Content-Type: multipart/mixed; boundary="-------------0411051126842" This is a multi-part message in MIME format. ---------------0411051126842 Content-Type: text/plain; name="04-362.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-362.keywords" Billiards, separatrix splitting, exponentially small phenomena, numeric experiments, homoclinic bifurcations ---------------0411051126842 Content-Type: application/postscript; name="circun.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="circun.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: paper.dvi %%Pages: 40 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times Helvetica %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips paper %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.11.05:1558 %%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: psfrag.pro %% %% This is file `psfrag.pro', %% generated with the docstrip utility. %% %% The original source files were: %% %% psfrag.dtx (with options: `filepro') %% %% Copyright (c) 1996 Craig Barratt, Michael C. 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b(p)-5 b(ar)g(ameter)p Fk(,)38 b(2)p Fh(n)h Fk(is)g(the)g Fj(de)-5 b(gr)g(e)g(e)40 b(of)h(the)g(p)-5 b(erturb)g(ation)p Fk(,)39 b Fh(a)g Fk(and)g Fh(b)g Fk(are)g(the)166 2103 y Fj(semi-axes)g(lengths)47 b Fk(of)39 b(the)g(unp)s(erturb)s(ed)h(ellipse)e Fh(C)2186 2118 y Fs(0)2225 2103 y Fk(,)h(and)g Fh(e)g Fk(is)g(the)g Fj(e)-5 b(c)g(c)g(entricity)48 b Fk(of)166 2224 y Fh(C)236 2239 y Fs(0)275 2224 y Fk(.)37 b(The)h(diameter)e(of)h Fh(C)1141 2239 y Fq(\017)1210 2224 y Fk(has)h(length)e(2)p Fh(a)p Fk(,)h(its)g(ends)h(are)f(the)g(p)s(oin)m(ts)g(\()p Fi(\006)p Fh(a;)17 b Fk(0\))37 b(and)g(is)166 2344 y(alw)m(a)m(ys)c(h)m (yp)s(erb)s(olic.)f(The)i Fj(hyp)-5 b(erb)g(olicity)34 b(p)-5 b(ar)g(ameter)43 b Fh(h)28 b(>)f Fk(0)33 b(determined)f(b)m(y) 361 2575 y Fh(a=b)c Fk(=)g(cosh)q(\()p Fh(h=)p Fk(2\))p Fh(;)211 b(a=\015)33 b Fk(=)27 b(sinh\()p Fh(h=)p Fk(2\))p Fh(;)212 b(e)27 b Fk(=)h(tanh\()p Fh(h=)p Fk(2\))166 2911 y(quan)m(ti\014es)33 b(ho)m(w)g(m)m(uc)m(h)g(h)m(yp)s(erb)s(olic)e 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Fh(:)166 5139 y Fk(The)26 b(sym)m(b)s(ol)691 5087 y Fs(as)684 5139 y Fk(=)f(has)g(b)s(een)g(used) h(to)e(emphasize)h(the)g(asymptotic)f(nature)h(of)f(this)g(series.)166 5259 y(In)38 b(other)f(w)m(ords,)i(if)d(w)m(e)i(retain)f(only)f (\014nitely)h(man)m(y)g(terms)h(of)e(the)i(righ)m(t-hand)e(side.)166 5380 y(the)d(error)g(will)d(b)s(e)j(of)f(the)h(order)f(of)g(the)h (\014rst)h(discarded)f(term.)1769 5712 y(3)p eop %%Page: 4 4 4 3 bop 166 83 a Fk(The)26 b(dominan)m(t)d(asymptotic)i(co)s(e\016cien) m(t)g Fh(\013)1792 47 y Fq(\017)1791 108 y Fs(0)1855 83 y Fk(do)s(es)h(not)e(v)-5 b(anish)25 b(in)f(the)h(range)g(0)j Fi(\024)g Fh(\017)g Fi(\024)g Fk(1)166 203 y(for)37 b Fh(n)f Fk(=)g(2)p Fh(;)17 b Fk(3)p Fh(;)g Fk(4.)36 b(Therefore,)j(the)e (homo)s(clinic)e(lengths)i(no)g(longer)g(coincide)g(and)g(the)166 324 y(separatrices)c(really)e(split)g(under)i(the)f(p)s(erturbations)g (of)g(degree)h(four,)f(six)g(and)g(eigh)m(t.)166 444 y(In)f(addition,)f(although)g(the)h(\014rst)h(co)s(e\016cien)m(ts)g(of) f(the)g(asymptotic)g(series)2996 378 y Fg(P)3084 465 y Fq(j)t Fr(\025)p Fs(0)3227 444 y Fh(\013)3290 408 y Fq(\017)3289 469 y(j)3326 444 y Fh(h)3382 408 y Fs(2)p Fq(j)166 565 y Fk(decrease,)d(an)f(accurate)g(computation)e(of)h(sev)m (eral)i(h)m(undreds)g(of)e(its)g(co)s(e\016cien)m(ts)i(sho)m(ws)166 685 y(that)35 b(it)f(is)h(Gevrey)900 649 y Fs(1)957 685 y Fk(,)g(and)g(so)h(it)e(div)m(erges)i(for)e(all)f Fh(h)g Fi(6)p Fk(=)e(0.)k(In)h(fact,)f(it)f(is)h(Gevrey-1)g(of)166 805 y(t)m(yp)s(e)f Fh(\032)28 b Fk(=)f(1)p Fh(=)p Fk(2)p Fh(\031)773 769 y Fs(2)812 805 y Fk(,)32 b(since)h(the)g(follo)m(wing)d (asymptotic)i(expansion)h(holds)371 1121 y(\026)-59 b Fh(\013)424 1080 y Fq(\017)423 1145 y(j)487 1121 y Fk(:=)634 1047 y(\(2)p Fh(\031)780 1010 y Fs(2)819 1047 y Fk(\))857 1010 y Fs(2)p Fq(j)929 1047 y Fh(\013)992 1010 y Fq(\017)991 1071 y(j)p 628 1097 407 4 v 628 1189 a Fh(\017)p Fk(\(2)p Fh(j)28 b Fk(+)22 b(2\)!)1078 1068 y Fs(as)1072 1121 y Fk(=)37 b(\026)-59 b Fh(\013)1238 1080 y Fq(\017)1237 1145 y Fr(1)1334 1121 y Fk(+)1432 1038 y Fg(X)1436 1222 y Fq(l)q Fr(\025)p Fs(2)1583 1094 y Fk(\026)1569 1121 y Fh(\014)1630 1080 y Fq(\017)1624 1145 y(l)1662 1121 y Fh(j)1708 1080 y Fr(\000)p Fq(l)1984 1121 y Fk(\()p Fh(j)33 b Fi(!)28 b Fk(+)p Fi(1)p Fk(\))p Fh(:)166 1574 y Fk(W)-8 b(e)36 b(ha)m(v)m(e)g(computed)g(the)g(limit)41 b(\026)-59 b Fh(\013)1486 1538 y Fq(\017)1485 1599 y Fr(1)1595 1574 y Fk(and)36 b(the)f(\014rst)h(co)s(e\016cien)m(ts)2673 1548 y(\026)2659 1574 y Fh(\014)2720 1538 y Fq(\017)2714 1599 y(l)2787 1574 y Fk(with)f(more)g(than)166 1695 y(fort)m(y)28 b(decimal)f(digits)g(for)g Fh(n)h Fk(=)g(2)p Fh(;)17 b Fk(3)p Fh(;)g Fk(4)27 b(and)h(for)g Fh(\017)g Fk(=)f(10)2207 1659 y Fr(\000)p Fq(k)2332 1695 y Fk(with)h Fh(k)j Fk(=)d(1)p Fh(;)17 b Fk(2)p Fh(;)g(:)g(:)g(:)e(;)i Fk(9)p Fh(;)g Fk(10)p Fh(;)g Fk(50.)166 1815 y(The)39 b(v)-5 b(alues)37 b(obtained)g(for)g Fh(\017)f Fk(=)g(10)1512 1779 y Fr(\000)p Fs(50)1679 1815 y Fk(are)i(a)f(v)m(ery)i(accurate)f(appro)m(ximation)d (of)i(the)166 1936 y(limits)44 b(\026)-59 b Fh(\013)505 1899 y Fs(0)504 1960 y Fr(1)614 1936 y Fk(=)36 b(lim)861 1951 y Fq(\017)p Fr(!)p Fs(0)1026 1936 y Fk(\026)-59 b Fh(\013)1079 1899 y Fq(\017)1078 1960 y Fr(1)1190 1936 y Fk(and)1398 1909 y(\026)1384 1936 y Fh(\014)1445 1899 y Fs(0)1439 1960 y Fq(l)1520 1936 y Fk(=)35 b(lim)1767 1951 y Fq(\017)p Fr(!)p Fs(0)1936 1909 y Fk(\026)1922 1936 y Fh(\014)1983 1899 y Fq(\017)1977 1960 y(l)2015 1936 y Fk(.)i(Surprisingly)-8 b(,)36 b(if)g(the)h(p)s(erturbation)166 2056 y(is)k(quartic:)h Fh(n)i Fk(=)f(2,)f(then)52 b(\026)-59 b Fh(\013)1276 2020 y Fs(0)1275 2081 y Fr(1)1393 2056 y Fk(=)44 b Fi(\000)p Fk(8)e(and)g(the)g(\014rst)g(limit)d(v)-5 b(alues)2823 2030 y(\026)2809 2056 y Fh(\014)2870 2020 y Fs(0)2864 2081 y Fq(l)2951 2056 y Fk(are)42 b(rational)166 2176 y(com)m(binations)31 b(of)h(p)s(o)m(w)m(ers)i(of)e Fh(\031)1364 2140 y Fs(4)1404 2176 y Fk(,)g(namely)375 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b([6])g(on)g(p)s(erturb)s(ed)i(w)m(eakly)f(h)m(yp)s(erb)s (olic)f(McMillan)f(maps.)h(W)-8 b(e)27 b(note)f(that)h(the)f(study)166 3262 y(of)g(billiards)d(is)j(computationally)d(more)j(exp)s(ensiv)m(e)i (\(b)m(y)g(a)e(factor)g(ten\))g(than)h(the)f(study)166 3382 y(of)35 b(McMillan)f(maps.)h(In)h(spite)f(of)g(this)g(dra)m(wbac)m (k,)i(the)f(exp)s(erimen)m(ts)g(in)f(the)h(curren)m(t)166 3503 y(pap)s(er)k(are)f(more)g(accurate)h(due)h(to)e(three)h(factors:)g (1\))f(the)h(hardw)m(are)h(is)e(faster)h(\(w)m(e)166 3623 y(ha)m(v)m(e)30 b(used)g(a)f(Beo)m(wulf)g(cluster)g(with)f(sev)m (eral)i(tens)f(of)g(pro)s(cessors,)h(instead)f(of)f(a)h(single)166 3744 y(computer\);)j(2\))f(the)h(soft)m(w)m(are)h(is)f(b)s(etter)g(\(w) m(e)h(ha)m(v)m(e)g(written)e(our)h(programs)f(using)g(the)166 3864 y(P)-8 b(ARI)32 b(system)g([1],)g(instead)f(of)g(hand-made)g (routines\);)g(and)h(3\))f(the)h(algorithms)c(ha)m(v)m(e)166 3984 y(b)s(een)33 b(tuned)h(in)e(sev)m(eral)h(tric)m(ky)g(w)m(a)m(ys.)i 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Fk(ha)m(v)m(e)i(the)e(same)h(asymptotic)e(b)s(eha)m(vior,)h(but)h(with) p 166 5037 299 4 v 166 5102 a Fs(1)257 5135 y Fv(A)d(series)606 5067 y Fo(P)702 5162 y Fq(j)754 5135 y Fu(f)799 5149 y Fq(j)835 5135 y Fu(x)887 5102 y Fq(j)957 5135 y Fv(is)f Fx(Gevr)-5 b(ey-)p Fu(r)38 b Fx(of)e(typ)-5 b(e)37 b Fu(\032)c Fv(when)g(there)h(exist)g(constan)m(ts)h Fu(C)q(;)15 b(`)31 b(>)g Fv(0)j(suc)m(h)166 5260 y(that)28 b Ft(j)p Fu(f)430 5274 y Fq(j)466 5260 y Ft(j)e(\024)f Fu(C)7 b(\032)732 5227 y Fq(r)r(j)802 5260 y Fv(\000\()p Fu(r)s(j)19 b Fv(+)14 b Fu(`)p Fv(\),)28 b(where)f(\000\()p Fu(z)t Fv(\))h(stands)f(for)g(the)h(Gamma)g(function.)e(When)h Fu(r)h Fv(=)d(1,)166 5373 y(the)31 b(Borel)f(transform)982 5305 y Fo(P)1078 5400 y Fq(j)1129 5373 y Fu(f)1174 5387 y Fq(j)1210 5373 y Fu(s)1253 5340 y Fq(j)t Fr(\000)p Fs(1)1380 5373 y Fu(=)p Fv(\()p Fu(j)c Ft(\000)20 b Fv(1\)!)32 b(con)m(v)m(erges)g(in)d(the)i(disk)d Ft(f)p Fu(s)e Ft(2)f Fd(C)49 b Fv(:)25 b Ft(j)p Fu(s)p Ft(j)g Fu(<)g Fv(1)p Fu(=\032)p Ft(g)p Fv(.)1769 5712 y Fk(4)p eop %%Page: 5 5 5 4 bop 166 83 a Fk(opp)s(osite)32 b(signs.)h(In)g(fact,)361 288 y Fh(!)426 247 y Fr(\006)519 235 y Fs(as)512 288 y Fk(=)28 b Fi(\006)p Fk(2)p Fh(\031)801 247 y Fs(2)840 288 y Fh(a\017h)986 247 y Fr(\000)p Fs(2)1098 285 y Fk(e)1141 247 y Fr(\000)p Fq(\031)1239 223 y Fp(2)1274 247 y Fq(=h)1372 205 y Fg(X)1370 388 y Fq(j)t Fr(\025)p Fs(0)1510 288 y Fh(\013)1573 247 y Fq(\017)1572 312 y(j)1608 288 y Fh(h)1664 247 y Fs(2)p Fq(j)1931 288 y Fk(\()p Fh(h)g Fi(!)f Fk(0)2229 247 y Fs(+)2288 288 y Fh(;)17 b(\017)33 b Fk(\014xed)q(\))166 681 y(where)443 615 y Fg(P)530 702 y Fq(j)t Fr(\025)p Fs(0)674 681 y Fh(\013)737 645 y Fq(\017)736 706 y(j)772 681 y Fh(h)828 645 y Fs(2)p Fq(j)927 681 y Fk(is)27 b(the)h(same)f(series)h(that)g(app)s(eared)f (in)g(the)h(expansion)g(of)f(\001)3206 696 y Fs(Length)3427 681 y Fk(.)166 802 y(Th)m(us,)37 b(it)d(is)g(quite)i(natural)e(to)g (study)j(the)e(asymptotic)f(b)s(eha)m(vior)h(of)g(\012)d(:=)g Fh(!)3121 766 y Fs(+)3204 802 y Fk(+)23 b Fh(!)3368 766 y Fr(\000)3427 802 y Fk(.)166 922 y(There)29 b(are)e(reasons)h(to)g (guess)g(that)f(\012)h(has)g(order)2025 919 y(e)2068 886 y Fr(\000)p Fs(2)p Fq(\031)2201 862 y Fp(2)2236 886 y Fq(=h)2316 922 y Fk(,)f(see)i Fi(x)p Fk(2.5.)e(W)-8 b(e)28 b(ha)m(v)m(e)h(c)m(hec)m(k)m(ed)166 1042 y(that)j(this)h(guest)g (is)f(correct,)h(since)361 1247 y(\012)28 b(:=)g Fh(!)655 1206 y Fs(+)735 1247 y Fk(+)22 b Fh(!)898 1206 y Fr(\000)991 1195 y Fs(as)984 1247 y Fk(=)28 b(16)p Fh(\031)1245 1206 y Fs(2)1284 1247 y Fh(a\017h)1430 1206 y Fr(\000)p Fs(2)1542 1244 y Fk(e)1585 1206 y Fr(\000)p Fs(2)p Fq(\031)1718 1183 y Fp(2)1753 1206 y Fq(=h)1851 1164 y Fg(X)1849 1347 y Fq(j)t Fr(\025)p Fs(0)1988 1247 y Fh(!)2053 1206 y Fq(\017)2049 1272 y(j)2086 1247 y Fh(h)2142 1206 y Fs(2)p Fq(j)2409 1247 y Fk(\()p Fh(h)g Fi(!)f Fk(0)2707 1206 y Fs(+)2766 1247 y Fh(;)17 b(\017)32 b Fk(\014xed)q(\))166 1641 y(for)42 b(some)g(new)i(series)1061 1574 y Fg(P)1148 1661 y Fq(j)t Fr(\025)p Fs(0)1292 1641 y Fh(!)1357 1605 y Fq(\017)1353 1665 y(j)1389 1641 y Fh(h)1445 1605 y Fs(2)p Fq(j)1517 1641 y Fk(,)e(whic)m(h)h(is)f(also)g(Gevrey-1)h(of)e (t)m(yp)s(e)j Fh(\032)h Fk(=)f(1)p Fh(=)p Fk(2)p Fh(\031)3388 1605 y Fs(2)3427 1641 y Fk(,)166 1761 y(although)32 b(the)h(analysis)e (of)h(this)h(asymptotic)f(series)h(has)g(b)s(een)g(more)f(delicate.)166 1981 y(As)41 b Fh(\017)f Fi(!)g Fk(0,)g(the)h(Gevrey)g(co)s(e\016cien)m (ts)g Fh(\013)1737 1945 y Fq(\017)1736 2006 y(j)1813 1981 y Fk(and)f Fh(!)2075 1945 y Fq(\017)2071 2006 y(j)2147 1981 y Fk(tend)h(to)e(the)i(T)-8 b(a)m(ylor)40 b(co)s(e\016cien)m(ts) 166 2101 y Fh(\013)229 2065 y Fs(0)228 2126 y Fq(j)305 2101 y Fk(and)c Fh(!)563 2065 y Fs(0)559 2126 y Fq(j)638 2101 y Fk(of)g(a)g(couple)h(of)f(analytic)f(functions)h(\(in)g(the)h(v) -5 b(ariable)34 b Fh(h)p Fk(\))j(whic)m(h)g(can)f(b)s(e)166 2222 y(explicitly)31 b(computed)i(with)f(a)g(discrete)h(Melnik)m(o)m(v) g(metho)s(d.)f(F)-8 b(or)32 b(instance,)361 2493 y Fh(\013)424 2452 y Fs(0)423 2518 y(0)491 2493 y Fk(=)27 b(\()p Fi(\000)p Fk(1\))796 2452 y Fq(n)844 2493 y Fk(4)p Fh(\031)952 2452 y Fs(2)1007 2385 y Fq(n)p Fr(\000)p Fs(2)1014 2410 y Fg(X)1013 2592 y Fq(j)t Fs(=0)1167 2426 y Fk(\()p Fi(\000)p Fk(1\))1369 2389 y Fq(j)1406 2426 y Fh(\031)1465 2389 y Fs(2)p Fq(j)p 1167 2470 370 4 v 1168 2561 a Fk(\(2)p Fh(j)h Fk(+)22 b(1\)!)1546 2493 y Fh(;)212 b(!)1850 2452 y Fs(0)1846 2518 y(0)1916 2493 y Fk(=)28 b(\()p Fi(\000)p Fk(1\))2222 2452 y Fq(n)2269 2493 y Fk(8)p Fh(\031)2377 2452 y Fs(2)2433 2385 y Fq(n)p Fr(\000)p Fs(2)2439 2410 y Fg(X)2438 2592 y Fq(j)t Fs(=0)2592 2426 y Fk(\()p Fi(\000)p Fk(1\))2794 2389 y Fq(j)2831 2426 y Fk(\(2)p Fh(\031)t Fk(\))3015 2389 y Fs(2)p Fq(j)p 2592 2470 495 4 v 2656 2561 a Fk(\(2)p Fh(j)g Fk(+)22 b(1\)!)3096 2493 y Fh(:)166 2885 y Fk(Ob)m(viously)-8 b(,)28 b(the)h(errors)g Fi(j)p Fh(\013)1167 2849 y Fq(\017)1166 2909 y(j)1215 2885 y Fi(\000)13 b Fh(\013)1368 2849 y Fs(0)1367 2909 y Fq(j)1408 2885 y Fi(j)28 b Fk(=)1567 2894 y(O)1643 2885 y(\()p Fh(\017)p Fk(\))h(and)f Fi(j)p Fh(!)2065 2849 y Fq(\017)2061 2909 y(j)2110 2885 y Fi(\000)13 b Fh(!)2265 2849 y Fs(0)2261 2909 y Fq(j)2305 2885 y Fi(j)27 b Fk(=)2464 2894 y(O)2540 2885 y(\()p Fh(\017)p Fk(\))h(are)h(not)f(uniform)e(in)166 3005 y(the)i(index)g Fh(j)6 b Fk(,)28 b(since)g(the)h(T)-8 b(a)m(ylor)27 b(co)s(e\016cien)m(ts)i(v)m(erify)f(some)g Fj(p)-5 b(otential)30 b(b)-5 b(ounds)p Fk(,)27 b(whereas)166 3126 y(the)33 b(Gevrey)h(co)s(e\016cien)m(ts)f Fj(gr)-5 b(ow)43 b Fk(at)32 b(a)h(factorial)d(rate,)i(see)i(remark)e(3.)166 3346 y(A)25 b(similar)c(understanding)26 b(of)e(the)h(next)h(exp)s (onen)m(tial)e(terms)h(seems)h(unreac)m(hable)f(with)166 3466 y(the)i(presen)m(t)h(tec)m(hniques,)h(and)e(not)f(for)h(computing) e(limitations.)d(F)-8 b(or)26 b(instance,)h(in)f(the)166 3586 y(same)38 b(w)m(a)m(y)i(that)e(the)h(sum)f(of)g(the)g(homo)s (clinic)d(in)m(v)-5 b(arian)m(ts)38 b Fh(!)2528 3550 y Fs(+)2624 3586 y Fk(and)h Fh(!)2885 3550 y Fr(\000)2981 3586 y 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Fs(4)2678 5363 y Fk(=)27 b(1)2830 5267 y Fg(o)2902 5363 y Fh(:)1769 5712 y Fk(5)p eop %%Page: 6 6 6 5 bop 166 83 a Fk(Once)46 b(\014xed)g Fh(\017)g Fk(and)f Fh(h)p Fk(,)g(w)m(e)h(lo)s(ok)e(for)h(c)m(hanges)h(in)f(the)g(top)s (ology)f(of)g(the)i(p)s(erturb)s(ed)166 203 y(in)m(v)-5 b(arian)m(t)41 b(curv)m(es)k(when)e(the)g Fj(bifur)-5 b(c)g(ation)44 b(p)-5 b(ar)g(ameter)52 b Fh(d)43 b Fk(v)-5 b(aries.)42 b(It)g(turns)i(out)e(that)166 324 y(there)33 b(exist)g(three)h(bifurcation)c(v)-5 b(alues)33 b Fh(d)1733 339 y Fs(+)1819 324 y Fh(<)28 b(d)1974 339 y Fs(0)2041 324 y Fh(<)f(d)2195 339 y Fr(\000)2286 324 y Fk(suc)m(h)34 b(that)361 531 y Fh(d)28 b Fk(=)f Fh(d)594 546 y Fr(\006)680 531 y Fi(\))h Fh(!)873 489 y Fr(\006)959 531 y Fk(=)f(0)p Fh(;)212 b(d)27 b Fk(=)h Fh(d)1583 546 y Fs(0)1650 531 y Fi(\))f Fk(\001)1858 546 y Fs(Length)2107 531 y Fk(=)g(0)p Fh(:)166 837 y Fk(A)m(t)k(the)g(bifurcation)e(v)-5 b(alue)31 b Fh(d)c Fk(=)h Fh(d)1449 852 y Fs(+)1538 837 y Fk(\(resp)s(ectiv)m (ely)-8 b(,)32 b Fh(d)27 b Fk(=)h Fh(d)2358 852 y Fr(\000)2416 837 y Fk(\))j(the)g(separatrices)h(ha)m(v)m(e)g(a)166 957 y(cubic)c(tangency)g(at)g(the)g(four)f Fh(y)t Fk(-axial)d(\(resp)s (ectiv)m(ely)-8 b(,)29 b Fh(x)p Fk(-axial\))c(p)s(ersisten)m(t)k(homo)s (clinic)166 1078 y(orbits)39 b(and)h(the)h(n)m(um)m(b)s(er)f(of)f (primary)g(homo)s(clinic)e(orbits)i(c)m(hanges.)i(In)f(the)h(w)m(eakly) 166 1198 y(h)m(yp)s(erb)s(olic)35 b(case,)h(these)h(bifurcations)d(are) i Fj(almost)h(invisible)p Fk(,)d(since)i(they)h(tak)m(e)f(place)166 1319 y(in)c(an)g(exp)s(onen)m(tially)g(small)e(\(with)i(resp)s(ect)i (to)e Fh(h)p Fk(\))h(range)g(of)f Fh(d)p Fk(.)g(Concretely)-8 b(,)361 1535 y(\001)442 1550 y Fs(Bifur)632 1535 y Fk(:=)28 b Fh(d)814 1550 y Fr(\000)895 1535 y Fi(\000)22 b Fh(d)1045 1550 y Fs(+)1138 1482 y(as)1132 1535 y Fk(=)1235 1532 y(e)1279 1494 y Fr(\000)p Fq(\031)1377 1470 y Fp(2)1411 1494 y Fq(=h)1509 1452 y Fg(X)1508 1635 y Fq(j)t Fr(\025)p Fs(0)1647 1535 y Fh(\016)1694 1494 y Fq(\017)1690 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b(forces)h(us)g(to)f(use)h(a)f(m)m(ultiple-precision)c (arithmetic,)i(to)i(expand)h(the)g(in)m(v)-5 b(arian)m(t)166 2632 y(curv)m(es)38 b(up)e(to)f(high)g(order,)h(and)f(to)h(tak)m(e)g (adv)-5 b(an)m(tage)36 b(of)f(symmetries.)g(The)i(metho)s(ds)166 2752 y(go)s(es)26 b(bac)m(k)g(to)f(C.)h(Sim\023)-49 b(o)24 b([19],)h(and)h(w)m(ere)h(\014rst)f(applied)e(in)h([8].)g(They)i(are)e (also)g(explained)166 2873 y(in)k([6].)h(The)h(main)d(obstacle)i(is)g (the)g(computation)f(of)g(exp)s(onen)m(tially)g(small)f(quan)m(tities) 166 2993 y(with)d(m)m(uc)m(h)h(more)e(precision)h(than)g(the)g(usual)g (one;)h(namely)-8 b(,)24 b(with)h(a)g(relativ)m(e)f(error)h(less)166 3113 y(than)30 b(10)489 3077 y Fr(\000)p Fs(1500)718 3113 y Fk(in)f(the)g(most)g(extreme)i(cases.)g(Hence,)f(the)g(use)h(of) e(a)g(m)m(ultiple-precision)166 3234 y(arithmetic)k(is)h(una)m(v)m (oidable,)h(due)g(to)g(the)g(requiremen)m(t)g(of)f(a)h(v)m(ery)h(high)e (precision)g(in)166 3354 y(the)23 b(\014nal)g(result,)g(and)g(the)g 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b(complete)e(this)h(in)m(tro)s(duction)f(with)h(a)g(note)g(on)g(the)h (organization)d(of)i(this)f(pap)s(er.)166 4417 y(In)39 b(section)h(2,)e(w)m(e)i(in)m(tro)s(duce)f(some)g(concepts)i(ab)s(out)d (the)i(splitting)d(of)h(separatrices)166 4537 y(for)e(area-preserving)g (maps)g(and)h(w)m(e)g(also)f(recall)e(some)j(previous)f(celebrated)h (results)166 4658 y(ab)s(out)44 b(exp)s(onen)m(tially)f(small)f (phenomena.)i(In)h(section)f(3,)g(w)m(e)h(presen)m(t)h(the)e(con)m(v)m (ex)166 4778 y(billiard)20 b(tables)j(studied)h(through)f(the)h(rest)g (of)f(the)h(pap)s(er.)f(Afterw)m(ards,)i(w)m(e)f(describ)s(e)g(in)166 4898 y(section)h(4)f(the)h(results)g(on)f(the)h(exp)s(onen)m(tially)e (small)f(splittings)h(that)h(tak)m(e)i(place)e(under)166 5019 y(monomial)40 b(p)s(erturbations.)j(The)i(study)g(of)e(the)h (primary)e(homo)s(clinic)f(bifurcations)166 5139 y(under)g(a)e (binomial)e(p)s(erturbation)i(is)h(con)m(tained)g(in)f(section)h(5.)g (The)h(details)d(on)i(the)166 5259 y(Melnik)m(o)m(v)27 b(and)g(n)m(umerical)e(computations)g(are)i(relegated)f(to)g(app)s (endixes)i Fi(x)p Fk(A)f(and)f Fi(x)p Fk(B,)166 5380 y(resp)s(ectiv)m(ely)-8 b(.)1769 5712 y(6)p eop %%Page: 7 7 7 6 bop 166 83 a Fl(2)112 b(Notations)37 b(and)h(a)g(bit)e(of)i (history)166 434 y Fk(W)-8 b(e)29 b(in)m(tro)s(duce)g(some)g(quan)m (tities)f(used)i(to)f(measure)g(the)g(size)g(of)g(the)g(splitting)e (and)i(w)m(e)166 554 y(repro)s(duce)34 b(sev)m(eral)f(celebrated)g (results)h(on)e(their)h(exp)s(onen)m(tially)e(smallness.)h(F)-8 b(or)32 b(the)166 674 y(sak)m(e)h(of)f(space)h(the)g(exp)s(osition)e (is)g(v)m(ery)j(compact,)e(sometimes)f(ev)m(en)i(without)f(precise)166 795 y(statemen)m(ts)24 b(of)f(the)h(theorems.)g(W)-8 b(e)23 b(are)h(mainly)d(in)m(terested)j(in)f(the)h(ideas.)f(More)h (details)166 915 y(can)33 b(b)s(e)g(found)f(in)g(the)h(surv)m(ey)i ([12].)166 1310 y Fj(2.1)99 b(Homo)-5 b(clinic)34 b(invariants)166 1661 y Fk(Let)f Fh(f)40 b Fk(:)28 b Fh(M)40 b Fi(!)28 b Fh(M)43 b Fk(b)s(e)34 b(an)f(analytic)f(exact)h(area-preserving)h (di\013eomorphism)c(de\014ned)166 1781 y(on)45 b(an)f(exact)i (symplectic)e(surface)h Fh(M)10 b Fk(.)46 b(Let)f Fe(f)j Fk(=)g Fi(\000)17 b Fk(d)p Fh(\022)48 b Fk(b)s(e)d(the)g(exact)g (symplectic)166 1902 y(t)m(w)m(o-form)35 b(suc)m(h)j(that)e Fh(f)1082 1866 y Fr(\003)1121 1902 y Fe(f)d Fk(=)h Fe(f)p Fk(.)i(Then)h(there)g(exists)g(a)f(function)g Fh(L)e Fk(:)f Fh(M)45 b Fi(!)33 b Fe(R)47 b Fk(suc)m(h)166 2022 y(that)32 b Fh(f)436 1986 y Fr(\003)476 2022 y Fh(\022)25 b Fi(\000)d Fh(\022)31 b Fk(=)44 b(d)p Fh(L)p Fk(.)34 b(This)e(function)g(is)h(the)g Fj(gener)-5 b(ating)34 b(function)39 b Fk(of)32 b(the)h(map)f Fh(f)11 b Fk(.)166 2253 y(Let)26 b Fh(m)419 2268 y Fr(1)519 2253 y Fk(b)s(e)f(a)g(saddle)h (p)s(oin)m(t)e(of)h Fh(f)11 b Fk(.)25 b(Then)i(the)e(eigen)m(v)-5 b(alues)26 b(of)f Fh(d)-16 b(f)11 b Fk(\()p Fh(m)2699 2268 y Fr(1)2773 2253 y Fk(\))25 b(are)g(of)g(the)h(form)166 2373 y Fh(\025)36 b Fk(and)h Fh(\025)510 2337 y Fr(\000)p Fs(1)640 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Fk(\()p Fh(m)633 3338 y Fr(1)708 3323 y Fk(\))h(=)f Fi(f)p Fh(m)28 b Fi(2)g Fh(M)39 b Fk(:)27 b(lim)1435 3338 y Fq(n)p Fr(!\0001)1694 3323 y Fk(dist\()p Fh(f)1949 3287 y Fq(n)1996 3323 y Fk(\()p Fh(m)p Fk(\))p Fh(;)17 b(m)2286 3338 y Fr(1)2361 3323 y Fk(\))27 b(=)h(0)p Fi(g)16 b Fh(:)166 3564 y Fk(They)36 b(are)f(analytic)e(immersions)g(of)h(the)h(real)f(line)f(in)h Fh(M)46 b Fk(without)34 b(self-in)m(tersections)166 3684 y(and)f(there)g(exist)g(some)f(analytic)g(parameterizations)e Fh(m)2328 3648 y Fs(u)p Fq(;)p Fs(s)2447 3684 y Fk(:)e Fe(R)39 b Fi(!)27 b Fh(W)2835 3648 y Fs(u)p Fq(;)p Fs(s)2958 3684 y Fk(suc)m(h)34 b(that)361 3924 y Fh(m)446 3883 y Fs(u)p Fq(;)p Fs(s)537 3924 y Fk(\(0\))28 b(=)f Fh(m)878 3939 y Fr(1)953 3924 y Fh(;)212 b(f)11 b Fk(\()p Fh(m)1374 3883 y Fs(u)1417 3924 y Fk(\()p Fh(r)s Fk(\)\))27 b(=)h Fh(m)1794 3883 y Fs(u)1838 3924 y Fk(\()p Fh(\025r)s Fk(\))p Fh(;)211 b(f)11 b Fk(\()p Fh(m)2438 3883 y Fs(s)2470 3924 y Fk(\()p Fh(r)s Fk(\)\))27 b(=)h Fh(m)2847 3883 y Fs(s)2879 3924 y Fk(\()p Fh(r)s(=\025)p Fk(\))p Fh(:)166 4275 y Fk(It)41 b(is)g(con)m(v)m(enien)m(t)h(to)f(pass)h(to)f(the)g (parameter)f Fh(t)j Fk(=)f(ln)15 b Fh(r)44 b Fk(\(resp)s(ectiv)m(ely)-8 b(,)42 b Fh(t)g Fk(=)g Fi(\000)17 b Fk(ln)f Fh(r)s Fk(\))166 4395 y(on)45 b(the)g(unstable)f(\(resp)s(ectiv)m(ely)-8 b(,)46 b(stable\))e(curv)m(e)i(b)m(y)g(setting)e Fh( )2648 4359 y Fs(u)2691 4395 y Fk(\()p Fh(t)p Fk(\))49 b(=)e Fh(m)3059 4359 y Fs(u)3103 4395 y Fk(\()3141 4392 y(e)3184 4359 y Fq(t)3214 4395 y Fk(\))e(and)166 4516 y Fh( )233 4479 y Fs(s)265 4516 y Fk(\()p Fh(t)p Fk(\))28 b(=)f Fh(m)592 4479 y Fs(s)625 4516 y Fk(\()663 4513 y(e)706 4479 y Fr(\000)p Fq(t)790 4516 y Fk(\).)33 b(Clearly)-8 b(,)32 b(the)h(functions)f Fh( )1901 4479 y Fs(u)p Fq(;)p Fs(s)1992 4516 y Fk(\()p Fh(t)p Fk(\))h(satisfy)f(the)h(conditions)404 4756 y(lim)361 4811 y Fq(t)p Fr(!\0001)599 4756 y Fh( )666 4715 y Fs(u)710 4756 y Fk(\()p Fh(t)p Fk(\))27 b(=)h Fh(m)1037 4771 y Fr(1)1112 4756 y Fh(;)255 b Fk(lim)1351 4811 y Fq(t)p Fr(!)p Fs(+)p Fr(1)1589 4756 y Fh( )1656 4715 y Fs(s)1688 4756 y Fk(\()p Fh(t)p Fk(\))28 b(=)f Fh(m)2015 4771 y Fr(1)2090 4756 y Fh(;)212 b(f)11 b Fk(\()p Fh( )2493 4715 y Fs(u)p Fq(;)p Fs(s)2583 4756 y Fk(\()p Fh(t)p Fk(\)\))28 b(=)g Fh( )2931 4715 y Fs(u)p Fq(;)p Fs(s)3022 4756 y Fk(\()p Fh(t)22 b Fk(+)g Fh(h)p Fk(\))p Fh(:)166 5139 y Fk(Finally)-8 b(,)41 b(let)i Fh(O)49 b Fk(=)e(\()p Fh(m)1050 5154 y Fq(n)1097 5139 y Fk(\))1135 5154 y Fq(n)p Fr(2)p Ff(Z)1317 5139 y Fk(b)s(e)d(a)f(homo)s(clinic)e (orbit)i(to)g Fh(m)2520 5154 y Fr(1)2639 5139 y Fk(passing)g(b)m(y)i(a) f(p)s(oin)m(t)166 5259 y Fh(m)251 5274 y Fs(0)332 5259 y Fk(\(that)e(is,)f Fh(m)810 5274 y Fq(n)901 5259 y Fk(=)i Fh(f)1079 5223 y Fq(n)1125 5259 y Fk(\()p Fh(m)1248 5274 y Fs(0)1288 5259 y Fk(\))f(and)f(lim)1702 5274 y Fq(n)p Fr(!\0061)1961 5259 y Fh(m)2046 5274 y Fq(n)2137 5259 y Fk(=)i Fh(m)2341 5274 y Fr(1)2416 5259 y Fk(\))f(suc)m(h)h(that)e Fh(m)3030 5223 y Fs(u)p Fq(;)p Fs(s)3121 5259 y Fk(\()p Fh(r)3206 5223 y Fs(u)p Fq(;)p Fs(s)3297 5259 y Fk(\))i(=)166 5380 y Fh( )233 5344 y Fs(u)p Fq(;)p Fs(s)324 5380 y Fk(\()p Fh(t)397 5344 y Fs(u)p Fq(;)p Fs(s)488 5380 y Fk(\))27 b(=)h Fh(m)742 5395 y Fs(0)814 5380 y Fk(for)k(some)h (parameters)f Fh(r)1759 5344 y Fs(u)p Fq(;)p Fs(s)1877 5380 y Fh(>)c Fk(0)k(and)h Fh(t)2287 5344 y Fs(u)p Fq(;)p Fs(s)2406 5380 y Fk(=)27 b(ln)16 b Fh(r)2654 5344 y Fs(u)p Fq(;)p Fs(s)2744 5380 y Fk(.)1769 5712 y(7)p eop %%Page: 8 8 8 7 bop 166 83 a Fk(Then)34 b(the)f Fj(L)-5 b(azutkin)-10 b('s)34 b(homo)-5 b(clinic)34 b(invariant)41 b Fk(of)32 b Fh(m)2157 98 y Fs(0)2229 83 y Fk(is)g(de\014ned)i(as)f(the)g(quan)m (tit)m(y)361 305 y Fh(!)t Fk(\()p Fh(m)549 320 y Fs(0)588 305 y Fk(\))28 b(:=)f Fe(f)p Fk(\()925 279 y(_)894 305 y Fh( )961 264 y Fs(u)1005 305 y Fk(\()p Fh(t)1078 264 y Fs(u)1121 305 y Fk(\))p Fh(;)1234 279 y Fk(_)1203 305 y Fh( )1270 264 y Fs(s)1302 305 y Fk(\()p Fh(t)1375 264 y Fs(s)1407 305 y Fk(\)\))g(=)h Fh(r)1661 264 y Fs(u)1704 305 y Fh(r)1751 264 y Fs(s)1783 305 y Fe(f)p Fk(\()h(_)-56 b Fh(m)1978 264 y Fs(u)2021 305 y Fk(\()p Fh(r)2106 264 y Fs(u)2149 305 y Fk(\))p Fh(;)46 b Fk(_)-56 b Fh(m)2316 264 y Fs(s)2348 305 y Fk(\()p Fh(r)2433 264 y Fs(s)2465 305 y Fk(\)\))p Fh(:)166 627 y Fk(It)35 b(do)s(es)h(not)f(dep)s(end)h (on)f(the)g(p)s(oin)m(t)g(of)f(the)i(homo)s(clinic)c(orbit:)i Fh(!)t Fk(\()p Fh(m)2813 642 y Fq(n)2859 627 y Fk(\))e(=)g Fh(!)t Fk(\()p Fh(m)3225 642 y Fs(0)3264 627 y Fk(\))j(for)166 748 y(all)d Fh(n)p Fk(,)h(so)h(that)g(w)m(e)g(can)g(write)f Fh(!)g Fk(=)c Fh(!)t Fk(\()p Fh(O)s Fk(\).)j(Moreo)m(v)m(er,)j Fh(!)i Fk(is)c(in)m(v)-5 b(arian)m(t)33 b(b)m(y)h(symplectic)166 868 y(c)m(hanges)k(of)e(v)-5 b(ariables)35 b(and)h(is)g(prop)s (ortional)e(to)i(the)h(splitting)d(angle.)i(In)g(particular,)166 988 y Fh(!)t Fk(\()p Fh(O)s Fk(\))26 b(=)i(0)g(if)g(and)h(only)f(if)g (the)h(in)m(v)-5 b(arian)m(t)27 b(curv)m(es)k(are)e(tangen)m(t)g(along) f Fh(O)s Fk(.)g(Therefore,)i(it)166 1109 y(is)f(a)h(v)m(ery)h(suitable) e(quan)m(tit)m(y)h(to)f(measure)h(the)h(splitting)c(size)j(on)g(a)f (homo)s(clinic)e(orbit.)166 1330 y(On)33 b(the)g(other)f(hand,)h(the)g Fj(homo)-5 b(clinic)33 b(action)40 b Fk(of)32 b(the)h(orbit)f Fh(O)i Fk(is)e(the)h(quan)m(tit)m(y)361 1552 y Fh(W)14 b Fk([)p Fh(O)s Fk(])27 b(:=)764 1469 y Fg(X)757 1653 y Fq(n)p Fr(2)p Ff(Z)907 1552 y Fk(\()p Fh(L)p Fk(\()p Fh(m)1134 1567 y Fq(n)1182 1552 y Fk(\))22 b Fi(\000)h Fh(L)p Fk(\()p Fh(m)1531 1567 y Fr(1)1606 1552 y Fk(\)\))p Fh(:)166 1945 y Fk(By)29 b(h)m(yp)s(erb)s(olicit)m(y)f(this)g(series)h (is)f(absolutely)f(con)m(v)m(ergen)m(t.)k(No)m(w,)e(let)f(us)h(supp)s (ose)h(that)166 2065 y Fh(O)244 2029 y Fs(+)348 2065 y Fk(=)46 b(\()p Fh(m)593 2029 y Fs(+)593 2090 y Fq(n)652 2065 y Fk(\))690 2080 y Fq(n)p Fr(2)p Ff(Z)871 2065 y Fk(and)e Fh(O)1150 2029 y Fr(\000)1254 2065 y Fk(=)h(\()p Fh(m)1498 2029 y Fr(\000)1498 2090 y Fq(n)1557 2065 y Fk(\))1595 2080 y Fq(n)p Fr(2)p Ff(Z)1777 2065 y Fk(are)e(a)g(couple)g 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Ff(Z)1561 2889 y Fk(\()p Fh(L)p Fk(\()p Fh(m)1788 2848 y Fr(\000)1788 2913 y Fq(n)1848 2889 y Fk(\))22 b Fi(\000)h Fh(L)p Fk(\()p Fh(m)2197 2848 y Fs(+)2197 2913 y Fq(n)2256 2889 y Fk(\)\))p Fh(:)166 3287 y Fk(Finally)-8 b(,)25 b(w)m(e)k(recall)d(that)i(a)f(homo)s(clinic)e(orbit)i Fh(O)j Fk(=)d(\()p Fh(m)2240 3302 y Fq(n)2287 3287 y Fk(\))2325 3302 y Fq(n)p Fr(2)p Ff(Z)2491 3287 y Fk(is)h(called)e Fj(primary)37 b Fk(when)166 3408 y(the)27 b(pieces)h(of)e(the)h(curv)m (es)i Fh(W)1271 3372 y Fs(u)p Fq(;)p Fs(s)1388 3408 y Fk(b)s(et)m(w)m(een)g(the)e(saddle)g(p)s(oin)m(t)f Fh(m)2548 3423 y Fr(1)2650 3408 y Fk(and)h(the)g(homo)s(clinic)166 3528 y(p)s(oin)m(t)37 b Fh(m)511 3543 y Fs(0)588 3528 y Fk(ha)m(v)m(e)i(only)e(their)f(ends)j(in)e(common.)f(These)j(orbits)e (are)h(v)m(ery)g(imp)s(ortan)m(t,)166 3649 y(b)s(ecause)29 b(if)d(there)h(is)g(one)g(homo)s(clinic)d(orbit,)j(there)g(are)h(an)f (in\014nit)m(y)f(of)h(them,)g(but)g(only)166 3769 y(a)34 b(\014nite)g(p)s(ortion)g(of)g(them)g(are)g(primary)-8 b(.)33 b(And)i(this)f(p)s(ortion)g(su\016ces)i(to)e(understand)166 3889 y(the)42 b(structure)h(of)e(the)h(homo)s(clinic)c(tangle.)j(The)h (homo)s(clinic)c(orbits)j(men)m(tioned)g(in)166 4010 y(this)32 b(pap)s(er)h(are)g(alw)m(a)m(ys)g(primary)-8 b(.)166 4355 y Fj(2.2)99 b(The)34 b(r^)-50 b(ole)35 b(of)g(the)f(r)-5 b(eversors)166 4696 y Fk(In)43 b(general,)f(all)e(celebrated)j (area-preserving)g(maps)f(ha)m(v)m(e)i(a)e(v)m(ery)i(useful)f(prop)s (ert)m(y)166 4817 y(whic)m(h)33 b(simpli\014es)d(v)m(ery)k(m)m(uc)m(h)e (the)h(searc)m(h)g(of)f(homo)s(clinic)d(p)s(oin)m(ts.)i(Let)i(us)f (explain)g(it.)166 5038 y(W)-8 b(e)29 b(supp)s(ose)h(that)f(the)g (di\013eomorphism)d Fh(f)40 b Fk(has)29 b(a)f(rev)m(ersor)j Fh(R)q Fk(,)e(that)f(is,)g(an)h(in)m(v)m(olutiv)m(e)166 5158 y(map)j(suc)m(h)i(that)e Fh(f)h Fi(\016)22 b Fh(R)29 b Fk(=)e Fh(R)d Fi(\016)d Fh(f)1401 5122 y Fr(\000)p Fs(1)1496 5158 y Fk(.)32 b(W)-8 b(e)33 b(also)f(assume)h(that)f(the)h Fj(symmetry)i(line)361 5380 y Fk(Fix)o Fi(f)p Fh(R)q Fi(g)28 b Fk(:=)f Fi(f)p Fh(m)h Fi(2)g Fh(M)39 b Fk(:)28 b Fh(R)q Fk(\()p Fh(m)p Fk(\))f(=)h Fh(m)p Fi(g)1769 5712 y Fk(8)p eop %%Page: 9 9 9 8 bop 166 83 a Fk(is)34 b(a)g(smo)s(oth)g(curv)m(e)i(in)e(the)h (surface)g Fh(M)10 b Fk(.)36 b(Finally)-8 b(,)31 b(w)m(e)36 b(supp)s(ose)g(that)e Fh(m)2906 98 y Fr(1)3012 83 y Fi(2)e Fk(Fix)o Fi(f)p Fh(R)q Fi(g)p Fk(.)166 203 y(Then)i(the)f(rev)m(ersor)h (in)m(terc)m(hanges)g(the)f(in)m(v)-5 b(arian)m(t)31 b(curv)m(es:)j Fh(R)q Fk(\()p Fh(W)2638 167 y Fs(u)p Fq(;)p Fs(s)2729 203 y Fk(\))28 b(=)f Fh(W)3004 167 y Fs(s)p Fq(;)p Fs(u)3094 203 y Fk(.)166 424 y(Under)f(these)g (assumptions,)f(the)h(p)s(oin)m(ts)f(on)g(the)g(in)m(tersection)h(of)e (the)i(in)m(v)-5 b(arian)m(t)24 b(curv)m(es)166 544 y(with)30 b(the)h(symmetry)g(line)f(are)g(a)h(sp)s(ecial)e(kind)i(of)f(homo)s (clinic)e(p)s(oin)m(ts,)i(usually)g(called)166 665 y Fj(symmetric)p Fk(.)h(F)-8 b(rom)31 b(a)g(n)m(umerical)g(p)s(oin)m(t)g (of)g(view,)h(the)g(symmetric)f(homo)s(clinic)e(p)s(oin)m(ts)166 785 y(are)g(easier)h(to)f(compute,)g(since)h(the)f(symmetry)h(lines)e (often)i(ha)m(v)m(e)g(closed)g(expressions.)166 905 y(F)-8 b(urthermore,)32 b(if)f(a)h(rev)m(ersible)g(di\013eomorphism)e(has)j (an)f(in)m(v)-5 b(arian)m(t)31 b(curv)m(e)i(transv)m(erse)166 1026 y(to)i(a)g(symmetry)g(line,)f(then)i(the)g(asso)s(ciated)f (symmetric)g(homo)s(clinic)d(p)s(oin)m(ts)j(p)s(ersist)166 1146 y(under)e(small)e(rev)m(ersible)i(p)s(erturbations)f(of)g(the)h (di\013eomorphism.)166 1367 y(On)42 b(the)h(other)f(hand,)g(if)f Fh(R)i Fk(is)f(a)f(rev)m(ersor)j(of)d Fh(f)11 b Fk(,)42 b(then)h Fh(f)c Fi(\016)28 b Fh(R)43 b Fk(is)f(another)g(rev)m(ersor.) 166 1487 y(Therefore,)i(the)f(rev)m(ersors)i(\(and)e(so)g(their)g (symmetric)f(homo)s(clinic)e(orbits\))i(app)s(ear)166 1607 y(in)d(natural)f(couples.)j(That)e(is,)h(if)e(a)i(rev)m(ersible)g (map)f(has)h(a)f(symmetric)g(homo)s(clinic)166 1728 y(orbit)33 b Fh(O)484 1692 y Fs(+)576 1728 y Fk(passing)h(b)m(y)g(a)g(p)s(oin)m(t) f Fh(m)1480 1687 y Fs(+)1480 1750 y(0)1569 1728 y Fi(2)d Fk(Fix)p Fi(f)p Fh(R)q Fi(g)p Fk(,)j(usually)g(it)g(has)i(another)e (symmetric)166 1848 y(homo)s(clinic)23 b(orbit)j Fh(O)961 1812 y Fr(\000)1045 1848 y Fk(passing)h(b)m(y)g(a)f(p)s(oin)m(t)g Fh(m)1920 1807 y Fr(\000)1920 1870 y Fs(0)2007 1848 y Fi(2)i Fk(Fix)o Fi(f)p Fh(f)21 b Fi(\016)10 b Fh(R)q Fi(g)p Fk(.)26 b(Then)h(the)g(Lazutkin's)166 1969 y(homo)s(clinic)19 b(in)m(v)-5 b(arian)m(ts)22 b Fh(!)1149 1932 y Fr(\006)1235 1969 y Fk(=)27 b Fh(!)t Fk(\()p Fh(O)1519 1932 y Fr(\006)1577 1969 y Fk(\))22 b(and)h(the)f(lob)s(e)g(area)g Fh(A)g Fk(enclosed)h(b)m(y)h(these)f(orbits)166 2089 y(are)28 b(the)g(most)f(natural)g(quan)m(tities)h(to)f(measure)h(the)g (splitting)e(size.)i(W)-8 b(e)28 b(presen)m(t)i(some)166 2209 y(examples)j(in)e(the)i(next)h(subsections.)166 2552 y Fj(2.3)99 b(The)34 b(standar)-5 b(d)35 b(map)f(and)g(the)h(H)n (\023)-47 b(enon)34 b(map)166 2893 y Fk(Probably)-8 b(,)32 b(the)h(most)f(celebrated)h(example)f(is)h(the)g Fj(standar)-5 b(d)34 b(map)k Fk(de\014ned)c(b)m(y)361 3114 y Fh(S)6 b(M)38 b Fk(:)28 b Fe(T)677 3073 y Fs(2)748 3114 y Fi(!)f Fe(T)938 3073 y Fs(2)981 3114 y Fh(;)212 b(S)6 b(M)k Fk(\()p Fh(x;)17 b(y)t Fk(\))27 b(=)h(\()p Fh(x)22 b Fk(+)g Fh(y)j Fk(+)d Fh(\017)17 b Fk(sin)g Fh(x;)g(y)25 b Fk(+)d Fh(\017)17 b Fk(sin)g Fh(x)p Fk(\))p Fh(:)166 3435 y Fk(It)32 b(is)e(area-preserving)i(and)f(rev)m(ersible.)h(The)h (map)d Fh(R)q Fk(\()p Fh(x;)17 b(y)t Fk(\))27 b(=)h(\(2)p Fh(\031)23 b Fi(\000)d Fh(x;)d(y)23 b Fk(+)d Fh(\017)d Fk(sin)f Fh(x)p Fk(\))32 b(is)166 3556 y(one)k(of)g(the)g(rev)m (ersors,)i(b)s(eing)d Fi(f)p Fh(x)e Fk(=)g Fh(\031)t Fi(g)j Fk(its)f(symmetry)h(line.)f(If)h Fh(\017)d(>)h Fk(0,)h(the)h(origin)e(is)166 3676 y(h)m(yp)s(erb)s(olic)24 b(and)h(its)f(c)m(haracteristic)g(exp)s(onen)m(t)i Fh(h)f Fk(is)f(determined)g(b)m(y)i Fh(\017)i Fk(=)f(4)17 b(sinh)3158 3633 y Fs(2)3197 3676 y Fk(\()p Fh(h=)p Fk(2\).)166 3797 y(Let)24 b Fh(!)397 3760 y Fs(+)480 3797 y Fk(and)g Fh(!)726 3760 y Fr(\000)808 3797 y Fk(b)s(e)h(the)f(Lazutkin's)h(in)m(v)-5 b(arian)m(ts)23 b(of)h(the)g(symmetric)f(homo)s(clinic)e(orbits)166 3917 y(passing)43 b(b)m(y)h(the)g(\014rst)g(in)m(tersection)f(of)f(the) i(in)m(v)-5 b(arian)m(t)42 b(curv)m(es)j(with)e(the)g(symmetry)166 4037 y(lines)38 b(of)g(the)h(rev)m(ersors)h Fh(R)g Fk(and)f Fh(S)6 b(M)36 b Fi(\016)26 b Fh(R)q Fk(,)39 b(resp)s(ectiv)m(ely)-8 b(.)40 b(Let)e Fh(A)h Fk(b)s(e)g(the)g(area)f(of)g(the)166 4158 y(lob)s(e)h(enclosed)i(b)s(et)m(w)m(een)h(the)f(couple)f(of)f (symmetric)h(homo)s(clinic)c(orbits.)k(Then)h(the)166 4278 y(follo)m(wing)30 b(asymptotic)i(form)m(ulae)f(hold)361 4499 y Fh(!)426 4458 y Fr(\006)519 4447 y Fs(as)512 4499 y Fk(=)d Fi(\006)p Fk(4)p Fh(\031)t(h)857 4458 y Fr(\000)p Fs(2)968 4496 y Fk(e)1011 4458 y Fr(\000)p Fq(\031)1109 4435 y Fp(2)1144 4458 y Fq(=h)1242 4416 y Fg(X)1240 4599 y Fq(j)t Fr(\025)p Fs(0)1380 4499 y Fh(!)1441 4514 y Fq(j)1477 4499 y Fh(h)1533 4458 y Fs(2)p Fq(j)1605 4499 y Fh(;)212 b(A)1951 4447 y Fs(as)1944 4499 y Fk(=)28 b(2)p Fh(\031)2156 4458 y Fr(\000)p Fs(1)2267 4496 y Fk(e)2310 4458 y Fr(\000)p Fq(\031)2408 4435 y Fp(2)2442 4458 y Fq(=h)2540 4416 y Fg(X)2539 4599 y Fq(j)t Fr(\025)p Fs(0)2678 4499 y Fh(!)2739 4514 y Fq(j)2775 4499 y Fh(h)2831 4458 y Fs(2)p Fq(j)166 4898 y Fk(for)38 b(some)g(real)f(co)s(e\016cien) m(ts)i Fh(!)1326 4913 y Fq(j)1401 4898 y Fk(that)f(can)g(b)s(e)h (determined)f(through)g(some)g(auxiliary)166 5019 y(problems)28 b(indep)s(enden)m(t)i(of)e Fh(\017)p Fk(.)i(These)g(asymptotic)e(form)m (ulae)g(w)m(ere)i(\014rst)f(stated)h(in)e([9].)166 5139 y(A)j(complete)g(pro)s(of)g(can)g(b)s(e)h(found)f(in)g([10].)g(The)i (fact)e(that)g(the)h(quan)m(tities)f(2)p Fh(\031)3164 5103 y Fs(2)3203 5139 y Fh(h)3259 5103 y Fr(\000)p Fs(2)3353 5139 y Fh(A)p Fk(,)166 5259 y Fh(!)231 5223 y Fs(+)289 5259 y Fk(,)d(and)g Fi(\000)p Fh(!)671 5223 y Fr(\000)758 5259 y Fk(ha)m(v)m(e)h(the)f(same)g(asymptotic)f(expansion)h(can)g(b)s (e)g(understo)s(o)s(d)g(with)g(an)166 5380 y(argumen)m(t)k(based)i(on)e (the)h(splitting)e(function,)h(see)i Fi(x)p Fk(2.5.)1769 5712 y(9)p eop %%Page: 10 10 10 9 bop 166 83 a Fk(The)38 b(\014rst)f(asymptotic)f(co)s(e\016cien)m (t)h Fh(!)1602 98 y Fs(0)1676 83 y Fi(')f Fk(1118)p Fh(:)p Fk(827706)e(is)i(the)h Fj(L)-5 b(azutkin)-10 b('s)39 b(c)-5 b(onstant)p Fk(.)166 203 y(Sev)m(eral)27 b(h)m(undreds)i(of)e (asymptotic)f(co)s(e\016cien)m(ts)i Fh(!)2062 218 y Fq(k)2132 203 y Fk(w)m(ere)g(computed)f(b)m(y)h(C.)g(Sim\023)-49 b(o.)25 b(His)166 324 y(exp)s(erimen)m(ts)30 b(suggest)h(that)e(the)h (series)1680 257 y Fg(P)1767 345 y Fq(j)t Fr(\025)p Fs(0)1911 324 y Fh(!)1972 339 y Fq(j)2008 324 y Fh(h)2064 288 y Fs(2)p Fq(j)2165 324 y Fk(is)g(Gevrey-1)g(of)f(t)m(yp)s(e)h Fh(\032)e Fk(=)g(1)p Fh(=)p Fk(2)p Fh(\031)3388 288 y Fs(2)3427 324 y Fk(.)166 544 y(Another)33 b(celebrated)g (area-preserving)g(map)f(is)g(the)h Fj(H)n(\023)-47 b(enon)33 b(map)361 756 y Fh(H)8 b(M)38 b Fk(:)28 b Fe(R)703 715 y Fs(2)776 756 y Fi(!)f Fe(R)969 715 y Fs(2)1015 756 y Fh(;)211 b(H)8 b(M)i Fk(\()p Fh(x;)17 b(y)t Fk(\))28 b(=)f(\()p Fh(x)c Fk(+)f Fh(y)j Fk(+)d Fh(\017x)p Fk(\(1)h Fi(\000)f Fh(x)p Fk(\))p Fh(;)17 b(y)26 b Fk(+)c Fh(\017x)p Fk(\(1)g Fi(\000)h Fh(x)p Fk(\)\))p Fh(:)166 1068 y Fk(As)30 b(b)s(efore,)g(the)g(origin)e(is)h(a)g(h)m(yp)s(erb)s(olic)g(\014xed)i (p)s(oin)m(t)e(if)f Fh(\017)g(>)g Fk(0.)h(The)i(relation)d(b)s(et)m(w)m (een)166 1188 y(the)35 b(c)m(haracteristic)f(exp)s(onen)m(t)h Fh(h)g Fk(and)f Fh(\017)h Fk(is)f(the)g(same)g(than)h(in)e(the)i (previous)f(sample.)166 1309 y(The)28 b(H)m(\023)-46 b(enon)28 b(map)e(is)h(rev)m(ersible,)h Fh(R)q Fk(\()p Fh(x;)17 b(y)t Fk(\))26 b(=)i(\()p Fh(x)11 b Fi(\000)g Fh(y)t(;)17 b Fi(\000)p Fh(y)t Fk(\))26 b(is)h(a)g(distinguished)f(rev) m(ersor,)166 1429 y(and)k Fi(f)p Fh(y)g Fk(=)e(0)p Fi(g)h Fk(is)g(the)h(symmetry)g(line.)e(The)i(Lazutkin's)g(in)m(v)-5 b(arian)m(ts)29 b(of)g(the)h(asso)s(ciated)166 1550 y(couple)j(of)f (symmetric)f(homo)s(clinic)f(orbits)i(v)m(erify)h(the)g(asymptotic)f (form)m(ulae)361 1762 y Fh(!)426 1721 y Fr(\006)519 1709 y Fs(as)512 1762 y Fk(=)c Fi(\006)p Fk(4)p Fh(\031)t(h)857 1721 y Fr(\000)p Fs(6)968 1759 y Fk(e)1011 1721 y Fr(\000)p Fs(2)p Fq(\031)1144 1697 y Fp(2)1179 1721 y Fq(=h)1277 1679 y Fg(X)1276 1862 y Fq(j)t Fr(\025)p Fs(0)1415 1762 y Fh(!)1476 1777 y Fq(j)1512 1762 y Fh(h)1568 1721 y Fs(2)p Fq(j)1640 1762 y Fh(:)166 2152 y Fk(This)37 b(asymptotic)e (series)i(has)g(nothing)e(to)h(do)h(with)f(the)g(one)h(of)f(the)h (standard)f(map.)166 2272 y(I)c(do)f(not)h(kno)m(w)g(references)i(with) d(a)g(complete)g(pro)s(of)g(of)g(this)g(asymptotic)g(expansion.)166 2393 y(The)44 b(\014rst)f(asymptotic)f(co)s(e\016cien)m(t)h Fh(!)1626 2408 y Fs(0)1710 2393 y Fi(')i Fk(2474425)p Fh(:)p Fk(5935525)39 b(can)k(b)s(e)g(found)g(in)f([3].)166 2513 y(The)g(fact)e(that)h Fh(!)856 2528 y Fs(0)936 2513 y Fi(6)p Fk(=)h(0)e(w)m(as)i(analytically)37 b(established)k(in)f ([11].)h(Sev)m(eral)g(h)m(undreds)166 2633 y(of)f(digits)f(of)i Fh(!)740 2648 y Fs(0)819 2633 y Fk(are)g(listed)f(in)g([20].)g(Some)g (n)m(umerical)f(exp)s(erimen)m(ts)j(p)s(erformed)e(b)m(y)166 2754 y(C.)33 b(Sim\023)-49 b(o)31 b(suggest)i(that)g(this)f(new)i (asymptotic)d(series)j(is)e(also)f(Gevrey-1.)166 3089 y Fj(2.4)99 b(McMil)5 b(lan)35 b(maps)f(and)g(Melnikov)g(metho)-5 b(ds)166 3429 y Fk(No)m(w)52 b(w)m(e)h(presen)m(t)h(a)d(qualitativ)m (ely)f(di\013eren)m(t)i(kind)g(of)f(example.)h(Concretely)-8 b(,)53 b(w)m(e)166 3549 y(study)47 b(p)s(erturbations)e(of)g(some)g(in) m(tegrable)g(standard-lik)m(e)f(maps)i(\014rst)f(in)m(tro)s(duced)166 3670 y(b)m(y)38 b(McMillan)e([18].)g(Although)h(these)h(p)s(erturb)s (ed)g(maps)f(are)g(b)m(y)h(far)f(less)g(celebrated)166 3790 y(than)45 b(the)h(standard)f(or)g(the)g(H)m(\023)-46 b(enon)46 b(maps,)f(they)h(ha)m(v)m(e)g(the)g(follo)m(wing)c(in)m (teresting)166 3910 y(prop)s(ert)m(y)-8 b(.)41 b(They)g(dep)s(end)g(on) f(t)m(w)m(o)h(parameters:)f(the)g(p)s(erturbation)g(strength)g Fh(\017)h Fk(and)166 4031 y(the)29 b(c)m(haracteristic)f(exp)s(onen)m (t)i Fh(h)f Fk(of)f(the)h(origin.)d(F)-8 b(or)28 b Fh(\017)g Fk(=)f(0,)i(they)g(are)g(in)m(tegrable)e(with)166 4151 y(a)37 b(separatrix)g(to)g(the)h(origin,)e(whereas)i(they)h(asymptote)e (to)g(\015o)m(ws)i(with)e(homo)s(clinic)166 4272 y(connections)29 b(as)g Fh(h)e Fi(!)h Fk(0)1064 4235 y Fs(+)1122 4272 y Fk(.)h(Moreo)m(v)m(er,)h(some)e(explicit)f(exp)s(onen)m(tially)g (small)f(estimates)166 4392 y(of)35 b(the)g(splitting)e(size)j(can)f(b) s(e)g(obtained)g(using)g(a)g(discrete)g(v)m(ersion)h(of)f(the)g(Melnik) m(o)m(v)166 4512 y(metho)s(d.)25 b(It)g(represen)m(ts)i(a)e(strong)g (coincidence)g(with)g(the)g(billiard)d(maps)i(here)i(studied.)166 4732 y(T)-8 b(o)33 b(b)s(egin)f(with,)g(let)g(us)h(consider)g(the)g (area-preserving)f(map)361 5001 y Fh(f)39 b Fk(:)27 b Fe(R)568 4960 y Fs(2)641 5001 y Fi(!)h Fe(R)835 4960 y Fs(2)880 5001 y Fh(;)212 b(f)11 b Fk(\()p Fh(x;)17 b(y)t Fk(\))26 b(=)1535 4855 y Fg( )1601 5001 y Fh(y)t(;)17 b Fi(\000)p Fh(x)22 b Fk(+)1989 4934 y(2)p Fh(\026)2097 4949 y Fs(0)2136 4934 y Fh(y)p 1959 4978 260 4 v 1959 5070 a Fk(1)f(+)h Fh(y)2179 5041 y Fs(2)2250 5001 y Fk(+)g Fh(\017V)2466 4960 y Fr(0)2489 5001 y Fk(\()p Fh(y)t Fk(\))2617 4855 y Fg(!)166 5380 y Fk(where)j Fh(V)d Fk(\()p Fh(y)t Fk(\))27 b(=)776 5313 y Fg(P)864 5401 y Fq(n)p Fr(\025)p Fs(1)1017 5380 y Fh(V)1074 5395 y Fq(n)1121 5380 y Fh(y)1173 5344 y Fs(2)p Fq(n)1279 5380 y Fk(is)c Fj(any)33 b Fk(ev)m(en)25 b(en)m(tire)f(p)s(erturbation.)g(If)g Fh(\026)j Fk(:=)h Fh(\026)2987 5395 y Fs(0)3030 5380 y Fk(+)5 b Fh(\017V)3207 5395 y Fs(1)3274 5380 y Fh(>)28 b Fk(1,)1745 5712 y(10)p eop %%Page: 11 11 11 10 bop 166 83 a Fk(the)30 b(origin)c(is)j(a)g(h)m(yp)s(erb)s(olic)f (p)s(oin)m(t)g(whose)j(c)m(haracteristic)e(exp)s(onen)m(t)h Fh(h)f Fk(is)g(determined)166 203 y(b)m(y)40 b(the)g(relation)d Fh(\026)i Fk(=)g(cosh)18 b Fh(h)p Fk(.)39 b(This)h(map)e(is)h(rev)m (ersible)h(and)f Fh(R)q Fk(\()p Fh(x;)17 b(y)t Fk(\))39 b(=)g(\()p Fh(y)t(;)17 b(x)p Fk(\))38 b(is)h(a)166 324 y(rev)m(ersor)f(whose)g(symmetry)e(line)f(is)h(the)h(bisectrix)f Fi(f)p Fh(y)i Fk(=)c Fh(x)p Fi(g)p Fk(.)i(Let)h Fh(A)f Fk(b)s(e)h(the)g(area)f(of)166 444 y(the)g(region)e(enclosed)i(b)m(y)g (the)g(in)m(v)-5 b(arian)m(t)33 b(curv)m(es)38 b(b)s(et)m(w)m(een)f (the)f(couple)f(of)f(symmetric)166 565 y(homo)s(clinic)29 b(orbits)j(con)m(tained)h(in)e(the)i(\014rst)g(quadran)m(t.)g(The)2502 574 y(O)2578 565 y(\()p Fh(\017)p Fk(\)-term)e(of)h Fh(A)h Fk(can)f(b)s(e)166 685 y(computed)h(using)f(standard)h(ideas)g(in)e (Melnik)m(o)m(v)i(theory:)361 907 y Fh(A)28 b Fk(=)f Fh(A)p Fk(\()p Fh(h;)17 b(\017)p Fk(\))28 b(=)g Fh(\017A)1097 922 y Fs(1)1137 907 y Fk(\()p Fh(h)p Fk(\))22 b(+)1389 916 y(O)1465 907 y(\()p Fh(\017)1542 866 y Fs(2)1582 907 y Fk(\))p Fh(;)211 b(A)1931 922 y Fs(1)1971 907 y Fk(\()p Fh(h)p Fk(\))28 b(=)2234 904 y(e)2278 866 y Fr(\000)p Fq(\031)2376 843 y Fp(2)2410 866 y Fq(=h)2507 811 y Fg(\020)2556 907 y Fk(8)p Fh(\031)2680 875 y Fg(b)2664 907 y Fh(V)22 b Fk(\(2)p Fh(\031)t Fk(\))f(+)3046 916 y(O)3122 907 y(\()p Fh(h)3216 866 y Fs(2)3256 907 y Fk(\))3294 811 y Fg(\021)166 1261 y Fk(where)465 1228 y Fg(b)449 1261 y Fh(V)g Fk(\()p Fh(\030)5 b Fk(\))28 b(=)784 1194 y Fg(P)872 1281 y Fq(n)p Fr(\025)p Fs(1)1025 1261 y Fh(V)1082 1276 y Fq(n)1129 1261 y Fh(\030)1177 1225 y Fs(2)p Fq(n)p Fr(\000)p Fs(1)1349 1261 y Fh(=)p Fk(\(2)p Fh(n)22 b Fi(\000)h Fk(1\)!)33 b(is)g(the)h(Borel)e(transform)h(of)f Fh(V)22 b Fk(\()p Fh(y)t Fk(\),)32 b(see)j([5].)166 1381 y(Th)m(us,)25 b(the)g(Melnik)m(o)m(v)f(term)f Fh(\017A)1338 1396 y Fs(1)1378 1381 y Fk(\()p Fh(h)p Fk(\))h(giv)m(es)g(the)g(righ)m (t)f(asymptotic)g(b)s(eha)m(vior)g(of)h Fh(A)g Fk(when)166 1501 y Fh(h)29 b Fk(is)g(\014xed,)i Fh(A)679 1516 y Fs(1)718 1501 y Fk(\()p Fh(h)p Fk(\))d Fi(6)p Fk(=)f(0,)j(and)f Fh(\017)f Fi(!)f Fk(0.)i(On)h(the)f(con)m(trary)-8 b(,)30 b(when)g Fh(h)e Fi(!)g Fk(0)2814 1465 y Fs(+)2902 1501 y Fk(the)h(Melnik)m(o)m(v)166 1622 y(prediction)j(is,)g(at)g(a)g (\014rst)h(glance,)g(useless,)h(unless)f Fh(\017)g Fk(is)f(exp)s(onen)m (tially)g(small)e(in)i Fh(h)p Fk(.)166 1843 y(The)i(pap)s(ers)f([5])f (and)h([6])f(are)h(dev)m(oted)h(to)e(clarify)f(the)i(asymptotic)f(b)s (eha)m(vior)g(of)g Fh(A)g Fk(as)166 1963 y Fh(h)c Fi(!)f Fk(0)426 1927 y Fs(+)485 1963 y Fk(.)33 b(In)g(the)g(\014rst)g(pap)s (er,)f(it)g(is)g(analytically)e(established)i(that)h(if)e Fh(p)d(>)f Fk(6)33 b(then)361 2186 y Fh(A)28 b Fk(=)f Fh(\017)621 2183 y Fk(e)665 2145 y Fr(\000)p Fq(\031)763 2121 y Fp(2)797 2145 y Fq(=h)894 2090 y Fg(\020)944 2186 y Fk(8)p Fh(\031)1067 2153 y Fg(b)1052 2186 y Fh(V)21 b Fk(\(2)p Fh(\031)t Fk(\))g(+)1433 2195 y(O)1509 2186 y(\()p Fh(h)1603 2145 y Fs(2)1643 2186 y Fk(\))1681 2090 y Fg(\021)1942 2186 y Fk(\()p Fh(h)28 b Fi(!)f Fk(0)2240 2145 y Fs(+)2299 2186 y Fh(;)17 b(\017)28 b Fk(=)f Fh(h)2569 2145 y Fq(p)2609 2186 y Fk(\))p Fh(:)166 2509 y Fk(On)h(the)g(other)g (hand,)g(the)h(main)d(conclusion)h(of)g(the)i(n)m(umeric)e(exp)s (erimen)m(ts)h(presen)m(ted)166 2629 y(in)k(the)h(second)h(pap)s(er)e (is)g(that)h(if)e Fh(V)1514 2593 y Fr(0)1537 2629 y Fk(\()p Fh(y)t Fk(\))c(=)g Fh(y)36 b Fk(or)c Fh(V)2077 2593 y Fr(0)2100 2629 y Fk(\()p Fh(y)t Fk(\))27 b(=)g Fh(y)2410 2593 y Fs(3)2481 2629 y Fk(then)361 2861 y Fh(A)468 2809 y Fs(as)462 2861 y Fk(=)g Fh(\017)621 2858 y Fk(e)665 2820 y Fr(\000)p Fq(\031)763 2797 y Fp(2)797 2820 y Fq(=h)895 2778 y Fg(X)894 2961 y Fq(j)t Fr(\025)p Fs(0)1033 2861 y Fh(\013)1096 2820 y Fq(\017)1095 2886 y(j)1132 2861 y Fh(h)1188 2820 y Fs(2)p Fq(j)1455 2861 y Fk(\()p Fh(h)g Fi(!)h Fk(0)1753 2820 y Fs(+)1812 2861 y Fh(;)17 b(\017)32 b Fk(\014xed)q(\))166 3286 y(where)448 3219 y Fg(P)535 3307 y Fq(j)t Fr(\025)p Fs(0)679 3286 y Fh(\013)742 3250 y Fq(\017)741 3310 y(j)777 3286 y Fh(h)833 3250 y Fs(2)p Fq(j)938 3286 y Fk(is)g(Gevrey-1)h(of)f(t)m(yp)s(e)h Fh(\032)28 b Fk(=)g(1)p Fh(=)p Fk(2)p Fh(\031)2172 3250 y Fs(2)2243 3286 y Fk(and)33 b Fh(\013)2496 3250 y Fq(\017)2495 3310 y Fs(0)2562 3286 y Fk(=)28 b(8)p Fh(\031)2789 3254 y Fg(b)2774 3286 y Fh(V)21 b Fk(\(2)p Fh(\031)t Fk(\))g(+)3156 3295 y(O)3231 3286 y(\()p Fh(\017)p Fk(\).)166 3633 y Fj(2.5)99 b(The)34 b(splitting)h(function)g(and)f(the)h(splitting)f(p) -5 b(otential)166 3974 y Fk(W)d(e)39 b(describ)s(e)h(brie\015y)f(some)f (ideas)h(originally)c(prop)s(osed)k(b)m(y)h(Lazutkin,)f(whic)m(h)g(w)m (ere)166 4095 y(the)28 b(\014rst)g(step)h(to)e(pro)m(v)m(e)i(the)f(exp) s(onen)m(tially)e(smallness)h(of)g(some)g(splitting)f(quan)m(tities.) 166 4215 y(Besides,)44 b(they)h(help)e(to)g(understand)h(the)g(close)f (relation)f(among)g(sev)m(eral)i(splitting)166 4335 y(quan)m(tities.)30 b(Although)g(these)i(ideas)e(are)g(semi-heuristic)f(in)h(the)h(most)f (general)f(frame,)166 4456 y(their)j(v)-5 b(alidit)m(y)31 b(has)i(b)s(een)g(established)g(rigorously)e(in)g(some)i(concrete)h (cases.)166 4677 y(After)46 b(Lazutkin,)h(the)f(standard)h(w)m(a)m(y)g (to)f(pro)m(v)m(e)i(the)e(exp)s(onen)m(tial)g(smallness)f(in)h Fh(h)166 4797 y Fk(of)41 b(the)h(ab)s(o)m(v)m(e-men)m(tioned)g(quan)m (tities)f(is)g(to)g(construct)i(a)e(real)f(analytic)h Fh(h)p Fk(-p)s(erio)s(dic)166 4918 y(function)32 b(\011\()p Fh(t)p Fk(\),)h(called)e(the)i Fj(splitting)i(function)p Fk(,)d(whose)i(main)d(prop)s(erties)h(are:)199 5139 y(\(1\))48 b(Its)24 b(ro)s(ots)g(are)f(in)g(1-to-1)f(corresp)s(ondence)k(with)e (the)g(primary)e(homo)s(clinic)f(p)s(oin)m(ts.)199 5259 y(\(2\))48 b(The)39 b(area)f(of)g(the)g(lob)s(e)g(enclosed)g(b)m(y)i (the)e(separatrices)h(b)s(et)m(w)m(een)h(t)m(w)m(o)f(primary)372 5380 y(homo)s(clinic)21 b(p)s(oin)m(ts)k(is)f(equal)g(to)g(its)g(in)m (tegral)f(b)s(et)m(w)m(een)k(the)e(corresp)s(onding)f(ro)s(ots.)1745 5712 y(11)p eop %%Page: 12 12 12 11 bop 199 83 a Fk(\(3\))48 b(The)27 b(Lazutkin's)g(in)m(v)-5 b(arian)m(t)25 b(of)h(a)g(primary)e(homo)s(clinic)g(orbit)h(is)g(equal) i(to)e(its)h(\014rst)372 203 y(deriv)-5 b(ativ)m(e)32 b(at)g(the)h(corresp)s(onding)g(ro)s(ot.)199 324 y(\(4\))48 b(It)33 b(has)g(zero)g(mean:)1145 253 y Fg(R)1200 279 y Fq(h)1184 349 y Fs(0)1262 324 y Fk(\011\()p Fh(t)p Fk(\))17 b(d)p Fh(t)27 b Fk(=)h(0.)166 559 y(Next,)44 b(this)e(function)h(is)f(analytically)e(extended)45 b(to)e(a)f(complex) h(strip)f(of)h(the)g(form)166 679 y(\005)239 694 y Fq(\016)319 679 y Fk(=)f Fi(f)p Fh(t)f Fi(2)i Fe(C)67 b Fk(:)42 b Fi(j=)p Fh(t)p Fi(j)g Fh(<)f(\016)t Fi(g)g Fk(for)f(some)h(p)s(ositiv)m (e)f(width)h Fh(\016)k Fk(and)c(it)f(is)g(b)s(ounded)i(in)e(a)166 799 y(bit)33 b(narro)m(w)m(er)i(strips,)f(usually)f(of)g(width)h Fh(\016)27 b Fi(\000)c Fh(h)p Fk(.)34 b(Hence,)i(its)d(F)-8 b(ourier)33 b(co)s(e\016cien)m(ts)i(are)166 920 y(exp)s(onen)m(tially)k (small,)e(and)j(so)g(the)g(splitting)d(quan)m(tities)j(also)f(are.)g (In)i(man)m(y)e(cases,)166 1040 y(these)32 b(quan)m(tities)e(ha)m(v)m (e)i(order)f Fh(h)1394 1004 y Fq(\014)1458 1037 y Fk(e)1501 1004 y Fr(\000)p Fs(2)p Fq(\031)r(\016)r(=h)1748 1040 y Fk(,)g(for)f(some)g Fh(\014)6 b Fk(.)30 b(W)-8 b(e)31 b(note)g(that)g Fh(\016)g Fk(=)d Fh(\031)34 b Fk(for)c(the)166 1160 y(H)m(\023)-46 b(enon)33 b(map,)g(whereas)h Fh(\016)e Fk(=)c Fh(\031)t(=)p Fk(2)k(for)h(the)g(standard)g(map,)g(the)g (McMillan)e(p)s(erturb)s(ed)166 1281 y(maps,)h(and)h(our)f(billiard)e (maps.)166 1516 y(Using)23 b(that)f(\011\()p Fh(t)p Fk(\))h(has)g(zero) h(mean,)e(w)m(e)i(\014nd)g(another)f Fh(h)p Fk(-p)s(erio)s(dic)e (function)h(\002\()p Fh(t)p Fk(\),)h(called)166 1636 y(the)39 b Fj(splitting)i(p)-5 b(otential)p Fk(,)38 b(suc)m(h)j(that)d (\011\()p Fh(t)p Fk(\))h(=)f(\002)2009 1600 y Fr(0)2032 1636 y Fk(\()p Fh(t)p Fk(\).)h(W)-8 b(e)39 b(can)g(c)m(hose)i(\002\()p Fh(t)p Fk(\))e(in)f(suc)m(h)i(a)166 1756 y(w)m(a)m(y)35 b(that)f(it)g(has)g(zero)h(mean.)e(Besides,)j(when)f(the)g(map)e(is)h (rev)m(ersible,)g(the)h(splitting)166 1877 y(p)s(oten)m(tial)47 b(is)h(ev)m(en)i(and,)f(normalizing)c(it,)i(one)i(can)g(imp)s(ose)e (that)i(the)f(symmetric)166 1997 y(primary)31 b(homo)s(clinic)f(orbits) i(are)g(lo)s(cated)g(at)g Fh(t)c Fk(=)g(0)k(and)h Fh(t)28 b Fk(=)f Fh(h=)p Fk(2.)32 b(Hence,)361 2247 y Fh(A)c Fk(=)f(\002\()p Fh(h=)p Fk(2\))22 b Fi(\000)h Fk(\002\(0\))p Fh(;)211 b(!)1497 2206 y Fs(+)1583 2247 y Fk(=)28 b(\002)1763 2206 y Fr(00)1805 2247 y Fk(\(0\))p Fh(;)211 b(!)2233 2206 y Fr(\000)2319 2247 y Fk(=)28 b(\002)2499 2206 y Fr(00)2541 2247 y Fk(\()p Fh(h=)p Fk(2\))p Fh(:)166 2611 y Fk(W)-8 b(e)33 b(write)f(the)h(F)-8 b(ourier)32 b(expansion)h(of)f (the)h(splitting)d(p)s(oten)m(tial)h(as)361 2860 y(\002\()p Fh(t)p Fk(\))d(=)f Fi(\000)766 2821 y Fs(1)p 766 2837 36 4 v 766 2895 a(2)829 2794 y Fg(P)916 2881 y Fq(n)p Fr(\025)p Fs(1)1070 2860 y Fk(\002)1146 2875 y Fq(n)1209 2860 y Fk(cos)q(\()p Fh(T)14 b(nt)p Fk(\))p Fh(;)166 3224 y Fk(where)44 b Fh(T)60 b Fk(=)45 b(2)p Fh(\031)t(=h)p Fk(.)e(In)h(what)f(follo)m(ws,)f(w)m(e)i(supp)s(ose)h(that)e(\002)2554 3239 y Fq(n)2644 3224 y Fk(has)g(order)3094 3221 y(e)3137 3188 y Fr(\000)p Fs(2)p Fq(\031)r(n\016)r(=h)3427 3224 y Fk(,)166 3344 y(whic)m(h)34 b(is)g(rather)g(natural,)f(since)h(the)g (splitting)e(p)s(oten)m(tial)g(is)h Fh(h)p Fk(-p)s(erio)s(dic)f(and)i (can)g(b)s(e)166 3465 y(extended)25 b(to)d(the)h(strip)f(\005)1129 3480 y Fq(\016)1167 3465 y Fk(.)g(Then)i(w)m(e)f(can)g(appro)m(ximate)e (the)i(\014rst)g(F)-8 b(ourier)22 b(co)s(e\016cien)m(ts)166 3585 y(of)32 b(\002\()p Fh(t)p Fk(\))h(in)e(terms)i(of)f(the)h (splitting)d(quan)m(tities)j Fh(!)2062 3549 y Fr(\006)2153 3585 y Fk(and)f Fh(A)p Fk(.)h(Firstly)-8 b(,)31 b(w)m(e)j(note)f(that) 361 3835 y Fh(A)28 b Fk(=)572 3752 y Fg(X)565 3935 y Fq(n)p Fr(\025)p Fs(0)715 3835 y Fk(\002)791 3850 y Fs(2)p Fq(n)p Fs(+1)963 3835 y Fh(;)212 b Fk(\006)28 b(=)g(4)1476 3752 y Fg(X)1469 3935 y Fq(n)p Fr(\025)p Fs(0)1619 3835 y Fh(n)1677 3794 y Fs(2)1716 3835 y Fk(\002)1792 3850 y Fs(2)p Fq(n)1875 3835 y Fh(;)211 b Fk(\001)28 b(=)2332 3752 y Fg(X)2326 3935 y Fq(n)p Fr(\025)p Fs(0)2459 3835 y Fk(\(2)p Fh(n)22 b Fk(+)g(1\))2811 3794 y Fs(2)2850 3835 y Fk(\002)2926 3850 y Fs(2)p Fq(n)p Fs(+1)3098 3835 y Fh(;)166 4285 y Fk(where)34 b(\006)28 b(:=)g(\()p Fh(!)780 4249 y Fs(+)860 4285 y Fk(+)22 b Fh(!)1023 4249 y Fr(\000)1081 4285 y Fk(\))p Fh(=T)1239 4249 y Fs(2)1311 4285 y Fk(and)32 b(\001)c(:=)g(\()p Fh(!)1843 4249 y Fs(+)1924 4285 y Fi(\000)22 b Fh(!)2088 4249 y Fr(\000)2147 4285 y Fk(\))p Fh(=T)2305 4249 y Fs(2)2343 4285 y Fk(.)33 b(Therefore,)h(w)m(e)f(get)g (that)361 4534 y(\002)437 4549 y Fs(1)504 4534 y Fi(\031)28 b Fk(\(9)p Fh(A)22 b Fi(\000)h Fk(\001\))p Fh(=)p Fk(8)p Fh(;)211 b Fk(\002)1422 4549 y Fs(2)1489 4534 y Fi(\031)29 b Fk(\006)p Fh(=)p Fk(4)p Fh(;)211 b Fk(\002)2077 4549 y Fs(3)2144 4534 y Fi(\031)29 b Fk(\(\001)22 b Fi(\000)h Fh(A)p Fk(\))p Fh(=)p Fk(8)p Fh(:)166 4898 y Fk(These)34 b(digressions)e(clarify)f(wh)m(y)j(the)e(quan)m(tities)g(2)p Fh(\031)2166 4862 y Fs(2)2206 4898 y Fh(h)2262 4862 y Fr(\000)p Fs(2)2356 4898 y Fh(A)p Fk(,)g Fh(!)2553 4862 y Fs(+)2612 4898 y Fk(,)g(and)h Fi(\000)p Fh(!)3003 4862 y Fr(\000)3094 4898 y Fk(ha)m(v)m(e)g(the)166 5019 y(same)k(asymptotic) f(b)s(eha)m(vior)g(as)h Fh(h)e Fi(!)g Fk(0)1722 4983 y Fs(+)1780 5019 y Fk(.)i(They)h(also)e(suggest)i(that)f(the)g(sum)f (\012)g(:=)166 5139 y Fh(!)231 5103 y Fs(+)312 5139 y Fk(+)22 b Fh(!)475 5103 y Fr(\000)561 5139 y Fi(\031)28 b Fk(16)p Fh(\031)823 5103 y Fs(2)862 5139 y Fh(h)918 5103 y Fr(\000)p Fs(2)1013 5139 y Fk(\002)1089 5154 y Fs(2)1161 5139 y Fk(and)k(the)i(com)m(bination)c Fh(h)2130 5103 y Fs(2)2170 5139 y Fk(\()p Fh(!)2273 5103 y Fs(+)2353 5139 y Fi(\000)23 b Fh(!)2518 5103 y Fr(\000)2576 5139 y Fk(\))p Fh(=)p Fk(4)p Fh(\031)2771 5103 y Fs(2)2832 5139 y Fi(\000)g Fh(A)33 b Fk(ha)m(v)m(e)h(sizes)166 5259 y(of)g(order)536 5256 y(e)579 5223 y Fr(\000)p Fs(4)p Fq(\031)r(\016)r(=h)861 5259 y Fk(and)1053 5256 y(e)1096 5223 y Fr(\000)p Fs(6)p Fq(\031)r(\016)r(=h)1343 5259 y Fk(,)h(resp)s(ectiv)m(ely)-8 b(.)35 b(Th)m(us,)i(w)m(e)e(ha)m(v)m(e)h (found)f(candidates)g(to)166 5380 y(study)f(the)f(second)h(and)e(third) g(exp)s(onen)m(tial)g(terms)h(that)f(go)m(v)m(ern)i(the)f(splitting.) 1745 5712 y(12)p eop %%Page: 13 13 13 12 bop 166 83 a Fl(3)112 b(Billiard)34 b(tables)166 433 y Fk(W)-8 b(e)28 b(collect)f(some)h(classical)f(results)h(on)g(con) m(v)m(ex)i(and)f(elliptic)c(billiards.)g(Most)j(of)g(them)166 554 y(can)f(b)s(e)g(found)f(in)g(the)h(monographes)f([14,22],)g (although)f(w)m(e)j(ha)m(v)m(e)g(tak)m(en)f(the)g(notation)166 674 y(from)k([13,)i Fi(x)p Fk(9.2].)f(W)-8 b(e)33 b(also)f(presen)m(t)i (the)f(p)s(erturbations)f(w)m(e)i(shall)d(study)j(later)e(on.)166 1067 y Fj(3.1)99 b(Convex)34 b(bil)5 b(liar)-5 b(d)34 b(tables)166 1417 y Fk(Let)28 b Fh(C)35 b Fk(b)s(e)28 b(a)g(closed)g(con)m(v)m(ex)j(curv)m(e)e(in)f(the)g(Euclidean)g(plane)f Fe(R)2541 1381 y Fs(2)2587 1417 y Fk(.)h(Let)g Fh(\015)33 b Fk(:)27 b Fe(T)k Fi(!)d Fh(C)35 b Fk(b)s(e)28 b(a)166 1537 y(parameterization)23 b(of)h(this)h(curv)m(e,)i(where)f Fe(T)31 b Fk(:=)d Fe(R)5 b Fh(=)p Fk(2)p Fh(\031)t Fe(Z)28 b Fk(stands)e(for)e(the)i(con\014guration)166 1658 y(space.)34 b(Finally)-8 b(,)30 b(let)i(us)h(consider)g(the)g(phase)g(space)361 1897 y Fe(A)56 b Fk(:=)27 b Fe(T)f Fi(\002)d Fk(\(0)p Fh(;)17 b(\031)t Fk(\))27 b(=)g Fi(f)p Fk(\()p Fh(\022)s(;)17 b(r)s Fk(\))27 b(:)h Fh(\022)j Fi(2)d Fe(T)p Fh(;)17 b Fk(0)30 b Fh(<)e(r)i(<)e(\031)t Fi(g)p Fh(:)1070 b Fk(\(2\))166 2246 y(Then)49 b(w)m(e)g(can)g(mo)s(del)d(the)j(billiard) 44 b(dynamics)k(inside)f Fh(C)55 b Fk(b)m(y)49 b(means)f(of)g(the)g (map)166 2367 y Fh(f)11 b Fk(\()p Fh(\022)s(;)17 b(r)s Fk(\))41 b(=)h(\()p Fh(\022)685 2331 y Fr(0)708 2367 y Fh(;)17 b(r)799 2331 y Fr(0)822 2367 y Fk(\))40 b(de\014ned)j(as)e (follo)m(ws.)e(If)i(the)h(particle)d(hits)i(the)g(curv)m(e)h(at)f(a)g (p)s(oin)m(t)166 2487 y Fh(c)32 b Fk(=)f Fh(\015)5 b Fk(\()p Fh(\022)s Fk(\))32 b Fi(2)g Fh(C)42 b Fk(under)36 b(an)e(angle)g(of)h(incidence)g Fh(r)s Fk(,)g(the)g(next)h(p)s(oin)m(t) e(is)g Fh(c)2881 2451 y Fr(0)2936 2487 y Fk(=)e Fh(\015)5 b Fk(\()p Fh(\022)3186 2451 y Fr(0)3209 2487 y Fk(\))32 b Fi(2)g Fh(C)166 2608 y Fk(and)h(the)g(next)g(angle)f(of)g(incidence)h (is)f Fh(r)1677 2571 y Fr(0)1700 2608 y Fk(.)g(This)h(map)f(has)h(the)g (follo)m(wing)d(prop)s(erties:)165 2837 y Fi(\017)49 b Fj(R)-5 b(e)g(gularity.)32 b Fk(If)h(the)g(curv)m(e)h(is)e(smo)s(oth) g(or)g(analytic,)f(so)i(is)f(the)h(map.)165 2958 y Fi(\017)49 b Fj(Hyp)-5 b(erb)g(olicity.)27 b Fk(Let)f Fh(c)1080 2973 y Fr(\006)1167 2958 y Fk(=)i Fh(\015)5 b Fk(\()p Fh(\022)1410 2973 y Fr(\006)1469 2958 y Fk(\))27 b(b)s(e)g(the)g(ends)h (of)e(a)g(c)m(hord)i(of)e Fh(C)33 b Fk(and)27 b Fh(`)h Fk(=)f Fi(j)p Fh(c)3117 2973 y Fs(+)3198 2958 y Fi(\000)c Fh(c)3340 2973 y Fr(\000)3399 2958 y Fi(j)p Fk(.)264 3078 y(Let)j Fh(\024)488 3093 y Fr(\006)574 3078 y Fk(b)s(e)h(the)g (curv)-5 b(ature)27 b(of)f Fh(C)33 b Fk(at)27 b Fh(c)1657 3093 y Fr(\006)1716 3078 y Fk(.)f(Then)i(the)f(c)m(hord)g(is)f(h)m(yp)s (erb)s(olic)g(if)g(and)g(only)264 3199 y(if)35 b(\()p Fh(\024)451 3214 y Fs(+)535 3199 y Fk(+)25 b Fh(\024)692 3214 y Fr(\000)751 3199 y Fk(\))p Fh(`)35 b(>)g Fk(4.)h(When)i(this)f (condition)e(holds,)h(the)i(c)m(haracteristic)e(exp)s(onen)m(t)264 3319 y Fh(h)27 b(>)h Fk(0)k(of)g(the)h(asso)s(ciated)g(t)m(w)m(o-p)s (erio)s(dic)e(p)s(oin)m(ts)h(is)g(determined)h(b)m(y)g(the)g(relation) 459 3578 y(2)17 b(cosh\()p Fh(h=)p Fk(2\))27 b(=)1070 3473 y Fg(q)p 1153 3473 468 4 v 105 x Fk(\()p Fh(\024)1247 3593 y Fs(+)1328 3578 y Fk(+)22 b Fh(\024)1482 3593 y Fr(\000)1541 3578 y Fk(\))p Fh(`:)165 3817 y Fi(\017)49 b Fj(R)-5 b(eversibility.)42 b Fk(If)h Fh(C)50 b Fk(is)42 b(symmetric)g(with)h(regard)f(to)h(b)s(oth)g(axes)h(of)e(co)s (ordinates,)264 3938 y(then)33 b(the)g(map)f(is)g(rev)m(ersible,)h Fh(R)q Fk(\()p Fh(\022)s(;)17 b(r)s Fk(\))27 b(=)h(\()p Fh(\031)e Fi(\000)c Fh(\022)s(;)17 b(r)s Fk(\))32 b(is)g(a)h(rev)m (ersor,)h(and)459 4177 y(Fix)o Fi(f)p Fh(R)q Fi(g)27 b Fk(=)h Fi(f)p Fk(\()p Fh(\022)s(;)17 b(r)s Fk(\))27 b Fi(2)h Fe(A)55 b Fk(:)28 b Fh(\022)j Fk(=)d Fh(\031)t(=)p Fk(2)99 b(\(mo)s(d)32 b Fh(\031)t Fk(\))p Fi(g)p Fh(:)165 4417 y Fi(\017)49 b Fj(Exactness.)22 b Fk(Let)h Fh(\032)28 b Fk(=)g Fi(j)14 b Fk(_)-41 b Fh(\015)5 b Fk(\()p Fh(\022)s Fk(\))p Fi(j)16 b Fk(cos)h Fh(r)26 b Fk(and)d Fh(\032)1780 4381 y Fr(0)1832 4417 y Fk(=)k Fi(j)14 b Fk(_)-41 b Fh(\015)5 b Fk(\()p Fh(\022)2105 4381 y Fr(0)2128 4417 y Fk(\))p Fi(j)17 b Fk(cos)g Fh(r)2405 4381 y Fr(0)2428 4417 y Fk(.)23 b(The)h(map)f(is)g(exact)h(in)e(the)264 4537 y(\()p Fh(\022)s(;)17 b(\032)p Fk(\))36 b(co)s(ordinates:)f Fh(\032)1116 4501 y Fr(0)1156 4537 y Fk(d)p Fh(\022)1258 4501 y Fr(0)1306 4537 y Fi(\000)25 b Fh(\032)17 b Fk(d)p Fh(\022)37 b Fk(=)50 b(d)p Fh(L)p Fk(\()p Fh(\022)s(;)17 b(\022)2035 4501 y Fr(0)2059 4537 y Fk(\),)36 b(where)h Fh(L)p Fk(\()p Fh(\022)s(;)17 b(\022)2689 4501 y Fr(0)2713 4537 y Fk(\))33 b(=)g Fi(j)p Fh(\015)5 b Fk(\()p Fh(\022)s Fk(\))22 b Fi(\000)h Fh(\015)5 b Fk(\()p Fh(\022)3365 4501 y Fr(0)3388 4537 y Fk(\))p Fi(j)264 4658 y Fk(is)32 b(the)h Fj(gener)-5 b(ating)34 b(function)39 b Fk(or)33 b Fj(L)-5 b(agr)g(angian)p Fk(.)165 4778 y Fi(\017)49 b Fj(Twist)30 b(char)-5 b(acter.)27 b Fk(Giv)m(en)g Fh(c)h Fk(=)f Fh(\015)5 b Fk(\()p Fh(\022)s Fk(\),)28 b(the)g(function)f(\(0)p Fh(;)17 b(\031)t Fk(\))27 b Fi(3)h Fh(r)j Fi(7!)c Fh(\022)2806 4742 y Fr(0)2829 4778 y Fk(\()p Fh(\022)s(;)17 b(r)s Fk(\))27 b Fi(2)i Fe(T)12 b Fi(n)g(f)p Fh(\022)s Fi(g)264 4898 y Fk(is)26 b(a)h(di\013eomorphism.)d(Hence,)29 b(giv)m(en)e(t)m(w) m(o)g(di\013eren)m(t)h(impact)d(p)s(oin)m(ts)i Fh(c)g Fk(and)g Fh(c)3160 4862 y Fr(0)3183 4898 y Fk(,)g(there)264 5019 y(exists)33 b(a)f(unique)h(billiard)c(tra)5 b(jectory)34 b(from)d Fh(c)h Fk(to)h Fh(c)2189 4983 y Fr(0)2212 5019 y Fk(.)165 5139 y Fi(\017)49 b Fj(L)-5 b(agr)g(angian)34 b(formulation.)f Fk(The)i(billiard)30 b(dynamics)j(can)h(b)s(e)g (expressed)i(b)m(y)f(means)264 5259 y(of)i(implicit)d(di\013erence)k (equations)g(of)f(second)i(order:)f(giv)m(en)f(three)i(impact)d(p)s (oin)m(ts)264 5380 y Fh(c)306 5395 y Fr(\000)365 5380 y Fh(;)17 b(c;)g(c)537 5395 y Fs(+)623 5380 y Fi(2)28 b Fh(C)39 b Fk(suc)m(h)34 b(that)e Fh(c)1299 5395 y Fr(\000)1385 5380 y Fk(=)c Fh(\015)5 b Fk(\()p Fh(\022)1628 5395 y Fr(\000)1687 5380 y Fk(\),)32 b Fh(c)c Fk(=)g Fh(\015)5 b Fk(\()p Fh(\022)s Fk(\))32 b(and)g Fh(c)2401 5395 y Fs(+)2488 5380 y Fk(=)27 b Fh(\015)5 b Fk(\()p Fh(\022)2730 5395 y Fs(+)2790 5380 y Fk(\),)32 b(there)h(exists)g(a)1745 5712 y(13)p eop %%Page: 14 14 14 13 bop 264 83 a Fk(billiard)29 b(tra)5 b(jectory)33 b(from)e Fh(c)1327 98 y Fr(\000)1419 83 y Fk(to)h Fh(c)1580 98 y Fs(+)1671 83 y Fk(passing)h(b)m(y)g Fh(c)g Fk(if)e(and)i(only)f (if)459 326 y Fh(@)510 341 y Fs(2)550 326 y Fh(L)p Fk(\()p Fh(\022)699 341 y Fr(\000)758 326 y Fh(;)17 b(\022)s Fk(\))22 b(+)g Fh(@)1059 341 y Fs(1)1099 326 y Fh(L)p Fk(\()p Fh(\022)s(;)17 b(\022)1340 341 y Fs(+)1400 326 y Fk(\))28 b(=)f(0)p Fh(:)165 570 y 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[]} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave 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copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 3.000 UL LT0 478 2520 M 4 -102 V 15 -101 V 24 -100 V 33 -100 V 43 -97 V 52 -96 V 61 -93 V 70 -91 V 78 -87 V 86 -84 V 94 -80 V 101 -76 V 109 -71 V 115 -67 V 121 -61 V 126 -57 V 132 -51 V 136 -45 V 140 -39 V 144 -33 V 147 -27 V 148 -21 V 151 -15 V 151 -8 V 153 -1 V 151 4 V 151 12 V 150 17 V 148 24 V 145 31 V 142 36 V 138 42 V 134 48 V 130 54 V 123 59 V 118 64 V 112 69 V 105 74 V 98 78 V 90 82 V 82 85 V 74 89 V 65 92 V 57 95 V 47 96 V 39 99 V 28 100 V 20 101 V 9 101 V 0 102 V -9 101 V -20 101 V -28 100 V -39 99 V -47 96 V -57 95 V -65 92 V -74 89 V -82 85 V -90 82 V -98 78 V -105 74 V -112 69 V -118 64 V -123 59 V -130 54 V -134 48 V -138 42 V -142 36 V -145 31 V -148 24 V -150 17 V -151 12 V -151 4 V -153 -1 V -151 -8 V -151 -15 V -148 -21 V -147 -27 V -144 -33 V -140 -39 V -136 -45 V -132 -51 V -126 -57 V -121 -61 V -115 -67 V -109 -71 V -101 -76 V -94 -80 V -86 -84 V -78 -87 V -70 -91 V -61 -93 V -52 -96 V -43 -97 V 521 2823 L 497 2723 L 482 2622 L -4 -102 V 1.000 UL LTb 1.000 UL LT0 4458 3723 M 518 2227 L 4750 353 V 478 2508 L 4791 14 V 478 2519 L 4458 3723 M 0 -2406 V 518 2813 L 5268 2460 L 478 2532 L 4791 -14 V -4791 3 V 1.000 UL LTb 1.000 UP 1.000 UL LT0 4458 2520 BoxF 1289 2520 BoxF stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 1532 w @beginspecial 50 @llx 50 @lly 328 @urx 302 @ury 1800 @rwi @setspecial %%BeginDocument: yaxial.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: yaxial.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Fri Sep 3 18:52:12 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 328 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 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3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -4 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -4 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 4 -4 V 3 -5 V 3 -5 V 4 -5 V 3 -4 V 4 -5 V 3 -5 V 3 -4 V 4 -5 V 3 -4 V 4 -5 V 3 -5 V 3 -4 V 4 -5 V 3 -4 V 4 -5 V 3 -4 V 3 -5 V 4 -4 V 3 -5 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 4 -4 V 3 -3 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -3 V 3 -4 V 4 -4 V 3 -4 V 4 -3 V 3 -4 V 4 -3 V 4 -4 V 3 -4 V 4 -3 V 3 -4 V 4 -3 V 3 -4 V 4 -3 V 4 -3 V 3 -4 V 4 -3 V 3 -3 V 4 -4 V 3 -3 V 4 -3 V 4 -4 V 3 -3 V 4 -3 V 3 -3 V 4 -3 V 4 -3 V 3 -3 V 4 -3 V 4 -3 V 3 -3 V 4 -3 V 3 -3 V 4 -3 V 4 -3 V 3 -3 V 4 -2 V 4 -3 V 3 -3 V 4 -3 V 4 -2 V 3 -3 V 4 -3 V 4 -2 V 3 -3 V 4 -2 V 4 -3 V 3 -2 V 4 -3 V 4 -2 V 3 -2 V 4 -3 V 4 -2 V 4 -2 V 3 -2 V 4 -3 V 4 -2 V 3 -2 V 4 -2 V 4 -2 V 4 -2 V 3 -2 V 4 -2 V 4 -2 V 3 -2 V 4 -2 V 4 -1 V 4 -2 V 3 -2 V 4 -2 V 4 -1 V 4 -2 V 3 -2 V 4 -1 V 4 -2 V 4 -1 V 3 -2 V 4 -1 V 4 -2 V 4 -1 V 3 -1 V 4 -2 V 4 -1 V 4 -1 V 3 -1 V 4 -1 V 4 -2 V 4 -1 V 3 -1 V 4 -1 V 4 -1 V 4 -1 V 3 0 V 4 -1 V 4 -1 V 4 -1 V 4 -1 V 3 0 V 4 -1 V 4 -1 V 4 0 V 3 -1 V 4 0 V 4 -1 V 4 0 V 4 -1 V 3 0 V 4 -1 V currentpoint stroke M 4 0 V 4 0 V 4 0 V 3 -1 V 4 0 V 4 0 V 4 0 V 3 0 V 4 0 V 4 0 V 4 0 V 4 0 V 3 0 V 4 0 V 4 0 V 4 1 V 4 0 V 3 0 V 4 0 V 4 1 V 4 0 V 3 1 V 4 0 V 4 0 V 4 1 V 4 1 V 3 0 V 4 1 V 4 0 V 4 1 V 4 1 V 3 1 V 4 0 V 4 1 V 4 1 V 3 1 V 4 1 V 4 1 V 4 1 V 3 1 V 4 1 V 4 1 V 4 1 V 4 1 V 3 1 V 4 2 V 4 1 V 4 1 V 3 1 V 4 2 V 4 1 V 4 1 V 3 2 V 4 1 V 4 2 V 4 1 V 3 2 V 4 1 V 4 2 V 4 1 V 3 2 V 4 2 V 4 1 V 4 2 V 3 2 V 4 1 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 1 V 4 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 3 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 3 V 4 2 V 3 2 V 4 2 V 4 3 V 4 2 V 3 2 V 4 3 V 4 2 V 3 3 V 4 2 V 4 2 V 3 3 V 4 2 V 4 3 V 3 2 V 4 3 V 4 2 V 3 3 V 4 2 V 4 3 V 3 3 V 4 2 V 4 3 V 3 2 V 4 3 V 4 3 V 3 2 V 4 3 V 4 3 V 3 2 V 4 3 V 4 3 V 3 2 V 4 3 V 4 3 V 3 3 V 4 2 V 4 3 V 3 3 V 4 3 V 4 3 V 3 2 V 4 3 V 4 3 V 3 3 V 4 3 V 3 2 V 4 3 V 4 3 V 3 3 V 4 3 V 4 3 V 3 3 V 4 2 V 4 3 V 3 3 V 4 3 V 4 3 V 3 3 V 4 3 V 3 3 V 4 2 V 4 3 V 3 3 V 4 3 V 4 3 V 3 3 V 4 3 V 4 3 V 3 3 V 4 2 V 3 3 V 4 3 V 4 3 V 3 3 V 4 3 V 4 3 V 3 3 V 4 2 V 4 3 V 3 3 V 4 3 V 3 3 V 4 3 V 4 3 V 3 2 V 4 3 V 4 3 V 3 3 V 4 3 V 4 2 V 3 3 V 4 3 V 3 3 V 4 3 V 4 2 V 3 3 V 4 3 V 4 3 V 3 2 V 4 3 V 4 3 V 3 3 V 4 2 V 4 3 V 3 3 V 4 2 V 4 3 V 3 3 V 4 2 V 4 3 V 3 2 V 4 3 V 4 3 V 3 2 V 4 3 V 4 2 V 3 3 V 4 2 V 4 3 V 3 2 V 4 3 V 4 2 V 3 3 V 4 2 V 4 3 V 3 2 V 4 3 V 4 2 V 3 2 V 4 3 V 4 2 V 3 3 V 4 2 V 4 2 V 3 2 V 4 3 V 4 2 V 3 2 V 4 3 V 4 2 V 3 2 V 4 2 V 4 2 V 3 3 V 4 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 3 V 3 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 4 2 V 3 2 V 4 1 V 4 2 V 3 2 V 4 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 1 V 4 2 V 3 2 V 4 2 V 4 2 V 4 2 V 3 1 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 1 V 4 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 1 V 3 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 3 V 4 2 V 3 2 V 3 2 V 3 2 V 3 2 V 3 2 V 2 1 V 2 2 V 3 2 V 2 1 V 2 2 V 2 2 V 2 1 V 2 2 V 2 1 V 1 1 V 2 2 V 2 1 V 1 2 V 2 1 V 1 1 V 2 1 V 1 2 V 2 1 V 1 1 V 1 1 V 2 2 V 1 1 V 1 1 V 1 1 V 2 1 V 1 2 V 1 1 V 1 1 V 1 1 V 1 1 V 1 1 V 1 1 V 1 1 V 1 1 V 1 2 V 1 1 V 1 1 V 1 1 V 1 1 V 1 1 V 0 1 V 1 1 V 1 1 V 1 1 V 1 1 V 0 1 V 1 1 V 1 1 V 1 1 V 0 1 V 1 1 V 1 1 V 1 1 V 0 1 V 1 1 V 1 0 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V currentpoint stroke M 0 1 V 1 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 2 V 0 1 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 1 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 1 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 1 V 0 2 V 1 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 3 V 0 2 V -1 2 V 0 2 V 0 2 V 0 3 V 0 2 V 0 2 V 0 3 V 0 2 V 0 3 V 0 2 V 0 3 V 0 2 V 0 3 V 0 3 V 0 2 V 0 3 V -1 3 V 0 3 V 0 3 V 0 3 V 0 3 V 0 3 V 0 3 V 0 4 V -1 3 V 0 3 V 0 4 V 0 4 V 0 3 V -1 4 V 0 4 V 0 4 V 0 4 V 0 5 V -1 4 V 0 5 V 0 5 V -1 5 V 0 5 V 0 6 V -1 6 V 0 6 V -1 7 V 0 7 V -1 8 V -1 8 V 0 9 V -1 10 V -1 11 V -1 10 V -1 10 V 0 10 V -1 10 V -1 10 V -1 10 V -1 10 V -1 11 V 0 10 V -1 10 V -1 10 V -1 9 V 0 7 V -1 6 V 0 5 V 0 5 V -1 6 V 0 4 V 0 4 V 0 5 V -1 3 V 0 4 V 0 4 V 0 3 V 0 3 V -1 4 V 0 3 V 0 3 V 0 2 V 0 3 V 0 3 V 0 2 V -1 3 V 0 2 V 0 3 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 3 V 0 1 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 1 V -1 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V currentpoint stroke M 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 1 1 V 1 1 V 1 1 V 1 1 V 1 0 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 0 1 V 1 0 V 1 1 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 1 1 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 1 -1 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 1 -1 V 1 0 V 1 0 V 1 0 V 1 -1 V 1 0 V 1 0 V 1 -1 V 1 0 V 1 0 V 1 -1 V 1 0 V 1 0 V 1 -1 V 1 0 V 1 -1 V 1 0 V 1 -1 V 1 0 V 1 -1 V 1 0 V 1 -1 V 1 0 V 1 -1 V 1 0 V 2 -1 V 1 -1 V 1 0 V 1 -1 V 1 -1 V 2 -1 V 1 0 V 1 -1 V 2 -1 V 1 -1 V 1 -1 V 2 -1 V 1 -1 V 2 -1 V 1 -1 V 2 -1 V 2 -2 V 2 -1 V 1 -1 V 2 -2 V 2 -1 V 2 -2 V 2 -1 V 2 -2 V 2 -2 V 2 -1 V 3 -2 V 2 -2 V 2 -3 V 3 -2 V 3 -2 V 2 -3 V 3 -2 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 4 -4 V 3 -4 V 4 -3 V 3 -4 V 4 -4 V 4 -3 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -3 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -3 V 3 -4 V 3 -3 V 2 -2 V 2 -2 V 1 -2 V 2 -2 V 1 -1 V 1 -1 V 2 -2 V 1 -1 V 1 -1 V 0 -1 V 1 0 V 1 -1 V 1 -1 V 1 -1 V 1 0 V 0 -1 V 1 0 V 0 -1 V 1 -1 V 1 0 V 0 -1 V 1 0 V 0 -1 V 1 0 V 1 0 V 0 -1 V 1 0 V 0 -1 V 1 0 V 1 0 V 0 -1 V 1 0 V 1 0 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 1 V 0 1 V currentpoint stroke M -1 1 V 0 1 V -1 1 V 0 2 V -1 1 V -1 2 V 0 1 V -1 2 V -1 2 V -1 2 V -1 2 V -1 2 V -1 3 V -1 2 V -1 3 V -2 4 V -1 3 V -2 4 V -2 4 V -2 4 V -3 5 V -2 5 V -3 6 V -3 7 V -4 6 V -3 6 V -3 7 V -3 6 V -3 6 V -3 6 V -4 7 V -3 6 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 5 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 5 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 5 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 5 V -3 6 V -3 6 V -3 6 V -3 5 V -3 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-6 V -4 -5 V -3 -6 V -3 -5 V -3 -6 V -4 -5 V -3 -6 V -3 -5 V -3 -6 V -4 -5 V -3 -5 V -3 -6 V -3 -5 V -4 -6 V -3 -5 V -3 -6 V -3 -5 V -4 -6 V -3 -5 V -3 -5 V -3 -6 V -4 -5 V -3 -6 V -3 -5 V -3 -5 V -4 -6 V -3 -5 V -3 -6 V -3 -5 V -4 -5 V -3 -6 V -3 -5 V -3 -6 V -4 -5 V -3 -5 V -3 -6 V -3 -5 V -4 -5 V -3 -6 V -3 -5 V -3 -5 V -4 -6 V -3 -5 V -3 -5 V -3 -5 V -4 -6 V -3 -5 V -3 -5 V -4 -5 V -3 -6 V -3 -5 V -3 -5 V -4 -5 V -3 -6 V -3 -5 V -4 -5 V -3 -5 V -3 -5 V -3 -5 V -4 -6 V -3 -5 V -3 -5 V -4 -5 V -3 -5 V -3 -5 V -4 -5 V -3 -6 V -3 -5 V -4 -5 V -3 -5 V -3 -5 V -4 -5 V -3 -5 V -3 -5 V -4 -5 V -3 -5 V -3 -5 V -4 -5 V -3 -5 V -3 -5 V -4 -5 V -3 -5 V -3 -4 V -4 -5 V -3 -5 V -3 -5 V -4 -5 V -3 -5 V -3 -5 V -4 -4 V -3 -5 V -3 -5 V -4 -5 V -3 -5 V -4 -4 V -3 -5 V -3 -5 V -4 -5 V -3 -4 V -4 -5 V -3 -5 V -3 -4 V -4 -5 V -3 -4 V -4 -5 V -3 -5 V -3 -4 V -4 -5 V -3 -4 V -4 -5 V -3 -4 V -3 -5 V -4 -4 V -3 -5 V -4 -4 V -3 -5 V -4 -4 V -3 -4 V -3 -5 V -4 -4 V -3 -4 V -4 -5 V -3 -4 V -4 -4 V -3 -4 V -4 -5 V -3 -4 V -4 -4 V -3 -4 V -4 -4 V -3 -5 V -4 -4 V -3 -4 V -3 -4 V -4 -4 V -3 -4 V -4 -4 V -3 -4 V -4 -4 V -3 -4 V -4 -4 V -4 -4 V -3 -3 V -4 -4 V -3 -4 V -4 -4 V -3 -4 V -4 -3 V -3 -4 V -4 -4 V -3 -4 V -4 -3 V -3 -4 V -4 -3 V -4 -4 V -3 -4 V -4 -3 V -3 -4 V -4 -3 V -3 -4 V -4 -3 V -4 -3 V -3 -4 V -4 -3 V -3 -3 V -4 -4 V -3 -3 V -4 -3 V -4 -4 V -3 -3 V -4 -3 V -3 -3 V -4 -3 V -4 -3 V -3 -3 V -4 -3 V -4 -3 V -3 -3 V -4 -3 V -3 -3 V -4 -3 V -4 -3 V -3 -3 V -4 -2 V -4 -3 V -3 -3 V -4 -3 V -4 -2 V -3 -3 V -4 -3 V -4 -2 V -3 -3 V -4 -2 V -4 -3 V -3 -2 V -4 -3 V -4 -2 V -3 -2 V -4 -3 V -4 -2 V -4 -2 V -3 -2 V -4 -3 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -1 V -4 -2 V -3 -2 V -4 -2 V -4 -1 V -4 -2 V -3 -2 V -4 -1 V -4 -2 V -4 -1 V -3 -2 V -4 -1 V -4 -2 V -4 -1 V -3 -1 V -4 -2 V -4 -1 V -4 -1 V -3 -1 V -4 -1 V -4 -2 V -4 -1 V -3 -1 V -4 -1 V -4 -1 V -4 -1 V -3 0 V -4 -1 V -4 -1 V -4 -1 V -4 -1 V -3 0 V -4 -1 V -4 -1 V -4 0 V -3 -1 V -4 0 V -4 -1 V -4 0 V -4 -1 V -3 0 V -4 -1 V currentpoint stroke M -4 0 V -4 0 V -4 0 V -3 -1 V -4 0 V -4 0 V -4 0 V -3 0 V -4 0 V -4 0 V -4 0 V -4 0 V -3 0 V -4 0 V -4 0 V -4 1 V -4 0 V -3 0 V -4 0 V -4 1 V -4 0 V -3 1 V -4 0 V -4 0 V -4 1 V -4 1 V -3 0 V -4 1 V -4 0 V -4 1 V -4 1 V -3 1 V -4 0 V -4 1 V -4 1 V -3 1 V -4 1 V -4 1 V -4 1 V -3 1 V -4 1 V -4 1 V -4 1 V -4 1 V -3 1 V -4 2 V -4 1 V -4 1 V -3 1 V -4 2 V -4 1 V -4 1 V -3 2 V -4 1 V -4 2 V -4 1 V -3 2 V -4 1 V -4 2 V -4 1 V -3 2 V -4 2 V -4 1 V -4 2 V -3 2 V -4 1 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -3 1 V -4 2 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -3 2 V -4 3 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -3 2 V -4 3 V -4 2 V -3 2 V -4 2 V -4 3 V -4 2 V -3 2 V -4 3 V -4 2 V -3 3 V -4 2 V -4 2 V -3 3 V -4 2 V -4 3 V -3 2 V -4 3 V -4 2 V -3 3 V -4 2 V -4 3 V -3 3 V -4 2 V -4 3 V -3 2 V -4 3 V -4 3 V -3 2 V -4 3 V -4 3 V -3 2 V -4 3 V -4 3 V -3 2 V -4 3 V -4 3 V -3 3 V -4 2 V -4 3 V -3 3 V -4 3 V -4 3 V -3 2 V -4 3 V -4 3 V -3 3 V -4 3 V -3 2 V -4 3 V -4 3 V -3 3 V -4 3 V -4 3 V -3 3 V -4 2 V -4 3 V -3 3 V -4 3 V -4 3 V -3 3 V -4 3 V -3 3 V -4 2 V -4 3 V -3 3 V -4 3 V -4 3 V -3 3 V -4 3 V -4 3 V -3 3 V -4 2 V -3 3 V -4 3 V -4 3 V -3 3 V -4 3 V -4 3 V -3 3 V -4 2 V -4 3 V -3 3 V -4 3 V -3 3 V -4 3 V -4 3 V -3 2 V -4 3 V -4 3 V -3 3 V -4 3 V -4 2 V -3 3 V -4 3 V -3 3 V -4 3 V -4 2 V -3 3 V -4 3 V -4 3 V -3 2 V -4 3 V -4 3 V -3 3 V -4 2 V -4 3 V -3 3 V -4 2 V -4 3 V -3 3 V -4 2 V -4 3 V -3 2 V -4 3 V -4 3 V -3 2 V -4 3 V -4 2 V -3 3 V -4 2 V -4 3 V -3 2 V -4 3 V -4 2 V -3 3 V -4 2 V -4 3 V -3 2 V -4 3 V -4 2 V -3 2 V -4 3 V -4 2 V -3 3 V -4 2 V -4 2 V -3 2 V -4 3 V -4 2 V -3 2 V -4 3 V -4 2 V -3 2 V -4 2 V -4 2 V -3 3 V -4 2 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -3 2 V -4 2 V -4 3 V -3 2 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -4 2 V -3 2 V -4 1 V -4 2 V -3 2 V -4 2 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -3 2 V -4 1 V -4 2 V -3 2 V -4 2 V -4 2 V -4 2 V -3 1 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -3 1 V -4 2 V -4 2 V -4 2 V -3 2 V -4 2 V -4 1 V -3 2 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -3 2 V -4 3 V -4 2 V -3 2 V -3 2 V -3 2 V -3 2 V -3 2 V -2 1 V -2 2 V -3 2 V -2 1 V -2 2 V -2 2 V -2 1 V -2 2 V -2 1 V -1 1 V -2 2 V -2 1 V -1 2 V -2 1 V -1 1 V -2 1 V -1 2 V -2 1 V -1 1 V -1 1 V -2 2 V -1 1 V -1 1 V -1 1 V -2 1 V -1 2 V -1 1 V -1 1 V -1 1 V -1 1 V -1 1 V -1 1 V -1 1 V -1 1 V -1 2 V -1 1 V -1 1 V -1 1 V -1 1 V -1 1 V 0 1 V -1 1 V -1 1 V -1 1 V -1 1 V 0 1 V -1 1 V -1 1 V -1 1 V 0 1 V -1 1 V -1 1 V -1 1 V 0 1 V -1 1 V -1 0 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 0 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V currentpoint stroke M 0 1 V -1 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 2 V 0 1 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V -1 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V -1 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 1 V 0 2 V -1 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 3 V 0 2 V 1 2 V 0 2 V 0 2 V 0 3 V 0 2 V 0 2 V 0 3 V 0 2 V 0 3 V 0 2 V 0 3 V 0 2 V 0 3 V 0 3 V 0 2 V 0 3 V 1 3 V 0 3 V 0 3 V 0 3 V 0 3 V 0 3 V 0 3 V 0 4 V 1 3 V 0 3 V 0 4 V 0 4 V 0 3 V 1 4 V 0 4 V 0 4 V 0 4 V 0 5 V 1 4 V 0 5 V 0 5 V 1 5 V 0 5 V 0 6 V 1 6 V 0 6 V 1 7 V 0 7 V 1 8 V 1 8 V 0 9 V 1 10 V 1 11 V 1 10 V 1 10 V 0 10 V 1 10 V 1 10 V 1 10 V 1 10 V 1 11 V 0 10 V 1 10 V 1 10 V 1 9 V 0 7 V 1 6 V 0 5 V 0 5 V 1 6 V 0 4 V 0 4 V 0 5 V 1 3 V 0 4 V 0 4 V 0 3 V 0 3 V 1 4 V 0 3 V 0 3 V 0 2 V 0 3 V 0 3 V 0 2 V 1 3 V 0 2 V 0 3 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 3 V 0 1 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 1 V 1 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V currentpoint stroke M 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V -1 1 V -1 1 V -1 1 V -1 1 V -1 0 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V 0 1 V -1 0 V -1 1 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 1 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 -1 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 -1 V -1 0 V -1 0 V -1 0 V -1 -1 V -1 0 V -1 0 V -1 -1 V -1 0 V -1 0 V -1 -1 V -1 0 V -1 0 V -1 -1 V -1 0 V -1 -1 V -1 0 V -1 -1 V -1 0 V -1 -1 V -1 0 V -1 -1 V -1 0 V -1 -1 V -1 0 V -2 -1 V -1 -1 V -1 0 V -1 -1 V -1 -1 V -2 -1 V -1 0 V -1 -1 V -2 -1 V -1 -1 V -1 -1 V -2 -1 V -1 -1 V -2 -1 V -1 -1 V -2 -1 V -2 -2 V -2 -1 V -1 -1 V -2 -2 V -2 -1 V -2 -2 V -2 -1 V -2 -2 V -2 -2 V -2 -1 V -3 -2 V -2 -2 V -2 -3 V -3 -2 V -3 -2 V -2 -3 V -3 -2 V -3 -3 V -3 -3 V -3 -3 V -4 -3 V -3 -3 V -4 -4 V -3 -4 V -4 -3 V -3 -4 V -4 -4 V -4 -3 V -3 -4 V -4 -4 V -3 -4 V -4 -4 V -3 -4 V -4 -3 V -3 -4 V -4 -4 V -3 -4 V -4 -4 V -4 -4 V -3 -4 V -4 -4 V -3 -4 V -4 -4 V -3 -4 V -4 -5 V -3 -4 V -4 -4 V -3 -4 V -4 -4 V -3 -4 V -4 -4 V -3 -4 V -4 -5 V -3 -4 V -4 -4 V -3 -4 V -4 -4 V -3 -5 V -4 -4 V -3 -4 V -3 -4 V -4 -5 V -3 -4 V -4 -4 V -3 -4 V -4 -5 V -3 -4 V -4 -4 V -3 -5 V -4 -4 V -3 -4 V -4 -4 V -3 -5 V -3 -4 V -4 -4 V -3 -5 V -4 -4 V -3 -4 V -4 -5 V -3 -4 V -4 -4 V -3 -5 V -4 -4 V -3 -4 V -3 -5 V -4 -4 V -3 -5 V -4 -4 V -3 -4 V -4 -5 V -3 -4 V -3 -4 V -4 -5 V -3 -4 V -4 -4 V -3 -5 V -4 -4 V -3 -5 V -4 -4 V -3 -4 V -3 -5 V -4 -4 V -3 -4 V -4 -5 V -3 -4 V -4 -5 V -3 -4 V -3 -4 V -4 -5 V -3 -4 V -4 -5 V -3 -4 V -3 -4 V -4 -5 V -3 -4 V -4 -5 V -3 -4 V -4 -4 V -3 -5 V -3 -4 V -4 -4 V -3 -5 V -4 -4 V -3 -5 V -3 -4 V -4 -4 V -3 -5 V -4 -4 V -3 -4 V -3 -5 V -4 -4 V -3 -5 V -4 -4 V -3 -4 V -3 -5 V -4 -4 V -3 -4 V -4 -5 V -3 -4 V -3 -4 V -4 -5 V -3 -4 V -4 -4 V -3 -4 V -3 -5 V -4 -4 V -3 -4 V -4 -4 V -3 -5 V -3 -4 V -4 -4 V -3 -4 V -3 -4 V -4 -4 V -3 -5 V -4 -4 V -3 -4 V -3 -4 V -4 -4 V -3 -3 V -3 -4 V -3 -3 V -2 -2 V -2 -2 V -1 -2 V -2 -2 V -1 -1 V -1 -1 V -2 -2 V -1 -1 V -1 -1 V 0 -1 V -1 0 V -1 -1 V -1 -1 V -1 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V -1 0 V 0 -1 V -1 0 V -1 0 V -1 0 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V currentpoint stroke M 1 1 V 0 1 V 1 1 V 0 2 V 1 1 V 1 2 V 0 1 V 1 2 V 1 2 V 1 2 V 1 2 V 1 2 V 1 3 V 1 2 V 1 3 V 2 4 V 1 3 V 2 4 V 2 4 V 2 4 V 3 5 V 2 5 V 3 6 V 3 7 V 4 6 V 3 6 V 3 7 V 3 6 V 3 6 V 3 6 V 4 7 V 3 6 V 3 6 V 3 6 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 6 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 5 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 4 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6 V 3 6 V 3 5 V 4 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 4 6 V 3 6 V 3 5 V 3 6 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6 V 3 6 V 3 5 V 4 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 4 6 V 3 6 V 3 5 V 3 6 V 3 6 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6 V 3 6 V 3 5 V 4 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 4 6 V 3 6 V 3 5 V 3 6 V 3 6 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6 V 3 6 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6 V 3 6 V 3 6 V 3 5 V 4 6 V 3 6 V 3 6 V 3 6 V 3 6 V 3 5 V 4 6 V 3 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 3 6 V 4 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 3 6 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 4 6 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 3 6 V 4 6 V 3 5 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 3 6 V 4 5 V 3 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 3 6 V 3 5 V 4 6 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 3 5 V 3 6 V 3 6 V 3 6 V 2 5 V 2 4 V 2 3 V 1 3 V 1 2 V 1 2 V 1 2 V 1 2 V 1 1 V 0 1 V 1 2 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 0 V -1 0 V 0 -1 V -1 0 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 -1 V -1 0 V 0 -1 V -1 -1 V -1 -1 V -1 -1 V -1 -1 V -1 -1 V -1 -2 V -1 -1 V -2 -2 V -1 -2 V -2 -2 V -1 -2 V -2 -3 V -2 -2 V -3 -3 V -2 -4 V -3 -3 V -3 -4 V -3 -5 V -3 -5 V -4 -5 V -3 -5 V -4 -5 V -3 -5 V -4 -5 V -3 -5 V -4 -5 V -3 -5 V -4 -5 V -3 -5 V -3 -5 V -4 -5 V -3 -5 V -4 -5 V -3 -6 V -3 -5 V -4 -5 V -3 -5 V -4 -5 V -3 -6 V -3 -5 V -4 -5 V currentpoint stroke M -3 -5 V -3 -5 V -4 -6 V -3 -5 V -3 -5 V -4 -5 V -3 -6 V -3 -5 V -4 -5 V -3 -6 V -3 -5 V -4 -5 V -3 -5 V -3 -6 V -4 -5 V -3 -5 V -3 -6 V -3 -5 V -4 -5 V -3 -6 V -3 -5 V -4 -6 V -3 -5 V -3 -5 V -3 -6 V -4 -5 V -3 -5 V -3 -6 V -4 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-5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 4 -4 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -4 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -4 V 3 -5 V 4 -5 V 3 -5 V 3 -4 V 4 -5 V 3 -5 V 4 -4 V 3 -5 V 3 -4 V 4 -5 V 3 -5 V 4 -4 V 3 -5 V 3 -4 V 4 -5 V 3 -4 V 4 -5 V 3 -4 V 4 -5 V 3 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -3 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -3 V 3 -4 V 4 -4 V 4 -4 V 3 -3 V 4 -4 V 3 -3 V 4 -4 V 3 -4 V 4 -3 V 3 -4 V 4 -3 V 4 -4 V 3 -3 V 4 -3 V 3 -4 V 4 -3 V 4 -3 V 3 -4 V 4 -3 V 3 -3 V 4 -4 V 4 -3 V 3 -3 V 4 -3 V 3 -3 V 4 -3 V 4 -3 V 3 -3 V 4 -3 V 4 -3 V 3 -3 V 4 -3 V 3 -3 V 4 -3 V 4 -3 V 3 -2 V 4 -3 V 4 -3 V 3 -3 V 4 -2 V 4 -3 V 3 -3 V 4 -2 V 4 -3 V 3 -2 V 4 -3 V 4 -2 V 3 -3 V 4 -2 V 4 -2 V 3 -3 V 4 -2 V 4 -2 V 4 -2 V 3 -3 V 4 -2 V 4 -2 V 3 -2 V 4 -2 V 4 -2 V 4 -2 V 3 -2 V 4 -2 V 4 -2 V 3 -2 V 4 -1 V 4 -2 V 4 -2 V 3 -2 V 4 -1 V 4 -2 V 4 -2 V 3 -1 V 4 -2 V 4 -1 V 4 -2 V 3 -1 V 4 -2 V 4 -1 V 4 -1 V 3 -2 V 4 -1 V 4 -1 V 4 -1 V 3 -1 V 4 -2 V 4 -1 V 4 -1 V 3 -1 V 4 -1 V 4 -1 V 4 0 V 4 -1 V 3 -1 V 4 -1 V 4 -1 V 4 0 V 3 -1 V 4 -1 V 4 0 V 4 -1 V 4 0 V 3 -1 V 4 0 V 4 -1 V 4 0 V 3 -1 V currentpoint stroke M 4 0 V 4 0 V 4 0 V 4 -1 V 3 0 V 4 0 V 4 0 V 4 0 V 4 0 V 3 0 V 4 0 V 4 0 V 4 0 V 3 0 V 4 0 V 4 1 V 4 0 V 4 0 V 3 0 V 4 1 V 4 0 V 4 1 V 4 0 V 3 0 V 4 1 V 4 1 V 4 0 V 3 1 V 4 0 V 4 1 V 4 1 V 4 1 V 3 0 V 4 1 V 4 1 V 4 1 V 3 1 V 4 1 V 4 1 V 4 1 V 4 1 V 3 1 V 4 1 V 4 1 V 4 1 V 3 2 V 4 1 V 4 1 V 4 1 V 3 2 V 4 1 V 4 1 V 4 2 V 3 1 V 4 2 V 4 1 V 4 2 V 3 1 V 4 2 V 4 1 V 4 2 V 3 2 V 4 1 V 4 2 V 4 2 V 3 1 V 4 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 1 V 3 2 V 4 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 3 V 4 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 3 V 4 2 V 4 2 V 3 2 V 4 3 V 4 2 V 4 2 V 3 3 V 4 2 V 4 3 V 3 2 V 4 2 V 4 3 V 3 2 V 4 3 V 4 2 V 3 3 V 4 2 V 4 3 V 3 2 V 4 3 V 4 3 V 3 2 V 4 3 V 4 2 V 3 3 V 4 3 V 4 2 V 3 3 V 4 3 V 4 2 V 3 3 V 4 3 V 4 2 V 3 3 V 4 3 V 4 3 V 3 2 V 4 3 V 4 3 V 3 3 V 4 3 V 4 2 V 3 3 V 4 3 V 3 3 V 4 3 V 4 2 V 3 3 V 4 3 V 4 3 V 3 3 V 4 3 V 4 3 V 3 2 V 4 3 V 4 3 V 3 3 V 4 3 V 3 3 V 4 3 V 4 3 V 3 2 V 4 3 V 4 3 V 3 3 V 4 3 V 4 3 V 3 3 V 4 3 V 3 3 V 4 2 V 4 3 V 3 3 V 4 3 V 4 3 V 3 3 V 4 3 V 4 3 V 3 2 V 4 3 V 3 3 V 4 3 V 4 3 V 3 3 V 4 3 V 4 2 V 3 3 V 4 3 V 4 3 V 3 3 V 4 2 V 3 3 V 4 3 V 4 3 V 3 3 V 4 2 V 4 3 V 3 3 V 4 3 V 4 2 V 3 3 V 4 3 V 4 3 V 3 2 V 4 3 V 4 3 V 3 2 V 4 3 V 4 3 V 3 2 V 4 3 V 3 2 V 4 3 V 4 3 V 3 2 V 4 3 V 4 2 V 3 3 V 4 2 V 4 3 V 3 2 V 4 3 V 4 2 V 3 3 V 4 2 V 4 3 V 3 2 V 4 3 V 4 2 V 3 2 V 4 3 V 4 2 V 4 3 V 3 2 V 4 2 V 4 2 V 3 3 V 4 2 V 4 2 V 3 3 V 4 2 V 4 2 V 3 2 V 4 2 V 4 3 V 3 2 V 4 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 3 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 4 2 V 3 1 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 1 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 4 1 V 3 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 1 V 3 2 V 4 2 V 4 2 V 4 2 V 3 2 V 4 1 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 2 V 4 2 V 4 2 V 3 3 V 4 2 V 3 2 V 4 2 V 2 2 V 3 2 V 3 2 V 2 1 V 3 2 V 2 2 V 2 1 V 2 2 V 2 2 V 2 1 V 2 2 V 2 1 V 2 1 V 1 2 V 2 1 V 2 2 V 1 1 V 2 1 V 1 1 V 2 2 V 1 1 V 1 1 V 2 1 V 1 2 V 1 1 V 2 1 V 1 1 V 1 1 V 1 2 V 1 1 V 1 1 V 1 1 V 1 1 V 1 1 V 1 1 V 1 1 V 1 1 V 1 2 V 1 1 V 1 1 V 1 1 V 1 1 V 1 1 V 1 1 V 1 1 V 0 1 V 1 1 V 1 1 V 1 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 1 1 V 1 1 V 0 1 V 1 0 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V currentpoint stroke M 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 1 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 1 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 1 1 V 0 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 3 V 0 2 V 0 2 V 0 2 V 0 2 V 0 3 V 0 2 V 0 2 V 0 3 V -1 2 V 0 3 V 0 2 V 0 3 V 0 2 V 0 3 V 0 3 V 0 2 V 0 3 V 0 3 V 0 3 V 0 3 V -1 3 V 0 3 V 0 3 V 0 3 V 0 4 V 0 3 V 0 3 V -1 4 V 0 4 V 0 3 V 0 4 V 0 4 V -1 4 V 0 4 V 0 5 V 0 4 V -1 5 V 0 5 V 0 5 V -1 5 V 0 6 V -1 6 V 0 6 V -1 7 V 0 7 V -1 8 V 0 8 V -1 9 V -1 10 V -1 11 V 0 10 V -1 10 V -1 10 V -1 10 V -1 10 V -1 10 V 0 10 V -1 11 V -1 10 V -1 10 V -1 10 V 0 9 V -1 7 V 0 6 V -1 5 V 0 5 V 0 6 V -1 4 V 0 4 V 0 5 V 0 3 V -1 4 V 0 4 V 0 3 V 0 3 V 0 4 V -1 3 V 0 3 V 0 2 V 0 3 V 0 3 V 0 2 V 0 3 V 0 2 V 0 3 V -1 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 3 V 0 1 V 0 2 V 0 2 V 0 2 V 0 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 1 V 0 2 V 0 2 V 0 1 V 0 2 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 1 V 0 2 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V currentpoint stroke M 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 1 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 1 1 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 1 0 V 0 1 V 1 0 V 1 1 V 1 0 V 0 1 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 1 0 V 1 1 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 0 1 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 1 -1 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 1 0 V 0 -1 V 1 0 V 1 0 V 1 0 V 1 0 V 0 -1 V 1 0 V 1 0 V 1 0 V 0 -1 V 1 0 V 1 0 V 1 0 V 0 -1 V 1 0 V 1 0 V 1 -1 V 1 0 V 1 0 V 1 -1 V 1 0 V 1 -1 V 1 -1 V 1 0 V 1 -1 V 2 0 V 1 -1 V 1 0 V 1 -1 V 1 -1 V 1 0 V 1 -1 V 2 -1 V 1 -1 V 1 0 V 1 -1 V 2 -1 V 1 -1 V 2 -1 V 1 -1 V 2 -1 V 1 -1 V 2 -1 V 1 -1 V 2 -2 V 2 -1 V 2 -1 V 1 -2 V 2 -1 V 2 -2 V 2 -1 V 2 -2 V 2 -2 V 3 -1 V 2 -2 V 2 -2 V 3 -3 V 2 -2 V 3 -2 V 3 -3 V 2 -2 V 3 -3 V 3 -3 V 3 -3 V 4 -3 V 3 -3 V 4 -4 V 3 -4 V 4 -3 V 4 -4 V 3 -4 V 4 -3 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 4 -4 V 3 -3 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -5 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -5 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -5 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 4 -4 V 3 -5 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 4 -4 V 3 -5 V 4 -4 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 4 -5 V 3 -4 V 3 -4 V 4 -4 V 3 -4 V 3 -3 V 3 -4 V 3 -3 V 2 -2 V 2 -2 V 2 -2 V 1 -2 V 1 -1 V 2 -1 V 1 -2 V 1 -1 V 1 -1 V 1 -1 V 1 0 V 0 -1 V 1 -1 V 1 0 V 0 -1 V 1 0 V 0 -1 V 1 0 V 0 -1 V 1 0 V 0 -1 V 1 0 V 1 -1 V 1 -1 V 1 0 V 0 -1 V 1 0 V 1 -1 V 1 0 V 1 0 V 0 -1 V 1 0 V 1 0 V 0 1 V 1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V currentpoint stroke M 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 1 V -1 1 V 0 2 V -1 1 V 0 2 V -1 1 V -1 2 V -1 2 V 0 2 V -1 2 V -1 2 V -2 3 V -1 2 V -1 3 V -2 4 V -1 3 V -2 4 V -2 4 V -2 4 V -2 5 V -3 5 V -3 6 V -3 7 V -3 6 V -3 6 V -4 7 V -3 6 V -3 6 V -3 6 V -3 7 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 5 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 5 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -4 5 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -4 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -4 6 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -4 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 6 V -3 5 V -4 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -4 6 V -3 5 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 5 V -3 6 V -3 6 V -3 6 V -3 5 V -2 4 V -1 3 V -2 3 V -1 2 V -1 2 V -1 2 V -1 2 V 0 1 V -1 1 V -1 2 V 0 1 V -1 1 V 0 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V -1 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 0 1 V 0 1 V -1 0 V 0 1 V 0 1 V 1 0 V 0 1 V 1 0 V 1 0 V 0 -1 V 1 0 V 0 -1 V 1 0 V 1 -1 V 1 0 V 0 -1 V 1 -1 V 1 -1 V 1 -1 V 1 -1 V 1 -1 V 1 -1 V 0 -1 V 1 0 V 1 -1 V 0 -1 V 1 -1 V 1 -1 V 1 -1 V 1 -1 V 1 -2 V 2 -1 V 1 -2 V 1 -2 V 2 -2 V 2 -2 V 2 -3 V 2 -2 V 2 -3 V 2 -4 V 3 -3 V 3 -4 V 3 -5 V 4 -5 V 3 -5 V 4 -5 V 3 -5 V 4 -5 V 3 -5 V 4 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -5 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V currentpoint stroke M 3 -6 V 4 -5 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -5 V 3 -6 V 3 -5 V 4 -6 V 3 -5 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 3 -5 V 4 -5 V 3 -6 V 3 -5 V 3 -6 V 4 -5 V 3 -6 V 3 -5 V 3 -5 V 4 -6 V 3 -5 V 3 -6 V 1 -1 V 1.000 UL LTb /Times findfont 100 scalefont setfont 713 4794 M (rho) Lshow /Helvetica findfont 140 scalefont setfont /Times findfont 100 scalefont setfont 6962 2389 M (theta) Lshow /Helvetica findfont 140 scalefont setfont 1.000 UL LT0 238 2520 M 6724 0 V -122 -33 R 122 33 V -122 32 V 1.000 UL LT0 618 168 M 0 4704 V 32 -122 R -32 122 V 585 4750 L 3.000 UL LT0 3600 2520 M 3 5 V 1 1 V 3 5 V 3 6 V 3 5 V 4 6 V 3 5 V 3 5 V 3 6 V 4 5 V 3 6 V 3 5 V 4 5 V 3 6 V 3 5 V 3 5 V 4 6 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V 1 0 V 1 1 V 2 1 V 1 1 V 2 1 V 1 1 V 2 1 V 1 1 V 2 1 V 1 1 V 2 2 V 2 1 V 2 1 V 1 2 V 2 1 V 2 2 V 2 1 V 2 2 V 2 2 V 3 1 V 2 2 V 2 2 V 3 3 V 2 2 V 3 2 V 3 3 V 2 2 V 3 3 V 3 3 V 3 3 V 4 3 V 3 3 V 4 4 V 3 4 V 4 3 V 4 4 V 3 4 V 4 3 V 3 4 V 4 4 V 3 4 V 4 4 V 4 4 V 3 3 V 4 4 V 3 4 V 4 4 V 3 4 V 4 4 V 3 4 V 4 4 V 3 4 V 4 4 V 3 4 V 4 5 V 3 4 V 4 4 V 3 4 V 4 4 V 3 4 V 4 4 V 3 4 V 4 5 V 3 4 V 4 4 V 3 4 V 4 4 V 3 5 V 4 4 V 3 4 V 4 4 V 3 5 V 4 4 V 3 4 V 4 4 V 3 5 V 4 4 V 3 4 V 3 5 V 4 4 V 3 4 V 4 4 V 3 5 V 4 4 V 3 4 V 4 5 V 3 4 V 4 4 V 3 5 V 3 4 V 4 4 V 3 5 V 4 4 V 3 4 V 4 5 V 3 4 V 4 5 V 3 4 V 3 4 V 4 5 V 3 4 V 4 4 V 3 5 V 4 4 V 3 4 V 4 5 V 3 4 V 3 5 V 4 4 V 3 4 V 4 5 V 3 4 V 4 4 V 3 5 V 3 4 V 4 5 V 3 4 V 4 4 V 3 5 V 3 4 V 4 5 V 3 4 V 4 4 V 3 5 V 4 4 V 3 5 V 3 4 V 4 4 V 3 5 V 4 4 V 3 4 V 3 5 V 4 4 V 3 5 V 4 4 V 3 4 V 4 5 V 3 4 V 3 4 V 4 5 V 3 4 V 4 5 V 3 4 V 3 4 V 4 5 V 3 4 V 4 4 V 3 5 V 3 4 V 4 4 V 3 5 V 4 4 V 3 4 V 3 4 V 4 5 V 3 4 V 3 4 V 4 4 V 3 5 V 4 4 V 3 4 V 3 4 V 4 4 V 3 4 V 4 5 V 3 4 V 3 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-4 4 V -3 4 V -4 5 V -3 4 V -4 4 V -3 5 V -4 4 V -3 4 V -3 4 V -4 5 V -3 4 V -4 4 V -3 4 V -4 4 V -3 5 V -4 4 V -3 4 V -4 4 V -3 4 V -4 4 V -3 4 V -4 4 V -3 4 V -4 4 V -3 4 V -4 4 V -3 3 V -4 4 V -3 4 V -4 4 V -3 4 V -4 3 V -3 4 V -4 4 V -4 4 V -3 3 V -4 4 V -3 3 V -4 4 V -3 4 V -4 3 V -3 4 V -4 3 V -4 4 V -3 3 V -4 3 V -3 4 V -4 3 V -4 3 V -3 4 V -4 3 V -3 3 V -4 4 V -4 3 V -3 3 V -4 3 V -3 3 V -4 3 V -4 3 V -3 3 V -4 3 V -4 3 V -3 3 V -4 3 V -3 3 V -4 3 V -4 3 V -3 2 V -4 3 V -4 3 V -3 3 V -4 2 V -4 3 V -3 3 V -4 2 V -4 3 V -3 2 V -4 3 V -4 2 V -3 3 V -4 2 V -4 2 V -3 3 V -4 2 V -4 2 V -4 2 V -3 3 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -4 2 V -3 2 V -4 2 V -4 2 V -3 2 V -4 1 V -4 2 V -4 2 V -3 2 V -4 1 V -4 2 V -4 2 V -3 1 V -4 2 V -4 1 V -4 2 V -3 1 V -4 2 V -4 1 V -4 1 V -3 2 V -4 1 V -4 1 V -4 1 V -3 1 V -4 2 V -4 1 V -4 1 V -3 1 V -4 1 V -4 1 V -4 0 V -4 1 V -3 1 V -4 1 V -4 1 V -4 0 V -3 1 V -4 1 V -4 0 V -4 1 V -4 0 V -3 1 V -4 0 V -4 1 V -4 0 V -3 1 V currentpoint stroke M -4 0 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V -3 -3 V -4 -3 V -3 -3 V -4 -3 V -4 -2 V -3 -3 V -4 -3 V -4 -3 V -3 -3 V -4 -3 V -4 -3 V -3 -2 V -4 -3 V -4 -3 V -3 -3 V -4 -3 V -3 -3 V -4 -3 V -4 -3 V -3 -2 V -4 -3 V -4 -3 V -3 -3 V -4 -3 V -4 -3 V -3 -3 V -4 -3 V -3 -3 V -4 -2 V -4 -3 V -3 -3 V -4 -3 V -4 -3 V -3 -3 V -4 -3 V -4 -3 V -3 -2 V -4 -3 V -3 -3 V -4 -3 V -4 -3 V -3 -3 V -4 -3 V -4 -2 V -3 -3 V -4 -3 V -4 -3 V -3 -3 V -4 -2 V -3 -3 V -4 -3 V -4 -3 V -3 -3 V -4 -2 V -4 -3 V -3 -3 V -4 -3 V -4 -2 V -3 -3 V -4 -3 V -4 -3 V -3 -2 V -4 -3 V -4 -3 V -3 -2 V -4 -3 V -4 -3 V -3 -2 V -4 -3 V -3 -2 V -4 -3 V -4 -3 V -3 -2 V -4 -3 V -4 -2 V -3 -3 V -4 -2 V -4 -3 V -3 -2 V -4 -3 V -4 -2 V -3 -3 V -4 -2 V -4 -3 V -3 -2 V -4 -3 V -4 -2 V -3 -2 V -4 -3 V -4 -2 V -4 -3 V -3 -2 V -4 -2 V -4 -2 V -3 -3 V -4 -2 V -4 -2 V -3 -3 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -3 V -3 -2 V -4 -2 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -3 -2 V -4 -3 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -4 -2 V -3 -1 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -3 -1 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -4 -1 V -3 -2 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -1 V -3 -2 V -4 -2 V -4 -2 V -4 -2 V -3 -2 V -4 -1 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -3 -2 V -4 -2 V -4 -2 V -3 -3 V -4 -2 V -3 -2 V -4 -2 V -2 -2 V -3 -2 V -3 -2 V -2 -1 V -3 -2 V -2 -2 V -2 -1 V -2 -2 V -2 -2 V -2 -1 V -2 -2 V -2 -1 V -2 -1 V -1 -2 V -2 -1 V -2 -2 V -1 -1 V -2 -1 V -1 -1 V -2 -2 V -1 -1 V -1 -1 V -2 -1 V -1 -2 V -1 -1 V -2 -1 V -1 -1 V -1 -1 V -1 -2 V -1 -1 V -1 -1 V -1 -1 V -1 -1 V -1 -1 V -1 -1 V -1 -1 V -1 -1 V -1 -2 V -1 -1 V -1 -1 V -1 -1 V -1 -1 V -1 -1 V -1 -1 V -1 -1 V 0 -1 V -1 -1 V -1 -1 V -1 -1 V -1 -1 V 0 -1 V -1 -1 V -1 -1 V 0 -1 V -1 -1 V -1 -1 V -1 -1 V 0 -1 V -1 0 V -1 -1 V 0 -1 V -1 -1 V -1 -1 V 0 -1 V -1 -1 V 0 -1 V -1 -1 V -1 -1 V 0 -1 V -1 -1 V -1 -1 V 0 -1 V -1 -1 V 0 -1 V -1 -1 V -1 -1 V 0 -1 V -1 0 V 0 -1 V -1 -1 V 0 -1 V -1 -1 V 0 -1 V -1 -1 V 0 -1 V -1 -1 V 0 -1 V -1 -1 V 0 -1 V -1 -1 V -1 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V -1 0 V 0 -1 V currentpoint stroke M 0 -1 V -1 -1 V 0 -1 V -1 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -2 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -2 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -2 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -2 V 0 -1 V 0 -1 V -1 -1 V 0 -2 V 0 -1 V 0 -1 V 0 -1 V 0 -2 V 0 -1 V 0 -1 V 0 -2 V 0 -1 V 0 -1 V 0 -2 V -1 -1 V 0 -1 V 0 -2 V 0 -1 V 0 -1 V 0 -2 V 0 -1 V 0 -2 V 0 -1 V 0 -2 V 0 -1 V 0 -2 V 0 -1 V 0 -2 V 0 -1 V 0 -2 V 0 -1 V 0 -2 V -1 -1 V 0 -2 V 0 -2 V 0 -1 V 0 -2 V 0 -2 V 0 -1 V 0 -2 V 0 -2 V 0 -2 V 0 -1 V 0 -2 V 0 -2 V 0 -2 V 0 -2 V 0 -2 V 0 -1 V 0 -2 V 0 -2 V 0 -2 V 0 -2 V 0 -2 V 0 -2 V 0 -2 V 0 -2 V 0 -3 V 0 -2 V 0 -2 V 0 -2 V 0 -2 V 0 -3 V 0 -2 V 0 -2 V 0 -3 V 1 -2 V 0 -3 V 0 -2 V 0 -3 V 0 -2 V 0 -3 V 0 -3 V 0 -2 V 0 -3 V 0 -3 V 0 -3 V 0 -3 V 1 -3 V 0 -3 V 0 -3 V 0 -3 V 0 -4 V 0 -3 V 0 -3 V 1 -4 V 0 -4 V 0 -3 V 0 -4 V 0 -4 V 1 -4 V 0 -4 V 0 -5 V 0 -4 V 1 -5 V 0 -5 V 0 -5 V 1 -5 V 0 -6 V 1 -6 V 0 -6 V 1 -7 V 0 -7 V 1 -8 V 0 -8 V 1 -9 V 1 -10 V 1 -11 V 0 -10 V 1 -10 V 1 -10 V 1 -10 V 1 -10 V 1 -10 V 0 -10 V 1 -11 V 1 -10 V 1 -10 V 1 -10 V 0 -9 V 1 -7 V 0 -6 V 1 -5 V 0 -5 V 0 -6 V 1 -4 V 0 -4 V 0 -5 V 0 -3 V 1 -4 V 0 -4 V 0 -3 V 0 -3 V 0 -4 V 1 -3 V 0 -3 V 0 -2 V 0 -3 V 0 -3 V 0 -2 V 0 -3 V 0 -2 V 0 -3 V 1 -2 V 0 -2 V 0 -2 V 0 -2 V 0 -2 V 0 -3 V 0 -1 V 0 -2 V 0 -2 V 0 -2 V 0 -2 V 0 -2 V 0 -1 V 0 -2 V 0 -2 V 0 -1 V 0 -2 V 0 -2 V 0 -1 V 0 -2 V 0 -1 V 0 -2 V 0 -1 V 0 -1 V 0 -2 V 0 -1 V 0 -1 V 0 -2 V 0 -1 V 0 -1 V 0 -1 V 0 -2 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -2 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V currentpoint stroke M 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V -1 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 -1 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V 0 -1 V -1 0 V -1 -1 V -1 0 V 0 -1 V -1 0 V -1 0 V 0 -1 V -1 0 V -1 0 V 0 -1 V -1 0 V -1 0 V 0 -1 V -1 0 V -1 0 V -1 0 V -1 -1 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V 0 -1 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 1 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V -1 1 V -1 0 V -1 0 V -1 1 V -1 0 V -1 1 V -1 1 V -1 0 V -1 1 V -2 0 V -1 1 V -1 0 V -1 1 V -1 1 V -1 0 V -1 1 V -2 1 V -1 1 V -1 0 V -1 1 V -2 1 V -1 1 V -2 1 V -1 1 V -2 1 V -1 1 V -2 1 V -1 1 V -2 2 V -2 1 V -2 1 V -1 2 V -2 1 V -2 2 V -2 1 V -2 2 V -2 2 V -3 1 V -2 2 V -2 2 V -3 3 V -2 2 V -3 2 V -3 3 V -2 2 V -3 3 V -3 3 V -3 3 V -4 3 V -3 3 V -4 4 V -3 4 V -4 3 V -4 4 V -3 4 V -4 3 V -3 4 V -4 4 V -3 4 V -4 4 V -4 4 V -3 3 V -4 4 V -3 4 V -4 4 V -3 4 V -4 4 V -3 4 V -4 4 V -3 4 V -4 4 V -3 4 V -4 5 V -3 4 V -4 4 V -3 4 V -4 4 V -3 4 V -4 4 V -3 4 V -4 5 V -3 4 V -4 4 V -3 4 V -4 4 V -3 5 V -4 4 V -3 4 V -4 4 V -3 5 V -4 4 V -3 4 V -4 4 V -3 5 V -4 4 V -3 4 V -3 5 V -4 4 V -3 4 V -4 4 V -3 5 V -4 4 V -3 4 V -4 5 V -3 4 V -4 4 V -3 5 V -3 4 V -4 4 V -3 5 V -4 4 V -3 4 V -4 5 V -3 4 V -4 5 V -3 4 V -3 4 V -4 5 V -3 4 V -4 4 V -3 5 V -4 4 V -3 4 V -4 5 V -3 4 V -3 5 V -4 4 V -3 4 V -4 5 V -3 4 V -4 4 V -3 5 V -3 4 V -4 5 V -3 4 V -4 4 V -3 5 V -3 4 V -4 5 V -3 4 V -4 4 V -3 5 V -4 4 V -3 5 V -3 4 V -4 4 V -3 5 V -4 4 V -3 4 V -3 5 V -4 4 V -3 5 V -4 4 V -3 4 V -4 5 V -3 4 V -3 4 V -4 5 V -3 4 V -4 5 V -3 4 V -3 4 V -4 5 V -3 4 V -4 4 V -3 5 V -3 4 V -4 4 V -3 5 V -4 4 V -3 4 V -3 4 V -4 5 V -3 4 V -3 4 V -4 4 V -3 5 V -4 4 V -3 4 V -3 4 V -4 4 V -3 4 V -4 5 V -3 4 V -3 4 V -4 4 V -3 4 V -3 3 V -3 4 V -3 3 V -2 2 V -2 2 V -2 2 V -1 2 V -1 1 V -2 1 V -1 2 V -1 1 V -1 1 V -1 1 V -1 0 V 0 1 V -1 1 V -1 0 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V -1 0 V 0 1 V -1 0 V -1 1 V -1 1 V -1 0 V 0 1 V -1 0 V -1 1 V -1 0 V -1 0 V 0 1 V -1 0 V -1 0 V 0 -1 V -1 0 V 0 -1 V 0 -1 V -1 0 V 0 -1 V 0 -1 V 1 0 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 1 0 V 0 -1 V 0 -1 V 0 -1 V 1 -1 V 0 -1 V 0 -1 V 1 -1 V 0 -1 V 0 -1 V 0 -1 V 1 0 V 0 -1 V 0 -1 V 1 0 V 0 -1 V 0 -1 V 0 -1 V 1 0 V 0 -1 V currentpoint stroke M 0 -1 V 0 -1 V 1 0 V 0 -1 V 0 -1 V 1 -1 V 0 -1 V 1 -1 V 0 -1 V 0 -1 V 1 -1 V 1 -1 V 0 -2 V 1 -1 V 0 -2 V 1 -1 V 1 -2 V 1 -2 V 0 -2 V 1 -2 V 1 -2 V 2 -3 V 1 -2 V 1 -3 V 2 -4 V 1 -3 V 2 -4 V 2 -4 V 2 -4 V 2 -5 V 3 -5 V 3 -6 V 3 -7 V 3 -6 V 3 -6 V 4 -7 V 3 -6 V 3 -6 V 3 -6 V 3 -7 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 4 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 4 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 4 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 4 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 4 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 4 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 4 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 4 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -5 V 3 -6 V 3 -6 V 3 -6 V 3 -5 V 2 -4 V 1 -3 V 2 -3 V 1 -2 V 1 -2 V 1 -2 V 1 -2 V 0 -1 V 1 -1 V 1 -2 V 0 -1 V 1 -1 V 0 -1 V 0 -1 V 1 -1 V 0 -1 V 1 -1 V 0 -1 V 1 -1 V 0 -1 V 0 -1 V 1 0 V 0 -1 V 0 -1 V 0 -1 V 1 0 V 0 -1 V 0 -1 V 0 -1 V 1 0 V 0 -1 V 0 -1 V 0 -1 V 0 -1 V 1 0 V 0 -1 V 0 -1 V -1 0 V 0 -1 V -1 0 V -1 0 V 0 1 V -1 0 V 0 1 V -1 0 V -1 1 V -1 0 V 0 1 V -1 1 V -1 1 V -1 1 V -1 1 V -1 1 V -1 1 V 0 1 V -1 0 V -1 1 V 0 1 V -1 1 V -1 1 V -1 1 V -1 1 V -1 2 V -2 1 V -1 2 V -1 2 V -2 2 V -2 2 V -2 3 V -2 2 V -2 3 V -2 4 V -3 3 V -3 4 V -3 5 V -4 5 V -3 5 V -4 5 V -3 5 V -4 5 V -3 5 V -4 5 V -3 5 V -4 5 V -3 5 V -3 5 V -4 5 V -3 5 V -4 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 5 V -4 5 V currentpoint stroke M -3 6 V -4 5 V -3 5 V -3 5 V -4 5 V -3 6 V -3 5 V -4 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -4 5 V -3 6 V -3 5 V -3 5 V -4 6 V -3 5 V -3 5 V -4 6 V -3 5 V -3 6 V -3 5 V -4 5 V -3 6 V -3 5 V -4 6 V -3 5 V -3 5 V -3 6 V -4 5 V -3 6 V -3 5 V -4 6 V -3 5 V -3 5 V -3 6 V -4 5 V -3 6 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -3 5 V -4 6 V -3 5 V -3 6 V -3 5 V -4 5 V -3 6 V -3 5 V -4 6 V -3 5 V -3 6 V -3 5 V -4 6 V -3 5 V -3 5 V -3 6 V -4 5 V -3 6 V -3 5 V -3 6 V -4 5 V -3 6 V -3 5 V -3 5 V -4 6 V -3 5 V -3 6 V -3 5 V -4 6 V -3 5 V -3 5 V -3 6 V -4 5 V -3 6 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -4 5 V -3 6 V -3 5 V -3 5 V -4 6 V -3 5 V -3 6 V -3 5 V -4 5 V -3 6 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -3 5 V -4 6 V -3 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -3 5 V -4 6 V -3 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -3 5 V -4 5 V -3 6 V -3 5 V -3 5 V -4 6 V -3 5 V -3 5 V -4 6 V -3 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -3 5 V -4 5 V -3 6 V -3 5 V -4 5 V -3 6 V -3 5 V -3 5 V -4 5 V -3 6 V -3 5 V -3 5 V -4 6 V -3 5 V -3 5 V -4 6 V -3 5 V -3 5 V -3 5 V -4 6 V -3 5 V -3 5 V -4 6 V -3 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -3 5 V -4 5 V -3 5 V -3 6 V -4 5 V -3 5 V -3 6 V -3 5 V -4 5 V -3 6 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Fg(P)912 926 y Fq(j)t Fr(\025)p Fs(0)1055 905 y Fh(\013)1118 869 y Fq(\017)1117 930 y(j)1154 905 y Fh(h)1210 869 y Fs(2)p Fq(j)1308 905 y Fj(and)1489 839 y Fg(P)1577 926 y Fq(j)t Fr(\025)p Fs(0)1720 905 y Fh(!)1785 869 y Fq(\017)1781 930 y(j)1818 905 y Fh(h)1874 869 y Fs(2)p Fq(j)1972 905 y Fj(such)f(that)h(the)g(lob)-5 b(e)25 b(ar)-5 b(e)g(a)26 b Fh(A)i Fk(=)g(\001)3203 920 y Fs(Length)3424 905 y Fj(,)166 1025 y(the)j(L)-5 b(azutkin)-10 b('s)31 b(homo)-5 b(clinic)30 b(invariants)g Fh(!)1783 989 y Fr(\006)1841 1025 y Fj(,)i(and)e(the)h(sum)g Fk(\012)e(=)e Fh(!)2716 989 y Fs(+)2788 1025 y Fk(+)14 b Fh(!)2943 989 y Fr(\000)3033 1025 y Fj(asso)-5 b(ciate)g(d)166 1146 y(to)35 b(the)g(monomial)e(p)-5 b(erturb)g(ations)35 b(\(1\))f(have)h(the)g(asymptotic)f(exp)-5 b(ansions)438 1405 y Fh(A)551 1353 y Fs(as)544 1405 y Fk(=)33 b Fh(a\017)760 1402 y Fk(e)804 1369 y Fr(\000)p Fq(\031)902 1346 y Fp(2)936 1369 y Fq(=h)1033 1339 y Fg(P)1121 1426 y Fq(j)t Fr(\025)p Fs(0)1264 1405 y Fh(\013)1327 1369 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y(;)2177 1575 y Fk(\()p Fh(h)28 b Fi(!)f Fk(0)2475 1534 y Fs(+)2534 1575 y Fh(;)17 b(\017)35 b Fj(\014xe)-5 b(d)o Fk(\))p Fh(:)374 b Fk(\(10\))166 2098 y Fj(In)34 b(addition,)g(the)h(fol)5 b(lowing)34 b(pr)-5 b(op)g(erties)34 b(hold:)193 2318 y(\(1\))49 b(If)30 b Fh(\013)533 2282 y Fs(0)573 2318 y Fk(\()p Fh(h)p Fk(\))d(=)836 2251 y Fg(P)923 2338 y Fq(j)t Fr(\025)p Fs(0)1067 2318 y Fh(\013)1130 2282 y Fs(0)1129 2342 y Fq(j)1169 2318 y Fh(h)1225 2282 y Fs(2)p Fq(j)1328 2318 y Fj(and)j Fh(!)1578 2282 y Fs(0)1617 2318 y Fk(\()p Fh(h)p Fk(\))e(=)1880 2251 y Fg(P)1968 2338 y Fq(j)t Fr(\025)p Fs(0)2111 2318 y Fh(!)2176 2282 y Fs(0)2172 2342 y Fq(j)2215 2318 y Fh(h)2271 2282 y Fs(2)p Fq(j)2374 2318 y Fj(ar)-5 b(e)31 b(the)g(T)-7 b(aylor)30 b(exp)-5 b(ansions)372 2438 y(of)29 b(the)g(even)g(analytic) g(functions)g(that)h(app)-5 b(e)g(ar)28 b(in)h(the)g(Melnikov)g(terms)g (\(6\),)g(then)372 2558 y Fh(\013)435 2522 y Fq(\017)434 2583 y(j)498 2558 y Fk(=)f Fh(\013)665 2522 y Fs(0)664 2583 y Fq(j)726 2558 y Fk(+)824 2567 y(O)900 2558 y(\()p Fh(\017)p Fk(\))35 b Fj(and)f Fh(!)1304 2522 y Fq(\017)1300 2583 y(j)1364 2558 y Fk(=)28 b Fh(!)1533 2522 y Fs(0)1529 2583 y Fq(j)1594 2558 y Fk(+)1692 2567 y(O)1768 2558 y(\()p Fh(\017)p Fk(\))35 b Fj(for)f(al)5 b(l)35 b Fh(j)e Fi(\025)c Fk(0)p Fj(.)193 2679 y(\(2\))49 b(The)34 b(series)846 2612 y Fg(P)934 2700 y Fq(j)t Fr(\025)p Fs(0)1077 2679 y Fh(\013)1140 2643 y Fq(\017)1139 2703 y(j)1176 2679 y Fh(h)1232 2643 y Fs(2)p Fq(j)1339 2679 y Fj(and)1528 2612 y Fg(P)1616 2700 y Fq(j)t Fr(\025)p Fs(0)1759 2679 y Fh(!)1824 2643 y Fq(\017)1820 2703 y(j)1856 2679 y Fh(h)1912 2643 y Fs(2)p Fq(j)2019 2679 y Fj(ar)-5 b(e)35 b(Gevr)-5 b(ey-1)35 b(of)f(typ)-5 b(e)35 b Fh(\032)28 b Fk(=)g(1)p Fh(=)p Fk(2)p Fh(\031)3304 2643 y Fs(2)3343 2679 y Fj(.)193 2799 y(\(3\))49 b(The)34 b(se)-5 b(quenc)g(e)34 b Fk(\()10 b(\026)-59 b Fh(\013)1068 2763 y Fq(\017)1067 2824 y(j)1104 2799 y Fk(\))1142 2814 y Fq(j)t Fr(\025)p Fs(0)1303 2799 y Fj(de\014ne)-5 b(d)34 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Fq(\017)2671 3509 y(l)2709 3484 y Fh(j)2755 3448 y Fr(\000)p Fq(l)2871 3484 y Fj(as)34 b Fh(j)g Fi(!)27 b Fk(+)p Fi(1)p Fj(.)193 3604 y(\(4\))49 b(The)34 b(se)-5 b(quenc)g(es)34 b Fk(\()8 b(\026)-57 b Fh(!)1110 3568 y Fs(0)1106 3629 y Fq(j)1149 3604 y Fk(\))1187 3619 y Fq(j)t Fr(\025)p Fs(0)1313 3604 y Fj(,)35 b Fk(\()8 b(\026)-57 b Fh(!)1481 3568 y Fs(1)1477 3629 y Fq(j)1520 3604 y Fk(\))1558 3619 y Fq(j)t Fr(\025)p Fs(0)1719 3604 y Fj(and)35 b Fk(\()8 b(^)-57 b Fh(!)2012 3568 y Fq(\017)2008 3629 y(j)2044 3604 y Fk(\))2082 3619 y Fq(j)t Fr(\025)p Fs(0)2243 3604 y Fj(de\014ne)-5 b(d)34 b(by)575 3900 y Fk(\026)-57 b Fh(!)632 3859 y Fs(0)628 3925 y Fq(j)693 3900 y Fk(+)22 b Fh(\017)8 b Fk(\026)-57 b Fh(!)895 3859 y Fs(1)891 3925 y Fq(j)956 3900 y Fk(+)22 b Fh(\017)1093 3859 y Fs(2)1133 3900 y Fh(j)1179 3859 y Fs(6)1226 3900 y Fk(^)-56 b Fh(!)1284 3859 y Fq(\017)1280 3925 y(j)1343 3900 y Fk(:=)1520 3833 y(\(2)p Fh(\031)1666 3797 y Fs(2)1705 3833 y Fk(\))1743 3797 y Fs(2)p Fq(j)p 1484 3877 367 4 v 1484 3968 a Fk(\(2)p Fh(j)28 b Fk(+)22 b(2\)!)1871 3826 y Fh(!)1936 3790 y Fq(\017)1932 3851 y(j)1990 3826 y Fi(\000)h Fh(!)2155 3790 y Fs(0)2151 3851 y Fq(j)p 1871 3877 323 4 v 2012 3968 a Fh(\017)3280 3900 y Fk(\(12\))372 4178 y Fj(have)h(some)h(\014nite)g(limits)32 b Fk(\026)-57 b Fh(!)1394 4141 y Fs(0)1390 4202 y Fr(1)1465 4178 y Fh(;)24 b Fk(\026)-56 b Fh(!)1574 4141 y Fs(1)1570 4202 y Fr(1)1644 4178 y Fh(;)25 b Fk(^)-57 b Fh(!)1753 4141 y Fq(\017)1749 4202 y Fr(1)1851 4178 y Fi(6)p Fk(=)27 b(0)e Fj(when)g Fh(j)33 b Fi(!)28 b Fk(+)p Fi(1)p Fj(.)c(In)h(fact,)g (ther)-5 b(e)25 b(exist)372 4298 y(some)g(asymptotic)g(c)-5 b(o)g(e\016cients)32 b Fk(\026)-56 b Fh(\021)1633 4262 y Fs(0)1629 4323 y Fq(l)1672 4298 y Fj(,)32 b Fk(\026)-56 b Fh(\021)1779 4262 y Fs(1)1775 4323 y Fq(l)1844 4298 y Fj(and)32 b Fk(^)-56 b Fh(\021)2076 4262 y Fq(\017)2072 4323 y(l)2134 4298 y Fj(such)25 b(that)34 b Fk(\026)-57 b Fh(!)2601 4262 y Fs(0)2597 4323 y Fq(j)2674 4245 y Fs(as)2668 4298 y Fk(=)35 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b(e)37 b(exist)g(some)g(values)47 b Fk(\026)-58 b Fh(\013)1494 4517 y Fs(0)1493 4578 y Fr(1)1567 4554 y Fh(;)25 b Fk(^)-57 b Fh(!)1676 4517 y Fs(0)1672 4578 y Fr(1)1746 4554 y Fh(;)1804 4527 y Fk(\026)1790 4554 y Fh(\014)1851 4517 y Fs(0)1845 4578 y Fq(l)1890 4554 y Fh(;)24 b Fk(^)-56 b Fh(\021)1986 4517 y Fs(0)1982 4578 y Fq(l)2058 4554 y Fi(2)33 b Fe(R)49 b Fj(such)37 b(that)48 b Fk(\026)-59 b Fh(\013)2756 4517 y Fq(\017)2755 4578 y Fr(1)2863 4554 y Fk(=)42 b(\026)-58 b Fh(\013)3035 4517 y Fs(0)3034 4578 y Fr(1)3133 4554 y Fk(+)3233 4563 y(O)3309 4554 y(\()p Fh(\017)p Fk(\))p Fj(,)380 4674 y Fk(^)h Fh(!)437 4638 y Fq(\017)433 4699 y Fr(1)535 4674 y Fk(=)35 b(^)-56 b Fh(!)704 4638 y Fs(0)700 4699 y Fr(1)796 4674 y Fk(+)894 4683 y(O)970 4674 y(\()p Fh(\017)p Fk(\))p Fj(,)1164 4648 y Fk(\026)1150 4674 y Fh(\014)1211 4638 y Fq(\017)1205 4699 y(l)1271 4674 y Fk(=)1389 4648 y(\026)1375 4674 y Fh(\014)1436 4638 y Fs(0)1430 4699 y Fq(l)1497 4674 y Fk(+)1595 4683 y(O)1671 4674 y(\()p 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y Fq(j)3416 5259 y Fk(\))166 5380 y(corresp)s(onding)31 b(to)g(the)g(\014rst)h(\014v)m(e)g(T)-8 b(a)m(ylor)31 b(co)s(e\016cien)m(ts)h(of)f(the)g(functions)g(\(8\).)g(It)g(turns)1745 5712 y(20)p eop %%Page: 21 21 21 20 bop 168 1138 a @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1951 @rwi @setspecial %%BeginDocument: alpha0.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: alpha0.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Tue Sep 28 19:15:52 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { 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mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall 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V 64 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 64 0 V 64 0 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 1658 w @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1951 @rwi @setspecial %%BeginDocument: omega0.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: omega0.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Tue Oct 5 09:32:10 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont end %%EndProlog gnudict begin gsave 50 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Fq(\017)747 1342 y Fs(0)787 1318 y Fu(=\013)890 1285 y Fs(0)890 1342 y(0)951 1318 y Ft(\000)20 b Fv(1)p Ft(j)31 b Fv(\(left\))g(and)e Ft(j)p Fu(!)1634 1285 y Fq(\017)1631 1342 y Fs(0)1671 1318 y Fu(=!)1776 1285 y Fs(0)1773 1342 y(0)1836 1318 y Ft(\000)19 b Fv(1)p Ft(j)31 b Fv(\(righ)m(t\))g(v)m(ersus)f Fu(\017)p Fv(,)h(for)f Fu(n)25 b Fv(=)g(2)p Fu(;)15 b Fv(3)p Fu(;)g Fv(4.)168 2548 y @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1951 @rwi @setspecial %%BeginDocument: alpha_quartic.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: alpha_quartic.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Mon Sep 27 20:55:17 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill 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bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 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/hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } 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4981 y Fq(\017)2011 5042 y(j)2080 5018 y Fk(for)f Fh(n)c Fk(=)g(2)p Fh(;)17 b Fk(3)p Fh(;)g Fk(4)31 b(and)i(sev)m(eral)h(v)-5 b(alues)166 5138 y(of)39 b Fh(\017)p Fk(.)h(T)-8 b(o)40 b(b)s(e)f(precise,)i(w)m(e)f(ha)m(v)m(e) h(p)s(erformed)e(the)h(computations)e(for)h Fh(\017)h Fk(=)f(10)3127 5102 y Fr(\000)p Fq(k)3264 5138 y Fk(with)166 5258 y Fh(k)d Fk(=)d(1)p Fh(;)17 b Fk(2)p Fh(;)g(:)g(:)g(:)e(;)i Fk(9)p Fh(;)g Fk(10)p Fh(;)g Fk(50.)34 b(All)g(the)i(considered)h(v)-5 b(alues)36 b(of)f Fh(\017)h Fk(ha)m(v)m(e)h(the)f(same)g(b)s(eha)m (vior.)166 5379 y(The)h(results)g(for)f Fh(\017)f Fk(=)1049 5339 y Fs(1)p 1031 5355 71 4 v 1031 5413 a(10)1148 5379 y Fk(are)i(sho)m(wn)h(in)d(\014gure)i(6.)f(They)i(strongly)f(suggest)g (that)f(the)1745 5712 y(21)p eop %%Page: 22 22 22 21 bop 168 1138 a @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1951 @rwi @setspecial %%BeginDocument: alphatilde.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: rho1_1.eps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Fri 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mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 574 280 M 63 0 V 6325 0 R -63 0 V 490 280 M (0.9) Rshow 574 739 M 63 0 V 6325 0 R -63 0 V 490 739 M (0.95) Rshow 574 1198 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1) Rshow 574 1658 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.05) Rshow 574 2117 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.1) Rshow 574 2576 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.15) Rshow 574 3035 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.2) Rshow 574 3494 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.25) Rshow 574 3954 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.3) Rshow 574 4413 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.35) Rshow 574 4872 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.4) Rshow 574 280 M 0 63 V 0 4529 R 0 -63 V 574 140 M (0) Cshow 1735 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (50) Cshow 2897 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (100) Cshow 4058 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (150) Cshow 5220 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (200) Cshow 6381 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (250) Cshow 1.000 UL LTa 574 280 M 0 4592 V 1.000 UL LTb 574 280 M 6388 0 V 0 4592 V -6388 0 V 574 280 L 0.600 UP 1.000 UL LT0 806 2609 CircleF 830 3370 CircleF 853 2984 CircleF 876 2942 CircleF 899 2863 CircleF 922 2792 CircleF 946 2726 CircleF 969 2666 CircleF 992 2611 CircleF 1015 2560 CircleF 1039 2513 CircleF 1062 2469 CircleF 1085 2428 CircleF 1108 2391 CircleF 1131 2355 CircleF 1155 2322 CircleF 1178 2291 CircleF 1201 2262 CircleF 1224 2235 CircleF 1248 2209 CircleF 1271 2184 CircleF 1294 2161 CircleF 1317 2139 CircleF 1341 2118 CircleF 1364 2099 CircleF 1387 2080 CircleF 1410 2062 CircleF 1433 2044 CircleF 1457 2028 CircleF 1480 2012 CircleF 1503 1997 CircleF 1526 1983 CircleF 1550 1969 CircleF 1573 1955 CircleF 1596 1942 CircleF 1619 1930 CircleF 1643 1918 CircleF 1666 1907 CircleF 1689 1896 CircleF 1712 1885 CircleF 1735 1875 CircleF 1759 1865 CircleF 1782 1855 CircleF 1805 1846 CircleF 1828 1836 CircleF 1852 1828 CircleF 1875 1819 CircleF 1898 1811 CircleF 1921 1803 CircleF 1945 1795 CircleF 1968 1787 CircleF 1991 1780 CircleF 2014 1773 CircleF 2037 1766 CircleF 2061 1759 CircleF 2084 1753 CircleF 2107 1746 CircleF 2130 1740 CircleF 2154 1734 CircleF 2177 1728 CircleF 2200 1722 CircleF 2223 1716 CircleF 2246 1711 CircleF 2270 1705 CircleF 2293 1700 CircleF 2316 1695 CircleF 2339 1690 CircleF 2363 1685 CircleF 2386 1680 CircleF 2409 1675 CircleF 2432 1671 CircleF 2456 1666 CircleF 2479 1662 CircleF 2502 1658 CircleF 2525 1653 CircleF 2548 1649 CircleF 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CircleF 3873 1499 CircleF 3896 1498 CircleF 3919 1496 CircleF 3942 1495 CircleF 3965 1493 CircleF 3989 1491 CircleF 4012 1490 CircleF 4035 1488 CircleF 4058 1487 CircleF 4082 1485 CircleF 4105 1484 CircleF 4128 1482 CircleF 4151 1481 CircleF 4175 1479 CircleF 4198 1478 CircleF 4221 1476 CircleF 4244 1475 CircleF 4267 1474 CircleF 4291 1472 CircleF 4314 1471 CircleF 4337 1469 CircleF 4360 1468 CircleF 4384 1467 CircleF 4407 1465 CircleF 4430 1464 CircleF 4453 1463 CircleF 4476 1462 CircleF 4500 1460 CircleF 4523 1459 CircleF 4546 1458 CircleF 4569 1457 CircleF 4593 1456 CircleF 4616 1454 CircleF 4639 1453 CircleF 4662 1452 CircleF 4686 1451 CircleF 4709 1450 CircleF 4732 1449 CircleF 4755 1447 CircleF 4778 1446 CircleF 4802 1445 CircleF 4825 1444 CircleF 4848 1443 CircleF 4871 1442 CircleF 4895 1441 CircleF 4918 1440 CircleF 4941 1439 CircleF 4964 1438 CircleF 4988 1437 CircleF 5011 1436 CircleF 5034 1435 CircleF 5057 1434 CircleF 5080 1433 CircleF 5104 1432 CircleF 5127 1431 CircleF 5150 1430 CircleF 5173 1429 CircleF 5197 1428 CircleF 5220 1427 CircleF 5243 1426 CircleF 5266 1425 CircleF 5290 1424 CircleF 5313 1424 CircleF 5336 1423 CircleF 5359 1422 CircleF 5382 1421 CircleF 5406 1420 CircleF 5429 1419 CircleF 5452 1418 CircleF 5475 1417 CircleF 5499 1417 CircleF 5522 1416 CircleF 5545 1415 CircleF 5568 1414 CircleF 5591 1413 CircleF 5615 1413 CircleF 5638 1412 CircleF 5661 1411 CircleF 5684 1410 CircleF 5708 1409 CircleF 5731 1409 CircleF 5754 1408 CircleF 5777 1407 CircleF 5801 1406 CircleF 5824 1406 CircleF 5847 1405 CircleF 5870 1404 CircleF 5893 1403 CircleF 5917 1403 CircleF 5940 1402 CircleF 5963 1401 CircleF 5986 1401 CircleF 6010 1400 CircleF 6033 1399 CircleF 6056 1398 CircleF 6079 1398 CircleF 6103 1397 CircleF 6126 1396 CircleF 6149 1396 CircleF 6172 1395 CircleF 6195 1394 CircleF 6219 1394 CircleF 6242 1393 CircleF 6265 1392 CircleF 6288 1392 CircleF 6312 1391 CircleF 6335 1391 CircleF 6358 1390 CircleF 6381 1389 CircleF 6405 1389 CircleF 6428 1388 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2242 TriUF 1666 2224 TriUF 1689 2206 TriUF 1712 2188 TriUF 1735 2172 TriUF 1759 2156 TriUF 1782 2140 TriUF 1805 2125 TriUF 1828 2111 TriUF 1852 2097 TriUF 1875 2083 TriUF 1898 2070 TriUF 1921 2057 TriUF 1945 2045 TriUF 1968 2033 TriUF 1991 2021 TriUF 2014 2010 TriUF 2037 1999 TriUF 2061 1989 TriUF 2084 1978 TriUF 2107 1968 TriUF 2130 1959 TriUF 2154 1949 TriUF 2177 1940 TriUF 2200 1931 TriUF 2223 1922 TriUF 2246 1914 TriUF 2270 1906 TriUF 2293 1897 TriUF 2316 1890 TriUF 2339 1882 TriUF 2363 1874 TriUF 2386 1867 TriUF 2409 1860 TriUF 2432 1853 TriUF 2456 1846 TriUF 2479 1839 TriUF 2502 1833 TriUF 2525 1826 TriUF 2548 1820 TriUF 2572 1814 TriUF 2595 1808 TriUF 2618 1802 TriUF 2641 1796 TriUF 2665 1791 TriUF 2688 1785 TriUF 2711 1780 TriUF 2734 1774 TriUF 2758 1769 TriUF 2781 1764 TriUF 2804 1759 TriUF 2827 1754 TriUF 2850 1749 TriUF 2874 1745 TriUF 2897 1740 TriUF 2920 1735 TriUF 2943 1731 TriUF 2967 1726 TriUF 2990 1722 TriUF 3013 1718 TriUF 3036 1714 TriUF 3060 1710 TriUF 3083 1706 TriUF 3106 1702 TriUF 3129 1698 TriUF 3152 1694 TriUF 3176 1690 TriUF 3199 1686 TriUF 3222 1683 TriUF 3245 1679 TriUF 3269 1676 TriUF 3292 1672 TriUF 3315 1669 TriUF 3338 1665 TriUF 3361 1662 TriUF 3385 1659 TriUF 3408 1656 TriUF 3431 1652 TriUF 3454 1649 TriUF 3478 1646 TriUF 3501 1643 TriUF 3524 1640 TriUF 3547 1637 TriUF 3571 1634 TriUF 3594 1632 TriUF 3617 1629 TriUF 3640 1626 TriUF 3663 1623 TriUF 3687 1620 TriUF 3710 1618 TriUF 3733 1615 TriUF 3756 1613 TriUF 3780 1610 TriUF 3803 1607 TriUF 3826 1605 TriUF 3849 1602 TriUF 3873 1600 TriUF 3896 1598 TriUF 3919 1595 TriUF 3942 1593 TriUF 3965 1591 TriUF 3989 1588 TriUF 4012 1586 TriUF 4035 1584 TriUF 4058 1582 TriUF 4082 1580 TriUF 4105 1577 TriUF 4128 1575 TriUF 4151 1573 TriUF 4175 1571 TriUF 4198 1569 TriUF 4221 1567 TriUF 4244 1565 TriUF 4267 1563 TriUF 4291 1561 TriUF 4314 1559 TriUF 4337 1557 TriUF 4360 1555 TriUF 4384 1554 TriUF 4407 1552 TriUF 4430 1550 TriUF 4453 1548 TriUF 4476 1546 TriUF 4500 1545 TriUF 4523 1543 TriUF 4546 1541 TriUF 4569 1539 TriUF 4593 1538 TriUF 4616 1536 TriUF 4639 1534 TriUF 4662 1533 TriUF 4686 1531 TriUF 4709 1529 TriUF 4732 1528 TriUF 4755 1526 TriUF 4778 1525 TriUF 4802 1523 TriUF 4825 1522 TriUF 4848 1520 TriUF 4871 1519 TriUF 4895 1517 TriUF 4918 1516 TriUF 4941 1514 TriUF 4964 1513 TriUF 4988 1511 TriUF 5011 1510 TriUF 5034 1509 TriUF 5057 1507 TriUF 5080 1506 TriUF 5104 1505 TriUF 5127 1503 TriUF 5150 1502 TriUF 5173 1501 TriUF 5197 1499 TriUF 5220 1498 TriUF 5243 1497 TriUF 5266 1495 TriUF 5290 1494 TriUF 5313 1493 TriUF 5336 1492 TriUF 5359 1490 TriUF 5382 1489 TriUF 5406 1488 TriUF 5429 1487 TriUF 5452 1486 TriUF 5475 1485 TriUF 5499 1483 TriUF 5522 1482 TriUF 5545 1481 TriUF 5568 1480 TriUF 5591 1479 TriUF 5615 1478 TriUF 5638 1477 TriUF 5661 1476 TriUF 5684 1474 TriUF 5708 1473 TriUF 5731 1472 TriUF 5754 1471 TriUF 5777 1470 TriUF 5801 1469 TriUF 5824 1468 TriUF 5847 1467 TriUF 5870 1466 TriUF 5893 1465 TriUF 5917 1464 TriUF 5940 1463 TriUF 5963 1462 TriUF 5986 1461 TriUF 6010 1460 TriUF 6033 1459 TriUF 6056 1458 TriUF 6079 1457 TriUF 6103 1456 TriUF 6126 1455 TriUF 6149 1455 TriUF 6172 1454 TriUF 6195 1453 TriUF 6219 1452 TriUF 6242 1451 TriUF 6265 1450 TriUF 6288 1449 TriUF 6312 1448 TriUF 6335 1447 TriUF 6358 1447 TriUF 6381 1446 TriUF 6405 1445 TriUF 6428 1444 TriUF 6451 1443 TriUF 6474 1442 TriUF 6497 1442 TriUF 6521 1441 TriUF 6544 1440 TriUF 6567 1439 TriUF 6590 1438 TriUF 6614 1437 TriUF 6637 1437 TriUF 6660 1436 TriUF 6683 1435 TriUF 6706 1434 TriUF 6730 1434 TriUF 6753 1433 TriUF 6776 1432 TriUF 6799 1431 TriUF 6823 1431 TriUF 6846 1430 TriUF 6869 1429 TriUF 6892 1428 TriUF 6916 1428 TriUF 6939 1427 TriUF 6962 1426 TriUF 0.600 UP 1.000 UL LT2 830 4606 DiaF 853 4034 DiaF 876 3788 DiaF 899 3684 DiaF 922 3548 DiaF 946 3434 DiaF 969 3330 DiaF 992 3235 DiaF 1015 3149 DiaF 1039 3071 DiaF 1062 2999 DiaF 1085 2933 DiaF 1108 2871 DiaF 1131 2815 DiaF 1155 2762 DiaF 1178 2713 DiaF 1201 2667 DiaF 1224 2624 DiaF 1248 2584 DiaF 1271 2546 DiaF 1294 2511 DiaF 1317 2477 DiaF 1341 2445 DiaF 1364 2415 DiaF 1387 2387 DiaF 1410 2360 DiaF 1433 2334 DiaF 1457 2309 DiaF 1480 2286 DiaF 1503 2263 DiaF 1526 2242 DiaF 1550 2222 DiaF 1573 2202 DiaF 1596 2183 DiaF 1619 2165 DiaF 1643 2148 DiaF 1666 2131 DiaF 1689 2115 DiaF 1712 2100 DiaF 1735 2085 DiaF 1759 2070 DiaF 1782 2057 DiaF 1805 2043 DiaF 1828 2030 DiaF 1852 2018 DiaF 1875 2006 DiaF 1898 1994 DiaF 1921 1983 DiaF 1945 1972 DiaF 1968 1961 DiaF 1991 1951 DiaF 2014 1941 DiaF 2037 1931 DiaF 2061 1921 DiaF 2084 1912 DiaF 2107 1903 DiaF 2130 1894 DiaF 2154 1886 DiaF 2177 1878 DiaF 2200 1870 DiaF 2223 1862 DiaF 2246 1854 DiaF 2270 1847 DiaF 2293 1839 DiaF 2316 1832 DiaF 2339 1825 DiaF 2363 1819 DiaF 2386 1812 DiaF 2409 1806 DiaF 2432 1799 DiaF 2456 1793 DiaF 2479 1787 DiaF 2502 1781 DiaF 2525 1775 DiaF 2548 1770 DiaF 2572 1764 DiaF 2595 1759 DiaF 2618 1753 DiaF 2641 1748 DiaF 2665 1743 DiaF 2688 1738 DiaF 2711 1733 DiaF 2734 1728 DiaF 2758 1724 DiaF 2781 1719 DiaF 2804 1715 DiaF 2827 1710 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64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 1658 w @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1951 @rwi @setspecial %%BeginDocument: omegatilde.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: rho2_1.eps %%Creator: gnuplot 3.7 patchlevel 1 %%CreationDate: Fri Aug 6 19:36:52 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 574 280 M 63 0 V 6325 0 R -63 0 V 490 280 M (0.9) Rshow 574 739 M 63 0 V 6325 0 R -63 0 V 490 739 M (0.95) Rshow 574 1198 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1) Rshow 574 1658 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.05) Rshow 574 2117 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.1) Rshow 574 2576 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.15) Rshow 574 3035 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.2) Rshow 574 3494 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.25) Rshow 574 3954 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.3) Rshow 574 4413 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.35) Rshow 574 4872 M 63 0 V 6325 0 R -63 0 V -6409 0 R (1.4) Rshow 574 280 M 0 63 V 0 4529 R 0 -63 V 574 140 M (0) Cshow 1735 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (50) Cshow 2897 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (100) Cshow 4058 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (150) Cshow 5220 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (200) Cshow 6381 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R (250) Cshow 1.000 UL LTa 574 280 M 0 4592 V 1.000 UL LTb 574 280 M 6388 0 V 0 4592 V -6388 0 V 574 280 L 0.600 UP 1.000 UL LT0 1317 4867 CircleF 1341 4783 CircleF 1364 4704 CircleF 1387 4628 CircleF 1410 4556 CircleF 1433 4487 CircleF 1457 4421 CircleF 1480 4358 CircleF 1503 4298 CircleF 1526 4240 CircleF 1550 4184 CircleF 1573 4131 CircleF 1596 4080 CircleF 1619 4030 CircleF 1643 3983 CircleF 1666 3937 CircleF 1689 3893 CircleF 1712 3850 CircleF 1735 3809 CircleF 1759 3769 CircleF 1782 3731 CircleF 1805 3694 CircleF 1828 3658 CircleF 1852 3623 CircleF 1875 3589 CircleF 1898 3556 CircleF 1921 3524 CircleF 1945 3493 CircleF 1968 3463 CircleF 1991 3434 CircleF 2014 3406 CircleF 2037 3379 CircleF 2061 3352 CircleF 2084 3326 CircleF 2107 3300 CircleF 2130 3276 CircleF 2154 3252 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3454 2471 CircleF 3478 2463 CircleF 3501 2455 CircleF 3524 2447 CircleF 3547 2439 CircleF 3571 2432 CircleF 3594 2424 CircleF 3617 2416 CircleF 3640 2409 CircleF 3663 2402 CircleF 3687 2394 CircleF 3710 2387 CircleF 3733 2380 CircleF 3756 2373 CircleF 3780 2366 CircleF 3803 2360 CircleF 3826 2353 CircleF 3849 2346 CircleF 3873 2340 CircleF 3896 2333 CircleF 3919 2327 CircleF 3942 2321 CircleF 3965 2314 CircleF 3989 2308 CircleF 4012 2302 CircleF 4035 2296 CircleF 4058 2290 CircleF 4082 2284 CircleF 4105 2279 CircleF 4128 2273 CircleF 4151 2267 CircleF 4175 2262 CircleF 4198 2256 CircleF 4221 2251 CircleF 4244 2245 CircleF 4267 2240 CircleF 4291 2235 CircleF 4314 2230 CircleF 4337 2224 CircleF 4360 2219 CircleF 4384 2214 CircleF 4407 2209 CircleF 4430 2204 CircleF 4453 2199 CircleF 4476 2195 CircleF 4500 2190 CircleF 4523 2185 CircleF 4546 2180 CircleF 4569 2176 CircleF 4593 2171 CircleF 4616 2167 CircleF 4639 2162 CircleF 4662 2158 CircleF 4686 2153 CircleF 4709 2149 CircleF 4732 2145 CircleF 4755 2140 CircleF 4778 2136 CircleF 4802 2132 CircleF 4825 2128 CircleF 4848 2124 CircleF 4871 2120 CircleF 4895 2116 CircleF 4918 2112 CircleF 4941 2108 CircleF 4964 2104 CircleF 4988 2100 CircleF 5011 2096 CircleF 5034 2092 CircleF 5057 2088 CircleF 5080 2085 CircleF 5104 2081 CircleF 5127 2077 CircleF 5150 2074 CircleF 5173 2070 CircleF 5197 2067 CircleF 5220 2063 CircleF 5243 2059 CircleF 5266 2056 CircleF 5290 2053 CircleF 5313 2049 CircleF 5336 2046 CircleF 5359 2042 CircleF 5382 2039 CircleF 5406 2036 CircleF 5429 2033 CircleF 5452 2029 CircleF 5475 2026 CircleF 5499 2023 CircleF 5522 2020 CircleF 5545 2017 CircleF 5568 2013 CircleF 5591 2010 CircleF 5615 2007 CircleF 5638 2004 CircleF 5661 2001 CircleF 5684 1998 CircleF 5708 1995 CircleF 5731 1992 CircleF 5754 1990 CircleF 5777 1987 CircleF 5801 1984 CircleF 5824 1981 CircleF 5847 1978 CircleF 5870 1975 CircleF 5893 1973 CircleF 5917 1970 CircleF 5940 1967 CircleF 5963 1964 CircleF 5986 1962 CircleF 6010 1959 CircleF 6033 1956 CircleF 6056 1954 CircleF 6079 1951 CircleF 6103 1948 CircleF 6126 1946 CircleF 6149 1943 CircleF 6172 1941 CircleF 6195 1938 CircleF 6219 1936 CircleF 6242 1933 CircleF 6265 1931 CircleF 6288 1928 CircleF 6312 1926 CircleF 6335 1924 CircleF 6358 1921 CircleF 6381 1919 CircleF 6405 1917 CircleF 6428 1914 CircleF 6451 1912 CircleF 6474 1910 CircleF 6497 1907 CircleF 6521 1905 CircleF 6544 1903 CircleF 6567 1901 CircleF 6590 1898 CircleF 6614 1896 CircleF 6637 1894 CircleF 6660 1892 CircleF 6683 1890 CircleF 6706 1887 CircleF 6730 1885 CircleF 6753 1883 CircleF 6776 1881 CircleF 6799 1879 CircleF 6823 1877 CircleF 6846 1875 CircleF 6869 1873 CircleF 6892 1871 CircleF 6916 1869 CircleF 6939 1867 CircleF 6962 1865 CircleF 0.600 UP 1.000 UL LT1 1201 4871 TriUF 1224 4782 TriUF 1248 4696 TriUF 1271 4615 TriUF 1294 4538 TriUF 1317 4465 TriUF 1341 4395 TriUF 1364 4329 TriUF 1387 4265 TriUF 1410 4204 TriUF 1433 4146 TriUF 1457 4090 TriUF 1480 4037 TriUF 1503 3986 TriUF 1526 3936 TriUF 1550 3889 TriUF 1573 3843 TriUF 1596 3799 TriUF 1619 3757 TriUF 1643 3716 TriUF 1666 3677 TriUF 1689 3639 TriUF 1712 3602 TriUF 1735 3566 TriUF 1759 3532 TriUF 1782 3499 TriUF 1805 3466 TriUF 1828 3435 TriUF 1852 3405 TriUF 1875 3375 TriUF 1898 3347 TriUF 1921 3319 TriUF 1945 3292 TriUF 1968 3266 TriUF 1991 3240 TriUF 2014 3216 TriUF 2037 3191 TriUF 2061 3168 TriUF 2084 3145 TriUF 2107 3123 TriUF 2130 3101 TriUF 2154 3080 TriUF 2177 3059 TriUF 2200 3039 TriUF 2223 3019 TriUF 2246 3000 TriUF 2270 2981 TriUF 2293 2963 TriUF 2316 2945 TriUF 2339 2927 TriUF 2363 2910 TriUF 2386 2893 TriUF 2409 2877 TriUF 2432 2861 TriUF 2456 2845 TriUF 2479 2830 TriUF 2502 2815 TriUF 2525 2800 TriUF 2548 2786 TriUF 2572 2771 TriUF 2595 2757 TriUF 2618 2744 TriUF 2641 2730 TriUF 2665 2717 TriUF 2688 2704 TriUF 2711 2692 TriUF 2734 2679 TriUF 2758 2667 TriUF 2781 2655 TriUF 2804 2644 TriUF 2827 2632 TriUF 2850 2621 TriUF 2874 2610 TriUF 2897 2599 TriUF 2920 2588 TriUF 2943 2577 TriUF 2967 2567 TriUF 2990 2557 TriUF 3013 2547 TriUF 3036 2537 TriUF 3060 2527 TriUF 3083 2517 TriUF 3106 2508 TriUF 3129 2499 TriUF 3152 2490 TriUF 3176 2481 TriUF 3199 2472 TriUF 3222 2463 TriUF 3245 2455 TriUF 3269 2446 TriUF 3292 2438 TriUF 3315 2430 TriUF 3338 2421 TriUF 3361 2413 TriUF 3385 2406 TriUF 3408 2398 TriUF 3431 2390 TriUF 3454 2383 TriUF 3478 2375 TriUF 3501 2368 TriUF 3524 2361 TriUF 3547 2354 TriUF 3571 2347 TriUF 3594 2340 TriUF 3617 2333 TriUF 3640 2326 TriUF 3663 2319 TriUF 3687 2313 TriUF 3710 2306 TriUF 3733 2300 TriUF 3756 2294 TriUF 3780 2287 TriUF 3803 2281 TriUF 3826 2275 TriUF 3849 2269 TriUF 3873 2263 TriUF 3896 2257 TriUF 3919 2252 TriUF 3942 2246 TriUF 3965 2240 TriUF 3989 2235 TriUF 4012 2229 TriUF 4035 2224 TriUF 4058 2218 TriUF 4082 2213 TriUF 4105 2208 TriUF 4128 2202 TriUF 4151 2197 TriUF 4175 2192 TriUF 4198 2187 TriUF 4221 2182 TriUF 4244 2177 TriUF 4267 2172 TriUF 4291 2167 TriUF 4314 2163 TriUF 4337 2158 TriUF 4360 2153 TriUF 4384 2149 TriUF 4407 2144 TriUF 4430 2140 TriUF 4453 2135 TriUF 4476 2131 TriUF 4500 2126 TriUF 4523 2122 TriUF 4546 2118 TriUF 4569 2113 TriUF 4593 2109 TriUF 4616 2105 TriUF 4639 2101 TriUF 4662 2097 TriUF 4686 2093 TriUF 4709 2089 TriUF 4732 2085 TriUF 4755 2081 TriUF 4778 2077 TriUF 4802 2073 TriUF 4825 2069 TriUF 4848 2066 TriUF 4871 2062 TriUF 4895 2058 TriUF 4918 2055 TriUF 4941 2051 TriUF 4964 2047 TriUF 4988 2044 TriUF 5011 2040 TriUF 5034 2037 TriUF 5057 2033 TriUF 5080 2030 TriUF 5104 2027 TriUF 5127 2023 TriUF 5150 2020 TriUF 5173 2016 TriUF 5197 2013 TriUF 5220 2010 TriUF 5243 2007 TriUF 5266 2004 TriUF 5290 2000 TriUF 5313 1997 TriUF 5336 1994 TriUF 5359 1991 TriUF 5382 1988 TriUF 5406 1985 TriUF 5429 1982 TriUF 5452 1979 TriUF 5475 1976 TriUF 5499 1973 TriUF 5522 1970 TriUF 5545 1967 TriUF 5568 1964 TriUF 5591 1962 TriUF 5615 1959 TriUF 5638 1956 TriUF 5661 1953 TriUF 5684 1950 TriUF 5708 1948 TriUF 5731 1945 TriUF 5754 1942 TriUF 5777 1940 TriUF 5801 1937 TriUF 5824 1934 TriUF 5847 1932 TriUF 5870 1929 TriUF 5893 1927 TriUF 5917 1924 TriUF 5940 1922 TriUF 5963 1919 TriUF 5986 1917 TriUF 6010 1914 TriUF 6033 1912 TriUF 6056 1909 TriUF 6079 1907 TriUF 6103 1904 TriUF 6126 1902 TriUF 6149 1900 TriUF 6172 1897 TriUF 6195 1895 TriUF 6219 1893 TriUF 6242 1890 TriUF 6265 1888 TriUF 6288 1886 TriUF 6312 1884 TriUF 6335 1881 TriUF 6358 1879 TriUF 6381 1877 TriUF 6405 1875 TriUF 6428 1873 TriUF 6451 1871 TriUF 6474 1868 TriUF 6497 1866 TriUF 6521 1864 TriUF 6544 1862 TriUF 6567 1860 TriUF 6590 1858 TriUF 6614 1856 TriUF 6637 1854 TriUF 6660 1852 TriUF 6683 1850 TriUF 6706 1848 TriUF 6730 1846 TriUF 6753 1844 TriUF 6776 1842 TriUF 6799 1840 TriUF 6823 1838 TriUF 6846 1836 TriUF 6869 1834 TriUF 6892 1832 TriUF 6916 1831 TriUF 6939 1829 TriUF 6962 1827 TriUF 0.600 UP 1.000 UL LT2 922 4761 DiaF 946 4606 DiaF 969 4472 DiaF 992 4356 DiaF 1015 4254 DiaF 1039 4164 DiaF 1062 4083 DiaF 1085 4009 DiaF 1108 3942 DiaF 1131 3879 DiaF 1155 3821 DiaF 1178 3767 DiaF 1201 3716 DiaF 1224 3667 DiaF 1248 3621 DiaF 1271 3577 DiaF 1294 3534 DiaF 1317 3494 DiaF 1341 3455 DiaF 1364 3418 DiaF 1387 3382 DiaF 1410 3348 DiaF 1433 3314 DiaF 1457 3282 DiaF 1480 3251 DiaF 1503 3221 DiaF 1526 3192 DiaF 1550 3164 DiaF 1573 3136 DiaF 1596 3110 DiaF 1619 3084 DiaF 1643 3059 DiaF 1666 3035 DiaF 1689 3012 DiaF 1712 2989 DiaF 1735 2967 DiaF 1759 2945 DiaF 1782 2924 DiaF 1805 2904 DiaF 1828 2884 DiaF 1852 2865 DiaF 1875 2846 DiaF 1898 2827 DiaF 1921 2809 DiaF 1945 2792 DiaF 1968 2775 DiaF 1991 2758 DiaF 2014 2742 DiaF 2037 2726 DiaF 2061 2710 DiaF 2084 2695 DiaF 2107 2680 DiaF 2130 2666 DiaF 2154 2651 DiaF 2177 2637 DiaF 2200 2624 DiaF 2223 2611 DiaF 2246 2598 DiaF 2270 2585 DiaF 2293 2572 DiaF 2316 2560 DiaF 2339 2548 DiaF 2363 2536 DiaF 2386 2525 DiaF 2409 2513 DiaF 2432 2502 DiaF 2456 2491 DiaF 2479 2480 DiaF 2502 2470 DiaF 2525 2460 DiaF 2548 2450 DiaF 2572 2440 DiaF 2595 2430 DiaF 2618 2420 DiaF 2641 2411 DiaF 2665 2402 DiaF 2688 2392 DiaF 2711 2383 DiaF 2734 2375 DiaF 2758 2366 DiaF 2781 2357 DiaF 2804 2349 DiaF 2827 2341 DiaF 2850 2333 DiaF 2874 2325 DiaF 2897 2317 DiaF 2920 2309 DiaF 2943 2301 DiaF 2967 2294 DiaF 2990 2287 DiaF 3013 2279 DiaF 3036 2272 DiaF 3060 2265 DiaF 3083 2258 DiaF 3106 2251 DiaF 3129 2245 DiaF 3152 2238 DiaF 3176 2231 DiaF 3199 2225 DiaF 3222 2218 DiaF 3245 2212 DiaF 3269 2206 DiaF 3292 2200 DiaF 3315 2194 DiaF 3338 2188 DiaF 3361 2182 DiaF 3385 2176 DiaF 3408 2170 DiaF 3431 2165 DiaF 3454 2159 DiaF 3478 2154 DiaF 3501 2148 DiaF 3524 2143 DiaF 3547 2138 DiaF 3571 2132 DiaF 3594 2127 DiaF 3617 2122 DiaF 3640 2117 DiaF 3663 2112 DiaF 3687 2107 DiaF 3710 2102 DiaF 3733 2097 DiaF 3756 2093 DiaF 3780 2088 DiaF 3803 2083 DiaF 3826 2079 DiaF 3849 2074 DiaF 3873 2070 DiaF 3896 2065 DiaF 3919 2061 DiaF 3942 2057 DiaF 3965 2052 DiaF 3989 2048 DiaF 4012 2044 DiaF 4035 2040 DiaF 4058 2036 DiaF 4082 2032 DiaF 4105 2028 DiaF 4128 2024 DiaF 4151 2020 DiaF 4175 2016 DiaF 4198 2012 DiaF 4221 2008 DiaF 4244 2005 DiaF 4267 2001 DiaF 4291 1997 DiaF 4314 1994 DiaF 4337 1990 DiaF 4360 1986 DiaF 4384 1983 DiaF 4407 1979 DiaF 4430 1976 DiaF 4453 1972 DiaF 4476 1969 DiaF 4500 1966 DiaF 4523 1962 DiaF 4546 1959 DiaF 4569 1956 DiaF 4593 1953 DiaF 4616 1949 DiaF 4639 1946 DiaF 4662 1943 DiaF 4686 1940 DiaF 4709 1937 DiaF 4732 1934 DiaF 4755 1931 DiaF 4778 1928 DiaF 4802 1925 DiaF 4825 1922 DiaF 4848 1919 DiaF 4871 1916 DiaF 4895 1913 DiaF 4918 1910 DiaF 4941 1908 DiaF 4964 1905 DiaF 4988 1902 DiaF 5011 1899 DiaF 5034 1896 DiaF 5057 1894 DiaF 5080 1891 DiaF 5104 1888 DiaF 5127 1886 DiaF 5150 1883 DiaF 5173 1881 DiaF 5197 1878 DiaF 5220 1876 DiaF 5243 1873 DiaF 5266 1871 DiaF 5290 1868 DiaF 5313 1866 DiaF 5336 1863 DiaF 5359 1861 DiaF 5382 1858 DiaF 5406 1856 DiaF 5429 1854 DiaF 5452 1851 DiaF 5475 1849 DiaF 5499 1847 DiaF 5522 1844 DiaF 5545 1842 DiaF 5568 1840 DiaF 5591 1838 DiaF 5615 1835 DiaF 5638 1833 DiaF 5661 1831 DiaF 5684 1829 DiaF 5708 1827 DiaF 5731 1825 DiaF 5754 1822 DiaF 5777 1820 DiaF 5801 1818 DiaF 5824 1816 DiaF 5847 1814 DiaF 5870 1812 DiaF 5893 1810 DiaF 5917 1808 DiaF 5940 1806 DiaF 5963 1804 DiaF 5986 1802 DiaF 6010 1800 DiaF 6033 1798 DiaF 6056 1796 DiaF 6079 1794 DiaF 6103 1792 DiaF 6126 1791 DiaF 6149 1789 DiaF 6172 1787 DiaF 6195 1785 DiaF 6219 1783 DiaF 6242 1781 DiaF 6265 1780 DiaF 6288 1778 DiaF 6312 1776 DiaF 6335 1774 DiaF 6358 1772 DiaF 6381 1771 DiaF 6405 1769 DiaF 6428 1767 DiaF 6451 1766 DiaF 6474 1764 DiaF 6497 1762 DiaF 6521 1761 DiaF 6544 1759 DiaF 6567 1757 DiaF 6590 1756 DiaF 6614 1754 DiaF 6637 1752 DiaF 6660 1751 DiaF 6683 1749 DiaF 6706 1747 DiaF 6730 1746 DiaF 6753 1744 DiaF 6776 1743 DiaF 6799 1741 DiaF 6823 1740 DiaF 6846 1738 DiaF 6869 1737 DiaF 6892 1735 DiaF 6916 1734 DiaF 6939 1732 DiaF 6962 1731 DiaF 1.000 UL LT3 574 1198 M 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V 64 0 V 65 0 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 178 1318 a Fv(Fig.)30 b(6.)h(The)f(co)s(e\016cien)m(ts)40 b(~)-54 b Fu(\013)1169 1285 y Fq(\017)1169 1343 y(j)1236 1318 y Fv(\(left\))31 b(and)37 b(~)-52 b Fu(!)1703 1285 y Fq(\017)1700 1343 y(j)1766 1318 y Fv(\(righ)m(t\))31 b(v)m(ersus)f Fu(j)5 b Fv(,)31 b(for)f Fu(\017)25 b Fv(=)2754 1282 y Fs(1)p 2736 1297 71 4 v 2736 1349 a(10)2847 1318 y Fv(and)30 b Fu(n)25 b Fv(=)g(2)p Fu(;)15 b Fv(3)p Fu(;)g Fv(4.)166 2167 y @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1288 @rwi @setspecial %%BeginDocument: alphabar2.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: ../tex/alphabar2.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Mon Oct 11 18:55:37 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 658 280 M 63 0 V 6241 0 R -63 0 V 574 280 M (-1.23) Rshow 658 936 M 63 0 V 6241 0 R -63 0 V 574 936 M (-1.22) Rshow 658 1592 M 63 0 V 6241 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64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 1107 w @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1288 @rwi @setspecial %%BeginDocument: alphabar3.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: ../tex/alphabar3.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Mon Oct 11 18:56:17 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 658 280 M 63 0 V 6241 0 R -63 0 V 574 280 M (-24) Rshow 658 854 M 63 0 V 6241 0 R -63 0 V 574 854 M (-23.5) Rshow 658 1428 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-23) Rshow 658 2002 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-22.5) Rshow 658 2576 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-22) Rshow 658 3150 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-21.5) Rshow 658 3724 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-21) Rshow 658 4298 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-20.5) Rshow 658 4872 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-20) Rshow 1610 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R ( 50) Cshow 2799 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R ( 100) Cshow 3988 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R ( 150) Cshow 5178 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R ( 200) Cshow 6367 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R ( 250) 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{lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } 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360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy 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{ vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { 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Fu(:)f(:)h(:)p 1412 1079 V 106 w Ft(\000)p Fv(27480)p Fu(:)p Fv(9190533494)g Fu(:)f(:)g(:)p 2411 1079 V 177 w Fv(32532)p Fu(:)p Fv(123248798)q(6)g Fu(:)h(:)f(:)229 1173 y Fv(\026)216 1197 y Fu(\014)272 1164 y Fs(0)267 1222 y(5)p 397 1248 V 413 1248 V 465 1197 a Ft(\000)p Fv(17079)p Fu(:)p Fv(3558289270)h Fu(:)f(:)h(:)p 1412 1248 V 106 w Ft(\000)p Fv(355090)p Fu(:)p Fv(495717297)g Fu(:)f(:)g(:)p 2411 1248 V 106 w Ft(\000)p Fv(239903)p Fu(:)p Fv(15088023)q(0)g Fu(:)h(:)f(:)166 1400 y Fk(p)s(erturbations)32 b(of)g(degree)i(six)e(and)h(eigh)m(t.)166 1620 y(W)-8 b(e)39 b(study)g(no)m(w)g(the)g(asymptotic)f(b)s(eha)m(vior)g(of)g(the) g(Gevrey)i(co)s(e\016cien)m(ts)f Fh(!)3106 1584 y Fq(\017)3102 1645 y(j)3139 1620 y Fk(.)f(T)-8 b(o)38 b(b)s(e)166 1741 y(more)i(precise,)h(w)m(e)h(presen)m(t)g(some)e(evidences)j(on)d(the)h (asymptotic)f(b)s(eha)m(vior)g(of)g(the)166 1861 y(sequences)g(\()8 b(\026)-57 b Fh(!)716 1825 y Fs(0)712 1886 y Fq(j)754 1861 y Fk(\))792 1876 y Fq(j)t Fr(\025)p Fs(0)919 1861 y Fk(,)37 b(\()8 b(\026)-57 b Fh(!)1086 1825 y Fs(1)1082 1886 y Fq(j)1124 1861 y Fk(\))1162 1876 y Fq(j)t Fr(\025)p Fs(0)1289 1861 y Fk(,)37 b(and)f(\()8 b(^)-57 b Fh(!)1649 1825 y Fq(\017)1645 1886 y(j)1681 1861 y Fk(\))1719 1876 y Fq(j)t Fr(\025)p Fs(0)1883 1861 y Fk(de\014ned)38 b(in)d(\(12\).)h(W) -8 b(e)37 b(ha)m(v)m(e)h(plotted)e(the)166 1981 y(\014rst)h(terms)f(of) g(these)i(sequences)i(in)35 b(\014gures)j(8,)e(9,)g(and)h(10,)f(resp)s (ectiv)m(ely)-8 b(,)37 b(for)f Fh(\017)f Fk(=)3391 1942 y Fs(1)p 3373 1958 71 4 v 3373 2016 a(10)166 2102 y Fk(and)41 b Fh(n)h Fk(=)f(2)p Fh(;)17 b Fk(3)p Fh(;)g Fk(4.)40 b(F)-8 b(or)40 b(eac)m(h)i(index)f Fh(j)47 b Fi(\025)42 b Fk(0,)f(the)g(terms)49 b(\026)-57 b Fh(!)2409 2066 y Fs(0)2405 2126 y Fq(j)2448 2102 y Fk(,)48 b(\026)-57 b Fh(!)2580 2066 y Fs(1)2576 2126 y Fq(j)2619 2102 y Fk(,)41 b(and)49 b(^)-57 b Fh(!)2950 2066 y Fq(\017)2946 2126 y(j)3023 2102 y Fk(ha)m(v)m(e)42 b(b)s(een)166 2222 y(obtained)32 b(b)m(y)h(means)g(of)f(an)g(extrap)s(olation)f(in)g(the)i (p)s(erturbativ)m(e)g(parameter)f Fh(\017)p Fk(,)h(from)166 2343 y(the)41 b(computed)g(v)-5 b(alues)40 b(of)h Fh(!)1283 2306 y Fq(\017)1279 2367 y(j)1355 2343 y Fk(in)f(the)h(net)g Fh(\017)h Fk(=)f(10)2125 2306 y Fr(\000)p Fs(1)2219 2343 y Fh(;)17 b Fk(10)2361 2306 y Fr(\000)p Fs(2)2455 2343 y Fh(;)g(:)g(:)g(:)f(;)h Fk(10)2772 2306 y Fr(\000)p Fs(10)2901 2343 y Fk(.)40 b(The)i(\014gures)166 2463 y(strongly)32 b(suggest)i(that)e(the)h(limits)369 2684 y(\026)-57 b Fh(!)426 2643 y Fs(0)422 2709 y Fr(1)524 2684 y Fk(=)74 b(lim)628 2742 y Fq(j)t Fr(!)p Fs(+)p Fr(1)881 2684 y Fk(\026)-57 b Fh(!)938 2643 y Fs(0)934 2709 y Fq(j)977 2684 y Fh(;)219 b Fk(\026)-57 b Fh(!)1280 2643 y Fs(1)1276 2709 y Fr(1)1379 2684 y Fk(=)74 b(lim)1482 2742 y Fq(j)t Fr(!)p Fs(+)p Fr(1)1735 2684 y Fk(\026)-57 b Fh(!)1792 2643 y Fs(1)1788 2709 y Fq(j)1831 2684 y Fh(;)219 b Fk(^)-56 b Fh(!)2135 2643 y Fq(\017)2131 2709 y Fr(1)2233 2684 y Fk(=)74 b(lim)2336 2742 y Fq(j)t Fr(!)p Fs(+)p Fr(1)2589 2684 y Fk(^)-57 b Fh(!)2646 2643 y Fq(\017)2642 2709 y(j)166 3041 y Fk(exist,)30 b(are)f(\014nite,)g(and)g(do)g(not)g (v)-5 b(anish.)29 b(As)g(b)s(efore,)h(these)g(limits)c(ha)m(v)m(e)k(b)s (een)g(obtained)166 3162 y(b)m(y)44 b(means)f(of)g(another)g(extrap)s (olation)e(algorithm)e(based)44 b(on)f(the)h(h)m(yp)s(othesis)g(that) 166 3282 y(there)33 b(exist)g(some)g(asymptotic)e(co)s(e\016cien)m(ts) 41 b(\026)-56 b Fh(\021)1938 3246 y Fs(0)1934 3307 y Fq(l)1977 3282 y Fk(,)40 b(\026)-56 b Fh(\021)2089 3246 y Fs(1)2085 3307 y Fq(l)2161 3282 y Fk(and)39 b(^)-56 b Fh(\021)2402 3246 y Fq(\017)2398 3307 y(l)2467 3282 y Fk(suc)m(h)34 b(that)369 3503 y(\026)-57 b Fh(!)426 3462 y Fs(0)422 3528 y Fq(j)499 3451 y Fs(as)493 3503 y Fk(=)35 b(\026)-57 b Fh(!)661 3462 y Fs(0)657 3528 y Fr(1)754 3503 y Fk(+)852 3420 y Fg(X)856 3605 y Fq(l)q Fr(\025)p Fs(2)995 3503 y Fk(\026)h Fh(\021)1040 3462 y Fs(0)1036 3528 y Fq(l)1079 3503 y Fh(j)1125 3462 y Fr(\000)p Fq(l)1206 3503 y Fh(;)220 b Fk(\026)-57 b Fh(!)1510 3462 y Fs(1)1506 3528 y Fq(j)1583 3451 y Fs(as)1576 3503 y Fk(=)36 b(\026)-57 b Fh(!)1745 3462 y Fs(1)1741 3528 y Fr(1)1838 3503 y Fk(+)1936 3420 y Fg(X)1940 3605 y Fq(l)q Fr(\025)p Fs(2)2079 3503 y Fk(\026)h Fh(\021)2124 3462 y Fs(1)2120 3528 y Fq(l)2163 3503 y Fh(j)2209 3462 y Fr(\000)p Fq(l)2290 3503 y Fh(;)220 b Fk(^)-57 b Fh(!)2594 3462 y Fq(\017)2590 3528 y(j)2660 3451 y Fs(as)2654 3503 y Fk(=)35 b(^)-57 b Fh(!)2822 3462 y Fq(\017)2818 3528 y Fr(1)2915 3503 y Fk(+)3013 3420 y Fg(X)3017 3605 y Fq(l)q Fr(\025)p Fs(1)3156 3503 y Fk(^)i Fh(\021)3202 3462 y Fq(\017)3198 3528 y(l)3234 3503 y Fh(j)3280 3462 y Fr(\000)p Fq(l)166 3912 y Fk(as)29 b Fh(j)34 b Fi(!)27 b Fk(+)p Fi(1)p Fk(.)i(F)-8 b(or)28 b(instance,)i(w)m(e)g(ha)m(v)m(e)g (listed)f(in)f(table)g(4)h(the)h(limits)k(\026)-57 b Fh(!)2835 3876 y Fs(0)2831 3937 y Fr(1)2934 3912 y Fk(and)30 b(the)f(\014rst)166 4033 y(asymptotic)37 b(co)s(e\016cien)m(ts)45 b(\026)-56 b Fh(\021)1223 3997 y Fs(0)1219 4057 y Fq(l)1299 4033 y Fk(for)37 b Fh(n)f Fk(=)f(2)p Fh(;)17 b Fk(3)p Fh(;)g Fk(4.)36 b(The)i(tables)f(with)g(the)h(limits)k(\026)-57 b Fh(!)3189 3997 y Fs(1)3185 4057 y Fr(1)3297 4033 y Fk(and)174 4153 y(^)g Fh(!)231 4117 y Fq(\017)227 4178 y Fr(1)335 4153 y Fk(join)m(tly)33 b(with)g(the)i(\014rst)f(asymptotic) f(co)s(e\016cien)m(ts)42 b(\026)-56 b Fh(\021)2288 4117 y Fs(1)2284 4178 y Fq(l)2361 4153 y Fk(and)41 b(^)-56 b Fh(\021)2604 4117 y Fq(\017)2600 4178 y(l)2670 4153 y Fk(ha)m(v)m(e)35 b(b)s(een)g(skipp)s(ed)166 4273 y(for)25 b(the)h(sak)m(e)h(of)e(brevit)m(y)-8 b(.)26 b(When)g(the)g(p)s (erturbation)f(is)f(quartic)i(\(that)f(is,)g(when)h Fh(n)i Fk(=)g(2\),)166 4394 y(w)m(e)34 b(see)f(that)40 b(\026)-56 b Fh(!)744 4358 y Fs(0)740 4418 y Fr(1)842 4394 y Fk(=)27 b(2)10 b(\026)-59 b Fh(\013)1057 4358 y Fs(0)1056 4418 y Fr(1)1163 4394 y Fk(and)40 b(\026)-56 b Fh(\021)1405 4358 y Fs(0)1401 4418 y Fq(l)1472 4394 y Fk(=)28 b(2)1639 4368 y(\026)1625 4394 y Fh(\014)1686 4358 y Fs(0)1680 4418 y Fq(l)1724 4394 y Fk(,)33 b(for)f(all)e Fh(l)g Fi(\025)f Fk(2,)j(compare)g(tables)g(3)h(and)f(4.)166 4798 y Fl(5)112 b(Almost)36 b(in)m(visible)e(homo)s(clinic)h (bifurcations)h(in)h(a)h(billiard)d(table)166 5139 y Fk(The)j(previous)g(section)g(dealed)f(with)g(some)g(splitting)f(quan)m (tities)h(asso)s(ciated)g(to)g(the)166 5259 y(eigh)m(t)30 b(axial)e(homo)s(clinic)f(tra)5 b(jectories)30 b(under)h(the)f (monomial)c(p)s(erturbations)k(\(1\).)f(F)-8 b(or)166 5380 y(more)35 b(general)f(p)s(erturbations)h(there)h(exist)f(other)h (primary)d(homo)s(clinic)g(tra)5 b(jectories)1745 5712 y(24)p eop %%Page: 25 25 25 24 bop 166 752 a @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1288 @rwi @setspecial %%BeginDocument: omegabar0_2.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: ../tex/omegabar2.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Fri Oct 8 18:17:36 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 658 280 M 63 0 V 6241 0 R -63 0 V 574 280 M (-16.1) Rshow 658 936 M 63 0 V 6241 0 R -63 0 V 574 936 M (-16) Rshow 658 1592 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-15.9) Rshow 658 2248 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-15.8) Rshow 658 2904 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-15.7) Rshow 658 3560 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-15.6) Rshow 658 4216 M 63 0 V 6241 0 R -63 0 V -6325 0 R 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64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 1107 w @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1288 @rwi @setspecial %%BeginDocument: omegabar0_3.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: ../tex/omegabar3.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Fri Oct 8 18:37:07 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 574 280 M 63 0 V 6325 0 R -63 0 V 490 280 M (-160) Rshow 574 1045 M 63 0 V 6325 0 R -63 0 V -6409 0 R (-159) Rshow 574 1811 M 63 0 V 6325 0 R -63 0 V -6409 0 R (-158) Rshow 574 2576 M 63 0 V 6325 0 R -63 0 V -6409 0 R (-157) Rshow 574 3341 M 63 0 V 6325 0 R -63 0 V -6409 0 R (-156) Rshow 574 4107 M 63 0 V 6325 0 R -63 0 V -6409 0 R (-155) Rshow 574 4872 M 63 0 V 6325 0 R -63 0 V -6409 0 R (-154) Rshow 846 280 M 0 63 V 0 4529 R 0 -63 V 846 140 M ( 50) Cshow 2205 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R ( 100) Cshow 3564 280 M 0 63 V 0 4529 R 0 -63 V 0 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PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V 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/CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 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def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke 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( 150) Cshow 5178 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R ( 200) Cshow 6367 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R ( 250) Cshow 1.000 UL LTb 658 280 M 6304 0 V 0 4592 V -6304 0 V 658 280 L 0.600 UP 1.000 UL LT0 1300 298 CircleF 1324 524 CircleF 1348 732 CircleF 1372 925 CircleF 1395 1103 CircleF 1419 1270 CircleF 1443 1424 CircleF 1467 1569 CircleF 1491 1703 CircleF 1514 1829 CircleF 1538 1947 CircleF 1562 2058 CircleF 1586 2162 CircleF 1610 2260 CircleF 1633 2352 CircleF 1657 2439 CircleF 1681 2521 CircleF 1705 2598 CircleF 1728 2671 CircleF 1752 2740 CircleF 1776 2806 CircleF 1800 2868 CircleF 1824 2927 CircleF 1847 2983 CircleF 1871 3037 CircleF 1895 3087 CircleF 1919 3136 CircleF 1943 3182 CircleF 1966 3226 CircleF 1990 3268 CircleF 2014 3308 CircleF 2038 3346 CircleF 2062 3383 CircleF 2085 3418 CircleF 2109 3452 CircleF 2133 3484 CircleF 2157 3515 CircleF 2180 3544 CircleF 2204 3573 CircleF 2228 3600 CircleF 2252 3626 CircleF 2276 3652 CircleF 2299 3676 CircleF 2323 3699 CircleF 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0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V 64 0 V 63 0 V 64 0 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 1106 w @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1288 @rwi @setspecial %%BeginDocument: omegabar1_4.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: ../tex/omegabar1_4.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Wed Oct 13 13:10:33 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def 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@setspecial %%BeginDocument: omegahat4.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: ../tex/omegahat4.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Wed Oct 13 16:12:52 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { 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2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V 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closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def 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currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V 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copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt 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stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } 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Fs(+)1810 5379 y Fv(.)g(Besides,)2226 5356 y(\026)2202 5379 y Fu(A)g Ft(\030)g(\006)7 b Fv(\026)-52 b Fu(!)2522 5346 y Fr(\006)2605 5379 y Fv(as)25 b Fu(\017)g Ft(!)g Fv(0)g(and)f Fu(h)i Ft(!)f Fv(0)3369 5346 y Fs(+)3428 5379 y Fv(.)1745 5712 y Fk(27)p eop %%Page: 28 28 28 27 bop 168 1138 a @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1951 @rwi @setspecial %%BeginDocument: I_2.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: ../tex/I_2.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Thu Oct 14 17:44:37 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def 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stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length 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-33 10 V -32 10 V -32 9 V -33 8 V -32 7 V -33 6 V -32 4 V -32 4 V -33 2 V -32 0 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 1658 w @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1951 @rwi @setspecial %%BeginDocument: I_1.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: ../tex/I_1.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Thu Oct 14 17:44:26 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 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b(12.)h(The)f(graphs)f(of)h(the)h (functions)1666 1295 y(\026)1642 1318 y Fu(A)p Fv(\()p Fu(d)p Fv(\))g(\(con)m(tin)m(uous)f(lines\),)38 b(\026)-52 b Fu(!)2682 1285 y Fr(\000)2741 1318 y Fv(\()p Fu(d)p Fv(\))33 b(\(dotted)g(lines\),)166 1430 y(and)k(\026)-52 b Fu(!)403 1397 y Fs(+)462 1430 y Fv(\()p Fu(d)p Fv(\))31 b(\(dashed)f(lines\),)f(for)h Fu(h)c Fv(=)f(2)31 b(\(left\))f(and)g Fu(h)c Fv(=)f(1)30 b(\(righ)m(t\).)h(Here,)h Fu(\017)25 b Fv(=)2986 1395 y Fs(1)p 2986 1410 36 4 v 2986 1462 a(2)3031 1430 y Fv(.)166 2311 y @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1288 @rwi @setspecial %%BeginDocument: d_1_10.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: d_1_10.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Thu Oct 14 16:15:06 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def 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-5 V 7 -5 V currentpoint stroke M 6 -5 V 7 -5 V 6 -5 V 6 -4 V 7 -5 V 6 -5 V 7 -4 V 6 -5 V 6 -5 V 7 -4 V 6 -5 V 6 -4 V 7 -5 V 6 -4 V 7 -4 V 6 -5 V 6 -4 V 7 -4 V 6 -5 V 7 -4 V 6 -4 V 6 -4 V 7 -4 V 6 -5 V 7 -4 V 6 -4 V 6 -4 V 7 -4 V 6 -4 V 6 -4 V 7 -4 V 6 -3 V 7 -4 V 6 -4 V 6 -4 V 7 -4 V 6 -3 V 7 -4 V 6 -4 V 6 -3 V 7 -4 V 6 -4 V 6 -3 V 7 -4 V 6 -3 V 7 -4 V 6 -3 V 6 -4 V 7 -3 V 6 -4 V 7 -3 V 6 -3 V 6 -4 V 7 -3 V 6 -3 V 7 -3 V 6 -4 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 7 -2 V 6 -3 V 6 -3 V 7 -3 V 6 -2 V 6 -3 V 7 -3 V 6 -2 V 7 -3 V 6 -3 V 6 -2 V 7 -3 V 6 -2 V 7 -3 V 6 -2 V 6 -3 V 7 -2 V 6 -3 V 7 -2 V 6 -2 V 6 -3 V 7 -2 V 6 -2 V 6 -3 V 7 -2 V 6 -2 V 7 -3 V 6 -2 V 6 -2 V 7 -2 V 6 -2 V 7 -3 V 6 -2 V 6 -2 V 7 -2 V 6 -2 V 6 -2 V 7 -2 V 6 -2 V 7 -2 V 6 -2 V 6 -2 V 7 -2 V 6 -2 V 7 -2 V 6 -2 V 6 -2 V 7 -2 V 6 -2 V 7 -1 V 6 -2 V 6 -2 V 7 -2 V 6 -2 V 6 -1 V 7 -2 V 6 -2 V 7 -2 V 6 -1 V 6 -2 V 7 -2 V 6 -1 V 7 -2 V 6 -2 V 6 -1 V 7 -2 V 6 -2 V 7 -1 V 6 -2 V 6 -1 V 7 -2 V 6 -1 V 6 -2 V 7 -1 V 6 -2 V 7 -1 V 6 -2 V 6 -1 V 7 -2 V 6 -1 V 7 -1 V 6 -2 V 6 -1 V 7 -2 V 6 -1 V 7 -1 V 6 -2 V 6 -1 V 7 -1 V 6 -1 V 6 -2 V 7 -1 V 6 -1 V 7 -1 V 6 -2 V 6 -1 V 7 -1 V 6 -1 V 7 -2 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -2 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 7 0 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 7 -1 V 6 -1 V 6 0 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 7 0 V 6 -1 V 6 -1 V 7 -1 V 6 0 V 7 -1 V 6 -1 V 6 -1 V 7 0 V 6 -1 V 6 -1 V 7 -1 V 6 0 V 7 -1 V 6 -1 V 6 0 V 7 -1 V 6 -1 V 7 0 V 6 -1 V 6 -1 V 7 0 V 6 -1 V 7 -1 V 6 0 V 6 -1 V 7 0 V 6 -1 V 6 -1 V 7 0 V 6 -1 V 7 0 V 6 -1 V 6 -1 V 7 0 V 6 -1 V 7 0 V 6 -1 V 6 0 V 7 -1 V 6 0 V 7 -1 V 6 -1 V 6 0 V 7 -1 V 6 0 V 6 -1 V 7 0 V 6 -1 V 7 0 V 6 -1 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 -1 V 6 0 V 7 -1 V 6 0 V 6 -1 V 7 0 V 6 0 V 6 -1 V 7 0 V 6 -1 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 -1 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 -1 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 -1 V 7 0 V 6 0 V 6 -1 V 7 0 V 6 0 V 7 -1 V 6 0 V 6 0 V 7 -1 V 6 0 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 7 -1 V 6 0 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 0 V 6 -1 V 7 0 V 6 0 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 0 V 6 -1 V 7 0 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 7 -1 V 6 0 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 -1 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 0 V currentpoint stroke M 6 0 V 7 0 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 -1 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 -1 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 -1 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 -1 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 -1 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 1.000 UL LT0 580 3541 M 7 0 V 6 0 V 7 0 V 6 0 V 6 -1 V 7 0 V 6 0 V 6 -1 V 7 0 V 6 -1 V 7 0 V 6 -1 V 6 -1 V 7 0 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 6 -2 V 7 -1 V 6 -1 V 7 -2 V 6 -1 V 6 -2 V 7 -1 V 6 -2 V 7 -2 V 6 -1 V 6 -2 V 7 -2 V 6 -2 V 7 -1 V 6 -2 V 6 -2 V 7 -2 V 6 -2 V 6 -2 V 7 -2 V 6 -3 V 7 -2 V 6 -2 V 6 -2 V 7 -2 V 6 -3 V 7 -2 V 6 -3 V 6 -2 V 7 -2 V 6 -3 V 7 -2 V 6 -3 V 6 -3 V 7 -2 V 6 -3 V 6 -3 V 7 -2 V 6 -3 V 7 -3 V 6 -3 V 6 -2 V 7 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -4 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -4 V 7 -3 V 6 -3 V 6 -3 V 7 -4 V 6 -3 V 6 -3 V 7 -4 V 6 -3 V 7 -3 V 6 -4 V 6 -3 V 7 -4 V 6 -3 V 7 -3 V 6 -4 V 6 -3 V 7 -3 V 6 -4 V 7 -3 V 6 -4 V 6 -3 V 7 -4 V 6 -3 V 6 -3 V 7 -4 V 6 -3 V 7 -4 V 6 -3 V 6 -4 V 7 -3 V 6 -3 V 7 -4 V 6 -3 V 6 -4 V 7 -3 V 6 -3 V 6 -4 V 7 -3 V 6 -4 V 7 -3 V 6 -3 V 6 -4 V 7 -3 V 6 -3 V 7 -4 V 6 -3 V 6 -3 V 7 -4 V 6 -3 V 7 -3 V 6 -4 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -4 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -4 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 7 -2 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 7 -2 V 6 -3 V 6 -3 V 7 -3 V 6 -2 V 7 -3 V 6 -2 V 6 -3 V 7 -3 V 6 -2 V 6 -3 V 7 -2 V 6 -3 V 7 -2 V 6 -3 V 6 -2 V 7 -3 V 6 -2 V 7 -2 V 6 -3 V 6 -2 V 7 -2 V 6 -3 V 6 -2 V 7 -2 V 6 -2 V 7 -2 V 6 -3 V 6 -2 V 7 -2 V 6 -2 V 7 -2 V 6 -2 V 6 -2 V 7 -2 V 6 -2 V 7 -2 V 6 -2 V 6 -2 V 7 -2 V 6 -1 V 6 -2 V 7 -2 V 6 -2 V 7 -2 V 6 -1 V 6 -2 V 7 -2 V 6 -1 V 7 -2 V 6 -2 V 6 -1 V 7 -2 V 6 -1 V 7 -2 V 6 -1 V 6 -2 V 7 -1 V 6 -2 V 6 -1 V 7 -2 V 6 -1 V 7 -1 V 6 -2 V 6 -1 V 7 -1 V 6 -2 V 7 -1 V 6 -1 V 6 -1 V 7 -2 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -2 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 0 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 0 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 0 V 7 -1 V 6 -1 V 6 0 V 7 -1 V 6 -1 V 7 0 V 6 -1 V 6 -1 V 7 0 V 6 -1 V 6 -1 V 7 0 V 6 -1 V 7 0 V 6 -1 V 6 0 V 7 -1 V 6 0 V 7 -1 V 6 0 V 6 -1 V 7 0 V 6 -1 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 -1 V 7 0 V 6 0 V 6 -1 V 7 0 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 -1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 -1 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 -1 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 -1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 1 V 6 0 V currentpoint stroke M 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 1 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0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 1 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V currentpoint stroke M 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 1.000 UL LT0 580 3541 M 7 0 V 6 0 V 7 0 V 6 0 V 6 -1 V 7 0 V 6 0 V 6 -1 V 7 0 V 6 -1 V 7 0 V 6 -1 V 6 -1 V 7 0 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 6 -2 V 7 -1 V 6 -1 V 7 -2 V 6 -1 V 6 -2 V 7 -1 V 6 -2 V 7 -2 V 6 -1 V 6 -2 V 7 -2 V 6 -2 V 7 -1 V 6 -2 V 6 -2 V 7 -2 V 6 -2 V 6 -2 V 7 -2 V 6 -3 V 7 -2 V 6 -2 V 6 -2 V 7 -2 V 6 -3 V 7 -2 V 6 -3 V 6 -2 V 7 -2 V 6 -3 V 7 -2 V 6 -3 V 6 -3 V 7 -2 V 6 -3 V 6 -3 V 7 -2 V 6 -3 V 7 -3 V 6 -3 V 6 -2 V 7 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -4 V 7 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -4 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -4 V 7 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -3 V 6 -3 V 6 -3 V 7 -2 V 6 -3 V 7 -3 V 6 -2 V 6 -3 V 7 -3 V 6 -2 V 7 -3 V 6 -2 V 6 -3 V 7 -2 V 6 -2 V 6 -3 V 7 -2 V 6 -2 V 7 -2 V 6 -2 V 6 -2 V 7 -2 V 6 -2 V 7 -1 V 6 -2 V 6 -2 V 7 -1 V 6 -2 V 7 -1 V 6 -1 V 6 -2 V 7 -1 V 6 -1 V 6 -1 V 7 -1 V 6 -1 V 7 0 V 6 -1 V 6 -1 V 7 0 V 6 -1 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 1 V 6 0 V 6 1 V 7 0 V 6 1 V 7 1 V 6 1 V 6 1 V 7 1 V 6 1 V 7 2 V 6 1 V 6 2 V 7 1 V 6 2 V 7 2 V 6 2 V 6 2 V 7 2 V 6 2 V 6 2 V 7 2 V 6 3 V 7 2 V 6 3 V 6 3 V 7 2 V 6 3 V 7 3 V 6 3 V 6 3 V 7 4 V 6 3 V 6 3 V 7 4 V 6 3 V 7 4 V 6 3 V 6 4 V 7 4 V 6 4 V 7 4 V 6 4 V 6 4 V 7 4 V 6 5 V 7 4 V 6 4 V 6 5 V 7 4 V 6 5 V 6 4 V 7 5 V 6 5 V 7 5 V 6 5 V 6 5 V 7 5 V 6 5 V 7 5 V 6 5 V 6 5 V 7 5 V 6 6 V 7 5 V 6 5 V 6 6 V 7 5 V 6 6 V 6 5 V 7 6 V 6 5 V 7 6 V 6 6 V 6 5 V 7 6 V 6 6 V 7 6 V 6 6 V 6 6 V 7 5 V 6 6 V 7 6 V 6 6 V 6 6 V 7 6 V 6 6 V 6 6 V 7 6 V 6 6 V 7 7 V 6 6 V 6 6 V 7 6 V 6 6 V 7 6 V 6 6 V 6 7 V 7 6 V 6 6 V 6 6 V 7 6 V 6 7 V 7 6 V 6 6 V 6 6 V 7 6 V 6 7 V 7 6 V 6 6 V 6 6 V 7 6 V 6 7 V 7 6 V 6 6 V 6 6 V 7 6 V 6 7 V 6 6 V 7 6 V 6 6 V 7 6 V 6 6 V 6 6 V 7 6 V 6 6 V 7 7 V 6 6 V 6 6 V 7 6 V 6 6 V 7 6 V 6 6 V 6 6 V 7 5 V 6 6 V 6 6 V 7 6 V 6 6 V 7 6 V 6 6 V 6 5 V 7 6 V 6 6 V 7 6 V 6 5 V 6 6 V 7 6 V 6 5 V 7 6 V 6 6 V 6 5 V 7 6 V 6 5 V 6 6 V 7 5 V 6 6 V 7 5 V 6 5 V 6 6 V 7 5 V 6 6 V 7 5 V 6 5 V 6 5 V 7 6 V 6 5 V 6 5 V 7 5 V 6 5 V 7 5 V 6 5 V 6 5 V 7 5 V 6 5 V 7 5 V 6 5 V 6 5 V 7 5 V 6 5 V 7 4 V 6 5 V 6 5 V 7 5 V 6 4 V 6 5 V 7 5 V 6 4 V 7 5 V 6 4 V 6 5 V 7 4 V 6 5 V 7 4 V 6 5 V 6 4 V 7 4 V 6 5 V 7 4 V 6 4 V 6 4 V 7 5 V 6 4 V 6 4 V 7 4 V 6 4 V 7 4 V 6 4 V 6 4 V 7 4 V 6 4 V 7 4 V 6 4 V 6 4 V 7 4 V 6 4 V 6 3 V 7 4 V 6 4 V 7 4 V 6 3 V 6 4 V 7 4 V 6 3 V 7 4 V 6 3 V 6 4 V 7 3 V 6 4 V 7 3 V 6 4 V 6 3 V 7 3 V 6 4 V 6 3 V 7 3 V 6 4 V 7 3 V 6 3 V 6 3 V 7 3 V 6 3 V currentpoint stroke M 7 4 V 6 3 V 6 3 V 7 3 V 6 3 V 7 3 V 6 3 V 6 3 V 7 3 V 6 3 V 6 2 V 7 3 V 6 3 V 7 3 V 6 3 V 6 2 V 7 3 V 6 3 V 7 3 V 6 2 V 6 3 V 7 3 V 6 2 V 7 3 V 6 2 V 6 3 V 7 2 V 6 3 V 6 2 V 7 3 V 6 2 V 7 3 V 6 2 V 6 2 V 7 3 V 6 2 V 7 2 V 6 3 V 6 2 V 7 2 V 6 3 V 6 2 V 7 2 V 6 2 V 7 2 V 6 2 V 6 3 V 7 2 V 6 2 V 7 2 V 6 2 V 6 2 V 7 2 V 6 2 V 7 2 V 6 2 V 6 2 V 7 2 V 6 2 V 6 2 V 7 1 V 6 2 V 7 2 V 6 2 V 6 2 V 7 2 V 6 1 V 7 2 V 6 2 V 6 2 V 7 1 V 6 2 V 7 2 V 6 1 V 6 2 V 7 2 V 6 1 V 6 2 V 7 2 V 6 1 V 7 2 V 6 1 V 6 2 V 7 1 V 6 2 V 7 1 V 6 2 V 6 1 V 7 2 V 6 1 V 7 2 V 6 1 V 6 2 V 7 1 V 6 1 V 6 2 V 7 1 V 6 2 V 7 1 V 6 1 V 6 1 V 7 2 V 6 1 V 7 1 V 6 2 V 6 1 V 7 1 V 6 1 V 6 2 V 7 1 V 6 1 V 7 1 V 6 1 V 6 1 V 7 2 V 6 1 V 7 1 V 6 1 V 6 1 V 7 1 V 6 1 V 7 1 V 6 1 V 6 2 V 7 1 V 6 1 V 6 1 V 7 1 V 6 1 V 7 1 V 6 1 V 6 1 V 7 1 V 6 1 V 7 1 V 6 1 V 6 0 V 7 1 V 6 1 V 7 1 V 6 1 V 6 1 V 7 1 V 6 1 V 6 1 V 7 0 V 6 1 V 7 1 V 6 1 V 6 1 V 7 1 V 6 0 V 7 1 V 6 1 V 6 1 V 7 1 V 6 0 V 7 1 V 6 1 V 6 1 V 7 0 V 6 1 V 6 1 V 7 1 V 6 0 V 7 1 V 6 1 V 6 0 V 7 1 V 6 1 V 7 1 V 6 0 V 6 1 V 7 1 V 6 0 V 6 1 V 7 0 V 6 1 V 7 1 V 6 0 V 6 1 V 7 1 V 6 0 V 7 1 V 6 0 V 6 1 V 7 1 V 6 0 V 7 1 V 6 0 V 6 1 V 7 0 V 6 1 V 6 0 V 7 1 V 6 1 V 7 0 V 6 1 V 6 0 V 7 1 V 6 0 V 7 1 V 6 0 V 6 1 V 7 0 V 6 1 V 7 0 V 6 1 V 6 0 V 7 0 V 6 1 V 6 0 V 7 1 V 6 0 V 7 1 V 6 0 V 6 1 V 7 0 V 6 0 V 7 1 V 6 0 V 6 1 V 7 0 V 6 1 V 6 0 V 7 0 V 6 1 V 7 0 V 6 1 V 6 0 V 7 0 V 6 1 V 7 0 V 6 0 V 6 1 V 7 0 V 6 0 V 7 1 V 6 0 V 6 1 V 7 0 V 6 0 V 6 1 V 7 0 V 6 0 V 7 1 V 6 0 V 6 0 V 7 0 V 6 1 V 7 0 V 6 0 V 6 1 V 7 0 V 6 0 V 7 1 V 6 0 V 6 0 V 7 1 V 6 0 V 6 0 V 7 0 V 6 1 V 7 0 V 6 0 V 6 0 V 7 1 V 6 0 V 7 0 V 6 0 V 6 1 V 7 0 V 6 0 V 7 0 V 6 1 V 6 0 V 7 0 V 6 0 V 6 1 V 7 0 V 6 0 V 7 0 V 6 1 V 6 0 V 7 0 V 6 0 V 7 1 V 6 0 V 6 0 V 7 0 V 6 0 V 6 1 V 7 0 V 6 0 V 7 0 V 6 0 V 6 1 V 7 0 V 6 0 V 7 0 V 6 0 V 6 1 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 1 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 1 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 1 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 1 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 1 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 1 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 1 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 1 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V currentpoint stroke M 7 0 V 6 0 V 7 0 V 6 1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 1 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 1 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 1 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 1 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 6 0 V 7 0 V 6 0 V 7 0 V 6 0 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 1107 w @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1288 @rwi @setspecial %%BeginDocument: d_1_2.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: d_1_2.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Thu Oct 14 09:19:22 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat Opaque stroke grestore } def /CircW { stroke [] 0 setdash hpt 0 360 arc Opaque stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /Symbol-Oblique /Symbol findfont [1 0 .167 1 0 0] makefont dup length dict begin {1 index /FID eq {pop pop} {def} ifelse} forall currentdict end definefont end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 140 scalefont setfont 1.000 UL LTb 658 280 M 63 0 V 6241 0 R -63 0 V 574 280 M (-1) Rshow 658 854 M 63 0 V 6241 0 R -63 0 V 574 854 M (-0.95) Rshow 658 1428 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-0.9) Rshow 658 2002 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-0.85) Rshow 658 2576 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-0.8) Rshow 658 3150 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-0.75) Rshow 658 3724 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-0.7) Rshow 658 4298 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-0.65) Rshow 658 4872 M 63 0 V 6241 0 R -63 0 V -6325 0 R (-0.6) Rshow 658 280 M 0 63 V 0 4529 R 0 -63 V 658 140 M ( 0) Cshow 2234 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R ( 0.5) Cshow 3810 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R ( 1) Cshow 5386 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R ( 1.5) Cshow 6962 280 M 0 63 V 0 4529 R 0 -63 V 0 -4669 R ( 2) Cshow 1.000 UL LTb 658 280 M 6304 0 V 0 4592 V -6304 0 V 658 280 L 1.000 UL LT0 658 4259 M 32 0 V 31 -1 V 32 0 V 31 -2 V 32 -1 V 31 -2 V 32 -2 V 31 -2 V 32 -3 V 31 -3 V 32 -4 V 31 -3 V 32 -4 V 31 -5 V 32 -4 V 31 -5 V 32 -6 V 31 -5 V 32 -6 V 31 -7 V 32 -6 V 31 -7 V 32 -7 V 31 -8 V 32 -7 V 32 -9 V 31 -8 V 32 -9 V 31 -9 V 32 -9 V 31 -9 V 32 -10 V 31 -10 V 32 -11 V 31 -11 V 32 -11 V 31 -11 V 32 -11 V 31 -12 V 32 -12 V 31 -13 V 32 -12 V 31 -13 V 32 -13 V 31 -14 V 32 -13 V 31 -14 V 32 -14 V 31 -15 V 32 -14 V 32 -15 V 31 -15 V 32 -16 V 31 -15 V 32 -16 V 31 -16 V 32 -16 V 31 -17 V 32 -16 V 31 -17 V 32 -17 V 31 -17 V 32 -18 V 31 -17 V 32 -18 V 31 -18 V 32 -18 V 31 -19 V 32 -18 V 31 -19 V 32 -19 V 31 -19 V 32 -19 V 31 -20 V 32 -19 V 32 -20 V 31 -20 V 32 -20 V 31 -20 V 32 -21 V 31 -20 V 32 -21 V 31 -21 V 32 -21 V 31 -21 V 32 -22 V 31 -21 V 32 -22 V 31 -21 V 32 -22 V 31 -23 V 32 -22 V 31 -22 V 32 -23 V 31 -23 V 32 -23 V 31 -23 V 32 -23 V 31 -24 V 32 -23 V 32 -24 V 31 -24 V 32 -24 V 31 -25 V 32 -24 V 31 -25 V 32 -25 V 31 -25 V 32 -26 V 31 -25 V 32 -26 V 31 -26 V 32 -27 V 31 -26 V 32 -27 V 31 -27 V 32 -27 V 31 -28 V 32 -28 V 31 -28 V 32 -28 V 31 -29 V 32 -29 V 31 -29 V 32 -29 V 32 -30 V 31 -30 V 32 -31 V 31 -30 V 32 -31 V 31 -32 V 32 -31 V 31 -32 V 32 -33 V 31 -32 V 32 -33 V 31 -33 V 32 -34 V 31 -34 V 32 -34 V 31 -35 V 32 -35 V 31 -35 V 32 -36 V 31 -36 V 32 -36 V 31 -37 V 32 -37 V 31 -38 V 32 -38 V 32 -38 V 31 -39 V 32 -39 V 31 -39 V 32 -40 V 31 -40 V 32 -40 V 31 -41 V 32 -41 V 31 -42 V 32 -42 V 31 -42 V 32 -43 V 31 -43 V 32 -43 V 31 -44 V 32 -44 V 31 -45 V 32 -45 V 31 -45 V 32 -45 V 31 -46 V 32 -47 V 31 -46 V 32 -47 V 31 -46 V 1.000 UL LT0 690 4259 M 31 -1 V 32 0 V 31 -2 V 32 -1 V 31 -2 V 32 -2 V 31 -2 V 32 -3 V 31 -3 V 32 -4 V 31 -3 V 32 -4 V 31 -5 V 32 -4 V 31 -5 V 32 -6 V 31 -5 V 32 -6 V 31 -7 V 32 -6 V 31 -7 V 32 -7 V 31 -8 V 32 -7 V 32 -9 V 31 -8 V 32 -9 V 31 -9 V 32 -9 V 31 -9 V 32 -10 V 31 -10 V 32 -11 V 31 -11 V 32 -11 V 31 -11 V 32 -11 V 31 -12 V 32 -12 V 31 -13 V 32 -12 V 31 -13 V 32 -13 V 31 -14 V 32 -13 V 31 -14 V 32 -14 V 31 -15 V 32 -14 V 32 -15 V 31 -15 V 32 -16 V 31 -15 V 32 -16 V 31 -16 V 32 -16 V 31 -17 V 32 -16 V 31 -17 V 32 -17 V 31 -17 V 32 -18 V 31 -17 V 32 -18 V 31 -18 V 32 -18 V 31 -19 V 32 -18 V 31 -19 V 32 -19 V 31 -19 V 32 -19 V 31 -19 V 32 -20 V 32 -19 V 31 -20 V 32 -20 V 31 -20 V 32 -20 V 31 -21 V 32 -20 V 31 -20 V 32 -21 V 31 -21 V 32 -21 V 31 -21 V 32 -21 V 31 -21 V 32 -21 V 31 -22 V 32 -21 V 31 -21 V 32 -22 V 31 -22 V 32 -21 V 31 -22 V 32 -22 V 31 -22 V 32 -22 V 32 -22 V 31 -22 V 32 -22 V 31 -22 V 32 -22 V 31 -22 V 32 -23 V 31 -22 V 32 -22 V 31 -22 V 32 -23 V 31 -22 V 32 -22 V 31 -22 V 32 -23 V 31 -22 V 32 -22 V 31 -22 V 32 -23 V 31 -22 V 32 -22 V 31 -22 V 32 -22 V 31 -22 V 32 -23 V 32 -22 V 31 -22 V 32 -21 V 31 -22 V 32 -22 V 31 -22 V 32 -22 V 31 -21 V 32 -22 V 31 -21 V 32 -22 V 31 -21 V 32 -22 V 31 -21 V 32 -21 V 31 -21 V 32 -21 V 31 -21 V 32 -21 V 31 -21 V 32 -20 V 31 -21 V 32 -20 V 31 -21 V 32 -20 V 32 -20 V 31 -20 V 32 -20 V 31 -20 V 32 -19 V 31 -20 V 32 -19 V 31 -20 V 32 -19 V 31 -19 V 32 -19 V 31 -19 V 32 -19 V 31 -18 V 32 -19 V 31 -18 V 32 -18 V 31 -18 V 32 -18 V 31 -18 V 32 -18 V 31 -17 V 32 -18 V 31 -17 V 32 -17 V 32 -17 V 31 -17 V 32 -17 V 31 -16 V 32 -17 V 31 -16 V 32 -16 V 31 -16 V 32 -16 V 31 -15 V 32 -16 V 31 -15 V 32 -15 V 31 -15 V 32 -15 V 31 -15 V 32 -15 V 31 -14 V 32 -14 V 31 -15 V 32 -14 V 31 -13 V 32 -14 V 31 -14 V 32 -13 V 1.000 UL LT0 690 4259 M 31 -1 V 32 0 V 31 -2 V 32 -1 V 31 -2 V 32 -2 V 31 -2 V 32 -3 V 31 -3 V 32 -4 V 31 -3 V 32 -4 V 31 -5 V 32 -4 V 31 -5 V 32 -6 V 31 -5 V 32 -6 V 31 -7 V 32 -6 V 31 -7 V 32 -7 V 31 -8 V 32 -7 V 32 -9 V 31 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31 -8 V 32 -8 V 31 -8 V 32 -7 V 31 -6 V 32 -6 V 31 -6 V 32 -5 V 31 -4 V 32 -4 V 31 -3 V 32 -3 V 32 -2 V 31 -1 V 32 -1 V 31 -1 V 32 1 V 31 0 V 32 2 V 31 2 V 32 2 V 31 4 V 32 3 V 31 5 V 32 5 V 31 5 V 32 7 V 31 6 V 32 8 V 31 8 V 32 9 V 31 9 V 32 10 V 31 10 V 32 11 V 31 12 V 32 12 V 32 13 V 31 13 V 32 14 V 31 15 V 32 15 V 31 16 V 32 16 V 31 17 V 32 18 V 31 18 V 32 18 V 31 19 V 32 20 V 31 20 V 32 21 V 31 21 V 32 22 V 31 23 V 32 22 V 31 24 V 32 24 V 31 24 V 32 25 V 31 25 V 32 26 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 1106 w @beginspecial 50 @llx 50 @lly 410 @urx 302 @ury 1288 @rwi @setspecial %%BeginDocument: d_1_1.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: d_1_1.eps %%Creator: gnuplot 3.7 patchlevel 2 %%CreationDate: Thu Oct 14 09:18:37 2004 %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 256 dict def gnudict begin /Color false def /Solid false def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -46 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke userlinewidth 2 mul setlinewidth } def /AL { stroke userlinewidth 2 div setlinewidth } def /UL { dup gnulinewidth mul /userlinewidth exch def dup 1 lt {pop 1} if 10 mul /udl exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 udl mul 2 udl mul] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 1 0 0 DL } def /LT1 { PL [4 dl 2 dl] 0 1 0 DL } def /LT2 { PL [2 dl 3 dl] 0 0 1 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /Opaque { gsave closepath 1 setgray fill grestore 0 setgray closepath } def /DiaW { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V Opaque stroke } def /BoxW { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V Opaque stroke } def /TriUW { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V Opaque stroke } def /TriDW { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V Opaque stroke } def /PentW 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64 -59 V 63 -58 V 63 -59 V 63 -59 V 63 -58 V 63 -59 V 63 -59 V 63 -58 V 63 -59 V 63 -58 V 63 -58 V 63 -58 V 63 -58 V stroke grestore end showpage %%Trailer %%DocumentFonts: Helvetica %%EndDocument @endspecial 166 2491 a(Fig.)40 b(13.)h(The)e(graphs)g(of)h(the)f (functions)g Fu(d)1743 2458 y Fq(\017)1743 2513 y Fr(\000)1802 2491 y Fv(\()p Fu(h)p Fv(\),)i Fu(d)2037 2458 y Fq(\017)2037 2513 y Fr(\003)2077 2491 y Fv(\()p Fu(h)p Fv(\),)f(and)f Fu(d)2497 2458 y Fq(\017)2497 2513 y Fs(+)2557 2491 y Fv(\()p Fu(h)p Fv(\),)i(for)e Fu(\017)i Fv(=)3110 2455 y Fs(1)p 3092 2470 71 4 v 3092 2522 a(10)3212 2491 y Fv(in)d(the)166 2603 y(ranges)31 b(0)25 b Fu(<)g(h)h(<)f Fv(10)31 b(\(left\),)g(0)26 b Fu(<)f(h)g(<)g Fv(2)31 b(\(cen)m(ter\),)i(and)c(0)d Fu(<)f(h)h(<)f Fv(1)30 b(\(righ)m(t\).)166 2807 y Fk(force)35 b(metho)s(ds.)g(Due)g(to)g(this,)f(as)i(a)e(\014rst) i(step)g(in)e(our)h(study)-8 b(,)36 b(w)m(e)g(shall)d(deriv)m(e)j(some) 166 2927 y(analytical)28 b(predictions)h(of)h(the)g(bifurcation)f(v)-5 b(alues)30 b(b)m(y)h(using)e(the)i(Melnik)m(o)m(v)f(metho)s(d)166 3047 y(describ)s(ed)k(in)d Fi(x)p Fk(4.)i(The)h(Melnik)m(o)m(v)f(p)s (oten)m(tial)e(asso)s(ciated)h(to)h(the)g(binomial)c(p)s(erturba-)166 3168 y(tion)j(\(14\))g(is)g(linear)f(in)g(the)i(parameter)f Fh(d)p Fk(.)h(Concretely)-8 b(,)33 b(it)f(is)g(the)h(elliptic)d (function)361 3403 y Fh(L)p Fk(\()p Fh(t)p Fk(;)17 b Fh(d)p Fk(\))28 b(=)f Fh(L)830 3418 y Fs(3)870 3403 y Fk(\()p Fh(t)p Fk(\))22 b(+)g Fh(dL)1218 3418 y Fs(2)1258 3403 y Fk(\()p Fh(t)p Fk(\))p Fh(;)166 3746 y Fk(where)43 b Fh(L)523 3761 y Fq(n)571 3746 y Fk(\()p Fh(t)p Fk(\))e(is)h (de\014ned)h(in)e(\(4\).)h(The)h(function)e Fh(L)2160 3761 y Fq(n)2207 3746 y Fk(\()p Fh(t)p Fk(\))h(is)g(the)g(Melnik)m(o)m (v)g(p)s(oten)m(tial)166 3867 y(asso)s(ciated)36 b(to)f(the)h(monomial) c(p)s(erturbation)j(\(1\).)h(Th)m(us,)h(if)e Fh(d)2564 3831 y Fq(\017)2564 3891 y Fr(\006)2622 3867 y Fk(\()p Fh(h)p Fk(\))e(=)g Fh(d)2947 3831 y Fs(0)2947 3891 y Fr(\006)3006 3867 y Fk(\()p Fh(h)p Fk(\))24 b(+)3263 3876 y(O)3338 3867 y(\()p Fh(\017)p Fk(\))166 3987 y(are)33 b(the)g(bifurcation)e(v)-5 b(alues,)32 b(their)g(Melnik)m(o)m(v)h (appro)m(ximations)e Fh(d)2703 3951 y Fs(0)2703 4012 y Fr(\006)2762 3987 y Fk(\()p Fh(h)p Fk(\))h(are)h(giv)m(en)f(b)m(y)361 4223 y Fh(d)412 4182 y Fs(0)412 4247 y Fr(\006)471 4223 y Fk(\()p Fh(h)p Fk(\))c(=)f Fi(\000)p Fh(L)877 4182 y Fr(00)877 4247 y Fs(3)920 4223 y Fk(\()p Fh(t)993 4238 y Fr(\006)1053 4223 y Fk(\))p Fh(=L)1206 4182 y Fr(00)1206 4247 y Fs(2)1248 4223 y Fk(\()p Fh(t)1321 4238 y Fr(\006)1380 4223 y Fk(\))p Fh(;)212 b Fk(where)34 b Fh(t)1974 4238 y Fs(+)2061 4223 y Fk(=)27 b(0)33 b(and)f Fh(t)2470 4238 y Fr(\000)2557 4223 y Fk(=)c Fh(h=)p Fk(2)p Fh(;)438 b Fk(\(15\))166 4566 y(b)s(ecause)50 b(they)g(are)f(the)g(ro)s(ots)g (of)f(the)h(functions)g Fh(L)2220 4530 y Fr(00)2263 4566 y Fk(\()p Fh(t)2336 4581 y Fr(\006)2395 4566 y Fk(;)17 b Fi(\001)p Fk(\).)48 b(In)i(the)f(same)g(w)m(a)m(y)-8 b(,)50 b(if)166 4686 y Fh(d)217 4650 y Fq(\017)217 4711 y Fr(\003)256 4686 y Fk(\()p Fh(h)p Fk(\))28 b(=)f Fh(d)570 4650 y Fs(0)570 4711 y Fr(\003)610 4686 y Fk(\()p Fh(h)p Fk(\))22 b(+)862 4695 y(O)938 4686 y(\()p Fh(\017)p Fk(\),)32 b(then)361 4977 y Fh(d)412 4936 y Fs(0)412 5001 y Fr(\003)451 4977 y Fk(\()p Fh(h)p Fk(\))c(=)724 4909 y Fh(L)790 4924 y Fs(3)830 4909 y Fk(\()p Fh(h=)p Fk(2\))22 b Fi(\000)h Fh(L)1248 4924 y Fs(3)1287 4909 y Fk(\(0\))p 724 4953 688 4 v 724 5045 a Fh(L)790 5060 y Fs(2)830 5045 y Fk(\(0\))f Fi(\000)h Fh(L)1143 5060 y Fs(2)1183 5045 y Fk(\()p Fh(h=)p Fk(2\))1422 4977 y Fh(;)166 5380 y Fk(since)36 b Fh(d)459 5344 y Fs(0)459 5404 y Fr(\003)498 5380 y Fk(\()p Fh(h)p Fk(\))g(is)f(the)h(ro)s(ot)e(of)h(the)h(function)f Fh(L)p Fk(\()p Fh(h=)p Fk(2;)17 b Fi(\001)p Fk(\))24 b Fi(\000)h Fh(L)p Fk(\(0;)17 b Fi(\001)p Fk(\).)35 b(W)-8 b(e)36 b(ha)m(v)m(e)g(computed)1745 5712 y(28)p eop %%Page: 29 29 29 28 bop 166 83 a Fk(these)34 b(Melnik)m(o)m(v)f(appro)m(ximations)e (in)g Fi(x)p Fk(A.3.)i(The)h(result)e(is:)388 361 y Fh(d)439 325 y Fs(0)439 386 y(+)497 361 y Fk(\()p Fh(h)p Fk(\))p Fh(;)17 b(d)724 325 y Fs(0)724 386 y Fr(\003)763 361 y Fk(\()p Fh(h)p Fk(\))p Fh(;)g(d)990 325 y Fs(0)990 386 y Fr(\000)1049 361 y Fk(\()p Fh(h)p Fk(\))33 b(=)g Fh(d)1374 325 y Fs(0)1413 361 y Fk(\()p Fh(h)p Fk(\))22 b(+)1665 370 y(O)1741 361 y(\()1779 358 y(e)1823 325 y Fr(\000)p Fq(\031)1921 302 y Fp(2)1955 325 y Fq(=h)2035 361 y Fk(\))p Fh(;)575 542 y(d)626 506 y Fs(0)626 567 y Fr(\000)685 542 y Fk(\()p Fh(h)p Fk(\))g Fi(\000)h Fh(d)990 506 y Fs(0)990 567 y(+)1049 542 y Fk(\()p Fh(h)p Fk(\))33 b(=)1323 539 y(e)1366 506 y Fr(\000)p Fq(\031)1464 482 y Fp(2)1499 506 y Fq(=h)1596 542 y Fh(\016)1643 506 y Fs(0)1682 542 y Fk(\()p Fh(h)p Fk(\))22 b(+)1934 551 y(O)2010 542 y(\()2048 539 y(e)2091 506 y Fr(\000)p Fs(2)p Fq(\031)2224 482 y Fp(2)2259 506 y Fq(=h)2339 542 y Fk(\))2394 241 y Fg(9)2394 316 y(>)2394 341 y(=)2394 490 y(>)2394 515 y(;)2679 440 y Fk(\()p Fh(h)28 b Fi(!)f Fk(0)2977 399 y Fs(+)3036 440 y Fk(\))p Fh(;)179 b Fk(\(16\))166 920 y(where)34 b Fh(d)499 883 y Fs(0)538 920 y Fk(\()p Fh(h)p Fk(\))f(and)f Fh(\016)939 883 y Fs(0)979 920 y Fk(\()p Fh(h)p Fk(\))g(are)h(the)g(ev)m(en)h(analytic)d(functions)361 1350 y Fh(d)412 1309 y Fs(0)451 1350 y Fk(\()p Fh(h)p Fk(\))17 b(=)905 1283 y(1)p 702 1327 454 4 v 702 1423 a(cosh)887 1380 y Fs(2)927 1423 y Fk(\()p Fh(h=)p Fk(2\))1188 1350 y Fi(\000)1298 1283 y Fk(2\()p Fh(\031)1444 1247 y Fs(2)1505 1283 y Fk(+)22 b Fh(h)1659 1247 y Fs(2)1699 1283 y Fk(\))17 b(tanh)1948 1240 y Fs(2)1988 1283 y Fk(\()p Fh(h=)p Fk(2\))p 1298 1327 920 4 v 1685 1419 a(3)p Fh(h)1790 1390 y Fs(2)2255 1350 y Fk(=)27 b(1)22 b Fi(\000)2539 1283 y Fh(\031)2598 1247 y Fs(2)p 2539 1327 99 4 v 2564 1419 a Fk(6)2669 1350 y(+)2777 1283 y Fh(\031)2836 1247 y Fs(2)2898 1283 y Fi(\000)h Fk(15)p 2777 1327 318 4 v 2887 1419 a(36)3105 1350 y Fh(h)3161 1309 y Fs(2)3223 1350 y Fk(+)f Fi(\001)17 b(\001)g(\001)365 1630 y Fh(\016)412 1589 y Fs(0)451 1630 y Fk(\()p Fh(h)p Fk(\))g(=)f(32)p Fh(\031)849 1589 y Fs(2)888 1630 y Fh(h)944 1589 y Fr(\000)p Fs(2)1055 1630 y Fk(tanh)1250 1587 y Fs(2)1290 1630 y Fk(\()p Fh(h=)p Fk(2\))27 b(=)h(8)p Fh(\031)1759 1589 y Fs(2)1814 1484 y Fg( )1880 1630 y Fk(1)22 b Fi(\000)2061 1563 y Fh(h)2117 1527 y Fs(2)p 2061 1607 96 4 v 2084 1699 a Fk(6)2188 1630 y(+)2296 1563 y(17)p Fh(h)2450 1527 y Fs(4)p 2296 1607 194 4 v 2320 1699 a Fk(720)2522 1630 y Fi(\000)2657 1563 y Fk(31)p Fh(h)2811 1527 y Fs(6)p 2631 1607 244 4 v 2631 1699 a Fk(10080)2907 1630 y(+)g Fi(\001)17 b(\001)g(\001)3121 1484 y Fg(!)3204 1630 y Fh(:)166 1941 y Fk(Hence,)37 b(w)m(e)g(guess)g(that)e(the)h (bifurcation)e(v)-5 b(alues)36 b Fh(d)2121 1905 y Fq(\017)2121 1966 y Fr(\006)2179 1941 y Fk(\()p Fh(h)p Fk(\))g(are)f(close)h(to)f Fh(d)2922 1905 y Fs(0)2922 1966 y(0)2994 1941 y Fk(:=)e Fh(d)3181 1905 y Fs(0)3220 1941 y Fk(\(0\))g(=)166 2062 y(1)d Fi(\000)h Fh(\031)412 2026 y Fs(2)451 2062 y Fh(=)p Fk(6)47 b Fi(')h(\000)p Fk(0)p Fh(:)p Fk(645,)d(for)e(small)f(enough)j (v)-5 b(alues)45 b(of)e Fh(h)i Fk(and)f Fh(\017)p Fk(.)h(See,)g(for)f (instance,)166 2182 y(\014gure)32 b(13.)e(W)-8 b(e)32 b(also)e(ha)m(v)m(e)j(the)e(follo)m(wing)e(exp)s(onen)m(tially)h(small) f(Melnik)m(o)m(v)j(prediction)166 2303 y(for)g(the)h(range)g(in)e(whic) m(h)j(the)f(primary)e(homo)s(clinic)e(bifurcations)j(tak)m(e)h(place:) 361 2547 y(\001)442 2562 y Fs(Bifur)632 2547 y Fk(:=)28 b Fh(d)814 2506 y Fq(\017)814 2572 y Fr(\000)872 2547 y Fk(\()p Fh(h)p Fk(\))23 b Fi(\000)f Fh(d)1177 2506 y Fq(\017)1177 2572 y Fs(+)1236 2547 y Fk(\()p Fh(h)p Fk(\))28 b Fi(\030)1501 2544 y Fk(e)1544 2506 y Fr(\000)p Fq(\031)1642 2483 y Fp(2)1677 2506 y Fq(=h)1773 2547 y Fh(\016)1820 2506 y Fs(0)1816 2572 y(0)3280 2547 y Fk(\(17\))166 2904 y(for)k(small)e(enough)j(v)-5 b(alues)33 b(of)f Fh(h)h Fk(and)f Fh(\017)p Fk(.)h(Here,)h Fh(\016)1990 2868 y Fs(0)1986 2928 y(0)2057 2904 y Fk(:=)27 b Fh(\016)2234 2868 y Fs(0)2274 2904 y Fk(\(0\))g(=)h(8)p Fh(\031)2638 2868 y Fs(2)2677 2904 y Fk(.)166 3136 y(W)-8 b(e)43 b(state)g(b)s(elo)m (w)f(a)g(conjecture)i(on)e(the)h(asymptotic)e(b)s(eha)m(vior)i(of)f (the)g(bifurcation)166 3256 y(range)31 b(\001)511 3271 y Fs(Bifur)673 3256 y Fk(.)g(If)g(it)f(w)m(as)h(true,)h(the)f (prediction)f(\(17\))g(w)m(ould)h(giv)m(e)g(the)g(righ)m(t)f(answ)m(er) i(in)166 3377 y(the)38 b(singular)e(p)s(erturbativ)m(e)i(case,)g (whereas)h(in)e(the)h(singular)e(nonp)s(erturbativ)m(e)i(case)166 3497 y(the)45 b(asymptotic)f(constan)m(t)h Fh(\016)1312 3461 y Fs(0)1308 3522 y(0)1399 3497 y Fk(=)j(8)p Fh(\031)1631 3461 y Fs(2)1714 3497 y Fk(w)m(ould)d(ha)m(v)m(e)g(to)f(b)s(e)h (substituted)g(b)m(y)h(a)e(new)166 3618 y(asymptotic)32 b(constan)m(t)h Fh(\016)1108 3581 y Fq(\017)1104 3642 y Fs(0)1172 3618 y Fk(=)27 b(8)p Fh(\031)1383 3581 y Fs(2)1444 3618 y Fk(+)1542 3627 y(O)1618 3618 y(\()p Fh(\017)p Fk(\).)166 3850 y Fl(Conjecture)38 b(8)48 b Fj(F)-7 b(or)39 b(any)g(smal)5 b(l)39 b(enough)f Fh(\017)f Fi(6)p Fk(=)f(0)p Fj(,)j(ther)-5 b(e)39 b(exists)g(a)g(series)3015 3783 y Fg(P)3103 3871 y Fq(j)t Fr(\025)p Fs(0)3246 3850 y Fh(\016)3293 3814 y Fq(\017)3289 3874 y(j)3326 3850 y Fh(h)3382 3814 y Fs(2)p Fq(j)166 3970 y Fj(such)c(that)g(the)g(fol)5 b(lowing)33 b(asymptotic)i(exp)-5 b(ansion)33 b(holds:)361 4215 y Fk(\001)442 4230 y Fs(Bifur)638 4162 y(as)632 4215 y Fk(=)736 4212 y(e)779 4174 y Fr(\000)p Fq(\031)877 4150 y Fp(2)911 4174 y Fq(=h)1009 4132 y Fg(X)1008 4315 y Fq(j)t Fr(\025)p Fs(0)1147 4215 y Fh(\016)1194 4174 y Fq(\017)1190 4239 y(j)1227 4215 y Fh(h)1283 4174 y Fs(2)p Fq(j)1554 4215 y Fk(\()p Fh(h)28 b Fi(!)f Fk(0)1852 4174 y Fs(+)1911 4215 y Fh(;)17 b(\017)35 b Fj(\014xe)-5 b(d)o Fk(\))p Fh(:)166 4666 y Fj(Besides,)32 b(if)639 4600 y Fg(P)727 4687 y Fq(j)t Fr(\025)p Fs(0)870 4666 y Fh(\016)917 4630 y Fs(0)913 4691 y Fq(j)957 4666 y Fh(h)1013 4630 y Fs(2)p Fq(j)1117 4666 y Fj(is)h(the)g(T)-7 b(aylor)32 b(exp)-5 b(ansion)31 b(of)i Fh(\016)2294 4630 y Fs(0)2333 4666 y Fk(\()p Fh(h)p Fk(\))28 b(=)g(32)p Fh(\031)2754 4630 y Fs(2)2793 4666 y Fh(h)2849 4630 y Fr(\000)p Fs(2)2960 4666 y Fk(tanh)3155 4623 y Fs(2)3194 4666 y Fk(\()p Fh(h=)p Fk(2\))p Fj(,)166 4786 y(then)35 b Fh(\016)430 4750 y Fq(\017)426 4811 y(j)490 4786 y Fk(=)28 b Fh(\016)641 4750 y Fs(0)637 4811 y Fq(j)702 4786 y Fk(+)800 4795 y(O)876 4786 y(\()p Fh(\017)p Fk(\))35 b Fj(for)g(al)5 b(l)34 b Fh(j)g Fi(\025)28 b Fk(0)p Fj(.)35 b(In)f(p)-5 b(articular,)35 b Fh(\016)2264 4750 y Fq(\017)2260 4811 y Fs(0)2327 4786 y Fk(=)27 b(8)p Fh(\031)2538 4750 y Fs(2)2600 4786 y Fk(+)2698 4795 y(O)2774 4786 y(\()p Fh(\017)p Fk(\))p Fj(.)166 5019 y Fk(W)-8 b(e)45 b(ha)m(v)m(e)i (limited)41 b(the)46 b(exp)s(osition)e(of)g(the)i(n)m(umerical)d (evidences)k(supp)s(orting)d(this)166 5139 y(conjecture,)37 b(for)e(the)h(sak)m(e)h(of)f(brevit)m(y)-8 b(,)36 b(to)f(one)h(table.)f (Namely)-8 b(,)35 b(table)g(5,)h(in)e(whic)m(h)j(it)166 5259 y(can)f(b)s(e)g(realized)f(v)m(ery)i(clearly)e(that)g Fh(\016)1638 5223 y Fq(\017)1634 5284 y Fs(0)1707 5259 y Fk(=)e(8)p Fh(\031)1924 5223 y Fs(2)1987 5259 y Fk(+)2087 5268 y(O)2163 5259 y(\()p Fh(\017)p Fk(\).)j(The)h(computation)d(of)i (other)166 5380 y(asymptotic)c(co)s(e\016cien)m(ts)i(giv)m(es)f(rise)f (to)g(similar)d(tables.)1745 5712 y(29)p eop %%Page: 30 30 30 29 bop 166 63 a Fv(T)-8 b(able)30 b(5)166 176 y(The)g(constan)m(t)i Fu(\016)763 143 y Fq(\017)760 200 y Fs(0)830 176 y Fv(for)e Fu(\017)25 b Fv(=)g(10)1217 143 y Fr(\000)p Fq(k)1315 176 y Fv(.)31 b(The)f(last)g(ro)m(w)h(con)m(tains)f(the)h(limit)d Fu(\016)2678 143 y Fs(0)2675 200 y(0)2743 176 y Fv(=)d(8)p Fu(\031)2939 143 y Fs(2)2979 176 y Fv(.)216 327 y Fu(k)p 355 378 4 170 v 1540 w(\016)1846 294 y Fq(\017)1843 352 y Fs(0)p 166 381 3164 4 v 216 500 a Fv(10)p 355 551 4 170 v 100 w(78)p Fu(:)p Fv(95683519783)q(302)q(44)q(91)q(82)q(50)q(679) q(63)q(68)q(19)q(03)q(490)q(80)q(91)q(43)q(25)q(143)q(55)q(28)q(50)q (38)q(189)16 b Fu(:)f(:)h(:)216 669 y Fv(20)p 355 720 V 100 w(78)p Fu(:)p Fv(95683520871)q(486)q(89)q(49)q(58)q(77)q(435)q (53)q(11)q(81)q(54)q(212)q(11)q(09)q(22)q(86)q(809)q(18)q(69)q(09)q(55) q(383)g Fu(:)f(:)h(:)216 839 y Fv(30)p 355 889 V 100 w(78)p Fu(:)p Fv(95683520871)q(486)q(89)q(50)q(67)q(59)q(279)q(98)q(90) q(03)q(90)q(637)q(92)q(04)q(89)q(77)q(088)q(63)q(98)q(08)q(36)q(546)g Fu(:)f(:)h(:)216 1008 y Fv(40)p 355 1059 V 100 w(78)p Fu(:)p Fv(95683520871)q(486)q(89)q(50)q(67)q(59)q(279)q(99)q(00)q(92)q (09)q(082)q(49)q(87)q(13)q(41)q(346)q(74)q(14)q(46)q(26)q(028)g Fu(:)f(:)h(:)216 1177 y Fv(50)p 355 1228 V 100 w(78)p Fu(:)p Fv(95683520871)q(486)q(89)q(50)q(67)q(59)q(279)q(99)q(00)q(92)q (09)q(082)q(50)q(95)q(95)q(25)q(792)q(52)q(36)q(82)q(68)q(609)g Fu(:)f(:)h(:)216 1347 y Ft(1)p 355 1398 V 99 w Fv(78)p Fu(:)p Fv(95683520871)q(486)q(89)q(50)q(67)q(59)q(279)q(99)q(00)q(92)q (09)q(082)q(50)q(95)q(95)q(25)q(792)q(63)q(25)q(01)q(13)q(068)g Fu(:)f(:)h(:)166 1610 y Fl(6)112 b(Conclusion)166 1966 y Fk(W)-8 b(e)23 b(ha)m(v)m(e)h(form)m(ulated)d(sev)m(eral)i (conjectures)h(related)e(to)g(the)h(singular)e(phenomena)i(that)166 2087 y(app)s(ear)40 b(in)f(some)h(billiards)d(tables.)j(W)-8 b(e)40 b(ha)m(v)m(e)h(also)e(presen)m(ted)k(some)d(v)m(ery)h(accurate) 166 2207 y(n)m(umerical)j(exp)s(erimen)m(ts,)h(whic)m(h)h(supp)s(ort)f (strongly)f(b)s(oth)h(conjectures.)i(The)f(next)166 2327 y(step)c(m)m(ust)g(b)s(e)f(to)g(pro)m(v)m(e)h(these)h(conjectures.)g(W) -8 b(e)41 b(hop)s(e)h(that)f(this)g(problem)f(w)m(ould)166 2448 y(b)s(e)c(a)g(stim)m(ulating)d(c)m(hallenge)i(for)g(some)h (readers,)h(whether)g(in)f(this)f(framew)m(ork)h(with)166 2568 y(billiard)19 b(maps)i(or)h(in)g(the)g(m)m(uc)m(h)h(more)e (celebrated)i(ab)s(out)f(generalized)f(standard)i(maps.)166 3076 y Fl(Ac)m(kno)m(wledgemen)m(ts)166 3432 y Fk(The)42 b(author)f(has)g(b)s(een)h(partially)d(supp)s(orted)j(b)m(y)g(the)f (Gran)m(t)g(CIRIT)h(2001SGR-70)166 3553 y(\(Catalonia\))c(and)j(the)g (MCyT-FEDER)h(Gran)m(t)e(BFM2003-9504)e(\(Spain\).)i(I)h(w)m(ould)166 3673 y(lik)m(e)f(to)h(thank)g(to)f(Leop)s(old)g(P)m(alomo,)f(P)m(ablo)h (S\023)-49 b(anc)m(hez,)43 b(and)e(Jaume)f(Timoneda)g(for)166 3793 y(their)e(e\013orts)h(to)f(sho)m(w)h(me)f(the)h(p)s(ossibilities)d (of)i(Beo)m(wulf)g(clusters.)i(I)e(am)g(also)f(v)m(ery)166 3914 y(grateful)25 b(to)h(R.)g(de)h(la)e(Lla)m(v)m(e)h(for)g(its)g(v)-5 b(aluable)25 b(advises)i(on)f(the)h(P)-8 b(ARI)26 b(system.)i(Besides,) 166 4034 y(I)d(am)f(indebted)h(to)g(A.)g(Delshams,)f(E.)h(F)-8 b(on)m(tic)m(h,)25 b(and)f(C.)i(Sim\023)-49 b(o)23 b(for)h(v)m(ery)j (useful)d(remarks)166 4155 y(and)34 b(commen)m(ts.)g(Finally)-8 b(,)31 b(it)i(is)g(a)h(pleasan)m(t)g(obligation)c(to)k(express)i(m)m(y) e(gratitude)f(to)166 4275 y(the)g(P)-8 b(ARI)33 b(dev)m(elop)s(ers)h (that)e(mak)m(e)h(p)s(ossible)f(the)h(existence)h(of)e(an)g(excellen)m (t)h(to)s(ol.)166 4783 y Fl(A)112 b(Some)37 b(details)f(of)i(the)f (Melnik)m(o)m(v)g(computations)166 5139 y Fk(W)-8 b(e)36 b(ha)m(v)m(e)g(adopted)g(a)f(v)m(ery)i(compact)e(st)m(yle)g(along)f (this)h(app)s(endix,)h(a)m(v)m(oiding)e(to)h(giv)m(e)166 5259 y(the)41 b(more)f(cum)m(b)s(ersome)g(and)h(less)g(crucial)e (details)g(in)h(the)h(computations,)e(b)s(ecause)166 5380 y(similar)29 b(ones)34 b(can)e(b)s(e)h(found)g(in)f(the)h (literature.)e(See,)j(for)e(instance,)h([4,5,7].)1745 5712 y(30)p eop %%Page: 31 31 31 30 bop 166 83 a Fj(A.1)100 b(Melnikov)34 b(p)-5 b(otential)34 b(for)h(monomial)e(p)-5 b(erturb)g(ations)166 426 y Fk(Here,)29 b(w)m(e)g(address)h(some)e(computations)f(related)g(to)h(the)h (elliptic)c(Melnik)m(o)m(v)j(p)s(oten)m(tial)166 547 y Fh(L)232 562 y Fq(n)279 547 y Fk(\()p Fh(t)p Fk(\))h(=)g Fh(ae)620 511 y Fs(2)p Fq(n)719 480 y Fg(P)807 568 y Fq(k)r Fr(2)p Ff(Z)958 547 y Fh(`)999 562 y Fq(n)1046 547 y Fk(\()p Fh(t)22 b Fk(+)h Fh(k)s(h)p Fk(\))33 b(in)m(tro)s(duced)h (in)e(\(4\).)h(All)f(the)i(computed)f(ob)5 b(jects)35 b(are)166 667 y(expressed)g(in)d(terms)h(of)f(the)h(ev)m(en)h(deriv)-5 b(ativ)m(es)33 b(of)f(the)h(function)361 893 y Fh( )t Fk(\()p Fh(t)p Fk(\))28 b(=)f(\(2)p Fh(K)r(=h)p Fk(\))985 852 y Fs(2)1041 893 y Fk(dn)1149 850 y Fs(2)1189 893 y Fk(\(2)p Fh(K)7 b(t=h;)17 b(k)s Fk(\))p Fh(;)166 1222 y Fk(where)27 b(dn\()p Fh(u)p Fk(\))g(=)h(dn\()p Fh(u;)17 b(k)s Fk(\))25 b(is)g(one)h(of)f(the)i(t)m(w)m(elv)m(e)g(Jacobian)e (elliptic)d(functions.)k(Here,)h Fh(k)166 1356 y Fk(is)g(the)h Fj(mo)-5 b(dulus)p Fk(,)28 b Fh(K)34 b Fk(=)1042 1285 y Fg(R)1097 1305 y Fq(\031)r(=)p Fs(2)1081 1378 y(0)1215 1356 y Fk(\(1)12 b Fi(\000)g Fh(k)1457 1320 y Fs(2)1513 1356 y Fk(sin)17 b Fh(u)p Fk(\))1744 1320 y Fr(\000)p Fs(1)p Fq(=)p Fs(2)1925 1356 y Fk(d)p Fh(u)27 b Fk(is)g(the)h Fj(c)-5 b(omplete)29 b(el)5 b(liptic)30 b(inte)-5 b(gr)g(al)30 b(of)166 1490 y(the)d(\014rst)g(kind)p Fk(,)c(and)h Fh(K)1016 1454 y Fr(0)1068 1490 y Fk(=)1171 1419 y Fg(R)1226 1439 y Fq(\031)r(=)p Fs(2)1210 1512 y(0)1344 1490 y Fk(\(1)5 b Fi(\000)g Fh(k)1572 1454 y Fr(0)p Fs(2)1647 1490 y Fk(sin)16 b Fh(u)p Fk(\))1877 1454 y Fr(\000)p Fs(1)p Fq(=)p Fs(2)2058 1490 y Fk(d)p Fh(u)24 b Fk(where)h Fh(k)2519 1454 y Fr(0)2566 1490 y Fk(is)e(the)i Fj(c)-5 b(omplementary)166 1610 y(mo)g(dulus)p Fk(:)36 b Fh(k)627 1574 y Fs(2)692 1610 y Fk(+)24 b Fh(k)846 1574 y Fr(0)p Fs(2)940 1610 y Fk(=)34 b(1.)i(Finally)-8 b(,)34 b(the)j(quan)m(tit)m(y)g Fh(q)i Fk(=)2275 1607 y(e)2318 1574 y Fr(\000)p Fq(\031)r(K)2481 1551 y Fb(0)2503 1574 y Fq(=K)2643 1610 y Fk(is)d(called)f(the)i Fj(nome)p Fk(.)166 1730 y(W)-8 b(e)33 b(refer)g(to)f([24])g(for)g(a)h (general)f(bac)m(kground)h(on)g(elliptic)d(functions.)166 1954 y(W)-8 b(e)30 b(recall)e(that)i Fj(el)5 b(liptic)31 b(functions)h(ar)-5 b(e)32 b(char)-5 b(acterize)g(d)31 b(\(mo)-5 b(dulo)31 b(additive)g(c)-5 b(onstants\))166 2074 y(by)38 b(their)g(p)-5 b(erio)g(ds,)37 b(p)-5 b(oles)37 b(and)h(princip)-5 b(al)37 b(p)-5 b(arts)8 b Fk(:)36 b(the)g(di\013erence)h(of)f(elliptic)d(functions)166 2194 y(with)c(the)h(same)f(p)s(erio)s(ds,)g(p)s(oles)g(and)h(principal) d(parts)j(is)f(a)g(b)s(ounded)h(en)m(tire)g(function,)166 2315 y(and)j(hence)h(constan)m(t)f(b)m(y)h(Liouville's)c(theorem.)166 2538 y(Th)m(us,)46 b(w)m(e)f(are)f(naturally)e(led)i(to)f(the)i(searc)m (h)g(for)e(the)i(p)s(oles)e(\(and)h(their)g(principal)166 2658 y(parts\))33 b(of)f Fh(L)631 2673 y Fq(n)678 2658 y Fk(\()p Fh(t)p Fk(\))c(=)g Fh(ae)1017 2622 y Fs(2)p Fq(n)1116 2592 y Fg(P)1203 2679 y Fq(k)r Fr(2)p Ff(Z)1354 2658 y Fh(`)1395 2673 y Fq(n)1442 2658 y Fk(\()p Fh(t)22 b Fk(+)g Fh(k)s(h)p Fk(\).)166 2881 y(The)32 b(function)e Fh(`)786 2896 y Fq(n)833 2881 y Fk(\()p Fh(t)p Fk(\))h(is)f (meromorphic)f(and)i Fh(\031)t Fk(i-p)s(erio)s(dic.)d(Its)j(p)s(oles)g (are)g(the)g(p)s(oin)m(ts)f(in)166 3001 y(the)38 b(sets)h Fh(\031)t Fk(i)p Fh(=)p Fk(2)24 b(+)h Fh(\031)t Fk(i)p Fe(Z)34 b Fk(and)k Fh(\031)t Fk(i)p Fh(=)p Fk(2)24 b Fi(\006)i Fh(h=)p Fk(2)f(+)g Fh(\031)t Fk(i)p Fe(Z)p Fk(.)34 b(F)-8 b(rom)36 b(no)m(w)j(on,)e Fh(a)2727 3016 y Fq(j)2764 3001 y Fk(\()p Fh(f)5 b(;)17 b(\034)11 b Fk(\))38 b(stands)g(for)166 3122 y(the)30 b(co)s(e\016cien)m(t)f(of)g (the)h(term)e(\()p Fh(t)15 b Fi(\000)g Fh(\034)c Fk(\))1555 3086 y Fq(j)1622 3122 y Fk(in)28 b(the)i(Lauren)m(t)f(expansion)h(of)f (a)g(meromorphic)166 3242 y(function)k Fh(f)11 b Fk(\()p Fh(t)p Fk(\))33 b(around)g Fh(t)d Fk(=)e Fh(\034)11 b Fk(.)34 b(The)h(p)s(oles)e Fh(t)1851 3201 y Fr(\006)1851 3264 y Fs(0)1939 3242 y Fi(2)c Fh(\031)t Fk(i)p Fh(=)p Fk(2)21 b Fi(\006)j Fh(h=)p Fk(2)e(+)h Fh(\031)t Fk(i)p Fe(Z)29 b Fk(are)34 b(simple)d(and,)166 3363 y(due)h(to)g(the)g (symmetry)-8 b(,)32 b Fh(a)1160 3378 y Fr(\000)p Fs(1)1254 3363 y Fk(\()p Fh(`)1333 3378 y Fq(n)1380 3363 y Fh(;)17 b(t)1459 3321 y Fs(+)1459 3384 y(0)1518 3363 y Fk(\))j(+)h Fh(a)1724 3378 y Fr(\000)p Fs(1)1818 3363 y Fk(\()p Fh(`)1897 3378 y Fq(n)1944 3363 y Fh(;)c(t)2023 3321 y Fr(\000)2023 3384 y Fs(0)2082 3363 y Fk(\))28 b(=)f(0.)32 b(The)g(p)s(oles)g Fh(t)2840 3378 y Fs(0)2907 3363 y Fi(2)c Fh(\031)t Fk(i)p Fh(=)p Fk(2)19 b(+)h Fh(\031)t Fk(i)p Fe(Z)166 3483 y Fk(ha)m(v)m(e)34 b(order)f(2)p Fh(n)22 b Fi(\000)g Fk(2)33 b(and)f Fh(a)1196 3498 y Fr(\000)p Fq(j)1288 3483 y Fk(\()p Fh(`)1367 3498 y Fq(n)1414 3483 y Fh(;)17 b(t)1493 3498 y Fs(0)1532 3483 y Fk(\))28 b(=)f(0)33 b(for)f(all)e(o)s(dd)j(in)m (tegers)g Fh(j)g Fi(\025)28 b Fk(1.)166 3706 y(Therefore,)40 b Fh(L)707 3721 y Fq(n)755 3706 y Fk(\()p Fh(t)p Fk(\))f(=)g Fh(ae)1116 3670 y Fs(2)p Fq(n)1215 3640 y Fg(P)1303 3727 y Fq(k)r Fr(2)p Ff(Z)1453 3706 y Fh(`)1494 3721 y Fq(n)1541 3706 y Fk(\()p Fh(t)27 b Fk(+)g Fh(k)s(h)p Fk(\))39 b(is)g(an)g (elliptic)d(function)j(c)m(haracterized)166 3826 y(\(mo)s(dulo)29 b(an)i(additiv)m(e)g(constan)m(t\))h(b)m(y)g(the)g(follo)m(wing)c(prop) s(erties:)j(\(1\))g(Its)h(p)s(erio)s(ds)f(are)166 3947 y Fh(h)k Fk(and)g Fh(\031)t Fk(i;)f(\(2\))g(Its)i(p)s(oles)e(are)h(the) g(p)s(oin)m(ts)g(in)f(the)h(set)h Fh(\031)t Fk(i)p Fh(=)p Fk(2)22 b(+)i Fh(h)p Fe(Z)d Fk(+)i Fh(\031)t Fk(i)p Fe(Z)p Fk(;)32 b(and)j(\(3\))f(Its)166 4067 y(principal)d(part)h(around)g(a)h (p)s(ole)e Fh(t)1439 4082 y Fs(0)1512 4067 y Fk(is)h Fh(ae)1706 4031 y Fs(2)p Fq(n)1805 4001 y Fg(P)1893 4027 y Fq(n)p Fr(\000)p Fs(1)1893 4092 y Fq(j)t Fs(=1)2046 4067 y Fh(a)2097 4082 y Fr(\000)p Fs(2)p Fq(j)2224 4067 y Fk(\()p Fh(`)2303 4082 y Fq(n)2350 4067 y Fh(;)17 b(\031)t Fk(i)p Fh(=)p Fk(2\)\()p Fh(t)k Fi(\000)h Fh(t)2845 4082 y Fs(0)2885 4067 y Fk(\))2923 4031 y Fr(\000)p Fs(2)p Fq(j)3049 4067 y Fk(.)166 4290 y(On)50 b(the)h(other)f(hand,)h(the)g (square)g(of)f(the)h(Jacobian)e(elliptic)f(function)h(dn)q(\()p Fh(u)p Fk(\))57 b(=)166 4411 y(dn\()p Fh(u;)17 b(k)s Fk(\))48 b(is)g(c)m(haracterized)h(\(mo)s(dulo)d(an)i(additiv)m(e)g (constan)m(t\))h(b)m(y)g(the)g(prop)s(erties:)166 4531 y(\(1'\))31 b(Its)g(p)s(erio)s(ds)g(are)g(2)p Fh(K)38 b Fk(and)31 b(2)p Fh(K)1490 4495 y Fr(0)1513 4531 y Fk(i;)f(\(2'\))h (Its)h(p)s(oles)e(are)i(the)f(p)s(oin)m(ts)g(in)f(the)h(set)h Fh(K)3308 4495 y Fr(0)3332 4531 y Fk(i)18 b(+)166 4651 y(2)p Fh(K)7 b Fe(Z)15 b Fk(+)j(2)p Fh(K)622 4615 y Fr(0)645 4651 y Fk(i)p Fe(Z)p Fk(;)27 b(and)j(\(3'\))g(The)i(principal)c(part)i (around)h(an)m(y)g(p)s(ole)e Fh(u)2750 4666 y Fs(0)2820 4651 y Fk(is)h Fi(\000)p Fk(\()p Fh(u)17 b Fi(\000)h Fh(u)3255 4666 y Fs(0)3294 4651 y Fk(\))3332 4615 y Fr(\000)p Fs(2)3427 4651 y Fk(,)166 4772 y(see)34 b([24,)e Fi(x)p Fk(22].)166 4995 y(Hence,)i(if)d(w)m(e)j(tak)m(e)f Fh(q)f Fk(=)1105 4992 y(e)1149 4959 y Fr(\000)p Fq(\031)1247 4935 y Fp(2)1281 4959 y Fq(=h)1361 4995 y Fk(,)h(then)g Fh(K)1733 4959 y Fr(0)1784 4995 y Fk(=)28 b Fh(K)7 b(\031)t(=h)32 b Fk(and)361 5293 y Fh(L)427 5308 y Fq(n)474 5293 y Fk(\()p Fh(t)p Fk(\))c(=)g(constan)m(t)17 b Fi(\000)p Fh(ae)1267 5251 y Fs(2)p Fq(n)1367 5185 y(n)p Fr(\000)p Fs(1)1374 5210 y Fg(X)1372 5392 y Fq(j)t Fs(=1)1574 5225 y Fh(\030)1617 5240 y Fq(n;j)1716 5225 y Fk(\()p Fh(h)p Fk(\))p 1527 5269 369 4 v 1527 5361 a(\(2)p Fh(j)27 b Fi(\000)c Fk(1\)!)1905 5293 y Fh( )1972 5251 y Fs(\(2)p Fq(j)t Fr(\000)p Fs(2\))2189 5293 y Fk(\()p Fh(t)p Fk(\))p Fh(:)902 b Fk(\(A.1\))1745 5712 y(31)p eop %%Page: 32 32 32 31 bop 166 83 a Fk(where)34 b Fh(\030)491 98 y Fq(n;j)589 83 y Fk(\()p Fh(h)p Fk(\))28 b(=)g Fh(a)904 98 y Fr(\000)p Fs(2)p Fq(j)1031 83 y Fk(\()p Fh(`)1110 98 y Fq(n)1156 83 y Fh(;)17 b(\031)t Fk(i)p Fh(=)p Fk(2\).)31 b(In)i(particular,)e (the)i(Melnik)m(o)m(v)g(terms)f(\(5\))h(are)361 375 y Fh(A)434 390 y Fs(1)474 375 y Fk(\()p Fh(h)p Fk(\))27 b(=)h Fh(ae)833 334 y Fs(2)p Fq(n)932 267 y(n)p Fr(\000)p Fs(1)939 292 y Fg(X)937 475 y Fq(j)t Fs(=1)1092 308 y Fh(\030)1135 323 y Fq(n;j)1233 308 y Fk(\()p Fh(h)p Fk(\))p Fh(\016)1408 323 y Fq(j)1445 308 y Fk(\()p Fh(h)p Fk(\))p 1092 352 486 4 v 1150 444 a(\(2)p Fh(j)g Fi(\000)23 b Fk(1\)!)1587 375 y Fh(;)212 b Fk(\012)1896 390 y Fs(1)1936 375 y Fk(\()p Fh(h)p Fk(\))27 b(=)h Fi(\000)p Fh(ae)2372 334 y Fs(2)p Fq(n)2472 267 y(n)p Fr(\000)p Fs(1)2478 292 y Fg(X)2477 475 y Fq(j)t Fs(=1)2631 308 y Fh(\030)2674 323 y Fq(n;j)2773 308 y Fk(\()p Fh(h)p Fk(\))p Fh(\033)2960 323 y Fq(j)2997 308 y Fk(\()p Fh(h)p Fk(\))p 2631 352 498 4 v 2696 444 a(\(2)p Fh(j)g Fi(\000)22 b Fk(1\)!)166 787 y(where)28 b Fh(\016)485 802 y Fq(j)522 787 y Fk(\()p Fh(h)p Fk(\))g(:=)f Fh( )879 750 y Fs(\(2)p Fq(j)t Fr(\000)p Fs(2\))1096 787 y Fk(\(0\))11 b Fi(\000)g Fh( )1387 750 y Fs(\(2)p Fq(j)t Fr(\000)p Fs(2\))1603 787 y Fk(\()p Fh(h=)p Fk(2\))27 b(and)g Fh(\033)2099 802 y Fq(j)2135 787 y Fk(\()p Fh(h)p Fk(\))h(:=)g Fh( )2493 750 y Fs(\(2)p Fq(j)t Fs(\))2620 787 y Fk(\(0\))11 b(+)g Fh( )2910 750 y Fs(\(2)p Fq(j)t Fs(\))3035 787 y Fk(\()p Fh(h=)p Fk(2\).)26 b(W)-8 b(e)166 907 y(need)34 b(t)m(w)m(o)f(lemmas)e(to)h(study)i(the)f (asymptotic)f(b)s(eha)m(vior)g(of)g(the)h(ab)s(o)m(v)m(e)g(quan)m (tities.)166 1127 y Fl(Lemma)k(9)49 b Fj(F)-7 b(or)44 b(any)h Fk(1)i Fi(\024)g Fh(j)52 b Fi(\024)47 b Fh(n)30 b Fi(\000)g Fk(1)p Fj(,)45 b(ther)-5 b(e)45 b(exists)g(an)g(even)f (analytic)h(function)176 1221 y Fk(\026)166 1247 y Fh(\030)209 1262 y Fq(n;j)308 1247 y Fk(\()p Fh(h)p Fk(\))34 b Fj(such)h(that)906 1221 y Fk(\026)895 1247 y Fh(\030)938 1262 y Fq(n;j)1037 1247 y Fk(\(0\))27 b(=)h(\()p Fi(\000)p Fk(1\))1495 1211 y Fq(n)1542 1247 y Fk(4)1591 1211 y Fq(n)p Fr(\000)p Fq(j)1760 1247 y Fj(and)34 b Fh(\030)1992 1262 y Fq(n;j)2091 1247 y Fk(\()p Fh(h)p Fk(\))27 b(=)h Fh(h)2410 1211 y Fs(2\()p Fq(j)t Fr(\000)p Fq(n)p Fs(\))2645 1221 y Fk(\026)2634 1247 y Fh(\030)2677 1262 y Fq(n;j)2776 1247 y Fk(\()p Fh(h)p Fk(\))p Fj(.)166 1467 y Fk(The)34 b(ab)s(o)m(v)m(e)f(lemma)d(is) i(obtained)g(using)h(a)f(tric)m(k)h(con)m(tained)f(in)g([4,)h Fi(x)p Fk(4.4.4].)166 1687 y Fl(Lemma)k(10)49 b Fj(If)34 b Fh(T)42 b Fk(=)27 b(2)p Fh(\031)t(=h)p Fj(,)35 b(then)126 1907 y(a\))50 b Fh( )t Fk(\()p Fh(t)p Fk(\))27 b(=)h(2)p Fh(T)695 1871 y Fs(2)734 1907 y Fk([1)p Fh(=)p Fk(8)21 b(+)h Fh(q)1074 1871 y Fs(2)1136 1907 y Fk(+)g Fh(q)e Fk(cos)e Fh(T)c(t)22 b Fk(+)g(2)p Fh(q)1767 1871 y Fs(2)1822 1907 y Fk(cos)17 b(2)p Fh(T)d(t)22 b Fk(+)2244 1916 y(O)2320 1907 y(\()p Fh(q)2405 1871 y Fs(3)2444 1907 y Fk(\)])p Fj(,)131 2028 y(b\))50 b Fh( )333 1992 y Fs(\(4\))427 2028 y Fk(\()p Fh(t)500 2043 y Fr(\006)559 2028 y Fk(\))p Fh(= )713 1992 y Fr(00)755 2028 y Fk(\()p Fh(t)828 2043 y Fr(\006)888 2028 y Fk(\))27 b(=)h Fi(\000)p Fh(T)1205 1992 y Fs(2)1244 2028 y Fk(\(1)22 b Fi(\006)h Fk(24)p Fh(q)i Fk(+)1717 2037 y(O)1793 2028 y(\()p Fh(q)1878 1992 y Fs(2)1917 2028 y Fk(\)\))p Fj(,)35 b(wher)-5 b(e)34 b Fh(t)2368 2043 y Fs(+)2455 2028 y Fk(=)28 b(0)34 b Fj(and)h Fh(t)2867 2043 y Fr(\000)2954 2028 y Fk(=)27 b Fh(h=)p Fk(2)p Fj(.)131 2148 y(c\))50 b Fh(\016)309 2163 y Fq(j)345 2148 y Fk(\()p Fh(h)p Fk(\))28 b(:=)g Fh( )703 2112 y Fs(\(2)p Fq(j)t Fr(\000)p Fs(2\))920 2148 y Fk(\(0\))21 b Fi(\000)i Fh( )1233 2112 y Fs(\(2)p Fq(j)t Fr(\000)p Fs(2\))1450 2148 y Fk(\()p Fh(h=)p Fk(2\))k(=)h(4)p Fh(T)1931 2112 y Fs(2)p Fq(j)2002 2148 y Fh(q)t Fk([\()p Fi(\000)p Fk(1\))2278 2112 y Fq(j)t Fr(\000)p Fs(1)2427 2148 y Fk(+)2525 2157 y(O)2601 2148 y(\()p Fh(q)t Fk(\)])p Fj(,)34 b(for)h(al)5 b(l)34 b Fh(j)g Fi(\025)28 b Fk(1)p Fj(.)126 2269 y(d\))50 b Fh(\033)321 2284 y Fq(j)358 2269 y Fk(\()p Fh(h)p Fk(\))27 b(:=)h Fh( )715 2232 y Fs(\(2)p Fq(j)t Fs(\))842 2269 y Fk(\(0\))21 b(+)h Fh( )1153 2232 y Fs(\(2)p Fq(j)t Fs(\))1280 2269 y Fk(\()p Fh(h=)p Fk(2\))27 b(=)h(2\(2)p Fh(T)14 b Fk(\))1886 2232 y Fs(2)p Fq(j)t Fs(+2)2047 2269 y Fh(q)2094 2232 y Fs(2)2133 2269 y Fk([\()p Fi(\000)p Fk(1\))2362 2232 y Fq(j)2421 2269 y Fk(+)2519 2278 y(O)2595 2269 y(\()p Fh(q)t Fk(\)])p Fj(,)35 b(for)f(al)5 b(l)35 b Fh(j)e Fi(\025)c Fk(1)p Fj(.)166 2607 y Fl(PR)m(OOF.)49 b Fk(The)36 b(\014rst)h(item)d(follo)m (ws)g(directly)h(from)g(the)h(de\014nition)f(of)g Fh( )t Fk(\()p Fh(t)p Fk(\))g(and)h(the)166 2727 y(F)-8 b(ourier)31 b(expansion)361 2984 y(dn)q(\(2)p Fh(K)7 b(t=h;)17 b(k)s Fk(\))27 b(=)1103 2917 y Fh(\031)p 1063 2961 139 4 v 1063 3053 a Fk(2)p Fh(K)1234 2984 y Fk(+)1342 2917 y(2)p Fh(\031)p 1342 2961 108 4 v 1351 3053 a(K)1483 2901 y Fg(X)1477 3084 y Fq(n)p Fr(\025)p Fs(1)1738 2917 y Fh(q)1785 2881 y Fq(n)p 1636 2961 299 4 v 1636 3053 a Fk(1)22 b(+)g Fh(q)1852 3024 y Fs(2)p Fq(n)1961 2984 y Fk(cos\()p Fh(nT)14 b(t)p Fk(\))p Fh(;)166 3383 y Fk(whic)m(h)29 b(can)g(b)s(e)g(found)g (in)e([24,)i(page)f(511].)g(The)i(others)f(follo)m(w)e(from)g(the)i (\014rst)g(one.)98 b Fa(2)166 3722 y Fk(No)m(w)42 b(w)m(e)h(are)f (ready)g(to)g(pro)m(v)m(e)h(the)f(form)m(ulae)e(for)h(the)h(constan)m (ts)i Fh(\013)2827 3686 y Fs(0)2826 3746 y(0)2909 3722 y Fk(=)f Fh(\013)3091 3686 y Fs(0)3130 3722 y Fk(\(0\))f(and)166 3842 y Fh(!)231 3806 y Fs(0)227 3867 y(0)297 3842 y Fk(=)28 b Fh(!)466 3806 y Fs(0)505 3842 y Fk(\(0\))j(giv)m(en)h(in)e(\(7\).)i (F)-8 b(or)30 b(instance,)i(using)f(the)h(couple)g(of)f(lemmas)f(w)m(e) i(see)h(that)361 4131 y Fh(\013)424 4090 y Fs(0)463 4131 y Fk(\()p Fh(h)p Fk(\))28 b(=)g(4)p Fh(e)821 4090 y Fs(2)p Fq(n)919 4024 y(n)p Fr(\000)p Fs(1)926 4048 y Fg(X)925 4231 y Fq(j)t Fs(=1)1079 4064 y Fk(\()p Fi(\000)p Fk(1\))1281 4028 y Fq(j)t Fr(\000)p Fs(1)1408 4064 y Fh(T)1479 4028 y Fs(2)p Fq(j)p 1079 4108 472 4 v 1130 4200 a Fk(\(2)p Fh(j)g Fi(\000)23 b Fk(1\)!)1560 4131 y Fh(\030)1603 4146 y Fq(n;j)1702 4131 y Fk(\()p Fh(h)p Fk(\))k(=)h(4\()p Fh(e=h)p Fk(\))2240 4090 y Fs(2)p Fq(n)2339 4024 y(n)p Fr(\000)p Fs(1)2345 4048 y Fg(X)2344 4231 y Fq(j)t Fs(=1)2498 4064 y Fk(\()p Fi(\000)p Fk(1\))2700 4028 y Fq(j)t Fr(\000)p Fs(1)2827 4064 y Fk(\(2)p Fh(\031)t Fk(\))3011 4028 y Fs(2)p Fq(j)p 2498 4108 585 4 v 2606 4200 a Fk(\(2)p Fh(j)g Fi(\000)23 b Fk(1\)!)3103 4105 y(\026)3092 4131 y Fh(\030)3135 4146 y Fq(n;j)3234 4131 y Fk(\()p Fh(h)p Fk(\))166 4560 y(and)37 b Fh(\013)423 4524 y Fs(0)422 4584 y(0)496 4560 y Fk(=)d Fh(\013)669 4524 y Fs(0)708 4560 y Fk(\(0\))g(=)g(4)1026 4524 y Fs(1)p Fr(\000)p Fq(n)1179 4493 y Fg(P)1267 4520 y Fq(n)p Fr(\000)p Fs(1)1267 4584 y Fq(j)t Fs(=1)1431 4513 y(\()p Fr(\000)p Fs(1\))1575 4490 y Fc(j)s Fb(\000)p Fp(1)1687 4513 y Fs(\(2)p Fq(\031)r Fs(\))1819 4490 y Fp(2)p Fc(j)p 1431 4537 453 4 v 1541 4594 a Fs(\(2)p Fq(j)t Fr(\000)p Fs(1\)!)1893 4560 y Fk(\()p Fi(\000)p Fk(1\))2095 4524 y Fq(n)2142 4560 y Fk(4)2191 4524 y Fq(n)p Fr(\000)p Fq(j)2359 4560 y Fk(=)g(\()p Fi(\000)p Fk(1\))2671 4524 y Fq(n)2718 4560 y Fk(4)p Fh(\031)2826 4524 y Fs(2)2882 4493 y Fg(P)2969 4520 y Fq(n)p Fr(\000)p Fs(2)2969 4584 y Fq(j)t Fs(=0)3133 4513 y(\()p Fr(\000)p Fs(1\))3277 4490 y Fc(j)3311 4513 y Fq(\031)3354 4490 y Fp(2)p Fc(j)p 3133 4537 284 4 v 3159 4594 a Fs(\(2)p Fq(j)t Fs(+1\)!)3427 4560 y Fk(.)166 4702 y(W)-8 b(e)38 b(recall)e(that)i Fh(e)e Fk(=)h(tanh\()p Fh(h=)p Fk(2\))f Fi(\030)g Fh(h=)p Fk(2)i(and)1986 4675 y(\026)1976 4702 y Fh(\030)2019 4717 y Fq(n;j)2117 4702 y Fk(\(0\))e(=)g(\()p Fi(\000)p Fk(1\))2592 4665 y Fq(n)2639 4702 y Fk(4)2688 4665 y Fq(n)p Fr(\000)p Fq(j)2822 4702 y Fk(.)i(The)h(constan)m(t)166 4822 y Fh(!)231 4786 y Fs(0)227 4847 y(0)302 4822 y Fk(is)32 b(obtained)g(in)g(the)h(same)g(w) m(a)m(y)-8 b(.)166 5042 y(Finally)g(,)24 b(w)m(e)29 b(are)e(going)e(to) i(c)m(hec)m(k)i(that)e(when)h(the)g(monomial)23 b(p)s(erturbation)j(is) h(quartic)166 5162 y(the)33 b(functions)g Fh(\013)818 5126 y Fs(0)857 5162 y Fk(\()p Fh(h)p Fk(\))f(and)h Fh(!)1276 5126 y Fs(0)1315 5162 y Fk(\()p Fh(h)p Fk(\))f(ha)m(v)m(e)i(the)f(form) e(giv)m(en)i(in)f(\(8\).)g(If)h Fh(n)28 b Fk(=)f(2,)33 b(then)361 5380 y Fh(\013)424 5339 y Fs(0)463 5380 y Fk(\()p Fh(h)p Fk(\))28 b(=)g(4)p Fh(e)821 5339 y Fs(4)860 5380 y Fh(T)931 5339 y Fs(2)970 5380 y Fh(\030)1013 5395 y Fs(2)p Fq(;)p Fs(1)1107 5380 y Fk(\()p Fh(h)p Fk(\))g(=)f(16)p Fh(\031)1527 5339 y Fs(2)1566 5380 y Fh(h)1622 5339 y Fr(\000)p Fs(2)1733 5380 y Fk(tanh)1928 5337 y Fs(4)1968 5380 y Fk(\()p Fh(h=)p Fk(2\))p Fh(a)2249 5395 y Fr(\000)p Fs(2)2343 5380 y Fk(\()p Fh(`)2422 5395 y Fs(2)2461 5380 y Fh(;)17 b(\031)t Fk(i)p Fh(=)p Fk(2\))p Fh(;)1745 5712 y Fk(32)p eop %%Page: 33 33 33 32 bop 166 83 a Fk(and)33 b(a)f(straigh)m(tforw)m(ard)g(computation) f(sho)m(ws)j(that)361 353 y Fh(a)412 368 y Fr(\000)p Fs(2)507 353 y Fk(\()p Fh(`)586 368 y Fs(2)625 353 y Fh(;)17 b(\031)t Fk(i)p Fh(=)p Fk(2\))26 b(=)1228 285 y(1)p 1031 330 443 4 v 1031 425 a(sinh)1205 382 y Fs(2)1245 425 y Fk(\()p Fh(h=)p Fk(2\))1484 353 y Fh(:)166 753 y Fk(The)34 b(pro)s(of)d(for)h Fh(!)835 717 y Fs(0)874 753 y Fk(\()p Fh(h)p Fk(\))h(follo)m(ws)e(the)i(same)f(lines.)166 1123 y Fj(A.2)100 b(Pr)-5 b(o)g(of)34 b(of)h(the)g(b)-5 b(ounds)34 b(on)h(the)g(numb)-5 b(er)34 b(of)h(primary)f(homo)-5 b(clinic)33 b(orbits)166 1468 y Fk(Here)g(w)m(e)h(shall)d(pro)m(v)m(e)j (the)f(three)g(claims)e(con)m(tained)i(in)e(lemma)g(7.)166 1694 y(The)k(\014rst)f(claim)e(is)h(the)i(lo)m(w)m(er)f(b)s(ound)g Fh(\032)c Fi(\025)h Fk(8,)j(whic)m(h)g(is)g(trivial.)d(It)j(do)s(es)h (not)e(require)166 1814 y(additional)22 b(commen)m(ts,)j(b)s(ecause)h (w)m(e)g(already)f(kno)m(w)h(that)f(the)g(eigh)m(t)g(axial)e(homo)s (clinic)166 1935 y(tra)5 b(jectories)33 b(p)s(ersist)f(under)i(small)c (enough)j(symmetric)f(p)s(erturbations.)166 2160 y(The)25 b(second)f(claim)d(is)i(the)h(upp)s(er)g(b)s(ound)g Fh(\032)k Fi(\024)g Fk(8\()p Fh(n)t Fi(\000)t Fk(1\).)23 b(W)-8 b(e)24 b(kno)m(w)h(that)e(in)g(our)g(billiard)166 2281 y(problem)36 b(there)i(exists)g(a)f(real)g(analytic)f Fh(h)p Fk(-p)s(erio)s(dic)f(o)s(dd)i(function)g(\011\()p Fh(t)p Fk(\),)g(called)f(the)166 2401 y(splitting)d(function,)j(whose)h (ro)s(ots)e(mo)s(dulo)e Fh(h)p Fe(Z)h Fk(are)h(in)g(one-to-four)g (corresp)s(ondence)166 2521 y(with)29 b(the)h(primary)e(homo)s(clinic)e (tra)5 b(jectories.)29 b(\(The)h(factor)f(four)g(has)h(b)s(een)g (explained)166 2642 y(at)25 b(the)h(b)s(eginning)d(of)i Fi(x)p Fk(5.\))g(W)-8 b(e)26 b(also)e(recall)g(that)h(\011\()p Fh(t)p Fk(\))j(=)f(\002)2365 2605 y Fr(0)2388 2642 y Fk(\()p Fh(t)p Fk(\))h(=)g Fh(\017L)2736 2605 y Fr(0)2760 2642 y Fk(\()p Fh(t)p Fk(\))7 b(+)2961 2651 y(O)3037 2642 y(\()p Fh(\017)p Fk(\),)25 b(where)166 2762 y Fh(L)232 2726 y Fr(0)256 2762 y Fk(\()p Fh(t)p Fk(\))33 b(is)g(the)g(deriv)-5 b(ativ)m(e)33 b(of)g(the)g(Melnik)m(o)m(v)h(p)s(oten)m(tial)e(asso)s (ciated)h(to)g(the)g(p)s(olynomial)166 2882 y(p)s(erturbation)h(\(3\).) g(Therefore,)h(to)f(pro)m(v)m(e)i(the)e(upp)s(er)h(b)s(ound,)g(it)e (su\016ces)j(to)e(see)i(that)166 3003 y Fh(L)232 2967 y Fr(0)256 3003 y Fk(\()p Fh(t)p Fk(\))c(has)h(at)g(the)g(most)f(2)p Fh(n)22 b Fi(\000)g Fk(2)33 b(ro)s(ots)f(in)g Fe(R)t Fh(=h)p Fe(Z)p Fk(,)k(coun)m(ted)e(with)e(m)m(ultiplicit)m(y)-8 b(.)166 3228 y(If)35 b Fh(P)14 b Fk(\()p Fh(s)p Fk(\))31 b(=)603 3162 y Fg(P)691 3188 y Fq(n)691 3253 y(j)t Fs(=2)834 3228 y Fh(p)883 3243 y Fq(j)920 3228 y Fh(s)966 3192 y Fq(j)1002 3228 y Fk(,)k(then)h Fh(L)p Fk(\()p Fh(t)p Fk(\))c(=)1605 3162 y Fg(P)1693 3188 y Fq(n)1693 3253 y(j)t Fs(=2)1836 3228 y Fh(p)1885 3243 y Fq(j)1922 3228 y Fh(L)1988 3243 y Fq(j)2024 3228 y Fk(\()p Fh(t)p Fk(\),)j(where)i (the)e(functions)g Fh(L)3141 3243 y Fq(j)3178 3228 y Fk(\()p Fh(t)p Fk(\))g(are)166 3349 y(de\014ned)41 b(in)d(\(4\).)h(F)-8 b(rom)38 b(the)h(prop)s(erties)h(of)e(the)i(functions)f Fh(L)2508 3364 y Fq(j)2545 3349 y Fk(\()p Fh(t)p Fk(\))g(listed)g(in)f Fi(x)p Fk(A.1,)i(w)m(e)166 3469 y(deduce)e(that)f(the)g(only)f(p)s (oles)g(of)h(the)g(elliptic)d(function)i Fh(L)p Fk(\()p Fh(t)p Fk(\))h(are)g(the)g(p)s(oin)m(ts)f(in)g(the)166 3589 y(set)30 b Fh(\031)t Fk(i)p Fh(=)p Fk(2)14 b(+)g Fe(Z)p Fh(h)g Fk(+)g Fe(Z)p Fh(\031)t Fk(i,)22 b(and)29 b(all)e(of)h(them)h(ha)m(v)m(e)h(order)f(2)p Fh(n)14 b Fi(\000)g Fk(2.)30 b(Its)f(deriv)-5 b(ativ)m(e)29 b Fh(L)3149 3553 y Fr(0)3172 3589 y Fk(\()p Fh(t)p Fk(\))g(has)166 3710 y(the)f(same)f(p)s(oles,)f(but)i(of)e(order)i(2)p Fh(n)11 b Fi(\000)g Fk(1.)27 b(Th)m(us,)i Fh(L)2009 3674 y Fr(0)2032 3710 y Fk(\()p Fh(t)p Fk(\))f(has)f(exactly)h(2)p Fh(n)11 b Fi(\000)g Fk(1)27 b(ro)s(ots)g(in)f(an)m(y)166 3830 y(complex)33 b(cell.)f(\(Non-constan)m(t)i(elliptic)c(functions)j (ha)m(v)m(e)i(the)f(same)f(n)m(um)m(b)s(er)g(of)g(p)s(oles)166 3951 y(and)39 b(ro)s(ots)f(in)g(a)g(cell,)g(coun)m(ted)i(with)e(m)m (ultiplicit)m(y)-8 b(.\))35 b(Besides,)40 b(using)e(the)h(symmetry)166 4071 y Fh(L)232 4035 y Fr(0)256 4071 y Fk(\()p Fi(\000)p Fh(t)p Fk(\))c(=)g Fi(\000)p Fh(L)733 4035 y Fr(0)758 4071 y Fk(\()p Fh(t)p Fk(\))i(and)g(the)g(p)s(erio)s(dicit)m(y)e Fh(L)1830 4035 y Fr(0)1854 4071 y Fk(\()p Fh(t)25 b Fk(+)g Fh(h)g Fk(+)g(2)p Fh(\031)t Fk(i\))34 b(=)h Fh(L)2620 4035 y Fr(0)2644 4071 y Fk(\()p Fh(t)p Fk(\),)i(w)m(e)h(obtain)e(that) 166 4191 y Fh(L)232 4155 y Fr(0)256 4191 y Fk(\()p Fh(h=)p Fk(2)19 b(+)h Fh(\031)t Fk(i\))26 b(=)i Fi(\000)p Fh(L)961 4155 y Fr(0)985 4191 y Fk(\()p Fh(h=)p Fk(2)19 b(+)h Fh(\031)t Fk(i\).)30 b(So,)i Fh(h=)p Fk(2)19 b(+)h Fh(\031)t Fk(i)30 b(is)h(a)g(complex)g(ro)s(ot)g(of)g Fh(L)2964 4155 y Fr(0)2987 4191 y Fk(\()p Fh(t)p Fk(\))h(and)f(the)166 4312 y(n)m(um)m(b)s(er)i(of)f(real)g(ro)s(ots)g(\(mo)s(dulo)e Fh(h)p Fe(Z)p Fk(\))g(of)j Fh(L)1827 4276 y Fr(0)1850 4312 y Fk(\()p Fh(t)p Fk(\))g(is)f(less)h(or)f(equal)g(than)h(2)p Fh(n)22 b Fi(\000)h Fk(2.)166 4537 y(The)h(last)e(claim)e(is)i(that)h (the)g(n)m(um)m(b)s(er)g(of)f(primary)g(homo)s(clinic)d(tra)5 b(jectories)23 b(c)m(hanges)h(in)166 4658 y(eigh)m(ts.)29 b(This)g(prop)s(ert)m(y)h(follo)m(ws)d(from)h(t)m(w)m(o)h(facts.)g (First,)f(due)i(to)e(the)i(symmetries,)e(the)166 4778 y(four)c(separatrices)h(ha)m(v)m(e)g(the)g(same)f(homo)s(clinic)d (bifurcations,)j(so)g(that)g(they)h(app)s(ear)f(in)166 4898 y(fours.)29 b(Second,)h(due)f(to)f(the)h(rev)m(ersibilit)m(y)-8 b(,)28 b(the)h Fh(h)p Fk(-p)s(erio)s(dic)d(splitting)h(function)h (\011\()p Fh(t)p Fk(\))g(is)166 5019 y(o)s(dd)c(and)h(the)f(axial)f (homo)s(clinic)e(p)s(oin)m(ts)j(are)g(lo)s(cated)f(at)h(the)h(p)s(oin)m (ts)f Fh(t)2761 5034 y Fs(+)2848 5019 y Fi(\021)k Fk(0)100 b(\(mo)s(d)32 b Fh(h)p Fk(\))166 5139 y(and)38 b Fh(t)396 5154 y Fr(\000)492 5139 y Fi(\021)g Fh(h=)p Fk(2)99 b(\(mo)s(d)32 b Fh(h)p Fk(\).)38 b(Then,)h(it)e(turns)i(out)f(that)g(an)m(y)g(c)m (hange)h(in)f(the)g(in)m(terv)-5 b(al)166 5259 y(\(0)p Fh(;)17 b(h=)p Fk(2\))38 b(generates)j(a)e(t)m(win)g(c)m(hange)h(in)f (\()p Fh(h=)p Fk(2)p Fh(;)17 b(h)p Fk(\))38 b(|see,)i(for)f(instance,)h (\014gure)g(11|,)166 5380 y(and)33 b(the)g(n)m(um)m(b)s(er)g(of)f(ro)s (ots)g(of)g(\011\()p Fh(t)p Fk(\))h(c)m(hanges)g(in)f(t)m(w)m(os.)i (Finally)-8 b(,)30 b(8)d(=)h(4)22 b Fi(\002)h Fk(2.)1745 5712 y(33)p eop %%Page: 34 34 34 33 bop 166 83 a Fj(A.3)100 b(Melnikov)34 b(c)-5 b(omputations)34 b(for)h(the)g(bifur)-5 b(c)g(ation)34 b(pr)-5 b(oblem)166 426 y Fk(W)d(e)39 b(recall)e(the)i(notation)e Fh(\030)1219 441 y Fq(n;j)1318 426 y Fk(\()p Fh(h)p Fk(\))h(=)f Fh(a)1652 441 y Fr(\000)p Fs(2)p Fq(j)1779 426 y Fk(\()p Fh(`)1858 441 y Fq(n)1905 426 y Fh(;)17 b(\031)t Fk(i)p Fh(=)p Fk(2\))37 b(in)m(tro)s(duced)h(in)g Fi(x)p Fk(A.1.)h(If)f(w)m(e)i(set) 166 547 y Fh(\021)218 511 y Fr(\000)p Fs(1)343 547 y Fk(=)31 b(sinh\()p Fh(h=)p Fk(2\),)j(it)g(is)g(easy)i(to)e(c)m(hec)m(k) j(that)d Fh(\030)1974 562 y Fs(2)p Fq(;)p Fs(1)2068 547 y Fk(\()p Fh(h)p Fk(\))d(=)g Fh(\021)2390 511 y Fs(2)2429 547 y Fk(,)k Fh(\030)2534 562 y Fs(3)p Fq(;)p Fs(1)2628 547 y Fk(\()p Fh(h)p Fk(\))c(=)g Fh(\021)2950 511 y Fs(2)2989 547 y Fk(\(2)p Fh(=)p Fk(3)23 b Fi(\000)h Fh(\021)3350 511 y Fs(2)3389 547 y Fk(\),)166 667 y(and)33 b Fh(\030)399 682 y Fs(3)p Fq(;)p Fs(2)493 667 y Fk(\()p Fh(h)p Fk(\))27 b(=)h Fi(\000)p Fh(\021)885 631 y Fs(2)925 667 y Fk(.)k(Th)m(us,)i (using)f(form)m(ula)d(\(A.1\))j(w)m(e)g(\014nd)g(the)g(expressions)361 909 y Fh(L)427 867 y Fr(00)427 933 y Fs(2)470 909 y Fk(\()p Fh(t)p Fk(\))28 b(=)f Fi(\000)p Fh(ae)885 867 y Fs(4)926 909 y Fh(\021)978 867 y Fs(2)1017 909 y Fh( )1084 867 y Fr(00)1126 909 y Fk(\()p Fh(t)p Fk(\))p Fh(;)212 b(L)1542 867 y Fr(00)1542 933 y Fs(3)1585 909 y Fk(\()p Fh(t)p Fk(\))28 b(=)f Fh(ae)1923 867 y Fs(6)1963 909 y Fh(\021)2015 867 y Fs(2)2071 812 y Fg(\020)2120 909 y Fk(\()p Fh(\021)2210 867 y Fs(2)2272 909 y Fi(\000)22 b Fk(2)p Fh(=)p Fk(3\))p Fh( )2623 867 y Fr(00)2665 909 y Fk(\()p Fh(t)p Fk(\))g(+)g Fh( )2963 867 y Fs(\(4\))3057 909 y Fk(\()p Fh(t)p Fk(\))p Fh(=)p Fk(6)3266 812 y Fg(\021)3332 909 y Fh(:)166 1237 y Fk(Finally)-8 b(,)25 b(from)h(the)i(second)h(item)d(con)m(tained)h (in)g(lemma)e(10)i(w)m(e)i(see)f(that)f(the)h(Melnik)m(o)m(v)166 1358 y(appro)m(ximations)f(of)h(the)h(bifurcation)e(v)-5 b(alues)28 b Fh(d)1951 1322 y Fq(\017)1951 1382 y Fr(\006)2010 1358 y Fk(\()p Fh(h)p Fk(\))g(=)f Fh(d)2324 1322 y Fs(0)2324 1382 y Fr(\006)2383 1358 y Fk(\()p Fh(h)p Fk(\))14 b(+)2619 1367 y(O)2695 1358 y(\()p Fh(\017)p Fk(\))28 b(de\014ned)i(in)e(\(15\)) 166 1478 y(v)m(erify)33 b(the)g(asymptotic)f(estimate)361 1765 y Fh(d)412 1724 y Fs(0)412 1789 y Fr(\006)471 1765 y Fk(\()p Fh(h)p Fk(\))c(=)f Fi(\000)821 1697 y Fh(L)887 1661 y Fr(00)887 1722 y Fs(3)930 1697 y Fk(\()p Fh(t)1003 1712 y Fr(\006)1063 1697 y Fk(\))p 821 1742 279 4 v 821 1833 a Fh(L)887 1799 y Fr(00)887 1855 y Fs(2)930 1833 y Fk(\()p Fh(t)1003 1848 y Fr(\006)1063 1833 y Fk(\))1138 1765 y(=)h Fh(e)1287 1724 y Fs(2)1343 1619 y Fg( )1409 1765 y Fh(\021)1461 1724 y Fs(2)1522 1765 y Fi(\000)1632 1697 y Fk(2)p 1632 1742 49 4 v 1632 1833 a(3)1712 1765 y(+)1820 1697 y Fh( )1887 1661 y Fs(\(4\))1982 1697 y Fk(\()p Fh(t)2055 1712 y Fr(\006)2114 1697 y Fk(\))p 1820 1742 332 4 v 1822 1833 a(6)p Fh( )1938 1804 y Fr(00)1980 1833 y Fk(\()p Fh(t)2053 1848 y Fr(\006)2112 1833 y Fk(\))2162 1619 y Fg(!)2255 1765 y Fk(=)2369 1697 y Fh(e)2414 1661 y Fs(2)2453 1697 y Fk(\(6)p Fh(\021)2592 1661 y Fs(2)2653 1697 y Fi(\000)23 b Fk(4)f Fi(\000)h Fh(T)2995 1661 y Fs(2)3034 1697 y Fk(\))p 2369 1742 703 4 v 2696 1833 a(6)3104 1765 y(+)3202 1774 y(O)3278 1765 y(\()p Fh(q)t Fk(\))p Fh(;)166 2150 y Fk(whereas)34 b(their)e(di\013erence)h(is)f (exp)s(onen)m(tially)g(small:)361 2437 y Fh(d)412 2396 y Fs(0)412 2461 y Fr(\000)471 2437 y Fk(\()p Fh(h)p Fk(\))22 b Fi(\000)h Fh(d)776 2396 y Fs(0)776 2461 y(+)834 2437 y Fk(\()p Fh(h)p Fk(\))28 b(=)1108 2369 y Fh(e)1153 2333 y Fs(2)p 1108 2414 85 4 v 1126 2505 a Fk(6)1219 2291 y Fg( )1295 2369 y Fh( )1362 2333 y Fs(\(4\))1456 2369 y Fk(\()p Fh(t)1529 2384 y Fr(\000)1588 2369 y Fk(\))p 1295 2414 332 4 v 1320 2505 a Fh( )1387 2476 y Fr(00)1430 2505 y Fk(\()p Fh(t)1503 2520 y Fr(\000)1562 2505 y Fk(\))1658 2437 y Fi(\000)1768 2369 y Fh( )1835 2333 y Fs(\(4\))1929 2369 y Fk(\()p Fh(t)2002 2384 y Fs(+)2061 2369 y Fk(\))p 1768 2414 V 1794 2505 a Fh( )1861 2476 y Fr(00)1903 2505 y Fk(\()p Fh(t)1976 2520 y Fs(+)2035 2505 y Fk(\))2109 2291 y Fg(!)2202 2437 y Fk(=)g(8)p Fh(T)2426 2396 y Fs(2)2465 2437 y Fh(e)2510 2396 y Fs(2)2550 2437 y Fh(q)e Fk(+)2717 2446 y(O)2792 2437 y(\()p Fh(q)2877 2396 y Fs(2)2917 2437 y Fk(\))p Fh(:)166 2832 y Fk(This)34 b(implies)e(that)h(the)i (functions)f Fh(d)1578 2796 y Fs(0)1617 2832 y Fk(\()p Fh(h)p Fk(\))g(and)g Fh(\016)2021 2796 y Fs(0)2060 2832 y Fk(\()p Fh(h)p Fk(\))g(de\014ned)i(implicitly)29 b(in)k(\(16\))h(are) 166 2952 y Fh(d)217 2916 y Fs(0)256 2952 y Fk(\()p Fh(h)p Fk(\))f(=)g Fh(e)575 2916 y Fs(2)614 2952 y Fk(\()p Fh(\021)704 2916 y Fs(2)768 2952 y Fi(\000)24 b Fk(2)p Fh(=)p Fk(3)g Fi(\000)g Fh(T)1212 2916 y Fs(2)1252 2952 y Fh(=)p Fk(6\))35 b(and)g Fh(\016)1662 2916 y Fs(0)1702 2952 y Fk(\()p Fh(h)p Fk(\))d(=)h(8)p Fh(T)2095 2916 y Fs(2)2134 2952 y Fh(e)2179 2916 y Fs(2)2219 2952 y Fk(.)i(T)-8 b(o)36 b(obtain)e(their)i(\014nal)e(forms,)166 3073 y(it)f(su\016ces)k(to)d (recall)f(that)h Fh(e)d Fk(=)g(tanh\()p Fh(h=)p Fk(2\),)j Fh(T)44 b Fk(=)31 b(2)p Fh(\031)t(=h)p Fk(,)j(and)g Fh(\021)2589 3037 y Fr(\000)p Fs(1)2714 3073 y Fk(=)d(sinh\()p Fh(h=)p Fk(2\).)j(The)166 3193 y(computation)d(of)h Fh(d)897 3157 y Fs(0)897 3218 y Fr(\003)936 3193 y Fk(\()p Fh(h)p Fk(\))h(follo)m(w)e(the)i(same)f(lines.)g(W)-8 b(e)33 b(skip)g(it.)166 3614 y Fl(B)112 b(Some)37 b(details)f(of)i(the)f(n)m (umerical)f(computations)166 3958 y Fk(In)g(this)e(app)s(endix)i(w)m(e) 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Fk(2.1.)g(This)f(di\013erence)h(causes)h(an)e(imp)s(ortan)m(t)e(loss)i (of)166 5380 y(signi\014can)m(t)c(digits,)f(ev)m(en)j(for)e(mo)s (derate)g(v)-5 b(alues)36 b(of)g Fh(h)p Fk(.)h(It)g(can)g(only)f(b)s(e) g(o)m(v)m(ercome)i(b)m(y)1745 5712 y(34)p eop %%Page: 35 35 35 34 bop 166 83 a Fk(computing)25 b(the)j(actions)e(with)g(more)g (correct)h(digits)e(than)i(the)g(lost)f(ones.)i(F)-8 b(or)25 b(sample,)166 203 y(let)32 b(us)h(set)g Fh(a)28 b Fk(=)g(1,)k Fh(\017)c Fk(=)1073 164 y Fs(1)p 1056 180 71 4 v 1056 238 a(10)1136 203 y Fk(,)33 b(and)g Fh(n)28 b Fk(=)f(2.)32 b(Under)i(this)e(setup,)i(the)f(actions)f(are)361 532 y Fh(W)14 b Fk([)p Fh(O)572 491 y Fs(+)630 532 y Fk(])j Fi(')g(\000)p Fk(0)p Fh(:)p Fk(0999250385227157127315613502752)o (3196)o(06013)o(5775)o(467)p Fh(;)361 671 y(W)d Fk([)p Fh(O)572 630 y Fr(\000)630 671 y Fk(])j Fi(')g(\000)p Fk(0)p Fh(:)p Fk(0999250385227157127315613502752)o(3196)o(06013)o(5716) o(286)p Fh(;)166 888 y Fk(for)43 b Fh(h)j Fk(=)h(10)649 852 y Fr(\000)p Fs(1)743 888 y Fk(.)c(Then)i Fh(A)h Fi(')h Fk(5)p Fh(:)p Fk(9181)29 b Fi(\001)g Fk(10)1778 852 y Fr(\000)p Fs(43)1950 888 y Fk(and)44 b(so)g(the)g(cancellation)d (causes)k(the)166 1009 y(loss)36 b(of)g(more)f(than)i(40)e(digits.)g (Smaller)f(v)-5 b(alues)36 b(of)g Fh(h)g Fk(cause)i(stronger)e (cancellations:)166 1129 y Fh(A)30 b Fi(')h Fk(1)p Fh(:)p Fk(0137)22 b Fi(\001)g Fk(10)819 1093 y Fr(\000)p Fs(428)1018 1129 y Fk(for)33 b Fh(h)d Fk(=)g(10)1458 1093 y Fr(\000)p Fs(2)1552 1129 y Fk(,)k(and)g Fh(A)c Fi(')g Fk(2)p Fh(:)p Fk(1022)22 b Fi(\001)h Fk(10)2457 1093 y Fr(\000)p Fs(4286)2691 1129 y Fk(for)33 b Fh(h)d Fk(=)g(10)3131 1093 y Fr(\000)p Fs(3)3225 1129 y Fk(.)k(The)166 1249 y(cancellations)d(are)h(ev)m(en)i (w)m(orse)f(in)f(the)g(computation)f(of)h(\012)c(=)g Fh(!)2621 1213 y Fs(+)2700 1249 y Fk(+)22 b Fh(!)2863 1213 y Fr(\000)2921 1249 y Fk(.)32 b(And)h(to)f(top)166 1370 y(it)d(all,)f(in)g(order)i(to)g(get)f(h)m(undreds)j(of)d(co)s 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b(main)d(n)m(umerical)g(di\016culties)h(that)g(app)s(ear) h(during)f(the)h(study)h(of)e(the)h(singular)166 2653 y(splitting)30 b(of)i(separatrices)i(of)e(our)g(billiard)d(maps)j(are)h (the)g(computation)e(of:)165 2873 y Fi(\017)49 b Fk(The)33 b(billiard)c(maps)j(and)h(their)f(di\013eren)m(tials)f(with)i(an)f (arbitrary)g(precision)g Fh(P)14 b Fk(;)165 2993 y Fi(\017)49 b Fk(The)33 b(T)-8 b(a)m(ylor)32 b(expansions)i(of)e(the)h(in)m(v)-5 b(arian)m(t)31 b(curv)m(es)k(up)e(to)f(an)g(arbitrary)g(order)h Fh(K)7 b Fk(;)165 3113 y Fi(\017)49 b Fk(The)33 b(homo)s(clinic)d(in)m (v)-5 b(arian)m(ts)31 b Fh(A)i Fk(and)g(\012)g(with)f(an)g(arbitrary)g (precision)g Fh(Q)p Fk(;)h(and)165 3234 y Fi(\017)49 b Fk(The)33 b(Gevrey)h(expansions)f(\(10\))f(up)h(to)f(an)h(arbitrary)e (order)i Fh(J)9 b Fk(.)166 3454 y(The)25 b(quan)m(tities)e Fh(Q)h Fk(and)f Fh(J)33 b Fk(m)m(ust)23 b(b)s(e)h(inputs)g(of)f(the)h (algorithm,)c(b)s(ecause)25 b(they)g(set)f(some)166 3574 y(prop)s(erties)40 b(of)f(the)h(ob)5 b(jects)41 b(w)m(e)g(are)f(lo)s (oking)e(for.)h(On)h(the)g(con)m(trary)-8 b(,)41 b Fh(P)53 b Fk(and)40 b Fh(K)47 b Fk(are)166 3694 y(determined)22 b(in)g(an)g(automatic)e(w)m(a)m(y)k(when)f(the)g(computation)e(b)s (egins.)h(F)-8 b(or)22 b(instance,)g(in)166 3815 y(the)h(computation)e (of)h(the)h(lob)s(e)f(area)g Fh(A)28 b Fk(=)f Fh(W)14 b Fk([)p Fh(O)1957 3779 y Fr(\000)2015 3815 y Fk(])r Fi(\000)r Fh(W)g Fk([)p Fh(O)2334 3779 y Fs(+)2392 3815 y Fk(])23 b(the)f(n)m(um)m(b)s(er)h(of)f(digits)f(lost)166 3935 y(b)m(y)34 b(cancellations)d(is)h(appro)m(ximately)f(equal)i(to)f Fh(R)23 b Fk(+)f Fh(S)6 b Fk(,)33 b(where)h Fh(R)29 b Fk(=)e Fh(R)q Fk(\()p Fh(\017)p Fk(\))h(=)g Fi(j)17 b Fk(log)3295 3959 y Fs(10)3387 3935 y Fh(\017)p Fi(j)166 4056 y Fk(and)28 b Fh(S)34 b Fk(=)27 b Fh(S)6 b Fk(\()p Fh(h)p Fk(\))28 b(=)f Fh(\031)936 4019 y Fs(2)976 4056 y Fh(h)1032 4019 y Fr(\000)p Fs(1)1143 4056 y Fk(log)1269 4079 y Fs(10)1360 4053 y Fk(e)1404 4056 y(.)h(Hence,)i(to)e(compute)g Fh(A)g Fk(with)g(precision)g Fh(Q)g Fk(w)m(e)i(m)m(ust)166 4176 y(tak)m(e)37 b Fh(P)48 b Fi(\031)35 b Fh(Q)26 b Fk(+)e Fh(R)i Fk(+)f Fh(S)6 b Fk(.)36 b(Analogously)-8 b(,)36 b(w)m(e)h(set)h Fh(P)48 b Fi(\031)35 b Fh(Q)25 b Fk(+)g Fh(R)g Fk(+)g(2)p Fh(S)42 b Fk(to)37 b(compute)f(the)166 4296 y(sum)f(\012)d(=)g Fh(!)650 4260 y Fs(+)733 4296 y Fk(+)23 b Fh(!)897 4260 y Fr(\000)991 4296 y Fk(with)34 b(the)i(same)f(precision.)f(On)i(the)f(other)g(hand,)h(the)f(order)h Fh(K)166 4417 y Fk(is)e(determined)g(under)i(the)f(follo)m(wing)d (optimization)f(criterion.)i(If)h Fh(K)42 b Fk(is)34 b(to)s(o)g(big,)f(the)166 4537 y(T)-8 b(a)m(ylor)36 b(expansions)h(b)s (ecome)g(to)s(o)e(exp)s(ensiv)m(e,)k(but)d(a)g(to)s(o)g(lo)m(w)g Fh(K)43 b Fk(is)36 b(also)g(exp)s(ensiv)m(e,)166 4658 y(b)s(ecause)48 b(then)g(the)g(n)m(um)m(b)s(er)f(of)g(iterates)f(to)h (reac)m(h)h(the)f(homo)s(clinic)d(p)s(oin)m(ts)j(\(from)166 4778 y(a)38 b(lo)s(cal)f(fundamen)m(tal)h(domain)f(in)h(whic)m(h)h(the) g(T)-8 b(a)m(ylor)38 b(expansion)h(giv)m(es)h(an)e(enough)166 4898 y(accurate)h(appro)m(ximation)e(of)h(the)h(in)m(v)-5 b(arian)m(t)37 b(curv)m(es\))j(gro)m(ws)g(to)s(o)d(m)m(uc)m(h.)i (Therefore,)166 5019 y(there)26 b(exists)f(some)g(optimal)d(order)j (for)g(whic)m(h)g(the)h(computations)e(b)s(ecome)h(the)g(fastest)166 5139 y(ones.)h(This)g(optimal)c(v)-5 b(alue)25 b(can)h(b)s(e)g (estimated,)e(see)j([6,)e Fi(x)p Fk(5D].)g(T)-8 b(o)26 b(acquain)m(t)f(the)h(reader)166 5259 y(with)38 b(the)g(magnitude)f(of) g(our)h(computations,)f(w)m(e)i(note)g(that)e(w)m(e)i(ha)m(v)m(e)h (reac)m(hed)f(the)166 5380 y(v)-5 b(alues)33 b Fh(Q)28 b Fk(=)f(1500)32 b(and)h Fh(J)k Fk(=)28 b(300,)k(with)g Fh(P)41 b Fk(=)28 b(7000)k(and)h Fh(K)i Fk(=)27 b(1100.)32 b(In)h(the)g(previous)1745 5712 y(35)p eop %%Page: 36 36 36 35 bop 166 83 a Fk(literature)32 b(there)i(are)g(not)f(so)h(extreme) g(computations,)e(b)s(eing)h([6])g(the)h(only)f(one)h(that)166 203 y(reac)m(hes)g(a)f(comparable)e(lev)m(el.)166 424 y(The)37 b(problem)e(of)h(the)g(computation)f(of)g(the)i(map)e(and)h (its)g(di\013eren)m(tial)e(is)i(trivial)d(for)166 544 y(the)c(standard)g(map,)f(the)h(H)m(\023)-46 b(enon)29 b(map,)f(the)h(p)s(erturb)s(ed)g(McMillan)e(maps,)h(and)h(others)166 665 y(generalized)34 b(standard)g(maps.)g(These)i(maps)e(ha)m(v)m(e)h (explicit)e(expressions)j(in)d(terms)h(of)166 785 y(p)s(olynomial)42 b(or)i(trigonometric)f(functions.)i(The)h(billiard)c(map)i(is)h(sligh)m (tly)e(harder,)166 905 y(since)i(w)m(e)i(ha)m(v)m(e)f(to)f(solv)m(e)g (a)g(nonlinear)e(equation)i(to)g(\014nd)g(the)h(in)m(tersection)f(of)f (the)166 1026 y(re\015ected)34 b(ra)m(y)f(with)f(the)h(con)m(v)m(ex)i (curv)m(e,)f(see)g Fi(x)p Fk(B.2.)166 1246 y(The)d(lo)s(cal)c(in)m(v)-5 b(arian)m(t)29 b(curv)m(es)j(of)d(w)m(eakly)i(h)m(yp)s(erb)s(olic)e(ob) 5 b(jects)31 b(m)m(ust)f(b)s(e)g(dev)m(elop)s(ed)g(up)166 1367 y(to)42 b(high)g(orders,)h(see)h([19])f(for)f(general)g(commen)m (ts.)g(Then)i(the)f(initial)c(iterates)j(can)166 1487 y(b)s(e)33 b(tak)m(en)h(far)f(enough)g(from)f(the)h(h)m(yp)s(erb)s (olic)g(ob)5 b(ject)33 b(and)h(so)f(the)g(homo)s(clinic)d(p)s(oin)m(ts) 166 1607 y(can)43 b(b)s(e)g(attained)f(in)g(a)h(few)g(iterations.)e (Here,)j(few)f(means)g(thousands,)h(instead)f(of)166 1728 y(millions.)24 b(In)k(this)f(w)m(a)m(y)-8 b(,)28 b(undesirable)g(accum)m(ulation)e(errors)i(due)g(to)f(the)h(large)e (amoun)m(t)166 1848 y(of)d(op)s(erations)f(is)h(a)m(v)m(oided)h(and)f (computing)g(time)e(is)i(reduced.)i(The)g(T)-8 b(a)m(ylor)23 b(co)s(e\016cien)m(ts)166 1968 y(of)j(the)i(in)m(v)-5 b(arian)m(t)25 b(curv)m(es)k(of)d(man)m(y)h(analytic)f(area-preserving) h(maps)f(can)h(b)s(e)g(obtained)166 2089 y(recursiv)m(ely)-8 b(.)30 b(The)f(recursiv)m(e)i(algorithm)25 b(for)j(our)h(billiard)c (maps)k(is)f(describ)s(ed)h(in)f Fi(x)p Fk(B.3.)166 2209 y(Its)33 b(deriv)-5 b(ation)31 b(is)h(less)h(direct)g(than)f(for)g (maps)g(giv)m(en)h(b)m(y)h(closed)e(explicit)g(form)m(ulae.)166 2430 y(The)37 b(metho)s(d)d(to)i(compute)f(the)h(symmetric)f(primary)f (homo)s(clinic)e(p)s(oin)m(ts)k(and)f(their)166 2550 y(homo)s(clinic)k(in)m(v)-5 b(arian)m(ts)42 b(do)s(es)h(not)f(dep)s (end)i(v)m(ery)g(m)m(uc)m(h)f(on)f(the)h(form)e(of)h(the)h(map.)166 2670 y(The)37 b(symmetric)d(homo)s(clinic)f(p)s(oin)m(ts)i(are)h(found) g(as)f(the)h(in)m(tersections)h(b)s(et)m(w)m(een)g(the)166 2791 y(unstable)42 b(in)m(v)-5 b(arian)m(t)40 b(curv)m(e)k(and)d(the)i (corresp)s(onding)e(symmetry)h(lines.)f(The)i(global)166 2911 y(unstable)29 b(curv)m(e)h(is)f(obtained)f(from)f(the)j(lo)s(cal)c (one)j(b)m(y)h(forw)m(ard)f(iteration)e(of)h(the)h(map.)166 3032 y(The)43 b(extrap)s(olation)d(metho)s(d)i(to)g(obtain)f(the)i (\014rst)g(co)s(e\016cien)m(ts)g(of)f(the)g(asymptotic)166 3152 y(expansion)35 b(of)f(the)h(homo)s(clinic)d(in)m(v)-5 b(arian)m(ts)33 b(is)h(v)m(ery)i(standard.)f(W)-8 b(e)35 b(refer)g(to)f([6])h(for)f(a)166 3272 y(general)e(bac)m(kground)i(on)e (these)i(metho)s(ds.)166 3493 y(The)47 b(main)e(principle)g(to)h (design)g(v)-5 b(alid)44 b(algorithms)g(for)i(the)h(ab)s(o)m(v)m(e)g (computations)166 3613 y(is)f(that,)f(since)i(the)f(use)h(of)e(a)h(m)m (ultiple-precision)d(arithmetic)g(is)j(una)m(v)m(oidable,)g(w)m(e)166 3733 y(ha)m(v)m(e)40 b(to)f(mitigate)c(its)k(cost)g(in)f(all)f(the)i(p) s(ossible)f(w)m(a)m(ys.)i(T)-8 b(o)39 b(men)m(tion)f(just)h(the)g(most) 166 3854 y(ob)m(vious)31 b(w)m(a)m(y)-8 b(,)32 b(w)m(e)g(shall)d(solv)m (e)i(an)m(y)h(nonlinear)d(equation)h(b)m(y)i(using)e(the)i (quadratically)166 3974 y(con)m(v)m(ergen)m(t)40 b(Newton's)g(metho)s (d.)e(Of)f(course,)j(w)m(e)f(will)d(b)s(egin)h(the)i(metho)s(d)e(in)h (single)166 4095 y(precision)c(and)h(later)e(w)m(e)j(will)c(re\014ne)k (the)f(result)f(b)m(y)i(doubling)d(the)i(n)m(um)m(b)s(er)g(of)f(digits) 166 4215 y(after)29 b(eac)m(h)i(Newton)f(iteration.)d(This)j(metho)s (dology)e(causes)j(a)e(cascade)i(of)e(c)m(hanges)h(in)166 4335 y(the)37 b(n)m(um)m(b)s(er)g(of)g(digits)e(used)j(along)d(a)h (concrete)i(computation,)e(b)s(ecause)i(sometimes)166 4456 y(eac)m(h)33 b(ev)-5 b(aluation)31 b(of)h(the)h(initial)c (nonlinear)i(terms)h(requires)i(the)f(solution)e(of)h(another)166 4576 y(nonlinear)f(equation)i(and)f(so)h(forth,)f(but)h(the)g(increase) g(in)f(sp)s(eed)i(is)e(sp)s(ectacular.)166 4919 y Fj(B.2)99 b(Bil)5 b(liar)-5 b(d)34 b(maps)g(and)g(their)h(di\013er)-5 b(entials)166 5259 y Fk(In)32 b Fi(x)p Fk(3.1)f(w)m(e)i(ha)m(v)m(e)g (mo)s(deled)d(billiards)f(inside)i(a)g(closed)h(con)m(v)m(ex)h(curv)m (e)h Fh(C)k Fk(b)m(y)33 b(means)e(of)166 5380 y(di\013eomorphisms)25 b(on)h(an)h(ann)m(ulus,)g(but)g(from)e(a)h(n)m(umerical)g(p)s(oin)m(t)f (of)i(view)g(it)e(is)h(b)s(etter)1745 5712 y(36)p eop %%Page: 37 37 37 36 bop 166 83 a Fk(to)32 b(mo)s(del)f(them)i(b)m(y)g(means)g(of)f (di\013eomorphisms)e(de\014ned)k(on)f(the)g(phase)g(space)361 332 y Fh(M)39 b Fk(=)27 b Fi(f)p Fh(m)h Fk(=)g(\()p Fh(q)t(;)17 b(p)p Fk(\))27 b Fi(2)h Fh(C)h Fi(\002)22 b Fe(S)f Fk(:)28 b Fh(p)33 b Fk(is)f(directed)h(out)m(w)m(ard)g Fh(C)39 b Fk(at)33 b Fh(q)t Fi(g)166 696 y Fk(consisting)i(of)g(p)s(oin)m(ts)g Fh(q)h Fk(=)c(\()p Fh(x;)17 b(y)t Fk(\))32 b Fi(2)h Fh(C)42 b Fk(and)36 b(v)m(elo)s(cities)e Fh(p)f Fk(=)f(\()p Fh(u;)17 b(v)t Fk(\))31 b Fi(2)i Fe(S)o Fk(.)d(That)36 b(is,)f(w)m(e)166 816 y(use)d(the)f(four)f(co)s(ordinates)h Fh(x)p Fk(,)g Fh(y)t Fk(,)f Fh(u)p Fk(,)g Fh(v)t Fk(,)g(restricted)i(to)e(the)h (conditions)f(\()p Fh(x;)17 b(y)t Fk(\))27 b Fi(2)h Fh(C)38 b Fk(and)166 936 y Fh(u)222 900 y Fs(2)283 936 y Fk(+)22 b Fh(v)432 900 y Fs(2)498 936 y Fk(=)28 b(1.)k(Then)i(the)f(billiard)c (map)i Fh(f)11 b Fk(\()p Fh(q)t(;)17 b(p)p Fk(\))27 b(=)h(\()p Fh(q)2176 900 y Fr(0)2199 936 y Fh(;)17 b(p)2292 900 y Fr(0)2315 936 y Fk(\))32 b(is)g(de\014ned)i(as)e(follo)m(ws.)g(The) 166 1057 y(new)f(v)m(elo)s(cit)m(y)e Fh(p)770 1021 y Fr(0)823 1057 y Fk(is)h(the)g(re\015ection)g(of)f Fh(p)h Fk(with)f(resp)s(ect)i(to)f(the)g(tangen)m(t)g(line)f Fh(T)3114 1072 y Fq(q)3152 1057 y Fh(C)7 b Fk(.)30 b(The)166 1177 y(new)i(p)s(oin)m(t)f Fh(q)666 1141 y Fr(0)720 1177 y Fk(is)g(determined)g(b)m(y)h(imp)s(osing)c(that)j Fh(q)2129 1141 y Fr(0)2180 1177 y Fk(=)d Fh(q)23 b Fi(\000)d Fh(\034)11 b(p)2549 1141 y Fr(0)2600 1177 y Fi(2)28 b Fh(C)38 b Fk(for)31 b(some)g Fh(\034)39 b(<)28 b Fk(0.)166 1298 y(The)k(existence)h(and)e(uniqueness)i(of)e Fh(q)1614 1261 y Fr(0)1668 1298 y Fk(follo)m(ws)f(from)g(the)i(con)m(v)m(exit)m (y)h(of)d(the)i(curv)m(e)h Fh(C)7 b Fk(.)166 1532 y(F)-8 b(or)26 b(brevit)m(y)-8 b(,)27 b(hereafter)h(w)m(e)f(restrict)g(the)g (study)h(to)f(the)g(monomial)c(p)s(erturbations)j(\(1\),)166 1653 y(whic)m(h)32 b(are)g(con)m(v)m(ex)h(for)e(all)e Fh(\017)g Fi(\025)f Fk(0)j(and)g(for)g(all)f(in)m(teger)h Fh(n)d Fi(\025)g Fk(2.)k(F)-8 b(or)30 b(further)i(reference,)166 1773 y(w)m(e)i(write)e(their)g(implicit)d(equations)j(as)361 2022 y Fh(x)416 1981 y Fs(2)484 2022 y Fk(=)27 b Fh(\026)646 2037 y Fs(0)707 2022 y Fk(+)c Fh(\026)865 2037 y Fs(1)904 2022 y Fh(y)956 1981 y Fs(2)1016 2022 y Fk(+)f Fh(\026)1173 2037 y Fq(n)1220 2022 y Fh(y)1272 1981 y Fs(2)p Fq(n)1353 2022 y Fh(;)1853 b Fk(\(B.1\))166 2386 y(where)30 b Fh(\026)503 2401 y Fs(0)570 2386 y Fk(=)e Fh(a)725 2349 y Fs(2)764 2386 y Fk(,)h Fh(\026)879 2401 y Fs(1)946 2386 y Fk(=)f Fi(\000)p Fh(a)1178 2349 y Fs(2)1218 2386 y Fh(=b)1308 2349 y Fs(2)1347 2386 y Fk(,)h(and)g Fh(\026)1648 2401 y Fq(n)1723 2386 y Fk(=)e Fi(\000)p Fh(\017a)1993 2349 y Fs(2)2034 2386 y Fh(=\015)2139 2349 y Fs(2)p Fq(n)2221 2386 y Fk(.)i(W)-8 b(e)29 b(lo)s(ok)f(for)g(an)h(algorithm)d(to)166 2506 y(compute)33 b(the)g(billiard)d(map)i Fh(f)43 b Fk(join)m(tly)32 b(with)g(its)h(di\013eren)m(tial)48 b(d)p Fh(f)43 b Fk(as)33 b(fast)g(as)g(p)s(ossible)166 2626 y(with)40 b(arbitrary)e(\(but)i(\014xed\))h(accuracy)-8 b(,)41 b(for)f(relativ)m(ely)e(small)g(v)-5 b(alues)40 b(of)f Fh(h)h Fk(and)g(not)166 2747 y(v)m(ery)34 b(big)e(v)-5 b(alues)32 b(of)g Fh(\017)p Fk(.)h(T)m(ypically)-8 b(,)32 b(10)1591 2711 y Fr(\000)p Fs(3)1713 2747 y 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b(e)43 b(write)166 3222 y Fh(q)213 3186 y Fr(0)264 3222 y Fk(=)28 b(\()p Fh(x)461 3186 y Fr(0)484 3222 y Fh(;)17 b(y)580 3186 y Fr(0)603 3222 y Fk(\))27 b Fi(2)i Fh(C)7 b Fk(,)32 b Fh(p)948 3186 y Fr(0)999 3222 y Fk(=)c(\()p Fh(u)1197 3186 y Fr(0)1220 3222 y Fh(;)17 b(v)1315 3186 y Fr(0)1338 3222 y Fk(\))28 b Fi(2)g Fe(S)o Fk(,)45 b(_)-45 b Fh(q)1660 3186 y Fr(0)1711 3222 y Fk(=)28 b(\()16 b(_)-43 b Fh(x)1908 3186 y Fr(0)1931 3222 y Fh(;)34 b Fk(_)-44 b Fh(y)2027 3186 y Fr(0)2049 3222 y Fk(\))28 b Fi(2)h Fh(T)2267 3237 y Fq(q)2301 3218 y Fb(0)2327 3222 y Fh(C)7 b Fk(,)33 b(and)52 b(_)-46 b Fh(p)2703 3186 y Fr(0)2754 3222 y Fk(=)27 b(\()17 b(_)-44 b Fh(u)2951 3186 y Fr(0)2974 3222 y Fh(;)31 b Fk(_)-41 b Fh(v)3069 3186 y Fr(0)3092 3222 y Fk(\))28 b Fi(2)g Fh(T)3309 3237 y Fq(p)3345 3218 y Fb(0)3371 3222 y Fe(S)p Fk(.)166 3343 y(Using)k(these)i(notations,)e(w)m(e)h(p)s(erform)f(the)h (computation)e(in)h(t)m(w)m(o)h(steps.)165 3577 y Fi(\017)49 b Fj(Computation)37 b(of)h(the)f(new)h(velo)-5 b(city:)41 b Fk(W)-8 b(e)36 b(set)h Fh(r)f Fk(=)d(\()p Fh(\013)q(;)17 b(\014)6 b Fk(\))35 b(and)51 b(_)-42 b Fh(r)35 b Fk(=)f(\()20 b(_)-47 b Fh(\013;)3068 3551 y Fk(_)3043 3577 y Fh(\014)6 b Fk(\),)35 b(where)264 3698 y Fh(\013)28 b Fk(=)f Fh(x)p Fk(,)52 b(_)-48 b Fh(\013)29 b Fk(=)44 b(_)-44 b Fh(x)q Fk(,)30 b Fh(\014)k Fk(=)27 b Fi(\000)p Fk(\()p Fh(\026)1243 3713 y Fs(1)1300 3698 y Fk(+)18 b Fh(n\026)1511 3713 y Fq(n)1558 3698 y Fh(y)1610 3661 y Fs(2)p Fq(n)p Fr(\000)p Fs(2)1782 3698 y Fk(\))p Fh(y)33 b Fk(and)2113 3671 y(_)2089 3698 y Fh(\014)g Fk(=)27 b Fi(\000)p Fk(\()p Fh(\026)2454 3713 y Fs(1)2512 3698 y Fk(+)17 b(\(2)p Fh(n)h Fi(\000)g Fk(1\))p Fh(n\026)3067 3713 y Fq(n)3114 3698 y Fh(y)3166 3661 y Fs(2)p Fq(n)p Fr(\000)p Fs(2)3338 3698 y Fk(\))f(_)-44 b Fh(y)s Fk(.)264 3818 y(Then)39 b(the)g(v)m(ector)g Fh(r)i Fk(is)d(normal)f(to)h(the)h(curv)m(e)h(\(B.1\))e(at)g(the)h(p)s (oin)m(t)e Fh(q)t Fk(.)h(Therefore,)264 3938 y Fh(p)313 3902 y Fr(0)364 3938 y Fk(=)27 b Fh(p)22 b Fi(\000)h Fh(\027)28 b Fi(\001)22 b Fh(r)35 b Fk(and)52 b(_)-46 b Fh(p)1082 3902 y Fr(0)1133 3938 y Fk(=)47 b(_)-46 b Fh(p)22 b Fi(\000)g Fh(\027)29 b Fi(\001)37 b Fk(_)-42 b Fh(r)25 b Fi(\000)39 b Fk(_)-44 b Fh(\027)29 b Fi(\001)22 b Fh(r)s Fk(,)32 b(where)i(the)f(quan)m(tities)459 4237 y Fh(\027)h Fk(=)27 b(2)703 4170 y Fi(h)p Fh(p;)17 b(r)s Fi(i)p 703 4214 217 4 v 707 4305 a(h)p Fh(r)m(;)g(r)s Fi(i)930 4237 y Fh(;)228 b Fk(_)-43 b Fh(\027)34 b Fk(=)1364 4170 y Fi(h)p Fh(p)1452 4133 y Fr(0)1475 4170 y Fh(;)i Fk(_)-46 b Fh(p)22 b Fi(\000)h Fh(\027)28 b Fi(\001)37 b Fk(_)-42 b Fh(r)s Fi(i)p 1364 4214 538 4 v 1513 4305 a(h)p Fh(p)1601 4276 y Fr(0)1624 4305 y Fh(;)17 b(r)s Fi(i)264 4540 y Fk(ha)m(v)m(e)36 b(b)s(een)h(determined)e(b)m(y)h(imp)s (osing)d(that)j Fh(p)2057 4504 y Fr(0)2112 4540 y Fi(2)d Fe(S)c Fk(and)55 b(_)-46 b Fh(p)2544 4504 y Fr(0)2600 4540 y Fi(2)33 b Fh(T)2756 4555 y Fq(p)2792 4536 y Fb(0)2818 4540 y Fe(S)o Fk(,)d(resp)s(ectiv)m(ely)-8 b(.)264 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Fq(n)2129 5394 y Fh(v)2180 5365 y Fr(0)2203 5354 y Fs(2)p Fq(n)2855 5325 y Fh(;)1745 5712 y Fk(37)p eop %%Page: 38 38 38 37 bop 264 100 a Fk(and)32 b Fh(\030)496 115 y Fq(j)560 100 y Fk(=)664 3 y Fg(\020)735 55 y Fs(2)p Fq(n)713 134 y(j)t Fs(+1)836 3 y Fg(\021)885 100 y Fk(\()p Fi(\000)p Fh(y)t(=v)1152 63 y Fr(0)1175 100 y Fk(\))1213 63 y Fs(2)p Fq(n)p Fr(\000)p Fq(j)t Fr(\000)p Fs(1)1505 100 y Fk(for)g Fh(j)h Fk(=)28 b(2)p Fh(;)17 b(:)g(:)g(:)f(;)h Fk(2)p Fh(n)k Fi(\000)i Fk(2,)32 b(whereas)j(the)e(quan)m(tit)m(y)475 399 y(_)-43 b Fh(\034)39 b Fk(=)653 331 y Fi(h)18 b Fk(_)-45 b Fh(q)26 b Fi(\000)d Fh(\034)33 b Fi(\001)41 b Fk(_)-46 b Fh(p)1035 295 y Fr(0)1058 331 y Fh(;)17 b(r)1149 295 y Fr(0)1172 331 y Fi(i)p 653 375 558 4 v 800 467 a(h)p Fh(p)888 438 y Fr(0)911 467 y Fh(;)g(r)1002 438 y Fr(0)1025 467 y Fi(i)264 672 y Fk(is)36 b(determined)h(b)m(y)h(imp)s(osing)c (that)55 b(_)-45 b Fh(q)1702 636 y Fr(0)1761 672 y Fi(2)35 b Fh(T)1919 687 y Fq(q)1953 668 y Fb(0)1980 672 y Fh(C)7 b Fk(:)37 b(If)g Fh(r)2270 636 y Fr(0)2330 672 y Fk(is)f(an)m(y)i (normal)d(v)m(ector)j(to)f Fh(C)264 792 y Fk(at)32 b Fh(q)430 756 y Fr(0)453 792 y Fk(,)h(then)51 b(_)-45 b Fh(q)782 756 y Fr(0)833 792 y Fi(2)28 b Fh(T)984 807 y Fq(q)1018 788 y Fb(0)1044 792 y Fh(C)35 b Fi(,)27 b(h)18 b Fk(_)-45 b Fh(q)1362 756 y Fr(0)1385 792 y Fh(;)17 b(r)1476 756 y Fr(0)1499 792 y Fi(i)27 b Fk(=)h(0.)166 1012 y(The)g(more)e(exp)s(ensiv)m(e)j(part)e(is)f(to)h(solv)m(e)g(the)h (p)s(olynomial)23 b(equation)k(\004)2797 1027 y Fq(n)2844 1012 y Fk(\()p Fh(\034)11 b Fk(\))28 b(=)f(0,)g(whic)m(h)166 1133 y(is)k(obtained)f(b)m(y)i(imp)s(osing)d(that)h(the)i(new)g(impact) d(p)s(oin)m(t)h Fh(q)2408 1096 y Fr(0)2459 1133 y Fk(=)e Fh(q)23 b Fi(\000)c Fh(\034)11 b(p)2827 1096 y Fr(0)2882 1133 y Fk(v)m(eri\014es)32 b(\(B.1\).)166 1253 y(The)e(ro)s(ot)e Fh(\034)40 b Fk(is)28 b(simple)f(and)i(negativ)m(e.)g(It)g(is)f (computed)h(b)m(y)h(Newton's)g(metho)s(d)e(taking)166 1373 y(as)33 b(initial)c(appro)m(ximation)h(its)i(v)-5 b(alue)32 b(for)g Fh(\017)c Fk(=)g(0,)k(namely)361 1646 y Fh(\034)403 1661 y Fs(0)471 1646 y Fk(=)27 b(2)655 1579 y Fh(xu)766 1542 y Fr(0)790 1579 y Fh(=a)890 1542 y Fs(2)951 1579 y Fk(+)22 b Fh(y)t(v)1152 1542 y Fr(0)1174 1579 y Fh(=b)1264 1542 y Fs(2)p 633 1623 694 4 v 633 1714 a Fk(\()p Fh(u)727 1686 y Fr(0)750 1714 y Fh(=a)p Fk(\))888 1686 y Fs(2)949 1714 y Fk(+)g(\()p Fh(v)1136 1686 y Fr(0)1159 1714 y Fh(=b)p Fk(\))1287 1686 y Fs(2)1364 1646 y Fk(=)28 b(2)1527 1579 y Fh(xu)1638 1542 y Fr(0)1683 1579 y Fi(\000)23 b Fh(\026)1842 1594 y Fs(1)1881 1579 y Fh(y)t(v)1984 1542 y Fr(0)p 1527 1623 480 4 v 1541 1715 a Fh(u)1597 1686 y Fr(0)1619 1675 y Fs(2)1681 1715 y Fi(\000)g Fh(\026)1840 1730 y Fs(1)1879 1715 y Fh(v)1930 1686 y Fr(0)1953 1675 y Fs(2)2016 1646 y Fh(:)166 2036 y Fj(B.3)99 b(T)-7 b(aylor)34 b(exp)-5 b(ansions)33 b(of)i(the)g (invariant)f(curves)166 2376 y Fk(The)f(dynamics)f(on)g(the)h(unstable) f(in)m(v)-5 b(arian)m(t)31 b(curv)m(e)j(can)e(b)s(e)h(linearized.)d (There)k(exists)166 2497 y(some)e(analytic)g(maps)g Fh(q)f Fk(=)d(\()p Fh(x;)17 b(y)t Fk(\))27 b(:)h Fe(R)38 b Fi(!)28 b Fh(C)39 b Fk(and)33 b Fh(p)27 b Fk(=)h(\()p Fh(u;)17 b(v)t Fk(\))26 b(:)i Fe(R)39 b Fi(!)27 b Fe(S)f Fk(suc)m(h)34 b(that)361 2716 y Fh(q)t Fk(\(0\))27 b(=)h(\()p Fh(a;)17 b Fk(0\))p Fh(;)211 b(p)p Fk(\(0\))27 b(=)h(\(1)p Fh(;)17 b Fk(0\))p Fh(;)211 b(f)11 b Fk(\()p Fh(q)t Fk(\()p Fh(r)s Fk(\))p Fh(;)17 b(p)p Fk(\()p Fh(r)s Fk(\)\))26 b(=)h Fi(\000)p Fk(\()p Fh(q)t Fk(\()p Fh(\025r)s Fk(\))p Fh(;)17 b(p)p Fk(\()p Fh(\025r)s Fk(\)\))p Fh(;)166 3034 y Fk(where)34 b Fh(\025)27 b Fk(=)636 3031 y(e)679 2998 y Fq(h)757 3034 y Fk(is)32 b(the)h(c)m(haracteristic)f(m)m(ultiplier)d(of)j(the)h (h)m(yp)s(erb)s(olic)f(p)s(erio)s(dic)f(orbit.)166 3254 y(Due)f(to)f(the)h(axial)e(symmetries)h(of)g(the)h(monomial)c(p)s (erturbations)j(\(1\),)h(the)g(functions)166 3375 y Fh(x)p Fk(\()p Fh(r)s Fk(\))g(and)h Fh(u)p Fk(\()p Fh(r)s 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Fk(b)p Fh(;)g Fk(c)p Fh(;)g Fk(d)p Fi(g)42 b Fk(and)h Fh(l)k Fi(2)e Fe(Z)1409 5395 y Fs(+)1466 5380 y Fk(,)e(w)m(e)g(denote)h(b)m(y)g(\()p Fh(?)p Fk(\))2285 5395 y Fq(l)2353 5380 y Fk(the)f(equation)g(obtained)f(b)m(y)1745 5712 y(38)p eop %%Page: 39 39 39 38 bop 166 83 a Fk(equating)46 b(the)760 92 y(O)836 83 y(\()p Fh(r)921 47 y Fq(l)947 83 y Fk(\)-terms)f(in)h(b)s(oth)g (sides)h(of)f(the)h(functional)e(equation)h(\()p Fh(?)p Fk(\).)g(F)-8 b(or)166 203 y(instance,)33 b(\(a\))698 218 y Fs(2)p Fq(k)808 203 y Fk(reads)g(as)1183 137 y Fg(P)1271 163 y Fq(k)1271 228 y(s)p Fs(=0)1414 203 y Fh(u)1470 218 y Fq(s)1507 203 y Fh(u)1563 218 y Fq(k)r Fr(\000)p Fq(s)1715 203 y Fk(+)1813 137 y Fg(P)1900 163 y Fq(k)1900 228 y(s)p Fs(=1)2044 203 y Fh(v)2091 218 y Fq(s)2128 203 y Fh(v)2175 218 y Fq(k)r Fs(+1)p Fr(\000)p Fq(s)2423 203 y Fk(=)27 b(0.)166 423 y(W)-8 b(e)29 b(kno)m(w)g(that)f Fh(x)842 438 y Fs(0)909 423 y Fk(=)g Fh(a)g Fk(and)g Fh(u)1333 438 y Fs(0)1400 423 y Fk(=)f(1.)h(Then)h(equations)g(\(b\)) 2420 438 y Fs(2)2459 423 y Fk(,)f(\(c\))2633 438 y Fs(1)2673 423 y Fk(,)g(and)g(\(d\))3043 438 y Fs(1)3111 423 y Fk(giv)m(e)g(rise) 166 544 y(to)39 b(a)g(system)h(whose)g(one-parametric)e(family)e(of)j (non-trivial)d(solutions)i(is:)h Fh(y)3185 559 y Fs(1)3263 544 y Fi(6)p Fk(=)g(0,)166 664 y Fh(v)213 679 y Fs(1)280 664 y Fk(=)28 b Fh(\025)441 628 y Fr(\000)p Fs(1)p Fq(=)p Fs(2)606 664 y Fh(y)654 679 y Fs(1)693 664 y Fh(=b)p Fk(,)d(and)h Fh(x)1073 679 y Fs(1)1140 664 y Fk(=)i Fi(\000)p Fh(ay)1420 679 y Fs(1)1459 664 y Fh(=b)p Fk(.)e(W)-8 b(e)25 b(tak)m(e)h Fh(y)2014 679 y Fs(1)2081 664 y Fk(=)i(2)p Fh(b)p Fk(,)d Fh(v)2374 679 y Fs(1)2441 664 y Fk(=)j(2)p Fh(\025)2651 628 y Fr(\000)p Fs(1)p Fq(=)p Fs(2)2841 664 y Fk(and)d Fh(x)3078 679 y Fs(1)3146 664 y Fk(=)i Fi(\000)p Fk(2)p Fh(a)p Fk(.)166 785 y(Besides,)34 b(the)f(co)s (e\016cien)m(t)g Fh(u)1219 800 y Fs(1)1286 785 y Fk(=)27 b Fi(\000)p Fk(2)p Fh(=\025)33 b Fk(is)f(found)g(using)h(equation)f (\(a\))2804 800 y Fs(2)2843 785 y Fk(.)166 1005 y(No)m(w,)g(let)f(us)i (supp)s(ose)g(that)e Fh(x)1310 1020 y Fs(0)1350 1005 y Fh(;)17 b(:)g(:)g(:)f(;)h(x)1624 1020 y Fq(k)1667 1005 y Fh(;)g(y)1759 1020 y Fs(1)1798 1005 y Fh(;)g(:)g(:)g(:)e(;)i(y)2064 1020 y Fq(k)2106 1005 y Fh(;)g(u)2206 1020 y Fs(0)2245 1005 y Fh(;)g(:)g(:)g(:)f(;)h(u)2520 1020 y Fq(k)2562 1005 y Fh(;)g(v)2653 1020 y Fs(1)2692 1005 y Fh(;)g(:)g(:)g(:)f(;)h(v) 2958 1020 y Fq(k)3032 1005 y Fk(ha)m(v)m(e)33 b(b)s(een)166 1125 y(computed)j(for)f(some)g(in)m(teger)h Fh(k)f Fi(\025)f Fk(1.)h(Then)i(using)e(the)h(equations)g(\(b\))2909 1140 y Fs(2)p Fq(k)r Fs(+2)3077 1125 y Fk(,)f(\(c\))3258 1140 y Fs(2)p Fq(k)r Fs(+1)3427 1125 y Fk(,)166 1245 y(and)e(\(d\))486 1260 y Fs(2)p Fq(k)r Fs(+1)654 1245 y Fk(,)g(w)m(e)g(get)g(the)g (linear)e(system)361 1372 y Fg(0)361 1518 y(B)361 1568 y(B)361 1618 y(B)361 1668 y(B)361 1717 y(B)361 1771 y(@)574 1498 y Fk(2)p Fh(a)478 b Fk(4)p Fh(a)1252 1462 y Fs(2)1291 1498 y Fh(=b)610 b Fk(0)450 1679 y(4\()p Fh(k)25 b Fk(+)d(1\))33 b(\(4)p Fh(k)25 b Fk(+)d(2\)\()p Fh(\025)g Fk(+)g(1\))p Fh(\025)1538 1643 y Fr(\000)p Fs(1)p Fq(=)p Fs(2)1736 1679 y Fk(2\()p Fh(\025)1880 1643 y Fs(2)p Fq(k)r Fs(+1)2069 1679 y Fk(+)g(1\))p Fh(b)600 1859 y Fk(0)383 b(\()p Fh(\025)1127 1823 y Fs(2)p Fq(k)r Fs(+1)1317 1859 y Fk(+)22 b(1\))312 b Fi(\000)p Fk(2)p Fh(\025)1997 1823 y Fs(2)p Fq(k)r Fs(+1)2166 1859 y Fh(a)2312 1372 y Fg(1)2312 1518 y(C)2312 1568 y(C)2312 1618 y(C)2312 1668 y(C)2312 1717 y(C)2312 1771 y(A)2402 1372 y(0)2402 1518 y(B)2402 1568 y(B)2402 1618 y(B)2402 1668 y(B)2402 1717 y(B)2402 1771 y(@)2491 1498 y Fh(x)2546 1513 y Fq(k)r Fs(+1)2495 1679 y Fh(y)2543 1694 y Fq(k)r Fs(+1)2495 1859 y Fh(v)2542 1874 y Fq(k)r Fs(+1)2696 1372 y Fg(1)2696 1518 y(C)2696 1568 y(C)2696 1618 y(C)2696 1668 y(C)2696 1717 y(C)2696 1771 y(A)2796 1668 y Fk(=)2900 1372 y Fg(0)2900 1518 y(B)2900 1568 y(B)2900 1618 y(B)2900 1668 y(B)2900 1717 y(B)2900 1771 y(@)2989 1498 y Fh(\014)3044 1513 y Fs(1)2989 1679 y Fh(\014)3044 1694 y Fs(2)2989 1859 y Fh(\014)3044 1874 y Fs(3)3100 1372 y Fg(1)3100 1518 y(C)3100 1568 y(C)3100 1618 y(C)3100 1668 y(C)3100 1717 y(C)3100 1771 y(A)166 2184 y Fk(whose)44 b(indep)s(enden)m(t)g(term)f(dep)s(ends)h(only)f(on) f(previously)h(computed)g(co)s(e\016cien)m(ts.)166 2304 y(The)33 b(determinan)m(t)e(of)g(this)g(system)i(is)e(4)p Fh(ab)p Fk(\()p Fh(\025)1870 2268 y Fs(2)p Fq(k)1969 2304 y Fi(\000)20 b Fk(1\)\(1)g Fi(\000)h Fh(\025)2415 2268 y Fs(2)p Fq(k)r Fs(+2)2583 2304 y Fk(\))27 b Fi(6)p Fk(=)h(0,)j(for)h(an)m(y)g Fh(k)f Fi(\025)d Fk(1.)166 2425 y(Th)m(us,)g(the)f(co)s(e\016cien)m(ts)h Fh(x)1139 2440 y Fq(k)r Fs(+1)1272 2425 y Fk(,)f Fh(y)1374 2440 y Fq(k)r Fs(+1)1533 2425 y Fk(and)f Fh(v)1763 2440 y Fq(k)r Fs(+1)1923 2425 y Fk(can)h(b)s(e)g(computed.)f(Next,)i(w)m(e)g (compute)166 2545 y(the)41 b(co)s(e\016cien)m(t)g Fh(u)861 2560 y Fq(k)r Fs(+1)1034 2545 y Fk(from)e(equation)i(\(a\))1804 2560 y Fs(2)p Fq(k)r Fs(+2)1972 2545 y Fk(.)f(Therefore,)i(this)e (algorithm)e(can)i(b)s(e)166 2666 y(applied)31 b(recursiv)m(ely)j(to)e (obtain)g(the)h(T)-8 b(a)m(ylor)32 b(expansions)i(up)e(to)h(an)m(y)g (order.)166 3033 y Fl(References)166 3366 y Fv([1])71 b(C.)32 b(Batut,)h(K.)f(Belabas,)h(D.)f(Bernardi,)f(H.)i(Cohen)e(and)h (M.)g(Olivier,)e Fx(User's)k(Guide)g(to)332 3479 y(P)-7 b(ARI/GP)73 b Fv(\(F)-8 b(reely)31 b(a)m(v)-5 b(ailable)29 b(from)h Fn(http://www.parigp-home.de)o(/)p Fv(\).)166 3653 y([2])71 b(G.D.)32 b(Birkho\013,)e Fx(Dynamic)-5 b(al)34 b(Systems)k Fv(\(AMS,)31 b(Pro)m(vidence,)g(1927\).)166 3828 y([3])71 b(V.L.)32 b(Cherno)m(v,)f(On)f(separatrix)h(splitting)e (of)i(some)h(quadratic)e(area-preserving)h(maps)332 3941 y(of)g(the)f(plane,)g Fx(R)-5 b(e)g(gul.)33 b(Chaotic)h(Dyn.)c Fw(3)h Fv(\(1998\))i(49{65.)166 4116 y([4])71 b(A.)62 b(Delshams)f(and)f(R.)i(Ram)-10 b(\023)-35 b(\020rez-Ros,)61 b(P)m(oincar)m(\023)-43 b(e-Melnik)m(o)m(v-Arnold)62 b(metho)s(d)f(for)332 4229 y(analytic)30 b(planar)f(maps,)i Fx(Nonline)-5 b(arity)39 b Fw(9)31 b Fv(\(1996\))i(1{26.)166 4404 y([5])71 b(A.)29 b(Delshams)e(and)g(R.)h(Ram)-10 b(\023)-35 b(\020rez-Ros,)28 b(Exp)s(onen)m(tially)e(small)g(splitting) g(of)i(separatrices)332 4517 y(for)21 b(p)s(erturb)s(ed)e(in)m (tegrable)h(standard-lik)m(e)g(maps,)h Fx(J.)j(Nonline)-5 b(ar)25 b(Sci.)c Fw(8)g Fv(\(1998\))i(317{352.)166 4691 y([6])71 b(A.)62 b(Delshams)e(and)h(R.)g(Ram)-10 b(\023)-35 b(\020rez-Ros,)62 b(Singular)d(separatrix)h(splitting)f(and)h(the)332 4804 y(Melnik)m(o)m(v)31 b(metho)s(d:)f(An)g(exp)s(erimen)m(tal)f (study)-8 b(,)30 b Fx(Exp.)j(Math.)e Fw(8)f Fv(\(1999\))j(29{48.)166 4979 y([7])71 b(A.)23 b(Delshams,)e(Y)-8 b(u.)22 b(F)-8 b(edoro)m(v)24 b(and)d(R.)h(Ram)-10 b(\023)-35 b(\020rez-Ros,)22 b(Homo)s(clinic)e(billiard)e(orbits)j(inside)332 5092 y(symmetrically)29 b(p)s(erturb)s(ed)f(ellipsoids,)f Fx(Nonline)-5 b(arity)40 b Fw(14)31 b Fv(\(2001\))h(1141{1195.)166 5267 y([8])71 b(E.)84 b(F)-8 b(on)m(tic)m(h)84 b(and)f(C.)g(Sim\023)-45 b(o,)82 b(The)h(splitting)d(of)k(separatrices)f(for)g(analytic)332 5380 y(di\013eomorphisms,)28 b Fx(Er)-5 b(go)g(dic)34 b(The)-5 b(ory)34 b(Dynam.)f(Systems)39 b Fw(10)31 b Fv(\(1990\))h(295{318.)1745 5712 y Fk(39)p eop %%Page: 40 40 40 39 bop 166 83 a Fv([9])71 b(V.G.)42 b(Gelfreic)m(h,)d(V.F.)j (Lazutkin)d(and)h(N.V.)h(Sv)-5 b(anidze,)39 b(A)i(re\014ned)e(form)m (ula)g(for)h(the)332 196 y(separatrix)30 b(splitting)e(for)i(the)h (standard)e(map,)i Fx(Phys.)i(D)38 b Fw(71)31 b Fv(\(1994\))i(82{101.) 166 382 y([10])26 b(V.G.)62 b(Gelfreic)m(h,)e(A)g(pro)s(of)g(of)g(the)g (exp)s(onen)m(tially)f(small)g(transv)m(ersalit)m(y)h(of)g(the)332 495 y(separatrices)31 b(for)f(the)h(standard)e(map,)h Fx(Comm.)k(Math.)f(Phys.)e Fw(201)g Fv(\(1999\))i(155{216.)166 682 y([11])26 b(V.G.)40 b(Gelfreic)m(h)d(and)h(D.)h(Sauzin,)e(Borel)h (summation)f(and)h(splitting)e(of)j(separatrices)332 795 y(for)31 b(the)f(H)m(\023)-43 b(enon)31 b(map,)g Fx(A)n(nn.)h(Inst.)h(F)-7 b(ourier)33 b(\(Gr)-5 b(enoble\))35 b Fw(51)c Fv(\(2001\))i(513{567.)166 981 y([12])26 b(V.G.)f(Gelfreic)m (h)d(and)h(V.F.)h(Lazutkin,)f(Splitting)d(of)k(separatrices:)f(p)s (erturbation)e(theory)332 1094 y(and)30 b(exp)s(onen)m(tial)g (smallness,)e Fx(R)n(ussian)33 b(Math.)g(Surveys)38 b Fw(56)31 b Fv(\(2001\))i(499{558.)166 1280 y([13])26 b(A.)j(Katok)h(and)d(B.)i(Hasselblatt,)g Fx(Intr)-5 b(o)g(duction)33 b(to)f(the)f(Mo)-5 b(dern)32 b(The)-5 b(ory)32 b(of)f(Dynamic)-5 b(al)332 1393 y(Systems)39 b Fv(\(Cam)m(bridge)30 b(Univ.)f(Press,)h (Cam)m(bridge,)g(1995\).)166 1579 y([14])c(V.V.)50 b(Kozlo)m(v)f(and)f (D.V.)i(T)-8 b(reshc)m(h)m(\177)-43 b(ev,)50 b Fx(Bil)5 b(liar)-5 b(ds:)51 b(A)e(Genetic)g(Intr)-5 b(o)g(duction)52 b(to)e(the)332 1692 y(Dynamics)55 b(of)g(Systems)h(with)f(Imp)-5 b(acts)7 b Fv(,)57 b(T)-8 b(rans.)54 b(Math.)h(Monographs)f Fw(89)h Fv(\(AMS,)332 1805 y(Pro)m(vidence,)31 b(1991\).)166 1992 y([15])26 b(V.F.)i(Lazutkin,)f(Splitting)d(of)j(separatrices)g 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Fv(\(SMF,)g(P)m(aris,)f(1995\).)166 4612 y([23])c(M.B.)50 b(T)-8 b(abano)m(v,)49 b(Separatrices)e(splitting)f(for)i(Birkho\013)7 b('s)47 b(billiard)d(in)j(a)h(symmetric)332 4725 y(con)m(v)m(ex)32 b(domain,)e(close)h(to)g(an)f(ellipse,)e Fx(Chaos)40 b Fw(4)30 b Fv(\(1994\))j(595{606.)166 4911 y([24])26 b(E.T.)34 b(Whittak)m(er)h(and)f(G.N.)h(W)-8 b(atson,)36 b Fx(A)f(Course)i(of)f(Mo)-5 b(dern)37 b(A)n(nalysis)42 b Fv(\(Cam)m(bridge)332 5024 y(Univ.)30 b(Press,)g(Cam)m(bridge,)g (1927\).)1745 5712 y Fk(40)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0411051126842--