%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: paper.dvi %%Pages: 10 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%DocumentPaperSizes: Letter %%EndComments %DVIPSCommandLine: dvips -o reed.ps paper %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1998.02.10:0630 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B 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Fg(1)1866 4206 y Fk(e)1913 4170 y Fd(\000)p Fj(t)p 1678 4250 337 4 v 1678 4341 a Fm(\()p Fk(n)23 b Fh(\000)g Fm(1\)!)2058 4273 y Fk(;)49 b(t)28 b(>)f Fm(0)33 b Fk(:)1872 5280 y Ff(3)p eop %%Page: 4 4 4 3 bop 0 83 a Fl(Lemma)44 b(1.)54 b Fe(Let)40 b Fh(f)p Fk(t)788 98 y Fj(n)832 83 y Fh(g)e Fe(b)s(e)h(a)f(sequence)h(of)f(n)m (um)m(b)s(ers)h(suc)m(h)g(that)g Fk(t)2579 98 y Fj(n)2660 83 y Fh(\030)d Fk(cn)j Fe(as)e Fk(n)f Fh(!)g(1)i Fe(for)f(some)0 216 y Fk(c)28 b Fh(2)g Fm(\(0)p Fk(;)17 b Fm(1\))p Fe(.)44 b(Then)1232 395 y Fi(P)17 b Fh(f)o Fk(S)1437 410 y Fj(n)1509 395 y Fk(<)27 b(t)1650 410 y Fj(n)1694 395 y Fh(g)h(\030)1969 327 y Fm(1)p 1887 372 215 4 v 1887 463 a(1)22 b Fh(\000)h Fk(c)2122 327 y(e)2169 291 y Fd(\000)p Fj(t)2245 301 y Fc(n)2282 327 y Fm(\()p Fk(t)2356 342 y Fj(n)2401 327 y Fm(\))2439 291 y Fj(n)p 2122 372 362 4 v 2259 463 a Fk(n)p Fm(!)2526 395 y Fk(:)0 752 y Fl(Pr)n(oof.)56 b Fm(By)32 b(in)m(tegration)i(b)m(y)f(parts,)758 982 y Fi(P)17 b Fh(f)o Fk(S)963 997 y Fj(n)1035 982 y Fk(<)27 b(t)1176 997 y Fj(n)1220 982 y Fh(g)h Fm(=)1403 846 y Fb(Z)1503 872 y Fj(t)1527 882 y Fc(n)1458 1071 y Fg(0)1614 915 y Fk(t)1650 878 y Fj(n)p Fd(\000)p Fg(1)1779 915 y Fk(e)1826 878 y Fd(\000)p Fj(t)p 1591 959 337 4 v 1591 1050 a Fm(\()p Fk(n)23 b Fh(\000)g Fm(1\)!)1954 982 y Fk(dt)1298 1255 y Fm(=)k Fk(e)1450 1214 y Fd(\000)p Fj(t)1526 1224 y Fc(n)1580 1114 y Fb(\022)1667 1187 y Fk(t)1703 1151 y Fj(n)1703 1212 y(n)p 1663 1232 88 4 v 1663 1323 a Fk(n)p Fm(!)1783 1255 y(+)1978 1187 y Fk(t)2014 1151 y Fj(n)p Fg(+1)2014 1212 y Fj(n)p 1892 1232 337 4 v 1892 1323 a Fm(\()p Fk(n)c Fm(+)e(1\)!)2261 1255 y(+)2456 1187 y Fk(t)2492 1151 y Fj(n)p Fg(+2)2492 1212 y Fj(n)p 2370 1232 V 2370 1323 a Fm(\()p Fk(n)i Fm(+)f(2\)!)2739 1255 y(+)g Fh(\001)17 b(\001)g(\001)2955 1114 y Fb(\023)1298 1527 y Fh(\030)1495 1460 y Fm(1)p 1413 1504 215 4 v 1413 1595 a(1)22 b Fh(\000)g Fk(c)1647 1460 y(e)1694 1423 y Fd(\000)p Fj(t)1770 1433 y Fc(n)1808 1460 y Fm(\()p Fk(t)1882 1475 y Fj(n)1926 1460 y Fm(\))1964 1423 y Fj(n)p 1647 1504 362 4 v 1785 1595 a Fk(n)p Fm(!)2052 1527 y Fk(:)p 2113 1473 65 4 v 2113 1531 4 59 v 2175 1531 V 2113 1534 65 4 v 0 1891 a Fl(Lemma)38 b(2.)55 b Fe(Let)35 b Fk(t)27 b Fh(2)h Fm(\(0)p Fk(;)17 b Fm(1\))34 b Fe(b)s(e)g(a)f (\014xed)g(constan)m(t.)46 b(Then)909 2052 y Fk(e)956 2016 y Fd(\000)p Fj(tn)1076 2052 y Fm(\()p Fk(tn)p Fm(\))1248 2016 y Fj(n)p 909 2096 384 4 v 1057 2188 a Fk(n)p Fm(!)1330 2119 y Fh(\024)29 b Fi(P)17 b Fh(f)o Fk(S)1641 2134 y Fj(n)1713 2119 y Fk(<)27 b(tn)p Fh(g)h(\024)2186 2052 y Fm(1)p 2107 2096 208 4 v 2107 2188 a(1)22 b Fh(\000)g Fk(t)2334 2052 y(e)2381 2016 y Fd(\000)p Fj(tn)2500 2052 y Fm(\()p Fk(tn)p Fm(\))2672 2016 y Fj(n)p 2334 2096 384 4 v 2482 2188 a Fk(n)p Fm(!)2761 2119 y Fk(:)p 2822 2065 65 4 v 2822 2123 4 59 v 2883 2123 V 2822 2126 65 4 v 0 2477 a Fl(Pr)n(oof.)56 b Fm(The)44 b(lo)m(w)m(er)f(b)s(ound)h (follo)m(ws)f(directly)g(b)m(y)h(in)m(tegration)g(b)m(y)g(parts)e(as)h (in)h(the)h(pro)s(of)e(of)0 2610 y(Lemma)34 b(1.)43 b(F)-8 b(or)32 b(the)j(upp)s(er)e(b)s(ound,)h(note)g(that)665 2835 y Fi(P)17 b Fh(f)o Fk(S)870 2850 y Fj(n)942 2835 y Fk(<)27 b(tn)p Fh(g)h(\024)g Fk(e)1373 2794 y Fd(\000)p Fj(tn)1508 2695 y Fb(\022)1592 2768 y Fm(\()p Fk(tn)p Fm(\))1764 2732 y Fj(n)p 1592 2812 218 4 v 1657 2904 a Fk(n)p Fm(!)1841 2835 y(+)1968 2768 y(\()p Fk(tn)p Fm(\))2140 2732 y Fj(n)p Fg(+1)p 1951 2812 337 4 v 1951 2904 a Fm(\()p Fk(n)22 b Fm(+)g(1\)!)2319 2835 y(+)2446 2768 y(\()p Fk(tn)p Fm(\))2618 2732 y Fj(n)p Fg(+2)p 2429 2812 V 2429 2904 a Fm(\()p Fk(n)g Fm(+)g(2\)!)2797 2835 y(+)g Fh(\001)17 b(\001)g(\001)3013 2695 y Fb(\023)1221 3137 y Fh(\024)1336 3070 y Fk(e)1383 3034 y Fd(\000)p Fj(tn)1502 3070 y Fm(\()p Fk(tn)p Fm(\))1674 3034 y Fj(n)p 1336 3115 384 4 v 1484 3206 a Fk(n)p Fm(!)1746 2967 y Fb( )1824 3137 y Fm(1)22 b(+)2074 3070 y Fk(tn)p 2006 3115 232 4 v 2006 3206 a(n)g Fm(+)f(1)2269 3137 y(+)2369 2997 y Fb(\022)2520 3070 y Fk(tn)p 2452 3115 V 2452 3206 a(n)h Fm(+)g(1)2694 2997 y Fb(\023)2767 3019 y Fg(2)2827 3137 y Fm(+)f Fh(\001)c(\001)g(\001)3042 2967 y Fb(!)1221 3440 y Fh(\024)1336 3372 y Fk(e)1383 3336 y Fd(\000)p Fj(tn)1502 3372 y Fm(\()p Fk(tn)p Fm(\))1674 3336 y Fj(n)p 1336 3417 384 4 v 1484 3508 a Fk(n)p Fm(!)1746 3299 y Fb(\022)1908 3372 y Fm(1)p 1829 3417 208 4 v 1829 3508 a(1)22 b Fh(\000)g Fk(t)2046 3299 y Fb(\023)2169 3440 y Fk(:)p 2230 3386 65 4 v 2230 3444 4 59 v 2292 3444 V 2230 3447 65 4 v 0 3815 a Fl(Lemma)38 b(3.)1231 3949 y Fk(A)28 b Fh(\024)1438 3872 y(p)p 1521 3872 60 4 v 77 x Fk(n)p Fm(2)1631 3908 y Fj(n)1675 3949 y Fi(P)17 b Fh(f)o Fk(S)1880 3964 y Fj(n)1952 3949 y Fk(<)27 b(n=\013)p Fh(g)h(\024)g Fk(B)38 b(;)0 4116 y Fe(where)33 b Fk(A)28 b Fm(=)g Fk(e)543 4080 y Fd(\000)p Fg(1)p Fj(=)p Fg(12)731 4116 y Fk(=)781 4034 y Fh(p)p 864 4034 111 4 v 82 x Fm(2)p Fk(\031)37 b Fe(and)d Fk(B)e Fm(=)27 b Fk(\013=)p Fm(\(\()p Fk(\013)d Fh(\000)e Fm(1\))1879 4034 y Fh(p)p 1963 4034 V 1963 4116 a Fm(2)p Fk(\031)t Fm(\))p Fe(.)0 4452 y Fl(Pr)n(oof.)56 b Fm(F)-8 b(rom)31 b(Lemma)j(2,)726 4617 y Fk(e)773 4581 y Fd(\000)p Fj(n=\013)944 4617 y Fm(\()p Fk(n=\013)p Fm(\))1194 4581 y Fj(n)p 726 4662 514 4 v 939 4753 a Fk(n)p Fm(!)1277 4685 y Fh(\024)28 b Fi(P)17 b Fh(f)o Fk(S)1587 4700 y Fj(n)1659 4685 y Fk(<)27 b(n=\013)p Fh(g)h(\024)2273 4617 y Fm(1)p 2131 4662 336 4 v 2131 4753 a(1)22 b Fh(\000)g Fm(1)p Fk(=\013)2486 4617 y(e)2533 4581 y Fd(\000)p Fj(n=\013)2704 4617 y Fm(\()p Fk(n=\013)p Fm(\))2954 4581 y Fj(n)p 2486 4662 514 4 v 2699 4753 a Fk(n)p Fm(!)3042 4685 y Fk(:)0 4928 y Fm(Use)34 b(the)g(fact)h(that)f Fk(n)p Fm(!)28 b(=)f(\()p Fk(n=e)p Fm(\))1229 4891 y Fj(n)1273 4845 y Fh(p)p 1356 4845 170 4 v 83 x Fm(2)p Fk(\031)t(ne)1573 4891 y Fj(\022)r(=)p Fg(\(12)p Fj(n)p Fg(\))1834 4928 y Fm(for)33 b(some)f Fk(\022)e Fh(2)f Fm(\(0)p Fk(;)17 b Fm(1\))33 b(and)h(the)g(de\014nition)h(of)f Fk(\013)p Fm(.)p 3720 4874 65 4 v 3720 4932 4 59 v 3781 4932 V 3720 4935 65 4 v 1872 5280 a Ff(4)p eop %%Page: 5 5 5 4 bop 0 83 a Fl(Lemma)38 b(4.)55 b Fe(There)33 b(exists)f(a)h(univ)m (ersal)f(constan)m(t)j Fk(C)40 b Fe(suc)m(h)34 b(that)1421 276 y Fi(P)17 b Fh(f)p Fk(S)1627 291 y Fj(n)1699 276 y Fh(\025)28 b Fk(C)7 b(n)p Fh(g)28 b(\024)g Fm(2)2175 235 y Fd(\000)p Fg(2)p Fj(n)2337 276 y Fk(:)0 470 y(C)35 b Fm(=)27 b(5)33 b Fe(will)f(do.)0 838 y Fl(Pr)n(oof.)56 b Fm(T)-8 b(ak)m(e)40 b Fk(C)47 b(>)39 b Fm(1.)66 b(By)40 b(Cherno\013)8 b('s)39 b(exp)s(onen)m(tial)i(b)s(ounding)h(metho)s(d)f (\(Cherno\013,)i(1952\),)0 971 y(for)32 b Fk(t)c(>)f Fm(0,)648 1114 y Fi(P)17 b Fh(f)p Fk(S)854 1129 y Fj(n)926 1114 y Fh(\025)28 b Fk(C)7 b(n)p Fh(g)27 b(\024)h Fi(E)p Fk(e)1473 1073 y Fj(tS)1538 1083 y Fc(n)1576 1114 y Fk(e)1623 1073 y Fd(\000)p Fj(tC)5 b(n)1822 1114 y Fm(=)27 b(\(1)c Fh(\000)f Fk(t)p Fm(\))2211 1073 y Fd(\000)p Fj(n)2308 1114 y Fk(e)2355 1073 y Fd(\000)p Fj(tC)5 b(n)2554 1114 y Fm(=)2659 1034 y Fb(\000)2705 1114 y Fk(C)i(e)2830 1073 y Fg(1)p Fd(\000)p Fj(C)2971 1034 y Fb(\001)3016 1053 y Fj(n)3110 1114 y Fk(;)0 1291 y Fm(where)33 b(w)m(e)h(to)s(ok)f (1)22 b Fh(\000)g Fk(t)28 b Fm(=)f(1)p Fk(=C)7 b Fm(.)44 b(F)-8 b(or)32 b Fk(C)40 b Fm(large)32 b(enough)i(\(e.g.,)f Fk(C)i Fh(\025)28 b Fm(5\),)34 b(this)f(is)f(less)g(than)j(4)3500 1255 y Fd(\000)p Fj(n)3595 1291 y Fm(.)p 3667 1237 65 4 v 3667 1295 4 59 v 3728 1295 V 3667 1298 65 4 v 0 1602 a Fl(Lemma)41 b(5.)55 b Fe(Let)36 b Fk(E)769 1617 y Fg(1)807 1602 y Fk(;)17 b(E)925 1617 y Fg(2)962 1602 y Fk(;)g(:)g(:)g(:)e(;)i(E) 1258 1617 y Fj(n)1337 1602 y Fe(b)s(e)36 b(i.i.d.)e(random)h(v)-6 b(ariables)33 b(with)j(a)g(densit)m(y)-8 b(,)36 b(and)g(let)g Fk(a)f Fe(b)s(e)h(a)0 1735 y(\014xed)e(constan)m(t.)45 b(Then)213 1928 y Fi(P)17 b Fh(f)p Fk(E)431 1943 y Fg(1)497 1928 y Fk(<)27 b(a;)17 b(E)773 1943 y Fg(1)832 1928 y Fm(+)k Fk(E)1004 1943 y Fg(2)1070 1928 y Fk(<)27 b Fm(2)p Fk(a;)17 b(:)g(:)g(:)d(;)j(E)1573 1943 y Fg(1)1632 1928 y Fm(+)22 b Fh(\001)17 b(\001)g(\001)j Fm(+)i Fk(E)2043 1943 y Fj(n)2115 1928 y Fk(<)27 b(na)g Fh(j)h Fk(E)2489 1943 y Fg(1)2549 1928 y Fm(+)21 b Fh(\001)c(\001)g(\001)k Fm(+)g Fk(E)2959 1943 y Fj(n)3032 1928 y Fk(<)27 b(na)p Fh(g)g(\025)3447 1861 y Fm(1)p 3442 1905 60 4 v 3442 1997 a Fk(n)3545 1928 y(:)0 2256 y Fl(Pr)n(oof.)56 b Fm(De\014ne)40 b Fk(F)777 2271 y Fj(i)840 2256 y Fm(=)c Fk(E)1027 2271 y Fj(i)1081 2256 y Fh(\000)26 b Fk(a)38 b Fm(for)g(all)g Fk(i)p Fm(.)60 b(De\014ne)39 b Fk(E)2086 2271 y Fj(r)2160 2256 y Fm(=)d Fk(E)2347 2271 y Fj(r)r Fd(\000)p Fj(n)2513 2256 y Fm(when)k Fk(n)d(<)f(r)k Fh(\024)d Fm(2)p Fk(n)p Fm(.)60 b(Then,)41 b(b)m(y)0 2389 y(symmetry)-8 b(,)138 2582 y Fi(P)17 b Fh(f)p Fk(E)356 2597 y Fg(1)422 2582 y Fk(<)27 b(a;)17 b(E)698 2597 y Fg(1)757 2582 y Fm(+)k Fk(E)929 2597 y Fg(2)995 2582 y Fk(<)27 b Fm(2)p Fk(a;)17 b(:)g(:)g(:)d(;)j(E)1498 2597 y Fg(1)1557 2582 y Fm(+)22 b Fh(\001)17 b(\001)g(\001)j Fm(+)i Fk(E)1968 2597 y Fj(n)2040 2582 y Fk(<)27 b(na)g Fh(j)g Fk(E)2413 2597 y Fg(1)2473 2582 y Fm(+)22 b Fh(\001)17 b(\001)g(\001)j Fm(+)i Fk(E)2884 2597 y Fj(n)2956 2582 y Fk(<)28 b(na)p Fh(g)244 2740 y Fm(=)f Fi(P)17 b Fh(f)p Fk(F)558 2755 y Fg(1)623 2740 y Fk(<)27 b Fm(0)p Fk(;)17 b(F)887 2755 y Fg(1)946 2740 y Fm(+)22 b Fk(F)1110 2755 y Fg(2)1175 2740 y Fk(<)27 b Fm(0)p Fk(;)17 b(:)g(:)g(:)d(;)j(F)1616 2755 y Fg(1)1675 2740 y Fm(+)k Fh(\001)c(\001)g(\001)k Fm(+)g Fk(F)2076 2755 y Fj(n)2148 2740 y Fk(<)27 b Fm(0)h Fh(j)f Fk(F)2450 2755 y Fg(1)2510 2740 y Fm(+)21 b Fh(\001)c(\001)g (\001)k Fm(+)g Fk(F)2911 2755 y Fj(n)2983 2740 y Fk(<)27 b Fm(0)p Fh(g)244 2961 y Fm(=)364 2894 y(1)p 359 2938 V 359 3029 a Fk(n)498 2837 y Fj(n)446 2867 y Fb(X)464 3076 y Fj(i)p Fg(=1)606 2961 y Fi(P)17 b Fh(f)p Fk(F)815 2976 y Fj(i)870 2961 y Fk(<)27 b Fm(0)p Fk(;)17 b(F)1134 2976 y Fj(i)1182 2961 y Fm(+)22 b Fk(F)1346 2976 y Fj(i)p Fg(+1)1485 2961 y Fk(<)28 b Fm(0)p Fk(;)17 b(:)g(:)g(:)d(;)j(F)1927 2976 y Fj(i)1975 2961 y Fm(+)22 b Fh(\001)17 b(\001)g(\001)j Fm(+)i Fk(F)2377 2976 y Fj(i)p Fg(+)p Fj(n)p Fd(\000)p Fg(1)2608 2961 y Fk(<)27 b Fm(0)h Fh(j)f Fk(F)2910 2976 y Fg(1)2969 2961 y Fm(+)22 b Fh(\001)17 b(\001)g(\001)j Fm(+)i Fk(F)3371 2976 y Fj(n)3443 2961 y Fk(<)27 b Fm(0)p Fh(g)244 3194 y Fm(=)g Fi(P)17 b Fh(f)p Fk(F)558 3209 y Fj(S)635 3194 y Fk(<)27 b Fm(0)p Fk(;)17 b(F)899 3209 y Fj(S)969 3194 y Fm(+)k Fk(F)1132 3209 y Fj(S)t Fg(+1)1294 3194 y Fk(<)27 b Fm(0)p Fk(;)17 b(:)g(:)g(:)d(;)j(F)1735 3209 y Fj(S)1805 3194 y Fm(+)22 b Fh(\001)17 b(\001)g(\001)j Fm(+)i Fk(F)2207 3209 y Fj(S)t Fg(+)p Fj(n)p Fd(\000)p Fg(1)2459 3194 y Fk(<)28 b Fm(0)f Fh(j)g Fk(F)2761 3209 y Fg(1)2821 3194 y Fm(+)22 b Fh(\001)17 b(\001)g(\001)j Fm(+)i Fk(F)3223 3209 y Fj(n)3294 3194 y Fk(<)27 b Fm(0)p Fh(g)50 b Fk(;)0 3388 y Fm(where)39 b Fk(S)44 b Fm(is)38 b(indep)s(enden)m(t)k(of)d(the)i Fk(E)1445 3403 y Fj(i)1472 3388 y Fm('s)d(and)h(uniformly)f(distributed)i(on)f Fh(f)p Fm(1)p Fk(;)17 b(:)g(:)g(:)d(;)j(n)p Fh(g)p Fm(.)59 b(No)m(w,)41 b(\014x)0 3521 y Fk(E)73 3536 y Fg(1)111 3521 y Fk(;)17 b(:)g(:)g(:)d(;)j(E)406 3536 y Fj(n)450 3521 y Fm(,)29 b(and)h(let)f Fk(s)f Fh(2)g(f)p Fm(1)p Fk(;)17 b(:)g(:)g(:)c(;)k(n)p Fh(g)28 b Fm(b)s(e)i(the)g(\(unique\))h(v)-6 b(alue)29 b(at)g(whic)m(h)2811 3446 y Fb(P)2916 3549 y Fj(i>)p Fg(0)p Fj(;i)38 b Fm(0.)62 b(If)40 b Fk(s)e(>)f Fm(1,)k(then,)h(as)2686 3588 y Fb(P)2791 3614 y Fj(n)2791 3691 y(i)p Fg(=1)2920 3663 y Fk(F)2984 3678 y Fj(i)3049 3663 y Fk(<)37 b Fm(0,)k(w)m(e)f(see) f(that)0 3721 y Fb(P)105 3747 y Fj(s)p Fg(+)p Fj(j)105 3824 y(i)p Fg(=)p Fj(s)240 3796 y Fk(F)304 3811 y Fj(i)359 3796 y Fm(=)464 3721 y Fb(P)569 3747 y Fj(s)p Fg(+)p Fj(j)569 3824 y(i)p Fg(=1)704 3796 y Fk(F)768 3811 y Fj(i)817 3796 y Fh(\000)917 3721 y Fb(P)1022 3747 y Fj(s)p Fd(\000)p Fg(1)1022 3824 y Fj(i)p Fg(=1)1158 3796 y Fk(F)1222 3811 y Fj(i)1277 3796 y Fk(<)27 b Fm(0)33 b(for)g(all)f Fk(j)i Fh(\025)28 b Fm(0.)44 b(Th)m(us,)318 3989 y Fi(P)17 b Fh(f)p Fk(F)527 4004 y Fj(S)603 3989 y Fk(<)27 b Fm(0)p Fk(;)17 b(F)867 4004 y Fj(S)937 3989 y Fm(+)22 b Fk(F)1101 4004 y Fj(S)t Fg(+1)1262 3989 y Fk(<)27 b Fm(0)p Fk(;)17 b(:)g(:)g(:)d(;)j(F)1703 4004 y Fj(S)1774 3989 y Fm(+)k Fh(\001)c(\001)g(\001)k Fm(+)g Fk(F)2175 4004 y Fj(S)t Fg(+)p Fj(n)p Fd(\000)p Fg(1)2428 3989 y Fk(<)27 b Fm(0)h Fh(j)f Fk(F)2730 4004 y Fg(1)2790 3989 y Fm(+)21 b Fh(\001)c(\001)g (\001)k Fm(+)g Fk(F)3191 4004 y Fj(n)3263 3989 y Fk(<)27 b Fm(0)p Fh(g)424 4179 y(\025)h Fi(P)p Fh(f)p Fk(S)34 b Fm(=)27 b Fk(s)p Fh(g)g Fm(=)1101 4111 y(1)p 1096 4156 V 1096 4247 a Fk(n)1199 4179 y(:)p 1260 4125 65 4 v 1260 4183 4 59 v 1322 4183 V 1260 4186 65 4 v 1872 5280 a Ff(5)p eop %%Page: 6 6 6 5 bop 0 83 a Fi(Pro)s(of)39 b(of)g(the)e(Theorem.)0 397 y Fl(Lemma)h(6.)55 b Fe(Consider)32 b(p)s(ositiv)m(e)h(in)m(tegers) g Fk(n)27 b(>)g(k)s Fe(.)44 b(Then)711 591 y Fi(P)17 b Fh(f)p Fk(H)939 606 y Fj(n)1010 591 y Fh(\025)28 b Fk(k)s Fh(g)g(\025)g Fi(P)17 b Fh(f9)33 b Fm(leaf)h Fk(y)d Fh(2)d Fk(R)2022 606 y Fj(k)2123 591 y Fm(:)61 b Fk(f)11 b Fm(\()p Fk(y)t Fm(\))28 b Fh(\025)g Fm(\()p Fk(k)e Fm(+)21 b(1\))p Fk(=n)p Fh(g)50 b Fk(:)0 909 y Fl(Pr)n(oof.)56 b Fm(This)32 b(follo)m(ws)g(immediately)h(from)g(facts)h(1)f(and)g(2.)p 2390 855 65 4 v 2390 913 4 59 v 2452 913 V 2390 916 65 4 v 0 1223 a Fl(Lemma)38 b(7.)55 b Fe(There)33 b(exists)f(a)h(constan)m (t)i Fk(d)28 b(>)f Fm(0)33 b Fe(suc)m(h)h(that)g(for)f(su\016cien)m (tly)h(large)d Fk(j)6 b Fe(,)391 1444 y Fi(P)17 b Fh(f9)33 b Fm(leaf)g Fk(y)f Fh(2)c Fk(R)1060 1459 y Fj(j)1156 1444 y Fm(:)60 b Fk(f)11 b Fm(\()p Fk(y)t Fm(\))28 b Fh(\025)g Fm(\()p Fk(j)h Fm(+)22 b(1\))p Fk(=)17 b Fm(exp\()p Fk(j)6 b(=\013)22 b Fm(+)g Fk(d)17 b Fm(log\()p Fk(j)6 b(=\013)p Fm(\)\))p Fh(g)29 b(\025)g Fm(1)21 b Fh(\000)3257 1376 y Fm(1)p 3240 1421 85 4 v 3240 1512 a Fk(j)3287 1483 y Fg(3)3367 1444 y Fk(:)0 1676 y Fe(W)-8 b(e)34 b(ma)m(y)e(pic)m(k)i Fk(d)27 b Fm(=)g Fk(\017)22 b Fm(+)g(15)p Fk(=)17 b Fm(log)f(2)33 b Fe(for)f(an)m(y)i(small)d Fk(\017)c(>)h Fm(0)p Fe(.)0 2048 y Fl(Pr)n(oof.)56 b Fm(T)-8 b(o)32 b(follo)m(w.)p 872 1994 65 4 v 872 2052 4 59 v 933 2052 V 872 2055 65 4 v 0 2362 a Fl(Lemma)38 b(8.)55 b Fe(Let)35 b Fk(d)e Fe(b)s(e)g(the)i(constan)m(t)g(of)e(Lemma)h(7.)44 b(Then,)33 b(for)g(su\016cien)m(tly)h(large)d Fk(n)p Fe(,)742 2582 y Fi(P)17 b Fh(f)p Fk(H)970 2597 y Fj(n)1041 2582 y Fh(\025)28 b Fk(\013)17 b Fm(log)g Fk(n)22 b Fh(\000)h Fk(d\013)16 b Fm(log)i(log)f Fk(n)22 b Fh(\000)g Fm(1)p Fh(g)28 b(\025)g Fm(1)22 b Fh(\000)2748 2515 y Fm(1)p 2572 2560 401 4 v 2572 2651 a(\()p Fk(\013)c Fm(log)f Fk(n)p Fm(\))2935 2622 y Fg(3)3016 2582 y Fk(:)0 2820 y Fe(W)-8 b(e)34 b(ma)m(y)e(c)m(ho)s(ose)h Fk(d)28 b Fm(=)f Fk(\017)22 b Fm(+)f(15)p Fk(=)c Fm(log)g(2)33 b Fe(for)f(an)m(y)h(small)f Fk(\017)27 b(>)g Fm(0)p Fe(.)0 3192 y Fl(Pr)n(oof.)56 b Fm(F)-8 b(ollo)m(ws)31 b(from)h(Lemmas)h(6)g (and)h(7)f(b)m(y)h(setting)g Fk(j)f Fm(=)27 b Fk(k)k Fm(=)c Fh(b)p Fk(\013)17 b Fm(log)g Fk(n)22 b Fh(\000)h Fk(d\013)16 b Fm(log)i(log)f Fk(n)p Fh(c)p Fm(.)p 3626 3138 65 4 v 3626 3196 4 59 v 3687 3196 V 3626 3199 65 4 v 0 3506 a Fl(Lemma)38 b(9.)1003 3688 y Fi(P)17 b Fh(f)o Fk(H)1230 3703 y Fj(n)1302 3688 y Fh(\025)28 b(d)p Fk(\013)17 b Fm(log)g Fk(n)22 b Fm(+)g Fk(i)p Fh(eg)27 b(\024)2120 3548 y Fb(\022)2211 3621 y Fm(2)p 2204 3666 V 2204 3757 a Fk(\013)2278 3548 y Fb(\023)2351 3570 y Fj(i)2428 3688 y Fk(;)49 b(i)27 b Fh(\025)h Fm(0)33 b Fk(:)0 4105 y Fl(Pr)n(oof.)56 b Fm(See)34 b(page)f(492)g(of)g(Devro)m(y)m(e)g (\(1986\).)p 1839 4051 V 1839 4109 4 59 v 1901 4109 V 1839 4112 65 4 v 299 4299 a(Note)i(that)g(the)f(Theorem)f(follo)m(ws)f (from)h(Lemmas)g(8)g(and)g(9)g(without)i(further)f(w)m(ork.)1872 5280 y Ff(6)p eop %%Page: 7 7 7 6 bop 0 83 a Fi(Pro)s(of)39 b(of)g(Lemma)d(7.)0 382 y Fl(Lemma)29 b(10.)55 b Fe(F)-8 b(or)23 b(ev)m(ery)g Fk(i)p Fe(,)i(with)h(probabilit)m(y)d(at)i(least)f Fm(1)5 b Fh(\000)g Fm(2)2314 346 y Fd(\000)p Fj(i)2392 382 y Fe(,)26 b(ev)m(ery)d(leaf)i(of)g Fk(R)3058 397 y Fj(i)3109 382 y Fe(has)f Fk(f)11 b Fm(\()p Fk(y)t Fm(\))28 b Fh(\025)g Fk(e)3647 346 y Fd(\000)p Fg(5)p Fj(i)3758 382 y Fe(.)0 736 y Fl(Pr)n(oof.)56 b Fm(The)37 b(probabilit)m(y)g(that)i(for)d(some) h(leaf)g Fk(y)k Fm(of)c Fk(R)2223 751 y Fj(i)2250 736 y Fm(,)h(w)m(e)f(ha)m(v)m(e)h Fk(f)11 b Fm(\()p Fk(y)t Fm(\))35 b Fk(<)f(e)3083 700 y Fd(\000)p Fg(5)p Fj(i)3231 736 y Fm(is)i(at)i(most)f(2)3759 700 y Fj(i)0 869 y Fm(times)c Fi(P)p Fh(f)p Fk(S)455 884 y Fj(i)510 869 y Fh(\025)28 b Fm(5)p Fk(i)p Fh(g)p Fm(,)k(where)h Fk(S)1159 884 y Fj(i)1219 869 y Fm(is)f(gamma)g Fk(i)h Fm(distributed.)44 b(By)32 b(Lemma)i(4,)e(this)h(do)s(es)g(not)h(exceed)0 1001 y(2)50 965 y Fj(i)77 1001 y Fk(=)p Fm(4)177 965 y Fj(i)231 1001 y Fm(=)27 b(2)386 965 y Fd(\000)p Fj(i)465 1001 y Fm(.)p 537 947 65 4 v 537 1005 4 59 v 598 1005 V 537 1008 65 4 v 0 1301 a Fl(Lemma)41 b(11.)55 b Fe(F)-8 b(or)35 b(su\016cien)m(tly)i(large)e Fk(k)s Fe(,)h(with)h(probabilit)m (y)f(at)h(least)f Fm(1)p Fk(=k)2880 1264 y Fg(3)2917 1301 y Fe(,)g(there)h(is)e(a)h(leaf)h Fk(y)j Fe(of)0 1433 y Fk(R)76 1448 y Fj(k)150 1433 y Fe(with)34 b Fk(f)11 b Fm(\()p Fk(y)t Fm(\))28 b Fh(\025)g Fk(e)746 1397 y Fd(\000)p Fj(k)r(=\013)914 1433 y Fe(.)299 1677 y Fm(Lemma)35 b(11)f(will)f(b)s(e)i(pro)m(v)m(ed)f(further)h(on.)47 b(If)35 b(Lemma)g(11)f(is)f(true,)i(then)h(w)m(e)f(can)g(pro)s(ceed)0 1809 y(with)h(the)h(pro)s(of)e(of)h(Lemma)g(7)g(as)e(follo)m(ws:)48 b(\014rst)35 b(note)i(that)g(w)m(e)f(can)g(obtain)h(a)e(cop)m(y)h(of)g Fk(R)3530 1824 y Fj(i)p Fg(+)p Fj(k)3681 1809 y Fm(b)m(y)0 1942 y(making)g(eac)m(h)i(leaf)f(of)g Fk(R)965 1957 y Fj(i)1028 1942 y Fm(a)g(ro)s(ot)e(of)i(a)g(cop)m(y)g(of)g Fk(R)1964 1957 y Fj(k)2004 1942 y Fm(,)h(where)e(all)g(these)i(trees)e (are)g(indep)s(enden)m(tly)0 2075 y(lab)s(elled.)43 b(De\014ne)30 b Fk(k)h Fm(=)c Fh(b)p Fk(j)21 b Fh(\000)15 b Fk(A)i Fm(log)h Fk(j)6 b Fh(c)29 b Fm(and)h Fk(i)d Fm(=)h Fh(d)p Fk(A)17 b Fm(log)g Fk(j)6 b Fh(e)30 b Fm(so)f(that)i Fk(j)i Fm(=)27 b Fk(k)18 b Fm(+)d Fk(i)28 b Fm(with)i(some)f(constan)m (t)0 2208 y Fk(A)34 b Fm(to)g(b)s(e)f(pic)m(k)m(ed)h(further)g(on.)44 b(Note)35 b(\014rst)d(that)j(for)e Fk(j)39 b Fm(large)31 b(enough,)j(if)g Fk(A)28 b(>)f(\013)p Fm(,)1031 2369 y Fk(k)p 1026 2413 V 1026 2505 a(\013)1123 2436 y Fm(+)21 b(5)p Fk(i)27 b Fh(\024)1458 2369 y Fk(j)p 1449 2413 V 1449 2505 a(\013)1545 2436 y Fm(+)22 b(5)p Fk(A)17 b Fm(log)1931 2296 y Fb(\022)2023 2369 y Fk(j)p 2015 2413 V 2015 2505 a(\013)2089 2296 y Fb(\023)2184 2436 y Fh(\000)23 b Fm(log)q(\()p Fk(j)28 b Fm(+)22 b(1\))34 b Fk(:)0 2665 y Fm(Then,)99 2854 y Fi(P)17 b Fh(f6)28 b(9)33 b Fm(leaf)h Fk(y)d Fh(2)d Fk(R)796 2869 y Fj(j)864 2854 y Fm(with)34 b Fk(f)11 b Fm(\()p Fk(y)t Fm(\))28 b Fh(\025)g Fm(1)p Fk(=)17 b Fm(exp\()p Fk(j)6 b(=\013)22 b Fm(+)g(5)p Fk(A)17 b Fm(log)q(\()p Fk(j)6 b(=\013)p Fm(\))23 b Fh(\000)g Fm(log\()p Fk(j)29 b Fm(+)22 b(1\)\))p Fh(g)293 3011 y(\024)28 b Fi(P)493 2931 y Fb(\010)551 3011 y Fh(9)34 b Fm(leaf)f Fk(y)e Fh(2)e Fk(R)1076 3026 y Fj(i)1135 3011 y Fm(with)34 b Fk(f)11 b Fm(\()p Fk(y)t Fm(\))28 b Fk(<)g(e)1732 2970 y Fd(\000)p Fg(5)p Fj(i)1843 2931 y Fb(\011)487 3169 y Fm(+)22 b Fi(P)17 b Fh(f)o(6)28 b(9)34 b Fm(leaf)f Fk(y)e Fh(2)d Fk(R)1283 3184 y Fj(j)1351 3169 y Fm(with)34 b Fk(f)11 b Fm(\()p Fk(y)t Fm(\))28 b Fh(\025)g Fm(1)p Fk(=)17 b Fm(exp\()p Fk(j)6 b(=\013)22 b Fm(+)g Fk(d)17 b Fm(log\()p Fk(j)6 b(=\013)p Fm(\))23 b Fh(\000)g Fm(log)q(\()p Fk(j)28 b Fm(+)22 b(1\)\))768 3327 y Fh(j)27 b(8)34 b Fm(leaf)f Fk(y)f Fh(2)c Fk(R)1348 3342 y Fj(i)1402 3327 y Fm(:)f Fk(f)11 b Fm(\()p Fk(y)t Fm(\))28 b Fh(\025)h Fk(e)1827 3286 y Fd(\000)p Fg(5)p Fj(i)1938 3246 y Fb(\011)293 3485 y Fh(\024)f Fm(2)448 3443 y Fd(\000)p Fj(i)549 3485 y Fm(+)22 b Fi(P)17 b Fh(f)o Fm(ev)m(ery)33 b(cop)m(y)g(of)g Fk(R)1478 3500 y Fj(k)1552 3485 y Fm(con)m(tains)h(no)f(leaf)h Fk(y)i Fm(with)e Fk(f)11 b Fm(\()p Fk(y)t Fm(\))28 b Fh(\025)g Fm(1)p Fk(=)17 b Fm(exp\()p Fk(k)s(=\013)p Fm(\))p Fh(g)465 3642 y Fm(\(b)m(y)34 b(Lemma)g(10\))293 3816 y Fh(\024)28 b Fm(2)448 3775 y Fd(\000)p Fj(i)549 3816 y Fm(+)649 3736 y Fb(\000)694 3816 y Fm(1)22 b Fh(\000)h Fk(k)921 3775 y Fd(\000)p Fg(3)1010 3736 y Fb(\001)1056 3757 y Fg(2)1090 3733 y Fc(i)1230 3816 y Fm(\(b)m(y)35 b(Lemma)e(11\))293 3974 y Fh(\024)28 b Fm(2)448 3933 y Fd(\000)p Fj(i)549 3974 y Fm(+)22 b(exp)817 3893 y Fb(\000)863 3974 y Fh(\000)p Fm(2)990 3933 y Fj(i)1017 3974 y Fk(k)1072 3933 y Fd(\000)p Fg(3)1161 3893 y Fb(\001)293 4132 y Fh(\024)28 b Fk(j)445 4091 y Fd(\000)p Fj(A)11 b Fg(log)g(2)714 4132 y Fm(+)22 b(exp)982 4051 y Fb(\000)1028 4132 y Fh(\000)p Fk(j)1152 4091 y Fj(A)11 b Fg(log)g(2)p Fd(\000)p Fg(3)1432 4051 y Fb(\001)293 4290 y Fh(\024)28 b Fk(j)445 4248 y Fd(\000)p Fg(3)0 4479 y Fm(for)34 b Fk(j)41 b Fm(large)33 b(enough,)j(pro)m (vided)f(that)h Fk(A)17 b Fm(log)h(2)30 b Fk(>)g Fm(3.)49 b(This)34 b(pro)m(v)m(es)g(Lemma)i(7.)48 b(W)-8 b(e)36 b(note)g(that)g(w)m(e)0 4611 y(can)e(pic)m(k)f Fk(d)27 b Fm(=)h(5)p Fk(A)p Fm(,)33 b(where)g Fk(A)c Fm(=)e Fk(\017)21 b Fm(+)h(max)o(\()p Fk(\013;)17 b Fm(3)p Fk(=)g Fm(log)g(2\))34 b(for)e(an)m(y)h(small)f Fk(\017)27 b(>)g Fm(0.)p 3033 4558 V 3033 4616 4 59 v 3095 4616 V 3033 4619 65 4 v 1872 5280 a Ff(7)p eop %%Page: 8 8 8 7 bop 0 83 a Fi(Pro)s(of)39 b(of)e(Lemma)f(11.)51 b Fm(Let)34 b Fk(P)46 b Fm(b)s(e)33 b(a)f(path)i(from)e(the)i(ro)s(ot)e (to)h(a)f(leaf)h Fk(y)j Fm(of)c Fk(R)3065 98 y Fj(k)3106 83 y Fm(.)44 b(The)33 b(condition)0 216 y Fk(f)11 b Fm(\()p Fk(y)t Fm(\))28 b Fh(\025)g Fm(1)p Fk(=e)469 180 y Fj(k)r(=\013)618 216 y Fm(is)k(equiv)-6 b(alen)m(t)35 b(to)1393 346 y Fb(X)1402 557 y Fj(e)p Fd(2)p Fj(P)1537 441 y Fm(\()p Fh(\000)17 b Fm(log)h Fk(L)p Fm(\()p Fk(e)p Fm(\)\))29 b Fh(\024)2189 373 y(j)p Fk(P)14 b Fh(j)p 2189 418 134 4 v 2223 509 a Fk(\013)2365 441 y(:)0 697 y Fm(W)-8 b(e)34 b(call)e(a)h(leaf)h Fk(y)i Fm(sp)s(ecial)d(if|in)g(addition)h(to)f(the) i(ab)s(o)m(v)m(e)e(condition|it)h(satis\014es)1413 822 y Fb(X)1412 1035 y Fj(e)p Fd(2)p Fj(P)1538 1015 y Fa(0)1557 916 y Fm(\()p Fh(\000)17 b Fm(log)h Fk(L)p Fm(\()p Fk(e)p Fm(\)\))29 b Fh(\024)2208 849 y(j)p Fk(P)2314 813 y Fd(0)2336 849 y Fh(j)p 2208 893 156 4 v 2254 985 a Fk(\013)0 1178 y Fm(for)40 b(ev)m(ery)g(subpath)j Fk(P)900 1142 y Fd(0)963 1178 y Fm(of)e Fk(P)55 b Fm(that)43 b(originates)c(at)j(a)f(terminal)g (v)m(ertex)f Fk(y)t Fm(.)67 b(Suc)m(h)43 b(subpaths)f(are)0 1311 y(called)k(terminal.)83 b(Let)47 b Fh(S)54 b Fm(b)s(e)47 b(the)g(collection)g(of)g(sp)s(ecial)e(lea)m(v)m(es)h(of)g Fk(R)2831 1326 y Fj(k)2872 1311 y Fm(.)83 b(By)45 b(Lemma)h(5,)j(the)0 1444 y(exp)s(ected)d(n)m(um)m(b)s(er)g(of)f(sp)s(ecial)e(lea)m(v)m(es)h (is)g(at)h(least)f(1)p Fk(=k)k Fm(times)c Fi(P)p Fh(f)p Fk(S)2713 1459 y Fj(k)2801 1444 y Fk(<)i(k)s(=\013)p Fh(g)e Fm(times)h(2)3516 1407 y Fj(k)3557 1444 y Fm(.)78 b(By)0 1576 y(Lemma)34 b(3,)1511 1757 y Fi(E)p Fh(jS)7 b(j)28 b(\025)1911 1690 y Fk(e)1958 1654 y Fd(\000)p Fg(1)p Fj(=)p Fg(12)p 1852 1735 352 4 v 1852 1754 a Fh(p)p 1935 1754 111 4 v 83 x Fm(2)p Fk(\031)t(k)2100 1808 y Fg(3)p Fj(=)p Fg(2)2247 1757 y Fk(:)299 1961 y Fm(Next,)38 b(w)m(e)f(consider)f(the)i(exp)s(ected)h(n)m(um)m(b)s(er)e(of)g(pairs)e 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Fg(2)1746 2492 y Fk(;)g(:)g(:)g(:)d(;)j(e)2015 2507 y Fj(k)2087 2492 y Fm(b)s(e)33 b(the)h(edges)f(on)g(the)h(path)g (from)e(the)i(ro)s(ot)0 2625 y(to)41 b Fk(z)t Fm(,)h(and)f(de\014ne)h Fk(Q)825 2640 y Fj(i)892 2625 y Fm(=)e Fh(f)p Fk(e)1107 2640 y Fg(1)1143 2625 y Fk(;)17 b(:)g(:)g(:)d(;)j(e)1412 2640 y Fj(i)1438 2625 y Fh(g)p Fm(.)66 b(F)-8 b(or)39 b(an)m(y)i Fk(j)6 b Fm(,)41 b(the)h(n)m(um)m(b)s(er)g(of)e(lea)m(v)m (es)g(of)h Fk(R)3240 2640 y Fj(k)3321 2625 y Fm(whose)g(join)0 2758 y(with)36 b Fk(P)293 2773 y Fj(z)367 2758 y Fm(is)f Fk(Q)549 2773 y Fj(j)620 2758 y Fm(is)f(2)772 2722 y Fj(k)r Fd(\000)p Fj(j)896 2758 y Fm(.)51 b(F)-8 b(urthermore,)35 b(the)i(probabilit)m(y)e(that)j(a)d(leaf)h Fk(w)f Fh(2)d Fk(R)3042 2773 y Fj(k)3118 2758 y Fm(is)j(a)h(sp)s(ecial)f(leaf)0 2891 y(giv)m(en)i(that)h Fk(z)f Fh(2)d(S)44 b Fm(and)38 b(that)g Fk(P)1252 2906 y Fj(z)1314 2891 y Fh(\\)25 b Fk(P)1469 2906 y Fj(w)1556 2891 y Fm(=)33 b Fk(Q)1746 2906 y Fj(j)1818 2891 y 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Fh(2)d(S)52 b Fm(without)45 b(m)m(uc)m(h)h(trouble.)76 b(By)43 b(Lemma)h(3,)i(the)0 3779 y(probabilit)m(y)32 b(of)i(the)h(ev)m(en)m(t)f(men)m(tioned)i(ab)s(o)m(v)m(e)d(is)f(at)i (most)1830 3894 y Fk(\013)p 1236 3938 1254 4 v 1236 4043 a Fm(\()p Fk(\013)23 b Fh(\000)f Fm(1\))1548 3958 y Fb(p)p 1648 3958 583 4 v 85 x Fm(2)p Fk(\031)t Fm(\()p Fk(k)k Fh(\000)d Fk(j)28 b Fh(\000)23 b Fm(1\)2)2281 4014 y Fj(k)r Fd(\000)p Fj(j)t Fd(\000)p Fg(1)2533 3961 y Fk(:)0 4201 y Fm(Th)m(us,)1096 4383 y Fi(E)17 b Fh(fjf)p Fk(w)30 b Fh(2)e(S)36 b Fm(:)27 b Fk(P)1726 4398 y Fj(w)1802 4383 y Fh(\\)22 b Fk(P)1954 4398 y Fj(z)2020 4383 y Fm(=)27 b Fk(Q)2204 4398 y Fj(j)2239 4383 y Fh(gj)g(j)g Fk(z)32 b Fh(2)c(S)7 b(g)1199 4594 y(\024)1822 4526 y Fk(\013)p Fm(2)1936 4490 y Fj(k)r Fd(\000)p Fj(j)p 1314 4571 1254 4 v 1314 4676 a Fm(\()p Fk(\013)23 b Fh(\000)g Fm(1\))1627 4591 y Fb(p)p 1727 4591 583 4 v 85 x Fm(2)p Fk(\031)t Fm(\()p Fk(k)j Fh(\000)d Fk(j)28 b Fh(\000)22 b Fm(1\))q(2)2360 4647 y Fj(k)r Fd(\000)p 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