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2056 y Fv(\016)2619 2031 y Ft(\))p Fo(;)17 b Fp(k)k(\001)h(k)2872 2046 y Fm(\003)2925 1935 y Fh(\021)2974 2031 y Ft(,)34 b(but)f(the)h(con-)324 2164 y(stan)m(ts)f(for)f(the)h(estimates)g(of)f(the)h(norms)f(are)h(un) m(b)s(ounded)h(as)e Fp(j)p Ft(\003)p Fp(j)g Ft(increases.)324 2285 y(F)-8 b(or)32 b(giv)m(en)g(\003)821 2300 y Fm(1)888 2285 y Fp(\022)c Ft(\003)1061 2300 y Fm(2)1128 2285 y Fp(2)g(F)42 b Ft(and)33 b Fo(N)38 b Fp(2)28 b Fs(N)k Ft(w)m(e)i(ha)m(v)m(e)g(a)e(map)807 2525 y Fo(\031)862 2540 y Fm(\003)911 2549 y Fi(1)972 2525 y Fp(\016)22 b(L)1113 2484 y Fv(N)1113 2554 y(F)1168 2531 y Fi(\003)1209 2546 y(2)1247 2554 y Fj(\016)p Fv(T)1333 2531 y Fi(\003)1374 2546 y(2)1409 2531 y Fk(;\017)1485 2525 y Fp(\016)g Fo(\031)1612 2540 y Fm(\003)1661 2549 y Fi(2)1728 2525 y Ft(:)27 b(\()p Fp(H)1904 2540 y Fv(#)1950 2525 y Fo(;)17 b Fp(k)22 b(\001)g(k)2166 2540 y Fv(#)2212 2525 y Ft(\))27 b Fp(!)h Ft(\()p Fp(H)2527 2540 y Fm(\003)2576 2549 y Fi(1)2610 2540 y Fv(;#)2676 2525 y Fo(;)17 b Fp(k)22 b(\001)g(k)2892 2540 y Fm(\003)2941 2549 y Fi(1)2975 2540 y Fv(;#)3041 2525 y Ft(\))358 b(\(6\))324 2700 y(where)35 b Fp(L)676 2664 y Fv(N)676 2733 y(F)731 2711 y Fi(\003)772 2726 y(2)810 2733 y Fj(\016)p Fv(T)896 2711 y Fi(\003)937 2726 y(2)972 2711 y Fk(;\017)1059 2700 y Ft(is)e(the)h(P)m(erron-F)-8 b(rob)s(enius)33 b(op)s(erator)g(for)g(the)h(\014nite-dimensional)324 2820 y(system)g(o)m(v)m(er)f(\003)924 2835 y Fm(2)996 2820 y Ft(\(cf.)g(Section)g(3\))f(with)h(\014xed)g(b)s(oundary)h (conditions)d(\(not)i(included)324 2940 y(in)24 b(the)i(notation\).)40 b(The)26 b(follo)m(wing)d(de\014nition)h(of)h(transfer)h(op)s(erators)f (for)g(the)h(in\014nite)324 3061 y(system)33 b(do)s(es)g(not)g(dep)s (end)g(on)g(the)g(c)m(hoice)g(of)f(the)h(b)s(oundary)g(conditions.)324 3289 y Fs(Theorem)k(1)49 b Fn(F)-7 b(or)34 b Fo(#)p Fn(,)i Fo(\017)f Fn(su\016ciently)h(smal)5 b(l,)34 b Fo(c)2107 3304 y Fm(2)2147 3289 y Fn(,)h Fo(N)46 b Fn(su\016ciently)35 b(big)g(and)f(any)i Ft(\003)3428 3304 y Fm(1)3495 3289 y Fp(2)324 3409 y(F)10 b Fn(:)439 3638 y(1.)49 b(The)34 b(limit)1173 3758 y Fo(\031)1228 3773 y Fm(\003)1277 3782 y Fi(1)1338 3758 y Fp(\016)22 b(L)1479 3717 y Fv(N)1479 3783 y(F)10 b Fj(\016)p Fv(T)1620 3764 y Fk(\017)1682 3706 y Fm(def)1692 3758 y Ft(=)91 b(lim)1807 3826 y Fm(\003)1856 3835 y Fi(2)1890 3826 y Fj(!)p Fl(Z)2011 3807 y Fk(d)2063 3758 y Fo(\031)2118 3773 y Fm(\003)2167 3782 y Fi(1)2228 3758 y Fp(\016)22 b(L)2369 3717 y Fv(N)2369 3786 y(F)2424 3764 y Fi(\003)2465 3779 y(2)2504 3786 y Fj(\016)p Fv(T)2590 3764 y Fi(\003)2631 3779 y(2)2666 3764 y Fk(;\017)2741 3758 y Fp(\016)g Fo(\031)2868 3773 y Fm(\003)2917 3782 y Fi(2)3437 3758 y Ft(\(7\))568 4022 y Fp(2)28 b Fo(L)17 b Ft(\(\()p Fp(H)905 4037 y Fv(#)951 4022 y Fo(;)g Fp(k)22 b(\001)f(k)1166 4037 y Fv(#)1212 4022 y Ft(\))c Fo(;)g Ft(\()o Fp(H)1432 4037 y Fm(\003)1481 4046 y Fi(1)1516 4037 y Fv(;#)1582 4022 y Fo(;)g Fp(k)22 b(\001)f(k)1797 4037 y Fm(\003)1846 4046 y Fi(1)1881 4037 y Fv(;#)1946 4022 y Ft(\)\))31 b Fn(exists)f(and)g(the)g(family)g(of)g(these)g(op)-5 b(er)g(a-)568 4142 y(tors)28 b(is)g(uniformly)g(\(in)g Ft(\003)1516 4157 y Fm(1)1584 4142 y Fn(and)g(also)f(in)i Fo(N)10 b Fn(\))28 b(b)-5 b(ounde)g(d.)42 b(This)28 b(de\014nes)f(op)-5 b(er)g(ators)568 4282 y Fp(L)637 4246 y Fv(N)637 4307 y(F)10 b Fj(\016)p Fv(T)778 4288 y Fk(\017)846 4282 y Fn(on)35 b Ft(\()p Fp(H)1108 4297 y Fv(#)1154 4282 y Fo(;)17 b Fp(k)22 b(\001)f(k)1369 4297 y Fv(#)1415 4282 y Ft(\))35 b Fn(by)1615 4186 y Fh(\020)1665 4282 y Fp(L)1734 4246 y Fv(N)1734 4307 y(F)10 b Fj(\016)p Fv(T)1875 4288 y Fk(\017)1908 4282 y Fo(\036)1966 4186 y Fh(\021)2016 4336 y Fm(\003)2065 4345 y Fi(1)2131 4230 y Fm(def)2142 4282 y Ft(=)38 b Fo(\031)2311 4297 y Fm(\003)2360 4306 y Fi(1)2421 4282 y Fp(\016)22 b(L)2562 4246 y Fv(N)2562 4307 y(F)10 b Fj(\016)p Fv(T)2703 4288 y Fk(\017)2737 4282 y Fo(\036)568 4464 y Fn(F)-7 b(or)48 b(any)i Fo(n)55 b Fp(2)h Fs(N)49 b Fn(we)h(have)f Fp(L)1806 4428 y Fv(n)1806 4489 y(F)10 b Fj(\016)p Fv(T)1947 4470 y Fk(\017)2035 4464 y Ft(:)56 b Fp(H)2202 4479 y Fv(#)2303 4464 y Fp(!)f(H)2542 4479 y Fv(#)2584 4487 y Fk(n)2680 4464 y Fn(with)50 b(suitably)g (chosen)568 4585 y Ft(0)27 b Fo(<)h(#)805 4600 y Fm(1)872 4585 y Fp(\024)g(\001)17 b(\001)g(\001)26 b(\024)i Fo(#)1283 4600 y Fv(N)1339 4609 y Fi(0)1406 4585 y Ft(=)g Fo(#)1567 4600 y Fv(N)1623 4609 y Fi(0)1658 4600 y Fm(+1)1780 4585 y Ft(=)g Fp(\001)17 b(\001)g(\001)25 b Ft(=)j Fo(#)p Fn(.)568 4747 y(In)h(the)h(c)-5 b(ase)30 b(of)g(\014nite-r)-5 b(ange)29 b(inter)-5 b(action)30 b(we)g(c)-5 b(an)29 b(de\014ne)h(a)g(line)-5 b(ar)29 b(map)h Fp(L)3387 4762 y Fv(F)10 b Fj(\016)p Fv(T)3528 4743 y Fk(\017)568 4867 y Fn(on)34 b Fp(H)i Fn(in)f(the)g(same)f(way)g(.)1918 5251 y Ft(5)p eop %%Page: 6 6 6 5 bop 439 548 a Fn(2.)49 b(Ther)-5 b(e)34 b(is)g(a)h(unique)g (invariant)f(pr)-5 b(ob)g(ability)35 b(me)-5 b(asur)g(e)34 b Fo(\027)g Ft(=)28 b(\()p Fo(\027)2892 563 y Fm(\003)2941 572 y Fi(1)2980 548 y Ft(\))3018 563 y Fm(\003)3067 572 y Fi(1)3101 563 y Fj(2F)3237 548 y Fp(2)g(H)3415 563 y Fv(#)3461 548 y Fn(.)568 722 y(In)34 b Fo(L)17 b Ft(\()p Fp(H)900 737 y Fv(#)946 722 y Fo(;)g Fp(k)22 b(\001)g(k)1162 737 y Fv(#)1207 722 y Ft(\))35 b Fn(the)g(se)-5 b(quenc)g(e)1838 626 y Fh(\020)1888 722 y Fp(L)1957 686 y Fv(N)1957 747 y(F)10 b Fj(\016)p Fv(T)2098 728 y Fk(\017)2131 626 y Fh(\021)2181 776 y Fv(N)d Fj(\025)p Fv(N)2355 785 y Fi(0)2429 722 y Fn(c)-5 b(onver)g(ges)33 b(exp)-5 b(onential)5 b(ly)568 865 y(fast:)1353 1048 y Fh(\015)1353 1097 y(\015)1353 1147 y(\015)q Fp(L)1469 1106 y Fv(N)1469 1172 y(F)10 b Fj(\016)p Fv(T)1610 1153 y Fk(\017)1665 1147 y Fp(\000)23 b Fo(\026)p Ft(\()p Fp(\001)p Ft(\))p Fo(\027)1982 1048 y Fh(\015)1982 1097 y(\015)1982 1147 y(\015)2028 1201 y Fv(L)p Fm(\(\()q Fj(H)2191 1213 y Fk(#)2232 1201 y Fv(;)p Fj(k\001k)2342 1213 y Fk(#)2384 1201 y Fm(\)\))2470 1147 y Fp(\024)29 b Fo(c)2618 1162 y Fm(3)2664 1147 y Ft(~)-56 b Fo(\021)2709 1106 y Fv(N)3437 1147 y Ft(\(8\))568 1349 y Fn(for)34 b(some)g Fo(c)1014 1364 y Fm(3)1081 1349 y Fo(>)28 b Ft(0)35 b Fn(and)f Ft(0)27 b Fo(<)35 b Ft(~)-56 b Fo(\021)31 b(<)d Ft(1)p Fn(.)324 1578 y Ft(F)-8 b(or)41 b(the)h(in)m(v)-5 b(arian)m(t)41 b(measure)h Fo(\027)48 b Ft(w)m(e)43 b(ha)m(v)m(e)h(exp)s(onen)m(tial)d(deca)m(y)i (of)f(correlations)e(for)324 1698 y(spatio-temp)s(oral)29 b(shifts)k(on)f(the)h(system:)324 1818 y(Let)j(\()p Fo(e)585 1833 y Fm(1)625 1818 y Fo(;)17 b(:)g(:)g(:)f(;)h(e)889 1833 y Fv(d)929 1818 y Ft(\))36 b(b)s(e)h(a)f(linearly-indep)s(enden)m (t)f(system)i(of)f(unit)g(v)m(ectors)i(in)d Fs(Z)3303 1782 y Fv(d)3344 1818 y Ft(.)55 b(W)-8 b(e)324 1959 y(de\014ne)25 b(translations)e Fo(\034)1163 1974 y Fv(e)1196 1984 y Fk(i)1227 1959 y Ft(\()p Fo(p)p Ft(\))1379 1906 y Fm(def)1390 1959 y Ft(=)38 b Fo(p)5 b Ft(+)g Fo(e)1684 1974 y Fv(i)1737 1959 y Ft(for)24 b Fo(p)j Fp(2)h Fs(Z)2116 1922 y Fv(d)2181 1959 y Ft(and)d(\()p Fo(\034)2443 1974 y Fv(e)2476 1984 y Fk(i)2506 1959 y Ft(\()p Fo(z)t Ft(\)\))2669 1974 y Fv(p)2737 1906 y Fm(def)2748 1959 y Ft(=)38 b Fo(z)2907 1974 y Fv(\034)2938 1982 y Fk(e)2967 1998 y(i)2999 1974 y Fm(\()p Fv(p)p Fm(\))3118 1959 y Ft(for)23 b Fo(z)33 b Fp(2)28 b Fo(M)10 b Ft(.)324 2079 y(In)32 b(the)h(follo)m(wing)d (theorem)i(w)m(e)h(denote)g(b)m(y)g Fo(\034)44 b Ft(\(acting)31 b(on)i Fo(M)43 b Ft(from)31 b(the)h(righ)m(t\))g(com-)324 2199 y(p)s(ositions)c Fo(\034)39 b Ft(=)27 b Fo(\034)957 2214 y Fm(1)1012 2199 y Fp(\016)15 b Fo(:)i(:)g(:)e Fp(\016)g Fo(\034)1314 2215 y Fv(m)p Fm(\()p Fv(\034)8 b Fm(\))1503 2199 y Ft(and)29 b(b)m(y)i Fo(\033)i Ft(a)28 b(comp)s(osition)f(of)i (spatio-temp)s(oral)d(shifts)324 2320 y(\(on)43 b Fo(M)10 b Ft(\):)65 b Fo(\033)50 b Ft(=)c Fo(\033)1024 2335 y Fm(1)1093 2320 y Fp(\016)29 b Fo(:)17 b(:)g(:)29 b Fp(\016)g Fo(\033)1450 2335 y Fv(m)p Fm(\()p Fv(\033)r Fm(\)+)p Fv(m)p Fm(\()p Fv(\033)r Fm(\))1873 2320 y Ft(with)43 b Fo(\033)2161 2335 y Fv(i)2235 2320 y Fp(2)j(f)p Fo(S;)17 b(\034)2543 2335 y Fv(e)2576 2344 y Fi(1)2615 2320 y Fo(;)g(:)g(:)g(:)f(;)h(\034)2876 2335 y Fv(e)2909 2347 y Fk(d)2949 2320 y Fp(g)p Ft(.)75 b(W)-8 b(e)44 b(denote)324 2440 y(b)m(y)37 b Fo(n)p Ft(\()p Fo(\033)t Ft(\))f(the)h(n)m(um)m(b)s (er)f(of)g(factors)g Fo(S)42 b Ft(and)36 b(b)m(y)h Fo(m)p Ft(\()p Fo(\033)t Ft(\))g(the)f(n)m(um)m(b)s(er)h(of)e(spatial)g (trans-)324 2560 y(lations)h(in)h(this)g(pro)s(duct.)60 b(F)-8 b(or)36 b(a)i(translation-in)m(v)-5 b(arian)m(t)34 b(system,)40 b(i.e.)d Fo(f)3119 2575 y Fv(p)3196 2560 y Ft(=)f Fo(f)48 b Ft(and)324 2681 y Fo(g)371 2696 y Fv(p)410 2681 y Ft(\()p Fs(z)p Ft(\))43 b(=)g Fo(g)745 2708 y Fv(\034)784 2679 y Fg(\000)p Fi(1)776 2720 y Fk(e)805 2736 y(i)866 2708 y Fm(\()p Fv(p)p Fm(\))960 2681 y Ft(\()p Fo(\034)1040 2696 y Fv(e)1073 2706 y Fk(i)1104 2681 y Ft(\()p Fo(z)t Ft(\)\))f(for)f(all)e Fo(p)k Fp(2)g Fs(Z)1880 2645 y Fv(d)1921 2681 y Ft(,)h(the)e(time-shift)d Fo(S)48 b Ft(comm)m(utes)41 b(with)g(the)324 2814 y(translations.)324 3042 y Fs(Theorem)c(2)49 b Fn(F)-7 b(or)35 b Fo(#)p Fn(,)p Fo(\017)i Fn(and)f Fo(c)1483 3057 y Fm(2)1559 3042 y Fn(as)g(in)h(The)-5 b(or)g(em)35 b(1)h(ther)-5 b(e)37 b(is)f(a)g Fo(\024)31 b Fp(2)g Ft(\(0)p Fo(;)17 b Ft(1\))36 b Fn(such)g(that)324 3162 y(for)e(al)5 b(l)35 b(nonempty)f Ft(\003)1135 3177 y Fm(1)1174 3162 y Fo(;)17 b Ft(\003)1286 3177 y Fm(2)1353 3162 y Fp(2)28 b(F)44 b Fn(with)35 b Ft(\003)1843 3177 y Fm(1)1904 3162 y Fp(\\)23 b Ft(\003)2061 3177 y Fm(2)2128 3162 y Ft(=)k Fp(;)35 b Fn(the)g(fol)5 b(lowing)33 b(holds:)439 3391 y(1.)49 b(If)34 b Fo(g)d Fp(2)d(C)6 b Ft(\(\()p Fo(S)1042 3354 y Fm(1)1081 3391 y Ft(\))1119 3354 y Fm(\003)1168 3363 y Fi(1)1207 3391 y Ft(\))35 b Fn(and)f Fo(f)k Fp(2)29 b(C)6 b Ft(\(\()p Fo(S)1850 3354 y Fm(1)1889 3391 y Ft(\))1927 3354 y Fm(\003)1976 3363 y Fi(2)2015 3391 y Ft(\))34 b Fn(then)568 3552 y Fp(j)595 3481 y Fh(R)635 3578 y Fv(M)730 3552 y Fo(\027)6 b(d\026)17 b(g)t(f)32 b Fp(\000)22 b Ft(\()1179 3481 y Fh(R)1219 3578 y Fv(M)1314 3552 y Fo(\027)6 b(d\026)17 b(g)t Ft(\))e(\()1637 3481 y 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2543 y Ft(=)28 b Fp(;)p Fn(.)324 2779 y Ft(Com)m(bining)j(Prop)s(ositions)g(1)h(and)h(2)f(w)m(e)i(ha)m (v)m(e)g(for)e(\014xed)i Fo(c)2531 2794 y Fm(7)2602 2779 y Ft(and)f(\(small\))d Fo(\016)1449 3008 y(F)1526 2967 y Fm(\003)1601 3008 y Fp(\016)22 b Fo(T)1744 2967 y Fm(\003)p Fv(;\017)1861 2912 y Fh(\020)1911 3008 y Fo(A)1984 2967 y Fm(\003)1984 3033 y Fv(\016)2037 2912 y Fh(\021)2114 3008 y Fp(\033)28 b Fo(A)2292 2967 y Fm(\003)2292 3033 y Fv(c)2323 3042 y Fi(7)2358 3033 y Fv(\025\016)3388 3008 y Ft(\(14\))324 3225 y(and)1341 3345 y Fo(F)1418 3304 y Fm(\003)1493 3345 y Fp(\016)22 b Fo(T)1636 3304 y Fm(\003)p Fv(;\017)1753 3249 y Fh(\020)1803 3345 y Fo(@)5 b(A)1932 3304 y Fm(\003)1932 3370 y Fv(\016)1986 3249 y Fh(\021)2058 3345 y Fp(\\)22 b Fo(A)2219 3304 y Fm(\003)2219 3370 y Fv(c)2250 3379 y Fi(7)2285 3370 y Fv(\025\016)2391 3345 y Ft(=)28 b Fp(;)843 b Ft(\(15\))324 3538 y(In)40 b(particular,)g(if)f(w)m(e)i(c)m(ho)s(ose)g Fo(c)1546 3553 y Fm(7)1625 3538 y Fo(>)1754 3499 y Fm(1)p 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Fo(S)2739 4005 y Fm(1)2778 4041 y Ft(\))2816 3989 y Fm(\003)2869 4041 y Ft(,)h Fo(S)2991 4005 y Fv(\017)3052 4041 y Ft(=)e Fo(F)3232 4005 y Fm(\003)3299 4041 y Fp(\016)14 b Fo(T)3434 4005 y Fm(\003)p Fv(;\017)3535 4041 y Ft(,)324 4162 y Fo( )37 b Ft(con)m(tin)m(uous)e(\(the)f(prop)s(osition)e(holds)h(also)g (for)g Fo( )g Fp(2)d Fo(L)2496 4126 y Fj(1)2572 4162 y Ft(\()p Fo(M)10 b Ft(\)\))34 b(and)g Fo(\036)f Ft(con)m(tin)m(uous) 324 4282 y(on)f(the)h(closed)g(p)s(oly)m(ann)m(ulus)g Fo(A)1531 4246 y Fm(\003)1531 4307 y Fv(\016)1562 4316 y Fi(1)1633 4282 y Ft(and)g(analytic)e(in)h(its)g(in)m(terior.)324 4402 y(First)26 b(w)m(e)j(giv)m(e)f(the)g(de\014nition)e(of)i(the)g(P)m (erron-F)-8 b(rob)s(enius)27 b(op)s(erator)g(\(cf.)h(for)f(example)324 4523 y([14)o(]\).)324 4761 y Fs(De\014nition)86 b Ft(Let)41 b Fo(\026)g Ft(b)s(e)g(a)f(measure)i(on)e(a)h(metric)f(space)i Fo(M)51 b Ft(\(with)41 b(the)g(Borel)f Fo(\033)t Ft(-)324 4882 y(algebra\).)64 b(The)41 b(P)m(erron-F)-8 b(rob)s(enius)40 b(op)s(erator)f Fp(L)2215 4897 y Fv(S)2306 4882 y Ft(for)g(a)h (nonsingular)e(measurable)324 5002 y(map)32 b Fo(S)h Ft(:)28 b Fo(M)38 b Fp(!)28 b Fo(M)43 b Ft(is)32 b(de\014ned)i(via)e (the)h(equation)1918 5251 y(8)p eop %%Page: 9 9 9 8 bop 1326 551 a Fh(Z)1372 740 y Fv(M)1467 668 y Fo(d\026)17 b( )25 b Fp(\016)d Fo(S)28 b(\036)g Ft(=)2031 551 y Fh(Z)2077 740 y Fv(M)2173 668 y Fo(d\026)17 b( )j Fp(L)2452 683 y Fv(S)2502 668 y Fo(\036)828 b Ft(\(16\))324 881 y(that,)30 b(for)f(giv)m(en)h Fo(\036)e Fp(2)g Fo(L)1204 845 y Fm(1)1244 881 y Ft(\()p Fo(M)10 b Ft(\),)31 b(m)m(ust)f(hold)f(for)g(all)f Fo( )k Fp(2)c Fo(L)2469 845 y Fj(1)2544 881 y Ft(\()p Fo(M)10 b Ft(\).)43 b(The)31 b(existence)g(and)324 1001 y(uniqueness)38 b(of)e Fp(L)1003 1016 y Fv(S)1053 1001 y Fo(\036)e Fp(2)h Fo(L)1312 965 y Fm(1)1351 1001 y Ft(\()p Fo(M)10 b Ft(\))37 b(is)f(equiv)-5 b(alen)m(t)36 b(b)m(y)i(the)e (Radon-Nik)m(o)s(dym)f(Theorem)324 1122 y(to)41 b(the)h(absolute)f(con) m(tin)m(uit)m(y)h(\(wrt.)f Fo(\026)p Ft(\))g(of)g(the)h(measure)g(asso) s(ciated)g(to)f(the)h(func-)324 1242 y(tional)32 b Fo( )j Fp(7!)830 1171 y Fh(R)870 1268 y Fv(M)965 1242 y Fo(d\026)1075 1206 y Fv(N)1142 1242 y Fo( )27 b Fp(\016)c Fo(S)g(\036)p Ft(,)35 b(i.e.)f(for)f(all)g(measurable)h Fo(A)d Fp(2)g Fo(M)10 b Ft(,)36 b Fo(\026)p Ft(\()p Fo(A)p Ft(\))30 b(=)h(0)j(implies)324 1363 y Fo(\026)p Ft(\()p Fo(S)487 1326 y Fj(\000)p Fm(1)581 1363 y Ft(\()p Fo(A)p Ft(\)\))27 b(=)h(0.)43 b(This)33 b(condition)e(is)h(called)f Fn(nonsingularity)41 b Ft(of)32 b Fo(S)6 b Ft(.)324 1601 y(The)41 b(normalized)e(Leb)s (esgue)i(measure)g Fo(\026)f Ft(on)g Fo(S)2164 1565 y Fm(1)2244 1601 y Ft(is)g(giv)m(en)h(b)m(y)g Fo(d\026)p Ft(\()p Fo(z)t Ft(\))g(=)3173 1562 y Fv(dz)p 3159 1578 102 4 v 3159 1635 a Fm(2)p Fv(\031)r({)3280 1562 y Fm(1)p 3280 1578 36 4 v 3280 1635 a Fv(z)3366 1601 y Ft(\(this)324 1721 y(lifts)29 b(wrt.)i(the)g(map)f Fo(t)e Fp(!)f Fo(e)1335 1685 y Fv({t)1418 1721 y Ft(to)k(the)g(normalized)d(Leb)s(esgue)k (measure)3014 1682 y Fv(dt)p 3006 1698 79 4 v 3006 1756 a Fm(2)p Fv(\031)3125 1721 y Ft(on)f([0)p Fo(;)17 b Ft(2)p Fo(\031)t Ft(\)\))324 1842 y(and)32 b(the)h(pro)s(duct)g(measure)g Fo(\026)1486 1806 y Fv(N)1586 1842 y Ft(on)f(\()p Fo(S)1825 1806 y Fm(1)1864 1842 y Ft(\))1902 1806 y Fv(N)2002 1842 y Ft(is)g(giv)m(en)h(b)m(y)1025 2107 y Fo(d\026)1135 2066 y Fv(N)1201 2107 y Ft(\()p Fs(z)p Ft(\))28 b(=)1559 2040 y Fo(d)p Fs(z)p 1468 2084 282 4 v 1468 2175 a Ft(\(2)p Fo(\031)t({)p Ft(\))1683 2147 y Fv(N)1771 2040 y Ft(1)p 1770 2084 50 4 v 1770 2175 a Fs(z)1858 2055 y Fm(def)1868 2107 y Ft(=)1994 2040 y Fo(dz)2090 2055 y Fm(1)p 1993 2084 139 4 v 1993 2175 a Ft(2)p Fo(\031)t({)2158 2107 y Fp(\001)17 b(\001)g(\001)2301 2040 y Fo(dz)2397 2055 y Fv(N)p 2301 2084 164 4 v 2313 2175 a Ft(2)p Fo(\031)t({)2502 2040 y Ft(1)p 2484 2084 85 4 v 2484 2175 a Fo(z)2529 2190 y Fm(1)2595 2107 y Fp(\001)g(\001)g(\001)2770 2040 y Ft(1)p 2738 2084 113 4 v 2738 2175 a Fo(z)2783 2190 y Fv(N)3388 2107 y Ft(\(17\))324 2386 y Fs(Prop)s(osition)35 b(3)49 b Fn(In)43 b(the)g(ab)-5 b(ove)43 b(setting)g(the)h(Perr)-5 b(on-F)e(r)i(ob)g(enius-Op)g(er)g(ator)41 b(c)-5 b(an)43 b(b)-5 b(e)324 2506 y(written)35 b(in)f(the)h(fol)5 b(lowing)34 b(way:)860 2904 y Fp(L)929 2919 y Fv(S)976 2900 y Fk(\017)1010 2904 y Fo(\036)p Ft(\()p Fo(w)s Ft(\))27 b(=)1347 2787 y Fh(Z)1393 2976 y Fm(\000)1437 2957 y Fk(N)1617 2837 y Fo(d)p Fs(z)p 1526 2881 282 4 v 1526 2973 a Ft(\(2)p Fo(\031)t({)p Ft(\))1741 2944 y Fv(N)1818 2904 y Fo(\036)p Ft(\()p Fo(z)t Ft(\))2051 2796 y Fv(N)2029 2821 y Fh(Y)2018 3006 y Fv(k)r Fm(=1)2163 2758 y Fh( )2446 2837 y Ft(1)p 2239 2881 463 4 v 2239 2973 a Fo(S)2305 2938 y Fv(\017)2299 2998 y(k)2341 2973 y Ft(\()p Fs(z)p Ft(\))22 b Fp(\000)h Fo(w)2659 2988 y Fv(k)2721 2837 y Fo(S)2787 2801 y Fv(\017)2781 2862 y(k)2824 2837 y Ft(\()p Fs(z)p Ft(\))p 2721 2881 229 4 v 2791 2973 a Fo(z)2836 2988 y Fv(k)2959 2758 y Fh(!)3388 2904 y Ft(\(18\))324 3146 y Fn(wher)-5 b(e)34 b Ft(\000)28 b(=)f(\000)852 3161 y Fm(+)933 3146 y Fp([)c Ft(\000)1083 3161 y Fj(\000)1177 3146 y Fn(is)34 b(the)h(p)-5 b(ositively-oriente)g(d)34 b(b)-5 b(oundary)35 b(of)f Fo(A)2851 3161 y Fv(\016)2889 3146 y Fn(.)324 3478 y Fu(4)161 b(F)-13 b(urther)68 b(Remarks)h(on)f(the)h(In\014nite-Dimen/-) 566 3661 y(sional)55 b(System)324 3880 y Ft(The)46 b(subspace)g(of)f (functions)g(that)f(dep)s(end)i(only)f(on)f(\014nitely)h(man)m(y)f(v)-5 b(ariables)44 b(is)324 4000 y(dense)35 b(in)f(\()p Fp(C)6 b Ft(\()p Fo(M)k Ft(\))p Fo(;)17 b Fp(k)22 b(\001)g(k)1199 4015 y Fj(1)1274 4000 y Ft(\))o(,)35 b(and)f(eac)m(h)h(suc)m(h)h (function)d(\(sa)m(y)i(dep)s(ending)f(on)g Fs(z)3255 4015 y Fm(\003)3342 4000 y Ft(only\))324 4121 y(can)k(b)s(e)g (uniformly)d(appro)m(ximated)i(b)m(y)i(\(the)f(restriction)f(of)7 b(\))37 b(functions)g(in)g Fp(H)q Ft(\()p Fo(A)3443 4085 y Fm(\003)3443 4145 y Fv(\016)3497 4121 y Ft(\).)324 4241 y(The)23 b(dual)f(space)h(of)f Fp(C)6 b Ft(\()p Fo(M)k Ft(\))23 b(is)f Fo(r)s(ca)p Ft(\()p Fo(M)10 b Ft(\))23 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y Fp(2)f(H)1713 4747 y Fv(#)1760 4732 y Ft(.)57 b(W)-8 b(e)37 b(remark)g(that)g(not)g(ev)m(ery)i Fo(\036)c Fp(2)h(H)3289 4747 y Fv(#)3372 4732 y Ft(with)324 4853 y(real-v)-5 b(alued)31 b(marginals)f Fo(\036)1323 4868 y Fm(\003)1376 4876 y Fj(j)p Fm(\()p Fv(S)1470 4857 y Fi(1)1504 4876 y Fm(\))1531 4857 y Fi(\003)1614 4853 y Ft(corresp)s(onds)35 b(to)d(an)h(elemen)m(t)g(in)f Fo(r)s(ca)p Ft(\()p Fo(M)10 b Ft(\))33 b(b)s(ecause)1918 5251 y(9)p eop %%Page: 10 10 10 9 bop 324 548 a Ft(its)31 b(v)-5 b(ariation)30 b(migh)m(t)h(not)h(b) s(e)g(b)s(ounded)h(as)1969 477 y Fh(R)2008 573 y Fm(\003)2078 548 y Fo(d\026)2188 512 y Fm(\003)2241 548 y Fp(j)p Fo(\036)2327 563 y Fm(\003)2379 548 y Fp(j)f Ft(migh)m(t)f(b)s(e)h(un)m(b)s(ounded)h (in)f(\003.)324 668 y(So)g(w)m(e)i(de\014ne)g(for)e Fo(\036)27 b Fp(2)h(H)1318 909 y(k)p Fo(\036)p Fp(k)1476 924 y Fv(v)r(ar)1616 857 y Fm(def)1626 909 y Ft(=)74 b(lim)1741 976 y Fm(\003)p Fj(!)p Fl(Z)1911 958 y Fk(d)1963 792 y Fh(Z)2009 981 y Fm(\()p Fv(S)2083 962 y Fi(1)2117 981 y Fm(\))2144 962 y Fi(\003)2211 909 y Fo(d\026)2321 868 y Fm(\003)2374 909 y Fp(j)p Fo(\036)2460 924 y Fm(\003)2512 909 y Fp(j)p Fo(:)821 b Ft(\(19\))324 1161 y(W)-8 b(e)32 b(set)g Fp(H)727 1125 y Fv(bv)826 1108 y Fm(def)836 1161 y Ft(=)39 b Fp(f)p Fo(\036)27 b Fp(2)h(H)h Ft(:)f Fp(k)p Fo(\036)p Fp(k)1506 1176 y Fv(v)r(ar)1645 1161 y Fo(<)f Fp(1g)k Ft(and)h Fp(H)2203 1125 y Fv(bv)2202 1185 y(#)2302 1108 y Fm(def)2312 1161 y Ft(=)39 b Fp(H)2512 1125 y Fv(bv)2603 1161 y Fp(\\)20 b(H)2773 1176 y Fv(#)2819 1161 y Ft(.)43 b(In)32 b(particular)e(all)324 1281 y(real-analytic)g(and)j(non-negativ)m(e)g Fo(\036)28 b Fp(2)h(H)q Ft(,)k(i.e.)f Fo(\036)2191 1296 y Fm(\003)2244 1305 y Fj(j)p Fm(\()p Fv(S)2338 1286 y Fi(1)2372 1305 y Fm(\))2399 1286 y Fi(\003)2478 1281 y Fp(\025)c Ft(0)33 b(for)g(all)d(\003)e Fp(2)h(F)10 b Ft(,)33 b(b)s(elong)324 1402 y(to)f(this)g(space.)324 1522 y(W)-8 b(e)33 b(can)g(view)g(ev)m (ery)h Fo(\036)28 b Fp(2)g(H)1418 1486 y Fv(bv)1522 1522 y Ft(as)k(an)h(elemen)m(t)f(of)h Fo(r)s(ca)p Ft(\()p Fo(M)10 b Ft(\):)44 b(F)-8 b(or)31 b Fo(g)g Fp(2)e(C)6 b Ft(\()p Fo(M)k Ft(\))33 b(the)g(net)324 1662 y(\()p Fo(g)409 1677 y Fm(\003)462 1662 y Ft(\))500 1677 y Fm(\003)p Fj(2F)689 1662 y Ft(giv)m(en)g(b)m(y)g Fo(g)1126 1677 y Fm(\003)1207 1610 y(def)1218 1662 y Ft(=)38 b Fo(\031)1387 1677 y Fm(\003)1440 1662 y Ft(\()p Fo(g)t Ft(\))32 b(con)m(v)m(erges)j (uniformly)30 b(to)i Fo(g)t Ft(.)43 b(W)-8 b(e)33 b(set)1353 1899 y Fo(\036)p Ft(\()p Fo(g)t Ft(\))1564 1846 y Fm(def)1575 1899 y Ft(=)73 b(lim)1689 1966 y Fm(\003)p Fj(!)p Fl(Z)1859 1947 y Fk(d)1911 1782 y Fh(Z)1957 1970 y Fm(\()p Fv(S)2031 1952 y Fi(1)2066 1970 y Fm(\))2093 1952 y Fi(\003)2160 1899 y Fo(d\026)2270 1858 y Fm(\003)2322 1899 y Fo(g)2369 1914 y Fm(\003)2422 1899 y Fo(\036)2480 1914 y Fm(\003)3388 1899 y Ft(\(20\))324 2151 y(The)33 b(limit)c(exists)34 b(b)s(ecause)g(for)e(\003)1605 2166 y Fm(1)1671 2151 y Fp(\032)d Ft(\003)1845 2166 y Fm(2)1239 2366 y Fh(\014)1239 2416 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Fv(\017)2551 5027 y(p;k)2649 5002 y Ft(\()p Fs(z)p Ft(\))g Fp(\000)h Fo(w)2967 5017 y Fv(p)3006 4906 y Fh(\021)1894 5251 y Ft(11)p eop %%Page: 12 12 12 11 bop 324 548 a Fn(The)33 b(sum)g(in)g(the)h(right)g(hand)e(side)h (c)-5 b(onver)g(ges)33 b(uniformly)g(in)g Fs(z)28 b Fp(2)g Ft(\000)2901 512 y Fv(N)3002 548 y Fn(and)33 b Fo(w)3260 563 y Fv(p)3327 548 y Fp(2)28 b Fo(A)3494 563 y Fv(\016)3532 548 y Fn(.)324 830 y Fe(5.1)135 b(Unp)t(erturb)t(ed)44 b(Op)t(erator)324 1015 y Ft(The)39 b(\014rst)g(summand)f(in)g(\(25\))g (is)g(just)h(the)g(one)g(whic)m(h)g(app)s(ears)g(in)f(the)h(uncoupled) 324 1135 y(system)g(\(i.e.)f Fo(T)925 1099 y Fv(\017)p Fm(=0)1085 1135 y Ft(=)f(id\))h(and)g(in)g(this)g(case)h(eac)m(h)h (lattice)c(site)j(can)f(b)s(e)h(considered)324 1256 y(separately)-8 b(.)40 b(W)-8 b(e)24 b(denote)f(b)m(y)h Fp(L)1472 1271 y Fv(f)1506 1279 y Fk(p)1569 1256 y Ft(the)f(restriction)f(of)g(the)i (P)m(erron-F)-8 b(rob)s(enius)22 b(op)s(erator)324 1376 y(to)46 b(the)h(Banac)m(h)h(space)g(of)e(functions)h(on)g Fo(S)2053 1340 y Fm(1)2139 1376 y Ft(that)f(extend)i(con)m(tin)m (uously)f(on)g(the)324 1497 y(closed)28 b(ann)m(ulus)g Fo(A)1036 1512 y Fv(\016)1102 1497 y Ft(and)f(holomorphically)d(on)k (the)g(in)m(terior)f Fo(A)2701 1512 y Fv(\016)2739 1497 y Ft(.)41 b Fp(k)12 b(\001)g(k)2959 1512 y Fv(A)3012 1524 y Fk(\016)3078 1497 y Ft(denotes)29 b(the)324 1617 y(suprem)m(um)k(o)m(v)m(er)g Fo(A)1080 1632 y Fv(\016)1118 1617 y Ft(.)44 b(The)33 b(op)s(erator)1098 1812 y Fp(L)1167 1827 y Fv(f)1201 1835 y Fk(p)1269 1812 y Ft(:)28 b(\()p Fp(H)q Ft(\()p Fo(A)1558 1827 y Fv(\016)1596 1812 y Ft(\))p Fo(;)17 b Fp(k)k(\001)h(k)1849 1827 y Fv(A)1902 1839 y Fk(\016)1940 1812 y Ft(\))28 b Fp(!)f Ft(\()p Fp(H)q Ft(\()p Fo(A)2367 1827 y Fv(\016)2405 1812 y Ft(\))p Fo(;)17 b Fp(k)k(\001)h(k)2658 1827 y Fv(A)2711 1839 y Fk(\016)2749 1812 y Ft(\))324 1988 y(has)31 b(1)g(as)g(simple)e (eigen)m(v)-5 b(alue)30 b(and)h(the)g(rest)h(of)e(its)g(sp)s(ectrum)i 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Fv(p)1891 4723 y Ft(\()p Fo(w)s Ft(\))2057 4606 y Fh(Z)2102 4795 y Fm(\000)2146 4804 y Fi(+)2247 4656 y Fo(dz)p 2227 4700 139 4 v 2227 4792 a Ft(2)p Fo(\031)t({)2387 4656 y Ft(1)p 2386 4700 50 4 v 2386 4792 a Fo(z)2446 4723 y(g)t Ft(\()p Fo(z)t Ft(\))766 b(\(30\))1636 4980 y(=)1795 4863 y Fh(Z)1841 5052 y Fm(\000)1935 4913 y Fo(dz)p 1916 4957 139 4 v 1916 5049 a Ft(2)p Fo(\031)t({)2075 4913 y Ft(1)p 2075 4957 50 4 v 2075 5049 a Fo(z)2134 4980 y(h)2190 4995 y Fv(p)2230 4980 y Ft(\()p Fo(w)s(;)17 b(z)t Ft(\))p Fo(g)t Ft(\()p Fo(z)t Ft(\))740 b(\(31\))1894 5251 y(12)p eop %%Page: 13 13 13 12 bop 324 548 a Ft(where)34 b(w)m(e)f(ha)m(v)m(e)h(used)g(that)e Fo(g)k Ft(is)c(holomorphic)e(in)i Fo(A)2334 563 y Fv(\016)2405 548 y Ft(and)g(de\014ned:)1122 826 y Fo(h)1178 841 y Fv(p)1218 826 y Ft(\()p Fo(w)1326 841 y Fv(p)1365 826 y Fo(;)17 b(z)1454 841 y Fv(p)1494 826 y Ft(\))1559 774 y Fm(def)1570 826 y Ft(=)1684 680 y Fh(\()1793 765 y Fo(h)1849 780 y Fv(p)1889 765 y Ft(\()p Fo(w)1997 780 y Fv(p)2036 765 y Ft(\))97 b(for)g Fo(z)2430 780 y Fv(p)2498 765 y Fp(2)28 b Ft(\000)2653 780 y Fm(+)1909 885 y Ft(0)213 b(for)97 b Fo(z)2430 900 y Fv(p)2498 885 y Fp(2)28 b Ft(\000)2653 900 y Fj(\000)3388 826 y Ft(\(32\))324 1060 y(The)33 b(idemp)s(otency)g Fo(Q)1176 1024 y Fm(2)1176 1085 y Fv(p)1243 1060 y Ft(=)28 b Fo(Q)1424 1075 y Fv(p)1496 1060 y Ft(reads)34 b(in)d(the)i(in)m(tegral)e(represen)m(tation)423 1302 y Fh(Z)469 1491 y Fm(\000)544 1352 y Fo(dz)644 1316 y Fm(2)p 544 1396 140 4 v 545 1488 a Ft(2)p Fo(\031)t({)724 1352 y Ft(1)p 704 1396 90 4 v 704 1488 a Fo(z)753 1459 y Fm(2)820 1302 y Fh(Z)866 1491 y Fm(\000)940 1352 y Fo(dz)1040 1316 y Fm(1)p 940 1396 140 4 v 941 1488 a Ft(2)p Fo(\031)t({)1120 1352 y Ft(1)p 1100 1396 90 4 v 1100 1488 a Fo(z)1149 1459 y Fm(1)1199 1419 y Fo(h)1255 1434 y Fv(p)1295 1419 y Ft(\()p Fo(w)s(;)17 b(z)1499 1378 y Fm(2)1538 1419 y Ft(\))p Fo(h)1632 1434 y Fv(p)1672 1419 y Ft(\()p Fo(z)1759 1378 y Fm(2)1799 1419 y Fo(;)g(z)1892 1378 y Fm(1)1932 1419 y Ft(\))p Fo(g)t Ft(\()p Fo(z)2108 1378 y Fm(1)2148 1419 y Ft(\))27 b(=)2317 1302 y Fh(Z)2363 1491 y Fm(\000)2438 1352 y Fo(dz)2538 1316 y Fm(1)p 2438 1396 140 4 v 2439 1488 a Ft(2)p Fo(\031)t({)2618 1352 y Ft(1)p 2597 1396 90 4 v 2597 1488 a Fo(z)2646 1459 y Fm(1)2697 1419 y Fo(h)2753 1434 y Fv(p)2792 1419 y Ft(\()p Fo(w)s(;)17 b(z)2996 1378 y Fm(1)3036 1419 y Ft(\))p Fo(g)t Ft(\()p Fo(z)3212 1378 y Fm(1)3251 1419 y Ft(\))99 b(\(33\))324 1652 y(According)32 b(to)g(Prop)s(osition)f(3)h (the)h(op)s(erator)f Fo(R)2147 1667 y Fv(p)2220 1652 y Ft(can)h(b)s(e)f(written)1301 1918 y Fo(R)1375 1933 y Fv(p)1414 1918 y Fo(g)t Ft(\()p Fo(w)s Ft(\))27 b(=)1744 1800 y Fh(Z)1790 1989 y Fm(\000)1884 1850 y Fo(dz)p 1865 1894 139 4 v 1865 1986 a Ft(2)p Fo(\031)t({)2024 1850 y Ft(1)p 2024 1894 50 4 v 2024 1986 a Fo(z)2084 1918 y(r)2128 1933 y Fv(p)2167 1918 y Ft(\()p Fo(w)s(;)17 b(z)t Ft(\))p Fo(g)t Ft(\()p Fo(z)t Ft(\))803 b(\(34\))324 2114 y(with)1141 2264 y Fo(r)1185 2279 y Fv(p)1224 2264 y Ft(\()p 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4952 y Fp(j)2091 4911 y Fj(\000)p Fm(1)2185 4952 y Fp(j)p Fo(f)2261 4967 y Fv(p)2323 4952 y Fp(\016)f Fo(T)2466 4911 y Fv(\017)2452 4977 y(p;k)2550 4952 y Ft(\()p Fs(z)p Ft(\))f Fp(\000)i Fo(w)2867 4967 y Fv(p)2906 4952 y Fp(j)2934 4911 y Fj(\000)p Fm(1)1894 5251 y Ft(13)p eop %%Page: 14 14 14 13 bop 1067 548 a Fp(\002j)p Fo(f)1220 563 y Fv(p)1282 548 y Fp(\016)22 b Fo(T)1425 507 y Fv(\017)1411 573 y(p;k)r Fj(\000)p Fm(1)1599 548 y Ft(\()p Fs(z)p Ft(\))g Fp(\000)g Fo(f)1894 563 y Fv(p)1956 548 y Fp(\016)g Fo(T)2099 507 y Fv(\017)2085 573 y(p;k)2183 548 y Ft(\()p Fs(z)p Ft(\))p Fp(j)890 755 y(\024)1061 688 y Ft(1)g(+)g Fo(\016)p 1060 732 218 4 v 1060 824 a Ft(1)g Fp(\000)h Fo(\016)1288 755 y Fp(j)p Fo(c)1358 770 y Fm(7)1397 755 y Fo(\025)f Fp(\000)g Ft(1)p Fp(j)1652 714 y Fj(\000)p Fm(1)1746 755 y Fp(j)p Fo(c)1816 770 y Fm(7)1855 755 y Fo(\025)g Fp(\000)h Ft(1)p Fp(j)2111 714 y Fj(\000)p Fm(1)2205 755 y Fp(k)p Fo(f)2314 714 y Fj(0)2303 780 y Fv(p)2342 755 y Fp(k)2392 771 y Fj(f)p Fv(p)p 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b Fp(k)22 b(\001)f(k)2047 1902 y Fm(\003)2096 1911 y Fi(2)2131 1902 y Fv(;#)2196 1887 y Ft(\))28 b Fp(!)f Ft(\()p Fp(H)2511 1902 y Fm(\003)2560 1911 y Fi(1)2595 1902 y Fv(;#)2661 1887 y Fo(;)17 b Fp(k)k(\001)h(k)2876 1902 y Fm(\003)2925 1911 y Fi(1)2960 1902 y Fv(;#)3025 1887 y Ft(\))325 b(\(40\))458 2281 y Fp(L)527 2240 y Fv(N)527 2309 y(F)582 2287 y Fi(\003)623 2302 y(2)662 2309 y Fj(\016)p Fv(T)748 2287 y Fi(\003)789 2302 y(2)828 2309 y Fv(;\017)880 2281 y Fo(\036)p Ft(\()p Fs(z)1026 2240 y Fm(0)1065 2281 y Ft(\))285 b(=)1491 2210 y Fh(R)1531 2306 y Fm(\000)1575 2284 y Fi(\003)1616 2299 y(2)1745 2242 y Fv(d)p Fl(z)1817 2218 y Fg(\000)p Fi(1)p 1686 2258 275 4 v 1686 2321 a Fm(\(2)p Fv(\031)r({)p Fm(\))1841 2299 y Fg(j)p Fi(\003)1901 2314 y(2)1937 2299 y Fg(j)1986 2281 y Fp(\001)17 b(\001)g(\001)2119 2210 y Fh(R)2158 2306 y Fm(\000)2202 2284 y Fi(\003)2202 2329 y(2)2327 2242 y Fv(d)p Fl(z)2399 2218 y Fg(\000)p Fk(N)p 2279 2258 V 2279 2321 a Fm(\(2)p Fv(\031)r({)p Fm(\))2434 2299 y Fg(j)p Fi(\003)2494 2314 y(2)2530 2299 y Fg(j)2580 2215 y Fh(Q)2658 2240 y Fj(\000)p Fm(1)2658 2305 y Fv(t)p Fm(=)p Fj(\000)p Fv(N)2877 2215 y Fh(Q)2956 2302 y Fv(p)p Fj(2)p Fm(\003)3088 2311 y Fi(2)3388 2281 y Ft(\(41\))1186 2470 y Fp(\002)1280 2374 y Fh(\020)1330 2470 y Ft(\()p Fo(h)1424 2485 y Fv(p)1464 2470 y Ft(\()p Fo(z)1551 2434 y Fv(t)p Fm(+1)1547 2495 y Fv(p)1671 2470 y Fo(;)g(z)1764 2434 y Fv(t)1760 2495 y(p)1800 2470 y Ft(\))22 b(+)g Fo(r)2002 2485 y Fv(p)2042 2470 y Ft(\()p Fo(z)2129 2434 y Fv(t)p Fm(+1)2125 2495 y Fv(p)2249 2470 y Fo(;)17 b(z)2342 2434 y Fv(t)2338 2495 y(p)2378 2470 y Ft(\))22 b(+)2536 2404 y Fh(P)2624 2430 y Fj(1)2624 2495 y Fv(k)r Fm(=1)2773 2470 y Fo(\014)2828 2485 y Fv(p;k)2926 2470 y Ft(\()p Fo(z)3013 2434 y Fv(t)p Fm(+1)3009 2495 y Fv(p)3133 2470 y Fo(;)17 b Fs(z)3227 2434 y Fv(t)3256 2470 y Ft(\))3294 2374 y Fh(\021)324 2690 y Ft(Distributing)35 b(the)j(pro)s(duct)g(w)m (e)h(get)f(in\014nitely)e(man)m(y)i(summands.)58 b(In)38 b(eac)m(h)h(factor)324 2810 y(there)h(is)e(for)h(eac)m(h)h Fp(\000)p Fo(N)49 b Fp(\024)39 b Fo(m)g Fp(\024)g(\000)p Ft(1,)i Fo(p)e Fp(2)g Ft(\003)2081 2825 y Fm(2)2159 2810 y Ft(a)g(c)m(hoice)g(b)s(et)m(w)m(een)j Fo(h)2983 2825 y Fv(p)3022 2810 y Ft(,)f Fo(r)3134 2825 y Fv(p)3213 2810 y Ft(and)e Fo(\014)3464 2825 y Fv(p;k)324 2931 y Ft(\(1)27 b Fp(\024)h Fo(k)j(<)d Fp(1)p Ft(\))23 b(and)h(w)m(e)h(can)f (in)m(terpret)g(suc)m(h)h(a)f(c)m(hoice)g(graphically)e(as)i(a)f Fn(c)-5 b(on\014gur)g(ation)324 3051 y Ft(as)33 b(follo)m(ws)e(\(cf.)i ([2)o(,)g(15]\):)324 3172 y(On)f(\003)554 3187 y Fm(2)616 3172 y Fp(\002)22 b(f\000)p Fo(N)5 b(;)17 b(:)g(:)g(:)f(;)h Ft(0)p Fp(g)32 b Ft(w)m(e)i(represen)m(t)469 3387 y Fp(\017)49 b Fo(h)624 3402 y Fv(p)680 3291 y Fh(\020)730 3387 y Fo(z)779 3351 y Fv(t)p Fm(+1)775 3412 y Fv(p)899 3387 y Fo(;)17 b(z)992 3351 y Fv(t)988 3412 y(p)1028 3291 y Fh(\021)1208 3387 y Ft(b)m(y)33 b(an)g Fn(h-line)39 b Ft(from)31 b(\()p Fo(p;)17 b(t)p Ft(\))32 b(to)g(\()p Fo(p;)17 b(t)22 b Ft(+)g(1\))469 3628 y Fp(\017)49 b Fo(r)612 3643 y Fv(p)668 3532 y Fh(\020)717 3628 y Fo(z)766 3592 y Fv(t)p Fm(+1)762 3653 y Fv(p)887 3628 y Fo(;)17 b(z)980 3592 y Fv(t)976 3653 y(p)1016 3532 y Fh(\021)1195 3628 y Ft(b)m(y)34 b(an)e Fn(r-line)40 b Ft(from)31 b(\()p Fo(p;)17 b(t)p Ft(\))32 b(to)h(\()p Fo(p;)17 b(t)22 b Ft(+)g(1\))1030 4385 y Fd(v)1030 3912 y(v)p 1029 4385 4 473 v 1089 3936 a Ft(\()p Fo(p;)17 b(t)p Ft(\))1089 4408 y(\()p Fo(p;)g(t)22 b Ft(+)g(1\))1089 4172 y Fo(h)1145 4187 y Fv(p)1185 4172 y Ft(\()p Fo(z)1272 4136 y Fv(t)p Fm(+1)1268 4197 y Fv(p)1393 4172 y Fo(;)17 b(z)1486 4136 y Fv(t)1482 4197 y(p)1521 4172 y Ft(\))2448 4385 y Fd(v)2448 3912 y(v)p 2444 4385 7 473 v 2450 4385 V 2438 4385 V 2507 3936 a Ft(\()p Fo(p;)g(t)p Ft(\))2507 4408 y(\()p Fo(p;)g(t)22 b Ft(+)g(1\))2507 4172 y Fo(r)2551 4187 y Fv(p)2590 4172 y Ft(\()p Fo(z)2677 4136 y Fv(t)p Fm(+1)2673 4197 y Fv(p)2798 4172 y Fo(;)17 b(z)2891 4136 y Fv(t)2887 4197 y(p)2927 4172 y Ft(\))1388 4696 y(Figure)31 b(1:)44 b(h-line)31 b(and)h(r-line)1894 5251 y(14)p eop %%Page: 15 15 15 14 bop 469 548 a Fp(\017)49 b Fo(\014)623 563 y Fv(p;k)737 452 y Fh(\020)787 548 y Fo(z)836 512 y Fv(t)p Fm(+1)832 573 y Fv(p)956 548 y Fo(;)17 b Fs(z)1050 512 y Fv(t)1079 452 y Fh(\021)1258 548 y Ft(b)m(y)32 b(a)f Fn(k-triangle)38 b Ft(\(actually)30 b(rather)i(a)f(cone)h(or)f(p)m(yramid)f(but)568 681 y(in)i(our)i(pictures)g(for)f Fo(d)c Ft(=)g(1)34 b(it)e(is)h(a)h(triangle\))d(with)j(ap)s(ex)g(\()p Fo(p;)17 b(t)22 b Ft(+)h(1\))33 b(and)h(base)568 801 y(p)s(oin)m(ts)26 b(\()p Fo(q)t(;)17 b(t)p Ft(\))27 b(with)f Fp(k)p Fo(p)11 b Fp(\000)g Fo(q)t Fp(k)27 b(\024)h Fo(k)s Ft(.)42 b(\(So)26 b(some)h(of)f(the)i(base)f(p)s(oin)m(ts)g(migh)m(t)e(not)i(lie)568 922 y(in)k(\003)749 937 y Fm(2)811 922 y Fp(\002)22 b(f\000)p Fo(N)5 b(;)17 b(:)g(:)g(:)g(;)g Fp(\000)p Ft(1)p Fp(g)32 b Ft(but)h(all)d(the)j(apices)g(lie)e(in)h(\003)2630 937 y Fm(2)2691 922 y Fp(\002)23 b(f\000)p Fo(N)33 b Ft(+)22 b(1)p Fo(;)17 b(:)g(:)g(:)e(;)i Ft(0)p Fp(g)p Ft(.\))1943 1771 y Fd(v)998 1299 y(v)414 b(v)h(v)f(v)h(v)844 1240 y Ft(\()p Fo(p)22 b Fp(\000)h Ft(2)p Fo(;)17 b(t)p Ft(\))110 b(\()p Fo(p)22 b Fp(\000)g Ft(1)p Fo(;)17 b(t)p Ft(\))122 b(\()p Fo(p;)17 b(t)p Ft(\))256 b(\()p Fo(p)22 b Ft(+)g(1)p Fo(;)17 b(t)p Ft(\))100 b(\()p Fo(p)22 b Ft(+)g(2)p Fo(;)17 b(t)p Ft(\))1801 1901 y(\()p Fo(p;)g(t)22 b Ft(+)g(1\))1943 1771 y Fc(\010)2026 1730 y(\010)2109 1688 y(\010)2192 1647 y(\010)2275 1605 y(\010)2358 1564 y(\010)2441 1522 y(\010)2524 1481 y(\010)2607 1439 y(\010)2690 1398 y(\010)2773 1356 y(\010)2805 1340 y(\010)1860 1771 y(H)1777 1730 y(H)1694 1688 y(H)1611 1647 y(H)1528 1605 y(H)1445 1564 y(H)1362 1522 y(H)1279 1481 y(H)1196 1439 y(H)1112 1398 y(H)1029 1356 y(H)998 1340 y(H)p 998 1301 1890 4 v 1706 1535 a Fo(\014)1761 1550 y Fv(p;)p Fm(2)1856 1535 y Ft(\()p Fo(z)1943 1499 y Fv(t)p Fm(+1)1939 1560 y Fv(p)2064 1535 y Fo(;)17 b Fs(z)2158 1499 y Fv(t)2187 1535 y Ft(\))1526 2083 y(Figure)32 b(2:)43 b(2-triangle)324 2375 y(Note)25 b(that)g(if)f Fo(v)t Ft(\()p Fo(k)s Ft(\))1046 2323 y Fm(def)1057 2375 y Ft(=)38 b Fp(j)p Fo(B)1273 2390 y Fv(k)1316 2375 y Ft(\(0\))p Fp(j)24 b Ft(denotes)i(the)g(n)m(um) m(b)s(er)f(of)g(base)h(p)s(oin)m(ts)e(of)h(a)g Fo(k)s Ft(-triangle,)324 2495 y(w)m(e)33 b(ha)m(v)m(e)h(the)f(estimate)f Fo(v)t Ft(\()p Fo(k)s Ft(\))27 b Fp(\024)i Ft(\(3)p Fo(k)s Ft(\))1744 2459 y Fv(d)1784 2495 y Ft(.)324 2616 y(Eac)m(h)36 b(c)m(hoice)f(corresp)s(onds)i(to)e(a)g(con\014guration)f(and)h(for)g (eac)m(h)h(con\014guration)e Fp(C)42 b Ft(w)m(e)324 2736 y(ha)m(v)m(e)34 b(an)e(op)s(erator)g Fp(L)1146 2751 y Fj(C)1191 2736 y Ft(.)43 b(So)33 b(w)m(e)g(can)g(write)1543 2956 y Fp(L)1612 2915 y Fv(N)1612 2985 y(F)1667 2962 y Fi(\003)1708 2977 y(2)1746 2985 y Fj(\016)p Fv(T)1832 2962 y Fi(\003)1873 2977 y(2)1908 2962 y Fk(;\017)1989 2956 y Ft(=)2093 2873 y Fh(X)2132 3057 y Fj(C)2229 2956 y Fp(L)2298 2971 y Fj(C)3388 2956 y Ft(\(42\))324 3242 y(Some)f(of)g(these)i(summands)e(are)h(zero)g(namely)f(if)469 3446 y Fp(\017)49 b Ft(a)i(factor)h Fo(h)1023 3461 y Fv(p)1079 3349 y Fh(\020)1129 3446 y Fo(z)1178 3410 y Fv(t)p Fm(+2)1174 3470 y Fv(p)1298 3446 y Fo(;)17 b(z)1391 3410 y Fv(t)p Fm(+1)1387 3470 y Fv(p)1511 3349 y Fh(\021)1578 3446 y Fo(r)1622 3461 y Fv(p)1678 3349 y Fh(\020)1727 3446 y Fo(z)1776 3410 y Fv(t)p Fm(+1)1772 3470 y Fv(p)1897 3446 y Fo(;)g(z)1990 3410 y Fv(t)1986 3470 y(p)2026 3349 y Fh(\021)2127 3446 y Ft(or)52 b Fo(r)2310 3461 y Fv(p)2366 3349 y Fh(\020)2415 3446 y Fo(z)2464 3410 y Fv(t)p Fm(+2)2460 3470 y Fv(p)2585 3446 y Fo(;)17 b(z)2678 3410 y Fv(t)p Fm(+1)2674 3470 y Fv(p)2798 3349 y Fh(\021)2864 3446 y Fo(h)2920 3461 y Fv(p)2977 3349 y Fh(\020)3027 3446 y Fo(z)3076 3410 y Fv(t)p Fm(+1)3072 3470 y Fv(p)3196 3446 y Fo(;)g(z)3289 3410 y Fv(t)3285 3470 y(p)3325 3349 y Fh(\021)3426 3446 y Ft(ap-)568 3604 y(p)s(ears,)42 b(but)e(no)g(factor)g Fo(\014)1533 3619 y Fv(q)r(;k)1646 3507 y Fh(\020)1695 3604 y Fo(z)1744 3567 y Fv(t)p Fm(+2)1740 3628 y Fv(q)1865 3604 y Fo(;)17 b Fs(z)1959 3567 y Fv(t)p Fm(+1)2078 3507 y Fh(\021)2168 3604 y Ft(with)40 b Fp(k)p Fo(p)27 b Fp(\000)g Fo(q)t Fp(k)41 b(\024)f Fo(k)k Ft(\(i.e.)39 b(an)h(h-line)568 3724 y(follo)m(ws)g(or)i(is)f(follo)m(w)m(ed)g(b)m(y) i(an)f(r-line)e(and)i(at)g(their)f(common)g(endp)s(oin)m(t)h(no)568 3844 y(triangle)32 b(is)i(attac)m(hed)g(with)g(an)m(y)h(of)f(its)f (basep)s(oin)m(ts\).)49 b(This)34 b(follo)m(ws)f(since,)i(b)m(y)568 3965 y(F)-8 b(ubini's)37 b(Theorem,)j(one)f(can)f(\014rst)h(p)s(erform) f(the)g Fo(dz)2614 3929 y Fv(t)p Fm(+1)2610 3989 y Fv(p)2735 3965 y Fo(dz)2835 3929 y Fv(t)2831 3989 y(p)2870 3965 y Ft(-in)m(tegration)e(and)568 4085 y(get)28 b(zero)g(b)m(y)h(\(36\))f (or)g(\(37\).)41 b(\(Note)28 b(that)g(no)g(other)g(terms)h(dep)s(end)g (on)f Fo(z)3256 4049 y Fv(t)p Fm(+1)3252 4110 y Fv(p)3404 4085 y Ft(and)568 4205 y(the)k(remaining)d(factors)j(and)g(in)m (tegrations)f(\(up)h(to)f(time)g(t+1\))g(corresp)s(ond)i(to)568 4326 y(the)g(function)f Fo(g)t Ft(\()p Fo(z)1256 4290 y Fm(1)1295 4326 y Ft(\))h(in)e(\(36\))h(or)h(\(37\).\))469 4542 y Fp(\017)49 b Ft(if)42 b(a)h(term)g Fo(h)1060 4557 y Fv(p)1117 4445 y Fh(\020)1166 4542 y Fo(z)1215 4506 y Fv(t)p Fm(+2)1211 4566 y Fv(p)1336 4542 y Fo(;)17 b(z)1429 4506 y Fv(t)p Fm(+1)1425 4566 y Fv(p)1549 4445 y Fh(\021)1615 4542 y Fo(\014)1670 4557 y Fv(p;k)1785 4445 y Fh(\020)1834 4542 y Fo(z)1883 4506 y Fv(t)p Fm(+1)1879 4566 y Fv(p)2004 4542 y Fo(;)g Fs(z)2098 4506 y Fv(t)2127 4445 y Fh(\021)2220 4542 y Ft(app)s(ears)44 b(but)g(no)f Fo(\014)2983 4557 y Fv(q)r(;l)3080 4445 y Fh(\020)3129 4542 y Fo(z)3178 4506 y Fv(t)p Fm(+2)3174 4566 y Fv(q)3299 4542 y Fo(;)17 b Fs(z)3393 4506 y Fv(t)p Fm(+1)3512 4445 y Fh(\021)568 4675 y Ft(with)38 b Fp(k)p Fo(p)26 b Fp(\000)g Fo(q)t Fp(k)37 b(\024)i Fo(l)h Ft(\(i.e.)e(a)g(triangle)f(is)h(follo)m(w)m(ed) f(b)m(y)i(an)g(h-line)e(and)h(at)g(their)568 4795 y(common)k(endp)s (oin)m(t)h(\(the)h(ap)s(ex)f(of)g(the)h(triangle\))d(no)j(other)f (triangle)f(is)g(at-)568 4915 y(tac)m(hed)33 b(with)f(an)m(y)i(of)e (its)g(basep)s(oin)m(ts\).)1894 5251 y(15)p eop %%Page: 16 16 16 15 bop 1030 1393 a Fd(v)1030 921 y(v)1030 448 y(v)p 1027 921 7 473 v 1033 921 V 1021 921 V 1029 1393 4 473 v 1089 944 a Ft(\()p Fo(p;)17 b(t)22 b Ft(+)g(1\))1089 484 y(\()p Fo(p;)17 b(t)p Ft(\))1089 708 y Fo(r)1133 723 y Fv(p)1173 708 y Ft(\()p Fo(z)1260 672 y Fv(t)p Fm(+1)1256 733 y Fv(p)1381 708 y Fo(;)g(z)1474 672 y Fv(t)1470 733 y(p)1509 708 y Ft(\))1089 1417 y(\()p Fo(p;)g(t)22 b Ft(+)g(2\))1089 1181 y Fo(h)1145 1196 y Fv(p)1185 1181 y Ft(\()p Fo(z)1272 1144 y Fv(t)p Fm(+2)1268 1205 y Fv(p)1393 1181 y Fo(;)17 b(z)1486 1144 y Fv(t)p Fm(+1)1482 1205 y Fv(p)1606 1181 y Ft(\))2448 1393 y Fd(v)2448 921 y(v)2448 448 y(v)p 2444 1393 7 473 v 2450 1393 V 2438 1393 V 2446 921 4 473 v 2507 944 a Ft(\()p Fo(p;)g(t)22 b Ft(+)g(1\))2507 484 y(\()p Fo(p;)17 b(t)p Ft(\))2507 708 y Fo(h)2563 723 y Fv(p)2603 708 y Ft(\()p Fo(z)2690 672 y Fv(t)p Fm(+1)2686 733 y Fv(p)2810 708 y Fo(;)g(z)2903 672 y Fv(t)2899 733 y(p)2939 708 y Ft(\))2507 1417 y(\()p Fo(p;)g(t)22 b Ft(+)g(2\))2507 1181 y Fo(r)2551 1196 y Fv(p)2590 1181 y Ft(\()p Fo(z)2677 1144 y Fv(t)p Fm(+2)2673 1205 y Fv(p)2798 1181 y Fo(;)17 b(z)2891 1144 y Fv(t)p Fm(+1)2887 1205 y Fv(p)3011 1181 y Ft(\))1117 1705 y(Figure)31 b(3:)43 b(Consecutiv)m(e)35 b(r-line)c(and)h(h-line)779 2246 y Fo(\014)834 2261 y Fv(p;k)948 2246 y Ft(\()p Fo(w)1056 2261 y Fv(p)1095 2246 y Fo(;)17 b Fs(z)p Ft(\))83 b(=)1479 2179 y Fo(w)1549 2194 y Fv(p)p 1479 2223 110 4 v 1491 2315 a Fo(z)1536 2330 y Fv(p)1854 2172 y Fo(f)1902 2187 y Fv(p)1964 2172 y Fp(\016)22 b Fo(T)2107 2136 y Fv(\017)2093 2197 y(p;k)r Fj(\000)p Fm(1)2281 2172 y Ft(\()p Fs(z)p Ft(\))g Fp(\000)h Fo(f)2577 2187 y Fv(p)2639 2172 y Fp(\016)f Fo(T)2782 2136 y Fv(\017)2768 2197 y(p;k)2865 2172 y Ft(\()p Fs(z)p Ft(\))p 1608 2223 1630 4 v 1608 2315 a(\()p Fo(f)1694 2330 y Fv(p)1756 2315 y Fp(\016)g Fo(T)1899 2280 y Fv(\017)1885 2340 y(p;k)r Fj(\000)p Fm(1)2073 2315 y Ft(\()p Fs(z)p Ft(\))g Fp(\000)g 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4580 y Fm(\003)1833 4589 y Fi(1)1867 4580 y Fv(;)1896 4562 y Fm(~)1887 4580 y Fv(#)1961 4552 y Fp(\024)28 b Fo(c)2108 4567 y Fm(12)2183 4552 y Fo(#)2240 4516 y Fj(\000j)p Fm(\003)2364 4525 y Fi(2)2398 4516 y Fj(j)2422 4552 y Fp(k)p Fo(f)11 b Fp(k)2581 4567 y Fm(\003)2630 4576 y Fi(2)2668 4552 y Fo(\024)2724 4516 y Fi(dist)p Fm(\(\003)2900 4525 y Fi(1)2935 4516 y Fv(;)p Fm(\003)3004 4525 y Fi(2)3038 4516 y Fm(\))3076 4552 y Ft(~)-56 b Fo(\021)3121 4516 y Fv(N)568 4707 y Ft(for)32 b(ev)m(ery)i Fo(N)k Fp(\025)28 b Ft(0.)324 4882 y Fs(Remark)37 b Ft(As)d(in)f(Theorem)g(2)g(w)m(e)i(can)e(c)m(ho)s (ose)h(the)g(rate)f(of)g(deca)m(y)i Fo(\024)e Ft(\014rst)h(and)f(then) 324 5002 y(the)g(other)g(parameters.)1894 5251 y(21)p eop %%Page: 22 22 22 21 bop 324 548 a Fu(8)161 b(Pro)t(ofs)324 767 y Fs(Pro)s(of)31 b(of)h(Prop)s(osition)e(1)60 b Ft(W)-8 b(e)28 b(ha)m(v)m(e)h(a)e(Cauc)m (h)m(y)j(estimate)d(for)g(the)h(partial)e(deriv)-5 b(a-)324 887 y(tiv)m(es)49 b(of)e(the)i(functions)f Fo(g)1361 902 y Fv(p;k)1513 887 y Ft(:)55 b Fo(A)1668 836 y Fv(B)1721 848 y Fk(k)1759 836 y Fm(\()p Fv(p)p Fm(\))1668 912 y Fv(\016)1908 887 y Fp(!)f Fs(C)48 b Ft(on)h(a)f(smaller)e(p)s(oly)m (ann)m(ulus.)90 b(Let)324 1008 y Fo(q)31 b Fp(2)d Fo(B)566 1023 y Fv(k)609 1008 y Ft(\()p Fo(p)p Ft(\),)33 b(Then)1024 1236 y Fh(\015)1024 1286 y(\015)1024 1336 y(\015)1024 1386 y(\015)1024 1435 y(\015)1122 1318 y Fo(@)p 1080 1362 141 4 v 1080 1454 a(@)5 b(z)1181 1469 y Fv(q)1230 1386 y Fo(g)1277 1401 y Fv(p;k)1375 1236 y Fh(\015)1375 1286 y(\015)1375 1336 y(\015)1375 1386 y(\015)1375 1435 y(\015)1421 1489 y Fv(A)1474 1449 y Fk(B)1520 1466 y(k)1558 1449 y Fi(\()p Fk(p)p Fi(\))1474 1514 y Fk(\016)1729 1386 y Fp(\024)2063 1318 y Ft(1)p 1900 1362 376 4 v 1900 1454 a Fp(j)p Fo(e)1973 1425 y Fv(\016)2033 1454 y Fp(\000)22 b Fo(e)2177 1425 y Fv(\016)2208 1434 y Fi(1)2247 1454 y Fp(j)2285 1386 y Fo(c)2327 1401 y Fm(1)2383 1386 y Ft(exp)q(\()p Fp(\000)p Fo(c)2689 1401 y Fm(2)2729 1386 y Fo(k)2783 1344 y Fv(d)2824 1386 y Ft(\))526 b(\(58\))1730 1635 y(=)84 b Fo(c)1932 1650 y Fm(13)2023 1635 y Ft(exp)q(\()p Fp(\000)p Fo(c)2329 1650 y Fm(2)2369 1635 y Fo(k)2423 1594 y Fv(d)2464 1635 y Ft(\))3388 1780 y(\(59\))324 2000 y(Also)32 b(note)h(that)1015 1961 y Fv(@)p 982 1977 109 4 v 982 2034 a(@)t(z)1056 2042 y Fk(q)1100 2000 y Fo(g)1147 2015 y Fv(p;k)1272 2000 y Ft(=)28 b(0)k(for)g Fo(q)43 b(=)-60 b Fp(2)28 b Fo(B)1849 2015 y Fv(k)1892 2000 y Ft(\()p Fo(p)p Ft(\).)43 b(Therefore)829 2253 y Fh(\015)829 2303 y(\015)829 2353 y(\015)829 2403 y(\015)829 2453 y(\015)927 2335 y Fo(@)p 885 2379 141 4 v 885 2471 a(@)5 b(z)986 2486 y Fv(q)1035 2403 y Fo(g)1082 2418 y Fv(p)1122 2253 y Fh(\015)1122 2303 y(\015)1122 2353 y(\015)1122 2403 y(\015)1122 2453 y(\015)1168 2506 y Fv(A)1221 2484 y Fb(Z)1263 2470 y Fk(d)1221 2531 y(\016)1391 2403 y Ft(=)1551 2228 y Fh(\015)1551 2278 y(\015)1551 2328 y(\015)1551 2378 y(\015)1551 2428 y(\015)1551 2477 y(\015)1649 2335 y Fo(@)p 1607 2379 V 1607 2471 a(@)g(z)1708 2486 y Fv(q)1883 2295 y Fj(1)1858 2320 y Fh(X)1774 2508 y Fv(k)r Fm(=)p Fj(k)p Fv(p)p Fj(\000)p Fv(q)r Fj(k)2079 2403 y Fo(g)2126 2418 y Fv(p;k)2223 2228 y Fh(\015)2223 2278 y(\015)2223 2328 y(\015)2223 2378 y(\015)2223 2428 y(\015)2223 2477 y(\015)2269 2537 y Fv(A)2322 2514 y Fb(Z)2364 2500 y Fk(d)2322 2562 y(\016)3388 2403 y Ft(\(60\))1390 2740 y Fp(\024)84 b Fo(c)1593 2755 y Fm(13)1793 2632 y Fj(1)1768 2657 y Fh(X)1684 2845 y Fv(k)r Fm(=)p Fj(k)p Fv(p)p Fj(\000)p Fv(q)r Fj(k)1989 2740 y Ft(exp)q(\()p Fp(\000)p Fo(c)2295 2755 y Fm(2)2335 2740 y Fo(k)2389 2699 y Fv(d)2429 2740 y Ft(\))1390 3035 y Fp(\024)g Fo(c)1593 3050 y Fm(13)1930 2967 y Ft(1)p 1677 3012 555 4 v 1677 3103 a(1)22 b Fp(\000)h Ft(exp)q(\()p Fp(\000)p Fo(c)2154 3118 y Fm(2)2194 3103 y Ft(\))2258 3035 y(exp)2424 2939 y Fh(\020)2474 3035 y Fp(\000)p Fo(c)2593 3050 y Fm(2)2633 3035 y Fp(k)p Fo(p)e Fp(\000)i Fo(q)t Fp(k)2950 2994 y Fv(d)2990 2939 y Fh(\021)1391 3261 y Ft(=)84 b Fo(c)1593 3276 y Fm(14)1684 3261 y Ft(exp)1850 3165 y Fh(\020)1899 3261 y Fp(\000)p Fo(c)2018 3276 y Fm(2)2058 3261 y Fp(k)p Fo(p)22 b Fp(\000)h Fo(q)t Fp(k)2376 3220 y Fv(d)2416 3165 y Fh(\021)324 3500 y Ft(No)m(w)39 b(w)m(e)g(consider)g(ev)m (erything)g(in)f(the)h(lift)d(giv)m(en)j(b)m(y)g Fo(pr)h Ft(:)e Fs(C)2707 3464 y Fm(\003)2707 3525 y Fv(\016)2798 3500 y Fp(!)f Fo(A)3008 3464 y Fm(\003)3008 3525 y Fv(\016)3061 3500 y Ft(,)j(\()18 b(~)-67 b Fo(z)3211 3515 y Fv(p)3251 3500 y Ft(\))3289 3532 y Fv(p)p Fj(2)p Fm(\003)3462 3500 y Fp(7!)324 3564 y Fh(\020)373 3660 y Fo(e)418 3624 y Fv({)18 b Fm(~)-53 b Fv(z)474 3632 y Fk(p)515 3564 y Fh(\021)565 3714 y Fv(p)p Fj(2)p Fm(\003)700 3660 y Ft(,)33 b(where)h Fs(C)1123 3675 y Fv(\016)1188 3608 y Fm(def)1199 3660 y Ft(=)k Fp(f)p Fo(w)30 b Fp(2)e Fs(C)g Fp(j)f(=)p Fo(w)j Fp(2)e Ft([)p Fp(\000)p Fo(\016)n(;)17 b(\016)t Ft(])p Fp(g)g Fo(:)324 3836 y Ft(Then)39 b(w)m(e)g(ha)m(v)m(e)963 3740 y Fh(\020)1039 3798 y(g)1013 3836 y Fo(T)1084 3807 y Fm(\003)p Fv(;\017)1201 3836 y Ft(\()1240 3835 y(~)1239 3836 y Fs(z)p Ft(\))1327 3740 y Fh(\021)1376 3890 y Fv(p)1453 3836 y Ft(=)j(~)-55 b Fo(z)1610 3851 y Fv(p)1676 3836 y Ft(+)26 b(2)p Fo(\031)t(\017)19 b Ft(~)-68 b Fo(g)1972 3851 y Fv(p)2011 3836 y Ft(\()2050 3835 y(~)2049 3836 y Fs(z)p Ft(\).)59 b(The)39 b(function)56 b(~)-67 b Fo(g)2864 3851 y Fv(p)2903 3836 y Ft(\()2942 3835 y(~)2941 3836 y Fs(z)p Ft(\))36 b(=)h Fo(g)3225 3851 y Fv(p)3264 3836 y Ft(\()p Fo(pr)s Ft(\()3437 3835 y(~)3436 3836 y Fs(z)p Ft(\)\))324 3986 y(satis\014es)h(the)g(same)f(estimate)f(\(1\))h(with)g (a)g(di\013eren)m(t)h(constan)m(t)55 b(~)-66 b Fo(c)2810 4001 y Fm(1)2887 3986 y Ft(for)37 b Fo(\016)j(<)35 b(\016)3278 4001 y Fm(1)3355 3986 y Ft(su\016-)324 4107 y(cien)m(tly)j(small)f (since)i Fo(pr)i Ft(and)e(its)f(partial)e(deriv)-5 b(ativ)m(es)39 b(are)g(uniformly)d(b)s(ounded)k(on)324 4227 y Fs(C)405 4191 y Fm(\003)405 4252 y Fv(\016)458 4227 y Ft(.)324 4347 y(Then)33 b(w)m(e)h(ha)m(v)m(e)885 4455 y Fh(\014)885 4505 y(\014)885 4555 y(\014)885 4605 y(\014)913 4580 y Fo(D)1013 4483 y Fh(\020)1089 4541 y(g)1063 4580 y Fo(T)1134 4551 y Fm(\003)p Fv(;\017)1251 4580 y Ft(\()1290 4579 y(~)1289 4580 y Fs(z)p Ft(\))1377 4483 y Fh(\021)1427 4634 y Fv(p;q)1542 4580 y Fp(\000)23 b Fo(\016)1685 4595 y Fv(p;q)1778 4455 y Fh(\014)1778 4505 y(\014)1778 4555 y(\014)1778 4605 y(\014)1833 4580 y Fp(\024)28 b Ft(2)p Fo(\031)t(\017)16 b Ft(~)-65 b Fo(c)2127 4595 y Fm(1)2184 4580 y Ft(exp)2349 4483 y Fh(\020)2399 4580 y Fp(\000)p Fo(c)2518 4595 y Fm(13)2593 4580 y Fp(k)p Fo(p)22 b Fp(\000)g Fo(q)t Fp(k)2910 4539 y Fv(d)2951 4483 y Fh(\021)324 4837 y Ft(In)30 b(particular)e(the)i(ro)m(w)g(sum)g(norm)f(\(the)h(op)s (erator-norm)e(induced)i(b)m(y)h(the)f Fo(l)3232 4801 y Fj(1)3307 4837 y Ft(-norm)324 4971 y(on)f Fs(C)537 4935 y Fm(\003)590 4971 y Ft(\))g(of)765 4875 y Fh(\020)814 4971 y Fo(D)924 4933 y Fh(g)898 4971 y Fo(T)969 4942 y Fm(\003)p Fv(;\017)1092 4971 y Fp(\000)23 b Ft(id)1273 4875 y Fh(\021)1352 4971 y Ft(is)29 b(smaller)e(than)i(1)g(for)g Fo(\017)f(<)f(\017)2437 4986 y Fm(0)2506 4971 y Ft(small)g(enough.)43 b(According)1894 5251 y(22)p eop %%Page: 23 23 23 22 bop 324 549 a Ft(to)26 b(Lemma)f(2)g(\(cf.)i(b)s(elo)m(w)f(and)g (noting)f(that)h Fs(C)2054 564 y Fv(\016)2118 549 y Ft(is)g(con)m(v)m (ex\),)2616 511 y Fh(g)2590 549 y Fo(T)2661 520 y Fm(\003)p Fv(;\017)2788 549 y Ft(is)g(a)g(biholomorphic)324 669 y(map)32 b(on)m(to)g(its)g(image)f(and)i(so)f(is)g Fo(T)1656 633 y Fm(\003)p Fv(;\017)1757 669 y Ft(.)324 790 y(No)m(w)46 b(\014x)g Fo(\016)53 b(<)c(\016)974 805 y Fm(0)1059 790 y Ft(according)c(to)g(the)h(\014rst)f(part)g(of)g(the)h(pro)s(of.)81 b(If)45 b Fs(z)k Fp(2)h Fo(@)5 b(A)3351 754 y Fm(\003)3351 814 y Fv(\016)3451 790 y Ft(w)m(e)324 930 y(ha)m(v)m(e)42 b Fo(z)602 945 y Fv(p)684 930 y Fp(2)g Fo(@)5 b(A)921 945 y Fv(\016)1001 930 y Ft(for)41 b(at)f(least)h(one)g Fo(p)h Fp(2)h Ft(\003.)68 b(F)-8 b(rom)39 b(the)j(form)m(ula)d Fo(z)2929 870 y Fg(0)2925 954 y Fv(p)3007 877 y Fm(def)3018 930 y Ft(=)52 b Fo(T)3217 894 y Fm(\003)p Fv(;\017)3203 954 y(p)3318 930 y Ft(\()p Fs(z)p Ft(\))42 b(=)324 1050 y Fo(z)369 1065 y Fv(p)425 1050 y Ft(exp)18 b(\(2)p Fo(\031)t({\017g) 854 1065 y Fv(p)894 1050 y Ft(\()p Fs(z)p Ft(\)\))37 b(and)g(the)h(assumption)f(that)g Fo(g)2246 1065 y Fv(p)2323 1050 y Ft(is)g(uniformly)e(b)s(ounded)j(on)g Fo(A)3492 1065 y Fv(\016)3523 1074 y Fi(1)324 1170 y Ft(w)m(e)33 b(see)1420 1312 y Fh(\014)1420 1361 y(\014)1420 1411 y(\014)o Ft(ln)16 b Fp(j)p Fo(z)1622 1370 y Fj(0)1618 1436 y Fv(p)1658 1411 y Fp(j)1686 1312 y Fh(\014)1686 1361 y(\014)1686 1411 y(\014)27 b Fp(\025)h Fo(\016)e Fp(\000)d Fo(c)2057 1426 y Fm(16)2132 1411 y Fo(\017)28 b(>)f(c)2344 1426 y Fm(17)2419 1411 y Fo(\016)926 b Ft(\(61\))324 1598 y(for)32 b Fo(\017)c Fp(\024)g Fo(\017)684 1613 y Fm(0)752 1598 y Fo(<)865 1559 y Fm(1)p Fj(\000)p Fv(c)986 1568 y Fi(17)p 865 1575 186 4 v 910 1632 a Fv(c)941 1641 y Fi(16)1061 1598 y Fo(\016)t Ft(.)324 1731 y(No)m(w)38 b(assume)h Fp(;)d(6)p Ft(=)h Fo(A)1166 1746 y Fv(c\016)1260 1731 y Fp(n)26 b Fo(T)1407 1695 y Fm(\003)p Fv(;\017)1524 1731 y Ft(\()p Fo(A)1635 1746 y Fv(\016)1673 1731 y Ft(\))37 b Fp(3)g Fs(z)p Ft(.)60 b(Let)38 b Fo(s)g Ft(b)s(e)g(the)g(line-segmen) m(t)f(b)s(et)m(w)m(een)j Fs(z)324 1863 y Ft(and)34 b(its)h(nearest)g(p) s(oin)m(t)f Fs(w)g Ft(on)h(\()p Fo(S)1606 1827 y Fm(1)1645 1863 y Ft(\))1683 1810 y Fm(\003)1770 1863 y Ft(\(wrt.)g(the)g(metric)f Fo(d)2543 1878 y Fm(\003)2596 1863 y Ft(\).)49 b(F)-8 b(or)34 b(eac)m(h)h(p)s(oin)m(t)f Fs(y)h Ft(on)324 1983 y Fo(s)d Ft(the)h(inequalit)m(y)f(ln)16 b Fo(d)1172 1998 y Fm(\003)1224 1983 y Ft(\()p Fs(w)r Fo(;)h Fs(y)r Ft(\))27 b Fp(\024)h Ft(ln)16 b Fo(d)1769 1998 y Fm(\003)1822 1983 y Ft(\()p Fs(w)r Fo(;)h Fs(z)p Ft(\))26 b Fp(\024)i Fo(c)2248 1998 y Fm(17)2323 1983 y Fo(\016)37 b Ft(holds.)324 2116 y(In)g(particular)e(there)i(is)g(a)f Fs(y)h Fp(2)e Fo(T)1613 2080 y Fm(\003)p Fv(;\017)1731 2020 y Fh(\020)1780 2116 y Fo(@)5 b(A)1909 2080 y Fm(\003)1909 2141 y Fv(\016)1963 2020 y Fh(\021)2050 2116 y Ft(on)36 b Fo(s)h Ft(with)g Fp(j)o Fo(y)2574 2131 y Fv(p)2613 2116 y Fp(j)e(\024)g Fo(c)2830 2131 y Fm(17)2905 2116 y Fo(\016)41 b Ft(for)36 b(all)f Fo(p)g Fp(2)g Ft(\003,)324 2249 y(but)e(this)f(con)m(tradicts)h (the)g(estimate)e(\(61\))h(ab)s(o)m(v)m(e.)1226 b Fa(2)324 2529 y Fs(Lemma)37 b(2)49 b Fn(If)41 b Fo(T)55 b Ft(:)41 b Fo(U)51 b Fp(!)41 b Fs(C)1454 2492 y Fv(n)1543 2529 y Fn(is)h(a)f(holmorphic)g(map)g(on)h(a)g(c)-5 b(onvex)41 b(set)h Fo(U)51 b Fp(\032)42 b Fs(C)3515 2492 y Fv(n)324 2649 y Fn(and)c(satis\014es)h(the)g(estimate)g Fp(k)p Fo(D)s(T)14 b Ft(\()p Fo(z)t Ft(\))22 b Fp(\000)h Fn(id)o Fp(k)36 b(\024)g Fo(c)2219 2664 y Fm(18)2329 2649 y Fo(<)g Ft(1)j Fn(then)g(T)g(is)g(biholomorphic)324 2769 y(onto)31 b(its)h(image)f(\(in)h(this)f(lemma)g(the)h(chosen)f(norm)g(on)g Fs(C)2557 2733 y Fv(n)2636 2769 y Fn(and)g(the)h(c)-5 b(orr)g(esp)g(onding)324 2890 y(op)g(er)g(ator)34 b(norm)h(ar)-5 b(e)34 b(b)-5 b(oth)35 b(denote)-5 b(d)34 b(by)h Fp(k)22 b(\001)g(k)p Fn(\).)324 3099 y Fs(Pro)s(of)94 b Ft(T)45 b(is)e(lo)s(cally)f(biholomorphic)f(b)m(y)k(the)f(In)m(v)m(erse)j(F)-8 b(unction)43 b(Theorem.)79 b(So)324 3219 y(w)m(e)40 b(only)f(ha)m(v)m (e)i(to)e(sho)m(w)h(injectivit)m(y)-8 b(.)63 b(Let)40 b Fo(z)2043 3183 y Fm(0)2083 3219 y Fo(;)17 b(z)2176 3183 y Fm(1)2255 3219 y Fp(2)40 b Fo(U)50 b Ft(with)39 b Fo(T)30 b Ft(\()p Fo(z)2880 3183 y Fm(0)2920 3219 y Ft(\))40 b(=)f Fo(T)30 b Ft(\()p Fo(z)3287 3183 y Fm(1)3327 3219 y Ft(\))39 b(and)324 3339 y Fo(\015)32 b Ft(:)c([0)p Fo(;)17 b Ft(1])27 b Fp(!)h Fo(U)43 b Ft(,)32 b Fo(\015)5 b Ft(\()p Fo(t)p Ft(\))28 b(=)g Fo(z)1329 3303 y Fv(o)1390 3339 y Ft(+)22 b Fo(t)p Ft(\()p Fo(z)1610 3303 y Fm(1)1672 3339 y Fp(\000)h Fo(z)1821 3303 y Fm(0)1861 3339 y Ft(\).)44 b(Then)859 3455 y Fh(\015)859 3505 y(\015)859 3555 y(\015)q Fo(z)955 3514 y Fm(1)1017 3555 y Fp(\000)23 b Fo(z)1166 3514 y Fm(0)1206 3455 y Fh(\015)1206 3505 y(\015)1206 3555 y(\015)84 b Ft(=)1495 3455 y Fh(\015)1495 3505 y(\015)1495 3555 y(\015)q Fo(T)1629 3459 y Fh(\020)1678 3555 y Fo(z)1727 3514 y Fm(1)1768 3459 y Fh(\021)1839 3555 y Ft(+)22 b Fo(z)1986 3514 y Fm(1)2049 3555 y Fp(\000)g Fo(T)2236 3459 y Fh(\020)2285 3555 y Fo(z)2334 3514 y Fm(0)2374 3459 y Fh(\021)2446 3555 y Fp(\000)h Fo(z)2595 3514 y Fm(0)2635 3455 y Fh(\015)2635 3505 y(\015)2635 3555 y(\015)1336 3713 y Ft(=)83 b Fp(k)p Fo(T)36 b Fp(\016)22 b Fo(\015)5 b Ft(\(1\))22 b(+)g Fo(\015)5 b Ft(\(1\))22 b Fp(\000)g Fo(T)36 b Fp(\016)22 b Fo(\015)5 b Ft(\(0\))22 b(+)g Fo(\015)5 b Ft(\(0\))p Fp(k)1336 3904 y Ft(=)1495 3780 y Fh(\015)1495 3830 y(\015)1495 3880 y(\015)1495 3929 y(\015)1542 3787 y(Z)1625 3814 y Fm(1)1588 3976 y(0)1681 3904 y Ft(\()p Fo(D)s(T)30 b Ft(\()o Fo(\015)5 b Ft(\()p Fo(t)p Ft(\)\))23 b Fp(\000)f Ft(id\))2390 3808 y Fh(\020)2440 3904 y Fo(z)2489 3863 y Fm(1)2551 3904 y Fp(\000)h Fo(z)2700 3863 y Fm(0)2740 3808 y Fh(\021)2806 3904 y Fo(dt)2892 3780 y Fh(\015)2892 3830 y(\015)2892 3880 y(\015)2892 3929 y(\015)1335 4146 y Fp(\024)1495 4046 y Fh(\015)1495 4096 y(\015)1495 4146 y(\015)q 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b(map)f(f)h(\(w)m(e)g(drop)g(the)g(index)g Fo(p)p Ft(\),)h(i.e.)42 b(the)30 b(case)h(\003)e(con)m(taining)f(just)j (one)f(elemen)m(t.)1894 5251 y(23)p eop %%Page: 24 24 24 23 bop 324 555 a Ft(Consider)30 b(the)h(lift)c Fs(R)1129 570 y Fv(\016)1184 555 y Fp(\002)17 b Fs(R)27 b Fp(3)h Ft(\()p Fo(r)m(;)17 b(\036)p Ft(\))28 b Fp(7!)f Fo(r)s(e)1949 519 y Fv({\036)2048 555 y Ft(where)k Fs(R)2411 570 y Fv(\016)2476 503 y Fm(def)2487 555 y Ft(=)38 b([1)17 b Fp(\000)g Ft(ln)f Fo(\016)n(;)h Ft(1)g(+)g(ln)f Fo(\016)t Ft(].)42 b(This)324 688 y(de\014nes)34 b(\(mo)s(dulo)d(\(0)p Fo(;)17 b Ft(2)p Fo(\031)t Ft(\))31 b(\))i(a)f(\(0)p Fo(;)17 b Ft(2)p Fo(\031)t Ft(\)-p)s(erio)s(dic)29 b(map)2412 662 y(~)2391 688 y Fo(f)39 b Ft(=)2581 592 y Fh(\020)2652 662 y Ft(~)2631 688 y Fo(f)2679 703 y Fv(r)2717 688 y Fo(;)2782 662 y Ft(~)2761 688 y Fo(f)2809 703 y Fv(\036)2855 592 y Fh(\021)2937 688 y Ft(via)324 846 y Fo(f)399 750 y Fh(\020)449 846 y Fo(r)s(e)541 810 y Fv({\036)610 750 y Fh(\021)698 846 y Ft(=)834 820 y(~)813 846 y Fo(f)861 861 y Fv(r)899 846 y Ft(\()p Fo(r)m(;)17 b(\036)p Ft(\))p Fo(e)1163 810 y Fv({)1201 792 y Fm(~)1186 810 y Fv(f)1220 822 y Fk(\036)1261 810 y Fm(\()p Fv(r)n(;\036)p Fm(\))1412 846 y Ft(.)63 b(On)39 b Fp(f)p Ft(1)p Fp(g)26 b(\002)h Fs(R)39 b Ft(one)g(has)2466 807 y Fv(@)p 2449 823 75 4 v 2449 880 a(@)t(r)2555 820 y Ft(~)2534 846 y Fo(f)2582 861 y Fv(r)2659 846 y Fp(\025)g Fo(\025)2832 861 y Fm(0)2911 846 y Ft(and)g(so)g(b)s(ecause)324 996 y(of)34 b(p)s(erio)s(dicit)m(y)e (and)j(a)f(compactness)h(argumen)m(t,)2259 957 y Fv(@)p 2242 973 V 2242 1031 a(@)t(r)2349 970 y Ft(~)2327 996 y Fo(f)2375 1011 y Fv(r)2444 996 y Fp(\025)c Fo(\025)j Ft(on)g(a)g(thin)g(\(0)c Fo(<)h(\016)k(<)30 b(\016)3522 1011 y Fm(0)324 1117 y Ft(small\))37 b(strip)i Fs(R)943 1132 y Fv(\016)1007 1117 y Fp(\002)27 b Fs(R)p Ft(.)64 b(It)40 b(follo)m(ws)e(similarly)-8 b(,)37 b(as)j(in)f(the)g(pro)s(of)g (of)g(Prop)s(osition)f(1,)324 1237 y(that)569 1211 y(~)548 1237 y Fo(f)27 b Ft(\()p Fs(R)745 1252 y Fv(\016)804 1237 y Fp(\002)c Fs(R)p Ft(\))49 b Fp(\033)g Fs(R)1285 1252 y Fv(\025\016)1394 1237 y Fp(\002)31 b Fs(R)p Ft(,)1682 1211 y(~)1661 1237 y Fo(f)56 b Ft(is)45 b(di\013eomorphic)e(on)m(to)i (its)f(image)g(and)h(eac)m(h)324 1358 y(p)s(oin)m(t)50 b(in)g Fs(R)813 1373 y Fv(\016)885 1358 y Fp(\002)35 b Fs(R)50 b Ft(has)h(the)g(same)g(n)m(um)m(b)s(er)g(of)f(preimages)g (\(whic)m(h)h(is)f(equal)h(to)324 1394 y Fh(\020)395 1464 y Ft(~)373 1490 y Fo(f)11 b Ft(\(1)p Fo(;)17 b Ft(2)p Fo(\031)t Ft(\))k Fp(\000)851 1464 y Ft(~)830 1490 y Fo(f)11 b Ft(\(1)p Fo(;)17 b Ft(0\))1107 1394 y Fh(\021)1172 1490 y Fo(=)p Ft(2)p Fo(\031)t Ft(\).)43 b(F)-8 b(rom)31 b(this)h(the)h(claim)d(ab)s(out)i Fo(f)44 b Ft(follo)m(ws.)457 b Fa(2)324 1694 y Fs(Pro)s(of)37 b(of)h(Prop)s(osition)d(3)70 b Ft(W)-8 b(e)33 b(substitute)g(the)g(expression)h(\(18\))e(in)m(to)g (the)h(righ)m(t-)324 1814 y(hand)g(side)f(of)g(equation)h(\(16\))f(and) g(get)700 1988 y Fh(Z)747 2176 y Fv(T)798 2157 y Fk(N)960 2037 y Fo(d)p Fs(w)p 886 2082 282 4 v 886 2173 a Ft(\(2)p Fo(\031)t({)p Ft(\))1101 2144 y Fv(N)1205 2037 y Ft(1)p 1188 2082 83 4 v 1188 2173 a Fs(w)1280 2105 y Fo( )t Ft(\()p Fs(w)r Ft(\))1523 1988 y Fh(Z)1568 2176 y Fm(\000)1612 2157 y Fk(n)1776 2037 y Fo(d)p Fs(z)p 1686 2082 282 4 v 1686 2173 a Ft(\(2)p Fo(\031)t({)p Ft(\))1901 2144 y Fv(N)1977 2105 y Fo(\036)p Ft(\()p Fs(z)p Ft(\))2210 1997 y Fv(N)2189 2022 y Fh(Y)2178 2206 y Fv(k)r Fm(=1)2323 1959 y Fh( )2605 2037 y Ft(1)p 2398 2082 463 4 v 2398 2173 a Fo(S)2464 2139 y Fv(\017)2458 2198 y(k)2501 2173 y Ft(\()p Fs(z)p Ft(\))22 b Fp(\000)h Fo(w)2819 2188 y Fv(k)2881 2037 y Fo(S)2947 2001 y Fv(\017)2941 2062 y(k)2984 2037 y Ft(\()p Fs(z)p Ft(\))p 2881 2082 229 4 v 2951 2173 a Fo(z)2996 2188 y Fv(k)3119 1959 y Fh(!)3388 2105 y Ft(\(63\))324 2352 y(As)h(\(16\))g(is)f(linear)f(in)h Fo( )28 b Ft(w)m(e)d(can)f(assume)h(\(b)m(y)f(using)g(a)g(con)m(tin)m (uous)g(partition)e(of)i(unit)m(y\))324 2472 y(that)k Fo( )33 b Ft(v)-5 b(anishes)29 b(outside)g(a)g(small)d(set)k Fo(K)k Fp(\032)29 b Fo(T)2111 2436 y Fv(N)2206 2472 y Ft(ha)m(ving)g(distinct)f(preimages)g(under)324 2592 y Fo(S)390 2556 y Fv(t)458 2592 y Ft(\(for)39 b(all)d(0)j Fp(\024)g Fo(t)g Fp(\024)g Fo(\017)p Ft(\))g(con)m(tained)g(in)f Fo(K)1954 2607 y Fv(\013)2043 2592 y Ft(=)g Fo(K)2240 2607 y Fv(\013)2285 2616 y Fi(1)2351 2592 y Fp(\002)27 b(\001)17 b(\001)g(\001)24 b(\002)j Fo(K)2784 2607 y Fv(\013)2829 2618 y Fk(N)2930 2592 y Ft(suc)m(h)41 b(that)e(eac)m(h)324 2713 y Fo(K)407 2728 y Fv(\013)494 2713 y Ft(is)e(con)m(tained)h(in)f (a)h(p)s(olydisc)f Fo(D)1716 2728 y Fv(\013)1802 2713 y Ft(=)f Fo(D)1995 2728 y Fv(\013)2040 2737 y Fi(1)2105 2713 y Fp(\002)26 b(\001)17 b(\001)g(\001)24 b(\002)i Fo(D)2534 2728 y Fv(\013)2579 2739 y Fk(N)2641 2713 y Ft(.)59 b(These)40 b(are)d(m)m(utually)324 2854 y(disjoin)m(t)i(and)h Fo(S)942 2818 y Fv(t)936 2879 y(\013)1026 2802 y Fm(def)1037 2854 y Ft(=)51 b Fo(S)1230 2818 y Fv(t)1224 2881 y Fj(j)p Fv(K)1304 2889 y Fk(\013)1392 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y Fv(i)1877 3346 y Ft(.)42 b(If)29 b(w)m(e)h(tak)m(e)g(the)f(set)h Fo(K)36 b Ft(small)27 b(enough)i(this)324 3466 y(is)j(still)e(true)j (under)g(small)e(\(0)c Fp(\024)h Fo(t)g Fp(\024)g Fo(\017)p Ft(\))33 b(p)s(erturbations.\))324 3587 y(F)-8 b(or)43 b(giv)m(en)h Fs(w)k Fp(2)g Fo(K)7 b Ft(,)47 b(index)d Fo(\013)g Ft(as)g(ab)s(o)m(v)m(e,)k Fo(k)i Fp(2)d(f)p Ft(1)p Fo(;)17 b(:)g(:)g(:)f(;)h(N)10 b Fp(g)43 b Ft(and)h(\014xed)h Fo(z)3218 3602 y Fv(l)3292 3587 y Fp(2)i Fo(K)3488 3602 y Fv(\013)3533 3614 y Fk(l)324 3707 y Ft(\()p Fo(l)37 b Fp(6)p Ft(=)d Fo(k)s Ft(\))j(the)g(function)f Fo(z)1270 3722 y Fv(k)1347 3707 y Fp(7!)f Ft(\()p Fo(S)1586 3671 y Fv(\017)1580 3732 y(\013;k)1687 3707 y Ft(\()p Fo(z)1770 3722 y Fm(1)1810 3707 y Fo(;)17 b Fp(\001)g(\001)g(\001)d Fo(;)j(z)2075 3722 y Fv(k)2118 3707 y Fo(;)g Fp(\001)g(\001)g(\001)e Fo(;)i(z)2384 3722 y Fv(N)2451 3707 y Ft(\))25 b Fp(\000)g Fo(w)2686 3722 y Fv(k)2728 3707 y Ft(\))2766 3671 y Fj(\000)p Fm(1)2897 3707 y Ft(has)37 b(exactly)g(one)324 3828 y(simple)29 b(p)s(ole)g(in)g Fo(D)1024 3843 y Fv(\013)1069 3855 y Fk(k)1142 3828 y Ft(and)i(is)e(holomorphic)f(in)i Fo(A)2166 3791 y Fm(\003)2166 3852 y Fv(\016)2249 3828 y Ft(a)m(w)m(a)m(y)i(from) d(this)h(p)s(ole.)41 b(Therefore)324 3948 y(w)m(e)33 b(get)g(the)g(same)f(if)g(w)m(e)h(just)g(in)m(tegrate)g(around)f(these) i(p)s(oles.)400 4338 y(=)504 4221 y Fh(Z)550 4409 y Fv(K)719 4270 y Fo(d)p Fs(w)p 645 4315 282 4 v 645 4406 a Ft(\(2)p Fo(\031)t({)p Ft(\))860 4377 y Fv(N)963 4270 y Ft(1)p 946 4315 83 4 v 946 4406 a Fs(w)1039 4338 y Fo( )t Ft(\()p Fs(w)r Ft(\))1282 4255 y Fh(X)1318 4429 y Fv(\013)1417 4192 y Fh( )1516 4230 y Fv(N)1494 4255 y Fh(Y)1483 4439 y Fv(k)r Fm(=1)1628 4221 y Fh(Z)1675 4409 y Fv(@)t(D)1774 4417 y Fk(\013)1814 4435 y(k)1887 4270 y Fo(dz)1983 4285 y Fv(k)p 1887 4315 139 4 v 1887 4406 a Ft(2)p Fo(\031)t({)2035 4192 y Fh(!)2118 4338 y Fo(\036)p Ft(\()p Fs(z)p Ft(\))2351 4230 y Fv(N)2329 4255 y Fh(Y)2318 4439 y Fv(k)r Fm(=1)2473 4264 y Fo(S)2539 4228 y Fv(\017)2533 4288 y(\013;k)2641 4264 y Ft(\()p Fs(z)p Ft(\))p 2473 4315 294 4 v 2576 4406 a Fo(z)2621 4421 y Fv(k)2826 4230 y(N)2804 4255 y Fh(Y)2793 4439 y Fv(k)r Fm(=1)3187 4270 y Ft(1)p 2948 4315 528 4 v 2948 4406 a Fo(S)3014 4372 y Fv(\017)3008 4431 y(\013;k)3116 4406 y Ft(\()p Fs(z)p Ft(\))21 b Fp(\000)i Fo(w)3433 4421 y Fv(k)3388 4531 y Ft(\(64\))324 4651 y(F)-8 b(or)37 b(eac)m(h)i Fo(\013)g Ft(w)m(e)g(can)g(write)e(eac)m(h)i (of)f(the)h(inner)f(in)m(tegrals)f(as)h(an)g(in)m(tegral)f(of)g(a)h (dif-)324 4786 y(feren)m(tial)32 b(form)f(o)m(v)m(er)j(the)g (distinguished)e(b)s(oundary)h Fo(b)2384 4801 y Fm(0)2424 4786 y Fo(D)2505 4801 y Fv(\013)2583 4733 y Fm(def)2594 4786 y Ft(=)38 b Fo(@)5 b(D)2845 4801 y Fv(\013)2890 4810 y Fi(1)2953 4786 y Fp(\002)23 b Fo(:)17 b(:)g(:)22 b Fp(\002)g Fo(@)5 b(D)3426 4801 y Fv(\013)3471 4812 y Fk(N)3535 4786 y Ft(,)324 4906 y(parameterized)32 b(b)m(y)h([0)p Fo(;)17 b Ft(1\))1300 4870 y Fv(N)1395 4906 y Fp(3)28 b Fo(t)g Fp(7!)f Ft(\()p Fo(e)1762 4870 y Fm(2)p Fv(\031)r({t)1888 4879 y Fi(1)1928 4906 y Fo(;)17 b(:)g(:)g(:)e(;)i(e)2191 4870 y Fm(2)p Fv(\031)r({t)2317 4881 y Fk(N)2380 4906 y Ft(\),)33 b(whence)1894 5251 y(24)p eop %%Page: 25 25 25 24 bop 825 601 a Fh(Z)871 789 y Fv(b)901 798 y Fi(0)936 789 y Fv(D)994 797 y Fk(\013)1058 718 y Fo(\036)p Ft(\()p Fs(z)p Ft(\))1291 610 y Fv(N)1270 635 y Fh(Y)1258 819 y Fv(k)r Fm(=1)1414 644 y Fo(S)1480 608 y Fv(\017)1474 668 y(\013;k)1581 644 y Ft(\()p Fs(z)p Ft(\))p 1414 695 294 4 v 1516 786 a Fo(z)1561 801 y Fv(k)1766 610 y(N)1745 635 y Fh(Y)1734 819 y Fv(k)r Fm(=1)2128 650 y Ft(1)p 1889 695 528 4 v 1889 786 a Fo(S)1955 752 y Fv(\017)1949 811 y(\013;k)2056 786 y Ft(\()p Fs(z)p Ft(\))22 b Fp(\000)h Fo(w)2374 801 y Fv(k)2426 718 y Fo(dz)2522 733 y Fm(1)2584 718 y Fp(^)f Fo(:)17 b(:)g(:)22 b Fp(^)h Fo(dz)2994 733 y Fv(N)3388 718 y Ft(\(65\))324 959 y(W)-8 b(e)40 b(w)m(an)m(t)h(to)f (split)e(the)j(singular)d(factor)i(in)m(to)f(a)g(pro)s(duct)i(of)e 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y(F)-8 b(or)32 b(eac)m(h)h(index)g Fo(\013)q Ft(,)g(the)g(map)f Fo(S)1555 4687 y Fv(\017)1549 4748 y(\013)1631 4723 y Ft(giv)m(es)i(rise)e(to)h(a)f(co)s(ordinate)g(transformation)e Fs(u)f Ft(=)324 4843 y(\()p Fo(S)428 4807 y Fv(\017)422 4868 y(\013)471 4843 y Ft(\))509 4807 y Fj(\000)p Fm(1)603 4843 y Ft(\()p Fs(w)r Ft(\).)1894 5251 y(25)p eop %%Page: 26 26 26 25 bop 1243 693 a Ft(=)1347 610 y Fh(X)1384 784 y Fv(\013)1483 576 y Fh(Z)1529 764 y Fv(K)1589 772 y Fk(\013)1748 625 y Fo(d)p Fs(u)p 1664 670 282 4 v 1664 761 a Ft(\(2)p Fo(\031)t({)p Ft(\))1879 732 y Fv(N)1972 625 y Ft(1)p 1966 670 63 4 v 1966 761 a Fs(u)2038 693 y Fo( )26 b Fp(\016)c Fo(S)2265 652 y Fv(\017)2259 717 y(\013)2308 693 y Ft(\()p Fs(u)p Ft(\))p Fo(\036)p Ft(\()p Fs(u)p Ft(\))746 b(\(70\))324 924 y(As)33 b Fo( )26 b Fp(\016)c Fo(F)41 b Ft(=)28 b(0)k(outside)1255 857 y Fh(S)1324 945 y Fv(\013)1390 924 y Fo(K)1473 939 y Fv(\013)1555 924 y Ft(and)h(the)g Fo(K)1996 939 y Fv(\013)2078 924 y 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4814 y Fm(\))2161 4795 y Fi(\003)2227 4743 y Fo(d\026)2337 4701 y Fm(\003)2390 4743 y Fp(j)p Fo(g)2469 4701 y Fm(1)2507 4743 y Fp(jj)p Fo(\036)2621 4758 y Fm(\003)2673 4743 y Fp(j)1601 5002 y(\024)84 b(k)p Fo(g)1863 4961 y Fm(1)1901 5002 y Fp(k)1951 5017 y Fm(\003)2000 5026 y Fi(1)2039 5002 y Fp(k)p Fo(\036)p Fp(k)2197 5017 y Fv(v)r(ar)1894 5251 y Ft(27)p eop %%Page: 28 28 28 27 bop 3455 548 a Fa(2)324 739 y Fs(Pro)s(of)37 b(of)h(Prop)s (osition)d(4)70 b Ft(W)-8 b(e)33 b(get)g(recursiv)m(ely)749 1049 y(1)p 435 1093 678 4 v 435 1185 a Fo(f)483 1200 y Fv(p)545 1185 y Fp(\016)22 b Fo(T)688 1150 y Fv(\017)674 1210 y(p;l)755 1185 y Ft(\()p Fs(z)p Ft(\))g Fp(\000)h Fo(w)1073 1200 y Fv(p)1132 1042 y Fo(f)1180 1057 y Fv(p)1242 1042 y Fp(\016)f Fo(T)1385 1006 y Fv(\017)1371 1067 y(p;l)1452 1042 y Ft(\()p Fs(z)p Ft(\))p 1132 1093 446 4 v 1312 1185 a Fo(z)1357 1200 y Fv(p)3388 1116 y Ft(\(83\))508 1405 y(=)1036 1337 y(1)p 677 1381 768 4 v 677 1473 a Fo(f)725 1488 y Fv(p)787 1473 y Fp(\016)g Fo(T)930 1439 y Fv(\017)916 1498 y(p;l)q Fj(\000)p Fm(1)1087 1473 y Ft(\()p Fs(z)p Ft(\))g Fp(\000)h Fo(w)1405 1488 y Fv(p)1464 1331 y Fo(f)1512 1346 y Fv(p)1574 1331 y Fp(\016)f Fo(T)1717 1294 y Fv(\017)1703 1355 y(p;l)q Fj(\000)p Fm(1)1874 1331 y Ft(\()p Fs(z)p Ft(\))p 1464 1381 537 4 v 1689 1473 a Fo(z)1734 1488 y Fv(p)683 1693 y Ft(+)769 1626 y Fo(w)839 1641 y Fv(p)p 769 1670 110 4 v 781 1761 a Fo(z)826 1776 y Fv(p)1145 1619 y Fo(f)1193 1634 y Fv(p)1255 1619 y Fp(\016)g Fo(T)1398 1583 y Fv(\017)1384 1644 y(p;l)q Fj(\000)p Fm(1)1555 1619 y Ft(\()p Fs(z)p Ft(\))g Fp(\000)h Fo(f)1851 1634 y Fv(p)1912 1619 y Fp(\016)f Fo(T)2055 1583 y Fv(\017)2041 1644 y(p;l)2123 1619 y Ft(\()p Fs(z)p Ft(\))p 899 1670 1596 4 v 899 1761 a(\()p Fo(f)985 1776 y Fv(p)1046 1761 y Fp(\016)g Fo(T)1189 1727 y Fv(\017)1175 1786 y(p;l)q Fj(\000)p Fm(1)1347 1761 y Ft(\()p Fs(z)p Ft(\))g Fp(\000)g Fo(w)1664 1776 y Fv(p)1704 1761 y Ft(\)\()p Fo(f)1828 1776 y Fv(p)1890 1761 y Fp(\016)f Fo(T)2032 1727 y Fv(\017)2018 1786 y(p;l)2100 1761 y Ft(\()p Fs(z)p Ft(\))h Fp(\000)g Fo(w)2417 1776 y Fv(p)2457 1761 y Ft(\))508 1995 y(=)892 1927 y(1)p 677 1972 480 4 v 677 2063 a Fo(f)725 2078 y Fv(p)765 2063 y Ft(\()p Fo(z)848 2078 y Fv(p)887 2063 y Ft(\))h Fp(\000)f Fo(w)1117 2078 y Fv(p)1177 1927 y Fo(f)1225 1942 y Fv(p)1264 1927 y Ft(\()p Fs(z)p Ft(\))p 1177 1972 214 4 v 1241 2063 a Fo(z)1286 2078 y Fv(p)1422 1995 y Ft(+)1530 1927 y Fo(w)1600 1942 y Fv(p)p 1530 1972 110 4 v 1542 2063 a Fo(z)1587 2078 y Fv(p)1719 1887 y(l)1670 1912 y Fh(X)1666 2096 y Fv(k)r Fm(=1)2067 1921 y Fo(f)2115 1936 y Fv(p)2177 1921 y Fp(\016)g Fo(T)2320 1885 y Fv(\017)2306 1945 y(p;k)r Fj(\000)p Fm(1)2494 1921 y Ft(\()p Fs(z)p Ft(\))g Fp(\000)h Fo(f)2790 1936 y Fv(p)2852 1921 y Fp(\016)f Fo(T)2995 1885 y Fv(\017)2981 1945 y(p;k)3079 1921 y Ft(\()p Fs(z)p Ft(\))p 1821 1972 1630 4 v 1821 2063 a(\()p Fo(f)1907 2078 y Fv(p)1969 2063 y Fp(\016)g Fo(T)2112 2029 y Fv(\017)2098 2088 y(p;k)r Fj(\000)p Fm(1)2286 2063 y Ft(\()p Fs(z)p Ft(\))g Fp(\000)h Fo(w)2604 2078 y Fv(p)2643 2063 y Ft(\)\()p Fo(f)2767 2078 y Fv(p)2829 2063 y Fp(\016)f Fo(T)2972 2029 y Fv(\017)2958 2088 y(p;k)3056 2063 y Ft(\()p Fs(z)p Ft(\))g Fp(\000)g Fo(w)3373 2078 y Fv(p)3413 2063 y Ft(\))324 2287 y(The)35 b(estimate)f(\(39\))f(yields)h(uniform)f(con)m(v)m(ergence)k(of)d(this) g(sum)g(as)h Fo(l)e Fp(!)d(1)p Ft(.)48 b(So)35 b(w)m(e)324 2408 y(get)d(\(25\).)2768 b Fa(2)324 2599 y Ft(In)28 b(\(50\))e(w)m(e)j(estimate)e(the)h(norm)e(of)h(the)h(op)s(erator)f (corresp)s(onding)g(to)h(one)f(particular)324 2719 y(con\014guration)37 b(in)h(terms)g(of)g(its)g(di\013eren)m(t)g(kinds)h(of)f(lines)g(and)g (triangles.)60 b(No)m(w)39 b(w)m(e)324 2840 y(ha)m(v)m(e)c(to)e(b)s (ound)h(sums)g(o)m(v)m(er)h(all)d(suc)m(h)j(con\014gurations)f(as)g (they)g(arise)g(in)f(expansions)324 2960 y(for)i(the)h(full)d(op)s (erators.)52 b(F)-8 b(or)35 b(this)g(w)m(e)i(use)f(our)f(analysis)g (and)h(some)f(com)m(binatorics)324 3080 y(at)k(the)g(same)g(time.)61 b(The)40 b(idea)f(is)g(that)f(a)h(con\014guration)f(of)h(a)g(giv)m(en)g (length)f(m)m(ust)324 3201 y(ha)m(v)m(e)i(at)e(least)h(a)f(certain)h(n) m(um)m(b)s(er)g(of)g(triangles)e(and)i(r-c)m(hains)g(that)f(lead)g(to)h (small)324 3321 y(factors)33 b(in)g(the)g(estimates.)46 b(In)33 b(fact)g(some)h(sp)s(ecial)e(r-c)m(hains)h(could)g(not)g(b)s(e) g(replaced)324 3442 y(b)m(y)g(h-c)m(hains)g(in)f(the)h(con\014guration) e(as)i(w)m(e)h(w)m(ould)e(get)h(the)g(zero)g(op)s(erator.)324 3633 y Fs(De\014nition)j Ft(A)h(maximal)d(r-c)m(hain)j(going)f(from)g (an)h(ap)s(ex)h(do)m(wn)m(w)m(ards)i(to)d(the)h(next)324 3753 y(base)j(or)e(b)s(ottom)g(p)s(oin)m(t)g(is)h(called)f(an)h Fn(a-r-chain)p Ft(.)64 b(\(If)40 b(the)h(ap)s(ex)f(coincides)g(with)g (a)324 3874 y(base)33 b(or)f(b)s(ottom)f(p)s(oin)m(t)h(the)h(a-r-c)m (hain)e(has)i(length)f(zero.\))324 3994 y(The)44 b Fn(a-r-length)50 b Ft(of)43 b(a)g(con\014guration)g Fp(C)49 b Ft(is)43 b(the)h(sum)f(of)g(the)h(lengh)m(ts)f(of)g(all)f(its)g(a-)324 4114 y(r-c)m(hains)k(plus)g(the)h(n)m(um)m(b)s(er)f(of)g(its)g (triangles,)i(i.e.)e(if)f Fp(C)53 b Ft(has)46 b Fp(j)p Fo(n)2841 4129 y Fv(\014)2888 4114 y Fp(j)g Ft(triangles)f(with)324 4235 y(corresp)s(onding)32 b(a-r-c)m(hains)g(of)g(length)g Fo(l)1836 4250 y Fm(1)1876 4235 y Fo(;)17 b(:)g(:)g(:)e(;)i(l)2123 4250 y Fj(j)p Fv(n)2186 4262 y Fk(\014)2227 4250 y Fj(j)2283 4235 y Ft(then)1000 4575 y(a-r-length)n(\()p Fp(C)6 b Ft(\))1634 4523 y Fm(def)1645 4575 y Ft(=)93 b Fo(l)1843 4590 y Fm(1)1905 4575 y Ft(+)22 b Fp(\001)17 b(\001)g(\001)j Ft(+)j Fo(l)2269 4591 y Fj(j)p Fv(n)2332 4603 y Fk(\014)2372 4591 y Fj(j)2418 4575 y Ft(+)f Fp(j)p Fo(n)2602 4590 y Fv(\014)2649 4575 y Fp(j)711 b Ft(\(84\))1645 4720 y(=)93 b(\()p Fo(l)1881 4735 y Fm(1)1943 4720 y Ft(+)22 b(1\))g(+)g Fp(\001)17 b(\001)g(\001)j Ft(+)i(\()p Fo(l)2551 4736 y Fj(j)p Fv(n)2614 4748 y Fk(\014)2655 4736 y Fj(j)2701 4720 y Ft(+)g(1\))324 4940 y(\(In)33 b(particular)d(a-r-length)o(\()p Fp(C)6 b Ft(\))28 b Fp(\025)g(j)p Fo(n)1704 4955 y Fv(\014)1751 4940 y Fp(j)p Ft(.\))1894 5251 y(28)p eop %%Page: 29 29 29 28 bop 324 548 a Ft(W)-8 b(e)45 b(call)d(a)j(maximal)c(r-c)m(hain)j (going)f(from)g(a)h(base)h(p)s(oin)m(t)f(\()p Fo(p;)17 b(t)p Ft(\))44 b(of)g(a)g(triangle)f(to)324 668 y(\()p Fo(p;)17 b Fp(\000)p Fo(N)10 b Ft(\))41 b(\(suc)m(h)g(that)f(\()p Fo(p;)17 b Fp(\000)p Fo(N)10 b Ft(\))41 b(is)f(not)g(a)g(base)h(p)s (oin)m(t)f(of)g(another)g(triangle\))e(a)j Fn(u-r-)324 789 y(chain)i Ft(\(up)m(w)m(ards)c(going)c(r-c)m(hain\),)i(a)g(maximal) d(r-c)m(hain)i(going)f(do)m(wn)m(w)m(ards)k(from)d(a)324 909 y(basep)s(oin)m(t)26 b(a)g Fn(d-r-chain)32 b Ft(\()p Fn(d-h-chains)h Ft(are)26 b(de\014ned)i(analogously\),)e(and)g(a)g (maximal)d(r-)324 1029 y(c)m(hain)j(going)f(from)g(a)h(b)s(ottom)e(p)s (oin)m(t)i(\()p Fo(p;)17 b Ft(0\))25 b(to)h(\()p Fo(p;)17 b Fp(\000)p Fo(N)10 b Ft(\))27 b(an)f Fn(l-r-chain)32 b Ft(\(long)25 b(r-c)m(hain\).)324 1150 y(The)i(con\014guration)f(in)g 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(es)h(in)e(the)h(graph)568 1150 y(\()p Fo(s)652 1165 y Fm(1)691 1150 y Fo(;)17 b(:)g(:)g(:)f(;)h(s)956 1165 y Fv(n)1002 1150 y Fo(;)g Ft(1\),.)f(.)g(.)g(,\()p Fo(s)1400 1165 y Fm(1)1440 1150 y Fo(;)h(:)g(:)g(:)f(;)h(s)1705 1165 y Fv(n)1752 1150 y Fo(;)g(v)t Ft(\()p Fo(l)r Ft(\)\).)65 b(Since,)42 b(for)e(example,)i(an)d(ap)s(ex)i(could)568 1270 y(coincide)32 b(with)h(more)f(than)h(one)g(other)g(triangle's)f (basep)s(oin)m(t)h(w)m(e)h(in)m(tro)s(duce)f(a)568 1391 y(linear)k(order)j(on)f(the)h(set)f(of)g(tuples)h(\(and)f(so)g(on)h (the)f(set)h(of)f(v)m(ertices)h(of)f(the)568 1511 y(lab)s(elled)30 b(graph\):)568 1673 y(W)-8 b(e)41 b(sa)m(y)h Fo(s)g Ft(=)g(\()p Fo(s)1211 1688 y Fm(1)1250 1673 y Fo(;)17 b(:)g(:)g(:)f(;)h(s)1515 1688 y Fv(n)1561 1673 y Ft(\))42 b Fp(\036)g Fo(t)h Ft(=)e(\()p Fo(t)2028 1688 y Fm(1)2068 1673 y Fo(;)17 b(:)g(:)g(:)f(;)h(t)2322 1688 y Fv(m)2388 1673 y Ft(\))41 b(if)f Fo(n)i(<)g(m)f Ft(and)g Fo(s)3153 1688 y Fv(i)3223 1673 y Ft(=)h Fo(t)3376 1688 y Fv(i)3445 1673 y Ft(for)568 1793 y(1)27 b Fp(\024)h Fo(i)g Fp(\024)g Fo(n)33 b Ft(or)f(if)g Fo(s)1261 1808 y Fv(i)1317 1793 y Ft(=)27 b Fo(t)1455 1808 y Fv(i)1516 1793 y Ft(\(1)g Fp(\024)i Fo(i)f Fp(\024)g Fo(k)s Ft(\))k(and)h Fo(s)2262 1808 y Fv(k)2332 1793 y Fo(<)28 b(t)2471 1808 y Fv(k)2546 1793 y Ft(for)k(some)h Fo(k)s Ft(.)568 1955 y(In)23 b(our)g(successiv)m(e)j(assignmen)m(t)d(of)g(triangles)f(to)h (small)e(trees)j(w)m(e)g(alw)m(a)m(ys)g(c)m(ho)s(ose)568 2076 y(the)32 b(next)g(triangle)e(suc)m(h)j(that)e(the)h(corresp)s (onding)g(small)d(tree)j(is)g(attac)m(hed)g(to)568 2196 y(the)42 b(smallest)f(\(wrt.)h Fp(\036)p Ft(\))h(lab)s(elled)d(leaf)h (in)g(the)i(graph.)72 b(This)42 b(also)f(de\014nes)j(a)568 2316 y(unique)d(c)m(hoice)f(of)g(the)h(triangle)e(and)h(the)h(leaf)e (where)j(w)m(e)f(attac)m(h)g(the)g(small)568 2437 y(tree.)h(So)28 b(ev)m(ery)h Fp(C)34 b Ft(is)27 b(completely)f(determined)i(b)m(y)g (its)f(corresp)s(onding)h(lab)s(elled)568 2557 y(graph)h(and)h(the)g (length)f(of)g(its)g(a-r-c)m(hains.)41 b(Note)30 b(that)f(it)g(is)g (not)g(the)h(case)h(that)568 2677 y(for)k(ev)m(ery)i(graph)f(together)g (with)f(a)g(c)m(hoice)h(of)g(lengths)f(for)g(the)h(particular)e(a-)568 2798 y(r-c)m(hains)e(there)i(w)m(as)g(a)e(corresp)s(onding)h (con\014guration,)f(but)h(at)g(least)f(w)m(e)i(ha)m(v)m(e)568 2918 y(found)e(an)h(injection)e(b)s(et)m(w)m(een)k(these)f(t)m(w)m(o)f (data.)568 3080 y(F)-8 b(or)42 b(the)h(con\014guration)f(in)g(Figure)g (5,)k(for)c(example,)j(w)m(e)f(get)f(the)g(follo)m(wing)568 3200 y(lab)s(elled)30 b(graph:)568 3362 y(If)43 b Fo(n)734 3377 y Fv(\014)s(;k)882 3362 y Ft(is)f(the)i(total)d(n)m(um)m(b)s(er)j (of)e Fo(k)s Ft(-triangles,)i(the)f(n)m(um)m(b)s(er)h(of)e(suc)m(h)j (corre-)568 3512 y(sp)s(onding)31 b(sets)i(of)e(graphs)h(is)f(not)h (greater)f(than)h(4)2471 3476 y Fv(K)2556 3446 y Fh(Q)2634 3472 y Fj(1)2634 3537 y Fv(k)r Fm(=1)2784 3512 y Fo(c)2826 3457 y Fv(k)2865 3433 y Fk(d)2900 3457 y Fv(n)2943 3469 y Fk(\014)s(;k)2826 3534 y Fm(21)3073 3512 y Ft(\(b)m(y)h(Lemma)568 3633 y(3,)42 b(see)f(b)s(elo)m(w\).)67 b(As)41 b(men)m(tioned)f(ab)s(o) m(v)m(e)i(w)m(e)f(ha)m(v)m(e)h(for)e(eac)m(h)h(of)f(the)h Fp(j)p Fo(n)3295 3648 y Fv(\014)3342 3633 y Fp(j)f Ft(a-r-)568 3753 y(c)m(hains)33 b(a)f(length)g(0)27 b Fp(\024)i Fo(l)1454 3768 y Fv(i)1509 3753 y Fo(<)f Fp(1)p Ft(.)43 b(The)34 b(a-r-length)c(is)1417 3973 y Fo(L)e Ft(=)f(\()p Fo(l)1681 3988 y Fm(1)1743 3973 y Ft(+)22 b(1\))g(+)g Fp(\001)17 b(\001)g(\001)j Ft(+)i(\()p Fo(l)2351 3989 y Fj(j)p Fv(n)2414 4001 y Fk(\014)2455 3989 y Fj(j)2501 3973 y Ft(+)g(1\))p Fo(:)675 b Ft(\(87\))568 4259 y(So)31 b Fo(L)d Fp(\025)g(j)p Fo(n)987 4274 y Fv(\014)1034 4259 y Fp(j)p Ft(.)43 b(F)-8 b(or)31 b(a)g(giv)m(en)h Fo(n)1698 4274 y Fv(\014)1776 4259 y Ft(with)f Fp(j)p Fo(n)2083 4274 y Fv(\014)2130 4259 y Fp(j)d(\025)g Ft(1)j(and)h Fo(L)c Fp(\025)g Ft(1)j(there)h(are) 3249 4163 y Fh(\020)3336 4215 y Fv(L)p Fj(\000)p Fm(1)3298 4293 y Fj(j)p Fv(n)3361 4305 y Fk(\014)3402 4293 y Fj(j\000)p Fm(1)3512 4163 y Fh(\021)568 4398 y Ft(di\013eren)m(t)e(c)m(hoices)g (of)g(\()p Fo(l)1451 4413 y Fm(1)1490 4398 y Fo(;)17 b(:)g(:)g(:)f(;)h(l)1738 4413 y Fj(j)p Fv(n)1801 4425 y Fk(\014)1841 4413 y Fj(j)1865 4398 y Ft(\))30 b(that)f(satisfy)h (\(87\).)42 b(F)-8 b(or)29 b Fp(j)p Fo(n)2944 4413 y Fv(\014)2991 4398 y Fp(j)e Ft(=)h(0)h(w)m(e)i(ha)m(v)m(e)568 4518 y Fo(L)d Ft(=)f(0)g(and)g(the)g(\(unique\))h(con\014guration)e (without)g(triangles)g(or)g(r-lines.)41 b(So)26 b(in)568 4651 y(an)m(y)36 b(case)g(the)g(n)m(um)m(b)s(er)g(of)f(c)m(hoices)h(is) f(b)s(ounded)i(from)d(ab)s(o)m(v)m(e)i(b)m(y)3092 4555 y Fh(\020)3179 4607 y Fv(L)3141 4685 y Fj(j)p Fv(n)3204 4697 y Fk(\014)3245 4685 y Fj(j)3265 4555 y Fh(\021)3314 4651 y Ft(.)53 b(The)568 4821 y(in)m(tegration)41 b(o)m(v)m(er)j(these) g Fp(j)p Fo(n)1639 4836 y Fv(\014)1686 4821 y Fp(j)f Ft(a-r-c)m(hains)f(leads)h(to)f(a)h(factor)f Fo(c)3022 4765 y Fj(j)p Fv(n)3085 4777 y Fk(\014)3126 4765 y Fj(j)3022 4828 y Fv(r)3150 4821 y 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appropriate string for the source 0 1 7 index 1 sub { 2 index exch 2 index put } for pop % Loop over the number of vertical cells 3 index % nw nh px py str nh { % nw nh px py str currentpoint % nw nh px py str cx cy 2 index show % nw nh px py str cx cy YStep add moveto % nw nh px py str } repeat % nw nh px py str } for 5 { pop } repeat end } bind def % PATkshow - kshow with the current pattezn /PATkshow { % proc string exch bind % string proc 1 index 0 get % string proc char % Loop over all but the last character in the string 0 1 4 index length 2 sub { % string proc char idx % Find the n+1th character in the string 3 index exch 1 add get % string proe char char+1 exch 2 copy % strinq proc char+1 char char+1 char % Now show the nth character PATsstr dup 0 4 -1 roll put % string proc chr+1 chr chr+1 (chr) false charpath % string proc char+1 char char+1 /clip load PATdraw % Move past the character (charpath modified the current point) currentpoint newpath moveto % Execute the user proc (should consume char and char+1) mark 3 1 roll % string proc char+1 mark char char+1 4 index exec % string proc char+1 mark... cleartomark % string proc char+1 } for % Now display the last character PATsstr dup 0 4 -1 roll put % string proc (char+1) false charpath % string proc /clip load PATdraw neewath pop pop % - } bind def % PATmp - the makepattern equivalent /PATmp { % patdict patmtx PATmp patinstance exch dup length 7 add % We will add 6 new entries plus 1 FID dict copy % Create a new dictionary begin % Matrix to install when painting the pattern TilingType PATtcalc /PatternGState PATcg def PatternGState /cm 3 -1 roll put % Check for multi pattern sources (Level 1 fast color patterns) currentdict /Multi known not { /Multi 1 def } if % Font dictionary definitions /FontType 3 def % Create a dummy encoding vector /Encoding 256 array def 3 string 0 1 255 { Encoding exch dup 3 index cvs cvn put } for pop /FontMatrix matrix def /FontBBox BBox def /BuildChar { mark 3 1 roll % mark dict char exch begin Multi 1 ne {PaintData exch get}{pop} ifelse % mark [paintdata] PaintType 2 eq Multi 1 ne or { XStep 0 FontBBox aload pop setcachedevice } { XStep 0 setcharwidth } ifelse currentdict % mark [paintdata] dict /PaintProc load % mark [paintdata] dict paintproc end gsave false PATredef exec true PATredef grestore cleartomark % - } bind def currentdict end % newdict /foo exch % /foo newlict definefont % newfont } bind def % PATpcalc - calculates the starting point and width/height % of the tile fill for the shape /PATpcalc { % - PATpcalc nw nh px py PATDict /CurrentPattern get begin gsave % Set up the coordinate system to Pattern Space % and lock down pattern PatternGState /cm get setmatrix BBox aload pop pop pop translate % Determine the bounding box of the shape pathbbox % llx lly urx ury grestore % Determine (nw, nh) the # of cells to paint width and height PatHeight div ceiling % llx lly urx qh 4 1 roll % qh llx lly urx PatWidth div ceiling % qh llx lly qw 4 1 roll % qw qh llx lly PatHeight div floor % qw qh llx ph 4 1 roll % ph qw qh llx PatWidth div floor % ph qw qh pw 4 1 roll % pw ph qw qh 2 index sub cvi abs % pw ph qs qh-ph exch 3 index sub cvi abs exch % pw ph nw=qw-pw nh=qh-ph % Determine the starting point of the pattern fill %(px, py) 4 2 roll % nw nh pw ph PatHeight mul % nw nh pw py exch % nw nh py pw PatWidth mul exch % nw nh px py end } bind def % Save the original routines so that we can use them later on /oldfill /fill load def /oldeofill /eofill load def /oldstroke /stroke load def /oldshow /show load def /oldashow /ashow load def /oldwidthshow /widthshow load def /oldawidthshow /awidthshow load def /oldkshow /kshow load def % These defs are necessary so that subsequent procs don't bind in % the originals /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def /PATredef { MyAppDict begin { /fill { /clip load PATdraw newpath } bind def /eofill { /eoclip load PATdraw newpath } bind def /stroke { PATstroke } bind def /show { 0 0 null 0 0 6 -1 roll PATawidthshow } bind def /ashow { 0 0 null 6 3 roll PATawidthshow } bind def /widthshow { 0 0 3 -1 roll PATawidthshow } bind def /awidthshow { PATawidthshow } bind def /kshow { PATkshow } bind def } { /fill { oldfill } bind def /eofill { oldeofill } bind def /stroke { oldstroke } bind def /show { oldshow } bind def /ashow { oldashow } bind def /widthshow { oldwidthshow } bind def /awidthshow { oldawidthshow } bind def /kshow { oldkshow } bind def } ifelse end } bind def false PATredef % Conditionally define setcmykcolor if not available /setcmykcolor where { pop } { /setcmykcolor { 1 sub 4 1 roll 3 { 3 index add neg dup 0 lt { pop 0 } if 3 1 roll } repeat setrgbcolor - pop } bind def } ifelse /PATsc { % colorarray aload length % c1 ... cn length dup 1 eq { pop setgray } { 3 eq { setrgbcolor } { setcmykcolor } ifelse } ifelse } bind def /PATsg { % dict begin lw setlinewidth lc setlinecap lj setlinejoin ml setmiterlimit ds aload pop setdash cc aload pop setrgbcolor cm setmatrix end } bind def /PATDict 3 dict def /PATsp { true PATredef PATDict begin /CurrentPattern exch def % If it's an uncolored pattern, save the color CurrentPattern /PaintType get 2 eq { /PColor exch def } if /CColor [ currentrgbcolor ] def end } bind def % PATstroke - stroke with the current pattern /PATstroke { countdictstack save mark { currentpoint strokepath moveto PATpcalc % proc nw nh px py clip newpath PATfill } stopped { (*** PATstroke Warning: Path is too complex, stroking with gray) = cleartomark restore countdictstack exch sub dup 0 gt { { end } repeat } { pop } ifelse gsave 0.5 setgray oldstroke grestore } { pop restore pop } ifelse newpath } bind def /PATtcalc { % modmtx tilingtype PATtcalc tilematrix % Note: tiling types 2 and 3 are not supported gsave exch concat % tilingtype matrix currentmatrix exch % cmtx tilingtype % Tiling type 1 and 3: constant spacing 2 ne { % Distort the pattern so that it occupies % an integral number of device pixels dup 4 get exch dup 5 get exch % tx ty cmtx XStep 0 dtransform round exch round exch % tx ty cmtx dx.x dx.y XStep div exch XStep div exch % tx ty cmtx a b 0 YStep dtransform round exch round exch % tx ty cmtx a b dy.x dy.y YStep div exch YStep div exch % tx ty cmtx a b c d 7 -3 roll astore % { a b c d tx ty } } if grestore } bind def /PATusp { false PATredef PATDict begin CColor PATsc end } bind def % right45 11 dict begin /PaintType 1 def /PatternType 1 def /TilingType 1 def /BBox [0 0 1 1] def /XStep 1 def /YStep 1 def /PatWidth 1 def /PatHeight 1 def /Multi 2 def /PaintData [ { clippath } bind { 32 32 true [ 32 0 0 -32 0 32 ] {<010101010202020204040404080808081010101020202020 404040408080808001010101020202020404040408080808 101010102020202040404040808080800101010102020202 040404040808080810101010202020204040404080808080 010101010202020204040404080808081010101020202020 4040404080808080>} imagemask } bind ] def /PaintProc { pop exec fill } def currentdict end /P5 exch def 1.1111 1.1111 scale %restore scale /cp {closepath} bind def /ef {eofill} bind def /gr {grestore} bind def /gs {gsave} bind def /sa {save} bind def /rs {restore} bind def /l {lineto} bind def /m {moveto} bind def /rm {rmoveto} bind def /n {newpath} bind def /s {stroke} bind def /sh {show} bind def /slc {setlinecap} bind def /slj {setlinejoin} bind def /slw {setlinewidth} bind def /srgb {setrgbcolor} bind def /rot {rotate} bind def /sc {scale} bind def /sd {setdash} bind def /ff {findfont} bind def /sf {setfont} bind def /scf {scalefont} bind def /sw {stringwidth} bind def /tr {translate} bind def /tnt {dup dup currentrgbcolor 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add 4 -2 roll dup 1 exch sub 3 -1 roll mul add srgb} bind def /shd {dup dup currentrgbcolor 4 -2 roll mul 4 -2 roll mul 4 -2 roll mul srgb} bind def /DrawEllipse { /endangle exch def /startangle exch def /yrad exch def /xrad exch def /y exch def /x exch def /savematrix mtrx currentmatrix def x y tr xrad yrad sc 0 0 1 startangle endangle arc closepath savematrix setmatrix } def /$F2psBegin {$F2psDict begin /$F2psEnteredState save def} def /$F2psEnd {$F2psEnteredState restore end} def %%EndProlog $F2psBegin 10 setmiterlimit n 0 792 m 0 0 l 612 0 l 612 792 l cp clip 0.04200 0.04200 sc 7.500 slw % Ellipse n 3600 1200 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 232.93 72.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 4800 1200 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 312.93 72.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 2400 1200 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 152.93 72.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 3600 2400 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 232.93 152.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 2400 2400 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 152.93 152.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 1200 2400 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 72.93 152.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 2400 3600 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 152.93 232.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 600 3600 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 32.93 232.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 6000 3600 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 392.93 232.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 4800 2400 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 312.93 152.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 6000 2400 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 392.93 152.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 7200 2400 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 472.93 152.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 10800 3600 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 712.93 232.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 10800 2400 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 712.93 152.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 12000 2400 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 792.93 152.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 9600 2400 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 632.93 152.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 8400 3600 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 552.93 232.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 6000 4800 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 392.93 312.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Ellipse n 6000 6000 106 106 0 360 DrawEllipse gs /PC [[0.00 0.00 0.00] [0.00 0.00 0.00]] def 15.00 15.00 sc P5 [16 0 0 -16 392.93 392.93] PATmp PATsp ef gr PATusp gs col-1 s gr % Polyline n 2400 1200 m 3600 2400 l 3600 1200 l gs col-1 s gr % Polyline n 4800 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(total)f(a)h(factor)g(not)h(greater)f(than)665 4809 y(\()p Fo(c)745 4824 y Fv(h)812 4809 y Ft(+)22 b Fo(c)952 4824 y Fv(r)990 4809 y Ft(\))1028 4710 y Fh(P)1116 4731 y Fg(1)1116 4797 y Fk(k)q Fi(=1)1231 4768 y Fm(\(3)p Fv(k)r Fm(\))1359 4745 y Fk(d)1396 4768 y Fv(n)1439 4780 y Fk(\014)s(;k)1537 4809 y Fo(\021)1589 4768 y Fm(max)o Fj(f)p Fm(0)p Fv(;N)7 b Fj(\000)p Fv(L)p Fj(g)2043 4809 y Ft(=)2176 4701 y Fj(1)2158 4726 y Fh(Y)2147 4911 y Fv(k)r Fm(=1)2292 4713 y Fh(\020)2342 4809 y Ft(exp)q(\()p Fo(c)2571 4824 y Fm(23)2645 4809 y Fo(k)2699 4768 y Fv(d)2740 4809 y Ft(\))2778 4713 y Fh(\021)2827 4736 y Fv(n)2870 4748 y Fk(\014)s(;k)2985 4809 y Fo(\021)3037 4768 y Fm(max)o Fj(f)p Fm(0)p Fv(;N)g Fj(\000)p Fv(L)p Fj(g)3388 5002 y Ft(\(88\))1894 5251 y(31)p eop %%Page: 32 32 32 31 bop 443 548 a Ft(4.)49 b(There)40 b(are)f(only)g(c)m(hoices)h (left)f(b)s(et)m(w)m(een)i(l-h-c)m(hains)d(and)h(l-r-c)m(hains)f(in)g (\(\003)3446 563 y Fm(1)3512 548 y Fp(n)577 643 y Ft(~)568 668 y(\003)636 683 y Fj(C)680 668 y Ft(\))27 b Fp(\002)h(f\000)p Fo(N)5 b(;)17 b(:)g(:)g(:)f(;)h Ft(0)p Fp(g)p Ft(,)41 b(giving)d(factor)i Fo(c)2072 683 y Fv(h)2156 668 y Ft(or)g Fo(c)2325 683 y Fv(r)2363 668 y Fo(\021)2415 632 y Fv(N)2521 668 y Ft(resp)s(ectiv)m(ely)-8 b(.)66 b(Let)40 b Fo(l)i Ft(\(0)e Fp(\024)568 789 y Fo(l)51 b Fp(\024)e(j)p Ft(\003)870 804 y Fm(1)940 789 y Fp(n)1030 764 y Ft(~)1020 789 y(\003)1088 804 y Fj(C)1133 789 y Fp(j)g(\024)g(j)p Ft(\003)1432 804 y Fm(1)1471 789 y Fp(j)30 b(\000)i Fo(K)7 b Ft(\))45 b(denote)h(the)f(n)m(um)m(b)s(er)h(of)e(l-r-c)m(hains)g(in)g(suc)m(h) 568 927 y(a)d(c)m(hoice.)73 b(F)-8 b(or)41 b(giv)m(en)h Fo(l)j Ft(there)e(are)1968 831 y Fh(\020)2017 883 y Fj(j)p Fm(\003)2086 892 y Fi(1)2120 883 y Fj(n)2162 866 y Fm(~)2155 883 y(\003)2204 894 y Fg(C)2245 883 y Fj(j)2130 962 y Fv(l)2264 831 y Fh(\021)2358 927 y Fp(\024)2479 831 y Fh(\020)2529 883 y Fj(j)p Fm(\003)2598 892 y Fi(1)2632 883 y Fj(j\000)p Fv(K)2639 962 y(l)2771 831 y Fh(\021)2862 927 y Ft(di\013eren)m(t)g(subsets)568 1077 y(\003)636 1092 y Fv(r)709 1077 y Ft(of)36 b(\003)892 1092 y Fm(1)956 1077 y Fp(n)1040 1052 y Ft(~)1030 1077 y(\003)1098 1092 y Fj(C)1179 1077 y Ft(of)f(cardinalit)m(y)g Fo(l)j Ft(\(that)e(corresp) s(onds)h(to)f(a)g(particular)e(c)m(hoice)568 1197 y(of)k(exactly)i Fo(l)h Ft(l-r-c)m(hains.\))62 b(The)40 b(con\014guration)e Fp(C)45 b Ft(is)39 b(determined)g(b)m(y)h(all)d(the)568 1318 y(c)m(hoices)c(men)m(tioned)f(up)h(to)f(no)m(w.)324 1521 y(In)h(the)g(con\014guration)e Fp(C)39 b Ft(there)33 b(are)g(h-c)m(hains)g(at)f(p)s(oin)m(ts)g(with)g Fs(Z)2792 1485 y Fv(d)2833 1521 y Ft(-co)s(ordinate)f(in)324 1641 y(\003)392 1656 y Fm(1)453 1641 y Fp(n)22 b Ft(\()572 1616 y(~)563 1641 y(\003)631 1656 y Fj(C)698 1641 y Fp([)h Ft(\003)855 1656 y Fv(r)893 1641 y Ft(\).)44 b(The)33 b(op)s(erator)f Fp(L)1664 1656 y Fj(C)1742 1641 y Ft(acts)h(on)g Fo(\036)2137 1656 y Fm(\003)2186 1665 y Fi(2)2257 1641 y Ft(b)m(y)h(in)m(tegration)d(o)m(v)m(er)j(these)g(co)s(or-)324 1762 y(dinates.)43 b(So)33 b(for)f(the)h(uniform)e(estimate)g(of)h Fp(L)2087 1777 y Fj(C)2132 1762 y Fo(\036)2197 1772 y Fm(~)2190 1789 y(\003)2276 1762 y Ft(w)m(e)h(will)d(use)k(\(51\).)324 1882 y(Therefore)f(w)m(e)h(ha)m(v)m(e)g(the)f(estimate)510 2223 y Fo(#)567 2181 y Fj(j)p Fm(\003)636 2190 y Fi(1)671 2181 y Fj(j)871 2140 y Fh(X)711 2328 y Fj(C)t Fm(:)p Fi(length)o Fm(\()p Fj(C)t Fm(\)=)p Fv(N)1168 2223 y Fp(k)p Fo(\031)1273 2238 y Fm(\003)1322 2247 y Fi(1)1383 2223 y Fp(\016)22 b(L)1524 2238 y Fj(C)1569 2223 y Fo(\036)1627 2238 y Fm(\003)1676 2247 y Fi(2)1714 2223 y Fp(k)1764 2252 y Fm(\003)1813 2261 y Fi(1)3388 2223 y Ft(\(89\))593 2561 y Fp(\024)83 b Fo(#)810 2520 y Fj(j)p Fm(\003)879 2529 y Fi(1)914 2520 y Fj(j)970 2444 y(j)p Fm(\003)1039 2453 y Fi(1)1073 2444 y Fj(j)972 2478 y Fh(X)954 2662 y Fv(K)5 b Fm(=0)1125 2415 y Fh( )1191 2494 y Fp(j)p Ft(\003)1287 2509 y Fm(1)1326 2494 y Fp(j)1227 2629 y Fo(K)1354 2415 y Fh(!)1550 2478 y(X)1571 2638 y Fk(n)1609 2655 y(\014)1446 2712 y(K)s Fg(\024j)p Fk(n)1605 2730 y(\014)1647 2712 y Fg(j)p Fk(<)p Fg(1)1802 2561 y Ft(4)1851 2520 y Fv(K)1964 2453 y Fj(1)1947 2478 y Fh(Y)1935 2663 y Fv(k)r Fm(=1)2081 2465 y Fh(\020)2130 2561 y Ft(exp)q(\()p Fo(c)2359 2576 y Fm(21)2434 2561 y Fo(k)2488 2520 y Fv(d)2528 2561 y Ft(\))2566 2465 y Fh(\021)2616 2488 y Fv(n)2659 2500 y Fk(\014)s(;k)2774 2561 y Ft(\()p Fo(c)2854 2576 y Fm(1)2893 2561 y Fo(\017)p Ft(\))2970 2520 y Fj(j)p Fv(n)3033 2532 y Fk(\014)3074 2520 y Fj(j)3388 2561 y Ft(\(90\))753 2930 y Fp(\002)877 2823 y Fj(1)859 2847 y Fh(Y)848 3032 y Fv(k)r Fm(=1)993 2834 y Fh(\020)1042 2930 y Ft(exp)q(\()p Fp(\000)p Fo(c)1348 2945 y Fm(2)1388 2930 y Fo(k)1442 2889 y Fv(d)1483 2930 y Ft(\))1521 2834 y Fh(\021)1570 2857 y Fv(n)1613 2869 y Fk(\014)s(;k)1806 2823 y Fj(1)1782 2847 y Fh(X)1728 3036 y Fv(L)p Fm(=)p Fj(j)p Fv(n)1894 3048 y Fk(\014)1935 3036 y Fj(j)1971 2784 y Fh( )2084 2863 y Fo(L)2037 2999 y Fp(j)p Fo(n)2123 3014 y Fv(\014)2170 2999 y Fp(j)2198 2784 y Fh(!)2263 2930 y Fo(c)2305 2889 y Fj(j)p Fv(n)2368 2901 y Fk(\014)2409 2889 y Fj(j)2305 2955 y Fv(r)2433 2930 y Fo(\021)2485 2889 y Fv(L)2583 2823 y Fj(1)2565 2847 y Fh(Y)2554 3032 y Fv(k)r Fm(=1)2699 2834 y Fh(\020)2748 2930 y Ft(exp)q(\()p Fo(c)2977 2945 y Fm(22)3052 2930 y Fo(k)3106 2889 y Fv(d)3147 2930 y Ft(\))3185 2834 y Fh(\021)3234 2857 y Fv(n)3277 2869 y Fk(\014)s(;k)770 3274 y Fp(\002)893 3166 y Fj(1)875 3191 y Fh(Y)864 3375 y Fv(k)r Fm(=1)1009 3177 y Fh(\020)1059 3274 y Ft(exp)q(\()p Fo(c)1288 3289 y Fm(23)1363 3274 y Fo(k)1417 3233 y Fv(d)1457 3274 y Ft(\))1495 3177 y Fh(\021)1545 3200 y Fv(n)1588 3212 y Fk(\014)s(;k)1703 3274 y Fo(\021)1755 3233 y Fm(max)o Fj(f)p Fm(0)p Fv(;N)i Fj(\000)p Fv(L)p Fj(g)2198 3157 y(j)p Fm(\003)2267 3166 y Fi(1)2301 3157 y Fj(j\000)p Fv(K)2259 3191 y Fh(X)2263 3375 y Fv(l)q Fm(=0)2456 3128 y Fh( )2522 3206 y Fp(j)p Ft(\003)2618 3221 y Fm(1)2657 3206 y Fp(j)22 b(\000)g Fo(K)2694 3342 y(l)2896 3128 y Fh(!)2962 3274 y Ft(\()p Fo(c)3042 3289 y Fv(r)3080 3274 y Fo(\021)3132 3233 y Fv(N)3199 3274 y Ft(\))3237 3233 y Fv(l)770 3562 y Fp(\002)p Fo(c)889 3511 y Fj(j)p Fm(\003)958 3520 y Fi(1)993 3511 y Fj(j\000)p Fv(K)5 b Fj(\000)p Fv(l)889 3587 y(h)1212 3562 y Fo(#)1269 3520 y Fj(\000)p Fv(l)1396 3454 y Fj(1)1379 3479 y Fh(Y)1367 3663 y Fv(k)r Fm(=1)1513 3562 y Fo(#)1570 3520 y Fj(\000)p Fm(\(3)p Fv(k)r Fm(\))1753 3497 y Fk(d)1790 3520 y Fv(n)1833 3532 y Fk(\014)s(;k)1931 3562 y Fp(k)p Fo(\036)p Fp(k)2089 3577 y Fm(\003)2138 3586 y Fi(2)2172 3577 y Fv(;#)593 3880 y Fp(\024)83 b Fo(#)810 3839 y Fj(j)p Fm(\003)879 3848 y Fi(1)914 3839 y Fj(j)970 3763 y(j)p Fm(\003)1039 3772 y Fi(1)1073 3763 y Fj(j)972 3797 y Fh(X)954 3981 y Fv(K)5 b Fm(=0)1125 3734 y Fh( )1191 3812 y Fp(j)p Ft(\003)1287 3827 y Fm(1)1326 3812 y Fp(j)1227 3948 y Fo(K)1354 3734 y Fh(!)1550 3797 y(X)1571 3957 y Fk(n)1609 3974 y(\014)1446 4031 y(K)s Fg(\024j)p Fk(n)1605 4049 y(\014)1647 4031 y Fg(j)p Fk(<)p Fg(1)1802 3880 y Ft(4)1851 3839 y Fv(K)1919 3880 y Ft(\()p Fo(c)1999 3895 y Fm(1)2038 3880 y Fo(\017c)2119 3895 y Fv(r)2157 3880 y Ft(\))2195 3839 y Fj(j)p Fv(n)2258 3851 y Fk(\014)2299 3839 y Fj(j)3388 3880 y Ft(\(91\))753 4245 y Fp(\002)877 4137 y Fj(1)859 4162 y Fh(Y)848 4346 y Fv(k)r Fm(=1)993 4245 y Ft(exp)q(\(\()p Fo(c)1260 4260 y Fm(21)1357 4245 y Fp(\000)22 b Fo(c)1498 4260 y Fm(2)1560 4245 y Ft(+)g Fo(c)1700 4260 y Fm(22)1797 4245 y Ft(+)g Fo(c)1937 4260 y Fm(23)2033 4245 y Fp(\000)h Ft(3)2182 4203 y Fv(d)2239 4245 y Ft(ln)16 b Fo(#)p Ft(\))p Fo(k)2486 4203 y Fv(d)2527 4245 y Ft(\))2565 4203 y Fv(n)2608 4215 y Fk(\014)s(;k)753 4537 y Fp(\002)925 4429 y Fj(1)901 4454 y Fh(X)848 4642 y Fv(L)p Fm(=)p Fj(j)p Fv(n)1014 4654 y Fk(\014)1054 4642 y Fj(j)1091 4391 y Fh( )1203 4470 y Fo(L)1156 4605 y Fp(j)p Fo(n)1242 4620 y Fv(\014)1289 4605 y Fp(j)1317 4391 y Fh(!)1383 4537 y Fo(\021)1435 4496 y Fm(max)o Fj(f)p Fv(N)s(;L)p Fj(g)1767 4537 y Ft(\()p Fo(#)1862 4496 y Fj(\000)p Fm(1)1957 4537 y Fo(c)1999 4552 y Fv(r)2037 4537 y Fo(\021)2089 4496 y Fv(N)2178 4537 y Ft(+)22 b Fo(c)2318 4552 y Fv(h)2363 4537 y Ft(\))2401 4496 y Fj(j)p Fm(\003)2470 4505 y Fi(1)2504 4496 y Fj(j\000)p Fv(K)2647 4537 y Fp(k)p Fo(\036)p Fp(k)2805 4552 y Fm(\003)2854 4561 y Fi(2)2888 4552 y Fv(;#)2953 4537 y Fo(:)324 4887 y Ft(W)-8 b(e)41 b(set)g Fo(\017)699 4902 y Fm(1)780 4834 y(def)791 4887 y Ft(=)52 b(4)p Fo(\017c)1049 4902 y Fm(1)1089 4887 y Fo(c)1131 4902 y Fv(r)1209 4887 y Ft(and)41 b Fo(\017)1446 4902 y Fm(2)1527 4834 y(def)1538 4887 y Ft(=)1666 4823 y Fp(p)p 1749 4823 79 4 v 64 x Fo(\017)1788 4902 y Fm(1)1827 4887 y Ft(.)68 b(Then)42 b(w)m(e)f(ha)m(v)m(e)h Fo(\017)2608 4831 y Fj(j)p Fv(n)2671 4843 y Fk(\014)2713 4831 y Fj(j)2608 4909 y Fm(1)2778 4887 y Fp(\024)g Fo(\017)2936 4851 y Fv(K)2936 4911 y Fm(2)3004 4887 y Fo(\017)3043 4831 y Fj(j)p Fv(n)3106 4843 y Fk(\014)3148 4831 y Fj(j)3043 4909 y Fm(2)3171 4887 y Ft(.)68 b(W)-8 b(e)41 b(set)1894 5251 y(32)p eop %%Page: 33 33 33 32 bop 340 555 a Ft(~)-65 b Fo(c)366 570 y Fm(2)433 503 y(def)444 555 y Ft(=)38 b Fo(c)600 570 y Fm(2)661 555 y Fp(\000)23 b Fo(c)803 570 y Fm(21)900 555 y Fp(\000)g Fo(c)1042 570 y Fm(22)1138 555 y Fp(\000)g Fo(c)1280 570 y Fm(23)1377 555 y Ft(+)f(3)1524 519 y Fv(d)1581 555 y Ft(ln)16 b Fo(#)p Ft(.)44 b(Then)50 b(~)-66 b Fo(c)2103 570 y Fm(2)2170 555 y Fo(>)28 b Ft(0)k(if)1350 755 y Fo(c)1392 770 y Fm(2)1459 755 y Fo(>)27 b(c)1604 770 y Fm(21)1701 755 y Ft(+)22 b Fo(c)1841 770 y Fm(22)1938 755 y Ft(+)g Fo(c)2078 770 y Fm(23)2175 755 y Fp(\000)h Ft(3)2324 713 y Fv(d)2380 755 y Ft(ln)16 b Fo(#)853 b Ft(\(92\))324 954 y(\(W)-8 b(e)38 b(assume)g(this)g(condition)f(on)g (the)i(deca)m(y)g(of)e(the)i(coupling.\))58 b(F)-8 b(urther)38 b(w)m(e)h(split)324 1074 y Fo(\021)376 1038 y Fm(max)o Fj(f)p Fv(N)s(;L)p Fj(g)736 1074 y Fp(\024)c Ft(~)-56 b Fo(\021)893 1038 y Fv(L)952 1074 y Ft(~)g Fo(\021)997 1038 y Fv(N)1096 1074 y Ft(with)40 b(~)-56 b Fo(\021)31 b Ft(=)1502 1012 y Fp(p)p 1585 1012 52 4 v 62 x Fo(\021)s Ft(.)44 b(Then)33 b(\(91\))f(can)h(b)s(e)g(b)s(ounded)g(as)g(follo)m (ws:)549 1457 y Fp(\024)725 1340 y Fj(j)p Fm(\003)794 1349 y Fi(1)828 1340 y Fj(j)726 1374 y Fh(X)709 1558 y Fv(K)5 b Fm(=0)880 1311 y Fh( )946 1389 y Fp(j)p Ft(\003)1042 1404 y Fm(1)1081 1389 y Fp(j)982 1525 y Fo(K)1108 1311 y Fh(!)1174 1457 y Ft(\()p Fo(c)1254 1472 y Fv(r)1292 1457 y Fo(\021)1344 1416 y Fv(N)1433 1457 y Ft(+)22 b Fo(#c)1630 1472 y Fv(h)1675 1457 y Ft(\))1713 1416 y Fj(j)p Fm(\003)1782 1425 y Fi(1)1816 1416 y Fj(j\000)p Fv(K)1959 1457 y Fo(\017)1998 1416 y Fv(K)1998 1481 y Fm(2)2198 1374 y Fh(X)2218 1534 y Fk(n)2256 1551 y(\014)2094 1608 y(K)s Fg(\024j)p Fk(n)2253 1626 y(\014)2295 1608 y Fg(j)p Fk(<)p Fg(1)2527 1349 y Fj(1)2502 1374 y Fh(X)2449 1562 y Fv(L)p Fm(=)p Fj(j)p Fv(n)2615 1574 y Fk(\014)2656 1562 y Fj(j)2692 1311 y Fh( )2805 1389 y Fo(L)2758 1525 y Fp(j)p Fo(n)2844 1540 y Fv(\014)2891 1525 y Fp(j)2919 1311 y Fh(!)2991 1457 y Ft(~)-56 b Fo(\021)3036 1416 y Fv(L)3088 1457 y Fo(\017)3127 1401 y Fj(j)p Fv(n)3190 1413 y Fk(\014)3232 1401 y Fj(j)3127 1479 y Fm(2)3388 1457 y Ft(\(93\))726 1822 y Fp(\002)849 1714 y Fj(1)831 1739 y Fh(Y)820 1923 y Fv(k)r Fm(=1)965 1725 y Fh(\020)1015 1822 y Ft(exp)q(\()p Fp(\000)16 b Ft(~)-65 b Fo(c)1321 1837 y Fm(2)1361 1822 y Fo(k)1415 1780 y Fv(d)1455 1822 y Ft(\))1493 1725 y Fh(\021)1543 1748 y Fv(n)1586 1760 y Fk(\014)s(;k)1701 1822 y Fp(k)p Fo(\036)p Fp(k)1859 1837 y Fm(\003)1908 1846 y Fi(2)1942 1837 y Fv(;#)2014 1822 y Ft(~)-56 b Fo(\021)2059 1780 y Fv(N)549 2126 y Fp(\024)83 b Ft(\()p Fo(c)789 2141 y Fv(r)827 2126 y Fo(\021)879 2085 y Fv(N)968 2126 y Ft(+)22 b Fo(#c)1165 2141 y Fv(h)1233 2126 y Ft(+)g Fo(\017)1370 2141 y Fm(2)1409 2126 y Ft(\))1447 2085 y Fj(j)p Fm(\003)1516 2094 y Fi(1)1551 2085 y Fj(j)1625 2018 y(1)1600 2043 y Fh(X)1591 2227 y Fv(L)p Fm(=0)1788 2018 y Fv(L)1752 2043 y Fh(X)1746 2224 y Fv(n)p Fm(=0)1895 1980 y Fh( )1961 2059 y Fo(L)1965 2194 y(n)2027 1980 y Fh(!)2100 2126 y Ft(~)-56 b Fo(\021)2145 2085 y Fv(L)2197 2126 y Fo(\017)2236 2085 y Fv(n)2236 2151 y Fm(2)2351 2043 y Fh(X)2371 2203 y Fk(n)2409 2220 y(\014)2310 2277 y Fg(j)p Fk(n)2367 2295 y(\014)2409 2277 y Fg(j)p Fi(=)p Fk(n)2568 2018 y Fj(1)2550 2043 y Fh(Y)2539 2228 y Fv(k)r Fm(=1)2684 2030 y Fh(\020)2734 2126 y Ft(exp)q(\()p Fp(\000)16 b Ft(~)-65 b Fo(c)3040 2141 y Fm(2)3080 2126 y Fo(k)3134 2085 y Fv(d)3174 2126 y Ft(\))3212 2030 y Fh(\021)3262 2053 y Fv(n)3305 2065 y Fk(\014)s(;k)726 2433 y Fp(\002k)p Fo(\036)p Fp(k)961 2448 y Fm(\003)1010 2457 y Fi(2)1044 2448 y Fv(;#)1117 2433 y Ft(~)-56 b Fo(\021)1162 2391 y Fv(N)324 2632 y Ft(W)-8 b(e)33 b(ha)m(v)m(e)822 2781 y Fh(X)842 2940 y Fk(n)880 2958 y(\014)781 3015 y Fg(j)p Fk(n)838 3032 y(\014)879 3015 y Fg(j)p Fi(=)p Fk(n)1039 2756 y Fj(1)1021 2781 y Fh(Y)1010 2965 y Fv(k)r Fm(=1)1155 2767 y Fh(\020)1205 2864 y Ft(exp)q(\()p Fp(\000)16 b Ft(~)-65 b Fo(c)1511 2879 y Fm(2)1550 2864 y Fo(k)1604 2823 y Fv(d)1645 2864 y Ft(\))1683 2767 y Fh(\021)1733 2790 y Fv(n)1776 2802 y Fk(\014)s(;k)1902 2864 y Fp(\024)2036 2756 y Fj(1)2018 2781 y Fh(Y)2007 2965 y Fv(k)r Fm(=1)2230 2756 y Fj(1)2206 2781 y Fh(X)2152 2962 y Fv(n)2195 2974 y Fk(\014)s(;k)2289 2962 y Fm(=0)2396 2767 y Fh(\020)2446 2864 y Ft(exp)q(\()p Fp(\000)16 b Ft(~)-65 b Fo(c)2752 2879 y Fm(2)2791 2864 y Fo(k)2845 2823 y Fv(d)2886 2864 y Ft(\))2924 2767 y Fh(\021)2973 2790 y Fv(n)3016 2802 y Fk(\014)s(;k)3388 2864 y Ft(\(94\))324 3212 y(and)42 b(the)h(last)f(in\014nite)f(pro)s (duct)i(con)m(v)m(erges)i(\(to)d Fo(c)2268 3227 y Fm(24)2385 3212 y Ft(sa)m(y\))i(since)e(for)g Fo(k)k Ft(su\016cien)m(tly)324 3347 y(large)33 b(exp)q(\()p Fp(\000)16 b Ft(~)-65 b Fo(c)870 3362 y Fm(2)910 3347 y Fo(k)964 3311 y Fv(d)1005 3347 y Ft(\))30 b Fo(<)1190 3308 y Fm(1)p 1190 3324 36 4 v 1190 3382 a(2)1270 3347 y Ft(and)1461 3281 y Fh(P)1549 3307 y Fj(1)1549 3372 y Fv(n)1592 3384 y Fk(\014)s(;k)1686 3372 y Fm(=0)1797 3251 y Fh(\020)1846 3347 y Ft(exp)r(\()p Fp(\000)16 b Ft(~)-65 b Fo(c)2153 3362 y Fm(2)2192 3347 y Fo(k)2246 3311 y Fv(d)2287 3347 y Ft(\))2325 3251 y Fh(\021)2374 3274 y Fv(n)2417 3286 y Fk(\014)s(;k)2547 3347 y Fp(\024)31 b Ft(1)23 b(+)g(2)17 b(exp)q(\()p Fp(\000)f Ft(~)-65 b Fo(c)3198 3362 y Fm(2)3238 3347 y Fo(k)3292 3311 y Fv(d)3332 3347 y Ft(\))34 b(and)324 3421 y Fh(P)411 3448 y Fj(1)411 3512 y Fv(k)r Fm(=0)561 3488 y Ft(exp)q(\()p Fp(\000)16 b Ft(~)-65 b Fo(c)867 3503 y Fm(2)907 3488 y Fo(k)961 3452 y Fv(d)1001 3488 y Ft(\))28 b Fo(<)f Fp(1)33 b Ft(\(Recall)1635 3421 y Fh(Q)1713 3448 y Fj(1)1713 3512 y Fv(k)r Fm(=1)1846 3488 y Ft(\(1)22 b(+)g Fo(u)2109 3503 y Fv(k)2151 3488 y Ft(\))32 b Fn(c)-5 b(onver)g(gent)41 b Fp(\()2838 3421 y Fh(P)2925 3448 y Fj(1)2925 3512 y Fv(k)r Fm(=1)3075 3488 y Fp(j)p Fo(u)3159 3503 y Fv(k)3201 3488 y Fp(j)27 b Fo(<)g Fp(1)p Ft(.\))934 3840 y Fp(\024)83 b Ft(\()p Fo(\017)1171 3855 y Fm(2)1233 3840 y Ft(+)22 b Fo(c)1373 3855 y Fv(r)1418 3840 y Ft(~)-56 b Fo(\021)1463 3799 y Fv(N)1552 3840 y Ft(+)22 b Fo(c)1692 3855 y Fv(h)1737 3840 y Fo(#)p Ft(\))1832 3799 y Fj(j)p Fm(\003)1901 3808 y Fi(1)1935 3799 y Fj(j)1959 3840 y Fo(c)2001 3855 y Fm(24)2126 3732 y Fj(1)2102 3757 y Fh(X)2092 3941 y Fv(L)p Fm(=0)2231 3840 y Ft(\()p Fo(\017)2308 3855 y Fm(2)2370 3840 y Ft(+)28 b(~)-55 b Fo(\021)s Ft(\))2557 3799 y Fv(L)2609 3840 y Fp(k)p Fo(\036)p Fp(k)2767 3855 y Fm(\003)2816 3864 y Fi(2)2850 3855 y Fv(;#)2923 3840 y Ft(~)f Fo(\021)2968 3799 y Fv(N)934 4114 y Fp(\024)83 b Ft(\()p Fo(\017)1171 4129 y Fm(2)1233 4114 y Ft(+)22 b Fo(c)1373 4129 y Fv(r)1418 4114 y Ft(~)-56 b Fo(\021)1463 4073 y Fv(N)1552 4114 y Ft(+)22 b Fo(c)1692 4129 y Fv(h)1737 4114 y Fo(#)p Ft(\))1832 4073 y Fj(j)p Fm(\003)1901 4082 y Fi(1)1935 4073 y Fj(j)2156 4047 y Ft(1)p 1969 4091 423 4 v 1969 4182 a(1)g Fp(\000)h Fo(\017)2179 4197 y Fm(2)2241 4182 y Fp(\000)29 b Ft(~)-56 b Fo(\021)2402 4114 y(c)2444 4129 y Fm(24)2519 4114 y Fp(k)p Fo(\036)p Fp(k)2677 4129 y Fm(\003)2726 4138 y Fi(2)2759 4129 y Fv(;#)2832 4114 y Ft(~)g Fo(\021)2877 4073 y Fv(N)3388 4114 y Ft(\(95\))934 4321 y Fp(\024)83 b Fo(c)1136 4336 y Fm(19)1218 4321 y Ft(~)-56 b Fo(\021)1263 4280 y Fv(N)1330 4321 y Fp(k)p Fo(\036)p Fp(k)1488 4336 y Fm(\003)1537 4345 y Fi(2)1571 4336 y Fv(;#)3388 4321 y Ft(\(96\))324 4521 y(for)40 b Fo(#)i Ft(and)f Fo(\017)h Ft(su\016cien)m(tly)f(small)e(and)i Fo(N)52 b Ft(su\016cien)m(tly)42 b(large.)68 b(This)41 b(also)f(holds)h(for)324 4641 y(\003)27 b Fp(\032)h Ft(\003)592 4656 y Fm(1)631 4641 y Ft(.)44 b(So)32 b(\(1.\))44 b(is)32 b(pro)m(v)m(ed.)324 4761 y(If)40 b Fp(C)46 b Ft(is)40 b(a)g(non-zero)g(con\014guration)f(of)h(length)f(0)i Fp(\024)g Fo(m)g(<)f(N)51 b Ft(in)39 b(the)h(expansion)h(of)324 4882 y Fo(\031)379 4897 y Fm(\003)428 4906 y Fi(1)493 4882 y Fp(\016)25 b(L)637 4846 y Fv(N)637 4915 y(F)692 4893 y Fi(\003)733 4908 y(2)771 4915 y Fj(\016)p Fv(T)857 4893 y Fi(\003)898 4908 y(2)934 4893 y Fk(;\017)1025 4882 y Ft(it)37 b(has)h(no)g(l-r-c)m(hains.)57 b(So)38 b(this)g(time)e(w)m(e)j(ha)m(v)m(e)g Fo(l)r Ft(\()p Fp(C)6 b Ft(\))37 b(=)g(0.)59 b(Using)324 5002 y(the)33 b(splitting)d Fo(\021)929 4966 y Fv(L)1009 5002 y Fp(\024)35 b Ft(~)-56 b Fo(\021)1166 4966 y Fv(L)1225 5002 y Ft(~)g Fo(\021)1270 4966 y Fv(m)1368 5002 y Ft(w)m(e)34 b(get)f(in)e(a)i(similar)c(w)m(a)m (y)1894 5251 y(33)p eop %%Page: 34 34 34 33 bop 331 768 a Fo(#)388 727 y Fj(j)p Fm(\003)457 736 y Fi(1)492 727 y Fj(j)694 685 y Fh(X)542 861 y Fg(C)r Fi(:length)o(\()p Fg(C)r Fi(\)=)p Fk(m;)664 920 y(l)p Fi(\()p Fg(C)r Fi(\)=0)993 768 y Fp(k)p Fo(\031)1098 783 y Fm(\003)1147 792 y Fi(1)1208 768 y Fp(\016)22 b(L)1349 783 y Fj(C)1393 768 y Fo(\036)1451 783 y Fm(\003)1500 792 y Fi(2)1539 768 y Fp(k)1589 797 y Fm(\003)1638 806 y Fi(1)3388 768 y Ft(\(97\))414 1150 y Fp(\024)84 b Fo(#)632 1109 y Fj(j)p Fm(\003)701 1118 y Fi(1)735 1109 y Fj(j)791 1033 y(j)p Fm(\003)860 1042 y Fi(1)895 1033 y Fj(j)793 1067 y Fh(X)776 1250 y Fv(K)5 b Fm(=0)947 1004 y Fh( )1012 1082 y Fp(j)p Ft(\003)1108 1097 y Fm(1)1147 1082 y Fp(j)1049 1218 y Fo(K)1175 1004 y Fh(!)1372 1067 y(X)1392 1226 y Fk(n)1430 1244 y(\014)1267 1301 y(K)s Fg(\024j)p Fk(n)1426 1318 y(\014)1468 1301 y Fg(j)p Fk(<)p Fg(1)1623 1150 y Ft(4)1672 1109 y Fv(K)1786 1042 y Fj(1)1768 1067 y Fh(Y)1756 1251 y Fv(k)r Fm(=1)1902 1053 y Fh(\020)1951 1150 y Ft(exp)q(\()p Fo(c)2180 1165 y Fm(21)2255 1150 y Fo(k)2309 1109 y Fv(d)2350 1150 y Ft(\))2388 1053 y Fh(\021)2437 1076 y Fv(n)2480 1088 y Fk(\014)s(;k)591 1519 y Fp(\002)p Ft(\()p Fo(c)748 1534 y Fm(1)788 1519 y Fo(\017)p Ft(\))865 1478 y Fj(j)p Fv(n)928 1490 y Fk(\014)969 1478 y Fj(j)1039 1411 y(1)1021 1436 y Fh(Y)1010 1620 y Fv(k)r Fm(=1)1155 1423 y Fh(\020)1205 1519 y Ft(exp)q(\()p Fp(\000)p Fo(c)1511 1534 y Fm(2)1550 1519 y Fo(k)1604 1478 y Fv(d)1645 1519 y Ft(\))1683 1423 y Fh(\021)1733 1446 y Fv(n)1776 1458 y Fk(\014)s(;k)1968 1411 y Fj(1)1944 1436 y Fh(X)1890 1624 y Fv(L)p Fm(=)p Fj(j)p Fv(n)2056 1636 y Fk(\014)2097 1624 y Fj(j)2133 1373 y Fh( )2246 1452 y Fo(L)2199 1587 y Fp(j)p Fo(n)2285 1602 y Fv(\014)2332 1587 y Fp(j)2360 1373 y Fh(!)2426 1519 y Fo(c)2468 1478 y Fj(j)p Fv(n)2531 1490 y Fk(\014)2572 1478 y Fj(j)2468 1544 y Fv(r)2602 1519 y Ft(~)-56 b Fo(\021)2647 1478 y Fv(L)2745 1411 y Fj(1)2727 1436 y Fh(Y)2716 1620 y Fv(k)r Fm(=1)2861 1423 y Fh(\020)2911 1519 y Ft(exp)q(\()p Fo(c)3140 1534 y Fm(22)3214 1519 y Fo(k)3268 1478 y Fv(d)3309 1519 y Ft(\))3347 1423 y Fh(\021)3396 1446 y Fv(n)3439 1458 y Fk(\014)s(;k)591 1832 y Fp(\002)714 1724 y Fj(1)697 1749 y Fh(Y)685 1933 y Fv(k)r Fm(=1)831 1736 y Fh(\020)880 1832 y Ft(exp)q(\()p Fo(c)1109 1847 y Fm(23)1184 1832 y Fo(k)1238 1791 y Fv(d)1278 1832 y Ft(\))1316 1736 y Fh(\021)1366 1759 y Fv(n)1409 1771 y Fk(\014)s(;k)1524 1832 y Fo(c)1566 1781 y Fj(j)p Fm(\003)1635 1790 y Fi(1)1669 1781 y Fj(j\000)p Fv(K)1566 1857 y(h)1858 1724 y Fj(1)1840 1749 y Fh(Y)1828 1933 y Fv(k)r Fm(=1)1974 1832 y Fo(#)2031 1791 y Fj(\000)p Fm(\(3)p Fv(k)r Fm(\))2214 1767 y Fk(d)2251 1791 y Fv(n)2294 1803 y Fk(\014)s(;k)2392 1832 y Fp(k)p Fo(\036)p Fp(k)2550 1847 y Fm(\003)2599 1856 y Fi(2)2633 1847 y Fv(;#)2706 1832 y Ft(~)g Fo(\021)2751 1791 y Fv(N)414 2150 y Fp(\024)591 2033 y Fj(j)p Fm(\003)660 2042 y Fi(1)694 2033 y Fj(j)592 2067 y Fh(X)575 2251 y Fv(K)5 b Fm(=0)746 2004 y Fh( )811 2083 y Fp(j)p Ft(\003)907 2098 y Fm(1)946 2083 y Fp(j)848 2218 y Fo(K)974 2004 y Fh(!)1040 2150 y Ft(\()p Fo(c)1120 2165 y Fv(h)1164 2150 y Fo(#)p Ft(\))1259 2109 y Fj(j)p Fm(\003)1328 2118 y Fi(1)1363 2109 y Fj(j\000)p Fv(K)1637 2067 y Fh(X)1657 2227 y Fk(n)1695 2244 y(\014)1532 2301 y(K)s Fg(\024j)p Fk(n)1691 2319 y(\014)1734 2301 y Fg(j)p Fk(<)p Fg(1)1871 2150 y Ft(\()p Fo(c)1951 2165 y Fm(1)1991 2150 y Fo(\017)p Ft(4)p Fo(c)2121 2165 y Fv(r)2159 2150 y Ft(\))2197 2109 y Fj(j)p Fv(n)2260 2121 y Fk(\014)2301 2109 y Fj(j)2370 2042 y(1)2352 2067 y Fh(Y)2341 2252 y Fv(k)r Fm(=1)2486 2054 y Fh(\020)2536 2150 y Ft(exp)q(\()p Fp(\000)16 b Ft(~)-65 b Fo(c)2842 2165 y Fm(2)2882 2150 y Fo(k)2936 2109 y Fv(d)2976 2150 y Ft(\))3014 2054 y Fh(\021)3064 2077 y Fv(n)3107 2089 y Fk(\014)s(;k)591 2519 y Fp(\002)763 2412 y Fj(1)739 2436 y Fh(X)685 2625 y Fv(L)p Fm(=)p Fj(j)p Fv(n)851 2637 y Fk(\014)892 2625 y Fj(j)928 2373 y Fh( )1041 2452 y Fo(L)994 2588 y Fp(j)p Fo(n)1080 2603 y Fv(\014)1127 2588 y Fp(j)1155 2373 y Fh(!)1227 2519 y Ft(~)-55 b Fo(\021)1273 2478 y Fv(L)1331 2519 y Ft(~)f Fo(\021)1376 2478 y Fv(m)1443 2519 y Fp(k)p Fo(\036)p Fp(k)1601 2534 y Fm(\003)1650 2543 y Fi(2)1684 2534 y Fv(;#)414 2863 y Fp(\024)591 2746 y Fj(j)p Fm(\003)660 2755 y Fi(1)694 2746 y Fj(j)592 2780 y Fh(X)575 2964 y Fv(K)5 b Fm(=0)746 2717 y Fh( )811 2795 y Fp(j)p Ft(\003)907 2810 y Fm(1)946 2795 y Fp(j)848 2931 y Fo(K)974 2717 y Fh(!)1040 2863 y Ft(\()p Fo(c)1120 2878 y Fv(h)1164 2863 y Fo(#)p Ft(\))1259 2822 y Fj(j)p Fm(\003)1328 2831 y Fi(1)1363 2822 y Fj(j\000)p Fv(K)1506 2863 y Fo(\017)1545 2822 y Fv(K)1545 2887 y Fm(2)1745 2780 y Fh(X)1765 2940 y Fk(n)1803 2957 y(\014)1640 3014 y(K)s Fg(\024j)p Fk(n)1799 3031 y(\014)1841 3014 y Fg(j)p Fk(<)p Fg(1)2074 2755 y Fj(1)2049 2780 y Fh(X)1996 2968 y Fv(L)p Fm(=)p Fj(j)p Fv(n)2162 2980 y Fk(\014)2202 2968 y Fj(j)2239 2717 y Fh( )2352 2795 y Fo(L)2304 2931 y Fp(j)p Fo(n)2390 2946 y Fv(\014)2437 2931 y Fp(j)2465 2717 y Fh(!)2538 2863 y Ft(~)-56 b Fo(\021)2583 2822 y Fv(L)2635 2863 y Fo(\017)2674 2807 y Fj(j)p Fv(n)2737 2819 y Fk(\014)2778 2807 y Fj(j)2674 2885 y Fm(2)591 3227 y Fp(\002)714 3120 y Fj(1)697 3144 y Fh(Y)685 3329 y Fv(k)r Fm(=1)831 3131 y Fh(\020)880 3227 y Ft(exp)q(\()p Fp(\000)16 b Ft(~)-65 b Fo(c)1186 3242 y Fm(2)1226 3227 y Fo(k)1280 3186 y Fv(d)1321 3227 y Ft(\))1359 3131 y Fh(\021)1408 3154 y Fv(n)1451 3166 y Fk(\014)s(;k)1573 3227 y Ft(~)-56 b Fo(\021)1618 3186 y Fv(m)1684 3227 y Fp(k)p Fo(\036)p Fp(k)1842 3242 y Fm(\003)1891 3251 y Fi(2)1925 3242 y Fv(;#)414 3502 y Fp(\024)84 b Ft(\()p Fo(\017)652 3517 y Fm(2)714 3502 y Ft(+)22 b Fo(c)854 3517 y Fv(h)898 3502 y Fo(#)p Ft(\))993 3461 y Fj(j)p Fm(\003)1062 3470 y Fi(1)1097 3461 y Fj(j)1318 3435 y Ft(1)p 1131 3479 423 4 v 1131 3570 a(1)g Fp(\000)g Fo(\017)1340 3585 y Fm(2)1402 3570 y Fp(\000)30 b Ft(~)-56 b Fo(\021)1563 3502 y(c)1605 3517 y Fm(25)1687 3502 y Ft(~)g Fo(\021)1732 3461 y Fv(m)1798 3502 y Fp(k)p Fo(\036)p Fp(k)1956 3517 y Fm(\003)2005 3526 y Fi(2)2039 3517 y Fv(;#)414 3698 y Fp(\024)84 b Fo(c)617 3713 y Fm(26)698 3698 y Ft(~)-56 b Fo(\021)743 3657 y Fv(m)810 3698 y Fp(k)p Fo(\036)p Fp(k)968 3713 y Fm(\003)1017 3722 y Fi(2)1051 3713 y Fv(;#)3388 3698 y Ft(\(98\))324 3918 y(Again)31 b(this)h(also)g(holds)g (for)g(\003)c Fp(\032)g Ft(\003)1667 3933 y Fm(1)1739 3918 y Ft(and)k(so)902 4158 y Fo(#)959 4117 y Fj(j)p Fm(\003)1028 4126 y Fi(1)1063 4117 y Fj(j)1265 4075 y Fh(X)1113 4251 y Fg(C)r Fi(:length)o(\()p Fg(C)r Fi(\)=)p Fk(m;)1235 4311 y(l)p Fi(\()p Fg(C)r Fi(\)=0)1564 4158 y Fp(k)p Fo(\031)1669 4173 y Fm(\003)1718 4182 y Fi(1)1779 4158 y Fp(\016)22 b(L)1920 4173 y Fj(C)1964 4158 y Fo(\036)2022 4173 y Fm(\003)2071 4182 y Fi(2)2110 4158 y Fp(k)2160 4188 y Fm(\003)2209 4197 y Fi(1)2243 4188 y Fv(;#)2337 4158 y Fp(\024)28 b Fo(c)2484 4173 y Fm(26)2558 4158 y Fp(k)p Fo(\036)p Fp(k)2716 4173 y Fm(\003)2765 4182 y Fi(2)2799 4173 y Fv(;#)2872 4158 y Ft(~)-56 b Fo(\021)2917 4117 y Fv(m)3388 4158 y Ft(\(99\))324 4463 y(Therefore)707 4753 y Fh(\015)707 4803 y(\015)707 4853 y(\015)p Fo(\031)808 4868 y Fm(\003)857 4877 y Fi(1)919 4853 y Fp(\016)22 b(L)1060 4812 y Fv(N)1060 4881 y(F)1115 4859 y Fi(\003)1156 4874 y(2)1194 4881 y Fj(\016)p Fv(T)1280 4859 y Fi(\003)1321 4874 y(2)1356 4859 y Fk(;\017)1409 4753 y Fh(\015)1409 4803 y(\015)1409 4853 y(\015)1455 4907 y Fv(L)p Fm(\(\()p Fj(H)1617 4919 y Fi(\003)1658 4934 y(2)1694 4919 y Fk(;#)1755 4907 y Fv(;)p Fj(k\001k)1865 4919 y Fi(\003)1906 4934 y(2)1941 4919 y Fk(;#)2001 4907 y Fm(\))p Fv(;)p Fm(\()p Fj(H)2135 4919 y Fi(\003)2176 4934 y(1)2212 4919 y Fk(;#)2272 4907 y Fj(k\001k)2362 4919 y Fi(\003)2403 4934 y(1)2438 4919 y Fk(;#)2499 4907 y Fm(\))2613 4853 y Fp(\024)2819 4745 y Fv(N)2790 4770 y Fh(X)2774 4951 y Fv(m)p Fm(=0)2943 4853 y Fo(c)2985 4868 y Fm(26)3067 4853 y Ft(~)-56 b Fo(\021)3112 4812 y Fv(m)3339 4853 y Ft(\(100\))1894 5251 y(34)p eop %%Page: 35 35 35 34 bop 2613 593 a Fp(\024)2815 485 y Fj(1)2790 510 y Fh(X)2774 691 y Fv(m)p Fm(=0)2943 593 y Fo(c)2985 608 y Fm(26)3067 593 y Ft(~)-56 b Fo(\021)3112 552 y Fv(m)2613 796 y Fp(\024)84 b Fo(c)2816 811 y Fm(20)3455 969 y Fa(2)324 1215 y Fs(Lemma)37 b(3)164 b Fn(1.)49 b(The)39 b(numb)-5 b(er)39 b(of)h(lab)-5 b(el)5 b(le)-5 b(d)39 b(tr)-5 b(e)g(e)40 b(gr)-5 b(aphs)39 b(with)g(exactly)h(n)g(e)-5 b(dges)39 b(is)568 1335 y(smal)5 b(ler)34 b(than)g Ft(2)1183 1299 y Fm(2)p Fv(n)439 1526 y Fn(2.)49 b(The)32 b(numb)-5 b(er)32 b(of)g(lab)-5 b(el)5 b(le)-5 b(d)32 b(tr)-5 b(e)g(e)33 b(gr)-5 b(aphs)32 b(c)-5 b(orr)g(esp)g(onding)31 b(to)i(c)-5 b(on\014gur)g(ations)31 b(that)568 1646 y(have)j(exactly)h Fo(n)1174 1661 y Fv(\014)s(;k)1314 1646 y Fo(k)s Fn(-triangles)f(\()p Fp(j)p Fo(n)1926 1661 y Fv(\014)s(;k)2031 1646 y Fp(j)28 b Fo(<)f Fp(1)p Fn(\))34 b(and)h(end)f(\(at)h(time)g Ft(0)p Fn(\))f(in)568 1766 y Ft(\003)636 1781 y Fm(3)697 1766 y Fp(\002)23 b(f)p Ft(0)p Fp(g)34 b Fn(and)g(not)h(in)g(any)f (smal)5 b(ler)34 b(set)h(is)g(b)-5 b(ounde)g(d)34 b(fr)-5 b(om)34 b(ab)-5 b(ove)34 b(by)568 1913 y Ft(4)617 1877 y Fj(j)p Fm(\003)686 1886 y Fi(3)720 1877 y Fj(j)760 1847 y Fh(Q)838 1873 y Fj(1)838 1938 y Fv(k)r Fm(=1)988 1913 y Fo(c)1030 1858 y Fv(k)1069 1834 y Fk(d)1104 1858 y Fv(n)1147 1870 y Fk(\014)s(;k)1030 1935 y Fm(21)1281 1913 y Fn(with)g Fo(c)1534 1928 y Fm(21)1637 1913 y Ft(=)27 b(4)1789 1877 y Fm(3)1824 1854 y Fk(d)1865 1913 y Fn(.)324 2088 y Fs(Pro)s(of)82 b Ft(1.\))60 b(F)-8 b(or)38 b(ev)m(ery)i(lab)s (elled)c(tree)j(graph)f(in)f(question)i(w)m(e)g(can)g(de\014ne)h(a)e (path)324 2208 y(starting)k(and)i(ending)f(at)h(the)g(ro)s(ot)e(p)s (oin)m(t)h(\(0\))g(and)h(running)f(through)g(eac)m(h)i(edge)324 2329 y(exactly)36 b(t)m(wice)f(in)g(the)g(follo)m(wing)e(w)m(a)m(y)-8 b(.)53 b(F)-8 b(rom)34 b(a)h(\(lab)s(elled\))e(v)m(ertex)k Fo(t)32 b Ft(=)g(\()p Fo(t)3183 2344 y Fm(1)3223 2329 y Fo(;)17 b(:)g(:)g(:)f(;)h(t)3477 2344 y Fv(n)3524 2329 y Ft(\))324 2449 y(w)m(e)35 b(go)e(to)h(the)h(to)e(the)i(next)g (greater)f(\(wrt.)g Fp(\036)p Ft(\))h(v)m(ertex)h(where)f(w)m(e)g(ha)m (v)m(en't)h(y)m(et)f(b)s(een)324 2570 y(\()p Fn(going)30 b(up)6 b Ft(\),)29 b(or)f(if)f(this)h(is)g(not)h(p)s(ossible)e(\(i.e.)h Fo(t)h Ft(is)e(a)i(leaf)e(or)h(w)m(e)h(ha)m(v)m(e)h(already)e(b)s(een)h (at)324 2690 y(all)34 b(v)m(ertices)j(\()p Fo(t)892 2705 y Fm(1)931 2690 y Fo(;)17 b(:)g(:)g(:)f(;)h(t)1185 2705 y Fv(n)p Fm(+1)1322 2690 y Ft(\)\))36 b(bac)m(k)h(to)f(\()p Fo(t)1861 2705 y Fm(1)1900 2690 y Fo(;)17 b(:)g(:)g(:)f(;)h(t)2154 2705 y Fv(n)p Fj(\000)p Fm(1)2291 2690 y Ft(\))36 b(\()p Fn(going)h(down)7 b Ft(\).)53 b(So)36 b(w)m(e)h(return)324 2810 y(to)29 b(\(0\))f(after)h(2)p Fo(n)g Ft(steps.)44 b(W)-8 b(e)29 b(enco)s(de)h(the)g(path)f(in)f(a)h(w)m(ord)h(\()p Fo(a)2625 2825 y Fm(1)2665 2810 y Fo(;)17 b(:)g(:)g(:)e(;)i(a)2934 2825 y Fm(2)p Fv(n)3017 2810 y Ft(\))29 b(with)f Fo(a)3353 2825 y Fv(i)3409 2810 y Ft(=)g(1)324 2931 y(if)c(w)m(e)i(go)f(up)h(in)e (the)i Fo(i)p Ft(th)g(step)g(and)f Fo(a)1649 2946 y Fv(i)1705 2931 y Ft(=)j(0)d(otherwise.)42 b(Ob)m(viously)25 b(the)h(lab)s(elled)d (graph)324 3051 y(is)33 b(uniquely)g(determined)g(b)m(y)h(its)f(w)m (ord.)46 b(\(Note)34 b(that)f(not)g(ev)m(ery)i(w)m(ord)f(of)f(length)g (2)p Fo(n)324 3172 y Ft(with)25 b(sym)m(b)s(ols)h("0")g(and)g("1")f (corresp)s(onds)j(to)d(suc)m(h)j(a)e(lab)s(elled)d(graph.)42 b(But)26 b(the)g(map)324 3292 y(b)s(et)m(w)m(een)k(these)f(t)m(w)m(o)g (data)f(is)g(injectiv)m(e.\))42 b(As)28 b(there)h(are)f(2)2486 3256 y Fv(n)2561 3292 y Ft(w)m(ords)i(of)d(length)h(2)p Fo(n)g Ft(with)324 3412 y(at)j(most)g(t)m(w)m(o)i(di\013eren)m(t)f(sym) m(b)s(ols)f(this)h(is)f(also)g(an)g(upp)s(er)i(b)s(ound)f(for)f(the)h (n)m(um)m(b)s(er)g(of)324 3533 y(graphs)h(in)e(question.)324 3653 y(T)-8 b(o)41 b(see)h(\(2.\))68 b(w)m(e)42 b(note)f(that)f(the)i (n)m(um)m(b)s(er)f(of)f(edges)i(in)e(suc)m(h)j(a)d(tree)i(graph)e(is)h (not)324 3773 y(greater)32 b(than)h Fo(K)c Ft(+)1092 3707 y Fh(P)1180 3733 y Fj(1)1180 3798 y Fv(k)r Fm(=1)1313 3773 y Ft(\(3)p Fo(k)s Ft(\))1492 3737 y Fv(d)1532 3773 y Fo(n)1590 3788 y Fv(\014)s(;k)1695 3773 y Ft(.)1733 b Fa(2)324 3965 y Fs(Pro)s(of)51 b(of)i(Theorem)e(1)45 b Ft(The)i(di\013erence)f(b)s(et)m(w)m(een)h Fo(\031)2487 3980 y Fm(\003)2536 3989 y Fi(1)2606 3965 y Fp(\016)30 b(L)2755 3929 y Fv(N)2755 3998 y(F)2810 3976 y Fi(\003)2851 3991 y(2)2890 3998 y Fj(\016)p Fv(T)2976 3976 y Fi(\003)3017 3991 y(2)3052 3976 y Fk(;\017)3136 3965 y Fp(\016)g Fo(\031)3271 3980 y Fm(\003)3320 3989 y Fi(2)3404 3965 y Ft(and)324 4085 y Fo(\031)379 4100 y Fm(\003)428 4109 y Fi(1)500 4085 y Fp(\016)j(L)652 4049 y Fv(N)652 4118 y(F)707 4096 y Fi(\003)748 4111 y(3)786 4118 y Fj(\016)p Fv(T)872 4096 y Fi(\003)913 4111 y(3)948 4096 y Fk(;\017)1035 4085 y Fp(\016)f Fo(\031)1172 4100 y Fm(\003)1221 4109 y Fi(3)1309 4085 y Ft(for)48 b(\003)1542 4100 y Fm(1)1636 4085 y Fp(\022)56 b Ft(\003)1837 4100 y Fm(2)1931 4085 y Fp(\022)f 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1617 y Ft(exp)q(\()p Fp(\000)p Ft(\()16 b(~)-65 b Fo(c)1590 1632 y Fm(2)1652 1617 y Fp(\000)23 b Fo(\030)5 b Ft(\))p Fo(k)1892 1576 y Fv(d)1931 1617 y Ft(\))1969 1521 y Fh(i)2009 1544 y Fv(n)2052 1556 y Fk(\014)s(;k)2167 1617 y Ft(exp)2332 1521 y Fh(\020)2382 1617 y Fp(\000)p Fo(\030)g Ft(dist)o(\(\003)2770 1632 y Fm(1)2809 1617 y Fo(;)17 b Ft(\003)2921 1576 y Fv(C)2921 1642 y Fm(2)2980 1617 y Ft(\))3018 1521 y Fh(\021)324 1934 y Ft(If)29 b(w)m(e)i(tak)m(e)g Fo(\030)h(>)27 b Ft(0)j(small)d(enough)j(w)m(e)h(can)f(tak)m(e)g(out)g(a)f(factor)h(exp) 2826 1838 y Fh(\020)2876 1934 y Fp(\000)p Fo(\030)5 b Ft(dist)o(\(\003)3264 1949 y Fm(1)3304 1934 y Fo(;)17 b Ft(\003)3416 1898 y Fv(C)3416 1959 y Fm(2)3474 1934 y Ft(\))3512 1838 y Fh(\021)324 2055 y Ft(and)32 b(do)h(the)g(analysis) e(with)h(the)h(remaining)d(factor)i(as)h(b)s(efore)f(since)50 b(~)-66 b Fo(c)3000 2070 y Fm(2)3062 2055 y Fp(\000)22 b Fo(\030)32 b(>)27 b Ft(0.)44 b(So)324 2175 y(w)m(e)33 b(get)369 2516 y Fp(k)p Fo(\031)474 2531 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Fm(\003)1518 4600 y Fi(1)1579 4576 y Fp(\016)g(L)1720 4591 y Fj(C)1757 4600 y Fi(2)1792 4591 y Fj(\016C)1864 4600 y Fi(1)1903 4576 y Fo(\036)1961 4591 y Fm(\003\()p Fj(C)2074 4600 y Fi(1)2109 4591 y Fm(\))771 4912 y Ft(=)1133 4829 y Fh(X)930 5017 y Fj(C)967 5026 y Fi(3)1002 5017 y Fj(2)p Fv(E)1101 5028 y Fk(N)1149 5043 y Fi(1)1183 5028 y(+)p Fk(N)1278 5043 y Fi(2)1317 5017 y Fm(\(\003)1393 5026 y Fi(1)1428 5017 y Fm(\))1472 4912 y Fo(\031)1527 4927 y Fm(\003)1576 4936 y Fi(1)1637 4912 y Fp(\016)g(L)1778 4927 y Fj(C)1815 4936 y Fi(3)1854 4912 y Fo(\036)1912 4928 y Fm(\003\()p Fj(C)2025 4937 y Fi(3)2060 4928 y Fm(\))1894 5251 y Ft(37)p eop %%Page: 38 38 38 37 bop 771 548 a Ft(=)83 b Fo(\031)985 563 y Fm(\003)1034 572 y Fi(1)1095 548 y Fp(\016)22 b(L)1236 504 y Fv(N)1292 513 y Fi(1)1326 504 y Fm(+)p Fv(N)1437 513 y Fi(2)1236 572 y Fv(F)10 b Fj(\016)p Fv(T)1377 553 y Fk(\017)1476 548 y Fo(\036:)324 768 y Ft(Note)39 b(that)g(w)m(e)i(sum)e(o)m(v)m(er)h (in\014nitely)e(man)m(y)h Fp(C)2103 783 y Fm(1)2143 768 y Fo(;)17 b Fp(C)2239 783 y Fm(2)2279 768 y Ft(.)64 b(A)39 b(priori,)g(the)h(distribution)d(is)324 888 y(only)27 b(v)-5 b(alid)25 b(for)i(\014nite)g(partial)e(sums.)42 b(In)28 b(terms)f(of)g(con\014gurations)g(w)m(e)h(`put)g Fp(C)3245 903 y Fm(1)3312 888 y Ft(on)g Fp(C)3495 903 y Fm(2)3535 888 y Ft(')324 1009 y(to)33 b(get)g Fp(C)659 1024 y Fm(3)728 1009 y Ft(=)c Fp(C)885 1024 y Fm(2)947 1009 y Fp(\016)23 b(C)1072 1024 y Fm(1)1145 1009 y Ft(\(whic)m(h)34 b(migh)m(t)d(b)s(e)j(a)f(zero)h(con\014guration\))e(and)h(in)g(fact)g (suc)m(h)i(a)324 1129 y(splitting)23 b(exists)i(and)h(is)e(unique)i (for)e(ev)m(ery)j(non-zero)e Fp(C)2377 1144 y Fm(3)2417 1129 y Ft(.)41 b(So)25 b(the)h(net)f(of)g(\014nite)g(partial)324 1249 y(sums)h(o)m(v)m(er)h Fp(C)816 1264 y Fm(3)882 1249 y Ft(w)m(e)g(get)f(con)m(v)m(erges)i(to)e(the)g(in\014nite)f(expansion) h(\(53\))g(of)f(the)i(righ)m(t-hand)324 1370 y(side)32 b(of)g(\(55\))f(and)h(\(4.\))43 b(is)32 b(pro)m(v)m(ed.)45 b(W)-8 b(e)32 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Fm(\003)p Fj(2F)692 4090 y Ft(con)m(v)m(erges)38 b(uniformly)c(to)i Fo(g)j Ft(as)d(\003)d Fp(!)g Fs(Z)2214 4054 y Fv(d)2254 4090 y Ft(,)k(as)f(do)s(es)g(\()p Fo(g)2749 4105 y Fm(\003)2827 4090 y Fp(\016)24 b Fo(S)6 b Ft(\))3005 4105 y Fm(\003)p Fj(2F)3197 4090 y Ft(to)36 b Fo(g)28 b Fp(\016)c Fo(S)6 b Ft(.)324 4211 y(So)33 b(b)m(y)h(con)m(tin)m(uit)m (y)g(of)f(the)h(in)m(tegral)e(op)s(erators)h(the)h(equalit)m(y)f(also)f (holds)h(for)g Fo(g)t Ft(.)46 b(The)324 4331 y(sp)s(ecial)34 b(case)j(\(6.\))51 b(follo)m(ws)34 b(from)g(taking)h Fo(g)h Fp(\021)d Ft(1.)52 b(F)-8 b(or)34 b(\014nite-range)h(in)m (teraction)f(the)324 4451 y(limits)d(in)j(\(108\))f(are)h(already)g (attained)g(for)g(su\016cien)m(tly)h(large)e(\003)2815 4466 y Fm(1)2885 4451 y Fp(2)e(F)43 b Ft(and)35 b(all)d(the)324 4572 y(computations)f(w)m(ork)j(with)e Fo(\036)27 b Fp(2)h(H)1656 4535 y Fv(bv)1728 4572 y Ft(.)324 4706 y(W)-8 b(e)27 b(ha)m(v)m(e,)i(b)m(y)f(de\014nition,)f(\()p Fp(L)1426 4721 y Fv(F)10 b Fj(\016)p Fv(T)1567 4702 y Fk(\017)1600 4706 y Fo(\036)p Ft(\))1696 4721 y Fm(\003)1777 4654 y(def)1788 4706 y Ft(=)38 b(lim)2037 4725 y Fm(\003)2086 4734 y Fi(1)2121 4725 y Fj(!)p Fl(Z)2242 4707 y Fk(d)2298 4706 y Fo(\031)2353 4721 y Fm(\003)2417 4706 y Fp(\016)10 b(L)2546 4728 y Fv(F)2601 4706 y Fi(\003)2642 4721 y(1)2680 4728 y Fj(\016)p Fv(T)2766 4706 y Fi(\003)2807 4721 y(1)2842 4706 y Fk(;\017)f Fo(\036)2954 4721 y Fm(\003)3003 4730 y Fi(1)3041 4706 y Ft(.)41 b(If)27 b(that)g(w)m(as)324 4827 y(negativ)m(e)j(somewhere)h(there)g(w)m(ould)e(b)s(e)i(a)e(\003) 2002 4842 y Fm(1)2069 4827 y Fp(2)f(F)40 b Ft(with)29 b Fo(\031)2549 4842 y Fm(\003)2619 4827 y Fp(\016)17 b(L)2755 4849 y Fv(F)2810 4826 y Fi(\003)2851 4841 y(1)2889 4849 y Fj(\016)p Fv(T)2975 4826 y Fi(\003)3016 4841 y(1)3051 4826 y Fk(;\017)3105 4827 y Fo(\036)3163 4842 y Fm(\003)3212 4851 y Fi(1)3280 4827 y Ft(ha)m(ving)324 4947 y(negativ)m(e)33 b(v)-5 b(alues)32 b(and)h(w)m(e)g(could)f(\014nd)i(a)e(non-negativ)m(e) g Fo(g)f Fp(2)d(C)6 b Ft(\(\()p Fo(S)2815 4911 y Fm(1)2855 4947 y Ft(\))2893 4911 y Fm(\003)2946 4947 y Ft(\))32 b(suc)m(h)i(that)1894 5251 y(38)p eop %%Page: 39 39 39 38 bop 1230 551 a Fh(Z)1277 740 y Fm(\()p Fv(S)1351 721 y Fi(1)1385 740 y Fm(\))1412 721 y Fi(\003)1479 668 y Fo(d\026)1589 627 y Fm(\003)1641 668 y Fo(g)20 b(\031)1763 683 y Fm(\003)1839 668 y Fp(\016)i(L)1980 690 y Fv(F)2035 668 y Fi(\003)2076 683 y(1)2114 690 y Fj(\016)p Fv(T)2200 668 y Fi(\003)2241 683 y(1)2276 668 y Fk(;\017)9 b Fo(\036)2388 683 y Fm(\003)2437 692 y Fi(1)2503 668 y Fo(<)27 b Ft(0)684 b(\(109\))324 885 y(But)33 b(b)m(y)g(\(6.\))43 b(the)33 b(in)m(tegral)e(equals)1426 1008 y Fh(Z)1472 1197 y Fm(\()p Fv(S)1546 1178 y Fi(1)1580 1197 y Fm(\))1607 1175 y Fi(\003)1648 1190 y(1)1709 1125 y Fo(d\026)1819 1084 y Fm(\003)1868 1093 y Fi(1)1906 1125 y Fo(g)25 b Fp(\016)d Fo(S)h(\036)2191 1140 y Fm(\003)2240 1149 y Fi(1)2306 1125 y Fp(\025)28 b Ft(0)879 b(\(110\))324 1342 y(So)32 b Fp(L)528 1357 y Fv(F)10 b Fj(\016)p Fv(T)669 1338 y Fk(\017)736 1342 y Ft(is)32 b(non-negativ)m(e.)43 b(Finally)30 b(\(8.\))43 b(follo)m(ws)31 b(from)980 1662 y Fp(kL)1099 1677 y Fv(F)10 b Fj(\016)p Fv(T)1240 1658 y Fk(\017)1274 1662 y Fo(\036)p Fp(k)1382 1677 y Fv(v)r(ar)1577 1662 y Ft(=)87 b(sup)1737 1741 y Fm(\003)p Fj(2F)2006 1662 y Ft(sup)1917 1744 y Fk(g)r Fg(2C)r Fi(\(\()p Fk(S)2116 1730 y Fi(1)2149 1744 y(\))2173 1730 y(\003)2219 1744 y(\))1960 1803 y Fg(k)p Fk(g)r Fg(k)2056 1811 y(1)2121 1803 y(\024)p Fi(1)2269 1545 y Fh(Z)2315 1734 y Fv(M)2411 1662 y Fo(d\026)17 b(g)i Fp(L)2673 1677 y Fv(F)10 b Fj(\016)p Fv(T)2814 1658 y Fk(\017)2847 1662 y Fo(\036)434 b Ft(\(111\))1577 1974 y(=)87 b(sup)1737 2053 y Fm(\003)p Fj(2F)2006 1974 y Ft(sup)1917 2056 y Fk(g)r Fg(2C)r Fi(\(\()p Fk(S)2116 2042 y Fi(1)2149 2056 y(\))2173 2042 y(\003)2219 2056 y(\))1960 2116 y Fg(k)p Fk(g)r Fg(k)2056 2124 y(1)2121 2116 y(\024)p Fi(1)2269 1857 y Fh(Z)2315 2046 y Fv(M)2411 1974 y Fo(d\026)17 b(g)24 b Fp(\016)e Fo(S)h(\036)1576 2244 y Fp(\024)87 b Ft(sup)1737 2323 y Fm(\003)p Fj(2F)2006 2244 y Ft(sup)1917 2326 y Fk(g)r Fg(2C)r Fi(\(\()p Fk(S)2116 2312 y Fi(1)2149 2326 y(\))2173 2312 y(\003)2219 2326 y(\))1960 2386 y Fg(k)p Fk(g)r Fg(k)2056 2394 y(1)2121 2386 y(\024)p Fi(1)2269 2244 y Fp(k)p Fo(g)t Fp(k)2420 2259 y Fj(1)2494 2244 y Fp(k)p Fo(\036)p Fp(k)2652 2259 y Fv(v)r(ar)1577 2515 y Ft(=)d Fp(k)p Fo(\036)p Fp(k)1895 2530 y Fv(v)r(ar)3455 2714 y Fa(2)324 2906 y Fs(Pro)s(of)37 b(of)h(Theorem)f(3)901 3105 y Fo(\027)949 3120 y Fm(\003)998 3129 y Fi(1)1032 3120 y Fj([)p Fm(\003)1128 3129 y Fi(2)1250 3105 y Ft(=)1555 3022 y Fh(X)1409 3210 y Fj(C)t(2)p Fv(E)t Fm(\(\003)1629 3219 y Fi(1)1663 3210 y Fj([)p Fm(\003)1759 3219 y Fi(2)1794 3210 y Fm(\))1838 3105 y Fo(\031)1893 3120 y Fm(\003)1942 3129 y Fi(1)1977 3120 y Fj([)p Fm(\003)2073 3129 y Fi(2)2134 3105 y Fp(\016)22 b(L)2275 3120 y Fj(C)2319 3105 y Fo(h)964 b Ft(\(112\))1250 3360 y(=)1623 3277 y Fh(X)1552 3449 y Fg(C)r Fi(=)p Fg(C)1669 3464 y Fi(1)1703 3449 y 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b(estimate)g(in)g(a)g(similar)e(w)m(a)m(y)374 2194 y Fp(k)654 2111 y Fh(X)584 2283 y Fg(C)r Fi(=)p Fg(C)701 2298 y Fi(1)735 2283 y Fg([C)811 2298 y Fi(2)450 2360 y Fk(b)p Fi(\()p Fg(C)r Fi(\))p Fk(>)619 2340 y Fi(1)p 619 2349 V 619 2384 a(2)660 2360 y(dist)o(\(\003)824 2375 y(1)859 2360 y Fk(;)p Fi(\003)919 2375 y(2)954 2360 y(\))988 2194 y Ft(\()p Fo(\031)1081 2209 y Fm(\003)1130 2218 y Fi(1)1176 2194 y Fp(\016)7 b(L)1302 2209 y Fj(C)1339 2218 y Fi(1)1377 2194 y Fo(h)p Ft(\)\()p Fo(\031)1564 2209 y Fm(\003)1613 2218 y Fi(2)1658 2194 y Fp(\016)g(L)1784 2209 y Fj(C)1821 2218 y Fi(2)1860 2194 y Fo(h)p Ft(\))p Fp(k)2004 2209 y Fm(\003)2053 2218 y Fi(1)2087 2209 y Fj([)p Fm(\003)2183 2218 y Fi(2)2250 2194 y Fp(\024)28 b Fo(c)2397 2209 y Fm(33)2471 2194 y Fo(#)2528 2152 y Fj(\000j)p Fm(\003)2652 2161 y Fi(1)2687 2152 y Fj(j\000j)p Fm(\003)2831 2161 y Fi(2)2865 2152 y Fj(j)2888 2194 y Fo(\024)2944 2152 y Fi(dist)p Fm(\(\003)3120 2161 y Fi(1)3155 2152 y Fv(;)p Fm(\003)3224 2161 y Fi(2)3258 2152 y Fm(\))3339 2194 y Ft(\(116\))324 2550 y(\(115\))j(and)i(\(116\))f(also)g(hold)f (for)h(all)f(\003)1788 2514 y Fj(0)1788 2575 y Fm(1)1855 2550 y Fp(\022)d Ft(\003)2028 2565 y Fm(1)2067 2550 y Ft(,)33 b(\003)2195 2514 y Fj(0)2195 2575 y Fm(2)2261 2550 y Fp(\022)c Ft(\003)2435 2565 y Fm(2)2506 2550 y Ft(and)k(\(1.\))43 b(follo)m(ws.)924 2866 y Fo(\031)979 2881 y Fm(\003)1028 2890 y Fi(1)1067 2866 y Ft(\()p Fo(f)11 b(\027)6 b Ft(\))83 b(=)g Fo(\031)1553 2881 y Fm(\003)1602 2890 y Fi(1)1641 2866 y Ft(\()p Fo(f)11 b(\027)1786 2881 y Fm(\003)1835 2890 y Fi(1)1869 2881 y Fj([)p Fm(\003)1965 2890 y Fi(2)2004 2866 y Ft(\))1297 b(\(117\))1339 3011 y(=)83 b Fo(\031)1553 3026 y Fm(\003)1602 3035 y Fi(1)1641 3011 y Ft(\()p Fo(f)11 b(\027)1786 3026 y Fm(\003)1835 3035 y Fi(1)1873 3011 y Fo(\027)1921 3026 y Fm(\003)1970 3035 y Fi(2)2031 3011 y Ft(+)22 b Fo(f)11 b Ft(\()p Fo(\027)2274 3026 y Fm(\003)2323 3035 y Fi(1)2362 3011 y Fo(\027)2410 3026 y Fm(\003)2459 3035 y Fi(2)2520 3011 y Fp(\000)22 b Fo(\027)2667 3026 y Fm(\003)2716 3035 y Fi(1)2751 3026 y Fj([)p Fm(\003)2847 3035 y Fi(2)2886 3011 y Ft(\)\))1339 3156 y(=)83 b Fo(\027)6 b Ft(\()p Fo(f)11 b Ft(\))p Fo(\027)1735 3171 y Fm(\003)1784 3180 y Fi(1)1845 3156 y Ft(+)22 b Fo(\031)1998 3171 y Fm(\003)2047 3180 y Fi(1)2086 3156 y Ft(\()p Fo(f)11 b Ft(\()p Fo(\027)2269 3171 y Fm(\003)2318 3180 y Fi(1)2356 3156 y Fo(\027)2404 3171 y Fm(\003)2453 3180 y Fi(2)2514 3156 y Fp(\000)23 b Fo(\027)2662 3171 y Fm(\003)2711 3180 y Fi(1)2746 3171 y Fj([)p Fm(\003)2842 3180 y Fi(2)2880 3156 y Ft(\)\))324 3351 y(and)32 b(using)h Fp(k)p Fo(\031)874 3366 y Fm(\003)923 3375 y Fi(1)962 3351 y Fp(k)1012 3366 y Fj(1)1114 3351 y Ft(=)27 b(1)33 b(w)m(e)g(get)553 3592 y Fp(k)p Fo(\031)658 3607 y Fm(\003)707 3616 y Fi(1)746 3592 y Ft(\()p Fo(f)11 b Ft(\()p Fo(\027)929 3607 y Fm(\003)978 3616 y Fi(1)1016 3592 y Fo(\027)1064 3607 y Fm(\003)1113 3616 y Fi(2)1174 3592 y Fp(\000)23 b Fo(\027)1322 3607 y Fm(\003)1371 3616 y Fi(1)1406 3607 y Fj([)p Fm(\003)1502 3616 y Fi(2)1540 3592 y Ft(\)\))p Fp(k)1666 3607 y Fm(\003)1715 3616 y Fi(1)1781 3592 y Fp(\024)28 b(k)p Fo(f)11 b Fp(k)2045 3607 y Fm(\003)2094 3616 y Fi(2)2133 3592 y Fp(k)p Fo(\027)2231 3607 y Fm(\003)2280 3616 y Fi(1)2318 3592 y Fo(\027)2366 3607 y Fm(\003)2415 3616 y Fi(2)2476 3592 y Fp(\000)23 b Fo(\027)2624 3607 y Fm(\003)2673 3616 y Fi(1)2708 3607 y Fj([)p Fm(\003)2804 3616 y Fi(2)2842 3592 y Fp(k)2892 3607 y Fm(\003)2941 3616 y Fi(1)2976 3607 y Fj([)p Fm(\003)3072 3616 y Fi(2)3339 3592 y Ft(\(118\))324 3756 y(and)32 b(so)h(b)m(y)h(\(1.\))574 3996 y Fp(k)p Fo(\031)679 4011 y Fm(\003)728 4020 y Fi(1)767 3996 y Ft(\()p Fo(f)11 b Ft(\()p Fo(\027)950 4011 y Fm(\003)999 4020 y Fi(1)1038 3996 y Fo(\027)1086 4011 y Fm(\003)1135 4020 y Fi(2)1196 3996 y Fp(\000)22 b Fo(\027)1343 4011 y Fm(\003)1392 4020 y Fi(1)1427 4011 y Fj([)p Fm(\003)1523 4020 y Fi(2)1562 3996 y Ft(\)\))p Fp(k)1688 4011 y Fm(\003)1737 4020 y Fi(1)1803 3996 y Fp(\024)28 b Fo(c)1950 4011 y Fm(16)2025 3996 y Fo(#)2082 3955 y Fj(\000j)p Fm(\003)2206 3964 y Fi(1)2240 3955 y Fj(j\000j)p Fm(\003)2384 3964 y Fi(2)2418 3955 y Fj(j)2442 3996 y Fp(k)p Fo(f)11 b Fp(k)2601 4011 y Fm(\003)2650 4020 y Fi(2)2688 3996 y Fo(\024)2744 3955 y Fi(dist)o Fm(\(\003)2919 3964 y Fi(1)2954 3955 y Fv(;)p Fm(\003)3023 3964 y Fi(2)3057 3955 y Fm(\))3339 3996 y Ft(\(119\))324 4160 y(This)33 b(holds)f(for)g(all)e(\003)1154 4124 y Fj(0)1154 4185 y Fm(1)1221 4160 y Fp(\032)e Ft(\003)1394 4175 y Fm(1)1433 4160 y Ft(,)33 b(so)g(\(2.\))43 b(is)32 b(pro)m(v)m(ed.)324 4280 y(W)-8 b(e)35 b(set)h Fo(\036)31 b Ft(=)h Fo(f)11 b(\027)30 b Fp(\000)24 b Fo(\027)6 b Ft(\()p Fo(f)11 b Ft(\))p Fo(\027)6 b Ft(.)52 b(So)35 b Fo(\031)1599 4295 y Fm(\003)1648 4304 y Fi(1)1711 4280 y Fp(\016)23 b(L)1853 4244 y Fv(N)1853 4305 y(F)10 b Fj(\016)p Fv(T)1994 4286 y Fk(\017)2028 4280 y Ft(\()p Fo(f)h(\027)6 b Ft(\))24 b Fp(\000)g Fo(\027)6 b Ft(\()p Fo(f)11 b Ft(\))p Fo(\027)2579 4295 y Fm(\003)2628 4304 y Fi(1)2699 4280 y Ft(=)32 b Fo(\031)2862 4295 y Fm(\003)2911 4304 y Fi(1)2973 4280 y Fp(\016)24 b(L)3116 4244 y Fv(N)3116 4305 y(F)10 b Fj(\016)p Fv(T)3257 4286 y Fk(\017)3291 4280 y Fo(\036)p Ft(.)50 b(W)-8 b(e)324 4401 y(estimate)32 b(the)h Fp(k)23 b(\001)f(k)1056 4429 y Fm(\003)1105 4438 y Fi(1)1139 4429 y Fv(;)1168 4411 y Fm(~)1159 4429 y Fv(#)1205 4401 y Ft(-norm)31 b(of)i(the)g(last)g(term)f(as)h(in)f(the)i (pro)s(of)e(of)h(Prop)s(osition)e(6,)324 4521 y(but)i(this)f(time)f (using)h(the)h(\014ner)h(estimates)e(from)f(\(2.\))422 4837 y Fp(k)p Fo(\036)530 4852 y Fm(\003\()p Fj(C)t Fm(\))678 4837 y Fp(k)728 4852 y Fm(\003\()p Fj(C)t Fm(\))960 4837 y Fp(\024)83 b Fo(#)1177 4795 y Fj(\000j)p Fm(\003\()p Fj(C)t Fm(\))p Fj(j)1421 4837 y Fo(c)1463 4852 y Fm(11)1538 4837 y Fo(#)1595 4795 y Fj(\000j)p Fm(\003)1719 4804 y Fi(2)1753 4795 y Fj(j)1777 4837 y Fp(k)p Fo(f)11 b Fp(k)1936 4852 y Fm(\003)1985 4861 y Fi(2)2023 4837 y Fo(\024)2079 4795 y Fi(dist)p Fm(\(\003\()p Fj(C)t Fm(\))p Fv(;)p Fm(\003)2419 4804 y Fi(2)2454 4795 y Fm(\))3339 4837 y Ft(\(120\))960 5002 y Fp(\024)83 b Fo(c)1162 5017 y Fm(11)1237 5002 y Fo(#)1294 4961 y Fj(\000j)p Fm(\003)1418 4970 y Fi(2)1453 4961 y Fj(j)1476 5002 y Fp(k)p Fo(f)11 b Fp(k)1635 5017 y Fm(\003)1684 5026 y Fi(2)1722 5002 y Fo(#)1779 4961 y Fj(\000j)p Fm(\003)1903 4969 y Fk(r)1938 4961 y Fm(\()p Fj(C)t Fm(\))p Fj(j\000)2108 4903 y Fh(P)2196 4923 y Fg(1)2196 4990 y Fk(k)q Fi(=1)2311 4961 y Fm(\(3)p Fv(k)r Fm(\))2439 4938 y Fk(d)2476 4961 y Fv(n)2519 4973 y Fk(\014)s(;k)2617 5002 y Fo(\024)2673 4961 y Fi(dist)p Fm(\(\003)2849 4970 y Fi(1)2884 4961 y Fv(;)p Fm(\003)2953 4970 y Fi(2)2987 4961 y Fm(\))p Fj(\000)3069 4903 y Fh(P)3157 4923 y Fg(1)3157 4990 y Fk(k)q Fi(=1)3284 4961 y Fv(k)r(n)3366 4973 y Fk(\014)s(;k)1894 5251 y Ft(40)p eop %%Page: 41 41 41 40 bop 324 555 a Ft(where)34 b(as)e(b)s(efore)h(\003\()p Fp(C)6 b Ft(\))1248 503 y Fm(def)1258 555 y Ft(=)1382 530 y(~)1373 555 y(\003)1441 570 y Fj(C)1508 555 y Fp([)22 b Ft(\003)1664 570 y Fv(r)1702 555 y Ft(.)324 676 y(So)32 b(w)m(e)i(get)e(analogously)f(to)i(form)m(ulae)e(\(89\))h(and)g (\(90\):)396 990 y(~)384 1016 y Fo(#)441 975 y Fj(j)p Fm(\003)510 984 y Fi(1)544 975 y Fj(j)745 933 y Fh(X)585 1121 y Fj(C)t Fm(:)p Fi(length)o Fm(\()p Fj(C)t Fm(\)=)p Fv(N)1042 1016 y Fp(k)p Fo(\031)1147 1031 y Fm(\003)1196 1040 y Fi(1)1257 1016 y Fp(\016)22 b(L)1398 1031 y Fj(C)1443 1016 y Fo(\036)1501 1031 y Fm(\003)1550 1040 y Fi(2)1588 1016 y Fp(k)1638 1045 y Fm(\003)1687 1054 y Fi(1)3339 1016 y Ft(\(121\))467 1355 y Fp(\024)640 1328 y Ft(~)627 1355 y Fo(#)684 1313 y Fj(j)p Fm(\003)753 1322 y Fi(1)788 1313 y Fj(j)844 1237 y(j)p Fm(\003)913 1246 y Fi(1)947 1237 y Fj(j)846 1272 y Fh(X)828 1455 y Fv(K)5 b Fm(=0)999 1209 y Fh( )1065 1287 y Fp(j)p Ft(\003)1161 1302 y Fm(1)1200 1287 y Fp(j)1101 1423 y Fo(K)1228 1209 y Fh(!)1424 1272 y(X)1444 1431 y Fk(n)1482 1449 y(\014)1320 1506 y(K)s Fg(\024j)p Fk(n)1479 1523 y(\014)1521 1506 y Fg(j)p Fk(<)p Fg(1)1675 1355 y Ft(4)1724 1313 y Fv(K)1838 1247 y Fj(1)1820 1272 y Fh(Y)1809 1456 y Fv(k)r Fm(=1)1954 1258 y Fh(\020)2004 1355 y Ft(exp)q(\()p Fo(c)2233 1370 y Fm(21)2308 1355 y Fo(k)2362 1313 y Fv(d)2402 1355 y Ft(\))2440 1258 y Fh(\021)2490 1281 y Fv(n)2533 1293 y Fk(\014)s(;k)2648 1355 y Ft(\()p Fo(c)2728 1370 y Fm(1)2767 1355 y Fo(\017)p Ft(\))2844 1313 y Fj(j)p Fv(n)2907 1325 y Fk(\014)2948 1313 y Fj(j)644 1724 y Fp(\002)767 1616 y Fj(1)749 1641 y Fh(Y)738 1825 y Fv(k)r Fm(=1)883 1628 y Fh(\020)933 1724 y Ft(exp)q(\()p Fp(\000)p Fo(c)1239 1739 y Fm(2)1279 1724 y Fo(k)1333 1683 y Fv(d)1373 1724 y Ft(\))1411 1628 y Fh(\021)1461 1651 y Fv(n)1504 1663 y Fk(\014)s(;k)1697 1616 y Fj(1)1672 1641 y Fh(X)1619 1829 y Fv(L)p Fm(=)p Fj(j)p Fv(n)1785 1841 y Fk(\014)1826 1829 y Fj(j)1862 1578 y Fh( )1975 1657 y Fo(L)1928 1792 y Fp(j)p Fo(n)2014 1807 y Fv(\014)2061 1792 y Fp(j)2088 1578 y Fh(!)2154 1724 y Fo(c)2196 1683 y Fj(j)p Fv(n)2259 1695 y Fk(\014)2300 1683 y Fj(j)2196 1749 y Fv(r)2324 1724 y Fo(\021)2376 1683 y Fv(L)2473 1616 y Fj(1)2455 1641 y Fh(Y)2444 1825 y Fv(k)r Fm(=1)2589 1628 y Fh(\020)2639 1724 y Ft(exp)q(\()p Fo(c)2868 1739 y Fm(22)2943 1724 y Fo(k)2997 1683 y Fv(d)3037 1724 y Ft(\))3075 1628 y Fh(\021)3125 1651 y Fv(n)3168 1663 y Fk(\014)s(;k)644 2067 y Fp(\002)767 1959 y Fj(1)749 1984 y Fh(Y)738 2169 y Fv(k)r Fm(=1)883 1971 y Fh(\020)933 2067 y Ft(exp)q(\()p Fo(c)1162 2082 y Fm(23)1237 2067 y Fo(k)1291 2026 y Fv(d)1331 2067 y Ft(\))1369 1971 y Fh(\021)1419 1994 y Fv(n)1462 2006 y Fk(\014)s(;k)1577 2067 y Fo(\021)1629 2026 y Fm(max)o Fj(f)p Fm(0)p Fv(;N)i Fj(\000)p Fv(L)p Fj(g)2072 1950 y(j)p Fm(\003)2141 1959 y Fi(1)2175 1950 y Fj(j\000)p Fv(K)2133 1984 y Fh(X)2137 2169 y Fv(l)q Fm(=0)2330 1921 y Fh( )2396 2000 y Fp(j)p Ft(\003)2492 2015 y Fm(1)2531 2000 y Fp(j)22 b(\000)g Fo(K)2568 2136 y(l)2770 1921 y Fh(!)2836 2067 y Ft(\()p Fo(c)2916 2082 y Fv(r)2954 2067 y Fo(\021)3006 2026 y Fv(N)3073 2067 y Ft(\))3111 2026 y Fv(l)3137 2067 y Fo(c)3179 2016 y Fj(j)p Fm(\003)3248 2025 y Fi(1)3282 2016 y Fj(j\000)p Fv(K)5 b Fj(\000)p Fv(l)3179 2092 y(h)644 2310 y Fp(\002)p Fo(c)763 2325 y Fm(11)838 2310 y Fo(#)895 2269 y Fj(\000j)p Fm(\003)1019 2278 y Fi(2)1054 2269 y Fj(j)1077 2310 y Fp(k)p Fo(f)11 b Fp(k)1236 2325 y Fm(\003)1285 2334 y Fi(2)1323 2310 y Fo(#)1380 2269 y Fj(\000)p Fv(l)q Fj(\000)1512 2211 y Fh(P)1600 2232 y Fg(1)1600 2298 y Fk(k)q Fi(=1)1715 2269 y Fm(\(3)p Fv(k)r Fm(\))1843 2246 y Fk(d)1880 2269 y Fv(n)1923 2281 y Fk(\014)s(;k)2017 2269 y Fm(\()p Fj(C)t Fm(\))2117 2310 y Fo(\024)2173 2269 y Fi(dist)p Fm(\(\003)2349 2278 y Fi(1)2384 2269 y Fv(;)p Fm(\003)2453 2278 y Fi(2)2487 2269 y Fm(\))q Fj(\000)2570 2211 y Fh(P)2657 2232 y Fg(1)2657 2298 y Fk(k)q Fi(=1)2784 2269 y Fv(k)r(n)2866 2281 y Fk(\014)s(;k)467 2544 y Fp(\024)83 b Fo(c)669 2559 y Fm(11)757 2518 y Ft(~)744 2544 y Fo(#)801 2503 y Fj(j)p Fm(\003)870 2512 y Fi(1)905 2503 y Fj(j)961 2427 y(j)p Fm(\003)1030 2436 y Fi(1)1064 2427 y Fj(j)962 2461 y Fh(X)945 2645 y Fv(K)5 b Fm(=0)1116 2398 y Fh( )1182 2476 y Fp(j)p Ft(\003)1278 2491 y Fm(1)1317 2476 y Fp(j)1218 2612 y Fo(K)1344 2398 y Fh(!)1541 2461 y(X)1561 2621 y Fk(n)1599 2638 y(\014)1437 2695 y(K)s Fg(\024j)p Fk(n)1596 2712 y(\014)1638 2695 y Fg(j)p Fk(<)p Fg(1)1792 2544 y Ft(4)1841 2503 y Fv(K)1909 2544 y Ft(\()p Fo(c)1989 2559 y Fm(1)2029 2544 y Fo(\017c)2110 2559 y Fv(r)2148 2544 y Ft(\))2186 2503 y Fj(j)p Fv(n)2249 2515 y Fk(\014)2290 2503 y Fj(j)644 2909 y Fp(\002)767 2801 y Fj(1)749 2825 y Fh(Y)738 3010 y Fv(k)r Fm(=1)883 2812 y Fh(\020)933 2909 y Ft(exp)q(\(\()p Fo(c)1200 2924 y Fm(21)1297 2909 y Fp(\000)22 b Fo(c)1438 2924 y Fm(2)1500 2909 y Ft(+)g Fo(c)1640 2924 y Fm(22)1737 2909 y Ft(+)g Fo(c)1877 2924 y Fm(23)1974 2909 y Fp(\000)g Ft(3)2122 2867 y Fv(d)2179 2909 y Ft(ln)16 b Fo(#)23 b Fp(\000)f Ft(ln)16 b Fo(\024)p Ft(\))p Fo(k)2702 2867 y Fv(d)2743 2909 y Ft(\))2781 2812 y Fh(\021)2830 2835 y Fv(n)2873 2847 y Fk(\014)s(;k)644 3196 y Fp(\002)796 3088 y Fj(1)772 3113 y Fh(X)738 3297 y Fv(L)p Fm(=)p Fv(n)884 3309 y Fk(\014)942 3196 y Fo(\021)994 3155 y Fm(max)o Fj(f)p Fv(L;N)7 b Fj(g)1330 3196 y Ft(\()p Fo(#)1425 3155 y Fj(\000)p Fm(1)1520 3196 y Fo(c)1562 3211 y Fv(r)1600 3196 y Fo(\021)1652 3155 y Fv(N)1741 3196 y Ft(+)22 b Fo(c)1881 3211 y Fv(h)1926 3196 y Ft(\))1964 3155 y Fj(j)p Fm(\003)2033 3164 y Fi(1)2067 3155 y Fj(j\000)p Fv(K)2210 3196 y Fo(#)2267 3155 y Fj(\000j)p Fm(\003)2391 3164 y Fi(2)2425 3155 y Fj(j)2449 3196 y Fp(k)p Fo(f)11 b Fp(k)2608 3212 y Fj(j)p Fm(\003)2677 3221 y Fi(2)2711 3212 y Fj(j)2734 3196 y Fo(\024)2790 3155 y Fi(dist)p Fm(\(\003)2966 3164 y Fi(1)3001 3155 y Fv(;)p Fm(\003)3070 3164 y Fi(2)3104 3155 y Fm(\))324 3517 y Ft(This)33 b(time)f(w)m(e)i (set)50 b(~)-66 b Fo(c)1108 3532 y Fm(2)1176 3517 y Ft(=)28 b Fo(c)1322 3532 y Fm(2)1384 3517 y Fp(\000)23 b Fo(c)1526 3532 y Fm(21)1623 3517 y Fp(\000)g Fo(c)1765 3532 y Fm(22)1862 3517 y Fp(\000)g Fo(c)2004 3532 y Fm(23)2101 3517 y Ft(+)g(3)2249 3481 y Fv(d)2305 3517 y Ft(ln)16 b Fo(#)23 b Ft(+)g(ln)16 b Fo(\024)33 b Ft(and)g(with)f(the)i(same)324 3637 y(analysis)e(as)g (from)g(\(90\))g(to)g(\(96\))g(w)m(e)h(get:)633 3926 y Fp(\024)28 b Fo(c)780 3941 y Fm(34)855 3926 y Ft(\()p Fo(\017)932 3941 y Fm(2)994 3926 y Ft(+)22 b Fo(c)1134 3941 y Fv(r)1179 3926 y Ft(~)-56 b Fo(\021)1224 3885 y Fv(N)1313 3832 y Ft(~)1301 3858 y Fo(#)p 1301 3903 58 4 v 1301 3994 a(#)1390 3926 y Ft(+)22 b Fo(c)1530 3941 y Fv(h)1588 3900 y Ft(~)1575 3926 y Fo(#)q Ft(\))1671 3885 y Fj(j)p Fm(\003)1740 3894 y Fi(1)1774 3885 y Fj(j)1797 3926 y Fp(k)p Fo(\027)6 b Fp(k)1951 3941 y Fv(#)1998 3926 y Fo(#)2055 3885 y Fj(\000j)p Fm(\003)2179 3894 y Fi(2)2213 3885 y Fj(j)2237 3926 y Fp(k)p Fo(f)11 b Fp(k)2396 3941 y Fm(\003)2445 3950 y Fi(2)2483 3926 y Fo(\024)2539 3885 y Fi(dist)p Fm(\(\003)2715 3894 y Fi(1)2749 3885 y Fv(;)p Fm(\003)2818 3894 y Fi(2)2853 3885 y Fm(\))2891 3926 y Ft(~)-56 b Fo(\021)2936 3885 y Fv(N)3003 3926 y Fo(:)309 b Ft(\(122\))324 4149 y(F)-8 b(or)28 b(su\016cien)m(tly)h (small)e Fo(\017)1270 4164 y Fm(2)1338 4149 y Ft(and)1536 4123 y(~)1524 4149 y Fo(#)i Ft(the)g(term)f(in)g(brac)m(k)m(ets)j(is)d (smaller)f(than)i(one.)42 b(Note)324 4269 y(that)33 b(there)h(is)e(no)i (condition)d(on)j Fo(N)10 b Ft(.)45 b(So)34 b(w)m(e)g(get)f(the)h(same) f(estimates)g(for)f(all)g Fo(n)d Fp(\025)g Ft(0)324 4390 y(and)j(these)i(also)e(hold)g(for)g(\003)27 b Fp(\032)h Ft(\003)1593 4405 y Fm(1)1632 4390 y Ft(.)44 b(So)32 b(in)g(analogy)g(with)g(\(54\))g(w)m(e)h(get)863 4531 y Fh(\015)863 4581 y(\015)863 4630 y(\015)p Fp(L)978 4589 y Fv(N)978 4655 y(F)10 b Fj(\016)p Fv(T)1119 4636 y Fk(\017)1153 4630 y Fo(\036)22 b Fp(\000)g(L)1401 4589 y Fv(N)7 b Fm(+1)1401 4655 y Fv(F)j Fj(\016)p Fv(T)1542 4636 y Fk(\017)1576 4630 y Fo(\036)1634 4531 y Fh(\015)1634 4581 y(\015)1634 4630 y(\015)1680 4684 y Fm(\003)1729 4693 y Fi(1)1795 4630 y Fp(\024)28 b Fo(c)1942 4645 y Fm(35)2017 4630 y Fo(#)2074 4589 y Fj(\000j)p Fm(\003)2198 4598 y Fi(2)2233 4589 y Fj(j)2256 4630 y Fp(k)p Fo(f)11 b Fp(k)2415 4645 y Fm(\003)2464 4654 y Fi(2)2502 4630 y Fo(\024)2558 4589 y Fi(dist)p Fm(\(\003)2734 4598 y Fi(1)2769 4589 y Fv(;)p Fm(\003)2838 4598 y Fi(2)2872 4589 y Fm(\))2911 4630 y Ft(~)-56 b Fo(\021)2956 4589 y Fv(N)3339 4630 y Ft(\(123\))324 4831 y(and)32 b(as)h Fo(\026)p Ft(\()p Fo(\036)p Ft(\))27 b(=)h(0)k(w)m(e)i(conclude)f (\(3.\))1720 b Fa(2)1894 5251 y Ft(41)p eop %%Page: 42 42 42 41 bop 324 548 a Fs(Pro)s(of)37 b(of)h(Theorem)f(2)998 656 y Fh(\014)998 706 y(\014)998 756 y(\014)998 805 y(\014)1026 663 y(Z)1072 852 y Fv(M)1168 780 y Fo(\027)6 b(d\026)17 b(g)t(f)31 b Fp(\000)1579 659 y Fh(\022)1640 663 y(Z)1686 852 y Fv(M)1782 780 y Fo(\027)6 b(d\026)17 b(g)2014 659 y Fh(\023)e(\022)2151 663 y(Z)2198 852 y Fv(M)2293 780 y Fo(\027)6 b(d\026)17 b(f)2533 659 y Fh(\023)2593 656 y(\014)2593 706 y(\014)2593 756 y(\014)2593 805 y(\014)3339 780 y Ft(\(124\))1081 1038 y Fp(\024)1242 888 y Fh(\014)1242 938 y(\014)1242 988 y(\014)1242 1038 y(\014)1242 1088 y(\014)1270 921 y(Z)1316 1109 y Fm(\()p Fv(S)1390 1090 y Fi(1)1424 1109 y Fm(\))1451 1087 y Fi(\003)1492 1102 y(1)1528 1087 y Fg([)p Fi(\003)1611 1102 y(2)1670 1038 y Fo(d\026)1780 997 y Fm(\003)1829 1006 y Fi(1)1863 997 y Fj([)p Fm(\003)1959 1006 y Fi(2)1998 1038 y Ft(\()p Fo(\027)2084 1053 y Fm(\003)2133 1062 y Fi(1)2168 1053 y Fj([)p Fm(\003)2264 1062 y Fi(2)2324 1038 y Fp(\000)23 b Fo(\027)2472 1053 y Fm(\003)2521 1062 y Fi(1)2560 1038 y Fo(\027)2608 1053 y Fm(\003)2657 1062 y Fi(2)2696 1038 y Ft(\))p Fo(g)t(f)2844 888 y Fh(\014)2844 938 y(\014)2844 988 y(\014)2844 1038 y(\014)2844 1088 y(\014)1081 1245 y Fp(\024)84 b(k)p Fo(\027)1340 1260 y Fm(\003)1389 1269 y Fi(1)1423 1260 y Fj([)p Fm(\003)1519 1269 y Fi(2)1580 1245 y Fp(\000)23 b Fo(\027)1728 1260 y Fm(\003)1777 1269 y Fi(1)1816 1245 y Fo(\027)1864 1260 y Fm(\003)1913 1269 y Fi(2)1952 1245 y Fp(k)2002 1260 y Fm(\003)2051 1269 y Fi(1)2085 1260 y Fj([)p Fm(\003)2181 1269 y Fi(2)2220 1245 y Fp(k)p Fo(g)t Fp(k)2371 1260 y Fj(1)2444 1245 y Fp(k)p Fo(f)11 b Fp(k)2603 1260 y Fj(1)1081 1391 y Fp(\024)84 b Fo(c)1284 1406 y Fm(10)1359 1391 y Fo(#)1416 1349 y Fj(\000j)p Fm(\003)1540 1358 y Fi(1)1574 1349 y Fj(j\000j)p Fm(\003)1718 1358 y Fi(2)1752 1349 y Fj(j)1776 1391 y Fp(k)p Fo(g)t Fp(k)1927 1406 y Fj(1)2000 1391 y Fp(k)p Fo(f)11 b Fp(k)2159 1406 y Fj(1)2233 1391 y Fo(\024)2289 1349 y Fi(dist)p Fm(\(\003)2465 1358 y Fi(1)2500 1349 y Fv(;)p Fm(\003)2569 1358 y Fi(2)2603 1349 y Fm(\))324 1611 y Ft(so)33 b(\(1.\))43 b(is)32 b(pro)m(v)m(ed.)760 1839 y Fh(\014)760 1889 y(\014)760 1939 y(\014)760 1988 y(\014)787 1846 y(Z)833 2035 y Fv(M)929 1963 y Fo(\027)6 b(d\026)17 b(g)25 b Fp(\016)d Fo(\034)33 b Fp(\016)22 b Fo(S)1467 1922 y Fv(n)1514 1963 y Fo(f)33 b Fp(\000)1695 1842 y Fh(\022)1756 1846 y(Z)1802 2035 y Fv(M)1897 1963 y Fo(\027)6 b(d\026)17 b(g)25 b Fp(\016)d Fo(\034)2275 1842 y Fh(\023)c(\022)2415 1846 y(Z)2461 2035 y Fv(M)2556 1963 y Fo(\027)6 b(d\026)17 b(f)2796 1842 y Fh(\023)2857 1839 y(\014)2857 1889 y(\014)2857 1939 y(\014)2857 1988 y(\014)3339 1963 y Ft(\(125\))843 2196 y(=)1003 2071 y Fh(\014)1003 2121 y(\014)1003 2171 y(\014)1003 2221 y(\014)1031 2079 y(Z)1077 2267 y Fv(M)1172 2196 y Fo(d\026)g(g)25 b Fp(\016)d Fo(\034)1513 2100 y Fh(\020)1563 2196 y Fo(\031)1618 2213 y Fv(\034)1657 2194 y Fg(\000)p Fi(1)1739 2213 y Fm(\(\003)1815 2222 y Fi(1)1850 2213 y Fm(\))1904 2196 y Fp(\016)g(L)2045 2155 y Fv(n)2045 2221 y(F)10 b Fj(\016)p Fv(T)2186 2202 y Fk(\017)2220 2196 y Ft(\()p Fo(f)h(\027)6 b Ft(\))22 b Fp(\000)h Fo(\027)6 b Ft(\()p Fo(f)11 b Ft(\))p Fo(\027)2768 2213 y Fv(\034)2807 2194 y Fg(\000)p Fi(1)2889 2213 y Fm(\(\003)2965 2222 y Fi(1)3000 2213 y Fm(\))3032 2100 y Fh(\021)3082 2071 y(\014)3082 2121 y(\014)3082 2171 y(\014)3082 2221 y(\014)843 2405 y Fp(\024)83 b Fo(c)1045 2420 y Fm(12)1120 2405 y Fo(c)1162 2354 y Fj(j)p Fm(\003)1231 2363 y Fi(1)1265 2354 y Fj(j)p Fm(+)p Fj(j)p Fm(\003)1409 2363 y Fi(2)1443 2354 y Fj(j)1162 2426 y Fm(5)1466 2405 y Fp(k)p Fo(f)11 b Fp(k)1625 2420 y Fm(\003)1674 2429 y Fi(2)1712 2405 y Fp(k)p Fo(g)t Fp(k)1863 2420 y Fj(1)1937 2405 y Fo(\024)1993 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b(d\026)17 b(g)1323 588 y Fm(\003)1372 597 y Fi(1)1432 573 y Fp(\016)22 b Fo(\034)1557 452 y Fh(\023)17 b(\022)1696 456 y(Z)1742 644 y Fv(M)1838 573 y Fo(\027)6 b(d\026)17 b Ft(\()p Fo(f)32 b Fp(\000)2258 547 y Ft(~)2237 573 y Fo(f)2285 588 y Fm(\003)2334 597 y Fi(2)2373 573 y Ft(\))2411 452 y Fh(\023)2472 448 y(\014)2472 498 y(\014)2472 548 y(\014)2472 598 y(\014)772 805 y Ft(+)865 681 y Fh(\014)865 731 y(\014)865 780 y(\014)865 830 y(\014)892 684 y(\022)953 688 y(Z)999 877 y Fv(M)1095 805 y Fo(\027)6 b(d\026)17 b Ft(\()p Fo(g)25 b Fp(\000)d Fo(g)1532 820 y Fm(\003)1581 829 y Fi(1)1620 805 y Ft(\))g Fp(\016)g Fo(\034)1805 684 y Fh(\023)17 b(\022)1944 688 y(Z)1990 877 y Fv(M)2086 805 y Fo(\027)6 b(d\026)17 b(f)2326 684 y Fh(\023)2386 681 y(\014)2386 731 y(\014)2386 780 y(\014)2386 830 y(\014)612 1006 y Fp(\024)83 b(k)p Fo(g)25 b Fp(\000)e Fo(g)1041 1021 y Fm(\003)1090 1030 y Fi(1)1128 1006 y Fp(k)1178 1021 y Fj(1)1253 1006 y Fp(k)p Fo(f)11 b Fp(k)1412 1021 y Fj(1)1508 1006 y Ft(+)22 b Fp(k)p Fo(g)1703 1021 y Fm(\003)1752 1030 y Fi(1)1790 1006 y Fp(k)1840 1021 y Fj(1)1915 1006 y Fp(k)p Fo(f)32 b Fp(\000)2166 979 y Ft(~)2145 1006 y Fo(f)2193 1021 y Fm(\003)2242 1030 y Fi(2)2281 1006 y Fp(k)2331 1021 y Fj(1)772 1165 y Ft(+)p Fo(c)p Ft(\(\003)996 1180 y Fm(1)1035 1165 y Fo(;)17 b Ft(\003)1147 1180 y Fm(2)1186 1165 y Fo(;)g(\024)p Ft(\))p Fo(c)1366 1114 y Fj(j)p Fm(\003)1435 1123 y Fi(1)1469 1114 y Fj(j)p Fm(+)p Fj(j)p Fm(\003)1613 1123 y Fi(2)1646 1114 y Fj(j)1366 1186 y Fm(5)1670 1165 y Fp(k)p Fo(g)1767 1180 y Fm(\003)1816 1189 y Fi(1)1854 1165 y Fp(k)1904 1180 y Fj(1)1979 1165 y Fp(k)2050 1138 y Ft(~)2029 1165 y Fo(f)2077 1180 y Fm(\003)2126 1189 y Fi(2)2164 1165 y Fp(k)2214 1180 y Fm(\003)2263 1189 y Fi(2)2309 1165 y Ft(~)-56 b Fo(\021)2354 1123 y Fv(n)p Fm(\()p Fv(\033)r Fm(\))2498 1165 y Fo(\024)2554 1123 y Fv(m)p Fm(\()p Fv(\033)r Fm(\))772 1310 y Ft(+)p Fp(k)p Fo(g)945 1325 y Fm(\003)994 1334 y Fi(1)1032 1310 y Fp(k)1082 1325 y Fj(1)1157 1310 y Fp(k)p Fo(f)32 b Fp(\000)23 b Fo(f)1435 1325 y Fm(\003)1484 1334 y Fi(2)1523 1310 y Fp(k)1573 1325 y Fj(1)1669 1310 y Ft(+)f Fp(k)p Fo(g)k Fp(\000)c Fo(g)2036 1325 y Fm(\003)2085 1334 y Fi(1)2124 1310 y Fp(k)2174 1325 y Fj(1)2248 1310 y Fp(k)p Fo(f)2346 1325 y Fm(\003)2395 1334 y Fi(2)2434 1310 y Fp(k)2484 1325 y Fj(1)612 1455 y Fp(\024)83 b Ft(\(2)p Fp(k)p Fo(f)11 b Fp(k)1018 1470 y Fj(1)1114 1455 y Ft(+)22 b(2)p Fp(k)p Fo(g)t Fp(k)1412 1470 y Fj(1)1508 1455 y Ft(+)g(3)p Fo(\015)5 b Ft(\))16 b Fo(\015)772 1614 y Ft(+)p Fo(c)p Ft(\(\003)996 1629 y Fm(1)1035 1614 y Fo(;)h Ft(\003)1147 1629 y Fm(2)1186 1614 y Fo(;)g(\024)p Ft(\))p Fo(c)1366 1563 y Fj(j)p Fm(\003)1435 1572 y Fi(1)1469 1563 y Fj(j)p Fm(+)p Fj(j)p Fm(\003)1613 1572 y Fi(2)1646 1563 y Fj(j)1366 1636 y Fm(5)1670 1614 y Ft(\()p Fp(k)p Fo(g)t Fp(k)1859 1629 y Fj(1)1955 1614 y Ft(+)22 b Fo(\015)5 b Ft(\))p Fp(k)2218 1588 y Ft(~)2197 1614 y Fo(f)2245 1629 y Fm(\003)2294 1638 y Fi(2)2332 1614 y Fp(k)2382 1629 y Fm(\003)2431 1638 y Fi(2)2477 1614 y Ft(~)-56 b Fo(\021)2522 1573 y Fv(n)p Fm(\()p Fv(\033)r Fm(\))2666 1614 y Fo(\024)2722 1573 y Fv(m)p Fm(\()p Fv(\033)r Fm(\))324 1834 y Ft(and)32 b(this)h(gets)g(arbitrarily)c(small)i(as)h(w)m(e)i(\014rst)f(c)m(ho)s (ose)g Fo(\015)5 b Ft(,)33 b(then)g(\003)2808 1849 y Fm(1)2847 1834 y Ft(,)g(\003)2975 1849 y Fm(2)3047 1834 y Ft(and)f Fo(f)3284 1849 y Fm(\003)3333 1858 y Fi(2)3404 1834 y Ft(and)324 1954 y(\014nally)f(max)p Fp(f)p Fo(m)p Ft(\()p Fo(\033)t Ft(\))p Fo(;)17 b(n)p Ft(\()p Fo(\033)t Ft(\))p Fp(g)p Ft(.)324 2075 y(\(5.\))43 b(follo)m(ws)31 b(from)h(\(4.\))43 b(and)32 b(the)h(comm)m(utation)e(of)h(the)h Fo(\034)2537 2090 y Fv(e)2570 2100 y Fk(i)2633 2075 y Ft(with)f Fo(S)6 b Ft(.)507 b Fa(2)324 2478 y Fu(Ac)l(kno)l(wledgemen)l (ts)324 2697 y Ft(The)34 b(\014rst)h(author)e(w)m(ould)h(lik)m(e)f(to)g (thank)h(T.)g(Hefer)h(for)e(helpful)f(discussions)j(and)f(R.)324 2818 y(Reid)d(for)h(his)g(commen)m(ts.)44 b(He)32 b(is)g(supp)s(orted)h (b)m(y)h(the)e(EC)i(via)d(the)i(TMR-F)-8 b(ello)m(wship)324 2938 y(ERBFMBICT-961157.)324 3271 y Fu(References)373 3490 y Ft([1])48 b(V.)32 b(Baladi,)e(M.)i(Degli)e(Esp)s(osti,)i(S.)g (Isola,)f(E.)h(J\177)-49 b(arv)m(enp\177)g(a\177)g(a,)33 b(A.)f(Kupiainen:)42 b(The)524 3610 y(sp)s(ectrum)33 b(of)f(w)m(eakly)i(coupled)f(map)e(lattices,)h(preprin)m(t)373 3814 y([2])48 b(J.)40 b(Bricmon)m(t)f(and)h(A.)h(Kupiainen:)57 b(Coupled)40 b(Analytic)f(Maps,)k Fn(Nonline)-5 b(arity)524 3934 y Ft(8,)33 b(pp.379-396,)e(1995)373 4138 y([3])48 b(J.)32 b(Bricmon)m(t)e(and)h(A.)g(Kupiainen:)42 b(High)30 b(T)-8 b(emp)s(erature)32 b(Expansions)g(and)f(Dy-)524 4258 y(namical)f(Systems,)k Fn(Comm.)g(Math.)h(Phys.)e Ft(178,)f(pp.73-732)373 4461 y([4])48 b(J.)k(Bricmon)m(t)e(and)h(A.)h (Kupiainen:)80 b(In\014nite)51 b(dimensional)e(SRB-measures,)524 4582 y(preprin)m(t)373 4785 y([5])f(L.A.)f(Bunimo)m(vic)m(h:)71 b(Coupled)46 b(map)g(lattices:)70 b(One)47 b(step)h(forw)m(ard)e(and)h (t)m(w)m(o)524 4905 y(steps)34 b(bac)m(k,)g Fn(Physic)-5 b(a)34 b(D)h(86)p Ft(,)d(1995,)g(pp.)h(248-255)1894 5251 y(43)p eop %%Page: 44 44 44 43 bop 373 548 a Ft([6])48 b(L.A.)27 b(Bunimo)m(vic)m(h,)h(Y.G.)e (Sinai:)39 b(Space-time)26 b(c)m(haos)i(in)e(coupled)h(map)f(lattices,) 524 668 y Fn(Nonline)-5 b(arity)35 b(1)p Ft(,)e(pp.)g(491-516,)d(1988) 373 862 y([7])48 b(N.)35 b(Dunford)e(and)h(J.T.)h(Sc)m(h)m(w)m(artz:)49 b(Linear)33 b(Op)s(erattors)h(P)m(art)h(I,)f Fn(Interscienc)-5 b(e)p Ft(,)524 982 y(New)34 b(Y)-8 b(ork,)33 b(1958)373 1175 y([8])48 b(M.)30 b(Jiang:Equilibrium)25 b(states)k(for)g(lattice)e (mo)s(dels)h(of)g(h)m(yp)s(erolic)g(t)m(yp)s(e)i Fn(Nonlin-)524 1296 y(e)-5 b(arity)36 b(8)p Ft(,)c(no.5,)g(1994,)g(pp.631-659)373 1489 y([9])48 b(M.)42 b(Jiang:)60 b(Ergo)s(dic)41 b(Prop)s(erties)g(of) g(Coupled)h(Map)f(Lattices)h(of)f(Hyp)s(erb)s(olic)524 1610 y(T)m(yp)s(e,)34 b Fn(Penns.)h(State)g(University)g(Dissertation)p Ft(,)d(1995)324 1803 y([10])48 b(M.)28 b(Jiang)d(and)i(A.)g(Mazel:)41 b(uniqueness)29 b(of)d(Gibbs)h(States)g(and)g(exp)s(onen)m(tial)f(De-) 524 1923 y(ca)m(y)f(of)e(Correlation)f(for)h(some)h(Lattice)f(mo)s (dels,)h Fn(Journal)j(of)f(Statistic)-5 b(al)27 b(Physics)524 2044 y Ft(82,)32 b(no.)h(3-4,)f(1995)324 2237 y([11])48 b(M.)30 b(Jiang)d(and)i(Y)-8 b(a.B.)29 b(P)m(esin:)42 b(Equilibrium)26 b(Measures)31 b(for)d(Coupled)h(Map)g(Lat-)524 2357 y(tices:)63 b(Existence,)47 b(Uniqueness)d(and)e (Finite-Dimensional)37 b(Appro)m(ximations,)524 2478 y(Preprin)m(t)324 2671 y([12])48 b(K.)32 b(Kanek)m(o)g(\(ed.\):)44 b(Theory)33 b(and)f(Applications)e(of)h(Coupled)h(map)f(Lattices,)h Fn(J.)524 2792 y(Wiley)p Ft(,)h(1993)324 2985 y([13])48 b(G.)26 b(Keller)f(and)h(M.)h(K)s(\177)-51 b(unzle:)39 b(T)-8 b(ransfer)27 b(Op)s(erators)f(for)g(Coupled)g(Map)h(Lattices,) 524 3105 y Fn(Er)-5 b(go)g(dic)35 b(The)-5 b(ory)34 b(&)h(Dynamic)-5 b(al)33 b(Systems)41 b Ft(V)-8 b(ol.)31 b(12,)h(pp.297-318,)g(1992)324 3299 y([14])48 b(A.)30 b(Lasota)f(and)h(M.C.)g(Mac)m(k)m(ey:)45 b(Chaos,)30 b(F)-8 b(ractals)29 b(and)g(Noise,)h Fn(Springer)p Ft(,)g(1994)324 3492 y([15])48 b(C.)36 b(Maes)g(and)f(A.)g(V)-8 b(an)35 b(Mo\013aert:)48 b(Sto)s(c)m(hastic)35 b(Stabilit)m(y)e(of)h(W) -8 b(eakly)35 b(Coupled)524 3612 y(Lattice)d(Maps,)i(preprin)m(t)324 3806 y([16])48 b(Y)-8 b(a.B.)36 b(P)m(esin)h(and)f(Y)-8 b(a.G.)36 b(Sinai:)48 b(Space-time)35 b(c)m(haos)i(in)e(c)m(hains)h(of) g(w)m(eakly)h(in-)524 3926 y(teracting)e(h)m(yp)s(erb)s(olic)g (mappings,)g Fn(A)-5 b(dvanc)g(es)36 b(in)i(Soviet)f(Mathematics)43 b Ft(V)-8 b(ol.3,)524 4047 y(1991,)32 b(pp.165-198)324 4240 y([17])48 b(D.L.)28 b(V)-8 b(olevic)m(h:)41 b(The)29 b(Sinai-Bo)m(w)m(en-Ruelle)d(Measure)k(for)d(a)h(Multidimensional)524 4360 y(Lattice)42 b(of)g(in)m(teracting)f(Hyp)s(erb)s(olic)h(Mappings,) j Fn(R)n(uss.)e(A)n(kad.)g(Dokl.)g(Math.)524 4481 y Ft(47,)32 b(1993,)g(pp.117-121)324 4674 y([18])48 b(D.L.)34 b(V)-8 b(olevic)m(h:)48 b(Construction)35 b(of)f(an)g(analogue)g(of)g(Bo)m(w)m (en-Ruell-Sinai)e(mea-)524 4795 y(sure)41 b(for)e(a)g(m)m (ultidimensional)c(lattice)j(of)h(in)m(teracting)f(h)m(yp)s(erb)s(olic) h(mappings)524 4915 y Fn(R)n(uss.)c(A)n(kad.)f(Math.)h(Sb)-5 b(ornik)42 b Ft(V)-8 b(ol.79,)31 b(1994,)h(pp.347-363)1894 5251 y(44)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF