The body of this paper (6426 lines of Postscript), from the line following "BODY" to the line preceding "ENDBODY", contains a total of 426991 characters. In the table below, this count is broken down by ASCII code; following one white space after the code is the corresponding character (intended for trouble shooting, in case things got garbled). 73138 lowercase letters 48416 uppercase letters 194894 digits 2 ASCII characters 9 6426 ASCII characters 10 36801 ASCII characters 32 68 ASCII characters 33 ! 8 ASCII characters 34 " 1 ASCII characters 35 # 125 ASCII characters 37 % 14 ASCII characters 39 ' 27256 ASCII characters 40 ( 27261 ASCII characters 41 ) 1 ASCII characters 42 * 566 ASCII characters 43 + 443 ASCII characters 44 , 307 ASCII characters 45 - 600 ASCII characters 46 . 140 ASCII characters 47 / 2029 ASCII characters 58 : 899 ASCII characters 59 ; 681 ASCII characters 60 < 677 ASCII characters 61 = 661 ASCII characters 62 > 15 ASCII characters 64 @ 155 ASCII characters 91 [ 4999 ASCII characters 92 \ 151 ASCII characters 93 ] 24 ASCII characters 94 ^ 20 ASCII characters 96 ` 105 ASCII characters 123 { 67 ASCII characters 125 } 41 ASCII characters 126 ~ For further information, please contact the authors: koch@math.utexas.edu wittwer@cgeuge11.bitnet BODY %!PS-Adobe-2.0 %%Creator: dvips 5.47 Copyright 1986-91 Radical Eye Software %%Pages: 32 1 %%BoundingBox: 0 0 612 792 %%EndComments %%BeginProcSet: tex.pro /TeXDict 200 dict def % define a working dictionary TeXDict begin % start using it. /N /def load def /B { bind def } N /S /exch load def /X { S N } B /TR /translate load N /isls false N /vsize 10 N /@rigin % -xps -yps @rigin - establishes dvips conventions { isls { [ 0 1 -1 0 0 0 ] concat } if 72 Resolution div 72 VResolution div neg scale Resolution VResolution vsize neg mul TR } B /@letter { /vsize 10 N } B /@landscape { /isls true N /vsize -1 N } B /@a4 { /vsize 10.6929133858 N } B /@legal { /vsize 13 N } B /@manualfeed { statusdict /manualfeed true put } B % n @copies - set number of copies /@copies { /#copies X } B /FMat [1 0 0 -1 0 0] N /FBB [0 0 0 0] N /df { /sf 1 N /fntrx FMat N df-tail } B /dfs { 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0 moveto } B /eop % - eop - -- end a page { % eop-aux % -- to observe VM usage clear SI restore showpage userdict /eop-hook known { eop-hook } if } B /@start % - @start - -- start everything { userdict /start-hook known { start-hook } if /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 0 1 255 {IE S 1 string dup 0 3 index put cvn put} for } B /p /show load N % the main character setting routine /DidV false def % Kludge for drawing vertical lines /RMat [ 1 0 0 -1 0 0 ] N % things we need for rules /BDot 8 string N /v { % can't use ...fill; it makes rules too big /ruley X /rulex X V } B /V { gsave TR -.1 -.1 TR rulex ruley scale 1 1 false RMat { BDot } imagemask grestore } B /a { moveto } B % absolute positioning /delta 0 N % we need a variable to hold space moves /tail { dup /delta X 0 rmoveto } B /M { S p delta add tail } B /b { S p tail } B % show and tail! /c { -4 M } B /d { -3 M } B /e { -2 M } B /f { -1 M } B /g { 0 M } B /h { 1 M } B /i { 2 M } B /j { 3 M } B /k { 4 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y(\026)1286 1008 y Fq(t)q(;)j(:)p Fv(\))p Fs(k)1385 1017 y Fu(\()p Fl(b)p Fu(\))1444 1008 y Fq(:)390 b Fv(\(2)p Fq(:)p Fv(8\))p 1807 1161 43 43 v 100 1261 a(Next,)21 b(w)o(e)e(consider)f(the)h(deriv)m(ativ)o(e)g(of)h Fs(N)26 b Fv(at)20 b(p)q(oin)o(ts)e Fq(f)25 b Fs(2)18 b(H)1316 1270 y Fu(\()p Fl(\014)r Fu(\))1394 1261 y Fv(for)h(whic)o(h)f(the)h(norm)f Fs(j)l(j)l(j)p Fq(f)5 b Fs(j)l(j)l(j)p Fv(,)0 1320 y(de\014ned)16 b(in)32 b(\(1.5\),)17 b(is)f(\014nite.)0 1415 y Fk(Prop)r(osition)26 b(2.2.)80 b Fj(Assume)22 b(that)h Fq(f)29 b Fj(is)22 b(a)g(function)h(in)f Fs(H)1256 1424 y Fu(\()p Fl(\014)r Fu(\))1337 1415 y Fj(with)g Fs(j)l(j)l(j)p Fq(f)5 b Fs(j)l(j)l(j)24 b Fq(<)g Fs(1)p Fj(.)41 b(Then)22 b(the)0 1474 y(op)q(erator)16 b Fq(A)p Fv(\()p Fq(f)5 b Fv(\))p Fj(,)18 b(de\014ned)e(b)o(y)g(the)h(equation)98 1575 y Fr(\000)121 1615 y Fq(A)p Fv(\()p Fq(f)5 b Fv(\))p Fq(g)251 1575 y Fr(\001)275 1615 y Fv(\()p Fq(t)p Fv(\))15 b(=)521 1581 y(1)p 404 1604 258 2 v 404 1614 a Fr(p)p 454 1614 208 2 v 42 x Fv(\(1)d Fs(\000)e Fq(\014)590 1642 y Fu(2)612 1656 y Fv(\))p Fq(\031)703 1537 y Fp(1)687 1547 y Fr(Z)665 1688 y Fp(\0001)747 1615 y Fq(ds)17 b(e)836 1587 y Fp(\000)907 1574 y Fo(1)p 873 1580 85 2 v 873 1605 a(1)p Fn(\000)p Fm(\014)938 1598 y Fo(2)964 1587 y Fu(\()p Fl(s)p Fp(\000)p Fl(\014)r(t)p Fu(\))1086 1572 y Fo(2)1106 1615 y Fq(f)5 b Fv(\()p Fq(s)p Fv(\))p Fq(g)r Fv(\()p Fq(s)p Fv(\))j Fq(;)61 b(g)15 b Fs(2)f(H)1494 1624 y Fu(\()p Fl(\014)r Fu(\))1561 1615 y Fq(;)24 b(t)14 b Fs(2)1700 1605 y Fn(j)1691 1615 y Fv(C)8 b Fq(;)99 b Fv(\(2)p Fq(:)p Fv(9\))0 1765 y Fj(is)17 b(a)g(b)q(ounded)f(linear)g(op)q(erator)h(on)g Fs(H)750 1774 y Fu(\()p Fl(\014)r Fu(\))808 1765 y Fj(,)g(whose)g(norm)f(is)h (less)g(than)g(or)f(equal)h(to)h Fs(j)l(j)l(j)p Fq(f)5 b Fs(j)l(j)l(j)p Fj(.)24 b(F)l(urther-)0 1824 y(more,)16 b(for)g(an)o(y)g Fq(g)r(;)8 b(h)13 b Fs(2)h(H)488 1833 y Fu(\()p Fl(\014)r Fu(\))546 1824 y Fj(,)j(the)f(follo)o(wing)g(holds.)507 1970 y Fs(h)p Fq(A)p Fv(\()p Fq(f)5 b Fv(\))p Fq(g)r(;)j(h)p Fs(i)726 1979 y Fu(\()p Fl(\014)r Fu(\))800 1970 y Fv(=)882 1936 y(1)p 858 1959 72 2 v 858 1969 a Fs(p)p 900 1969 31 2 v 36 x Fq(\031)971 1892 y Fp(1)955 1902 y Fr(Z)933 2043 y Fp(\0001)1015 1970 y Fq(dt)17 b(e)1099 1950 y Fp(\000)p Fl(t)1145 1935 y Fo(2)1167 1970 y Fq(f)5 b Fv(\()p Fq(t)p Fv(\))p Fq(g)r Fv(\()p Fq(t)p Fv(\))p 1334 1927 86 2 v Fq(h)p Fv(\()p Fq(t)p Fv(\))10 b Fq(:)380 b Fv(\(2)p Fq(:)p Fv(10\))0 2139 y Fk(Pro)r(of.)17 b Fv(Let)g Fq(f)287 2146 y Fm(H)r(T)359 2139 y Fs(\021)c Fv(1.)22 b(Then)16 b(for)g(all)h Fq(z)r(;)8 b(t)14 b Fs(2)896 2129 y Fn(j)887 2139 y Fv(C)j(w)o(e)f(ha)o(v)o(e)195 2200 y Fp(1)179 2215 y Fr(X)177 2320 y Fl(n)p Fu(=0)267 2228 y Fq(z)292 2210 y Fl(n)p 267 2250 53 2 v 271 2296 a Fq(n)p Fv(!)325 2221 y Fr(\000)348 2262 y Fq(A)p Fv(\()p Fq(f)428 2269 y Fm(H)r(T)486 2262 y Fv(\))p Fq(p)530 2269 y Fl(n)558 2221 y Fr(\001)581 2262 y Fv(\()p Fq(t)p Fv(\))e(=)704 2221 y Fr(\000)727 2262 y Fq(A)p Fv(\()p Fq(f)807 2269 y Fm(H)r(T)865 2262 y Fv(\))p Fq(G)923 2269 y Fl(z)946 2221 y Fr(\001)969 2262 y Fv(\()p Fq(t)p Fv(\))651 2443 y(=)826 2409 y(1)p 710 2431 258 2 v 710 2441 a Fr(p)p 760 2441 208 2 v 43 x Fv(\(1)d Fs(\000)g Fq(\014)896 2469 y Fu(2)918 2484 y Fv(\))p Fq(\031)1008 2365 y Fp(1)992 2375 y Fr(Z)971 2515 y Fp(\0001)1053 2443 y Fq(ds)16 b(e)1141 2415 y Fp(\000)1212 2401 y Fo(1)p 1179 2407 85 2 v 1179 2432 a(1)p Fn(\000)p Fm(\014)1244 2425 y Fo(2)1269 2415 y Fu(\()p Fl(s)p Fp(\000)p Fl(\014)r(t)p Fu(\))1391 2400 y Fo(2)1409 2415 y Fu(+)1440 2386 y Fp(p)p 1472 2386 20 2 v 1472 2415 a Fu(2)p Fl(z)q(s)p Fp(\000)p Fl(z)1582 2400 y Fo(2)1602 2415 y Fl(=)p Fu(2)651 2627 y Fv(=)e Fq(G)743 2634 y Fl(\014)r(z)790 2627 y Fv(\()p Fq(t)p Fv(\))h(=)931 2565 y Fp(1)915 2579 y Fr(X)913 2685 y Fl(n)p Fu(=0)1003 2593 y Fq(z)1028 2575 y Fl(n)p 1003 2615 53 2 v 1007 2661 a Fq(n)p Fv(!)1061 2627 y Fq(\014)1092 2606 y Fl(n)1119 2627 y Fq(p)1144 2634 y Fl(n)1171 2627 y Fv(\()p Fq(t)p Fv(\))8 b Fq(;)1823 2447 y Fv(\(2)p Fq(:)p Fv(11\))963 2790 y(6)p eop %%Page: 7 8 bop 0 50 a Fv(whic)o(h)16 b(implies)f(that)i Fq(A)p Fv(\()p Fq(f)500 57 y Fm(H)r(T)558 50 y Fv(\))p Fq(p)602 57 y Fl(n)643 50 y Fv(=)d Fq(\014)727 32 y Fl(n)754 50 y Fq(p)779 57 y Fl(n)822 50 y Fv(for)j(all)f Fq(n)p Fv(.)22 b(Hence)472 158 y Fs(h)p Fq(A)p Fv(\()p Fq(f)571 165 y Fm(H)r(T)630 158 y Fv(\))p Fq(p)674 165 y Fl(n)701 158 y Fq(;)8 b(p)748 165 y Fl(k)773 158 y Fs(i)792 167 y Fu(\()p Fl(\014)r Fu(\))864 158 y Fv(=)14 b Fq(\014)948 138 y Fl(n)975 158 y Fs(h)p Fq(p)1019 165 y Fl(n)1046 158 y Fq(;)8 b(p)1093 165 y Fl(k)1118 158 y Fs(i)1137 167 y Fu(\()p Fl(\014)r Fu(\))1209 158 y Fv(=)14 b Fs(h)p Fq(p)1306 165 y Fl(n)1333 158 y Fq(;)8 b(p)1380 165 y Fl(k)1405 158 y Fs(i)1424 167 y Fu(\(1\))1823 158 y Fv(\(2)p Fq(:)p Fv(12\))0 267 y(for)18 b(all)g Fq(n)h Fv(and)f Fq(k)r Fv(.)27 b(Let)19 b(no)o(w)f Fq(f)s(;)8 b(g)r(;)g(h)18 b Fv(b)q(e)h(arbitrary)e(p)q(olynomials.)26 b(Then)37 b(\(2.10\))19 b(is)f(obtained)g(from)0 326 y(\(2.12\))f(b)o(y)f 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Fq(>)f Fv(0)i Fj(for)g(all)0 1300 y Fq(t)e Fs(2)g Fv(I)-10 b(R)p Fj(.)22 b(Then)16 b Fq(D)q Fs(N)7 b Fv(\()p Fq(f)e Fv(\))19 b Fj(is)d(a)g(p)q(ositiv)o(e)g(trace{class)g(op)q (erator)g(on)g Fs(H)1303 1309 y Fu(\()p Fl(\014)r Fu(\))1361 1300 y Fj(,)h(and)f Fv(tr\()p Fq(D)q Fs(N)7 b Fv(\()p Fq(f)e Fv(\)\))16 b(=)e(2)p Fs(j)l(j)l(j)p Fq(f)5 b Fs(j)l(j)l(j)p Fj(.)0 1414 y Fk(Pro)r(of.)18 b Fv(Note)h(\014rst)e(that)h Fq(D)q Fs(N)7 b Fv(\()p Fq(f)e Fv(\))18 b(=)d(2)p Fq(A)p Fv(\()p Fq(f)5 b Fv(\).)28 b(It)18 b(is)f(clear)g(from)35 b(Prop)q(osition)16 b(2.2)i(that,)g(giv)o(en)f(our)0 1474 y(assumptions)e(on)j Fq(f)5 b Fv(,)19 b Fq(A)p Fv(\()p Fq(f)5 b Fv(\))20 b(is)d(self{adjoin)o(t)g (and)g(p)q(ositiv)o(e.)26 b(The)17 b(trace)h(of)g Fq(A)p Fv(\()p Fq(f)5 b Fv(\))20 b(can)e(b)q(e)g(computed)0 1533 y(b)o(y)e(using)g(equation) 33 b(\(2.3\):)166 1697 y(tr)205 1657 y Fr(\000)227 1697 y Fq(A)p Fv(\()p Fq(f)5 b Fv(\))331 1657 y Fr(\001)370 1697 y Fv(=)440 1635 y Fp(1)424 1650 y Fr(X)423 1756 y 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Fl(\014)887 1897 y Fv(\()p Fq(t;)j(t)p Fv(\))15 b(=)f Fs(j)l(j)l(j)p Fq(f)5 b Fs(j)l(j)l(j)j Fq(:)1823 1802 y Fv(\(2)p Fq(:)p Fv(15\))p 1807 2110 43 43 v 100 2221 a(F)l(rom)14 b(no)o(w)h(on)g(w)o(e)h(restrict)f (our)g(atten)o(tion)g(to)h(a)g(subspace)e(of)i Fs(H)1352 2230 y Fu(\()p Fl(\014)r Fu(\))1426 2221 y Fv(whic)o(h)f(is)g(in)o(v)m(arian)o(t)f (under)0 2280 y(the)j(action)f(of)h Fs(N)7 b Fv(.)0 2375 y Fk(De\014nition)17 b(2.4.)42 b Fj(Denote)14 b(b)o(y)f Fs(H)672 2357 y Fl(e)672 2391 y Fu(\()p Fl(\014)r Fu(\))744 2375 y Fj(the)h(real)f (subspace)f(of)i Fs(H)1222 2384 y Fu(\()p Fl(\014)r Fu(\))1294 2375 y Fj(consisting)e(of)i(all)f(ev)o(en)h(functions)0 2439 y(in)j Fs(H)101 2448 y Fu(\()p Fl(\014)r Fu(\))177 2439 y Fj(whic)o(h)g(tak)o (e)h(real)f(v)m(alues)h(when)f(restricted)g(to)h(the)g(real)f(axis.)25 b(T)l(o)18 b(ev)o(ery)f(suc)o(h)g(function)g Fq(f)0 2499 y Fj(w)o(e)f(asso)q(ciate)h(an)f Fq(`)373 2481 y Fu(2)412 2499 y Fj(sequence)g Fv(\()p Fq(f)661 2506 y Fu(0)684 2499 y Fq(;)8 b(f)730 2506 y Fu(1)753 2499 y Fq(;)g(:)g(:)g(:)p Fv(\))17 b Fj(b)o(y)g(de\014ning)494 2634 y Fq(f)518 2641 y Fl(n)560 2634 y Fv(=)668 2600 y Fq(\014)699 2582 y Fl(n)p 618 2622 158 2 v 618 2632 a Fr(p)p 668 2632 108 2 v 43 x Fv(\(2)p Fq(n)p Fv(\)!)782 2634 y Fs(h)p Fq(f)s(;)8 b(p)875 2641 y Fu(2)p Fl(n)922 2634 y Fs(i)941 2643 y Fu(\()p Fl(\014)r Fu(\))1008 2634 y Fq(;)108 b(n)13 b Fv(=)h(0)p Fq(;)8 b Fv(1)p Fq(;)g Fv(2)p Fq(;)g(:)g(:)g(:)17 b(:)367 b Fv(\(2)p Fq(:)p Fv(16\))963 2790 y(7)p eop %%Page: 8 9 bop 0 50 a Fk(Lemma)18 b(2.5.)51 b Fj(Assume)17 b(that)h Fq(f)s(;)8 b(g)19 b Fj(are)f(functions)e(in)i Fs(H)1134 32 y Fl(e)1134 66 y Fu(\()p Fl(\014)r Fu(\))1192 50 y Fj(,)g(and)f(that)h Fs(j)l(j)l(j)p Fq(f)5 b Fs(j)l(j)l(j)16 b Fq(<)f Fs(1)p Fj(.)25 b(Then)18 b Fq(A)p Fv(\()p Fq(f)5 b Fv(\))p Fq(g)0 114 y Fj(lies)16 b(in)g Fs(H)186 96 y Fl(e)186 130 y Fu(\()p Fl(\014)r Fu(\))244 114 y Fj(,)h(and)439 240 y Fr(\000)462 281 y Fq(A)p Fv(\()p Fq(f)5 b Fv(\))p Fq(g)592 240 y Fr(\001)616 301 y Fl(k)655 281 y Fv(=)748 218 y Fp(1)732 233 y Fr(X)707 339 y Fl(m;n)p Fu(=0)838 281 y Fq(A)875 255 y Fu(\()p Fl(m)p Fu(\))875 296 y Fl(k)q(;n)944 281 y Fq(f)968 288 y Fl(m)1006 281 y Fq(g)1030 288 y Fl(n)1065 281 y Fq(;)108 b(k)16 b Fv(=)d(0)p Fq(;)8 b Fv(1)p Fq(;)g Fv(2)p Fq(;)g(:)g(:)g(:)17 b(;)312 b Fv(\(2)p Fq(:)p Fv(17\))0 462 y Fj(where)16 b Fq(A)181 436 y Fu(\()p Fl(m)p Fu(\))181 477 y Fl(k)q(;n)264 462 y Fv(=)e(0)i Fj(if)h(either)f Fq(k)g(>)d(m)e Fv(+)g Fq(n)p Fj(,)16 b(or)g Fq(m)e(>)f(n)e Fv(+)g Fq(k)r Fj(,)16 b(or)g Fq(n)e(>)f(k)g Fv(+)e Fq(m)p Fj(;)16 b(otherwise)491 608 y Fq(A)528 582 y Fu(\()p Fl(m)p Fu(\))528 623 y Fl(k)q(;n)612 608 y Fv(=)d Fq(\014)695 587 y Fl(k)q Fu(+)p Fl(m)p Fu(+)p Fl(n)841 548 y Fr(q)p 890 548 569 2 v 890 567 a(\000)964 585 y Fu(2)p Fl(k)913 625 y(k)q Fu(+)p Fl(m)p Fp(\000)p Fl(n)1057 567 y Fr(\001\000)1147 585 y Fu(2)p Fl(m)1103 625 y(m)p Fu(+)p Fl(n)p Fp(\000)p Fl(k)1246 567 y Fr(\001\000)1342 585 y Fu(2)p Fl(n)1292 625 y(n)p Fu(+)p Fl(k)q Fp(\000)p Fl(m)1436 567 y Fr(\001)612 736 y Fs(\024)g Fq(\014)695 716 y Fl(k)q Fu(+)p Fl(m)p Fu(+)p Fl(n)847 659 y Fr(p)p 896 659 105 2 v 896 702 a Fv(\(2)p Fq(k)r Fv(\)!)p 847 725 155 2 v 903 770 a Fq(k)r Fv(!)1016 666 y Fr(\022)1052 703 y Fq(m)e Fv(+)g Fq(n)1106 770 y(k)1187 666 y Fr(\023)1232 736 y Fq(:)1823 676 y Fv(\(2)p Fq(:)p Fv(18\))0 906 y Fk(Pro)r(of.)17 b Fv(F)l(or)f Fq(k)r(;)8 b(m;)g(n)13 b Fv(=)g(0)p Fq(;)8 b Fv(1)p Fq(;)g Fv(2)p Fq(;)g(:)g(:)g(:)h Fv(,)16 b(de\014ne)400 1052 y Fq(A)437 1026 y Fu(\()p Fl(m)p Fu(\))437 1067 y Fl(k)q(;n)520 1052 y Fv(=)629 1018 y Fq(\014)660 1000 y Fl(k)p 579 1040 V 579 1050 a Fr(p)p 629 1050 105 2 v 43 x Fv(\(2)p Fq(k)r Fv(\)!)805 1018 y Fq(\014)836 1000 y Fl(m)p 754 1040 172 2 v 754 1050 a Fr(p)p 804 1050 122 2 v 43 x Fv(\(2)p Fq(m)p Fv(\)!)995 1018 y Fq(\014)1026 1000 y Fl(n)p 945 1040 158 2 v 945 1050 a Fr(p)p 995 1050 108 2 v 43 x Fv(\(2)p Fq(n)p Fv(\)!)1108 1052 y Fs(h)p Fq(p)1152 1059 y Fu(2)p Fl(k)1197 1052 y Fq(;)8 b(A)p Fv(\()p Fq(p)1300 1059 y Fu(2)p Fl(m)1359 1052 y Fv(\))p Fq(p)1403 1059 y Fu(2)p Fl(n)1450 1052 y Fs(i)1469 1061 y Fu(\()p Fl(\014)r Fu(\))1536 1052 y Fq(:)273 b Fv(\(2)p Fq(:)p Fv(19\))0 1205 y(Then,)13 b(with)g(the)g(exception)g(of)g(the)g(last)g (inequalit)o(y)f(in)26 b(\(2.18\),)13 b(the)h(assertion)d(follo)o(ws)h(from) 25 b(Prop)q(o-)0 1264 y(sition)16 b(2.2,)g(since)127 1397 y(1)p 103 1420 72 2 v 103 1430 a Fs(p)p 145 1430 31 2 v 36 x Fq(\031)216 1353 y Fp(1)200 1363 y Fr(Z)178 1504 y Fp(\0001)260 1431 y Fq(dt)8 b(e)335 1411 y Fp(\000)p Fl(t)381 1396 y Fo(2)404 1431 y Fq(p)429 1438 y Fu(2)p Fl(k)473 1431 y Fv(\()p Fq(t)p Fv(\))p Fq(p)554 1438 y Fu(2)p Fl(m)613 1431 y Fv(\()p Fq(t)p Fv(\))p Fq(p)694 1438 y Fu(2)p Fl(n)742 1431 y Fv(\()p Fq(t)p Fv(\))14 b(=)1100 1397 y(\(2)p Fq(k)r Fv(\)!)j(\(2)p Fq(m)p Fv(\)!)g(\(2)p Fq(n)p Fv(\)!)p 871 1420 827 2 v 871 1465 a(\()p Fq(k)c Fv(+)d Fq(m)h Fs(\000)g Fq(n)p Fv(\)!\()p Fq(m)g Fv(+)g Fq(n)g Fs(\000)g Fq(k)r Fv(\)!\()p Fq(n)g Fv(+)g Fq(k)h Fs(\000)f Fq(m)p Fv(\)!)1711 1431 y Fq(:)98 b Fv(\(2)p Fq(:)p Fv(20\))0 1610 y(Here,)17 b Fs(j)p Fq(m)10 b Fs(\000)h Fq(n)p Fs(j)i(\024)h Fq(k)h Fs(\024)f Fq(m)c Fv(+)h Fq(n)d Fv(;)17 b(otherwise)e(the)i(in)o(tegral)e(is)h(zero.)22 b(T)l(o)16 b(pro)o(v)o(e)g(the)g(last)g(inequalit)o(y)g(in)0 1679 y(\(2.18\),)h(it)g(su\016ces)e(to)i(sho)o(w)e(that)i Fq(A)706 1654 y Fu(\()p Fl(m)p Fu(\))706 1694 y Fl(k)q(;n)784 1679 y Fv(,)f(if)h(nonzero,)e(is)h(b)q(ounded)g(b)o(y)594 1825 y Fq(X)639 1799 y Fu(\()p Fl(m)p Fu(\))635 1840 y Fl(k)q(;n)722 1825 y Fs(\021)d Fq(\014)805 1804 y Fl(k)q Fu(+)p Fl(m)p Fu(+)p Fl(n)951 1765 y Fr(q)p 1000 1765 190 2 v 1000 1784 a(\000)1074 1803 y Fu(2)p Fl(k)1023 1842 y(k)q Fu(+)p Fl(m)p Fp(\000)p Fl(n)1167 1784 y Fr(\001)8 b(\000)1221 1803 y Fl(m)p Fu(+)p Fl(n)1255 1842 y(k)1311 1784 y Fr(\001)1343 1825 y Fq(:)466 b Fv(\(2)p Fq(:)p Fv(21\))0 1946 y(But)177 2036 y Fr(\002)198 2077 y Fq(A)235 2051 y Fu(\()p Fl(m)p Fu(\))235 2092 y Fl(k)q(;n)304 2036 y Fr(\003)325 2047 y Fu(2)361 2077 y Fv(=)414 2036 y Fr(\002)434 2077 y Fq(X)479 2051 y Fu(\()p Fl(m)p Fu(\))475 2092 y Fl(k)q(;n)549 2036 y Fr(\003)569 2047 y Fu(2)670 2043 y Fv(\(2)p Fq(m)p Fv(\)!)17 b(\(2)p Fq(n)p Fv(\)!)p 606 2065 375 2 v 606 2111 a(\()p Fq(m)11 b Fv(+)g Fq(n)p Fv(\)!\()p Fq(m)g Fv(+)g Fq(n)p Fv(\)!)1251 2043 y Fq(k)r Fv(!)16 b Fq(k)r Fv(!)p 1025 2065 551 2 v 1025 2111 a(\()p Fq(k)d Fv(+)e Fq(m)g Fs(\000)g Fq(n)p Fv(\)!\()p Fq(k)i Fv(+)e Fq(n)f Fs(\000)h Fq(m)p Fv(\)!)361 2241 y(=)414 2200 y Fr(\002)434 2241 y Fq(X)479 2215 y Fu(\()p Fl(m)p Fu(\))475 2256 y Fl(k)q(;n)549 2200 y Fr(\003)569 2211 y Fu(2)600 2174 y Fp(j)p Fl(m)p Fp(\000)p Fl(n)p Fp(j)625 2193 y Fr(Y)623 2300 y Fl(j)r Fu(=1)841 2207 y Fv(\()p Fq(m)g Fv(+)g Fq(n)p Fv(\))g(+)g Fq(j)p 729 2229 481 2 v 729 2275 a Fv(\()p Fq(m)g Fv(+)g Fq(n)p Fv(\))g Fs(\000)g(j)p Fq(m)g Fs(\000)g Fq(n)p Fs(j)f Fv(+)h Fq(j)1255 2207 y(k)h Fs(\000)f(j)p Fq(m)g Fs(\000)g Fq(n)p Fs(j)f Fv(+)h Fq(j)p 1255 2229 335 2 v 1366 2275 a(k)i Fv(+)e Fq(j)361 2429 y Fv(=)414 2389 y Fr(\002)434 2429 y Fq(X)479 2403 y Fu(\()p Fl(m)p Fu(\))475 2444 y Fl(k)q(;n)549 2389 y Fr(\003)569 2399 y Fu(2)600 2362 y Fp(j)p Fl(m)p Fp(\000)p Fl(n)p Fp(j)625 2382 y Fr(Y)623 2488 y Fl(j)r Fu(=1)729 2395 y Fv(\()p Fq(m)g Fv(+)g Fq(n)g Fv(+)g Fq(j)s Fv(\)\()p Fq(k)h Fv(+)f Fq(j)s Fv(\))g Fs(\000)g(j)p Fq(m)g Fs(\000)f Fq(n)p Fs(j)p Fv(\()p Fq(m)h Fv(+)g Fq(n)g Fv(+)g Fq(j)s Fv(\))p 729 2417 889 2 v 782 2463 a(\()p Fq(m)h Fv(+)e Fq(n)h Fv(+)g Fq(j)s Fv(\)\()p Fq(k)i Fv(+)e Fq(j)s Fv(\))g Fs(\000)f(j)p Fq(m)h Fs(\000)g Fq(n)p Fs(j)p Fv(\()p Fq(k)i Fv(+)d Fq(j)s Fv(\))1632 2429 y Fq(;)1823 2265 y Fv(\(2)p Fq(:)p Fv(22\))0 2596 y(and)16 b(eac)o(h)g(factor)h(in)f(the)h(last)f(pro)q(duct)g(is)g(clearly)24 b Fs(\024)14 b Fv(1,)i(whenev)o(er)g Fs(j)p Fq(m)11 b Fs(\000)g Fq(n)p Fs(j)i(\024)h Fq(k)h Fs(\024)f Fq(m)c Fv(+)h Fq(n)p Fv(.)p 1807 2674 43 43 v 963 2790 a(8)p eop %%Page: 9 10 bop 0 52 a Fk(Remark.)48 b Fv(Since)18 b(the)h(co)q(e\016cien)o(ts)f Fq(A)775 26 y Fu(\()p Fl(m)p Fu(\))775 67 y Fl(k)q(;n)863 52 y Fv(are)g(symmetric)g(under)f(p)q(erm)o(utations)g(of)h(the)h(indices,)0 111 y(the)i(\014xed)f(p)q(oin)o(t)g(equation)g(for)g Fs(N)28 b Fv(is)20 b(formally)g(the)g(equation)h(for)f(the)h(stationary)f(p)q(oin)o (ts)f(of)i(the)0 171 y(functional,)694 272 y Fq(f)e Fs(7!)859 209 y Fp(1)843 224 y Fr(X)801 332 y Fl(k)q(;m;n)p Fu(=0)965 272 y Fq(A)1002 246 y Fu(\()p Fl(m)p Fu(\))1002 287 y Fl(k)q(;n)1072 272 y Fq(f)1096 279 y Fl(k)1120 272 y Fq(f)1144 279 y Fl(m)1183 272 y Fq(f)1207 279 y Fl(n)1243 272 y Fq(;)566 b Fv(\(2)p Fq(:)p Fv(23\))0 428 y(on)16 b(a)h(sphere)266 390 y Fr(P)319 443 y Fl(k)351 428 y Fs(j)p Fq(f)389 435 y Fl(k)414 428 y Fs(j)428 410 y Fu(2)464 428 y Fv(=)d Fq(const:)100 521 y Fv(Since)d(it)i(is)e (di\016cult)g(in)h(general)f(to)i(\014nd)e(go)q(o)q(d)h(estimates)g(for)f (the)i(norm)d(of)j(a)f(b)q(ounded)f(op)q(erator)0 580 y(on)16 b Fq(`)90 562 y Fu(2)112 580 y Fv(,)h(w)o(e)f(will)g(con)o(tin)o(ue)f(our)h (analysis)f(of)i Fs(N)24 b Fv(on)16 b(w)o(eigh)o(ted)g Fq(`)1205 562 y Fu(1)1243 580 y Fv(spaces.)0 677 y Fk(De\014nition)22 b(2.6.)52 b Fj(F)l(or)17 b(an)o(y)h(giv)o(en)f Fq(\032)g Fs(\025)f Fv(1)p Fj(,)i(denote)g(b)o(y)f Fs(B)1149 684 y Fl(\032)1190 677 y Fj(the)i(v)o(ector)f(space)f(of)i(all)e(functions)g(in)0 736 y Fs(H)42 718 y Fl(e)42 752 y Fu(\()p Fl(\014)r Fu(\))100 736 y Fj(,)g(for)f(whic)o(h)f(the)i(sum)783 852 y Fs(k)p Fq(f)5 b Fs(k)862 859 y Fl(\032)899 852 y Fs(\021)970 790 y Fp(1)953 805 y Fr(X)952 911 y Fl(n)p Fu(=0)1035 852 y Fs(j)p Fq(f)1073 859 y Fl(n)1101 852 y Fs(j)p Fq(\032)1141 832 y Fl(n)1823 852 y Fv(\(2)p Fq(:)p Fv(24\))0 997 y Fj(is)16 b(\014nite.)22 b(Equipp)q(ed)16 b(with)g(the)h(norm)e Fs(k)p Fq(:)p Fs(k)815 1004 y Fl(\032)838 997 y Fj(,)h Fs(B)901 1004 y Fl(\032)941 997 y Fj(is)g(a)g(Banac)o(h)g (space.)0 1126 y Fk(Lemma)k(2.7.)56 b Fj(Assume)17 b(that)i Fq(r)h Fs(\025)d Fv(1)h Fj(and)g Fq(\032)g Fs(\021)f Fq(\014)e Fv(+)d(2)p Fq(\014)1138 1108 y Fu(2)1160 1126 y Fq(r)19 b Fs(\025)f Fv(1)p Fj(.)28 b(If)19 b Fq(f)24 b Fj(is)18 b(a)h(function)f(in)h Fs(B)1812 1133 y Fl(\032)1853 1126 y Fj(then)0 1185 y Fq(A)p Fv(\()p Fq(f)5 b Fv(\))18 b Fj(is)f(a)f(b)q(ounded)f(linear)h(op)q(erator)g (from)g Fs(B)908 1192 y Fl(\032)947 1185 y Fj(to)h Fs(B)1041 1192 y Fl(r)1063 1185 y Fj(,)f(and)574 1309 y Fs(k)p Fq(A)p Fv(\()p Fq(f)5 b Fv(\))p Fq(g)r Fs(k)754 1316 y Fl(r)791 1309 y Fs(\024)14 b(k)p Fq(f)5 b Fs(k)923 1316 y Fl(\032)955 1309 y Fs(k)p Fq(g)r Fs(k)1031 1316 y Fl(\032)1062 1309 y Fq(;)108 b Fs(8)p Fq(g)14 b Fs(2)g(B)1331 1316 y Fl(\032)1362 1309 y Fq(:)447 b Fv(\(2)p Fq(:)p Fv(25\))0 1472 y Fk(Pro)r(of.)17 b Fv(By)g(using)f(the)g(b)q(ound)33 b(\(2.18\))17 b(and)e(the)i(fact)g(that) 1171 1432 y Fr(\000)1194 1450 y Fu(2)p Fl(k)1204 1489 y(k)1236 1432 y Fr(\001)1273 1472 y Fs(\024)c Fv(4)1350 1454 y Fl(k)1375 1472 y Fv(,)j(for)g(all)h Fq(k)e Fs(\025)f Fv(0,)i(w)o(e)g(obtain)354 1641 y Fs(k)p Fq(A)p Fv(\()p Fq(f)5 b Fv(\))p Fq(g)r Fs(k)534 1648 y Fl(r)571 1641 y Fv(=)640 1578 y Fp(1)624 1593 y Fr(X)623 1701 y Fl(k)q Fu(=0)710 1598 y Fr(\014)710 1628 y(\014)740 1600 y(\000)763 1641 y Fq(A)p Fv(\()p Fq(f)g Fv(\))p Fq(g)893 1600 y Fr(\001)917 1661 y Fl(k)956 1598 y Fr(\014)956 1628 y(\014)986 1641 y Fq(r)1009 1620 y Fl(k)571 1814 y Fs(\024)665 1752 y Fp(1)649 1767 y Fr(X)623 1873 y Fl(m;n)p Fu(=0)754 1752 y Fl(m)p Fu(+)p Fl(n)763 1767 y Fr(X)763 1874 y Fl(k)q Fu(=0)853 1814 y Fq(\014)884 1794 y Fl(k)q Fu(+)p Fl(m)p Fu(+)p Fl(n)1029 1752 y Fr(q)p 1079 1752 88 2 v 22 x(\000)1102 1792 y Fu(2)p Fl(k)1112 1832 y(k)1144 1774 y Fr(\001)j(\000)1198 1792 y Fl(m)p Fu(+)p Fl(n)1232 1832 y(k)1288 1774 y Fr(\001)1311 1814 y Fs(j)p Fq(f)1349 1821 y Fl(m)1387 1814 y Fs(j)17 b(j)p Fq(g)1456 1821 y Fl(n)1482 1814 y Fs(j)8 b Fq(r)1527 1794 y Fl(k)571 1993 y Fs(\024)665 1931 y Fp(1)649 1946 y Fr(X)623 2051 y Fl(m;n)p Fu(=0)754 1993 y Fs(j)p Fq(f)792 2000 y Fl(m)830 1993 y Fs(j)17 b(j)p Fq(g)899 2000 y Fl(n)925 1993 y Fs(j)947 1931 y Fl(m)p Fu(+)p Fl(n)957 1946 y Fr(X)956 2053 y Fl(k)q Fu(=0)1046 1923 y Fr(\022)1083 1959 y Fq(m)11 b Fv(+)f Fq(n)1136 2027 y(k)1217 1923 y Fr(\023)1254 1993 y Fq(\014)1285 1972 y Fl(m)p Fu(+)p Fl(n)p Fp(\000)p Fl(k)1431 1993 y Fv(\(2)p Fq(\014)1506 1972 y Fu(2)1528 1993 y Fq(r)q Fv(\))1570 1972 y Fl(k)571 2165 y Fv(=)665 2103 y Fp(1)649 2118 y Fr(X)623 2223 y Fl(m;n)p Fu(=0)754 2165 y Fs(j)p Fq(f)792 2172 y Fl(m)830 2165 y Fs(j)17 b(j)p Fq(g)899 2172 y Fl(n)925 2165 y Fs(j)p Fv(\()p Fq(\014)d Fv(+)d(2)p Fq(\014)1106 2145 y Fu(2)1128 2165 y Fq(r)q Fv(\))1170 2145 y Fl(m)p Fu(+)p Fl(n)1278 2165 y Fv(=)j Fs(k)p Fq(f)5 b Fs(k)1410 2172 y Fl(\032)1442 2165 y Fs(k)p Fq(g)r Fs(k)1518 2172 y Fl(\032)1549 2165 y Fq(:)1823 1909 y Fv(\(2)p Fq(:)p Fv(26\))p 1807 2384 43 43 v 963 2790 a(9)p eop %%Page: 10 11 bop 0 50 a Fw(3.)28 b(Existence)e(of)h(a)h(Non{Gaussian)e(Fixed)h(P)n(oin)n (t)0 121 y Fv(In)17 b(this)g(section)g(w)o(e)h(pro)o(v)o(e)e(the)i(existence) 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Fp(\003)1034 1524 y Fv(\))15 b(=)e Fq(h)1149 1503 y Fp(\003)1172 1524 y Fv(\()p Fq(g)1217 1503 y Fp(\003)1240 1524 y Fv(\))e(+)g Fq(g)1346 1503 y Fp(\003)1382 1524 y Fv(=)j Fq(f)1464 1503 y Fp(\003)1487 1524 y Fq(:)1848 1484 y Fv(\(3)p Fq(:)p Fv(5\))0 1624 y Fk(Remark.)32 b Fv(F)l(or)11 b(a)h(purely)e(n)o(umerical)g(computation)g(of)i Fq(f)1108 1606 y Fp(\003)1132 1624 y Fv(,)g(the)g(pro)q(cedure)f(outlined)g(ab)q(o)o(v) o(e)g(already)0 1683 y(w)o(orks)16 b(for)h Fq(d)d Fv(=)h(4.)23 b(Numerically)l(,)16 b(the)h(three)g(largest)f(eigen)o(v)m(alues)g(of)i Fq(D)q Fs(N)7 b Fv(\()p Fq(f)1494 1665 y Fp(\003)1518 1683 y Fv(\))18 b(are)f(appro)o(ximately)0 1743 y(2,)e(1)p Fq(:)p Fv(427,)g(and)f(0)p Fq(:)p Fv(859.)21 b(Th)o(us,)14 b(ev)o(en)h(a)g(pro)s (jection)f Fq(P)22 b Fv(of)16 b(rank)e(t)o(w)o(o)h(\(whic)o(h)f(mak)o(es)g Fq(B)k Fv(a)d(map)f(on)h(I)-10 b(R)1908 1721 y Fu(2)1931 1743 y Fv(\))0 1802 y(should)15 b(do)h(the)h(job,)g(if)f(c)o(hosen)g 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Fj(Let)17 b Fq(\024)c Fv(=)h(0)p Fq(:)p Fv(46)8 b Fj(.)22 b(Then)16 b(for)g(an)o(y)g(t)o(w)o(o)g (functions)g Fq(h;)1314 2192 y Fv(~)1314 2205 y Fq(h)g Fj(in)g Fv(\(I)c Fs(\000)e Fq(P)d Fv(\))p Fs(B)1606 2212 y Fl(\032)1646 2205 y Fj(w)o(e)17 b(ha)o(v)o(e)19 2277 y Fv(\()p Fq(i)p Fv(\))100 2234 y Fr(\014)100 2264 y(\014)130 2236 y(\000)153 2277 y Fq(A)p Fv(\()p Fq(h)p Fv(\))258 2263 y(~)257 2277 y Fq(h)286 2236 y Fr(\001)309 2296 y Fl(k)348 2234 y Fr(\014)348 2264 y(\014)364 2277 y Fs(\024)d Fq(\032)443 2258 y Fp(\000)p Fu(14)516 2277 y Fq(A)553 2251 y Fu(\(7\))553 2292 y Fl(k)q(;)p Fu(7)618 2277 y Fs(k)p Fq(h)p Fs(k)697 2284 y Fl(\032)719 2277 y Fs(k)745 2263 y Fv(~)744 2277 y Fq(h)p Fs(k)798 2284 y Fl(\032)829 2277 y Fq(;)58 b(k)16 b Fv(=)d(0)p Fq(;)8 b Fv(1)p Fq(;)g(:)g(:)g(:)h(;)f Fv(6)p Fj(.)2 2350 y Fv(\()p Fq(ii)p Fv(\))26 b Fs(k)p Fv(\(I)11 b Fs(\000)g Fq(P)c Fv(\))p Fq(A)p Fv(\()p Fq(h)p Fv(\))386 2337 y(~)385 2350 y Fq(h)q Fs(k)440 2357 y Fl(\032)476 2350 y Fs(\024)14 b Fq(\024)p Fs(k)p Fq(h)p Fs(k)637 2357 y Fl(\032)659 2350 y Fs(k)685 2337 y Fv(~)684 2350 y Fq(h)p Fs(k)738 2357 y Fl(\032)769 2350 y Fq(:)0 2462 y Fk(Pro)r(of.)h Fv(Without)g(loss)f(of)h (generalit)o(y)g(w)o(e)f(ma)o(y)h(assume)e(that)j Fq(h)e Fv(and)1342 2449 y(~)1341 2462 y Fq(h)h Fv(are)g(of)g(norm)e(one.)22 b(T)l(o)14 b(pro)o(v)o(e)0 2522 y(\()p Fq(i)p Fv(\),)j(w)o(e)g(use)f(that)h(for)f Fq(k)f Fs(\024)f Fv(6)i(and)g Fq(i)e Fs(\025)g Fv(14,)240 2643 y Fq(\032)266 2622 y Fp(\000)p Fu(\()p Fl(i)p Fu(+1\))395 2643 y Fq(\014)426 2622 y Fl(k)q Fu(+\()p Fl(i)p Fu(+1\))576 2572 y Fr(\022)613 2609 y Fq(i)d Fv(+)g(1)651 2677 y Fq(k)716 2572 y Fr(\023)766 2643 y Fv(=)j Fq(\032)845 2622 y Fp(\000)p Fl(i)892 2643 y Fq(\014)923 2622 y Fl(k)q Fu(+)p Fl(i)992 2572 y Fr(\022)1034 2609 y Fq(i)1029 2677 y(k)1057 2572 y Fr(\023)1143 2609 y Fq(i)e Fv(+)e(1)p 1099 2631 192 2 v 1099 2677 a Fq(i)h Fv(+)g(1)g Fs(\000)g Fq(k)1319 2609 y(\014)p 1319 2631 31 2 v 1322 2677 a(\032)1370 2643 y Fs(\024)i Fq(\032)1448 2622 y Fp(\000)p Fl(i)1496 2643 y Fq(\014)1527 2622 y Fl(k)q Fu(+)p Fl(i)1596 2572 y Fr(\022)1638 2609 y Fq(i)1632 2677 y(k)1660 2572 y Fr(\023)1697 2643 y Fq(:)137 b Fv(\(3)p Fq(:)p Fv(6\))950 2790 y(10)p eop %%Page: 11 12 bop 0 50 a Fv(The)16 b(last)h(inequalit)o(y)f(in)33 b(\(3.6\))17 b(follo)o(ws)e(since)643 199 y Fq(\032)f(>)795 165 y Fv(2)p Fq(\014)p 741 187 164 2 v 741 233 a Fv(2)d Fs(\000)g Fv(4)p Fq(\014)883 218 y Fu(2)925 199 y Fv(=)1042 165 y(2)p Fq(\014)p 983 187 174 2 v 983 234 a Fv(2)g Fs(\000)g Fv(2)1094 219 y Fu(1)p Fl(=)p Fu(3)1177 199 y Fq(>)i Fv(2)p Fq(\014)e(:)541 b Fv(\(3)p Fq(:)p Fv(7\))0 348 y(By)17 b(com)o(bining)e(the)h(b)q(ounds)32 b(\(2.18\))17 b(and)33 b(\(3.6\),)16 b(w)o(e)h(\014nd)e(that)323 452 y Fr(\014)323 482 y(\014)353 454 y(\000)376 494 y Fq(A)p Fv(\()p Fq(h)p Fv(\))481 481 y(~)480 494 y Fq(h)510 454 y Fr(\001)532 514 y Fl(k)571 452 y Fr(\014)571 482 y(\014)601 494 y Fs(\024)38 b Fv(sup)654 536 y Fl(m;n)p Fp(\025)p Fu(7)785 494 y Fq(\032)811 474 y Fp(\000)p Fu(\()p Fl(m)p Fu(+)p Fl(n)p Fu(\))966 494 y Fq(A)1003 468 y Fu(\()p Fl(m)p Fu(\))1003 509 y Fl(k)q(;n)1086 494 y Fs(\024)19 b Fv(sup)1139 536 y Fl(i)p Fp(\025)p Fu(14)1240 494 y Fq(\032)1266 474 y Fp(\000)p Fl(i)1314 494 y Fq(\014)1345 474 y Fl(k)q Fu(+)p Fl(i)1414 432 y Fr(q)p 1464 432 88 2 v 22 x(\000)1486 472 y Fu(2)p Fl(k)1496 511 y(k)1528 454 y Fr(\001)8 b(\000)1586 472 y Fl(i)1582 511 y(k)1604 454 y Fr(\001)601 634 y Fs(\024)14 b Fq(\032)680 613 y Fp(\000)p Fu(14)753 634 y Fq(\014)784 613 y Fl(k)q Fu(+14)879 572 y Fr(q)p 928 572 V 928 593 a(\000)951 612 y Fu(2)p Fl(k)961 651 y(k)993 593 y Fr(\001)8 b(\000)1047 612 y Fu(14)1056 651 y Fl(k)1087 593 y Fr(\001)1123 634 y Fv(=)14 b Fq(\032)1202 613 y Fp(\000)p Fu(14)1275 634 y Fq(A)1312 608 y Fu(\(7\))1312 649 y Fl(k)q(;)p Fu(7)1377 634 y Fq(;)1848 556 y Fv(\(3)p Fq(:)p Fv(8\))0 765 y(and)i(\()p Fq(i)p Fv(\))h(follo)o(ws.)k(T)l(o)c(pro)o(v)o(e)e(\()p Fq(ii)p Fv(\),)i(w)o(e)g(use)f(again)g(the)h(b)q(ound)32 b(\(2.18\):)264 934 y Fs(k)p Fv(\(I)11 b Fs(\000)g Fq(P)c Fv(\))p Fq(A)p Fv(\()p Fq(h)p Fv(\))550 921 y(~)549 934 y Fq(h)q Fs(k)604 941 y Fl(\032)641 934 y Fs(\024)37 b Fv(sup)693 976 y Fl(m;n)p Fp(\025)p Fu(7)824 872 y Fl(m)p Fu(+)p Fl(n)833 887 y Fr(X)833 994 y Fl(k)q Fu(=7)923 934 y Fq(\032)949 914 y Fl(k)q Fp(\000)p Fl(m)p Fp(\000)p Fl(n)1096 934 y Fq(A)1133 908 y Fu(\()p Fl(m)p Fu(\))1133 949 y Fl(k)q(;n)641 1109 y Fs(\024)18 b Fv(sup)693 1151 y Fl(i)p Fp(\025)p Fu(14)786 1109 y Fv(\()p Fq(\014)s(=\032)p Fv(\))906 1089 y Fl(i)961 1047 y(i)932 1062 y Fr(X)931 1169 y Fl(k)q Fu(=7)1004 1109 y Fv(\()p Fq(\014)s(\032)p Fv(\))1099 1089 y Fl(k)1124 1047 y Fr(q)p 1174 1047 V 22 x(\000)1196 1087 y Fu(2)p Fl(k)1206 1126 y(k)1238 1069 y Fr(\001)8 b(\000)1296 1087 y Fl(i)1292 1126 y(k)1314 1069 y Fr(\001)641 1284 y Fs(\024)693 1223 y Fr(q)p 743 1223 164 2 v 61 x Fv(4)768 1270 y Fp(\000)p Fu(7)821 1244 y Fr(\000)844 1262 y Fu(14)854 1302 y(7)884 1244 y Fr(\001)928 1284 y Fv(sup)923 1326 y Fl(i)p Fp(\025)p Fu(14)1017 1284 y Fv(\()p Fq(\014)s(=\032)p Fv(\))1137 1264 y Fl(i)1191 1222 y(i)1162 1237 y Fr(X)1162 1344 y Fl(k)q Fu(=0)1234 1284 y Fv(\()p Fq(\014)s(\032)p Fv(\))1329 1264 y Fl(k)1354 1284 y Fv(2)1379 1264 y Fl(k)1403 1244 y Fr(\000)1430 1262 y Fl(i)1426 1302 y(k)1448 1244 y Fr(\001)641 1434 y Fv(=)693 1373 y Fr(q)p 743 1373 V 61 x Fv(4)768 1420 y Fp(\000)p Fu(7)821 1394 y Fr(\000)844 1412 y Fu(14)854 1452 y(7)884 1394 y Fr(\001)928 1434 y Fv(sup)923 1476 y Fl(i)p Fp(\025)p Fu(14)1017 1434 y Fv(\()p Fq(\014)s(=\032)p Fv(\))1137 1414 y Fl(i)1153 1434 y Fv(\(1)k(+)e(2)p Fq(\014)s(\032)p Fv(\))1359 1414 y Fl(i)1390 1434 y Fq(<)1442 1373 y Fr(q)p 1492 1373 V 61 x Fv(4)1517 1420 y Fp(\000)p Fu(7)1570 1394 y Fr(\000)1593 1412 y Fu(14)1603 1452 y(7)1633 1394 y Fr(\001)1672 1434 y Fq(:)1848 1179 y Fv(\(3)p Fq(:)p Fv(9\))0 1602 y(The)16 b(last)g(square)f(ro)q(ot)h(in)32 b(\(3.9\))17 b(equals)801 1560 y Fr(p)p 851 1560 200 2 v 42 x Fv(429)p Fq(=)p Fv(2048)o(,)f(whic)o(h)f (is)h(smaller)e(than)i(0)p Fq(:)p Fv(46,)g(as)f(can)h(easily)0 1662 y(b)q(e)h(c)o(hec)o(k)o(ed)e(b)o(y)i(v)o(erifying)f(e.g.)21 b(that)c(2048)11 b Fs(\002)g Fv(0)p Fq(:)p Fv(46)f Fs(\002)h Fv(0)p Fq(:)p Fv(46)j Fq(>)f Fv(429.)p 1807 1741 43 43 v 100 1861 a(In)22 b(order)g(to)h(estimate)f(mixed)h(terms)f(of)g(the)h(form)f Fq(A)p Fv(\()p Fq(g)r Fv(\))p Fq(h)p Fv(,)j(w)o(e)d(will)h(need)f(b)q(ounds)f (on)i(the)0 1921 y(follo)o(wing)15 b(sums)g Fq(S)366 1928 y Fl(m;n)440 1921 y Fv(.)734 1995 y Fq(S)765 2002 y Fl(m;n)853 1995 y Fs(\021)906 1948 y Fr(X)905 2055 y Fl(k)q Fp(\025)p Fu(7)986 1995 y Fq(\032)1012 1975 y Fl(k)q Fp(\000)p Fu(7)1088 1995 y Fq(A)1125 1970 y Fu(\()p Fl(m)p Fu(\))1125 2010 y Fl(k)q(;n)1202 1995 y Fq(:)607 b Fv(\(3)p Fq(:)p Fv(10\))0 2188 y Fk(Lemma)22 b(3.2.)69 b Fj(De\014ne)20 b Fq(\033)553 2195 y Fl(m)613 2188 y Fs(\021)g Fv(max)p Fs(f)p Fq(S)821 2195 y Fl(m;)p Fu(7)898 2188 y Fq(;)8 b(\032)946 2170 y Fp(\000)p Fu(1)1000 2188 y Fq(S)1031 2195 y Fl(m;)p Fu(8)1100 2188 y Fs(g)p Fj(.)34 b(If)21 b Fq(g)i Fj(is)d(a)h(function)f(in)h Fq(P)7 b Fs(B)1708 2195 y Fl(\032)1739 2188 y Fj(,)21 b(and)g Fq(h)f Fj(a)0 2247 y(function)c(in)g Fv(\(I)c Fs(\000)e Fq(P)d Fv(\))p Fs(B)442 2254 y Fl(\032)474 2247 y Fj(,)16 b(then)19 2319 y Fv(\()p Fq(i)p Fv(\))100 2277 y Fr(\014)100 2307 y(\014)130 2279 y(\000)153 2319 y Fq(A)p Fv(\()p Fq(g)r Fv(\))p Fq(h)283 2279 y Fr(\001)306 2339 y Fl(k)344 2277 y Fr(\014)344 2307 y(\014)361 2319 y Fs(\024)e(k)p Fq(h)p Fs(k)493 2326 y Fl(\032)515 2319 y Fq(\032)541 2301 y Fp(\000)p Fu(7)603 2282 y Fr(P)655 2294 y Fu(6)655 2334 y Fl(m)p Fu(=0)752 2319 y Fq(A)789 2293 y Fu(\()p Fl(m)p Fu(\))789 2334 y Fl(k)q(;)p Fu(7)858 2319 y Fs(j)p Fq(g)896 2326 y Fl(m)934 2319 y Fs(j)8 b Fq(;)58 b(k)15 b Fv(=)f(0)p Fq(;)8 b Fv(1)p Fq(;)g(:)g(:)g(:)g(;)g Fv(6)p Fj(.)2 2395 y Fv(\()p Fq(ii)p Fv(\))26 b Fs(k)p Fv(\(I)11 b Fs(\000)g Fq(P)c Fv(\))p Fq(A)p Fv(\()p Fq(g)r Fv(\))p Fq(h)p Fs(k)436 2402 y Fl(\032)473 2395 y Fs(\024)14 b(k)p Fq(h)p Fs(k)605 2402 y Fl(\032)636 2357 y Fr(P)688 2370 y Fu(6)688 2410 y Fl(m)p Fu(=0)785 2395 y Fq(\033)813 2402 y Fl(m)851 2395 y Fs(j)p Fq(g)889 2402 y Fl(m)927 2395 y Fs(j)8 b Fq(:)0 2519 y Fk(Pro)r(of.)17 b Fv(Clearly)l(,)459 2584 y Fr(\014)459 2614 y(\014)490 2586 y(\000)513 2627 y Fq(A)p Fv(\()p Fq(g)r Fv(\))p Fq(h)643 2586 y Fr(\001)666 2647 y Fl(k)704 2584 y Fr(\014)704 2614 y(\014)721 2627 y Fs(\024)c(k)p Fq(h)p Fs(k)852 2634 y Fl(\032)916 2565 y Fu(6)890 2579 y Fr(X)883 2685 y Fl(m)p Fu(=0)977 2627 y Fs(j)p Fq(g)1015 2634 y Fl(m)1053 2627 y Fs(j)59 b Fv(max)1075 2659 y Fu(7)p Fp(\024)p Fl(n)p Fp(\024)p Fl(k)q Fu(+)p Fl(m)1278 2627 y Fq(\032)1304 2606 y Fp(\000)p Fl(n)1362 2627 y Fq(A)1399 2601 y Fu(\()p Fl(m)p Fu(\))1399 2642 y Fl(k)q(;n)1477 2627 y Fq(:)332 b Fv(\(3)p Fq(:)p Fv(11\))950 2790 y(11)p eop %%Page: 12 13 bop 0 50 a Fv(The)12 b(follo)o(wing)g(b)q(ound)f(sho)o(ws)g(that)i(the)f (maxim)o(um)f(in)24 b(\(3.11\))13 b(is)f(tak)o(en)h(for)f Fq(n)h Fv(=)h(7.)21 b(If)12 b Fq(j)k Fs(\021)e(j)p Fq(k)t Fs(\000)s Fq(m)p Fs(j)f Fq(<)0 109 y(n)k Fv(and)e Fq(`)f Fs(\021)g Fv(max)o Fs(f)p Fq(k)r(;)8 b(m)p Fs(g)13 b Fq(<)h(n)p Fv(,)i(then)82 222 y Fq(\032)108 204 y Fp(\000)p Fu(\()p Fl(n)p Fu(+1\))247 222 y Fq(A)284 196 y Fu(\()p Fl(m)p Fu(\))284 237 y Fl(k)q(;n)p Fu(+1)p 82 252 315 2 v 144 310 a Fq(\032)170 295 y Fp(\000)p Fl(n)228 310 y Fq(A)265 284 y Fu(\()p Fl(m)p Fu(\))265 325 y Fl(k)q(;n)416 263 y Fv(=)474 230 y Fq(\014)p 474 252 31 2 v 477 298 a(\032)546 186 y Fr(p)p 596 186 359 2 v 43 x Fv(\(2)p Fq(n)11 b Fv(+)g(2\)\(2)p Fq(n)g Fv(+)g(1\)\()p Fq(k)i Fv(+)e Fq(m)g Fs(\000)g Fq(n)p Fv(\))p 534 252 695 2 v 534 298 a(\()p Fq(n)g Fv(+)g(1)g(+)g Fq(m)f Fs(\000)h Fq(k)r Fv(\)\()p Fq(n)g Fv(+)g(1)g Fs(\000)g Fq(m)g Fv(+)g Fq(k)r Fv(\))1248 263 y Fs(\024)1307 230 y Fv(2)p Fq(\014)p 1307 252 56 2 v 1322 298 a(\032)1423 230 y Fv(\()p Fq(n)g Fv(+)g(1\)\(2)p Fq(`)g Fs(\000)g Fq(n)g Fs(\000)g Fq(j)s Fv(\))p 1391 252 478 2 v 1391 298 a(\()p Fq(n)g Fv(+)g(1)g(+)g Fq(j)s Fv(\)\()p Fq(n)g Fv(+)g(1)g Fs(\000)g Fq(j)s Fv(\))416 423 y(=)474 389 y(2)p Fq(\014)p 474 411 56 2 v 489 457 a(\032)559 389 y Fv(\()p Fq(n)g Fv(+)g(1\)\(2)p Fq(`)g Fs(\000)g Fq(n)p Fv(\))g Fs(\000)g Fv(\()p Fq(n)g Fv(+)g(1\))p Fq(j)p 559 411 569 2 v 701 457 a Fv(\()p Fq(n)g Fv(+)g(1\))855 442 y Fu(2)889 457 y Fs(\000)g Fq(j)963 442 y Fu(2)1147 423 y Fs(\024)1205 389 y Fv(2)p Fq(\014)p 1205 411 56 2 v 1220 457 a(\032)1281 423 y(<)i Fv(2)e Fs(\000)g Fv(2)1444 402 y Fu(1)p Fl(=)p Fu(3)1520 423 y Fq(<)j Fv(1)p Fq(:)1823 512 y Fv(\(3)p Fq(:)p Fv(12\))0 571 y(This)i(pro)o(v)o(es)f(\()p Fq(i)p Fv(\).)23 b(In)16 b(order)f(to)i(v)o(erify)g(\()p Fq(ii)p Fv(\),)g(w)o(e)f(note)h(\014rst)f(that)h(since)350 713 y Fq(A)387 687 y Fu(\()p Fl(m)p Fu(\))387 728 y Fl(k)q Fu(+1)p Fl(;n)p Fu(+1)563 713 y Fv(=)d Fq(\014)647 693 y Fu(2)675 636 y Fr(p)p 725 636 357 2 v 42 x Fv(\(2)p Fq(k)f Fv(+)e(2\)\(2)p Fq(n)g Fv(+)g(2\))1081 636 y Fr(p)p 1131 636 V 42 x Fv(\(2)p Fq(k)i Fv(+)e(1\)\(2)p Fq(n)g Fv(+)g(1\))p 675 702 813 2 v 734 747 a(\()p Fq(k)i Fv(+)e Fq(n)g Fv(+)f(2)h Fs(\000)g Fq(m)p Fv(\)\()p Fq(k)i Fv(+)e Fq(n)g Fv(+)g(1)g Fs(\000)g Fq(m)p Fv(\))1493 713 y Fq(A)1530 687 y Fu(\()p Fl(m)p Fu(\))1530 728 y Fl(k)q(;n)563 851 y Fs(\024)j Fq(\014)647 830 y Fu(2)780 817 y Fv(\()p Fq(k)f Fv(+)d Fq(n)h Fv(+)g(2\)\()p Fq(k)i Fv(+)e Fq(n)g Fv(+)g(1\))p 675 839 695 2 v 675 885 a(\()p Fq(k)i Fv(+)e Fq(n)g Fv(+)g(2)g Fs(\000)f Fq(m)p Fv(\)\()p Fq(k)j Fv(+)e Fq(n)g Fv(+)g(1)g Fs(\000)g Fq(m)p Fv(\))1376 851 y Fq(A)1413 825 y Fu(\()p Fl(m)p Fu(\))1413 866 y Fl(k)q(;n)1491 851 y Fq(;)1823 777 y Fv(\(3)p Fq(:)p Fv(13\))0 980 y(w)o(e)16 b(ha)o(v)o(e)147 1081 y Fq(S)178 1088 y Fl(m;n)p Fu(+1)316 1081 y Fv(=)369 1034 y Fr(X)369 1141 y Fl(k)q Fp(\025)p Fu(7)450 1081 y Fq(\032)476 1061 y Fl(k)q Fp(\000)p Fu(7)551 1081 y Fq(A)588 1055 y Fu(\()p Fl(m)p Fu(\))588 1096 y Fl(k)q(;n)p Fu(+1)714 1081 y Fv(=)d Fq(A)803 1055 y Fu(\()p Fl(m)p Fu(\))803 1095 y(7)p Fl(;n)p Fu(+1)924 1081 y Fv(+)e Fq(\032)1008 1034 y Fr(X)1008 1141 y Fl(k)q Fp(\025)p Fu(7)1089 1081 y Fq(\032)1115 1061 y Fl(k)q Fp(\000)p Fu(7)1190 1081 y Fq(A)1227 1055 y Fu(\()p Fl(m)p Fu(\))1227 1096 y Fl(k)q Fu(+1)p Fl(;n)p Fu(+1)316 1244 y Fs(\024)j Fq(A)406 1218 y Fu(\()p Fl(m)p Fu(\))406 1257 y(7)p Fl(;n)p Fu(+1)526 1244 y Fv(+)d Fq(\032\014)633 1223 y Fu(2)765 1210 y Fv(\(7)h(+)f Fq(n)g Fv(+)f(2\)\(7)i(+)f Fq(n)g Fv(+)g(1\))p 661 1232 690 2 v 661 1278 a(\(7)g(+)g Fq(n)g Fv(+)g(2)g Fs(\000)g Fq(m)p Fv(\)\(7)g(+)g Fq(n)g Fv(+)g(1)g Fs(\000)f Fq(m)p Fv(\))1365 1196 y Fr(X)1365 1304 y Fl(k)q Fp(\025)p Fu(7)1446 1244 y Fq(\032)1472 1223 y Fl(k)q Fp(\000)p Fu(7)1547 1244 y Fq(A)1584 1218 y Fu(\()p Fl(m)p Fu(\))1584 1259 y Fl(k)q(;n)1662 1244 y Fq(;)1823 1184 y Fv(\(3)p Fq(:)p Fv(14\))0 1398 y(and)16 b(th)o(us)369 1474 y Fq(S)400 1481 y Fl(m;n)p Fu(+1)538 1474 y Fs(\024)d Fq(A)627 1448 y Fu(\()p Fl(m)p Fu(\))627 1487 y(7)p Fl(;n)p Fu(+1)748 1474 y Fv(+)e Fq(\032\014)855 1454 y Fu(2)883 1440 y Fv(\(9)g(+)g Fq(n)p Fv(\)\(8)g(+)g Fq(n)p Fv(\))p 883 1463 309 2 v 883 1508 a(\(3)g(+)g Fq(n)p Fv(\)\(2)g(+)g Fq(n)p Fv(\))1198 1474 y Fq(S)1229 1481 y Fl(m;n)1311 1474 y Fq(;)107 b(m)14 b Fs(\024)g Fv(6)p Fq(:)241 b Fv(\(3)p Fq(:)p Fv(15\))0 1591 y(If)17 b Fq(m)c(<)h Fv(7)g Fq(<)f(n)p Fv(,)k(then)f(b)o(y)g(com)o(bining)31 b(\(3.15\))17 b(with)g(the)f(b)q(ound)98 1704 y Fq(\032)124 1684 y Fu(7)p Fp(\000)p Fu(\()p Fl(n)p Fu(+1\))283 1704 y Fq(A)320 1678 y Fu(\()p Fl(m)p Fu(\))320 1717 y(7)p Fl(;n)p Fu(+1)444 1704 y Fv(=)d Fq(\032)522 1684 y Fp(\000)p Fu(2\()p Fl(n)p Fp(\000)p Fu(6\))682 1704 y Fq(\032)708 1684 y Fu(\()p Fl(n)p Fu(+1\))p Fp(\000)p Fu(7)868 1704 y Fq(A)905 1678 y Fu(\()p Fl(m)p Fu(\))905 1717 y Fl(n)p Fu(+1)p Fl(;)p Fu(7)1028 1704 y Fs(\024)h Fq(\032)1107 1684 y Fp(\000)p Fu(4)1169 1657 y Fr(X)1168 1764 y Fl(k)q Fp(\025)p Fu(7)1249 1704 y Fq(\032)1275 1684 y Fl(k)q Fp(\000)p Fu(7)1351 1704 y Fq(A)1388 1678 y Fu(\()p Fl(m)p Fu(\))1388 1719 y Fl(k)q(;)p Fu(7)1471 1704 y Fv(=)f Fq(\032)1549 1684 y Fp(\000)p Fu(4)1603 1704 y Fq(S)1634 1711 y Fl(m;)p Fu(7)1711 1704 y Fq(;)98 b Fv(\(3)p Fq(:)p Fv(16\))0 1859 y(w)o(e)16 b(\014nd)g(that)344 1973 y Fq(\032)370 1952 y Fu(7)p Fp(\000)p Fu(\()p Fl(n)p Fu(+1\))529 1973 y Fq(S)560 1980 y Fl(m;n)p Fu(+1)698 1973 y Fs(\024)e Fq(\032)777 1952 y Fp(\000)p Fu(4)830 1973 y Fq(S)861 1980 y Fl(m;)p Fu(7)941 1973 y Fv(+)d Fq(\032)1017 1952 y Fu(7)p Fp(\000)p Fl(n)1095 1973 y Fq(\014)1126 1952 y Fu(2)1162 1939 y Fv(17)g Fs(\002)g Fv(16)p 1162 1961 161 2 v 1162 2007 a(11)g Fs(\002)g Fv(10)1337 1973 y Fq(S)1368 1980 y Fl(m;n)698 2097 y Fs(\024)751 2042 y Fr(h)774 2097 y Fq(\032)800 2076 y Fp(\000)p Fu(4)865 2097 y Fv(+)920 2063 y(136)p 920 2086 75 2 v 933 2131 a(55)1001 2097 y Fq(\014)1032 2076 y Fu(2)1054 2042 y Fr(i)1103 2097 y Fv(max)o Fs(f)p Fq(S)1251 2104 y Fl(m;)p Fu(7)1329 2097 y Fq(;)d(\032)1377 2076 y Fu(7)p Fp(\000)p Fl(n)1455 2097 y Fq(S)1486 2104 y Fl(m;n)1559 2097 y Fs(g)g Fq(:)1823 2032 y Fv(\(3)p Fq(:)p Fv(17\))0 2217 y(The)16 b(assertion)g(\()p Fq(ii)p Fv(\))h(no)o(w)f (follo)o(ws)g(since)450 2368 y Fs(k)p Fv(\(I)11 b Fs(\000)g Fq(P)c Fv(\))p Fq(A)p Fv(\()p Fq(g)r Fv(\))p Fq(h)p Fs(k)786 2375 y Fl(\032)823 2368 y Fs(\024)14 b(k)p Fq(h)p Fs(k)955 2375 y Fl(\032)1003 2368 y Fv(sup)1002 2410 y Fl(n)p Fp(\025)p Fu(7)1119 2305 y(6)1093 2320 y Fr(X)1086 2426 y Fl(m)p Fu(=0)1180 2368 y Fq(\032)1206 2347 y Fu(7)p Fp(\000)p Fl(n)1284 2368 y Fq(S)1315 2375 y Fl(m;n)1389 2368 y Fs(j)p Fq(g)1427 2375 y Fl(m)1464 2368 y Fs(j)8 b Fq(;)323 b Fv(\(3)p Fq(:)p Fv(18\))0 2517 y(and)16 b(since)g(the)h(term)f([)p Fq(:)8 b(:)g(:)p Fv(])16 b(in)33 b(\(3.17\))17 b(is)f(b)q(ounded)f(b)o(y)289 2646 y([)p Fq(:)8 b(:)g(:)p Fv(])14 b Fq(<)441 2591 y Fr(\020)544 2612 y Fq(\014)p 477 2635 164 2 v 477 2680 a Fv(1)d Fs(\000)g Fv(2)p Fq(\014)619 2666 y Fu(2)647 2591 y Fr(\021)677 2601 y Fp(\000)p Fu(4)741 2646 y Fv(+)797 2612 y(136)p 797 2635 75 2 v 809 2680 a(55)878 2646 y Fq(\014)909 2625 y Fu(2)945 2646 y Fv(=)i(24)e Fs(\000)g Fv(30)g Fs(\002)g Fv(2)1244 2625 y Fp(\000)p Fu(1)p Fl(=)p Fu(3)1348 2646 y Fv(+)1404 2612 y(34)p 1404 2635 50 2 v 1404 2680 a(55)1460 2646 y(2)1485 2625 y Fu(1)p Fl(=)p Fu(3)1561 2646 y Fq(<)j Fv(1)8 b Fq(;)162 b Fv(\(3)p Fq(:)p Fv(19\))950 2790 y(12)p eop %%Page: 13 14 bop 0 50 a Fv(as)16 b(can)g(easily)h(b)q(e)g(v)o(eri\014ed)e(b)o(y)h(using)g (the)g(inequalities)g(\(I1\))h(b)q(elo)o(w.)p 1807 130 43 43 v 100 252 a(The)i(follo)o(wing)f(inequalities)h(will)g(b)q(e)h(used)f(later)g (on,)h(in)f(order)f(to)i(estimate)g(p)q(o)o(w)o(ers)e(of)h(2)1887 234 y Fu(1)p Fl(=)p Fu(6)0 312 y Fv(and)d Fq(\032)123 294 y Fp(\006)p Fu(1)176 312 y Fv(.)0 408 y Fk(Inequalities)22 b(3.3.)43 b Fj(If)17 b Fq(\032)11 b Fs(\000)g Fq(\014)s(=)p Fv(\(1)g Fs(\000)f Fv(2)p Fq(\014)794 390 y Fu(2)816 408 y Fv(\))17 b Fj(is)g(a)f(su\016cien)o(tly)f(small)h(p)q(ositiv)o(e)g(n)o(um)o(b)q(er,)e (then)0 501 y Fv(\(I1\))100 b(0)p Fq(:)p Fv(793700)16 b Fq(<)30 b Fv(2)480 483 y Fp(\000)p Fu(1)p Fl(=)p Fu(3)590 501 y Fq(<)h Fv(0)p Fq(:)p Fv(793701)8 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Fl(k)q(;n)643 2297 y Fs(\000)10 b Fq(Q)731 2271 y Fu(\()p Fl(m)p Fu(\))731 2312 y Fl(k)q(;n)815 2254 y Fr(\014)815 2284 y(\014)831 2297 y Fs(\024)919 2277 y Fu(1)p 890 2285 80 2 v 890 2314 a(2000)983 2297 y Fq(;)108 b Fv(0)14 b Fs(\024)f Fq(k)r(;)8 b(m;)g(n)14 b Fs(\024)f Fv(6)8 b Fq(:)0 2464 y Fk(De\014nition)21 b(3.5.)45 b Fj(If)17 b Fq(g)h Fj(is)e(a)h(v)o(ector)g(in)f Fq(P)7 b Fs(B)867 2471 y Fl(\032)898 2464 y Fj(,)17 b(denote)g(b)o(y)f Fq(Q)p Fv(\()p Fq(g)r Fv(\))i Fj(b)q(e)f(the)g(symmetric)f Fv(7)11 b Fs(\002)g Fv(7)17 b Fj(matrix)0 2523 y(with)f(en)o(tries)520 2627 y Fq(Q)p Fv(\()p Fq(g)r Fv(\))623 2634 y Fl(k)q(;n)699 2627 y Fv(=)784 2565 y Fu(6)758 2579 y Fr(X)751 2685 y Fl(m)p Fu(=0)845 2627 y Fq(Q)884 2601 y Fu(\()p Fl(m)p Fu(\))884 2642 y Fl(k)q(;n)954 2627 y Fq(g)978 2634 y Fl(m)1024 2627 y Fq(;)107 b Fv(0)14 b Fs(\024)g Fq(k)r(;)8 b(n)13 b Fs(\024)h Fv(6)8 b Fq(:)393 b Fv(\(3)p Fq(:)p Fv(25\))950 2790 y(14)p eop %%Page: 15 16 bop 0 50 a Fj(De\014ne)16 b(v)o(ectors)h Fq(u;)8 b(v)r(;)g(w)17 b Fj(in)f Fq(P)7 b Fs(B)606 57 y Fl(\032)643 50 y Fs(\031)13 b Fv(I)-10 b(R)740 28 y Fu(7)779 50 y Fj(b)o(y)16 b(the)h(follo)o(wing)e (equations.)486 186 y Fq(u)515 193 y Fl(k)553 186 y Fv(=)641 166 y Fu(1)p 611 174 80 2 v 611 203 a(2000)738 123 y(6)712 138 y Fr(X)705 244 y Fl(m)p Fu(=0)791 186 y Fv(\()p Fq(p)835 193 y Fl(m)884 186 y Fv(+)c Fq(\016)956 193 y Fl(m)994 186 y Fv(\))1021 119 y Fu(min)q Fp(f)p Fl(k)q Fu(+)p Fl(m;)p Fu(6)p Fp(g)1099 138 y Fr(X)1051 248 y Fl(n)p Fu(=)p Fp(j)p Fl(k)q Fp(\000)p Fl(m)p Fp(j)1248 186 y Fv(\()p Fq(p)1292 193 y Fl(n)1331 186 y Fv(+)f Fq(\016)1402 193 y Fl(n)1430 186 y Fv(\))e Fq(;)490 371 y(v)514 378 y Fl(k)553 371 y Fv(=)621 351 y Fu(1)p 611 359 40 2 v 611 388 a(10)665 371 y Fq(\032)691 350 y Fp(\000)p Fu(7)786 309 y(6)760 324 y Fr(X)753 429 y Fl(m)p Fu(=0)847 371 y Fq(A)884 345 y Fu(\()p Fl(m)p Fu(\))884 386 y Fl(k)q(;)p Fu(7)953 371 y Fv(\()p Fq(p)997 378 y Fl(m)1047 371 y Fv(+)i Fq(\016)1118 378 y Fl(m)1156 371 y Fv(\))e 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Fq(g)1385 715 y Fl(k)1423 708 y Fs(\024)e Fq(f)1500 715 y Fl(k)1541 708 y Fv(for)i(all)h Fq(k)r Fv(.)0 803 y Fk(Prop)r(osition)k(3.6.)56 b Fj(Assume)17 b(that)i Fv(\(N1{3\))h Fj(hold,)e(and)g(let)h Fq(\015)1250 810 y Fl(k)1292 803 y Fv(=)e(3)12 b Fs(\002)g Fv(10)1486 785 y Fp(\000)p Fu(5)1539 803 y Fj(,)19 b(for)g Fv(0)e Fs(\024)g Fq(k)i Fs(\024)e Fv(6)p Fj(.)28 b(If)0 862 y(there)16 b(is)h(a)f(symmetric)g Fv(7)11 b Fs(\002)f Fv(7)17 b Fj(matrix)f Fq(M)5 b Fj(,)17 b(suc)o(h)e(that)0 941 y Fv(\(N4\))423 898 y Fr(\014)423 928 y(\014)453 900 y(\000)476 941 y Fv(I)c Fs(\000)g Fq(M)5 b Fv(\(2)p Fq(Q)p Fv(\()p Fq(p)p Fv(\))13 b Fs(\000)e Fv(I\))854 900 y Fr(\001)877 961 y Fl(k)q(;n)952 898 y Fr(\014)952 928 y(\014)969 941 y Fs(\024)1057 921 y Fu(1)p 1027 929 80 2 v 1027 958 a(2000)1121 941 y Fq(;)108 b Fv(0)14 b Fs(\024)f Fq(k)r(;)8 b(n)13 b Fs(\024)h Fv(6)8 b Fq(;)0 1023 y Fv(\(N5\))346 b Fs(j)p Fq(M)5 b Fs(j)527 982 y Fr(\000)551 1023 y Fs(j)p Fq(Q)p Fv(\()p Fq(p)p Fv(\))p Fq(p)12 b Fs(\000)f Fq(p)p Fs(j)g Fv(+)f Fq(Q)p Fv(\()p Fq(\016)r Fv(\))p Fq(\016)15 b Fv(+)c Fq(u)f Fv(+)h(2)p Fq(v)i Fv(+)e Fq(w)1279 982 y Fr(\001)1313 1023 y Fv(+)f Fq(\015)17 b Fs(\024)c Fq(\016)e(;)0 1101 y Fj(then)16 b Fq(B)j Fj(has)d(a)h(\014xed)f(p)q(oin)o(t)g(in)g Fq(G)p Fj(.)0 1202 y Fk(Pro)r(of.)h Fv(First,)f(w)o(e)h(note)g(that)g Fq(M)23 b Fv(is)16 b(nonsingular.)k(This)c(follo)o(ws)g(since)h(\(I)11 b Fs(\000)g Fv(\(2)p Fq(Q)p Fv(\()p Fq(p)p Fv(\))i Fs(\000)e Fv(I\))p Fq(M)5 b Fv(\))p Fq(g)17 b Fv(=)d Fq(g)0 1261 y Fv(whenev)o(er)i Fq(M)5 b(g)17 b Fv(=)d(0,)j(whic)o(h)f(b)o(y)g(\(N4\))i(implies)e(that)h Fq(g)f Fv(=)e(0.)23 b(Consequen)o(tly)l(,)16 b(w)o(e)h(ma)o(y)f(consider)g (the)0 1320 y(\014xed)g(p)q(oin)o(t)g(problem)f(for)i(the)f(map)g Fq(g)f Fs(7!)h Fv(~)-27 b Fq(g)r Fv(,)735 1399 y(~)g Fq(g)15 b Fv(=)f Fq(g)f Fs(\000)d Fq(M)5 b Fv(\()p Fq(B)r Fv(\()p Fq(g)r Fv(\))13 b Fs(\000)e Fq(g)r Fv(\))d Fq(;)606 b Fv(\(3)p Fq(:)p Fv(27\))0 1477 y(instead)16 b(of)h Fq(B)r Fv(.)22 b(Assume)15 b(no)o(w)h(that)h Fq(g)f Fs(2)e Fq(G)p Fv(.)22 b(Then)142 1553 y(~)-26 b Fq(g)12 b Fs(\000)f Fq(p)j Fv(=)g(\()p Fq(g)f Fs(\000)d Fq(p)p Fv(\))i Fs(\000)f Fq(M)583 1513 y Fr(\002)604 1553 y Fq(P)c(A)p Fv(\()p Fq(g)14 b Fv(+)c Fq(h)815 1532 y Fp(\003)838 1553 y Fv(\()p Fq(g)r Fv(\)\)\()p Fq(g)k Fv(+)c Fq(h)1056 1532 y Fp(\003)1079 1553 y Fv(\()p Fq(g)r Fv(\)\))i Fs(\000)f Fq(g)1250 1513 y Fr(\003)266 1633 y Fv(=)319 1592 y Fr(\000)341 1633 y Fv(I)h Fs(\000)e Fq(M)5 b Fv(\(2)p Fq(Q)p Fv(\()p Fq(p)p Fv(\))14 b Fs(\000)c Fv(I\))719 1592 y Fr(\001)743 1633 y Fv(\()p Fq(g)j Fs(\000)d Fq(p)p Fv(\))i Fs(\000)f Fq(M)1007 1592 y Fr(\002)1028 1633 y Fv(\()p Fq(Q)p Fv(\()p Fq(p)p Fv(\))p Fq(p)i Fs(\000)e Fq(p)p Fv(\))g(+)g Fq(Q)p Fv(\()p Fq(g)i Fs(\000)e Fq(p)p Fv(\)\()p Fq(g)j Fs(\000)d Fq(p)p Fv(\))263 1712 y(+)g(\()p Fq(P)c(A)12 b Fs(\000)f Fq(Q)p Fv(\)\()p Fq(g)r Fv(\))p Fq(g)i Fv(+)e(2)p Fq(P)c(A)p Fv(\()p Fq(g)r Fv(\))p Fq(h)873 1692 y Fp(\003)896 1712 y Fv(\()p Fq(g)r Fv(\))12 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(follo)o(ws)f(b)o(y)h(Bro)o(w)o(er's)f(\014xed)h(p)q(oin)o(t)g(theorem.)p 1807 2329 43 43 v 100 2426 a(As)23 b(sho)o(wn)f(earlier)g(in)47 b(\(3.5\),)25 b(the)f(conclusions)d(of)j(the)g(last)f(t)o(w)o(o)g(prop)q (ositions)e(imply)i(the)0 2485 y(follo)o(wing)16 b(theorem.)22 b(The)16 b(pro)q(of)h(of)g(this)f(theorem)h(is)f(completed)g(in)h(Section)f (5,)h(b)o(y)f(v)o(erifying)h(that)0 2544 y(the)g(n)o(umerical)d(b)q(ounds)h (\(N1-5\))j(are)e(satis\014ed.)0 2639 y Fk(Theorem)h(3.7.)44 b Fs(N)23 b Fj(has)16 b(a)h(\014xed)f(p)q(oin)o(t)g(in)g Fq(G)c Fs(\010)e Fq(H)t Fj(.)950 2790 y Fv(15)p eop %%Page: 16 17 bop 0 50 a Fw(4.)28 b(Pro)r(of)g(of)g(Theorems)f(1.1)h(and)f(1.4.)0 124 y Fv(In)20 b(this)h(section)f(w)o(e)h(sho)o(w)e(that)i(an)o(y)g(nonzero)e (\014xed)i(p)q(oin)o(t)f Fq(f)1231 106 y Fp(\003)1275 124 y Fv(of)h Fs(N)7 b Fv(,)22 b(that)f(lies)f(in)h Fs(B)1718 131 y Fl(\032)1761 124 y Fv(for)g(some)0 183 y Fq(\032)14 b(>)f(\014)s(=)p Fv(\(1)r Fs(\000)r Fv(2)p Fq(\014)291 165 y Fu(2)313 183 y Fv(\),)h(and)d(is)h(di\013eren)o(t)f(from)g Fq(f)827 190 y Fm(H)r(T)898 183 y Fs(\021)j Fv(1,)f(has)e(the)i(prop)q(erties)e(describ)q (ed)f(in)24 b(Theorem)11 b(1.1)0 242 y(and)35 b(Theorem)16 b(1.4.)25 b(An)18 b(example)f(of)h(suc)o(h)f(a)g(\014xed)h(p)q(oin)o(t)f (\(and)g(probably)g(the)g(only)h(example\))g(is)0 302 y(describ)q(ed)d(in)g (Section)h(3.)21 b(W)l(e)16 b(note)g(that)g(the)g(Gaussian)f(\014xed)g(p)q (oin)o(t)h Fq(f)1385 309 y Fm(U)s(V)1457 302 y Fv(lies)f(in)h Fs(B)1633 309 y Fl(\032)1671 302 y Fv(if)g(and)f(only)h(if)0 361 y Fq(\032)e(<)f(\014)s(=)p Fv(\(1)e Fs(\000)g Fv(2)p Fq(\014)309 343 y Fu(2)331 361 y Fv(\).)0 456 y Fk(Lemma)21 b(4.1.)62 b Fj(Assume)19 b(that)h Fq(f)687 438 y Fp(\003)730 456 y Fs(2)f(B)815 463 y Fl(\032)858 456 y Fj(for)g(some)g Fq(\032)g(>)g(\014)s(=)p Fv(\(1)13 b Fs(\000)g Fv(2)p Fq(\014)1389 438 y Fu(2)1411 456 y Fv(\))p Fj(,)21 b(and)e(that)h Fq(f)1705 438 y Fp(\003)1748 456 y Fj(is)g(a)f(\014xed)0 515 y(p)q(oin)o(t)d(for)g Fs(N)7 b Fj(.)22 b(Then)17 b(there)f(are)g(constan)o(ts)g Fq(K)q(;)8 b(`)16 b Fj(suc)o(h)g(that)633 642 y Fs(j)p Fq(f)676 621 y Fp(\003)699 642 y Fv(\()p Fq(t)p Fv(\))p Fs(j)f(\024)e Fq(K)t(e)905 621 y Fl(`)p Fp(j)p Fl(t)p Fp(j)961 606 y Fo(6)p Fm(=)p Fo(5)1026 642 y Fq(;)108 b Fs(8)p Fq(t)12 b Fs(2)1276 631 y Fn(j)1268 642 y Fv(C)p Fq(:)530 b Fv(\(4)p Fq(:)p Fv(1\))0 787 y Fk(Pro)r(of.)14 b Fv(W)l(e)g(ma)o(y)f(assume)g(that)h Fq(f)663 769 y Fp(\003)701 787 y Fs(6\021)f Fv(0.)21 b(Let)15 b Fq(r)h Fv(=)d Fq(\014)s(=)p Fv(\(1)6 b Fs(\000)g Fv(2)p Fq(\014)1197 769 y Fu(2)1218 787 y Fv(\),)15 b(and)e(de\014ne)g Fq(\036)h Fv(:)g(I)-10 b(R)1617 794 y Fu(+)1664 787 y Fs(!)14 b Fv(I)-10 b(R)1773 794 y Fu(+)1811 787 y Fs([)6 b(f1g)0 847 y Fv(b)o(y)16 b(the)h(equation)f Fq(\036)p Fv(\()p Fq(s)p Fv(\))f(=)f Fs(k)p Fq(f)571 828 y Fp(\003)594 847 y Fs(k)619 854 y Fl(r)q Fu(+)p Fl(s)699 847 y Fv(.)22 b(Then,)16 b(as)g(a)g(consequence)g(of)34 b(Lemma)16 b(2.7,)g(w)o(e)g(ha)o(v)o(e)560 977 y Fq(\036)p Fv(\()p Fq(s)p Fv(\))e Fs(\024)g Fq(\036)p Fv(\(2)p Fq(\014)823 956 y Fu(2)845 977 y Fq(s)p Fv(\))887 956 y Fu(2)924 977 y Fs(\024)f Fq(:)8 b(:)g(:)14 b Fs(\024)g Fq(\036)1131 937 y Fr(\000)1153 977 y Fv(\(2)p Fq(\014)1228 956 y Fu(2)1251 977 y Fv(\))1270 956 y Fl(n)1297 977 y Fq(s)1320 937 y Fr(\001)1344 947 y Fu(2)1364 932 y Fm(n)1848 977 y Fv(\(4)p Fq(:)p Fv(2\))0 1099 y(for)i(all)g Fq(n)p Fv(.)22 b(This)16 b(sho)o(ws)f(that)i(in)f(fact)h Fq(f)761 1081 y Fp(\003)799 1099 y Fs(2)d(B)879 1106 y Fl(\032)918 1099 y Fv(for)j(all)f Fq(\032)d(>)h(r)q Fv(.)23 b(Let)17 b Fq(I)h Fv(=)1397 1059 y Fr(\002)1418 1099 y Fv(1)p Fq(;)1493 1079 y Fu(1)p 1471 1087 64 2 v 1471 1116 a(2)p Fl(\014)1516 1106 y Fo(2)1541 1059 y Fr(\003)1562 1099 y Fv(,)e(and)g(de\014ne)727 1232 y Fq(c)d Fs(\021)h Fv(max)826 1264 y Fl(x)p Fp(2)p Fl(I)924 1232 y Fq(x)952 1211 y Fp(\000)p Fu(3)p Fl(=)p Fu(2)1054 1232 y Fv(ln)8 b Fq(\036)p Fv(\()p Fq(x)p Fv(\))g Fq(:)626 b Fv(\(4)p Fq(:)p Fv(3\))0 1378 y(F)l(or)16 b(an)o(y)h Fq(s)e Fs(\025)g Fv(1,)i(there)g(exists)g Fq(n)e Fs(\025)f Fv(0)k(and)e Fq(x)f Fs(2)g Fq(I)t Fv(,)i(suc)o(h)f(that)i Fq(s)d Fv(=)g(\(2)p Fq(\014)1374 1360 y Fu(2)1396 1378 y Fv(\))1415 1360 y Fp(\000)p Fl(n)1474 1378 y Fq(x)p Fv(.)24 b(Since)17 b(2)p Fq(\014)1727 1360 y Fu(2)1764 1378 y Fv(=)e(2)1843 1360 y Fp(\000)p Fu(2)p Fl(=)p Fu(3)1936 1378 y Fv(,)0 1437 y(w)o(e)h(ha)o(v)o(e)g (th)o(us)569 1496 y Fq(\036)p Fv(\()p Fq(s)p Fv(\))e Fs(\024)g Fq(\036)p Fv(\()p Fq(x)p Fv(\))823 1476 y Fu(2)843 1461 y Fm(n)884 1496 y Fv(=)g Fq(\036)p Fv(\()p Fq(x)p Fv(\))1033 1476 y Fu(\()p Fl(s=x)p Fu(\))1127 1461 y Fo(3)p Fm(=)p Fo(2)1198 1496 y Fs(\024)f Fq(e)1273 1476 y Fl(cs)1310 1461 y Fo(3)p Fm(=)p Fo(2)1367 1496 y Fq(:)467 b Fv(\(4)p Fq(:)p Fv(4\))0 1589 y(Let)20 b(no)o(w)f Fq(t)g Fv(b)q(e)h(an)o(y)e(complex)h(n)o(um)o(b)q(er)e(of)j(mo)q(dulus)e(at)h (least)h(\()p Fq(r)15 b Fv(+)d(1\))1383 1571 y Fu(5)p Fl(=)p Fu(4)1446 1589 y Fv(,)20 b(and)f(de\014ne)f Fq(\032)h Fv(=)f Fs(j)p Fq(t)p Fs(j)1874 1571 y Fu(4)p Fl(=)p Fu(5)1936 1589 y Fv(.)0 1649 y(F)l(rom)d(the)i(b)q(ound)32 b(\(4.4\))17 b(and)32 b(Lemma)16 b(2.1)g(it)h(follo)o(ws)f(that)360 1806 y Fs(k)p Fq(f)414 1785 y Fp(\003)437 1806 y Fs(k)462 1785 y Fu(2)462 1819 y(\()p Fl(\014)r(=\032)p Fu(\))575 1806 y Fv(=)646 1743 y Fp(1)629 1758 y Fr(X)628 1864 y Fl(n)p Fu(=0)717 1772 y Fv(\()p Fq(\014)s(=\032)p Fv(\))837 1754 y Fu(2)p Fl(n)p 717 1794 168 2 v 747 1840 a Fv(\(2)p Fq(n)p Fv(\)!)890 1806 y Fs(jh)p Fq(f)952 1785 y Fp(\003)976 1806 y Fq(;)8 b(p)1023 1813 y Fu(2)p Fl(n)1070 1806 y Fs(i)1089 1815 y Fu(\()p Fl(\014)r(=\032)p Fu(\))1189 1806 y Fs(j)1203 1785 y Fu(2)575 1971 y Fv(=)646 1909 y Fp(1)629 1924 y Fr(X)628 2029 y Fl(n)p Fu(=0)717 1937 y Fv(\()p Fq(\014)s(=\032)p Fv(\))837 1919 y Fu(2)p Fl(n)p 717 1959 V 747 2005 a Fv(\(2)p Fq(n)p Fv(\)!)899 1971 y Fq(\032)925 1950 y Fu(4)p Fl(n)971 1971 y Fs(jh)p Fq(f)1033 1950 y Fp(\003)1057 1971 y Fq(;)g(p)1104 1978 y Fu(2)p Fl(n)1151 1971 y Fs(i)1170 1980 y Fu(\()p Fl(\014)r Fu(\))1229 1971 y Fs(j)1243 1950 y Fu(2)1279 1971 y Fv(=)1349 1909 y Fp(1)1333 1924 y Fr(X)1331 2029 y Fl(n)p Fu(=0)1415 1971 y Fq(\032)1441 1950 y Fu(2)p Fl(n)1487 1971 y Fs(j)p Fq(f)1530 1950 y Fp(\003)1525 1983 y Fl(n)1554 1971 y Fs(j)1568 1950 y Fu(2)575 2108 y Fs(\024)14 b(k)p Fq(f)682 2088 y Fp(\003)705 2108 y Fs(k)730 2088 y Fu(2)730 2121 y Fl(\032)767 2108 y Fs(\024)g Fq(e)843 2088 y Fu(2)p Fl(c)p Fu(\()p Fl(\032)p Fp(\000)p Fl(r)q Fu(\))984 2073 y Fo(3)p Fm(=)p Fo(2)1049 2108 y Fq(;)1848 1938 y Fv(\(4)p Fq(:)p Fv(5\))0 2224 y(and)i(th)o(us,)f(using)h(again)32 b(Lemma)16 b(2.1,)371 2362 y Fs(j)p Fq(f)414 2341 y Fp(\003)438 2362 y Fv(\()p Fq(t)p Fv(\))p Fs(j)e(\024)g Fq(e)598 2341 y Fl(c)p Fu(\()p Fl(\032)p Fp(\000)p Fl(r)q Fu(\))719 2326 y Fo(3)p Fm(=)p Fo(2)910 2328 y Fv(1)p 782 2350 282 2 v 795 2384 a Fo(4)785 2360 y Fr(p)p 834 2360 229 2 v 834 2403 a Fv(1)d Fs(\000)g Fv(\()p Fq(\014)s(=\032)p Fv(\))1040 2388 y Fu(2)1069 2362 y Fq(e)1120 2323 y Fm(\014)p 1098 2332 67 2 v 1098 2352 a(\032)p Fn(\000)p Fm(\014)1170 2340 y Fp(j)p Fl(t)p Fp(j)1209 2324 y Fo(2)1244 2362 y Fs(\024)j Fv(2)p Fq(e)1345 2341 y Fu(\()p Fl(c)p Fu(+2)p Fl(\014)r Fu(\))p Fp(j)p Fl(t)p Fp(j)1510 2326 y Fo(6)p Fm(=)p Fo(5)1565 2362 y Fq(:)269 b Fv(\(4)p Fq(:)p Fv(6\))p 1807 2556 43 43 v 100 2670 a(As)16 b(men)o(tioned)f(in)i(the)f(in)o (tro)q(duction,)f Fs(N)24 b Fv(has)16 b(the)h(follo)o(wing)e(prop)q(ert)o(y)l (.)950 2790 y(16)p eop %%Page: 17 18 bop 0 50 a Fk(Lemma)18 b(4.2.)50 b Fj(If)18 b Fq(g)h Fj(is)e(a)h(p)q (olynomial)e(whose)h(zeros)g(lie)g(in)h(a)f(strip)g Fs(j)p Fv(Re)o(\()p Fq(z)r Fv(\))p Fs(j)g(\024)e Fq(a)p Fj(,)j(then)g Fs(N)7 b Fv(\()p Fq(g)r Fv(\))18 b Fj(is)f(a)0 109 y(p)q(olynomial)e(whose)h (zeros)g(lie)h(in)f(the)h(strip)e Fs(j)p Fv(Re\()p Fq(z)r Fv(\))p Fs(j)g(\024)e Fq(\014)s(a)p Fj(.)0 225 y Fk(Pro)r(of.)k Fv(Let)g Fq(d)g Fv(b)q(e)g(the)g(degree)f(of)g Fq(g)r Fv(.)22 b(W)l(e)17 b(use)f(the)h(follo)o(wing)e(form)o(ula)g(for)h Fs(N)7 b Fv(\()p Fq(g)r Fv(\).)523 306 y Fr(\000)546 347 y Fs(N)g Fv(\()p Fq(g)r Fv(\))658 306 y Fr(\001)682 347 y Fv(\()p Fq(\014)732 326 y Fp(\000)p Fu(1)785 347 y Fq(:)p Fv(\))14 b(=)g Fq(e)914 313 y Fo(1)p 914 319 17 2 v 914 338 a(4)937 326 y Fu(\(1)p Fp(\000)p Fl(\014)1029 311 y Fo(2)1047 326 y Fu(\))p Fl(D)1097 311 y Fo(2)1119 347 y Fq(g)1145 326 y Fu(2)1180 347 y Fv(=)31 b(lim)1233 377 y Fl(n)p Fp(!1)1354 347 y Fq(g)1378 354 y Fl(n)1413 347 y Fq(;)421 b Fv(\(4)p Fq(:)p Fv(7\))0 474 y(where)448 560 y Fq(g)472 567 y Fl(n)513 560 y Fs(\021)566 505 y Fr(\020)595 560 y Fv(1)11 b(+)g Fq(i)704 483 y Fr(p)p 754 483 139 2 v 43 x Fv(1)g Fs(\000)g Fq(\014)871 512 y Fu(2)p 704 549 189 2 v 751 594 a Fv(2)776 559 y Fs(p)p 817 559 30 2 v 817 594 a Fq(n)899 560 y(D)941 505 y Fr(\021)972 515 y Fl(n)999 505 y Fr(\020)1028 560 y Fv(1)g Fs(\000)g Fq(i)1137 483 y Fr(p)p 1187 483 139 2 v 43 x Fv(1)g Fs(\000)g Fq(\014)1304 512 y Fu(2)p 1137 549 189 2 v 1184 594 a Fv(2)1209 559 y Fs(p)p 1250 559 30 2 v 1250 594 a Fq(n)1332 560 y(D)1374 505 y Fr(\021)1404 515 y Fl(n)1432 560 y Fq(g)1458 540 y Fu(2)1488 560 y Fq(;)346 b Fv(\(4)p Fq(:)p Fv(8\))0 680 y(and)20 b(where)f Fq(D)j Fv(denotes)e(di\013eren)o(tiation.)31 b(Equation)40 b(\(4.7\))21 b(can)f(b)q(e)g(regarded)f(as)h(an)g(iden)o(tit)o (y)f(in)0 740 y(the)i(v)o(ector)g(space)f Fs(V)25 b Fv(of)c(p)q(olynomials)e (of)i(degree)f Fs(\024)h Fv(2)p Fq(d)p Fv(,)h(equipp)q(ed)e(with)h(some)f (norm,)g(where)g Fq(D)0 799 y Fv(acts)15 b(as)g(a)g(b)q(ounded)f(linear)h(op) q(erator.)20 b(Since)15 b(con)o(v)o(ergence)f(in)h Fs(V)k Fv(implies)14 b(uniform)f(con)o(v)o(ergence)h(on)0 858 y(compact)i(subsets)e(of)448 848 y Fn(j)440 858 y Fv(C,)i(the)g(assertion)e(will)i(follo)o(w)f(if)h(w)o(e) g(sho)o(w)f(that)h(all)g(zeros)f(of)h Fq(g)1656 865 y Fu(1)1678 858 y Fq(;)8 b(g)1724 865 y Fu(2)1746 858 y Fq(;)g(:)g(:)g(:)17 b Fv(lie)e(in)0 918 y(the)j(strip)g Fs(j)p Fv(Re)o(\()p Fq(z)r Fv(\))p Fs(j)g(\024)e Fq(a)p Fv(.)27 b(But)19 b(this)f(is)f(a)i(consequence)e (of)i(the)f(follo)o(wing)f(theorem)h(b)o(y)g(T)l(ak)m(agi)g([10]:)0 977 y(If)f Fq(p)g Fv(is)f(a)h(p)q(olynomial)e(of)i(degree)f Fq(n)p Fv(,)h Fq(Z)j Fv(the)d(set)g(of)g(its)g(zeros,)f(and)g Fq(s)h Fv(a)f(complex)h(n)o(um)o(b)q(er,)d(then)j(the)0 1036 y(zeros)f(of)h(the)f(p)q(olynomial)g Fq(t)e Fs(7!)f Fq(p)p Fv(\()p Fq(t)p Fv(\))f Fs(\000)f Fq(sp)811 1018 y Fp(0)825 1036 y Fv(\()p Fq(t)p Fv(\))18 b(lie)e(in)g(the)h(con)o(v)o(ex)f(h)o(ull)g (of)g(the)h(set)g Fq(Z)d Fs([)e Fv(\()p Fq(Z)j Fv(+)c Fq(ns)p Fv(\).)p 1807 1114 43 43 v 100 1225 a(That)h(this)g(prop)q(ert)o(y)g(of)h Fs(N)20 b Fv(yields)12 b(some)g(information)f(ab)q(out)h(the)h(\014xed)g(p)q (oin)o(ts)e(of)i Fs(N)20 b Fv(is)12 b(eviden)o(t)0 1285 y(from)k(the)h(follo) o(wing)e(fact.)0 1380 y Fk(Lemma)j(4.3.)48 b Fj(Assume)17 b(that)h Fq(f)666 1362 y Fp(\003)704 1380 y Fs(2)e(H)795 1362 y Fl(e)795 1396 y Fu(\()p Fl(\014)r Fu(\))870 1380 y Fj(is)h(a)h(\014xed)f(p)q(oin)o(t)g (of)g Fs(N)7 b Fj(,)18 b(and)f(that)g Fs(j)l(j)l(j)p Fq(f)1622 1362 y Fp(\003)1646 1380 y Fs(j)l(j)l(j)d Fq(<)h Fs(1)p Fj(.)25 b(Then)0 1447 y(there)16 b(exists)h(a)f(p)q(olynomial)g Fq(g)i Fj(suc)o(h)d(that)i Fs(N)7 b Fv(\()p Fq(g)r Fv(\))15 b Fs(!)e Fq(f)1045 1429 y Fp(\003)1086 1447 y Fj(in)j Fs(H)1186 1429 y Fl(e)1186 1463 y Fu(\()p Fl(\014)r Fu(\))1244 1447 y Fj(,)g(as)g Fq(n)e Fs(!)g(1)p Fj(.)0 1571 y Fk(Pro)r(of.)k Fv(W)l(e)g(ma)o(y)e(assume)g (that)i Fq(f)681 1553 y Fp(\003)720 1571 y Fs(6\021)d Fv(0,)i(whic)o(h)f (actually)i(implies)e(that)h Fq(f)1471 1553 y Fp(\003)1495 1571 y Fv(\()p Fq(t)p Fv(\))f Fq(>)f Fv(0)i(for)g(all)g Fq(t)e Fs(2)g Fv(I)-10 b(R.)0 1630 y(Let)23 b Fq(N)k Fv(b)q(e)c(the)f(map)f(on)h Fs(H)560 1612 y Fl(e)560 1646 y Fu(\()p Fl(\014)r Fu(\))618 1630 y Fv(,)h(de\014ned)e(b)o(y)h Fq(N)5 b Fv(\()p Fq(f)g Fv(\))25 b(=)d Fs(N)7 b Fv(\()p Fq(f)1201 1612 y Fp(\003)1240 1630 y Fv(+)15 b Fq(f)5 b Fv(\))16 b Fs(\000)e Fq(f)1440 1612 y Fp(\003)1464 1630 y Fv(.)38 b(This)22 b(map)f(is)g(clearly)0 1700 y Fq(C)40 1682 y Fu(1)62 1700 y Fv(,)h(and)f(b)o(y)g(assumption)e Fq(N)5 b Fv(\(0\))23 b(=)f(0.)36 b(By)43 b(Corollary)20 b(2.3)h(the)h(deriv)m(ativ)o (e)f Fq(D)q(N)1629 1707 y Fu(0)1674 1700 y Fv(=)h(2)p Fq(A)p Fv(\()p Fq(f)1845 1682 y Fp(\003)1869 1700 y Fv(\))g(of)0 1760 y Fq(N)28 b Fv(at)c(zero)e(is)h(p)q(ositiv)o(e)g(and)f(compact.)41 b(Denote)23 b(b)o(y)g Fs(H)1131 1742 y Fl(s)1176 1760 y Fv(and)f Fs(H)1321 1742 y Fl(u)1370 1760 y Fv(the)i(sp)q(ectral)e(subspaces)f(for)0 1819 y Fq(D)q(N)82 1826 y Fu(0)124 1819 y Fv(asso)q(ciated)d(with)h(the)g(sp) q(ectrum)f(lying)h(in)f([0)p Fq(;)8 b Fv(1\))19 b(and)g([1)p Fq(;)8 b Fs(1)p Fv(\),)19 b(resp)q(ectiv)o(ely)l(.)29 b(By)19 b(the)g(stable)0 1878 y(manifold)d(theorem)g([12],)h Fq(N)22 b Fv(has)17 b(a)g(lo)q(cal)g(stable)g(manifold)f Fs(W)1227 1860 y Fl(s)1263 1878 y Fs(3)f Fv(0,)i(whic)o(h)f(is)h(the)g(graph)f(of)h(a)g Fq(C)1928 1860 y Fu(1)0 1937 y Fv(map)f Fq(\036)d Fv(:)h Fq(U)i Fs(\\)c(H)319 1919 y Fl(s)354 1937 y Fs(!)i(H)460 1919 y Fl(u)487 1937 y Fv(,)i(where)g Fq(U)22 b Fv(is)16 b(some)g(op)q(en)g(neigh)o(b)q(orho) q(o)q(d)f(of)i(zero)f(in)g Fs(H)1587 1919 y Fl(e)1587 1953 y Fu(\()p Fl(\014)r Fu(\))1645 1937 y Fv(.)100 2002 y Fs(W)153 1984 y Fl(s)193 2002 y Fv(can)j(also)f(b)q(e)i(c)o(haracterized)d(as)i(the)g (in)o(v)o(erse)f(image)g(of)h(zero)g(of)g(the)g(map)g Fq( )h Fv(:)d Fq(U)24 b Fs(!)18 b(H)1910 1984 y Fl(u)1936 2002 y Fv(,)0 2061 y(de\014ned)f(b)o(y)h Fq( )r Fv(\()p Fq(f)5 b Fv(\))18 b(=)e Fq(\036)p Fv(\()p Fq(f)496 2043 y Fl(s)518 2061 y Fv(\))c Fs(\000)g Fq(f)629 2043 y Fl(u)656 2061 y Fv(,)18 b(where)f Fq(f)862 2043 y Fl(s)902 2061 y Fv(and)h Fq(f)1030 2043 y Fl(u)1074 2061 y Fv(are)g(the)g(comp)q(onen)o(ts)f(of)h Fq(f)24 b Fv(in)17 b Fs(H)1729 2043 y Fl(s)1769 2061 y Fv(and)h Fs(H)1910 2043 y Fl(u)1936 2061 y Fv(,)0 2120 y(resp)q(ectiv)o(ely)l(.)26 b(This)18 b(map)f Fq( )j Fv(is)e(clearly)g Fq(C)824 2102 y Fu(1)845 2120 y Fv(,)h(and)e(the)h(rank)g(of)g Fq(D)q( )1312 2127 y Fu(0)1354 2120 y Fv(is)f(equal)h(to)h Fq(n)d Fs(\021)g Fv(dim)o Fs(H)1827 2102 y Fl(u)1854 2120 y Fv(,)i(i.e.)0 2180 y(if)f(w)o(e)f(c)o(ho)q(ose)g(a)h(basis)e(\(linear)h(co)q(ordinates\))g(in)g Fs(H)983 2162 y Fl(u)1026 2180 y Fv(and)g(denote)h(b)o(y)f Fq( )1385 2187 y Fl(k)1410 2180 y Fv(\()p Fq(f)5 b Fv(\))18 b(the)f Fq(k)1609 2162 y Fu(th)1665 2180 y Fv(co)q(ordinate)f(of)0 2239 y Fq( )r Fv(\()p Fq(f)5 b Fv(\),)23 b(then)d(the)g(gradien)o(ts)e Fq(h)591 2246 y Fl(k)635 2239 y Fs(\021)i(r)p Fq( )768 2246 y Fl(k)792 2239 y Fv(\(0\),)i(1)e Fs(\024)f Fq(k)i Fs(\024)f Fq(n)p Fv(,)h(are)e(linearly)h(indep)q(enden)o(t.)31 b(Th)o(us,)19 b(for)0 2298 y Fq(h)29 2305 y Fu(0)65 2298 y Fv(=)13 b Fq(f)146 2280 y Fp(\003)170 2298 y Fv(,)k(the)f(map)448 2448 y Fq(F)480 2455 y Fl(h)503 2460 y Fo(0)523 2455 y Fl(;)p Fp(\001\001\001)p Fl(;h)606 2460 y Fm(n)646 2448 y Fv(:)e(\()p Fq(\025)722 2455 y Fu(1)745 2448 y Fq(;)8 b(:)g(:)g(:)g(;)g(\025)884 2455 y Fl(n)912 2448 y Fv(\))14 b Fs(7!)g Fq( )1043 2407 y Fr(\000)1066 2448 y Fq(h)1095 2455 y Fu(0)1128 2448 y Fv(+)1202 2385 y Fl(n)1178 2400 y Fr(X)1178 2508 y Fl(k)q Fu(=1)1258 2448 y Fq(\025)1287 2455 y Fl(k)1312 2448 y Fq(h)1341 2455 y Fl(k)1376 2448 y Fs(\000)d Fq(f)1455 2427 y Fp(\003)1479 2407 y Fr(\001)1848 2448 y Fv(\(4)p Fq(:)p Fv(9\))0 2611 y(is)18 b(a)g Fq(C)135 2593 y Fu(1)174 2611 y Fv(di\013eomorphism)d(from)i(some)h(op)q(en)g(neigh)o(b)q(orho)q(o)q (d)e Fq(V)27 b Fs(3)17 b Fv(0)h(in)g(I)-10 b(R)1461 2589 y Fl(n)1507 2611 y Fv(to)18 b(some)f(op)q(en)h(neigh-)0 2670 y(b)q(orho)q(o)q(d)13 b(of)h(zero)f(in)g Fs(H)448 2652 y Fl(u)475 2670 y Fv(.)21 b(By)14 b(the)g(implicit)f(function)g(theorem,)g(the)h (equation)g Fq(F)1561 2677 y Fl(g)1580 2682 y Fo(0)1599 2677 y Fl(;:::;g)1678 2682 y Fm(n)1705 2670 y Fv(\()p Fq(\025)p Fv(\))h(=)e(0)h(has)950 2790 y(17)p eop %%Page: 18 19 bop 0 50 a Fv(a)18 b(solution)f Fq(\025)f Fs(2)g Fq(V)29 b Fv(for)17 b(an)o(y)h(\()p Fq(g)601 57 y Fu(0)623 50 y Fq(;)8 b(:)g(:)g(:)h(;)f(g)758 57 y Fl(n)785 50 y Fv(\))18 b(in)882 12 y Fr(L)937 25 y Fl(n)937 65 y(k)q Fu(=0)1020 50 y Fs(H)1062 32 y Fl(e)1062 66 y Fu(\()p Fl(\014)r Fu(\))1138 50 y Fv(su\016cien)o(tly)f (close)g(to)h(\()p Fq(h)1619 57 y Fu(0)1642 50 y Fq(;)8 b(:)g(:)g(:)g(;)g(h) 1781 57 y Fl(n)1808 50 y Fv(\).)27 b(But)0 117 y(since)16 b(p)q(olynomials)g (are)h(dense)f(in)g Fs(H)720 99 y Fl(e)720 133 y Fu(\()p Fl(\014)r Fu(\))778 117 y Fv(,)h(this)g(implies)f(that)h Fq( )r Fv(\()p Fq(g)c Fs(\000)e Fq(f)1354 99 y Fp(\003)1378 117 y Fv(\))k(=)f(0)j(for)g (some)f(p)q(olynomial)0 184 y Fq(g)r Fv(.)22 b(Since)16 b Fq(g)c Fs(\000)f Fq(f)307 166 y Fp(\003)345 184 y Fs(2)j(W)445 166 y Fl(s)466 184 y Fv(,)j(w)o(e)f(ha)o(v)o(e)g Fq(N)730 166 y Fl(n)757 184 y Fv(\()p Fq(g)d Fs(\000)e Fq(f)892 166 y Fp(\003)916 184 y Fv(\))j Fs(!)g Fv(0)i(as)g Fq(n)e Fs(!)g(1)p Fv(,)i(and)g(the)h (assertion)e(follo)o(ws.)p 1807 264 43 43 v 0 423 a Fk(Corollary)i(4.4.)42 b Fj(Assume)14 b(that)g Fq(f)701 405 y Fp(\003)739 423 y Fs(2)g(H)828 405 y Fl(e)828 439 y Fu(\()p Fl(\014)r Fu(\))900 423 y Fj(is)g(a)h(\014xed)f (p)q(oin)o(t)g(of)g Fs(N)7 b Fj(,)15 b(and)f(that)g Fs(j)l(j)l(j)p Fq(f)1628 405 y Fp(\003)1652 423 y Fs(j)l(j)l(j)f Fq(<)g Fs(1)p Fj(.)22 b(Then)0 487 y(all)16 b(zeros)g(of)h Fq(f)280 469 y Fp(\003)320 487 y Fj(lie)g(on)f(the)h(imaginary)d(axis.)0 614 y Fk(Pro)r(of.)f Fv(By)28 b(Lemma)12 b(4.3)h(there)h(is)e(a)i(sequence)f(of)h (p)q(olynomials)d Fq(g)r(;)d Fs(N)f Fv(\()p Fq(g)r Fv(\))p Fq(;)h Fs(N)1480 596 y Fu(2)1503 614 y Fv(\()p Fq(g)r Fv(\))p Fq(;)g(:)g(:)g(:)14 b Fv(con)o(v)o(erging)e(to)0 674 y Fq(f)29 656 y Fp(\003)53 674 y Fv(,)k(and)g(b)o(y)33 b(Lemma)16 b(4.2)g(the)h(zeros)f (of)g Fs(N)839 656 y Fl(n)867 674 y Fv(\()p Fq(g)r Fv(\))h(lie)f(in)g(the)h (strip)f Fs(j)p Fv(Re)o(\()p Fq(z)r Fv(\))p Fs(j)f(\024)f Fq(\014)1523 656 y Fl(n)1550 674 y Fq(a)i Fv(for)h(some)f Fq(a)e(<)f Fs(1)p Fv(.)0 733 y(Th)o(us,)20 b(since)f(con)o(v)o(ergence)h(in)g Fs(H)654 715 y Fl(e)654 749 y Fu(\()p Fl(\014)r Fu(\))732 733 y Fv(implies)f(uniform)f(con)o(v)o(ergence)i(on)g(compact)g(subsets)f(of)1909 723 y Fn(j)1900 733 y Fv(C,)0 800 y Fq(f)29 782 y Fp(\003)71 800 y Fv(cannot)e(ha)o(v)o(e)h(an)o(y)f(zeros)g(o\013)h(the)g(imaginary)f (axis)g(\(using)h(e.g.)26 b(the)18 b(standard)e(form)o(ula)g(for)i(the)0 859 y(n)o(um)o(b)q(er)c(of)j(zeros)f(inside)f(a)i(circle\).)p 1807 939 V 0 1062 a Fk(Pro)r(of)c(of)h(Theorem)f(1.1.)36 b Fv(W)l(e)13 b(tak)o(e)g Fq(f)790 1069 y Fm(I)r(R)849 1062 y Fv(to)h(b)q(e)f(the)g(\014xed)g(p)q(oin)o(t)f(of)i Fs(N)20 b Fv(describ)q(ed)12 b(in)g(Section)h(3,)h(or)0 1121 y(an)o(y)d(other)h (non{constan)o(t)e(\014xed)i(p)q(oin)o(t)f Fq(f)786 1103 y Fp(\003)822 1121 y Fv(that)h(lies)g(in)f(one)h(of)g(the)g(spaces)f Fs(B)1461 1128 y Fl(\032)1496 1121 y Fv(with)g Fq(\032)j(>)g(\014)s(=)p Fv(\(1)r Fs(\000)r Fv(2)p Fq(\014)1896 1103 y Fu(2)1917 1121 y Fv(\).)0 1181 y(Then,)j(b)o(y)34 b(Lemma)17 b(4.1,)g(the)g(logarithm)f(of)i Fq(f)907 1163 y Fp(\003)930 1181 y Fv(\()p Fq(t)p 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Fv(is)f(giv)o(en)g(b)o(y)g(a)h(canonical)0 1477 y(pro)q(duct)33 b(\(1.2\).)22 b(This)16 b(implies)f(in)h(particular)f (that)i Fq(f)1050 1459 y Fp(\003)1090 1477 y Fv(is)f(con)o(v)o(ex,)g(and)g (th)o(us)263 1657 y Fq(f)292 1636 y Fp(\003)316 1657 y Fv(\()p Fq(\014)366 1636 y Fp(\000)p Fu(1)419 1657 y Fq(t)p Fv(\))f(=)645 1623 y(1)p 529 1645 258 2 v 529 1655 a Fr(p)p 579 1655 208 2 v 43 x Fv(\(1)c Fs(\000)g Fq(\014)715 1684 y Fu(2)737 1698 y Fv(\))p Fq(\031)828 1579 y Fp(1)812 1589 y Fr(Z)790 1730 y Fp(\0001)872 1657 y Fq(ds)d(e)952 1629 y Fp(\000)1023 1616 y Fo(1)p 990 1622 85 2 v 990 1647 a(1)p Fn(\000)p Fm(\014)1055 1640 y Fo(2)1080 1629 y Fl(s)1099 1614 y Fo(2)1127 1637 y Fu(1)p 1127 1645 20 2 v 1127 1674 a(2)1153 1617 y Fr(\002)1173 1657 y Fq(f)1202 1636 y Fp(\003)1226 1657 y Fv(\()p Fq(t)j Fv(+)g Fq(s)p Fv(\))1366 1636 y Fu(2)1400 1657 y Fv(+)g Fq(f)1479 1636 y Fp(\003)1503 1657 y Fv(\()p Fq(t)g Fs(\000)g Fq(s)p Fv(\))1643 1636 y Fu(2)1666 1617 y Fr(\003)471 1856 y Fs(\025)645 1823 y Fv(1)p 529 1845 258 2 v 529 1855 a Fr(p)p 579 1855 208 2 v 43 x Fv(\(1)g Fs(\000)g Fq(\014)715 1883 y Fu(2)737 1898 y Fv(\))p Fq(\031)828 1779 y Fp(1)812 1789 y Fr(Z)790 1929 y Fp(\0001)872 1856 y Fq(ds)d(e)952 1829 y Fp(\000)1023 1815 y Fo(1)p 990 1821 85 2 v 990 1846 a(1)p Fn(\000)p Fm(\014)1055 1839 y Fo(2)1080 1829 y Fl(s)1099 1814 y Fo(2)1121 1856 y Fq(f)1150 1836 y Fp(\003)1173 1856 y Fv(\()p Fq(t)p Fv(\))1229 1836 y Fu(2)1266 1856 y Fv(=)14 b Fq(f)1348 1836 y Fp(\003)1372 1856 y Fv(\()p Fq(t)p Fv(\))1428 1836 y Fu(2)1459 1856 y Fq(;)1823 1761 y Fv(\(4)p Fq(:)p Fv(10\))0 2052 y(for)i(all)h Fq(t)d Fs(2)g Fv(I)-10 b(R.)23 b(The)16 b(inequalit)o(y)34 b(\(4.10\))17 b(sho)o(ws)e(that)i(the)g(sequence)g Fq(n)d Fs(7!)g Fv(\()p Fq(\014)1504 2034 y Fp(\000)p Fl(n)1562 2052 y Fq(t)p Fv(\))1599 2034 y Fp(\000)p Fu(6)p Fl(=)p Fu(5)1702 2052 y Fv(ln)8 b Fq(f)1781 2034 y Fp(\003)1804 2052 y Fv(\()p Fq(\014)1854 2034 y Fp(\000)p Fl(n)1913 2052 y Fq(t)p Fv(\))0 2111 y(is)19 b(non{decreasing,)e(and)h(hence) h(con)o(v)o(ergen)o(t,)f(for)h(an)o(y)f(giv)o(en)h Fq(t)e Fs(2)i Fv(I)-10 b(R)o(.)29 b(In)19 b(particular,)f(if)h Fq(f)5 b Fv(\()p Fq(t)p Fv(\))19 b Fq(>)f Fv(1)0 2171 y(\(suc)o(h)f(a)h Fq(t)g Fv(exists)g(since)f Fq(f)503 2153 y Fp(\003)545 2171 y Fv(is)g(not)h(constan) o(t\),)g(then)g(the)g(limit)f(is)h(p)q(ositiv)o(e,)g(pro)o(ving)e(that)i Fq(`)1807 2178 y Fu(+)1856 2171 y Fq(>)e Fv(0.)0 2230 y(The)j(inequalit)o(y) 39 b(\(4.10\))19 b(also)g(sho)o(ws)f(that)i(the)f(function)g Fq(t)g Fs(7!)f Fq(t)1274 2212 y Fp(\000)p Fu(6)p Fl(=)p Fu(5)1376 2230 y Fv(ln)8 b Fq(f)1455 2212 y Fp(\003)1479 2230 y Fv(\()p Fq(t)p Fv(\))20 b(v)m(aries)f(b)o(y)g(no)g(more)0 2289 y(than)d(a)g(factor)h (of)f(t)o(w)o(o)g(on)g(an)o(y)g(in)o(terv)m(al)g([)p Fq(t;)8 b(\014)878 2271 y Fp(\000)p Fu(1)931 2289 y Fq(t)p Fv(].)22 b(Hence)17 b Fq(`)1168 2296 y Fp(\000)1215 2289 y Fs(\025)d Fq(`)1289 2296 y Fu(+)1322 2289 y Fq(=)p Fv(2.)21 b(This)16 b(completes)g(the)g(pro)q(of)0 2349 y(of)33 b(Theorem)16 b(1.1.)p 1807 2428 43 43 v 0 2551 a Fk(Pro)r(of)21 b(of)h(Theorem)g(1.4.)55 b Fv(Let)20 b Fq(f)740 2558 y Fm(I)r(R)806 2551 y Fv(b)q(e)h(the)f(\014xed)g (p)q(oin)o(t)f(of)i Fs(N)27 b Fv(describ)q(ed)19 b(in)40 b(Theorem)19 b(1.1,)0 2611 y(or)e(an)o(y)f(other)h(\014xed)g(p)q(oin)o(t)g(with)g(the)g (prop)q(erties:)22 b Fq(f)1018 2618 y Fm(I)r(R)1081 2611 y Fv(is)17 b(en)o(tire)g(analytic,)g(and)f(real{v)m(alued)h(when)0 2670 y(restricted)k(to)h(the)f(real)g(axis,)h(and)f Fq(t)h Fs(7!)g Fq(f)844 2677 y Fm(I)r(R)890 2670 y Fv(\()p Fq(t)p Fv(\))8 b(exp)q(\()p Fs(\000)p Fq(")p Fs(j)p Fq(t)p Fs(j)1158 2652 y Fu(2)1180 2670 y Fv(\))22 b(is)f(a)g(b)q(ounded)g(function)g(on)1826 2660 y Fn(j)1818 2670 y Fv(C,)h(for)950 2790 y(18)p eop %%Page: 19 20 bop 0 50 a Fv(ev)o(ery)17 b Fq(")d(>)f Fv(0.)22 b(Giv)o(en)16 b(an)o(y)g(p)q(ositiv)o(e)h(in)o(teger)f Fq(k)r Fv(,)g(let)h Fq(d)c Fv(=)h(2)1117 32 y Fl(k)q Fp(\000)p Fu(1)1192 50 y Fv(.)22 b(By)17 b(using)f(that)h Fq(f)1569 57 y Fm(I)r(R)1629 50 y Fv(=)c Fs(N)1729 32 y Fl(k)1754 50 y Fv(\()p Fq(f)1797 57 y Fm(I)r(R)1843 50 y Fv(\),)k(w)o(e)0 109 y(obtain)f(a)g(represen)o(tation)f (for)h Fq(f)622 116 y Fm(I)r(R)685 109 y Fv(of)h(the)g(form)573 213 y Fq(f)597 220 y Fm(I)r(R)643 213 y Fv(\()p Fq(t)p Fv(\))e(=)e(exp)842 173 y Fr(\000)865 213 y Fs(\000)p Fv(2)929 193 y Fl(k)q Fp(\000)p Fu(1)1004 213 y Fq(\014)1035 193 y Fu(2)p Fl(k)1123 194 y Fu(1)p 1085 202 95 2 v 1085 230 a(1)p Fp(\000)p Fl(\014)1161 220 y Fo(2)1186 213 y Fq(t)1204 193 y Fu(2)1226 173 y Fr(\001)1249 213 y Fq(f)1273 220 y Fl(k)1298 213 y Fv(\()p Fq(t)p Fv(\))8 b Fq(;)447 b Fv(\(4)p Fq(:)p Fv(11\))0 317 y(with)25 465 y Fq(f)49 472 y Fl(k)74 465 y Fv(\()p Fq(t)p Fv(\))14 b(=)197 397 y Fr(Z)247 465 y Fq(d\026)303 472 y Fl(k)327 465 y Fv(\()p Fq(s)p Fv(\))8 b Fq(e)419 445 y Fl(tL)461 450 y Fm(k)485 445 y Fu(\()p Fl(s)p Fu(\))570 403 y Fl(d)548 418 y Fr(Y)546 524 y Fl(j)r Fu(=1)623 465 y Fq(f)647 472 y Fm(I)r(R)693 465 y Fv(\()p Fq(s)735 472 y Fl(j)757 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b(\001)g(\001)f(\001)h(\001)g(\001)101 b(\000)p Fg(1)p Fh(:)p Fg(258)1629 1075 y Fr(1)1629 1163 y(C)1629 1193 y(C)1629 1223 y(C)1629 1253 y(C)1629 1285 y(A)0 1472 y Fv(The)15 b(table)f(b)q(elo)o(w)h(giv)o(es)f(the)h(comp)q(onen)o(ts)f(of)g Fq(g)i Fs(\021)d Fq(p)8 b Fv(+)g Fq(\016)r Fv(,)14 b(where)h Fq(p)g Fv(and)f Fq(\016)i Fv(are)f(the)g(v)o(ectors)f(\(p)q(olyno-)0 1532 y(mials\))h(de\014ned)g(earlier)f(in)31 b(\(3.21\),)17 b(and)e(b)q(ounds)f(on)h(p)q(o)o(w)o(ers)g(of)h Fq(\032)p Fv(.)21 b(These)15 b(quan)o(tities)h(will)f(b)q(e)h(used)0 1592 y(frequen)o(tly)g (later)h(on.)324 1711 y Fh(p)345 1717 y Fe(0)378 1711 y Fg(=)12 b(0)p Fh(:)p Fg(909)i Fh(;)49 b(\016)613 1717 y Fe(0)646 1711 y Fg(=)11 b(0)p Fh(:)p Fg(005)k Fh(;)49 b(g)882 1717 y Fe(0)915 1711 y Fg(=)11 b(0)p Fh(:)p Fg(914)k Fh(;)49 b(\032)1153 1697 y Fe(1)1185 1711 y Ff(\024)12 b Fg(1)p Fh(:)p Fg(517)j Fh(;)49 b(\032)1424 1697 y Fd(\000)p Fe(1)1485 1711 y Ff(\024)11 b Fg(0)p Fh(:)p Fg(660)324 1756 y Fh(p)345 1762 y Fe(1)378 1756 y Fg(=)h(0)p Fh(:)p Fg(492)i Fh(;)49 b(\016)613 1762 y Fe(1)646 1756 y Fg(=)11 b(0)p Fh(:)p Fg(010)k Fh(;)49 b(g)882 1762 y Fe(1)915 1756 y Fg(=)11 b(0)p Fh(:)p Fg(502)k Fh(;)49 b(\032)1153 1742 y Fe(2)1185 1756 y Ff(\024)12 b Fg(2)p Fh(:)p Fg(301)j Fh(;)49 b(\032)1424 1742 y Fd(\000)p Fe(2)1485 1756 y Ff(\024)11 b Fg(0)p Fh(:)p Fg(435)324 1801 y Fh(p)345 1807 y Fe(2)378 1801 y Fg(=)h(0)p Fh(:)p Fg(251)i Fh(;)49 b(\016)613 1807 y Fe(2)646 1801 y Fg(=)11 b(0)p Fh(:)p Fg(004)k Fh(;)49 b(g)882 1807 y Fe(2)915 1801 y Fg(=)11 b(0)p Fh(:)p Fg(255)k Fh(;)49 b(\032)1153 1787 y Fe(3)1185 1801 y Ff(\024)12 b Fg(3)p Fh(:)p Fg(489)j Fh(;)49 b(\032)1424 1787 y Fd(\000)p Fe(3)1485 1801 y Ff(\024)11 b Fg(0)p Fh(:)p Fg(287)324 1846 y Fh(p)345 1852 y Fe(3)378 1846 y Fg(=)h(0)p Fh(:)p Fg(112)i Fh(;)49 b(\016)613 1852 y Fe(3)646 1846 y Fg(=)11 b(0)p Fh(:)p Fg(008)k Fh(;)49 b(g)882 1852 y Fe(3)915 1846 y Fg(=)11 b(0)p Fh(:)p Fg(120)k Fh(;)49 b(\032)1153 1832 y Fe(4)1185 1846 y Ff(\024)12 b Fg(5)p Fh(:)p Fg(292)j Fh(;)49 b(\032)1424 1832 y Fd(\000)p Fe(4)1485 1846 y Ff(\024)11 b Fg(0)p Fh(:)p Fg(189)324 1890 y Fh(p)345 1896 y Fe(4)378 1890 y Fg(=)h(0)p Fh(:)p Fg(045)i Fh(;)49 b(\016)613 1896 y Fe(4)646 1890 y Fg(=)11 b(0)p Fh(:)p Fg(007)k Fh(;)49 b(g)882 1896 y Fe(4)915 1890 y Fg(=)11 b(0)p Fh(:)p Fg(052)k Fh(;)49 b(\032)1153 1876 y Fe(5)1185 1890 y Ff(\024)12 b Fg(8)p Fh(:)p Fg(026)j Fh(;)49 b(\032)1424 1876 y Fd(\000)p Fe(5)1485 1890 y Ff(\024)11 b Fg(0)p Fh(:)p Fg(125)324 1935 y Fh(p)345 1941 y Fe(5)378 1935 y Fg(=)h(0)p Fh(:)p Fg(016)i Fh(;)49 b(\016)613 1941 y Fe(5)646 1935 y Fg(=)11 b(0)p Fh(:)p Fg(005)k Fh(;)49 b(g)882 1941 y Fe(5)915 1935 y Fg(=)11 b(0)p Fh(:)p Fg(021)k Fh(;)49 b(\032)1153 1921 y Fe(6)1185 1935 y Ff(\024)12 b Fg(12)p Fh(:)p Fg(18)j Fh(;)49 b(\032)1424 1921 y Fd(\000)p Fe(6)1485 1935 y Ff(\024)11 b Fg(0)p Fh(:)p Fg(083)324 1980 y Fh(p)345 1986 y Fe(6)378 1980 y Fg(=)h(0)p Fh(:)p Fg(005)i Fh(;)49 b(\016)613 1986 y Fe(6)646 1980 y Fg(=)11 b(0)p Fh(:)p Fg(003)k Fh(;)49 b(g)882 1986 y Fe(6)915 1980 y Fg(=)11 b(0)p Fh(:)p Fg(008)k Fh(;)49 b(\032)1153 1966 y Fe(7)1185 1980 y Ff(\024)12 b Fg(18)p Fh(:)p Fg(47)j Fh(;)49 b(\032)1424 1966 y Fd(\000)p Fe(7)1485 1980 y Ff(\024)11 b Fg(0)p Fh(:)p Fg(055)423 2060 y Fc(T)l(able)32 b(1.)j Fg(The)13 b(v)o(ectors)h Fh(p;)7 b(\016)o(;)g(g)q Fg(,)14 b(and)g(b)q(ounds)e(on)i Fh(\032)1311 2046 y Fd(\000)p Fe(7)1360 2060 y Fh(;)7 b(:)g(:)f(:)h(;)g(\032)1476 2046 y Fe(7)1496 2060 y Fg(.)0 2194 y Fv(The)19 b(ab)q(o)o(v)o(e)f(b)q(ounds)g(of)h(the)g (form)f Fq(\032)717 2176 y Fl(k)760 2194 y Fs(\024)f Fq(r)838 2201 y Fl(k)882 2194 y Fv(w)o(ere)i(obtained)f(iterativ)o(ely)h(b)o(y)g (setting)f Fq(r)1705 2201 y Fl(k)1748 2194 y Fv(=)g Fq(r)1827 2201 y Fl(k)q Fp(\000)p Fl(i)1897 2194 y Fq(r)1919 2201 y Fl(i)1936 2194 y Fv(,)0 2254 y(rounded)h(up)g(to)i(5)f(digits)g(after)h(the)f(decimal)g (p)q(oin)o(t,)g(where)g Fq(i)h Fv(is)f(the)g(in)o(teger)g(part)g(of)g Fq(k)r(=)p Fv(2.)33 b(The)0 2314 y(iteration)16 b(w)o(as)h(started)f(with)h (the)g(upp)q(er)f(b)q(ounds)f Fq(r)1006 2321 y Fp(\006)p Fu(1)1077 2314 y Fv(in)i(the)g(inequalities)f(\(I3\))h(of)g(Section)g(3,)g(and)0 2374 y(with)e Fq(r)134 2381 y Fu(0)157 2374 y Fv(=1.)21 b(A)o(t)15 b(the)g(end,)f(the)h(n)o(um)o(b)q(ers)e Fq(r)822 2381 y Fl(k)862 2374 y Fv(w)o(ere)h(rounded)f(up)h(to)h(the)g(n)o(um)o(b)q(er)e(of)i (decimals)f(sho)o(wn)0 2433 y(in)i(T)l(able)g(1.)950 2790 y(20)p eop %%Page: 21 22 bop 0 50 a Ft(5.1.)54 b(Pro)r(of)24 b(of)g(\(N1\))0 124 y Fv(Most)17 b(of)h(the)f(w)o(ork)g(in)g(pro)o(ving)f(that)i Fq(\033)f Fs(\024)e Fv(0)p Fq(:)p Fv(157)i(and)g Fq(\034)k Fs(\024)15 b Fv(0)p Fq(:)p Fv(064)i(go)q(es)g(in)o(to)g(estimating)g(62)g(of)h(the)0 191 y(co)q(e\016cien)o(ts)e Fq(A)289 165 y Fu(\()p Fl(m)p Fu(\))289 206 y Fl(k)q(;n)358 191 y Fv(.)22 b(The)17 b(follo)o(wing)e(b)q(ounds)g(will) h(turn)g(out)h(to)g(b)q(e)g(su\016cien)o(t.)254 333 y Fh(A)286 312 y Fe(\(0\))286 347 y(7)p Fb(;)p Fe(7)377 333 y Fg(=)12 b(2)443 319 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 333 y Ff(\002)588 317 y Fe(1000)p 588 324 74 2 v 588 349 a(4096)676 333 y Ff(\002)737 292 y Fd(p)p 767 292 19 2 v 25 x Fe(2)p 724 324 74 2 v 724 349 a(1000)932 333 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(9)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(3)h Ff(\002)f Fg(0)p Fh(:)p Fg(002)184 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(001)254 402 y Fh(A)286 381 y Fe(\(1\))286 416 y(6)p Fb(;)p Fe(7)377 402 y Fg(=)12 b(2)443 388 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 402 y Ff(\002)588 386 y Fe(1000)p 588 393 V 588 418 a(4096)676 402 y Ff(\002)724 361 y Fd(p)p 754 361 55 2 v 25 x Fe(182)p 724 393 85 2 v 730 418 a(1000)932 402 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(891)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(245)h Ff(\002)g Fg(0)p Fh(:)p Fg(0135)78 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(003)254 471 y Fh(A)286 450 y Fe(\(1\))286 484 y(7)p Fb(;)p Fe(7)377 471 y Fg(=)437 454 y Fe(700)p 428 461 74 2 v 428 486 a(2048)516 471 y Ff(\002)568 429 y Fd(p)p 598 429 19 2 v 25 x Fe(1)p 565 461 55 2 v 565 486 a(100)932 471 y Ff(\024)g Fg(0)p Fh(:)p Fg(4)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(01)311 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(004)254 540 y Fh(A)286 518 y Fe(\(1\))286 553 y(7)p Fb(;)p Fe(8)377 540 y Fg(=)12 b(2)443 525 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 540 y Ff(\002)588 523 y Fe(1000)p 588 530 74 2 v 588 555 a(4096)676 540 y Ff(\002)728 498 y Fd(p)p 758 498 37 2 v 25 x Fe(30)p 724 530 74 2 v 724 555 a(1000)932 540 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(8)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(25)h Ff(\002)f Fg(0)p Fh(:)p Fg(01)184 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(002)254 608 y Fh(A)286 587 y Fe(\(2\))286 622 y(5)p Fb(;)p Fe(7)377 608 y Fg(=)12 b(2)443 594 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 608 y Ff(\002)588 592 y Fe(1000)p 588 599 V 588 624 a(4096)676 608 y Ff(\002)724 567 y Fd(p)p 754 567 V 25 x Fe(2002)p 724 599 103 2 v 739 624 a(1000)932 608 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(90)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(245)h Ff(\002)f Fg(0)p Fh(:)p Fg(045)121 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(010)254 677 y Fh(A)286 656 y Fe(\(2\))286 690 y(6)p Fb(;)p Fe(7)377 677 y Fg(=)428 661 y Fe(1000)p 428 668 74 2 v 428 693 a(1024)516 677 y Ff(\002)565 636 y Fd(p)p 594 636 55 2 v 594 661 a Fe(546)p 565 668 85 2 v 570 693 a(1000)932 677 y Ff(\024)g Fg(0)p Fh(:)p Fg(98)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(0234)248 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(023)254 746 y Fh(A)286 724 y Fe(\(2\))286 759 y(6)p Fb(;)p Fe(8)377 746 y Fg(=)12 b(2)443 732 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 746 y Ff(\002)588 729 y Fe(1000)p 588 736 74 2 v 588 761 a(4096)676 746 y Ff(\002)724 704 y Fd(p)p 754 704 55 2 v 25 x Fe(455)p 724 736 85 2 v 730 761 a(1000)932 746 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(80)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(25)h Ff(\002)f Fg(0)p Fh(:)p Fg(022)142 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(005)254 814 y Fh(A)286 793 y Fe(\(2\))286 828 y(7)p Fb(;)p Fe(7)377 814 y Fg(=)12 b(2)443 800 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 814 y Ff(\002)597 798 y Fe(910)p 588 805 74 2 v 588 830 a(8192)676 814 y Ff(\002)724 773 y Fd(p)p 754 773 19 2 v 25 x Fe(6)p 724 805 49 2 v 730 830 a(10)932 814 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(80)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(112)h Ff(\002)f Fg(0)p Fh(:)p Fg(245)121 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(022)254 883 y Fh(A)286 862 y Fe(\(2\))286 897 y(7)p Fb(;)p Fe(8)377 883 y Fg(=)12 b(2)443 869 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 883 y Ff(\002)597 867 y Fe(700)p 588 874 74 2 v 588 899 a(2048)676 883 y Ff(\002)724 842 y Fd(p)p 754 842 37 2 v 25 x Fe(10)p 724 874 67 2 v 730 899 a(100)932 883 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(891)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(35)h Ff(\002)f Fg(0)p Fh(:)p Fg(032)121 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(010)254 952 y Fh(A)286 931 y Fe(\(2\))286 965 y(7)p Fb(;)p Fe(9)377 952 y Fg(=)437 936 y Fe(3000)p 428 943 92 2 v 428 967 a(16384)534 952 y Ff(\002)586 911 y Fd(p)p 616 911 37 2 v 25 x Fe(85)p 583 943 74 2 v 583 967 a(1000)932 952 y Ff(\024)g Fg(0)p Fh(:)p Fg(2)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(01)311 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(002)254 1021 y Fh(A)286 999 y Fe(\(3\))286 1034 y(4)p Fb(;)p Fe(7)377 1021 y Fg(=)12 b(2)443 1007 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 1021 y Ff(\002)588 1004 y Fe(1000)p 588 1011 V 588 1036 a(4096)676 1021 y Ff(\002)724 979 y Fd(p)p 754 979 V 25 x Fe(6006)p 724 1011 103 2 v 739 1036 a(1000)932 1021 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(891)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(2442)h Ff(\002)g Fg(0)p Fh(:)p Fg(078)78 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(017)254 1089 y Fh(A)286 1068 y Fe(\(3\))286 1103 y(5)p Fb(;)p Fe(7)377 1089 y Fg(=)428 1073 y Fe(1000)p 428 1080 74 2 v 428 1105 a(2048)516 1089 y Ff(\002)565 1048 y Fd(p)p 594 1048 92 2 v 594 1073 a Fe(15015)p 565 1080 122 2 v 589 1105 a(1000)932 1089 y Ff(\024)g Fg(0)p Fh(:)p Fg(489)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(1226)227 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(060)254 1158 y Fh(A)286 1137 y Fe(\(3\))286 1172 y(5)p Fb(;)p Fe(8)377 1158 y Fg(=)12 b(2)443 1144 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 1158 y Ff(\002)588 1142 y Fe(1000)p 588 1149 74 2 v 588 1174 a(4096)676 1158 y Ff(\002)724 1117 y Fd(p)p 754 1117 V 25 x Fe(2002)p 724 1149 103 2 v 739 1174 a(1000)932 1158 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(80)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(245)h Ff(\002)f Fg(0)p Fh(:)p Fg(045)121 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(009)254 1227 y Fh(A)286 1206 y Fe(\(3\))286 1240 y(6)p Fb(;)p Fe(7)377 1227 y Fg(=)12 b(2)443 1213 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 1227 y Ff(\002)588 1210 y Fe(3300)p 588 1217 74 2 v 588 1242 a(8192)676 1227 y Ff(\002)724 1185 y Fd(p)p 754 1185 55 2 v 25 x Fe(910)p 724 1217 85 2 v 739 1242 a(100)932 1227 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(794)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(403)h Ff(\002)g Fg(0)p Fh(:)p Fg(302)99 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(097)254 1295 y Fh(A)286 1274 y Fe(\(3\))286 1309 y(6)p Fb(;)p Fe(8)377 1295 y Fg(=)12 b(2)443 1281 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 1295 y Ff(\002)597 1279 y Fe(300)p 588 1286 74 2 v 588 1311 a(2048)676 1295 y Ff(\002)724 1254 y Fd(p)p 754 1254 55 2 v 25 x Fe(546)p 724 1286 85 2 v 739 1311 a(100)932 1295 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(891)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(147)h Ff(\002)g Fg(0)p Fh(:)p Fg(234)99 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(031)254 1364 y Fh(A)286 1343 y Fe(\(3\))286 1378 y(6)p Fb(;)p Fe(9)377 1364 y Fg(=)428 1348 y Fe(10000)p 428 1355 92 2 v 428 1380 a(16384)534 1364 y Ff(\002)583 1323 y Fd(p)p 613 1323 74 2 v 25 x Fe(4641)p 583 1355 103 2 v 589 1380 a(10000)932 1364 y Ff(\024)g Fg(0)p Fh(:)p Fg(7)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(007)290 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(005)254 1433 y Fh(A)286 1412 y Fe(\(3\))286 1446 y(7)p Fb(;)p Fe(7)377 1433 y Fg(=)12 b(2)443 1419 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 1433 y Ff(\002)597 1417 y Fe(910)p 588 1424 74 2 v 588 1449 a(2048)676 1433 y Ff(\002)724 1392 y Fd(p)p 754 1392 19 2 v 25 x Fe(5)p 724 1424 49 2 v 730 1449 a(10)932 1433 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(891)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(445)h Ff(\002)g Fg(0)p Fh(:)p Fg(224)99 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(089)254 1502 y Fh(A)286 1480 y Fe(\(3\))286 1515 y(7)p Fb(;)p Fe(8)377 1502 y Fg(=)437 1485 y Fe(4550)p 428 1492 92 2 v 428 1517 a(16384)534 1502 y Ff(\002)583 1460 y Fd(p)p 613 1460 19 2 v 25 x Fe(3)p 583 1492 49 2 v 589 1517 a(10)932 1502 y Ff(\024)g Fg(0)p Fh(:)p Fg(278)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(174)248 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(049)254 1570 y Fh(A)286 1549 y Fe(\(3\))286 1584 y(7)p Fb(;)p Fe(9)377 1570 y Fg(=)12 b(2)443 1556 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 1570 y Ff(\002)588 1554 y Fe(2100)p 588 1561 74 2 v 588 1586 a(8192)676 1570 y Ff(\002)724 1529 y Fd(p)p 754 1529 37 2 v 25 x Fe(51)p 724 1561 67 2 v 730 1586 a(100)932 1570 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(80)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(257)h Ff(\002)f Fg(0)p Fh(:)p Fg(072)121 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(015)254 1639 y Fh(A)286 1618 y Fe(\(3\))286 1653 y(7)p Fb(;)p Fe(10)377 1639 y Fg(=)12 b(2)443 1625 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 1639 y Ff(\002)588 1623 y Fe(10000)p 588 1630 92 2 v 588 1655 a(32768)694 1639 y Ff(\002)743 1598 y Fd(p)p 772 1598 74 2 v 772 1623 a Fe(4845)p 743 1630 103 2 v 748 1655 a(10000)932 1639 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(9)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(31)h Ff(\002)f Fg(0)p Fh(:)p Fg(007)163 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(002)254 1708 y Fh(A)286 1687 y Fe(\(4\))286 1721 y(4)p Fb(;)p Fe(7)377 1708 y Fg(=)428 1692 y Fe(100)p 428 1699 55 2 v 428 1723 a(256)498 1708 y Ff(\002)546 1666 y Fd(p)p 576 1666 V 26 x Fe(429)p 546 1699 85 2 v 561 1723 a(100)932 1708 y Ff(\024)g Fg(0)p Fh(:)p Fg(3907)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(2072)205 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(081)254 1777 y Fh(A)286 1755 y Fe(\(4\))286 1790 y(4)p Fb(;)p Fe(8)377 1777 y Fg(=)12 b(2)443 1762 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 1777 y Ff(\002)588 1760 y Fe(3000)p 588 1767 74 2 v 588 1792 a(8192)676 1777 y Ff(\002)724 1735 y Fd(p)p 754 1735 V 25 x Fe(1430)p 724 1767 103 2 v 739 1792 a(1000)932 1777 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(7938)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(3663)g Ff(\002)f Fg(0)p Fh(:)p Fg(03782)15 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(011)254 1845 y Fh(A)286 1824 y Fe(\(4\))286 1859 y(5)p Fb(;)p Fe(7)377 1845 y Fg(=)12 b(2)443 1831 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 1845 y Ff(\002)588 1829 y Fe(2100)p 588 1836 74 2 v 588 1861 a(4096)676 1845 y Ff(\002)724 1804 y Fd(p)p 754 1804 V 25 x Fe(2145)p 724 1836 103 2 v 748 1861 a(100)932 1845 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(794)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(513)h Ff(\002)g Fg(0)p Fh(:)p Fg(464)99 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(189)254 1914 y Fh(A)286 1893 y Fe(\(4\))286 1928 y(5)p Fb(;)p Fe(8)377 1914 y Fg(=)12 b(2)443 1900 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 1914 y Ff(\002)597 1898 y Fe(500)p 588 1905 74 2 v 588 1930 a(1024)676 1914 y Ff(\002)724 1873 y Fd(p)p 754 1873 55 2 v 25 x Fe(143)p 724 1905 85 2 v 739 1930 a(100)932 1914 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(90)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(49)h Ff(\002)f Fg(0)p Fh(:)p Fg(120)142 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(053)254 1983 y Fh(A)286 1962 y Fe(\(4\))286 1996 y(5)p Fb(;)p Fe(9)377 1983 y Fg(=)428 1966 y Fe(30000)p 428 1973 92 2 v 428 1998 a(32768)534 1983 y Ff(\002)583 1941 y Fd(p)p 613 1941 74 2 v 25 x Fe(4862)p 583 1973 103 2 v 589 1998 a(10000)932 1983 y Ff(\024)g Fg(1)p Fh(:)p Fg(0)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(007)290 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(007)254 2051 y Fh(A)286 2030 y Fe(\(4\))286 2065 y(6)p Fb(;)p Fe(7)377 2051 y Fg(=)12 b(2)443 2037 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 2051 y Ff(\002)597 2035 y Fe(770)p 588 2042 74 2 v 588 2067 a(2048)676 2051 y Ff(\002)724 2010 y Fd(p)p 754 2010 37 2 v 25 x Fe(65)p 724 2042 67 2 v 739 2067 a(10)932 2051 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(891)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(376)h Ff(\002)g Fg(0)p Fh(:)p Fg(807)99 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(271)254 2120 y Fh(A)286 2099 y Fe(\(4\))286 2134 y(6)p Fb(;)p Fe(8)377 2120 y Fg(=)437 2104 y Fe(770)p 428 2111 74 2 v 428 2136 a(4096)516 2120 y Ff(\002)565 2079 y Fd(p)p 594 2079 37 2 v 594 2104 a Fe(39)p 565 2111 67 2 v 579 2136 a(10)932 2120 y Ff(\024)g Fg(0)p Fh(:)p Fg(1880)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(625)226 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(118)254 2189 y Fh(A)286 2168 y Fe(\(4\))286 2202 y(6)p Fb(;)p Fe(9)377 2189 y Fg(=)12 b(2)443 2175 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 2189 y Ff(\002)597 2173 y Fe(300)p 588 2180 74 2 v 588 2205 a(2048)676 2189 y Ff(\002)724 2148 y Fd(p)p 754 2148 55 2 v 25 x Fe(663)p 724 2180 85 2 v 739 2205 a(100)932 2189 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(7938)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(1465)g Ff(\002)f Fg(0)p Fh(:)p Fg(2575)36 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(030)254 2258 y Fh(A)286 2236 y Fe(\(4\))286 2271 y(6)p Fb(;)p Fe(10)377 2258 y Fg(=)12 b(2)443 2244 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 2258 y Ff(\002)588 2241 y Fe(10000)p 588 2248 92 2 v 588 2273 a(65536)694 2258 y Ff(\002)743 2216 y Fd(p)p 772 2216 V 772 2241 a Fe(62985)p 743 2248 122 2 v 758 2273 a(10000)932 2258 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(90)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(16)h Ff(\002)f Fg(0)p Fh(:)p Fg(026)142 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(004)254 2326 y Fh(A)286 2305 y Fe(\(4\))286 2340 y(7)p Fb(;)p Fe(7)377 2326 y Fg(=)428 2310 y Fe(10010)p 428 2317 92 2 v 428 2342 a(32768)534 2326 y Ff(\002)583 2285 y Fd(p)p 613 2285 37 2 v 25 x Fe(70)p 583 2317 67 2 v 598 2342 a(10)932 2326 y Ff(\024)g Fg(0)p Fh(:)p Fg(3055)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(837)226 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(256)254 2395 y Fh(A)286 2374 y Fe(\(4\))286 2409 y(7)p Fb(;)p Fe(8)377 2395 y Fg(=)12 b(2)443 2381 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 2395 y Ff(\002)597 2379 y Fe(910)p 588 2386 74 2 v 588 2411 a(2048)676 2395 y Ff(\002)724 2354 y Fd(p)p 754 2354 37 2 v 25 x Fe(21)p 724 2386 67 2 v 739 2411 a(10)932 2395 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(794)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(4444)h Ff(\002)g Fg(0)p Fh(:)p Fg(459)78 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(162)254 2464 y Fh(A)286 2443 y Fe(\(4\))286 2477 y(7)p Fb(;)p Fe(9)377 2464 y Fg(=)12 b(2)443 2450 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 2464 y Ff(\002)597 2448 y Fe(9100)p 588 2455 92 2 v 588 2479 a(32768)694 2464 y Ff(\002)743 2422 y Fd(p)p 772 2422 55 2 v 772 2448 a Fe(714)p 743 2455 85 2 v 758 2479 a(100)932 2464 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(891)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(278)h Ff(\002)g Fg(0)p Fh(:)p Fg(27)120 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(067)254 2533 y Fh(A)286 2511 y Fe(\(4\))286 2546 y(7)p Fb(;)p Fe(10)377 2533 y Fg(=)428 2516 y Fe(10000)p 428 2523 92 2 v 428 2548 a(16384)534 2533 y Ff(\002)583 2491 y Fd(p)p 613 2491 V 25 x Fe(67830)p 583 2523 122 2 v 598 2548 a(10000)932 2533 y Ff(\024)g Fg(0)p Fh(:)p Fg(611)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(0261)227 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(016)254 2601 y Fh(A)286 2580 y Fe(\(4\))286 2615 y(7)p Fb(;)p Fe(11)377 2601 y Fg(=)12 b(2)443 2587 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 2601 y Ff(\002)597 2585 y Fe(30000)p 588 2592 110 2 v 588 2617 a(262144)713 2601 y Ff(\002)761 2560 y Fd(p)p 791 2560 92 2 v 25 x Fe(35530)p 761 2592 122 2 v 776 2617 a(10000)932 2601 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(8)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(12)h Ff(\002)f Fg(0)p Fh(:)p Fg(02)184 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(002)254 2670 y Fh(A)286 2649 y Fe(\(5\))286 2684 y(5)p Fb(;)p Fe(7)377 2670 y Fg(=)12 b(2)443 2656 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 2670 y Ff(\002)597 2654 y Fe(150)p 588 2661 74 2 v 588 2686 a(1024)676 2670 y Ff(\002)724 2629 y Fd(p)p 754 2629 55 2 v 25 x Fe(858)p 724 2661 85 2 v 748 2686 a(10)932 2670 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(8909)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(1465)g Ff(\002)f Fg(2)p Fh(:)p Fg(93)78 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(383)950 2790 y Fv(21)p eop %%Page: 22 23 bop 254 50 a Fh(A)286 29 y Fe(\(5\))286 63 y(5)p Fb(;)p Fe(8)377 50 y Fg(=)428 34 y Fe(13500)p 428 41 92 2 v 428 65 a(32768)534 50 y Ff(\002)583 8 y Fd(p)p 613 8 74 2 v 26 x Fe(1430)p 583 41 103 2 v 607 65 a(100)932 50 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(4120)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(3782)205 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(156)254 119 y Fh(A)286 97 y Fe(\(5\))286 132 y(5)p Fb(;)p Fe(9)377 119 y Fg(=)12 b(2)443 104 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 119 y Ff(\002)597 102 y Fe(5000)p 588 109 92 2 v 588 134 a(16384)694 119 y Ff(\002)743 77 y Fd(p)p 772 77 V 772 102 a Fe(24310)p 743 109 122 2 v 767 134 a(1000)932 119 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(794)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(306)h Ff(\002)g Fg(0)p Fh(:)p Fg(156)99 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(038)254 187 y Fh(A)286 166 y Fe(\(5\))286 201 y(5)p Fb(;)p Fe(10)377 187 y Fg(=)12 b(2)443 173 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 187 y Ff(\002)588 171 y Fe(10000)p 588 178 92 2 v 588 203 a(65536)694 187 y Ff(\002)743 146 y Fd(p)p 772 146 V 772 171 a Fe(92378)p 743 178 122 2 v 758 203 a(10000)932 187 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(90)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(16)h Ff(\002)f Fg(0)p Fh(:)p Fg(031)142 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(005)254 255 y Fh(A)286 234 y Fe(\(5\))286 269 y(6)p Fb(;)p Fe(7)377 255 y Fg(=)437 239 y Fe(3465)p 428 246 92 2 v 428 271 a(32768)534 255 y Ff(\002)577 222 y(p)p 612 222 43 2 v 33 x Fg(26)278 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(1058)10 b Ff(\002)g Fg(5)p Fh(:)p Fg(10)247 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(540)254 324 y Fh(A)286 303 y Fe(\(5\))286 337 y(6)p Fb(;)p Fe(8)377 324 y Fg(=)12 b(2)443 310 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 324 y Ff(\002)597 308 y Fe(550)p 588 315 74 2 v 588 340 a(2048)676 324 y Ff(\002)724 283 y Fd(p)p 754 283 55 2 v 25 x Fe(195)p 724 315 85 2 v 748 340 a(10)932 324 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(7938)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(2686)g Ff(\002)f Fg(1)p Fh(:)p Fg(397)57 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(298)254 393 y Fh(A)286 371 y Fe(\(5\))286 406 y(6)p Fb(;)p Fe(9)377 393 y Fg(=)12 b(2)443 379 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 393 y Ff(\002)597 376 y Fe(9900)p 588 383 92 2 v 588 408 a(65536)694 393 y Ff(\002)743 351 y Fd(p)p 772 351 74 2 v 772 376 a Fe(6630)p 743 383 103 2 v 767 408 a(100)932 393 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(891)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(1511)h Ff(\002)g Fg(0)p Fh(:)p Fg(815)78 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(110)254 461 y Fh(A)286 440 y Fe(\(5\))286 475 y(6)p Fb(;)p Fe(10)377 461 y Fg(=)437 445 y Fe(5000)p 428 452 92 2 v 428 477 a(32768)534 461 y Ff(\002)583 420 y Fd(p)p 613 420 V 25 x Fe(25194)p 583 452 122 2 v 607 477 a(1000)932 461 y Ff(\024)g Fg(0)p Fh(:)p Fg(153)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(16)269 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(025)254 530 y Fh(A)286 509 y Fe(\(5\))286 544 y(6)p Fb(;)p Fe(11)377 530 y Fg(=)12 b(2)443 516 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 530 y Ff(\002)588 514 y Fe(100000)p 588 521 110 2 v 588 546 a(262144)713 530 y Ff(\002)761 489 y Fd(p)p 791 489 V 25 x Fe(646646)p 761 521 140 2 v 776 546 a(100000)932 530 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(80)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(39)h Ff(\002)f Fg(0)p Fh(:)p Fg(009)142 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(003)254 598 y Fh(A)286 577 y Fe(\(5\))286 611 y(7)p Fb(;)p Fe(7)377 598 y Fg(=)12 b(2)443 584 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 598 y Ff(\002)597 582 y Fe(3003)p 588 589 92 2 v 588 614 a(16384)694 598 y Ff(\002)737 565 y(p)p 772 565 43 2 v 33 x Fg(14)118 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(7938)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(1833)g Ff(\002)f Fg(3)p Fh(:)p Fg(742)57 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(545)254 667 y Fh(A)286 646 y Fe(\(5\))286 680 y(7)p Fb(;)p Fe(8)377 667 y Fg(=)12 b(2)443 653 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 667 y Ff(\002)588 650 y Fe(10010)p 588 657 92 2 v 588 682 a(32768)694 667 y Ff(\002)743 625 y Fd(p)p 772 625 55 2 v 772 650 a Fe(210)p 743 657 85 2 v 767 682 a(10)932 667 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(8909)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(3055)g Ff(\002)f Fg(1)p Fh(:)p Fg(45)78 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(395)254 735 y Fh(A)286 714 y Fe(\(5\))286 749 y(7)p Fb(;)p Fe(9)377 735 y Fg(=)428 719 y Fe(3900)p 428 726 74 2 v 428 751 a(8192)516 735 y Ff(\002)565 694 y Fd(p)p 594 694 V 594 719 a Fe(1785)p 565 726 103 2 v 589 751 a(100)932 735 y Ff(\024)g Fg(0)p Fh(:)p Fg(477)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(423)248 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(202)254 804 y Fh(A)286 783 y Fe(\(5\))286 818 y(7)p Fb(;)p Fe(10)377 804 y Fg(=)12 b(2)443 790 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 804 y Ff(\002)588 788 y Fe(195000)p 588 795 110 2 v 588 820 a(262144)713 804 y Ff(\002)761 763 y Fd(p)p 791 763 92 2 v 25 x Fe(13566)p 761 795 122 2 v 785 820 a(1000)932 804 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(794)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(744)h Ff(\002)g Fg(0)p Fh(:)p Fg(1165)78 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(069)254 873 y Fh(A)286 852 y Fe(\(5\))286 886 y(7)p Fb(;)p Fe(11)377 873 y Fg(=)12 b(2)443 859 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 873 y Ff(\002)597 857 y Fe(35000)p 588 864 110 2 v 588 889 a(131072)713 873 y Ff(\002)761 832 y Fd(p)p 791 832 74 2 v 25 x Fe(3553)p 761 864 103 2 v 776 889 a(1000)932 873 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(9)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(27)h Ff(\002)f Fg(0)p Fh(:)p Fg(06)184 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(015)254 942 y Fh(A)286 920 y Fe(\(5\))286 955 y(7)p Fb(;)p Fe(12)377 942 y Fg(=)428 925 y Fe(100000)p 428 932 110 2 v 428 957 a(524288)553 942 y Ff(\002)601 900 y Fd(p)p 631 900 V 25 x Fe(490314)p 601 932 140 2 v 616 957 a(100000)932 942 y Ff(\024)g Fg(0)p Fh(:)p Fg(2)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(01)311 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(002)254 1010 y Fh(A)286 989 y Fe(\(6\))286 1024 y(6)p Fb(;)p Fe(7)377 1010 y Fg(=)12 b(2)443 996 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 1010 y Ff(\002)597 994 y Fe(990)p 588 1001 74 2 v 588 1026 a(2048)676 1010 y Ff(\002)724 969 y Fd(p)p 754 969 55 2 v 25 x Fe(429)p 724 1001 85 2 v 748 1026 a(10)932 1010 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(7938)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(4834)g Ff(\002)f Fg(2)p Fh(:)p Fg(0713)36 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(795)254 1079 y Fh(A)286 1058 y Fe(\(6\))286 1093 y(6)p Fb(;)p Fe(8)377 1079 y Fg(=)12 b(2)443 1065 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 1079 y Ff(\002)588 1063 y Fe(14850)p 588 1070 92 2 v 588 1095 a(65536)694 1079 y Ff(\002)743 1038 y Fd(p)p 772 1038 55 2 v 772 1063 a Fe(715)p 743 1070 85 2 v 767 1095 a(10)932 1079 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(8909)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(2266)g Ff(\002)f Fg(2)p Fh(:)p Fg(674)57 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(540)254 1148 y Fh(A)286 1127 y Fe(\(6\))286 1161 y(6)p Fb(;)p Fe(9)377 1148 y Fg(=)437 1132 y Fe(5500)p 428 1139 92 2 v 428 1163 a(32768)534 1148 y Ff(\002)583 1106 y Fd(p)p 613 1106 V 26 x Fe(24310)p 583 1139 122 2 v 616 1163 a(100)932 1148 y Ff(\024)g Fg(0)p Fh(:)p Fg(1679)10 b Ff(\002)g Fg(1)p Fh(:)p Fg(56)247 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(262)254 1217 y Fh(A)286 1195 y Fe(\(6\))286 1230 y(6)p Fb(;)p Fe(10)377 1217 y Fg(=)12 b(2)443 1202 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 1217 y Ff(\002)588 1200 y Fe(33000)p 588 1207 92 2 v 588 1232 a(65536)694 1217 y Ff(\002)743 1175 y Fd(p)p 772 1175 V 772 1200 a Fe(46189)p 743 1207 122 2 v 767 1232 a(1000)932 1217 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(7938)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(5036)g Ff(\002)f Fg(0)p Fh(:)p Fg(2150)36 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(086)254 1285 y Fh(A)286 1264 y Fe(\(6\))286 1299 y(6)p Fb(;)p Fe(11)377 1285 y Fg(=)12 b(2)443 1271 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 1285 y Ff(\002)588 1269 y Fe(30000)p 588 1276 92 2 v 588 1301 a(65536)694 1285 y Ff(\002)743 1244 y Fd(p)p 772 1244 110 2 v 772 1269 a Fe(176358)p 743 1276 140 2 v 767 1301 a(10000)932 1285 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(9)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(46)h Ff(\002)f Fg(0)p Fh(:)p Fg(042)163 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(018)254 1354 y Fh(A)286 1333 y Fe(\(6\))286 1367 y(6)p Fb(;)p Fe(12)377 1354 y Fg(=)428 1338 y Fe(100000)p 428 1345 110 2 v 428 1370 a(524288)553 1354 y Ff(\002)601 1313 y Fd(p)p 631 1313 V 25 x Fe(676039)p 601 1345 140 2 v 616 1370 a(100000)932 1354 y Ff(\024)g Fg(0)p Fh(:)p Fg(2)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(01)311 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(002)254 1423 y Fh(A)286 1401 y Fe(\(6\))286 1436 y(7)p Fb(;)p Fe(7)377 1423 y Fg(=)12 b(2)443 1409 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 1423 y Ff(\002)588 1406 y Fe(30030)p 588 1413 92 2 v 588 1438 a(65536)694 1423 y Ff(\002)743 1381 y Fd(p)p 772 1381 55 2 v 772 1406 a Fe(462)p 743 1413 85 2 v 767 1438 a(10)932 1423 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(8909)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(4583)g Ff(\002)f Fg(2)p Fh(:)p Fg(150)57 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(878)254 1491 y Fh(A)286 1470 y Fe(\(6\))286 1505 y(7)p Fb(;)p Fe(8)377 1491 y Fg(=)437 1475 y Fe(4290)p 428 1482 92 2 v 428 1507 a(16384)534 1491 y Ff(\002)583 1450 y Fd(p)p 613 1450 55 2 v 25 x Fe(770)p 583 1482 85 2 v 607 1507 a(10)932 1491 y Ff(\024)g Fg(0)p Fh(:)p Fg(2619)10 b Ff(\002)g Fg(2)p Fh(:)p Fg(775)226 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(727)254 1560 y Fh(A)286 1539 y Fe(\(6\))286 1574 y(7)p Fb(;)p Fe(9)377 1560 y Fg(=)12 b(2)443 1546 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 1560 y Ff(\002)588 1544 y Fe(128700)p 588 1551 110 2 v 588 1576 a(262144)713 1560 y Ff(\002)761 1519 y Fd(p)p 791 1519 92 2 v 25 x Fe(13090)p 761 1551 122 2 v 794 1576 a(100)932 1560 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(7938)10 b Ff(\002)g Fg(0)p Fh(:)p Fg(4910)g Ff(\002)f Fg(1)p Fh(:)p Fg(1442)36 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(446)254 1629 y Fh(A)286 1608 y Fe(\(6\))286 1642 y(7)p Fb(;)p Fe(10)377 1629 y Fg(=)12 b(2)443 1615 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 1629 y Ff(\002)588 1613 y Fe(65000)p 588 1620 92 2 v 588 1645 a(65536)694 1629 y Ff(\002)743 1588 y Fd(p)p 772 1588 V 772 1613 a Fe(49742)p 743 1620 122 2 v 767 1645 a(1000)932 1629 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(891)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(992)h Ff(\002)g Fg(0)p Fh(:)p Fg(224)99 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(198)254 1698 y Fh(A)286 1676 y Fe(\(6\))286 1711 y(7)p Fb(;)p Fe(11)377 1698 y Fg(=)428 1681 y Fe(100100)p 428 1688 110 2 v 428 1713 a(524288)553 1698 y Ff(\002)601 1656 y Fd(p)p 631 1656 55 2 v 25 x Fe(969)p 601 1688 85 2 v 616 1713 a(100)932 1698 y Ff(\024)g Fg(0)p Fh(:)p Fg(191)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(312)248 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(060)254 1766 y Fh(A)286 1745 y Fe(\(6\))286 1780 y(7)p Fb(;)p Fe(12)377 1766 y Fg(=)12 b(2)443 1752 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)539 1766 y Ff(\002)597 1750 y Fe(21000)p 588 1757 110 2 v 588 1782 a(131072)713 1766 y Ff(\002)761 1725 y Fd(p)p 791 1725 74 2 v 25 x Fe(7429)p 761 1757 103 2 v 776 1782 a(1000)932 1766 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(794)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(1603)h Ff(\002)g Fg(0)p Fh(:)p Fg(0862)57 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(011)254 1835 y Fh(A)286 1814 y Fe(\(6\))286 1849 y(7)p Fb(;)p Fe(13)377 1835 y Fg(=)12 b(2)443 1821 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)539 1835 y Ff(\002)597 1819 y Fe(500000)p 588 1826 128 2 v 588 1851 a(2097152)731 1835 y Ff(\002)779 1794 y Fd(p)p 809 1794 110 2 v 25 x Fe(193154)p 779 1826 140 2 v 794 1851 a(100000)932 1835 y Ff(\024)26 b Fg(0)p Fh(:)p Fg(90)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(239)h Ff(\002)f Fg(0)p Fh(:)p Fg(0044)100 b Ff(\024)26 b Fg(0)p Fh(:)p Fg(001)323 1938 y Fc(T)l(able)32 b(2.)k Fg(Bounds)12 b(on)i Fh(A)789 1917 y Fe(\()p Fb(m)p Fe(\))789 1953 y Fb(k)q(;n)867 1938 y Fg(for)f(0)f Ff(\024)g Fh(m)g Ff(\024)g Fg(6)i(and)g Fh(m)e Ff(\024)f Fh(k)i Ff(\024)f Fg(7)g Ff(\024)g Fh(n)f Ff(\024)h Fg(13.)0 2132 y Fv(The)j(equalities)g(in)f(this)h(table)g(are)g(obtained)f(b) o(y)h(simplifying)e(\(in)i(an)g(ob)o(vious)e(w)o(a)o(y\))i(the)h(expression)0 2202 y(\(2.18\))j(for)f(eac)o(h)g(of)g(the)h(co)q(e\016cien)o(ts)f Fq(A)775 2176 y Fu(\()p Fl(m)p Fu(\))775 2217 y Fl(k)q(;n)844 2202 y Fv(.)28 b(Then,)18 b(in)g(order)f(to)i(v)o(erify)f(e.g.)27 b(that)19 b Fq(A)1676 2176 y Fu(\(5\))1676 2215 y(7)p Fl(;)p Fu(11)1767 2202 y Fs(\024)e Fv(0)p Fq(:)p Fv(015,)0 2269 y(it)24 b(su\016ces)e(to)i(c)o(hec)o(k)f(that)h(0)p Fq(:)p Fv(27)15 b Fs(\002)h Fv(131072)25 b Fs(\025)g Fv(35000)e(and)g(\(1000)16 b Fs(\002)f Fv(0)p Fq(:)p Fv(06\))1504 2251 y Fu(2)1552 2269 y Fs(\025)25 b Fv(3553,)g(and)e(that)0 2329 y(0)p Fq(:)p Fv(9)13 b Fs(\002)h Fv(0)p Fq(:)p Fv(27)f Fs(\002)h Fv(0)p Fq(:)p Fv(06)20 b Fs(\024)g Fv(0)p Fq(:)p Fv(015.)34 b(Here,)21 b(the)g(b)q(ound)f(2)1024 2311 y Fp(\000)p Fu(1)p Fl(=)p Fu(6)1138 2329 y Fs(\024)g Fv(0)p Fq(:)p Fv(9)h(follo)o(ws)e(from)h(inequalit)o(y)g(\(I2\))h(in)0 2388 y(Section)16 b(3.)100 2448 y(By)j(using)e(the)i(inequalities)e(from)h (the)g(last)h(t)o(w)o(o)f(tables,)g(w)o(e)g(get)h(immediately)f(the)g(follo)o (wing)0 2515 y(b)q(ounds)12 b(on)i(the)g(sums)e Fq(S)473 2522 y Fl(m;n)555 2515 y Fv(,)i(de\014ned)f(in)g(\(3.10\).)21 b(Note)15 b(that)f(the)g(co)q(e\016cien)o(ts)f Fq(A)1561 2489 y Fu(\()p Fl(m)p Fu(\))1561 2530 y Fl(k)q(;n)1644 2515 y Fv(are)g(symmetric)0 2575 y(under)h(p)q(erm)o(utations)f(of)i(the)g(indices)f Fq(k)r(;)8 b(m;)g(n)p Fv(,)15 b(and)f(that)i(man)o(y)e(of)h(these)g(co)q(e\016cien)o(ts) f(are)h(zero;)g(see)0 2634 y(Lemma)h(2.5.)950 2790 y(22)p eop %%Page: 23 24 bop 264 103 a Fh(S)290 108 y Fb(m;n)360 103 y Fg(=)11 b Fh(A)436 82 y Fe(\()p Fb(m)p Fe(\))436 117 y(7)p Fb(;n)500 103 y Fg(+)g Fh(\032A)598 82 y Fe(\()p Fb(m)p Fe(\))598 117 y(8)p Fb(;n)672 103 y Fg(+)p Fh(\032)727 89 y Fe(2)748 103 y Fh(A)780 82 y Fe(\()p Fb(m)p Fe(\))780 117 y(9)p Fb(;n)844 103 y Fg(+)p Fh(\032)899 89 y Fe(3)919 103 y Fh(A)951 82 y Fe(\()p Fb(m)p Fe(\))951 117 y(10)p Fb(;n)1016 103 y Fg(+)p Fh(\032)1071 89 y Fe(4)1091 103 y Fh(A)1123 82 y Fe(\()p Fb(m)p Fe(\))1123 117 y(11)p Fb(;n)1187 103 y Fg(+)p Fh(\032)1242 89 y Fe(5)1263 103 y Fh(A)1295 82 y Fe(\()p Fb(m)p Fe(\))1295 117 y(12)p Fb(;n)1359 103 y Fg(+)s Fh(\032)1417 89 y Fe(6)1437 103 y Fh(A)1469 82 y Fe(\()p Fb(m)p Fe(\))1469 117 y(13)p Fb(;n)264 175 y Fh(S)290 181 y Fe(0)p Fb(;)p Fe(7)359 175 y Ff(\024)h Fg(0)p Fh(:)p Fg(001)1045 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(001)264 225 y Fh(S)290 231 y Fe(1)p Fb(;)p Fe(6)359 225 y Ff(\024)h Fg(0)p Fh(:)p Fg(003)1045 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(003)264 274 y Fh(S)290 280 y Fe(1)p Fb(;)p Fe(7)359 274 y Ff(\024)h Fg(0)p Fh(:)p Fg(004+0)p Fh(:)p Fg(00304)874 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(008)264 324 y Fh(S)290 330 y Fe(2)p Fb(;)p Fe(5)359 324 y Ff(\024)h Fg(0)p Fh(:)p Fg(010)1045 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(010)264 374 y Fh(S)290 380 y Fe(2)p Fb(;)p Fe(6)359 374 y Ff(\024)h Fg(0)p Fh(:)p Fg(023+0)p Fh(:)p Fg(00759)874 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(031)264 424 y Fh(S)290 430 y Fe(2)p Fb(;)p Fe(7)359 424 y Ff(\024)h Fg(0)p Fh(:)p Fg(022+0)p Fh(:)p Fg(01517)q(+0)p Fh(:)p Fg(00461)702 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(042)264 474 y Fh(S)290 480 y Fe(3)p Fb(;)p Fe(4)359 474 y Ff(\024)h Fg(0)p Fh(:)p Fg(017)1045 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(017)264 524 y Fh(S)290 530 y Fe(3)p Fb(;)p Fe(5)359 524 y Ff(\024)h Fg(0)p Fh(:)p Fg(060+0)p Fh(:)p Fg(01366)874 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(074)264 573 y Fh(S)290 579 y Fe(3)p Fb(;)p Fe(6)359 573 y Ff(\024)h Fg(0)p Fh(:)p Fg(097+0)p Fh(:)p Fg(04703)q(+0)p Fh(:)p Fg(01151)702 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(156)264 623 y Fh(S)290 629 y Fe(3)p Fb(;)p Fe(7)359 623 y Ff(\024)h Fg(0)p Fh(:)p Fg(089+0)p Fh(:)p Fg(07434)q(+0)p Fh(:)p Fg(03452)q(+0)p Fh(:)p Fg(00698)530 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(205)264 673 y Fh(S)290 679 y Fe(4)p Fb(;)p Fe(4)359 673 y Ff(\024)h Fg(0)p Fh(:)p Fg(081+0)p Fh(:)p Fg(01669)874 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(098)264 723 y Fh(S)290 729 y Fe(4)p Fb(;)p Fe(5)359 723 y Ff(\024)h Fg(0)p Fh(:)p Fg(189+0)p Fh(:)p Fg(08041)q(+0)p Fh(:)p Fg(01611)702 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(286)264 773 y Fh(S)290 779 y Fe(4)p Fb(;)p Fe(6)359 773 y Ff(\024)h Fg(0)p Fh(:)p Fg(271+0)p Fh(:)p Fg(17901)q(+0)p Fh(:)p Fg(06903)q(+0)p Fh(:)p Fg(01396)530 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(533)264 822 y Fh(S)290 828 y Fe(4)p Fb(;)p Fe(7)359 822 y Ff(\024)h Fg(0)p Fh(:)p Fg(256+0)p Fh(:)p Fg(24576)q(+0)p Fh(:)p Fg(15417)q(+0)p Fh(:)p Fg(05583)q(+0)p Fh(:)p Fg(01059)358 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(723)264 872 y Fh(S)290 878 y Fe(5)p Fb(;)p Fe(5)359 872 y Ff(\024)h Fg(0)p Fh(:)p Fg(383+0)p Fh(:)p Fg(23666)q(+0)p Fh(:)p Fg(08744)q(+0)p Fh(:)p Fg(01745)530 b Ff(\024)11 b Fg(0)p Fh(:)p Fg(725)264 922 y Fh(S)290 928 y Fe(5)p Fb(;)p Fe(6)359 922 y Ff(\024)h Fg(0)p Fh(:)p Fg(540+0)p Fh(:)p Fg(45207)q(+0)p Fh(:)p Fg(25311)q(+0)p Fh(:)p Fg(08723)q(+0)p Fh(:)p Fg(01588)358 b Ff(\024)11 b Fg(1)p Fh(:)p Fg(349)264 972 y Fh(S)290 978 y Fe(5)p Fb(;)p Fe(7)359 972 y Ff(\024)h Fg(0)p Fh(:)p Fg(545+0)p Fh(:)p Fg(59922)q(+0)p Fh(:)p Fg(46481)q(+0)p Fh(:)p Fg(24075)q(+0)p Fh(:)p Fg(07938+0)p Fh(:)p Fg(01606)187 b Ff(\024)11 b Fg(1)p Fh(:)p Fg(946)264 1022 y Fh(S)290 1028 y Fe(6)p Fb(;)p Fe(6)359 1022 y Ff(\024)h Fg(0)p Fh(:)p Fg(795+0)p Fh(:)p Fg(81918)q(+0)p Fh(:)p Fg(60287)q(+0)p Fh(:)p Fg(30006)q(+0)p Fh(:)p Fg(09526+0)p Fh(:)p Fg(01606)187 b Ff(\024)11 b Fg(2)p Fh(:)p Fg(629)264 1071 y Fh(S)290 1077 y Fe(6)p Fb(;)p Fe(7)359 1071 y Ff(\024)h Fg(0)p Fh(:)p Fg(878+1)p Fh(:)p Fg(10286)q(+1)p Fh(:)p Fg(02625)q(+0)p Fh(:)p Fg(69083)q(+0)p Fh(:)p Fg(31752+0)p Fh(:)p Fg(08829)q(+0)p Fh(:)p Fg(01218)j Ff(\024)c Fg(4)p Fh(:)p Fg(116)235 1172 y Fc(T)l(able)32 b(3.)k Fg(Bounds)12 b(on)i(the)g(sums)e Fh(S)874 1177 y Fb(m;n)950 1172 y Fg(,)i(for)g(0)e Ff(\024)g Fh(m)g Ff(\024)g Fh(n)f Ff(\024)h Fg(7,)i(whic)o(h)e(are)i(not)g(zero.)0 1307 y Fv(In)k(order)g(to)h(b)q(ound)e Fq(\032)441 1289 y Fp(\000)p Fu(1)495 1307 y Fq(S)526 1314 y Fl(m;)p Fu(8)603 1307 y Fv(,)i(and)f(then)h Fq(\033)879 1314 y Fl(m)934 1307 y Fv(=)e(max)p Fs(f)p Fq(S)1139 1314 y Fl(m;)p Fu(7)1216 1307 y Fq(;)g(\032)1273 1289 y Fp(\000)p Fu(1)1326 1307 y Fq(S)1357 1314 y Fl(m;)p Fu(8)1426 1307 y Fs(g)p Fv(,)i(w)o(e)f(use)h(the)g(inequalit)o(y)0 1367 y(\(3.15\))e(with)f Fq(n)e Fv(=)g(7,)i(together)h(with)f(the)h(b)q(ounds)e(from)h(T)l(able)g(2)g (and)g(T)l(able)g(3.)367 1540 y Fh(\032)389 1526 y Fd(\000)p Fe(1)438 1540 y Fh(S)464 1546 y Fb(m;)p Fe(8)539 1540 y Ff(\024)c Fh(\032)606 1526 y Fd(\000)p Fe(1)679 1540 y Ff(\002)d Fh(A)753 1519 y Fe(\()p Fb(m)p Fe(\))753 1553 y(8)p Fb(;)p Fe(7)807 1540 y Fg(+)856 1524 y Fe(8)p 856 1531 19 2 v 856 1556 a(3)880 1540 y Fh(\014)906 1526 y Fe(2)936 1540 y Ff(\002)g Fh(S)1004 1546 y Fb(m;)p Fe(7)367 1617 y Fh(\032)389 1603 y Fd(\000)p Fe(1)438 1617 y Fh(S)464 1623 y Fe(0)p Fb(;)p Fe(8)539 1617 y Ff(\024)j Fg(0)p Fh(:)p Fg(66)e Ff(\002)f Fg(0)p Fh(:)p Fg(000+0)p Fh(:)p Fg(84)h Ff(\002)f Fg(0)p Fh(:)p Fg(001)15 b Ff(\024)d Fg(0)p Fh(:)p Fg(001)i(,)138 b Fh(\033)1407 1623 y Fe(0)1442 1617 y Ff(\024)12 b Fg(0)p Fh(:)p Fg(001)367 1667 y Fh(\032)389 1653 y Fd(\000)p Fe(1)438 1667 y Fh(S)464 1673 y Fe(1)p Fb(;)p Fe(8)539 1667 y Ff(\024)g Fg(0)p Fh(:)p Fg(66)e Ff(\002)f Fg(0)p Fh(:)p Fg(002+0)p Fh(:)p Fg(84)h Ff(\002)f Fg(0)p Fh(:)p Fg(008)15 b Ff(\024)d Fg(0)p Fh(:)p Fg(009)i(,)138 b Fh(\033)1407 1673 y Fe(1)1442 1667 y Ff(\024)12 b Fg(0)p Fh(:)p Fg(009)367 1717 y Fh(\032)389 1703 y Fd(\000)p Fe(1)438 1717 y Fh(S)464 1723 y Fe(2)p Fb(;)p Fe(8)539 1717 y Ff(\024)g Fg(0)p Fh(:)p Fg(66)e Ff(\002)f Fg(0)p Fh(:)p Fg(010+0)p Fh(:)p Fg(84)h Ff(\002)f Fg(0)p Fh(:)p Fg(042)15 b Ff(\024)d Fg(0)p Fh(:)p Fg(042)i(,)138 b Fh(\033)1407 1723 y Fe(2)1442 1717 y Ff(\024)12 b Fg(0)p Fh(:)p Fg(042)367 1767 y Fh(\032)389 1753 y Fd(\000)p Fe(1)438 1767 y Fh(S)464 1773 y Fe(3)p Fb(;)p Fe(8)539 1767 y Ff(\024)g Fg(0)p Fh(:)p Fg(66)e Ff(\002)f Fg(0)p Fh(:)p Fg(049+0)p Fh(:)p Fg(84)h Ff(\002)f Fg(0)p Fh(:)p Fg(205)15 b Ff(\024)d Fg(0)p Fh(:)p Fg(205)i(,)138 b Fh(\033)1407 1773 y Fe(3)1442 1767 y Ff(\024)12 b Fg(0)p Fh(:)p Fg(205)367 1817 y Fh(\032)389 1803 y Fd(\000)p Fe(1)438 1817 y Fh(S)464 1823 y Fe(4)p Fb(;)p Fe(8)539 1817 y Ff(\024)g Fg(0)p Fh(:)p Fg(66)e Ff(\002)f Fg(0)p Fh(:)p Fg(162+0)p Fh(:)p Fg(84)h Ff(\002)f Fg(0)p Fh(:)p Fg(723)15 b Ff(\024)d Fg(0)p Fh(:)p Fg(715)i(,)138 b Fh(\033)1407 1823 y Fe(4)1442 1817 y Ff(\024)12 b Fg(0)p Fh(:)p Fg(723)367 1867 y Fh(\032)389 1852 y Fd(\000)p Fe(1)438 1867 y Fh(S)464 1873 y Fe(5)p Fb(;)p Fe(8)539 1867 y Ff(\024)g Fg(0)p Fh(:)p Fg(66)e Ff(\002)f Fg(0)p Fh(:)p Fg(395+0)p Fh(:)p Fg(84)h Ff(\002)f Fg(1)p Fh(:)p Fg(946)15 b Ff(\024)d Fg(1)p Fh(:)p Fg(896)i(,)138 b Fh(\033)1407 1873 y Fe(5)1442 1867 y Ff(\024)12 b Fg(1)p Fh(:)p Fg(946)367 1916 y Fh(\032)389 1902 y Fd(\000)p Fe(1)438 1916 y Fh(S)464 1922 y Fe(6)p Fb(;)p Fe(8)539 1916 y Ff(\024)g Fg(0)p Fh(:)p Fg(66)e Ff(\002)f Fg(0)p Fh(:)p Fg(727+0)p Fh(:)p Fg(84)h Ff(\002)f Fg(4)p Fh(:)p Fg(116)15 b Ff(\024)d Fg(3)p Fh(:)p Fg(938)i(,)138 b Fh(\033)1407 1922 y Fe(6)1442 1916 y Ff(\024)12 b Fg(4)p Fh(:)p Fg(116)647 2012 y Fc(T)l(able)33 b(4.)i Fg(Bounds)12 b(on)i Fh(\033)1105 2018 y Fe(0)1126 2012 y Fh(;)7 b(:)g(:)f(:)h(;)g(\033)1244 2018 y Fe(6)1272 2012 y Fg(.)0 2147 y Fv(F)l(rom)15 b(these)h(b)q(ounds,)g(using)f(the)i(v)m(alues) f(of)h Fq(g)891 2154 y Fu(0)913 2147 y Fq(;)8 b(:)g(:)g(:)g(;)g(g)1047 2154 y Fu(6)1086 2147 y Fv(giv)o(en)16 b(in)g(T)l(able)g(1,)h(it)g(follo)o (ws)e(that)128 2261 y Fq(\033)47 b Fv(=)67 b Fq(\033)337 2268 y Fu(0)360 2261 y Fq(g)384 2268 y Fu(0)460 2261 y Fv(+)46 b Fq(\033)573 2268 y Fu(1)595 2261 y Fq(g)619 2268 y Fu(1)687 2261 y Fv(+)g Fq(\033)800 2268 y Fu(2)822 2261 y Fq(g)846 2268 y Fu(2)914 2261 y Fv(+)f Fq(\033)1026 2268 y Fu(3)1049 2261 y Fq(g)1073 2268 y Fu(3)1141 2261 y Fv(+)g Fq(\033)1253 2268 y Fu(4)1276 2261 y Fq(g)1300 2268 y Fu(4)1368 2261 y Fv(+)g Fq(\033)1480 2268 y Fu(5)1503 2261 y Fq(g)1527 2268 y Fu(5)1595 2261 y Fv(+)g Fq(\033)1707 2268 y Fu(6)1730 2261 y Fq(g)1754 2268 y Fu(6)203 2325 y Fs(\024)30 b Fv(0)p Fq(:)p Fv(000914)o(+0)p Fq(:)p Fv(004518)o(+0)p Fq(:)p Fv(010710)o(+0)p Fq(:)p Fv(024600)o(+0)p Fq(:)p Fv(037596)o(+0)p Fq(:)p Fv(040866)o(+0)p Fq(:)p Fv(032928)203 2390 y Fs(\024)g Fv(0)p Fq(:)p Fv(153)8 b Fq(;)0 2524 y Fv(whic)o(h)16 b(completes)f(the)i(pro)q(of)f(of)h(\(N1\).)950 2790 y(23)p eop %%Page: 24 25 bop 0 50 a Ft(5.2.)54 b(Pro)r(of)24 b(of)g(\(N2\))0 125 y Fv(By)17 b(de\014nition,)e Fq(\034)22 b Fv(is)16 b(obtained)g(b)o(y)g(adding)f(up)h (all)g(the)h(pro)q(ducts)e Fq(\032)1303 106 y Fu(7)1326 125 y Fq(g)1350 132 y Fl(m)1387 125 y Fq(S)1418 132 y Fl(m;n)1492 125 y Fq(g)1516 132 y Fl(n)1551 125 y Fv(,)i(for)f(0)d Fs(\024)h Fq(m;)8 b(n)14 b Fs(\024)f Fv(6.)0 184 y(By)h(using)f(the)h(v)m(alues)g(from) f(T)l(able)g(1)h(and)f(T)l(able)g(3,)h(the)g(sum)f(o)o(v)o(er)g(all)g Fq(n)h Fs(\024)g Fv(6)f(of)h Fq(S)1577 191 y Fl(m;n)1651 184 y Fq(g)1675 191 y Fl(n)1716 184 y Fv(is)f(b)q(ounded)0 244 y(as)j(follo)o(ws.)221 349 y Fg(\()p Fh(S)r(g)q Fg(\))302 354 y Fb(m)348 349 y Ff(\024)26 b Fh(S)433 355 y Fb(m;)p Fe(1)498 349 y Fh(g)518 355 y Fe(1)567 349 y Fg(+)7 b Fh(S)633 355 y Fb(m;)p Fe(2)697 349 y Fh(g)717 355 y Fe(2)760 349 y Fg(+)g Fh(S)826 355 y Fb(m;)p Fe(3)890 349 y Fh(g)910 355 y Fe(3)953 349 y Fg(+)g Fh(S)1019 355 y Fb(m;)p Fe(4)1083 349 y Fh(g)1103 355 y Fe(4)1145 349 y Fg(+)h Fh(S)1212 355 y Fb(m;)p Fe(5)1276 349 y Fh(g)1296 355 y Fe(5)1338 349 y Fg(+)15 b Fh(S)1412 355 y Fb(m;)p Fe(6)1476 349 y Fh(g)1496 355 y Fe(6)221 408 y Fg(\()p Fh(S)r(g)q Fg(\))302 414 y Fe(1)348 408 y Ff(\024)990 b Fg(0)p Fh(:)p Fg(000024)15 b Ff(\024)d Fg(0)p Fh(:)p Fg(00003)221 453 y(\()p Fh(S)r(g)q Fg(\))302 459 y Fe(2)348 453 y Ff(\024)797 b Fg(0)p Fh(:)p Fg(000210)q(+0)p Fh(:)p Fg(000248)15 b Ff(\024)d Fg(0)p Fh(:)p Fg(00046)221 498 y(\()p Fh(S)r(g)q Fg(\))302 504 y Fe(3)348 498 y Ff(\024)605 b Fg(0)p Fh(:)p Fg(000884+0)p Fh(:)p Fg(001554)q(+0)p Fh(:)p Fg(001248)15 b Ff(\024)d Fg(0)p Fh(:)p Fg(00369)221 543 y(\()p Fh(S)r(g)q Fg(\))302 549 y Fe(4)348 543 y Ff(\024)412 b Fg(0)p Fh(:)p Fg(002040)q(+0)p Fh(:)p Fg(005096+0)p Fh(:)p Fg(006006)q(+0)p Fh(:)p Fg(004264)15 b Ff(\024)d Fg(0)p Fh(:)p Fg(01741)221 588 y(\()p Fh(S)r(g)q Fg(\))302 594 y Fe(5)348 588 y Ff(\024)219 b Fg(0)p Fh(:)p Fg(002550)q(+0)p Fh(:)p Fg(008880)q(+0)p Fh(:)p Fg(014872+0)p Fh(:)p Fg(015225)q(+0)p Fh(:)p Fg(010792)15 b Ff(\024)d Fg(0)p Fh(:)p Fg(05232)221 633 y(\()p Fh(S)r(g)q Fg(\))302 639 y Fe(6)348 633 y Ff(\024)i Fg(0)p Fh(:)p Fg(001506)f(+0)p Fh(:)p Fg(007905)q(+0)p Fh(:)p Fg(018720)q(+0)p Fh(:)p Fg(027716+0)p Fh(:)p Fg(028329)q(+0)p Fh(:)p Fg(021032)i Ff(\024)d Fg(0)p Fh(:)p Fg(10521)425 702 y Fc(T)l(able)32 b(5.)j Fg(Bounds)13 b(on)h(\()p Fh(S)r(g)q Fg(\))940 707 y Fb(m)983 702 y Fg(,)g(for)g(1)e Ff(\024)f Fh(m)i Ff(\024)e Fg(6)d(;)28 b(\()p Fh(S)r(g)q Fg(\))1395 708 y Fe(0)1429 702 y Fg(=)11 b(0.)0 837 y Fv(Giv)o(en)16 b(these)h (inequalities,)e(and)h(from)g(T)l(able)g(1)h(the)f(v)m(alues)h(of)f Fq(g)1255 844 y Fl(n)1299 837 y Fv(and)g(the)h(b)q(ound)e(on)h Fq(\032)1730 819 y Fu(7)1752 837 y Fv(,)h(w)o(e)f(\014nd)159 944 y Fq(\034)53 b Fv(=)63 b Fq(\032)362 926 y Fu(7)408 904 y Fr(\002)450 944 y Fq(g)474 951 y Fu(1)496 944 y Fv(\()p Fq(S)s(g)r Fv(\))594 951 y Fu(1)627 944 y Fv(+)11 b Fq(g)701 951 y Fu(2)723 944 y Fv(\()p Fq(S)s(g)r Fv(\))821 951 y Fu(2)854 944 y Fv(+)g Fq(g)928 951 y Fu(3)950 944 y Fv(\()p Fq(S)s(g)r Fv(\))1048 951 y Fu(3)1081 944 y Fv(+)g Fq(g)1155 951 y Fu(4)1177 944 y Fv(\()p Fq(S)s(g)r Fv(\))1275 951 y Fu(4)1308 944 y Fv(+)g Fq(g)1382 951 y Fu(5)1404 944 y Fv(\()p Fq(S)s(g)r Fv(\))1502 951 y Fu(5)1535 944 y Fv(+)f Fq(g)1608 951 y Fu(6)1631 944 y Fv(\()p Fq(S)s(g)r Fv(\))1729 951 y Fu(6)1770 904 y Fr(\003)234 1009 y Fs(\024)k Fv(18)p Fq(:)p Fv(47)408 968 y Fr(\002)439 1009 y Fv(0)p Fq(:)p Fv(000016)o(+0)p Fq(:)p Fv(000118)o(+0)p Fq(:)p Fv(000443)o(+0)p Fq(:)p Fv(000906)o(+0)p Fq(:)p Fv(001099)o(+0)p Fq(:)p Fv(000842)1770 968 y Fr(\003)234 1068 y Fv(=)g(18)p Fq(:)p Fv(47)o Fs(\002)p Fv(0)p Fq(:)p Fv(003424)234 1128 y Fs(\024)g Fv(0)p Fq(:)p Fv(064)8 b Fq(;)0 1218 y Fv(whic)o(h)16 b(completes)f(the)i(pro)q(of)f(of)h(\(N2\).)0 1397 y Ft(5.3.)54 b(Pro)r(of)24 b(of)g(\(N3\))100 1484 y Fv(In)18 b(the)h(next)g(table)f(w)o(e) h(de\014ne)f(the)h(n)o(um)o(b)q(ers)d Fq(Q)1044 1458 y Fu(\()p Fl(m)p Fu(\))1044 1499 y Fl(k)q(;n)1140 1484 y Fv(and)i(v)o(erify)g(\(N3\),)i (b)o(y)f(giving)f(b)q(ounds)f(of)0 1544 y(the)g(form)e Fq(A)240 1518 y Fu(\()p Fl(m)p Fu(\))240 1559 y Fl(k)q(;n)323 1544 y Fs(2)f Fq(Q)409 1518 y Fu(\()p Fl(m)p Fu(\))409 1559 y Fl(k)q(;n)489 1544 y Fs(\006)d Fq(\017)p Fv(,)16 b(where)g Fq(\017)h Fv(denotes)e(the)i(in) o(terv)m(al)f([0)p Fq(;)8 b Fv(1)p Fq(=)p Fv(2000].)21 b(Here,)16 b Fq(a)e Fs(2)h Fq(q)d Fv(+)f Fq(\017)16 b Fv(means)0 1604 y(that)i Fq(a)12 b Fs(\000)f Fq(q)17 b Fs(2)e Fq(\017)p Fv(,)j(and)e Fq(a)g Fs(2)f Fq(q)f Fs(\000)d Fq(\017)17 b Fv(means)g(that)g Fq(q)d Fs(\000)d Fq(a)16 b Fs(2)f Fq(\017)p Fv(.)24 b(The)17 b(v)m(alues)g Fq(Q)1433 1578 y Fu(\()p Fl(m)p Fu(\))1433 1619 y Fl(k)q(;n)1520 1604 y Fv(whic)o(h)f(are)h(not)h(giv)o(en)0 1663 y(b)q(elo)o(w)e(are)g(either)h(zero)f(\(if)h Fq(m)11 b Fv(+)g Fq(k)k(<)e(n)p Fv(\),)k(or)f(determined)f(b)o(y)i(symmetry)l(.)127 1754 y Fh(A)159 1733 y Fe(\()p Fb(m)p Fe(\))159 1769 y Fb(k)q(;n)1598 1754 y Fh(Q)1631 1733 y Fe(\()p Fb(m)p Fe(\))1631 1769 y Fb(k)q(;n)127 1834 y Fh(A)159 1813 y Fe(\(0\))159 1847 y(0)p Fb(;)p Fe(0)250 1834 y Fg(=)12 b(1)e Ff(\002)368 1801 y(p)p 403 1801 22 2 v 33 x Fg(1)292 b Ff(2)26 b Fg(1)9 b Ff(\002)h Fg(1)677 b Ff(2)26 b Fg(1)p Fh(:)p Fg(000)10 b Ff(\000)g Fh(\017)127 1897 y(A)159 1876 y Fe(\(0\))159 1910 y(1)p Fb(;)p Fe(1)250 1897 y Fg(=)i(2)316 1883 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 1897 y Ff(\002)461 1880 y Fe(1)p 461 1887 19 2 v 461 1912 a(4)494 1897 y Ff(\002)537 1864 y(p)p 572 1864 22 2 v 33 x Fg(2)123 b Ff(2)26 b Fg(0)p Fh(:)p Fg(890[89)p Fh(;)7 b Fg(90])j Ff(\002)f Fg(0)p Fh(:)p Fg(25)h Ff(\002)f Fg(1)p Fh(:)p Fg(414[2)p Fh(;)e Fg(3])188 b Ff(2)26 b Fg(0)p Fh(:)p Fg(315)10 b Ff(\000)g Fh(\017)127 1960 y(A)159 1938 y Fe(\(0\))159 1973 y(2)p Fb(;)p Fe(2)250 1960 y Fg(=)i(2)316 1946 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 1960 y Ff(\002)461 1943 y Fe(1)p 461 1950 19 2 v 461 1975 a(8)494 1960 y Ff(\002)537 1927 y(p)p 572 1927 22 2 v 33 x Fg(1)123 b Ff(2)26 b Fg(0)p Fh(:)p Fg(79[3)p Fh(;)7 b Fg(4])i Ff(\002)g Fg(0)p Fh(:)p Fg(125)h Ff(\002)g Fg(1)390 b Ff(2)26 b Fg(0)p Fh(:)p Fg(099)10 b(+)g Fh(\017)127 2023 y(A)159 2002 y Fe(\(0\))159 2037 y(3)p Fb(;)p Fe(3)250 2023 y Fg(=)301 2007 y Fe(10)p 301 2014 37 2 v 301 2039 a(32)353 2023 y Ff(\002)401 1982 y Fd(p)p 431 1982 19 2 v 25 x Fe(1)p 401 2014 49 2 v 407 2039 a(10)716 2023 y Ff(2)26 b Fg(0)p Fh(:)p Fg(31[2)p Fh(;)7 b Fg(3])i Ff(\002)g Fg(0)p Fh(:)p Fg(1)506 b Ff(2)26 b Fg(0)p Fh(:)p Fg(031)10 b(+)g Fh(\017)127 2087 y(A)159 2066 y Fe(\(0\))159 2101 y(4)p Fb(;)p Fe(4)250 2087 y Fg(=)i(2)316 2073 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 2087 y Ff(\002)461 2071 y Fe(100)p 461 2078 55 2 v 461 2103 a(128)531 2087 y Ff(\002)583 2046 y Fd(p)p 612 2046 19 2 v 612 2071 a Fe(2)p 579 2078 55 2 v 579 2103 a(100)716 2087 y Ff(2)26 b Fg(0)p Fh(:)p Fg(89[0)p Fh(;)7 b Fg(1])i Ff(\002)g Fg(0)p Fh(:)p Fg(7[8)p Fh(;)e Fg(9])j Ff(\002)f Fg(0)p Fh(:)p Fg(014[1)p Fh(;)e Fg(2])188 b Ff(2)26 b Fg(0)p Fh(:)p Fg(010)10 b Ff(\000)g Fh(\017)127 2151 y(A)159 2130 y Fe(\(0\))159 2164 y(5)p Fb(;)p Fe(5)250 2151 y Fg(=)i(2)316 2137 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 2151 y Ff(\002)461 2135 y Fe(100)p 461 2142 V 461 2167 a(256)531 2151 y Ff(\002)583 2110 y Fd(p)p 612 2110 19 2 v 612 2135 a Fe(1)p 579 2142 55 2 v 579 2167 a(100)716 2151 y Ff(2)26 b Fg(0)p Fh(:)p Fg([79)p Fh(;)7 b Fg(80])i Ff(\002)g Fg(0)p Fh(:)p Fg([39)p Fh(;)e Fg(40])j Ff(\002)f Fg(0)p Fh(:)p Fg(01)273 b Ff(2)26 b Fg(0)p Fh(:)p Fg(003)10 b(+)g Fh(\017)127 2215 y(A)159 2193 y Fe(\(0\))159 2228 y(6)p Fb(;)p Fe(6)250 2215 y Fg(=)301 2198 y Fe(1000)p 301 2205 74 2 v 301 2230 a(1024)389 2215 y Ff(\002)450 2173 y Fd(p)p 480 2173 19 2 v 25 x Fe(1)p 438 2205 74 2 v 438 2230 a(1000)716 2215 y Ff(2)26 b Fg([0)p Fh(:)p Fg(9)p Fh(;)6 b Fg(1)p Fh(:)p Fg(0])k Ff(\002)f Fg(0)p Fh(:)p Fg(001)473 b Ff(2)26 b Fg(0)p Fh(:)p Fg(001)10 b Ff(\000)g Fh(\017)127 2278 y(A)159 2256 y Fe(\(1\))159 2291 y(1)p Fb(;)p Fe(1)250 2278 y Fg(=)301 2261 y Fe(1)p 301 2268 19 2 v 301 2293 a(2)335 2278 y Ff(\002)377 2244 y(p)p 412 2244 22 2 v 34 x Fg(1)283 b Ff(2)26 b Fg(0)p Fh(:)p Fg(5)9 b Ff(\002)h Fg(1)644 b Ff(2)26 b Fg(0)p Fh(:)p Fg(500)10 b Ff(\000)g Fh(\017)127 2340 y(A)159 2319 y Fe(\(1\))159 2354 y(1)p Fb(;)p Fe(2)250 2340 y Fg(=)i(2)316 2326 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 2340 y Ff(\002)461 2324 y Fe(1)p 461 2331 19 2 v 461 2356 a(8)494 2340 y Ff(\002)537 2307 y(p)p 572 2307 22 2 v 33 x Fg(6)123 b Ff(2)26 b Fg(0)p Fh(:)p Fg(793[7)p Fh(;)7 b Fg(8])i Ff(\002)h Fg(0)p Fh(:)p Fg(125)f Ff(\002)h Fg(2)p Fh(:)p Fg(449[4)p Fh(;)d Fg(5])209 b Ff(2)26 b Fg(0)p Fh(:)p Fg(243)10 b(+)g Fh(\017)127 2403 y(A)159 2382 y Fe(\(1\))159 2417 y(2)p Fb(;)p Fe(2)250 2403 y Fg(=)i(2)316 2389 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 2403 y Ff(\002)461 2387 y Fe(1)p 461 2394 19 2 v 461 2419 a(4)494 2403 y Ff(\002)537 2370 y(p)p 572 2370 22 2 v 33 x Fg(2)123 b Ff(2)26 b Fg(0)p Fh(:)p Fg(890[89)p Fh(;)7 b Fg(90])j Ff(\002)f Fg(0)p Fh(:)p Fg(25)h Ff(\002)f Fg(1)p Fh(:)p Fg(414[2)p Fh(;)e Fg(3])188 b Ff(2)26 b Fg(0)p Fh(:)p Fg(315)10 b Ff(\000)g Fh(\017)127 2467 y(A)159 2446 y Fe(\(1\))159 2481 y(2)p Fb(;)p Fe(3)250 2467 y Fg(=)301 2451 y Fe(10)p 301 2458 37 2 v 301 2483 a(32)353 2467 y Ff(\002)401 2426 y Fd(p)p 431 2426 V 25 x Fe(15)p 401 2458 67 2 v 416 2483 a(10)716 2467 y Ff(2)26 b Fg(0)p Fh(:)p Fg(3125)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(387[2)p Fh(;)e Fg(3])421 b Ff(2)26 b Fg(0)p Fh(:)p Fg(121)10 b(+)g Fh(\017)127 2530 y(A)159 2509 y Fe(\(1\))159 2544 y(3)p Fb(;)p Fe(3)250 2530 y Fg(=)i(2)316 2516 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 2530 y Ff(\002)470 2514 y Fe(3)p 461 2521 37 2 v 461 2546 a(16)513 2530 y Ff(\002)555 2497 y(p)p 590 2497 22 2 v 33 x Fg(1)105 b Ff(2)26 b Fg(0)p Fh(:)p Fg(79[3)p Fh(;)7 b Fg(4])i Ff(\002)g Fg(0)p Fh(:)p Fg(1875)h Ff(\002)g Fg(1)369 b Ff(2)26 b Fg(0)p Fh(:)p Fg(149)10 b Ff(\000)g Fh(\017)127 2594 y(A)159 2573 y Fe(\(1\))159 2607 y(3)p Fb(;)p Fe(4)250 2594 y Fg(=)i(2)316 2580 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 2594 y Ff(\002)461 2578 y Fe(10)p 461 2585 37 2 v 461 2609 a(64)513 2594 y Ff(\002)561 2552 y Fd(p)p 591 2552 V 26 x Fe(14)p 561 2585 67 2 v 576 2609 a(10)716 2594 y Ff(2)26 b Fg(0)p Fh(:)p Fg(890[8)p Fh(;)7 b Fg(9])i Ff(\002)h Fg(0)p Fh(:)p Fg(156[2)p Fh(;)d Fg(3])i Ff(\002)h Fg(0)p Fh(:)p Fg(37[4)p Fh(;)d Fg(5])145 b Ff(2)26 b Fg(0)p Fh(:)p Fg(052)10 b(+)g Fh(\017)127 2658 y(A)159 2636 y Fe(\(1\))159 2671 y(4)p Fb(;)p Fe(4)250 2658 y Fg(=)301 2641 y Fe(10)p 301 2648 37 2 v 301 2673 a(16)353 2658 y Ff(\002)401 2616 y Fd(p)p 431 2616 19 2 v 25 x Fe(1)p 401 2648 49 2 v 407 2673 a(10)716 2658 y Ff(2)26 b Fg(0)p Fh(:)p Fg(625)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(1)569 b Ff(2)26 b Fg(0)p Fh(:)p Fg(063)10 b Ff(\000)g Fh(\017)127 2721 y(A)159 2700 y Fe(\(1\))159 2735 y(4)p Fb(;)p Fe(5)250 2721 y Fg(=)i(2)316 2707 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 2721 y Ff(\002)470 2705 y Fe(30)p 461 2712 55 2 v 461 2737 a(256)531 2721 y Ff(\002)579 2680 y Fd(p)p 609 2680 19 2 v 25 x Fe(5)p 579 2712 49 2 v 585 2737 a(10)716 2721 y Ff(2)26 b Fg(0)p Fh(:)p Fg(79[3)p Fh(;)7 b Fg(4])i Ff(\002)g Fg(0)p Fh(:)p Fg(11[7)p Fh(;)e Fg(8])j Ff(\002)f Fg(0)p Fh(:)p Fg(22[3)p Fh(;)e Fg(4])188 b Ff(2)26 b Fg(0)p Fh(:)p Fg(021)10 b Ff(\000)g Fh(\017)950 2864 y Fv(24)p eop %%Page: 25 26 bop 127 50 a Fh(A)159 29 y Fe(\(1\))159 63 y(5)p Fb(;)p Fe(5)250 50 y Fg(=)12 b(2)316 36 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 50 y Ff(\002)470 34 y Fe(50)p 461 41 55 2 v 461 65 a(256)531 50 y Ff(\002)579 8 y Fd(p)p 609 8 19 2 v 26 x Fe(2)p 579 41 49 2 v 585 65 a(10)716 50 y Ff(2)26 b Fg(0)p Fh(:)p Fg(89[0)p Fh(;)7 b Fg(1])i Ff(\002)g Fg(0)p Fh(:)p Fg(195[3)p Fh(;)f Fg(4])h Ff(\002)g Fg(0)p Fh(:)p Fg(14[1)p Fh(;)e Fg(2])167 b Ff(2)26 b Fg(0)p Fh(:)p Fg(025)10 b Ff(\000)g Fh(\017)127 114 y(A)159 92 y Fe(\(1\))159 127 y(5)p Fb(;)p Fe(6)250 114 y Fg(=)301 97 y Fe(1000)p 301 104 74 2 v 301 129 a(1024)389 114 y Ff(\002)441 72 y Fd(p)p 471 72 37 2 v 25 x Fe(66)p 438 104 74 2 v 438 129 a(1000)716 114 y Ff(2)26 b Fg(0)p Fh(:)p Fg(97[6)p Fh(;)7 b Fg(7])i Ff(\002)g Fg(0)p Fh(:)p Fg(0081[2)p Fh(;)f Fg(3])357 b Ff(2)26 b Fg(0)p Fh(:)p Fg(008)10 b Ff(\000)g Fh(\017)127 177 y(A)159 156 y Fe(\(1\))159 191 y(6)p Fb(;)p Fe(6)250 177 y Fg(=)i(2)316 163 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 177 y Ff(\002)470 161 y Fe(30)p 461 168 55 2 v 461 193 a(256)531 177 y Ff(\002)579 136 y Fd(p)p 609 136 19 2 v 25 x Fe(1)p 579 168 49 2 v 585 193 a(10)716 177 y Ff(2)26 b Fg(0)p Fh(:)p Fg([79)p Fh(;)7 b Fg(80])i Ff(\002)g Fg(0)p Fh(:)p Fg(11[7)p Fh(;)e Fg(8])j Ff(\002)f Fg(0)p Fh(:)p Fg(1)294 b Ff(2)26 b Fg(0)p Fh(:)p Fg(009)10 b(+)g Fh(\017)127 240 y(A)159 219 y Fe(\(2\))159 254 y(2)p Fb(;)p Fe(2)250 240 y Fg(=)310 224 y Fe(3)p 301 231 37 2 v 301 256 a(16)353 240 y Ff(\002)395 207 y(p)p 430 207 22 2 v 33 x Fg(6)265 b Ff(2)26 b Fg(0)p Fh(:)p Fg(1875)10 b Ff(\002)f Fg(2)p Fh(:)p Fg(4[49)p Fh(;)e Fg(50])421 b Ff(2)26 b Fg(0)p Fh(:)p Fg(459)10 b(+)g Fh(\017)127 303 y(A)159 282 y Fe(\(2\))159 317 y(2)p Fb(;)p Fe(3)250 303 y Fg(=)i(2)316 289 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 303 y Ff(\002)461 287 y Fe(1)p 461 294 19 2 v 461 319 a(8)494 303 y Ff(\002)537 270 y(p)p 572 270 43 2 v 33 x Fg(10)102 b Ff(2)26 b Fg(0)p Fh(:)p Fg(793[7)p Fh(;)7 b Fg(8])i Ff(\002)h Fg(0)p Fh(:)p Fg(125)f Ff(\002)h Fg(3)p Fh(:)p Fg(16[2)p Fh(;)d Fg(3])230 b Ff(2)26 b Fg(0)p Fh(:)p Fg(314)10 b Ff(\000)g Fh(\017)127 367 y(A)159 346 y Fe(\(2\))159 380 y(2)p Fb(;)p Fe(4)250 367 y Fg(=)i(2)316 353 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 367 y Ff(\002)461 351 y Fe(10)p 461 358 37 2 v 461 383 a(64)513 367 y Ff(\002)561 326 y Fd(p)p 591 326 V 25 x Fe(35)p 561 358 67 2 v 576 383 a(10)716 367 y Ff(2)26 b Fg(0)p Fh(:)p Fg(89[0)p Fh(;)7 b Fg(1])i Ff(\002)g Fg(0)p Fh(:)p Fg(156[2)p Fh(;)f Fg(3])h Ff(\002)g Fg(0)p Fh(:)p Fg(59[1)p Fh(;)e Fg(2])167 b Ff(2)26 b Fg(0)p Fh(:)p Fg(082)10 b(+)g Fh(\017)127 430 y(A)159 409 y Fe(\(2\))159 443 y(3)p Fb(;)p Fe(3)250 430 y Fg(=)i(2)316 416 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 430 y Ff(\002)461 414 y Fe(15)p 461 421 37 2 v 461 445 a(64)513 430 y Ff(\002)555 397 y(p)p 590 397 22 2 v 33 x Fg(3)105 b Ff(2)26 b Fg(0)p Fh(:)p Fg(890[89)p Fh(;)7 b Fg(90])j Ff(\002)f Fg(0)p Fh(:)p Fg(234[3)p Fh(;)e Fg(4])j Ff(\002)f Fg(1)p Fh(:)p Fg(73[2)p Fh(;)e Fg(3])103 b Ff(2)26 b Fg(0)p Fh(:)p Fg(362)10 b Ff(\000)g Fh(\017)127 494 y(A)159 472 y Fe(\(2\))159 507 y(3)p Fb(;)p Fe(4)250 494 y Fg(=)301 477 y Fe(10)p 301 484 37 2 v 301 509 a(32)353 494 y Ff(\002)401 452 y Fd(p)p 431 452 V 25 x Fe(42)p 401 484 67 2 v 416 509 a(10)716 494 y Ff(2)26 b Fg(0)p Fh(:)p Fg(3125)10 b Ff(\002)f Fg(0)p Fh(:)p Fg(64[8)p Fh(;)e Fg(9])442 b Ff(2)26 b Fg(0)p Fh(:)p Fg(203)10 b Ff(\000)g Fh(\017)127 557 y(A)159 536 y Fe(\(2\))159 571 y(3)p Fb(;)p Fe(5)250 557 y Fg(=)i(2)316 543 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 557 y Ff(\002)461 541 y Fe(100)p 461 548 55 2 v 461 573 a(256)531 557 y Ff(\002)579 516 y Fd(p)p 609 516 V 25 x Fe(210)p 579 548 85 2 v 594 573 a(100)716 557 y Ff(2)26 b Fg(0)p Fh(:)p Fg(793[7)p Fh(;)7 b Fg(8])i Ff(\002)h Fg(0)p Fh(:)p Fg(390[6)p Fh(;)d Fg(7])i Ff(\002)h Fg(0)p Fh(:)p Fg(14[4)p Fh(;)d Fg(5])145 b Ff(2)26 b Fg(0)p Fh(:)p Fg(045)10 b Ff(\000)g Fh(\017)127 620 y(A)159 599 y Fe(\(2\))159 634 y(4)p Fb(;)p Fe(4)250 620 y Fg(=)i(2)316 606 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 620 y Ff(\002)470 604 y Fe(7)p 461 611 37 2 v 461 636 a(64)513 620 y Ff(\002)555 587 y(p)p 590 587 22 2 v 33 x Fg(6)105 b Ff(2)26 b Fg(0)p Fh(:)p Fg(793[7)p Fh(;)7 b Fg(8])i Ff(\002)h Fg(0)p Fh(:)p Fg(1093[7)p Fh(;)d Fg(8])i Ff(\002)h Fg(2)p Fh(:)p Fg(4[49)p Fh(;)d Fg(50])103 b Ff(2)26 b Fg(0)p Fh(:)p Fg(213)10 b Ff(\000)g Fh(\017)127 684 y(A)159 663 y Fe(\(2\))159 698 y(4)p Fb(;)p Fe(5)250 684 y Fg(=)i(2)316 670 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 684 y Ff(\002)461 668 y Fe(10)p 461 675 37 2 v 461 700 a(32)513 684 y Ff(\002)561 643 y Fd(p)p 591 643 V 25 x Fe(15)p 561 675 67 2 v 576 700 a(10)716 684 y Ff(2)26 b Fg(0)p Fh(:)p Fg(890[8)p Fh(;)7 b Fg(9])i Ff(\002)h Fg(0)p Fh(:)p Fg(3125)g Ff(\002)f Fg(0)p Fh(:)p Fg(387[2)p Fh(;)e Fg(3])188 b Ff(2)26 b Fg(0)p Fh(:)p Fg(108)10 b Ff(\000)g Fh(\017)127 748 y(A)159 727 y Fe(\(2\))159 761 y(4)p Fb(;)p Fe(6)250 748 y Fg(=)310 731 y Fe(300)p 301 738 74 2 v 301 763 a(1024)389 748 y Ff(\002)438 706 y Fd(p)p 468 706 37 2 v 25 x Fe(55)p 438 738 67 2 v 443 763 a(100)716 748 y Ff(2)26 b Fg(0)p Fh(:)p Fg(29[2)p Fh(;)7 b Fg(3])i Ff(\002)g Fg(0)p Fh(:)p Fg(07[4)p Fh(;)e Fg(5])400 b Ff(2)26 b Fg(0)p Fh(:)p Fg(022)10 b Ff(\000)g Fh(\017)127 811 y(A)159 790 y Fe(\(2\))159 825 y(5)p Fb(;)p Fe(5)250 811 y Fg(=)310 795 y Fe(450)p 301 802 74 2 v 301 827 a(1024)389 811 y Ff(\002)438 770 y Fd(p)p 468 770 19 2 v 25 x Fe(6)p 438 802 49 2 v 443 827 a(10)716 811 y Ff(2)26 b Fg(0)p Fh(:)p Fg(4[39)p Fh(;)7 b Fg(40])i Ff(\002)h Fg(0)p Fh(:)p Fg(24[49)p Fh(;)d Fg(50])336 b Ff(2)26 b Fg(0)p Fh(:)p Fg(108)10 b Ff(\000)g Fh(\017)127 875 y(A)159 854 y Fe(\(2\))159 889 y(5)p Fb(;)p Fe(6)250 875 y Fg(=)i(2)316 861 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 875 y Ff(\002)470 859 y Fe(50)p 461 866 55 2 v 461 891 a(256)531 875 y Ff(\002)579 834 y Fd(p)p 609 834 37 2 v 25 x Fe(11)p 579 866 67 2 v 594 891 a(10)716 875 y Ff(2)26 b Fg(0)p Fh(:)p Fg(793[7)p Fh(;)7 b Fg(8])i Ff(\002)h Fg(0)p Fh(:)p Fg(195[3)p Fh(;)d Fg(4])i Ff(\002)h Fg(0)p Fh(:)p Fg(33[1)p Fh(;)d Fg(2])145 b Ff(2)26 b Fg(0)p Fh(:)p Fg(051)10 b(+)g Fh(\017)127 939 y(A)159 918 y Fe(\(2\))159 952 y(6)p Fb(;)p Fe(6)250 939 y Fg(=)i(2)316 925 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 939 y Ff(\002)470 923 y Fe(330)p 461 930 74 2 v 461 955 a(1024)549 939 y Ff(\002)597 898 y Fd(p)p 627 898 19 2 v 25 x Fe(3)p 597 930 49 2 v 603 955 a(10)716 939 y Ff(2)26 b Fg(0)p Fh(:)p Fg(890[8)p Fh(;)7 b Fg(9])i Ff(\002)h Fg(0)p Fh(:)p Fg(322[2)p Fh(;)d Fg(3])i Ff(\002)h Fg(0)p Fh(:)p Fg(17[3)p Fh(;)d Fg(4])145 b Ff(2)26 b Fg(0)p Fh(:)p Fg(050)10 b Ff(\000)g Fh(\017)127 1002 y(A)159 981 y Fe(\(3\))159 1015 y(3)p Fb(;)p Fe(3)250 1002 y Fg(=)310 986 y Fe(5)p 301 993 37 2 v 301 1018 a(32)353 1002 y Ff(\002)395 969 y(p)p 430 969 43 2 v 33 x Fg(10)244 b Ff(2)26 b Fg(0)p Fh(:)p Fg(15625)10 b Ff(\002)f Fg(3)p Fh(:)p Fg(16[2)p Fh(;)e Fg(3])421 b Ff(2)26 b Fg(0)p Fh(:)p Fg(494)10 b(+)g Fh(\017)127 1066 y(A)159 1044 y Fe(\(3\))159 1079 y(3)p Fb(;)p Fe(4)250 1066 y Fg(=)i(2)316 1052 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 1066 y Ff(\002)461 1049 y Fe(150)p 461 1056 55 2 v 461 1081 a(256)531 1066 y Ff(\002)579 1024 y Fd(p)p 609 1024 37 2 v 25 x Fe(70)p 579 1056 67 2 v 594 1081 a(10)716 1066 y Ff(2)26 b Fg(0)p Fh(:)p Fg(793[7)p Fh(;)7 b Fg(8])i Ff(\002)h Fg(0)p Fh(:)p Fg(58[59)p Fh(;)d Fg(60])i Ff(\002)h Fg(0)p Fh(:)p Fg(836[6)p Fh(;)d Fg(7])103 b Ff(2)26 b Fg(0)p Fh(:)p Fg(389)10 b(+)g Fh(\017)127 1129 y(A)159 1108 y Fe(\(3\))159 1143 y(3)p Fb(;)p Fe(5)250 1129 y Fg(=)i(2)316 1115 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 1129 y Ff(\002)470 1113 y Fe(90)p 461 1120 55 2 v 461 1145 a(128)531 1129 y Ff(\002)579 1088 y Fd(p)p 609 1088 19 2 v 25 x Fe(7)p 579 1120 49 2 v 585 1145 a(10)716 1129 y Ff(2)26 b Fg(0)p Fh(:)p Fg(890[8)p Fh(;)7 b Fg(9])i Ff(\002)h Fg(0)p Fh(:)p Fg(70[3)p Fh(;)d Fg(4])i Ff(\002)g Fg(0)p Fh(:)p Fg(264[5)p Fh(;)f Fg(6])145 b Ff(2)26 b Fg(0)p Fh(:)p Fg(166)10 b Ff(\000)g Fh(\017)127 1193 y(A)159 1172 y Fe(\(3\))159 1207 y(3)p Fb(;)p Fe(6)250 1193 y Fg(=)301 1177 y Fe(100)p 301 1184 55 2 v 301 1209 a(512)371 1193 y Ff(\002)419 1152 y Fd(p)p 449 1152 V 25 x Fe(231)p 419 1184 85 2 v 434 1209 a(100)716 1193 y Ff(2)26 b Fg(0)p Fh(:)p Fg(195[3)p Fh(;)7 b Fg(4])i Ff(\002)h Fg(0)p Fh(:)p Fg(15[19)p Fh(;)d Fg(20])336 b Ff(2)26 b Fg(0)p Fh(:)p Fg(030)10 b Ff(\000)g Fh(\017)127 1256 y(A)159 1235 y Fe(\(3\))159 1270 y(4)p Fb(;)p Fe(4)250 1256 y Fg(=)i(2)316 1242 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 1256 y Ff(\002)470 1240 y Fe(7)p 461 1247 37 2 v 461 1272 a(32)513 1256 y Ff(\002)555 1223 y(p)p 590 1223 22 2 v 33 x Fg(5)105 b Ff(2)26 b Fg(0)p Fh(:)p Fg(890[89)p Fh(;)7 b Fg(90])j Ff(\002)f Fg(0)p Fh(:)p Fg(21875)h Ff(\002)g Fg(2)p Fh(:)p Fg(23[6)p Fh(;)d Fg(7])145 b Ff(2)26 b Fg(0)p Fh(:)p Fg(436)10 b Ff(\000)g Fh(\017)127 1319 y(A)159 1298 y Fe(\(3\))159 1333 y(4)p Fb(;)p Fe(5)250 1319 y Fg(=)301 1303 y Fe(105)p 301 1310 55 2 v 301 1335 a(512)371 1319 y Ff(\002)413 1286 y(p)p 449 1286 22 2 v 449 1319 a Fg(2)246 b Ff(2)26 b Fg(0)p Fh(:)p Fg(2050[7)p Fh(;)7 b Fg(8])i Ff(\002)h Fg(1)p Fh(:)p Fg(414[2)p Fh(;)d Fg(3])336 b Ff(2)26 b Fg(0)p Fh(:)p Fg(290)10 b(+)g Fh(\017)127 1383 y(A)159 1362 y Fe(\(3\))159 1396 y(4)p Fb(;)p Fe(6)250 1383 y Fg(=)i(2)316 1369 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 1383 y Ff(\002)470 1366 y Fe(30)p 461 1373 55 2 v 461 1398 a(128)531 1383 y Ff(\002)579 1341 y Fd(p)p 609 1341 37 2 v 25 x Fe(33)p 579 1373 67 2 v 594 1398 a(10)716 1383 y Ff(2)26 b Fg(0)p Fh(:)p Fg(793[7)p Fh(;)7 b Fg(8])i Ff(\002)h Fg(0)p Fh(:)p Fg(234[3)p Fh(;)d Fg(4])i Ff(\002)h Fg(0)p Fh(:)p Fg(57[4)p Fh(;)d Fg(5])145 b Ff(2)26 b Fg(0)p Fh(:)p Fg(107)10 b Ff(\000)g Fh(\017)127 1446 y(A)159 1424 y Fe(\(3\))159 1459 y(5)p Fb(;)p Fe(5)250 1446 y Fg(=)i(2)316 1432 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 1446 y Ff(\002)470 1429 y Fe(15)p 461 1436 55 2 v 461 1461 a(128)531 1446 y Ff(\002)573 1413 y(p)p 609 1413 43 2 v 609 1446 a Fg(10)65 b Ff(2)26 b Fg(0)p Fh(:)p Fg(7937[0)p Fh(;)7 b Fg(1])i Ff(\002)h Fg(0)p Fh(:)p Fg(1171[8)p Fh(;)d Fg(9])j Ff(\002)f Fg(3)p Fh(:)p Fg(16[2)p Fh(;)e Fg(3])103 b Ff(2)26 b Fg(0)p Fh(:)p Fg(294)10 b(+)g Fh(\017)127 1509 y(A)159 1488 y Fe(\(3\))159 1523 y(5)p Fb(;)p Fe(6)250 1509 y Fg(=)i(2)316 1495 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 1509 y Ff(\002)470 1493 y Fe(450)p 461 1500 74 2 v 461 1525 a(4096)549 1509 y Ff(\002)597 1468 y Fd(p)p 627 1468 55 2 v 25 x Fe(330)p 597 1500 85 2 v 622 1525 a(10)716 1509 y Ff(2)26 b Fg(0)p Fh(:)p Fg(890[8)p Fh(;)7 b Fg(9])i Ff(\002)h Fg(0)p Fh(:)p Fg(109[8)p Fh(;)d Fg(9])i Ff(\002)h Fg(1)p Fh(:)p Fg(81[6)p Fh(;)d Fg(7])145 b Ff(2)26 b Fg(0)p Fh(:)p Fg(178)10 b Ff(\000)g Fh(\017)127 1573 y(A)159 1552 y Fe(\(3\))159 1587 y(6)p Fb(;)p Fe(6)250 1573 y Fg(=)310 1557 y Fe(550)p 301 1564 74 2 v 301 1589 a(1024)389 1573 y Ff(\002)438 1532 y Fd(p)p 468 1532 37 2 v 25 x Fe(10)p 438 1564 67 2 v 453 1589 a(10)716 1573 y Ff(2)26 b Fg(0)p Fh(:)p Fg(537[1)p Fh(;)7 b Fg(2])i Ff(\002)h Fg(0)p Fh(:)p Fg(316[2)p Fh(;)d Fg(3])357 b Ff(2)26 b Fg(0)p Fh(:)p Fg(170)10 b Ff(\000)g Fh(\017)127 1637 y(A)159 1616 y Fe(\(4\))159 1650 y(4)p Fb(;)p Fe(4)250 1637 y Fg(=)301 1621 y Fe(350)p 301 1628 55 2 v 301 1653 a(512)371 1637 y Ff(\002)419 1596 y Fd(p)p 449 1596 37 2 v 25 x Fe(70)p 419 1628 67 2 v 434 1653 a(10)716 1637 y Ff(2)26 b Fg(0)p Fh(:)p Fg(683[5)p Fh(;)7 b Fg(6])i Ff(\002)h Fg(0)p Fh(:)p Fg(836[6)p Fh(;)d Fg(7])357 b Ff(2)26 b Fg(0)p Fh(:)p Fg(572)10 b Ff(\000)g Fh(\017)127 1700 y(A)159 1679 y Fe(\(4\))159 1713 y(4)p Fb(;)p Fe(5)250 1700 y Fg(=)i(2)316 1686 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 1700 y Ff(\002)470 1684 y Fe(21)p 461 1691 55 2 v 461 1715 a(128)531 1700 y Ff(\002)573 1667 y(p)p 609 1667 43 2 v 609 1700 a Fg(14)65 b Ff(2)26 b Fg(0)p Fh(:)p Fg(7937[0)p Fh(;)7 b Fg(1])i Ff(\002)h Fg(0)p Fh(:)p Fg(1640[6)p Fh(;)d Fg(7])j Ff(\002)f Fg(3)p Fh(:)p Fg(74[1)p Fh(;)e Fg(2])103 b Ff(2)26 b Fg(0)p Fh(:)p Fg(487)10 b(+)g Fh(\017)127 1764 y(A)159 1742 y Fe(\(4\))159 1777 y(4)p Fb(;)p Fe(6)250 1764 y Fg(=)i(2)316 1750 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 1764 y Ff(\002)470 1747 y Fe(70)p 461 1754 55 2 v 461 1779 a(512)531 1764 y Ff(\002)579 1722 y Fd(p)p 609 1722 V 25 x Fe(462)p 579 1754 85 2 v 603 1779 a(10)716 1764 y Ff(2)26 b Fg(0)p Fh(:)p Fg(890[8)p Fh(;)7 b Fg(9])i Ff(\002)h Fg(0)p Fh(:)p Fg(1367[1)p Fh(;)d Fg(2])i Ff(\002)h Fg(2)p Fh(:)p Fg(1[49)p Fh(;)d Fg(50])103 b Ff(2)26 b Fg(0)p Fh(:)p Fg(262)10 b Ff(\000)g Fh(\017)127 1827 y(A)159 1805 y Fe(\(4\))159 1840 y(5)p Fb(;)p Fe(5)250 1827 y Fg(=)i(2)316 1812 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 1827 y Ff(\002)470 1810 y Fe(105)p 461 1817 74 2 v 461 1842 a(1024)549 1827 y Ff(\002)592 1793 y(p)p 627 1793 43 2 v 34 x Fg(35)47 b Ff(2)26 b Fg(0)p Fh(:)p Fg(890[89)p Fh(;)7 b Fg(90])j Ff(\002)f Fg(0)p Fh(:)p Fg(1025[39)p Fh(;)f Fg(40])h Ff(\002)g Fg(5)p Fh(:)p Fg(916[0)p Fh(;)f Fg(1])18 b Ff(2)26 b Fg(0)p Fh(:)p Fg(540)10 b(+)g Fh(\017)127 1890 y(A)159 1869 y Fe(\(4\))159 1904 y(5)p Fb(;)p Fe(6)250 1890 y Fg(=)310 1874 y Fe(30)p 301 1881 55 2 v 301 1906 a(256)371 1890 y Ff(\002)419 1849 y Fd(p)p 449 1849 74 2 v 25 x Fe(1155)p 419 1881 103 2 v 453 1906 a(10)716 1890 y Ff(2)26 b Fg(0)p Fh(:)p Fg(1171[8)p Fh(;)7 b Fg(9])i Ff(\002)h Fg(3)p Fh(:)p Fg(39[8)p Fh(;)d Fg(9])357 b Ff(2)26 b Fg(0)p Fh(:)p Fg(398)10 b(+)g Fh(\017)127 1954 y(A)159 1933 y Fe(\(4\))159 1968 y(6)p Fb(;)p Fe(6)250 1954 y Fg(=)i(2)316 1940 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 1954 y Ff(\002)461 1938 y Fe(4950)p 461 1945 74 2 v 461 1970 a(8192)549 1954 y Ff(\002)597 1913 y Fd(p)p 627 1913 37 2 v 25 x Fe(70)p 597 1945 67 2 v 612 1970 a(10)716 1954 y Ff(2)26 b Fg(0)p Fh(:)p Fg(793[7)p Fh(;)7 b Fg(8])i Ff(\002)h Fg(0)p Fh(:)p Fg(604[2)p Fh(;)d Fg(3])i Ff(\002)h Fg(0)p Fh(:)p Fg(836[6)p Fh(;)d Fg(7])124 b Ff(2)26 b Fg(0)p Fh(:)p Fg(401)10 b(+)g Fh(\017)127 2017 y(A)159 1996 y Fe(\(5\))159 2030 y(5)p Fb(;)p Fe(5)250 2017 y Fg(=)310 2001 y Fe(189)p 301 2008 74 2 v 301 2033 a(1024)389 2017 y Ff(\002)432 1984 y(p)p 467 1984 43 2 v 33 x Fg(14)207 b Ff(2)26 b Fg(0)p Fh(:)p Fg(18457[0)p Fh(;)7 b Fg(1])j Ff(\002)f Fg(3)p Fh(:)p Fg(741[6)p Fh(;)e Fg(7])315 b Ff(2)26 b Fg(0)p Fh(:)p Fg(691)10 b Ff(\000)g Fh(\017)127 2081 y(A)159 2059 y Fe(\(5\))159 2094 y(5)p Fb(;)p Fe(6)250 2081 y Fg(=)i(2)316 2067 y Fd(\000)p Fe(1)p Fb(=)p Fe(3)412 2081 y Ff(\002)461 2064 y Fe(1050)p 461 2071 74 2 v 461 2096 a(2048)549 2081 y Ff(\002)597 2039 y Fd(p)p 627 2039 55 2 v 25 x Fe(231)p 597 2071 85 2 v 622 2096 a(10)716 2081 y Ff(2)26 b Fg(0)p Fh(:)p Fg(7937[0)p Fh(;)7 b Fg(1])i Ff(\002)h Fg(0)p Fh(:)p Fg(512[69)p Fh(;)d Fg(70])j Ff(\002)f Fg(1)p Fh(:)p Fg(5198[6)p Fh(;)e Fg(7])40 b Ff(2)26 b Fg(0)p Fh(:)p Fg(618)10 b(+)g Fh(\017)127 2144 y(A)159 2122 y Fe(\(5\))159 2157 y(6)p Fb(;)p Fe(6)250 2144 y Fg(=)i(2)316 2130 y Fd(\000)p Fe(1)p Fb(=)p Fe(6)412 2144 y Ff(\002)470 2127 y Fe(297)p 461 2134 74 2 v 461 2159 a(1024)549 2144 y Ff(\002)592 2110 y(p)p 627 2110 22 2 v 34 x Fg(7)68 b Ff(2)26 b Fg(0)p Fh(:)p Fg(890[89)p Fh(;)7 b Fg(90])j Ff(\002)f Fg(0)p Fh(:)p Fg(2900[3)p Fh(;)e Fg(4])j Ff(\002)f Fg(2)p Fh(:)p Fg(645[7)p Fh(;)e Fg(8])61 b Ff(2)26 b Fg(0)p Fh(:)p Fg(684)10 b Ff(\000)g Fh(\017)127 2207 y(A)159 2186 y Fe(\(6\))159 2221 y(6)p Fb(;)p Fe(6)250 2207 y Fg(=)301 2191 y Fe(2310)p 301 2198 74 2 v 301 2223 a(4096)389 2207 y Ff(\002)438 2166 y Fd(p)p 468 2166 55 2 v 25 x Fe(231)p 438 2198 85 2 v 462 2223 a(10)716 2207 y Ff(2)26 b Fg(0)p Fh(:)p Fg(56[39)p Fh(;)7 b Fg(40])i Ff(\002)h Fg(1)p Fh(:)p Fg(519[8)p Fh(;)d Fg(9])336 b Ff(2)26 b Fg(0)p Fh(:)p Fg(857)10 b(+)g Fh(\017)257 2290 y Fc(T)l(able)32 b(6.)j Fg(De\014nition)13 b(of)i Fh(Q)758 2269 y Fe(\()p Fb(m)p Fe(\))758 2305 y Fb(k)q(;n)843 2290 y Fg(and)e(b)q(ound)g(on)h Fh(A)1147 2269 y Fe(\()p Fb(m)p Fe(\))1147 2305 y Fb(k)q(;n)1217 2290 y Fg(,)g(for)g(0)e Ff(\024)g Fh(m)g Ff(\024)g Fh(k)g Ff(\024)g Fh(n)g Ff(\024)f Fg(6.)0 2440 y Fv(In)20 b(the)g(third)f(column)g(of)h(this)g(table)g(w)o(e)g(ha)o(v)o(e)f (used)h(an)f(abbreviated)h(notation)f(for)h(in)o(terv)m(als)f(b)o(y)0 2499 y(writing)k(e.g.)44 b(0)p Fq(:)p Fv(29[2)p Fq(;)8 b Fv(3])24 b(instead)f(of)h([0)p Fq(:)p Fv(292)p Fq(;)8 b Fv(0)p Fq(:)p Fv(293])g(;)27 b(and)c(the)h(pro)q(duct)g(of)g(t)o(w)o(o)f(in)o(terv)m(als)h ([)p Fq(x;)8 b(y)r Fv(])0 2559 y(and)20 b([)p Fq(s;)8 b(t)p Fv(])21 b(in)g(I)-10 b(R)320 2566 y Fu(+)374 2559 y Fv(is)21 b(de\014ned)f(to)h(b)q(e)g(the)g(in)o(terv)m(al)f([)p Fq(xs;)8 b(y)r(t)p Fv(].)36 b(No)o(w,)22 b(in)e(order)g(to)h(v)o(erify)g(e.g.)35 b(that)0 2629 y Fq(A)37 2603 y Fu(\(2\))37 2642 y(4)p Fl(;)p Fu(6)108 2629 y Fs(2)17 b Fv(0)p Fq(:)p Fv(022)11 b Fs(\000)h Fq(\017)p Fv(,)19 b(it)f(su\016ces)f(to)i(c)o(hec)o(k)f(that)g(1024)12 b Fs(\002)g Fv(0)p Fq(:)p Fv(292)k Fs(\024)g Fv(300)h Fs(\024)f Fv(1024)c Fs(\002)g Fv(0)p Fq(:)p Fv(293)17 b(and)h(\(100)12 b Fs(\002)0 2694 y Fv(0)p Fq(:)p Fv(074\))133 2676 y Fu(2)169 2694 y Fs(\024)i Fv(55)f Fs(\024)h Fv(\(100)9 b Fs(\002)f Fv(0)p Fq(:)p Fv(075\))621 2676 y Fu(2)643 2694 y Fv(,)15 b(and)g(that)h(0)p Fq(:)p Fv(292)8 b Fs(\002)g Fv(0)p Fq(:)p Fv(074)13 b Fs(\025)h Fv(0)p Fq(:)p Fv(0215)h(and)f(0)p Fq(:)p Fv(293)8 b Fs(\002)h Fv(0)p Fq(:)p Fv(075)k Fs(\024)h Fv(0)p Fq(:)p Fv(022.)950 2864 y(25)p eop %%Page: 26 27 bop 0 50 a Ft(5.4.)54 b(Pro)r(of)24 b(of)g(\(N4\))0 137 y Fv(W)l(e)18 b(\014rst)f(compute)h(the)g(matrix)g Fq(Q)p Fv(\()p Fq(p)p Fv(\),)h(as)f(de\014ned)f(in)g(\(3.25\),)i(using)d(the)j(v)m(alues)e(of)h Fq(p)1687 144 y Fl(m)1743 137 y Fv(and)g Fq(Q)1881 111 y Fu(\()p Fl(m)p Fu(\))1881 152 y Fl(k)q(;n)0 196 y Fv(from)e(T)l(able)g(1)g(and)g(T)l (able)g(6,)g(resp)q(ectiv)o(ely)l(.)109 332 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))195 339 y Fb(k)q(;n)254 332 y Fg(=)i Fh(Q)338 311 y Fe(\(0\))338 347 y Fb(k)q(;n)396 332 y Fh(p)417 338 y Fe(0)458 332 y Fg(+)c Fh(Q)538 311 y Fe(\(1\))538 347 y Fb(k)q(;n)596 332 y Fh(p)617 338 y Fe(1)651 332 y Fg(+)g Fh(Q)731 311 y Fe(\(2\))731 347 y Fb(k)q(;n)789 332 y Fh(p)810 338 y Fe(2)844 332 y Fg(+)g Fh(Q)924 311 y Fe(\(3\))924 347 y Fb(k)q(;n)982 332 y Fh(p)1003 338 y Fe(3)1037 332 y Fg(+)g Fh(Q)1117 311 y Fe(\(4\))1117 347 y Fb(k)q(;n)1175 332 y Fh(p)1196 338 y Fe(4)1230 332 y Fg(+)g Fh(Q)1310 311 y Fe(\(5\))1310 347 y Fb(k)q(;n)1367 332 y Fh(p)1388 338 y Fe(5)1423 332 y Fg(+)g Fh(Q)1503 311 y Fe(\(6\))1503 347 y Fb(k)q(;n)1560 332 y Fh(p)1581 338 y Fe(6)102 405 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 411 y Fe(0)p Fb(;)p Fe(0)254 405 y Fg(=)d(0)p Fh(:)p Fg(909000)1173 b(=)12 b(0)p Fh(:)p Fg(909000)102 452 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 458 y Fe(1)p Fb(;)p Fe(0)254 452 y Fg(=)204 b(0)p Fh(:)p Fg(154980)980 b(=)12 b(0)p Fh(:)p Fg(154980)102 500 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 506 y Fe(1)p Fb(;)p Fe(1)254 500 y Fg(=)f(0)p Fh(:)p Fg(286335)q(+0)p Fh(:)p Fg(246000)q(+0)p Fh(:)p Fg(060993)787 b(=)12 b(0)p Fh(:)p Fg(593328)102 547 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 553 y Fe(2)p Fb(;)p Fe(0)254 547 y Fg(=)397 b(0)p Fh(:)p Fg(024849)787 b(=)12 b(0)p Fh(:)p Fg(024849)102 594 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 600 y Fe(2)p Fb(;)p Fe(1)254 594 y Fg(=)204 b(0)p Fh(:)p Fg(119556)q(+0)p Fh(:)p Fg(079065)q(+0)p Fh(:)p Fg(013552)594 b(=)12 b(0)p Fh(:)p Fg(212173)102 642 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 648 y Fe(2)p Fb(;)p Fe(2)254 642 y Fg(=)f(0)p Fh(:)p Fg(089991)q(+0)p Fh(:)p Fg(154980)q(+0)p Fh(:)p Fg(115209)q(+0)p Fh(:)p Fg(035168)q(+0)p Fh(:)p Fg(003690)401 b(=)12 b(0)p Fh(:)p Fg(399038)102 689 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 695 y Fe(3)p Fb(;)p Fe(0)254 689 y Fg(=)590 b(0)p Fh(:)p Fg(003472)k(=)12 b(0)p Fh(:)p Fg(003472)102 736 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 742 y Fe(3)p Fb(;)p Fe(1)254 736 y Fg(=)397 b(0)p Fh(:)p Fg(030371)q(+0)p Fh(:)p Fg(016688)q(+0)p Fh(:)p Fg(002340)k(=)12 b(0)p Fh(:)p Fg(049399)102 784 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 790 y Fe(3)p Fb(;)p Fe(2)254 784 y Fg(=)204 b(0)p Fh(:)p Fg(059532)q(+0)p Fh(:)p Fg(078814)q(+0)p Fh(:)p Fg(040544)q(+0)p Fh(:)p Fg(009135)q(+0)p Fh(:)p Fg(000720)k(=)12 b(0)p Fh(:)p Fg(188745)102 831 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 837 y Fe(3)p Fb(;)p Fe(3)254 831 y Fg(=)f(0)p Fh(:)p Fg(028179)q(+0)p Fh(:)p Fg(073308)q(+0)p Fh(:)p Fg(090862)q(+0)p Fh(:)p Fg(055328)q(+0)p Fh(:)p Fg(017505)q(+0)p Fh(:)p Fg(002656)q(+0)p Fh(:)p Fg(000150)k(=)d(0)p Fh(:)p Fg(267988)102 878 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 884 y Fe(4)p Fb(;)p Fe(0)254 878 y Fg(=)783 b(0)p Fh(:)p Fg(000450)401 b(=)12 b(0)p Fh(:)p Fg(000450)102 926 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 932 y Fe(4)p Fb(;)p Fe(1)254 926 y Fg(=)590 b(0)p Fh(:)p Fg(005824)q(+0)p Fh(:)p Fg(002835)q(+0)p Fh(:)p Fg(000336)208 b(=)12 b(0)p Fh(:)p Fg(008995)102 973 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 979 y Fe(4)p Fb(;)p Fe(2)254 973 y Fg(=)397 b(0)p Fh(:)p Fg(020582)q(+0)p Fh(:)p Fg(022736)q(+0)p Fh(:)p Fg(009585)q(+0)p Fh(:)p Fg(001728)q(+0)p Fh(:)p Fg(000110)15 b(=)d(0)p Fh(:)p Fg(054741)102 1020 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1026 y Fe(4)p Fb(;)p Fe(3)254 1020 y Fg(=)204 b(0)p Fh(:)p Fg(025584)q(+0)p Fh(:)p Fg(050953)q(+0)p Fh(:)p Fg(043568)q(+0)p Fh(:)p Fg(019620)q(+0)p Fh(:)p Fg(004640)q(+0)p Fh(:)p Fg(000535)15 b(=)d(0)p Fh(:)p Fg(144900)102 1068 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1074 y Fe(4)p Fb(;)p Fe(4)254 1068 y Fg(=)f(0)p Fh(:)p Fg(009090)q(+0)p Fh(:)p Fg(030996)q(+0)p Fh(:)p Fg(053463)q(+0)p Fh(:)p Fg(048832)q(+0)p Fh(:)p Fg(025740)q(+0)p Fh(:)p Fg(007792)q(+0)p Fh(:)p Fg(001310)k(=)d(0)p Fh(:)p Fg(177223)102 1115 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1121 y Fe(5)p Fb(;)p Fe(0)254 1115 y Fg(=)976 b(0)p Fh(:)p Fg(000048)208 b(=)12 b(0)p Fh(:)p Fg(000048)102 1162 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1168 y Fe(5)p Fb(;)p Fe(1)254 1162 y Fg(=)783 b(0)p Fh(:)p Fg(000945)q(+0)p Fh(:)p Fg(000400)q(+0)p Fh(:)p Fg(000040)15 b(=)d(0)p Fh(:)p Fg(001385)102 1210 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1216 y Fe(5)p Fb(;)p Fe(2)254 1210 y Fg(=)590 b(0)p Fh(:)p Fg(005040)q(+0)p Fh(:)p Fg(004860)q(+0)p Fh(:)p Fg(001728)q(+0)p Fh(:)p Fg(000255)15 b(=)d(0)p Fh(:)p Fg(011883)102 1257 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1263 y Fe(5)p Fb(;)p Fe(3)254 1257 y Fg(=)397 b(0)p Fh(:)p Fg(011295)q(+0)p Fh(:)p Fg(018592)q(+0)p Fh(:)p Fg(013050)q(+0)p Fh(:)p Fg(004704)q(+0)p Fh(:)p Fg(000890)15 b(=)d(0)p Fh(:)p Fg(048531)102 1304 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1310 y Fe(5)p Fb(;)p Fe(4)254 1304 y Fg(=)204 b(0)p Fh(:)p Fg(010332)q(+0)p Fh(:)p Fg(027108)q(+0)p Fh(:)p Fg(032480)q(+0)p Fh(:)p Fg(021915)q(+0)p Fh(:)p Fg(008640)q(+0)p Fh(:)p Fg(001990)15 b(=)d(0)p Fh(:)p Fg(102465)102 1352 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1358 y Fe(5)p Fb(;)p Fe(5)254 1352 y Fg(=)f(0)p Fh(:)p Fg(002727)q(+0)p Fh(:)p Fg(012300)q(+0)p Fh(:)p Fg(027108)q(+0)p Fh(:)p Fg(032928)q(+0)p Fh(:)p Fg(024300)q(+0)p Fh(:)p Fg(011056)q(+0)p Fh(:)p Fg(003090)k(=)d(0)p Fh(:)p Fg(113509)102 1399 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1405 y Fe(6)p Fb(;)p Fe(0)254 1399 y Fg(=)1169 b(0)p Fh(:)p Fg(000005)15 b(=)d(0)p Fh(:)p Fg(000005)102 1446 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1452 y Fe(6)p Fb(;)p Fe(1)254 1446 y Fg(=)976 b(0)p Fh(:)p Fg(000128)q(+0)p Fh(:)p Fg(000045)15 b(=)d(0)p Fh(:)p Fg(000173)102 1494 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1500 y Fe(6)p Fb(;)p Fe(2)254 1494 y Fg(=)783 b(0)p Fh(:)p Fg(000990)q(+0)p Fh(:)p Fg(000816)q(+0)p Fh(:)p Fg(000250)15 b(=)d(0)p Fh(:)p Fg(002056)102 1541 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1547 y Fe(6)p Fb(;)p Fe(3)254 1541 y Fg(=)590 b(0)p Fh(:)p Fg(003360)q(+0)p Fh(:)p Fg(004815)q(+0)p Fh(:)p Fg(002848)q(+0)p Fh(:)p Fg(000850)15 b(=)d(0)p Fh(:)p Fg(011873)102 1588 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1594 y Fe(6)p Fb(;)p Fe(4)254 1588 y Fg(=)397 b(0)p Fh(:)p Fg(005522)q(+0)p Fh(:)p Fg(011984)q(+0)p Fh(:)p Fg(011790)q(+0)p Fh(:)p Fg(006368)q(+0)p Fh(:)p Fg(002005)15 b(=)d(0)p Fh(:)p Fg(037669)102 1635 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))188 1641 y Fe(6)p Fb(;)p Fe(5)254 1635 y Fg(=)204 b(0)p Fh(:)p 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Ff(\000)p Fg(0)p Fh(:)p Fg(170813756)p 1772 369 V 50 w(1)p Fh(:)p Fg(081374284)-50 396 y(\()p Fh(M)t(T)e Fg(\))58 403 y Fb(k)q(;n)p 137 413 V 172 396 a Fg(0)p Fh(:)p Fg(000155742)p 409 413 V 50 w(0)p Fh(:)p Fg(000242204)p 682 413 V 51 w(0)p Fh(:)p Fg(000168466)p 955 413 V 50 w(0)p Fh(:)p Fg(000063712)p 1227 413 V 51 w(0)p Fh(:)p Fg(000035224)p 1500 413 V 18 w Ff(\000)p Fg(0)p Fh(:)p Fg(000099718)p 1772 413 V 50 w(1)p Fh(:)p Fg(000046122)661 508 y Fc(T)l(able)33 b(8.)i Fg(The)13 b(matrix)g(pro)q(duct)g Fh(M)t(T)6 b Fg(.)0 643 y Fv(By)18 b(lo)q(oking)f(at)h(the)g(v)m(alues)f(of)g (\()p Fq(M)5 b(T)i Fv(\))736 650 y Fl(k)q(;n)816 643 y Fv(in)17 b(T)l(able)g(8,)h(it)f(is)g(easy)h(to)f(see)h(that)g Fs(j)p Fq(\016)1570 650 y Fl(k)q(;n)1642 643 y Fs(\000)12 b Fv(\()p Fq(M)5 b(T)i Fv(\))1820 650 y Fl(k)q(;n)1882 643 y Fs(j)15 b(\024)0 702 y Fv(0)p Fq(:)p Fv(0005,)h(for)g(0)e Fs(\024)f Fq(k)r(;)8 b(n)14 b Fs(\024)f Fv(6.)22 b(This)16 b(completes)g(the)h(pro)q (of)f(of)h(\(N4\).)0 882 y Ft(5.5.)54 b(Pro)r(of)24 b(of)g(\(N5\))0 969 y Fv(W)l(e)15 b(b)q(egin)f(with)g(the)h(matrix)f Fq(Q)p Fv(\()p Fq(\016)r Fv(\),)i(as)e(de\014ned)g(in)g(\(3.25\),)h(using)f(the)g(v) m(alues)h(of)f Fq(\016)1578 976 y Fl(m)1631 969 y Fv(and)g Fq(Q)1765 943 y Fu(\()p Fl(m)p Fu(\))1765 984 y Fl(k)q(;n)1849 969 y Fv(from)0 1028 y(T)l(able)i(1)g(and)g(T)l(able)g(6,)h(resp)q(ectiv)o (ely)l(.)102 1197 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1204 y Fb(k)q(;n)267 1197 y Fg(=)j Fh(Q)353 1175 y Fe(\(0\))353 1212 y Fb(k)q(;n)411 1197 y Fh(\016)430 1203 y Fe(0)472 1197 y Fg(+)8 b Fh(Q)546 1175 y Fe(\(1\))546 1212 y Fb(k)q(;n)603 1197 y Fh(\016)622 1203 y Fe(1)665 1197 y Fg(+)g Fh(Q)739 1175 y Fe(\(2\))739 1212 y Fb(k)q(;n)796 1197 y Fh(\016)815 1203 y Fe(2)858 1197 y Fg(+)g Fh(Q)932 1175 y Fe(\(3\))932 1212 y Fb(k)q(;n)989 1197 y Fh(\016)1008 1203 y Fe(3)1051 1197 y Fg(+)g Fh(Q)1125 1175 y Fe(\(4\))1125 1212 y Fb(k)q(;n)1182 1197 y Fh(\016)1201 1203 y Fe(4)1244 1197 y Fg(+)f Fh(Q)1317 1175 y Fe(\(5\))1317 1212 y Fb(k)q(;n)1375 1197 y Fh(\016)1394 1203 y Fe(5)1437 1197 y Fg(+)g Fh(Q)1510 1175 y Fe(\(6\))1510 1212 y Fb(k)q(;n)1568 1197 y Fh(\016)1587 1203 y Fe(6)102 1267 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1273 y Fe(0)p Fb(;)p Fe(0)267 1267 y Fg(=)12 b(0)p Fh(:)p Fg(005000)1173 b(=)11 b(0)p Fh(:)p Fg(005000)102 1312 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1318 y Fe(1)p Fb(;)p Fe(0)267 1312 y Fg(=)205 b(0)p Fh(:)p Fg(003150)980 b(=)11 b(0)p Fh(:)p Fg(003150)102 1357 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1363 y Fe(1)p Fb(;)p Fe(1)267 1357 y Fg(=)h(0)p Fh(:)p Fg(001575)q(+0)p Fh(:)p Fg(005000)q(+0)p Fh(:)p Fg(000972)787 b(=)11 b(0)p Fh(:)p Fg(007547)102 1402 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1408 y Fe(2)p Fb(;)p Fe(0)267 1402 y Fg(=)398 b(0)p Fh(:)p Fg(000396)787 b(=)11 b(0)p Fh(:)p Fg(000396)102 1446 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1452 y Fe(2)p Fb(;)p Fe(1)267 1446 y Fg(=)205 b(0)p Fh(:)p Fg(002430)q(+0)p Fh(:)p Fg(001260)q(+0)p Fh(:)p Fg(000968)594 b(=)11 b(0)p Fh(:)p Fg(004658)102 1491 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1497 y Fe(2)p Fb(;)p Fe(2)267 1491 y Fg(=)h(0)p Fh(:)p Fg(000495)q(+0)p Fh(:)p Fg(003150)q(+0)p Fh(:)p Fg(001836)q(+0)p Fh(:)p Fg(002512)q(+0)p Fh(:)p Fg(000574)401 b(=)11 b(0)p Fh(:)p Fg(008567)102 1536 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1542 y Fe(3)p Fb(;)p Fe(0)267 1536 y Fg(=)591 b(0)p Fh(:)p Fg(000248)j(=)11 b(0)p Fh(:)p Fg(000248)102 1581 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1587 y Fe(3)p Fb(;)p Fe(1)267 1581 y Fg(=)398 b(0)p Fh(:)p Fg(000484)q(+0)p Fh(:)p Fg(001192)q(+0)p Fh(:)p Fg(000364)j(=)11 b(0)p Fh(:)p Fg(002040)102 1626 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1632 y Fe(3)p Fb(;)p Fe(2)267 1626 y Fg(=)205 b(0)p Fh(:)p Fg(001210)q(+0)p Fh(:)p Fg(001256)q(+0)p Fh(:)p Fg(002896)q(+0)p Fh(:)p Fg(001421)q(+0)p Fh(:)p Fg(000225)j(=)11 b(0)p Fh(:)p Fg(007008)102 1671 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1677 y Fe(3)p Fb(;)p Fe(3)267 1671 y Fg(=)h(0)p Fh(:)p Fg(000155)q(+0)p Fh(:)p Fg(001490)q(+0)p Fh(:)p Fg(001448)q(+0)p Fh(:)p Fg(003952)q(+0)p Fh(:)p Fg(002723)q(+0)p Fh(:)p Fg(000830)q(+0)p Fh(:)p Fg(000090)j(=)c(0)p Fh(:)p Fg(010688)102 1715 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1721 y Fe(4)p Fb(;)p Fe(0)267 1715 y Fg(=)784 b(0)p Fh(:)p Fg(000070)401 b(=)11 b(0)p Fh(:)p Fg(000070)102 1760 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1766 y Fe(4)p Fb(;)p Fe(1)267 1760 y Fg(=)591 b(0)p Fh(:)p Fg(000416)q(+0)p Fh(:)p Fg(000441)q(+0)p Fh(:)p Fg(000105)208 b(=)11 b(0)p Fh(:)p Fg(000962)102 1805 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1811 y Fe(4)p Fb(;)p Fe(2)267 1805 y Fg(=)398 b(0)p Fh(:)p Fg(000328)q(+0)p Fh(:)p Fg(001624)q(+0)p Fh(:)p Fg(001491)q(+0)p Fh(:)p Fg(000540)q(+0)p Fh(:)p Fg(000066)15 b(=)c(0)p Fh(:)p Fg(004049)102 1850 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1856 y Fe(4)p Fb(;)p Fe(3)267 1850 y Fg(=)205 b(0)p Fh(:)p Fg(000520)q(+0)p Fh(:)p Fg(000812)q(+0)p Fh(:)p Fg(003112)q(+0)p Fh(:)p Fg(003052)q(+0)p Fh(:)p Fg(001450)q(+0)p Fh(:)p Fg(000321)15 b(=)c(0)p Fh(:)p Fg(009267)102 1895 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1901 y Fe(4)p Fb(;)p Fe(4)267 1895 y Fg(=)h(0)p Fh(:)p Fg(000050)q(+0)p Fh(:)p Fg(000630)q(+0)p Fh(:)p Fg(000852)q(+0)p Fh(:)p Fg(003488)q(+0)p Fh(:)p Fg(004004)q(+0)p Fh(:)p Fg(002435)q(+0)p Fh(:)p Fg(000786)j(=)c(0)p Fh(:)p Fg(012245)102 1940 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1946 y Fe(5)p Fb(;)p Fe(0)267 1940 y Fg(=)977 b(0)p Fh(:)p Fg(000015)208 b(=)11 b(0)p Fh(:)p Fg(000015)102 1984 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 1990 y Fe(5)p Fb(;)p Fe(1)267 1984 y Fg(=)784 b(0)p Fh(:)p Fg(000147)q(+0)p Fh(:)p Fg(000125)q(+0)p Fh(:)p Fg(000024)15 b(=)c(0)p Fh(:)p Fg(000296)102 2029 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 2035 y Fe(5)p Fb(;)p Fe(2)267 2029 y Fg(=)591 b(0)p Fh(:)p Fg(000360)q(+0)p Fh(:)p Fg(000756)q(+0)p Fh(:)p Fg(000540)q(+0)p Fh(:)p Fg(000153)15 b(=)c(0)p Fh(:)p Fg(001809)102 2074 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 2080 y Fe(5)p Fb(;)p Fe(3)267 2074 y Fg(=)398 b(0)p Fh(:)p Fg(000180)q(+0)p Fh(:)p Fg(001328)q(+0)p Fh(:)p Fg(002030)q(+0)p Fh(:)p Fg(001470)q(+0)p 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Fg(=)784 b(0)p Fh(:)p Fg(000154)q(+0)p Fh(:)p Fg(000255)q(+0)p Fh(:)p Fg(000150)15 b(=)c(0)p Fh(:)p Fg(000559)102 2343 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 2349 y Fe(6)p Fb(;)p Fe(3)267 2343 y Fg(=)591 b(0)p Fh(:)p Fg(000240)q(+0)p Fh(:)p Fg(000749)q(+0)p Fh(:)p Fg(000890)q(+0)p Fh(:)p Fg(000510)15 b(=)c(0)p Fh(:)p Fg(002389)102 2388 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 2394 y Fe(6)p Fb(;)p Fe(4)267 2388 y Fg(=)398 b(0)p Fh(:)p Fg(000088)q(+0)p Fh(:)p Fg(000856)q(+0)p Fh(:)p Fg(001834)q(+0)p Fh(:)p Fg(001990)q(+0)p Fh(:)p Fg(001203)15 b(=)c(0)p Fh(:)p Fg(005971)102 2433 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 2439 y Fe(6)p Fb(;)p Fe(5)267 2433 y Fg(=)205 b(0)p Fh(:)p Fg(000080)q(+0)p Fh(:)p Fg(000204)q(+0)p Fh(:)p Fg(001424)q(+0)p Fh(:)p Fg(002786)q(+0)p Fh(:)p Fg(003090)q(+0)p Fh(:)p Fg(002052)15 b(=)c(0)p Fh(:)p Fg(009636)102 2478 y Fh(Q)p Fg(\()p Fh(\016)q Fg(\))187 2484 y Fe(6)p Fb(;)p Fe(6)267 2478 y Fg(=)h(0)p Fh(:)p Fg(000005)q(+0)p Fh(:)p Fg(000090)q(+0)p Fh(:)p Fg(000200)q(+0)p Fh(:)p Fg(001360)q(+0)p Fh(:)p Fg(002807)q(+0)p Fh(:)p Fg(003420)q(+0)p Fh(:)p Fg(002571)j(=)c(0)p Fh(:)p Fg(010453)590 2557 y Fc(T)l(able)32 b(9.)j Fg(The)13 b(symmetric)f(matrix)i Fh(Q)p Fg(\()p Fh(\016)q Fg(\).)0 2692 y Fv(Then,)i(from)g(the)g(data)h(in)f (T)l(ables)g(1,7,9,)f(w)o(e)i(can)f(determine)g(the)h(t)o(w)o(o)f(v)o(ectors) g Fq(Q)p Fv(\()p Fq(p)p Fv(\))p Fq(p)i Fv(and)e Fq(Q)p Fv(\()p Fq(\016)r Fv(\))p Fq(\016)r Fv(.)950 2889 y(28)p eop %%Page: 29 30 bop 187 50 2 55 v 264 32 a Fh(k)13 b Fg(=)f(0)p 440 50 V 153 w Fh(k)h Fg(=)f(1)p 694 50 V 153 w Fh(k)h Fg(=)e(2)p 948 50 V 153 w Fh(k)i Fg(=)f(3)p 1202 50 V 153 w Fh(k)h Fg(=)f(4)p 1455 50 V 153 w Fh(k)h Fg(=)e(5)p 1709 50 V 153 w Fh(k)i Fg(=)f(6)0 82 y Fh(Q)p Fg(\()p Fh(p)p Fg(\))86 89 y Fb(k)q(;)p Fe(0)140 82 y Fh(p)161 88 y Fe(0)p 187 100 V 203 82 a Fg(0)p Fh(:)p Fg(826281000)p 440 100 V 32 w(0)p Fh(:)p Fg(140876820)p 694 100 V 31 w(0)p Fh(:)p Fg(022587741)p 948 100 V 32 w(0)p Fh(:)p Fg(003156048)p 1202 100 V 32 w(0)p Fh(:)p 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Fh(g)375 1553 y Fe(5)p 410 1564 V 600 1564 V 790 1564 V 980 1564 V 1171 1564 V 1361 1564 V 1377 1547 a Fg(0)p Fh(:)p Fg(038388)p 1551 1564 V 293 1592 a(2)p Fh(g)334 1598 y Fe(0)355 1592 y Fh(g)375 1598 y Fe(6)p 410 1609 V 600 1609 V 790 1609 V 980 1609 V 1171 1609 V 1361 1609 V 1551 1609 V 1567 1592 a Fg(0)p Fh(:)p Fg(014624)314 1636 y Fh(g)334 1642 y Fe(1)355 1636 y Fh(g)375 1642 y Fe(1)p 410 1654 V 426 1636 a Fg(0)p Fh(:)p Fg(252004)p 600 1654 V 31 w(0)p Fh(:)p Fg(252004)p 790 1654 V 31 w(0)p Fh(:)p Fg(252004)p 980 1654 V 1171 1654 V 1361 1654 V 1551 1654 V 293 1681 a(2)p Fh(g)334 1687 y Fe(1)355 1681 y Fh(g)375 1687 y Fe(2)p 410 1699 V 600 1699 V 616 1681 a Fg(0)p Fh(:)p Fg(256020)p 790 1699 V 31 w(0)p Fh(:)p Fg(256020)p 980 1699 V 31 w(0)p Fh(:)p Fg(256020)p 1171 1699 V 1361 1699 V 1551 1699 V 293 1726 a(2)p Fh(g)334 1732 y Fe(1)355 1726 y Fh(g)375 1732 y Fe(3)p 410 1743 V 600 1743 V 790 1743 V 806 1726 a Fg(0)p Fh(:)p Fg(120480)p 980 1743 V 31 w(0)p Fh(:)p Fg(120480)p 1171 1743 V 32 w(0)p Fh(:)p Fg(120480)p 1361 1743 V 1551 1743 V 293 1771 a(2)p Fh(g)334 1777 y Fe(1)355 1771 y Fh(g)375 1777 y Fe(4)p 410 1788 V 600 1788 V 790 1788 V 980 1788 V 996 1771 a Fg(0)p Fh(:)p Fg(052208)p 1171 1788 V 32 w(0)p Fh(:)p Fg(052208)p 1361 1788 V 31 w(0)p Fh(:)p Fg(052208)p 1551 1788 V 293 1816 a(2)p Fh(g)334 1822 y Fe(1)355 1816 y Fh(g)375 1822 y Fe(5)p 410 1833 V 600 1833 V 790 1833 V 980 1833 V 1171 1833 V 1187 1816 a Fg(0)p Fh(:)p Fg(021084)p 1361 1833 V 31 w(0)p Fh(:)p Fg(021084)p 1551 1833 V 31 w(0)p Fh(:)p Fg(021084)293 1861 y(2)p Fh(g)334 1867 y Fe(1)355 1861 y Fh(g)375 1867 y Fe(6)p 410 1878 V 600 1878 V 790 1878 V 980 1878 V 1171 1878 V 1361 1878 V 1377 1861 a Fg(0)p Fh(:)p Fg(008032)p 1551 1878 V 31 w(0)p Fh(:)p Fg(008032)314 1905 y Fh(g)334 1911 y Fe(2)355 1905 y Fh(g)375 1911 y Fe(2)p 410 1923 V 426 1905 a Fg(0)p Fh(:)p Fg(065025)p 600 1923 V 31 w(0)p Fh(:)p Fg(065025)p 790 1923 V 31 w(0)p Fh(:)p Fg(065025)p 980 1923 V 31 w(0)p Fh(:)p Fg(065025)p 1171 1923 V 32 w(0)p Fh(:)p Fg(065025)p 1361 1923 V 1551 1923 V 293 1950 a(2)p Fh(g)334 1956 y Fe(2)355 1950 y Fh(g)375 1956 y Fe(3)p 410 1968 V 600 1968 V 616 1950 a Fg(0)p Fh(:)p Fg(061200)p 790 1968 V 31 w(0)p Fh(:)p Fg(061200)p 980 1968 V 31 w(0)p Fh(:)p Fg(061200)p 1171 1968 V 32 w(0)p Fh(:)p Fg(061200)p 1361 1968 V 31 w(0)p Fh(:)p Fg(061200)p 1551 1968 V 293 1995 a(2)p Fh(g)334 2001 y Fe(2)355 1995 y Fh(g)375 2001 y Fe(4)p 410 2012 V 600 2012 V 790 2012 V 806 1995 a Fg(0)p Fh(:)p Fg(026520)p 980 2012 V 31 w(0)p Fh(:)p Fg(026520)p 1171 2012 V 32 w(0)p Fh(:)p Fg(026520)p 1361 2012 V 31 w(0)p Fh(:)p Fg(026520)p 1551 2012 V 31 w(0)p Fh(:)p Fg(026520)293 2040 y(2)p Fh(g)334 2046 y Fe(2)355 2040 y Fh(g)375 2046 y Fe(5)p 410 2057 V 600 2057 V 790 2057 V 980 2057 V 996 2040 a Fg(0)p Fh(:)p Fg(010710)p 1171 2057 V 32 w(0)p Fh(:)p Fg(010710)p 1361 2057 V 31 w(0)p Fh(:)p Fg(010710)p 1551 2057 V 31 w(0)p Fh(:)p Fg(010710)293 2085 y(2)p Fh(g)334 2091 y Fe(2)355 2085 y Fh(g)375 2091 y Fe(6)p 410 2102 V 600 2102 V 790 2102 V 980 2102 V 1171 2102 V 1187 2085 a Fg(0)p Fh(:)p Fg(004080)p 1361 2102 V 31 w(0)p Fh(:)p Fg(004080)p 1551 2102 V 31 w(0)p Fh(:)p Fg(004080)314 2130 y Fh(g)334 2136 y Fe(3)355 2130 y Fh(g)375 2136 y Fe(3)p 410 2147 V 426 2130 a Fg(0)p Fh(:)p Fg(014400)p 600 2147 V 31 w(0)p Fh(:)p Fg(014400)p 790 2147 V 31 w(0)p Fh(:)p Fg(014400)p 980 2147 V 31 w(0)p Fh(:)p Fg(014400)p 1171 2147 V 32 w(0)p Fh(:)p Fg(014400)p 1361 2147 V 31 w(0)p Fh(:)p Fg(014400)p 1551 2147 V 31 w(0)p Fh(:)p Fg(014400)293 2174 y(2)p Fh(g)334 2180 y Fe(3)355 2174 y Fh(g)375 2180 y Fe(4)p 410 2192 V 600 2192 V 616 2174 a Fg(0)p Fh(:)p Fg(012480)p 790 2192 V 31 w(0)p Fh(:)p Fg(012480)p 980 2192 V 31 w(0)p Fh(:)p Fg(012480)p 1171 2192 V 32 w(0)p Fh(:)p Fg(012480)p 1361 2192 V 31 w(0)p Fh(:)p Fg(012480)p 1551 2192 V 31 w(0)p Fh(:)p Fg(012480)293 2219 y(2)p Fh(g)334 2225 y Fe(3)355 2219 y Fh(g)375 2225 y Fe(5)p 410 2237 V 600 2237 V 790 2237 V 806 2219 a Fg(0)p Fh(:)p Fg(005040)p 980 2237 V 31 w(0)p Fh(:)p Fg(005040)p 1171 2237 V 32 w(0)p Fh(:)p Fg(005040)p 1361 2237 V 31 w(0)p Fh(:)p Fg(005040)p 1551 2237 V 31 w(0)p Fh(:)p Fg(005040)293 2264 y(2)p Fh(g)334 2270 y Fe(3)355 2264 y Fh(g)375 2270 y Fe(6)p 410 2281 V 600 2281 V 790 2281 V 980 2281 V 996 2264 a Fg(0)p Fh(:)p Fg(001920)p 1171 2281 V 32 w(0)p Fh(:)p Fg(001920)p 1361 2281 V 31 w(0)p Fh(:)p Fg(001920)p 1551 2281 V 31 w(0)p Fh(:)p Fg(001920)314 2309 y Fh(g)334 2315 y Fe(4)355 2309 y Fh(g)375 2315 y Fe(4)p 410 2326 V 426 2309 a Fg(0)p Fh(:)p Fg(002704)p 600 2326 V 31 w(0)p Fh(:)p Fg(002704)p 790 2326 V 31 w(0)p Fh(:)p Fg(002704)p 980 2326 V 31 w(0)p Fh(:)p Fg(002704)p 1171 2326 V 32 w(0)p Fh(:)p Fg(002704)p 1361 2326 V 31 w(0)p Fh(:)p Fg(002704)p 1551 2326 V 31 w(0)p Fh(:)p Fg(002704)293 2354 y(2)p Fh(g)334 2360 y Fe(4)355 2354 y Fh(g)375 2360 y Fe(5)p 410 2371 V 600 2371 V 616 2354 a Fg(0)p Fh(:)p Fg(002184)p 790 2371 V 31 w(0)p Fh(:)p Fg(002184)p 980 2371 V 31 w(0)p Fh(:)p Fg(002184)p 1171 2371 V 32 w(0)p Fh(:)p Fg(002184)p 1361 2371 V 31 w(0)p Fh(:)p Fg(002184)p 1551 2371 V 31 w(0)p Fh(:)p Fg(002184)293 2399 y(2)p Fh(g)334 2405 y Fe(4)355 2399 y Fh(g)375 2405 y Fe(6)p 410 2416 V 600 2416 V 790 2416 V 806 2399 a Fg(0)p Fh(:)p Fg(000832)p 980 2416 V 31 w(0)p Fh(:)p Fg(000832)p 1171 2416 V 32 w(0)p Fh(:)p Fg(000832)p 1361 2416 V 31 w(0)p Fh(:)p Fg(000832)p 1551 2416 V 31 w(0)p Fh(:)p Fg(000832)314 2443 y Fh(g)334 2449 y Fe(5)355 2443 y Fh(g)375 2449 y Fe(5)p 410 2461 V 426 2443 a Fg(0)p Fh(:)p Fg(000441)p 600 2461 V 31 w(0)p Fh(:)p Fg(000441)p 790 2461 V 31 w(0)p Fh(:)p Fg(000441)p 980 2461 V 31 w(0)p Fh(:)p Fg(000441)p 1171 2461 V 32 w(0)p Fh(:)p Fg(000441)p 1361 2461 V 31 w(0)p Fh(:)p Fg(000441)p 1551 2461 V 31 w(0)p Fh(:)p Fg(000441)293 2488 y(2)p Fh(g)334 2494 y Fe(5)355 2488 y Fh(g)375 2494 y Fe(6)p 410 2506 V 600 2506 V 616 2488 a Fg(0)p Fh(:)p Fg(000336)p 790 2506 V 31 w(0)p Fh(:)p Fg(000336)p 980 2506 V 31 w(0)p Fh(:)p Fg(000336)p 1171 2506 V 32 w(0)p Fh(:)p Fg(000336)p 1361 2506 V 31 w(0)p Fh(:)p Fg(000336)p 1551 2506 V 31 w(0)p Fh(:)p Fg(000336)314 2533 y Fh(g)334 2539 y Fe(6)355 2533 y Fh(g)375 2539 y Fe(6)p 410 2550 V 426 2533 a Fg(0)p Fh(:)p Fg(000064)p 600 2550 V 31 w(0)p Fh(:)p Fg(000064)p 790 2550 V 31 w(0)p Fh(:)p Fg(000064)p 980 2550 V 31 w(0)p Fh(:)p Fg(000064)p 1171 2550 V 32 w(0)p Fh(:)p Fg(000064)p 1361 2550 V 31 w(0)p Fh(:)p Fg(000064)p 1551 2550 V 31 w(0)p Fh(:)p Fg(000064)264 2578 y(2000)p Fh(u)373 2585 y Fb(k)p 410 2595 V 426 2578 a Fg(1)p Fh(:)p Fg(170034)p 600 2595 V 31 w(1)p Fh(:)p Fg(584514)p 790 2595 V 31 w(1)p Fh(:)p Fg(285870)p 980 2595 V 31 w(0)p Fh(:)p Fg(851924)p 1171 2595 V 32 w(0)p Fh(:)p Fg(496764)p 1361 2595 V 31 w(0)p Fh(:)p Fg(262623)p 1551 2595 V 31 w(0)p Fh(:)p Fg(125451)675 2665 y Fc(T)l(able)32 b(11.)j Fg(The)13 b(v)o(ector)h(2000)p Fh(u)p Fg(.)950 2889 y Fv(29)p eop %%Page: 30 31 bop 100 50 a Fv(In)16 b(order)g(to)i(b)q(ound)e(the)h(v)o(ector)g(10)p Fq(\032)821 32 y Fu(7)843 50 y Fq(v)r Fv(,)g(w)o(e)g(use)g(again)f(the)i (comp)q(onen)o(ts)d(of)j Fq(g)g Fv(from)f(T)l(able)f(1,)0 117 y(together)h(with)f(the)h(b)q(ounds)e(on)h Fq(A)675 91 y Fu(\()p Fl(m)p Fu(\))675 132 y Fl(k)q(;)p Fu(7)761 117 y Fv(from)g(T)l(able)g(2.)195 285 y Fg(10)p Fh(\032)259 271 y Fe(7)279 285 y Fh(v)300 292 y Fb(k)353 285 y Fg(=)29 b Fh(A)447 263 y Fe(\(1\))447 300 y Fb(k)q(;)p Fe(7)500 285 y Fh(g)520 291 y Fe(1)572 285 y Fg(+)10 b Fh(A)647 263 y Fe(\(2\))647 300 y Fb(k)q(;)p Fe(7)700 285 y Fh(g)720 291 y Fe(2)765 285 y Fg(+)g Fh(A)840 263 y Fe(\(3\))840 300 y Fb(k)q(;)p Fe(7)892 285 y Fh(g)912 291 y Fe(3)958 285 y Fg(+)g Fh(A)1033 263 y Fe(\(4\))1033 300 y Fb(k)q(;)p Fe(7)1085 285 y Fh(g)1105 291 y Fe(4)1151 285 y Fg(+)g Fh(A)1226 263 y Fe(\(5\))1226 300 y Fb(k)q(;)p Fe(7)1278 285 y Fh(g)1298 291 y Fe(5)1344 285 y Fg(+)17 b Fh(A)1426 263 y Fe(\(6\))1426 300 y Fb(k)q(;)p Fe(7)1478 285 y Fh(g)1498 291 y Fe(6)195 359 y Fg(10)p Fh(\032)259 345 y Fe(7)279 359 y Fh(v)300 365 y Fe(1)353 359 y Ff(\024)991 b Fg(0)p Fh(:)p Fg(000024)15 b Ff(\024)c Fg(0)p Fh(:)p Fg(000024)195 404 y(10)p Fh(\032)259 389 y Fe(7)279 404 y Fh(v)300 410 y Fe(2)353 404 y Ff(\024)798 b Fg(0)p Fh(:)p Fg(000210)q(+0)p Fh(:)p Fg(000184)15 b Ff(\024)c Fg(0)p Fh(:)p Fg(000394)195 448 y(10)p Fh(\032)259 434 y Fe(7)279 448 y Fh(v)300 454 y Fe(3)353 448 y Ff(\024)605 b Fg(0)p Fh(:)p Fg(000884)q(+0)p Fh(:)p Fg(001260)q(+0)p Fh(:)p Fg(000776)15 b Ff(\024)c Fg(0)p Fh(:)p Fg(002920)195 493 y(10)p Fh(\032)259 479 y Fe(7)279 493 y Fh(v)300 499 y Fe(4)353 493 y Ff(\024)412 b Fg(0)p Fh(:)p Fg(002040)q(+0)p Fh(:)p Fg(004212)q(+0)p Fh(:)p Fg(003969)q(+0)p Fh(:)p Fg(002168)15 b Ff(\024)c Fg(0)p Fh(:)p Fg(012389)195 538 y(10)p Fh(\032)259 524 y Fe(7)279 538 y Fh(v)300 544 y Fe(5)353 538 y Ff(\024)219 b Fg(0)p Fh(:)p Fg(002550)q(+0)p Fh(:)p Fg(007200)q(+0)p Fh(:)p Fg(009828)q(+0)p Fh(:)p Fg(008043)q(+0)p Fh(:)p Fg(004320)15 b Ff(\024)c Fg(0)p Fh(:)p Fg(031941)195 583 y(10)p Fh(\032)259 569 y Fe(7)279 583 y Fh(v)300 589 y Fe(6)353 583 y Ff(\024)j Fg(0)p Fh(:)p Fg(001506)f(+0)p Fh(:)p Fg(005865)q(+0)p Fh(:)p Fg(011640)q(+0)p Fh(:)p Fg(014092)q(+0)p Fh(:)p Fg(011340)q(+0)p Fh(:)p Fg(006360)i Ff(\024)c Fg(0)p Fh(:)p Fg(050803)586 658 y Fc(T)l(able)32 b(12.)j Fg(Bound)13 b(on)h(the)g(v)o(ector)g(10)p Fh(\032)1291 644 y Fe(7)1312 658 y Fh(v)q Fg(.)0 792 y Fv(Next,)24 b(w)o(e)d(consider)f(the)h(v)o(ector)h Fq(\025)f Fs(\021)h(j)p Fq(Q)p Fv(\()p Fq(p)p Fv(\))p Fq(p)15 b Fs(\000)f Fq(p)p Fs(j)h Fv(+)f Fq(Q)p Fv(\()p Fq(\016)r Fv(\))p Fq(\016)j Fv(+)d Fq(u)g Fv(+)g(2)p Fq(v)i Fv(+)e Fq(w)q Fv(.)36 b(The)21 b(comp)q(onen)o(ts)0 862 y Fq(w)36 869 y Fl(k)78 862 y Fv(=)d(0)p Fq(:)p Fv(01)p Fq(\032)250 843 y Fp(\000)p Fu(7)303 862 y Fq(\032)329 843 y Fp(\000)p Fu(7)382 862 y Fq(A)419 836 y Fu(\(7\))419 877 y Fl(k)q(;)p Fu(7)495 862 y Fv(of)h Fq(w)h Fv(are)f(easy)g(to)g(b)q(ound)f(using)g(T)l(able)h(1)g(and)f(T)l(able)h(2;)h (only)f(t)o(w)o(o)f(digit)0 921 y(precision)e(is)g(needed.)24 b(W)l(e)17 b(also)g(ha)o(v)o(e)f(to)i(compute)e Fs(j)p Fq(Q)p Fv(\()p Fq(p)p Fv(\))p Fq(p)d Fs(\000)e Fq(p)p Fs(j)d Fv(,)p Fq(u)p Fv(,)17 b(and)f(2)p Fq(v)r Fv(,)h(using)f(the)i(data)f(from)0 981 y(T)l(able)k(1)h(and)f(T)l(ables)g(10-12.)37 b(The)22 b(v)o(ector)g Fq(Q)p Fv(\()p Fq(\016)r Fv(\))p Fq(\016)j Fv(can)c(b)q(e)h(found)f(in)h(T)l (able)f(10.)38 b(Putting)21 b(it)h(all)0 1041 y(together,)16 b(w)o(e)h(obtain)f(the)h(follo)o(wing.)217 1195 y Fh(\025)242 1202 y Fb(k)293 1195 y Ff(\021)j(j)p Fh(Q)p Fg(\()p Fh(p)p Fg(\))p Fh(p)11 b Ff(\000)e Fh(p)p Ff(j)551 1202 y Fb(k)582 1195 y Fg(+)19 b(\()p Fh(Q)p Fg(\()p Fh(\016)q Fg(\))p Fh(\016)q Fg(\))771 1202 y Fb(k)815 1195 y Fg(+)77 b Fh(u)950 1202 y Fb(k)1048 1195 y Fg(+)68 b(2)p Fh(v)1191 1202 y Fb(k)1281 1195 y Fg(+)73 b Fh(w)1417 1202 y Fb(k)217 1255 y Fh(\025)242 1261 y Fe(0)293 1255 y Ff(\024)54 b Fg(0)p Fh(:)p Fg(000179)43 b(+)14 b(0)p Fh(:)p Fg(000061)27 b(+)14 b(0)p Fh(:)p Fg(000586)27 b(+)14 b(0)p Fh(:)p Fg(000000)27 b(+)14 b(0)p Fh(:)p Fg(000001)27 b(=)f(0)p Fh(:)p Fg(000827)217 1300 y Fh(\025)242 1306 y Fe(1)293 1300 y Ff(\024)54 b Fg(0)p Fh(:)p Fg(000011)43 b(+)14 b(0)p Fh(:)p Fg(000135)27 b(+)14 b(0)p Fh(:)p Fg(000793)27 b(+)14 b(0)p Fh(:)p Fg(000002)27 b(+)14 b(0)p Fh(:)p Fg(000001)27 b(=)f(0)p Fh(:)p Fg(000942)217 1345 y Fh(\025)242 1351 y Fe(2)293 1345 y Ff(\024)54 b Fg(0)p Fh(:)p Fg(000062)43 b(+)14 b(0)p Fh(:)p Fg(000178)27 b(+)14 b(0)p Fh(:)p Fg(000643)27 b(+)14 b(0)p Fh(:)p Fg(000006)27 b(+)14 b(0)p Fh(:)p Fg(000001)27 b(=)f(0)p Fh(:)p Fg(000890)217 1390 y Fh(\025)242 1396 y Fe(3)293 1390 y Ff(\024)54 b Fg(0)p Fh(:)p Fg(000207)43 b(+)14 b(0)p Fh(:)p Fg(000235)27 b(+)14 b(0)p Fh(:)p Fg(000426)27 b(+)14 b(0)p Fh(:)p Fg(000034)27 b(+)14 b(0)p Fh(:)p Fg(000003)27 b(=)f(0)p Fh(:)p Fg(000905)217 1434 y Fh(\025)242 1440 y Fe(4)293 1434 y Ff(\024)54 b Fg(0)p Fh(:)p Fg(000394)43 b(+)14 b(0)p Fh(:)p Fg(000256)27 b(+)14 b(0)p Fh(:)p Fg(000249)27 b(+)14 b(0)p Fh(:)p Fg(000138)27 b(+)14 b(0)p Fh(:)p Fg(000008)27 b(=)f(0)p Fh(:)p Fg(001045)217 1479 y Fh(\025)242 1485 y Fe(5)293 1479 y Ff(\024)54 b Fg(0)p Fh(:)p Fg(000091)43 b(+)14 b(0)p Fh(:)p Fg(000217)27 b(+)14 b(0)p Fh(:)p Fg(000132)27 b(+)14 b(0)p Fh(:)p Fg(000352)27 b(+)14 b(0)p Fh(:)p Fg(000017)27 b(=)f(0)p Fh(:)p Fg(000809)217 1524 y Fh(\025)242 1530 y Fe(6)293 1524 y Ff(\024)54 b Fg(0)p Fh(:)p Fg(000068)43 b(+)14 b(0)p Fh(:)p Fg(000144)27 b(+)14 b(0)p Fh(:)p Fg(000063)27 b(+)14 b(0)p Fh(:)p Fg(000560)27 b(+)14 b(0)p Fh(:)p Fg(000027)27 b(=)f(0)p Fh(:)p Fg(000862)627 1609 y Fc(T)l(able)32 b(13.)j Fg(Bound)13 b(on)h(the)g(v)o(ector)g Fh(\025)p Fg(.)0 1743 y Fv(The)j(pro)q(duct)g(of)h(the)g(matrix)f Fs(j)p Fq(M)5 b Fs(j)18 b Fv(with)g(the)f(ab)q(o)o(v)o(e)h(b)q(ound)e(on)i Fq(\025)26 b Fv(\(m)o(ultiplied)16 b(b)o(y)h(1000\))g(is)h(giv)o(en)0 1803 y(in)e(the)h(next)g(table.)p 442 1977 2 55 v 487 1960 a Fh(k)c Fg(=)f(0)p 632 1977 V 89 w Fh(k)h Fg(=)f(1)p 822 1977 V 90 w Fh(k)g Fg(=)g(2)p 1012 1977 V 90 w Fh(k)h Fg(=)e(3)p 1202 1977 V 90 w Fh(k)i Fg(=)f(4)p 1392 1977 V 89 w Fh(k)h Fg(=)f(5)p 1583 1977 V 89 w Fh(k)h Fg(=)f(6)191 2010 y(1000)p Ff(j)p Fh(M)t Ff(j)344 2017 y Fb(k)q(;)p Fe(0)397 2010 y Fh(\025)422 2016 y Fe(0)p 442 2027 V 458 2010 a Fg(0)p Fh(:)p Fg(870831)p 632 2027 V 31 w(0)p Fh(:)p Fg(491238)p 822 2027 V 31 w(0)p Fh(:)p Fg(635136)p 1012 2027 V 31 w(0)p Fh(:)p Fg(822038)p 1202 2027 V 31 w(0)p Fh(:)p Fg(568976)p 1392 2027 V 32 w(0)p Fh(:)p Fg(292758)p 1583 2027 V 31 w(0)p Fh(:)p Fg(121569)191 2060 y(1000)p Ff(j)p Fh(M)t Ff(j)344 2067 y Fb(k)q(;)p Fe(1)397 2060 y Fh(\025)422 2066 y Fe(1)p 442 2077 V 458 2060 a Fg(0)p Fh(:)p Fg(559548)p 632 2077 V 31 w(1)p Fh(:)p Fg(852914)p 822 2077 V 31 w(1)p Fh(:)p Fg(965012)p 1012 2077 V 31 w(2)p Fh(:)p Fg(498184)p 1202 2077 V 31 w(1)p Fh(:)p Fg(730454)p 1392 2077 V 32 w(0)p Fh(:)p Fg(891132)p 1583 2077 V 31 w(0)p Fh(:)p Fg(370206)191 2109 y(1000)p Ff(j)p Fh(M)t Ff(j)344 2116 y Fb(k)q(;)p Fe(2)397 2109 y Fh(\025)422 2115 y Fe(2)p 442 2127 V 458 2109 a Fg(0)p Fh(:)p Fg(683520)p 632 2127 V 31 w(1)p Fh(:)p Fg(856540)p 822 2127 V 31 w(0)p Fh(:)p Fg(355110)p 1012 2127 V 31 w(0)p Fh(:)p Fg(147740)p 1202 2127 V 31 w(0)p Fh(:)p Fg(070310)p 1392 2127 V 32 w(0)p Fh(:)p Fg(035600)p 1583 2127 V 31 w(0)p Fh(:)p Fg(015130)191 2159 y(1000)p Ff(j)p Fh(M)t Ff(j)344 2166 y Fb(k)q(;)p Fe(3)397 2159 y Fh(\025)422 2165 y Fe(3)p 442 2177 V 458 2159 a Fg(0)p Fh(:)p Fg(899570)p 632 2177 V 31 w(2)p Fh(:)p Fg(400060)p 822 2177 V 31 w(0)p Fh(:)p Fg(150230)p 1012 2177 V 31 w(2)p Fh(:)p Fg(144850)p 1202 2177 V 31 w(1)p Fh(:)p Fg(086905)p 1392 2177 V 32 w(0)p Fh(:)p Fg(587345)p 1583 2177 V 31 w(0)p Fh(:)p Fg(245255)191 2209 y(1000)p Ff(j)p Fh(M)t Ff(j)344 2216 y Fb(k)q(;)p Fe(4)397 2209 y Fh(\025)422 2215 y Fe(4)p 442 2226 V 458 2209 a Fg(0)p Fh(:)p Fg(718960)p 632 2226 V 31 w(1)p Fh(:)p Fg(919665)p 822 2226 V 31 w(0)p Fh(:)p Fg(082555)p 1012 2226 V 31 w(1)p Fh(:)p Fg(255045)p 1202 2226 V 31 w(2)p Fh(:)p Fg(433805)p 1392 2226 V 32 w(0)p Fh(:)p Fg(861080)p 1583 2226 V 31 w(0)p Fh(:)p Fg(382470)191 2259 y(1000)p Ff(j)p Fh(M)t Ff(j)344 2266 y Fb(k)q(;)p Fe(5)397 2259 y Fh(\025)422 2265 y Fe(5)p 442 2276 V 458 2259 a Fg(0)p Fh(:)p Fg(286386)p 632 2276 V 31 w(0)p Fh(:)p Fg(765314)p 822 2276 V 31 w(0)p Fh(:)p Fg(032360)p 1012 2276 V 31 w(0)p Fh(:)p Fg(525041)p 1202 2276 V 31 w(0)p Fh(:)p Fg(666616)p 1392 2276 V 32 w(1)p Fh(:)p Fg(335659)p 1583 2276 V 31 w(0)p Fh(:)p Fg(283150)191 2309 y(1000)p Ff(j)p Fh(M)t Ff(j)344 2316 y Fb(k)q(;)p Fe(6)397 2309 y Fh(\025)422 2315 y Fe(6)p 442 2326 V 458 2309 a Fg(0)p Fh(:)p Fg(126714)p 632 2326 V 31 w(0)p Fh(:)p Fg(338766)p 822 2326 V 31 w(0)p Fh(:)p Fg(014654)p 1012 2326 V 31 w(0)p Fh(:)p Fg(233602)p 1202 2326 V 31 w(0)p Fh(:)p Fg(315492)p 1392 2326 V 32 w(0)p Fh(:)p Fg(301700)p 1583 2326 V 31 w(1)p Fh(:)p Fg(084396)191 2358 y(1000\()p Ff(j)p Fh(M)t Ff(j)p Fh(\025)p Fg(\))401 2365 y Fb(k)p 442 2376 V 458 2358 a Fg(4)p Fh(:)p Fg(145529)p 632 2376 V 31 w(9)p Fh(:)p Fg(624497)p 822 2376 V 31 w(3)p Fh(:)p Fg(235057)p 1012 2376 V 31 w(7)p Fh(:)p Fg(626500)p 1202 2376 V 31 w(6)p Fh(:)p Fg(872558)p 1392 2376 V 32 w(4)p Fh(:)p Fg(305274)p 1583 2376 V 31 w(2)p Fh(:)p Fg(502176)534 2446 y Fc(T)l(able)32 b(14.)k Fg(Bound)13 b(on)h(the)g(pro)q(duct)e(1000)p Ff(j)p Fh(M)t Ff(j)p Fh(\025)p Fg(.)0 2580 y Fv(By)20 b(using)f(these)g(b)q(ounds,)g (and)g(the)h(v)m(alues)g(of)g Fq(\016)959 2587 y Fl(k)1011 2580 y Fv(from)f(T)l(able)g(1,)i(it)f(is)f(no)o(w)g(easy)g(to)h(c)o(hec)o(k)g (that)0 2640 y(\()p Fs(j)p Fq(M)5 b Fs(j)p Fq(\025)p Fv(\))148 2647 y Fl(k)185 2640 y Fv(+)11 b(3)g Fs(\002)f Fv(10)370 2622 y Fp(\000)p Fu(5)438 2640 y Fs(\024)j Fq(\016)512 2647 y Fl(k)545 2640 y Fv(,)k(for)f(0)e Fs(\024)f Fq(k)i Fs(\024)f Fv(6.)22 b(This)16 b(completes)g(the)g(pro)q(of)h(of)f(\(N5\).)950 2839 y(30)p eop %%Page: 31 32 bop 0 50 a Fk(References)22 169 y Fv([1])25 b(F.J.)11 b(Dyson,)h Fa(Existenc)m(e)i(of)h(a)f(Phase)h(T)l(r)m(ansition)h(in)e(a)g (One-Dimensional)h(Ising)g(F)l(err)m(omagnet.)100 229 y Fv(Comm)o(un.)k (Math.)j(Ph)o(ys.)f Fk(12)p Fv(,)16 b(91)g(\(1969\).)22 289 y([2])25 b(G.A.)16 b(Bak)o(er,)h Fa(Ising)h(Mo)m(del)h(with)f(a)g(Sc)m(aling) h(Inter)m(action.)k Fv(Ph)o(ys.)e(Rev.)h Fk(B5)p Fv(,)16 b(2622)g(\(1972\).) 22 349 y([3])25 b(P)l(.M.)e(Bleher,)j(Y)l(a.G.)e(Sinai,)h Fa(Critic)m(al)g (Indic)m(es)h(for)g(Dyson)-5 b('s)26 b(Asymptotic)m(al)s(ly)f(Hier)m(ar)m (chic)m(al)100 408 y(Mo)m(dels.)e Fv(Comm)o(un.)d(Math.)i(Ph)o(ys.)f Fk(45)p Fv(,)15 b(347)i(\(1975\).)22 468 y([4])25 b(P)l(.)h(Collet,)i(J.-P)l (.)d(Ec)o(kmann,)j Fa(A)e(R)m(enormalization)k(Gr)m(oup)d(A)o(nalysis)h(of)f (the)g(Hier)m(ar)m(chic)m(al)100 528 y(Mo)m(del)14 b(in)h(Statistic)m(al)f (Me)m(chanics.)23 b Fv(Lecture)12 b(Notes)h(in)f(Ph)o(ysics,)g(V)l(olume)g Fk(74)p Fv(,)h(Springer{V)l(erlag)100 588 y(\(1978\).)22 648 y([5])25 b(G.)11 b(Galla)o(v)o(otti,)i Fa(Some)g(Asp)m(e)m(cts)i(of)f(the)g (R)m(enormalization)i(Pr)m(oblems)e(in)f(Statistic)m(al)h(Me)m(chanics.)100 707 y Fv(Memorie)h(dell')h(Accademia)g(dei)g(Lincei)h Fk(15)p Fv(,)e(23)i(\(1978\).)22 767 y([6])25 b(H.)15 b(Ko)q(c)o(h,)g(P)l(.)g(Witt)o (w)o(er,)h Fa(A)g(Non{Gaussian)j(R)m(enormalization)h(Gr)m(oup)d(Fixe)m(d)h (Point)e(for)i(Hier-)100 827 y(ar)m(chic)m(al)i(Sc)m(alar)e(L)m(attic)m(e)h (Field)f(The)m(ories.)24 b Fv(Comm)o(un.)c(Math.)h(Ph)o(ys.)g Fk(106)p Fv(,)16 b(495)g(\(1986\).)22 887 y([7])25 b(H.)17 b(Ko)q(c)o(h,)f(P)l(.)g(Witt)o(w)o(er,)g Fa(Computing)i(Bounds)h(on)f(Critic) m(al)g(Indic)m(es.)25 b Fv(In:)c(Non{Linear)16 b(Ev)o(olu-)100 946 y(tion)h(and)g(Chaotic)h(Phenomena.)24 b(G.)18 b(Galla)o(v)o(otti,)g(P)l (.)f(Zw)o(eifel)h(\(eds.\).)26 b(NA)l(TO)18 b(ASI)g(Series)f(B:)100 1006 y(Ph)o(ysics)e(V)l(olume)h Fk(176)p Fv(,)f(Plen)o(um)g(Press)h (\(1988\).)22 1066 y([8])25 b(T.)c(Balaban,)i Fa(Ultr)m(aviolet)f(Stability)h (in)g(Field)g(The)m(ory.)38 b(The)23 b Fq(\036)1375 1048 y Fu(4)1375 1078 y(3)1397 1066 y Fa({mo)m(del.)40 b Fv(In:)32 b(Scaling)21 b(and)100 1126 y(Self{Similarit)o(y)14 b(in)i(Ph)o(ysics.)21 b(J)q(\177)-26 b(urg)15 b(F)l(r\177)-25 b(ohlic)o(h)15 b(\(ed.\).)22 b(Birkh\177)-25 b(auser,)15 b(Boston)h(\(1983\).)22 1186 y([9])25 b(K.)18 b(Ga)o(w\030)-22 b(edzki,)18 b(A.)g(Kupiainen,)f Fa(R)o(igor)m(ous)k (R)m(enormalization)h(Gr)m(oup)e(and)h(Asymptotic)f(F)l(r)m(e)m(e-)100 1245 y(dom.)38 b Fv(In:)32 b(Scaling)20 b(and)h(Self{Similarit)o(y)f(in)h(Ph) o(ysics.)36 b(J)q(\177)-26 b(urg)20 b(F)l(r\177)-25 b(ohlic)o(h)19 b(\(ed.\).)38 b(Birkh\177)-25 b(auser,)100 1305 y(Boston)16 b(\(1983\).)-3 1365 y([10])25 b(See)16 b(e.g.)22 b(Theorem)16 b(\(18.2a\))g(in:)100 1425 y(M.)h(Marden,)f Fa(Ge)m(ometry)j(of)g (Polynomials.)27 b Fv(Mathematical)17 b(Surv)o(eys,)g(Num)o(b)q(er)f Fk(3)p Fv(,)i(American)100 1484 y(Mathematical)d(So)q(ciet)o(y)i(\(1966\).)-3 1544 y([11])25 b(See)16 b(e.g.)22 b(Theorems)15 b(\(4.1.8\))i(and)f (\(4.2.1\))g(in:)100 1604 y(R.P)l(.)f(Boas,)i Fa(Entir)m(e)g(F)l(unctions.)23 b Fv(Academic)16 b(Press)g(\(1954\).)-3 1664 y([12])25 b(See)16 b(e.g.)22 b(Theorem)16 b(\(5.1\))h(in:)100 1724 y(M.W.)c(Hirsc)o(h,)g(C.C.)g (Pugh,)h(M.)f(Sh)o(ub,)f Fa(Invariant)k(Manifolds.)24 b Fv(Lecture)13 b(Notes)i(in)e(Mathemat-)100 1783 y(ics,)j(V)l(olume)g Fk(583)p Fv(,)f(Springer{V)l(erlag)f(\(1977\).)-3 1843 y([13])25 b(See)16 b(e.g.)22 b(Theorem)16 b(\(IX.9\))h(in:)100 1903 y(M.)12 b(Reed,)h(B.)g (Simon,)f Fa(Metho)m(ds)k(of)f(Mo)m(dern)h(Mathematic)m(al)f(Physics,)h(I:)f (F)l(unctional)g(A)o(nalysis.)100 1963 y Fv(Academic)h(Press)f(\(1980\).)950 2790 y(31)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF