The body of this paper (3375 lines of Postscript), from the line following "BODY" to the line preceding "ENDBODY", contains a total of 230320 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). 32071 lowercase letters 32591 uppercase letters 122667 digits 2 ASCII characters 9 3375 ASCII characters 10 13768 ASCII characters 32 40 ASCII characters 33 ! 39 ASCII characters 34 " 3 ASCII characters 35 # 89 ASCII characters 37 % 4 ASCII characters 39 ' 10274 ASCII characters 40 ( 10279 ASCII characters 41 ) 1 ASCII characters 42 * 61 ASCII characters 43 + 293 ASCII characters 44 , 137 ASCII characters 45 - 339 ASCII characters 46 . 134 ASCII characters 47 / 150 ASCII characters 58 : 274 ASCII characters 59 ; 481 ASCII characters 60 < 158 ASCII characters 61 = 502 ASCII characters 62 > 39 ASCII characters 64 @ 47 ASCII characters 91 [ 2247 ASCII characters 92 \ 46 ASCII characters 93 ] 4 ASCII characters 95 _ 51 ASCII characters 96 ` 87 ASCII characters 123 { 67 ASCII characters 125 } For further information, please contact the author: koch@math.utexas.edu BODY %!PS-Adobe-2.0 %%Creator: dvips 5.47 Copyright 1986-91 Radical Eye Software %%Pages: 14 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 { div /sf X /fntrx [ sf 0 0 sf neg 0 0 ] N df-tail } B /df-tail % id numcc maxcc df-tail -- initialize a new font dictionary { /nn 8 dict N % allocate new font dictionary nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar {CharBuilder} N /Encoding IE N end dup { /foo setfont } % dummy macro to be filled in 2 array copy cvx N % have to allocate a new one load % now we change it 0 nn put /ctr 0 N % go, count, and etc. [ % start next char definition } B /E { pop nn dup definefont setfont } B /ch-image {ch-data dup type /stringtype ne { ctr get /ctr ctr 1 add N } if } B % the hex string image, or array of same /ch-width {ch-data dup length 5 sub get} B % the number of pixels across /ch-height {ch-data dup length 4 sub get} B % the number of pixels tall /ch-xoff {128 ch-data dup length 3 sub get sub} B % num pixels right of origin /ch-yoff {ch-data dup length 2 sub get 127 sub} B % number of pixels below origin /ch-dx {ch-data dup length 1 sub get} B % number of pixels to next character /ctr 0 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-- 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 M } B /l { p -4 w } B /m { p -3 w } B /n { p -2 w } B 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y(are)c(giv)o(en)g(appro)o (ximately)f(b)o(y)g(the)i(eigen)o(v)m(alues)e(of)i Fl(H)1061 1899 y Fj(\()p Fn(1)p Fj(\))1057 1931 y Fk(osc)1135 1925 y Fo(.)24 b(These)17 b(eigen)o(v)m(alues)f(can)h(b)q(e)h(computed)0 1992 y(explicitly)l(,)e(since)f Fl(H)391 1966 y Fj(\()p Fn(1)p Fj(\))387 1997 y Fk(osc)474 1992 y Fo(+)528 1972 y Fj(1)p 528 1980 20 2 v 528 2009 a(2)554 1992 y Fo(tr\()p Fl(B)652 1974 y Fj(\()p Fn(1)p Fj(\))726 1992 y Fo(\))h(is)f(the)h(Hamiltonian)f(of)h Fl(d)f Fo(uncoupled)g(harmonic)e(oscillators)0 2058 y(whose)i(frequencies)g (are)g(the)h(eigen)o(v)m(alues)e(of)i(the)f(matrix)g Fl(B)1164 2040 y Fj(\()p Fn(1)p Fj(\))1254 2058 y Fo(\(whic)o(h)f(b)o(y)h(\(A3\))i(are) e(all)g(p)q(ositiv)o(e\).)0 2118 y(F)l(or)j(more)g(details,)h(including)e (the)i(pro)q(ofs)g(of)38 b(Theorem)18 b(1.1)g(and)38 b(Prop)q(osition)17 b(1.2,)i(w)o(e)g(refer)g(to)0 2178 y(Section)d(2.)0 2261 y Fe(Remarks.)30 2333 y Fo(1\))26 b(What)16 b(w)o(e)f(call)h(\\mean)f(\014eld)h 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Fg(X)1351 2675 y Fk(i)p Fn(2B)1418 2680 y Ff(N)q(;k)1488 2615 y Fl(S)1519 2622 y Fk(i)1535 2615 y Fo(\()p Fl(n)p Fo(\))8 b Fl(:)198 b Fo(\(1)p Fl(:)p Fo(19\))963 2790 y(4)p eop %%Page: 5 6 bop 100 50 a Fl(Z)138 32 y Fj(\()p Fk(N)t Fj(\))229 50 y Fo(represen)o(ts)22 b(the)h(\015uctuation)g(\(or)g(\014nite)g(size)g(correction\))g(of)h(the)f (mean)g(\014eld)f Fl(M)1866 32 y Fj(\()p Fk(N)t Fj(\))1936 50 y Fo(,)100 110 y(at)i(a)h(p)q(oin)o(t)f Fl(y)382 92 y Fj(\()p Fk(N)t Fj(\))475 110 y Fo(in)g(\()p Fh(\000)p Fo(1)p Fl(;)8 b Fo(1\))690 92 y Fk(d)738 110 y Fo(whic)o(h)24 b(minimizes)e(lo)q(cally)j (the)f(free)h(energy)f(densit)o(y)g Fl(h)1868 92 y Fj(\()p Fk(N)t Fj(\))1936 110 y Fo(.)100 169 y(Theorem)17 b(1.1)i(compares)e Fl(Z)654 151 y Fj(\()p Fk(N)t Fj(\))722 169 y Fo(\()p Fl(tN)5 b Fo(\))20 b(with)f(a)f(Mark)o(o)o(v)g(pro)q(cess)f Fl(Z)k Fo(=)c Fl(Z)t Fo(\()p Fl(t)p Fo(\))i(that)g(satis\014es)f(the)100 229 y(sto)q(c)o(hastic)e(di\013eren)o(tial)f(equation)720 350 y(_)704 362 y Fl(Z)i 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b(exp)q(ect)i(that)f Fl(Y)1078 662 y Fj(\()p Fk(N)t Fj(\))1147 680 y Fo(\()p Fl(tN)5 b Fo(\))23 b Fh(\031)e Fl(Y)11 b Fo(\()p Fl(t)p Fo(\))22 b(and)f Fl(Z)1589 662 y Fj(\()p Fk(N)t Fj(\))1657 680 y Fo(\()p Fl(tN)5 b Fo(\))23 b Fh(\031)f Fl(Z)t Fo(\()p Fl(t)p Fo(\),)100 740 y(with)16 b Fl(Y)25 b Fo(=)13 b Fl(Y)e Fo(\()p Fl(t)p Fo(\))18 b(and)e Fl(Z)h Fo(=)d Fl(Z)t Fo(\()p Fl(t)p Fo(\))j(satisfying)f(the)g(equation)h([12])636 859 y(_)623 871 y Fl(Y)652 878 y Fk(k)690 871 y Fo(=)d(tanh)843 831 y Fg(\000)866 871 y Fl(\014)s Fo(\()p Fl(`)937 845 y Fj(\()p Fk(N)t Fj(\))937 886 y Fk(k)1005 871 y Fo(\))1024 851 y Fn(\000)p Fj(1)1078 871 y Fo(\()p Fl(P)1136 851 y Fj(\()p Fk(N)t Fj(\))1205 871 y Fl(Y)d Fo(\))1264 878 y Fk(k)1289 831 y Fg(\001)1323 871 y Fh(\000)g Fl(Y)1402 878 y Fk(k)1823 871 y Fo(\(1)p Fl(:)p Fo(21\))100 993 y(and)21 b(\(1.20\),)12 b(resp)q(ectiv)o(ely)l(.)20 b(In)11 b(this)g(case,)h(the)f(\(no)o(w)f(time)h(dep)q(enden)o(t\))g(co)q (e\016cien)o(ts)g(in)f(equation)100 1053 y(\(1.20\))17 b(are)f(obtained)f(b)o (y)i(substituting)e Fl(Y)22 b Fo(+)992 1040 y(_)979 1053 y Fl(Y)27 b Fo(for)17 b Fl(y)1138 1035 y Fj(\()p Fk(N)t Fj(\))1223 1053 y Fo(in)g(the)f(de\014nition)32 b(\(1.12\).)30 1129 y(3\))26 b(If)16 b(the)g(free)g(energy)g(densit)o(y)g Fl(h)689 1111 y Fj(\()p Fk(N)t Fj(\))773 1129 y Fo(has)g(lo)q(cal)g(minima)e(at)j(more)e (than)g(one)h(p)q(oin)o(t,)g(then)g(almost)100 1189 y(ev)o(ery)j(tra)s (jectory)469 1149 y Fg(\000)491 1189 y Fl(M)544 1171 y Fj(\()p Fk(N)t Fj(\))614 1189 y Fo(\(0\))p Fl(;)8 b(M)752 1171 y Fj(\()p Fk(N)t Fj(\))822 1189 y Fo(\(1\))p Fl(;)g(:)g(:)g(:)966 1149 y Fg(\001)989 1189 y Fo(,)20 b(whic)o(h)d(starts)i(in)f(the)h(\\free)g (energy)g(v)m(alley")g(of)100 1249 y(one)h(of)g(these)g(p)q(oin)o(ts,)g(will) g(ev)o(en)o(tually)g(escap)q(e)g(that)h(v)m(alley)l(.)33 b(Ho)o(w)o(ev)o(er,) 20 b(this)g(happ)q(ens)f(on)h(a)100 1309 y(time)c(scale)h(whic)o(h)f(is)g(m)o (uc)o(h)f(larger)h(than)h(the)g(one)f(w)o(e)h(ha)o(v)o(e)f(considered)f (here:)22 b(The)17 b(exp)q(ected)100 1368 y(v)m(alue)i(of)g(the)h(\014rst)e (exit)i(time)f Fl(T)730 1375 y Fk(N)788 1368 y Fo(should)e(b)q(e)j(roughly)e (the)h(exp)q(onen)o(tial)g(of)h(the)f(free)g(energy)100 1428 y(di\013erence)14 b(b)q(et)o(w)o(een)h(the)h(b)q(ottom)f(and)g(the)g(lo)o(w)o (est)g(passage)f(out)h(of)g(the)h(v)m(alley)l(.)22 b(Since)15 b(it)g(tak)o(es)100 1488 y(a)j(large)h(n)o(um)o(b)q(er)d(of)j(\(essen)o (tially)g(indep)q(enden)o(t\))e(trials)h(b)q(efore)h(one)g(of)g(the)g(escap)q (e)g(attempts)100 1548 y(succeeds,)d(the)h(distribution)e(of)i Fl(T)758 1555 y Fk(N)796 1548 y Fl(=)p Fo(I)-9 b(E[)p Fl(T)907 1555 y Fk(N)945 1548 y Fo(])16 b(should)g(b)q(e)h(exp)q(onen)o(tial)g(with)f (mean)h(one,)f(in)h(the)100 1607 y(limit)d Fl(N)19 b Fh(!)13 b(1)p Fo(.)22 b(Suc)o(h)13 b(a)h(result)g(has)g(b)q(een)g(pro)o(v)o(ed)f(for) h(the)h(Curie{W)l(eiss)e(mo)q(del)h(\(the)h(Hop\014eld)100 1667 y(mo)q(del)h(with)g(one)g(stored)g(pattern\))h(at)g(lo)o(w)f(temp)q (erature,)g(b)o(y)g(Cassandro)e Fd(et.)j(al.)22 b Fo([13].)0 1860 y Fi(2.)28 b(Pro)r(ofs)0 1937 y Fo(W)l(e)20 b(start)g(with)f(some)g (immediate)g(consequences)g(of)h(\(A1\){\(A3\).)33 b(F)l(or)19 b(an)o(y)g(giv)o(en)g(pair)g(\()p Fl(`;)8 b(P)f Fo(\))20 b(in)0 1996 y(I)-10 b(R)45 1975 y Fk(d)72 1996 y Fh(\002)t Fo(L\(I)g(R)210 1975 y Fk(d)234 1996 y Fl(;)8 b Fo(I)-10 b(R)300 1975 y Fk(d)324 1996 y Fo(\))13 b(consider)f(the)h(equation)g Fl(D)q(h)900 2003 y Fk(`;P)962 1996 y Fo(\()p Fl(y)r Fo(\))i(=)f(0,)f(where)g Fl(h)1316 2003 y Fk(`;P)1391 1996 y Fo(is)f(the)i(function)f(on)f(\()p Fh(\000)p Fo(1)p Fl(;)c Fo(1\))1926 1978 y Fk(d)0 2056 y Fo(de\014ned)16 b(b)o(y)293 2233 y Fl(h)322 2240 y Fk(`;P)383 2233 y Fo(\()p Fl(y)r Fo(\))f Fh(\021)f(\000)560 2199 y Fo(1)p 560 2222 25 2 v 560 2267 a(2)648 2171 y Fk(d)622 2186 y Fg(X)598 2293 y Fk(k)q(;m)p Fj(=1)726 2233 y Fl(P)758 2240 y Fk(k)q(m)818 2233 y Fl(y)842 2240 y Fk(k)867 2233 y Fl(y)891 2240 y Fk(m)940 2233 y Fo(+)999 2199 y(1)p 996 2222 31 2 v 996 2267 a Fl(\014)1067 2171 y Fk(d)1042 2186 y Fg(X)1041 2293 y Fk(k)q Fj(=1)1122 2233 y Fl(`)1143 2240 y Fk(k)1191 2153 y(y)1211 2158 y Ff(k)1176 2165 y Fg(Z)1180 2307 y Fj(0)1225 2233 y Fl(dt)8 b Fo(tanh)1377 2211 y Fn(\000)p Fj(1)1431 2233 y Fo(\()p Fl(t)p Fo(\))k Fh(\000)1554 2199 y Fo(ln)c(2)p 1554 2222 75 2 v 1576 2267 a Fl(\014)1643 2233 y(:)191 b Fo(\(2)p Fl(:)p Fo(1\))0 2423 y(By)17 b(Assumption)d(\(A3\),)j (this)f(equation)g(has)f(a)h(solution)f Fl(y)h Fo(=)e Fl(y)1220 2405 y Fj(\()p Fn(1)p Fj(\))1310 2423 y Fo(for)h(\()p Fl(`;)8 b(P)f Fo(\))15 b(=)e(\()p Fl(`)1612 2405 y Fj(\()p Fn(1)p Fj(\))1686 2423 y Fl(;)8 b(P)1747 2405 y Fj(\()p Fn(1)p Fj(\))1820 2423 y Fo(\),)17 b(and)0 2483 y Fl(D)42 2465 y Fj(2)65 2483 y Fl(h)94 2494 y Fk(`)111 2484 y Fb(\()p Fc(1)p Fb(\))174 2494 y Fk(;P)216 2484 y Fb(\()p Fc(1)p Fb(\))282 2483 y Fo(\()p Fl(y)327 2465 y Fj(\()p Fn(1)p Fj(\))401 2483 y Fo(\))22 b(is)e(p)q(ositiv)o(e.)35 b(Th)o(us,)21 b(b)o(y)g(the)g(implicit)f(function)h(theorem,)g(there)g (exists)g(an)0 2550 y(op)q(en)g(neigh)o(b)q(orho)q(o)q(d)e Fh(O)j Fo(of)f(\()p Fl(`)605 2532 y Fj(\()p Fn(1)p Fj(\))678 2550 y Fl(;)8 b(P)739 2532 y Fj(\()p Fn(1)p Fj(\))813 2550 y Fo(\),)22 b(and)e(a)h Fl(C)1055 2532 y Fj(1)1097 2550 y Fo(map)f(\()p Fl(`;)8 b(P)f Fo(\))21 b Fh(7!)g Fl(y)r Fo(\()p Fl(`;)8 b(P)f Fo(\))22 b(on)e Fh(O)q Fo(,)i(suc)o(h)e(that)0 2610 y Fl(y)r Fo(\()p Fl(`)66 2592 y Fj(\()p Fn(1)p Fj(\))140 2610 y Fl(;)8 b(P)201 2592 y Fj(\()p Fn(1)p Fj(\))274 2610 y Fo(\))23 b(=)f Fl(y)403 2592 y Fj(\()p Fn(1)p Fj(\))498 2610 y Fo(and)f Fl(D)q(h)671 2617 y Fk(`;P)733 2610 y Fo(\()p Fl(y)r Fo(\()p Fl(`;)8 b(P)f Fo(\)\))24 b(=)f(0)e(for)h(all)f(\()p Fl(`;)8 b(P)f Fo(\))23 b Fh(2)g(O)q Fo(.)37 b(Assuming)20 b(also)h(\(A2\),)0 2670 y(it)j(follo)o(ws)e(that)h(if)h Fl(N)29 b Fo(is)22 b(su\016cien)o(tly)g (large)h(then)g Fl(y)1050 2652 y Fj(\()p Fk(N)t Fj(\))1144 2670 y Fh(\021)i Fl(y)r Fo(\()p Fl(`)1274 2652 y Fj(\()p Fk(N)t Fj(\))1343 2670 y Fl(;)8 b(P)1404 2652 y Fj(\()p Fk(N)t Fj(\))1473 2670 y Fo(\))24 b(lies)f(in)g(\()p Fh(\000)p Fo(1)p Fl(;)8 b Fo(1\))1823 2652 y Fk(d)1870 2670 y Fo(and)963 2790 y(5)p eop %%Page: 6 7 bop 0 50 a Fl(D)42 32 y Fj(2)65 50 y Fl(h)94 32 y Fj(\()p Fk(N)t Fj(\))163 50 y Fo(\()p Fl(y)208 32 y Fj(\()p Fk(N)t Fj(\))277 50 y Fo(\))20 b(=)f Fl(D)416 32 y Fj(2)439 50 y Fl(h)468 61 y Fk(`)485 51 y Fb(\()p Ff(N)s Fb(\))544 61 y Fk(;P)586 51 y Fb(\()p Ff(N)s Fb(\))648 50 y Fo(\()p Fl(y)693 32 y Fj(\()p Fk(N)t Fj(\))762 50 y Fo(\))i(is)e(p)q(ositiv)o(e.)32 b(By)21 b(using)d(the)j(fact)f(that)h(\()p Fl(Q)1645 32 y Fj(\()p Fk(N)t Fj(\))1714 50 y Fo(\))1733 32 y Fn(\000)p Fj(1)1787 50 y Fl(B)1827 32 y Fj(\()p Fk(N)t Fj(\))1917 50 y Fo(is)0 117 y(the)c(Hessian)f(matrix)g (for)g Fl(D)548 99 y Fj(2)571 117 y Fl(h)600 99 y Fj(\()p Fk(N)t Fj(\))668 117 y Fo(\()p Fl(y)713 99 y Fj(\()p Fk(N)t Fj(\))783 117 y Fo(\),)h(together)f(with)h(the)g(trivial)f(iden)o(tit)o(y)248 212 y Fg(\000)271 252 y Fl(Q)310 231 y Fj(\()p Fk(N)t Fj(\))379 212 y Fg(\001)402 222 y Fn(\000)p Fj(1)p Fk(=)p Fj(2)496 252 y Fl(B)536 231 y Fj(\()p Fk(N)t Fj(\))605 212 y Fg(\000)628 252 y Fl(Q)667 231 y Fj(\()p Fk(N)t Fj(\))736 212 y Fg(\001)759 222 y Fj(1)p Fk(=)p Fj(2)836 252 y Fo(=)888 212 y Fg(\000)911 252 y Fl(Q)950 231 y Fj(\()p Fk(N)t Fj(\))1019 212 y Fg(\001)1042 222 y Fj(1)p Fk(=)p Fj(2)1105 212 y Fg(\002\000)1148 252 y Fl(Q)1187 231 y Fj(\()p Fk(N)t Fj(\))1257 212 y Fg(\001)1280 222 y Fn(\000)p Fj(1)1333 252 y Fl(B)1373 231 y Fj(\()p Fk(N)t Fj(\))1442 212 y Fg(\003\000)1486 252 y Fl(Q)1525 231 y Fj(\()p Fk(N)t Fj(\))1594 212 y Fg(\001)1617 222 y Fj(1)p Fk(=)p Fj(2)1688 252 y Fl(;)146 b Fo(\(2)p Fl(:)p Fo(2\))0 380 y(w)o(e)13 b(see)g(that)g(the)h (matrix)e Fl(B)532 362 y Fj(\()p Fk(N)t Fj(\))614 380 y Fo(is)h (diagonalizable,)f(and)g(that)h(all)g(eigen)o(v)m(alues)f(of)i Fl(B)1607 362 y Fj(\()p Fk(N)t Fj(\))1689 380 y Fo(are)e(p)q(ositiv)o(e.)0 439 y(The)22 b(iden)o(tit)o(y)44 b(\(2.2\))23 b(also)e(sho)o(ws)g(that)i(the) f(matrix)g Fl(C)t Fo(\()p Fl(t)p Fo(\))1168 421 y Fj(\()p Fk(N)t Fj(\))1259 439 y Fo(is)f(p)q(ositiv)o(e)h(for)g(all)g Fl(t)h(>)g Fo(0;)i(hence)0 499 y(the)g(function)f Fl(\036)327 481 y Fj(\()p Fk(N)t Fj(\))420 499 y Fo(is)g(w)o(ell)g(de\014ned)f(on)h(\(0)p Fl(;)8 b Fh(1)p Fo(\))17 b Fh(\002)f Fo(I)-10 b(R)1095 477 y Fk(d)1118 499 y Fo(.)46 b(Here,)26 b(w)o(e)e(ha)o(v)o(e)g(assumed)f(that)i Fl(N)30 b Fo(is)0 559 y(either)19 b(\014nite)g(and)g(su\016cien)o(tly)f (large,)h(or)g Fl(N)k Fo(=)c Fh(1)p Fo(.)30 b(Under)18 b(the)i(same)e (assumption,)g(and)h(for)g(an)o(y)0 619 y(\()p Fl(u;)8 b(s)p Fo(\))15 b Fh(2)f Fo(I)-10 b(R)218 597 y Fk(d)253 619 y Fh(\002)10 b Fo(\(0)p Fl(;)e Fh(1)p Fo(\),)17 b(w)o(e)g(can)f(de\014ne)440 751 y Fl( )474 731 y Fj(\()p Fk(N)t Fj(\))472 764 y Fk(u;s)543 751 y Fo(\()p Fl(x)p Fo(\))f Fh(\021)f Fl(\036)707 731 y Fj(\()p Fk(N)t Fj(\))775 711 y Fg(\000)798 751 y Fl(s;)8 b(x)k Fh(\000)f Fl(e)956 731 y Fn(\000)p Fk(sB)1038 716 y Fb(\()p Ff(N)s Fb(\))1099 751 y Fl(u)1128 711 y Fg(\001)1159 751 y Fl(;)158 b(x)14 b Fh(2)g Fo(I)-10 b(R)1465 730 y Fk(d)1496 751 y Fl(:)338 b Fo(\(2)p Fl(:)p Fo(3\))0 919 y Fe(Lemma)21 b(2.1.)66 b Fd(Giv)o(en)20 b(an)o(y)g(compact)g(subset)f Fh(U)25 b Fd(of)c Fo(I)-10 b(R)1130 897 y Fk(d)1153 919 y Fd(,)21 b(and)e(constan)o(ts)h Fl(\016)o(;)8 b(s)1579 926 y Fk(o)1621 919 y Fl(>)20 b Fo(0)p Fd(,)h(there)f(is)g(a)0 979 y(p)q(ositiv)o(e)c(in)o(teger)g Fl(N)389 986 y Fk(o)428 979 y Fd(suc)o(h)f(that)452 1084 y Fk(d)427 1098 y Fg(X)403 1206 y Fk(k)q(;m)p Fj(=1)531 1103 y Fg(\014)531 1133 y(\014)547 1146 y Fl(C)t Fo(\()p Fl(s)p Fo(\))648 1120 y Fj(\()p Fk(N)t Fj(\))648 1161 y Fk(k)q(m)729 1146 y Fh(\000)c Fl(C)t Fo(\()p Fl(s)p Fo(\))880 1120 y Fj(\()p Fn(1)p Fj(\))880 1161 y Fk(k)q(m)953 1103 y Fg(\014)953 1133 y(\014)986 1146 y Fl(;)1025 1103 y Fg(\015)1025 1133 y(\015)1053 1146 y Fl( )1087 1125 y Fj(\()p Fk(N)t Fj(\))1085 1158 y Fk(u;s)1167 1146 y Fh(\000)g Fl( )1251 1125 y Fj(\()p Fn(1)p Fj(\))1249 1158 y Fk(u;s)1324 1103 y Fg(\015)1324 1133 y(\015)1352 1166 y Fk(L)1379 1156 y Fb(1)1432 1146 y Fh(\024)30 b Fl(\016)10 b(;)301 b Fo(\(2)p Fl(:)p Fo(4\))0 1318 y Fd(whenev)o(er)16 b Fl(u)d Fh(2)h(U)5 b Fd(,)17 b Fl(s)d Fh(\025)f Fl(s)487 1325 y Fk(o)510 1318 y Fd(,)j(and)g Fl(N)k Fh(\025)13 b Fl(N)789 1325 y Fk(o)811 1318 y Fd(.)0 1440 y Fe(Pro)r(of.)22 b Fo(The)g(b)q(ound)e(on)h Fl(C)t Fo(\()p Fl(s)p Fo(\))620 1422 y Fj(\()p Fk(N)t Fj(\))703 1440 y Fh(\000)14 b Fl(C)t Fo(\()p Fl(s)p Fo(\))857 1422 y Fj(\()p Fn(1)p Fj(\))952 1440 y Fo(follo)o(ws)20 b(since)h(the)g(map)f(\()p Fl(`;)8 b(P)f Fo(\))22 b Fh(7!)g Fl(y)r Fo(\()p Fl(`;)8 b(P)f Fo(\))22 b(is)e Fl(C)1928 1422 y Fj(1)0 1509 y Fo(near)e(\()p Fl(`)153 1491 y Fj(\()p Fn(1)p Fj(\))227 1509 y Fl(;)8 b(P)288 1491 y Fj(\()p Fn(1)p Fj(\))361 1509 y Fo(\).)30 b(T)l(o)18 b(pro)o(v)o(e)g(the)h (second)g(b)q(ound,)f(w)o(e)h(de\014ne)f(functions)g Fl( )1531 1483 y Fj([)p Fk(r)q Fj(])1529 1515 y Fk(u;s)1605 1509 y Fo(as)h(in)37 b(\(2.3\))20 b(and)0 1569 y(\(1.14\),)25 b(but)e(with)h(the)f(matrices)g Fl(C)t Fo(\()p Fl(t)p Fo(\))780 1551 y Fj(\()p Fk(N)t Fj(\))872 1569 y Fo(and)g Fl(B)1016 1551 y Fj(\()p Fk(N)t Fj(\))1108 1569 y Fo(replaced)g(b)o(y)g(the)g(in)o(terp)q(olated)g(matrices)0 1629 y Fl(C)t Fo(\()p Fl(t)p Fo(\))96 1611 y Fj([)p Fk(r)q Fj(])163 1629 y Fh(\021)f Fo(\(1)15 b Fh(\000)f Fl(r)q Fo(\))p Fl(C)t Fo(\()p Fl(t)p Fo(\))474 1611 y Fj(\()p Fn(1)p Fj(\))563 1629 y Fo(+)g Fl(r)q(C)t Fo(\()p Fl(t)p Fo(\))735 1611 y Fj(\()p Fk(N)t Fj(\))826 1629 y Fo(and)21 b Fl(B)968 1611 y Fj([)p Fk(r)q Fj(])1035 1629 y Fh(\021)h Fo(\(1)15 b Fh(\000)f Fl(r)q Fo(\))p Fl(B)1290 1611 y Fj(\()p Fn(1)p Fj(\))1380 1629 y Fo(+)g Fl(r)q(B)1496 1611 y Fj(\()p Fk(N)t Fj(\))1566 1629 y Fo(,)23 b(resp)q(ectiv)o(ely)l(,)f(for)0 1699 y(0)c Fh(\024)h Fl(r)h Fh(\024)e Fo(1.)31 b(F)l(rom)17 b(the)j(de\014nition)37 b(\(1.13\))20 b(it)g(is)f(clear)g(that)g(the)h(matrices)e Fl(C)t Fo(\()p Fl(s)p Fo(\))1612 1681 y Fj([)p Fk(r)q Fj(])1657 1699 y Fo(,)1691 1658 y Fg(\000)1714 1699 y Fl(C)t Fo(\()p Fl(s)p Fo(\))1815 1681 y Fj([)p Fk(r)q Fj(])1860 1658 y Fg(\001)1883 1669 y Fn(\000)p Fj(1)1936 1699 y Fo(,)0 1758 y(and)i(their)h Fl(s)p Fo({deriv)m(ativ)o(es,)h (are)e(uniformly)g(b)q(ounded)g(in)g(the)i(region)e(0)h Fh(\024)g Fl(r)i Fh(\024)e Fo(1,)h Fl(s)g Fh(\025)f Fl(s)1775 1765 y Fk(o)1798 1758 y Fo(.)35 b(As)21 b(a)0 1818 y(consequence,)14 b(w)o(e)g(\014nd)g(e.g.)g(that,)h(giv)o(en)f(an)o(y)f(in)o(teger)h Fl(n)g Fh(\025)f Fo(0,)i(there)f(are)g(p)q(ositiv)o(e)g(constan)o(ts)f Fl(K)19 b Fo(and)0 1878 y Fl(\034)6 b Fo(,)16 b(suc)o(h)f(that)105 1965 y Fg(\014)105 1995 y(\014)105 2024 y(\014)139 1988 y Fl(d)165 1970 y Fk(n)p 127 2010 77 2 v 127 2056 a Fl(ds)176 2042 y Fk(n)210 2022 y Fl( )244 2001 y Fj([)p Fk(r)q Fj(])242 2034 y Fk(u;s)299 2022 y Fo(\()p Fl(z)r Fo(\))362 1965 y Fg(\014)362 1995 y(\014)362 2024 y(\014)396 2022 y Fl(;)435 1965 y Fg(\014)435 1995 y(\014)435 2024 y(\014)569 1988 y Fl(@)598 1970 y Fk(n)p 458 2010 279 2 v 458 2056 a Fl(@)s(z)512 2034 y Fk(n)537 2039 y Fb(1)510 2069 y Fj(1)567 2056 y Fh(\001)8 b(\001)g(\001)g Fl(@)s(z)687 2034 y Fk(n)712 2039 y Ff(d)685 2071 y Fk(d)742 2022 y Fl( )776 2001 y Fj([)p Fk(r)q Fj(])774 2034 y Fk(u;s)831 2022 y Fo(\()p Fl(z)r Fo(\))894 1965 y Fg(\014)894 1995 y(\014)894 2024 y(\014)942 2022 y Fh(\024)30 b Fl(K)t(e)1080 2001 y Fn(\000)p Fk(\034)t Fn(k)p Fk(z)q Fn(k)1193 1986 y Fb(2)1217 2022 y Fl(;)108 b(n)1369 2029 y Fj(1)1402 2022 y Fo(+)11 b Fl(:)d(:)g(:)j Fo(+)g Fl(n)1601 2029 y Fk(d)1638 2022 y Fo(=)j Fl(n)8 b(;)105 b Fo(\(2)p Fl(:)p Fo(5\))0 2183 y(for)19 b(all)g Fl(r)h Fh(2)f Fo([0)p Fl(;)8 b Fo(1],)19 b Fl(s)g Fh(2)f Fo([)p Fl(s)508 2190 y Fk(o)531 2183 y Fl(;)8 b Fh(1)p Fo(\),)20 b Fl(u)e Fh(2)g(U)5 b Fo(,)20 b Fl(z)h Fh(2)d Fo(I)-10 b(R)964 2161 y Fk(d)987 2183 y Fo(,)20 b(and)f(for)f Fl(N)25 b Fo(su\016cien)o(tly)18 b(large.)30 b(If)19 b Fl(u)f Fo(=)g(0,)i(the)0 2253 y(di\013erence)c Fl( )257 2227 y Fj(\()p Fk(N)t Fj(\))255 2258 y Fk(u;s)337 2253 y Fh(\000)11 b Fl( )421 2227 y Fj(\()p Fn(1)p Fj(\))419 2258 y Fk(u;s)511 2253 y Fo(can)16 b(no)o(w)g(b)q(e)h(b)q(ounded)e(as)i(claimed)e(in)33 b(\(2.4\),)17 b(b)o(y)f(using)f(the)i(iden)o(tit)o(y)423 2389 y Fl(d)p 411 2411 50 2 v 411 2457 a(dr)467 2423 y(\036)497 2397 y Fj([)p Fk(r)q Fj(])497 2436 y(0)p Fk(;s)550 2423 y Fo(\()p Fl(z)r Fo(\))e(=)730 2360 y Fk(d)704 2375 y Fg(X)680 2483 y Fk(k)q(;m)p Fj(=1)800 2367 y Fg(h)829 2389 y Fo(1)p 829 2411 25 2 v 829 2457 a(2)878 2389 y Fl(d)p 866 2411 50 2 v 866 2457 a(dr)922 2382 y Fg(\000)945 2423 y Fl(C)t Fo(\()p Fl(s)p Fo(\))1046 2402 y Fj([)p Fk(r)q Fj(])1091 2382 y Fg(\001)1113 2443 y Fk(k)q(m)1173 2367 y Fg(i)1261 2389 y Fl(@)1290 2371 y Fj(2)p 1203 2411 168 2 v 1203 2457 a Fl(@)s(z)1255 2464 y Fk(k)1280 2457 y Fl(@)s(z)1332 2464 y Fk(m)1376 2423 y Fl(\036)1406 2397 y Fj([)p Fk(r)q Fj(])1406 2436 y(0)p Fk(;s)1459 2423 y Fo(\()p Fl(z)r Fo(\))8 b Fl(:)304 b Fo(\(2)p Fl(:)p Fo(6\))0 2595 y(The)16 b(extension)h(to)g Fl(u)c Fh(2)h(U)22 b Fo(is)16 b(straigh)o(tforw)o(ard.)p 1807 2674 43 43 v 963 2790 a(6)p eop %%Page: 7 8 bop 100 50 a Fo(W)l(e)20 b(con)o(tin)o(ue)e(b)o(y)h(estimating)g(the)h (elemen)o(ts)f(of)h(the)g(matrix)f Fl(N)5 b Fo(\()p 1384 10 54 2 v Fl(W)1438 57 y Fk(N)1489 50 y Fh(\000)13 b Fo(I\),)21 b(whic)o(h)e(will)g(turn)0 119 y(out)g(to)h(b)q(e)g(a)f(discrete)g(appro)o (ximation)e(for)i Fl(H)915 93 y Fj(\()p Fk(N)t Fj(\))911 125 y Fk(osc)985 119 y Fo(.)30 b(A)20 b(straigh)o(tforw)o(ard)d(computation,)i (using)f(the)0 179 y(de\014nition)d(of)p 278 139 V 17 w Fl(W)332 186 y Fk(N)370 179 y Fo(,)h(sho)o(ws)f(that)i(the)g(o\013{diagonal)e(elemen)o (ts)h(of)p 1276 139 V 17 w Fl(W)1330 186 y Fk(N)1384 179 y Fo(are)g(giv)o(en)g(b)o(y)p 400 289 V 400 329 a Fl(W)454 336 y Fk(N)492 329 y Fo(\()p Fl(z)r(;)8 b(z)583 309 y Fn(0)598 329 y Fo(\))14 b(=)684 244 y Fg(\()732 293 y Fl(w)769 272 y Fn(\006)768 308 y Fk(N)r(;k)837 253 y Fg(\000)890 274 y Fj(1)p 866 282 69 2 v 866 287 a Fn(p)p 899 287 36 2 v 30 x Fk(N)940 293 y Fl(;)8 b(z)987 253 y Fg(\001)1019 293 y Fl(;)49 b Fo(if)17 b Fl(z)1153 275 y Fn(0)1181 293 y Fo(=)d Fl(z)f Fh(\006)1388 274 y Fj(2)p 1326 282 144 2 v 1326 291 a Fn(p)p 1359 291 36 2 v 30 x Fk(N)s(`)1411 302 y Fb(\()p Ff(N)s Fb(\))1411 334 y Ff(k)1475 293 y Fl(\016)1497 300 y Fk(k)1530 293 y Fo(,)732 374 y(0)p Fl(;)311 b Fo(otherwise,)1848 329 y(\(2)p Fl(:)p Fo(7\))0 479 y(where)16 b Fl(\016)166 486 y Fk(k)207 479 y Fo(denotes)g(the)h(unit)f(v)o(ector)h(in)f(the)h(direction)f(of)g(the)h Fl(k)r Fo(-th)f(co)q(ordinate)g(axis,)g(and)g(where)76 651 y Fl(w)113 630 y Fn(\006)112 667 y Fk(N)r(;k)181 651 y Fo(\()p Fl(";)8 b(x)p Fo(\))15 b Fh(\021)657 580 y Fj(1)p 657 588 20 2 v 657 617 a(2)683 600 y Fl(`)704 574 y Fj(\()p Fk(N)t Fj(\))704 615 y Fk(k)773 544 y Fg(h)796 600 y Fo(1)c Fh(\006)g Fl(y)908 574 y Fj(\()p Fk(N)t Fj(\))906 615 y Fk(k)988 600 y Fh(\006)g Fl("x)1089 607 y Fk(k)1126 600 y Fo(+)f(2)p Fl(")1223 582 y Fj(2)1246 559 y Fg(\000)1269 600 y Fl(`)1290 574 y Fj(\()p Fk(N)t Fj(\))1290 615 y Fk(k)1358 559 y Fg(\001)1381 570 y Fn(\000)p Fj(1)1435 544 y Fg(i)p 366 640 1379 2 v 366 703 a Fo(1)h(+)f(exp)528 648 y Fg(h)551 703 y Fh(\006)p Fo(2)p Fl(\014)646 663 y Fg(\000)668 703 y Fl(`)689 677 y Fj(\()p Fk(N)t Fj(\))689 718 y Fk(k)758 663 y Fg(\001)781 673 y Fn(\000)p Fj(1)834 663 y Fg(\002)855 703 y Fl(P)894 689 y Fj(\()p Fk(N)t Fj(\))963 663 y Fg(\000)986 703 y Fl(y)1012 689 y Fj(\()p Fk(N)t Fj(\))1092 703 y Fo(+)h Fl("x)1193 663 y Fg(\001\003)1237 723 y Fk(k)1273 703 y Fo(+)f(2)p Fl(")1370 689 y Fj(2)1393 703 y Fl(\014)1424 663 y Fg(\000)1447 703 y Fl(`)1468 677 y Fj(\()p Fk(N)t Fj(\))1468 718 y Fk(k)1536 663 y Fg(\001)1559 673 y Fn(\000)p Fj(2)1613 703 y Fl(P)1652 677 y Fj(\()p Fk(N)t Fj(\))1645 718 y Fk(k)q(k)1720 648 y Fg(i)1758 651 y Fl(;)76 b Fo(\(2)p Fl(:)p Fo(8\))0 835 y(for)19 b(all)f Fl(x)g Fh(2)g Fo(I)-10 b(R)292 813 y Fk(d)334 835 y Fo(and)18 b Fl(")g Fh(2)g Fo(I)-10 b(R.)29 b(The)18 b(diagonal)g(en)o (tries)g(of)p 1140 795 54 2 v 19 w Fl(W)1194 842 y Fk(N)1250 835 y Fo(are)h(determined)f(b)o(y)g(the)h(fact)h(that)p 0 855 V 0 895 a Fl(W)54 902 y Fk(N)108 895 y Fo(is)c(a)h(matrix)f(of)h(transition)e (probabilities:)p 30 1009 V 30 1049 a Fl(W)84 1056 y Fk(N)122 1049 y Fo(\()p Fl(z)r(;)8 b(z)r Fo(\))16 b(=)d(1)e Fh(\000)412 987 y Fk(d)386 1002 y Fg(X)386 1109 y Fk(k)q Fj(=1)467 1049 y Fl(w)504 1028 y Fj(+)503 1064 y Fk(N)r(;k)572 1009 y Fg(\000)632 1015 y Fo(1)p 601 1038 87 2 v 601 1048 a Fh(p)p 642 1048 46 2 v 642 1090 a Fl(N)694 1049 y(;)d(z)13 b Fh(\000)884 1015 y Fo(2)p 808 1038 177 2 v 808 1054 a Fh(p)p 849 1054 46 2 v 849 1096 a Fl(N)6 b(`)916 1070 y Fj(\()p Fk(N)t Fj(\))916 1111 y Fk(k)991 1049 y Fl(\016)1013 1056 y Fk(k)1037 1009 y Fg(\001)1071 1049 y Fh(\000)1147 987 y Fk(d)1121 1002 y Fg(X)1121 1109 y Fk(k)q Fj(=1)1202 1049 y Fl(w)1239 1028 y Fn(\000)1238 1064 y Fk(N)r(;k)1307 1009 y Fg(\000)1367 1015 y Fo(1)p 1336 1038 87 2 v 1336 1048 a Fh(p)p 1377 1048 46 2 v 1377 1090 a Fl(N)1429 1049 y(;)i(z)13 b Fo(+)1619 1015 y(2)p 1543 1038 177 2 v 1543 1054 a Fh(p)p 1584 1054 46 2 v 1584 1096 a Fl(N)6 b(`)1651 1070 y Fj(\()p Fk(N)t Fj(\))1651 1111 y Fk(k)1726 1049 y Fl(\016)1748 1056 y Fk(k)1772 1009 y Fg(\001)1803 1049 y Fl(:)31 b Fo(\(2)p Fl(:)p Fo(9\))0 1211 y(Here,)14 b(as)e(w)o(ell)g(as)g(in)25 b(\(2.7\),)14 b(the)f(argumen)o(ts)e(of)p 911 1171 54 2 v 13 w Fl(W)965 1218 y Fk(N)1015 1211 y Fo(are)i(assumed)d(to)k(b)q(e)e(p)q(oin)o (ts)g(in)h Fh(Z)1648 1218 y Fk(N)1685 1211 y Fo(,)h(i.e.)20 b(v)o(ectors)0 1281 y(in)11 b(I)-10 b(R)98 1260 y Fk(d)133 1281 y Fo(whose)12 b(comp)q(onen)o(ts)e Fl(z)568 1288 y Fk(k)605 1281 y Fo(range)h(b)q(et)o(w)o(een)923 1239 y Fh(p)p 965 1239 46 2 v 42 x Fl(N)1010 1241 y Fg(\000)1033 1281 y Fh(\000)p Fl(y)1098 1255 y Fj(\()p Fk(N)t Fj(\))1096 1296 y Fk(k)1169 1281 y Fh(\006)r Fo(1)1235 1241 y Fg(\001)1257 1281 y Fo(,)i(in)e(incremen)o (ts)f(of)i(2)1660 1241 y Fg(\000)1683 1239 y Fh(p)p 1725 1239 V 42 x Fl(N)5 b(`)1791 1255 y Fj(\()p Fk(N)t Fj(\))1791 1296 y Fk(k)1860 1241 y Fg(\001)1883 1251 y Fn(\000)p Fj(1)1936 1281 y Fo(.)0 1341 y(If)17 b Fl(f)22 b Fo(is)16 b(a)h(function)f(on)g Fh(Z)488 1348 y Fk(N)526 1341 y Fo(,)g(de\014ne)86 1430 y Fg(\000)108 1470 y Fl(D)149 1477 y Fk(N)r(;k)219 1470 y Fl(f)248 1430 y Fg(\001)272 1470 y Fo(\()p Fl(z)r Fo(\))f Fh(\021)408 1436 y Fo(1)p 408 1458 25 2 v 408 1504 a(4)439 1425 y Fh(p)p 481 1425 46 2 v 45 x Fl(N)5 b(`)547 1444 y Fj(\()p Fk(N)t Fj(\))547 1485 y Fk(k)616 1415 y Fg(h)639 1470 y Fl(f)668 1430 y Fg(\000)692 1470 y Fl(z)13 b Fo(+)860 1436 y(2)p 784 1458 177 2 v 784 1474 a Fh(p)p 826 1474 46 2 v 42 x Fl(N)5 b(`)892 1491 y Fj(\()p Fk(N)t Fj(\))892 1531 y Fk(k)967 1470 y Fl(\016)989 1477 y Fk(k)1013 1430 y Fg(\001)1047 1470 y Fh(\000)11 b Fl(f)1126 1430 y Fg(\000)1150 1470 y Fl(z)i Fh(\000)1318 1436 y Fo(2)p 1242 1458 177 2 v 1242 1474 a Fh(p)p 1283 1474 46 2 v 1283 1516 a Fl(N)6 b(`)1350 1491 y Fj(\()p Fk(N)t Fj(\))1350 1531 y Fk(k)1424 1470 y Fl(\016)1446 1477 y Fk(k)1471 1430 y Fg(\001)1494 1415 y(i)1517 1470 y Fl(;)85 1577 y Fg(\000)108 1617 y Fo(\001)150 1624 y Fk(N)r(;k)219 1617 y Fl(f)248 1577 y Fg(\001)272 1617 y Fo(\()p Fl(z)r Fo(\))15 b Fh(\021)408 1583 y Fo(1)p 408 1606 25 2 v 408 1651 a(4)439 1617 y Fl(N)484 1577 y Fg(\000)507 1617 y Fl(`)528 1591 y Fj(\()p Fk(N)t Fj(\))528 1632 y Fk(k)597 1577 y Fg(\001)620 1587 y Fj(2)642 1562 y Fg(h)666 1617 y Fl(f)695 1577 y Fg(\000)718 1617 y Fl(z)f Fo(+)886 1583 y(2)p 811 1606 177 2 v 811 1622 a Fh(p)p 852 1622 46 2 v 852 1664 a Fl(N)6 b(`)919 1638 y Fj(\()p Fk(N)t Fj(\))919 1679 y Fk(k)993 1617 y Fl(\016)1015 1624 y Fk(k)1040 1577 y Fg(\001)1074 1617 y Fo(+)11 b Fl(f)1153 1577 y Fg(\000)1176 1617 y Fl(z)j Fh(\000)1344 1583 y Fo(2)p 1268 1606 177 2 v 1268 1622 a Fh(p)p 1310 1622 46 2 v 42 x Fl(N)5 b(`)1376 1638 y Fj(\()p Fk(N)t Fj(\))1376 1679 y Fk(k)1451 1617 y Fl(\016)1473 1624 y Fk(k)1498 1577 y Fg(\001)1531 1617 y Fh(\000)11 b Fo(2)p Fl(f)5 b Fo(\()p Fl(z)r Fo(\))1698 1562 y Fg(i)1724 1617 y Fl(;)1823 1554 y Fo(\(2)p Fl(:)p Fo(10\))0 1769 y(with)17 b(the)g(con)o(v)o(en)o(tion)e(that)j Fl(f)5 b Fo(\()p Fl(x)p Fo(\))16 b(=)e(0)j(whenev)o(er)f Fl(x)f Fh(62)g(Z)1110 1776 y Fk(N)1148 1769 y Fo(.)23 b(Then,)16 b(b)o(y)h(using)e (the)j(iden)o(tities)33 b(\(2.7\))0 1829 y(and)f(\(2.9\),)17 b(w)o(e)g(\014nd)e(the)i(follo)o(wing)e(represen)o(tation)g(for)p 1117 1789 54 2 v 16 w Fl(W)1171 1836 y Fk(N)1209 1829 y Fl(f)5 b Fo(.)100 1946 y Fg(\000)p 123 1946 V 40 x Fl(W)177 1993 y Fk(N)215 1986 y Fl(f)244 1946 y Fg(\001)267 1986 y Fo(\()p Fl(z)r Fo(\))15 b(=)f Fl(f)5 b Fo(\()p Fl(z)r Fo(\))13 b(+)569 1953 y(1)p 559 1975 46 2 v 559 2021 a Fl(N)644 1924 y Fk(d)618 1939 y Fg(X)618 2046 y Fk(k)q Fj(=1)691 1931 y Fg(h)714 1986 y Fl(a)740 1993 y Fk(k)765 1946 y Fg(\000)788 1986 y Fo(1)p Fl(=)838 1942 y Fh(p)p 879 1942 V 879 1986 a Fl(N)6 b(;)i(z)972 1946 y Fg(\001)995 1986 y Fl(f)d Fo(\()p Fl(z)r Fo(\))13 b(+)e Fl(b)1171 1993 y Fk(k)1196 1946 y Fg(\000)1218 1986 y Fo(1)p Fl(=)1268 1942 y Fh(p)p 1310 1942 V 44 x Fl(N)5 b(;)j(z)1402 1946 y Fg(\001)q(\000)1448 1986 y Fl(D)1489 1993 y Fk(N)r(;k)1559 1986 y Fl(f)1588 1946 y Fg(\001)1612 1986 y Fo(\()p Fl(z)r Fo(\))1100 2129 y(+)j Fl(q)1172 2136 y Fk(k)1196 2088 y Fg(\000)1219 2129 y Fo(1)p Fl(=)1269 2084 y Fh(p)p 1311 2084 V 45 x Fl(N)5 b(;)j(z)1403 2088 y Fg(\001\000)1449 2129 y Fo(\001)1491 2136 y Fk(N)r(;k)1560 2129 y Fl(f)1589 2088 y Fg(\001)1613 2129 y Fo(\()p Fl(z)r Fo(\))1676 2073 y Fg(i)1709 2129 y Fl(;)1823 2040 y Fo(\(2)p Fl(:)p Fo(11\))0 2249 y(where)16 b(the)h(co)q(e\016cien)o(ts) f Fl(a)508 2256 y Fk(k)533 2249 y Fl(;)8 b(b)576 2256 y Fk(k)601 2249 y Fl(;)g(q)645 2256 y Fk(k)686 2249 y Fo(are)16 b(giv)o(en)h(b)o(y)f (the)h(equations)48 2378 y Fl(a)74 2385 y Fk(k)99 2378 y Fo(\()p Fl(";)8 b(x)p Fo(\))16 b Fh(\021)d Fl(")301 2357 y Fn(\000)p Fj(2)355 2322 y Fg(h)378 2378 y Fl(w)415 2356 y Fj(+)414 2393 y Fk(N)r(;k)484 2378 y Fo(\()p Fl(";)8 b(x)p Fo(\))k Fh(\000)f Fl(w)694 2356 y Fn(\000)693 2393 y Fk(N)r(;k)762 2337 y Fg(\000)785 2378 y Fl(";)d(x)k Fo(+)947 2344 y(2)p Fl(")p 926 2366 90 2 v 926 2424 a(`)947 2398 y Fj(\()p Fk(N)t Fj(\))947 2439 y Fk(k)1021 2378 y Fl(\016)1043 2385 y Fk(k)1068 2337 y Fg(\001)1102 2378 y Fo(+)f Fl(w)1189 2356 y Fn(\000)1188 2393 y Fk(N)r(;k)1257 2378 y Fo(\()p Fl(";)d(x)p Fo(\))13 b Fh(\000)d Fl(w)1467 2356 y Fj(+)1466 2393 y Fk(N)r(;k)1536 2337 y Fg(\000)1558 2378 y Fl(";)e(x)k Fh(\000)1720 2344 y Fo(2)p Fl(")p 1699 2366 V 1699 2424 a(`)1720 2398 y Fj(\()p Fk(N)t Fj(\))1720 2439 y Fk(k)1795 2378 y Fl(\016)1817 2385 y Fk(k)1841 2337 y Fg(\001)1864 2322 y(i)1888 2378 y Fl(;)53 2512 y(b)74 2519 y Fk(k)99 2512 y Fo(\()p Fl(";)c(x)p Fo(\))16 b Fh(\021)d Fo(2)303 2471 y Fg(\000)326 2512 y Fl("`)370 2486 y Fj(\()p Fk(N)t Fj(\))370 2527 y Fk(k)439 2471 y Fg(\001)462 2482 y Fn(\000)p Fj(1)515 2471 y Fg(\002)536 2512 y Fl(w)573 2490 y Fj(+)572 2527 y Fk(N)r(;k)641 2512 y Fo(\()p Fl(";)8 b(x)p Fo(\))13 b Fh(\000)e Fl(w)852 2490 y Fn(\000)851 2527 y Fk(N)r(;k)920 2512 y Fo(\()p Fl(";)d(x)p Fo(\))1031 2471 y Fg(\003)1053 2512 y Fl(;)53 2606 y(q)75 2613 y Fk(k)99 2606 y Fo(\()p Fl(";)g(x)p Fo(\))16 b Fh(\021)d Fo(2)303 2565 y Fg(\000)326 2606 y Fl(`)347 2580 y Fj(\()p Fk(N)t Fj(\))347 2621 y Fk(k)416 2565 y Fg(\001)439 2576 y Fn(\000)p Fj(2)492 2565 y Fg(\002)513 2606 y Fl(w)550 2585 y Fj(+)549 2621 y Fk(N)r(;k)618 2606 y Fo(\()p Fl(";)8 b(x)p Fo(\))13 b(+)d Fl(w)828 2585 y Fn(\000)827 2621 y Fk(N)r(;k)896 2606 y Fo(\()p Fl(";)e(x)p Fo(\))1007 2565 y Fg(\003)1030 2606 y Fl(:)1823 2670 y Fo(\(2)p Fl(:)p Fo(12\))963 2790 y(7)p eop %%Page: 8 9 bop 0 50 a Fe(Lemma)17 b(2.2.)45 b Fd(There)17 b(is)f(a)h(p)q(olynomial)e Fl(p)i Fd(suc)o(h)f(that)h(for)f Fo(0)f Fl(<)f Fh(j)p Fl(")p Fh(j)g Fl(<)g Fo(1)p Fd(,)i Fo(1)f Fh(\024)e Fl(k)j Fh(\024)e Fl(d)p Fd(,)j Fl(x)e Fh(2)f Fo(I)-10 b(R)1816 28 y Fk(d)1839 50 y Fd(,)17 b(and)0 110 y(for)f Fl(N)k(<)13 b Fh(1)k Fd(su\016cien)o(tly)e (large,)164 207 y Fg(\014)164 237 y(\014)180 249 y Fl(a)206 256 y Fk(k)231 249 y Fo(\()p Fl(";)8 b(x)p Fo(\))13 b Fh(\000)e Fl(B)445 224 y Fj(\()p Fk(N)t Fj(\))443 265 y Fk(k)q(k)514 207 y Fg(\014)514 237 y(\014)547 249 y Fl(;)586 207 y Fg(\014)586 237 y(\014)602 249 y Fl(b)623 256 y Fk(k)648 249 y Fo(\()p Fl(";)d(x)p Fo(\))13 b Fh(\000)822 209 y Fg(\000)845 249 y Fl(B)885 229 y Fj(\()p Fk(N)t Fj(\))954 249 y Fl(x)982 209 y Fg(\001)1005 269 y Fk(k)1030 207 y Fg(\014)1030 237 y(\014)1063 249 y Fl(;)1102 207 y Fg(\014)1102 237 y(\014)1118 249 y Fl(q)1140 256 y Fk(k)1165 249 y Fo(\()p Fl(";)8 b(x)p Fo(\))13 b Fh(\000)d Fl(Q)1377 224 y Fj(\()p Fk(N)t Fj(\))1377 265 y Fk(k)q(k)1447 207 y Fg(\014)1447 237 y(\014)1477 249 y Fh(\024)k Fl("p)p Fo(\()p Fl(x)p Fo(\))p Fl(:)165 b Fo(\(2)p Fl(:)p Fo(13\))0 426 y Fe(Pro)r(of.)19 b Fo(The)f(estimate)g(for)f Fl(a)584 433 y Fk(k)627 426 y Fo(is)h(obtained)f(b)o(y)h(using)f(a)h(T)l(a)o(ylor)f (expansion)g(ab)q(out)h Fl(")e Fo(=)g(0)i(for)g(the)0 486 y(functions)g Fl(w)254 464 y Fn(\006)253 501 y Fk(N)r(;k)322 486 y Fo(,)h(with)g(a)g (remainder)e(of)i(the)g(order)e Fl(")1055 468 y Fj(3)1078 486 y Fo(.)29 b(After)19 b(substituting)f(this)g(expansion)g(in)o(to)0 545 y(the)e(de\014nition)30 b(\(2.12\))15 b(of)h Fl(a)545 552 y Fk(k)570 545 y Fo(\()p Fl(";)8 b(x)p Fo(\),)17 b(one)e(\014nds)f(that)i (the)g(negativ)o(e)f(p)q(o)o(w)o(ers)f(of)i Fl(")g Fo(cancel,)f(due)g(to)h (the)0 605 y(iden)o(tit)o(y)323 720 y(exp)399 664 y Fg(h)422 720 y Fo(2)p Fl(\014)478 679 y Fg(\000)501 720 y Fl(`)522 694 y Fj(\()p Fk(N)t Fj(\))522 735 y Fk(k)590 679 y Fg(\001)613 690 y Fn(\000)p Fj(1)667 679 y Fg(\000)690 720 y Fl(P)729 699 y Fj(\()p Fk(N)t Fj(\))797 720 y Fl(y)823 699 y Fj(\()p Fk(N)t Fj(\))892 679 y Fg(\001)915 740 y Fk(k)940 664 y Fg(i)977 720 y Fo(=)1036 685 y(1)11 b(+)f Fl(y)1147 659 y Fj(\()p Fk(N)t Fj(\))1145 700 y Fk(k)p 1036 708 181 2 v 1036 766 a Fo(1)h Fh(\000)f Fl(y)1147 740 y Fj(\()p Fk(N)t Fj(\))1145 781 y Fk(k)1223 720 y Fl(;)157 b Fo(1)14 b Fh(\024)f Fl(k)j Fh(\024)d Fl(d)8 b(;)196 b Fo(\(2)p Fl(:)p Fo(14\))0 879 y(whic)o(h)18 b(expresses)g(the)h (fact)g(that)h Fl(y)688 861 y Fj(\()p Fk(N)t Fj(\))776 879 y Fo(is)e(a)h(critical)g(p)q(oin)o(t)f(of)h Fl(h)1261 861 y Fj(\()p Fk(N)t Fj(\))1330 879 y Fo(.)29 b(The)19 b(co)q(e\016cien)o(t)f(of)i Fl(")1795 861 y Fj(0)1836 879 y Fo(turns)0 946 y(out)d(to)g(b)q(e)h Fl(B)259 920 y Fj(\()p Fk(N)t Fj(\))257 961 y Fk(k)q(k)328 946 y Fo(,)f(and)f(the)i(mo)q(dulus)d(of)i(the)h(remainder)d(satis\014es)33 b(\(2.13\))17 b(since)g(all)f(deriv)m(ativ)o(es)h(of)0 1005 y Fl(w)37 984 y Fn(\006)36 1020 y Fk(N)r(;k)122 1005 y Fo(are)f(p)q (olynomially)f(b)q(ounded.)21 b(The)c(estimates)f(for)g Fl(b)1144 1012 y Fk(k)1185 1005 y Fo(and)g Fl(q)1304 1012 y Fk(k)1346 1005 y Fo(are)g(obtained)g(similarly)l(.)p 1807 1094 43 43 v 100 1232 a(Next,)i(w)o(e)f(compare)f(the)h(t)o(w)o(o)g(maps)p 822 1192 54 2 v 16 w Fl(W)876 1239 y Fk(N)931 1232 y Fo(and)f Fl(T)1064 1206 y Fj(\()p Fk(N)t Fj(\))1057 1249 y(1)p Fk(=)n(N)1133 1232 y Fo(,)h(when)g(applied)f(to)h(the)h(functions)e Fl( )1867 1206 y Fj(\()p Fk(N)t Fj(\))1865 1238 y Fk(u;s)1936 1232 y Fo(.)0 1309 y(F)l(or)f(an)o(y)h(function)h Fl(f)22 b Fo(on)16 b(I)-10 b(R)539 1287 y Fk(d)562 1309 y Fo(,)16 b(denote)h(b)o(y)f Fl(R)860 1316 y Fk(N)898 1309 y Fl(f)22 b Fo(the)17 b(restriction)e(of)i Fl(f)23 b Fo(to)16 b(the)h(set)g Fh(Z)1632 1316 y Fk(N)1670 1309 y Fo(.)0 1409 y Fe(Lemma)f(2.3.)43 b Fd(Giv)o(en)15 b(a)h(compact)f (subset)g Fh(U)21 b Fd(of)15 b Fo(I)-10 b(R)1021 1387 y Fk(d)1044 1409 y Fd(,)16 b(and)f Fl(s)1193 1416 y Fk(o)1229 1409 y Fl(>)f Fo(0)p Fd(,)h(there)h(are)f(constan)o(ts)g Fl(K)q(;)8 b(\024)14 b(>)f Fo(0)0 1469 y Fd(suc)o(h)i(that)i(if)g Fl(N)22 b Fd(is)16 b(su\016cien)o(tly)f(large,)392 1566 y Fg(\014)392 1596 y(\014)409 1568 y(\000)p 431 1569 V 431 1609 a Fl(W)485 1616 y Fk(N)523 1609 y Fl(R)561 1616 y Fk(N)599 1609 y Fl( )633 1588 y Fj(\()p Fk(N)t Fj(\))631 1621 y Fk(u;s)702 1568 y Fg(\001)725 1609 y Fo(\()p Fl(z)r Fo(\))d Fh(\000)f Fl( )884 1583 y Fj(\()p Fk(N)t Fj(\))882 1626 y Fk(u;s)p Fj(+1)p Fk(=)n(N)1043 1609 y Fo(\()p Fl(z)r Fo(\))1106 1566 y Fg(\014)1106 1596 y(\014)1137 1609 y Fh(\024)j Fl(K)t(N)1281 1588 y Fn(\000)p Fj(3)p Fk(=)p Fj(2)1375 1609 y Fl(e)1398 1588 y Fn(\000)p Fk(\024)p Fn(k)p Fk(z)q Fn(k)1512 1573 y Fb(2)1544 1609 y Fl(;)265 b Fo(\(2)p Fl(:)p Fo(15\))0 1747 y Fd(for)16 b(all)g Fl(u)e Fh(2)g(U)5 b Fd(,)16 b Fl(s)f Fh(\025)e Fl(s)414 1754 y Fk(o)437 1747 y Fd(,)j(and)g Fl(z)g Fh(2)e(Z)686 1754 y Fk(N)724 1747 y Fd(.)0 1889 y Fe(Pro)r(of.)j Fo(By)g(using)f(T)l(a)o(ylor's)f(form)o(ula,)f (together)j(with)f(the)h(b)q(ound)32 b(\(2.5\),)17 b(w)o(e)g(obtain)21 2012 y Fg(\014)21 2041 y(\014)37 2014 y(\000)60 2054 y Fl(D)101 2061 y Fk(N)r(;k)171 2054 y Fl(R)209 2061 y Fk(N)247 2054 y Fl( )281 2033 y Fj(\()p Fk(N)t Fj(\))279 2066 y Fk(u;s)350 2014 y Fg(\001)373 2054 y Fo(\()p Fl(z)r Fo(\))12 b Fh(\000)530 2020 y Fl(@)p 504 2042 83 2 v 504 2088 a(@)s(x)561 2095 y Fk(k)592 2054 y Fl( )626 2033 y Fj(\()p Fk(N)t Fj(\))624 2066 y Fk(u;s)695 2054 y Fo(\()p Fl(z)r Fo(\))758 2012 y Fg(\014)758 2041 y(\014)784 2054 y Fl(;)814 2012 y Fg(\014)814 2041 y(\014)831 2014 y(\000)854 2054 y Fo(\001)896 2061 y Fk(N)r(;k)965 2054 y Fl(R)1003 2061 y Fk(N)1040 2054 y Fl( )1074 2033 y Fj(\()p Fk(N)t Fj(\))1072 2066 y Fk(u;s)1143 2014 y Fg(\001)1166 2054 y Fo(\()p Fl(z)r Fo(\))g Fh(\000)1313 2020 y Fl(@)1342 2002 y Fj(2)p 1297 2042 V 1297 2088 a Fl(@)s(x)1354 2071 y Fj(2)1354 2103 y Fk(k)1385 2054 y Fl( )1419 2033 y Fj(\()p Fk(N)t Fj(\))1417 2066 y Fk(u;s)1489 2054 y Fo(\()p Fl(z)r Fo(\))1552 2012 y Fg(\014)1552 2041 y(\014)1583 2054 y Fh(\024)1655 2020 y Fl(K)1701 2002 y Fn(0)p 1642 2042 87 2 v 1642 2052 a Fh(p)p 1683 2052 46 2 v 1683 2095 a Fl(N)1735 2054 y(e)1758 2033 y Fn(\000)p Fj(2)p Fk(\024)p Fn(k)p Fk(z)q Fn(k)1892 2018 y Fb(2)1915 2054 y Fl(;)1823 2145 y Fo(\(2)p Fl(:)p Fo(16\))0 2205 y(uniformly)g(on)h Fh(f)p Fo(\()p Fl(u;)8 b(s)p Fo(\))15 b(:)f Fl(u)f Fh(2)h(U)5 b Fl(;)j(s)14 b Fh(\025)g Fl(s)730 2212 y Fk(o)752 2205 y Fh(g)p Fo(,)g(for)g(some)f(constan)o(ts)f Fl(K)1264 2187 y Fn(0)1278 2205 y Fl(;)c(\024)14 b(>)f Fo(0.)21 b(Here,)14 b(w)o(e)g(ha)o(v)o(e)f(assumed)0 2275 y(that)18 b Fl(z)i Fo(is)d(an)g(in)o(terior)f(p)q(oin)o(t)h(of)g Fh(Z)674 2282 y Fk(N)712 2275 y Fo(,)h(i.e.)f(a)g(p)q(oin)o(t)g(in)g Fh(Z)1093 2282 y Fk(N)1149 2275 y Fo(for)g(whic)o(h)f Fl(z)e Fh(\006)e Fo(2)1482 2235 y Fg(\000)1504 2233 y Fh(p)p 1546 2233 V 42 x Fl(N)5 b(`)1612 2249 y Fj(\()p Fk(N)t Fj(\))1612 2290 y Fk(k)1681 2235 y Fg(\001)1704 2245 y Fn(\000)p Fj(1)1757 2275 y Fl(\016)1779 2282 y Fk(k)1822 2275 y Fo(lies)16 b(in)0 2335 y Fh(Z)36 2342 y Fk(N)92 2335 y Fo(for)i(all)g Fl(k)r Fo(.)27 b(If)18 b Fl(z)j Fo(is)d(a)g(non{in)o(terior)e(p)q(oin)o(t)i(of)g Fh(Z)1009 2342 y Fk(N)1047 2335 y Fo(,)h(then)f(w)o(e)g(ha)o(v)o(e)g Fh(k)p Fl(z)r Fh(k)1462 2317 y Fj(2)1501 2335 y Fl(>)e(cN)24 b Fo(for)18 b(some)f(\014xed)0 2395 y(constan)o(t)f Fl(c)d(>)h Fo(0,)i(so)g(that)h(the)g(righ)o(t)f(hand)f(side)h(of)33 b(\(2.5\))17 b(ma)o(y)f(b)q(e)h(replaced)f(b)o(y)735 2537 y Fl(const)11 b Fh(\001)g Fl(e)911 2516 y Fn(\000)p Fk(\024cN)1021 2537 y Fl(e)1044 2516 y Fn(\000)p Fj(2)p Fk(\024)p Fn(k)p Fk(z)q Fn(k)1178 2501 y Fb(2)1202 2537 y Fl(:)607 b Fo(\(2)p Fl(:)p Fo(17\))0 2670 y(The)19 b(resulting)e(inequalit)o(y)h(sho)o(ws)f(that)38 b(\(2.16\))19 b(also)f(holds)g(for)g(non{in)o(terior)e(p)q(oin)o(ts)i(of)h Fh(Z)1808 2677 y Fk(N)1846 2670 y Fo(.)28 b(By)963 2790 y(8)p eop %%Page: 9 10 bop 0 50 a Fo(using)15 b(the)i(iden)o(tit)o(y)33 b(\(2.11\),)17 b(the)f(left)h(hand)f(side)g(of)33 b(\(2.15\))17 b(can)f(b)q(e)h(written)g (as)f(follo)o(ws)60 124 y Fg(\014)60 154 y(\014)76 126 y(\000)p 99 126 54 2 v 40 x Fl(W)153 173 y Fk(N)191 166 y Fl(R)229 173 y Fk(N)267 166 y Fl( )301 146 y Fj(\()p Fk(N)t Fj(\))299 179 y Fk(u;s)370 126 y Fg(\001)393 166 y Fo(\()p Fl(z)r Fo(\))c Fh(\000)f Fl( )552 140 y Fj(\()p Fk(N)t Fj(\))550 183 y Fk(u;s)p Fj(+1)p Fk(=)n(N)711 166 y Fo(\()p Fl(z)r Fo(\))774 124 y Fg(\014)774 154 y(\014)113 303 y Fo(=)166 246 y Fg(\014)166 276 y(\014)166 306 y(\014)199 270 y Fo(1)p 188 292 46 2 v 188 338 a Fl(N)240 248 y Fg(n)299 241 y Fk(d)273 256 y Fg(X)273 363 y Fk(k)q Fj(=1)354 303 y Fl(a)380 310 y Fk(k)405 263 y Fg(\000)427 303 y Fo(1)p Fl(=)477 259 y Fh(p)p 519 259 V 44 x Fl(N)5 b(;)j(z)611 263 y Fg(\001)635 303 y Fl( )669 283 y Fj(\()p Fk(N)t Fj(\))667 316 y Fk(u;s)738 303 y Fo(\()p Fl(z)r Fo(\))k(+)888 241 y Fk(d)863 256 y Fg(X)863 363 y Fk(k)q Fj(=1)943 303 y Fl(b)964 310 y Fk(k)989 263 y Fg(\000)1012 303 y Fo(1)p Fl(=)1062 259 y Fh(p)p 1103 259 V 1103 303 a Fl(N)6 b(;)i(z)1196 263 y Fg(\001\000)1242 303 y Fl(D)1283 310 y Fk(N)r(;k)1353 303 y Fl( )1387 283 y Fj(\()p Fk(N)t Fj(\))1385 316 y Fk(u;s)1456 263 y Fg(\001)1479 303 y Fo(\()p Fl(z)r Fo(\))110 480 y(+)186 417 y Fk(d)160 432 y Fg(X)160 540 y Fk(k)q Fj(=1)241 480 y Fl(q)263 487 y Fk(k)288 439 y Fg(\000)311 480 y Fo(1)p Fl(=)361 435 y Fh(p)p 402 435 V 402 480 a Fl(N)d(;)j(z)494 439 y Fg(\001)q(\000)540 480 y Fo(\001)582 487 y Fk(N)r(;k)651 480 y Fl( )685 459 y Fj(\()p Fk(N)t Fj(\))683 492 y Fk(u;s)755 439 y Fg(\001)777 480 y Fo(\()p Fl(z)r Fo(\))13 b(+)902 439 y Fg(\000)925 480 y Fl(H)970 459 y Fj(\()p Fk(N)t Fj(\))966 492 y Fk(osc)1040 480 y Fl( )1074 459 y Fj(\()p Fk(N)t Fj(\))1072 492 y Fk(u;s)1143 439 y Fg(\001)1166 480 y Fo(\()p Fl(z)r Fo(\))1229 424 y Fg(o)110 633 y Fh(\000)160 578 y Fg(h)184 633 y Fl( )218 607 y Fj(\()p Fk(N)t Fj(\))216 650 y Fk(u;s)p Fj(+1)p Fk(=)n(N)377 633 y Fo(\()p Fl(z)r Fo(\))f Fh(\000)f Fl( )536 612 y Fj(\()p Fk(N)t Fj(\))534 645 y Fk(u;s)605 633 y Fo(\()p Fl(z)r Fo(\))h Fh(\000)746 599 y Fo(1)p 736 621 V 736 667 a Fl(N)805 599 y(d)p 793 621 50 2 v 793 667 a(ds)849 633 y( )883 612 y Fj(\()p Fk(N)t Fj(\))881 645 y Fk(u;s)952 633 y Fo(\()p Fl(z)r Fo(\))1015 578 y Fg(i)1050 633 y Fh(\000)1117 599 y Fo(1)p 1106 621 46 2 v 1106 667 a Fl(N)1158 578 y Fg(\020)1187 633 y Fl(H)1232 612 y Fj(\()p Fk(N)t Fj(\))1228 645 y Fk(osc)1302 633 y Fl( )1336 612 y Fj(\()p Fk(N)t Fj(\))1334 645 y Fk(u;s)1416 633 y Fo(+)1483 599 y Fl(d)p 1472 621 50 2 v 1472 667 a(ds)1527 633 y( )1561 612 y Fj(\()p Fk(N)t Fj(\))1559 645 y Fk(u;s)1630 578 y Fg(\021)1660 633 y Fo(\()p Fl(z)r Fo(\))1723 576 y Fg(\014)1723 605 y(\014)1723 635 y(\014)1749 633 y Fl(:)1823 403 y Fo(\(2)p Fl(:)p Fo(18\))0 756 y(The)k(term)h(in)f(curly)g(brac)o(k)o(ets)f(can)i(b)q (e)f(b)q(ounded)g(b)o(y)g(using)32 b(\(2.16\),)h(\(2.5\),)17 b(and)33 b(Lemma)15 b(2.2)i(:)378 829 y Fg(\014)378 859 y(\014)378 889 y(\014)411 853 y Fo(1)p 401 875 46 2 v 401 921 a Fl(N)452 831 y Fg(n)486 887 y Fl(:)8 b(:)g(:)544 831 y Fg(o)577 829 y(\014)577 859 y(\014)577 889 y(\014)607 887 y Fh(\024)674 853 y Fl(const)p 666 875 133 2 v 666 927 a(N)711 885 y Fh(p)p 753 885 46 2 v 42 x Fl(N)804 887 y(p)p Fo(\()p Fl(z)r Fo(\))p Fl(e)915 866 y Fn(\000)p Fj(2)p Fk(\024)p Fn(k)p Fk(z)q Fn(k)1049 851 y Fb(2)1088 887 y Fh(\024)14 b Fl(constN)1303 866 y Fn(\000)p Fj(3)p Fk(=)p Fj(2)1397 887 y Fl(e)1420 866 y Fn(\000)p Fk(\024)p Fn(k)p Fk(z)q Fn(k)1534 851 y Fb(2)1558 887 y Fl(:)251 b Fo(\(2)p Fl(:)p Fo(19\))0 1022 y(T)l(o)19 b(b)q(ound)f(the)i(term)f([)p Fl(:)8 b(:)g(:)p Fo(])19 b(in)38 b(\(2.18\),)21 b(w)o(e)e(apply)g(T)l(a)o (ylor's)e(form)o(ula,)h(together)i(with)38 b(\(2.5\),)21 b(and)0 1082 y(obtain)680 1100 y Fg(\014)680 1130 y(\014)697 1102 y(\002)717 1142 y Fl(:)8 b(:)g(:)776 1102 y Fg(\003)796 1100 y(\014)796 1130 y(\014)827 1142 y Fh(\024)13 b Fl(constN)1041 1121 y Fn(\000)p Fj(2)1095 1142 y Fl(e)1118 1121 y Fn(\000)p Fk(\024)p Fn(k)p Fk(z)q Fn(k)1232 1106 y Fb(2)1256 1142 y Fl(:)553 b Fo(\(2)p Fl(:)p Fo(20\))0 1246 y(The)23 b(assertion)e(no)o(w)h(follo)o(ws,)i(since)747 1205 y Fg(\000)770 1246 y Fl(H)815 1220 y Fj(\()p Fk(N)t Fj(\))811 1252 y Fk(osc)899 1246 y Fo(+)15 b Fl(d=ds)1053 1205 y Fg(\001)1076 1246 y Fl( )1110 1220 y Fj(\()p Fk(N)t Fj(\))1108 1252 y Fk(u;s)1204 1246 y Fo(=)24 b(0,)g(as)e(can)h(b)q(e)g(v)o(eri\014ed)f(b)o(y)g(direct)0 1305 y(computation;)15 b(see)i(also)f(Lemma)g(2.6.)p 1807 1383 43 43 v 100 1493 a(Due)21 b(to)g(the)h(factor)f Fl(N)557 1475 y Fn(\000)p Fj(3)p Fk(=)p Fj(2)673 1493 y Fo(in)42 b(\(2.15\),)22 b(w)o(e)f(can)g(rep)q(eat)h(this)e(b)q(ound)h Fl(n)g Fo(times)g(to)g (estimate)0 1512 y Fg(\000)p 23 1512 54 2 v 40 x Fl(W)77 1559 y Fk(N)115 1512 y Fg(\001)137 1522 y Fk(n)181 1552 y Fo(for)16 b Fl(n)e Fo(=)g Fn(O)p Fo(\()p Fl(N)450 1534 y Fj(3)p Fk(=)p Fj(2)514 1552 y Fo(\).)0 1655 y Fe(Lemma)f(2.4.)40 b Fd(Let)14 b Fl(N)19 b Fh(7!)14 b Fl(n)p Fo(\()p Fl(N)5 b Fo(\))13 b Fd(b)q(e)g(a)g (sequence)f(of)h(p)q(ositiv)o(e)f(in)o(tegers)g(suc)o(h)f(that)i Fl(N)1639 1637 y Fn(\000)p Fj(3)p Fk(=)p Fj(2)1734 1655 y Fl(n)p Fo(\()p Fl(N)5 b Fo(\))14 b Fh(!)g Fo(0)0 1715 y Fd(as)j Fl(N)k Fh(!)14 b(1)p Fd(.)25 b(Then)16 b(for)h(ev)o(ery)h(compact)f(subset)f Fh(U)23 b Fd(of)17 b Fo(I)-10 b(R)1125 1693 y Fk(d)1148 1715 y Fd(,)18 b(and)e(for)h(ev)o(ery)h Fl(\016)o(;)8 b(s)1553 1722 y Fk(o)1591 1715 y Fl(>)15 b Fo(0)p Fd(,)i(there)g(exists)0 1775 y(a)g(constan)o(t)e Fl(N)282 1782 y Fk(o)318 1775 y Fl(>)f Fo(0)j Fd(suc)o(h)e(that)465 1848 y Fg(\015)465 1878 y(\015)493 1850 y(\000)p 516 1850 V 40 x Fl(W)570 1897 y Fk(N)607 1850 y Fg(\001)630 1860 y Fk(n)p Fj(\()p Fk(N)t Fj(\))724 1890 y Fl(J)752 1897 y Fk(N)789 1890 y Fl( )823 1870 y Fj(\()p Fn(1)p Fj(\))821 1903 y Fk(u;s)908 1890 y Fh(\000)c Fl(J)986 1897 y Fk(N)1023 1890 y Fl(T)1059 1864 y Fj(\()p Fn(1)p Fj(\))1052 1907 y Fk(n)p Fj(\()p Fk(N)t Fj(\))p Fk(=)n(N)1199 1890 y Fl( )1233 1870 y Fj(\()p Fn(1)p Fj(\))1231 1903 y Fk(u;s)1307 1848 y Fg(\015)1307 1878 y(\015)1334 1910 y Fk(N)1386 1890 y Fh(\024)j Fl(\016)c(;)338 b Fo(\(2)p Fl(:)p Fo(21\))0 2003 y Fd(whenev)o(er)16 b Fl(u)d Fh(2)h(U)5 b Fd(,)17 b Fl(s)d Fh(\025)f Fl(s)487 2010 y Fk(o)527 2003 y Fd(and)i Fl(N)20 b Fh(\025)13 b Fl(N)775 2010 y Fk(o)798 2003 y Fd(.)0 2124 y Fe(Pro)r(of.)k Fo(First,)e(w)o(e)i(note) f(that)h Fl(T)637 2098 y Fj(\()p Fk(N)t Fj(\))630 2136 y Fk(t)706 2124 y Fl( )740 2098 y Fj(\()p Fk(N)t Fj(\))738 2129 y Fk(u;s)823 2124 y Fo(=)d Fl( )910 2098 y Fj(\()p Fk(N)t Fj(\))908 2136 y Fk(u;s)p Fj(+)p Fk(t)1027 2124 y Fo(for)i Fl(s;)8 b(t)15 b(>)e Fo(0)k(and)f(for)g Fl(N)22 b Fo(large)16 b(\(including)f Fh(1)p Fo(\).)0 2183 y(By)i(using)f(the)g(triangle)g(inequalit)o(y)l(,)g(w)o (e)g(estimate)h(the)g(left)g(hand)e(side)h(of)33 b(\(2.21\))17 b(as)g(follo)o(ws.)89 2258 y Fg(\015)89 2287 y(\015)116 2260 y(\000)p 139 2260 V 40 x Fl(W)193 2307 y Fk(N)231 2260 y Fg(\001)254 2270 y Fk(n)281 2300 y Fl(J)309 2307 y Fk(N)346 2300 y Fl( )380 2279 y Fj(\()p Fn(1)p Fj(\))378 2312 y Fk(u;s)465 2300 y Fh(\000)11 b Fl(J)543 2307 y Fk(N)580 2300 y Fl( )614 2274 y Fj(\()p Fn(1)p Fj(\))612 2317 y Fk(s)p Fj(+)p Fk(n=)n(N)743 2258 y Fg(\015)743 2287 y(\015)770 2320 y Fk(N)822 2300 y Fh(\024)874 2258 y Fg(\015)874 2287 y(\015)902 2260 y(\000)p 925 2260 V 40 x Fl(W)979 2307 y Fk(N)1017 2260 y Fg(\001)1040 2270 y Fk(n)1067 2300 y Fl(J)1095 2307 y Fk(N)1132 2260 y Fg(\000)1155 2300 y Fl( )1189 2279 y Fj(\()p Fn(1)p Fj(\))1187 2312 y Fk(u;s)1274 2300 y Fh(\000)f Fl( )1357 2279 y Fj(\()p Fk(N)t Fj(\))1355 2312 y Fk(u;s)1426 2260 y Fg(\001)1449 2258 y(\015)1449 2287 y(\015)1477 2320 y Fk(N)292 2397 y Fo(+)342 2355 y Fg(\015)342 2385 y(\015)369 2397 y Fl(J)397 2404 y Fk(N)435 2357 y Fg(\000)458 2397 y Fl( )492 2372 y Fj(\()p Fk(N)t Fj(\))490 2414 y Fk(u;s)p Fj(+)p Fk(n=)n(N)667 2397 y Fh(\000)g Fl( )750 2372 y Fj(\()p Fn(1)p Fj(\))748 2414 y Fk(u;s)p Fj(+)p Fk(n=)n(N)914 2357 y Fg(\001)937 2355 y(\015)937 2385 y(\015)965 2417 y Fk(N)1014 2397 y Fo(+)1063 2355 y Fg(\015)1063 2385 y(\015)1091 2357 y(\000)p 1114 2357 V 40 x Fl(W)1168 2404 y Fk(N)1206 2357 y Fg(\001)1229 2367 y Fk(n)1256 2357 y Fg(\000)1278 2397 y Fl(J)1306 2404 y Fk(N)1355 2397 y Fh(\000)h Fl(R)1443 2404 y Fk(N)1480 2357 y Fg(\001)1503 2397 y Fl( )1537 2377 y Fj(\()p Fk(N)t Fj(\))1535 2410 y Fk(u;s)1606 2355 y Fg(\015)1606 2385 y(\015)1634 2417 y Fk(N)292 2495 y Fo(+)342 2453 y Fg(\015)342 2483 y(\015)369 2455 y(\000)392 2495 y Fl(R)430 2502 y Fk(N)479 2495 y Fh(\000)g Fl(J)557 2502 y Fk(N)594 2455 y Fg(\001)617 2495 y Fl( )651 2469 y Fj(\()p Fk(N)t Fj(\))649 2512 y Fk(u;s)p Fj(+)p Fk(n=)n(N)815 2453 y Fg(\015)815 2483 y(\015)842 2515 y Fk(N)891 2495 y Fo(+)941 2453 y Fg(\015)941 2483 y(\015)969 2455 y(\000)p 992 2455 V 40 x Fl(W)1046 2502 y Fk(N)1083 2455 y Fg(\001)1106 2465 y Fk(n)1133 2495 y Fl(R)1171 2502 y Fk(N)1209 2495 y Fl( )1243 2475 y Fj(\()p Fk(N)t Fj(\))1241 2507 y Fk(u;s)1323 2495 y Fh(\000)g Fl(R)1411 2502 y Fk(N)1449 2495 y Fl( )1483 2469 y Fj(\()p Fk(N)t Fj(\))1481 2512 y Fk(u;s)p Fj(+)p Fk(n=)n(N)1646 2453 y Fg(\015)1646 2483 y(\015)1674 2515 y Fk(N)1720 2495 y Fl(;)1823 2397 y Fo(\(2)p Fl(:)p Fo(22\))0 2610 y(where)16 b Fl(n)e Fo(=)f Fl(n)p Fo(\()p Fl(N)5 b Fo(\).)23 b(The)16 b(\014rst)g(t)o(w)o(o)g(terms)g(on)g(the)g(righ)o(t)g(hand)f(side)h(of)33 b(\(2.22\))16 b(can)h(b)q(e)f(b)q(ounded)f(as)0 2670 y(desired)d(b)o(y)i (using)26 b(Lemma)13 b(2.1,)h(together)f(with)h(the)g(fact)g(that)g(the)g (maps)f Fl(J)1467 2677 y Fk(N)1518 2670 y Fo(:)22 b Fl(L)1588 2652 y Fj(1)1610 2670 y Fo(\(I)-10 b(R)1674 2648 y Fk(d)1697 2670 y Fo(\))15 b Fh(!)e Fl(`)1815 2652 y Fj(1)1837 2670 y Fo(\()p Fh(Z)1892 2677 y Fk(N)1931 2670 y Fo(\))963 2790 y(9)p eop %%Page: 10 11 bop 0 50 a Fo(and)p 94 10 54 2 v 13 w Fl(W)148 57 y Fk(N)200 50 y Fo(:)22 b Fl(`)257 32 y Fj(1)279 50 y Fo(\()p Fh(Z)334 57 y Fk(N)372 50 y Fo(\))15 b Fh(!)e Fl(`)490 32 y Fj(1)512 50 y Fo(\()p Fh(Z)567 57 y Fk(N)606 50 y Fo(\))h(are)g(of)g(norm)f Fh(\024)h Fo(1.)21 b(T)l(o)14 b(get)g(the)h(necessary)e(estimates)h(for)g (the)g(third)0 110 y(and)19 b(fourth)g(term,)h(w)o(e)g(can)f(apply)g(the)h (mean)f(v)m(alue)h(theorem)f(and)g(then)h(b)q(ound)f(the)h(deriv)m(ativ)o(es) 0 169 y(using)32 b(\(2.5\).)22 b(The)17 b(last)f(term)g(in)33 b(\(2.22\))17 b(is)f(con)o(trolled)f(b)o(y)h(using)32 b(Lemma)16 b(2.3)g(:)113 258 y Fg(\015)113 288 y(\015)140 260 y(\000)p 163 260 V 40 x Fl(W)217 307 y Fk(N)255 260 y Fg(\001)278 270 y Fk(n)305 300 y Fl(R)343 307 y Fk(N)381 300 y Fl( )415 280 y Fj(\()p Fk(N)t Fj(\))413 312 y Fk(u;s)495 300 y Fh(\000)11 b Fl(R)583 307 y Fk(N)620 300 y Fl( )654 274 y Fj(\()p Fk(N)t Fj(\))652 317 y Fk(u;s)p Fj(+)p Fk(n=)n(N)818 258 y Fg(\015)818 288 y(\015)846 320 y Fk(N)319 428 y Fo(=)371 371 y Fg(\015)371 401 y(\015)371 431 y(\015)438 366 y Fk(n)414 381 y Fg(X)407 486 y Fk(m)p Fj(=1)493 388 y Fg(\000)p 516 388 V 40 x Fl(W)570 435 y Fk(N)608 388 y Fg(\001)631 398 y Fk(n)p Fn(\000)p Fk(m)724 388 y Fg(\002)p 745 388 V 40 x Fl(W)799 435 y Fk(N)837 428 y Fl(R)875 435 y Fk(N)912 428 y Fl( )946 402 y Fj(\()p Fk(N)t Fj(\))944 445 y Fk(s)p Fj(+\()p Fk(m)p Fn(\000)p Fj(1\))p Fk(=)n(N)1179 428 y Fh(\000)g Fl(R)1267 435 y Fk(N)1304 428 y Fl( )1338 402 y Fj(\()p Fk(N)t Fj(\))1336 445 y Fk(s)p Fj(+)p Fk(m=)n(N)1477 388 y Fg(\003)1498 371 y(\015)1498 401 y(\015)1498 431 y(\015)1526 463 y Fk(N)319 593 y Fh(\024)402 531 y Fk(n)378 546 y Fg(X)371 652 y Fk(m)p Fj(=1)465 551 y Fg(\015)465 581 y(\015)p 493 553 V 12 x Fl(W)547 600 y Fk(N)585 593 y Fl(R)623 600 y Fk(N)661 593 y Fl( )695 568 y Fj(\()p Fk(N)t Fj(\))693 610 y Fk(s)p Fj(+\()p Fk(m)p Fn(\000)p Fj(1\))p Fk(=)n(N)927 593 y Fh(\000)g Fl(R)1015 600 y Fk(N)1052 593 y Fl( )1086 568 y Fj(\()p Fk(N)t Fj(\))1084 610 y Fk(s)p Fj(+)p Fk(m=)n(N)1226 551 y Fg(\015)1226 581 y(\015)1253 613 y Fk(N)1305 593 y Fh(\024)i Fl(n)e Fh(\001)g Fl(const)d(N)1593 573 y Fn(\000)p Fj(3)p Fk(=)p Fj(2)1696 593 y Fl(:)1823 465 y Fo(\(2)p Fl(:)p Fo(23\))p 1807 802 43 43 v 0 920 a Fe(Pro)r(of)17 b(of)g(Theorem)g(1.1.)40 b Fo(Assume)15 b(that)h(the)g(h)o(yp)q(othesis)f(of)i(Theorem)e(1.1.)21 b(are)16 b(satis\014ed,)f(and)0 987 y(let)i Fl(\016)g(>)d Fo(0.)23 b(Since)17 b(b)o(y)33 b(Prop)q(osition)16 b(1.2,)807 945 y Fg(\015)807 975 y(\015)834 987 y Fl(T)870 961 y Fj(\()p Fn(1)p Fj(\))863 993 y Fk(s)944 987 y Fl(f)h Fh(\000)11 b Fl(f)1064 945 y Fg(\015)1064 975 y(\015)1092 1007 y Fk(L)1119 997 y Fb(1)1156 987 y Fh(!)j Fo(0)j(as)g Fl(s)e Fh(#)f Fo(0,)j(w)o(e)g(can)g(\014nd)f Fl(s)f(>)f Fo(0,)j(and)0 1064 y(functions)f Fl(c)237 1071 y Fj(1)259 1064 y Fl( )293 1038 y Fj(\()p Fn(1)p Fj(\))291 1070 y Fk(u)315 1075 y Fb(1)334 1070 y Fk(;s)367 1064 y Fl(;)8 b(:)g(:)g(:)h(;)f(c)500 1071 y Fk(m)537 1064 y Fl( )571 1038 y Fj(\()p Fn(1)p Fj(\))569 1070 y Fk(u)593 1075 y Ff(m)626 1070 y Fk(;s)660 1064 y Fo(,)16 b(suc)o(h)f(that)704 1168 y Fg(\015)704 1198 y(\015)704 1228 y(\015)731 1225 y Fl(f)i Fh(\000)840 1163 y Fk(m)822 1178 y Fg(X)823 1284 y Fk(j)r Fj(=1)902 1225 y Fl(c)924 1232 y Fk(j)945 1225 y Fl( )979 1205 y Fj(\()p Fn(1)p Fj(\))977 1237 y Fk(u)1001 1242 y Ff(j)1019 1237 y Fk(;s)1052 1168 y Fg(\015)1052 1198 y(\015)1052 1228 y(\015)1080 1260 y Fk(L)1107 1250 y Fb(1)1143 1225 y Fl(<)1202 1191 y(\016)p 1202 1214 25 2 v 1202 1259 a Fo(4)1233 1225 y Fl(:)576 b Fo(\(2)p Fl(:)p Fo(24\))0 1397 y(By)34 b(Lemma)16 b(2.4,)g(there)g(exists)h Fl(N)672 1404 y Fk(o)708 1397 y Fl(>)c Fo(0)k(suc)o(h)e(that)395 1541 y Fh(j)p Fl(c)431 1548 y Fk(j)451 1541 y Fh(j)c(\001)501 1499 y Fg(\015)501 1528 y(\015)529 1501 y(\000)p 551 1501 54 2 v 551 1541 a Fl(W)605 1548 y Fk(N)643 1501 y Fg(\001)666 1511 y Fk(n)p Fj(\()p Fk(N)t Fj(\))759 1541 y Fl(J)787 1548 y Fk(N)825 1541 y Fl( )859 1520 y Fj(\()p Fn(1)p Fj(\))857 1553 y Fk(u)881 1558 y Ff(j)900 1553 y Fk(;s)944 1541 y Fh(\000)g Fl(J)1022 1548 y Fk(N)1059 1541 y Fl(T)1095 1515 y Fj(\()p Fn(1)p Fj(\))1088 1558 y Fk(n)p Fj(\()p Fk(N)t Fj(\))p Fk(=)n(N)1235 1541 y Fl( )1269 1520 y Fj(\()p Fn(1)p Fj(\))1267 1553 y Fk(u)1291 1558 y Ff(j)1310 1553 y Fk(;s)1343 1499 y Fg(\015)1343 1528 y(\015)1371 1561 y Fk(N)1422 1541 y Fh(\024)1503 1507 y Fl(\016)p 1481 1529 69 2 v 1481 1575 a Fo(2)p Fl(m)1823 1541 y Fo(\(2)p Fl(:)p Fo(25\))0 1676 y(for)16 b(1)e Fh(\024)g Fl(j)i Fh(\024)d Fl(m)p Fo(,)k(and)e Fl(N)20 b Fh(\025)13 b Fl(N)580 1683 y Fk(o)602 1676 y Fo(.)22 b(As)17 b(a)g(consequence,)f(w)o(e)g(ha)o(v)o(e)141 1766 y Fg(\015)141 1796 y(\015)169 1768 y(\000)p 192 1768 54 2 v 40 x Fl(W)246 1815 y Fk(N)284 1768 y Fg(\001)306 1778 y Fk(n)p Fj(\()p Fk(N)t Fj(\))400 1808 y Fl(J)428 1815 y Fk(N)465 1808 y Fl(f)h Fh(\000)11 b Fl(J)584 1815 y Fk(N)621 1808 y Fl(T)657 1782 y Fj(\()p Fn(1)p Fj(\))650 1825 y Fk(n)p Fj(\()p Fk(N)t Fj(\))p Fk(=)n(N)798 1808 y Fl(f)827 1766 y Fg(\015)827 1796 y(\015)855 1828 y Fk(N)509 1936 y Fh(\024)562 1879 y Fg(\015)562 1909 y(\015)562 1939 y(\015)589 1881 y(h)613 1896 y(\000)p 636 1896 V 40 x Fl(W)690 1943 y Fk(N)727 1896 y Fg(\001)750 1906 y Fk(n)p Fj(\()p Fk(N)t Fj(\))844 1936 y Fl(J)872 1943 y Fk(N)920 1936 y Fh(\000)g Fl(J)998 1943 y Fk(N)1035 1936 y Fl(T)1071 1910 y Fj(\()p Fn(1)p Fj(\))1064 1953 y Fk(n)p Fj(\()p Fk(N)t Fj(\))p Fk(=)n(N)1212 1881 y Fg(i)o(\020)1265 1936 y Fl(f)17 b Fh(\000)1374 1874 y Fk(m)1356 1889 y Fg(X)1357 1995 y Fk(j)r Fj(=1)1436 1936 y Fl(c)1458 1943 y Fk(j)1478 1936 y Fl( )1512 1916 y Fj(\()p Fn(1)p Fj(\))1510 1948 y Fk(u)1534 1953 y Ff(j)1553 1948 y Fk(;s)1586 1881 y Fg(\021)1616 1879 y(\015)1616 1909 y(\015)1616 1939 y(\015)1644 1971 y Fk(N)506 2109 y Fo(+)574 2047 y Fk(m)556 2061 y Fg(X)558 2168 y Fk(j)r Fj(=1)636 2109 y Fh(j)p Fl(c)672 2116 y Fk(j)693 2109 y Fh(j)11 b(\001)743 2066 y Fg(\015)743 2096 y(\015)770 2068 y(\000)p 793 2069 V 41 x Fl(W)847 2116 y Fk(N)885 2068 y Fg(\001)908 2079 y Fk(n)p Fj(\()p Fk(N)t Fj(\))1001 2109 y Fl(J)1029 2116 y Fk(N)1067 2109 y Fl( )1101 2088 y Fj(\()p Fn(1)p Fj(\))1099 2121 y Fk(u)1123 2126 y Ff(j)1141 2121 y Fk(;s)1185 2109 y Fh(\000)g Fl(J)1263 2116 y Fk(N)1301 2109 y Fl(T)1337 2083 y Fj(\()p Fn(1)p Fj(\))1330 2126 y Fk(n)p Fj(\()p Fk(N)t Fj(\))p Fk(=)n(N)1477 2109 y Fl( )1511 2088 y Fj(\()p Fn(1)p Fj(\))1509 2121 y Fk(u)1533 2126 y Ff(j)1551 2121 y Fk(;s)1585 2066 y Fg(\015)1585 2096 y(\015)1612 2129 y Fk(N)509 2268 y Fh(\024)j Fo(2)d Fh(\001)629 2234 y Fl(\016)p 628 2256 25 2 v 628 2302 a Fo(4)670 2268 y(+)g Fl(m)g Fh(\001)828 2234 y Fl(\016)p 806 2256 69 2 v 806 2302 a Fo(2)p Fl(m)894 2268 y Fo(=)j Fl(\016)c(:)1823 2039 y Fo(\(2)p Fl(:)p Fo(26\))0 2403 y(for)16 b(all)g Fl(N)k Fh(\025)13 b Fl(N)297 2410 y Fk(o)319 2403 y Fo(.)22 b(Since)17 b Fl(\016)e(>)f Fo(0)i(w)o(as)g(arbitrary)l(,)f(the)i(assertion)e(follo)o (ws.)p 1807 2482 43 43 v 100 2600 a(In)k(what)i(follo)o(ws,)e Fl(N)26 b Fo(is)20 b(assumed)e(to)j(\014xed)f(and)f(su\016cien)o(tly)g(large) g(\(or)h Fh(1)p Fo(\),)h(suc)o(h)e(that)i(the)0 2670 y(matrices)15 b Fl(Q)238 2652 y Fj(\()p Fk(N)t Fj(\))324 2670 y Fo(and)420 2630 y Fg(\000)443 2670 y Fl(Q)482 2652 y Fj(\()p Fk(N)t Fj(\))551 2630 y Fg(\001)574 2640 y Fn(\000)p Fj(1)628 2670 y Fl(B)668 2652 y Fj(\()p Fk(N)t Fj(\))753 2670 y Fo(are)h(w)o(ell)f(de\014ned)g(and)h (p)q(ositiv)o(e.)21 b(T)l(o)16 b(simplify)f(notation,)h(the)950 2790 y(10)p eop %%Page: 11 12 bop 0 50 a Fo(index)138 32 y Fj(\()p Fk(N)t Fj(\))227 50 y Fo(will)20 b(b)q(e)g(omitted)g(from)g(no)o(w)f(on.)32 b(Since)20 b Fl(Q)h Fo(and)e Fl(Q)1246 32 y Fn(\000)p Fj(1)1300 50 y Fl(B)j Fo(are)e(p)q(ositiv)o(e,)h(w)o(e)f(can)g(\014nd)f(a)0 110 y(p)q(ositiv)o(e)d (diagonal)g(matrix)g(\003)g(and)g(an)g(orthogonal)f(matrix)h Fl(V)28 b Fo(suc)o(h)15 b(that)601 234 y Fl(V)641 214 y Fn(\003)664 234 y Fo(\003)p Fl(V)24 b Fo(=)805 194 y Fg(\000)828 234 y Fl(Q)867 214 y Fn(\000)p Fj(1)921 234 y Fl(B)961 194 y Fg(\001)984 204 y Fj(1)p Fk(=)p Fj(2)1046 234 y Fl(Q)1085 194 y Fg(\000)1109 234 y Fl(Q)1148 214 y Fn(\000)p Fj(1)1201 234 y Fl(B)1241 194 y Fg(\001)1265 204 y Fj(1)p Fk(=)p Fj(2)1335 234 y Fl(:)474 b Fo(\(2)p Fl(:)p Fo(27\))100 348 y(In)18 b(order)f(to)i(discuss)e(the)h (prop)q(erties)g(of)g(the)h(time)f(ev)o(olution)g Fl(f)23 b Fh(7!)17 b Fl(T)1474 355 y Fk(t)1491 348 y Fl(f)5 b Fo(,)20 b(it)f(is)f(con)o(v)o(enien)o(t)f(to)0 408 y(in)o(tro)q(duce)f(the)g(Hilb)q (ert)h(space)f Fh(H)f(\021)e Fl(L)755 390 y Fj(2)777 367 y Fg(\000)800 408 y Fo(I)-10 b(R)845 386 y Fk(d)868 408 y Fl(;)8 b Fo(\012)926 415 y Fj(0)949 408 y Fo(\()p Fl(x)p Fo(\))1015 390 y Fn(\000)p Fj(1)1069 408 y Fl(d)1095 390 y Fk(d)1118 408 y Fl(x)1146 367 y Fg(\001)1186 408 y Fo(with)17 b(the)g(inner)e(pro)q(duct) 443 552 y Fh(h)p Fl(f)s(;)8 b(g)r Fh(i)14 b(\021)623 484 y Fg(Z)672 552 y Fl(d)698 531 y Fk(d)722 552 y Fl(x)8 b Fo(\012)794 559 y Fj(0)817 552 y Fo(\()p Fl(x)p Fo(\))883 531 y Fn(\000)p Fj(1)937 552 y Fl(f)d Fo(\()p Fl(x)p Fo(\))p 1032 509 93 2 v Fl(g)r Fo(\()p Fl(x)p Fo(\))11 b Fl(;)158 b(f)s(;)8 b(g)15 b Fh(2)g(H)8 b Fl(;)316 b Fo(\(2)p Fl(:)p Fo(28\))0 688 y(where)16 b(\012)180 695 y Fj(0)219 688 y Fo(is)g(the)h(function)f(giv)o(en)g(b)o(y)184 823 y(\012)220 830 y Fj(0)242 823 y Fo(\()p Fl(x)p Fo(\))f(=)376 782 y Fg(\002)397 823 y Fo(2)p Fl(\031)e Fh(\001)e Fo(det)557 782 y Fg(\000)580 823 y Fl(B)620 802 y Fn(\000)p Fj(1)674 823 y Fl(Q)713 782 y Fg(\001\003)757 793 y Fn(\000)p Fj(1)p Fk(=)p Fj(2)859 823 y Fo(exp)935 767 y Fg(\020)965 823 y Fh(\000)1010 789 y Fo(1)p 1010 811 25 2 v 1010 857 a(2)1040 782 y Fg(\012)1064 823 y Fl(x;)d(Q)1153 802 y Fn(\000)p Fj(1)1207 823 y Fl(B)r(x)1275 782 y Fg(\013)1299 767 y(\021)1337 823 y Fl(;)108 b(x)15 b Fh(2)f Fo(I)-10 b(R)1593 801 y Fk(d)1625 823 y Fl(:)184 b Fo(\(2)p Fl(:)p Fo(29\))0 953 y(Denote)21 b(b)o(y)g Fh(D)i Fo(the)e(\(dense\))g (subspace)e(of)j Fh(H)f Fo(spanned)f(b)o(y)g(the)h(function)g(\012)1514 960 y Fj(0)1557 953 y Fo(and)f(all)h(its)g(partial)0 1013 y(deriv)m(ativ)o (es,)16 b(and)g(de\014ne)g(op)q(erators)f Fl(A)760 1020 y Fj(1)783 1013 y Fl(;)8 b(:)g(:)g(:)h(;)f(A)931 1020 y Fk(d)971 1013 y Fo(on)16 b Fh(D)i Fo(as)f(follo)o(ws.)153 1132 y Fg(\000)176 1173 y Fl(A)213 1180 y Fk(m)251 1173 y Fl(f)280 1132 y Fg(\001)304 1173 y Fo(\()p Fl(x)p Fo(\))e Fh(\021)464 1111 y Fk(d)438 1125 y Fg(X)438 1233 y Fk(k)q Fj(=1)518 1173 y Fl(M)566 1180 y Fk(mk)627 1117 y Fg(h)683 1139 y Fl(@)p 656 1161 83 2 v 656 1207 a(@)s(x)713 1214 y Fk(k)755 1173 y Fo(+)805 1132 y Fg(\000)828 1173 y Fl(Q)867 1152 y Fn(\000)p Fj(1)921 1173 y Fl(B)r(x)989 1132 y Fg(\001)1012 1193 y Fk(k)1037 1117 y Fg(i)1060 1173 y Fl(f)5 b Fo(\()p Fl(x)p Fo(\))j Fl(;)110 b(f)20 b Fh(2)14 b(D)q Fl(;)59 b(x)14 b Fh(2)g Fo(I)-10 b(R)1624 1151 y Fk(d)1655 1173 y Fl(;)154 b Fo(\(2)p Fl(:)p Fo(30\))0 1353 y(where)16 b Fl(M)k Fh(\021)13 b Fl(V)304 1312 y Fg(\000)327 1353 y Fl(Q)366 1335 y Fn(\000)p Fj(1)420 1353 y Fl(B)460 1312 y Fg(\001)483 1323 y Fn(\000)p Fj(1)p Fk(=)p Fj(2)577 1353 y Fo(.)22 b(A)17 b(short)f(calculation)f(sho)o(ws)g (that)i(the)g(adjoin)o(t)f(of)h Fl(A)1646 1360 y Fk(m)1701 1353 y Fo(satis\014es)363 1480 y Fg(\000)386 1520 y Fl(A)423 1500 y Fn(\003)423 1533 y Fk(m)461 1520 y Fl(f)490 1480 y Fg(\001)514 1520 y Fo(\()p Fl(x)p Fo(\))e(=)f Fh(\000)721 1458 y Fk(d)695 1473 y Fg(X)695 1580 y Fk(k)q Fj(=1)775 1520 y Fl(M)823 1527 y Fk(mk)916 1487 y Fl(@)p 890 1509 V 890 1555 a(@)s(x)947 1562 y Fk(k)978 1520 y Fl(f)5 b Fo(\()p Fl(x)p Fo(\))j Fl(;)110 b(f)19 b Fh(2)14 b(D)q Fl(;)59 b(x)14 b Fh(2)h Fo(I)-10 b(R)1541 1499 y Fk(d)1573 1520 y Fl(:)236 b Fo(\(2)p Fl(:)p Fo(31\))0 1715 y Fe(Lemma)17 b(2.5.)44 b Fd(F)l(or)15 b(ev)o(ery)h Fl(f)k Fh(2)14 b(D)q Fd(,)j(and)f(for)g Fo(1)e Fh(\024)g Fl(m;)8 b(k)15 b Fh(\024)f Fl(d)p Fd(,)i(the)h(follo)o(wing)e(holds.)10 1775 y Fo(\()p Fl(a)p Fo(\))100 1734 y Fg(\002)120 1775 y Fl(A)157 1782 y Fk(m)196 1775 y Fl(;)8 b(A)255 1782 y Fk(k)280 1734 y Fg(\003)300 1775 y Fl(f)20 b Fo(=)397 1734 y Fg(\002)417 1775 y Fl(A)454 1757 y Fn(\003)454 1787 y Fk(m)492 1775 y Fl(;)8 b(A)551 1757 y Fn(\003)551 1789 y Fk(k)576 1734 y Fg(\003)597 1775 y Fl(f)20 b Fo(=)13 b(0)p Fd(.)15 1839 y Fo(\()p Fl(b)p Fo(\))100 1799 y Fg(\002)120 1839 y Fl(A)157 1846 y Fk(m)196 1839 y Fl(;)8 b(A)255 1821 y Fn(\003)255 1853 y Fk(k)280 1799 y Fg(\003)300 1839 y Fl(f)20 b Fo(=)14 b Fl(\016)419 1846 y Fk(mk)479 1839 y Fl(f)5 b Fd(.)14 1911 y Fo(\()p Fl(c)p Fo(\))26 b Fl(H)141 1918 y Fk(osc)200 1911 y Fl(f)19 b Fo(=)296 1874 y Fg(P)349 1886 y Fk(d)349 1926 y(k)q Fj(=1)432 1911 y Fo(\003)467 1918 y Fk(k)q(k)513 1911 y Fl(A)550 1893 y Fn(\003)550 1925 y Fk(k)575 1911 y Fl(A)612 1918 y Fk(k)637 1911 y Fl(f)5 b Fd(.)0 2028 y Fe(Pro)r(of.)16 b Fo(P)o(art)f(\()p Fl(a)p Fo(\))h(follo)o(ws)f (immediately)f(from)30 b(\(2.31\),)16 b(and)f(the)h(other)f(t)o(w)o(o)g(iden) o(tities)f(are)h(easy)h(to)0 2088 y(pro)o(v)o(e)f(b)o(y)i(using)e(that)i Fl(M)495 2070 y Fn(\003)518 2088 y Fo(\003)p Fl(M)j Fo(=)13 b Fl(Q)k Fo(and)f Fl(M)5 b(Q)918 2070 y Fn(\000)p Fj(1)973 2088 y Fl(B)r(M)1066 2070 y Fn(\003)1103 2088 y Fo(=)14 b(I.)p 1807 2166 43 43 v 100 2279 a(This)h(sho)o(ws)g(that)i Fl(H)505 2286 y Fk(osc)581 2279 y Fo(is)f(\(up)g(to)g(a)h(constan)o(t\))f(the)h (Hamiltonian)e(of)h Fl(d)h Fo(harmonic)d(oscillators)0 2345 y(with)j(frequencies)f(\003)404 2352 y Fk(k)q(k)450 2345 y Fo(.)22 b(The)16 b(eigen)o(v)m(alues)g(of)h Fl(H)944 2352 y Fk(osc)1020 2345 y Fo(are)f(linear)g(com)o(binations)e Fl(E)1582 2352 y Fk(\013)1624 2345 y Fh(\021)1677 2308 y Fg(P)1729 2320 y Fk(d)1729 2360 y(k)q Fj(=1)1813 2345 y Fl(\013)1845 2352 y Fk(k)1869 2345 y Fo(\003)1904 2352 y Fk(k)q(k)0 2405 y Fo(of)g(these)g (frequencies,)g(with)g(nonnegativ)o(e)f(in)o(teger)g(co)q(e\016cien)o(ts)h Fl(\013)1275 2412 y Fk(k)1299 2405 y Fo(,)h(and)e(the)h(corresp)q(onding)e (eigen-)0 2465 y(v)o(ectors)k(are)g(the)h(Hermite)g(functions)e(\012)781 2472 y Fk(\013)810 2465 y Fo(,)754 2625 y(\012)790 2632 y Fk(\013)833 2625 y Fh(\021)911 2563 y Fk(d)889 2578 y Fg(Y)885 2685 y Fk(k)q Fj(=1)958 2585 y Fg(\000)980 2625 y Fl(A)1017 2604 y Fn(\003)1017 2637 y Fk(k)1042 2585 y Fg(\001)1065 2595 y Fk(\013)1091 2600 y Ff(k)1115 2625 y Fo(\012)1151 2632 y Fj(0)1182 2625 y Fl(;)627 b Fo(\(2)p Fl(:)p Fo(32\))950 2790 y(11)p eop %%Page: 12 13 bop 0 50 a Fo(whic)o(h)16 b(form)f(an)h(orthogonal)g(basis)f(for)h Fh(H)p Fo(.)0 146 y Fe(Lemma)h(2.6.)44 b Fd(On)16 b Fh(H)h Fd(w)o(e)f(ha)o(v)o(e)g Fl(T)696 153 y Fk(t)727 146 y Fo(=)e Fl(e)803 128 y Fn(\000)p Fk(tH)882 133 y Ff(osc)954 146 y Fd(for)i(all)g Fl(t)e(>)g Fo(0)p Fd(.)0 265 y Fe(Pro)r(of.)22 b Fo(Let)f Fl(t)f(>)g Fo(0.)33 b(As)21 b(the)f(discussion)e(ab)q(o)o(v)o(e)i(sho)o(ws,)g Fl(H)1193 272 y Fk(osc)1273 265 y Fo(is)f(essen)o(tially)h(self{adjoin)o(t)f (on)h Fh(D)0 325 y Fo(and)f(p)q(ositiv)o(e.)30 b(Th)o(us)18 b Fl(e)464 307 y Fn(\000)p Fk(tH)543 312 y Ff(osc)618 325 y Fo(is)h(w)o(ell)g(de\014ned)g(and)g(b)q(ounded.)29 b(F)l(urthermor,)18 b(b)o(y)h(using)f(H\177)-25 b(older's)0 385 y(inequalit)o(y)l(,)16 b(w)o(e)g(\014nd)g(that)268 485 y Fg(\014)268 515 y(\014)285 527 y Fh(h)p Fl(g)r(;)8 b(T)381 534 y Fk(t)399 527 y Fl(f)d Fh(i)447 485 y Fg(\014)447 515 y(\014)464 497 y Fj(2)500 527 y Fh(\024)14 b(k)p Fl(g)r Fh(k)629 507 y Fj(2)651 527 y Fh(k)p Fl(f)5 b Fh(k)730 507 y Fj(2)761 459 y Fg(Z)811 527 y Fl(d)837 507 y Fk(d)860 527 y Fl(x)888 459 y Fg(Z)938 527 y Fl(d)964 507 y Fk(d)987 527 y Fl(u)j(\036)1054 487 y Fg(\000)1077 527 y Fl(t;)g(x)j Fh(\000)g Fl(e)1229 507 y Fn(\000)p Fk(tB)1310 527 y Fl(u)1339 487 y Fg(\001)1361 497 y Fj(2)1384 527 y Fo(\012)1420 534 y Fj(0)1442 527 y Fo(\()p Fl(u)p Fo(\))p Fl(=)p Fo(\012)1570 534 y Fj(0)1592 527 y Fo(\()p Fl(x)p Fo(\))d Fl(;)143 b Fo(\(2)p Fl(:)p Fo(33\))0 670 y(for)19 b(ev)o(ery)g Fl(g)r(;)8 b(f)24 b Fh(2)19 b(H)p Fo(.)30 b(This)19 b(pro)o(v)o(es)e(that)j Fl(T)860 677 y Fk(t)897 670 y Fo(de\014nes)e(a)i(b)q(ounded)e(linear)g(op)q(erator)h (on)f Fh(H)p Fo(;)k(in)c(fact)0 729 y Fh(k)p Fl(T)54 736 y Fk(t)72 729 y Fh(k)23 b(\024)g Fo(1)f(since)f(the)i(double)e(in)o(tegral)g (in)44 b(\(2.33\))22 b(equals)g(1.)39 b(Th)o(us,)22 b(it)g(su\016ces)f(to)i (v)o(erify)f(that)0 789 y Fl(T)29 796 y Fk(t)47 789 y Fo(\012)83 796 y Fk(\013)128 789 y Fo(=)16 b Fl(e)206 771 y Fn(\000)p Fk(tH)285 776 y Ff(osc)340 789 y Fo(\012)376 796 y Fk(\013)423 789 y Fo(for)i(all)g Fl(\013)p Fo(.)27 b(The)18 b(case)g Fl(\013)f Fo(=)f(0)i(is)g(easy)g(to)h(c)o(hec)o(k.)26 b(T)l(o)18 b(get)h(the)g(other)f (cases)f(w)o(e)0 849 y(pro)q(ceed)f(b)o(y)g(induction.)21 b(By)c(using)f(the) g(iden)o(tit)o(y)g Fl(M)5 b(B)1065 831 y Fn(\003)1089 849 y Fl(M)1142 831 y Fn(\000)p Fj(1)1210 849 y Fo(=)14 b(\003,)i(w)o(e)g(obtain) 156 975 y Fg(\000)179 1015 y Fl(T)208 1022 y Fk(t)226 1015 y Fl(A)263 994 y Fn(\003)263 1027 y Fk(m)301 1015 y Fo(\012)337 1022 y Fk(\013)365 975 y Fg(\001)388 1015 y Fo(\()p Fl(x)p Fo(\))f(=)f Fh(\000)595 953 y Fk(d)569 968 y Fg(X)569 1075 y Fk(k)q Fj(=1)649 1015 y Fl(M)697 1022 y Fk(mk)766 947 y Fg(Z)816 1015 y Fl(d)842 994 y Fk(d)865 1015 y Fl(u)8 b(\036)932 975 y Fg(\000)954 1015 y Fl(t;)g(x)k Fh(\000)f Fl(e)1107 994 y Fn(\000)p Fk(tB)1187 1015 y Fl(u)1216 975 y Fg(\001)1271 981 y Fl(@)p 1245 1004 83 2 v 1245 1049 a(@)s(u)1303 1056 y Fk(k)1333 1015 y Fo(\012)1369 1022 y Fk(\013)1397 1015 y Fo(\()p Fl(u)p Fo(\))469 1191 y(=)j Fh(\000)607 1129 y Fk(d)581 1144 y Fg(X)569 1251 y Fk(k)q(;l)p Fj(=1)674 1191 y Fl(M)722 1198 y Fk(mk)782 1151 y Fg(\000)805 1191 y Fl(e)828 1171 y Fn(\000)p Fk(tB)909 1151 y Fg(\001)932 1211 y Fk(lk)997 1158 y Fl(@)p 975 1180 73 2 v 975 1225 a(@)s(x)1032 1232 y Fk(l)1062 1123 y Fg(Z)1112 1191 y Fl(d)1138 1171 y Fk(d)1161 1191 y Fl(u)8 b(\036)1228 1151 y Fg(\000)1250 1191 y Fl(t;)g(x)k Fh(\000)f Fl(e)1403 1171 y Fn(\000)p Fk(tB)1484 1191 y Fl(u)1513 1151 y Fg(\001)1535 1191 y Fo(\012)1571 1198 y Fk(\013)1599 1191 y Fo(\()p Fl(u)p Fo(\))469 1374 y(=)572 1312 y Fk(d)547 1327 y Fg(X)522 1433 y Fk(m;n)p Fj(=1)644 1334 y Fg(\000)667 1374 y Fl(M)5 b(e)743 1354 y Fn(\000)p Fk(tB)821 1339 y Fc(\003)845 1374 y Fl(M)898 1354 y Fn(\000)p Fj(1)952 1334 y Fg(\001)975 1394 y Fk(mn)1038 1334 y Fg(\000)1060 1374 y Fl(A)1097 1354 y Fn(\003)1097 1387 y Fk(n)1125 1374 y Fl(T)1154 1381 y Fk(t)1171 1374 y Fo(\012)1207 1381 y Fk(\013)1236 1334 y Fg(\001)1259 1374 y Fo(\()p Fl(x)p Fo(\))469 1509 y(=)14 b Fl(e)545 1488 y Fn(\000)p Fk(t)p Fj(\003)618 1493 y Ff(mm)684 1509 y Fl(e)707 1488 y Fn(\000)p Fk(tE)782 1493 y Ff(\013)811 1468 y Fg(\000)834 1509 y Fl(A)871 1488 y Fn(\003)871 1521 y Fk(m)909 1509 y Fo(\012)945 1516 y Fk(\013)973 1468 y Fg(\001)996 1509 y Fo(\()p Fl(x)p Fo(\))h(=)1130 1468 y Fg(\000)1152 1509 y Fl(e)1175 1488 y Fn(\000)p Fk(tH)1254 1493 y Ff(osc)1310 1509 y Fl(A)1347 1488 y Fn(\003)1347 1521 y Fk(m)1385 1509 y Fo(\012)1421 1516 y Fk(\013)1449 1468 y Fg(\001)1472 1509 y Fo(\()p Fl(x)p Fo(\))8 b Fl(;)1823 1237 y Fo(\(2)p Fl(:)p Fo(34\))0 1637 y(for)16 b Fl(x)f Fh(2)f Fo(I)-10 b(R)210 1615 y Fk(d)234 1637 y Fo(,)16 b(1)e Fh(\024)f Fl(m)h Fh(\024)f Fl(d)p Fo(,)k(and)f(for)g(ev)o(ery)g(m)o(ulti{index)f Fl(\013)p Fo(.)22 b(This)16 b(pro)o(v)o(es)f(the)i(assertion.)p 1807 1716 43 43 v 0 1831 a Fe(Pro)r(of)j(of)h(Prop)r(osition)h(1.2.)51 b Fo(The)19 b(smo)q(othness)f(of)h(the)g(function)g(\()p Fl(t;)8 b(x)p Fo(\))20 b Fh(7!)1595 1791 y Fg(\000)1617 1831 y Fl(T)1646 1838 y Fk(t)1664 1831 y Fl(f)1693 1791 y Fg(\001)1717 1831 y Fo(\()p Fl(x)p Fo(\))g(follo)o(ws)0 1900 y(from)f(the)i(de\014nition)39 b(\(1.17\))20 b(of)h Fl(T)693 1907 y Fk(t)710 1900 y Fo(,)g(giv)o(en)f(the)g (fact)h(that)g(at)f(ev)o(ery)g(p)q(oin)o(t)g(in)g(\(0)p Fl(;)8 b Fh(1)p Fo(\))14 b Fh(\002)f Fo(I)-10 b(R)1823 1878 y Fk(d)1846 1900 y Fo(,)21 b(the)0 1959 y(partial)15 b(deriv)m(ativ)o(es)h(of)g(the)g (functions)f Fl(\036)793 1966 y Fk(u)833 1959 y Fo(:)f(\()p Fl(t;)8 b(x)p Fo(\))15 b Fh(7!)e Fl(\036)1075 1919 y Fg(\000)1098 1959 y Fl(t;)8 b(x)j Fh(\000)e Fl(e)1248 1941 y Fn(\000)p Fk(tB)1329 1959 y Fl(u)1358 1919 y Fg(\001)1396 1959 y Fo(are)16 b(b)q(ounded)f (uniformly)f(in)0 2028 y Fl(u)p Fo(.)30 b(F)l(urthermore,)17 b(b)o(y)38 b(Lemma)19 b(2.6,)g(w)o(e)g(ha)o(v)o(e)942 1987 y Fg(\000)965 2028 y Fl(H)1006 2035 y Fk(osc)1078 2028 y Fo(+)12 b Fl(@)s(=@)s(t)1230 1987 y Fg(\001)1253 2028 y Fl(\036)1283 2035 y Fk(u)1328 2028 y Fo(=)18 b(0)h(for)g(ev)o(ery)g Fl(u)f Fh(2)h Fo(I)-10 b(R)1785 2006 y Fk(d)1809 2028 y Fo(.)30 b(This)0 2088 y(pro)o(v)o(es)14 b(assertion)h(\()p Fl(a)p Fo(\).)23 b(T)l(o)15 b(pro)o(v)o(e)g(\()p Fl(b)p Fo(\),)i(\014x)e Fl(")g(>)e Fo(0.)22 b(Since)15 b Fh(H)h Fo(can)g(b)q(e)g(iden)o(ti\014ed)f(with)g(a)h (dense)f(linear)0 2147 y(subspace)g(of)i Fl(L)299 2129 y Fj(1)321 2107 y Fg(\000)344 2147 y Fo(I)-10 b(R)389 2126 y Fk(d)412 2147 y Fl(;)8 b(d)460 2129 y Fk(d)483 2147 y Fl(x)p Fo(\),)17 b(w)o(e)g(can)f(\014nd)g(a)g(function)g Fl(g)g Fh(2)e(H)j Fo(suc)o(h)e(that)i Fh(k)p Fl(f)g Fh(\000)11 b Fl(g)r Fh(k)1595 2156 y Fk(L)1622 2146 y Fb(1)1657 2147 y Fl(<)j(\017=)p Fo(3.)22 b(Th)o(us,)393 2230 y Fg(\015)393 2260 y(\015)420 2232 y(\000)443 2272 y Fl(T)472 2279 y Fk(t)501 2272 y Fh(\000)11 b Fo(I)569 2232 y Fg(\001)591 2272 y Fl(f)620 2230 y Fg(\015)620 2260 y(\015)649 2292 y Fk(L)676 2282 y Fb(1)712 2272 y Fh(\024)765 2230 y Fg(\015)765 2260 y(\015)792 2232 y(\000)815 2272 y Fl(T)844 2279 y Fk(t)873 2272 y Fh(\000)g Fo(I)941 2232 y Fg(\001)963 2272 y Fl(g)989 2230 y Fg(\015)989 2260 y(\015)1017 2292 y Fk(L)1044 2282 y Fb(1)1077 2272 y Fo(+)1127 2230 y Fg(\015)1127 2260 y(\015)1154 2232 y(\000)1177 2272 y Fl(T)1206 2279 y Fk(t)1235 2272 y Fh(\000)g Fo(I)1303 2232 y Fg(\001)1326 2272 y Fo(\()p Fl(f)17 b Fh(\000)11 b Fl(g)r Fo(\))1481 2230 y Fg(\015)1481 2260 y(\015)1508 2292 y Fk(L)1535 2282 y Fb(1)712 2356 y Fh(\024)765 2314 y Fg(\015)765 2344 y(\015)792 2316 y(\000)815 2356 y Fl(e)838 2336 y Fn(\000)p Fk(tH)917 2341 y Ff(osc)983 2356 y Fh(\000)g Fo(I)1051 2316 y Fg(\001)1074 2356 y Fl(g)1100 2314 y Fg(\015)1100 2344 y(\015)1127 2376 y Fn(H)1175 2356 y Fo(+)g(2)g Fh(\001)g Fl("=)p Fo(3)22 b Fh(\024)13 b Fl(")8 b(;)1823 2315 y Fo(\(2)p Fl(:)p Fo(35\))0 2476 y(if)20 b Fl(t)e(>)h Fo(0)g(is)g(su\016cien)o(tly)f(small.)30 b(Here,)20 b(w)o(e)f(ha)o(v)o(e)g(used)f(that)i Fl(T)1236 2483 y Fk(t)1273 2476 y Fo(is)f(a)h(b)q(ounded)e(liner)h(op)q(erator)f(on)0 2535 y Fl(L)34 2517 y Fj(1)56 2495 y Fg(\000)79 2535 y Fo(I)-10 b(R)124 2514 y Fk(d)147 2535 y Fl(;)8 b(d)195 2517 y Fk(d)218 2535 y Fl(x)p Fo(\))20 b(of)g(norm)d(one,)j(and)e(that)i Fh(k)p Fl(g)r Fh(k)872 2544 y Fk(L)899 2534 y Fb(1)938 2535 y Fo(=)e Fh(h)p Fo(\012)1050 2542 y Fj(0)1073 2535 y Fl(;)8 b Fh(j)p Fl(g)r Fh(ji)18 b(\024)f(k)p Fl(g)r Fh(k)1318 2542 y Fn(H)1354 2535 y Fo(,)j(for)f(ev)o(ery)g(function)g Fl(g)g Fh(2)g(H)p Fo(.)0 2595 y(Since)d Fl(")e(>)g Fo(0)i(w)o(as)g(arbitrary)l(,)f(\()p Fl(b)p Fo(\))j(follo)o(ws.)i(The)d(pro)q(of)f(of)h(\()p Fl(c)p Fo(\))g(is)f(similar.)p 1807 2674 V 950 2790 a(12)p eop %%Page: 13 14 bop 0 50 a Fe(References)22 179 y Fo([1])25 b(J.L.)18 b(v)m(an)g(Hemmen,)h (R.)f(K)q(\177)-26 b(uhn,)18 b Fa(Nonline)m(ar)j(Neur)m(al)f(Networks)p Fo(,)f(Ph)o(ys.)28 b(Rev.)g(Lett.)h Fe(57)p Fo(,)18 b(913)100 244 y(\(1986\).)22 309 y([2])25 b(J.J.)15 b(Hop\014eld,)h Fa(Neur)m(al)j (Networks)f(and)h(Physic)m(al)g(Systems)g(with)f(Emer)m(gent)f(Col)s(le)m (ctive)g(Com-)100 374 y(putational)h(A)o(bilities)p Fo(,)f(Pro)q(c.)k(Natl.)i (Acad.)f(Sci.)g(USA)17 b Fe(79)p Fo(,)e(2554)i(\(1982\).)22 438 y([3])25 b(L.)14 b(P)o(ersonnaz,)f(I.)i(Guy)o(on,)f(G.)h(Dreyfus,)f Fa(Information)k(Stor)m(age)e(and)h(R)m(etrieval)g(in)f(Spin{Glass)100 503 y(like)h(Neur)m(al)h(Networks)p Fo(,)f(J.)f(Ph)o(ysique)g(Lett.)23 b Fe(46)p Fo(,)16 b(L-359)g(\(1985\).)22 568 y([4])25 b(S.)17 b(Diederic)o(h,)h(M.)g(Opp)q(er,)f Fa(L)m(e)m(arning)22 b(of)e(Corr)m(elate)m (d)g(Patterns)f(in)h(Spin{Glass)h(Networks)f(by)100 633 y(L)m(o)m(c)m(al)g(L) m(e)m(arning)f(R)o(ules)p Fo(,)d(Ph)o(ys.)22 b(Rev.)g(Lett.)h Fe(58)p Fo(,)15 b(949)i(\(1987\).)22 697 y([5])25 b(K.W.)17 b(Berryman,)g(M.E.)h(Inc)o(hiosa,)e(A.M.)i(Ja\013e,)g(S.A.)f(Jano)o(wsky)l(,) g Fa(Conver)m(genc)m(e)j(of)g(an)g(Iter-)100 762 y(ative)f(Neur)m(al)h (Network)g(L)m(e)m(arning)h(A)o(lgorithm)f(for)g(Line)m(arly)h(Dep)m(endent)f (Patterns)p Fo(,)f(Harv)m(ard)100 827 y(Univ)o(ersit)o(y)c(Preprin)o(t)g (HUTMP)i(89/B237)f(\(1989\).)22 892 y([6])25 b(H.)16 b(Ko)q(c)o(h,)f(J.)g (Piask)o(o,)g Fa(Some)i(R)o(igor)m(ous)i(R)m(esults)f(on)f(the)g(Hop\014eld)i (Neur)m(al)e(Network)g(Mo)m(del)p Fo(,)g(J.)100 956 y(Stat.)22 b(Ph)o(ys.)f Fe(55)p Fo(,)16 b(903)g(\(1989\).)22 1021 y([7])25 b(D.J.)15 b(Amit,)i Fa(The)h(Pr)m(op)m(erties)h(of)g(Mo)m(dels)f(of)h(Simple) f(Neur)m(al)g(Networks)p Fo(,)f(in)f(\\Heidelb)q(erg)g(Col-)100 1086 y(lo)q(quium)i(on)h(Glassy)f(Dynamics)h(\(1986\)",)h(J.L.)e(v)m(an)i (Hemmen)e(and)h(I.)g(Morgenstern)e(\(eds.\),)100 1151 y(Lecture)f(Notes)h(in) f(Ph)o(ysics)g Fe(275)p Fo(,)f(430)i(\(1987\).)22 1215 y([8])25 b(J.L.)16 b(v)m(an)g(Hemmen,)g(D.)h(Grensing,)e(A.)h(Hub)q(er,)h(R.)f(K)q (\177)-26 b(uhn,)15 b Fa(Nonline)m(ar)k(Neur)m(al)f(Networks:)100 1280 y(I.)g(Gener)m(al)g(The)m(ory)p Fo(,)f(J.)f(Stat.)23 b(Ph)o(ys.)e Fe(50)p Fo(,)16 b(231)g(\(1987\).)100 1345 y Fa(II.)i(Information)h(Pr)m(o)m (c)m(essing)p Fo(,)g(J.)d(Stat.)23 b(Ph)o(ys.)e Fe(50)p Fo(,)15 b(259)i(\(1987\).)22 1410 y([9])25 b(R.J.)11 b(McEliece,)j(E.C.)e(P)o(osner,) g(E.R.)g(Ro)q(demic)o(h,)g(S.S.)g(V)l(enk)m(atesh,)i Fa(The)g(Cap)m(acity)i (of)f(the)f(Hop-)100 1474 y(\014eld)j(Asso)m(ciative)j(Memory)p Fo(,)c(IEEE)g(T)l(ransactions)f(on)g(Information)g(Theory)h Fe(33)p Fo(,)g(461)g(\(1986\).)-3 1539 y([10])25 b(C.M.)11 b(Newman,)i Fa(Memory)h(Cap)m(acity)h(in)g(Neur)m(al)f(Network)g(Mo)m(dels:) 22 b(R)o(igor)m(ous)16 b(L)m(ower)f(Bounds)p Fo(,)100 1604 y(Neural)h(Net)o(w)o(orks)g Fe(1)p Fo(,)g(223)g(\(1988\).)-3 1669 y([11])25 b(J.)18 b(Koml\023)-25 b(os,)17 b(R.)h(P)o(aturi,)g Fa(Conver)m(genc)m(e)i(R)m(esults)g(in)g(the)f(Hop\014eld)i(Mo)m(del)p Fo(,)f(Preprin)o(t,)d(UC)i(San)100 1733 y(Diego)d(\(1987\).)-3 1798 y([12])25 b(U.)12 b(Riedel,)h(R.)f(K)q(\177)-26 b(uhn,)12 b(J.L.)g(v)m(an)g(Hemmen,)h Fa(T)l(emp)m(or)m(al)i(Se)m(quenc)m(es)g(and)g (Chaos)h(in)e(Neur)m(al)g(Nets)p Fo(,)100 1863 y(Ph)o(ys.)21 b(Rev.)h(A)17 b Fe(38)p Fo(,)e(1105)i(\(1988\).)-3 1928 y([13])25 b(M.)17 b(Cassandro,)e(A.)j(Galv)o(es,)f(E.)g(Olivieri,)g(M.E.)g(V)l(ares,)g Fa(Metastable)h(Behavior)i(of)g(Sto)m(chastic)100 1993 y(Dynamics:)j(A)18 b(Pathwise)h(Appr)m(o)m(ach)p Fo(,)h(J.)c(Stat.)22 b(Ph)o(ys.)f Fe(35)p Fo(,)16 b(603)g(\(1984\).)950 2790 y(13)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ENDBODY