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-1.55415 -2.0722 curveto pgfstr restore restore restore restore newpath restore pgfc 0 TeXcolorgray 1385 827 a Fv(Figure)30 b(2:)41 b(Substitution)28 b(c)m(hannel)p 0 TeXcolorgray 0 TeXcolorgray 0 1103 a(is)37 b(to)i(prev)m(en)m(t)g(correct)g(deco)s(ding)e(of)h(the) h(message)g(b)m(y)f(in\015icting)d(substitution)h(errors)i(using)e(his) h(kno)m(wledge)0 1238 y(ab)s(out)29 b(the)g(enco)s(ding)f(and)g(deco)s (ding)g(algorithms)g(and)g(the)h(message)h Ft(m)p Fv(.)40 b(W)-8 b(e)30 b(are)f(considering)f(blo)s(c)m(k)g(enco)s(ding)0 1374 y(where)f(w)m(e)h(denote)h(the)f(blo)s(c)m(klength)f(b)m(y)g Ft(n)p Fv(.)40 b(W)-8 b(e)29 b(denote)f(the)g Ft(i)p Fv(th)g(sym)m(b)s(ol)e(the)i(enco)s(der)g(sends)e(o)m(v)m(er)k(the)d(c) m(hannel)0 1509 y(b)m(y)43 b Ft(c)178 1523 y Fr(i)253 1509 y Fs(2)k(f)p Fv(0)p Ft(;)15 b Fv(1)p Fs(g)p Fv(.)81 b(The)43 b(adv)m(ersary)g(tak)m(es)i Ft(c)1597 1523 y Fr(i)1669 1509 y Fv(and)d(is)h(able)f(to)i(in\015ict)e(a)h (substitution)e(error)i(to)h(create)h Ft(y)3855 1523 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Fv(\))j(is)d(of)h(the)g(form)g Ft(dec)h Fv(:)g Fs(f)p Fv(0)p Ft(;)15 b Fv(1)p Fs(g)2797 2289 y Fr(n)2871 2322 y Fs(!)25 b Fv([)p Ft(M)10 b Fv(].)40 b(W)-8 b(e)26 b(require)e(the)h(total)0 2458 y(n)m(um)m(b)s(er)e(of)i (errors)f(pro)s(duced)f(b)m(y)i(the)g(adv)m(ersary)g(to)g(b)s(e)f(at)h (most)g Ft(t)p Fv(,)h(i.e.,)g Ft(d)2589 2472 y Fr(H)2657 2458 y Fv(\()p Fn(c)p Ft(;)15 b Fn(y)s Fv(\))26 b Fs(\024)f Ft(t)p Fv(,)h(where)e Fn(c)h Fv(:=)g(\()p Ft(c)3601 2472 y Fp(1)3641 2458 y Ft(;)15 b(:)g(:)g(:)i(;)e(c)3882 2472 y Fr(n)3929 2458 y Fv(\).)0 2593 y(Let)32 b Ft(M)252 2607 y Fr(s)289 2593 y Fv(\()p Ft(n;)15 b(t)p Fv(\))32 b(b)s(e)e(the)i(maximal)e(n)m(um)m(b)s(er)g(of)h(messages)i(the)e (sender)g(can)g(transmit)g(to)h(the)f(receiv)m(er)h(and)f(de\014ne)0 2729 y(the)24 b(maximal)e(asymptotic)j(rate)f(of)g(feedbac)m(k)g(co)s (des)g(capable)g(of)g(correcting)g(a)g(fraction)f(of)h Ft(\034)34 b Fv(substitution)21 b(errors)0 2864 y(to)31 b(b)s(e)1335 3000 y Ft(R)1404 3014 y Fr(s)1441 3000 y Fv(\()p Ft(\034)10 b Fv(\))26 b(:=)f(lim)15 b(sup)1755 3072 y Fr(n)p Fm(!1)2011 2938 y Fv(log)i Ft(M)2232 2952 y Fr(s)2269 2938 y Fv(\()p Ft(n;)e Fs(b)p Ft(\034)10 b(n)p Fs(c)p Fv(\))p 2011 2979 609 4 v 2288 3062 a Ft(n)2630 3000 y(:)0 3203 y SDict begin H.S end 0 3203 a 0 3203 a SDict begin 16.31996 H.A end 0 3203 a 0 3203 a SDict begin [/View [/XYZ H.V]/Dest (subsection.2.2) cvn /DEST pdfmark end 0 3203 a 77 x Fo(2.2)112 b(Construction)38 b(of)f(a)h(substitution)g (co)s(de)g(from)g(an)g(insertion-deletion)h(co)s(de)0 3475 y Fv(T)-8 b(o)44 b(pro)m(v)m(e)g(that)g(\()650 3475 y SDict begin H.S end 650 3475 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 696 3416 a SDict begin H.R end 696 3416 a 696 3475 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.2) cvn H.B /ANN pdfmark end 696 3475 a Fv(\))g(is)e(an)h(upp)s(er)f(b) s(ound)f(on)j(the)f(maximal)f(rate)j(of)e(feedbac)m(k)h(co)s(des)g(for) f(the)h(adv)m(ersarial)0 3610 y(insertion-deletion)30 b(c)m(hannel,)j(w)m(e)g(com)m(bine)g(t)m(w)m(o)h(ideas.)47 b(First,)33 b(w)m(e)g(pro)m(v)m(e)g(that)h(a)f(feedbac)m(k)g (insertion-deletion)0 3746 y(co)s(de)e(can)f(b)s(e)g(used)g(to)h(tac)m (kle)g(substitution)d(errors.)0 3928 y SDict begin H.S end 0 3928 a 0 3928 a SDict begin 16.31996 H.A end 0 3928 a 0 3928 a SDict begin [/View [/XYZ H.V]/Dest (theorem.2) cvn /DEST pdfmark end 0 3928 a 0 TeXcolorgray Fj(Lemma)22 b(2.)p 0 TeXcolorgray 34 w Fq(Supp)-5 b(ose)26 b(that)f(a)g(fe)-5 b(e)g(db)g(ack)25 b(insertion-deletion)g(c)-5 b(o)g(de)25 b(of)g(length)f Ft(n)g Fq(and)h(size)f Ft(M)34 b Fq(c)-5 b(ap)g(able)26 b(of)e(c)-5 b(orr)g(e)g(cting)0 4064 y Ft(t)32 b Fq(err)-5 b(ors)34 b(is)e(given.)40 b(Then)33 b(ther)-5 b(e)33 b(exists)f(a)h(fe)-5 b(e)g(db)g(ack)32 b(c)-5 b(o)g(de)34 b(of)e(length)h Ft(n)18 b Fv(+)h Ft(t)32 b Fq(and)h(size)e Ft(M)43 b Fq(c)-5 b(ap)g(able)33 b(of)g(c)-5 b(orr)g(e)g(cting)33 b Ft(t)0 4199 y Fq(substitution)g(err)-5 b(ors.)141 4382 y Fv(Second,)30 b(w)m(e)h(mak)m(e)h(use)e(of)g(an)h (upp)s(er)d(b)s(ound)g(on)i(the)h(rate)g Ft(R)2312 4396 y Fr(s)2349 4382 y Fv(\()p Ft(\034)10 b Fv(\))31 b(pro)m(vided)e(b)m(y) h(Berlek)-5 b(amp)30 b([)3476 4382 y SDict begin H.S end 3476 4382 a 0 TeXcolorgray Fv(8)p 0 TeXcolorgray 3522 4323 a SDict begin H.R end 3522 4323 a 3522 4382 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.berlekamp1964block) cvn H.B /ANN pdfmark end 3522 4382 a Fv(].)0 4564 y SDict begin H.S end 0 4564 a 0 4564 a SDict begin 16.31996 H.A end 0 4564 a 0 4564 a SDict begin [/View [/XYZ H.V]/Dest (theorem.3) cvn /DEST pdfmark end 0 4564 a 0 TeXcolorgray Fj(Theorem)g(3)g Fv(\(F)-8 b(ollo)m(ws)27 b(from)f([)1112 4564 y SDict begin H.S end 1112 4564 a 0 TeXcolorgray Fv(8)p 0 TeXcolorgray 1158 4506 a SDict begin H.R end 1158 4506 a 1158 4564 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.berlekamp1964block) cvn H.B /ANN pdfmark end 1158 4564 a Fv(,)i(Chapter)d(IV])i(and)f([)1911 4564 y SDict begin H.S end 1911 4564 a 0 TeXcolorgray Fv(9)p 0 TeXcolorgray 1957 4506 a SDict begin H.R end 1957 4506 a 1957 4564 a SDict begin [/Color [0 1 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (cite.zigangirov1976number) cvn H.B /ANN pdfmark end 1957 4564 a Fv(]\))p Fj(.)p 0 TeXcolorgray 39 w Fq(The)j(maximal)i(asymptotic)g(r)-5 b(ate)30 b(for)g(the)g(adversarial)0 4700 y(substitution)j(channel)h (satis\014es)782 5056 y Ft(R)851 5070 y Fr(s)888 5056 y Fv(\()p Ft(\034)10 b Fv(\))26 b(=)1130 4788 y Fk(8)1130 4870 y(>)1130 4897 y(>)1130 4924 y(>)1130 4951 y(<)1130 5115 y(>)1130 5142 y(>)1130 5170 y(>)1130 5197 y(:)1210 4898 y Fv(1)21 b Fs(\000)f Ft(h)p Fv(\()p Ft(\034)10 b Fv(\))p Ft(;)596 b Fq(for)33 b Fv(0)25 b Fs(\024)g Ft(\034)36 b Fs(\024)25 b Fv(\(3)c Fs(\000)2835 4823 y(p)p 2910 4823 46 4 v 2910 4898 a Fv(5)q(\))p Ft(=)p Fv(4)p Ft(;)1210 5061 y Fv(\(1)g Fs(\000)f Fv(3)p Ft(\034)10 b Fv(\))15 b(log)1681 4960 y Fk(\020)1745 5024 y Fp(1+)1835 4969 y Fm(p)p 1894 4969 36 3 v 55 x Fp(5)p 1745 5040 185 4 v 1820 5092 a(2)1939 4960 y Fk(\021)2009 5061 y Ft(;)126 b Fq(for)33 b Fv(\(3)21 b Fs(\000)2497 4985 y(p)p 2572 4985 46 4 v 2572 5061 a Fv(5)q(\))p Ft(=)p Fv(4)26 b Ft(<)f(\034)36 b Fs(\024)25 b Fv(1)p Ft(=)p Fv(3)p Ft(;)1210 5233 y Fv(0)p Ft(;)847 b Fq(otherwise.)p 0 TeXcolorgray 1972 5486 a Fv(6)p 0 TeXcolorgray eop end end %%Page: 7 7 TeXDict begin HPSdict begin 7 6 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -174 a SDict begin H.S end -8 -174 a -8 -174 a SDict begin H.R end -8 -174 a -8 -174 a SDict begin [/View [/XYZ H.V]/Dest (page.7) cvn /DEST pdfmark end -8 -174 a 0 TeXcolorgray 0 TeXcolorgray 141 108 a Fv(By)35 b(Lemma)617 108 y SDict begin H.S end 617 108 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 662 49 a SDict begin H.R end 662 49 a 662 108 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2) cvn H.B /ANN pdfmark end 662 108 a Fv(,)h(w)m(e)f(ha)m(v)m(e)h Ft(M)1163 122 y Fr(s)1200 108 y Fv(\()p Ft(n)23 b Fv(+)g Ft(t;)15 b(t)p Fv(\))32 b Fs(\025)g Ft(M)1771 123 y Fr(id)1836 108 y Fv(\()p Ft(n;)15 b(t)p Fv(\))35 b(and,)g(th)m(us,)h(\(1)24 b(+)e Ft(\034)10 b Fv(\))p Ft(R)2858 122 y Fr(s)2896 108 y Fv(\()p Ft(\034)g(=)p Fv(\(1)24 b(+)f Ft(\034)10 b Fv(\)\))33 b Fs(\025)f Ft(R)3549 123 y Fr(id)3614 108 y Fv(\()p Ft(\034)10 b Fv(\).)54 b(This)0 243 y(b)s(ound)22 b(and)i(Theorem)823 243 y SDict begin H.S end 823 243 a 0 TeXcolorgray Fv(3)p 0 TeXcolorgray 868 185 a SDict begin H.R end 868 185 a 868 243 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.3) cvn H.B /ANN pdfmark end 868 243 a 25 w Fv(yield)e(that)j(the)g (asymptotic)f(rate)h(sp)s(eci\014ed)e(in)g(Theorem)2929 243 y SDict begin H.S end 2929 243 a 0 TeXcolorgray Fv(1)p 0 TeXcolorgray 2974 185 a SDict begin H.R end 2974 185 a 2974 243 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1) cvn H.B /ANN pdfmark end 2974 243 a 24 w Fv(is)h(indeed)f(an)h (upp)s(er)e(b)s(ound)0 379 y(on)30 b(the)h(maximal)e(ac)m(hiev)-5 b(able)31 b(rate)g(of)g(the)g(adv)m(ersarial)e(insertion-deletion)f(c)m (hannel.)41 b(So,)31 b(it)f(remains)f(to)i(c)m(hec)m(k)0 514 y(the)g(v)-5 b(alidit)m(y)28 b(of)j(Lemma)910 514 y SDict begin H.S end 910 514 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 956 456 a SDict begin H.R end 956 456 a 956 514 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2) cvn H.B /ANN pdfmark end 956 514 a Fv(.)p 0 TeXcolorgray 0 725 a Fq(Pr)-5 b(o)g(of)34 b(of)f(L)-5 b(emma)673 725 y SDict begin H.S end 673 725 a 0 TeXcolorgray Fq(2)p 0 TeXcolorgray 719 666 a SDict begin H.R end 719 666 a 719 725 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.2) cvn H.B /ANN pdfmark end 719 725 a Fq(.)p 0 TeXcolorgray 46 w Fv(Let)36 b(the)h(enco)s(ding)e(functions)f(of)j(the)f(giv)m(en)g (feedbac)m(k)h(insertion-deletion)d(co)s(de)i(b)s(e)g(de-)0 860 y(termined)g(b)m(y)i Ft(c)564 874 y Fr(i)630 860 y Fv(:)g([)p Ft(M)10 b Fv(])25 b Fs(\002)g Fv(\()p Fs(f)p Fv(0)p Ft(;)15 b Fv(1)p Fs(g)1217 827 y Fm(\003)1259 860 y Fv(\))1294 818 y Fr(i)p Fm(\000)p Fp(1)1450 860 y Fs(!)38 b(f)p Fv(0)p Ft(;)15 b Fv(1)p Fs(g)39 b Fv(for)f Ft(i)g Fs(2)f Fv([)p Ft(n)p Fv(])g(and)g(let)h(the)g(deco)s(ding)f (function)f(b)s(e)h(giv)m(en)h(as)0 996 y Ft(dec)26 b Fv(:)g Fs(f)p Fv(0)p Ft(;)15 b Fv(1)p Fs(g)425 963 y Fm(\003)492 996 y Fs(!)25 b Fv([)p Ft(M)10 b Fv(].)40 b(W)-8 b(e)28 b(shall)c(de\014ne)i(enco)s(ding)g(functions)f Ft(e)2249 1010 y Fr(j)2311 996 y Fv(:)g([)p Ft(M)10 b Fv(])j Fs(\002)g(f)p Fv(0)p Ft(;)i Fv(1)p Fs(g)2826 963 y Fr(j)t Fm(\000)p Fp(1)2980 996 y Fs(!)25 b(f)p Fv(0)p Ft(;)15 b Fv(1)p Fs(g)29 b Fv(for)d(all)g Ft(j)k Fs(2)25 b Fv([)p Ft(n)13 b Fv(+)g Ft(t)p Fv(])0 1131 y(for)31 b(the)g(substitution)e(c)m(hannel)i(using)f(the)h(enco)s(ding)f (functions)g Ft(c)2346 1145 y Fr(i)2405 1131 y Fv(as)i(comp)s(onen)m (ts.)43 b(Using)31 b(suc)m(h)f(functions)g Ft(e)3953 1145 y Fr(j)0 1267 y Fv(the)d(sender)f(w)m(ould)f(b)s(e)h(able)g(to)i (transmit)d(the)i(message)h(o)m(v)m(er)g(the)f(c)m(hannel)f(with)f(at)j (most)e Ft(t)h Fv(substitution)d(errors.)141 1402 y(W)-8 b(e)31 b(denote)g(the)f(n)m(um)m(b)s(er)e(of)i(correctly)h(receiv)m(ed) f(sym)m(b)s(ols)e(b)m(y)i Ft(i)p Fv(,)h(the)f(n)m(um)m(b)s(er)e(of)i (transmitted)g(sym)m(b)s(ols)e(b)m(y)0 1538 y Ft(j)5 b Fv(,)32 b(the)f(index)e(of)i(the)g(last)g(correctly)g(receiv)m(ed)g (sym)m(b)s(ol)f(b)m(y)h 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b Ft(j)c Fv(+)23 b(1.)53 b(Let)35 b Ft(z)2971 3177 y Fr(j)3043 3163 y Fv(b)s(e)e(the)i(receiv)m(ed)g(sym)m(b)s(ol.)0 3299 y(Up)s(date)30 b Fn(z)g Fs( )25 b Fv(\()p Fn(z)t Fs(jj)p Ft(z)700 3313 y Fr(j)738 3299 y Fv(\).)41 b(Rep)s(eat)31 b(Step)f(2.)141 3434 y(The)38 b(enco)s(ding)g(algorithm)f(successfully) g(terminates)i(after)g Ft(n)25 b Fv(+)g Ft(t)39 b Fv(transmissions)d(b) s(ecause)j(w)m(e)g(will)d(ha)m(v)m(e)0 3570 y Ft(n)e Fv(correct)h(receptions)f(within)e(a)j(blo)s(c)m(k)f(of)g Ft(n)23 b Fv(+)f Ft(t)34 b Fv(sym)m(b)s(ols.)52 b(The)33 b(deco)s(ding)h(sc)m(heme)h(can)f(just)g(b)s(e)g(tak)m(en)h(from)0 3705 y(the)d(insertion-deletion)d(co)s(de)j(b)s(ecause)f(the)h(output)f (string)g Fn(z)36 b Fv(is)30 b(a)i(p)s(ossible)e(output)h(of)h(the)f (insertion-deletion)0 3841 y(c)m(hannel)22 b(with)f(at)i(most)f Ft(t)h Fv(errors)e(for)h(enco)s(ding)g(functions)f Ft(c)2073 3855 y Fr(i)2101 3841 y Fv(,)j Ft(i)i Fs(2)f Fv([)p Ft(n)p Fv(].)38 b(Indeed,)23 b(if)e(w)m(e)i(de\014ne)3242 3840 y(^)3231 3841 y Fn(y)3288 3808 y Fp(\()p Fr(i)p Fp(\))3396 3841 y Fv(:=)i(\()p Fn(y)3606 3855 y Fp(1)3646 3841 y Ft(;)15 b(:)g(:)g(:)h(;)f Fn(y)3901 3855 y Fr(i)3929 3841 y Fv(\),)0 3976 y(w)m(e)31 b(obtain)676 4022 y Fr(n)632 4050 y Fk(X)640 4245 y Fr(i)p Fp(=1)778 4136 y Fv(\001\()p Ft(c)928 4150 y Fr(i)957 4136 y Fv(\()p Ft(m;)1124 4135 y Fv(^)1112 4136 y Fn(y)1169 4099 y Fp(\()p Fr(i)p Fp(\))1252 4136 y Fv(\))p Ft(;)15 b Fn(y)1381 4150 y Fr(i)1410 4136 y Fv(\))25 b(=)1611 4022 y Fr(n)1566 4050 y Fk(X)1575 4245 y Fr(i)p Fp(=1)1698 4136 y Fv(\()p Fs(j)p Fn(y)1812 4150 y Fr(i)1840 4136 y Fs(j)c(\000)f Fv(1\))26 b(=)2223 4022 y Fr(n)2179 4050 y Fk(X)2187 4245 y Fr(i)p Fp(=1)2325 4136 y Fs(j)p Fn(y)2404 4150 y Fr(i)2432 4136 y Fs(j)21 b(\000)f Ft(n)k Fv(=)h(\()p Ft(n)20 b Fv(+)g Ft(t)p Fv(\))h Fs(\000)f Ft(n)k Fv(=)h Ft(t:)0 4385 y Fv(Therefore,)34 b(the)g(deco)s(der)f(of)h(the)g(insertion-deletion)d(co)s(de)j(outputs) f(the)h(correct)h(message,)g(i.e.,)g Ft(dec)p Fv(\()p Fn(z)t Fv(\))e(=)d Ft(m)p Fv(.)0 4520 y(This)f(completes)h(the)h(pro)s (of.)p 3924 4520 4 62 v 3928 4462 55 4 v 3928 4520 V 3982 4520 4 62 v 0 4675 a SDict begin H.S end 0 4675 a 0 4675 a SDict begin 16.31996 H.A end 0 4675 a 0 4675 a SDict begin [/View [/XYZ H.V]/Dest (section.3) cvn /DEST pdfmark end 0 4675 a 161 x Fu(3)135 b(Lo)l(w)l(er)46 b(b)t(ound)d(on)i Fi(R)1291 4854 y Fz(id)1367 4836 y Fh(\()p Fi(\034)13 b Fh(\))0 5062 y Fv(In)40 b(this)g(section,)45 b(w)m(e)c(pro)m(vide)g (a)g(feedbac)m(k)h(insertion-deletion)d(co)s(de)i(with)f(an)h (asymptotic)h(rate)g(arbitrarily)0 5197 y(close)33 b(to)h(\()369 5197 y SDict begin H.S end 369 5197 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 415 5139 a SDict begin H.R end 415 5139 a 415 5197 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.1.2) cvn H.B /ANN pdfmark end 415 5197 a Fv(\))q(.)49 b(W)-8 b(e)34 b(adapt)f(an)g(algorithm)f(originally)f(suggested)j(b)m (y)f(Horstein)g([)2832 5197 y 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1054 y Fp(1)348 1040 y Fv(\))26 b Fs(\025)f Ft(\016)s Fq(.)42 b(F)-7 b(urthermor)i(e,)35 b Ft(g)s Fv(\(0)p Ft(;)15 b(n)1392 1054 y Fp(1)1433 1040 y Fv(\))26 b Ft(<)f(")1632 1054 y Fp(2)1671 1040 y Ft(n)20 b Fs(\000)g Fv(log)d Ft(t)p Fv(\(0\))33 b Fq(and)h Ft(g)s Fv(\()p Ft(n)2464 1054 y Fp(1)2504 1040 y Ft(;)15 b(n)p Fv(\))25 b Ft(>)g Fv(\()p Ft(")2832 1054 y Fp(1)2893 1040 y Fs(\000)20 b Ft(")3026 1054 y Fp(2)3065 1040 y Fv(\))p Ft(n)p Fq(.)141 1236 y Fv(The)42 b(follo)m(wing)f(lemma)i(sho)m(ws)f(that)h(after)g(sp) s(ending)d(enough)j(energy)g(the)f(true)h(segmen)m(t)h(will)c(b)s (ecome)0 1372 y(cen)m(tral.)0 1568 y SDict begin H.S end 0 1568 a 0 1568 a SDict begin 16.31996 H.A end 0 1568 a 0 1568 a SDict begin [/View [/XYZ H.V]/Dest (theorem.7) cvn /DEST pdfmark end 0 1568 a 0 TeXcolorgray Fj(Lemma)34 b(7.)p 0 TeXcolorgray 43 w Fq(L)-5 b(et)34 b Ft(\013)h Fq(and)g Ft(\014)k Fq(satisfy)j Fv(\()1342 1568 y SDict begin H.S end 1342 1568 a 0 TeXcolorgray Fv(3)p 0 TeXcolorgray 1388 1509 a SDict begin H.R end 1388 1509 a 1388 1568 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.3) cvn H.B /ANN pdfmark end 1388 1568 a Fv(\))q Fq(.)j(Then)35 b(ther)-5 b(e)34 b(exists)h(a)f(c)-5 b(onstant)36 b Ft(c)28 b(>)f Fv(0)35 b Fq(dep)-5 b(ending)35 b(only)g(on)f Ft(\013;)15 b(\014)40 b Fq(and)0 1703 y Ft(t)p Fv(\()p Ft(n)123 1717 y Fp(1)162 1703 y Fv(\))27 b Fq(such)h(that)g(if)e Ft(g)s Fv(\()p Ft(n)824 1717 y Fp(1)864 1703 y Ft(;)15 b(n)p Fv(\))26 b Ft(>)f(c)p Fq(,)j(it)f(fol)5 b(lows)28 b(that)h Ft(T)13 b Fv(\()p Ft(n)p Fv(\))26 b Fq(is)h(c)-5 b(entr)g(al.)42 b(Mor)-5 b(e)g(over,)29 b(the)e(length)h(of)f(the)g(true)g(se)-5 b(gment)0 1839 y(at)33 b(moment)h Ft(n)e Fq(is)h Fv(1)20 b Fs(\000)g Ft(o)p Fv(\(1\))34 b Fq(as)f Ft(c)26 b Fs(!)f(1)p Fq(.)141 2035 y Fv(Lemmas)503 2035 y SDict begin H.S end 503 2035 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 549 1976 a SDict begin H.R end 549 1976 a 549 2035 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6) cvn H.B /ANN pdfmark end 549 2035 a 32 w Fv(and)760 2035 y SDict begin H.S end 760 2035 a 0 TeXcolorgray Fv(7)p 0 TeXcolorgray 805 1976 a SDict begin H.R end 805 1976 a 805 2035 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.7) cvn H.B /ANN pdfmark end 805 2035 a 32 w Fv(are)33 b(pro)m(v)m(ed)g(in)d(Section)1717 2035 y SDict begin H.S end 1717 2035 a 0 TeXcolorgray Fv(4.4)p 0 TeXcolorgray 1833 1976 a SDict begin H.R end 1833 1976 a 1833 2035 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (subsection.4.4) cvn H.B /ANN pdfmark end 1833 2035 a Fv(.)46 b(Finally)-8 b(,)31 b(w)m(e)i(com)m(bine)f(these)h(t)m(w)m(o)g(lemmas)f (to)h(obtain)f(the)0 2170 y(main)d(result.)0 2367 y SDict begin H.S end 0 2367 a 0 2367 a SDict begin 16.31996 H.A end 0 2367 a 0 2367 a SDict begin [/View [/XYZ H.V]/Dest (theorem.8) cvn /DEST pdfmark end 0 2367 a 0 TeXcolorgray Fj(Theorem)40 b(8.)p 0 TeXcolorgray 45 w Fq(L)-5 b(et)37 b(the)h(maximal)h(fr)-5 b(action)39 b(of)f(substitution)g(err)-5 b(ors)39 b(b)-5 b(e)37 b Ft(\034)44 b Fs(2)34 b Fv([0)p Ft(;)15 b Fv(1)p Ft(=)p Fv(3\))p Fq(.)59 b(F)-7 b(or)38 b(any)g Ft(")d(>)e Fv(0)p Fq(,)39 b(ther)-5 b(e)0 2502 y(exists)28 b Ft(\013)e Fs(2)f Fv([0)p Ft(;)15 b Fv(\(3)10 b Fs(\000)695 2427 y(p)p 773 2427 46 4 v 773 2502 a Fv(5)q(\))p Ft(=)p Fv(2])29 b Fq(such)g(that)g(for)g Ft(n)e Fq(su\016ciently)i(lar) -5 b(ge,)30 b(the)e(enc)-5 b(o)g(ding)29 b(pr)-5 b(o)g(c)g(e)g(dur)g(e) 32 b(over)c(the)h(adversarial)0 2638 y(substitution)e(channel)g(with)g (fe)-5 b(e)g(db)g(ack)26 b(c)-5 b(an)27 b(c)-5 b(orr)g(e)g(ct)28 b(up)e(to)h Ft(\034)10 b(n)25 b Fq(err)-5 b(ors)28 b(with)f(tr)-5 b(ansmission)29 b(r)-5 b(ate)27 b(at)f(le)-5 b(ast)28 b Ft(R)3680 2652 y Fr(s)3716 2638 y Fv(\()p Ft(\034)10 b Fv(\))c Fs(\000)g Ft(")p Fq(,)0 2773 y(wher)-5 b(e)782 2972 y Ft(R)851 2986 y Fr(s)888 2972 y Fv(\()p Ft(\034)10 b Fv(\))26 b(=)1130 2785 y Fk(8)1130 2867 y(<)1130 3031 y(:)1210 2895 y Fv(1)21 b Fs(\000)f Ft(h)p Fv(\()p Ft(\034)10 b Fv(\))p Ft(;)596 b Fq(for)33 b Fv(0)25 b Fs(\024)g Ft(\034)36 b Fs(\024)25 b Fv(\(3)c Fs(\000)2835 2820 y(p)p 2910 2820 V 2910 2895 a Fv(5)q(\))p Ft(=)p Fv(4)p Ft(;)1210 3058 y Fv(\(1)g Fs(\000)f Fv(3)p Ft(\034)10 b Fv(\))15 b(log)1681 2957 y Fk(\020)1745 3021 y Fp(1+)1835 2966 y Fm(p)p 1894 2966 36 3 v 55 x Fp(5)p 1745 3037 185 4 v 1820 3089 a(2)1939 2957 y Fk(\021)2009 3058 y Ft(;)126 b Fq(for)33 b Fv(\(3)21 b Fs(\000)2497 2983 y(p)p 2572 2983 46 4 v 2572 3058 a Fv(5)q(\))p Ft(=)p Fv(4)26 b Ft(<)f(\034)36 b Fs(\024)25 b Fv(1)p Ft(=)p Fv(3)p Ft(:)p 0 TeXcolorgray 0 3269 a Fq(Pr)-5 b(o)g(of.)p 0 TeXcolorgray 47 w Fv(W)d(e)23 b(c)m(ho)s(ose)h Ft(M)32 b Fv(and)22 b(accordingly)f Ft(t)p Fv(\(0\))26 b(=)f(1)p Ft(=)-5 b(M)33 b Fv(suc)m(h)22 b(that)h(it)f(satis\014es)g(2)2794 3236 y Fp(\()p Fr(R)2874 3244 y Fe(s)2908 3236 y Fp(\()p Fr(\034)8 b Fp(\))p Fm(\000)p Fr(")p Fp(\))p Fr(n)3189 3269 y Ft(<)25 b(M)35 b(<)25 b Fv(2)3549 3236 y Fp(\()p Fr(R)3629 3244 y Fe(s)3664 3236 y Fp(\()p Fr(\034)8 b Fp(\))p Fm(\000)p Fr("=)p Fp(2\))p Fr(n)0 3404 y Fv(and)27 b Fs(\000)15 b Fv(log)h Ft(t)p Fv(\(0\))26 b Ft(<)f Fv(\()p Ft(R)766 3418 y Fr(s)803 3404 y Fv(\()p Ft(\034)10 b Fv(\))k Fs(\000)g Ft("=)p Fv(2\))p Ft(n)p Fv(.)41 b(By)28 b(de\014nition,)d(w)m(e)j(ha)m(v)m(e)g(that)g Ft(e)p Fv(\(0)p Ft(;)15 b(n)p Fv(\))27 b Fs(\024)e Ft(\034)10 b(n)26 b Fv(and)h Ft(f)10 b Fv(\(0)p Ft(;)15 b(n)p Fv(\))26 b Fs(\025)f Fv(\(1)14 b Fs(\000)g Ft(\034)c Fv(\))p Ft(n)p Fv(.)39 b(F)-8 b(or)0 3540 y(giv)m(en)30 b Ft(\013)h Fv(and)f Ft(\014)5 b Fv(,)31 b(w)m(e)g(can)f(compute)725 3749 y Ft(g)s Fv(\(0)p Ft(;)15 b(n)p Fv(\))27 b(=)e Ft(e)p Fv(\(0)p Ft(;)15 b(n)p Fv(\))g(log)j Ft(\013)j Fv(+)f Ft(f)10 b Fv(\(0)p Ft(;)15 b(n)p Fv(\))g(log)i Ft(\014)31 b Fs(\025)25 b Fv(\()p Ft(\034)g Fv(log)17 b Ft(\013)j Fv(+)g(\(1)h Fs(\000)f Ft(\034)10 b Fv(\))15 b(log)i Ft(\014)5 b Fv(\))p Ft(n:)0 3959 y Fv(Sub)g(ject)35 b(to)i(the)f (constrain)m(ts)f Ft(\013)25 b Fv(+)e Ft(\014)40 b Fv(=)34 b(2)i(and)f Ft(\013\014)1833 3926 y Fp(2)1907 3959 y Fs(\024)f Fv(1,)k(the)e(function)e Ft(\034)25 b Fv(log)17 b Ft(\013)24 b Fv(+)g(\(1)g Fs(\000)g Ft(\034)10 b Fv(\))15 b(log)i Ft(\014)41 b Fv(reac)m(hes)c(the)0 4095 y(maxim)m(um)28 b Ft(R)492 4109 y Fr(s)529 4095 y Fv(\()p Ft(\034)10 b Fv(\))29 b(when)g Ft(\013)c Fv(=)g(2)15 b(min)o(\()p Ft(\034)5 b(;)15 b Fv(\(3)20 b Fs(\000)1614 4020 y(p)p 1690 4020 V 75 x Fv(5\))p Ft(=)p Fv(4\))31 b(and)d Ft(\014)j Fv(=)25 b(2)18 b Fs(\000)f Ft(\013)p Fv(.)41 b(T)-8 b(o)29 b(mak)m(e)h(use)f(of)g(Lemma)3503 4095 y SDict begin H.S end 3503 4095 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 3549 4036 a SDict begin H.R end 3549 4036 a 3549 4095 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6) cvn H.B /ANN pdfmark end 3549 4095 a Fv(,)g(w)m(e)h(tak)m(e)h Ft(\013)0 4230 y Fv(sligh)m(tly)e(smaller)g(than)h(2)15 b(min)o(\()p Ft(\034)5 b(;)15 b Fv(\(3)22 b Fs(\000)1369 4155 y(p)p 1445 4155 46 4 v 75 x Fv(5)q(\))p Ft(=)p Fv(4\))32 b(to)f(ha)m(v)m(e)561 4456 y Ft(g)s Fv(\(0)p Ft(;)15 b(n)p Fv(\))27 b(=)e Ft(e)p Fv(\(0)p Ft(;)15 b(n)p Fv(\))g(log)j Ft(\013)j Fv(+)e Ft(f)10 b Fv(\(0)p Ft(;)15 b(n)p Fv(\))g(log)j Ft(\014)30 b Fs(\025)2101 4355 y Fk(\020)2155 4456 y Ft(R)2224 4470 y Fr(s)2261 4456 y Fv(\()p Ft(\034)10 b Fv(\))21 b Fs(\000)2527 4394 y Ft(")p 2503 4435 91 4 v 2503 4518 a Fv(10)2604 4355 y Fk(\021)2673 4456 y Ft(n)k(>)2859 4394 y Fv(2)p 2859 4435 46 4 v 2859 4518 a(5)2914 4456 y Ft("n)c Fs(\000)e Fv(log)e Ft(t)p Fv(\(0\))p Ft(:)0 4666 y Fv(By)34 b(applying)d(Lemma)848 4666 y SDict begin H.S end 848 4666 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 893 4607 a SDict begin H.R end 893 4607 a 893 4666 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6) cvn H.B /ANN pdfmark end 893 4666 a 34 w Fv(with)h Ft(")1179 4680 y Fp(1)1250 4666 y Fv(=)e(2)p Ft("=)p Fv(5)35 b(and)e Ft(")1785 4680 y Fp(2)1856 4666 y Fv(=)d Ft("=)p Fv(5,)36 b(there)e(exists)f(a)h(constan)m(t)h Ft(\016)f(>)c Fv(0)k(suc)m(h)g (that)g(for)f(all)0 4801 y(but)d(\014nite)f Ft(\013)d Fs(2)f Fv(\(0)p Ft(;)15 b Fv(1\),)32 b(there)f(is)e(a)i(\014rst)f (momen)m(t)h Ft(n)1820 4815 y Fp(1)1889 4801 y Fv(satisfying)e Ft(t)p Fv(\()p Ft(n)2417 4815 y Fp(1)2456 4801 y Fv(\))d Fs(\025)f Ft(\016)34 b Fv(and)c(furthermore)665 5027 y Ft(g)s Fv(\()p Ft(n)801 5041 y Fp(1)841 5027 y Ft(;)15 b(n)p Fv(\))25 b(=)g Ft(g)s Fv(\(0)p Ft(;)15 b(n)p Fv(\))22 b Fs(\000)e Ft(g)s Fv(\(0)p Ft(;)15 b(n)1682 5041 y Fp(1)1723 5027 y Fv(\))25 b(=)g Ft(e)p Fv(\()p Ft(n)2011 5041 y Fp(1)2051 5027 y Ft(;)15 b(n)p Fv(\))g(log)i Ft(\013)k Fv(+)f Ft(f)10 b Fv(\()p Ft(n)2644 5041 y Fp(1)2682 5027 y Ft(;)15 b(n)p Fv(\))g(log)i Ft(\014)31 b(>)3148 4965 y Fv(1)p 3148 5006 46 4 v 3148 5089 a(5)3203 5027 y Ft("n:)0 5237 y Fv(Th)m(us,)f(for)g Ft(n)g Fv(su\016cien)m(tly)f(large,)h(w)m(e) h(conclude)f(b)m(y)g(Lemma)2141 5237 y SDict begin H.S end 2141 5237 a 0 TeXcolorgray Fv(7)p 0 TeXcolorgray 2186 5178 a SDict begin H.R end 2186 5178 a 2186 5237 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.7) cvn H.B /ANN pdfmark end 2186 5237 a 30 w Fv(that)h Ft(T)13 b Fv(\()p Ft(n)p Fv(\))31 b(is)e(cen)m(tral.)p 3924 5237 4 62 v 3928 5179 55 4 v 3928 5237 V 3982 5237 4 62 v 0 TeXcolorgray 1950 5486 a(15)p 0 TeXcolorgray eop end end %%Page: 16 16 TeXDict begin HPSdict begin 16 15 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -174 a SDict begin H.S end -8 -174 a -8 -174 a SDict begin H.R end -8 -174 a -8 -174 a SDict begin [/View [/XYZ H.V]/Dest (page.16) cvn /DEST pdfmark end -8 -174 a 0 TeXcolorgray 0 TeXcolorgray 141 108 a Fv(The)39 b(k)m(ey)h(ingredien)m(t)d(in)h(the)h(pro)s(of)g(of)g (Theorem)1976 108 y SDict begin H.S end 1976 108 a 0 TeXcolorgray Fv(1)p 0 TeXcolorgray 2022 49 a SDict begin H.R end 2022 49 a 2022 108 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.1) cvn H.B /ANN pdfmark end 2022 108 a 39 w Fv(is)f(Lemma)2494 108 y SDict begin H.S end 2494 108 a 0 TeXcolorgray Fv(7)p 0 TeXcolorgray 2540 49 a SDict begin H.R end 2540 49 a 2540 108 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.7) cvn H.B /ANN pdfmark end 2540 108 a Fv(.)66 b(In)39 b(order)f(to)i(pro)m(v)m(e)g(it)e(w)m(e)i (in)m(tro)s(duce)0 243 y(sev)m(eral)31 b(auxiliary)d(functions)1122 451 y Ft(u)1174 465 y Fp(1)1213 451 y Fv(\()p Ft(i)p Fv(\))f(:=)e(log)q(\()p Ft(t)p Fv(\()p Ft(i)p Fv(\))p Ft(=x)p Fv(\()p Ft(i)p 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2.98883 ] 0.0 setdash 0.29 0.145 0.71 setrgbcolor 0.79701 pgfw 0.0 10.52545 moveto 0.95366 10.52545 lineto 1.90732 10.52545 lineto 2.86101 10.52545 lineto 3.81485 10.52545 lineto 4.76852 10.52545 lineto 5.72218 10.52545 lineto 6.67584 10.52545 lineto 7.6295 10.52545 lineto 8.58318 10.52545 lineto 9.53685 10.52545 lineto 10.49052 10.52545 lineto pgfstr restore save [ 2.98883 2.98883 ] 0.0 setdash [ 5.47952 2.98883 ] 0.0 setdash 0.29 0.145 0.71 setrgbcolor 0.79701 pgfw 10.52539 0.0 moveto 10.52539 10.52539 lineto pgfstr restore save [ 2.98883 2.98883 ] 0.0 setdash [ 5.47952 1.99255 ] 0.0 setdash 0.22 0.11 0.78 setrgbcolor 0.79701 pgfw 0.0 8.69345 moveto 0.95366 8.69345 lineto 1.90732 8.69345 lineto 2.86101 8.69345 lineto 3.81485 8.69345 lineto 4.76852 8.69345 lineto 5.72218 8.69345 lineto 6.67584 8.69345 lineto 7.6295 8.69345 lineto 8.58318 8.69345 lineto pgfstr restore save [ 2.98883 2.98883 ] 0.0 setdash [ 5.47952 1.99255 ] 0.0 setdash 0.22 0.11 0.78 setrgbcolor 0.79701 pgfw 8.69351 0.0 moveto 8.69351 8.69351 lineto pgfstr restore save [ 2.98883 2.98883 ] 0.0 setdash [ 5.47952 1.49442 ] 0.0 setdash 0.15 0.075 0.85 setrgbcolor 0.79701 pgfw 0.0 7.18047 moveto 0.95366 7.18047 lineto 1.90732 7.18047 lineto 2.86101 7.18047 lineto 3.81485 7.18047 lineto 4.76852 7.18047 lineto 5.72218 7.18047 lineto 6.67584 7.18047 lineto pgfstr restore save [ 2.98883 2.98883 ] 0.0 setdash [ 5.47952 1.49442 ] 0.0 setdash 0.15 0.075 0.85 setrgbcolor 0.79701 pgfw 7.18047 0.0 moveto 7.18047 7.18047 lineto pgfstr restore save [ 2.98883 2.98883 ] 0.0 setdash [ 5.47952 0.99628 ] 0.0 setdash 0.08 0.04 0.92 setrgbcolor 0.79701 pgfw 0.0 5.93076 moveto 0.95366 5.93076 lineto 1.90732 5.93076 lineto 2.86101 5.93076 lineto 3.81485 5.93076 lineto 4.76852 5.93076 lineto 5.72218 5.93076 lineto pgfstr restore save [ 2.98883 2.98883 ] 0.0 setdash [ 5.47952 0.99628 ] 0.0 setdash 0.08 0.04 0.92 setrgbcolor 0.79701 pgfw 5.93076 0.0 moveto 5.93076 5.93076 lineto pgfstr restore restore save save [1.0 0.0 0.0 1.0 86.15817 -26.51886 ] concat pgfs 663 1384 a 0 setgray 663 1384 a Ft(r)s Fv(\()p Ft(i)p Fv(\))663 1384 y pgfr restore restore save save [0.0 1.0 -1.0 0.0 -27.97343 87.0357 ] concat pgfs 663 1384 a 0 setgray 663 1384 a Ft(l)r Fv(\()p Ft(i)p Fv(\))663 1384 y pgfr restore restore save /pgffc{1 setgray}def /pgfsc{0 setgray}def save /pgffc{1 setgray}def /pgfsc{0 setgray}def 82.87518 123.26454 60.62743 101.01677 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 63.61626 107.91216 ] concat pgfs 666 811 a 666 811 a 666 811 a pgfo save /pgffc{0 setgray}def save [ 2.98883 1.99255 0.3985 1.99255 ] 0.0 setdash /pgfsc{}def/pgffc{}def 0 setgray 0.79701 pgfw 0.0 0.0 moveto 8.50412 0.0 lineto 17.00826 0.0 lineto pgfstr restore restore pgfc 810 811 a 810 811 a 810 811 a pgfo save save save [0.7 0.0 0.0 0.7 1.53424 -2.72835 ] concat pgfs 810 811 a /pgfsc{}def/pgffc{}def 810 811 a 810 811 a 0 setgray 810 811 a Fv(b)s(order)29 b(function)g(for)i Ft(t)p Fv(\(0\))26 b(=)1896 775 y Fp(1)p 1877 790 75 4 v 1877 842 a Fr(M)810 811 y pgfr restore restore restore pgfc 666 896 a 666 896 a 666 896 a pgfo save /pgffc{0 setgray}def save [ ] 0.0 setdash /pgfsc{}def/pgffc{}def 0 setgray 0.79701 pgfw 0.0 0.0 moveto 8.50412 0.0 lineto 17.00826 0.0 lineto pgfstr restore restore pgfc 810 896 a 810 896 a 810 896 a pgfo save save save [0.7 0.0 0.0 0.7 1.53424 -2.36264 ] concat pgfs 810 896 a /pgfsc{}def/pgffc{}def 810 896 a 810 896 a 0 setgray 810 896 a Ft(u)862 910 y Fp(1)902 896 y Fv(\()p Ft(i)p Fv(\))g(=)f Ft(u)1177 863 y Fm(00)1177 920 y Fp(1)810 896 y pgfr restore restore restore pgfc 666 977 a 666 977 a 666 977 a pgfo save /pgffc{0 setgray}def save 1 setlinecap [ 0.0 2.83466 ] 0.0 setdash /pgfsc{}def/pgffc{}def 1 0 0 setrgbcolor 0.79701 pgfw 0.0 0.0 moveto 8.50412 0.0 lineto 17.00826 0.0 lineto pgfstr restore restore pgfc 810 977 a 810 977 a 810 977 a pgfo save save save [0.7 0.0 0.0 0.7 1.53424 -2.29092 ] concat pgfs 810 977 a /pgfsc{}def/pgffc{}def 810 977 a 810 977 a 0 setgray 810 977 a Ft(v)854 991 y Fp(1)894 977 y Fv(\()p Ft(i)p Fv(\))31 b(for)f Ft(u)1217 991 y Fp(1)1257 977 y Fv(\()p Ft(i)p Fv(\))c Fs(\034)f Fv(1)810 977 y pgfr restore restore restore pgfc 666 1058 a 666 1058 a 666 1058 a pgfo save /pgffc{0 setgray}def save 1 setlinecap [ 0.0 1.13387 ] 0.0 setdash /pgfsc{}def/pgffc{}def 0.07 0.93 0 setrgbcolor 0.79701 pgfw 0.0 0.0 moveto 8.50412 0.0 lineto 17.00826 0.0 lineto pgfstr restore restore pgfc 810 1058 a 810 1058 a 810 1058 a pgfo save save save [0.7 0.0 0.0 0.7 1.53424 -2.36264 ] concat pgfs 810 1058 a /pgfsc{}def/pgffc{}def 810 1058 a 810 1058 a 0 setgray 810 1058 a Ft(v)854 1072 y Fp(1)894 1058 y Fv(\()p Ft(i)p Fv(\))31 b(for)f Ft(u)1217 1072 y Fp(1)1257 1058 y Fv(\()p Ft(i)p Fv(\))c(=)f Ft(u)1532 1025 y Fm(00)1532 1082 y Fp(1)1595 1058 y Fs(\000)20 b Ft(")810 1058 y pgfr restore restore restore pgfc 666 1139 a 666 1139 a 666 1139 a pgfo save /pgffc{0 setgray}def save [ 2.98883 2.98883 ] 0.0 setdash [ 2.98883 0.24907 ] 0.0 setdash /pgfsc{}def/pgffc{}def 1 0 0 setrgbcolor 0.79701 pgfw 0.0 0.0 moveto 8.50412 0.0 lineto 17.00826 0.0 lineto pgfstr restore restore pgfc 810 1139 a 810 1139 a 810 1139 a pgfo save save save [0.7 0.0 0.0 0.7 1.53424 -2.29092 ] concat pgfs 810 1139 a /pgfsc{}def/pgffc{}def 810 1139 a 810 1139 a 0 setgray 810 1139 a Ft(v)854 1153 y Fp(2)894 1139 y Fv(\()p Ft(i)p Fv(\))31 b(for)f Ft(u)1217 1153 y Fp(1)1257 1139 y Fv(\()p Ft(i)p Fv(\))c Fs(\034)f Fv(1)810 1139 y pgfr restore restore restore pgfc 666 1220 a 666 1220 a 666 1220 a pgfo save /pgffc{0 setgray}def save [ 2.98883 2.98883 ] 0.0 setdash [ 2.98883 7.97021 ] 0.0 setdash /pgfsc{}def/pgffc{}def 0.07 0.93 0 setrgbcolor 0.79701 pgfw 0.0 0.0 moveto 8.50412 0.0 lineto 17.00826 0.0 lineto pgfstr restore restore pgfc 810 1220 a 810 1220 a 810 1220 a pgfo save save save [0.7 0.0 0.0 0.7 1.53424 -2.36264 ] concat pgfs 810 1220 a /pgfsc{}def/pgffc{}def 810 1220 a 810 1220 a 0 setgray 810 1220 a Ft(v)854 1234 y Fp(2)894 1220 y Fv(\()p Ft(i)p Fv(\))31 b(for)f Ft(u)1217 1234 y Fp(1)1257 1220 y Fv(\()p Ft(i)p Fv(\))c(=)f Ft(u)1532 1187 y Fm(00)1532 1245 y Fp(1)1595 1220 y Fs(\000)20 b Ft(")810 1220 y pgfr restore restore restore pgfc 666 1302 a 666 1302 a 666 1302 a pgfo save /pgffc{0 setgray}def save [ 2.98883 1.99255 0.3985 1.99255 ] 0.0 setdash [ 5.47952 7.97021 ] 0.0 setdash /pgfsc{}def/pgffc{}def 1 0.5 0 setrgbcolor 0.79701 pgfw 0.0 0.0 moveto 8.50412 0.0 lineto 17.00826 0.0 lineto pgfstr restore restore pgfc 810 1302 a 810 1302 a 810 1302 a pgfo save save save [0.7 0.0 0.0 0.7 1.53424 -2.36264 ] concat pgfs 810 1302 a /pgfsc{}def/pgffc{}def 810 1302 a 810 1302 a 0 setgray 810 1302 a Ft(v)854 1316 y Fp(2)894 1302 y Fv(\()p Ft(i)p Fv(\))31 b(for)f Ft(u)1217 1316 y Fp(1)1257 1302 y Fv(\()p Ft(i)p Fv(\))c(=)f Ft(u)1532 1269 y Fm(00)1532 1326 y Fp(1)1595 1302 y Fv(+)20 b Ft(")810 1302 y pgfr restore restore restore pgfc 666 1384 a 666 1384 a 666 1384 a pgfo save /pgffc{0 setgray}def save 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pgfc 2701 1139 a 2701 1139 a 2701 1139 a pgfo save /pgffc{0 setgray}def save 1 setlinecap [ 0.0 4.72444 ] 0.0 setdash /pgfsc{}def/pgffc{}def 0.91 0.09 0 setrgbcolor 0.79701 pgfw 0.0 0.0 moveto 8.50412 0.0 lineto 17.00826 0.0 lineto pgfstr restore restore pgfc 2845 1139 a 2845 1139 a 2845 1139 a pgfo save save save [0.7 0.0 0.0 0.7 1.53424 -2.36264 ] concat pgfs 2845 1139 a /pgfsc{}def/pgffc{}def 2845 1139 a 2845 1139 a 0 setgray 2845 1139 a Ft(u)p Fv(\()p Ft(i)p Fv(\))32 b(for)e Ft(u)3221 1153 y Fp(1)3260 1139 y Fv(\()p Ft(i)p Fv(\))d Fs(\034)e Ft(u)3556 1106 y Fm(0)3556 1163 y Fp(1)2845 1139 y pgfr restore restore restore pgfc 2701 1220 a 2701 1220 a 2701 1220 a pgfo save /pgffc{0 setgray}def save 1 setlinecap [ 0.0 1.13387 ] 0.0 setdash /pgfsc{}def/pgffc{}def 0.05 0.95 0 setrgbcolor 0.79701 pgfw 0.0 0.0 moveto 8.50412 0.0 lineto 17.00826 0.0 lineto pgfstr restore restore pgfc 2845 1220 a 2845 1220 a 2845 1220 a pgfo save save save [0.7 0.0 0.0 0.7 1.53424 -2.36264 ] concat pgfs 2845 1220 a /pgfsc{}def/pgffc{}def 2845 1220 a 2845 1220 a 0 setgray 2845 1220 a Ft(u)p Fv(\()p Ft(i)p Fv(\))32 b(for)e Ft(u)3221 1234 y Fp(1)3260 1220 y Fv(\()p Ft(i)p Fv(\))d(=)d Ft(u)3535 1187 y Fm(0)3535 1245 y Fp(1)3595 1220 y Fs(\000)c Ft(")2845 1220 y pgfr restore restore restore pgfc 2701 1302 a 2701 1302 a 2701 1302 a pgfo save /pgffc{0 setgray}def save [ 2.98883 1.99255 0.3985 1.99255 ] 0.0 setdash [ 5.47952 7.97021 ] 0.0 setdash /pgfsc{}def/pgffc{}def 1 0.5 0 setrgbcolor 0.79701 pgfw 0.0 0.0 moveto 8.50412 0.0 lineto 17.00826 0.0 lineto pgfstr restore restore pgfc 2845 1302 a 2845 1302 a 2845 1302 a pgfo save save save [0.7 0.0 0.0 0.7 1.53424 -2.36264 ] concat pgfs 2845 1302 a /pgfsc{}def/pgffc{}def 2845 1302 a 2845 1302 a 0 setgray 2845 1302 a Ft(u)p Fv(\()p Ft(i)p Fv(\))32 b(for)e Ft(u)3221 1316 y Fp(1)3260 1302 y Fv(\()p Ft(i)p Fv(\))d(=)d Ft(u)3535 1269 y Fm(0)3535 1326 y Fp(1)3595 1302 y Fv(+)c Ft(")2845 1302 y pgfr restore restore restore pgfc 2701 1384 a 2701 1384 a 2701 1384 a pgfo save /pgffc{0 setgray}def save [ 2.98883 1.99255 0.3985 1.99255 ] 0.0 setdash [ 5.47952 0.24907 ] 0.0 setdash /pgfsc{}def/pgffc{}def 0.1 0.05 0.9 setrgbcolor 0.79701 pgfw 0.0 0.0 moveto 8.50412 0.0 lineto 17.00826 0.0 lineto pgfstr restore restore pgfc 2845 1384 a 2845 1384 a 2845 1384 a pgfo save save save [0.7 0.0 0.0 0.7 1.53424 -2.36264 ] concat pgfs 2845 1384 a /pgfsc{}def/pgffc{}def 2845 1384 a 2845 1384 a 0 setgray 2845 1384 a Ft(u)p Fv(\()p Ft(i)p Fv(\))32 b(for)e Ft(u)3221 1398 y Fp(1)3260 1384 y Fv(\()p Ft(i)p Fv(\))d Fs(\035)e Ft(u)3556 1351 y Fm(0)3556 1408 y Fp(1)2845 1384 y pgfr restore restore restore pgfc 2698 1384 a pgfr restore restore restore newpath restore pgfc 2366 1658 a pgfr restore save restore save restore restore newpath restore pgfc 0 TeXcolorgray 2035 2110 a Fv(Figure)h(6:)40 b(Illustration)24 b(for)j(lev)m(el)g (sets)g(of)g(the)g(function)2035 2245 y Ft(u)p Fv(\()p Ft(i)p Fv(\))k(for)f Ft(\034)36 b Fv(=)25 b(0)p Ft(:)p Fv(15,)32 b Ft(\013)26 b Fv(=)f(2)p Ft(\034)10 b Fv(,)31 b(and)e Ft(\014)i Fv(=)25 b(2)20 b Fs(\000)g Ft(\013)p Fv(.)p 0 TeXcolorgray 0 TeXcolorgray 0 2524 a SDict begin H.S end 0 2524 a 0 2524 a SDict begin 16.31996 H.A end 0 2524 a 0 2524 a SDict begin [/View [/XYZ H.V]/Dest (proposition.4) cvn /DEST pdfmark end 0 2524 a 0 TeXcolorgray Fj(Prop)s(osition)39 b(4.)p 0 TeXcolorgray 43 w Fq(If)c(ther)-5 b(e)35 b(is)g(a)g(cr)-5 b(ossing)36 b(and)g Ft(u)1782 2538 y Fp(1)1821 2524 y Fv(\()p Ft(i)23 b Fs(\000)e Fv(1\))30 b Fs(\025)f Ft(u)2264 2491 y Fm(00)2264 2548 y Fp(1)2306 2524 y Fq(,)35 b(then)h Ft(v)2618 2538 y Fp(2)2657 2524 y Fv(\()p Ft(i)p Fv(\))23 b Fs(\000)e Ft(v)2917 2538 y Fp(2)2957 2524 y Fv(\()p Ft(i)h Fs(\000)g Fv(1\))30 b Fs(\025)f(\000)15 b Fv(log)h Ft(\014)5 b Fq(.)48 b(If)35 b(ther)-5 b(e)0 2660 y(is)33 b(no)g(cr)-5 b(ossing,)33 b(then)74 3016 y Ft(v)118 3030 y Fp(2)157 3016 y Fv(\()p Ft(i)p Fv(\))22 b Fs(\000)e Ft(v)415 3030 y Fp(2)454 3016 y Fv(\()p Ft(i)h Fs(\000)f Fv(1\))26 b Fs(\025)834 2748 y Fk(8)834 2830 y(>)834 2857 y(>)834 2885 y(>)834 2912 y(<)834 3076 y(>)834 3103 y(>)834 3130 y(>)834 3157 y(:)915 2864 y Fv(log)16 b Ft(\014)5 b(;)109 b Fq(if)32 b(the)h(tr)-5 b(ansmission)35 b(is)e(c)-5 b(orr)g(e)g(ct)r Ft(;)915 3026 y Fv(log)16 b Ft(\013;)109 b Fq(if)32 b(the)h(tr)-5 b(ansmission)36 b(is)c(inc)-5 b(orr)g(e)g(ct)34 b(and)g Ft(v)2706 3040 y Fp(2)2746 3026 y Fv(\()p Ft(i)20 b Fs(\000)g Fv(1\))26 b Ft(<)f Fv(log)17 b Ft(\014)5 b(;)915 3189 y Fs(\000)15 b Fv(log)h Ft(\014)38 b Fv(\()p Fs(\025)25 b Fv(log)17 b Ft(\013)p Fv(\))p Ft(;)109 b Fq(if)32 b(the)h(tr)-5 b(ansmission)35 b(is)e(inc)-5 b(orr)g(e)g(ct)34 b(and)g Ft(v)3180 3203 y Fp(2)3219 3189 y Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))26 b Fs(\025)f Fv(log)17 b Ft(\014)5 b(:)141 3373 y Fv(Plen)m(t)m(y)30 b(of)g(incorrect)g(transmissions)d(implies)g(that) j(w)m(e)h(ha)m(v)m(e)g(a)f(lot)f(of)h(energy)g(left.)41 b(No)m(w)30 b(w)m(e)g(can)g(use)g(again)0 3509 y(Prop)s(osition)496 3509 y SDict begin H.S end 496 3509 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 542 3450 a SDict begin H.R end 542 3450 a 542 3509 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.2) cvn H.B /ANN pdfmark end 542 3509 a Fv(,)39 b(since)d(it)h(is)f (not)h(p)s(ossible)e(that)j(a)f(crossing)f(o)s(ccurs)h(when)f(the)i (true)f(segmen)m(t)h(is)e(cen)m(tral)i(and)0 3644 y(th)m(us,)30 b(conclude)g(that)h(the)g(function)e Ft(v)k Fv(tak)m(es)f(a)f(large)f (v)-5 b(alue)30 b(after)h Ft(n)f Fv(transmissions.)0 3831 y SDict begin H.S end 0 3831 a 0 3831 a SDict begin 16.31996 H.A end 0 3831 a 0 3831 a SDict begin [/View [/XYZ H.V]/Dest (example.2) cvn /DEST pdfmark end 0 3831 a 0 TeXcolorgray Fj(Example)37 b(2.)p 0 TeXcolorgray 44 w Fq(Figur)-5 b(es)886 3831 y SDict begin H.S end 886 3831 a 0 TeXcolorgray Fq(5)p 0 TeXcolorgray 932 3772 a SDict begin H.R end 932 3772 a 932 3831 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (figure.caption.5) cvn H.B /ANN pdfmark end 932 3831 a 35 w Fq(and)1147 3831 y SDict begin H.S end 1147 3831 a 0 TeXcolorgray Fq(6)p 0 TeXcolorgray 1193 3772 a SDict begin H.R end 1193 3772 a 1193 3831 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (figure.caption.5) cvn H.B /ANN pdfmark end 1193 3831 a 36 w Fq(il)5 b(lustr)-5 b(ate)36 b(di\013er)-5 b(ent)36 b(level)g(sets)f(of)h(the)g(functions)f Ft(v)s Fv(\()p Ft(i)p Fv(\))i Fq(and)g Ft(u)p Fv(\()p Ft(i)p Fv(\))f Fq(if)f(they)h(ar)-5 b(e)0 3966 y(tr)g(e)g(ate)g(d)26 b(as)e(function)g(on)g Ft(l)r Fv(\()p Ft(i)p Fv(\))h Fq(and)g Ft(r)s Fv(\()p Ft(i)p Fv(\))p Fq(.)39 b(A)23 b(level)h(set)f(of)h Ft(v)s Fv(\()p Ft(i)p Fv(\))h Fq(is)f(a)g(set)g (of)g(p)-5 b(airs)25 b Fv(\()p Ft(l)r Fv(\()p Ft(i)p Fv(\))p Ft(;)15 b(r)s Fv(\()p Ft(i)p Fv(\)\))26 b Fq(such)e(that)h Ft(v)s Fv(\()p Ft(i)p Fv(\))i(=)e Ft(const)p Fq(.)0 4102 y(Note)g(that)i(we)f(use)f(the)h(same)g(c)-5 b(olor)27 b(for)f(the)g(same)g(level)g(sets)g(in)f(the)h(gr)-5 b(aphs.)41 b(F)-7 b(or)27 b(high)f(level)f(sets)h(of)g(the)g(function)0 4237 y Ft(v)s Fv(\()p Ft(i)p Fv(\))p Fq(,)38 b(we)e(ar)-5 b(e)37 b(close)g(to)f(the)h(origin)f(of)g(the)h(gr)-5 b(aph,)38 b(which)f(c)-5 b(orr)g(esp)g(onds)40 b(to)c Ft(l)r Fv(\()p Ft(i)p Fv(\))h Fq(and)g Ft(r)s Fv(\()p Ft(i)p Fv(\))g Fq(b)-5 b(eing)36 b(smal)5 b(l)37 b(and)g(the)0 4373 y Ft(t)p Fv(\()p Ft(i)p Fv(\))j Fq(very)e(lar)-5 b(ge.)62 b(In)39 b(c)-5 b(ontr)g(ast,)42 b(for)e(high)f(level)g(sets)g (of)g(the)g(function)g Ft(u)p Fv(\()p Ft(i)p Fv(\))p Fq(,)i(we)e(c)-5 b(an)40 b(only)g(claim)f(that)h(we)f(ar)-5 b(e)0 4508 y(close)33 b(to)h(an)g(axis,)f(i.e.,)g(either)g Ft(l)r Fv(\()p Ft(i)p Fv(\))h Fq(or)g Ft(r)s Fv(\()p Ft(i)p Fv(\))f Fq(is)g(smal)5 b(l.)44 b(Notic)-5 b(e)34 b(that)g(in)f(c)-5 b(ase)33 b Ft(T)13 b Fv(\()p Ft(i)p Fv(\))34 b Fq(is)f(c)-5 b(entr)g(al,)35 b(we)e(ne)-5 b(e)g(d)34 b(to)f(have)0 4644 y Ft(l)r Fv(\()p Ft(i)p Fv(\))p Ft(;)15 b(r)s Fv(\()p Ft(i)p Fv(\))32 b Ft(<)457 4608 y Fp(1)p 457 4623 36 4 v 457 4675 a(2)503 4644 y Fq(.)49 b(Combining)35 b(the)h(two)g(\014gur)-5 b(es)36 b(for)f(the)h(level)f(sets)g(of)h Ft(v)s Fv(\()p Ft(i)p Fv(\))g Fq(and)g Ft(u)p Fv(\()p Ft(i)p Fv(\))p Fq(,)h(it)e(b)-5 b(e)g(c)g(omes)36 b(cle)-5 b(ar)36 b(that)h(we)0 4779 y(ne)-5 b(e)g(d)36 b(to)f(pr)-5 b(ove)36 b(that)g(the)f(function)g Ft(v)s Fv(\()p Ft(n)p Fv(\))g Fq(is)g(lar)-5 b(ge)35 b(in)g(or)-5 b(der)36 b(to)g(c)-5 b(onclude)35 b(that)h(after)g Ft(n)e Fq(tr)-5 b(ansmissions)37 b Ft(T)13 b Fv(\()p Ft(n)p Fv(\))35 b Fq(is)0 4915 y(c)-5 b(entr)g(al.)141 5101 y Fv(Man)m(y)27 b(tec)m(hnical)e(details)g(w)m(ere)h(hidden)e(in)g (this)h(discussion.)36 b(In)25 b(order)h(to)g(giv)m(e)g(a)g(formal)f (pro)s(of)g(of)h(Lemma)3919 5101 y SDict begin H.S end 3919 5101 a 0 TeXcolorgray Fv(7)p 0 TeXcolorgray 3965 5043 a SDict begin H.R end 3965 5043 a 3965 5101 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.7) cvn H.B /ANN pdfmark end 3965 5101 a Fv(,)0 5237 y(w)m(e)31 b(need)f(some)h(additional)d(prop) s(ositions.)p 0 TeXcolorgray 1950 5486 a(17)p 0 TeXcolorgray eop end end %%Page: 18 18 TeXDict begin HPSdict begin 18 17 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -174 a SDict begin H.S end -8 -174 a -8 -174 a SDict begin H.R end -8 -174 a -8 -174 a SDict begin [/View [/XYZ H.V]/Dest (page.18) cvn /DEST pdfmark end -8 -174 a 0 TeXcolorgray 0 TeXcolorgray 0 108 a SDict begin H.S end 0 108 a 0 108 a SDict begin 16.31996 H.A end 0 108 a 0 108 a SDict begin [/View [/XYZ H.V]/Dest (proposition.5) cvn /DEST pdfmark end 0 108 a 0 TeXcolorgray Fj(Prop)s(osition)36 b(5.)p 0 TeXcolorgray 42 w Fq(We)c(have)h(the)g (fol)5 b(lowing)34 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/pgffc{0.3 setgray}def 5.45462 16.36386 -8.18193 -13.63657 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 -8.18193 -10.90926 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 5.45462 16.36386 13.63658 -13.63657 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 13.63658 -10.90926 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 32.72775 66.50159 -33.2508 -101.4043 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 -27.27313 -87.70712 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 80 w Fv(Prop.)1238 365 y SDict begin H.S end 1238 365 a 0 TeXcolorgray Fv(8)p 0 TeXcolorgray 1284 306 a SDict begin H.R end 1284 306 a 1284 365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.8) cvn H.B /ANN pdfmark end 1284 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def 5.45462 16.36386 -8.18193 -98.67699 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 -8.18193 -95.94968 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 32.72775 66.50159 68.79744 -44.71068 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 74.7751 -31.0135 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 80 w Fv(Prop.)1238 365 y SDict begin H.S end 1238 365 a 0 TeXcolorgray Fv(3)p 0 TeXcolorgray 1284 306 a SDict begin H.R end 1284 306 a 1284 365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.3) cvn H.B /ANN pdfmark end 1284 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.7 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.7 setgray}def 5.45462 16.36386 72.04779 -41.98337 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 72.04779 -39.25606 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def 5.45462 16.36386 93.86632 -41.98337 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 93.86632 -39.25606 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 5.45462 16.36386 115.68483 -41.98337 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 115.68483 -39.25606 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 32.72775 66.50159 68.79744 -101.4043 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 74.7751 -87.70712 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 58 w Fv(Prop.)1216 365 y SDict begin H.S end 1216 365 a 0 TeXcolorgray Fv(10)p 0 TeXcolorgray 1306 306 a SDict begin H.R end 1306 306 a 1306 365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.10) cvn H.B /ANN pdfmark end 1306 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.7 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.7 setgray}def 5.45462 16.36386 72.04779 -98.67699 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 72.04779 -95.94968 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def 5.45462 16.36386 93.86632 -98.67699 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 93.86632 -95.94968 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 5.45462 16.36386 115.68483 -98.67699 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 115.68483 -95.94968 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 32.72775 66.50159 68.79744 -158.09792 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 74.7751 -144.40074 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 80 w Fv(Prop.)1238 365 y SDict begin H.S end 1238 365 a 0 TeXcolorgray Fv(9)p 0 TeXcolorgray 1284 306 a SDict begin H.R end 1284 306 a 1284 365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.9) cvn H.B /ANN pdfmark end 1284 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def 5.45462 16.36386 93.86632 -155.3706 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 93.86632 -152.6433 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 5.45462 16.36386 115.68483 -155.3706 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 115.68483 -152.6433 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 32.72775 66.50159 68.79744 -214.79155 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 74.7751 -201.09436 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 80 w Fv(Prop.)1238 365 y SDict begin H.S end 1238 365 a 0 TeXcolorgray Fv(4)p 0 TeXcolorgray 1284 306 a SDict begin H.R end 1284 306 a 1284 365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.4) cvn H.B /ANN pdfmark end 1284 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def 5.45462 16.36386 93.86632 -212.06422 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 93.86632 -209.33691 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 32.72775 66.50159 179.35025 -163.7672 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 185.32791 -150.07002 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 80 w Fv(Prop.)1238 365 y SDict begin H.S end 1238 365 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 1284 306 a SDict begin H.R end 1284 306 a 1284 365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.2) cvn H.B /ANN pdfmark end 1284 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def 5.45462 16.36386 204.41913 -161.03989 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 204.41913 -158.31258 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 5.45462 16.36386 226.23764 -161.03989 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 226.23764 -158.31258 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.7 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.7 setgray}def 32.72775 66.50159 -33.2508 -172.27132 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 -27.27313 -158.63475 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a 1 .pgfsetfillopacityalpha 909 365 a 940 365 a 1 .pgfsetstrokeopacityalpha 940 365 a 116 w Fv(Eq.)g(\()1257 365 y SDict begin H.S end 1257 365 a 0 TeXcolorgray Fv(3)p 0 TeXcolorgray 1303 306 a SDict begin H.R end 1303 306 a 1303 365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.3) cvn H.B /ANN pdfmark end 1303 365 a Fv(\))909 365 y pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def 32.72775 66.50159 -33.2508 -207.70483 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 -27.27313 -194.00766 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 80 w Fv(Prop.)1238 365 y SDict begin H.S end 1238 365 a 0 TeXcolorgray Fv(5)p 0 TeXcolorgray 1284 306 a SDict begin H.R end 1284 306 a 1284 365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.5) cvn H.B /ANN pdfmark end 1284 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 32.72775 66.50159 -33.2508 -243.13835 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 -27.27313 -229.44118 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a 1 .pgfsetfillopacityalpha 909 365 a 940 365 a 1 .pgfsetstrokeopacityalpha 940 365 a 110 w Fv(Prop.)1299 365 y SDict begin H.S end 1299 365 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 1344 306 a SDict begin H.R end 1344 306 a 1344 365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.6) cvn H.B /ANN pdfmark end 1344 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 32.72775 66.50159 306.9109 -101.4043 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 312.88857 -88.76773 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a 1 .pgfsetfillopacityalpha 909 365 a 940 365 a 1 .pgfsetstrokeopacityalpha 940 365 a 72 w Fv(Lemma)1337 365 y SDict begin H.S end 1337 365 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 1382 306 a SDict begin H.R end 1382 306 a 1382 365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.6) cvn H.B /ANN pdfmark end 1382 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 32.72775 66.50159 306.9109 -158.09792 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 312.88857 -145.46135 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a 1 .pgfsetfillopacityalpha 909 365 a 940 365 a 1 .pgfsetstrokeopacityalpha 940 365 a 72 w Fv(Lemma)1337 365 y SDict begin H.S end 1337 365 a 0 TeXcolorgray Fv(7)p 0 TeXcolorgray 1382 306 a SDict begin H.R end 1382 306 a 1382 365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.7) cvn H.B /ANN pdfmark end 1382 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.7 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.7 setgray}def 5.45462 16.36386 310.16125 -155.3706 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 310.16125 -152.6433 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{0.3 setgray}def 5.45462 16.36386 331.97978 -155.3706 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 331.97978 -152.6433 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 5.45462 16.36386 353.7983 -155.3706 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 353.7983 -152.6433 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 909 365 a pgfr restore restore save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def save /pgfsc{0 setgray}def 0.79701 pgfw /pgffc{1 setgray}def 32.72775 66.50159 391.95134 -129.75111 pgfe gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath restore save [1.0 0.0 0.0 1.0 397.929 -117.17516 ] concat pgfs 909 365 a /pgfsc{}def/pgffc{}def 909 365 a 909 365 a 0 setgray 909 365 a 13 w Fv(Theorem)1306 365 y SDict begin H.S end 1306 365 a 0 TeXcolorgray Fv(8)p 0 TeXcolorgray 1351 306 a SDict begin H.R end 1351 306 a 1351 365 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (theorem.8) cvn H.B /ANN pdfmark end 1351 365 a 909 365 a pgfr restore restore save 0.79701 pgfw 33.64928 8.38118 moveto 152.7058 8.38118 lineto 152.7058 -139.02211 lineto 178.24437 -139.02211 lineto pgfstr save [1.0 0.0 0.0 1.0 178.24437 -139.02211 ] concat save 0.6376 pgfw [ ] 0.0 setdash 1 setlinecap 1 setlinejoin -1.55415 2.0722 moveto -1.42464 1.29512 0.0 0.1295 0.38852 0.0 curveto 0.0 -0.1295 -1.42464 -1.29512 -1.55415 -2.0722 curveto pgfstr restore restore restore save 0.79701 pgfw 33.64928 -76.65923 moveto 39.31856 -76.65923 lineto 39.31856 -36.72798 lineto 67.69156 -36.72798 lineto pgfstr save [1.0 0.0 0.0 1.0 67.69156 -36.72798 ] concat save 0.6376 pgfw [ ] 0.0 setdash 1 setlinecap 1 setlinejoin -1.55415 2.0722 moveto -1.42464 1.29512 0.0 0.1295 0.38852 0.0 curveto 0.0 -0.1295 -1.42464 -1.29512 -1.55415 -2.0722 curveto pgfstr restore restore restore save 0.79701 pgfw 33.64928 -93.4216 moveto 67.69156 -93.4216 lineto pgfstr save [1.0 0.0 0.0 1.0 67.69156 -93.4216 ] concat save 0.6376 pgfw [ ] 0.0 setdash 1 setlinecap 1 setlinejoin -1.55415 2.0722 moveto -1.42464 1.29512 0.0 0.1295 0.38852 0.0 curveto 0.0 -0.1295 -1.42464 -1.29512 -1.55415 -2.0722 curveto pgfstr restore restore restore save 0.79701 pgfw 33.64929 0.0 moveto 53.49197 0.0 lineto 53.49197 -19.9656 lineto 67.69156 -19.9656 lineto pgfstr save [1.0 0.0 0.0 1.0 67.69156 -19.9656 ] concat save 0.6376 pgfw [ ] 0.0 setdash 1 setlinecap 1 setlinejoin -1.55415 2.0722 moveto -1.42464 1.29512 0.0 0.1295 0.38852 0.0 curveto 0.0 -0.1295 -1.42464 -1.29512 -1.55415 -2.0722 curveto pgfstr restore restore restore save 0.79701 pgfw 33.64928 -8.38118 moveto 47.82268 -8.38118 lineto 47.82268 -76.65923 lineto 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257.138 lineto 95.78712 217.14519 lineto 162.83829 149.15749 lineto 56.8355 253.56789 lineto 96.62056 211.076 lineto 164.25504 138.83997 lineto 59.24406 236.02812 lineto 100.71486 181.25833 lineto 171.21516 88.1497 lineto 71.07635 149.85454 lineto 21.3229 254.75269 lineto 36.2489 213.09016 lineto 61.62311 142.26396 lineto 104.75925 42.67917 lineto 31.42766 72.55467 lineto 9.42844 123.343 lineto 2.8284 209.68314 lineto 4.80832 136.47162 lineto 8.17448 40.94156 lineto 13.8964 12.2823 lineto 4.16881 20.88011 lineto pgfstr restore save 0.5 0 0.5 TeXcolorrgb 243.3141 70.92375 moveto 244.31038 70.92375 moveto 244.31038 71.474 243.86433 71.92003 243.3141 71.92003 curveto 242.76387 71.92003 242.31783 71.474 242.31783 70.92375 curveto 242.31783 70.37352 242.76387 69.92747 243.3141 69.92747 curveto 243.86433 69.92747 244.31038 70.37352 244.31038 70.92375 curveto closepath 243.3141 70.92375 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 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119.57413 193.64464 119.57413 curveto 194.19487 119.57413 194.64091 120.02017 194.64091 120.5704 curveto closepath 193.64464 120.5704 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 109.20654 204.9697 moveto 110.20282 204.9697 moveto 110.20282 205.51993 109.75677 205.96597 109.20654 205.96597 curveto 108.6563 205.96597 108.21027 205.51993 108.21027 204.9697 curveto 108.21027 204.41946 108.6563 203.97342 109.20654 203.97342 curveto 109.75677 203.97342 110.20282 204.41946 110.20282 204.9697 curveto closepath 109.20654 204.9697 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 185.6511 128.45915 moveto 186.64737 128.45915 moveto 186.64737 129.0094 186.20134 129.45543 185.6511 129.45543 curveto 185.10086 129.45543 184.65482 129.0094 184.65482 128.45915 curveto 184.65482 127.90892 185.10086 127.46288 185.6511 127.46288 curveto 186.20134 127.46288 186.64737 127.90892 186.64737 128.45915 curveto closepath 185.6511 128.45915 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 95.61757 218.38058 moveto 96.61385 218.38058 moveto 96.61385 218.93082 96.1678 219.37686 95.61757 219.37686 curveto 95.06732 219.37686 94.62129 218.93082 94.62129 218.38058 curveto 94.62129 217.83034 95.06732 217.38431 95.61757 217.38431 curveto 96.1678 217.38431 96.61385 217.83034 96.61385 218.38058 curveto closepath 95.61757 218.38058 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 162.54985 151.25763 moveto 163.54613 151.25763 moveto 163.54613 151.80786 163.10008 152.2539 162.54985 152.2539 curveto 161.99962 152.2539 161.55357 151.80786 161.55357 151.25763 curveto 161.55357 150.70738 161.99962 150.26135 162.54985 150.26135 curveto 163.10008 150.26135 163.54613 150.70738 163.54613 151.25763 curveto closepath 162.54985 151.25763 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 56.3454 257.138 moveto 57.34167 257.138 moveto 57.34167 257.68825 56.89563 258.13428 56.3454 258.13428 curveto 55.79517 258.13428 55.34912 257.68825 55.34912 257.138 curveto 55.34912 256.58777 55.79517 256.14172 56.3454 256.14172 curveto 56.89563 256.14172 57.34167 256.58777 57.34167 257.138 curveto closepath 56.3454 257.138 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 95.78717 217.14523 moveto 96.78345 217.14523 moveto 96.78345 217.69547 96.3374 218.14151 95.78717 218.14151 curveto 95.23694 218.14151 94.7909 217.69547 94.7909 217.14523 curveto 94.7909 216.595 95.23694 216.14896 95.78717 216.14896 curveto 96.3374 216.14896 96.78345 216.595 96.78345 217.14523 curveto closepath 95.78717 217.14523 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 162.8382 149.15758 moveto 163.83447 149.15758 moveto 163.83447 149.70781 163.38844 150.15385 162.8382 150.15385 curveto 162.28796 150.15385 161.84192 149.70781 161.84192 149.15758 curveto 161.84192 148.60735 162.28796 148.1613 162.8382 148.1613 curveto 163.38844 148.1613 163.83447 148.60735 163.83447 149.15758 curveto closepath 162.8382 149.15758 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 56.83559 253.56789 moveto 57.83186 253.56789 moveto 57.83186 254.11812 57.38582 254.56416 56.83559 254.56416 curveto 56.28535 254.56416 55.83931 254.11812 55.83931 253.56789 curveto 55.83931 253.01764 56.28535 252.57161 56.83559 252.57161 curveto 57.38582 252.57161 57.83186 253.01764 57.83186 253.56789 curveto closepath 56.83559 253.56789 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 96.62051 211.0761 moveto 97.61679 211.0761 moveto 97.61679 211.62634 97.17075 212.07237 96.62051 212.07237 curveto 96.07028 212.07237 95.62424 211.62634 95.62424 211.0761 curveto 95.62424 210.52586 96.07028 210.07982 96.62051 210.07982 curveto 97.17075 210.07982 97.61679 210.52586 97.61679 211.0761 curveto closepath 96.62051 211.0761 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 164.2549 138.84001 moveto 165.25117 138.84001 moveto 165.25117 139.39024 164.80513 139.83629 164.2549 139.83629 curveto 163.70465 139.83629 163.25862 139.39024 163.25862 138.84001 curveto 163.25862 138.28978 163.70465 137.84373 164.2549 137.84373 curveto 164.80513 137.84373 165.25117 138.28978 165.25117 138.84001 curveto closepath 164.2549 138.84001 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 59.24397 236.02798 moveto 60.24025 236.02798 moveto 60.24025 236.57823 59.7942 237.02426 59.24397 237.02426 curveto 58.69374 237.02426 58.2477 236.57823 58.2477 236.02798 curveto 58.2477 235.47775 58.69374 235.03171 59.24397 235.03171 curveto 59.7942 235.03171 60.24025 235.47775 60.24025 236.02798 curveto closepath 59.24397 236.02798 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 100.71472 181.25829 moveto 101.711 181.25829 moveto 101.711 181.80852 101.26497 182.25456 100.71472 182.25456 curveto 100.16449 182.25456 99.71844 181.80852 99.71844 181.25829 curveto 99.71844 180.70805 100.16449 180.26201 100.71472 180.26201 curveto 101.26497 180.26201 101.711 180.70805 101.711 181.25829 curveto closepath 100.71472 181.25829 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 171.21507 88.1497 moveto 172.21135 88.1497 moveto 172.21135 88.69994 171.7653 89.14598 171.21507 89.14598 curveto 170.66484 89.14598 170.2188 88.69994 170.2188 88.1497 curveto 170.2188 87.59946 170.66484 87.15343 171.21507 87.15343 curveto 171.7653 87.15343 172.21135 87.59946 172.21135 88.1497 curveto closepath 171.21507 88.1497 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 71.07625 149.85449 moveto 72.07253 149.85449 moveto 72.07253 150.40474 71.62648 150.85077 71.07625 150.85077 curveto 70.52602 150.85077 70.07997 150.40474 70.07997 149.85449 curveto 70.07997 149.30426 70.52602 148.85822 71.07625 148.85822 curveto 71.62648 148.85822 72.07253 149.30426 72.07253 149.85449 curveto closepath 71.07625 149.85449 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 21.32286 254.75269 moveto 24.31169 254.75269 moveto 24.31169 256.40338 22.97357 257.74152 21.32286 257.74152 curveto 19.67216 257.74152 18.33403 256.40338 18.33403 254.75269 curveto 18.33403 253.10197 19.67216 251.76385 21.32286 251.76385 curveto 22.97357 251.76385 24.31169 253.10197 24.31169 254.75269 curveto closepath 21.32286 254.75269 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb /pgffc{1 setgray}def save /pgfsc{}def/pgffc{}def 0.5 0 0.5 TeXcolorrgb /pgffc{1 setgray}def 48.79323 254.75269 moveto 48.79323 261.24588 43.5296 266.50952 37.03639 266.50952 curveto 30.54318 266.50952 25.27954 261.24588 25.27954 254.75269 curveto 25.27954 248.25948 30.54318 242.99583 37.03639 242.99583 curveto 43.5296 242.99583 48.79323 248.25948 48.79323 254.75269 curveto closepath 37.03639 254.75269 moveto gsave pgffc pgffill grestore newpath 0.5 0 0.5 TeXcolorrgb restore save [1.0 0.0 0.0 1.0 26.38173 252.262 ] concat pgfs 1292 2133 a /pgfsc{}def/pgffc{}def 1292 2133 a 1292 2133 a 0.5 0 0.5 setrgbcolor 1292 2133 a 0.5 0 0.5 TeXcolorrgb Fz(t)p FA(\(16\))p 0.5 0 0.5 TeXcolorrgb 1292 2133 a pgfr restore 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 36.2489 213.09026 moveto 37.24518 213.09026 moveto 37.24518 213.64049 36.79913 214.08653 36.2489 214.08653 curveto 35.69865 214.08653 35.25262 213.64049 35.25262 213.09026 curveto 35.25262 212.54002 35.69865 212.09398 36.2489 212.09398 curveto 36.79913 212.09398 37.24518 212.54002 37.24518 213.09026 curveto closepath 36.2489 213.09026 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 61.62311 142.26407 moveto 62.61938 142.26407 moveto 62.61938 142.81432 62.17336 143.26035 61.62311 143.26035 curveto 61.07288 143.26035 60.62683 142.81432 60.62683 142.26407 curveto 60.62683 141.71384 61.07288 141.26779 61.62311 141.26779 curveto 62.17336 141.26779 62.61938 141.71384 62.61938 142.26407 curveto closepath 61.62311 142.26407 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 104.7593 42.67921 moveto 105.75557 42.67921 moveto 105.75557 43.22945 105.30952 43.67549 104.7593 43.67549 curveto 104.20905 43.67549 103.76302 43.22945 103.76302 42.67921 curveto 103.76302 42.12898 104.20905 41.68294 104.7593 41.68294 curveto 105.30952 41.68294 105.75557 42.12898 105.75557 42.67921 curveto closepath 104.7593 42.67921 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 31.42775 72.55467 moveto 32.42403 72.55467 moveto 32.42403 73.1049 31.978 73.55095 31.42775 73.55095 curveto 30.87752 73.55095 30.43147 73.1049 30.43147 72.55467 curveto 30.43147 72.00444 30.87752 71.5584 31.42775 71.5584 curveto 31.978 71.5584 32.42403 72.00444 32.42403 72.55467 curveto closepath 31.42775 72.55467 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb 9.42833 123.34296 moveto 12.41716 123.34296 moveto 12.41716 124.99367 11.07904 126.33179 9.42833 126.33179 curveto 7.77763 126.33179 6.4395 124.99367 6.4395 123.34296 curveto 6.4395 121.69226 7.77763 120.35413 9.42833 120.35413 curveto 11.07904 120.35413 12.41716 121.69226 12.41716 123.34296 curveto closepath 9.42833 123.34296 moveto gsave pgffc pgffill grestore gsave pgfsc pgfstr grestore newpath 0 TeXcolorgray restore save 0.5 0 0.5 TeXcolorrgb /pgffc{1 setgray}def save /pgfsc{}def/pgffc{}def 0.5 0 0.5 TeXcolorrgb /pgffc{1 setgray}def 36.89871 123.34296 moveto 36.89871 129.83617 31.63507 135.09981 25.14186 135.09981 curveto 18.64865 135.09981 13.38501 129.83617 13.38501 123.34296 curveto 13.38501 116.84976 18.64865 111.58612 25.14186 111.58612 curveto 31.63507 111.58612 36.89871 116.84976 36.89871 123.34296 curveto closepath 25.14186 123.34296 moveto gsave pgffc pgffill grestore newpath 0.5 0 0.5 TeXcolorrgb restore save [1.0 0.0 0.0 1.0 14.4872 120.85226 ] concat pgfs 1292 2133 a /pgfsc{}def/pgffc{}def 1292 2133 a 1292 2133 a 0.5 0 0.5 setrgbcolor 1292 2133 a 0.5 0 0.5 TeXcolorrgb Fz(t)p FA(\(21\))p 0.5 0 0.5 TeXcolorrgb 1292 2133 a pgfr restore 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Ft(v)s Fv(\()p Ft(n)p Fv(\))26 b Fs(\035)f Fv(1)31 b(w)m(e)g(conclude)f(that)h Ft(T)13 b Fv(\()p Ft(n)p Fv(\))30 b(is)f(cen)m(tral)i(\(see)h(Fig.)3883 3138 y SDict begin H.S end 3883 3138 a 0 TeXcolorgray Fv(5)p 0 TeXcolorgray 3929 3079 a SDict begin H.R end 3929 3079 a 3929 3138 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (figure.caption.5) cvn H.B /ANN pdfmark end 3929 3138 a Fv(\).)0 3273 y Fj(Case)e Ft(m)25 b(>)g(m)527 3240 y Fm(0)550 3273 y Fv(.)41 b(Let)31 b Ft(n)834 3287 y Fp(2)903 3273 y Fv(b)s(e)f(the)g Ft(m)1263 3240 y Fm(0)1286 3273 y Fv(th)h(smallest)e(elemen)m(t)i(of)g Ft(I)7 b Fv(.)40 b(F)-8 b(rom)31 b(Prop)s(osition)3033 3273 y SDict begin H.S end 3033 3273 a 0 TeXcolorgray Fv(5)p 0 TeXcolorgray 3078 3215 a SDict begin H.R end 3078 3215 a 3078 3273 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.5) cvn H.B /ANN pdfmark end 3078 3273 a Fv(.)3103 3273 y SDict begin H.S end 3103 3273 a 0 TeXcolorgray Fv(5)p 0 TeXcolorgray 3149 3215 a SDict begin H.R end 3149 3215 a 3149 3273 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.5) cvn H.B /ANN pdfmark end 3149 3273 a 30 w Fv(and)f(the)h(de\014nition) d(of)0 3409 y(function)33 b Ft(u)g Fv(it)h(follo)m(ws)f(that)h Ft(u)1094 3423 y Fp(2)1134 3409 y Fv(\()p Ft(n)1224 3423 y Fp(2)1263 3409 y Fv(\))e Fs(\025)f Ft(u)p Fv(\()p Ft(n)1574 3423 y Fp(2)1613 3409 y Fv(\),)k(and)f Ft(u)p Fv(\()p Ft(n)2031 3423 y Fp(1)2070 3409 y Fv(\))e Fs(\025)e Ft(u)2290 3423 y Fp(1)2330 3409 y Fv(\()p Ft(n)2420 3423 y Fp(1)2459 3409 y Fv(\))i(=)f(log)q(\()p Ft(t)p Fv(\()p Ft(n)2903 3423 y Fp(1)2943 3409 y Fv(\))p Ft(=x)p Fv(\()p Ft(n)3165 3423 y Fp(1)3205 3409 y Fv(\)\)\))h Fs(\025)f Fv(log)q(\(2)p Ft(t)p Fv(\()p Ft(n)3764 3423 y Fp(1)3804 3409 y Fv(\)\))k(as)0 3544 y Ft(x)p Fv(\()p Ft(n)142 3558 y Fp(1)181 3544 y Fv(\))26 b Fs(\024)f Fv(1)p Ft(=)p Fv(2.)42 b(Com)m(bining)28 b(this)h(together)j(with)d(Prop)s(osition)2235 3544 y SDict begin H.S end 2235 3544 a 0 TeXcolorgray Fv(3)p 0 TeXcolorgray 2280 3486 a SDict begin H.R end 2280 3486 a 2280 3544 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.3) cvn H.B /ANN pdfmark end 2280 3544 a 30 w Fv(w)m(e)i(get)642 3771 y Ft(u)694 3785 y Fp(2)733 3771 y Fv(\()p Ft(n)823 3785 y Fp(2)862 3771 y Fv(\))26 b Fs(\025)f Ft(u)p Fv(\()p Ft(n)1161 3785 y Fp(2)1200 3771 y Fv(\))h Fs(\025)f Ft(u)p Fv(\()p Ft(n)1499 3785 y Fp(1)1538 3771 y Fv(\))c(+)f Ft(m)1765 3733 y Fm(0)1803 3771 y Fv(log)d Ft(\014)30 b Fs(\025)25 b Fv(log)r(\(2)p Ft(t)p Fv(\()p Ft(n)2434 3785 y Fp(1)2474 3771 y Fv(\)\))20 b(+)g Ft(m)2735 3733 y Fm(0)2774 3771 y Fv(log)c Ft(\014)31 b Fv(=)25 b(\012\()p Ft(m)3265 3733 y Fm(0)3288 3771 y Fv(\))p Ft(:)0 3998 y Fv(The)34 b(last)h(equalit)m(y)f(holds)g(due)g(to)h(our)g(c)m(hoice)g (of)g Ft(m)1881 3965 y Fm(0)1905 3998 y Fv(.)54 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pdfmark end 535 4133 a Fv(.)560 4133 y SDict begin H.S end 560 4133 a 0 TeXcolorgray Fv(3)p 0 TeXcolorgray 606 4075 a SDict begin H.R end 606 4075 a 606 4133 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.9) cvn H.B /ANN pdfmark end 606 4133 a Fv(,)c Ft(T)13 b Fv(\()p Ft(n)817 4147 y Fp(2)856 4133 y Fv(\))31 b(is)e(balanced.)40 b(By)31 b(Prop)s(ositions)2095 4133 y SDict begin H.S end 2095 4133 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 2141 4075 a SDict begin H.R end 2141 4075 a 2141 4133 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.2) cvn H.B /ANN pdfmark end 2141 4133 a Fv(,)2196 4133 y SDict begin H.S end 2196 4133 a 0 TeXcolorgray Fv(5)p 0 TeXcolorgray 2242 4075 a SDict begin H.R end 2242 4075 a 2242 4133 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.5) cvn H.B /ANN pdfmark end 2242 4133 a Fv(.)2267 4133 y SDict begin H.S end 2267 4133 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 2313 4075 a SDict begin H.R end 2313 4075 a 2313 4133 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.6) cvn H.B /ANN pdfmark end 2313 4133 a 30 w Fv(and)2520 4133 y SDict begin H.S end 2520 4133 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 2565 4075 a SDict begin H.R end 2565 4075 a 2565 4133 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.6) cvn H.B /ANN pdfmark end 2565 4133 a Fv(.)2590 4133 y SDict begin H.S end 2590 4133 a 0 TeXcolorgray Fv(1)p 0 TeXcolorgray 2636 4075 a SDict begin H.R end 2636 4075 a 2636 4133 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.7) cvn H.B /ANN pdfmark end 2636 4133 a Fv(,)f(w)m(e)h(kno)m(w)g(that)666 4360 y Ft(g)s Fv(\()p Ft(n)802 4374 y Fp(1)842 4360 y Ft(;)15 b(n)937 4374 y Fp(2)976 4360 y Fv(\))26 b Fs(\024)f Ft(v)s Fv(\()p Ft(n)1270 4374 y Fp(2)1310 4360 y Fv(\))20 b Fs(\000)g Ft(v)s Fv(\()p Ft(n)1593 4374 y Fp(1)1633 4360 y Fv(\))g(+)g(2)p Ft(m)1904 4323 y Fm(0)1943 4360 y Fv(log)c Ft(\014)31 b Fs(\024)25 b Ft(v)2297 4374 y Fp(2)2336 4360 y Fv(\()p Ft(n)2426 4374 y Fp(2)2466 4360 y Fv(\))20 b Fs(\000)g Ft(v)2656 4374 y Fp(1)2696 4360 y Fv(\()p Ft(n)2786 4374 y Fp(1)2825 4360 y Fv(\))h(+)f(2)p Ft(m)3097 4323 y Fm(0)3135 4360 y Fv(log)d Ft(\014)1037 4554 y Fs(\024)1143 4493 y Fv(1)p 1143 4533 46 4 v 1143 4616 a(2)1198 4554 y Ft(u)1250 4568 y Fp(2)1290 4554 y Fv(\()p Ft(n)1380 4568 y Fp(2)1419 4554 y Fv(\))k Fs(\000)f Fv(log)q(\(2)p Ft(t)p Fv(\()p Ft(n)1886 4568 y Fp(1)1926 4554 y Fv(\)\))h(+)f(2)p Ft(m)2233 4517 y Fm(0)2272 4554 y Fv(log)c Ft(\014)31 b Fs(\024)2591 4493 y Fv(1)p 2591 4533 V 2591 4616 a(2)2647 4554 y Ft(u)2699 4568 y Fp(2)2738 4554 y Fv(\()p Ft(n)2828 4568 y Fp(2)2868 4554 y Fv(\))20 b(+)g Ft(O)s Fv(\()p Ft(m)3201 4517 y Fm(0)3225 4554 y Fv(\))p Ft(:)3874 4554 y SDict begin H.S end 3874 4554 a 3874 4554 a SDict begin 19.31996 H.A end 3874 4554 a 3874 4554 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.5) cvn /DEST pdfmark end 3874 4554 a Fv(\(5\))0 4781 y(Let)i Ft(n)209 4795 y Fp(3)270 4781 y Fv(b)s(e)f(the)h(last)f(elemen)m(t)i(of)f Ft(I)7 b Fv(.)37 b(W)-8 b(e)23 b(kno)m(w)f(that)g Ft(T)13 b Fv(\()p Ft(n)1949 4795 y Fp(2)1989 4781 y Fv(\))22 b(is)e(cen)m(tral)i (and)g(balanced,)h(and)e Ft(u)3206 4795 y Fp(2)3245 4781 y Fv(\()p Ft(n)3335 4795 y Fp(2)3375 4781 y Fv(\))k(=)g(\012\()p Ft(m)3712 4748 y Fm(0)3735 4781 y Fv(\).)38 b(The)0 4916 y(same)c(conclusion)e(holds)g(for)i Ft(T)13 b Fv(\()p Ft(n)1214 4930 y Fp(3)1253 4916 y Fv(\),)35 b(in)d(particular,)h Ft(T)13 b Fv(\()p Ft(n)2062 4930 y Fp(3)2101 4916 y Fv(\))34 b(is)f(cen)m(tral)h(and)f(balanced,)h(and)f Ft(u)3389 4930 y Fp(2)3428 4916 y Fv(\()p Ft(n)3518 4930 y Fp(3)3558 4916 y Fv(\))e(=)f(\012\()p Ft(m)3906 4883 y Fm(0)3929 4916 y Fv(\).)0 5052 y(By)h(Prop)s(osition)632 5052 y SDict begin H.S end 632 5052 a 0 TeXcolorgray Fv(10)p 0 TeXcolorgray 723 4993 a SDict begin H.R end 723 4993 a 723 5052 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.10) cvn H.B /ANN pdfmark end 723 5052 a Fv(,)g(w)m(e)f(obtain)1437 5187 y SDict begin H.S end 1437 5187 a 1437 5187 a SDict begin 16.31996 H.A end 1437 5187 a 1437 5187 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.6) cvn /DEST pdfmark end 1437 5187 a Ft(g)s Fv(\()p Ft(n)1573 5201 y Fp(2)1613 5187 y Ft(;)15 b(n)1708 5201 y Fp(3)1748 5187 y Fv(\))25 b Fs(\024)g Ft(u)1956 5201 y Fp(2)1996 5187 y Fv(\()p Ft(n)2086 5201 y Fp(3)2125 5187 y Fv(\))20 b Fs(\000)g Ft(u)2323 5201 y Fp(2)2363 5187 y Fv(\()p Ft(n)2453 5201 y Fp(2)2492 5187 y Fv(\))p Ft(:)1322 b Fv(\(6\))p 0 TeXcolorgray 1950 5486 a(21)p 0 TeXcolorgray eop end end %%Page: 22 22 TeXDict begin HPSdict begin 22 21 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -174 a SDict begin H.S end -8 -174 a -8 -174 a SDict begin H.R end -8 -174 a -8 -174 a SDict begin [/View [/XYZ H.V]/Dest (page.22) cvn /DEST pdfmark end -8 -174 a 0 TeXcolorgray 0 TeXcolorgray 0 108 a Fv(By)31 b(Prop)s(osition)632 108 y SDict begin H.S end 632 108 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 678 49 a SDict begin H.R end 678 49 a 678 108 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.6) cvn H.B /ANN pdfmark end 678 108 a Fv(.)703 108 y SDict begin H.S end 703 108 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 748 49 a SDict begin H.R end 748 49 a 748 108 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.8) cvn H.B /ANN pdfmark end 748 108 a Fv(,)g(w)m(e)g(deriv)m(e)f(that)1074 324 y Ft(v)1118 338 y Fp(2)1158 324 y Fv(\()p Ft(n)1248 338 y Fp(3)1287 324 y Fv(\))c Fs(\025)1453 263 y Fv(1)p 1453 303 46 4 v 1453 386 a(2)1509 324 y Ft(u)1561 338 y Fp(2)1600 324 y Fv(\()p Ft(n)1690 338 y Fp(3)1730 324 y Fv(\))20 b Fs(\000)1886 263 y Fv(1)p 1886 303 V 1886 386 a(2)1957 324 y(log)q(\()p Ft(\014)5 b(=\013)p Fv(\))27 b(=)e(\012\()p Ft(m)2607 287 y Fm(0)2630 324 y Fv(\))h Fs(\035)f Ft(v)2854 287 y Fm(0)2851 347 y Fp(2)2891 324 y Ft(:)0 530 y Fv(If)39 b Ft(v)144 544 y Fp(2)184 530 y Fv(\()p Ft(n)p Fv(\))i Fs(\025)f Ft(v)508 497 y Fm(0)505 554 y Fp(2)545 530 y Fv(,)i(w)m(e)e(are)g(done.)68 b(Otherwise,)41 b(let)f Ft(n)1859 544 y Fp(4)1939 530 y Ft(>)g(n)2105 544 y Fp(3)2184 530 y Fv(b)s(e)f(a)h(\014rst)f(momen)m(t)h(when)f Ft(v)3250 544 y Fp(2)3289 530 y Fv(\()p Ft(n)3379 544 y Fp(4)3418 530 y Fv(\))i Ft(<)g(v)3653 497 y Fm(0)3650 554 y Fp(2)3689 530 y Fv(.)69 b(Since)0 665 y Ft(v)44 679 y Fp(2)84 665 y Fv(\()p Ft(i)p Fv(\))30 b Ft(>)g Fv(0)j(for)g(all)f Ft(i)e Fs(2)f Fv([)p Ft(n)896 679 y Fp(3)936 665 y Ft(;)15 b(n)1031 679 y Fp(4)1070 665 y Fv(\),)34 b(from)f(the)g(de\014nition)e (of)i(the)g(function)f Ft(v)2616 679 y Fp(2)2688 665 y Fv(it)h(follo)m(ws)f(that)i Ft(T)13 b Fv(\()p Ft(i)p Fv(\))34 b(is)e(cen)m(tral)h(for)0 801 y(all)e Ft(i)d Fs(2)g Fv([)p Ft(n)356 815 y Fp(3)395 801 y Ft(;)15 b(n)490 815 y Fp(4)530 801 y Fv(\).)46 b(By)32 b(the)g(de\014nition)e(of)i Ft(I)7 b Fv(,)33 b(there)f(are)g(no)g(crossings)g(b)s(et)m(w)m(een)g Ft(n)2859 815 y Fp(3)2930 801 y Fv(and)g Ft(n)3164 815 y Fp(4)3203 801 y Fv(.)45 b(By)33 b(Prop)s(ositions)3945 801 y SDict begin H.S end 3945 801 a 0 TeXcolorgray Fv(4)p 0 TeXcolorgray 3990 742 a SDict begin H.R end 3990 742 a 3990 801 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.4) cvn H.B /ANN pdfmark end 3990 801 a 0 936 a Fv(and)177 936 y SDict begin H.S end 177 936 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 222 878 a SDict begin H.R end 222 878 a 222 936 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.6) cvn H.B /ANN pdfmark end 222 936 a Fv(.)247 936 y SDict begin H.S end 247 936 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 293 878 a SDict begin H.R end 293 878 a 293 936 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.8) cvn H.B /ANN pdfmark end 293 936 a 30 w Fv(and)d(the)h(fact)g Ft(\013\014)g(<)25 b Fv(1,)31 b(w)m(e)g(obtain)494 1142 y Ft(g)s Fv(\()p Ft(n)630 1156 y Fp(3)670 1142 y Ft(;)15 b(n)765 1156 y Fp(4)805 1142 y Fv(\))25 b(=)g Ft(e)p Fv(\()p Ft(n)1093 1156 y Fp(3)1133 1142 y Ft(;)15 b(n)1228 1156 y Fp(4)1268 1142 y Fv(\))g(log)i Ft(\013)j Fv(+)g Ft(f)10 b Fv(\()p Ft(n)1765 1156 y Fp(3)1804 1142 y Ft(;)15 b(n)1899 1156 y Fp(4)1938 1142 y Fv(\))g(log)i Ft(\014)31 b Fs(\024)25 b Fv(2\()p Fs(\000)p Ft(e)p Fv(\()p Ft(n)2582 1156 y Fp(3)2622 1142 y Ft(;)15 b(n)2717 1156 y Fp(4)2756 1142 y Fv(\))21 b(+)f Ft(f)10 b Fv(\()p Ft(n)3048 1156 y Fp(3)3086 1142 y Ft(;)15 b(n)3181 1156 y Fp(4)3221 1142 y Fv(\)\))g(log)i Ft(\014)865 1302 y Fs(\024)25 b Fv(2)p Ft(v)1050 1316 y Fp(2)1090 1302 y Fv(\()p Ft(n)1180 1316 y Fp(4)1220 1302 y Fv(\))20 b Fs(\000)g Fv(2)p Ft(v)1455 1316 y Fp(2)1495 1302 y Fv(\()p Ft(n)1585 1316 y Fp(3)1625 1302 y Fv(\))25 b Ft(<)g Fs(\000)p Ft(u)1904 1316 y Fp(2)1943 1302 y Fv(\()p Ft(n)2033 1316 y Fp(3)2073 1302 y Fv(\))20 b(+)g Ft(O)s Fv(\()p Ft(v)2373 1265 y Fm(0)2370 1325 y Fp(2)2410 1302 y Fv(\))p Ft(:)3874 1302 y SDict begin H.S end 3874 1302 a 3874 1302 a SDict begin 19.31996 H.A end 3874 1302 a 3874 1302 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.7) cvn /DEST pdfmark end 3874 1302 a Fv(\(7\))0 1508 y(By)38 b(the)g(de\014nition)d(of)j Ft(I)7 b Fv(,)40 b(there)e(is)f(no)g Ft(i)h Fs(2)f Fv(\()p Ft(n)1677 1522 y Fp(4)1717 1508 y Ft(;)15 b(n)p Fv(])37 b(suc)m(h)h(that)g Ft(T)13 b Fv(\()p Ft(i)26 b Fs(\000)e Fv(1\))39 b(is)e(cen)m(tral)h (and)f(there)h(is)f(a)h(crossing)0 1643 y(b)s(et)m(w)m(een)31 b(momen)m(ts)g Ft(i)20 b Fs(\000)g Fv(1)31 b(and)f Ft(i)p Fv(.)41 b(By)30 b(Prop)s(osition)1862 1643 y SDict begin H.S end 1862 1643 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 1907 1585 a SDict begin H.R end 1907 1585 a 1907 1643 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.2) cvn H.B /ANN pdfmark end 1907 1643 a Fv(,)h(w)m(e)g(ha)m(v)m(e)1521 1849 y SDict begin H.S end 1521 1849 a 1521 1849 a SDict begin 16.31996 H.A end 1521 1849 a 1521 1849 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.8) cvn /DEST pdfmark end 1521 1849 a Ft(g)s Fv(\()p Ft(n)1657 1863 y Fp(4)1697 1849 y Ft(;)15 b(n)p Fv(\))25 b Fs(\024)g Ft(v)s Fv(\()p Ft(n)p Fv(\))c Fs(\000)f Ft(v)s Fv(\()p Ft(n)2369 1863 y Fp(4)2409 1849 y Fv(\))p Ft(:)1405 b Fv(\(8\))0 2055 y(Since)20 b(there)i(is)e(no)i(crossing)e(b)s(et)m(w)m(een)i Ft(n)1384 2069 y Fp(4)1425 2055 y Fs(\000)r Fv(1)g(and)f Ft(n)1788 2069 y Fp(4)1827 2055 y Fv(,)i(b)m(y)e(Prop)s(osition)2473 2055 y SDict begin H.S end 2473 2055 a 0 TeXcolorgray Fv(4)p 0 TeXcolorgray 2518 1996 a SDict begin H.R end 2518 1996 a 2518 2055 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.4) cvn H.B /ANN pdfmark end 2518 2055 a Fv(,)j(w)m(e)d(ha)m(v)m (e)i Ft(v)2936 2069 y Fp(2)2976 2055 y Fv(\()p Ft(n)3066 2069 y Fp(4)3105 2055 y Fv(\))i Fs(\025)g Ft(v)3305 2069 y Fp(2)3345 2055 y Fv(\()p Ft(n)3435 2069 y Fp(4)3476 2055 y Fs(\000)r Fv(1\))r Fs(\000)r Fv(log)18 b Ft(\014)30 b Fs(\025)0 2190 y Ft(v)47 2157 y Fm(0)44 2214 y Fp(2)110 2190 y Fs(\000)d Fv(log)16 b Ft(\014)47 b Fs(\035)42 b Fv(1.)71 b(By)40 b(the)g(de\014nition)e(of)j Ft(v)1602 2204 y Fp(2)1641 2190 y Fv(\()p Ft(n)1731 2204 y Fp(4)1771 2190 y Fv(\),)i(w)m(e)e(kno)m(w)f(that)h Ft(y)s Fv(\()p Ft(n)2610 2204 y Fp(4)2649 2190 y Fv(\))h Fs(\034)f Fv(1)g(and)f(so)g Ft(v)3296 2204 y Fp(1)3336 2190 y Fv(\()p Ft(n)3426 2204 y Fp(4)3465 2190 y Fv(\))i Ft(>)f Fv(0,)j(hence)0 2326 y Ft(v)s Fv(\()p Ft(n)137 2340 y Fp(4)177 2326 y Fv(\))25 b Fs(\025)g Fv(0.)41 b(Adding)29 b(up)g(\()931 2326 y SDict begin H.S end 931 2326 a 0 TeXcolorgray Fv(5)p 0 TeXcolorgray 977 2267 a SDict begin H.R end 977 2267 a 977 2326 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.5) cvn H.B /ANN pdfmark end 977 2326 a Fv(\))q(-\()1078 2326 y SDict begin H.S end 1078 2326 a 0 TeXcolorgray Fv(8)p 0 TeXcolorgray 1124 2267 a SDict begin H.R end 1124 2267 a 1124 2326 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.8) cvn H.B /ANN pdfmark end 1124 2326 a Fv(\))h(and)g(the)h(fact)g (that)g Ft(v)s Fv(\()p Ft(n)2036 2340 y Fp(4)2076 2326 y Fv(\))25 b Fs(\025)g Fv(0)31 b(yield)1242 2531 y Ft(v)s Fv(\()p Ft(n)p Fv(\))26 b Fs(\025)f Ft(g)s Fv(\()p Ft(n)1672 2545 y Fp(1)1712 2531 y Ft(;)15 b(n)p Fv(\))20 b Fs(\000)g Ft(O)s Fv(\()p Ft(m)2140 2494 y Fm(0)2163 2531 y Fv(\))h Fs(\000)f Ft(O)s Fv(\()p Ft(v)2464 2494 y Fm(0)2461 2554 y Fp(2)2501 2531 y Fv(\))25 b Fs(\035)g Fv(1)p Ft(;)0 2737 y Fv(whic)m(h)k(implies)f(that)j Ft(T)13 b Fv(\()p Ft(n)p Fv(\))30 b(is)f(cen)m(tral.)p 3924 2737 4 62 v 3928 2679 55 4 v 3928 2737 V 3982 2737 4 62 v 0 2884 a SDict begin H.S end 0 2884 a 0 2884 a SDict begin 16.31996 H.A end 0 2884 a 0 2884 a SDict begin [/View [/XYZ H.V]/Dest (subsection.4.5) cvn /DEST pdfmark end 0 2884 a 116 x Fo(4.5)112 b(Pro)s(ofs)38 b(of)f(prop)s(ositions)p 0 TeXcolorgray 0 3194 a Fq(Pr)-5 b(o)g(of)34 b(of)f(Pr)-5 b(op)g(osition)834 3194 y SDict begin H.S end 834 3194 a 0 TeXcolorgray Fq(5)p 0 TeXcolorgray 881 3136 a SDict begin H.R end 881 3136 a 881 3194 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.5) cvn H.B /ANN pdfmark end 881 3194 a Fq(.)p 0 TeXcolorgray 954 3194 a SDict begin H.S end 954 3194 a 0 TeXcolorgray Fv(1)p 0 TeXcolorgray 999 3136 a SDict begin H.R end 999 3136 a 999 3194 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.1) cvn H.B /ANN pdfmark end 999 3194 a Fv(:)58 b(If)39 b(there)g(is)f(a)h(crossing,)i(then)d(the) h(smaller)f(segmen)m(t)i(among)f Fs(f)p Ft(L)p Fv(\()p Ft(i)p Fv(\))p Ft(;)15 b(R)q Fv(\()p Ft(i)p Fv(\))p Fs(g)42 b Fv(has)0 3330 y(b)s(een)30 b(increased.)40 b(It)30 b(happ)s(ens)f(only)g(when)h(the)g(transmission)e(is)i(correct.)0 3465 y SDict begin H.S end 0 3465 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 45 3407 a SDict begin H.R end 45 3407 a 45 3465 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.2) cvn H.B /ANN pdfmark end 45 3465 a Fv(:)41 b(W)-8 b(e)30 b(only)f(pro)s(of)g(the)g(assertion)g(for)h(correct)g (transmission)d(b)s(ecause)j(the)g(case)g(for)f(incorrect)h (transmission)d(is)0 3601 y(pro)m(v)m(ed)f(analogously)-8 b(.)39 b(In)25 b(case)h(that)g Ft(T)13 b Fv(\()p Ft(i)d Fs(\000)g Fv(1\))27 b(is)d(non-cen)m(tral)i(the)g(statemen)m(t)h(follo) m(ws)d(b)s(ecause)i Ft(t)p Fv(\()p Ft(i)p Fv(\))g(=)f Ft(t)p Fv(\()p Ft(i)10 b Fs(\000)g Fv(1\))p Ft(\014)5 b Fv(.)0 3736 y(In)30 b(case)h Ft(T)13 b Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))31 b(is)e(cen)m(tral)i(w)m(e)g(ha)m(v)m(e)g(that) 1367 3942 y Ft(t)p Fv(\()p Ft(i)p Fv(\))26 b(=)f(1)c Fs(\000)f Ft(\014)5 b(x)p Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))h Fs(\000)f Ft(\013y)s Fv(\()p Ft(i)h Fs(\000)f Fv(1\))0 4148 y(and)30 b(therefore)338 4353 y Ft(t)p Fv(\()p Ft(i)p Fv(\))21 b Fs(\000)f Ft(t)p Fv(\()p Ft(i)h Fs(\000)f Fv(1\))26 b(=)f(\(1)c Fs(\000)f Ft(\014)5 b Fv(\))p Ft(x)p Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))h(+)f(\(1)h Fs(\000)f Ft(\013)p Fv(\))p Ft(y)s Fv(\()p Ft(i)h Fs(\000)f Fv(1\))26 b(=)2425 4292 y Ft(\014)g Fs(\000)20 b Ft(\013)p 2425 4332 226 4 v 2515 4416 a Fv(2)2661 4353 y(\()p Ft(y)s Fv(\()p Ft(i)h Fs(\000)f Fv(1\))h Fs(\000)f Ft(x)p Fv(\()p Ft(i)h Fs(\000)f 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b(w)m(e)g(ha)m(v)m(e)g(a)f(lo)s(ok)g(at)h(the)0 2134 y(case)31 b(that)g Ft(T)13 b Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))31 b(is)e(not)i(cen)m(tral.)41 b(Then)1435 2297 y Ft(t)p Fv(\()p Ft(i)p Fv(\))p 1426 2338 154 4 v 1426 2421 a Ft(x)p Fv(\()p Ft(i)p Fv(\))1615 2359 y Fs(\025)1730 2297 y Ft(t)p Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))p Ft(\014)p 1721 2338 367 4 v 1721 2421 a(x)p Fv(\()p Ft(i)h Fs(\000)e Fv(1\))p Ft(\014)2123 2359 y Fv(=)2238 2297 y Ft(t)p Fv(\()p Ft(i)i Fs(\000)f Fv(1\))p 2229 2338 311 4 v 2229 2421 a Ft(x)p Fv(\()p Ft(i)g Fs(\000)g Fv(1\))2549 2359 y Ft(;)0 2584 y Fv(where)37 b(the)h(inequalit)m(y)e(follo)m(w)m(ed)h(b) s(ecause)h Ft(\014)5 b(x)p Fv(\()p Ft(i)26 b Fs(\000)f Fv(1\))38 b Fs(\025)f Ft(x)p Fv(\()p Ft(i)p Fv(\))h(with)f(equalit)m(y) g(if)f(there)i(is)f(no)h(crossing)f(and)g(a)0 2720 y(strict)30 b(inequalit)m(y)f(if)g(there)i(is)e(a)i(crossing.)40 b(Therefore,)30 b(w)m(e)h(ha)m(v)m(e)h(sho)m(wn)e(the)g(assertion)g (for)g(b)s(oth)g(cases.)141 2855 y(Next)h(w)m(e)g(analyze)g(the)g(case) g(where)f Ft(T)13 b Fv(\()p Ft(i)20 b Fs(\000)g Fv(1\))31 b(is)f(cen)m(tral)h(and)e(there)i(is)e(no)i(crossing)1304 3018 y Ft(t)p Fv(\()p Ft(i)p Fv(\))p 1294 3059 154 4 v 1294 3142 a Ft(x)p Fv(\()p Ft(i)p Fv(\))1484 3080 y Fs(\025)1627 3018 y Ft(t)p Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))p 1590 3059 367 4 v 1590 3142 a Ft(\014)5 b(x)p Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))1992 3080 y Fs(\025)2097 3018 y Ft(\014)26 b Fs(\000)20 b Ft(\013)p 2097 3059 226 4 v 2153 3142 a(\013\014)2358 3080 y(>)25 b(\014)h Fs(\000)20 b Ft(\013:)0 3300 y Fv(The)42 b(\014rst)g(inequalit)m(y)g (follo)m(ws)g(b)s(ecause)g(correct)i(transmission)d(can)i(nev)m(er)g (reduce)g(the)g(length)f(of)h(the)g(true)0 3435 y(segmen)m(t.)e(The)28 b(second)g(one)g(follo)m(ws)f(b)s(ecause)h(otherwise)f(b)m(y)h(Prop)s (osition)2669 3435 y SDict begin H.S end 2669 3435 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 2715 3377 a SDict begin H.R end 2715 3377 a 2715 3435 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.6) cvn H.B /ANN pdfmark end 2715 3435 a Fv(.)2740 3435 y SDict begin H.S end 2740 3435 a 0 TeXcolorgray Fv(4)p 0 TeXcolorgray 2786 3377 a SDict begin H.R end 2786 3377 a 2786 3435 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.10) cvn H.B /ANN pdfmark end 2786 3435 a 27 w Fv(there)h(w)m(ould)d(b)s(e)i(a)g(crossing)g(and)0 3571 y(the)j(last)f(inequalit)m(y)f(follo)m(ws)g(b)s(ecause)h Ft(\013\014)h(<)25 b Fv(1.)141 3706 y(Next)32 b(w)m(e)g(consider)e(the) i(case)g(that)g Ft(T)13 b Fv(\()p Ft(i)21 b Fs(\000)g Fv(1\))32 b(is)e(cen)m(tral)i(and)f(there)g(is)g(a)g(crossing.)43 b(W)-8 b(e)33 b(observ)m(e)f(that)g(since)0 3842 y(there)f(is)e(a)i (crossing)e Ft(T)13 b Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))31 b(is)f(balanced)f(b)m(y)i(Prop)s(osition)2185 3842 y SDict begin H.S end 2185 3842 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 2230 3783 a SDict begin H.R end 2230 3783 a 2230 3842 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.6) cvn H.B /ANN pdfmark end 2230 3842 a Fv(.)2255 3842 y SDict begin H.S end 2255 3842 a 0 TeXcolorgray Fv(3)p 0 TeXcolorgray 2301 3783 a SDict begin H.R end 2301 3783 a 2301 3842 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.9) cvn H.B /ANN pdfmark end 2301 3842 a Fv(.)41 b(Therefore,)30 b Ft(y)s Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))p Ft(=x)p Fv(\()p Ft(i)i Fs(\000)e Fv(1\))26 b Ft(<)f(\014)5 b(=\013)31 b Fv(and)915 4113 y SDict begin H.S end 915 4113 a 915 4113 a SDict begin 16.31996 H.A end 915 4113 a 915 4113 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.11) cvn /DEST pdfmark end 915 4113 a 934 4051 a Ft(t)p Fv(\()p Ft(i)p Fv(\))p 925 4092 154 4 v 925 4175 a Ft(x)p Fv(\()p Ft(i)p Fv(\))1114 4113 y(=)1220 4051 y(1)20 b Fs(\000)g Ft(x)p Fv(\()p Ft(i)h Fs(\000)f Fv(1\))p Ft(\014)26 b Fs(\000)20 b Ft(y)s Fv(\()p Ft(i)h Fs(\000)f Fv(1\))p Ft(\013)p 1220 4092 1000 4 v 1537 4175 a(y)s Fv(\()p Ft(i)h Fs(\000)f Fv(1\))p Ft(\013)2254 4113 y Fv(=)2360 4051 y(1)h Fs(\000)f Ft(x)p Fv(\()p Ft(i)h Fs(\000)f Fv(1\))p Ft(\014)p 2360 4092 524 4 v 2439 4175 a(y)s Fv(\()p Ft(i)h Fs(\000)f Fv(1\))p Ft(\013)2914 4113 y Fs(\000)g Fv(1)p Ft(:)753 b Fv(\(11\))0 4316 y(Since)29 b Ft(u)289 4330 y Fp(1)329 4316 y Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))26 b Fs(\025)f Ft(u)761 4283 y Fm(0)761 4341 y Fp(1)830 4316 y Fv(w)m(e)31 b(obtain)1500 4481 y(1)20 b Fs(\000)g Ft(x)p Fv(\()p Ft(i)h Fs(\000)f Fv(1\))h Fs(\000)f Ft(y)s Fv(\()p Ft(i)h Fs(\000)e Fv(1\))p 1500 4522 885 4 v 1787 4605 a Ft(x)p Fv(\()p Ft(i)i Fs(\000)e Fv(1\))2420 4542 y Fs(\025)24 b Ft(\014)i Fs(\000)20 b Ft(\013;)1510 4756 y Fv(1)g Fs(\000)g Ft(x)p Fv(\()p Ft(i)h Fs(\000)f Fv(1\))h Fs(\000)f Ft(y)s Fv(\()p Ft(i)h Fs(\000)e Fv(1\))27 b Fs(\025)d Fv(\()p Ft(\014)i Fs(\000)20 b Ft(\013)p Fv(\))p Ft(x)p Fv(\()p Ft(i)i Fs(\000)e Fv(1\))p Ft(;)842 4954 y Fv(1)h Fs(\000)f Ft(x)p Fv(\()p Ft(i)h Fs(\000)f Fv(1\))h Fs(\000)1430 4893 y Ft(\014)26 b Fs(\000)20 b Ft(\013)p 1430 4933 226 4 v 1521 5016 a Fv(2)1666 4954 y Ft(x)p Fv(\()p Ft(i)h Fs(\000)f Fv(1\))h Fs(\000)f Ft(y)s Fv(\()p Ft(i)h Fs(\000)e Fv(1\))27 b Fs(\025)2525 4893 y Ft(\014)f Fs(\000)20 b Ft(\013)p 2525 4933 V 2616 5016 a Fv(2)2761 4954 y Ft(x)p Fv(\()p Ft(i)h Fs(\000)f Fv(1\))p Ft(;)1453 5174 y Fv(1)h Fs(\000)f Ft(\014)5 b(x)p Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))h Fs(\000)f Ft(y)s Fv(\()p Ft(i)h Fs(\000)e Fv(1\))27 b Fs(\025)2525 5113 y Ft(\014)f Fs(\000)20 b Ft(\013)p 2525 5153 V 2616 5237 a Fv(2)2761 5174 y Ft(x)p Fv(\()p Ft(i)h Fs(\000)f Fv(1\))p Ft(:)p 0 TeXcolorgray 1950 5486 a Fv(26)p 0 TeXcolorgray eop end end %%Page: 27 27 TeXDict begin HPSdict begin 27 26 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -174 a SDict begin H.S end -8 -174 a -8 -174 a SDict begin H.R end -8 -174 a -8 -174 a SDict begin [/View [/XYZ H.V]/Dest (page.27) cvn /DEST pdfmark end -8 -174 a 0 TeXcolorgray 0 TeXcolorgray 0 108 a Fv(Therefore,)30 b(w)m(e)h(get)792 379 y SDict begin H.S end 792 379 a 792 379 a SDict begin 16.31996 H.A end 792 379 a 792 379 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.12) cvn /DEST pdfmark end 792 379 a 802 318 a Fv(1)21 b Fs(\000)f Ft(x)p Fv(\()p Ft(i)h Fs(\000)f Fv(1\))p Ft(\014)p 802 358 524 4 v 882 442 a(y)s Fv(\()p Ft(i)g Fs(\000)g Fv(1\))p Ft(\013)1361 379 y Fs(\025)1477 271 y Fr(\014)s Fm(\000)p Fr(\013)p 1477 292 144 4 v 1531 344 a Fp(2)1630 312 y Ft(x)p Fv(\()p Ft(i)h Fs(\000)f Fv(1\))h(+)f Ft(y)s Fv(\()p Ft(i)g Fs(\000)g Fv(1\))p 1467 358 892 4 v 1730 442 a Ft(y)s Fv(\()p Ft(i)h Fs(\000)f Fv(1\))p Ft(\013)2393 379 y Fv(=)2506 318 y(1)p 2499 358 59 4 v 2499 442 a Ft(\013)2588 379 y Fv(+)2689 312 y Ft(x)p Fv(\()p Ft(i)h Fs(\000)e Fv(1\))3008 271 y Fr(\014)s Fm(\000)p Fr(\013)p 3008 292 144 4 v 3063 344 a Fp(2)p 2689 358 474 4 v 2743 442 a Ft(y)s Fv(\()p Ft(i)i Fs(\000)f Fv(1\))p Ft(\013)3172 379 y(:)631 b Fv(\(12\))0 632 y(By)31 b(using)e(the)h(fact)h(that)g Ft(T)13 b Fv(\()p Ft(i)21 b Fs(\000)f Fv(1\))31 b(is)e(balanced)h(w)m(e)h(get)432 909 y SDict begin H.S end 432 909 a 432 909 a SDict begin 16.31996 H.A end 432 909 a 432 909 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.13) cvn /DEST pdfmark end 432 909 a 449 847 a Fv(1)p 442 888 59 4 v 442 971 a Ft(\013)531 909 y Fv(+)632 842 y Ft(x)p Fv(\()p Ft(i)20 b Fs(\000)g Fv(1\))951 801 y Fr(\014)s Fm(\000)p Fr(\013)p 951 821 144 4 v 1006 873 a Fp(2)p 632 888 474 4 v 686 971 a Ft(y)s Fv(\()p Ft(i)h Fs(\000)f Fv(1\))p Ft(\013)1141 909 y(>)1253 847 y Fv(1)p 1246 888 59 4 v 1246 971 a Ft(\013)1335 909 y Fv(+)1436 847 y Ft(\014)25 b Fs(\000)20 b Ft(\013)p 1436 888 226 4 v 1498 971 a Fv(2)p Ft(\014)1697 909 y Fv(=)1803 847 y(2)p Ft(\014)26 b Fv(+)20 b Ft(\013\014)26 b Fs(\000)20 b Ft(\013)2300 814 y Fp(2)p 1803 888 537 4 v 1991 971 a Fv(2)p Ft(\013\014)2375 909 y Fv(=)2481 847 y(\()p Ft(\013)h Fv(+)f Ft(\014)5 b Fv(\))p Ft(\014)26 b Fv(+)19 b Ft(\013\014)26 b Fv(+)20 b Ft(\013)3228 814 y Fp(2)3288 847 y Fs(\000)g Fv(2)p Ft(\013)3482 814 y Fp(2)p 2481 888 1043 4 v 2922 971 a Fv(2)p Ft(\013\014)3533 909 y(;)270 b Fv(\(13\))0 1156 y(whic)m(h)29 b(w)m(e)i(simplify)c(to) 697 1349 y(\()p Ft(\013)21 b Fv(+)f Ft(\014)5 b Fv(\))993 1316 y Fp(2)1053 1349 y Fs(\000)20 b Fv(2)p Ft(\013)1247 1316 y Fp(2)p 697 1390 591 4 v 912 1473 a Fv(2)p Ft(\013\014)1322 1411 y Fv(=)1428 1349 y(2)h Fs(\000)f Ft(\013)1643 1316 y Fp(2)p 1428 1390 255 4 v 1498 1473 a Ft(\013\014)1718 1411 y Fv(=)25 b(1)c(+)1980 1349 y(2)g Fs(\000)f Fv(2)p Ft(\013)p 1980 1390 261 4 v 2053 1473 a(\013\014)2276 1411 y Fv(=)25 b(1)c(+)2539 1349 y Ft(\014)k Fs(\000)20 b Ft(\013)p 2539 1390 226 4 v 2594 1473 a(\013\014)2800 1411 y(>)25 b(\014)g Fs(\000)20 b Ft(\013)h Fv(+)f(1)p Ft(;)3828 1411 y SDict begin H.S end 3828 1411 a 3828 1411 a SDict begin 19.31996 H.A end 3828 1411 a 3828 1411 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.14) cvn /DEST pdfmark end 3828 1411 a Fv(\(14\))0 1659 y(where)28 b(the)g(last)g (inequalit)m(y)e(follo)m(ws)h(b)s(ecause)h(1)e Ft(>)f(\013\014)5 b Fv(.)41 b(Com)m(bining)25 b(equation)j(\()2844 1659 y SDict begin H.S end 2844 1659 a 0 TeXcolorgray Fv(11)p 0 TeXcolorgray 2936 1600 a SDict begin H.R end 2936 1600 a 2936 1659 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.11) cvn H.B /ANN pdfmark end 2936 1659 a Fv(\))g(and)f(the)i (inequalities)c(\()3838 1659 y SDict begin H.S end 3838 1659 a 0 TeXcolorgray Fv(12)p 0 TeXcolorgray 3929 1600 a SDict begin H.R end 3929 1600 a 3929 1659 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.12) cvn H.B /ANN pdfmark end 3929 1659 a Fv(\))q(,)0 1794 y(\()35 1794 y SDict begin H.S end 35 1794 a 0 TeXcolorgray Fv(13)p 0 TeXcolorgray 126 1736 a SDict begin H.R end 126 1736 a 126 1794 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.13) cvn H.B /ANN pdfmark end 126 1794 a Fv(\))q(,)30 b(\()252 1794 y SDict begin H.S end 252 1794 a 0 TeXcolorgray Fv(14)p 0 TeXcolorgray 343 1736 a SDict begin H.R end 343 1736 a 343 1794 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.14) cvn H.B /ANN pdfmark end 343 1794 a Fv(\))h(completes)g(the)f(pro)s (of)g(of)h(this)e(part.)141 1930 y(No)m(w)44 b(w)m(e)f(pro)s(of)f(that) h(if)f Ft(u)1114 1944 y Fp(1)1153 1930 y Fv(\()p Ft(i)30 b Fs(\000)e Fv(1\))46 b Fs(\025)g Ft(u)1643 1897 y Fm(00)1643 1954 y Fp(1)1685 1930 y Fv(,)h(then)42 b Ft(u)2028 1944 y Fp(1)2067 1930 y Fv(\()p Ft(i)p Fv(\))47 b Fs(\025)f Ft(u)2384 1897 y Fm(00)2384 1954 y 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(true)g(b)s(ecause)g Ft(T)13 b Fv(\()p Ft(i)p Fv(\))31 b(is)f(not)g(cen)m(tral)h(and)f(the)h(transmission)d(w)m(as)i (incorrect.)141 2324 y(Finally)37 b(w)m(e)i(consider)e(the)i(case)h (that)f Ft(T)13 b Fv(\()p Ft(i)26 b Fs(\000)g Fv(1\))39 b(is)f(not)h(cen)m(tral)g(but)e Ft(T)13 b Fv(\()p Ft(i)p Fv(\))40 b(is)d(cen)m(tral.)66 b(In)38 b(this)g(case,)k(the)0 2459 y(transmission)28 b(is)h(correct)j(since)e(otherwise)f(it)h(holds) f Ft(y)s Fv(\()p Ft(i)p Fv(\))d Fs(\025)f Ft(y)s Fv(\()p Ft(i)c Fs(\000)f Fv(1\))26 b Fs(\025)2587 2423 y Fp(1)p 2587 2438 36 4 v 2587 2490 a(2)2632 2459 y Fv(.)41 b(F)-8 b(urthermore,)30 b(recall)g(that)1382 2646 y Fs(\000)15 b Fv(log)q(\(2)p Ft(y)s Fv(\()p Ft(i)p Fv(\)\))27 b Fs(\025)e Fv(log)r(\(2\(1)d Fs(\000)d Ft(y)s Fv(\()p Ft(i)p Fv(\)\)\))p Ft(:)0 2834 y Fv(Therefore,)30 b(w)m(e)h(obtain)338 3044 y Ft(v)382 3058 y Fp(2)422 3044 y Fv(\()p Ft(i)p Fv(\))21 b Fs(\000)f Ft(v)679 3058 y Fp(2)718 3044 y Fv(\()p Ft(i)h Fs(\000)f Fv(1\))26 b(=)f Fs(\000)15 b Fv(log)r(\(2)p Ft(y)s Fv(\()p Ft(i)p Fv(\)\))22 b Fs(\000)e Fv(log)q(\(2\(1)i Fs(\000)e Ft(y)s Fv(\()p Ft(i)g Fs(\000)g Fv(1\)\)\))27 b Fs(\025)e Fv(log)2700 2915 y Fk(\022)2855 2982 y Fv(1)c Fs(\000)e Ft(y)s Fv(\()p Ft(i)p Fv(\))p 2777 3023 463 4 v 2777 3106 a(1)h Fs(\000)g Ft(y)s Fv(\()p Ft(i)h Fs(\000)f Fv(1\))3250 2915 y Fk(\023)3342 3044 y Fv(=)25 b(log)16 b Ft(\014)5 b(;)0 3253 y Fv(whic)m(h)29 b(concludes)h(the)g(pro)s(of.)p 3924 3253 4 62 v 3928 3195 55 4 v 3928 3253 V 3982 3253 4 62 v 0 TeXcolorgray 0 3457 a Fq(Pr)-5 b(o)g(of)34 b(of)f(Pr)-5 b(op)g(osition)834 3457 y SDict begin H.S end 834 3457 a 0 TeXcolorgray Fq(10)p 0 TeXcolorgray 927 3398 a SDict begin H.R end 927 3398 a 927 3457 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.10) cvn H.B /ANN pdfmark end 927 3457 a Fq(.)p 0 TeXcolorgray 45 w Fv(F)d(rom)31 b(Prop)s(osition)1729 3457 y SDict begin H.S end 1729 3457 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 1774 3398 a SDict begin H.R end 1774 3398 a 1774 3457 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.6) cvn H.B /ANN pdfmark end 1774 3457 a Fv(.)1799 3457 y SDict begin H.S end 1799 3457 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 1845 3398 a SDict begin H.R end 1845 3398 a 1845 3457 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.8) cvn H.B /ANN pdfmark end 1845 3457 a 30 w Fv(and)f(the)h(inequalit)m(y)f(\()2667 3457 y SDict begin H.S end 2667 3457 a 0 TeXcolorgray Fv(3)p 0 TeXcolorgray 2712 3398 a SDict begin H.R end 2712 3398 a 2712 3457 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.3) cvn H.B /ANN pdfmark end 2712 3457 a Fv(\))h(w)m(e)g(deduce)f(that)683 3670 y SDict begin H.S end 683 3670 a 683 3670 a SDict begin 16.31996 H.A end 683 3670 a 683 3670 a SDict begin [/View [/XYZ H.V]/Dest (equation.4.15) cvn /DEST pdfmark end 683 3670 a Ft(v)727 3684 y Fp(2)767 3670 y Fv(\()p Ft(i)833 3684 y Fp(0)873 3670 y Fv(\))c Ft(>)1039 3608 y(u)1091 3622 y Fp(2)1131 3608 y Fv(\()p Ft(i)1197 3622 y Fp(0)1237 3608 y Fv(\))p 1039 3649 233 4 v 1133 3732 a(2)1303 3670 y Fs(\000)1403 3608 y Fv(log)r(\()p Ft(\014)5 b(=\013)p Fv(\))p 1403 3649 349 4 v 1555 3732 a(2)1787 3670 y Fs(\025)25 b(\000)1964 3608 y Fv(1)p 1964 3649 46 4 v 1964 3732 a(2)2034 3670 y(log)r(\()p Ft(\013)2245 3632 y Fp(4)2285 3670 y Ft(\014)2341 3632 y Fp(6)2380 3670 y Fv(\))h Fs(\025)f(\000)2618 3608 y Fv(1)p 2618 3649 V 2618 3732 a(2)2688 3670 y(log)17 b Ft(\014)2877 3632 y Fm(\000)p Fp(2)2997 3670 y Fv(=)25 b(log)16 b Ft(\014)5 b(:)522 b Fv(\(15\))0 3857 y(Since)29 b Ft(v)281 3871 y Fp(2)321 3857 y Fv(\()p Ft(i)387 3871 y Fp(0)427 3857 y Fv(\))d(=)f Fs(\000)15 b Fv(log)q(\(2)p Ft(y)s Fv(\()p Ft(i)981 3871 y Fp(0)1022 3857 y Fv(\)\))26 b Ft(>)f Fv(log)16 b Ft(\014)5 b Fv(,)31 b(w)m(e)g(ha)m(v)m(e)g Ft(y)s Fv(\()p Ft(i)1915 3871 y Fp(0)1955 3857 y Fv(\))26 b Ft(<)f Fv(1)p Ft(=)p Fv(\(2)p Ft(\014)5 b Fv(\).)43 b(F)-8 b(rom)31 b Ft(y)s Fv(\()p Ft(i)2794 3871 y Fp(0)2834 3857 y Fv(\))25 b Ft(<)g Fv(1)p Ft(=)p Fv(\(2)p Ft(\014)5 b Fv(\))33 b(w)m(e)e(obtain)e(that)831 4072 y Ft(u)883 4086 y Fp(1)923 4072 y Fv(\()p Ft(i)989 4086 y Fp(0)1029 4072 y Fv(\))c(=)g(log)1318 3944 y Fk(\022)1395 4010 y Fv(1)20 b Fs(\000)g Ft(y)s Fv(\()p Ft(i)1665 4024 y Fp(0)1705 4010 y Fv(\))h Fs(\000)f Ft(x)p Fv(\()p Ft(i)1970 4024 y Fp(0)2010 4010 y Fv(\))p 1395 4051 651 4 v 1623 4134 a Ft(x)p Fv(\()p Ft(i)1741 4148 y Fp(0)1781 4134 y Fv(\))2055 3944 y Fk(\023)2147 4072 y Fs(\025)25 b Fv(log)2376 3944 y Fk(\022)2453 4010 y Fv(1)20 b Fs(\000)g Fv(2)p Ft(y)s Fv(\()p Ft(i)2768 4024 y Fp(0)2809 4010 y Fv(\))p 2453 4051 392 4 v 2554 4134 a Ft(y)s Fv(\()p Ft(i)2668 4148 y Fp(0)2708 4134 y Fv(\))2854 3944 y Fk(\023)2946 4072 y Ft(>)25 b(u)3094 4034 y Fm(0)3094 4094 y Fp(1)3134 4072 y Ft(:)0 4288 y Fv(F)-8 b(rom)33 b(Prop)s(osition)732 4288 y SDict begin H.S end 732 4288 a 0 TeXcolorgray Fv(7)p 0 TeXcolorgray 777 4230 a SDict begin H.R end 777 4230 a 777 4288 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.7) cvn H.B /ANN pdfmark end 777 4288 a 32 w Fv(it)f(follo)m(ws)f (that)h Ft(u)1452 4302 y Fp(1)1492 4288 y Fv(\()p Ft(i)p Fv(\))d Fs(\025)f Ft(u)1773 4255 y Fm(0)1773 4313 y Fp(1)1844 4288 y Fv(for)k(all)f Ft(i)d Fs(\025)g Ft(i)2302 4302 y Fp(0)2342 4288 y Fv(.)45 b(Prop)s(osition)2903 4288 y SDict begin H.S end 2903 4288 a 0 TeXcolorgray Fv(8)p 0 TeXcolorgray 2949 4230 a SDict begin H.R end 2949 4230 a 2949 4288 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.8) cvn H.B /ANN pdfmark end 2949 4288 a 32 w Fv(implies)29 b(that)k Ft(u)3542 4302 y Fp(2)3581 4288 y Fv(\()p Ft(i)p Fv(\))c Fs(\025)f Ft(c)k Fv(for)0 4424 y(all)j Ft(i)h Fs(\025)f Ft(i)336 4438 y Fp(0)376 4424 y Fv(.)59 b(Using)36 b(argumen)m(ts)h(as)f(in)f(\()1435 4424 y SDict begin H.S end 1435 4424 a 0 TeXcolorgray Fv(15)p 0 TeXcolorgray 1527 4365 a SDict begin H.R end 1527 4365 a 1527 4424 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.15) cvn H.B /ANN pdfmark end 1527 4424 a Fv(\),)j(w)m(e)f(get)h Ft(v)1967 4438 y Fp(2)2006 4424 y Fv(\()p Ft(i)p Fv(\))f Ft(>)e Fv(log)17 b Ft(\014)41 b Fv(for)c(all)e Ft(i)i Fv(suc)m(h)f(that)h Ft(T)13 b Fv(\()p Ft(i)p Fv(\))37 b(is)f(cen)m(tral)h(and)0 4559 y(balanced.)j(Note)32 b(that)f(since)f Ft(c)25 b Fs(\025)g Ft(u)1266 4526 y Fm(0)1266 4584 y Fp(2)1306 4559 y Fv(,)30 b(the)h(inequalit)m(y)d Ft(u)1991 4573 y Fp(2)2031 4559 y Fv(\()p Ft(i)p Fv(\))e Fs(\025)f Ft(u)2306 4526 y Fm(0)2306 4584 y Fp(2)2376 4559 y Fv(holds.)141 4695 y(Sa)m(y)33 b(that)g(there)g(are)g(crossings) f(at)i(the)e(momen)m(ts)i Ft(j)1985 4709 y Fp(1)2024 4695 y Ft(;)15 b(:)g(:)g(:)i(;)e(j)2263 4710 y Fr(k)2307 4695 y Fv(,)33 b Ft(i)2396 4709 y Fp(0)2465 4695 y Fs(\024)c Ft(j)2602 4709 y Fp(1)2670 4695 y Ft(<)g(:)15 b(:)g(:)30 b(<)f(j)3042 4710 y Fr(k)3114 4695 y Ft(<)g(i)3245 4709 y Fp(1)3285 4695 y Fv(.)47 b(P)m(artition)32 b([)p Ft(i)3803 4709 y Fp(0)3843 4695 y Ft(;)15 b(i)3914 4709 y Fp(1)3955 4695 y Fv(\))0 4830 y(in)m(to)23 b(the)g(union)d([)p Ft(i)626 4844 y Fp(0)666 4830 y Ft(;)15 b(j)743 4844 y Fp(1)784 4830 y Fv(\))5 b Fs([)g Fv([)p Ft(j)952 4844 y Fp(1)991 4830 y Ft(;)15 b(j)1068 4844 y Fp(1)1113 4830 y Fv(+)5 b(1\))g Fs([)g Fv([)p Ft(j)1402 4844 y Fp(1)1447 4830 y Fv(+)g(1)p Ft(;)15 b(j)1645 4844 y Fp(2)1685 4830 y Fv(])5 b Fs([)g Ft(:)15 b(:)g(:)6 b Fs([)f Fv([)p Ft(j)2020 4845 y Fr(k)2062 4830 y Ft(;)15 b(i)2133 4844 y Fp(1)2174 4830 y Fv(\).)38 b(By)23 b(Prop)s(osition)2889 4830 y SDict begin H.S end 2889 4830 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 2934 4772 a SDict begin H.R end 2934 4772 a 2934 4830 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.6) cvn H.B /ANN pdfmark end 2934 4830 a Fv(.)2959 4830 y SDict begin H.S end 2959 4830 a 0 TeXcolorgray Fv(3)p 0 TeXcolorgray 3005 4772 a SDict begin H.R end 3005 4772 a 3005 4830 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.9) cvn H.B /ANN pdfmark end 3005 4830 a 23 w Fv(and)3197 4830 y SDict begin H.S end 3197 4830 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 3242 4772 a SDict begin H.R end 3242 4772 a 3242 4830 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.6) cvn H.B /ANN pdfmark end 3242 4830 a Fv(.)3267 4830 y SDict begin H.S end 3267 4830 a 0 TeXcolorgray Fv(7)p 0 TeXcolorgray 3313 4772 a SDict begin H.R end 3313 4772 a 3313 4830 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.13) cvn H.B /ANN pdfmark end 3313 4830 a 22 w Fv(the)g(true)g(segmen)m(t)0 4966 y(is)32 b(cen)m(tral)i(and)e(balanced)h(at)h(all)e(the)h(endp)s(oin)m(ts)e(of)j (the)f(segmen)m(ts.)50 b(Th)m(us,)33 b(all)f(conditions)g(of)h(Prop)s (osition)3899 4966 y SDict begin H.S end 3899 4966 a 0 TeXcolorgray Fv(10)p 0 TeXcolorgray 3990 4907 a SDict begin H.R end 3990 4907 a 3990 4966 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.10) cvn H.B /ANN pdfmark end 3990 4966 a 0 5101 a Fv(are)j(satis\014ed.)54 b(Moreo)m(v)m(er,)38 b(if)c(w)m(e)i(pro)m(v)m(e)g(the)f(inequalit)m(y)g(\()2087 5101 y SDict begin H.S end 2087 5101 a 0 TeXcolorgray Fv(4)p 0 TeXcolorgray 2133 5043 a SDict begin H.R end 2133 5043 a 2133 5101 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.4) cvn H.B /ANN pdfmark end 2133 5101 a Fv(\))g(for)g(these)h (segmen)m(ts,)h(it)e(w)m(ould)f(also)h(b)s(e)f(true)h(for)0 5237 y(their)29 b(union.)39 b(Therefore,)31 b(it)f(is)f(su\016cien)m(t) h(to)h(consider)e(only)h(t)m(w)m(o)h(cases:)p 0 TeXcolorgray 1950 5486 a(30)p 0 TeXcolorgray eop end end %%Page: 31 31 TeXDict begin HPSdict begin 31 30 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -174 a SDict begin H.S end -8 -174 a -8 -174 a SDict begin H.R end -8 -174 a -8 -174 a SDict begin [/View [/XYZ H.V]/Dest (page.31) cvn /DEST pdfmark end -8 -174 a 0 TeXcolorgray 0 TeXcolorgray 0 17 a SDict begin H.S end 0 17 a 0 17 a SDict begin 16.31996 H.A end 0 17 a 0 17 a SDict begin [/View [/XYZ H.V]/Dest (Item.14) cvn /DEST pdfmark end 0 17 a 0 TeXcolorgray 111 108 a Fv(1.)p 0 TeXcolorgray 46 w Ft(i)258 122 y Fp(1)323 108 y Fv(=)25 b Ft(i)450 122 y Fp(0)510 108 y Fv(+)20 b(1)31 b(and)f(there)g(is)g(a)g(crossing.)0 163 y SDict begin H.S end 0 163 a 0 163 a SDict begin 16.31996 H.A end 0 163 a 0 163 a SDict begin [/View [/XYZ H.V]/Dest (Item.15) cvn /DEST pdfmark end 0 163 a 0 TeXcolorgray 111 318 a Fv(2.)p 0 TeXcolorgray 46 w(there)h(is)e(no)i(crossing)e(b)s(et)m (w)m(een)i Ft(i)1404 332 y Fp(0)1474 318 y Fv(and)f Ft(i)1682 332 y Fp(1)1722 318 y Fv(.)141 528 y(The)35 b(\014rst)f(case)i(is)e (trivial.)52 b(There)35 b(is)f(a)h(crossing)f(b)s(et)m(w)m(een)i Ft(i)2317 542 y Fp(0)2392 528 y Fv(and)e Ft(i)2604 542 y Fp(1)2677 528 y Fv(=)e Ft(i)2811 542 y Fp(0)2874 528 y Fv(+)23 b(1,)37 b(hence)e(the)g(transmission)0 664 y(is)f(correct)i(b)m(y)f(Prop)s(osition)1029 664 y SDict begin H.S end 1029 664 a 0 TeXcolorgray Fv(5)p 0 TeXcolorgray 1074 605 a SDict begin H.R end 1074 605 a 1074 664 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.5) cvn H.B /ANN pdfmark end 1074 664 a Fv(.)1099 664 y SDict begin H.S end 1099 664 a 0 TeXcolorgray Fv(1)p 0 TeXcolorgray 1145 605 a SDict begin H.R end 1145 605 a 1145 664 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.1) cvn H.B /ANN pdfmark end 1145 664 a Fv(.)54 b(In)34 b(this)g(case)i Ft(g)s Fv(\()p Ft(i)1833 678 y Fp(0)1874 664 y Ft(;)15 b(i)1945 678 y Fp(1)1985 664 y Fv(\))33 b(=)g(log)16 b Ft(\014)5 b Fv(.)55 b(F)-8 b(rom)35 b(the)g(other)h(hand)d Ft(u)3355 678 y Fp(2)3395 664 y Fv(\()p Ft(i)3461 678 y Fp(1)3501 664 y Fv(\))24 b Fs(\000)f Ft(u)3706 678 y Fp(2)3745 664 y Fv(\()p Ft(i)3811 678 y Fp(0)3851 664 y Fv(\))33 b(=)0 799 y Fs(\000)15 b Fv(log)q(\()p Ft(\013\014)5 b Fv(\))32 b(b)m(y)e(Prop)s(osition)1035 799 y SDict begin H.S end 1035 799 a 0 TeXcolorgray Fv(8)p 0 TeXcolorgray 1080 741 a SDict begin H.R end 1080 741 a 1080 799 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.8) cvn H.B /ANN pdfmark end 1080 799 a Fv(,)h(whic)m(h)e(is)g(at)i(least)g(log)16 b Ft(\014)36 b Fv(b)s(ecause)30 b(of)h(the)f(inequalit)m(y)g(\()3084 799 y SDict begin H.S end 3084 799 a 0 TeXcolorgray Fv(3)p 0 TeXcolorgray 3130 741 a SDict begin H.R end 3130 741 a 3130 799 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.3) cvn H.B /ANN pdfmark end 3130 799 a Fv(\))q(.)141 935 y(Pro)s(ceed)36 b(to)h(the)f(second)g(case,)i(where)d(there)h(is)f (no)h(crossing)f(b)s(et)m(w)m(een)h Ft(i)2768 949 y Fp(0)2844 935 y Fv(and)f Ft(i)3057 949 y Fp(1)3097 935 y Fv(.)57 b(The)35 b(smallest)g(part)h(is)0 1070 y(alw)m(a)m(ys)31 b(the)f(same,)i(therefore)903 1319 y Ft(g)s Fv(\()p Ft(i)1015 1333 y Fp(0)1055 1319 y Ft(;)15 b(i)1126 1333 y Fp(1)1166 1319 y Fv(\))26 b(=)f Ft(e)p Fv(\()p Ft(i)1431 1333 y Fp(0)1471 1319 y Ft(;)15 b(i)1542 1333 y Fp(1)1583 1319 y Fv(\))g(log)i Ft(\013)j Fv(+)g Ft(f)10 b Fv(\()p Ft(i)2056 1333 y Fp(0)2096 1319 y Ft(;)15 b(i)2167 1333 y Fp(1)2207 1319 y Fv(\))g(log)i Ft(\014)30 b Fv(=)25 b(log)2700 1191 y Fk(\022)2777 1258 y Ft(x)p Fv(\()p Ft(i)2895 1272 y Fp(1)2935 1258 y Fv(\))p 2777 1299 194 4 v 2777 1382 a Ft(x)p Fv(\()p Ft(i)2895 1396 y Fp(0)2935 1382 y Fv(\))2980 1191 y Fk(\023)3062 1319 y Ft(:)0 1574 y Fv(Denote)37 b Ft(v)361 1588 y Fp(2)400 1574 y Fv(\()p Ft(i)466 1588 y Fp(1)506 1574 y Fv(\))24 b Fs(\000)g Ft(v)704 1588 y Fp(2)743 1574 y Fv(\()p Ft(i)809 1588 y Fp(0)849 1574 y Fv(\))36 b(and)f Ft(u)1154 1588 y Fp(2)1193 1574 y Fv(\()p Ft(i)1259 1588 y Fp(1)1299 1574 y Fv(\))24 b Fs(\000)f Ft(u)1504 1588 y Fp(2)1544 1574 y Fv(\()p Ft(i)1610 1588 y Fp(0)1650 1574 y Fv(\))35 b(as)h(\001)p Ft(v)1957 1588 y Fp(2)2032 1574 y Fv(and)e(\001)p Ft(u)2341 1588 y Fp(2)2416 1574 y Fv(corresp)s(ondingly)-8 b(.)53 b(Since)35 b Ft(T)13 b Fv(\()p Ft(i)3491 1588 y Fp(0)3530 1574 y Fv(\))36 b(and)f Ft(T)13 b Fv(\()p Ft(i)3915 1588 y Fp(1)3955 1574 y Fv(\))0 1709 y(are)31 b(cen)m(tral,)g(w)m(e)g(ha)m(v)m(e)1135 1936 y(\001)p Ft(u)1263 1950 y Fp(2)1327 1936 y Fv(=)25 b Fs(\000)15 b Fv(log)r(\()p Ft(x)p Fv(\()p Ft(i)1780 1950 y Fp(1)1820 1936 y Fv(\))p Ft(y)s Fv(\()p Ft(i)1969 1950 y Fp(1)2009 1936 y Fv(\)\))21 b(+)f(log)r(\()p Ft(x)p Fv(\()p Ft(i)2462 1950 y Fp(0)2502 1936 y Fv(\))p Ft(y)s Fv(\()p Ft(i)2651 1950 y Fp(0)2691 1936 y Fv(\)\))1327 2142 y(=)25 b Fs(\000)15 b Fv(log)i Ft(y)s Fv(\()p Ft(i)1756 2156 y Fp(1)1796 2142 y Fv(\))j(+)g(log)d Ft(y)s Fv(\()p Ft(i)2189 2156 y Fp(0)2229 2142 y Fv(\))j Fs(\000)g Fv(log)2508 2013 y Fk(\022)2585 2080 y Ft(x)p Fv(\()p Ft(i)2703 2094 y Fp(1)2743 2080 y Fv(\))p 2585 2121 V 2585 2204 a Ft(x)p Fv(\()p Ft(i)2703 2218 y Fp(0)2743 2204 y Fv(\))2788 2013 y Fk(\023)1327 2356 y Fv(=)25 b(\001)p Ft(v)1543 2370 y Fp(2)1603 2356 y Fs(\000)20 b Ft(g)s Fv(\()p Ft(i)1806 2370 y Fp(0)1847 2356 y Ft(;)15 b(i)1918 2370 y Fp(1)1958 2356 y Fv(\))p Ft(;)0 2583 y Fv(whic)m(h)29 b(is)h(equiv)-5 b(alen)m(t)29 b(to)1553 2718 y Ft(g)s Fv(\()p Ft(i)1665 2732 y Fp(0)1706 2718 y Ft(;)15 b(i)1777 2732 y Fp(1)1817 2718 y Fv(\))26 b(=)f(\001)p Ft(v)2094 2732 y Fp(2)2153 2718 y Fs(\000)20 b Fv(\001)p Ft(u)2372 2732 y Fp(2)2411 2718 y Ft(:)0 2908 y Fv(Using)30 b(Prop)s(osition)745 2908 y SDict begin H.S end 745 2908 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 790 2849 a SDict begin H.R end 790 2849 a 790 2908 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.6) cvn H.B /ANN pdfmark end 790 2908 a Fv(.)815 2908 y SDict begin H.S end 815 2908 a 0 TeXcolorgray Fv(1)p 0 TeXcolorgray 861 2849 a SDict begin H.R end 861 2849 a 861 2908 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.7) cvn H.B /ANN pdfmark end 861 2908 a 30 w Fv(and)1068 2908 y SDict begin H.S end 1068 2908 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 1114 2849 a SDict begin H.R end 1114 2849 a 1114 2908 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.6) cvn H.B /ANN pdfmark end 1114 2908 a Fv(.)1139 2908 y SDict begin H.S end 1139 2908 a 0 TeXcolorgray Fv(2)p 0 TeXcolorgray 1184 2849 a SDict begin H.R end 1184 2849 a 1184 2908 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.8) cvn H.B /ANN pdfmark end 1184 2908 a 31 w Fv(for)g(momen)m(ts)h Ft(i)1772 2922 y Fp(1)1842 2908 y Fv(and)e Ft(i)2049 2922 y Fp(0)2120 2908 y Fv(resp)s(ectiv)m(ely)-8 b(,)30 b(w)m(e)h(obtain)653 3224 y Ft(g)s Fv(\()p Ft(i)765 3238 y Fp(0)806 3224 y Ft(;)15 b(i)877 3238 y Fp(1)917 3224 y Fv(\))25 b(=)g(\001)p Ft(v)1193 3238 y Fp(2)1253 3224 y Fs(\000)20 b Fv(\001)p Ft(u)1472 3238 y Fp(2)1536 3224 y Fs(\024)1642 3130 y Fv(\001)p Ft(u)1770 3144 y Fp(2)1830 3130 y Fv(+)g(log)2053 3029 y Fk(\020)2118 3089 y Fr(\014)p 2117 3109 46 4 v 2117 3161 a(\013)2173 3029 y Fk(\021)p 1642 3203 585 4 v 1912 3287 a Fv(2)2257 3224 y Fs(\000)g Fv(\001)p Ft(u)2476 3238 y Fp(2)2540 3224 y Fv(=)2646 3130 y Fs(\000)p Fv(\001)p Ft(u)2845 3144 y Fp(2)2905 3130 y Fv(+)f(log)3128 3029 y Fk(\020)3193 3089 y Fr(\014)p 3192 3109 46 4 v 3192 3161 a(\013)3248 3029 y Fk(\021)p 2646 3203 656 4 v 2951 3287 a Fv(2)3312 3224 y Ft(:)0 3465 y Fv(Recall)44 b(that)h Ft(y)s Fv(\()p Ft(i)613 3479 y Fp(0)653 3465 y Fv(\))50 b Ft(<)e Fv(1)p Ft(=)p Fv(\(2)p Ft(\014)5 b Fv(\),)51 b(whic)m(h)43 b(implies)f(that)j Ft(y)s Fv(\()p Ft(i)2116 3479 y Fp(0)2186 3465 y Fv(+)29 b(1\))50 b Ft(<)2545 3429 y Fp(1)p 2545 3444 36 4 v 2545 3496 a(2)2590 3465 y Fv(,)f(i.e.,)f Ft(T)13 b Fv(\()p Ft(i)2985 3479 y Fp(0)3055 3465 y Fv(+)29 b(1\))45 b(is)f(cen)m(tral.) 84 b(Using)0 3600 y(Prop)s(osition)489 3600 y SDict begin H.S end 489 3600 a 0 TeXcolorgray Fv(8)p 0 TeXcolorgray 535 3542 a SDict begin H.R end 535 3542 a 535 3600 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.8) cvn H.B /ANN pdfmark end 535 3600 a Fv(,)30 b(w)m(e)h(deduce)794 3827 y(\001)p Ft(u)922 3841 y Fp(2)987 3827 y Fv(=)25 b Ft(u)1135 3841 y Fp(2)1174 3827 y Fv(\()p Ft(i)1240 3841 y Fp(1)1280 3827 y Fv(\))c Fs(\000)f Ft(u)1479 3841 y Fp(2)1518 3827 y Fv(\()p Ft(i)1584 3841 y Fp(0)1644 3827 y Fv(+)g(1\))h(+)f Ft(u)1979 3841 y Fp(2)2019 3827 y Fv(\()p Ft(i)2085 3841 y Fp(0)2145 3827 y Fv(+)g(1\))h Fs(\000)f Ft(u)2480 3841 y Fp(2)2519 3827 y Fv(\()p Ft(i)2585 3841 y Fp(0)2625 3827 y Fv(\))26 b Fs(\025)f(\000)15 b Fv(log)q(\()p Ft(\013\014)5 b Fv(\))p Ft(:)0 4054 y Fv(Using)30 b(this)f(inequalit)m(y)g(and)g(the)i(inequalit)m(y)f(\() 1646 4054 y SDict begin H.S end 1646 4054 a 0 TeXcolorgray Fv(3)p 0 TeXcolorgray 1692 3995 a SDict begin H.R end 1692 3995 a 1692 4054 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (equation.4.3) cvn H.B /ANN pdfmark end 1692 4054 a Fv(\),)h(w)m(e)f(ha)m (v)m(e)450 4375 y Ft(g)s Fv(\()p Ft(i)562 4389 y Fp(0)602 4375 y Ft(;)15 b(i)673 4389 y Fp(1)713 4375 y Fv(\))26 b Fs(\024)880 4281 y(\000)p Fv(\001)p Ft(u)1079 4295 y Fp(2)1138 4281 y Fv(+)20 b(log)1362 4180 y Fk(\020)1427 4240 y Fr(\014)p 1426 4260 46 4 v 1426 4312 a(\013)1481 4180 y Fk(\021)p 880 4354 656 4 v 1185 4438 a Fv(2)1571 4375 y Fs(\024)1677 4281 y Fv(log)q(\()p Ft(\013\014)5 b Fv(\))22 b(+)d(log)2223 4180 y Fk(\020)2288 4240 y Fr(\014)p 2287 4260 46 4 v 2287 4312 a(\013)2343 4180 y Fk(\021)p 1677 4354 721 4 v 2014 4438 a Fv(2)2432 4375 y(=)25 b(log)17 b Ft(\014)30 b Fs(\024)25 b(\000)15 b Fv(log)q(\()p Ft(\013\014)5 b Fv(\))27 b Fs(\024)e Fv(\001)p Ft(u)3476 4389 y Fp(2)3515 4375 y Ft(:)p 3924 4602 4 62 v 3928 4544 55 4 v 3928 4602 V 3982 4602 4 62 v 0 TeXcolorgray 0 4812 a Fq(Pr)-5 b(o)g(of)34 b(of)f(Pr)-5 b(op)g(osition)834 4812 y SDict begin H.S end 834 4812 a 0 TeXcolorgray Fq(3)p 0 TeXcolorgray 881 4754 a SDict begin H.R end 881 4754 a 881 4812 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.3) cvn H.B /ANN pdfmark end 881 4812 a Fq(.)p 0 TeXcolorgray 45 w Fv(Recall)30 b(that)h Ft(u)1477 4826 y Fp(2)1516 4812 y Fv(\()p Ft(i)p Fv(\))c Fs(\025)e Ft(u)1792 4826 y Fp(1)1831 4812 y Fv(\()p Ft(i)p Fv(\))32 b(for)e(an)m(y)g Ft(x)p Fv(\()p Ft(i)p Fv(\),)i Ft(y)s Fv(\()p Ft(i)p Fv(\))f(and)f Ft(t)p Fv(\()p Ft(i)p Fv(\))h(b)m(y)f (Prop)s(osition)3622 4812 y SDict begin H.S end 3622 4812 a 0 TeXcolorgray Fv(5)p 0 TeXcolorgray 3667 4754 a SDict begin H.R end 3667 4754 a 3667 4812 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.5) cvn H.B /ANN pdfmark end 3667 4812 a Fv(.)3692 4812 y SDict begin H.S end 3692 4812 a 0 TeXcolorgray Fv(5)p 0 TeXcolorgray 3738 4754 a SDict begin H.R end 3738 4754 a 3738 4812 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (Item.5) cvn H.B /ANN pdfmark end 3738 4812 a Fv(.)141 4948 y(If)j Ft(u)287 4962 y Fp(1)326 4948 y Fv(\()p Ft(i)23 b Fs(\000)f Fv(1\))31 b Ft(<)e(u)771 4915 y Fm(0)771 4972 y Fp(1)811 4948 y Fv(,)34 b(then)f Ft(u)1132 4962 y Fp(1)1171 4948 y Fv(\()p Ft(i)p Fv(\))e Fs(\025)f Ft(u)1456 4962 y Fp(1)1495 4948 y Fv(\()p Ft(i)23 b Fs(\000)f Fv(1\))34 b(b)m(y)f(Prop)s(osition)2412 4948 y SDict begin H.S end 2412 4948 a 0 TeXcolorgray Fv(7)p 0 TeXcolorgray 2457 4889 a SDict begin H.R end 2457 4889 a 2457 4948 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.7) cvn H.B /ANN pdfmark end 2457 4948 a Fv(,)h(therefore,)h Ft(u)p Fv(\()p Ft(i)p Fv(\))c Fs(\025)f Ft(u)3264 4962 y Fp(1)3303 4948 y Fv(\()p Ft(i)p Fv(\))h Fs(\025)f Ft(u)3588 4962 y Fp(1)3627 4948 y Fv(\()p Ft(i)23 b Fs(\000)f Fv(1\))30 b(=)0 5083 y Ft(u)p Fv(\()p Ft(i)17 b Fs(\000)f Fv(1\).)41 b(If)27 b Ft(T)13 b Fv(\()p Ft(i)k Fs(\000)f Fv(1\))29 b(is)e(cen)m(tral)i(and)f(there)g(is)g(a)h(crossing,)f(then)g Ft(u)2383 5097 y Fp(1)2422 5083 y Fv(\()p Ft(i)p Fv(\))18 b Fs(\000)d Ft(u)2679 5097 y Fp(1)2719 5083 y Fv(\()p Ft(i)i Fs(\000)f Fv(1\))26 b Fs(\025)f Fv(log)16 b Ft(\014)34 b Fv(b)m(y)28 b(Prop)s(osition)3919 5083 y SDict begin H.S end 3919 5083 a 0 TeXcolorgray Fv(7)p 0 TeXcolorgray 3965 5025 a SDict begin H.R end 3965 5025 a 3965 5083 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.7) cvn H.B /ANN pdfmark end 3965 5083 a Fv(,)0 5219 y(hence,)j Ft(u)p Fv(\()p Ft(i)p Fv(\))21 b Fs(\000)f Ft(u)p Fv(\()p Ft(i)h Fs(\000)f Fv(1\))26 b Fs(\025)f Fv(log)16 b Ft(\014)5 b Fv(.)p 0 TeXcolorgray 1950 5486 a(31)p 0 TeXcolorgray eop end end %%Page: 32 32 TeXDict begin HPSdict begin 32 31 bop 0 TeXcolorgray 0 0 a SDict begin /product where{pop product(Distiller)search{pop pop pop version(.)search{exch pop exch pop(3011)eq{gsave newpath 0 0 moveto closepath clip/Courier findfont 10 scalefont setfont 72 72 moveto(.)show grestore}if}{pop}ifelse}{pop}ifelse}if end 0 0 a -8 -174 a SDict begin H.S end -8 -174 a -8 -174 a SDict begin H.R end -8 -174 a -8 -174 a SDict begin [/View [/XYZ H.V]/Dest (page.32) cvn /DEST pdfmark 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1>>/Subtype /Link/Dest (proposition.8) cvn H.B /ANN pdfmark end 3587 108 a Fv(.)39 b(It)23 b(means)0 243 y(that)28 b Ft(u)p Fv(\()p Ft(i)p Fv(\))e(=)f Ft(u)521 257 y Fp(2)561 243 y Fv(\()p Ft(i)p Fv(\))h Fs(\025)f Ft(u)836 257 y Fp(2)875 243 y Fv(\()p Ft(i)14 b Fs(\000)g Fv(1\))27 b(=)e Ft(u)p Fv(\()p Ft(i)14 b Fs(\000)g Fv(1\).)41 b(If)27 b Ft(T)13 b Fv(\()p Ft(i)h Fs(\000)g Fv(1\))28 b(is)e(cen)m(tral)i(and)f(there)g(is)f(a)i (crossing,)g(then)f Ft(T)13 b Fv(\()p Ft(i)p Fv(\))28 b(is)e(also)0 379 y(cen)m(tral)g(b)m(y)f(Prop)s(osition)902 379 y SDict begin H.S end 902 379 a 0 TeXcolorgray Fv(6)p 0 TeXcolorgray 947 320 a SDict begin H.R end 947 320 a 947 379 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 1]BorderArrayPatch/BS <</S/S/W 1>>/Subtype /Link/Dest (proposition.6) cvn H.B /ANN pdfmark end 947 379 a Fv(.)972 379 y SDict begin H.S end 972 379 a 0 TeXcolorgray Fv(5)p 0 TeXcolorgray 1018 320 a SDict begin H.R end 1018 320 a 1018 379 a SDict begin [/Color [1 0 0]/H /I/Border [0 0 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