(with)118 3842 y(just)h(a)f(few)h (di\013erences)g(of)f(notation.)37 b(More)26 b(precisely)h(one)g(has) 269 3986 y Fq(\014)269 4036 y(\014)269 4086 y(\014)376 4003 y(Y)311 4185 y Fz(v)r FA(2)p Fz(V)430 4193 y Fm(1)463 4185 y Fu(\()p Fz(\022)r Fu(\))562 4082 y Fv(F)615 4094 y Fz(v)654 3986 y Fq(\014)654 4036 y(\014)654 4086 y(\014)696 3986 y(\014)696 4036 y(\014)696 4086 y(\014)802 4003 y(Y)737 4185 y Fz(`)p FA(2)p Fz(L)856 4193 y Fm(1)888 4185 y Fu(\()p Fz(\022)r Fu(\))987 4082 y Fv(g)1027 4094 y Fz(`)1059 3986 y Fq(\014)1059 4036 y(\014)1059 4086 y(\014)1109 4082 y Fs(\024)1262 4003 y Fq(Y)1197 4185 y Fz(`)p FA(2)p Fz(L)1316 4193 y Fm(1)1347 4185 y Fu(\()p Fz(\022)r Fu(\))1447 4082 y Fs(j)p Ft(!)s Fs(j)13 b(j)p Ft(\027)1640 4094 y Fz(`)1672 4082 y Fs(j)h Fv(;)1912 3986 y Fq(\014)1912 4036 y(\014)1912 4086 y(\014)2018 4003 y(Y)1954 4185 y Fz(`)p FA(2)p Fz(L)2073 4193 y Fm(1)2104 4185 y Fu(\()p Fz(\022)r Fu(\))2203 4082 y Fv(g)2243 4094 y Fz(`)2275 3986 y Fq(\014)2275 4036 y(\014)2275 4086 y(\014)2325 4082 y Fs(\024)2478 4003 y Fq(Y)2413 4185 y Fz(`)p FA(2)p Fz(L)2532 4193 y Fm(1)2564 4185 y Fu(\()p Fz(\022)r Fu(\))2776 4026 y Fy(1)p 2673 4063 249 4 v 2673 4139 a Fs(j)p Ft(!)21 b Fs(\001)d Ft(\027)2866 4151 y Fz(`)2898 4139 y Fs(j)2954 4082 y(\024)23 b Fv(C)3107 4046 y FA(\000)p Fu(1)3101 4104 y(0)3196 4082 y Fs(j)p Ft(\027)3267 4094 y Fz(`)3299 4082 y Fs(j)3322 4048 y Fz(\034)3364 4082 y Fv(;)269 4289 y Fq(\014)269 4339 y(\014)269 4389 y(\014)376 4306 y(Y)311 4488 y Fz(v)r FA(2)p Fz(V)430 4496 y Fm(0)463 4488 y Fu(\()p Fz(\022)r Fu(\))562 4385 y Fv(F)615 4397 y Fz(v)654 4289 y Fq(\014)654 4339 y(\014)654 4389 y(\014)705 4385 y Fs(\024)793 4267 y Fq(\022)901 4328 y Fy(1)p 864 4365 115 4 v 864 4441 a(2)p Fv(c)942 4453 y Fu(0)989 4267 y Fq(\023)1050 4285 y FA(j)p Fz(V)1109 4293 y Fm(0)1141 4285 y Fu(\()p Fz(\022)r Fu(\))p FA(j)1264 4385 y Fv(;)1467 4289 y Fq(\014)1467 4339 y(\014)1467 4389 y(\014)1595 4306 y(Y)1508 4488 y Fz(v)r FA(2)p Fz(E)1637 4496 y Fn(W)1702 4488 y Fu(\()p Fz(\022)r Fu(\))1801 4385 y Fv(F)1854 4397 y Fz(v)1894 4289 y Fq(\014)1894 4339 y(\014)1894 4389 y(\014)1944 4385 y Fs(\024)g Fv(c)2068 4341 y FA(j)p Fz(E)2137 4349 y Fn(W)2201 4341 y Fu(\()p Fz(\022)r Fu(\))p FA(j)2068 4407 y Fu(0)2310 4385 y Fv(;)269 4544 y Fq(\014)269 4593 y(\014)269 4643 y(\014)389 4560 y(Y)311 4742 y Fz(v)r FA(2)p Fz(E)440 4750 y Fn(B)489 4742 y Fu(\()p Fz(\022)r Fu(\))588 4639 y Fv(F)641 4651 y Fz(v)681 4544 y Fq(\014)681 4593 y(\014)681 4643 y(\014)731 4639 y Fs(\024)g Fv(F)884 4605 y FA(j)p Fz(E)953 4613 y Fn(B)1002 4605 y Fu(\()p Fz(\022)r Fu(\))p FA(j)1203 4560 y Fq(Y)1125 4742 y Fz(v)r FA(2)p Fz(E)1254 4750 y Fn(B)1303 4742 y Fu(\()p Fz(\022)r Fu(\))1402 4639 y Fy(e)1439 4605 y FA(\000)p Fz(\030)r FA(j)p Fp(\027)1581 4613 y Fn(v)1616 4605 y FA(j)1640 4639 y Fv(;)3538 4389 y Fy(\(6)p Fv(:)p Fy(4\))118 4937 y(where)k(the)h(only)f(b)r(ound)h(whic)n(h)g(in)n(tro)r(duces)f(a)g (real)f(di\016cult)n(y)i(with)g(resp)r(ect)g(to)f(the)h(case)f(of)g(p)r (erio)r(dic)h(forcing)118 5043 y(terms)h(is)h(the)f(second)g(one)g(in)h (the)g(\014rst)f(line.)42 b(Indeed)30 b(it)g(is)f(the)h(source)e(of)h (a)g(small)g(divisors)f(problem,)i(whic)n(h)118 5150 y(can)d(not)h(b)r(e)g(set)g(only)f(through)g(the)h(Diophan)n(tine)f (condition)h(\(6.1\).)1872 5349 y(17)p eop %%Page: 18 18 18 17 bop 189 319 a Fy(T)-7 b(o)40 b(eac)n(h)g(order)f Fv(k)k Fy(w)n(e)d(obtain)g(for)g Fv(x)1433 289 y Fu(\()p Fz(k)q Fu(\))1526 319 y Fy(\()p Fv(t)p Fy(\))h(a)f(b)r(ound)h(lik)n(e)f Fv(AB)2306 289 y Fz(k)2347 319 y Fv(k)s Fy(!)2416 289 y Fu(max)p FA(f)p Fu(1)p Fz(;\034)7 b FA(g)2701 319 y Fy(,)44 b(where)c(the)h(factor)e(1)h(arises)118 426 y(from)e(the)g (propagators)c(of)k(the)g(lines)g(in)g Fv(L)1567 438 y Fu(1)1604 426 y Fy(\()p Fv(\022)r Fy(\))g(and)g(the)g(factor)f Fv(\034)48 b Fy(from)37 b(those)g(of)h(the)g(lines)g(in)g Fv(L)3459 438 y Fu(2)3496 426 y Fy(\()p Fv(\022)r Fy(\))g(in)118 532 y(\(6.3\).)50 b(The)32 b(last)g(assertion)f(is)h(easily)f(pro)n(v)n (ed)f(b)n(y)i(reasoning)e(as)i(in)g(\(4.3\),)h(with)g(max)o Fs(fj)p Ft(!)s Fs(j)14 b(j)p Ft(\027)3159 544 y Fz(`)3190 532 y Fs(j)p Fv(;)g(C)3315 496 y FA(\000)p Fu(1)3309 554 y(0)3405 532 y Fs(j)p Ft(\027)3476 544 y Fz(`)3507 532 y Fs(j)3530 502 y Fz(\034)3572 532 y Fs(g)30 b(\024)118 638 y Fy(max)p Fs(f)p Fv(C)380 603 y FA(\000)p Fu(1)374 660 y(0)469 638 y Fv(;)14 b Fs(j)p Ft(!)s Fs(jgj)p Ft(\027)728 650 y Fz(`)759 638 y Fs(j)782 608 y Fu(max)o FA(f)p Fu(1)p Fz(;\034)7 b FA(g)1095 638 y Fy(replacing)27 b Fv(\027)1492 650 y Fz(`)1524 638 y Fy(.)39 b(In)29 b(particular)e(only)h(for)f Fv(d)e Fy(=)f(2)k(and)g Fv(\034)34 b Fy(=)24 b(1)k(w)n(e)g(obtain)g (the)g(same)118 744 y(b)r(ound)33 b(prop)r(ortional)f(to)h Fv(k)s Fy(!)g(as)f(in)h(the)g(case)g(of)f(p)r(erio)r(dic)h(solution)g (\(of)g(course)f(with)h(di\013eren)n(t)g(constan)n(ts)f Fv(A)118 851 y Fy(and)g Fv(B)t Fy(\).)49 b(Note)32 b(that)g(the)g(v)n (ectors)e(satisfying)h(the)h(Diophan)n(tine)g(condition)f(\(6.1\))h (with)g Fv(\034)39 b Fy(=)30 b(1)h(for)g Fv(d)f Fy(=)g(2)h(is)118 957 y(of)36 b(zero)e(measure)g(but)j(ev)n(erywhere)c(dense.)61 b(An)36 b(example)f(of)g(v)n(ector)f(of)i(this)g(kind)f(is)h Ft(!)j Fy(=)c(\(1)p Fv(;)14 b(\015)3366 969 y Fu(0)3403 957 y Fy(\),)38 b(where)118 1063 y Fv(\015)161 1075 y Fu(0)221 1063 y Fy(=)23 b(\()341 995 y Fs(p)p 411 995 42 4 v 411 1063 a Fy(5)17 b Fs(\000)h Fy(1\))p Fv(=)p Fy(2)27 b(is)g(the)h(golden)f(section.)189 1170 y(Ho)n(w)n(ev)n(er,)39 b(to)f(deal)g(with)h(the)f(problem)g(of)g(accum)n(ulation)g(of)g(small) g(divisors)e(and)j(discuss)e(the)i(issue)f(of)118 1276 y(con)n(v)n(ergence)e(of)i(the)g(series,)i(w)n(e)e(need)g (Renormalization)f(Group)g(tec)n(hniques.)69 b(The)38 b(\014rst)g(step)g(is)g(just)h(to)118 1382 y(in)n(tro)r(duce)21 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Fi(-plane)g(for)f Fh(\025)e Fi(=)g(tan)12 b Fh(\031)r(=)p Fi(6)24 b(\(a\))g(and)g(for)f Fh(\025)c Fi(=)h(1)j(\(b\).)32 b(One)472 3131 y(can)25 b(write)e Fh(\025)d Fi(=)f(tan)13 b Fh(')p Fi(,)23 b(where)h Fh(')g Fi(is)f(the)h(angle)g(b)r(et)n(w)n (een)i(the)e(imaginary)f(axis)g(and)i(the)f(line)f Fh(a)d Fi(=)g Fh(\025b)p Fi(.)118 3484 y Fk(Lemma)32 b(6.)42 b Fw(Given)32 b Fy(0)26 b Fv(<)g(R)h(<)f Fy(1)p Fv(=)p Fy(4)p Fv(C)1345 3496 y Fu(0)1412 3484 y Fw(let)32 b Fs(C)1575 3496 y Fz(R)1661 3484 y Fw(b)l(e)f(de\014ne)l(d)h(as)g(in)g (L)l(emma)f(4.)45 b(F)-6 b(or)32 b(al)t(l)g Fv(")26 b Fs(2)h(C)3133 3496 y Fz(R)3219 3484 y Fw(and)32 b(al)t(l)g Fv(x)g Fw(one)118 3591 y(has)f Fs(j)p Fv(F)343 3603 y Fu(0)380 3591 y Fy(\()p Fv(x)p Fy(\))p Fs(j)24 b(\025)f Fy(min)p Fs(f)p Fv(C)865 3603 y Fu(0)903 3591 y Fv(;)14 b Fs(j)p Fv(x)p Fs(jg)p Fv(=)p Fy(2)p Fw(,)29 b(while)i(for)g(al)t(l)f Fv(")23 b Fs(2)g(D)1880 3603 y Fz(R;\025)2024 3591 y Fw(one)30 b(has)g Fs(j)p Fv(F)2405 3603 y Fu(0)2443 3591 y Fy(\()p Fv(x)p Fy(\))p Fs(j)24 b(\025)f Fv(\025)p Fs(j)p Fv(x)p Fs(j)p Fv(=)p Fy(2)p Fw(.)118 3768 y(Pr)l(o)l(of.)38 b Fy(W)-7 b(rite)28 b Fv(")22 b Fy(=)h Fv(a)17 b Fy(+)f Fv(ib)p Fy(,)27 b(so)f(that)h Fs(j)p Fv(F)1366 3780 y Fu(0)1404 3768 y Fy(\()p Fv(x)p Fy(\))p Fs(j)d Fy(=)f Fs(j)p Fv(x)p Fs(j)1743 3697 y Fq(p)p 1826 3697 623 4 v 71 x Fy(\(1)c Fs(\000)f Fv(bx)p Fy(\))2117 3744 y Fu(2)2173 3768 y Fy(+)g(\()p Fv(ax)p Fy(\))2411 3744 y Fu(2)2449 3768 y Fy(.)37 b(F)-7 b(or)26 b Fv(")d Fs(2)g(C)2841 3780 y Fz(R)2922 3768 y Fy(set)k Fv(A)c Fy(=)g(1)17 b Fs(\000)3364 3704 y(p)p 3433 3704 161 4 v 64 x Fv(C)3492 3780 y Fu(0)3530 3768 y Fv(R)q Fy(.)36 b(If)118 3874 y Fs(j)p Fv(x)p Fs(j)24 b(\025)e Fv(C)381 3886 y Fu(0)419 3874 y Fy(,)k(for)f Fs(j)p Fy(1)14 b Fs(\000)g Fv(bx)p Fs(j)22 b(\024)g Fv(A)k Fy(one)f(has)g Fs(j)p Fv(F)1426 3886 y Fu(0)1463 3874 y Fy(\()p Fv(x)p Fy(\))p Fs(j)f(\025)f(j)p Fv(ax)1823 3844 y Fu(2)1861 3874 y Fs(j)g(\025)f Fv(b)2030 3844 y Fu(2)2067 3874 y Fv(x)2114 3844 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Fv(x)p Fs(j)p Fv(=)p Fy(2.)66 b(F)-7 b(or)36 b Fv(")j Fs(2)g(D)2677 4099 y Fz(R;\025)2828 4087 y Fy(set)e Fv(A)j Fy(=)e(1)p Fv(=)p Fy(2:)55 b(one)37 b(\014nds)118 4193 y Fs(j)p Fv(F)194 4205 y Fu(0)232 4193 y Fy(\()p Fv(x)p Fy(\))p Fs(j)24 b(\025)f Fv(\025)p Fs(j)p Fv(x)p Fs(j)p Fv(=)p Fy(2.)p 3667 4185 42 42 v 189 4370 a(Then)28 b(the)g(follo)n (wing)e(result)i(holds.)118 4547 y Fk(Lemma)c(7.)35 b Fw(Set)25 b Fv(x)f Fy(=)e Ft(!)12 b Fs(\001)d Ft(\027)32 b Fw(and)26 b(assume)g Fs(j)p Fv(x)p Fs(j)d(\024)g Fv(C)1748 4559 y Fu(0)1785 4547 y Fw(.)38 b(Then)26 b(if)g Fv(R)h Fw(is)f(smal)t(l)g(enough)g(one)g(has)g Fs(j)p Fv(F)12 b Fy(\()p Fv(x)p Fy(\))p Fs(j)24 b(\025)f Fv(\025\015)5 b Fs(j)p Fv(x)p Fs(j)p Fv(=)p Fy(8)118 4654 y Fw(for)31 b(al)t(l)g Fv(")22 b Fs(2)i(D)574 4666 y Fz(R;\025)687 4654 y Fw(.)118 4831 y(Pr)l(o)l(of.)57 b Fy(Set)34 b Fv(F)594 4843 y Fu(1)632 4831 y Fy(\()p Fv(x)p Fy(\))g(=)f Fv(F)928 4843 y Fu(0)966 4831 y Fy(\()p Fv(x)p Fy(\))24 b(+)e Fv(c")33 b Fy(and)h Fv(")f 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5019 y Fu(2)525 5043 y Fv(=)p Fy(2)40 b Fs(\021)g(j)p Fv(c")p Fs(j)p Fv(=)p Fy(2,)g(so)d(that)h Fs(j)p Fv(F)12 b Fy(\()p Fv(x)p Fy(\))p Fs(j)42 b(\025)e(j)p Fv(c")p Fs(j)p Fv(=)p Fy(4)g Fs(\025)g(j)p Fv(cb)p Fs(j)p Fv(=)p Fy(4)g Fs(\025)g(j)p Fv(x)p Fs(j)p Fy(.)69 b(If)39 b Fs(j)p Fv(x)26 b Fy(+)f Fv(b)p Fy(\()p Fv(c)g Fs(\000)g Fv(x)3121 5013 y Fu(2)3159 5043 y Fy(\))p Fs(j)41 b(\025)f(j)p Fv(x)p Fs(j)p Fv(=)p Fy(2)e(and)118 5150 y Fs(j)p Fv(bc)p Fs(j)25 b(\024)g Fy(4)p Fs(j)p Fv(x)p Fs(j)30 b Fy(one)e(has)h Fs(j)p Fv(F)895 5162 y Fu(1)932 5150 y Fy(\()p Fv(x)p Fy(\))p Fs(j)e(\025)e Fv(\015)18 b Fy(max)p Fs(f)1441 5080 y(p)p 1510 5080 268 4 v 70 x Fv(x)1557 5126 y Fu(2)1613 5150 y Fy(+)g Fv(a)1740 5126 y Fu(2)1777 5150 y Fv(;)c Fs(j)p Fv(")p Fs(j)p Fv(=)p Fy(4)p Fs(g)p Fv(=)p Fy(2,)27 b(so)h(that)i Fs(j)p Fv(F)12 b Fy(\()p Fv(x)p Fy(\))p Fs(j)26 b(\025)f Fv(\015)2830 5080 y Fs(p)p 2899 5080 V 70 x Fv(x)2946 5126 y Fu(2)3002 5150 y Fy(+)18 b Fv(a)3129 5126 y Fu(2)3166 5150 y Fv(=)p Fy(4)25 b Fs(\025)g Fv(\015)5 b Fs(j)p Fv(x)p Fs(j)p Fv(=)p Fy(4.)40 b(If)1872 5349 y(20)p eop %%Page: 21 21 21 20 bop 118 319 a Fs(j)p Fv(x)10 b Fy(+)g Fv(b)p Fy(\()p Fv(c)g Fs(\000)g Fv(x)509 289 y Fu(2)547 319 y Fy(\))p Fs(j)24 b(\024)e(j)p Fv(x)p Fs(j)p Fv(=)p Fy(2)h(one)g(has)g Fs(j)p Fv(b)p Fy(\()p Fv(c)10 b Fs(\000)g Fv(x)1464 289 y Fu(2)1502 319 y Fy(\))p Fs(j)23 b(\025)g(j)p Fv(x)p Fs(j)p Fv(=)p Fy(2)g(and)g Fs(j)p Fv(bc)p Fs(j)g(\024)g Fy(3)p Fs(j)p Fv(x)p Fs(j)p Fy(,)h(whic)n(h)g(giv)n(e)e Fs(j)p Fv(")p Fs(j)2921 289 y Fu(2)2981 319 y Fs(\024)h Fy(3)p Fs(j)p Fv(")p Fs(j)3196 250 y(p)p 3265 250 268 4 v 69 x Fv(a)3309 295 y Fu(2)3364 319 y Fy(+)18 b Fv(x)3494 295 y Fu(2)3532 319 y Fv(=\015)27 b Fs(\024)118 426 y Fy(3)p Fv(\025R)q Fy(\()p Fs(j)p Fv(a)p Fs(j)20 b Fy(+)g Fs(j)p Fv(x)p Fs(j)p Fy(\))p Fv(=\015)5 b Fy(,)32 b(and)e Fs(j)p Fv(F)1009 438 y Fu(1)1047 426 y Fy(\()p Fv(x)p Fy(\))p Fs(j)f(\025)e(j)p Fv(a)p Fy(\()p Fv(c)21 b Fs(\000)f Fv(x)1590 395 y Fu(2)1627 426 y Fy(\))p Fs(j)28 b(\025)g(j)p Fv(a)p Fy(\()p Fv(c)20 b Fs(\000)g Fv(x)2090 395 y Fu(2)2128 426 y Fy(\))p Fs(j)p Fv(=)p Fy(2)g(+)g(\()p Fv(\025)p Fs(j)p Fv(x)p Fs(j)p Fv(=)p Fy(2\))p Fv(=)p Fy(2)27 b Fs(\025)h Fv(\015)5 b(\025)p Fy(\()p Fs(j)p Fv(a)p Fs(j)20 b Fy(+)g Fs(j)p Fv(x)p Fs(j)p Fy(\))p Fv(=)p Fy(4,)32 b(so)d(that)118 532 y Fs(j)p Fv(F)12 b Fy(\()p Fv(x)p Fy(\))p Fs(j)24 b(\025)f Fv(\015)5 b(\025)p Fs(j)p Fv(x)p Fs(j)p Fv(=)p Fy(8.)p 3667 524 42 42 v 189 709 a(Then)29 b(w)n(e)g(can)g(come)g(bac)n(k)g(to)g(the)g(b)r(ounds)h(of)f(the)h (renormalized)e(propagators,)e(and)j(pro)n(v)n(e)f(the)i(follo)n(wing) 118 815 y(result.)118 992 y Fk(Lemma)40 b(8.)65 b Fw(If)39 b Fv(R)g Fw(is)f(smal)t(l)i(enough)e(for)i(al)t(l)f Fv(n)g Fs(\025)f Fy(0)g Fw(and)h(al)t(l)g Fv(")g Fs(2)g(D)2496 1004 y Fz(R;\025)2648 992 y Fw(the)f(r)l(enormalize)l(d)i(pr)l(op)l (agators)118 1099 y Fv(g)161 1069 y Fu([)p Fz(n)p Fu(])243 1099 y Fy(\()p Fv(x)p Fy(;)14 b Fv(")p Fy(\))31 b Fw(satisfy)g(the)f(b) l(ounds)f(\(6.11\))j(with)e Fv(\014)d Fy(=)c(1)29 b Fw(and)h Fv(C)2015 1111 y Fu(1)2076 1099 y Fy(=)23 b Fv(\025C)2271 1111 y Fu(4)2309 1099 y Fw(,)30 b(with)g(a)g Fv(\025)p Fw(-indep)l(endent)h(c)l(onstant)e Fv(C)3545 1111 y Fu(4)3582 1099 y Fw(.)118 1276 y(Pr)l(o)l(of.)62 b Fy(The)36 b(pro)r(of)f(can)g (b)r(e)h(done)f(b)n(y)g(induction)h(on)g Fv(n)p Fy(.)60 b(F)-7 b(or)35 b Fv(n)h Fy(=)g(0)f(the)h(b)r(ound)g(is)f(trivially)g (satis\014ed)g(b)n(y)118 1382 y(Lemma)29 b(6.)43 b(Assuming)29 b(that)h(the)g(b)r(ounds)g(hold)f(for)g(all)g Fv(n)2007 1352 y FA(0)2057 1382 y Fv(<)d(n)j Fy(then)h(w)n(e)f(can)h(apply)f (Lemma)g(5)g(and)h(deduce)118 1489 y(the)23 b(b)r(ounds)g(\(6.13\).)35 b(In)23 b(turns)g(this)g(implies)g(that)g(the)g(renormalized)f (propagators)d(on)k(scale)f Fv(n)h Fy(can)f(b)r(e)i(written)118 1595 y(as)34 b Fv(g)270 1565 y Fu([)p Fz(n)p Fu(])353 1595 y Fy(\()p Fv(x)p Fy(;)14 b Fv(")p Fy(\))36 b(=)f(1)p Fv(=F)12 b Fy(\()p Fv(x)p Fy(\),)37 b(with)e Fv(F)12 b Fy(\()p Fv(x)p 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Fy(is)f(b)r(etter,)i(whic)n(h)118 1688 y(means)e(that)h(the)f(domain)g Fs(C)1057 1700 y Fz(R)1145 1688 y Fy(con)n(tained)g(inside)g Fs(D)1832 1700 y Fz(R)1920 1688 y Fy(in)h(the)g(\014rst)f(case)f(is)h(larger)f (than)h(the)h(domain)f Fs(C)3554 1700 y Fz(R)3642 1688 y Fy(of)118 1794 y(the)28 b(second)f(case.)37 b(The)28 b(adv)-5 b(an)n(tage)26 b(of)h(Lemma)h(10)f(is)g(that)h(it)h(allo)n(ws) d(to)i(obtain)f(uniform)h(b)r(ounds)g(inside)f(the)118 1901 y(corresp)r(onding)f(domain)h Fs(C)991 1913 y Fz(R)1045 1901 y Fy(.)37 b(Nev)n(ertheless,)27 b(b)r(ecause)g(of)h(the)g(factor)f Fv(\014)h Fy(=)22 b(2,)28 b(a)f(b)r(ound)h Fv(AB)3099 1871 y Fz(k)3140 1901 y Fv(k)s Fy(!)3209 1871 y Fu(2)p Fz(\034)3311 1901 y Fy(is)g(obtained)118 2007 y(for)33 b(the)i(co)r(e\016cien)n(ts)e Fv(x)874 1977 y Fu(\()p Fz(k)q Fu(\))967 2007 y Fy(\()p Fv(t)p Fy(\))i(of)f(the)g(formal)f (solution,)i(and)f(a)f(result)h(analogous)d(to)j(Prop)r(osition)e(2)i (can)f(b)r(e)118 2113 y(pro)n(v)n(ed)f(also)g(for)g(the)i(presen)n(t)f (case,)g(with)h Fv(N)9 b Fy(!)1641 2083 y Fu(2)p Fz(\034)1749 2113 y Fy(replacing)32 b Fv(N)9 b Fy(!;)36 b(w)n(e)c(do)h(not)g(giv)n (e)g(the)g(details)g(as)g(the)g(pro)r(of)118 2220 y(is)f(iden)n(tical.) 48 b(Hence)32 b(the)g(b)r(ounds)g(that)g(w)n(e)f(ha)n(v)n(e)f(are)h (not)h(go)r(o)r(d)f(enough)g(to)g(obtain)g(Borel-summabilit)n(y)f(in) 118 2326 y(the)38 b(case)e(of)i(quasi-p)r(erio)r(dic)e(forcing)g (terms,)k(a)d(situation)g(strongly)f(reminiscen)n(t)h(of)g(that)h (encoun)n(tered)e(in)118 2432 y(Ref.)c([6].)47 b(In)31 b(fact)h(at)f(b)r(est)g(one)g(can)g(set)g Fv(\034)39 b Fy(=)28 b(1)j(for)g Fv(d)e Fy(=)f(2)j(\(whic)n(h,)h(as)f(noted)g(ab)r (o)n(v)n(e,)g(corresp)r(onds)e(to)i(a)g(set)118 2539 y(of)k(Diophan)n(tine)f(v)n(ectors)g(of)g(zero)g(measure)f(but)j(ev)n (erywhere)d(dense\),)j(but)f(this)g(in)g(turn)g(implies)g(a)f(b)r(ound) 118 2645 y(prop)r(ortional)26 b(to)h Fv(N)9 b Fy(!)796 2615 y Fu(2)833 2645 y Fy(,)28 b(whic)n(h)g(is)f(not)h(enough)f(to)g (apply)h(Nev)-5 b(anlinna's)27 b(theorem.)189 2757 y(The)h(conclusion)e (is)i(that)g(the)g(resummed)f(series)1553 3019 y Fv(x)p Fy(\()p Fv(t)p Fy(\))d(=)1833 2915 y FA(1)1806 2940 y Fq(X)1805 3119 y Fz(k)q Fu(=0)1940 3019 y Fv(\026)1990 2984 y Fz(k)2031 3019 y Fv(x)2078 2984 y Fu([)p Fz(k)q Fu(])2157 3019 y Fy(\()p Fv(t)p Fy(\))p Fv(;)1222 b Fy(\(6)p Fv(:)p Fy(15\))118 3325 y(where)27 b(the)h(co)r(e\016cien)n(ts)f Fv(x)968 3295 y Fu([)p Fz(k)q Fu(])1047 3325 y Fy(\()p Fv(t)p Fy(\))h(are)f(giv)n(en)g(b)n(y)1529 3558 y Fv(x)1576 3524 y Fu([)p Fz(k)q Fu(])1655 3558 y Fy(\()p Fv(t)p Fy(\))1788 3479 y Fq(X)1763 3674 y Fp(\027)5 b FA(2)p Fo(Z)1898 3646 y Fn(d)1946 3558 y Fy(e)1983 3524 y Fz(i)p Fp(\027)g FA(\001)p Fp(!)r Fz(t)2149 3558 y Fv(x)2196 3515 y Fu([)p Fz(k)q Fu(])2196 3572 y Fp(\027)2275 3558 y Fv(;)1198 b Fy(\(6)p Fv(:)p Fy(16\))118 3898 y(with)39 b Fv(x)365 3855 y Fu([)p Fz(k)q Fu(])365 3912 y Fp(\027)482 3898 y Fy(de\014ned)f(b)n(y)g(\(6.6\),)i(is)e(w)n(ell)g(de\014ned)g (and)g(con)n(v)n(erges.)65 b(In)39 b(general)d(it)j(is)e(not)h(ob)n (vious)f({)h(ev)n(en)f(if)118 4005 y(exp)r(ected,)j({)d(that)g (\(6.15\))f(solv)n(es)g(the)i(equation)e(of)h(motion)g(\(1.1\).)65 b(Indeed,)40 b(unlik)n(e)d(the)h(case)e(of)h(p)r(erio)r(dic)118 4111 y(forcing)24 b(terms,)i(w)n(e)f(ha)n(v)n(e)e(no)i(result,)h(suc)n (h)f(as)f(Nev)-5 b(anlinna's)25 b(theorem)g(on)g(Borel)f(summabilit)n (y)-7 b(,)25 b(whic)n(h)g(w)n(e)g(can)118 4217 y(rely)34 b(up)r(on)g(in)h(order)e(to)h(link)g(the)h(resummed)f(series)f(to)h (the)h(formal)f(series.)55 b(Therefore)33 b(w)n(e)h(ha)n(v)n(e)f(to)i (c)n(hec)n(k)118 4323 y(b)n(y)29 b(hand)h(that)f(b)n(y)g(expanding)g (in)h(p)r(o)n(w)n(ers)e(of)h Fv(")g Fy(the)h(resummed)f(series)f(w)n(e) h(reco)n(v)n(er)e(the)j(formal)f(p)r(o)n(w)n(er)f(series)118 4430 y(\(6.3\).)39 b(This)29 b(means)f(that)h(the)f(resummed)h(series,) e(whic)n(h)i(in)f(principle)h(could)f(b)r(e)h(unrelated)f(to)g(the)h (equation)118 4536 y(of)c(motion)f(\(b)r(ecause)g(of)h(the)g(w)n(a)n(y) e(it)i(has)f(b)r(een)h(de\014ned\),)h(in)f(fact)g(solv)n(es)e(suc)n(h)h (an)h(equation.)35 b(Suc)n(h)25 b(a)f(prop)r(ert)n(y)118 4642 y(can)i(b)r(e)h(pro)n(v)n(ed)d(b)n(y)i(reasoning)f(as)g(in)i(Ref.) f([8],)h(Section)f(8.)36 b(Again)26 b(w)n(e)g(omit)g(the)h(details,)f (whic)n(h)g(can)g(b)r(e)h(easily)118 4749 y(w)n(ork)n(ed)f(out.)189 4861 y(W)-7 b(e)28 b(can)f(summarize)g(our)g(results)g(in)h(the)g (follo)n(wing)e(statemen)n(t.)118 5043 y Fk(Theorem)36 b(3.)49 b Fw(Given)35 b(the)f(e)l(quation)g(\(1.1\))h(with)f Fv(f)42 b Fw(analytic)36 b(in)e(its)f(ar)l(gument)g(and)i Ft(!)h Fw(satisfying)f(the)f(Dio-)118 5150 y(phantine)27 b(c)l(ondition)f(\(6.1\),)i(ther)l(e)e(exists)f Fv(")1509 5162 y Fu(0)1572 5150 y Fw(such)h(that)f(for)i(al)t(l)g(r)l(e)l(al)f Fv(")f Fw(with)h Fs(j)p Fv(")p Fs(j)d Fv(<)g(")2795 5162 y Fu(0)2857 5150 y Fw(ther)l(e)j(is)g(a)g(quasi-p)l(erio)l(dic)1872 5349 y Fy(22)p eop %%Page: 23 23 23 22 bop 118 319 a Fw(solution)32 b(with)h(the)f(same)g(fr)l(e)l (quency)g(ve)l(ctor)g(of)h(the)f(for)l(cing)h(term.)45 b(Such)32 b(a)g(solution)g(extends)f(to)h(a)h(function)118 426 y(analytic)e(in)f(a)g(domain)h Fs(D)967 438 y Fz(R)1051 426 y Fw(like)g(in)f(Figur)l(e)g(6.3,)h(with)g Fv(R)24 b Fy(=)e Fv(")2126 438 y Fu(0)2163 426 y Fw(.)189 603 y Fy(The)i(conclusion)f(is)g(that)h(the)g(summation)g(criterion)f (describ)r(ed)g(here)g(giv)n(es)g(a)g(w)n(ell)h(de\014ned)g(function,)h (whic)n(h)118 709 y(is)f(quasi-p)r(erio)r(dic)g(and)g(solv)n(es)f(the)i (equation)e(of)i(motion)f(\(1.1\),)h(but)g(the)f(criterion)g(is)g(not)g (equiv)-5 b(alen)n(t)25 b(to)f(Borel)118 815 y(summabilit)n(y)35 b(an)n(y)f(more.)58 b(In)34 b(particular)g(the)h(issue)g(if)g(suc)n(h)f (quasi-p)r(erio)r(dic)g(solutions)g(are)g(unique)h(or)f(not)118 922 y(remains)27 b(op)r(en,)h(as)e(in)i(Ref.)g([6].)851 1170 y FB(7.)50 b(Extension)36 b(to)h(more)g(general)g(nonlinearities) 118 1347 y Fy(When)30 b(considering)e(the)h(equation)g(\(1.5\))g(the)h (formal)e(analysis)g(of)h(Section)g(2)g(\(and)g(of)h(Section)f(6)g(in)g (the)h(case)118 1453 y(of)e(quasi-p)r(erio)r(dic)e(forcing)h(terms\))g (can)h(b)r(e)g(p)r(erformed)f(essen)n(tially)f(in)i(the)g(same)f(w)n(a) n(y)-7 b(.)36 b(If)28 b(w)n(e)f(write)1032 1697 y Fv(g)s Fy(\()p Fv(x)p Fy(\))d(=)1324 1593 y FA(1)1297 1618 y Fq(X)1298 1794 y Fz(p)p Fu(=0)1453 1641 y Fy(1)p 1441 1678 65 4 v 1441 1754 a Fv(p)p Fy(!)1516 1697 y Fv(g)1556 1709 y Fz(p)1608 1697 y Fy(\()p Fv(x)19 b Fs(\000)f Fv(c)1825 1709 y Fu(0)1862 1697 y Fy(\))1894 1663 y Fz(p)1933 1697 y Fv(;)180 b(g)2176 1709 y Fz(p)2237 1697 y Fy(=)2337 1641 y(d)2383 1611 y Fz(p)2421 1641 y Fv(g)p 2334 1678 132 4 v 2334 1754 a Fy(d)p Fv(x)2427 1730 y Fz(p)2476 1697 y Fy(\()p Fv(c)2544 1709 y Fu(0)2582 1697 y Fy(\))p Fv(;)893 1984 y Fy([)p Fv(g)s Fy(\()p Fv(x)p Fy(\)])1093 1950 y Fu(\()p Fz(k)q Fu(\))1093 2005 y Fz(\027)1210 1984 y Fy(=)1324 1880 y FA(1)1297 1905 y Fq(X)1298 2081 y Fz(p)p Fu(=0)1453 1928 y Fy(1)p 1441 1965 65 4 v 1441 2041 a Fv(p)p Fy(!)1516 1984 y Fv(g)1556 1996 y Fz(p)1757 1905 y Fq(X)1624 2084 y Fz(k)1659 2092 y Fm(1)1691 2084 y Fu(+)p Fz(:::)p Fu(+)p Fz(k)1888 2092 y Fn(p)1923 2084 y Fu(=)p Fz(k)1618 2146 y Fp(\027)1656 2154 y Fm(1)1688 2146 y Fu(+)p Fz(:::)o Fu(+)p Fp(\027)1887 2154 y Fn(p)1923 2146 y Fu(=)p Fp(\027)2040 1984 y Fv(x)2087 1941 y Fu(\()p Fz(k)2148 1949 y Fm(1)2181 1941 y Fu(\))2087 1998 y Fp(\027)2125 2006 y Fm(1)2225 1984 y Fv(:)14 b(:)g(:)g(x)2383 1939 y Fu(\()p Fz(k)2444 1947 y Fn(p)2479 1939 y Fu(\))2383 1998 y Fp(\027)2421 2006 y Fn(p)2510 1984 y Fv(;)180 b(k)25 b Fs(\025)e Fy(0)p Fv(;)3538 1886 y Fy(\(7)p Fv(:)p Fy(1\))118 2315 y(then)28 b(the)g(recursiv)n(e)e(equations)h(for)g Ft(\027)i Fs(6)p Fy(=)22 b Fk(0)28 b Fy(are)941 2504 y Fv(x)988 2461 y Fu(\(0\))988 2518 y Fp(\027)1100 2504 y Fy(=)23 b(0)p Fv(;)941 2680 y(x)988 2637 y Fu(\(1\))988 2694 y Fp(\027)1100 2680 y Fy(=)1257 2624 y Fv(f)1298 2636 y Fp(\027)p 1198 2661 205 4 v 1198 2737 a Fv(i)p Ft(!)e Fs(\001)d Ft(\027)1413 2680 y Fv(;)937 2880 y(x)984 2837 y Fu(\()p Fz(k)q Fu(\))984 2894 y Fp(\027)1100 2880 y Fy(=)23 b Fs(\000)p Fy(\()p Fv(i)p Ft(!)e Fs(\001)d Ft(\027)6 b Fy(\))14 b Fv(x)1583 2837 y Fu(\()p Fz(k)q FA(\000)p Fu(1\))1583 2894 y Fp(\027)1779 2880 y Fs(\000)1954 2824 y Fy(1)p 1872 2861 V 1872 2937 a Fv(i)p Ft(!)21 b Fs(\001)e Ft(\027)2087 2880 y Fy([)p Fv(g)s Fy(\()p Fv(x)p Fy(\)])2287 2837 y Fu(\()p Fz(k)q FA(\000)p Fu(1\))2287 2894 y Fp(\027)2466 2880 y Fv(;)180 b(k)25 b Fs(\025)e Fy(2)p Fv(;)3538 2698 y Fy(\(7)p Fv(:)p Fy(2\))118 3110 y(while)40 b(the)g(compatibilit)n(y)f(condition)h(b)r(ecomes)f([)p Fv(g)s Fy(\()p Fv(x)p Fy(\)])1940 3067 y Fu(\()p Fz(k)q Fu(\))1940 3132 y Fb(0)2077 3110 y Fy(=)j Fv(f)2225 3122 y Fb(0)2267 3110 y Fv(\016)2304 3122 y Fz(k)q(;)p Fu(0)2437 3110 y Fy(for)d Fv(k)46 b Fs(\025)d Fy(0.)72 b(The)40 b(latter)f(for)g Fv(k)46 b Fy(=)d(0)118 3216 y(giv)n(es)c Fv(g)s Fy(\()p Fv(c)445 3228 y Fu(0)482 3216 y Fy(\))44 b(=)f Fv(f)707 3228 y Fu(0)744 3216 y Fy(,)g(while)d(for)g Fv(k)46 b Fs(\025)e Fy(1)39 b(giv)n(es)g Fv(g)1717 3186 y FA(0)1740 3216 y Fy(\()p Fv(c)1808 3228 y Fu(0)1845 3216 y Fy(\))14 b Fv(c)1927 3228 y Fz(k)1995 3216 y Fy(+)26 b Fv(R)q Fy(\()p Fv(c)2218 3228 y Fu(0)2255 3216 y Fv(;)14 b(c)2328 3228 y Fu(1)2365 3216 y Fv(;)g(:)g(:)g(:)g(;)g(c)2586 3228 y Fz(k)q FA(\000)p Fu(1)2711 3216 y Fy(\))44 b(=)g(0,)e(where)e (the)g(function)118 3322 y Fv(R)q Fy(\()p Fv(c)250 3334 y Fu(0)287 3322 y Fv(;)14 b(c)360 3334 y Fu(1)397 3322 y Fv(;)g(:)g(:)g(:)g(;)g(c)618 3334 y Fz(k)q FA(\000)p Fu(1)744 3322 y Fy(\))40 b(dep)r(end)g(on)g(the)g(co)r(e\016cien)n(ts)f (to)h(all)f(orders)f Fv(k)2380 3292 y FA(0)2447 3322 y Fv(<)43 b(k)s Fy(,)f(hence,)h(in)d(particular,)i(on)d(the)118 3429 y(constan)n(ts)27 b Fv(c)522 3441 y Fu(0)559 3429 y Fv(;)14 b(:)g(:)g(:)f(;)h(c)779 3441 y Fz(k)q FA(\000)p Fu(1)905 3429 y Fy(.)37 b(Therefore)26 b(the)i(constan)n(ts)f Fv(c)1888 3441 y Fz(k)1957 3429 y Fy(can)g(b)r(e)h(\014xed)f(iterativ)n (ely)g(as)1335 3644 y Fv(c)1371 3656 y Fz(k)1435 3644 y Fy(=)c Fs(\000)1678 3588 y Fy(1)p 1598 3625 204 4 v 1598 3701 a Fv(g)1641 3677 y FA(0)1663 3701 y Fy(\()p Fv(c)1731 3713 y Fu(0)1769 3701 y Fy(\))1811 3644 y Fv(R)q Fy(\()p Fv(c)1943 3656 y Fu(0)1980 3644 y Fv(;)14 b(c)2053 3656 y Fu(1)2090 3644 y Fv(;)g(:)g(:)g(:)g(;)g(c)2311 3656 y Fz(k)q FA(\000)p Fu(1)2436 3644 y Fy(\))p Fv(;)1047 b Fy(\(7)p Fv(:)p Fy(3\))118 3870 y(pro)n(vided)30 b(that)i(one)f(has)g Fv(g)998 3840 y FA(0)1021 3870 y Fy(\()p Fv(c)1089 3882 y Fu(0)1126 3870 y Fy(\))f Fs(6)p Fy(=)f(0,)j(so)e(that)i(under)f(the)h (conditions)f(\(1.6\))g(one)g(has)g(the)h(formal)e(solubilit)n(y)118 3977 y(of)e(the)h(equations)e(of)h(motion)g(\(1.1\).)38 b(Note)29 b(that)f(the)h(\014srt)f(condition)g(in)g(\(1.6\))g(requires) f Fv(f)3051 3989 y Fu(0)3112 3977 y Fs(2)d Fy(Ran\()p Fv(g)s Fy(\),)k(and)g(if)118 4083 y(suc)n(h)d(a)g(condition)g(is)g (satis\014ed)g(then)h(the)f(condition)g(on)g(the)h(deriv)-5 b(ativ)n(e)25 b(is)g(a)g(genericit)n(y)f(condition.)36 b(Note)25 b(also)118 4189 y(that)k(the)h(class)e(of)h(functions)g Fv(g)s Fy(\()p Fv(x)p Fy(\))h(whic)n(h)f(are)f(not)h(allo)n(w)n(ed)f (dep)r(ends)h(on)g Fv(f)38 b Fy(\(more)28 b(precisely)h(on)f(its)i(a)n (v)n(erage)118 4296 y Fv(f)159 4308 y Fu(0)196 4296 y Fy(\).)37 b(F)-7 b(or)26 b(instance)h(an)f(explicit)i(example)e(of)h (function)g(whic)n(h)g(do)r(es)f(not)h(satisfy)g(\(1.6\))f(is)h Fv(g)s Fy(\()p Fv(x)p Fy(\))d(=)e(3)p Fv(x)3371 4265 y Fu(2)3426 4296 y Fs(\000)16 b Fy(2)p Fv(x)3596 4265 y Fu(3)3660 4296 y Fy(if)118 4402 y Fv(f)159 4414 y Fu(0)219 4402 y Fy(=)23 b(1.)189 4508 y(The)28 b(graphical)e(represen)n(tation)g (di\013ers)h(from)g(that)h(of)g(the)g(previous)f(section)g(as)g(no)n(w) g(the)h(n)n(um)n(b)r(er)f(of)h(lines)118 4615 y(en)n(tering)e(a)h(v)n (ertex)f Fv(v)31 b Fy(can)c(assume)f(an)n(y)g(v)-5 b(alue)27 b Fv(s)1676 4627 y Fz(v)1739 4615 y Fs(2)c Fr(N)15 b Fy(,)28 b(and)f(if)g Fv(v)36 b(=)-52 b Fs(2)24 b Fv(V)2369 4627 y Fu(0)2407 4615 y Fy(\()p Fv(\022)r Fy(\))k(the)f(corresp)r (onding)e(no)r(de)j(factor)e(is)1652 4813 y Fv(F)1705 4825 y Fz(v)1768 4813 y Fy(=)c Fs(\000)1961 4757 y Fv(")p 1930 4794 102 4 v 1930 4870 a(s)1969 4882 y Fz(v)2008 4870 y Fy(!)2041 4813 y Fv(g)2081 4825 y Fz(s)2112 4833 y Fn(v)2152 4813 y Fv(;)1363 b Fy(\(7)p Fv(:)p Fy(4\))118 5031 y(whic)n(h)34 b(is)h(b)r(ounded)g(prop)r(ortionally)d(to)i(some)g (constan)n(t)g Fv(G)h Fy(to)f(the)h(p)r(o)n(w)n(er)e Fv(s)2661 5043 y Fz(v)2700 5031 y Fy(.)58 b(Since)3004 4969 y Fq(P)3092 5056 y Fz(v)r FA(2)p Fz(V)14 b Fu(\()p Fz(\022)r Fu(\))3315 5031 y Fy(\()p Fv(s)3386 5043 y Fz(v)3448 5031 y Fs(\000)23 b Fy(1\))34 b(=)118 5150 y Fs(j)p Fv(E)5 b Fy(\()p Fv(\022)r Fy(\))p Fs(j)20 b(\000)e Fy(1)23 b Fs(\024)g Fv(k)f Fs(\000)c Fy(1)28 b(\(b)n(y)g(Lemma)f(3\))h (this)h(pro)r(duces)e(an)g(o)n(v)n(erall)f(constan)n(t)h Fv(G)2648 5119 y Fu(2)p 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b(Ph)n(ys.)f Fk(21)284 4274 y Fy(\(1980\),)27 b(no.)36 b(2,)27 b(261{263.)1872 5349 y(24)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0501280249931--