This is a multi-part message in MIME format. ---------------9906011803861 Content-Type: text/plain; name="99-205.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-205.keywords" tiling, hyperbolic plane ---------------9906011803861 Content-Type: application/postscript; name="text2.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="text2.ps" %!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: text2.dvi %%Pages: 8 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -o text2.ps text2 %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1999.06.01:1745 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 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%%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: Letter %%EndSetup %%Page: 1 1 1 0 bop 477 478 a Fq(CIR)m(CLE)57 b(P)-10 b(A)m(CKING)58 b(IN)f(THE)g(HYPERBOLIC)h(PLANE)1561 949 y Fp(Lewis)37 b(Bo)n(wen)1275 1190 y Fo(Departmen)m(t)c(of)g(Mathematics)1143 1309 y(The)h(Univ)m(ersit)m(y)f(of)g(T)-8 b(exas)33 b(at)g(Austin)1383 1429 y(Austin,)g(TX)g(78712-1082)120 1979 y(By)f(a)h Fn(cir)-5 b(cle)35 b(p)-5 b(acking)41 b Fo(w)m(e)34 b(will)e(mean)h(is) g(a)g(\014nite)h(collection)e(of)i(circles)g(of)f(some)g(\014xed)h (diame-)0 2159 y(ter)h(in)h(a)f(Riemannian)f(manifold)h(suc)m(h)i(that) d(no)i(t)m(w)m(o)f(circles)h(o)m(v)m(erlap)f(except)h(p)s(ossibly)f(in) g(their)0 2338 y(b)s(oundaries.)44 b(A)30 b(circle)g(pac)m(king)g(is)g Fn(optimal)41 b Fo(if)30 b(it)g(maximizes)e(the)i(n)m(um)m(b)s(er)h(of) f(touc)m(hes)i(\(b)s(et)m(w)m(een)0 2517 y(circles\))38 b(for)g(a)g(\014xed)h(n)m(um)m(b)s(er)f(of)h(circles.)58 b(Heitmann)37 b(and)i(Radin)f([1])f(sho)m(w)m(ed,)k(follo)m(wing)c (Har-)0 2697 y(b)s(orth)g([2],)g(that)g(an)m(y)g(optimal)e(circle)i (pac)m(king)h(in)f(the)g(Euclidean)h(plane)f(has)h(all)e(of)i(its)f (cen)m(ters)0 2876 y(lo)s(cated)25 b(on)h(the)g(v)m(ertices)g(of)f(a)h (triangulation)e(of)i(the)g(Euclidean)g(plane)g(b)m(y)f(congruen)m(t)i (equilateral)0 3055 y(triangles.)42 b(This)30 b(pap)s(er)g(will)f(sho)m (w)h(that)g(the)f(same)g(result)h(holds)h(in)e(the)h(h)m(yp)s(erb)s (olic)g(plane)g(if)g(the)0 3235 y(diameter)35 b Fm(d)g Fo(is)h(suc)m(h)h(that)e(an)g(equilateral)g(triangle)g(in)g(the)h(h)m (yp)s(erb)s(olic)g(plane)g(of)f(side)h(length)g Fm(d)0 3414 y Fo(has)e(eac)m(h)g(of)f(its)g(angles)h(is)f(equal)g(to)1450 3375 y Fl(2)p Fk(\031)p 1450 3391 89 4 v 1459 3448 a(N)1584 3414 y Fo(for)h(some)f Fm(N)38 b Fj(2)28 b(f)p Fo(7)p Fm(;)17 b Fo(8)p Fm(;)g Fo(9)p Fm(;)g(:)g(:)g(:)12 b Fj(g)p Fo(.)120 3602 y(F)-8 b(rom)46 b(no)m(w)j(on)f(\014x)g(suc)m(h)h (a)f(diameter)f Fm(d)p Fo(.)89 b(Let)48 b Fm(\032)g Fo(b)s(e)g(the)h (distance)g(function)g(in)f Fi(H)3504 3566 y Fl(2)3603 3602 y Fo(\(the)0 3781 y(h)m(yp)s(erb)s(olic)28 b(plane\).)43 b(An)28 b Fn(admissible)i(gr)-5 b(aph)36 b Fm(G)p Fo(,)29 b(is)f(a)g(graph)h(in)f Fi(H)2497 3745 y Fl(2)2576 3781 y Fo(suc)m(h)i(that)d(if)i Fm(v)t(;)17 b(w)30 b Fo(are)f(v)m(ertices)0 3961 y(of)34 b Fm(G)g Fo(then)g Fm(\032)p Fo(\()p Fm(v)t(;)17 b(w)s Fo(\))27 b Fj(\025)i Fm(d)k Fo(and)h(the)g(set)g(of)g(edges)h(of) e Fm(G)h Fo(is)g(the)g(set)g(of)g(geo)s(desic)g(arcs)g(b)s(et)m(w)m (een)h(an)m(y)0 4140 y(t)m(w)m(o)41 b(v)m(ertices)h Fm(v)t(;)17 b(w)44 b Fo(suc)m(h)f(that)e Fm(\032)p Fo(\()p Fm(v)t(;)17 b(w)s Fo(\))40 b(=)i Fm(d)p Fo(.)68 b(An)42 b Fn(optimal)h(gr)-5 b(aph)49 b Fm(G)42 b Fo(is)f(an)h(admissible)f(graph)0 4319 y(suc)m(h)h(that)e(the)h(n)m(um)m(b)s(er)g(of)g(edges)g(of)g Fm(G)g Fo(is)f(at)g(least)h(as)f(large)g(as)h(the)g(n)m(um)m(b)s(er)g (of)g(edges)g(of)g Fm(H)0 4499 y Fo(for)e(an)m(y)g(admissible)f(graph)h Fm(H)47 b Fo(with)39 b(the)g(n)m(um)m(b)s(er)g(of)h(v)m(ertices)f(of)g Fm(G)g Fo(equal)g(to)f(the)h(n)m(um)m(b)s(er)h(of)0 4678 y(v)m(ertices)f(of)g Fm(H)8 b Fo(.)60 b(There)39 b(is)g(a)f(bijection)i (b)s(et)m(w)m(een)g(circle)e(pac)m(kings)h(\(of)g(diameter)f Fm(d)p Fo(\))g(in)h Fi(H)3530 4642 y Fl(2)3619 4678 y Fo(and)0 4857 y(admissible)33 b(graphs)h(whic)m(h)g(maps)f(the)g(cen)m (ters)i(of)f(a)f(giv)m(en)g(circle)h(pac)m(king)f(bijectiv)m(ely)g(on)m (to)h(the)0 5037 y(v)m(ertices)h(of)g(an)f(admissible)g(graph.)48 b(This)35 b(bijection)g(restricts)g(to)f(a)g(bijection)h(b)s(et)m(w)m (een)h(optimal)p 0 5184 598 4 v 120 5280 a Fh(Researc)n(h)27 b(supp)r(orted)j(in)e(part)h(b)n(y)e(T)-7 b(exas)29 b(ARP)f(Gran)n(t)h (003658-152.)2959 5519 y(T)n(yp)r(eset)f(b)n(y)f Fg(A)3453 5536 y(M)3543 5519 y(S)p Fh(-T)3669 5536 y(E)3717 5519 y(X)1869 5758 y(1)p eop %%Page: 2 2 2 1 bop 0 -139 a Fh(2)1526 b(LEWIS)37 b(BO)n(WEN)0 100 y Fo(circle)30 b(pac)m(kings)h(and)g(optimal)d(graphs.)43 b(Let)31 b Fm(\013)f Fo(b)s(e)h(an)f(in)m(terior)g(angle)g(of)g(an)h (equilateral)e(triangle)0 279 y(of)k(side)h(length)g Fm(d)p Fo(.)43 b(The)34 b(main)e(result)i(follo)m(ws)f(immediately)d (from)0 533 y Fq(Theorem)38 b(1.)57 b Fn(If)749 494 y Fl(2)p Fk(\031)p 749 510 89 4 v 768 567 a(\013)878 533 y Fj(2)28 b(f)p Fo(7)p Fm(;)17 b Fo(8)p Fm(;)g(:)g(:)g(:)c Fj(g)35 b Fn(then)201 747 y Fo(1.)51 b Fn(A)m(l)5 b(l)37 b(the)f(fac)-5 b(es)34 b(of)h(any)h(optimal)g(gr)-5 b(aph)35 b(ar)-5 b(e)35 b(triangles.)201 926 y Fo(2.)51 b Fn(A)m(l)5 b(l)37 b(spir)-5 b(al)35 b(gr)-5 b(aphs)35 b(ar)-5 b(e)35 b(optimal.)120 1215 y Fo(Here,)47 b(a)e Fn(fac)-5 b(e)52 b Fo(of)45 b(an)g(admissible)g(graph)g Fm(G)g Fo(is)g(the)g(closure)h (of)f(a)g(b)s(ounded)i(comp)s(onen)m(t)e(of)0 1395 y Fi(H)71 1359 y Fl(2)144 1395 y Fj(\000)23 b Fm(G)p Fo(.)44 b(Spiral)33 b(graphs)h(are)f(de\014ned)j(later.)0 1649 y Fn(Notation.)50 b Fo(Let)d Fm(G)g Fo(b)s(e)h(a)e(\014nite)h(graph)h (of)f Fi(H)1787 1613 y Fl(2)1838 1649 y Fo(.)85 b(De\014ne)48 b Fm(V)22 b Fo(\()p Fm(G)p Fo(\))50 b(=)h Fj(f)p Fo(v)m(ertices)34 b(of)f Fm(G)p Fj(g)p Fo(,)50 b Fm(E)6 b Fo(\()p Fm(G)p Fo(\))50 b(=)0 1828 y Fj(f)p Fo(edges)34 b(of)g Fm(G)p Fj(g)p Fo(,)k Fm(F)14 b Fo(\()p Fm(G)p Fo(\))35 b(=)g Fj(f)p Fo(faces)g(of)e Fm(G)p Fj(g)p Fo(.)57 b(Let)38 b Fm(f)46 b Fj(2)35 b Fm(F)14 b Fo(\()p Fm(G)p Fo(\).)57 b(De\014ne)39 b Fm(A)p Fo(\()p Fm(f)11 b Fo(\))37 b(to)g(b)s(e)h(the)g (h)m(yp)s(erb)s(olic)0 2007 y(area)e(of)g Fm(f)11 b Fo(,)35 b(and)h Fm(A)p Fo(\()p Fm(G)p Fo(\))31 b(=)1026 1933 y Ff(P)1131 2037 y Fk(f)8 b Fe(2)p Fk(F)j Fl(\()p Fk(G)p Fl(\))1438 2007 y Fm(A)p Fo(\()p Fm(f)g Fo(\).)50 b Fm(@)6 b(G)36 b Fo(will)e(b)s(e)i(the)g(subgraph)h(of)f Fm(G)g Fo(con)m(tained)h(in)f(the)0 2187 y(closure)42 b(of)g(the)g(un)m(b)s (ounded)h(comp)s(onen)m(t)f(of)g Fi(H)1876 2151 y Fl(2)1954 2187 y Fj(\000)28 b Fm(G)p Fo(.)69 b(F)-8 b(or)41 b(an)m(y)h(\014nite)f (set)h Fm(S)6 b Fo(,)43 b(let)e Fj(j)p Fm(S)6 b Fj(j)41 b Fo(b)s(e)h(the)0 2366 y(n)m(um)m(b)s(er)d(of)f(elemen)m(ts)g(of)g Fm(S)6 b Fo(.)59 b(Let)38 b Fm(v)h Fj(2)d Fm(@)6 b(G)p Fo(.)59 b(If)38 b Fj(jf)p Fm(e)e Fj(2)g Fm(E)6 b Fo(\()p Fm(G)p Fo(\))34 b Fj(j)i Fm(e)26 b Fj(\\)f(f)p Fm(v)t Fj(g)36 b(6)p Fo(=)g Fj(;gj)f Fo(=)h(1)i(then)g(de\014ne)0 2545 y Fm(a)p Fo(\()p Fm(v)t Fo(\))27 b(=)i(0.)44 b(Otherwise)35 b(let)e Fj(f)p Fm(e)1150 2560 y Fl(1)1195 2545 y Fm(;)17 b(e)1286 2560 y Fl(2)1330 2545 y Fj(g)28 b Fo(=)h Fj(f)p Fm(e)f Fj(2)h Fm(E)6 b Fo(\()p Fm(@)g(G)p Fo(\))26 b Fj(j)i Fm(e)23 b Fj(\\)g(f)p Fm(v)t Fj(g)k(6)p Fo(=)i Fj(;g)k Fo(and)h(de\014ne)h Fm(a)p Fo(\()p Fm(v)t Fo(\))e(to)g(b)s(e)h (the)0 2725 y(angle)e(b)s(et)m(w)m(een)i Fm(e)689 2740 y Fl(1)766 2725 y Fo(and)f Fm(e)1005 2740 y Fl(2)1082 2725 y Fo(in)m(terior)g(to)e(the)i(face)g(of)g Fm(G)f Fo(con)m(taining)h Fm(v)t Fo(.)43 b(De\014ne)34 b Fm(\013)e Fo(to)g(b)s(e)h(an)f(angle)0 2904 y(in)m(terior)27 b(to)g(an)g (equilateral)f(triangle)h(of)h(side)g(length)f Fm(d)p Fo(.)42 b(Let)27 b Fm(N)39 b Fo(=)28 b(2)p Fm(\031)t(=\013)p Fo(.)41 b(F)-8 b(rom)26 b(no)m(w)i(on,)g(assume)0 3083 y Fm(N)39 b Fj(2)28 b(f)p Fo(7)p Fm(;)17 b Fo(8)p Fm(;)g Fo(9)p Fm(;)g(:)g(:)g(:)12 b Fj(g)p Fo(.)43 b(F)-8 b(or)33 b(an)m(y)g(sets)h Fm(A;)17 b(B)31 b Fj(2)d Fi(H)1754 3047 y Fl(2)1805 3083 y Fo(,)33 b(let)f Fm(\032)p Fo(\()p Fm(A;)17 b(B)5 b Fo(\))26 b(=)i(inf)8 b Fj(f)p Fm(\032)p Fo(\()p Fm(a;)17 b(b)p Fo(\))25 b Fj(j)j Fm(a)f Fj(2)h Fm(A;)49 b(b)28 b Fj(2)g Fm(B)5 b Fj(g)p Fo(.)0 3337 y Fq(Spiral)37 b(Graphs.)120 3522 y Fo(Let)c Fm(S)359 3537 y Fk(n)447 3522 y Fo(denote)h(the)f Fn(spir)-5 b(al)43 b Fo(graph)34 b(on)f Fm(n)g Fo(v)m(ertices)h(constructed)g(as)g(follo)m (ws:)120 3706 y(Let)i Fm(v)349 3721 y Fl(1)394 3706 y Fm(;)17 b(:)g(:)g(:)30 b(;)17 b(v)680 3721 y Fk(N)7 b Fl(+1)892 3706 y Fo(b)s(e)36 b(distinct)g(p)s(oin)m(ts)g(in)f Fi(H)1885 3670 y Fl(2)1971 3706 y Fo(suc)m(h)i(that)f Fm(\032)p Fo(\()p Fm(v)2556 3721 y Fl(1)2600 3706 y Fm(;)17 b(v)2693 3721 y Fk(i)2726 3706 y Fo(\))31 b(=)h Fm(d)k Fo(for)g(2)c Fj(\024)g Fm(i)g Fj(\024)g Fm(N)j Fo(+)24 b(1)0 3885 y(and)34 b Fm(\032)p Fo(\()p Fm(v)333 3900 y Fk(i)365 3885 y Fm(;)17 b(v)458 3900 y Fk(i)p Fl(+1)592 3885 y Fo(\))27 b(=)i Fm(d)k Fo(for)g(2)27 b Fj(\024)i Fm(i)f Fj(\024)g Fm(N)11 b Fo(.)44 b(Let)33 b Fm(S)1753 3900 y Fk(n)1840 3885 y Fo(\(for)h(1)27 b Fj(\024)h Fm(n)g Fj(\024)g Fm(N)33 b Fo(+)22 b(1\))33 b(b)s(e)h(the)f(admissible)g (graph)0 4065 y(with)j(v)m(ertex)g(set)h Fj(f)p Fm(v)792 4080 y Fk(i)825 4065 y Fj(g)875 4028 y Fk(n)875 4091 y(i)p Fl(=1)1009 4065 y Fo(.)53 b(Supp)s(ose)38 b Fm(S)1545 4080 y Fk(n)1635 4065 y Fo(\()p Fm(n)32 b Fj(\025)i Fm(N)h Fo(+)24 b(1\))36 b(has)h(b)s(een)g(constructed)h(in)e(suc)m(h)i(a)e(w)m (a)m(y)0 4244 y(that)d(prop)s(erties)h(a{g)e(\(b)s(elo)m(w\))h(are)g (satis\014ed.)190 4458 y(a\))50 b(All)32 b(the)i(faces)g(of)f Fm(S)1079 4473 y Fk(n)1167 4458 y Fo(are)g(triangles.)0 4672 y Fm(V)22 b Fo(\()p Fm(S)180 4687 y Fk(n)234 4672 y Fo(\))33 b(ma)m(y)f(b)s(e)i(ordered)g(\()p Fm(v)1105 4687 y Fl(1)1150 4672 y Fm(;)17 b(:)g(:)g(:)30 b(;)17 b(v)1436 4687 y Fk(n)1490 4672 y Fo(\))33 b(suc)m(h)i(that)185 4886 y(b\))50 b(F)-8 b(or)33 b(all)f Fm(m)c(<)g(n)p Fo(,)33 b Fm(V)22 b Fo(\()p Fm(S)1167 4901 y Fk(m)1243 4886 y Fo(\))27 b(=)h Fj(f)p Fm(v)1512 4901 y Fl(1)1557 4886 y Fm(;)17 b(:)g(:)g(:)31 b(;)17 b(v)1844 4901 y Fk(m)1919 4886 y Fj(g)196 5066 y Fo(c\))50 b(F)-8 b(or)33 b(1)27 b Fj(\024)h Fm(i)g Fj(\024)g Fm(n)22 b Fj(\000)h Fo(1)33 b(there)h(exists)f(an)g(edge)h(in)f Fm(E)6 b Fo(\()p Fm(S)2313 5081 y Fk(n)2367 5066 y Fo(\))33 b(with)g(endp)s(oin)m(ts)h Fm(v)3168 5081 y Fk(i)3235 5066 y Fo(and)f Fm(v)3476 5081 y Fk(i)p Fl(+1)3611 5066 y Fo(.)0 5280 y(Let)e Fm(m)g Fo(b)s(e)h(the)f(smallest)f(in)m(teger)h(suc)m(h)h(that)f(there)g (exists)g Fm(x)c Fj(2)i Fi(H)2516 5244 y Fl(2)2584 5280 y Fj(\000)18 b Fm(S)2740 5295 y Fk(n)2825 5280 y Fo(with)31 b Fm(d)c Fo(=)h Fm(\032)p Fo(\()p Fm(v)3373 5295 y Fk(m)3449 5280 y Fm(;)17 b(x)p Fo(\))29 b(and)p eop %%Page: 3 3 3 2 bop 864 -139 a Fh(CIR)n(CLE)37 b(P)-7 b(A)n(CKING)38 b(IN)f(THE)g(HYPERBOLIC)h(PLANE)822 b(3)0 100 y Fm(\032)p Fo(\()p Fm(v)t(;)17 b(x)p Fo(\))26 b Fj(\025)i Fm(d)33 b Fo(for)g(all)g Fm(v)e Fj(2)d Fm(V)23 b Fo(\()p Fm(S)1145 115 y Fk(n)1199 100 y Fo(\).)43 b(Then)185 368 y(d\))50 b Fj(f)p Fm(V)437 383 y Fk(m)512 368 y Fm(;)17 b(:)g(:)g(:)31 b(;)17 b(V)809 383 y Fk(n)863 368 y Fj(g)27 b Fo(=)h Fm(V)23 b Fo(\()p Fm(@)6 b(S)1285 383 y Fk(n)1338 368 y Fo(\).)196 548 y(e\))50 b(2)p Fm(\013)27 b Fj(\024)i Fm(a)p Fo(\()p Fm(v)716 563 y Fk(m)791 548 y Fo(\))e Fj(\024)h Fo(2)p Fm(\031)e Fj(\000)c Fo(2)p Fm(\013)p Fo(.)202 727 y(f)8 b(\))50 b(F)-8 b(or)33 b Fm(m)28 b(<)g(i)g(<)g(n)p Fo(,)33 b(2)p Fm(\013)27 b Fj(\024)h Fm(a)p Fo(\()p Fm(v)1402 742 y Fk(i)1435 727 y Fo(\))g Fj(\024)g Fo(3)p Fm(\013)p Fo(.)190 906 y(g\))50 b Fm(\013)28 b Fj(\024)g Fm(a)p Fo(\()p Fm(v)666 921 y Fk(n)720 906 y Fo(\))f Fj(\024)h Fo(2)p Fm(\013)33 b Fo(and)h Fm(a)p Fo(\()p Fm(v)1372 921 y Fk(n)1426 906 y Fo(\))27 b(=)i(2)p Fm(\013)k Fo(if)g(and)h(only)e (if)i Fm(a)p Fo(\()p Fm(v)2480 921 y Fk(n)2534 906 y Fo(\))27 b(=)h(2)p Fm(\031)d Fj(\000)e Fo(2)p Fm(\013)p Fo(.)0 1175 y Fj(j)p Fm(v)76 1190 y Fk(n)152 1175 y Fj(\000)g Fm(v)300 1190 y Fk(m)376 1175 y Fj(j)k Fo(=)h Fm(d)33 b Fo(b)m(y)h(prop)s(erties)g(c\))f(and)h(d\).)0 1414 y(Let)28 b Fm(r)218 1429 y Fk(n)272 1414 y Fm(;)17 b(r)362 1429 y Fk(m)p Fl(+1)566 1414 y Fo(b)s(e)28 b(the)h(half)f(lines)g(with) g(endp)s(oin)m(t)h Fm(v)1956 1429 y Fk(m)2059 1414 y Fo(whic)m(h)g(con)m(tain)f Fm(v)2733 1429 y Fk(n)2788 1414 y Fm(;)17 b(v)2881 1429 y Fk(m)p Fl(+1)3085 1414 y Fo(resp)s(ectiv)m(ely)-8 b(.)42 b(By)0 1593 y(f)8 b(\))29 b(and)g(g\),)g Fm(S)502 1608 y Fk(n)586 1593 y Fo(is)f(con)m(tained)i (in)f(a)g(single)g(comp)s(onen)m(t)g(of)g Fi(H)2277 1557 y Fl(2)2342 1593 y Fj(\000)14 b Fo(\()p Fm(r)2517 1608 y Fk(n)2584 1593 y Fj([)g Fm(r)2709 1608 y Fk(m)p Fl(+1)2886 1593 y Fo(\).)42 b(By)28 b(e\),)i(there)f(exists)0 1773 y(a)37 b(unique)i(p)s(oin)m(t)e Fm(v)728 1788 y Fk(n)p Fl(+1)918 1773 y Fj(2)f Fi(H)1091 1736 y Fl(2)1179 1773 y Fo(suc)m(h)k(that)d Fm(\032)p Fo(\()p Fm(v)1768 1788 y Fk(m)1843 1773 y Fm(;)17 b(v)1936 1788 y Fk(n)p Fl(+1)2091 1773 y Fo(\))34 b(=)i Fm(\032)p Fo(\()p Fm(v)2416 1788 y Fk(n)2470 1773 y Fm(;)17 b(v)2563 1788 y Fk(n)p Fl(+1)2717 1773 y Fo(\))35 b(=)g Fm(\032)p Fo(\()p Fm(V)22 b Fo(\()p Fm(S)3174 1788 y Fk(n)3228 1773 y Fo(\))p Fm(;)17 b(v)3360 1788 y Fk(n)p Fl(+1)3514 1773 y Fo(\))35 b(=)g Fm(d)p Fo(.)0 1952 y(Let)27 b Fm(S)233 1967 y Fk(n)p Fl(+1)414 1952 y Fo(b)s(e)g(the)g(admissible)f(graph)g(with)g(v)m(ertex)g(set)h Fj(f)p Fm(v)2221 1967 y Fl(1)2266 1952 y Fm(;)17 b(:)g(:)g(:)31 b(;)17 b(v)2553 1967 y Fk(n)p Fl(+1)2707 1952 y Fj(g)p Fo(.)42 b(a\){g\))25 b(are)i(seen)g(to)f(hold.)120 2191 y(Before)33 b(pro)m(ving)g(Theorem)g(1)g(w)m(e)h(require)f(t)m(w)m(o)g (tec)m(hnical)h(lemmas.)0 2608 y Fq(Lemma)k(1.)58 b Fn(L)-5 b(et)36 b Fm(G)g Fn(b)-5 b(e)35 b(an)h(admissable)f(gr)-5 b(aph.)45 b(L)-5 b(et)36 b Fm(`)2129 2623 y Fk(G)2224 2608 y Fo(=)28 b(min)o Fj(f)p Fm(\032)p Fo(\()p Fm(e;)17 b(v)t Fo(\))26 b Fj(j)i Fm(e)g Fj(2)g Fm(E)6 b Fo(\()p Fm(G)p Fo(\))p Fn(,)35 b Fm(v)d Fj(2)c Fm(V)22 b Fo(\()p Fm(G)p Fo(\))0 2788 y Fn(and)35 b Fm(v)42 b(=)-61 b Fj(2)28 b Fm(e)p Fj(g)p Fn(.)46 b(L)-5 b(et)35 b Fm(`)27 b Fo(=)h(inf)8 b Fj(f)p Fm(`)1097 2803 y Fk(G)1191 2788 y Fj(j)28 b Fm(G)35 b Fn(is)f(an)h(admissable)f(gr)-5 b(aph)p Fj(g)p Fn(.)44 b(Then)35 b Fm(`)g Fn(is)f(the)g(length)h(of)g(the)f(ar)-5 b(c)0 2967 y(depicte)g(d)34 b(in)i(Figur)-5 b(e)35 b(1:)1537 3786 y @beginspecial 217 @llx 487 @lly 362 @urx 647 @ury 706 @rwi @setspecial %%BeginDocument: fig1.eps %AI5_FileFormat 2.0 %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Macintosh_ColorPic %AI5_ArtSize: 612 792 %AI5_RulerUnits: 0 %AI5_ArtFlags: 1 0 0 1 0 0 1 1 0 %AI5_TargetResolution: 800 %AI5_NumLayers: 1 %AI5_OpenToView: 30 660 2 794 557 58 1 1 6 41 %AI5_OpenViewLayers: 7 userdict /Adobe_level2_AI5 23 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { 5 packedarray } bind def /setcustomcolor { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def } if /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? level2? { gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore add add add 4 eq } { 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and } ifelse put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking currentpacking true setpacking userdict /Adobe_typography_AI5 54 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def pop pop findfont _wv type /arraytype eq { _wv makeblendedfont } if dup length 2 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { /Tx { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { /Tx { dup currentpoint 4 2 roll gsave tr _psf grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave trj _pjsf grestore 3 1 roll moveto tr jsp } ddef } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { /Tx { dup currentpoint 4 2 roll currentpoint gsave newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll currentpoint gsave newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat 0 _rise translate _hs 1 scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { dup 1000 div /_fScl exch ddef % selectfont } def /Tl { pop 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { /_rise exch ddef currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { 100 div /_hs exch ddef iTm } def /TA { pop } def /Tq { pop } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { exch pop _fScl mul neg 0 rmoveto } def /TK { 2 npop } def /T* { _leading aload pop neg Td } def /T*- { _leading aload pop Td } def /T- { _ax neg 0 rmoveto _hyphen Tx } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ _fScl 1000 mul selectfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 17 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 14 dict def } if Adobe_ColorImage_AI6_Vars begin /channelcount 0 def /sourcecount 0 def /sourcearray 4 array def /plateindex -1 def /XIMask 0 def /XIBinary 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIBuffer null def /XIDataProc null def end /WalkRGBString null def /WalkCMYKString null def /StuffRGBIntoGrayString null def /RGBToGrayImageProc null def /StuffCMYKIntoGrayString null def /CMYKToGrayImageProc null def /ColorImageCompositeEmulator null def /SeparateCMYKImageProc null def /FourEqual null def /TestPlateIndex null def currentdict /_colorimage known not { /colorimage where { /colorimage get /_colorimage exch def } { /_colorimage null def } ifelse } if /_currenttransfer systemdict /currenttransfer get def /colorimage null def /XI null def /WalkRGBString { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval { } forall 5 index exec 3 index } for 5 { pop } repeat } def /WalkCMYKString { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval { } forall 6 index exec 3 index } for 5 { pop } repeat } def /StuffRGBIntoGrayString { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /RGBToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 3 idiv string dup 3 1 roll /StuffRGBIntoGrayString load exch WalkRGBString end } def /StuffCMYKIntoGrayString { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /CMYKToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 4 idiv string dup 3 1 roll /StuffCMYKIntoGrayString load exch WalkCMYKString end } def /ColorImageCompositeEmulator { pop true eq { Adobe_ColorImage_AI6_Vars /sourcecount get 5 add { pop } repeat } { Adobe_ColorImage_AI6_Vars /channelcount get 1 ne { Adobe_ColorImage_AI6_Vars begin sourcearray 0 3 -1 roll put channelcount 3 eq { /RGBToGrayImageProc } { /CMYKToGrayImageProc } ifelse load end } if image } ifelse } def /SeparateCMYKImageProc { Adobe_ColorImage_AI6_Vars begin sourcecount 0 ne { sourcearray plateindex get exec } { sourcearray 0 get exec dup length 4 idiv string 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop } ifelse end } def /FourEqual { 4 index ne { pop pop pop false } { 4 index ne { pop pop false } { 4 index ne { pop false } { 4 index eq } ifelse } ifelse } ifelse } def /TestPlateIndex { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 FourEqual { /plateindex 0 def } { 0 1 0 0 FourEqual { /plateindex 1 def } { 0 0 1 0 FourEqual { /plateindex 2 def } { 0 0 0 1 FourEqual { /plateindex 3 def } { 0 0 0 0 FourEqual { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /colorimage { Adobe_ColorImage_AI6_Vars begin /channelcount 1 index def /sourcecount 2 index 1 eq { channelcount 1 sub } { 0 } ifelse def 4 sourcecount add index dup 8 eq exch 1 eq or not end { /_colorimage load null ne { _colorimage } { Adobe_ColorImage_AI6_Vars /sourcecount get 7 add { pop } repeat } ifelse } { dup 3 eq TestPlateIndex dup -1 eq exch 5 eq or or { /_colorimage load null eq { ColorImageCompositeEmulator } { dup 1 eq { pop pop image } { Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { gsave 0 _currenttransfer exec 1 _currenttransfer exec eq { 0 _currenttransfer exec 0.5 lt } { 0 _currenttransfer exec 1 _currenttransfer exec gt } ifelse { { pop 0 } } { { pop 1 } } ifelse systemdict /settransfer get exec } if _colorimage Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { grestore } if } ifelse } ifelse } { dup 1 eq { pop pop image } { pop pop Adobe_ColorImage_AI6_Vars begin sourcecount -1 0 { exch sourcearray 3 1 roll put } for /SeparateCMYKImageProc load end systemdict /image get exec } ifelse } ifelse } ifelse } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIMask exch 0 ne def /XIBinary exch 0 ne def pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi } { XIImageWidth XIChannelCount mul } ifelse /XIBuffer exch string def XIBinary { /XIDataProc { currentfile XIBuffer readstring pop } def currentfile 128 string readline pop pop } { /XIDataProc { currentfile XIBuffer readhexstring pop } def } ifelse 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale XIMask { XIImageWidth XIImageHeight false [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load /_lp /null ddef _fc /_lp /imagemask ddef imagemask } { XIImageWidth XIImageHeight XIBitsPerPixel [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load XIChannelCount 1 eq { gsave 0 setgray image grestore } { false XIChannelCount colorimage } ifelse } ifelse grestore end } def end currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 81 dict dup begin put /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_rise 0 def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_wv 0 def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 91 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /sw { dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add } def /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 ddef { 1 index eq { /_cnt _cnt 1 add ddef } if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /sp { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer readline not { stop } if endString eq { cleartomark stop } if } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer readline not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 4 npop 6 1 roll pop 4 1 roll pop pop pop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 3 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end end setpacking Adobe_level2_AI5 /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron TE %AI3_BeginEncoding: _Times-Italic Times-Italic [/_Times-Italic/Times-Italic 0 0 1 TZ %AI3_EndEncoding TrueType %AI3_BeginEncoding: _Times-Roman Times-Roman [/_Times-Roman/Times-Roman 0 0 1 TZ %AI3_EndEncoding TrueType %AI5_Begin_NonPrinting Np 8 Bn %AI5_BeginGradient: (Black & White) (Black & White) 0 2 Bd [ < 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%AI5_EndLayer-- gsave annotatepage grestore showpage Adobe_Illustrator_AI5 /terminate get exec Adobe_ColorImage_AI6 /terminate get exec Adobe_typography_AI5 /terminate get exec Adobe_level2_AI5 /terminate get exec %%EndDocument @endspecial 1673 3966 a Fp(Figure)i(1)0 4383 y Fn(Pr)-5 b(o)g(of.)49 b Fo(Let)35 b Fm(G)h Fo(b)s(e)f(an)g(admissible)f(graph.) 49 b(Supp)s(ose)37 b(that)d(there)i(exists)f(\()p Fm(v)t(;)17 b(e)p Fo(\))29 b Fj(2)i Fm(V)22 b Fo(\()p Fm(G)p Fo(\))h Fj(\002)h Fm(E)6 b Fo(\()p Fm(G)p Fo(\))0 4563 y(suc)m(h)46 b(that)d Fm(v)61 b(=)-61 b Fj(2)46 b Fm(e)f Fo(and)f Fm(\032)p Fo(\()p Fm(v)t(;)17 b(e)p Fo(\))45 b Fm(<)h(`)p Fo(.)76 b(Let)44 b Fm(v)1791 4578 y Fl(1)1836 4563 y Fm(;)17 b(v)1929 4578 y Fl(2)2018 4563 y Fo(b)s(e)44 b(the)h(endp)s(oin)m(ts)g(of)f Fm(e)p Fo(.)77 b(Without)45 b(loss)f(of)0 4742 y(generalit)m(y)-8 b(,)30 b(assume)h Fm(\032)p Fo(\()p Fm(v)958 4757 y Fl(1)1002 4742 y Fm(;)17 b(v)t Fo(\))27 b Fj(\024)h Fm(\032)p Fo(\()p Fm(v)1409 4757 y Fl(2)1453 4742 y Fm(;)17 b(v)t Fo(\).)42 b(Let)31 b Fm(p)d Fj(2)g Fm(e)j Fo(suc)m(h)i(that)d Fm(\032)p Fo(\()p Fm(p;)17 b(v)t Fo(\))26 b Fj(\024)i Fm(\032)p Fo(\()p Fm(x;)17 b(v)t Fo(\))29 b(for)i(all)f Fm(x)e Fj(2)g Fm(e)p Fo(.)0 4921 y(Then)k Fm(\032)p Fo(\()p Fm(p;)17 b(v)t Fo(\))26 b Fm(<)i(`)f(<)i(d)e(<)h(\032)p Fo(\()p Fm(v)t(;)17 b(v)1261 4936 y Fl(2)1305 4921 y Fo(\).)43 b(So)31 b(there)h(exists)f(at)g(p)s(oin)m(t)g Fm(v)2507 4885 y Fe(0)2503 4946 y Fl(2)2575 4921 y Fj(2)e Fm(e)i Fo(suc)m(h)i(that)e Fm(p)g Fo(is)g(in)g(the)h(arc)0 5101 y(from)k Fm(v)287 5116 y Fl(1)368 5101 y Fo(to)g Fm(v)545 5065 y Fe(0)541 5125 y Fl(2)622 5101 y Fo(and)h Fm(\032)p Fo(\()p Fm(v)t(;)17 b(v)1059 5065 y Fe(0)1055 5125 y Fl(2)1098 5101 y Fo(\))33 b(=)g Fm(\032)p Fo(\()p Fm(v)t(;)17 b(v)1516 5116 y Fl(1)1559 5101 y Fo(\).)53 b(See)37 b(Figure)g(2.)53 b(Let)36 b Fm(m)h Fo(b)s(e)g(the)f(midp)s (oin)m(t)g(of)g(the)h(arc)0 5280 y(from)c Fm(v)284 5295 y Fl(1)362 5280 y Fo(to)f Fm(v)535 5244 y Fe(0)531 5305 y Fl(2)576 5280 y Fo(.)p eop %%Page: 4 4 4 3 bop 0 -139 a Fh(4)1526 b(LEWIS)37 b(BO)n(WEN)1396 900 y @beginspecial 212 @llx 470 @lly 446 @urx 683 @ury 988 @rwi @setspecial %%BeginDocument: fig2.eps %AI5_FileFormat 2.0 %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Macintosh_ColorPic %AI5_ArtSize: 612 792 %AI5_RulerUnits: 0 %AI5_ArtFlags: 1 0 0 1 0 0 1 1 0 %AI5_TargetResolution: 800 %AI5_NumLayers: 1 %AI5_OpenToView: 66 660 2 794 557 58 1 1 3 40 %AI5_OpenViewLayers: 7 userdict /Adobe_level2_AI5 23 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { 5 packedarray } bind def /setcustomcolor { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def } if /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? level2? { gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore add add add 4 eq } { 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and } ifelse put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking currentpacking true setpacking userdict /Adobe_typography_AI5 54 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def pop pop findfont _wv type /arraytype eq { _wv makeblendedfont } if dup length 2 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { /Tx { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { /Tx { dup currentpoint 4 2 roll gsave tr _psf grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave trj _pjsf grestore 3 1 roll moveto tr jsp } ddef } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { /Tx { dup currentpoint 4 2 roll currentpoint gsave newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll currentpoint gsave newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat 0 _rise translate _hs 1 scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { dup 1000 div /_fScl exch ddef % selectfont } def /Tl { pop 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { /_rise exch ddef currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { 100 div /_hs exch ddef iTm } def /TA { pop } def /Tq { pop } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { exch pop _fScl mul neg 0 rmoveto } def /TK { 2 npop } def /T* { _leading aload pop neg Td } def /T*- { _leading aload pop Td } def /T- { _ax neg 0 rmoveto _hyphen Tx } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ _fScl 1000 mul selectfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 17 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 14 dict def } if Adobe_ColorImage_AI6_Vars begin /channelcount 0 def /sourcecount 0 def /sourcearray 4 array def /plateindex -1 def /XIMask 0 def /XIBinary 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIBuffer null def /XIDataProc null def end /WalkRGBString null def /WalkCMYKString null def /StuffRGBIntoGrayString null def /RGBToGrayImageProc null def /StuffCMYKIntoGrayString null def /CMYKToGrayImageProc null def /ColorImageCompositeEmulator null def /SeparateCMYKImageProc null def /FourEqual null def /TestPlateIndex null def currentdict /_colorimage known not { /colorimage where { /colorimage get /_colorimage exch def } { /_colorimage null def } ifelse } if /_currenttransfer systemdict /currenttransfer get def /colorimage null def /XI null def /WalkRGBString { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval { } forall 5 index exec 3 index } for 5 { pop } repeat } def /WalkCMYKString { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval { } forall 6 index exec 3 index } for 5 { pop } repeat } def /StuffRGBIntoGrayString { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /RGBToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 3 idiv string dup 3 1 roll /StuffRGBIntoGrayString load exch WalkRGBString end } def /StuffCMYKIntoGrayString { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /CMYKToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 4 idiv string dup 3 1 roll /StuffCMYKIntoGrayString load exch WalkCMYKString end } def /ColorImageCompositeEmulator { pop true eq { Adobe_ColorImage_AI6_Vars /sourcecount get 5 add { pop } repeat } { Adobe_ColorImage_AI6_Vars /channelcount get 1 ne { Adobe_ColorImage_AI6_Vars begin sourcearray 0 3 -1 roll put channelcount 3 eq { /RGBToGrayImageProc } { /CMYKToGrayImageProc } ifelse load end } if image } ifelse } def /SeparateCMYKImageProc { Adobe_ColorImage_AI6_Vars begin sourcecount 0 ne { sourcearray plateindex get exec } { sourcearray 0 get exec dup length 4 idiv string 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop } ifelse end } def /FourEqual { 4 index ne { pop pop pop false } { 4 index ne { pop pop false } { 4 index ne { pop false } { 4 index eq } ifelse } ifelse } ifelse } def /TestPlateIndex { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 FourEqual { /plateindex 0 def } { 0 1 0 0 FourEqual { /plateindex 1 def } { 0 0 1 0 FourEqual { /plateindex 2 def } { 0 0 0 1 FourEqual { /plateindex 3 def } { 0 0 0 0 FourEqual { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /colorimage { Adobe_ColorImage_AI6_Vars begin /channelcount 1 index def /sourcecount 2 index 1 eq { channelcount 1 sub } { 0 } ifelse def 4 sourcecount add index dup 8 eq exch 1 eq or not end { /_colorimage load null ne { _colorimage } { Adobe_ColorImage_AI6_Vars /sourcecount get 7 add { pop } repeat } ifelse } { dup 3 eq TestPlateIndex dup -1 eq exch 5 eq or or { /_colorimage load null eq { ColorImageCompositeEmulator } { dup 1 eq { pop pop image } { Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { gsave 0 _currenttransfer exec 1 _currenttransfer exec eq { 0 _currenttransfer exec 0.5 lt } { 0 _currenttransfer exec 1 _currenttransfer exec gt } ifelse { { pop 0 } } { { pop 1 } } ifelse systemdict /settransfer get exec } if _colorimage Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { grestore } if } ifelse } ifelse } { dup 1 eq { pop pop image } { pop pop Adobe_ColorImage_AI6_Vars begin sourcecount -1 0 { exch sourcearray 3 1 roll put } for /SeparateCMYKImageProc load end systemdict /image get exec } ifelse } ifelse } ifelse } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIMask exch 0 ne def /XIBinary exch 0 ne def pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi } { XIImageWidth XIChannelCount mul } ifelse /XIBuffer exch string def XIBinary { /XIDataProc { currentfile XIBuffer readstring pop } def currentfile 128 string readline pop pop } { /XIDataProc { currentfile XIBuffer readhexstring pop } def } ifelse 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale XIMask { XIImageWidth XIImageHeight false [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load /_lp /null ddef _fc /_lp /imagemask ddef imagemask } { XIImageWidth XIImageHeight XIBitsPerPixel [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load XIChannelCount 1 eq { gsave 0 setgray image grestore } { false XIChannelCount colorimage } ifelse } ifelse grestore end } def end currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 81 dict dup begin put /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_rise 0 def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_wv 0 def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 91 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /sw { dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add } def /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 ddef { 1 index eq { /_cnt _cnt 1 add ddef } if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /sp { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer readline not { stop } if endString eq { cleartomark stop } if } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer readline not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 4 npop 6 1 roll pop 4 1 roll pop pop pop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 3 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end end setpacking Adobe_level2_AI5 /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand 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501.2247 431.3081 499.0086 434.0417 499.0086 c b LB %AI5_EndLayer-- gsave annotatepage grestore showpage Adobe_Illustrator_AI5 /terminate get exec Adobe_ColorImage_AI6 /terminate get exec Adobe_typography_AI5 /terminate get exec Adobe_level2_AI5 /terminate get exec %%EndDocument @endspecial 1673 1079 a Fp(Figure)g(2)0 1259 y Fo(Without)k(loss)h(of) g(generalit)m(y)f(assume)g Fm(p)h Fo(is)f(in)g(the)h(arc)g(from)e Fm(v)2519 1274 y Fl(1)2606 1259 y Fo(to)h Fm(m)p Fo(.)69 b(By)40 b(symmetry)f(there)0 1438 y(exists)33 b(a)h(p)s(oin)m(t)f Fm(p)669 1402 y Fe(0)731 1438 y Fo(on)h(the)g(arc)g(from)f Fm(m)h Fo(to)f Fm(v)1740 1402 y Fe(0)1736 1463 y Fl(2)1814 1438 y Fo(suc)m(h)i(that)f Fm(\032)p Fo(\()p Fm(v)t(;)17 b(p)p Fo(\))26 b(=)j Fm(\032)p Fo(\()p Fm(v)t(;)17 b(p)2903 1402 y Fe(0)2930 1438 y Fo(\).)45 b(See)34 b(Figure)g(3.)45 b(By)0 1617 y(the)34 b(la)m(w)e(of)i(sines,)0 1840 y(\()p Fj(\003)p Fo(\))1323 1773 y(sinh)18 b Fm(\032)p Fo(\()p Fm(m;)f(v)t Fo(\))p 1323 1817 507 4 v 1348 1909 a(sin)g Fm(a)p Fo(\()p Fm(v)t(pm)p Fo(\))1870 1840 y(=)1987 1773 y(sinh)g Fm(\032)p Fo(\()p Fm(v)t(;)g(p)p Fo(\))p 1987 1817 470 4 v 2197 1909 a(1)658 2963 y @beginspecial 212 @llx 477 @lly 372 @urx 683 @ury 699 @rwi @setspecial %%BeginDocument: fig3.eps %AI5_FileFormat 2.0 %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Macintosh_ColorPic %AI5_ArtSize: 612 792 %AI5_RulerUnits: 0 %AI5_ArtFlags: 1 0 0 1 0 0 1 1 0 %AI5_TargetResolution: 800 %AI5_NumLayers: 1 %AI5_OpenToView: 102 684 2 794 557 58 1 1 3 40 %AI5_OpenViewLayers: 7 userdict /Adobe_level2_AI5 23 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { 5 packedarray } bind def /setcustomcolor { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def } if /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? level2? { gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore add add add 4 eq } { 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and } ifelse put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking currentpacking true setpacking userdict /Adobe_typography_AI5 54 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def pop pop findfont _wv type /arraytype eq { _wv makeblendedfont } if dup length 2 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { /Tx { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { /Tx { dup currentpoint 4 2 roll gsave tr _psf grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave trj _pjsf grestore 3 1 roll moveto tr jsp } ddef } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { /Tx { dup currentpoint 4 2 roll currentpoint gsave newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll currentpoint gsave newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat 0 _rise translate _hs 1 scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { dup 1000 div /_fScl exch ddef % selectfont } def /Tl { pop 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { /_rise exch ddef currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { 100 div /_hs exch ddef iTm } def /TA { pop } def /Tq { pop } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { exch pop _fScl mul neg 0 rmoveto } def /TK { 2 npop } def /T* { _leading aload pop neg Td } def /T*- { _leading aload pop Td } def /T- { _ax neg 0 rmoveto _hyphen Tx } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ _fScl 1000 mul selectfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 17 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 14 dict def } if Adobe_ColorImage_AI6_Vars begin /channelcount 0 def /sourcecount 0 def /sourcearray 4 array def /plateindex -1 def /XIMask 0 def /XIBinary 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIBuffer null def /XIDataProc null def end /WalkRGBString null def /WalkCMYKString null def /StuffRGBIntoGrayString null def /RGBToGrayImageProc null def /StuffCMYKIntoGrayString null def /CMYKToGrayImageProc null def /ColorImageCompositeEmulator null def /SeparateCMYKImageProc null def /FourEqual null def /TestPlateIndex null def currentdict /_colorimage known not { /colorimage where { /colorimage get /_colorimage exch def } { /_colorimage null def } ifelse } if /_currenttransfer systemdict /currenttransfer get def /colorimage null def /XI null def /WalkRGBString { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval { } forall 5 index exec 3 index } for 5 { pop } repeat } def /WalkCMYKString { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval { } forall 6 index exec 3 index } for 5 { pop } repeat } def /StuffRGBIntoGrayString { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /RGBToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 3 idiv string dup 3 1 roll /StuffRGBIntoGrayString load exch WalkRGBString end } def /StuffCMYKIntoGrayString { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /CMYKToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 4 idiv string dup 3 1 roll /StuffCMYKIntoGrayString load exch WalkCMYKString end } def /ColorImageCompositeEmulator { pop true eq { Adobe_ColorImage_AI6_Vars /sourcecount get 5 add { pop } repeat } { Adobe_ColorImage_AI6_Vars /channelcount get 1 ne { Adobe_ColorImage_AI6_Vars begin sourcearray 0 3 -1 roll put channelcount 3 eq { /RGBToGrayImageProc } { /CMYKToGrayImageProc } ifelse load end } if image } ifelse } def /SeparateCMYKImageProc { Adobe_ColorImage_AI6_Vars begin sourcecount 0 ne { sourcearray plateindex get exec } { sourcearray 0 get exec dup length 4 idiv string 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop } ifelse end } def /FourEqual { 4 index ne { pop pop pop false } { 4 index ne { pop pop false } { 4 index ne { pop false } { 4 index eq } ifelse } ifelse } ifelse } def /TestPlateIndex { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 FourEqual { /plateindex 0 def } { 0 1 0 0 FourEqual { /plateindex 1 def } { 0 0 1 0 FourEqual { /plateindex 2 def } { 0 0 0 1 FourEqual { /plateindex 3 def } { 0 0 0 0 FourEqual { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /colorimage { Adobe_ColorImage_AI6_Vars begin /channelcount 1 index def /sourcecount 2 index 1 eq { channelcount 1 sub } { 0 } ifelse def 4 sourcecount add index dup 8 eq exch 1 eq or not end { /_colorimage load null ne { _colorimage } { Adobe_ColorImage_AI6_Vars /sourcecount get 7 add { pop } repeat } ifelse } { dup 3 eq TestPlateIndex dup -1 eq exch 5 eq or or { /_colorimage load null eq { ColorImageCompositeEmulator } { dup 1 eq { pop pop image } { Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { gsave 0 _currenttransfer exec 1 _currenttransfer exec eq { 0 _currenttransfer exec 0.5 lt } { 0 _currenttransfer exec 1 _currenttransfer exec gt } ifelse { { pop 0 } } { { pop 1 } } ifelse systemdict /settransfer get exec } if _colorimage Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { grestore } if } ifelse } ifelse } { dup 1 eq { pop pop image } { pop pop Adobe_ColorImage_AI6_Vars begin sourcecount -1 0 { exch sourcearray 3 1 roll put } for /SeparateCMYKImageProc load end systemdict /image get exec } ifelse } ifelse } ifelse } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIMask exch 0 ne def /XIBinary exch 0 ne def pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi } { XIImageWidth XIChannelCount mul } ifelse /XIBuffer exch string def XIBinary { /XIDataProc { currentfile XIBuffer readstring pop } def currentfile 128 string readline pop pop } { /XIDataProc { currentfile XIBuffer readhexstring pop } def } ifelse 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale XIMask { XIImageWidth XIImageHeight false [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load /_lp /null ddef _fc /_lp /imagemask ddef imagemask } { XIImageWidth XIImageHeight XIBitsPerPixel [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load XIChannelCount 1 eq { gsave 0 setgray image grestore } { false XIChannelCount colorimage } ifelse } ifelse grestore end } def end currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 81 dict dup begin put /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_rise 0 def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_wv 0 def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 91 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /sw { dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add } def /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 ddef { 1 index eq { /_cnt _cnt 1 add ddef } if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /sp { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer readline not { stop } if endString eq { cleartomark stop } if } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer readline not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 4 npop 6 1 roll pop 4 1 roll pop pop pop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 3 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end end setpacking Adobe_level2_AI5 /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron TE %AI3_BeginEncoding: _Times-Italic Times-Italic [/_Times-Italic/Times-Italic 0 0 1 TZ %AI3_EndEncoding TrueType %AI5_Begin_NonPrinting Np 8 Bn %AI5_BeginGradient: (Black & White) (Black & White) 0 2 Bd [ < FFFEFDFCFBFAF9F8F7F6F5F4F3F2F1F0EFEEEDECEBEAE9E8E7E6E5E4E3E2E1E0DFDEDDDCDBDAD9D8 D7D6D5D4D3D2D1D0CFCECDCCCBCAC9C8C7C6C5C4C3C2C1C0BFBEBDBCBBBAB9B8B7B6B5B4B3B2B1B0 AFAEADACABAAA9A8A7A6A5A4A3A2A1A09F9E9D9C9B9A999897969594939291908F8E8D8C8B8A8988 87868584838281807F7E7D7C7B7A797877767574737271706F6E6D6C6B6A69686766656463626160 5F5E5D5C5B5A595857565554535251504F4E4D4C4B4A494847464544434241403F3E3D3C3B3A3938 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Fo(suc)m(h)h(that)f(the)g(triangles)g Fm(v)1743 5116 y Fl(1)1788 5101 y Fm(v)1836 5116 y Fl(2)1881 5101 y Fm(v)1929 5116 y Fl(3)1974 5101 y Fo(,)h Fm(w)2102 5116 y Fl(1)2147 5101 y Fm(w)2218 5116 y Fl(2)2263 5101 y Fm(w)2334 5116 y Fl(3)2407 5101 y Fo(ha)m(v)m(e)g(in)m(terior)f (angles)g(at)f Fm(v)3437 5116 y Fl(1)3482 5101 y Fm(;)17 b(v)3575 5116 y Fl(2)3620 5101 y Fm(;)g(w)3736 5116 y Fl(1)0 5280 y Fo(and)34 b Fm(w)265 5295 y Fl(2)343 5280 y Fo(equal)f(to)g Fm(\013=)p Fo(2.)p eop %%Page: 5 5 5 4 bop 864 -139 a Fh(CIR)n(CLE)37 b(P)-7 b(A)n(CKING)38 b(IN)f(THE)g(HYPERBOLIC)h(PLANE)822 b(5)965 720 y @beginspecial 65 @llx 487 @lly 556 @urx 678 @ury 1850 @rwi @setspecial %%BeginDocument: fig4.eps %AI5_FileFormat 2.0 %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Macintosh_ColorPic %AI5_ArtSize: 612 792 %AI5_RulerUnits: 0 %AI5_ArtFlags: 1 0 0 1 0 0 1 1 0 %AI5_TargetResolution: 800 %AI5_NumLayers: 2 %AI5_OpenToView: -126 828 1 794 557 58 0 1 3 40 %AI5_OpenViewLayers: 77 userdict /Adobe_level2_AI5 23 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { 5 packedarray } bind def /setcustomcolor { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def } if /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? level2? { gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore add add add 4 eq } { 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and } ifelse put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking currentpacking true setpacking userdict /Adobe_typography_AI5 54 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def pop pop findfont _wv type /arraytype eq { _wv makeblendedfont } if dup length 2 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { /Tx { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { /Tx { dup currentpoint 4 2 roll gsave tr _psf grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave trj _pjsf grestore 3 1 roll moveto tr jsp } ddef } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { /Tx { dup currentpoint 4 2 roll currentpoint gsave newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll currentpoint gsave newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat 0 _rise translate _hs 1 scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { dup 1000 div /_fScl exch ddef % selectfont } def /Tl { pop 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { /_rise exch ddef currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { 100 div /_hs exch ddef iTm } def /TA { pop } def /Tq { pop } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { exch pop _fScl mul neg 0 rmoveto } def /TK { 2 npop } def /T* { _leading aload pop neg Td } def /T*- { _leading aload pop Td } def /T- { _ax neg 0 rmoveto _hyphen Tx } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ _fScl 1000 mul selectfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 17 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 14 dict def } if Adobe_ColorImage_AI6_Vars begin /channelcount 0 def /sourcecount 0 def /sourcearray 4 array def /plateindex -1 def /XIMask 0 def /XIBinary 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIBuffer null def /XIDataProc null def end /WalkRGBString null def /WalkCMYKString null def /StuffRGBIntoGrayString null def /RGBToGrayImageProc null def /StuffCMYKIntoGrayString null def /CMYKToGrayImageProc null def /ColorImageCompositeEmulator null def /SeparateCMYKImageProc null def /FourEqual null def /TestPlateIndex null def currentdict /_colorimage known not { /colorimage where { /colorimage get /_colorimage exch def } { /_colorimage null def } ifelse } if /_currenttransfer systemdict /currenttransfer get def /colorimage null def /XI null def /WalkRGBString { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval { } forall 5 index exec 3 index } for 5 { pop } repeat } def /WalkCMYKString { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval { } forall 6 index exec 3 index } for 5 { pop } repeat } def /StuffRGBIntoGrayString { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /RGBToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 3 idiv string dup 3 1 roll /StuffRGBIntoGrayString load exch WalkRGBString end } def /StuffCMYKIntoGrayString { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /CMYKToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 4 idiv string dup 3 1 roll /StuffCMYKIntoGrayString load exch WalkCMYKString end } def /ColorImageCompositeEmulator { pop true eq { Adobe_ColorImage_AI6_Vars /sourcecount get 5 add { pop } repeat } { Adobe_ColorImage_AI6_Vars /channelcount get 1 ne { Adobe_ColorImage_AI6_Vars begin sourcearray 0 3 -1 roll put channelcount 3 eq { /RGBToGrayImageProc } { /CMYKToGrayImageProc } ifelse load end } if image } ifelse } def /SeparateCMYKImageProc { Adobe_ColorImage_AI6_Vars begin sourcecount 0 ne { sourcearray plateindex get exec } { sourcearray 0 get exec dup length 4 idiv string 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop } ifelse end } def /FourEqual { 4 index ne { pop pop pop false } { 4 index ne { pop pop false } { 4 index ne { pop false } { 4 index eq } ifelse } ifelse } ifelse } def /TestPlateIndex { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 FourEqual { /plateindex 0 def } { 0 1 0 0 FourEqual { /plateindex 1 def } { 0 0 1 0 FourEqual { /plateindex 2 def } { 0 0 0 1 FourEqual { /plateindex 3 def } { 0 0 0 0 FourEqual { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /colorimage { Adobe_ColorImage_AI6_Vars begin /channelcount 1 index def /sourcecount 2 index 1 eq { channelcount 1 sub } { 0 } ifelse def 4 sourcecount add index dup 8 eq exch 1 eq or not end { /_colorimage load null ne { _colorimage } { Adobe_ColorImage_AI6_Vars /sourcecount get 7 add { pop } repeat } ifelse } { dup 3 eq TestPlateIndex dup -1 eq exch 5 eq or or { /_colorimage load null eq { ColorImageCompositeEmulator } { dup 1 eq { pop pop image } { Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { gsave 0 _currenttransfer exec 1 _currenttransfer exec eq { 0 _currenttransfer exec 0.5 lt } { 0 _currenttransfer exec 1 _currenttransfer exec gt } ifelse { { pop 0 } } { { pop 1 } } ifelse systemdict /settransfer get exec } if _colorimage Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { grestore } if } ifelse } ifelse } { dup 1 eq { pop pop image } { pop pop Adobe_ColorImage_AI6_Vars begin sourcecount -1 0 { exch sourcearray 3 1 roll put } for /SeparateCMYKImageProc load end systemdict /image get exec } ifelse } ifelse } ifelse } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIMask exch 0 ne def /XIBinary exch 0 ne def pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi } { XIImageWidth XIChannelCount mul } ifelse /XIBuffer exch string def XIBinary { /XIDataProc { currentfile XIBuffer readstring pop } def currentfile 128 string readline pop pop } { /XIDataProc { currentfile XIBuffer readhexstring pop } def } ifelse 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale XIMask { XIImageWidth XIImageHeight false [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load /_lp /null ddef _fc /_lp /imagemask ddef imagemask } { XIImageWidth XIImageHeight XIBitsPerPixel [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load XIChannelCount 1 eq { gsave 0 setgray image grestore } { false XIChannelCount colorimage } ifelse } ifelse grestore end } def end currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 81 dict dup begin put /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_rise 0 def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_wv 0 def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 91 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /sw { dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add } def /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 ddef { 1 index eq { /_cnt _cnt 1 add ddef } if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /sp { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer readline not { stop } if endString eq { cleartomark stop } if } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer readline not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 4 npop 6 1 roll pop 4 1 roll pop pop pop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 3 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end end setpacking Adobe_level2_AI5 /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash 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/_Times-Italic 21 Tf (f) Tx (\r) TX TO 0 To 1 0 0 1 437 607 0 Tp TP 0 Tr (i) Tx (\r) TX TO 0 To 1 0 0 1 209 654 0 Tp TP 0 Tr (f) Tx (\r) TX TO 0 To 1 0 0 1 202 519 0 Tp TP 0 Tr (v) Tx (\r) TX TO 0 To 1 0 0 1 212 511 0 Tp TP 0 Tr /_Times-Roman 18 Tf (3) Tx (\r) TX TO LB %AI5_EndLayer-- gsave annotatepage grestore showpage Adobe_Illustrator_AI5 /terminate get exec Adobe_ColorImage_AI6 /terminate get exec Adobe_typography_AI5 /terminate get exec Adobe_level2_AI5 /terminate get exec %%EndDocument @endspecial 1673 899 a Fp(Figure)37 b(4)120 1079 y Fo(Supp)s(ose)f (for)g(a)f(con)m(tradiction)g(that)g(the)h(triangles)f Fm(v)2207 1094 y Fl(1)2252 1079 y Fm(v)2300 1094 y Fl(2)2345 1079 y Fm(v)2393 1094 y Fl(3)2473 1079 y Fo(and)h Fm(w)2740 1094 y Fl(1)2785 1079 y Fm(w)2856 1094 y Fl(2)2901 1079 y Fm(w)2972 1094 y Fl(3)3052 1079 y Fo(in)m(tersect)h(in)e(their)0 1258 y(in)m(teriors.)57 b(See)38 b(Figure)f(4.)57 b(Since)38 b(In)m(t)c Fm(e)1548 1273 y Fl(1)1618 1258 y Fj(\\)25 b Fo(In)m(t)34 b 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Fo(in)m(tersects)h Fm(m)3396 1580 y Fe(0)3424 1617 y Fm(w)3495 1632 y Fl(3)3586 1617 y Fo(then)0 1796 y(without)36 b(loss)g(of)h(generalit)m(y)f Fm(v)1194 1811 y Fl(1)1239 1796 y Fm(v)1287 1811 y Fl(2)1356 1796 y Fj(\\)25 b Fm(w)1518 1811 y Fl(1)1563 1796 y Fm(w)1634 1811 y Fl(3)1711 1796 y Fj(6)p Fo(=)34 b Fj(;)p Fo(.)53 b(Let)37 b Fj(f)p Fm(i)p Fj(g)32 b Fo(=)i Fm(v)2460 1811 y Fl(1)2505 1796 y Fm(v)2553 1811 y Fl(2)2622 1796 y Fj(\\)25 b Fm(w)2784 1811 y Fl(1)2829 1796 y Fm(w)2900 1811 y Fl(3)2945 1796 y Fo(.)53 b(Then)37 b Fm(\032)p Fo(\()p Fm(w)3451 1811 y Fl(1)3496 1796 y Fm(;)17 b(e)3587 1811 y Fl(1)3631 1796 y Fo(\))33 b Fj(\024)0 1975 y Fm(\032)p Fo(\()p Fm(w)162 1990 y Fl(1)206 1975 y Fm(;)17 b(i)p Fo(\))37 b Fm(<)g(\032)p Fo(\()p Fm(w)637 1990 y Fl(1)682 1975 y Fm(;)17 b(w)798 1990 y Fl(3)842 1975 y Fo(\))37 b Fm(<)g(`)p Fo(.)61 b(This)39 b(con)m(tradicts)h(the)f(previous)g (lemma.)59 b(So)39 b Fm(v)3063 1990 y Fl(1)3108 1975 y Fm(v)3156 1990 y Fl(2)3227 1975 y Fj(\\)26 b Fm(m)3406 1939 y Fe(0)3435 1975 y Fm(w)3506 1990 y Fl(3)3588 1975 y Fo(=)38 b Fj(;)p Fo(.)0 2155 y(So)31 b(either)g Fm(v)464 2170 y Fl(1)509 2155 y Fm(v)557 2170 y Fl(3)633 2155 y Fo(or)g Fm(v)801 2170 y Fl(2)846 2155 y Fm(v)894 2170 y Fl(3)969 2155 y Fo(in)m(tersects)h Fm(m)1495 2118 y Fe(0)1524 2155 y Fm(w)1595 2170 y Fl(3)1640 2155 y Fo(.)43 b(Without)31 b(loss)g(of)g(generalit)m(y)f Fm(v)2913 2170 y Fl(1)2958 2155 y Fm(v)3006 2170 y Fl(3)3082 2155 y Fo(in)m(tersects)i Fm(m)3608 2118 y Fe(0)3636 2155 y Fm(w)3707 2170 y Fl(3)3752 2155 y Fo(.)0 2334 y(Let)k Fj(f)p Fm(i)p Fj(g)31 b Fo(=)g Fm(v)502 2349 y Fl(1)547 2334 y Fm(v)595 2349 y Fl(3)664 2334 y Fj(\\)24 b Fm(m)841 2298 y Fe(0)869 2334 y Fm(w)940 2349 y Fl(3)985 2334 y Fo(.)50 b(Then)36 b Fm(\032)p Fo(\()p Fm(v)1464 2349 y Fl(1)1509 2334 y Fm(;)17 b(e)1600 2349 y Fl(2)1644 2334 y Fo(\))31 b Fj(\024)g Fm(\032)p Fo(\()p Fm(v)1961 2349 y Fl(1)2006 2334 y Fm(;)17 b(i)p Fo(\))23 b(+)h Fm(\032)p Fo(\()p Fm(i;)17 b(m)2505 2298 y Fe(0)2532 2334 y Fo(\))31 b Fm(<)g(\032)p Fo(\()p Fm(v)2849 2349 y Fl(1)2894 2334 y Fm(;)17 b(v)2987 2349 y Fl(3)3031 2334 y Fo(\))23 b(+)h Fm(\032)p Fo(\()p Fm(w)3356 2349 y Fl(3)3400 2334 y Fm(;)17 b(m)3532 2298 y Fe(0)3560 2334 y Fo(\))31 b(=)g Fm(`)0 2513 y Fo(con)m(tradicting)38 b(the)h(previous)f(lemma.)56 b(Th)m(us)39 b Fm(v)1868 2528 y Fl(1)1913 2513 y Fm(v)1961 2528 y Fl(2)2006 2513 y Fm(v)2054 2528 y Fl(3)2137 2513 y Fo(and)g Fm(w)2407 2528 y Fl(1)2452 2513 y Fm(w)2523 2528 y Fl(2)2568 2513 y Fm(w)2639 2528 y Fl(3)2722 2513 y Fo(do)f(not)g(in)m(tersect)h(in)f (their)0 2693 y(in)m(teriors.)120 2872 y(F)-8 b(or)28 b(eac)m(h)j(edge)f Fm(e)g Fo(of)f Fm(f)40 b Fo(form)29 b(the)g(triangle)g Fm(v)1821 2887 y Fl(1)1866 2872 y Fm(v)1914 2887 y Fl(2)1959 2872 y Fm(v)2007 2887 y Fl(3)2081 2872 y Fo(as)h(ab)s(o)m(v)m(e)f(so)h(that)f(this)g(triangle)g(lies)g (in)h(the)0 3051 y(in)m(terior)k(of)g Fm(f)11 b Fo(.)45 b(Since)35 b(none)f(of)h(these)f(triangles)g(o)m(v)m(erlap)g(and)g(eac) m(h)h(has)f(area)g(=)3154 3012 y Fl(1)p 3154 3028 40 4 v 3154 3086 a(3)3206 3051 y Fo(\()p Fm(\031)25 b Fj(\000)e Fo(3)p Fm(\013)p Fo(\),)34 b(the)0 3231 y(lemma)d(is)i(pro)m(v)m(ed.)0 3469 y Fn(Pr)-5 b(o)g(of)36 b(of)g(The)-5 b(or)g(em)37 b(1.)50 b Fo(If)35 b Fm(G)g Fo(is)f(a)h(graph)g(of)f(one)h(v)m(ertex,)f (then)i Fm(G)e Fo(is)h(spiral)f(and)h(eac)m(h)h(face)f(of)g Fm(G)0 3648 y Fo(is)k(a)f(triangle.)60 b(Let)39 b Fm(n)d Fj(\025)h Fo(1.)60 b(Assume)39 b(for)g(induction)g(that)f(if)h Fm(G)g Fo(is)g(an)f(admissible)h(graph)f(suc)m(h)0 3828 y(that)31 b Fj(j)p Fm(V)22 b Fo(\()p Fm(G)p Fo(\))p Fj(j)27 b(\024)h Fm(n)j Fo(then)g(if)g Fm(G)h Fo(is)e(spiral)h(then)h Fm(G)f Fo(is)g(optimal)e(and)i(if)g Fm(G)h Fo(is)f(optimal)e(then)i (eac)m(h)h(face)0 4007 y(of)h Fm(G)h Fo(is)f(a)g(triangle.)120 4186 y(Let)f Fm(O)j Fo(b)s(e)e(an)g(optimal)d(graph)j(and)f(let)h Fm(S)38 b Fo(b)s(e)32 b(a)h(spiral)f(graph)g(suc)m(h)i(that)e Fj(j)p Fm(V)22 b Fo(\()p Fm(O)s Fo(\))p Fj(j)27 b Fo(=)h Fj(j)p Fm(V)22 b Fo(\()p Fm(S)6 b Fo(\))p Fj(j)27 b Fo(=)0 4366 y Fm(n)g Fo(+)g(1.)66 b(De\014ne)42 b Fm(F)715 4381 y Fk(k)805 4366 y Fo(as)e(the)h(n)m(um)m(b)s(er)h(of)e Fm(k)s Fo(-gon)h(faces)h(of)f Fm(O)s Fo(.)65 b(Then)2698 4291 y Ff(P)2803 4316 y Fe(1)2803 4396 y Fk(k)r Fe(\000)p Fl(3)2971 4366 y Fm(k)s(F)3090 4381 y Fk(k)3179 4366 y Fo(=)40 b(2\()p Fj(j)p Fm(E)6 b Fo(\()p Fm(O)s Fo(\))p Fj(j)25 b(\000)0 4545 y(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))p Fj(j)p Fo(\))21 b(+)j Fj(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))p Fj(j)p Fo(.)49 b(Com)m(bined)35 b(with)h(the)g (Euler)g(c)m(haracteristic)g(form)m(ula)f Fm(V)46 b Fj(\000)24 b Fm(E)29 b Fo(+)c Fm(F)45 b Fo(=)33 b(1,)0 4724 y(this)g(giv)m(es:)796 5036 y(3)p Fj(j)p Fm(V)21 b Fo(\()p Fm(O)s Fo(\))p Fj(j)g(\000)i Fo(3)f Fj(\000)1431 4895 y Ff(\022)1504 5036 y Fj(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))20 b(+)1980 4911 y Fe(1)1948 4941 y Ff(X)1947 5155 y Fk(k)r Fl(=4)2092 5036 y Fo(\()p Fm(k)25 b Fj(\000)e Fo(3\))p Fm(F)2461 5051 y Fk(k)2509 4895 y Ff(\023)0 5036 y Fo(\(1\))1022 5280 y(=)29 b Fj(j)p Fm(E)6 b Fo(\()p Fm(O)s Fo(\))p Fj(j)25 b(\025)j(j)p Fm(E)6 b Fo(\()p Fm(S)g Fo(\))p Fj(j)26 b Fo(=)i(3)p Fj(j)p Fm(V)22 b Fo(\()p Fm(S)6 b Fo(\))p Fj(j)21 b(\000)i Fo(3)f Fj(\000)g(j)p Fm(E)6 b Fo(\()p Fm(@)g(S)g Fo(\))p Fj(j)31 b Fm(:)p eop %%Page: 6 6 6 5 bop 0 -139 a Fh(6)1526 b(LEWIS)37 b(BO)n(WEN)0 100 y Fo(The)j(last)e(equalit)m(y)g(holds)i(b)s(ecause)h(all)d(faces)i(of)g Fm(S)45 b Fo(are)39 b(triangles.)61 b(Th)m(us)41 b Fj(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))p Fj(j)35 b(\024)j(j)p Fm(E)6 b Fo(\()p Fm(@)g(S)g Fo(\))p Fj(j)0 279 y Fo(with)31 b(equalit)m(y)g(if)g(and)i(only)e(if)g Fm(S)38 b Fo(is)31 b(optimal)f(and)i(ev)m(ery)g(face)g(of)g Fm(O)i Fo(is)e(a)f(triangle.) 43 b(Assume)31 b(for)h(a)0 458 y(con)m(tradiction)h(that)g Fj(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))p Fj(j)25 b Fm(<)j Fj(j)p Fm(E)6 b Fo(\()p Fm(@)g(S)g Fo(\))p Fj(j)p Fo(.)120 638 y(Let)30 b Fm(O)374 601 y Fe(0)431 638 y Fo(b)s(e)g(the)g(admissible)g(graph)g(with)f Fm(V)23 b Fo(\()p Fm(O)1912 601 y Fe(0)1939 638 y Fo(\))k(=)i Fm(V)22 b Fo(\()p Fm(O)s Fo(\))15 b Fj(\000)g Fm(V)22 b Fo(\()p Fm(@)6 b(O)s Fo(\).)42 b(If)30 b Fm(O)2996 601 y Fe(0)3054 638 y Fo(is)f(empt)m(y)g(then)i(b)m(y)0 817 y(\(1\))26 b(the)h(theorem)f(is)g(true.)42 b(So)27 b(assume)g Fm(O)1589 781 y Fe(0)1643 817 y Fo(is)f(not)h(empt)m(y)-8 b(.)40 b(Order)28 b(the)f(v)m(ertices)g(of)g Fm(S)6 b Fo(,)27 b(\()p Fm(v)3356 832 y Fl(1)3401 817 y Fm(;)17 b(:)g(:)g(:)30 b(;)17 b(v)3687 832 y Fk(n)3741 817 y Fo(\))0 996 y(as)33 b(in)f(the)h(de\014nition)g(of)f(spiral.)44 b(Let)33 b Fm(S)1518 960 y Fe(0)1578 996 y Fo(b)s(e)g(the)f(admissible) g(graph)h(with)f Fm(V)22 b Fo(\()p Fm(S)3059 960 y Fe(0)3087 996 y Fo(\))27 b(=)i Fj(f)p Fm(v)3357 1011 y Fl(1)3401 996 y Fm(;)17 b(:)g(:)g(:)31 b(;)17 b(v)3688 1011 y Fk(s)3730 996 y Fj(g)0 1176 y Fo(and)34 b Fm(s)27 b Fo(=)h Fj(j)p Fm(V)22 b Fo(\()p Fm(O)599 1139 y Fe(0)627 1176 y Fo(\))p Fj(j)p Fo(.)43 b(Then)34 b Fm(S)1092 1139 y Fe(0)1153 1176 y Fo(is)f(spiral.)44 b(By)32 b(induction,)i Fm(S)2258 1139 y Fe(0)2319 1176 y Fo(is)f(optimal)e(so)j Fj(j)p Fm(E)6 b Fo(\()p Fm(O)3133 1139 y Fe(0)3159 1176 y Fo(\))p Fj(j)27 b(\024)h(j)p Fm(E)6 b Fo(\()p Fm(S)3572 1139 y Fe(0)3599 1176 y Fo(\))p Fj(j)p Fo(.)120 1355 y(By)32 b(Gauss-Bonet,)876 1596 y Fm(A)p Fo(\()p Fm(O)s Fo(\))21 b Fj(\000)i Fm(A)p Fo(\()p Fm(S)6 b Fo(\))26 b(=)1611 1501 y Ff(X)1580 1715 y Fk(v)s Fe(2)p Fk(@)t(O)1803 1596 y Fm(\031)f Fj(\000)e Fm(a)p Fo(\()p Fm(v)t Fo(\))e Fj(\000)2323 1501 y Ff(X)2289 1715 y Fk(w)r Fe(2)p Fk(@)t(S)2518 1596 y Fm(\031)k Fj(\000)e Fm(a)p Fo(\()p Fm(w)s Fo(\))-2905 b(\(2\))859 1781 y Ff(X)828 1995 y Fk(v)s Fe(2)p Fk(@)t(O)1050 1875 y Fm(\031)26 b Fj(\000)c Fm(a)p Fo(\()p Fm(v)t Fo(\))27 b Fj(\024)h(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))p Fj(j)p Fm(\031)23 b Fj(\000)g Fo(\()p Fj(j)p Fm(E)6 b Fo(\()p Fm(O)s Fo(\))p Fj(j)20 b(\000)i(j)p Fm(E)6 b Fo(\()p Fm(O)2756 1834 y Fe(0)2783 1875 y Fo(\))p Fj(j)p Fo(\))p Fm(\013)-2953 b Fo(\(3\))0 2127 y(b)s(ecause)43 b(there)f(are)g(at)f (least)g Fj(j)p Fm(E)6 b Fo(\()p Fm(O)s Fo(\))p Fj(j)26 b(\000)i(j)p Fm(E)6 b Fo(\()p Fm(O)1836 2091 y Fe(0)1862 2127 y Fo(\))p Fj(j)41 b Fo(pairs)h(of)f(adjacen)m(t)i(edges)g(touc)m (hing)f Fm(@)6 b(O)43 b Fo(and)0 2306 y(eac)m(h)34 b(pair)f(con)m (tributes)i(at)d(least)h Fm(\013)h Fo(to)e(the)i(sum.)689 2422 y Ff(X)655 2637 y Fk(w)r Fe(2)p Fk(@)t(S)884 2517 y Fm(\031)26 b Fj(\000)c Fm(a)p Fo(\()p Fm(w)s Fo(\))27 b(=)1532 2422 y Ff(X)1403 2637 y Fk(w)r Fe(2)p Fk(@)t(S)t Fe(\000)p Fk(@)t(S)1779 2617 y Fc(0)1821 2517 y Fm(\031)f Fj(\000)c Fm(a)p Fo(\()p Fm(w)s Fo(\))g(+)2454 2422 y Ff(X)2329 2637 y Fk(w)r Fe(2)p Fk(@)t(S)t Fe(\\)p Fk(@)t(S)2697 2617 y Fc(0)2739 2517 y Fm(\031)j Fj(\000)e Fm(a)p Fo(\()p Fm(w)s Fo(\))-3126 b(\(4\))1298 2775 y(=)28 b Fm(\031)t Fj(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))p Fj(j)19 b(\000)k Fo(\()p Fj(j)p Fm(E)6 b Fo(\()p Fm(S)g Fo(\))p Fj(j)19 b(\000)k(j)p Fm(E)6 b Fo(\()p Fm(S)2588 2734 y Fe(0)2615 2775 y Fo(\))p Fj(j)21 b(\000)i Fo(1\))p Fm(\013)-2956 b Fo(\(5\))1691 2984 y(+)1915 2889 y Ff(X)1791 3104 y Fk(w)r Fe(2)p Fk(@)t(S)t Fe(\\)p Fk(@)t(S)2159 3084 y Fc(0)2200 2984 y Fm(\031)26 b Fj(\000)c Fm(a)p Fo(\()p Fm(w)s Fo(\))0 3235 y(b)s(ecause)36 b Fj(j)p Fm(V)22 b Fo(\()p Fm(@)6 b(S)g Fo(\))22 b Fj(\000)h Fm(V)f Fo(\()p Fm(@)6 b(S)1049 3199 y Fe(0)1076 3235 y Fo(\))p Fj(j)29 b Fo(=)i Fj(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))p Fj(j)p Fo(,)31 b(there)36 b(are)e(exactly)g(\()p Fj(j)p Fm(E)6 b Fo(\()p Fm(S)g Fo(\))p Fj(j)20 b(\000)k(j)p Fm(E)6 b Fo(\()p Fm(S)3115 3199 y Fe(0)3141 3235 y Fo(\))p Fj(j)23 b(\000)g Fo(1\))34 b(pairs)h(of)0 3415 y(adjacen)m(t)41 b(edges)h(radiating)d(from)h(the)g(v)m(ertices)h(in)f Fm(V)22 b Fo(\()p Fm(@)6 b(S)g Fo(\))26 b Fj(\000)h Fm(V)22 b Fo(\()p Fm(@)6 b(S)2683 3379 y Fe(0)2710 3415 y Fo(\))40 b(\(since)h(if)f Fm(a)p Fo(\()p Fm(v)3317 3430 y Fl(1)3362 3415 y Fo(\))f(=)g Fm(\013)i Fo(for)0 3594 y Fm(v)48 3609 y Fk(i)109 3594 y Fj(2)28 b Fm(@)6 b(S)36 b Fo(then)c Fm(i)c Fo(=)g Fm(n)j Fo(and)g(the)g(edge)h(connecting)g Fm(v)1971 3609 y Fk(n)p Fe(\000)p Fl(1)2158 3594 y Fo(to)e Fm(v)2325 3609 y Fk(m)2432 3594 y Fo(is)h(not)g(coun)m(ted)h(t)m (wice\),)f(and)g(eac)m(h)0 3773 y(pair)i(con)m(tributes)h(exactly)f Fm(\013)g Fo(to)g(the)g(sum)g(since)h(ev)m(ery)f(face)h(of)g Fm(S)39 b Fo(is)33 b(a)g(triangle.)0 3984 y(\(6\))463 3889 y Ff(X)338 4104 y Fk(w)r Fe(2)p Fk(@)t(S)t Fe(\\)p Fk(@)t(S)706 4084 y Fc(0)748 3984 y Fm(\031)25 b Fj(\000)e Fm(a)p Fo(\()p Fm(w)s Fo(\))k Fj(\025)h Fm(\031)d Fj(\000)e Fo(\(2)p Fm(\031)i Fj(\000)e Fo(2)p Fm(\013)p Fo(\))f(+)g(\()p Fm(\031)j Fj(\000)e Fo(3)p Fm(\013)p Fo(\))2368 3874 y Ff(h)2415 3984 y Fj(j)p Fm(E)6 b Fo(\()p Fm(@)g(S)g Fo(\))p Fj(j)19 b(\000)k(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))p Fj(j)19 b(\000)k Fo(1)3395 3874 y Ff(i)0 4236 y Fo(since)46 b Fm(a)p Fo(\()p Fm(v)396 4251 y Fk(m)471 4236 y Fo(\))h Fj(\024)g Fo(\(2)p Fm(\031)33 b Fj(\000)d Fo(2)p Fm(\013)p Fo(\))45 b(and)g Fm(a)p Fo(\()p Fm(v)1510 4251 y Fk(i)1543 4236 y Fo(\))h Fj(\024)i Fo(3)p Fm(\013)c Fo(for)h Fm(m)j(<)f(i)g(<)h(n)p Fo(,)f(and)e Fj(j)p Fm(V)22 b Fo(\()p Fm(@)6 b(S)g Fo(\))29 b Fj(\\)h Fm(V)22 b Fo(\()p Fm(@)6 b(S)3562 4199 y Fe(0)3589 4236 y Fo(\))p Fj(j)47 b Fo(=)0 4415 y Fj(j)p Fm(E)6 b Fo(\()p Fm(@)g(S)g Fo(\))p Fj(j)20 b(\000)i(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))p Fj(j)p Fo(.)120 4594 y(By)32 b(lemma)f(2,)966 4805 y Fm(A)p Fo(\()p Fm(O)s Fo(\))21 b Fj(\000)i Fm(A)p Fo(\()p Fm(S)6 b Fo(\))27 b(=)1742 4710 y Ff(X)1671 4929 y Fk(f)8 b Fe(2)p Fk(F)j Fl(\()p Fk(O)r Fl(\))1974 4805 y Fm(A)p Fo(\()p Fm(f)g Fo(\))21 b Fj(\000)2374 4710 y Ff(X)2308 4929 y Fk(f)8 b Fe(2)p Fk(F)j Fl(\()p Fk(S)t Fl(\))2601 4805 y Fm(A)p Fo(\()p Fm(f)g Fo(\))-2814 b(\(7\))1566 5071 y Fj(\025)28 b Fo(\()p Fj(j)p Fm(F)14 b Fo(\()p Fm(O)s Fo(\))p Fj(j)20 b(\000)j(j)p Fm(F)14 b Fo(\()p Fm(S)6 b Fo(\))p Fj(j)p Fo(\)\()p Fm(\031)24 b Fj(\000)e Fo(3)p Fm(\013)p Fo(\))1566 5280 y(=)28 b(\()p Fj(j)p Fm(E)6 b Fo(\()p Fm(O)s Fo(\))p Fj(j)20 b(\000)i(j)p Fm(E)6 b Fo(\()p Fm(S)g Fo(\))p Fj(j)p Fo(\)\()p Fm(\031)23 b Fj(\000)g Fo(3)p Fm(\013)p Fo(\))p eop %%Page: 7 7 7 6 bop 864 -139 a Fh(CIR)n(CLE)37 b(P)-7 b(A)n(CKING)38 b(IN)f(THE)g(HYPERBOLIC)h(PLANE)822 b(7)0 100 y Fo(and)41 b(equalit)m(y)f(holds)h(if)g(and)g(only)g(if)g(all)e(faces)j(of)f Fm(O)j Fo(are)d(triangles.)66 b(The)42 b(last)e(equalit)m(y)g(holds)0 279 y(from)33 b(the)g(Euler)g(c)m(haracteristic)h(form)m(ula.)120 466 y(Equations)e(\(2\){\(7\))g(and)i Fm(\013)27 b Fo(=)h(2)p Fm(\031)t(=)-6 b(N)44 b Fo(imply)244 788 y Fj(j)p Fm(E)6 b Fo(\()p Fm(O)s Fo(\))p Fj(j)20 b(\000)i(j)p Fm(E)6 b Fo(\()p Fm(S)g Fo(\))p Fj(j)20 b Fo(+)j Fj(j)p Fm(E)6 b Fo(\()p Fm(S)1271 747 y Fe(0)1297 788 y Fo(\))p Fj(j)22 b(\000)h(j)p Fm(E)6 b Fo(\()p Fm(O)1712 747 y Fe(0)1738 788 y Fo(\))p Fj(j)-1805 b Fo(\(8\))371 998 y Fj(\024)28 b Fm(N)33 b Fj(\000)23 b Fo(6)f Fj(\000)860 857 y Ff(\022)946 930 y Fm(N)33 b Fj(\000)22 b Fo(6)p 946 975 263 4 v 1052 1066 a(2)1220 857 y Ff(\023)1310 998 y Fo(\()p Fj(j)p Fm(E)6 b Fo(\()p Fm(@)g(S)g Fo(\))p Fj(j)19 b(\000)k(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))p Fj(j)p Fo(\))19 b Fj(\000)2318 857 y Ff(\022)2403 930 y Fm(N)33 b Fj(\000)23 b Fo(6)p 2403 975 V 2509 1066 a(2)2677 857 y Ff(\023)2767 998 y Fo(\()p Fj(j)p Fm(E)6 b Fo(\()p Fm(O)s Fo(\))p Fj(j)20 b(\000)j(j)p Fm(E)6 b Fo(\()p Fm(S)g Fo(\))p Fj(j)p Fo(\))0 1320 y(with)33 b(equalit)m(y)g(holding)g(only)h(if)f (all)g(faces)i(of)f Fm(O)i Fo(are)e(triangles.)45 b(Hence)34 b Fj(j)p Fm(E)6 b Fo(\()p Fm(@)g(S)g Fo(\))p Fj(j)20 b(\000)j(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))p Fj(j)25 b(\024)k Fo(2)0 1499 y(with)42 b(equalit)m(y)g(holding)g(only)g(if)h Fj(j)p Fm(E)6 b Fo(\()p Fm(O)s Fo(\))p Fj(j)41 b Fo(=)j Fj(j)p Fm(E)6 b Fo(\()p Fm(S)g Fo(\))p Fj(j)40 b Fo(and)j(all)f(faces)h (of)g Fm(O)j Fo(are)c(triangles.)72 b(This)0 1679 y(con)m(tradicts)34 b(\(1\).)43 b(So)33 b(w)m(e)h(ma)m(y)e(assume)h Fj(j)p Fm(E)6 b Fo(\()p Fm(@)g(S)g Fo(\))p Fj(j)20 b(\000)i(j)p Fm(E)6 b Fo(\()p Fm(@)g(O)s Fo(\))p Fj(j)25 b Fo(=)k(1.)120 1866 y(By)j(\(8\),)0 2188 y(\(9\))478 b(\()p Fj(j)p Fm(E)6 b Fo(\()p Fm(O)s Fo(\))p Fj(j)20 b(\000)i(j)p Fm(E)6 b Fo(\()p Fm(S)g Fo(\))p Fj(j)p Fo(\))1394 2048 y Ff(\022)1477 2121 y Fm(N)33 b Fj(\000)23 b Fo(4)p 1477 2165 V 1583 2257 a(2)1751 2048 y Ff(\023)1852 2188 y Fj(\024)1969 2121 y Fm(N)34 b Fj(\000)22 b Fo(6)p 1969 2165 V 2076 2257 a(2)2266 2188 y(+)h Fj(j)p Fm(E)6 b Fo(\()p Fm(O)2592 2147 y Fe(0)2618 2188 y Fo(\))p Fj(j)22 b(\000)g(j)p Fm(E)6 b Fo(\()p Fm(S)3020 2147 y Fe(0)3047 2188 y Fo(\))p Fj(j)32 b Fm(:)0 2511 y Fo(since)i Fj(j)p Fm(E)6 b Fo(\()p Fm(O)470 2475 y Fe(0)496 2511 y Fo(\))p Fj(j)28 b(\024)g(j)p Fm(E)6 b Fo(\()p Fm(S)910 2475 y Fe(0)936 2511 y Fo(\))p Fj(j)p Fo(,)33 b Fj(j)p Fm(E)6 b Fo(\()p Fm(O)s Fo(\))p Fj(j)25 b Fo(=)j Fj(j)p Fm(E)6 b Fo(\()p Fm(S)g Fo(\))p Fj(j)p Fo(,)31 b(i.e.,)h Fm(S)39 b Fo(is)33 b(optimal.)120 2698 y(By)26 b(\(1\),)i(there)g(exists)g(exactly)f(one)h(non)m (triangular)f(face,)j Fm(f)38 b Fo(of)28 b Fm(O)i Fo(and)e(it)g(is)f(a) h(4-gon.)42 b(Supp)s(ose,)0 2877 y(for)36 b(a)g(con)m(tradiction,)g (that)g(there)g(exists)g(a)g(v)m(ertex)g Fm(v)j Fo(in)d Fm(f)47 b Fo(whic)m(h)36 b(is)g(not)g(in)g Fm(@)6 b(O)s Fo(.)51 b(Let)36 b Fm(k)j Fo(b)s(e)e(the)0 3056 y(n)m(um)m(b)s(er)d(of) f(triangular)g(faces)h(of)g Fm(O)h Fo(con)m(taining)f Fm(v)t Fo(.)44 b(Since)34 b Fm(v)j Fo(is)c(in)g(the)h(in)m(terior)f(of) g Fm(O)s Fo(,)0 3379 y(\(10\))1105 b Fm(f)11 b Fo(\()p Fm(v)t Fo(\))26 b(=)i(2)p Fm(\031)e Fj(\000)c Fm(k)s(\013)28 b Fo(=)g(\()p Fm(N)33 b Fj(\000)23 b Fm(k)s Fo(\))p Fm(\013)0 3701 y Fo(where)34 b Fm(f)11 b Fo(\()p Fm(v)t Fo(\))27 b(=)33 b(the)h(in)m(terior)f(angle)g(of)h Fm(f)43 b Fo(at)33 b Fm(v)t Fo(.)44 b(But)33 b(this)g(implies)1331 4599 y @beginspecial 210 @llx 508 @lly 429 @urx 649 @ury 1118 @rwi @setspecial %%BeginDocument: fig5.eps %AI5_FileFormat 2.0 %AI3_ColorUsage: Black&White %AI3_TemplateBox: 306 396 306 396 %AI3_TileBox: 30 31 582 761 %AI3_DocumentPreview: Macintosh_ColorPic %AI5_ArtSize: 612 792 %AI5_RulerUnits: 0 %AI5_ArtFlags: 1 0 0 1 0 0 1 1 0 %AI5_TargetResolution: 800 %AI5_NumLayers: 2 %AI5_OpenToView: 114 684 2 794 557 58 0 1 3 40 %AI5_OpenViewLayers: 77 userdict /Adobe_level2_AI5 23 dict dup begin put /packedarray where not { userdict begin /packedarray { array astore readonly } bind def /setpacking /pop load def /currentpacking false def end 0 } if pop userdict /defaultpacking currentpacking put true setpacking /initialize { Adobe_level2_AI5 begin } bind def /terminate { currentdict Adobe_level2_AI5 eq { end } if } bind def mark /setcustomcolor where not { /findcmykcustomcolor { 5 packedarray } bind def /setcustomcolor { exch aload pop pop 4 { 4 index mul 4 1 roll } repeat 5 -1 roll pop setcmykcolor } def } if /gt38? mark {version cvr cvx exec} stopped {cleartomark true} {38 gt exch pop} ifelse def userdict /deviceDPI 72 0 matrix defaultmatrix dtransform dup mul exch dup mul add sqrt put userdict /level2? systemdict /languagelevel known dup { pop systemdict /languagelevel get 2 ge } if put /level2ScreenFreq { begin 60 HalftoneType 1 eq { pop Frequency } if HalftoneType 2 eq { pop GrayFrequency } if HalftoneType 5 eq { pop Default level2ScreenFreq } if end } bind def userdict /currentScreenFreq level2? {currenthalftone level2ScreenFreq} {currentscreen pop pop} ifelse put level2? not { /setcmykcolor where not { /setcmykcolor { exch .11 mul add exch .59 mul add exch .3 mul add 1 exch sub setgray } def } if /currentcmykcolor where not { /currentcmykcolor { 0 0 0 1 currentgray sub } def } if /setoverprint where not { /setoverprint /pop load def } if /selectfont where not { /selectfont { exch findfont exch dup type /arraytype eq { makefont } { scalefont } ifelse setfont } bind def } if /cshow where not { /cshow { [ 0 0 5 -1 roll aload pop ] cvx bind forall } bind def } if } if cleartomark /anyColor? { add add add 0 ne } bind def /testColor { gsave setcmykcolor currentcmykcolor grestore } bind def /testCMYKColorThrough { testColor anyColor? } bind def userdict /composite? level2? { gsave 1 1 1 1 setcmykcolor currentcmykcolor grestore add add add 4 eq } { 1 0 0 0 testCMYKColorThrough 0 1 0 0 testCMYKColorThrough 0 0 1 0 testCMYKColorThrough 0 0 0 1 testCMYKColorThrough and and and } ifelse put composite? not { userdict begin gsave /cyan? 1 0 0 0 testCMYKColorThrough def /magenta? 0 1 0 0 testCMYKColorThrough def /yellow? 0 0 1 0 testCMYKColorThrough def /black? 0 0 0 1 testCMYKColorThrough def grestore /isCMYKSep? cyan? magenta? yellow? black? or or or def /customColor? isCMYKSep? not def end } if end defaultpacking setpacking currentpacking true setpacking userdict /Adobe_typography_AI5 54 dict dup begin put /initialize { begin begin Adobe_typography_AI5 begin Adobe_typography_AI5 { dup xcheck { bind } if pop pop } forall end end end Adobe_typography_AI5 begin } def /terminate { currentdict Adobe_typography_AI5 eq { end } if } def /modifyEncoding { /_tempEncode exch ddef /_pntr 0 ddef { counttomark -1 roll dup type dup /marktype eq { pop pop exit } { /nametype eq { _tempEncode /_pntr dup load dup 3 1 roll 1 add ddef 3 -1 roll put } { /_pntr exch ddef } ifelse } ifelse } loop _tempEncode } def /TE { StandardEncoding 256 array copy modifyEncoding /_nativeEncoding exch def } def % /TZ { dup type /arraytype eq { /_wv exch def } { /_wv 0 def } ifelse /_useNativeEncoding exch def pop pop findfont _wv type /arraytype eq { _wv makeblendedfont } if dup length 2 add dict begin mark exch { 1 index /FID ne { def } if cleartomark mark } forall pop /FontName exch def counttomark 0 eq { 1 _useNativeEncoding eq { /Encoding _nativeEncoding def } if cleartomark } { /Encoding load 256 array copy modifyEncoding /Encoding exch def } ifelse FontName currentdict end definefont pop } def /tr { _ax _ay 3 2 roll } def /trj { _cx _cy _sp _ax _ay 6 5 roll } def /a0 { /Tx { dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss } ddef /Tj { dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss } ddef } def /a1 { /Tx { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll tr _psf newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave dup currentpoint 3 2 roll trj _pjsf newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /e0 { /Tx { tr _psf } ddef /Tj { trj _pjsf } ddef } def /e1 { /Tx { dup currentpoint 4 2 roll gsave tr _psf grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll gsave trj _pjsf grestore 3 1 roll moveto tr jsp } ddef } def /i0 { /Tx { tr sp } ddef /Tj { trj jsp } ddef } def /i1 { W N } def /o0 { /Tx { tr sw rmoveto } ddef /Tj { trj swj rmoveto } ddef } def /r0 { /Tx { tr _ctm _pss } ddef /Tj { trj _ctm _pjss } ddef } def /r1 { /Tx { dup currentpoint 4 2 roll currentpoint gsave newpath moveto tr _ctm _pss grestore 3 1 roll moveto tr sp } ddef /Tj { dup currentpoint 4 2 roll currentpoint gsave newpath moveto trj _ctm _pjss grestore 3 1 roll moveto tr jsp } ddef } def /To { pop _ctm currentmatrix pop } def /TO { iTe _ctm setmatrix newpath } def /Tp { pop _tm astore pop _ctm setmatrix _tDict begin /W { } def /h { } def } def /TP { end iTm 0 0 moveto } def /Tr { _render 3 le { currentpoint newpath moveto } if dup 8 eq { pop 0 } { dup 9 eq { pop 1 } if } ifelse dup /_render exch ddef _renderStart exch get load exec } def /iTm { _ctm setmatrix _tm concat 0 _rise translate _hs 1 scale } def /Tm { _tm astore pop iTm 0 0 moveto } def /Td { _mtx translate _tm _tm concatmatrix pop iTm 0 0 moveto } def /iTe { _render -1 eq { } { _renderEnd _render get dup null ne { load exec } { pop } ifelse } ifelse /_render -1 ddef } def /Ta { pop } def /Tf { dup 1000 div /_fScl exch ddef % selectfont } def /Tl { pop 0 exch _leading astore pop } def /Tt { pop } def /TW { 3 npop } def /Tw { /_cx exch ddef } def /TC { 3 npop } def /Tc { /_ax exch ddef } def /Ts { /_rise exch ddef currentpoint iTm moveto } def /Ti { 3 npop } def /Tz { 100 div /_hs exch ddef iTm } def /TA { pop } def /Tq { pop } def /Th { pop pop pop pop pop } def /TX { pop } def /Tk { exch pop _fScl mul neg 0 rmoveto } def /TK { 2 npop } def /T* { _leading aload pop neg Td } def /T*- { _leading aload pop Td } def /T- { _ax neg 0 rmoveto _hyphen Tx } def /T+ { } def /TR { _ctm currentmatrix pop _tm astore pop iTm 0 0 moveto } def /TS { currentfont 3 1 roll /_Symbol_ _fScl 1000 mul selectfont 0 eq { Tx } { Tj } ifelse setfont } def /Xb { pop pop } def /Tb /Xb load def /Xe { pop pop pop pop } def /Te /Xe load def /XB { } def /TB /XB load def currentdict readonly pop end setpacking userdict /Adobe_ColorImage_AI6 known not { userdict /Adobe_ColorImage_AI6 17 dict put } if userdict /Adobe_ColorImage_AI6 get begin /initialize { Adobe_ColorImage_AI6 begin Adobe_ColorImage_AI6 { dup type /arraytype eq { dup xcheck { bind } if } if pop pop } forall } def /terminate { end } def currentdict /Adobe_ColorImage_AI6_Vars known not { /Adobe_ColorImage_AI6_Vars 14 dict def } if Adobe_ColorImage_AI6_Vars begin /channelcount 0 def /sourcecount 0 def /sourcearray 4 array def /plateindex -1 def /XIMask 0 def /XIBinary 0 def /XIChannelCount 0 def /XIBitsPerPixel 0 def /XIImageHeight 0 def /XIImageWidth 0 def /XIImageMatrix null def /XIBuffer null def /XIDataProc null def end /WalkRGBString null def /WalkCMYKString null def /StuffRGBIntoGrayString null def /RGBToGrayImageProc null def /StuffCMYKIntoGrayString null def /CMYKToGrayImageProc null def /ColorImageCompositeEmulator null def /SeparateCMYKImageProc null def /FourEqual null def /TestPlateIndex null def currentdict /_colorimage known not { /colorimage where { /colorimage get /_colorimage exch def } { /_colorimage null def } ifelse } if /_currenttransfer systemdict /currenttransfer get def /colorimage null def /XI null def /WalkRGBString { 0 3 index dup length 1 sub 0 3 3 -1 roll { 3 getinterval { } forall 5 index exec 3 index } for 5 { pop } repeat } def /WalkCMYKString { 0 3 index dup length 1 sub 0 4 3 -1 roll { 4 getinterval { } forall 6 index exec 3 index } for 5 { pop } repeat } def /StuffRGBIntoGrayString { .11 mul exch .59 mul add exch .3 mul add cvi 3 copy put pop 1 add } def /RGBToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 3 idiv string dup 3 1 roll /StuffRGBIntoGrayString load exch WalkRGBString end } def /StuffCMYKIntoGrayString { exch .11 mul add exch .59 mul add exch .3 mul add dup 255 gt { pop 255 } if 255 exch sub cvi 3 copy put pop 1 add } def /CMYKToGrayImageProc { Adobe_ColorImage_AI6_Vars begin sourcearray 0 get exec dup length 4 idiv string dup 3 1 roll /StuffCMYKIntoGrayString load exch WalkCMYKString end } def /ColorImageCompositeEmulator { pop true eq { Adobe_ColorImage_AI6_Vars /sourcecount get 5 add { pop } repeat } { Adobe_ColorImage_AI6_Vars /channelcount get 1 ne { Adobe_ColorImage_AI6_Vars begin sourcearray 0 3 -1 roll put channelcount 3 eq { /RGBToGrayImageProc } { /CMYKToGrayImageProc } ifelse load end } if image } ifelse } def /SeparateCMYKImageProc { Adobe_ColorImage_AI6_Vars begin sourcecount 0 ne { sourcearray plateindex get exec } { sourcearray 0 get exec dup length 4 idiv string 0 2 index plateindex 4 2 index length 1 sub { get 255 exch sub 3 copy put pop 1 add 2 index } for pop pop exch pop } ifelse end } def /FourEqual { 4 index ne { pop pop pop false } { 4 index ne { pop pop false } { 4 index ne { pop false } { 4 index eq } ifelse } ifelse } ifelse } def /TestPlateIndex { Adobe_ColorImage_AI6_Vars begin /plateindex -1 def /setcmykcolor where { pop gsave 1 0 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 1 0 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 1 0 setcmykcolor systemdict /currentgray get exec 1 exch sub 0 0 0 1 setcmykcolor systemdict /currentgray get exec 1 exch sub grestore 1 0 0 0 FourEqual { /plateindex 0 def } { 0 1 0 0 FourEqual { /plateindex 1 def } { 0 0 1 0 FourEqual { /plateindex 2 def } { 0 0 0 1 FourEqual { /plateindex 3 def } { 0 0 0 0 FourEqual { /plateindex 5 def } if } ifelse } ifelse } ifelse } ifelse pop pop pop pop } if plateindex end } def /colorimage { Adobe_ColorImage_AI6_Vars begin /channelcount 1 index def /sourcecount 2 index 1 eq { channelcount 1 sub } { 0 } ifelse def 4 sourcecount add index dup 8 eq exch 1 eq or not end { /_colorimage load null ne { _colorimage } { Adobe_ColorImage_AI6_Vars /sourcecount get 7 add { pop } repeat } ifelse } { dup 3 eq TestPlateIndex dup -1 eq exch 5 eq or or { /_colorimage load null eq { ColorImageCompositeEmulator } { dup 1 eq { pop pop image } { Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { gsave 0 _currenttransfer exec 1 _currenttransfer exec eq { 0 _currenttransfer exec 0.5 lt } { 0 _currenttransfer exec 1 _currenttransfer exec gt } ifelse { { pop 0 } } { { pop 1 } } ifelse systemdict /settransfer get exec } if _colorimage Adobe_ColorImage_AI6_Vars /plateindex get 5 eq { grestore } if } ifelse } ifelse } { dup 1 eq { pop pop image } { pop pop Adobe_ColorImage_AI6_Vars begin sourcecount -1 0 { exch sourcearray 3 1 roll put } for /SeparateCMYKImageProc load end systemdict /image get exec } ifelse } ifelse } ifelse } def /XI { Adobe_ColorImage_AI6_Vars begin gsave /XIMask exch 0 ne def /XIBinary exch 0 ne def pop pop /XIChannelCount exch def /XIBitsPerPixel exch def /XIImageHeight exch def /XIImageWidth exch def pop pop pop pop /XIImageMatrix exch def XIBitsPerPixel 1 eq { XIImageWidth 8 div ceiling cvi } { XIImageWidth XIChannelCount mul } ifelse /XIBuffer exch string def XIBinary { /XIDataProc { currentfile XIBuffer readstring pop } def currentfile 128 string readline pop pop } { /XIDataProc { currentfile XIBuffer readhexstring pop } def } ifelse 0 0 moveto XIImageMatrix concat XIImageWidth XIImageHeight scale XIMask { XIImageWidth XIImageHeight false [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load /_lp /null ddef _fc /_lp /imagemask ddef imagemask } { XIImageWidth XIImageHeight XIBitsPerPixel [ XIImageWidth 0 0 XIImageHeight neg 0 0 ] /XIDataProc load XIChannelCount 1 eq { gsave 0 setgray image grestore } { false XIChannelCount colorimage } ifelse } ifelse grestore end } def end currentpacking true setpacking userdict /Adobe_Illustrator_AI5_vars 81 dict dup begin put /_eo false def /_lp /none def /_pf { } def /_ps { } def /_psf { } def /_pss { } def /_pjsf { } def /_pjss { } def /_pola 0 def /_doClip 0 def /cf currentflat def /_tm matrix def /_renderStart [ /e0 /r0 /a0 /o0 /e1 /r1 /a1 /i0 ] def /_renderEnd [ null null null null /i1 /i1 /i1 /i1 ] def /_render -1 def /_rise 0 def /_ax 0 def /_ay 0 def /_cx 0 def /_cy 0 def /_leading [ 0 0 ] def /_ctm matrix def /_mtx matrix def /_sp 16#020 def /_hyphen (-) def /_fScl 0 def /_cnt 0 def /_hs 1 def /_nativeEncoding 0 def /_useNativeEncoding 0 def /_tempEncode 0 def /_pntr 0 def /_tDict 2 dict def /_wv 0 def /Tx { } def /Tj { } def /CRender { } def /_AI3_savepage { } def /_gf null def /_cf 4 array def /_if null def /_of false def /_fc { } def /_gs null def /_cs 4 array def /_is null def /_os false def /_sc { } def /_pd 1 dict def /_ed 15 dict def /_pm matrix def /_fm null def /_fd null def /_fdd null def /_sm null def /_sd null def /_sdd null def /_i null def /discardSave null def /buffer 256 string def /beginString null def /endString null def /endStringLength null def /layerCnt 1 def /layerCount 1 def /perCent (%) 0 get def /perCentSeen? false def /newBuff null def /newBuffButFirst null def /newBuffLast null def /clipForward? false def end userdict /Adobe_Illustrator_AI5 known not { userdict /Adobe_Illustrator_AI5 91 dict put } if userdict /Adobe_Illustrator_AI5 get begin /initialize { Adobe_Illustrator_AI5 dup begin Adobe_Illustrator_AI5_vars begin discardDict { bind pop pop } forall dup /nc get begin { dup xcheck 1 index type /operatortype ne and { bind } if pop pop } forall end newpath } def /terminate { end end } def /_ null def /ddef { Adobe_Illustrator_AI5_vars 3 1 roll put } def /xput { dup load dup length exch maxlength eq { dup dup load dup length 2 mul dict copy def } if load begin def end } def /npop { { pop } repeat } def /sw { dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add } def /swj { dup 4 1 roll dup length exch stringwidth exch 5 -1 roll 3 index mul add 4 1 roll 3 1 roll mul add 6 2 roll /_cnt 0 ddef { 1 index eq { /_cnt _cnt 1 add ddef } if } forall pop exch _cnt mul exch _cnt mul 2 index add 4 1 roll 2 index add 4 1 roll pop pop } def /ss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put pop gsave false charpath currentpoint 4 index setmatrix stroke grestore moveto 2 copy rmoveto } exch cshow 3 npop } def /jss { 4 1 roll { 2 npop (0) exch 2 copy 0 exch put gsave _sp eq { exch 6 index 6 index 6 index 5 -1 roll widthshow currentpoint } { false charpath currentpoint 4 index setmatrix stroke } ifelse grestore moveto 2 copy rmoveto } exch cshow 6 npop } def /sp { { 2 npop (0) exch 2 copy 0 exch put pop false charpath 2 copy rmoveto } exch cshow 2 npop } def /jsp { { 2 npop (0) exch 2 copy 0 exch put _sp eq { exch 5 index 5 index 5 index 5 -1 roll widthshow } { false charpath } ifelse 2 copy rmoveto } exch cshow 5 npop } def /pl { transform 0.25 sub round 0.25 add exch 0.25 sub round 0.25 add exch itransform } def /setstrokeadjust where { pop true setstrokeadjust /c { curveto } def /C /c load def /v { currentpoint 6 2 roll curveto } def /V /v load def /y { 2 copy curveto } def /Y /y load def /l { lineto } def /L /l load def /m { moveto } def } { /c { pl curveto } def /C /c load def /v { currentpoint 6 2 roll pl curveto } def /V /v load def /y { pl 2 copy curveto } def /Y /y load def /l { pl lineto } def /L /l load def /m { pl moveto } def } ifelse /d { setdash } def /cf { } def /i { dup 0 eq { pop cf } if setflat } def /j { setlinejoin } def /J { setlinecap } def /M { setmiterlimit } def /w { setlinewidth } def /XR { 0 ne /_eo exch ddef } def /H { } def /h { closepath } def /N { _pola 0 eq { _doClip 1 eq { _eo {eoclip} {clip} ifelse /_doClip 0 ddef } if newpath } { /CRender { N } ddef } ifelse } def /n { N } def /F { _pola 0 eq { _doClip 1 eq { gsave _pf grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _fc /_doClip 0 ddef } { _pf } ifelse } { /CRender { F } ddef } ifelse } def /f { closepath F } def /S { _pola 0 eq { _doClip 1 eq { gsave _ps grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { _ps } ifelse } { /CRender { S } ddef } ifelse } def /s { closepath S } def /B { _pola 0 eq { _doClip 1 eq gsave F grestore { gsave S grestore _eo {eoclip} {clip} ifelse newpath /_lp /none ddef _sc /_doClip 0 ddef } { S } ifelse } { /CRender { B } ddef } ifelse } def /b { closepath B } def /W { /_doClip 1 ddef } def /* { count 0 ne { dup type /stringtype eq { pop } if } if newpath } def /u { } def /U { } def /q { _pola 0 eq { gsave } if } def /Q { _pola 0 eq { grestore } if } def /*u { _pola 1 add /_pola exch ddef } def /*U { _pola 1 sub /_pola exch ddef _pola 0 eq { CRender } if } def /D { pop } def /*w { } def /*W { } def /` { /_i save ddef clipForward? { nulldevice } if 6 1 roll 4 npop concat pop userdict begin /showpage { } def 0 setgray 0 setlinecap 1 setlinewidth 0 setlinejoin 10 setmiterlimit [] 0 setdash /setstrokeadjust where {pop false setstrokeadjust} if newpath 0 setgray false setoverprint } def /~ { end _i restore } def /O { 0 ne /_of exch ddef /_lp /none ddef } def /R { 0 ne /_os exch ddef /_lp /none ddef } def /g { /_gf exch ddef /_fc { _lp /fill ne { _of setoverprint _gf setgray /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /G { /_gs exch ddef /_sc { _lp /stroke ne { _os setoverprint _gs setgray /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /k { _cf astore pop /_fc { _lp /fill ne { _of setoverprint _cf aload pop setcmykcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /K { _cs astore pop /_sc { _lp /stroke ne { _os setoverprint _cs aload pop setcmykcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /x { /_gf exch ddef findcmykcustomcolor /_if exch ddef /_fc { _lp /fill ne { _of setoverprint _if _gf 1 exch sub setcustomcolor /_lp /fill ddef } if } ddef /_pf { _fc _eo {eofill} {fill} ifelse } ddef /_psf { _fc ashow } ddef /_pjsf { _fc awidthshow } ddef /_lp /none ddef } def /X { /_gs exch ddef findcmykcustomcolor /_is exch ddef /_sc { _lp /stroke ne { _os setoverprint _is _gs 1 exch sub setcustomcolor /_lp /stroke ddef } if } ddef /_ps { _sc stroke } ddef /_pss { _sc ss } ddef /_pjss { _sc jss } ddef /_lp /none ddef } def /A { pop } def /annotatepage { userdict /annotatepage 2 copy known {get exec} {pop pop} ifelse } def /XT { pop pop } def /discard { save /discardSave exch store discardDict begin /endString exch store gt38? { 2 add } if load stopped pop end discardSave restore } bind def userdict /discardDict 7 dict dup begin put /pre38Initialize { /endStringLength endString length store /newBuff buffer 0 endStringLength getinterval store /newBuffButFirst newBuff 1 endStringLength 1 sub getinterval store /newBuffLast newBuff endStringLength 1 sub 1 getinterval store } def /shiftBuffer { newBuff 0 newBuffButFirst putinterval newBuffLast 0 currentfile read not { stop } if put } def 0 { pre38Initialize mark currentfile newBuff readstring exch pop { { newBuff endString eq { cleartomark stop } if shiftBuffer } loop } { stop } ifelse } def 1 { pre38Initialize /beginString exch store mark currentfile newBuff readstring exch pop { { newBuff beginString eq { /layerCount dup load 1 add store } { newBuff endString eq { /layerCount dup load 1 sub store layerCount 0 eq { cleartomark stop } if } if } ifelse shiftBuffer } loop } if } def 2 { mark { currentfile buffer readline not { stop } if endString eq { cleartomark stop } if } loop } def 3 { /beginString exch store /layerCnt 1 store mark { currentfile buffer readline not { stop } if dup beginString eq { pop /layerCnt dup load 1 add store } { endString eq { layerCnt 1 eq { cleartomark stop } { /layerCnt dup load 1 sub store } ifelse } if } ifelse } loop } def end userdict /clipRenderOff 15 dict dup begin put { /n /N /s /S /f /F /b /B } { { _doClip 1 eq { /_doClip 0 ddef _eo {eoclip} {clip} ifelse } if newpath } def } forall /Tr /pop load def /Bb {} def /BB /pop load def /Bg {12 npop} def /Bm {6 npop} def /Bc /Bm load def /Bh {4 npop} def end /Lb { 4 npop 6 1 roll pop 4 1 roll pop pop pop 0 eq { 0 eq { (%AI5_BeginLayer) 1 (%AI5_EndLayer--) discard } { /clipForward? true def /Tx /pop load def /Tj /pop load def currentdict end clipRenderOff begin begin } ifelse } { 0 eq { save /discardSave exch store } if } ifelse } bind def /LB { discardSave dup null ne { restore } { pop clipForward? { currentdict end end begin /clipForward? false ddef } if } ifelse } bind def /Pb { pop pop 0 (%AI5_EndPalette) discard } bind def /Np { 0 (%AI5_End_NonPrinting--) discard } bind def /Ln /pop load def /Ap /pop load def /Ar { 72 exch div 0 dtransform dup mul exch dup mul add sqrt dup 1 lt { pop 1 } if setflat } def /Mb { q } def /Md { } def /MB { Q } def /nc 3 dict def nc begin /setgray { pop } bind def /setcmykcolor { 4 npop } bind def /setcustomcolor { 2 npop } bind def currentdict readonly pop end end setpacking Adobe_level2_AI5 /initialize get exec Adobe_Illustrator_AI5_vars Adobe_Illustrator_AI5 Adobe_typography_AI5 /initialize get exec Adobe_ColorImage_AI6 /initialize get exec Adobe_Illustrator_AI5 /initialize get exec [ 39/quotesingle 96/grave 128/Adieresis/Aring/Ccedilla/Eacute/Ntilde/Odieresis /Udieresis/aacute/agrave/acircumflex/adieresis/atilde/aring/ccedilla/eacute /egrave/ecircumflex/edieresis/iacute/igrave/icircumflex/idieresis/ntilde /oacute/ograve/ocircumflex/odieresis/otilde/uacute/ugrave/ucircumflex /udieresis/dagger/degree/cent/sterling/section/bullet/paragraph/germandbls /registered/copyright/trademark/acute/dieresis/.notdef/AE/Oslash /.notdef/plusminus/.notdef/.notdef/yen/mu/.notdef/.notdef /.notdef/.notdef/.notdef/ordfeminine/ordmasculine/.notdef/ae/oslash /questiondown/exclamdown/logicalnot/.notdef/florin/.notdef/.notdef /guillemotleft/guillemotright/ellipsis/.notdef/Agrave/Atilde/Otilde/OE/oe /endash/emdash/quotedblleft/quotedblright/quoteleft/quoteright/divide /.notdef/ydieresis/Ydieresis/fraction/currency/guilsinglleft/guilsinglright /fi/fl/daggerdbl/periodcentered/quotesinglbase/quotedblbase/perthousand /Acircumflex/Ecircumflex/Aacute/Edieresis/Egrave/Iacute/Icircumflex /Idieresis/Igrave/Oacute/Ocircumflex/.notdef/Ograve/Uacute/Ucircumflex /Ugrave/dotlessi/circumflex/tilde/macron/breve/dotaccent/ring/cedilla /hungarumlaut/ogonek/caron TE %AI3_BeginEncoding: _Times-Italic Times-Italic 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