Content-Type: multipart/mixed; boundary="-------------0405271430842" This is a multi-part message in MIME format. ---------------0405271430842 Content-Type: text/plain; name="04-168.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-168.keywords" Runaway particle, Vlasov equation ---------------0405271430842 Content-Type: application/postscript; name="vlasov.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="vlasov.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: vlasov.dvi %%Pages: 14 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips vlasov -o %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.05.27:1631 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul N/hsize 8.5 72 mul 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(electric)i(\014elds)g(and)g(sho)n(w)f(the)i Fm(runaway)i(e\013e)l(ct)d Fq(for)f(an)n(y)g(singular)515 2914 y(but)i(in)n(tegrable)e (particle/\015uid)h(in)n(teraction.)639 3014 y(It)e(w)n(ould)e(b)r(e)h (reasonable)e(to)i(study)g(more)f(sophisticated)h(mo)r(dels)g(enric)n (hed)f(in)h(three)515 3113 y(directions:)62 b(i\))41 b(taking)f(in)n(to)g(accoun)n(t)g(the)h(full)g(particle/\015uid)f(in)n (teraction;)46 b(ii\))41 b(con-)515 3213 y(sidering)35 b(more)h(general)f(initial)i(data,)h(p)r(ossibly)f(in)n(tro)r(ducing)f (an)g(external)g(con\014ning)515 3313 y(p)r(oten)n(tial;)g(iii\))e (considering)e(a)i(singular)e(in)n(teraction)g(among)h(the)h(\015uid)g (particles)e(\(for)515 3412 y(instance)27 b(Coulom)n(b-lik)n(e)f(at)h (short)g(distance\).)37 b(T)-7 b(o)27 b(giv)n(e)f(an)h(idea)g(of)h(the) f(di\016culties,)h(w)n(e)515 3512 y(remark)f(that)j(the)f(existence)g (of)g(the)h(Vlaso)n(v)e(\015uid)h(with)h(in\014nite)g(mass)e(and)h (Coulom)n(b-)515 3611 y(lik)n(e)21 b(in)n(teraction)g(is)h(a)g(w)n (ell-kno)n(wn)e(and)i(unsolv)n(ed)f(problem)h(\(for)f(b)r(ounded)i (mass)e(see)h(the)515 3711 y(celebrated)28 b(pap)r(ers)g([10)o(,)h(11)o (,)g(13)o(]\).)41 b(A)30 b(result)e(for)h(a)f(particular)g(state)g (with)i(un)n(b)r(ounded)515 3811 y(mass)21 b(has)h(b)r(een)g(recen)n (tly)f(obtained)h([6].)35 b(The)22 b(existence)g(of)g(the)g(full)h (dynamics)f([c)n(harged)515 3910 y(particle])k(+)h([Vlaso)n(v)f (\015uid])i(is)f(another)f(problem)h(that)h(needs)f(atten)n(tion)g(and) g(a)g(careful)515 4010 y(statemen)n(t,)g(since,)h(a)f(priori,)g(blo)n (w-up)g(of)g(the)h(solution)f(ma)n(y)g(o)r(ccur.)639 4159 y(The)d(plan)g(of)g(the)g(pap)r(er)g(is)g(the)g(follo)n(wing.)35 b(In)24 b(Section)g(2)g(the)g(mo)r(del)g(is)g(de\014ned)g(and)515 4259 y(the)k(main)f(results)g(are)g(stated.)37 b(The)27 b(pro)r(ofs)g(are)g(giv)n(en)g(in)g(the)h(remaining)f(sections.)515 4534 y Fp(2)134 b(De\014nitions)46 b(and)f(results)515 4715 y Fq(W)-7 b(e)28 b(consider)e(a)h Fm(Vlasov)k(\015uid)d Fq(in)g(three)f(dimensions,)h(namely)-7 b(,)27 b(the)h(pair)f(of)g (functions:)934 4898 y(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(\))24 b Fj(7!)f Fq(\()p Fl(X)6 b Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g Fk(t)p Fq(\))p Fk(;)g Fl(V)22 b Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g Fk(t)p Fq(\)\))p Fk(;)2218 4876 y Fq(^)2200 4898 y Fk(f)2241 4910 y Fi(0)2278 4898 y Fq(\()p Fl(x)p Fk(;)g Fl(v)s Fq(\))23 b Fj(7!)2631 4876 y Fq(^)2614 4898 y Fk(f)8 b Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g Fk(t)p Fq(\))p Fk(;)1926 5255 y Fq(2)p eop %%Page: 3 3 3 2 bop 515 523 a Fq(where)27 b(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(\))23 b Fj(2)h Fh(R)1117 493 y Fi(3)1178 523 y Fj(\002)18 b Fh(R)1315 493 y Fi(3)1358 523 y Fq(,)28 b Fk(t)23 b Fj(2)h Fh(R)p Fq(,)33 b(solution)27 b(to:)1191 644 y Fg(8)1191 719 y(>)1191 744 y(>)1191 769 y(<)1191 918 y(>)1191 943 y(>)1191 968 y(:)1337 686 y Fq(_)1306 707 y Fl(X)6 b Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g Fk(t)p Fq(\))24 b(=)e Fl(V)g Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g Fk(t)p Fq(\))p Fk(;)1334 795 y Fq(_)1306 817 y Fl(V)22 b Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g Fk(t)p Fq(\))23 b(=)g(\()p Fl(F)32 b Fj(\003)1967 795 y Fq(^)1949 817 y Fk(f)8 b Fq(\))28 b(\()p Fl(X)7 b Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g Fk(t)p Fq(\);)g Fk(t)p Fq(\))p Fk(;)1306 850 y Fg(\000)1344 917 y Fl(X)7 b Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g(0\))p Fk(;)g Fl(V)21 b Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g(0\))2114 850 y Fg(\001)2174 917 y Fq(=)23 b(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(\))p Fk(;)1324 1005 y Fq(^)1306 1027 y Fk(f)9 b Fq(\()p Fl(X)d Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g Fk(t)p Fq(\))p Fk(;)g Fl(V)22 b Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g Fk(t)p Fq(\);)g Fk(t)p Fq(\))23 b(=)2362 1005 y(^)2344 1027 y Fk(f)2385 1039 y Fi(0)2422 1027 y Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(\))p Fk(;)3208 864 y Fq(\(2.1\))515 1223 y(where)24 b(\()p Fl(F)j Fj(\003)941 1201 y Fq(^)923 1223 y Fk(f)9 b Fq(\)\()p Fl(y)t Fq(;)14 b Fk(t)p Fq(\))1233 1176 y Fk(:)1212 1223 y Fq(=)1299 1156 y Fg(R)1355 1223 y Fk(d)p Fl(v)1448 1193 y Ff(0)1485 1156 y Fg(R)1541 1223 y Fk(d)p Fl(x)1638 1193 y Ff(0)1676 1223 y Fl(F)f Fq(\()p Fl(y)k Fj(\000)c Fl(x)1977 1193 y Ff(0)2000 1223 y Fq(\))2050 1201 y(^)2032 1223 y Fk(f)c Fq(\()p Fl(x)2169 1193 y Ff(0)2192 1223 y Fk(;)14 b Fl(v)2279 1193 y Ff(0)2302 1223 y Fq(;)g Fk(t)p Fq(\))26 b(and)e(the)i(force)e(\014eld)i Fl(F)13 b Fq(\()p Fl(x)p Fq(\))23 b(=)515 1322 y Fj(\000r)p Fq(\010\()p Fj(j)p Fl(x)p Fj(j)p Fq(\))37 b(with)h(the)g(p)r(oten)n (tial)f(\010)j Fj(2)g Fk(C)1883 1292 y Fi(2)1920 1322 y Fq(\()p Fh(R)p Fq(\))k(a)37 b(non-negativ)n(e,)h(ev)n(en)f(function)h (suc)n(h)515 1422 y(that:)1376 1522 y(\010\(0\))23 b Fk(>)f Fq(0)27 b(and)h(\010\()p Fk(r)r Fq(\))c(=)f(0)k(if)h Fk(r)e(>)d Fq(1)p Fk(:)515 1677 y Fq(In)28 b(the)h(case)e(of)i(smo)r (oth)f(initial)g(data)1777 1655 y(^)1759 1677 y Fk(f)8 b Fq(\()p Fl(X)f Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g Fk(t)p Fq(\))p Fk(;)g Fl(V)21 b Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g Fk(t)p Fq(\);)g 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Fq(W)-7 b(e)22 b(are)g(not)g(able)g (to)g(pro)n(v)n(e)e(this)i(conjecture)g(in)h(its)f(generalit)n(y)-7 b(,)22 b(mainly)g(b)r(ecause)f(w)n(e)h(are)515 5006 y(not)k(able)f(to)h (con)n(trol)f(the)h(time)h(ev)n(olution)e(of)h(spatial)g(regions)e (with)j(v)n(ery)d(high)n(t)i(densit)n(y)1926 5255 y(3)p eop %%Page: 4 4 4 3 bop 515 523 a Fq(of)26 b(the)g(Vlaso)n(v)e(\015uid.)37 b(W)-7 b(e)26 b(can)g(pro)n(v)n(e)e(the)i(conjecture)g(only)f(for)g (the)i(particular)d(class)h(of)515 623 y(initial)j(data)f(with)h Fm(planar)j(symmetry)p Fq(,)d(i.e.)f(when:)1066 777 y(^)1048 799 y Fk(f)1089 811 y Fi(0)1126 799 y Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(\))24 b(=)1461 777 y(^)1443 799 y Fk(f)1484 811 y Fi(0)1521 799 y Fq(\(\()p Fk(x)1632 811 y Fi(1)1670 799 y Fk(;)14 b Fq(0)p Fk(;)g Fq(0\))p Fk(;)g Fl(v)s Fq(\))166 b Fj(8)27 b Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(\))23 b Fj(2)g Fh(R)2580 764 y Fi(3)2642 799 y Fj(\002)18 b Fh(R)2779 764 y Fi(3)2822 799 y Fk(;)363 b Fq(\(2.5\))515 974 y(where)27 b Fl(x)c Fq(=)f(\()p Fk(x)999 986 y Fi(1)1037 974 y Fk(;)14 b(x)1121 986 y Fi(2)1159 974 y Fk(;)g(x)1243 986 y Fi(3)1280 974 y Fq(\).)639 1074 y(The)35 b(planar)f(symmetry)g(is)h(readily)f(seen)g(to)h(b)r(e)g (preserv)n(ed)f(b)n(y)g(Vlaso)n(v)g(equation:)533 1152 y(^)515 1174 y Fk(f)556 1186 y Fi(0)593 1174 y Fq(\()p Fl(x)p Fk(;)14 b Fl(v)s Fq(;)g Fk(t)p Fq(\))23 b(=)994 1152 y(^)977 1174 y Fk(f)8 b Fq(\(\()p Fk(x)1137 1186 y Fi(1)1175 1174 y Fk(;)14 b Fq(0)p Fk(;)g Fq(0\))p Fk(;)g Fl(v)r Fq(;)g Fk(t)p Fq(\).)38 b(In)27 b(particular,)g(the)h(function) 760 1392 y Fk(f)9 b Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))1113 1345 y Fk(:)1092 1392 y Fq(=)1180 1279 y Fg(Z)1263 1392 y Fk(dv)1346 1404 y Fi(2)1397 1392 y Fk(dv)1480 1404 y Fi(3)1550 1370 y Fq(^)1532 1392 y Fk(f)8 b Fq(\(\()p Fk(x)1692 1404 y Fi(1)1731 1392 y Fk(;)14 b Fq(0)p Fk(;)g Fq(0\))p Fk(;)g Fq(\()p Fk(v)s(;)g(v)2110 1404 y Fi(2)2147 1392 y Fk(;)g(v)2224 1404 y Fi(3)2261 1392 y Fq(\);)g Fk(t)p Fq(\))p Fk(;)180 b Fq(\()p Fk(x;)14 b(v)s Fq(\))24 b Fj(2)g Fh(R)g Fj(\002)18 b Fh(R)p Fk(;)515 1606 y Fq(is)27 b(the)h(unique)g(solution)f(to)h(the)g(one-dimensional)e(Vlaso)n(v)g (equation:)1223 1714 y Fg(8)1223 1788 y(>)1223 1813 y(>)1223 1838 y(<)1223 1988 y(>)1223 2012 y(>)1223 2037 y(:)1371 1762 y Fq(_)1338 1783 y Fk(X)6 b Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))24 b(=)f Fk(V)c Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))p Fk(;)1360 1868 y Fq(_)1338 1889 y Fk(V)19 b Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))24 b(=)e(\()p Fk(F)31 b Fj(\003)18 b Fk(f)9 b Fq(\))28 b(\()p Fk(X)7 b Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\);)g Fk(t)p Fq(\))p Fk(;)1338 1922 y Fg(\000)1376 1990 y Fk(X)7 b Fq(\()p Fk(x;)14 b(v)s Fq(;)g(0\))p Fk(;)g(V)k Fq(\()p Fk(x;)c(v)s Fq(;)g(0\))2095 1922 y Fg(\001)2157 1990 y Fq(=)23 b(\()p Fk(x;)14 b(v)s Fq(\))p Fk(;)1338 2089 y(f)9 b Fq(\()p Fk(X)e Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))p Fk(;)g(V)19 b Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\);)g Fk(t)p Fq(\))24 b(=)f Fk(f)2368 2101 y Fi(0)2405 2089 y Fq(\()p Fk(x;)14 b(v)s Fq(\))p Fk(;)3208 1934 y Fq(\(2.6\))515 2271 y(where)34 b(\()p Fk(F)i Fj(\003)23 b Fk(f)9 b Fq(\)\()p Fk(y)s Fq(;)14 b Fk(t)p Fq(\))1261 2224 y Fk(:)1240 2271 y Fq(=)1340 2204 y Fg(R)1395 2271 y Fk(dv)1481 2241 y Ff(0)1519 2204 y Fg(R)1574 2271 y Fk(dx)1664 2241 y Ff(0)1702 2271 y Fk(F)e Fq(\()p Fk(y)26 b Fj(\000)d Fk(x)2001 2241 y Ff(0)2025 2271 y Fq(\))p Fk(f)9 b Fq(\()p Fk(x)2186 2241 y Ff(0)2210 2271 y Fk(;)14 b(v)2290 2241 y Ff(0)2313 2271 y Fq(;)g Fk(t)p Fq(\))35 b(and)g Fk(F)12 b Fq(\()p Fk(x)p Fq(\))36 b(=)f Fj(\000)p Fk(')3047 2241 y Ff(0)3071 2271 y Fq(\()p Fk(x)p Fq(\))h(with)515 2388 y Fk(')p Fq(\()p Fk(x)p Fq(\))725 2341 y Fk(:)704 2388 y Fq(=)23 b(2)p Fk(\031)897 2321 y Fg(R)953 2342 y Fi(1)937 2418 y(0)990 2388 y Fk(dy)17 b(y)f Fq(\010\()1240 2315 y Fg(p)p 1324 2315 268 4 v 1324 2388 a Fk(x)1371 2364 y Fi(2)1427 2388 y Fq(+)i Fk(y)1554 2364 y Fi(2)1591 2388 y Fq(\).)37 b(Moreo)n(v)n(er,)25 b(b)n(y)j(\(2.4\),)g(for)f(some)g Fk(K)2815 2400 y Fi(0)2875 2388 y Fk(>)22 b Fq(0,)1507 2588 y(0)h Fj(\024)f Fk(f)1700 2600 y Fi(0)1737 2588 y Fq(\()p Fk(x;)14 b(v)s Fq(\))24 b Fj(\024)f Fk(K)2111 2600 y Fi(0)2162 2588 y Fk(e)2201 2554 y Ff(\000)p Fe(\025v)2327 2529 y Fc(2)2363 2588 y Fk(:)822 b Fq(\(2.7\))515 2764 y(Under)19 b(the)h(planar)e(symmetry)h(the)h(c)n(harged)d(particle)i (mo)n(v)n(es)f(along)g(the)i(\014rst)f(co)r(ordinate)515 2864 y(axis)27 b(\(recall)f Fl(\027)j Fq(=)23 b(\()p Fk(\027)q(;)14 b Fq(0)p Fk(;)g Fq(0\))27 b(and)g Fl(E)h Fq(=)22 b(\()p Fk(E)5 b(;)14 b Fq(0)p Fk(;)g Fq(0\)\),)28 b Fl(\030)s Fq(\()p Fk(t)p Fq(\))23 b(=)g(\()p Fk(\030)t Fq(\()p Fk(t)p Fq(\))p Fk(;)14 b Fq(0)p Fk(;)g Fq(0\),)28 b(via)f(the)h(equation:)1218 3060 y(\177)1209 3082 y Fk(\030)5 b Fq(\()p Fk(t)p Fq(\))23 b(=)g Fk(E)g Fj(\000)1622 2969 y Fg(Z)1705 3082 y Fk(dx)14 b(dv)k( )1967 3048 y Ff(0)1991 3082 y Fq(\()p Fk(\030)t Fq(\()p Fk(t)p Fq(\))h Fj(\000)f Fk(x)p Fq(\))c Fk(f)9 b Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))q Fk(;)1209 3231 y(\030)t Fq(\(0\))24 b(=)e(0)p Fk(;)1729 3209 y Fq(_)1711 3231 y Fk(\030)t Fq(\(0\))h(=)g Fk(\027)q(:)3208 3136 y Fq(\(2.8\))515 3429 y(where)k Fk( )s Fq(\()p Fk(\030)t Fq(\))960 3382 y Fk(:)940 3429 y Fq(=)22 b(2)p Fk(\031)1133 3362 y Fg(R)1188 3383 y Fi(1)1172 3459 y(0)1226 3429 y Fk(d\021)17 b(\021)g Fq(\011\()1482 3356 y Fg(p)p 1565 3356 261 4 v 73 x Fk(\030)1605 3405 y Fi(2)1661 3429 y Fq(+)h Fk(\021)1788 3405 y Fi(2)1825 3429 y Fq(\).)639 3529 y(W)-7 b(e)28 b(can)f(no)n(w)g(state)h(our)f (main)g(results:)515 3705 y Fr(Prop)s(osition)j(2.1)40 b Fm(F)-6 b(or)30 b Fk(f)1390 3717 y Fi(0)1427 3705 y Fq(\()p Fk(x;)14 b(v)s Fq(\))31 b Fm(satisfying)39 b Fq(\(2.7\))29 b Fm(ther)l(e)h(exist)f Fk(a;)14 b(K)2778 3717 y Fi(1)2838 3705 y Fk(>)23 b Fq(0)29 b Fm(such)h(that:)1230 3881 y Fj(j)p Fk(V)19 b Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))p Fj(j)24 b(\024)f Fk(K)1784 3893 y Fi(1)1839 3881 y Fq(+)18 b(2)p Fj(j)p Fk(v)s Fj(j)h Fq(+)f Fk(at)169 b Fj(8)14 b Fk(t)22 b Fj(\025)h Fq(0)p Fk(:)544 b Fq(\(2.9\))515 4057 y Fr(Theorem)30 b(2.2)41 b Fm(If)34 b Fk( )1232 4027 y Ff(0)1288 4057 y Fm(is)g(inte)l(gr)l(able,)h(ther)l(e)f(exist)f Fk(K)2262 4069 y Fi(2)2299 4057 y Fk(;)14 b(K)2407 4069 y Fi(3)2473 4057 y Fk(>)29 b Fq(0)k Fm(and)h Fk(\027)2848 4069 y Fi(0)2886 4057 y Fk(;)14 b(E)2984 4069 y Fi(0)3051 4057 y Fk(>)29 b Fq(0)k Fm(such)515 4156 y(that,)d(for)h(any)f Fk(\027)e Fj(\025)23 b Fk(\027)1200 4168 y Fi(0)1267 4156 y Fm(and)30 b Fk(E)e Fj(\025)23 b Fk(E)1666 4168 y Fi(0)1703 4156 y Fm(,)1443 4332 y Fj(j)1484 4310 y Fq(_)1466 4332 y Fk(\030)5 b Fq(\()p Fk(t)p Fq(\))19 b Fj(\000)f Fk(E)5 b(t)18 b Fj(\000)g Fk(\027)5 b Fj(j)24 b(\024)e Fk(K)2151 4344 y Fi(2)2188 4332 y Fk(t)c Fq(+)g Fk(K)2390 4344 y Fi(3)2427 4332 y Fk(:)717 b Fq(\(2.10\))515 4508 y Fr(Remark)31 b(2.3)40 b Fq(It)c(is)g(natural)f(to)g(ask)g(what)h (happ)r(ens)g(if)g Fk(\027)41 b Fq(and/or)34 b Fk(E)41 b Fq(are)35 b(small.)61 b(It)515 4608 y(is)36 b(trivial)g(to)g (consider)g(small)g(initial)h(v)n(elo)r(cities)e(and)i(corresp)r (ondingly)d(high)i(electric)515 4707 y(\014elds.)53 b(Indeed,)34 b(the)g(b)r(oundedness)f(of)f(the)i(force)e(due)h(to)g(the)g(\015uid)h (implies)f(that,)i(for)515 4807 y(large)19 b(enough)i(electric)g (\014elds,)h(the)g(particle)e(sp)r(eed)i(b)r(ecomes)e(large)g(within)i (a)f(\014nite)h(time;)515 4907 y(as)i(a)g(result,)h(Eq.)f(\(2.10\))f (holds)i(with)g(a)f(suitable)g(constan)n(t)g Fk(K)2492 4919 y Fi(3)2553 4907 y Fq(dep)r(ending)h(on)f Fk(\027)5 b Fq(.)37 b(Other)515 5006 y(cases)26 b(are)h(harder.)1926 5255 y(4)p eop %%Page: 5 5 5 4 bop 515 523 a Fp(3)134 b(Pro)t(of)45 b(of)h(Prop)t(osition)f(2.1) 515 705 y Fq(Prop)r(osition)34 b(2.1)h(is)g(obtained)h(pro)n(ving)e (\014rst)h(an)h(analogous)d(result)j(in)g(the)g(case)f(of)g(a)515 805 y(cut-o\013)27 b(v)n(ersion)f(of)i(the)g(Vlaso)n(v)e(equation)h (and)h(then)g(remo)n(ving)e(the)i(cut-o\013.)639 904 y(F)-7 b(or)31 b Fk(N)t(;)14 b(M)39 b(>)29 b Fq(0)i(let)h Fk(f)1361 864 y Fe(N)s(;M)1352 926 y Fi(0)1509 904 y Fq(\()p Fk(x;)14 b(v)s Fq(\))1752 857 y Fk(:)1731 904 y Fq(=)29 b Fk(f)1866 916 y Fi(0)1903 904 y Fq(\()p Fk(x;)14 b(v)s Fq(\))p Fk(\037)p Fq(\()p Fj(j)p Fk(x)p Fj(j)32 b(\024)d Fk(M)9 b Fq(\))p Fk(\037)p Fq(\()p Fj(j)p Fk(v)s Fj(j)30 b(\024)f Fk(N)9 b Fq(\))32 b(where)f Fk(\037)p Fq(\()p Fk(A)p Fq(\))515 1004 y(denotes)e(the)h(c)n(haracteristic)e (function)i(of)g(the)g(set)g Fk(A)p Fq(.)43 b(W)-7 b(e)30 b(in)n(tro)r(duce)g(the)g(sequence)f(of)515 1103 y(cut-o\013)e (problems:)696 1223 y Fg(8)696 1298 y(>)696 1323 y(>)696 1347 y(<)696 1497 y(>)696 1522 y(>)696 1547 y(:)844 1266 y Fq(_)811 1287 y Fk(X)887 1257 y Fe(N)s(;M)1035 1287 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))24 b(=)f Fk(V)1472 1257 y Fe(N)s(;M)1620 1287 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))p Fk(;)833 1373 y Fq(_)811 1394 y Fk(V)878 1363 y Fe(N)s(;M)1027 1394 y Fq(\()p Fk(x;)g(v)s Fq(;)g Fk(t)p Fq(\))24 b(=)e(\()p Fk(F)31 b Fj(\003)18 b Fk(f)1622 1363 y Fe(N)s(;M)1770 1394 y Fq(\))28 b(\()p Fk(X)1938 1363 y Fe(N)s(;M)2087 1394 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\);)g Fk(t)p Fq(\))p Fk(;)811 1427 y Fg(\000)849 1494 y Fk(X)925 1464 y Fe(N)s(;M)1073 1494 y Fq(\()p Fk(x;)g(v)s Fq(;)g(0\))p Fk(;)g(V)1448 1464 y Fe(N)s(;M)1596 1494 y Fq(\()p Fk(x;)g(v)s Fq(;)g(0\))1866 1427 y Fg(\001)1928 1494 y Fq(=)23 b(\()p Fk(x;)14 b(v)s Fq(\))p Fk(;)98 b Fj(j)p Fk(x)p Fj(j)23 b(\024)g Fk(M)t(;)41 b Fj(j)p Fk(v)s Fj(j)24 b(\024)e Fk(N)t(;)811 1604 y(f)861 1573 y Fe(N)s(;M)1009 1604 y Fq(\()p Fk(X)1117 1573 y Fe(N)s(;M)1266 1604 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))p Fk(;)g(V)1629 1573 y Fe(N)s(;M)1777 1604 y Fq(\()p Fk(x;)g(v)s Fq(;)g Fk(t)p Fq(\);)g Fk(t)p Fq(\))24 b(=)f Fk(f)2296 1564 y Fe(N)s(;M)2287 1626 y Fi(0)2444 1604 y Fq(\()p Fk(x;)14 b(v)s Fq(\))p Fk(:)3208 1443 y Fq(\(3.1\))515 1782 y(W)-7 b(e)28 b(next)g(set:)1120 1964 y Fj(V)1178 1930 y Fe(N)s(;M)1327 1964 y Fq(\()p Fk(t)p Fq(\))1465 1917 y Fk(:)1444 1964 y Fq(=)56 b(sup)1532 2038 y Fe(s)p Ff(2)p Fi([0)p Fe(;t)p Fi(])1944 1964 y Fq(sup)1793 2038 y Ff(j)p Fe(x)p Ff(j\024)p Fe(M)s(;)26 b Ff(j)p Fe(v)r Ff(j\024)p Fe(N)2234 1894 y Fg(\014)2234 1943 y(\014)2262 1964 y Fk(V)2329 1930 y Fe(N)s(;M)2478 1964 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(s)p Fq(\))2745 1894 y Fg(\014)2745 1943 y(\014)3208 1964 y Fq(\(3.2\))515 2210 y(and)1396 2346 y Fk(R)1460 2312 y Fe(N)s(;M)1608 2346 y Fq(\()p Fk(t)p Fq(\))1747 2299 y Fk(:)1726 2346 y Fq(=)23 b(1)18 b(+)1957 2233 y Fg(Z)2040 2254 y Fe(t)2003 2422 y Fi(0)2069 2346 y Fk(ds)c Fj(V)2223 2312 y Fe(N)s(;M)2371 2346 y Fq(\()p Fk(s)p Fq(\))p Fk(:)711 b Fq(\(3.3\))639 2542 y(Giv)n(en)32 b(a)f(densit)n(y)h Fk(f)9 b Fq(\()p Fk(x;)14 b(v)s Fq(\),)34 b Fk(\026)c Fj(2)h Fh(R)37 b Fq(and)32 b Fk(R)f(>)f Fq(0,)i(w)n(e)g(call)f Fm(smo)l(othe)l(d)k(ener) l(gy)d Fq(of)g(the)515 2642 y(ball)27 b(with)h(cen)n(ter)f Fk(\026)h Fq(and)f(radius)g Fk(R)i Fq(the)e(quan)n(tit)n(y:)964 2862 y Fj(W)7 b Fq(\()p Fk(f)i Fq(;)14 b Fk(\026;)g(R)q Fq(\))1398 2815 y Fk(:)1378 2862 y Fq(=)1475 2806 y(1)p 1475 2843 42 4 v 1475 2919 a(2)1540 2749 y Fg(Z)1624 2862 y Fk(dx)g(dv)j(g)1871 2828 y Fe(\026;R)1985 2862 y Fq(\()p Fk(x)p Fq(\))d Fk(f)9 b Fq(\()p Fk(x;)14 b(v)s Fq(\))g Fk(v)2408 2828 y Fi(2)1176 3080 y Fq(+)1283 3024 y(1)p 1283 3061 V 1283 3137 a(2)1348 2967 y Fg(Z)1431 3080 y Fk(dx)g(dv)k(g)1679 3046 y Fe(\026;R)1793 3080 y Fq(\()p Fk(x)p Fq(\))c Fk(f)9 b Fq(\()p Fk(x;)14 b(v)s Fq(\))2173 2967 y Fg(Z)2272 3080 y Fk(dy)i(dw)h(f)9 b Fq(\()p Fk(y)s(;)14 b(w)r Fq(\))g Fk(')p Fq(\()p Fk(x)20 b Fj(\000)e Fk(y)s Fq(\))p Fk(;)112 b Fq(\(3.4\))515 3313 y(where)20 b Fk(g)791 3282 y Fe(\026;R)905 3313 y Fq(\()p Fk(x)p Fq(\))1061 3265 y Fk(:)1040 3313 y Fq(=)j Fk(g)s Fq(\()p Fj(j)p Fk(x)5 b Fj(\000)g Fk(\026)p Fj(j)p Fk(=R)q Fq(\))19 b(with)i Fk(g)26 b Fj(2)d Fk(C)1969 3282 y Ff(1)2040 3313 y Fq(\()p Fh(R)2126 3325 y Fi(+)2187 3313 y Fq(\))e(suc)n(h)f(that)h Fk(g)s Fq(\()p Fk(x)p Fq(\))j(=)f(1)d(for)g Fk(x)j Fj(2)h Fq([0)p Fk(;)14 b Fq(1],)515 3412 y Fk(g)s Fq(\()p Fk(x)p Fq(\))29 b(=)f(0)j(for)g Fk(x)e Fj(2)g Fq([2)p Fk(;)14 b Fj(1)p Fq(\),)32 b(and)e Fk(g)1634 3382 y Ff(0)1657 3412 y Fq(\()p Fk(x)p Fq(\))g Fj(2)f Fq([)p Fj(\000)p Fq(2)p Fk(;)14 b Fq(0].)46 b(Note)31 b(that)g(b)n(y)h(\(2.7\))f(and)g(since)g(the)515 3512 y(p)r(oten)n(tial)c Fk(')h Fq(is)g(b)r(ounded,)g(there)f(exists)g Fk(Q)1900 3524 y Fi(0)1960 3512 y Fk(>)c Fq(0)k(suc)n(h)g(that:)1269 3694 y(sup)1312 3760 y Fe(\026)1422 3694 y Fj(W)7 b Fq(\()p Fk(f)1584 3706 y Fi(0)1621 3694 y Fq(;)14 b Fk(\026;)g(R)q Fq(\))37 b Fj(\024)f Fk(Q)2045 3706 y Fi(0)2096 3694 y Fk(R)167 b Fj(8)14 b Fk(R)22 b(>)h Fq(0)p Fk(:)583 b Fq(\(3.5\))515 3934 y(Lemma)26 b(3.1)g(b)r(elo)n(w,)h(the)g (one-dimensional)f(v)n(ersion)f(of)h([5,)h(Prop.)f(4.2],)g(sho)n(ws)g (that)h(the)515 4033 y(energy)h(at)i(time)h Fk(t)f Fq(of)f(a)h(region)e (of)i(size)g Fk(R)1878 4003 y Fe(M)s(;N)2026 4033 y Fq(\()p Fk(t)p Fq(\))h(is)e(b)r(ounded)i(in)f(terms)f(of)h(the)h(size)e(of)515 4133 y(the)24 b(region.)34 b(In)24 b(Lemma)f(3.2)f(w)n(e)i(then)g(use)f (this)h(result)f(to)g(con)n(trol)g(the)h(maxim)n(um)f(sp)r(eed)515 4232 y(of)k(the)h(Vlaso)n(v)f(\015uid,)h(sho)n(wing)e(it)i(cannot)f (gro)n(w)f(more)h(than)h(linearly)e(in)i(time.)515 4415 y Fr(Lemma)h(3.1)41 b Fm(F)-6 b(or)26 b Fk(f)1209 4427 y Fi(0)1246 4415 y Fq(\()p Fk(x;)14 b(v)s Fq(\))26 b Fm(satisfying)35 b Fq(\(2.7\))25 b Fm(ther)l(e)h(exists)g Fk(K)2530 4427 y Fi(4)2589 4415 y Fk(>)d Fq(0)i Fm(such)h(that,)h(for)g (any)515 4515 y Fk(N)t(;)14 b(M)31 b(>)23 b Fq(0)p Fm(,)1158 4614 y Fq(sup)1200 4680 y Fe(\026)1297 4614 y Fj(W)7 b Fq(\()p Fk(f)1468 4580 y Fe(N)s(;M)1616 4614 y Fq(\()p Fk(t)p Fq(\))p Fk(;)14 b(\026;)g(R)1898 4580 y Fe(N)s(;M)2047 4614 y Fq(\()p Fk(t)p Fq(\)\))24 b Fj(\024)e Fk(K)2355 4626 y Fi(4)2406 4614 y Fk(R)2470 4580 y Fe(N)s(;M)2618 4614 y Fq(\()p Fk(t)p Fq(\))p Fk(:)515 4849 y Fq(The)27 b(pro)r(of)g(of)h(Lemma)f(3.1)g(is)h(giv)n(en)e(in)i(the)g(App)r (endix.)1926 5255 y(5)p eop %%Page: 6 6 6 5 bop 515 523 a Fr(Lemma)29 b(3.2)41 b Fm(F)-6 b(or)26 b Fk(f)1209 535 y Fi(0)1246 523 y Fq(\()p Fk(x;)14 b(v)s Fq(\))26 b Fm(satisfying)35 b Fq(\(2.7\))25 b Fm(ther)l(e)h(exists)g Fk(K)2530 535 y Fi(5)2589 523 y Fk(>)d Fq(0)i Fm(such)h(that,)h(for)g (any)515 623 y Fk(N)32 b Fj(\025)22 b Fk(K)772 635 y Fi(5)809 623 y Fm(,)30 b Fk(M)i(>)22 b Fq(0)p Fm(,)30 b(and)g Fk(t)23 b Fj(\025)g Fq(0)p Fm(,)1516 797 y Fj(V)1574 762 y Fe(N)s(;M)1722 797 y Fq(\()p Fk(t)p Fq(\))h Fk(<)f Fq(2\()p Fk(N)k Fq(+)18 b Fk(K)2256 762 y Fi(2)2250 817 y(5)2292 797 y Fk(t)p Fq(\))p Fk(:)831 b Fq(\(3.6\))515 970 y Fr(Pro)s(of.)52 b Fq(W)-7 b(e)27 b(pro)n(v)n(e)f(the)i(lemma)f(b) n(y)g(con)n(tradiction.)35 b(W)-7 b(e)28 b(assume)f(that)g(the)h(v)n (elo)r(cit)n(y)e(of)515 1070 y(a)g(\015uid)h(particle)f(b)r(ecomes)h (larger)d(than)j(2\()p Fk(N)e Fq(+)16 b Fk(K)2151 1040 y Fi(2)2145 1091 y(5)2188 1070 y Fk(t)p Fq(\))27 b(at)g(a)f(giv)n(en)g (time)h Fk(t)p Fq(.)36 b(W)-7 b(e)27 b(ev)-5 b(aluate)515 1170 y(the)37 b(force)f(acting)g(on)g(this)h(particle)f(\(during)h(its) g(bac)n(kw)n(ard)d(time-ev)n(olution\))i(due)h(to)515 1269 y(\\fast")28 b(and)h(\\slo)n(w")f(particles,)h(resp)r(ectiv)n(ely) -7 b(.)41 b(By)29 b(Lemma)g(3.1)g(w)n(e)g(can)g(sho)n(w)f(that)i(the) 515 1369 y(mass)c(of)h(the)h(fast)f(particles)f(at)h(eac)n(h)f(time)i (is)f(\014nite)g(so)g(that)g(they)g(pro)r(duce)g(a)g(b)r(ounded)515 1469 y(force.)45 b(On)30 b(the)h(other)f(hand,)i(slo)n(w)e(particles)f (in)n(teract)h(with)i(the)f(tagged)e(one)h(only)h(for)515 1568 y(a)c(short)f(time,)i(so)f(that)g(w)n(e)g(can)g(b)r(ound)h(their)f (a)n(v)n(erage)d(\(in)k(time\))g(force)f(b)n(y)g(a)g(constan)n(t.)515 1668 y(Therefore,)f(b)n(y)h(c)n(ho)r(osing)g Fk(K)1444 1638 y Fi(2)1438 1688 y(5)1508 1668 y Fq(large)f(enough)h(w)n(e)g(get)h (the)g(con)n(tradiction.)639 1791 y(De\014ne)g Fj(U)8 b Fq(\()p Fk(t)p Fq(\))1095 1744 y Fk(:)1074 1791 y Fq(=)23 b(2\()p Fk(N)k Fq(+)18 b Fk(K)1490 1760 y Fi(2)1484 1811 y(5)1527 1791 y Fk(t)p Fq(\))28 b(and)f(supp)r(ose,)g(ab)h(absurdo,)e (that)i(for)f(ev)n(ery)f Fk(K)3163 1803 y Fi(5)3223 1791 y Fk(>)d Fq(0,)1252 1965 y Fk(\034)1342 1917 y(:)1321 1965 y Fq(=)g(inf)7 b Fj(f)p Fk(t)22 b(>)h Fq(0)g(:)37 b Fj(V)1875 1930 y Fe(N)s(;M)2023 1965 y Fq(\()p Fk(t)p Fq(\))24 b Fj(\025)e(U)8 b Fq(\()p Fk(t)p Fq(\))p Fj(g)24 b Fk(<)e Fj(1)p Fk(;)567 b Fq(\(3.7\))515 2138 y(for)33 b(some)g Fk(M)42 b Fq(and)33 b Fk(N)42 b Fj(\025)33 b Fk(K)1436 2108 y Fi(2)1430 2159 y(5)1472 2138 y Fq(.)55 b(By)34 b(con)n(tin)n(uit)n(y)-7 b(,)34 b(w)n(e)f(can)h(pic)n(k)f(\()p Fk(x;)14 b(v)s Fq(\),)36 b Fj(j)p Fk(x)p Fj(j)e(\024)e Fk(M)9 b Fq(,)35 b Fj(j)p Fk(v)s Fj(j)e(\024)515 2238 y Fk(N)9 b Fq(,)39 b(suc)n(h)d(that)h Fj(j)p Fk(V)1128 2208 y Fe(N)s(;M)1277 2238 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(\034)9 b Fq(\))p Fj(j)40 b Fq(=)e Fj(U)8 b Fq(\()p Fk(\034)h Fq(\).)66 b(Moreo)n(v)n(er,)37 b(w)n(e)f(assume,)j(without)e (loss)f(of)515 2338 y(generalit)n(y)-7 b(,)26 b(that)i Fk(V)1161 2308 y Fe(N)s(;M)1309 2338 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(\034)9 b Fq(\))25 b(=)e Fj(U)8 b Fq(\()p Fk(\034)h Fq(\).)38 b(Let)1172 2551 y Fk(\034)1217 2516 y Ff(\003)1300 2504 y Fk(:)1279 2551 y Fq(=)23 b(sup)1505 2458 y Fg(n)1561 2551 y Fk(t)g(>)g Fq(0)f(:)37 b Fk(V)1893 2516 y Fe(N)s(;M)2042 2551 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))24 b Fk(<)2421 2494 y Fq(3)p 2421 2532 42 4 v 2421 2608 a(4)2473 2551 y Fj(U)8 b Fq(\()p Fk(\034)h Fq(\))2642 2458 y Fg(o)2699 2551 y Fk(:)486 b Fq(\(3.8\))515 2748 y(W)-7 b(e)28 b(simplify)g(notation)f(b)n(y)g(calling:)1154 2922 y Fk(X)1223 2934 y Fe(s)1302 2875 y Fk(:)1281 2922 y Fq(=)c Fk(X)1445 2892 y Fe(N)s(;M)1593 2922 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(s)p Fq(\))p Fk(;)194 b(V)2125 2934 y Fe(s)2205 2875 y Fk(:)2184 2922 y Fq(=)23 b Fk(V)2339 2887 y Fe(N)s(;M)2487 2922 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(s)p Fq(\))p Fk(;)1157 3046 y(Y)1205 3058 y Fe(s)1284 2999 y Fk(:)1263 3046 y Fq(=)23 b Fk(X)1427 3016 y Fe(N)s(;M)1575 3046 y Fq(\()p Fk(y)s(;)14 b(w)r Fq(;)g Fk(s)p Fq(\))p Fk(;)197 b(W)2155 3058 y Fe(s)2235 2999 y Fk(:)2214 3046 y Fq(=)23 b Fk(V)2369 3012 y Fe(N)s(;M)2517 3046 y Fq(\()p Fk(y)s(;)14 b(w)r Fq(;)g Fk(s)p Fq(\))p Fk(:)515 3220 y Fq(By)28 b(our)f(h)n(yp)r(otheses)g(on)h(the)h(initial)f(condition,)g Fk(\034)33 b(>)24 b Fq(0)k(and,)g(b)n(y)g(con)n(tin)n(uit)n(y)-7 b(,)27 b Fk(\034)3087 3190 y Ff(\003)3154 3220 y Fq(is)h(w)n(ell)515 3320 y(de\014ned)e(and)g(strictly)g(smaller)f(than)h Fk(\034)9 b Fq(.)37 b(Th)n(us,)26 b(under)g(the)h(h)n(yp)r(othesis)e (that)i Fk(\034)36 b Fq(is)25 b(\014nite,)515 3419 y(w)n(e)i(ha)n(v)n (e:)599 3570 y Fk(N)h Fq(+)18 b Fk(K)854 3539 y Fi(2)848 3590 y(5)890 3570 y Fk(\034)p 599 3607 337 4 v 747 3683 a Fq(2)1029 3626 y(=)83 b Fk(V)1225 3638 y Fe(\034)1285 3626 y Fj(\000)18 b Fk(V)1416 3638 y Fe(\034)1454 3622 y Fb(\003)1530 3626 y Fq(=)1631 3505 y Fg(\014)1631 3555 y(\014)1631 3605 y(\014)1631 3655 y(\014)1659 3513 y(Z)1742 3533 y Fe(\034)1705 3701 y(\034)1743 3685 y Fb(\003)1783 3626 y Fk(ds)1879 3513 y Fg(Z)1962 3626 y Fk(dy)f(dw)g(f)2232 3586 y Fe(N)s(;M)2223 3648 y Fi(0)2380 3626 y Fq(\()p Fk(y)s(;)d(w)r Fq(\))g Fk(')2654 3591 y Ff(0)2678 3626 y Fq(\()p Fk(X)2779 3638 y Fe(s)2833 3626 y Fj(\000)k Fk(Y)2964 3638 y Fe(s)3000 3626 y Fq(\))p Fj(j)3055 3505 y Fg(\014)3055 3555 y(\014)3055 3605 y(\014)3055 3655 y(\014)1029 3855 y Fj(\024)83 b(jj)p Fk(')1277 3821 y Ff(0)1300 3855 y Fj(jj)1346 3867 y Ff(1)1431 3742 y Fg(Z)1514 3855 y Fk(dy)16 b(dw)h(f)1783 3815 y Fe(N)s(;M)1774 3877 y Fi(0)1931 3855 y Fq(\()p Fk(y)s(;)d(w)r Fq(\))2151 3742 y Fg(Z)2235 3763 y Fe(\034)2198 3931 y(\034)2236 3914 y Fb(\003)2277 3855 y Fk(ds)g(\037)p Fq(\()p Fj(j)p Fk(X)2549 3867 y Fe(s)2603 3855 y Fj(\000)k Fk(Y)2734 3867 y Fe(s)2769 3855 y Fj(j)23 b(\024)g Fq(1\))p Fk(:)208 b Fq(\(3.9\))515 4071 y(W)-7 b(e)28 b(in)n(tro)r(duce)f(the)h(sets:)873 4245 y(\001)942 4211 y Fi(+)942 4266 y Fe(y)r(;w)1155 4198 y Fk(:)1134 4245 y Fq(=)1282 4153 y Fg(n)1337 4245 y Fk(s)23 b Fj(2)h Fq([)p Fk(\034)1546 4211 y Ff(\003)1585 4245 y Fk(;)14 b(\034)9 b Fq(])23 b(:)37 b Fj(j)p Fk(W)1874 4257 y Fe(s)1910 4245 y Fj(j)23 b Fk(>)g(N)k Fq(+)18 b Fk(K)2298 4211 y Fi(2)2292 4266 y(5)2335 4245 y Fk(\034)5 b(;)42 b Fj(j)p Fk(X)2533 4257 y Fe(s)2586 4245 y Fj(\000)18 b Fk(Y)2717 4257 y Fe(s)2753 4245 y Fj(j)23 b(\024)g Fq(1)2929 4153 y Fg(o)2984 4245 y Fk(;)160 b Fq(\(3.10\))873 4428 y(\001)942 4393 y Ff(\000)942 4448 y Fe(y)r(;w)1155 4381 y Fk(:)1134 4428 y Fq(=)1282 4336 y Fg(n)1337 4428 y Fk(s)23 b Fj(2)h Fq([)p Fk(\034)1546 4393 y Ff(\003)1585 4428 y Fk(;)14 b(\034)9 b Fq(])23 b(:)37 b Fj(j)p Fk(W)1874 4440 y Fe(s)1910 4428 y Fj(j)23 b(\024)g Fk(N)k Fq(+)18 b Fk(K)2298 4393 y Fi(2)2292 4448 y(5)2335 4428 y Fk(\034)5 b(;)42 b Fj(j)p Fk(X)2533 4440 y Fe(s)2586 4428 y Fj(\000)18 b Fk(Y)2717 4440 y Fe(s)2753 4428 y Fj(j)23 b(\024)g Fq(1)2929 4336 y Fg(o)2984 4428 y Fk(:)160 b Fq(\(3.11\))515 4622 y(Clearly:)1145 4844 y Fk(V)1193 4856 y Fe(s)1247 4844 y Fj(\000)18 b Fk(W)1408 4856 y Fe(s)1527 4844 y Fj(\025)1685 4788 y Fk(N)27 b Fq(+)18 b Fk(K)1939 4758 y Fi(2)1933 4809 y(5)1976 4788 y Fk(\034)p 1685 4825 V 1832 4901 a Fq(2)2363 4844 y(if)28 b Fk(s)23 b Fj(2)h Fq(\001)2649 4810 y Ff(\000)2649 4865 y Fe(y)r(;w)2758 4844 y Fk(;)386 b Fq(\(3.12\))1099 5006 y Fj(j)p Fk(V)1170 5018 y Fe(s)1224 5006 y Fj(\000)18 b Fk(W)1385 5018 y Fe(s)1421 5006 y Fj(j)83 b(\024)g Fq(4\()p Fk(N)27 b Fq(+)18 b Fk(K)2003 4972 y Fi(2)1997 5027 y(5)2039 5006 y Fk(\034)9 b Fq(\))250 b(if)29 b Fk(s)23 b Fj(2)g Fq(\001)2652 4972 y Fi(+)2652 5027 y Fe(y)r(;w)2761 5006 y Fk(:)383 b Fq(\(3.13\))1926 5255 y(6)p eop %%Page: 7 7 7 6 bop 515 523 a Fq(Let)35 b Fk(\016)s Fq(\()p Fk(t)p Fq(\))863 476 y Fk(:)842 523 y Fq(=)h Fk(X)1012 535 y Fe(t)1065 523 y Fj(\000)23 b Fk(Y)1201 535 y Fe(t)1230 523 y Fq(,)38 b Fk(t)1321 535 y Fi(1)1415 476 y Fk(:)1394 523 y Fq(=)e(inf)7 b Fj(f)p Fk(t)36 b Fj(2)g Fq([)p Fk(\034)1863 493 y Ff(\003)1902 523 y Fk(;)14 b(\034)9 b Fq(])37 b(:)50 b Fj(j)p Fk(\016)s Fq(\()p Fk(t)p Fq(\))p Fj(j)37 b(\024)f Fq(1)p Fj(g)p Fq(,)h Fk(t)2609 535 y Fi(2)2703 476 y Fk(:)2682 523 y Fq(=)f(sup)p Fj(f)p Fk(t)g Fj(2)g Fq([)p Fk(\034)3175 493 y Ff(\003)3214 523 y Fk(;)14 b(\034)9 b Fq(])37 b(:)515 623 y Fj(j)p Fk(\016)s Fq(\()p Fk(t)p Fq(\))p Fj(j)24 b(\024)e Fq(1)p Fj(g)p Fq(.)36 b(By)29 b(\(3.12\))d(w)n(e)i(ha)n(v)n(e)1647 601 y(_)1634 623 y Fk(\016)s Fq(\()p Fk(t)p Fq(\))c Fk(>)e Fq(0)28 b(inside)f(\001)2253 593 y Ff(\000)2253 643 y Fe(y)r(;w)2363 623 y Fq(,)g(so)g(that:)994 872 y(2)83 b Fj(\025)1266 752 y Fg(\014)1266 801 y(\014)1266 851 y(\014)1266 901 y(\014)1294 759 y(Z)1377 779 y Fe(t)1402 787 y Fc(2)1340 948 y Fe(t)1365 956 y Fc(1)1439 872 y Fk(ds)1548 850 y Fq(_)1535 872 y Fk(\016)s Fq(\()p Fk(s)p Fq(\))1678 752 y Fg(\014)1678 801 y(\014)1678 851 y(\014)1678 901 y(\014)1743 872 y Fq(=)1844 727 y Fg(\014)1844 776 y(\014)1844 826 y(\014)1844 876 y(\014)1844 926 y(\014)1872 759 y(Z)1918 948 y Fi(\001)1973 921 y Fb(\000)1973 957 y Fa(y)q(;w)2075 872 y Fk(ds)2184 850 y Fq(_)2171 872 y Fk(\016)s Fq(\()p Fk(s)p Fq(\))19 b(+)2416 759 y Fg(Z)2462 948 y Fi(\001)2517 921 y Fc(+)2517 957 y Fa(y)q(;w)2618 872 y Fk(ds)2728 850 y Fq(_)2714 872 y Fk(\016)s Fq(\()p Fk(s)p Fq(\))2857 727 y Fg(\014)2857 776 y(\014)2857 826 y(\014)2857 876 y(\014)2857 926 y(\014)1119 1122 y Fj(\025)1266 1009 y Fg(Z)1312 1198 y Fi(\001)1367 1171 y Fb(\000)1367 1207 y Fa(y)q(;w)1469 1122 y Fk(ds)1578 1100 y Fq(_)1565 1122 y Fk(\016)s Fq(\()p Fk(s)p Fq(\))g Fj(\000)1810 1009 y Fg(Z)1856 1198 y Fi(\001)1911 1171 y Fc(+)1911 1207 y Fa(y)q(;w)2013 1122 y Fk(ds)14 b Fj(j)2145 1100 y Fq(_)2132 1122 y Fk(\016)s Fq(\()p Fk(s)p Fq(\))p Fj(j)p Fk(;)515 1349 y Fq(whence,)27 b(from)h(\(3.12\))e(and)i (\(3.13\))f(w)n(e)g(get:)1333 1566 y(2)c Fj(\025)1495 1510 y Fk(N)k Fq(+)18 b Fk(K)1749 1480 y Fi(2)1743 1530 y(5)1786 1510 y Fk(\034)p 1495 1547 337 4 v 1643 1623 a Fq(2)1856 1499 y Fg(\000)1894 1566 y Fj(j)p Fq(\001)1986 1532 y Ff(\000)1986 1586 y Fe(y)r(;w)2095 1566 y Fj(j)h(\000)f Fq(8)p Fj(j)p Fq(\001)2354 1532 y Fi(+)2354 1586 y Fe(y)r(;w)2463 1566 y Fj(j)2486 1499 y Fg(\001)2538 1566 y Fk(;)515 1753 y Fq(namely:)1406 1887 y Fj(j)p Fq(\001)1498 1852 y Ff(\000)1498 1907 y Fe(y)r(;w)1607 1887 y Fj(j)23 b(\024)1898 1831 y Fq(4)p 1751 1868 V 1751 1944 a Fk(N)k Fq(+)18 b Fk(K)2005 1915 y Fi(2)1999 1966 y(5)2042 1944 y Fk(\034)2116 1887 y Fq(+)g(8)p Fj(j)p Fq(\001)2333 1852 y Fi(+)2333 1907 y Fe(y)r(;w)2442 1887 y Fj(j)p Fk(:)515 2083 y Fq(W)-7 b(e)29 b(can)f(then)h(decomp)r(ose)f(\001)1490 2053 y Ff(\000)1490 2103 y Fe(y)r(;w)1624 2083 y Fq(=)d(\001)1783 2053 y Ff(y)1783 2103 y Fe(y)r(;w)1911 2083 y Fj([)20 b Fq(\001)2055 2053 y Ff(\003)2055 2103 y Fe(y)r(;w)2164 2083 y Fq(,)29 b(with)h(\001)2476 2053 y Ff(y)2476 2103 y Fe(y)r(;w)2604 2083 y Fj(\\)20 b Fq(\001)2748 2053 y Ff(\003)2748 2103 y Fe(y)r(;w)2882 2083 y Fq(=)k Fj(;)p Fq(,)29 b Fj(j)p Fq(\001)3157 2053 y Ff(y)3157 2103 y Fe(y)r(;w)3267 2083 y Fj(j)24 b(\024)515 2183 y Fq(8)p Fj(j)p Fq(\001)649 2152 y Fi(+)649 2203 y Fe(y)r(;w)758 2183 y Fj(j)p Fq(,)k(and)f Fj(j)p Fq(\001)1085 2152 y Ff(\003)1085 2203 y Fe(y)r(;w)1194 2183 y Fj(j)d(\024)e Fq(4)p Fk(=)p Fq(\()p Fk(N)27 b Fq(+)18 b Fk(K)1698 2152 y Fi(2)1692 2203 y(5)1734 2183 y Fk(\034)9 b Fq(\).)639 2293 y(W)-7 b(e)23 b(no)n(w)f(notice)g(that,)i(b)n(y)f(\(3.3\),)g(\001) 1780 2262 y Fi(+)1780 2313 y Fe(y)r(;w)1898 2293 y Fj([)8 b Fq(\001)2030 2262 y Ff(\000)2030 2313 y Fe(y)r(;w)2162 2293 y Fj(6)p Fq(=)23 b Fj(;)f Fq(only)g(if)h Fk(y)j Fj(2)d Fq([)p Fk(x)8 b Fj(\000)g Fq(2)p Fk(R)2964 2262 y Fe(N)s(;M)3112 2293 y Fq(\()p Fk(\034)h Fq(\))p Fk(;)14 b(x)8 b Fq(+)515 2402 y(2)p Fk(R)621 2372 y Fe(N)s(;M)769 2402 y Fq(\()p Fk(\034)h Fq(\)])31 b(and)g(that,)g(b)n(y)f (de\014nition)h(\(3.7\))o(,)g Fk(R)2081 2372 y Fe(N)s(;M)2230 2402 y Fq(\()p Fk(\034)9 b Fq(\))29 b Fj(\024)e Fq(1)20 b(+)g(2\()p Fk(N)28 b Fq(+)20 b Fk(K)2938 2372 y Fi(2)2932 2423 y(5)2975 2402 y Fk(\034)9 b Fq(\))p Fk(\034)g Fq(.)47 b(Then,)515 2502 y(using)27 b Fk(N)32 b Fj(\025)22 b Fk(K)989 2514 y Fi(5)1054 2502 y Fq(and)27 b(setting)1510 2481 y(\026)1488 2502 y Fk(K)1559 2514 y Fi(0)1639 2455 y Fk(:)1619 2502 y Fq(=)22 b Fk(K)1777 2514 y Fi(0)1814 2431 y Fg(p)p 1897 2431 141 4 v 71 x Fk(\031)s(=\025)p Fq(,)28 b(w)n(e)f(ha)n(v)n(e:)622 2613 y Fg(Z)706 2726 y Fk(dy)16 b(dw)h(f)975 2686 y Fe(N)s(;M)966 2748 y Fi(0)1123 2726 y Fq(\()p Fk(y)s(;)d(w)r Fq(\))g Fj(j)p Fq(\001)1435 2691 y Ff(\003)1435 2746 y Fe(y)r(;w)1545 2726 y Fj(j)83 b(\024)1809 2669 y Fq(16)1914 2648 y(\026)1893 2669 y Fk(K)1964 2681 y Fi(0)2000 2669 y Fk(R)2064 2639 y Fe(N)s(;M)2212 2669 y Fq(\()p Fk(\034)9 b Fq(\))p 1809 2706 514 4 v 1897 2783 a Fk(N)28 b Fq(+)18 b Fk(K)2152 2754 y Fi(2)2146 2805 y(5)2188 2783 y Fk(\034)2369 2726 y Fj(\024)37 b Fq(32)2576 2705 y(\026)2555 2726 y Fk(K)2626 2738 y Fi(0)2671 2669 y Fq(1)18 b(+)g(\()p Fk(N)28 b Fq(+)18 b Fk(K)3101 2639 y Fi(2)3095 2690 y(5)3138 2669 y Fk(\034)9 b Fq(\))p Fk(\034)p 2671 2706 590 4 v 2798 2783 a(N)27 b Fq(+)18 b Fk(K)3052 2754 y Fi(2)3046 2805 y(5)3089 2783 y Fk(\034)1651 2958 y Fj(\024)83 b Fq(32)1904 2937 y(\026)1883 2958 y Fk(K)1954 2970 y Fi(0)2004 2841 y Fg(\022)2108 2902 y Fq(1)p 2075 2939 108 4 v 2075 3015 a Fk(K)2146 3027 y Fi(5)2211 2958 y Fq(+)18 b Fk(\034)2339 2841 y Fg(\023)2415 2958 y Fk(:)729 b Fq(\(3.14\))515 3189 y(On)27 b(the)i(other)e(hand,)h (letting)g Fk(N)1584 3159 y Fe(>)1640 3189 y Fq(\()p Fk(s)p Fq(\))1788 3142 y Fk(:)1767 3189 y Fq(=)1855 3122 y Fg(R)1911 3189 y Fk(dy)16 b(dw)h(f)2180 3149 y Fe(N)s(;M)2171 3211 y Fi(0)2328 3189 y Fq(\()p Fk(y)s(;)d(w)r Fq(\))g Fk(\037)p Fq(\(\001)2701 3159 y Fi(+)2701 3209 y Fe(y)r(;w)2812 3189 y Fq(\),)28 b(b)n(y)g(Liouville's)515 3289 y(theorem,)f(Lemma)g (3.1,)g(and)g(de\014nition)h(\(3.10\),)679 3449 y(\()p Fk(N)f Fq(+)18 b Fk(K)965 3419 y Fi(2)959 3470 y(5)1002 3449 y Fk(s)p Fq(\))1073 3419 y Fi(2)p 679 3486 432 4 v 874 3562 a Fq(2)1120 3505 y Fk(N)1196 3471 y Fe(>)1252 3505 y Fq(\()p Fk(s)p Fq(\))84 b Fj(\024)1586 3392 y Fg(Z)1669 3505 y Fk(d)5 b(Y)33 b(dW)26 b(f)1995 3471 y Fe(N)s(;M)2143 3505 y Fq(\()p Fk(Y)5 b(;)14 b(W)e Fq(;)i Fk(s)p Fq(\))2487 3449 y Fk(W)2577 3419 y Fi(2)p 2487 3486 128 4 v 2530 3562 a Fq(2)2638 3505 y Fk(\037)p Fq(\()p Fj(j)p Fk(X)2814 3517 y Fe(s)2868 3505 y Fj(\000)k Fk(Y)g Fj(j)23 b(\024)g Fq(1\))1439 3684 y Fj(\024)82 b(W)7 b Fq(\()p Fk(f)1757 3650 y Fe(N)s(;M)1906 3684 y Fq(\()p Fk(s)p Fq(\);)14 b Fk(X)2115 3696 y Fe(s)2150 3684 y Fk(;)g(R)2251 3650 y Fe(N)s(;M)2400 3684 y Fq(\()p Fk(s)p Fq(\)\))37 b Fj(\024)g Fk(K)2745 3696 y Fi(4)2781 3684 y Fk(R)2845 3650 y Fe(N)s(;M)2994 3684 y Fq(\()p Fk(s)p Fq(\))p Fk(;)515 3847 y Fq(and)27 b(hence)h(\(using)f(again)g Fk(N)32 b Fj(\025)22 b Fk(K)1635 3859 y Fi(5)1672 3847 y Fq(\):)702 3945 y Fg(Z)785 4058 y Fk(dy)17 b(dw)g(f)1055 4018 y Fe(N)s(;M)1046 4080 y Fi(0)1203 4058 y Fq(\()p Fk(y)s(;)d(w)r Fq(\))g Fj(j)p Fq(\001)1515 4023 y Fi(+)1515 4078 y Fe(y)r(;w)1625 4058 y Fj(j)23 b Fq(=)1759 3945 y Fg(Z)1842 3965 y Fe(\034)1805 4133 y Fi(0)1884 4058 y Fk(ds)14 b(N)2056 4023 y Fe(>)2111 4058 y Fq(\()p Fk(s)p Fq(\))1113 4293 y Fj(\024)1211 4236 y Fq(2)p Fk(K)1324 4248 y Fi(4)1360 4236 y Fk(\034)9 b(R)1469 4206 y Fe(N)s(;M)1618 4236 y Fq(\()p Fk(\034)g Fq(\))p 1211 4274 518 4 v 1250 4366 a(\()p Fk(N)28 b Fq(+)18 b Fk(K)1537 4338 y Fi(2)1531 4389 y(5)1573 4366 y Fk(\034)9 b Fq(\))1651 4321 y Fi(2)1761 4293 y Fj(\024)23 b Fq(4)p Fk(K)1962 4305 y Fi(4)1998 4293 y Fk(\034)2054 4236 y Fq(1)18 b(+)g(\()p Fk(N)27 b Fq(+)18 b Fk(K)2483 4206 y Fi(2)2477 4257 y(5)2520 4236 y Fk(\034)9 b Fq(\))p Fk(\034)p 2054 4274 590 4 v 2129 4366 a Fq(\()q Fk(N)27 b Fq(+)18 b Fk(K)2416 4338 y Fi(2)2410 4389 y(5)2453 4366 y Fk(\034)9 b Fq(\))2531 4321 y Fi(2)2677 4293 y Fj(\024)2774 4236 y Fq(8)p Fk(K)2887 4248 y Fi(4)p 2774 4274 150 4 v 2792 4350 a Fk(K)2869 4321 y Fi(2)2863 4372 y(5)2933 4293 y Fk(\034)c(:)170 b Fq(\(3.15\))515 4523 y(By)27 b(using)h(\(3.14\))f(and)h(\(3.15\))f (in)g(\(3.9\),)h(w)n(e)f(get:)835 4683 y Fk(N)g Fq(+)18 b Fk(K)1089 4653 y Fi(2)1083 4704 y(5)1125 4683 y Fk(\034)p 835 4721 337 4 v 982 4797 a Fq(2)1264 4740 y Fj(\024)83 b(jj)p Fk(')1512 4705 y Ff(0)1535 4740 y Fj(jj)1581 4752 y Ff(1)1666 4627 y Fg(Z)1749 4740 y Fk(dy)17 b(dw)r(f)2004 4700 y Fe(N)s(;M)1995 4762 y Fi(0)2153 4740 y Fq(\()p Fk(y)s(;)d(w)r Fq(\))2387 4672 y Fg(\000)2425 4740 y Fj(j)p Fq(\001)2517 4705 y Ff(\003)2517 4760 y Fe(y)r(;w)2627 4740 y Fj(j)k Fq(+)g(9)p Fj(j)p Fq(\001)2885 4705 y Fi(+)2885 4760 y Fe(y)r(;w)2994 4740 y Fj(j)3017 4672 y Fg(\001)1264 4969 y Fj(\024)83 b Fq(8)14 b Fj(jj)p Fk(')1568 4934 y Ff(0)1591 4969 y Fj(jj)1637 4981 y Ff(1)1721 4852 y Fg(\024)1775 4912 y Fq(4)1839 4891 y(\026)1817 4912 y Fk(K)1888 4924 y Fi(0)p 1775 4950 150 4 v 1796 5026 a Fk(K)1867 5038 y Fi(5)1953 4969 y Fq(+)2036 4852 y Fg(\022)2097 4969 y Fq(4)2160 4948 y(\026)2139 4969 y Fk(K)2210 4981 y Fi(0)2264 4969 y Fq(+)2357 4912 y(9)p Fk(K)2470 4924 y Fi(4)p 2357 4950 V 2375 5026 a Fk(K)2452 4997 y Fi(2)2446 5048 y(5)2517 4852 y Fg(\023)2592 4969 y Fk(\034)2637 4852 y Fg(\025)2695 4969 y Fk(;)1926 5255 y Fq(7)p eop %%Page: 8 8 8 7 bop 515 523 a Fq(whence)27 b(a)g(con)n(tradiction)g(for)g Fk(K)1578 535 y Fi(5)1642 523 y Fq(su\016cien)n(tly)h(large.)p 3331 477 49 4 v 3331 527 4 50 v 3376 527 V 3331 530 49 4 v 515 644 a Fr(Pro)s(of)42 b(of)f(Prop)s(osition)g(2.1.)78 b Fq(F)-7 b(ollo)n(wing)35 b([5)o(])i(the)g(solution)f(to)g(the)h (Vlaso)n(v)e(equa-)515 744 y(tion)g(\(2.6\))g(can)g(b)r(e)h(obtained)f (from)h(the)f(solutions)g(to)g(the)h(cut-o\013)g(problem)f(\(3.1\))g (as)515 844 y(\()p Fk(N)t(;)14 b(M)9 b Fq(\))32 b Fj(!)g(1)h Fq(in)g(a)f(suitable)h(w)n(a)n(y)-7 b(.)52 b(Giv)n(en)32 b(\()p Fk(x)2065 856 y Fi(0)2103 844 y Fk(;)14 b(v)2180 856 y Fi(0)2218 844 y Fq(\))32 b Fj(2)g Fh(R)2423 813 y Fi(2)2499 844 y Fq(let)h Fk(k)2667 856 y Fi(0)2737 844 y Fq(=)e Fj(bj)p Fk(x)2940 856 y Fi(0)2978 844 y Fj(jc)22 b Fq(+)f(1.)53 b(W)-7 b(e)515 943 y(c)n(ho)r(ose)35 b Fk(M)46 b Fq(=)37 b Fk(k)1059 955 y Fi(0)1121 943 y Fq(+)23 b Fk(N)1285 913 y Fi(5)1359 943 y Fq(and)36 b(use)g(the)h (shorthand)e(notation,)j(with)f(e.g.)f Fk(X)3035 913 y Fe(N)3097 943 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\),)515 1054 y Fk(V)582 1024 y Fe(N)645 1054 y Fq(\()p Fk(x;)g(v)s Fq(;)g Fk(t)p Fq(\))35 b(in)g(place)g(of)f Fk(X)1439 1024 y Fe(N)s(;k)1550 1032 y Fc(0)1581 1024 y Fi(+)p Fe(N)1691 999 y Fc(5)1727 1054 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\),)38 b Fk(V)2113 1024 y Fe(N)s(;k)2224 1032 y Fc(0)2256 1024 y Fi(+)p Fe(N)2366 999 y Fc(5)2402 1054 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\).)59 b(W)-7 b(e)35 b(de\014ne)g(\(b)r(elo)n(w)515 1154 y Fk(k)26 b Fj(2)d Fq([)p Fk(k)728 1166 y Fi(0)766 1154 y Fk(;)14 b(k)846 1166 y Fi(0)901 1154 y Fq(+)k Fk(N)1060 1124 y Fi(5)1097 1154 y Fq(]\):)597 1320 y Fk(\016)637 1286 y Fe(N)700 1320 y Fq(\()p Fk(x;)c(v)s Fq(;)g Fk(t)p Fq(\))1003 1273 y Fk(:)983 1320 y Fq(=)22 b Fj(j)p Fk(V)1160 1286 y Fe(N)6 b Fi(+1)1307 1320 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))20 b Fj(\000)e Fk(V)1734 1286 y Fe(N)1797 1320 y Fq(\()p Fk(x;)c(v)s Fq(;)g Fk(t)p Fq(\))p Fj(j)p Fk(;)98 b(u)2247 1286 y Fe(N)2247 1341 y(k)2310 1320 y Fq(\()p Fk(t)p Fq(\))2448 1273 y Fk(:)2428 1320 y Fq(=)151 b(sup)2515 1394 y Ff(j)p Fe(x)p Ff(j\024)p Fe(k)q(;)11 b Ff(j)p Fe(v)r Ff(j\024)p Fe(N)2911 1320 y Fk(\016)2951 1286 y Fe(N)3014 1320 y Fq(\()p Fk(x;)j(v)s Fq(;)g Fk(t)p Fq(\))p Fk(:)3167 1487 y Fq(\(3.16\))515 1587 y(W)-7 b(e)28 b(ha)n(v)n(e:)546 1790 y Fk(\016)586 1756 y Fe(N)649 1790 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))84 b Fj(\024)1139 1677 y Fg(Z)1222 1697 y Fe(t)1185 1865 y Fi(0)1251 1790 y Fk(ds)1347 1669 y Fg(\014)1347 1719 y(\014)1347 1769 y(\014)1347 1819 y(\014)1389 1677 y(Z)1472 1790 y Fk(dy)16 b(dw)h(f)1741 1754 y Fe(N)6 b Fi(+1)1732 1812 y(0)1888 1790 y Fq(\()p Fk(y)s(;)14 b(w)r Fq(\))g Fk(')2162 1756 y Ff(0)2186 1790 y Fq(\()p Fk(X)2294 1756 y Fe(N)6 b Fi(+1)2441 1790 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(s)p Fq(\))19 b Fj(\000)f Fk(X)2886 1756 y Fe(N)6 b Fi(+1)3032 1790 y Fq(\()p Fk(y)s(;)14 b(w)r Fq(;)g Fk(s)p Fq(\)\))1139 2022 y Fj(\000)1218 1909 y Fg(Z)1300 2022 y Fk(dy)j(dw)g(f)1570 1988 y Fe(N)1561 2043 y Fi(0)1632 2022 y Fq(\()p Fk(y)s(;)d(w)r Fq(\))g Fk(')1906 1988 y Ff(0)1931 2022 y Fq(\()p Fk(X)2039 1988 y Fe(N)2101 2022 y Fq(\()p Fk(x;)g(v)s Fq(;)g Fk(s)p Fq(\))20 b Fj(\000)e Fk(X)2547 1988 y Fe(N)2609 2022 y Fq(\()p Fk(y)s(;)c(w)r Fq(;)g Fk(s)p Fq(\)\))2923 1902 y Fg(\014)2923 1952 y(\014)2923 2002 y(\014)2923 2051 y(\014)2952 2022 y Fk(:)192 b Fq(\(3.17\))515 2246 y(By)38 b(Lemma)h(3.2)f(the)i Fk(dy)s Fq(-in)n(tegrals)d(are)h (restricted)g(to)h Fj(j)p Fk(y)29 b Fj(\000)d Fk(x)p Fj(j)42 b(\024)g Fk(p)2814 2216 y Fe(N)2814 2266 y(s)2916 2246 y Fq(where)d Fk(p)3210 2216 y Fe(N)3210 2266 y(s)3335 2199 y Fk(:)3314 2246 y Fq(=)515 2346 y(1)14 b(+)g(2\()p Fk(N)20 b Fq(+)14 b(1)g(+)g Fk(K)1102 2315 y Fi(2)1096 2366 y(5)1137 2346 y Fk(s)p Fq(\))p Fk(s)p Fq(.)36 b(The)25 b(con)n(tribution)g(at)g(time)h Fk(s)f Fq(in)g(\(3.17\))g(can)g(th)n (us)g(b)r(e)h(b)r(ounded)515 2445 y(b)n(y:)693 2536 y Fg(Z)776 2649 y Fk(dy)16 b(dw)h(f)1045 2615 y Fe(N)1036 2670 y Fi(0)1108 2649 y Fq(\()p Fk(y)s(;)d(w)r Fq(\))1328 2579 y Fg(\014)1328 2629 y(\014)1356 2649 y Fk(')1410 2615 y Ff(0)1434 2649 y Fq(\()p Fk(X)1542 2615 y Fe(N)6 b Fi(+1)1688 2649 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(s)p Fq(\))20 b Fj(\000)e Fk(X)2134 2615 y Fe(N)6 b Fi(+1)2280 2649 y Fq(\()p Fk(y)s(;)14 b(w)r Fq(;)g Fk(s)p Fq(\)\))1597 2828 y Fj(\000)31 b Fk(')1747 2794 y Ff(0)1771 2828 y Fq(\()p Fk(X)1879 2794 y Fe(N)1942 2828 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(s)p Fq(\))19 b Fj(\000)f Fk(X)2387 2794 y Fe(N)2449 2828 y Fq(\()p Fk(y)s(;)c(w)r Fq(;)g Fk(s)p Fq(\)\))2763 2757 y Fg(\014)2763 2807 y(\014)794 3003 y Fq(+)877 2890 y Fg(Z)960 3003 y Fk(dy)j(dw)1180 2933 y Fg(\014)1180 2983 y(\014)1207 3003 y Fk(f)1257 2968 y Fe(N)6 b Fi(+1)1248 3025 y(0)1404 3003 y Fq(\()p Fk(y)s(;)14 b(w)r Fq(\))19 b Fj(\000)f Fk(f)1762 2969 y Fe(N)1753 3024 y Fi(0)1825 3003 y Fq(\()p Fk(y)s(;)c(w)r Fq(\))2031 2933 y Fg(\014)2031 2983 y(\014)2073 2933 y(\014)2073 2983 y(\014)2101 3003 y Fk(')2155 2969 y Ff(0)2178 3003 y Fq(\()p Fk(X)2286 2969 y Fe(N)6 b Fi(+1)2433 3003 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(s)p Fq(\))19 b Fj(\000)f Fk(X)2878 2969 y Fe(N)6 b Fi(+1)3025 3003 y Fq(\()p Fk(y)s(;)14 b(w)r Fq(;)g Fk(s)p Fq(\)\))3339 2933 y Fg(\014)3339 2983 y(\014)693 3225 y Fj(\024)22 b Fq(2)844 3205 y(\026)822 3225 y Fk(K)893 3237 y Fi(0)943 3225 y Fj(jj)p Fk(')1043 3191 y Ff(00)1086 3225 y Fj(jj)1132 3237 y Ff(1)1216 3225 y Fk(p)1258 3191 y Fe(N)1258 3246 y(s)1435 3225 y Fq(sup)1345 3309 y Ff(j)p Fe(y)r Ff(\000)p Fe(x)p Ff(j\024)p Fe(p)1597 3284 y Fa(N)1597 3325 y(s)1397 3379 y Ff(j)p Fe(w)r Ff(j\024)p Fe(N)1673 3112 y Fg(Z)1756 3133 y Fe(s)1719 3301 y Fi(0)1791 3225 y Fk(d\034)i(\016)1934 3191 y Fe(N)1997 3225 y Fq(\()p Fk(y)s(;)14 b(w)r(;)g(\034)9 b Fq(\))794 3553 y(+)32 b Fk(K)962 3565 y Fi(0)999 3553 y Fj(k)p Fk(')1095 3518 y Ff(0)1118 3553 y Fj(k)1160 3565 y Ff(1)1230 3553 y Fk(p)1272 3518 y Fe(N)1272 3573 y(s)1348 3440 y Fg(Z)1431 3460 y Ff(1)1394 3628 y Fe(N)1502 3553 y Fk(dw)16 b(e)1659 3518 y Ff(\000)p Fe(\025w)1800 3493 y Fc(2)1836 3553 y Fk(;)1308 b Fq(\(3.18\))515 3781 y(where)27 b(in)h(the)g(last)f(inequalit)n(y)g(w)n(e)g(used)h(\(2.7\))f(and)2231 3760 y(\026)2209 3781 y Fk(K)2280 3793 y Fi(0)2339 3781 y Fq(=)c Fk(K)2498 3793 y Fi(0)2535 3710 y Fg(p)p 2618 3710 141 4 v 71 x Fk(\031)s(=\025)p Fq(.)37 b(Let)28 b(us)f(de\014ne:)1523 3947 y Fk(N)1590 3959 y Fe(\015)1632 3947 y Fq(\()p Fk(t)p Fq(\))1771 3900 y Fk(:)1750 3947 y Fq(=)c Fj(j)p Fk(v)1901 3959 y Fi(0)1938 3947 y Fj(j)c Fq(+)f Fk(\015)5 b Fq(\(1)17 b(+)h Fk(t)p Fq(\))p Fk(;)797 b Fq(\(3.19\))515 4113 y(where)28 b(the)h(parameter)e Fk(\015)i Fj(\025)c Fq(1)j(will)h(b)r(e)g(c)n(hosen)e(later)h(on.)40 b(W)-7 b(e)29 b(consider)e Fk(N)34 b Fj(\025)24 b Fk(N)3139 4125 y Fe(\015)3182 4113 y Fq(\()p Fk(t)p Fq(\))29 b(so)515 4213 y(that)f Fk(t)c Fj(\024)f Fk(N)r(=\015)32 b Fq(and)c Fk(p)1227 4183 y Fe(N)1227 4234 y(t)1313 4213 y Fj(\024)23 b Fk(C)1460 4225 y Fi(1)1498 4213 y Fk(N)1574 4183 y Fi(2)1611 4213 y Fk(=\015)32 b Fq(for)27 b(a)h(suitable)g(p)r(ositiv)n (e)f(constan)n(t)g Fk(C)2936 4225 y Fi(1)2974 4213 y Fq(.)38 b(By)29 b(\(3.17\))515 4313 y(and)e(\(3.18\))g(w)n(e)g(get,)h (for)f(some)g Fk(C)1594 4325 y Fi(2)1631 4313 y Fk(;)14 b(C)1727 4325 y Fi(3)1788 4313 y Fk(>)23 b Fq(0,)741 4537 y Fk(u)789 4502 y Fe(N)789 4557 y(k)824 4565 y Fc(0)861 4537 y Fq(\()p Fk(t)p Fq(\))g Fj(\024)1076 4480 y Fk(C)1135 4492 y Fi(2)1173 4480 y Fk(N)1249 4450 y Fi(3)p 1076 4517 210 4 v 1138 4593 a Fk(\015)1186 4570 y Fi(2)1296 4537 y Fk(e)1335 4502 y Ff(\000)p Fe(\025N)1485 4477 y Fc(2)1539 4537 y Fq(+)1632 4480 y Fk(C)1691 4492 y Fi(3)1729 4480 y Fk(N)1805 4450 y Fi(2)p 1632 4517 V 1713 4593 a Fk(\015)1866 4424 y Fg(Z)1949 4444 y Fe(t)1912 4612 y Fi(0)1978 4537 y Fk(dt)2051 4549 y Fi(1)2102 4424 y Fg(Z)2185 4444 y Fe(t)2210 4452 y Fc(1)2148 4612 y Fi(0)2247 4537 y Fk(dt)2320 4549 y Fi(2)2371 4537 y Fk(u)2419 4502 y Fe(N)2419 4559 y(k)2454 4567 y Fc(0)2486 4559 y Fi(+)p Fe(C)2585 4567 y Fc(1)2617 4559 y Fe(N)2676 4543 y Fc(2)2709 4559 y Fe(=\015)2785 4537 y Fq(\()p Fk(t)2847 4549 y Fi(2)2885 4537 y Fq(\))p Fk(:)227 b Fq(\(3.20\))515 4749 y(W)-7 b(e)28 b(iterate)f(\(3.20\))g Fk(`)g Fq(times)h(with:)1418 4969 y Fk(`)1497 4922 y(:)1476 4969 y Fq(=)1564 4852 y Fg(\026)1622 4913 y Fk(\015)5 b Fq(\()p Fk(M)27 b Fj(\000)18 b Fk(k)1936 4925 y Fi(0)1974 4913 y Fq(\))p 1622 4950 384 4 v 1709 5026 a Fk(C)1768 5038 y Fi(1)1806 5026 y Fk(N)1882 5002 y Fi(2)2016 4852 y Fg(\027)2087 4969 y Fq(=)2175 4852 y Fg(\026)2233 4913 y Fk(\015)5 b(N)2357 4883 y Fi(3)p 2233 4950 161 4 v 2265 5026 a Fk(C)2324 5038 y Fi(1)2404 4852 y Fg(\027)2452 4969 y Fk(;)692 b Fq(\(3.21\))1926 5255 y(8)p eop %%Page: 9 9 9 8 bop 515 523 a Fq(getting:)583 783 y Fk(u)631 749 y Fe(N)631 803 y(k)666 811 y Fc(0)702 783 y Fq(\()p Fk(t)p Fq(\))24 b Fj(\024)918 727 y Fk(C)977 739 y Fi(2)1014 727 y Fk(N)1090 697 y Fi(3)p 918 764 210 4 v 980 840 a Fk(\015)1028 816 y Fi(2)1137 783 y Fk(e)1176 749 y Ff(\000)p Fe(\025N)1326 724 y Fc(2)1380 679 y Fe(`)p Ff(\000)p Fi(1)1376 704 y Fg(X)1382 881 y Fe(i)p Fi(=0)1520 727 y Fq(\()p Fk(C)1611 739 y Fi(3)1649 727 y Fk(N)1725 697 y Fi(2)1762 727 y Fk(t)1792 697 y Fi(2)1829 727 y Fq(\))1861 697 y Fe(i)p 1520 764 370 4 v 1588 840 a Fk(\015)1636 816 y Fe(i)1663 840 y Fq(\(2)p Fk(i)p Fq(\)!)1917 783 y(+)18 b(4\()p Fk(N)27 b Fq(+)18 b(1)g(+)g Fk(K)2471 749 y Fi(2)2465 803 y(5)2508 783 y Fk(t)p Fq(\))2580 727 y(\()p Fk(C)2671 739 y Fi(3)2709 727 y Fk(N)2785 697 y Fi(2)2822 727 y Fq(\))2854 697 y Fe(`)2886 727 y Fk(t)2916 697 y Fi(2)p Fe(`)p Fi(+1)p 2580 764 486 4 v 2701 840 a Fk(\015)2749 816 y Fe(`)2781 840 y Fq(\(2)p Fk(`)p Fq(\)!)3076 783 y Fk(;)68 b Fq(\(3.22\))515 1050 y(where)27 b(w)n(e)h(used)h(that)f Fk(u)1297 1020 y Fe(N)1297 1074 y(k)1333 1057 y Fb(0)1360 1050 y Fq(\()p Fk(s)p Fq(\))d Fj(\024)e Fq(4\()p Fk(N)28 b Fq(+)18 b(1)h(+)f Fk(K)2049 1020 y Fi(2)2043 1071 y(5)2086 1050 y Fk(t)p Fq(\))28 b(for)g(an)n(y)g Fk(k)2508 1020 y Ff(0)2555 1050 y Fj(\024)c Fk(M)37 b Fq(and)28 b Fk(s)c Fj(2)g Fq([0)p Fk(;)14 b(t)p Fq(],)28 b(see)515 1150 y(Lemma)f(3.2.)36 b(Recalling)27 b Fk(t)c Fj(\024)g Fk(N)r(=\015)5 b Fq(,)26 b(the)i(\014rst)g(term)f(in)h(\(3.22\))f(is)g(b)r(ounded)h(b)n(y:)1085 1361 y Fk(C)1144 1373 y Fi(2)1182 1361 y Fk(N)1258 1331 y Fi(3)p 1085 1399 210 4 v 1148 1475 a Fk(\015)1196 1451 y Fi(2)1319 1418 y Fq(exp)1460 1276 y Fg(")1508 1418 y Fj(\000)p Fk(\025N)1697 1383 y Fi(2)1752 1418 y Fq(+)1835 1269 y Fg(s)p 1918 1269 117 4 v 1928 1361 a Fk(C)1987 1373 y Fi(3)p 1928 1399 97 4 v 1934 1475 a Fk(\015)1982 1451 y Fi(3)2035 1418 y Fk(N)2111 1383 y Fi(2)2148 1276 y Fg(#)2219 1418 y Fj(\024)23 b Fq(exp)2448 1301 y Fg(\024)2492 1418 y Fj(\000)2567 1361 y Fk(\025)p 2567 1399 49 4 v 2570 1475 a Fq(2)2625 1418 y Fk(N)2701 1383 y Fi(2)2738 1301 y Fg(\025)2795 1418 y Fk(;)515 1675 y Fq(for)31 b(su\016cien)n(tly)g(large)f Fk(\015)5 b Fq(.)49 b(By)31 b(our)g(c)n(hoice)g(\(3.21\))f(of)i Fk(`)f Fq(and)h(using)f(Stirling's) g(form)n(ula,)515 1775 y(the)e(last)f(term)h(in)f(\(3.22\))g(is)h (easily)e(seen)i(to)f(b)r(e)h(b)r(ounded)g(b)n(y)f(exp[)p Fj(\000)p Fk(C)2797 1787 y Fi(4)2834 1775 y Fk(N)2910 1744 y Fi(3)2947 1775 y Fq(])h(for)f Fk(\015)33 b Fq(large)515 1874 y(enough)k(and)h(a)f(p)r(ositiv)n(e)h(constan)n(t)f Fk(C)1784 1886 y Fi(4)1861 1874 y Fk(>)j Fq(0.)67 b(Hence,)41 b Fk(u)2429 1844 y Fe(N)2429 1898 y(k)2464 1906 y Fc(0)2500 1874 y Fq(\()p Fk(t)p Fq(\))g Fj(!)f Fq(0)e(as)f Fk(N)49 b Fj(!)40 b(1)e Fq(in)515 1984 y(a)33 b(summable)g(w)n(a)n(y)-7 b(.)53 b(Therefore,)34 b Fk(V)1672 1954 y Fe(N)1735 1984 y Fq(\()p Fk(x)1814 1996 y Fi(0)1852 1984 y Fk(;)14 b(v)1929 1996 y Fi(0)1967 1984 y Fk(;)g(t)p Fq(\))33 b(con)n(v)n(erges)e(as)i Fk(N)41 b Fj(!)33 b(1)g Fq(and,)i(for)e(an)n(y)515 2084 y Fk(N)f Fj(\025)22 b Fk(N)768 2096 y Fe(\015)810 2084 y Fq(\()p Fk(t)p Fq(\),)689 2294 y Fj(j)p Fk(V)779 2260 y Fe(N)842 2294 y Fq(\()p Fk(x)921 2306 y Fi(0)959 2294 y Fk(;)14 b(v)1036 2306 y Fi(0)1073 2294 y Fk(;)g(t)p Fq(\))19 b Fj(\000)f Fk(V)1341 2260 y Fe(N)1394 2268 y Fa(\015)1431 2260 y Fi(\()p Fe(t)p Fi(\))1512 2294 y Fq(\()p Fk(x)1591 2306 y Fi(0)1629 2294 y Fk(;)c(v)1706 2306 y Fi(0)1744 2294 y Fk(;)g(t)p Fq(\))p Fj(j)23 b(\024)2047 2216 y Fg(X)1977 2397 y Fe(n)p Ff(\025)p Fe(N)2123 2405 y Fa(\015)2160 2397 y Fi(\()p Fe(t)p Fi(\))2251 2202 y Fg(\020)2301 2294 y Fk(e)2340 2260 y Ff(\000)p Fe(\025n)2472 2235 y Fc(2)2504 2260 y Fe(=)p Fi(2)2594 2294 y Fq(+)18 b Fk(e)2716 2260 y Ff(\000)p Fe(C)2816 2268 y Fc(4)2847 2260 y Fe(n)2888 2235 y Fc(3)2925 2202 y Fg(\021)2998 2294 y Fj(\024)k Fk(C)3144 2306 y Fi(5)3182 2294 y Fk(;)515 2576 y Fq(for)27 b(a)g(suitable)g Fk(C)1079 2588 y Fi(5)1140 2576 y Fk(>)c Fq(0.)36 b(By)28 b(Lemma)f(3.2)g(and)g(\(3.19\))o(,)h(w)n (e)f(\014nally)h(get:)1100 2758 y Fj(j)p Fk(V)1190 2724 y Fe(N)1253 2758 y Fq(\()p Fk(x)1332 2770 y Fi(0)1370 2758 y Fk(;)14 b(v)1447 2770 y Fi(0)1484 2758 y Fk(;)g(t)p Fq(\))p Fj(j)23 b(\024)g Fk(C)1776 2770 y Fi(5)1832 2758 y Fq(+)18 b(2)p Fk(\015)23 b Fq(+)18 b(2)p Fj(j)p Fk(v)2211 2770 y Fi(0)2248 2758 y Fj(j)g Fq(+)g(2\()p Fk(\015)23 b Fq(+)18 b Fk(K)2672 2724 y Fi(2)2666 2779 y(5)2709 2758 y Fq(\))p Fk(t:)515 2941 y Fq(T)-7 b(aking)26 b(the)i(limit)h Fk(N)j Fj(!)23 b(1)k Fq(the)h(estimate)g(\(2.9\))f(follo)n(ws.)p 3331 2895 V 3331 2945 4 50 v 3376 2945 V 3331 2948 49 4 v 515 3216 a Fp(4)134 b(Pro)t(of)45 b(of)h(Theorem)f(2.2)515 3397 y Fq(By)28 b(\(2.8\))g(and)f(Liouville's)g(theorem,)g(for)g(an)n (y)g Fk(t)c Fj(\025)g Fq(0,)k(w)n(e)g(ha)n(v)n(e:)1350 3580 y Fj(j)1391 3558 y Fq(_)1373 3580 y Fk(\030)5 b Fq(\()p Fk(t)p Fq(\))19 b Fj(\000)f Fk(E)5 b(t)18 b Fj(\000)g Fk(\027)5 b Fj(j)37 b(\024)g(H)2085 3592 y Fi(1)2122 3580 y Fq(\()p Fk(t)p Fq(\))19 b(+)f Fj(H)2388 3592 y Fi(2)2426 3580 y Fq(\()p Fk(t)p Fq(\))p Fk(;)665 b Fq(\(4.1\))515 3763 y(with:)923 3982 y Fj(H)993 3994 y Fi(1)1030 3982 y Fq(\()p Fk(t)p Fq(\))84 b(=)1355 3869 y Fg(Z)1439 3890 y Fe(t)1402 4058 y Fi(0)1468 3982 y Fk(ds)1578 3869 y Fg(Z)1624 4058 y Ff(j)p Fe(v)r Ff(j)p Fe(>as)1822 3982 y Fk(dv)1922 3869 y Fg(Z)2005 3982 y Fk(dx)14 b(f)2150 3994 y Fi(0)2187 3982 y Fq(\()p Fk(x;)g(v)s Fq(\))g Fj(j)p Fk( )2472 3948 y Ff(0)2497 3982 y Fq(\()p Fk(\030)t Fq(\()p Fk(s)p Fq(\))19 b Fj(\000)f Fk(X)2843 3994 y Fe(s)2879 3982 y Fq(\))p Fj(j)p Fk(;)923 4234 y Fj(H)993 4246 y Fi(2)1030 4234 y Fq(\()p Fk(t)p Fq(\))84 b(=)1355 4121 y Fg(Z)1439 4141 y Fe(t)1402 4310 y Fi(0)1468 4234 y Fk(ds)1578 4121 y Fg(Z)1624 4310 y Ff(j)p Fe(v)r Ff(j\024)p Fe(as)1822 4234 y Fk(dv)1922 4121 y Fg(Z)2005 4234 y Fk(dx)14 b(f)2150 4246 y Fi(0)2187 4234 y Fq(\()p Fk(x;)g(v)s Fq(\))g Fj(j)p Fk( )2472 4200 y Ff(0)2497 4234 y Fq(\()p Fk(\030)t Fq(\()p Fk(s)p Fq(\))19 b Fj(\000)f Fk(X)2843 4246 y Fe(s)2879 4234 y Fq(\))p Fj(j)p Fk(;)515 4479 y Fq(where)32 b Fk(X)829 4491 y Fe(s)917 4432 y Fk(:)896 4479 y Fq(=)g Fk(X)7 b Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(s)p Fq(\).)53 b(Recalling)33 b(\(2.7\),)h(\(2.9\),)g(and)f(setting)g Fk(V)2728 4491 y Fe(s)2817 4432 y Fk(:)2796 4479 y Fq(=)e Fk(V)19 b Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(s)p Fq(\),)35 b(w)n(e)515 4579 y(ha)n(v)n(e,)26 b(for)h Fj(j)p Fk(v)s Fj(j)d Fk(>)e(as)p Fq(,)669 4806 y Fk(f)9 b Fq(\()p Fk(X)820 4818 y Fe(s)855 4806 y Fk(;)14 b(V)940 4818 y Fe(s)976 4806 y Fq(;)g Fk(s)p Fq(\))23 b(=)g Fk(f)1236 4818 y Fi(0)1273 4806 y Fq(\()p Fk(x;)14 b(v)s Fq(\))24 b Fj(\024)e Fk(G)p Fq(\()p Fk(V)1720 4818 y Fe(s)1757 4806 y Fq(\))1833 4759 y Fk(:)1812 4806 y Fq(=)h Fk(K)1971 4818 y Fi(0)2022 4806 y Fq(min)2174 4739 y Fg(\010)2222 4806 y Fk(e)2261 4768 y Ff(\000)2323 4746 y Fa(\025)p 2323 4755 35 3 v 2326 4788 a Fc(4)2368 4701 y Fg(\000)2406 4768 y Ff(j)p Fe(V)2465 4776 y Fa(s)2496 4768 y Ff(j\000)p Fe(K)2624 4776 y Fc(1)2656 4768 y Ff(\000)p Fe(as)2775 4701 y Fg(\001)2814 4718 y Fc(2)2850 4806 y Fq(;)14 b Fk(e)2926 4772 y Ff(\000)p Fe(\025a)3053 4747 y Fc(2)3085 4772 y Fe(s)3116 4747 y Fc(2)3153 4739 y Fg(\011)3202 4806 y Fk(:)1926 5255 y Fq(9)p eop %%Page: 10 10 10 9 bop 515 523 a Fq(Then,)28 b(b)n(y)f(Liouville's)g(theorem,)g(for)g (a)g(suitable)h(constan)n(t)f Fk(C)2512 535 y Fi(6)2572 523 y Fk(>)c Fq(0)k(and)h(an)n(y)e Fk(t)d Fj(\025)g Fq(0,)585 759 y Fj(H)655 771 y Fi(1)692 759 y Fq(\()p Fk(t)p Fq(\))84 b Fj(\024)1018 646 y Fg(Z)1101 666 y Fe(t)1064 834 y Fi(0)1130 759 y Fk(ds)1240 646 y Fg(Z)1323 759 y Fk(dX)20 b(dV)33 b(G)p Fq(\()p Fk(V)19 b Fq(\))14 b Fj(j)p Fk( )1869 724 y Ff(0)1893 759 y Fq(\()p Fk(\030)t Fq(\()p Fk(s)p Fq(\))19 b Fj(\000)f Fk(X)7 b Fq(\))p Fj(j)37 b(\024)f Fk(K)2510 771 y Fi(0)2561 759 y Fj(k)p Fk( )2660 724 y Ff(0)2683 759 y Fj(k)2725 771 y Fi(1)2789 646 y Fg(Z)2872 666 y Fe(t)2835 834 y Fi(0)2901 759 y Fk(ds)1018 988 y Fj(\002)1097 871 y Fg(\024)1140 988 y Fq(4\()p Fk(K)1285 1000 y Fi(1)1340 988 y Fq(+)18 b Fk(as)p Fq(\))p Fk(e)1577 954 y Ff(\000)p Fe(\025a)1704 929 y Fc(2)1736 954 y Fe(s)1767 929 y Fc(2)1823 988 y Fq(+)1906 875 y Fg(Z)1952 1064 y Ff(j)p Fe(V)c Ff(j)p Fe(>)p Fi(2\()p Fe(K)2212 1072 y Fc(1)2244 1064 y Fi(+)p Fe(as)p Fi(\))2392 988 y Fk(dV)33 b(e)2555 950 y Ff(\000)2617 928 y Fa(\025)p 2617 937 35 3 v 2620 970 a Fc(4)2662 883 y Fg(\000)2700 950 y Ff(j)p Fe(V)14 b Ff(j\000)p Fe(K)2901 958 y Fc(1)2933 950 y Ff(\000)p Fe(as)3052 883 y Fg(\001)3090 900 y Fc(2)3126 871 y Fg(\025)870 1174 y Fj(\024)83 b Fk(C)1077 1186 y Fi(6)1114 1174 y Fk(:)2071 b Fq(\(4.2\))639 1356 y(W)-7 b(e)31 b(no)n(w)e(in)n(v)n(estigate)g Fj(H)1446 1368 y Fi(2)1484 1356 y Fq(\()p Fk(t)p Fq(\),)i(i.e.)f(the)h(con)n (tribution)f(of)g(the)g(slo)n(w)f(particles)h(\(w.r.t.)515 1456 y(the)g(c)n(harged)e(one\).)42 b(The)30 b(main)f(p)r(oin)n(t)h(is) g(that)f(the)h(time)g(a)n(v)n(erage)d(of)j(the)g(force)e(can)i(b)r(e) 515 1556 y(con)n(trolled)d(b)n(y)h(the)g Fk(L)1217 1568 y Fi(1)1254 1556 y Fq(-norm)f(of)h(the)h(force)e(itself)i(divided)f(b)n (y)g(the)h(v)n(elo)r(cit)n(y)e(gap.)38 b(More)515 1655 y(precisely)-7 b(,)27 b(assume:)1550 1755 y Fk(\027)h Fj(\025)23 b Fq(8)p Fk(K)1820 1767 y Fi(1)1856 1755 y Fk(;)180 b(E)28 b Fj(\025)23 b Fq(8)p Fk(a;)863 b Fq(\(4.3\))515 1904 y(and)27 b(de\014ne:)1140 2033 y Fk(T)1244 1986 y(:)1224 2033 y Fq(=)22 b(inf)1426 1916 y Fg(\032)1488 2033 y Fk(t)h(>)g Fq(0)g(:)37 b Fj(j)1794 2011 y Fq(_)1777 2033 y Fk(\030)t Fq(\()p Fk(t)p Fq(\))19 b Fj(\000)f Fk(E)5 b(t)18 b Fj(\000)g Fk(\027)5 b Fj(j)24 b Fk(>)2400 1977 y(E)5 b(t)19 b Fq(+)f Fk(\027)p 2400 2014 244 4 v 2501 2090 a Fq(8)2654 1916 y Fg(\033)2730 2033 y Fk(:)455 b Fq(\(4.4\))515 2232 y(By)24 b(con)n(tin)n(uit)n(y)g Fk(T)34 b(>)22 b Fq(0.)36 b(W)-7 b(e)25 b(lo)r(ok)e(for)h(an)g(upp)r (er)h(b)r(ound)f(to)h Fj(H)2510 2244 y Fi(2)2547 2232 y Fq(\()p Fk(t)p Fq(\))g(when)g Fk(t)e Fj(2)g Fq([0)p Fk(;)14 b(T)e Fq(\).)35 b(W)-7 b(e)515 2332 y(write:)959 2468 y Fj(H)1029 2480 y Fi(2)1066 2468 y Fq(\()p Fk(t)p Fq(\))23 b(=)1271 2355 y Fg(Z)1317 2544 y Ff(j)p Fe(v)r Ff(j\024)p Fe(at)1509 2468 y Fk(dv)1609 2355 y Fg(Z)1692 2468 y Fk(dx)14 b(f)1837 2480 y Fi(0)1875 2468 y Fq(\()p Fk(x;)g(v)s Fq(\))2080 2355 y Fg(Z)2164 2376 y Fe(t)2127 2544 y Ff(j)p Fe(v)r Ff(j)p Fe(=a)2276 2468 y Fk(ds)g Fj(j)p Fk( )2452 2434 y Ff(0)2475 2468 y Fq(\()p Fk(\030)t Fq(\()p Fk(s)p Fq(\))20 b Fj(\000)e Fk(X)2822 2480 y Fe(s)2857 2468 y Fq(\))p Fj(j)p Fk(:)273 b Fq(\(4.5\))515 2679 y(By)28 b(\(2.9\))g(w)n(e)f(ha)n(v)n(e,)f(for)h(an)n(y)g Fk(s)c Fj(2)g Fq([0)p Fk(;)14 b(T)e Fq(\),)666 2853 y(7\()p Fk(E)5 b(s)18 b Fq(+)g Fk(\027)5 b Fq(\))p 666 2890 359 4 v 824 2966 a(8)1053 2909 y Fj(\000)18 b Fk(K)1207 2921 y Fi(1)1262 2909 y Fj(\000)g Fk(as)g Fj(\000)g Fq(2)p Fj(j)p Fk(v)s Fj(j)23 b(\024)1789 2887 y Fq(_)1771 2909 y Fk(\030)t Fq(\()p Fk(s)p Fq(\))c Fj(\000)f Fk(V)2064 2921 y Fe(s)2123 2909 y Fj(\024)2221 2853 y Fq(9\()p Fk(E)5 b(s)18 b Fq(+)g Fk(\027)5 b Fq(\))p 2221 2890 V 2379 2966 a(8)2608 2909 y(+)18 b Fk(K)2762 2921 y Fi(1)2817 2909 y Fq(+)g Fk(as)g Fq(+)g(2)p Fj(j)p Fk(v)s Fj(j)p Fk(;)515 3120 y Fq(and)27 b(therefore,)g(b)n(y)h(\(4.3\),)998 3294 y Fk(E)5 b(s)18 b Fq(+)g Fk(\027)p 998 3331 253 4 v 1103 3407 a Fq(2)1283 3350 y Fj(\024)1389 3329 y Fq(_)1371 3350 y Fk(\030)t Fq(\()p Fk(s)p Fq(\))h Fj(\000)f Fk(V)1664 3362 y Fe(s)1723 3350 y Fj(\024)1821 3294 y Fq(3\()p Fk(E)5 b(s)18 b Fq(+)g Fk(\027)5 b Fq(\))p 1821 3331 359 4 v 1979 3407 a(2)2356 3350 y Fj(8)14 b Fk(s)22 b Fj(2)h Fq([)p Fj(j)p Fk(v)s Fj(j)p Fk(=a;)14 b(T)e Fq(\))p Fk(:)301 b Fq(\(4.6\))515 3563 y(Let)38 b Fk(r)r Fq(\()p Fk(s)p Fq(\))879 3516 y Fk(:)858 3563 y Fq(=)j Fk(\030)t Fq(\()p Fk(s)p Fq(\))26 b Fj(\000)f Fk(X)1292 3575 y Fe(s)1366 3563 y Fq(and)38 b Fk(\034)1574 3575 y Fe(x;v)1733 3516 y Fk(:)1712 3563 y Fq(=)i(inf)7 b Fj(f)p Fk(t)41 b Fj(2)g Fq([0)p Fk(;)14 b(T)e Fq(\))40 b(:)54 b Fj(j)p Fk(\030)t Fq(\()p Fk(t)p Fq(\))27 b Fj(\000)e Fk(X)2782 3575 y Fe(t)2811 3563 y Fj(j)41 b(\024)f Fq(1)p Fj(g)e Fq(\(setting)515 3672 y Fk(\034)551 3684 y Fe(x;v)692 3625 y Fk(:)671 3672 y Fq(=)23 b Fj(1)k Fq(if)g(the)h(ab)r(o)n(v)n(e)e (set)h(is)g(empt)n(y\).)37 b(Since)j(_)-36 b Fk(r)s Fq(\()p Fk(s)p Fq(\))23 b(=)2340 3650 y(_)2322 3672 y Fk(\030)t Fq(\()p Fk(s)p Fq(\))18 b Fj(\000)f Fk(V)2613 3684 y Fe(s)2672 3672 y Fj(\025)23 b Fq(\()p Fk(E)5 b(\034)2894 3684 y Fe(x;v)3009 3672 y Fq(+)17 b Fk(\027)5 b Fq(\))p Fk(=)p Fq(2)27 b(for)515 3772 y Fk(s)c Fj(2)g Fq([)p Fj(j)p Fk(v)s Fj(j)p Fk(=a;)14 b(T)e Fq(\),)27 b(w)n(e)g(ha)n(v)n(e:) 926 3901 y Fg(Z)1009 3921 y Fe(t)972 4090 y Ff(j)p Fe(v)r Ff(j)p Fe(=a)1121 4014 y Fk(ds)14 b Fj(j)p Fk( )1297 3980 y Ff(0)1320 4014 y Fq(\()p Fk(\030)t Fq(\()p Fk(s)p Fq(\))20 b Fj(\000)e Fk(X)1667 4026 y Fe(s)1702 4014 y Fq(\))p Fj(j)24 b(\024)1868 3901 y Fg(Z)1951 3921 y Fi(1)1914 4090 y Ff(\000)p Fi(1)2003 4014 y Fk(dr)2147 3958 y Fq(2)p Fj(j)p Fk( )2269 3928 y Ff(0)2292 3958 y Fq(\()p Fk(r)r Fq(\))p Fj(j)p 2110 3995 347 4 v 2110 4071 a Fk(E)5 b(\034)2212 4083 y Fe(x;v)2328 4071 y Fq(+)18 b Fk(\027)2490 4014 y Fq(=)2640 3958 y(2)p Fj(k)p Fk( )2781 3928 y Ff(0)2804 3958 y Fj(k)2846 3970 y Fi(1)p 2588 3995 V 2588 4071 a Fk(E)5 b(\034)2690 4083 y Fe(x;v)2805 4071 y Fq(+)18 b Fk(\027)2945 4014 y(:)515 4258 y Fq(Then,)28 b(b)n(y)g(\(2.7\))f(and)h(\(4.5\))o(,)1021 4488 y Fj(H)1091 4500 y Fi(2)1129 4488 y Fq(\()p Fk(t)p Fq(\))23 b Fj(\024)g Fq(2)p Fk(K)1447 4500 y Fi(0)1483 4488 y Fj(k)p Fk( )1582 4454 y Ff(0)1605 4488 y Fj(k)1647 4500 y Fi(1)1698 4375 y Fg(Z)1744 4564 y Ff(j)p Fe(v)r Ff(j\024)p Fe(at)1936 4488 y Fk(dv)17 b(e)2075 4454 y Ff(\000)p Fe(\025v)2201 4429 y Fc(2)2252 4375 y Fg(Z)2335 4488 y Fk(dx)2449 4432 y(\037)p Fq(\()p Fk(\034)2569 4444 y Fe(x;v)2689 4432 y Fj(\024)23 b Fk(t)p Fq(\))p 2449 4469 391 4 v 2471 4545 a Fk(E)5 b(\034)2573 4557 y Fe(x;v)2688 4545 y Fq(+)18 b Fk(\027)2849 4488 y(:)515 4732 y Fq(By)28 b(\(4.6\))g(w)n(e)f(ha)n(v) n(e)f Fk(\034)1194 4744 y Fe(x;v)1314 4732 y Fk(<)d(T)39 b Fq(for)27 b Fj(j)p Fk(v)s Fj(j)c(\024)g Fk(at)k Fq(only)h(if:)955 4979 y Fj(j)p Fk(x)p Fj(j)c(\024)1169 4915 y Fq(3)p Fk(E)5 b(\034)1322 4884 y Fi(2)1313 4935 y Fe(x;v)p 1169 4960 241 4 v 1269 5036 a Fq(4)1438 4979 y(+)1531 4923 y(3)p Fk(\027)g(\034)1655 4935 y Fe(x;v)p 1531 4960 221 4 v 1621 5036 a Fq(2)1781 4979 y(+)18 b(1)k(=)2026 4923 y(3\()p Fk(E)5 b(\034)2202 4935 y Fe(x;v)2317 4923 y Fq(+)18 b Fk(\027)5 b Fq(\))2478 4892 y Fi(2)p 2026 4960 491 4 v 2217 5036 a Fq(4)p Fk(E)2545 4979 y Fj(\000)2638 4923 y Fq(3)p Fk(\027)2726 4892 y Fi(2)p 2638 4960 126 4 v 2647 5036 a Fq(4)p Fk(E)2791 4979 y Fq(+)18 b(1)p Fk(:)1905 5255 y Fq(10)p eop %%Page: 11 11 11 10 bop 515 523 a Fq(In)28 b(particular,)e(if)i Fj(j)p Fk(x)p Fj(j)c Fk(>)e Fq(4)27 b(then)i Fk(E)5 b(\034)1666 535 y Fe(x;v)1781 523 y Fq(+)18 b Fk(\027)29 b Fj(\025)2021 452 y Fg(p)p 2104 452 345 4 v 71 x Fk(E)5 b Fj(j)p Fk(x)p Fj(j)19 b Fq(+)f Fk(\027)2411 499 y Fi(2)2449 523 y Fq(,)28 b(so)f(that:)727 645 y Fg(Z)810 758 y Fk(dx)924 702 y(\037)p Fq(\()p Fk(\034)1044 714 y Fe(x;v)1165 702 y Fj(\024)22 b Fk(t)p Fq(\))p 924 739 391 4 v 946 815 a Fk(E)5 b(\034)1048 827 y Fe(x;v)1164 815 y Fq(+)18 b Fk(\027)1408 758 y Fj(\024)1555 645 y Fg(Z)1648 702 y Fk(dx)p 1648 739 91 4 v 1670 815 a(\027)1763 758 y(\037)p Fq(\()p Fj(j)p Fk(x)p Fj(j)23 b(\024)g Fq(4\))1555 992 y(+)1647 879 y Fg(Z)1909 936 y Fk(dx)p 1740 973 428 4 v 1740 990 a Fg(p)p 1823 990 345 4 v 71 x Fk(E)5 b Fj(j)p Fk(x)p Fj(j)19 b Fq(+)f Fk(\027)2130 1037 y Fi(2)2192 992 y Fk(\037)2244 875 y Fg(\022)2305 992 y Fk(E)5 b Fj(j)p Fk(x)p Fj(j)24 b(\024)e Fq(\()p Fk(E)5 b(t)19 b Fq(+)f Fk(\027)5 b Fq(\))2883 958 y Fi(2)2939 992 y Fj(\000)18 b Fk(\027)3068 958 y Fi(2)3106 875 y Fg(\023)1408 1229 y Fj(\024)96 b Fq(4)p Fk(t)18 b Fq(+)1754 1173 y(8)p 1752 1210 47 4 v 1752 1286 a Fk(\027)1808 1229 y(:)515 1440 y Fq(Therefore,)26 b(b)n(y)j(\(4.3\))o(,)f(for)f(suitable)h(constan)n(ts)e Fk(C)2115 1452 y Fi(7)2153 1440 y Fk(;)14 b(C)2249 1452 y Fi(8)2309 1440 y Fk(>)23 b Fq(0)k(and)h(an)n(y)e Fk(t)d Fj(2)h Fq([0)p Fk(;)14 b(T)e Fq(\),)1610 1622 y Fj(H)1680 1634 y Fi(2)1717 1622 y Fq(\()p Fk(t)p Fq(\))24 b Fj(\024)e Fk(C)1981 1634 y Fi(7)2033 1622 y Fk(t)c Fq(+)g Fk(C)2223 1634 y Fi(8)2261 1622 y Fk(:)924 b Fq(\(4.7\))515 1805 y(By)28 b(\(4.1\),)g(\(4.2\))o(,)g(and)f(\(4.7\),)h(w)n(e)f(\014nally)g (obtain)h(that,)g(if)g Fk(\027)33 b Fq(and)27 b Fk(E)33 b Fq(satisfy)28 b(\(4.3\))g(then:)1059 1988 y Fj(j)1100 1966 y Fq(_)1082 1988 y Fk(\030)t Fq(\()p Fk(t)p Fq(\))19 b Fj(\000)f Fk(E)5 b(t)19 b Fj(\000)f Fk(\027)5 b Fj(j)37 b(\024)g Fk(C)1783 2000 y Fi(7)1834 1988 y Fk(t)19 b Fq(+)f Fk(C)2025 2000 y Fi(6)2081 1988 y Fq(+)g Fk(C)2223 2000 y Fi(8)2426 1988 y Fj(8)c Fk(t)22 b Fj(2)i Fq([0)p Fk(;)14 b(T)e Fq(\))p Fk(:)372 b Fq(\(4.8\))515 2170 y(No)n(w)30 b(\014x)h Fk(\027)869 2182 y Fi(0)934 2170 y Fk(>)d Fq(8)14 b(max)o Fj(f)p Fk(K)1350 2182 y Fi(1)1386 2170 y Fq(;)g Fk(C)1482 2182 y Fi(6)1540 2170 y Fq(+)20 b Fk(C)1684 2182 y Fi(8)1721 2170 y Fj(g)31 b Fq(and)f Fk(E)2019 2182 y Fi(0)2085 2170 y Fk(>)d Fq(8)14 b(max)o Fj(f)p Fk(a)p Fq(;)g Fk(C)2569 2182 y Fi(7)2606 2170 y Fj(g)p Fq(.)46 b(By)30 b(con)n(tin)n(uit)n(y)g(and)515 2270 y(de\014nition)i(\(4.4\),)h(if)f Fk(\027)j Fj(\025)30 b Fk(\027)1407 2282 y Fi(0)1476 2270 y Fq(and)i Fk(E)j Fj(\025)30 b Fk(E)1894 2282 y Fi(0)1963 2270 y Fq(then)j Fk(T)41 b Fq(=)29 b Fj(1)p Fq(.)50 b(Bound)32 b(\(4.8\))f(is)h (therefore)515 2369 y(v)-5 b(alid)34 b(in)h(this)f(case)g(for)f(an)n(y) h(p)r(ositiv)n(e)g(time)h(and)f(\(2.10\))f(follo)n(ws)h(with)g Fk(K)2944 2381 y Fi(2)3015 2369 y Fq(=)g Fk(C)3173 2381 y Fi(7)3245 2369 y Fq(and)515 2469 y Fk(K)586 2481 y Fi(3)646 2469 y Fq(=)22 b Fk(C)792 2481 y Fi(6)848 2469 y Fq(+)c Fk(C)990 2481 y Fi(8)1028 2469 y Fq(.)p 3331 2423 49 4 v 3331 2473 4 50 v 3376 2473 V 3331 2476 49 4 v 515 2744 a Fp(App)t(endix)515 2926 y Fr(Pro)s(of)40 b(of)g(Lemma)e(3.1.)92 b Fq(The)35 b(main)g(to)r(ol)f(here)h(is)f (energy)g(conserv)-5 b(ation.)57 b(Let)35 b(us)515 3025 y(de\014ne:)1293 3162 y Fk(R)q Fq(\()p Fk(t;)14 b(s)p Fq(\))1571 3115 y Fk(:)1550 3162 y Fq(=)23 b Fk(R)1702 3127 y Fe(N)s(;M)1850 3162 y Fq(\()p Fk(t)p Fq(\))c(+)2046 3049 y Fg(Z)2129 3069 y Fe(t)2092 3237 y(s)2158 3162 y Fk(d\034)24 b Fj(V)2319 3127 y Fe(N)s(;M)2467 3162 y Fq(\()p Fk(\034)9 b Fq(\))p Fk(;)589 b Fq(\(A.1\))515 3353 y(so)27 b(that:)651 3536 y Fk(R)q Fq(\()p Fk(t;)14 b(t)p Fq(\))23 b(=)g Fk(R)1051 3501 y Fe(N)s(;M)1199 3536 y Fq(\()p Fk(t)p Fq(\))p Fk(;)98 b(R)q Fq(\()p Fk(t;)14 b Fq(0\))22 b Fj(\024)h Fq(2)p Fk(R)1867 3501 y Fe(N)s(;M)2015 3536 y Fq(\()p Fk(t)p Fq(\))p Fk(;)97 b(@)2273 3548 y Fe(s)2309 3536 y Fk(R)q Fq(\()p Fk(t;)14 b(s)p Fq(\))23 b(=)g Fj(\000V)2777 3501 y Fe(N)s(;M)2925 3536 y Fq(\()p Fk(s)p Fq(\))p Fk(:)137 b Fq(\(A.2\))515 3718 y(In)26 b(order)g(to)g(simplify)h(notation,)g(in)f(the)h(sequel)g(w)n(e)f(drop) g(the)h(cut-o\013)f(indexes)h(and)f(set:)1169 3901 y Fk(X)1238 3913 y Fe(t)1311 3854 y Fk(:)1290 3901 y Fq(=)d Fk(X)1454 3871 y Fe(N)s(;M)1601 3901 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))p Fk(;)195 b(V)2125 3913 y Fe(t)2199 3854 y Fk(:)2178 3901 y Fq(=)23 b Fk(V)2333 3867 y Fe(N)s(;M)2481 3901 y Fq(\()p Fk(x;)14 b(v)s Fq(;)g Fk(t)p Fq(\))p Fk(;)1172 4026 y(Y)1220 4038 y Fe(t)1293 3978 y Fk(:)1272 4026 y Fq(=)23 b Fk(X)1436 3995 y Fe(N)s(;M)1584 4026 y Fq(\()p Fk(y)s(;)14 b(w)r Fq(;)g Fk(t)p Fq(\))p Fk(;)197 b(W)2155 4038 y Fe(t)2229 3978 y Fk(:)2208 4026 y Fq(=)23 b Fk(V)2363 3991 y Fe(N)s(;M)2511 4026 y Fq(\()p Fk(y)s(;)14 b(w)r Fq(;)g Fk(t)p Fq(\))p Fk(:)515 4333 y Fq(By)27 b(de\014nition)h(\(3.4\))f(of)h Fk(W)40 b Fq(and)27 b(Liouville's)g(theorem,)719 4558 y Fj(W)7 b Fq(\()p Fk(f)i Fq(\()p Fk(s)p Fq(\);)14 b Fk(\026;)g(R)q Fq(\()p Fk(t;)g(s)p Fq(\)\))23 b(=)1504 4502 y(1)p 1504 4539 42 4 v 1504 4615 a(2)1569 4445 y Fg(Z)1652 4558 y Fk(dx)14 b(dv)k(g)1900 4524 y Fe(\026;R)p Fi(\()p Fe(t;s)p Fi(\))2142 4558 y Fq(\()p Fk(X)2243 4570 y Fe(s)2279 4558 y Fq(\))p Fk(f)2352 4570 y Fi(0)2389 4558 y Fq(\()p Fk(x;)c(v)s Fq(\))p Fk(V)2648 4524 y Fi(2)2628 4578 y Fe(s)1014 4776 y Fq(+)1121 4719 y(1)p 1121 4756 V 1121 4832 a(2)1186 4663 y Fg(Z)1297 4776 y Fk(dx)g(dv)k(dy)e(dw)h(f)1762 4788 y Fi(0)1799 4776 y Fq(\()p Fk(x;)d(v)s Fq(\))g Fk(f)2045 4788 y Fi(0)2083 4776 y Fq(\()p Fk(y)s(;)g(w)r Fq(\))g Fk(g)2346 4741 y Fe(\026;R)p Fi(\()p Fe(t;s)p Fi(\))2589 4776 y Fq(\()p Fk(X)2690 4788 y Fe(s)2725 4776 y Fq(\))g Fk(')p Fq(\()p Fk(X)2926 4788 y Fe(s)2981 4776 y Fj(\000)k Fk(Y)3112 4788 y Fe(s)3147 4776 y Fq(\))p Fk(:)1905 5255 y Fq(11)p eop %%Page: 12 12 12 11 bop 515 523 a Fq(F)-7 b(or)27 b Fk(s)c(<)f(t)p Fq(,)28 b(w)n(e)f(ha)n(v)n(e:)970 744 y Fk(@)1014 756 y Fe(s)1050 744 y Fj(W)7 b Fq(\()p Fk(f)i Fq(\()p Fk(s)p Fq(\);)14 b Fk(\026;)g(R)q Fq(\()p Fk(t;)g(s)p Fq(\)\))23 b(=)1825 631 y Fg(Z)1908 744 y Fk(dx)14 b(dv)k(f)2154 756 y Fi(0)2191 744 y Fq(\()p Fk(x;)c(v)s Fq(\))g Fk(g)2439 709 y Fe(\026;R)p Fi(\()p Fe(t;s)p Fi(\))2682 744 y Fq(\()p Fk(X)2783 756 y Fe(s)2818 744 y Fq(\))1099 961 y Fj(\002)1196 869 y Fg(n)1252 961 y Fk(V)1300 973 y Fe(s)1357 940 y Fq(_)1336 961 y Fk(V)1384 973 y Fe(s)1438 961 y Fq(+)1531 905 y(1)p 1531 942 42 4 v 1531 1018 a(2)1596 848 y Fg(Z)1679 961 y Fk(dy)j(dw)g(f)1940 973 y Fi(0)1977 961 y Fq(\()p Fk(y)s(;)d(w)r Fq(\))28 b(\()p Fk(V)2291 973 y Fe(s)2346 961 y Fj(\000)18 b Fk(W)2507 973 y Fe(s)2543 961 y Fq(\))28 b Fk(')2657 927 y Ff(0)2680 961 y Fq(\()p Fk(X)2781 973 y Fe(s)2835 961 y Fj(\000)18 b Fk(Y)2966 973 y Fe(s)3002 961 y Fq(\))3034 869 y Fg(o)1099 1179 y Fq(+)1206 1123 y(1)p 1206 1160 V 1206 1236 a(2)1272 1066 y Fg(Z)1355 1179 y Fk(dx)c(dv)j(f)1600 1191 y Fi(0)1637 1179 y Fq(\()p Fk(x;)d(v)s Fq(\))g Fk(@)1886 1191 y Fe(s)1923 1179 y Fk(g)1966 1145 y Fe(\026;R)p Fi(\()p Fe(t;s)p Fi(\))2208 1179 y Fq(\()p Fk(X)2309 1191 y Fe(s)2344 1179 y Fq(\))1099 1397 y Fj(\002)1196 1304 y Fg(n)1252 1397 y Fk(V)1318 1362 y Fi(2)1300 1417 y Fe(s)1374 1397 y Fq(+)1457 1284 y Fg(Z)1540 1397 y Fk(dy)j(dw)g(f)1801 1409 y Fi(0)1838 1397 y Fq(\()p Fk(y)s(;)d(w)r Fq(\))g Fk(')p Fq(\()p Fk(X)2213 1409 y Fe(s)2268 1397 y Fj(\000)k Fk(Y)2399 1409 y Fe(s)2434 1397 y Fq(\))2466 1304 y Fg(o)2522 1397 y Fk(:)643 b Fq(\(A.3\))515 1633 y(It)30 b(is)f(easy)f(to)i(see)f(that) h Fk(@)1342 1645 y Fe(s)1377 1633 y Fk(g)1420 1603 y Fe(\026;R)p Fi(\()p Fe(t;s)p Fi(\))1662 1633 y Fq(\()p Fk(X)1763 1645 y Fe(s)1799 1633 y Fq(\))d Fj(\024)e Fq(0,)30 b(whence)f(the)h(last)f(term)h(on)f(the)h(r.h.s.)f(of)515 1733 y(\(A.3\))f(is)f(non)h(p)r(ositiv)n(e)f(and)g(can)h(b)r(e)g (neglected.)36 b(Th)n(us,)28 b(b)n(y)g(\(3.1\))f(and)h(\(A.3\))g(w)n(e) f(get:)611 1958 y Fk(@)655 1970 y Fe(s)690 1958 y Fj(W)7 b Fq(\()p Fk(f)i Fq(\()p Fk(s)p Fq(\);)14 b Fk(\026;)g(R)q Fq(\()p Fk(t;)g(s)p Fq(\)\))84 b Fj(\024)e(\000)1674 1902 y Fq(1)p 1674 1939 V 1674 2015 a(2)1739 1845 y Fg(Z)1822 1958 y Fk(dx)14 b(dv)k(dy)f(dw)f(f)2287 1970 y Fi(0)2324 1958 y Fq(\()p Fk(x;)e(v)s Fq(\))g Fk(f)2570 1970 y Fi(0)2608 1958 y Fq(\()p Fk(y)s(;)g(w)r Fq(\))g Fk(g)2871 1924 y Fe(\026;R)p Fi(\()p Fe(t;s)p Fi(\))3114 1958 y Fq(\()p Fk(X)3215 1970 y Fe(s)3251 1958 y Fq(\))1585 2129 y Fj(\002)28 b Fq(\()p Fk(V)1758 2141 y Fe(s)1812 2129 y Fq(+)18 b Fk(W)1973 2141 y Fe(s)2009 2129 y Fq(\))c Fk(')2109 2095 y Ff(0)2133 2129 y Fq(\()p Fk(X)2234 2141 y Fe(s)2288 2129 y Fj(\000)k Fk(Y)2419 2141 y Fe(s)2455 2129 y Fq(\))1438 2296 y(=)1595 2240 y(1)p 1595 2277 V 1595 2353 a(2)1661 2183 y Fg(Z)1744 2296 y Fk(dx)c(dv)k(dy)e(dw)h(f)2209 2308 y Fi(0)2246 2296 y Fq(\()p Fk(x;)d(v)s Fq(\))g Fk(f)2492 2308 y Fi(0)2530 2296 y Fq(\()p Fk(y)s(;)g(w)r Fq(\))g Fk(')2804 2262 y Ff(0)2828 2296 y Fq(\()p Fk(X)2929 2308 y Fe(s)2983 2296 y Fj(\000)k Fk(Y)3114 2308 y Fe(s)3150 2296 y Fq(\))p Fk(V)3230 2308 y Fe(s)1585 2496 y Fj(\002)1664 2404 y Fg(\020)1713 2496 y Fk(g)1756 2462 y Fe(\026;R)p Fi(\()p Fe(t;s)p Fi(\))1999 2496 y Fq(\()p Fk(Y)2079 2508 y Fe(s)2115 2496 y Fq(\))g Fj(\000)g Fk(g)2291 2462 y Fe(\026;R)p Fi(\()p Fe(t;s)p Fi(\))2533 2496 y Fq(\()p Fk(X)2634 2508 y Fe(s)2670 2496 y Fq(\))2702 2404 y Fg(\021)2766 2496 y Fk(:)399 b Fq(\(A.4\))515 2711 y(No)n(w,)35 b(since)f Fj(j)p Fk(g)s Fq(\()p Fk(x)p Fq(\))23 b Fj(\000)g Fk(g)s Fq(\()p Fk(y)s Fq(\))p Fj(j)33 b(\024)h Fq(2)p Fj(j)p Fk(x)23 b Fj(\000)f Fk(y)s Fj(j)p Fq(,)35 b(b)n(y)f(using)g Fj(j)p Fk(V)2303 2723 y Fe(s)2339 2711 y Fj(j)g(\024)f(V)2552 2681 y Fe(N)s(;M)2734 2711 y Fq(=)h Fj(\000)p Fk(@)2942 2723 y Fe(s)2977 2711 y Fk(R)q Fq(\()p Fk(t;)14 b(s)p Fq(\))34 b(and)515 2811 y(\(2.6\))o(,)28 b(w)n(e)f(ha)n(v)n(e:)638 3042 y Fj(W)7 b Fq(\()p Fk(f)i Fq(\()p Fk(s)p Fq(\);)14 b Fk(\026;)g(R)q Fq(\()p Fk(t;)g(s)p Fq(\)\))23 b Fj(\024)g(\000)1488 2985 y Fk(@)1532 2997 y Fe(s)1567 2985 y Fk(R)q Fq(\()p Fk(t;)14 b(s)p Fq(\))p 1488 3022 314 4 v 1527 3098 a Fk(R)q Fq(\()p Fk(t;)g(s)p Fq(\))1811 3042 y Fj(k)p Fk(')1907 3007 y Ff(0)1930 3042 y Fj(k)1972 3054 y Ff(1)2056 2929 y Fg(Z)2139 3042 y Fk(dx)g(dv)k(dy)e(dw)h(f)2604 3054 y Fi(0)2641 3042 y Fq(\()p Fk(x;)d(v)s Fq(\))g Fk(f)2887 3054 y Fi(0)2925 3042 y Fq(\()p Fk(y)s(;)g(w)r Fq(\))g Fk(\037)3197 3054 y Fe(s)3233 3042 y Fk(;)515 3275 y Fq(where)29 b Fk(\037)809 3287 y Fe(s)893 3228 y Fk(:)872 3275 y Fq(=)e Fk(\037)p Fq(\()p Fj(j)p Fk(X)1140 3287 y Fe(s)1195 3275 y Fj(\000)20 b Fk(Y)1328 3287 y Fe(s)1364 3275 y Fj(j)27 b(\024)g Fq(1\))14 b([)p Fk(\037)p Fq(\()p Fj(j)p Fk(X)1793 3287 y Fe(s)1848 3275 y Fj(\000)20 b Fk(\026)p Fj(j)27 b(\024)g Fq(2)p Fk(R)q Fq(\()p Fk(t;)14 b(s)p Fq(\)\))20 b(+)f Fk(\037)p Fq(\()p Fj(j)p Fk(Y)2692 3287 y Fe(s)2748 3275 y Fj(\000)h Fk(\026)p Fj(j)27 b(\024)g Fq(2)p Fk(R)q Fq(\()p Fk(t;)14 b(s)p Fq(\)\)].)515 3374 y(By)27 b(symmetry)g(and)h(Liouville's)f(theorem,)g(w)n(e)g(get:)805 3600 y Fj(W)7 b Fq(\()p Fk(f)i Fq(\()p Fk(s)p Fq(\);)14 b Fk(\026;)g(R)q Fq(\()p Fk(t;)g(s)p Fq(\)\))37 b Fj(\024)f(\000)14 b Fq(2)1751 3544 y Fk(@)1795 3556 y Fe(s)1831 3544 y Fk(R)q Fq(\()p Fk(t;)g(s)p Fq(\))p 1751 3581 V 1791 3657 a Fk(R)q Fq(\()p Fk(t;)g(s)p Fq(\))2075 3600 y Fj(k)p Fk(')2171 3566 y Ff(0)2194 3600 y Fj(k)2236 3612 y Ff(1)2319 3487 y Fg(Z)2402 3600 y Fk(dx)g(dv)k(dy)f(dw)g(f)9 b Fq(\()p Fk(x;)14 b(y)s Fq(;)g Fk(s)p Fq(\))1127 3782 y Fj(\002)k Fk(f)9 b Fq(\()p Fk(y)s(;)14 b(w)r Fq(;)g Fk(s)p Fq(\))g Fk(\037)1608 3714 y Fg(\000)1646 3782 y Fj(j)p Fk(x)19 b Fj(\000)f Fk(y)s Fj(j)23 b(\024)g Fq(1)2038 3714 y Fg(\001)2089 3782 y Fk(\037)2141 3714 y Fg(\000)2179 3782 y Fj(j)p Fk(x)c Fj(\000)f Fk(\026)p Fj(j)23 b(\024)g Fq(2)p Fk(R)q Fq(\()p Fk(t;)14 b(s)p Fq(\))2811 3714 y Fg(\001)2849 3782 y Fk(:)316 b Fq(\(A.5\))515 3964 y(Consider)31 b(no)n(w)g(the)i(partition)f(of)g Fh(R)38 b Fq(in)n(to)32 b(in)n(terv)-5 b(als)31 b(\001)2312 3976 y Fe(i)2371 3964 y Fq(=)f([)p Fk(i\016)o(;)14 b Fq(\()p Fk(i)20 b Fq(+)h(1\))p Fk(\016)s Fq(\),)34 b Fk(i)c Fj(2)h Fh(Z)o Fq(,)d(with)515 4064 y Fk(\016)599 4017 y(:)578 4064 y Fq(=)22 b(min)q Fj(f)p Fk(x)h(>)f Fq(0)h(:)37 b Fk(')p Fq(\()p Fk(x)p Fq(\))24 b Fj(\024)f Fk(')p Fq(\(0\))p Fk(=)p Fq(2)p Fj(g)k Fq(and)g(de\014ne:)725 4289 y Fk(\032)768 4301 y Fe(s)803 4289 y Fq(\()p Fk(x)p Fq(\))959 4242 y Fk(:)939 4289 y Fq(=)1026 4176 y Fg(Z)1109 4289 y Fk(dv)17 b(f)9 b Fq(\()p Fk(x;)14 b(y)s Fq(;)g Fk(s)p Fq(\))p Fk(;)180 b(\032)1773 4255 y Fe(R)1773 4310 y(s)1828 4289 y Fq(\()p Fk(x)p Fq(\))1984 4242 y Fk(:)1963 4289 y Fq(=)23 b Fk(\032)2094 4301 y Fe(s)2129 4289 y Fq(\()p Fk(x)p Fq(\))14 b Fk(\037)p Fq(\()p Fj(j)p Fk(x)20 b Fj(\000)e Fk(\026)p Fj(j)23 b(\024)g Fq(2)p Fk(R)q Fq(\()p Fk(t;)14 b(s)p Fq(\))k(+)g(1\))p Fk(:)1905 5255 y Fq(12)p eop %%Page: 13 13 13 12 bop 515 523 a Fq(The)27 b(in)n(tegral)g(on)g(the)h(r.h.s.)f(of)34 b(\(A.5\))28 b(can)g(b)r(e)g(b)r(ounded)g(in)f(the)h(follo)n(wing)f(w)n (a)n(y:)712 608 y Fg(Z)795 721 y Fk(dx)14 b(dv)k(dy)f(dw)g(f)9 b Fq(\()p Fk(x;)14 b(y)s Fq(;)g Fk(s)p Fq(\))g Fk(f)9 b Fq(\()p Fk(y)s(;)14 b(w)r Fq(;)g Fk(s)p Fq(\))g Fk(\037)1950 654 y Fg(\000)1987 721 y Fj(j)p Fk(x)19 b Fj(\000)f Fk(y)s Fj(j)23 b(\024)g Fq(1)2379 654 y Fg(\001)2430 721 y Fk(\037)2482 654 y Fg(\000)2520 721 y Fj(j)p Fk(x)c Fj(\000)f Fk(\026)p Fj(j)23 b(\024)g Fq(2)p Fk(R)q Fq(\()p Fk(t;)14 b(s)p Fq(\))3152 654 y Fg(\001)791 938 y Fq(=)892 860 y Fg(X)915 1036 y Fe(i;j)1026 825 y Fg(Z)1072 1014 y Fi(\001)1127 1022 y Fa(i)1157 938 y Fk(dx)1261 825 y Fg(Z)1308 1014 y Fi(\001)1363 1022 y Fa(j)1397 938 y Fk(dy)j(\032)1541 904 y Fe(R)1541 959 y(s)1595 938 y Fq(\()p Fk(x)p Fq(\))d Fk(\032)1763 904 y Fe(R)1763 959 y(s)1819 938 y Fq(\()p Fk(y)s Fq(\))g Fk(\037)1993 871 y Fg(\000)2031 938 y Fj(j)p Fk(x)19 b Fj(\000)f Fk(y)s Fj(j)23 b(\024)f Fq(1)2422 871 y Fg(\001)791 1259 y Fj(\024)902 1203 y Fq(1)p 902 1240 42 4 v 902 1316 a(2)981 1181 y Fg(X)1004 1357 y Fe(i;j)1115 1093 y Fg(2)1115 1242 y(4)1170 1142 y(\022)1231 1146 y(Z)1278 1335 y Fi(\001)1333 1343 y Fa(i)1363 1259 y Fk(dx)14 b(\032)1510 1225 y Fe(R)1510 1280 y(s)1564 1259 y Fq(\()p Fk(x)p Fq(\))1675 1142 y Fg(\023)1738 1160 y Fi(2)1793 1259 y Fq(+)1876 1117 y Fg( )1942 1146 y(Z)1988 1335 y Fi(\001)2043 1343 y Fa(j)2078 1259 y Fk(dy)j(\032)2222 1225 y Fe(R)2222 1280 y(s)2276 1259 y Fq(\()p Fk(y)s Fq(\))2384 1117 y Fg(!)2450 1135 y Fi(2)2487 1093 y Fg(3)2487 1242 y(5)2556 1259 y Fk(\037)2608 1192 y Fg(\000)2646 1259 y Fj(j)p Fk(i)i Fj(\000)f Fk(j)5 b Fj(j)23 b(\024)f Fq(1)c(+)g Fk(\016)3155 1225 y Ff(\000)p Fi(1)3244 1192 y Fg(\001)791 1559 y Fq(=)892 1442 y Fg(\022)953 1559 y Fq(3)g(+)1106 1502 y(2)p 1106 1540 V 1107 1616 a Fk(\016)1158 1442 y Fg(\023)1246 1480 y(X)1295 1657 y Fe(i)1380 1442 y Fg(\022)1441 1446 y(Z)1487 1634 y Fi(\001)1542 1642 y Fa(i)1572 1559 y Fk(dx)c(\032)1719 1524 y Fe(R)1719 1579 y(s)1774 1559 y Fq(\()p Fk(x)p Fq(\))1885 1442 y Fg(\023)1947 1459 y Fi(2)791 1818 y Fj(\024)962 1762 y Fq(2)p 902 1799 161 4 v 902 1875 a Fk(')p Fq(\(0\))1086 1701 y Fg(\022)1147 1818 y Fq(3)k(+)1300 1762 y(2)p 1300 1799 42 4 v 1301 1875 a Fk(\016)1352 1701 y Fg(\023)1427 1740 y(X)1475 1916 y Fe(i)1561 1705 y Fg(Z)1607 1894 y Fi(\001)1662 1902 y Fa(i)1692 1818 y Fk(dx)1796 1705 y Fg(Z)1842 1894 y Fi(\001)1897 1902 y Fa(i)1927 1818 y Fk(dy)f(\032)2071 1784 y Fe(R)2071 1839 y(s)2125 1818 y Fq(\()p Fk(x)p Fq(\))d Fk(\032)2293 1784 y Fe(R)2293 1839 y(s)2349 1818 y Fq(\()p Fk(y)s Fq(\))g Fk(')p Fq(\()p Fk(x)19 b Fj(\000)f Fk(y)s Fq(\))791 2078 y Fj(\024)962 2022 y Fq(2)p 902 2059 161 4 v 902 2135 a Fk(')p Fq(\(0\))1086 1961 y Fg(\022)1147 2078 y Fq(3)g(+)1300 2022 y(2)p 1300 2059 42 4 v 1301 2135 a Fk(\016)1352 1961 y Fg(\023)1427 1965 y(Z)1510 2078 y Fk(dx)c(dy)j(\032)1758 2090 y Fe(s)1793 2078 y Fq(\()p Fk(x)p Fq(\))d Fk(\032)1961 2090 y Fe(s)1998 2078 y Fq(\()p Fk(y)s Fq(\))g Fk(')p Fq(\()p Fk(x)19 b Fj(\000)f Fk(y)s Fq(\))c Fk(\037)2497 2011 y Fg(\000)2535 2078 y Fj(j)p Fk(x)19 b Fj(\000)f Fk(\026)p Fj(j)23 b(\024)g Fq(2)p Fk(R)q Fq(\()p Fk(t;)14 b(s)p Fq(\))k(+)g(1)3310 2011 y Fg(\001)791 2311 y Fj(\024)962 2255 y Fq(2)p 902 2292 161 4 v 902 2368 a Fk(')p Fq(\(0\))1086 2194 y Fg(\022)1147 2311 y Fq(3)g(+)1300 2255 y(2)p 1300 2292 42 4 v 1301 2368 a Fk(\016)1352 2194 y Fg(\023)1441 2311 y Fj(W)7 b Fq(\()p Fk(f)i Fq(\()p Fk(s)p Fq(\);)14 b Fk(\026;)g Fq(3)p Fk(R)q Fq(\()p Fk(t;)g(s)p Fq(\)\))p Fk(:)515 2511 y Fq(On)27 b(the)h(other)f(hand,)h(b)n(y)f(the)h(p)r(ositivit)n(y) f(of)h(the)g(p)r(oten)n(tial,)921 2741 y Fj(W)7 b Fq(\()p Fk(f)i Fq(;)14 b Fk(\026;)g Fq(3)p Fk(R)q Fq(\))22 b Fj(\024)1533 2637 y Fi(2)1490 2662 y Fg(X)1463 2841 y Fe(k)q Fi(=)p Ff(\000)p Fi(2)1650 2741 y Fj(W)7 b Fq(\()p Fk(f)i Fq(;)14 b Fk(\026)k Fq(+)g Fk(k)s(R)q(;)c(R)q Fq(\))22 b Fj(\024)h Fq(5)k(sup)2476 2807 y Fe(\021)2570 2741 y Fj(W)7 b Fq(\()p Fk(f)t(;)14 b(\021)s(;)g(R)q Fq(\))p Fk(;)515 2982 y Fq(and)27 b(hence,)h(b)n(y)g(\(A.5\))g(and)g (the)g(previous)e(estimates,)i(for)f(some)g Fk(C)2683 2994 y Fi(9)2743 2982 y Fk(>)c Fq(0,)944 3185 y Fk(@)988 3197 y Fe(s)1024 3185 y Fj(W)7 b Fq(\()p Fk(f)i Fq(\()p Fk(s)p Fq(\);)14 b Fk(\026;)g(R)q Fq(\()p Fk(t;)g(s)p Fq(\)\))37 b Fj(\024)g(\000)p Fk(C)1951 3197 y Fi(9)2012 3129 y Fk(@)2056 3141 y Fe(s)2091 3129 y Fk(R)q Fq(\()p Fk(t;)14 b(s)p Fq(\))p 2012 3166 314 4 v 2051 3242 a Fk(R)q Fq(\()p Fk(t;)g(s)p Fq(\))2349 3185 y Fj(W)7 b Fq(\()p Fk(f)i Fq(\()p Fk(s)p Fq(\);)14 b Fk(R)q Fq(\()p Fk(t;)g(s)p Fq(\)\))p Fk(;)515 3391 y Fq(where)36 b Fj(W)7 b Fq(\()p Fk(f)i Fq(;)14 b Fk(R)q Fq(\))1127 3344 y Fk(:)1106 3391 y Fq(=)38 b(sup)1334 3411 y Fe(\021)1402 3391 y Fj(W)7 b Fq(\()p Fk(f)i Fq(;)14 b Fk(\021)s(;)g(R)q Fq(\).)64 b(By)36 b(in)n(tegrating)g(the)h(di\013eren)n(tial)g(inequalit)n(y)515 3491 y(and)27 b(taking)g(the)h(suprem)n(um)f(o)n(v)n(er)f Fk(\026)i Fq(w)n(e)f(get:)550 3688 y Fj(W)7 b Fq(\()p Fk(f)i Fq(\()p Fk(s)p Fq(\);)14 b Fk(R)q Fq(\()p Fk(t;)g(s)p Fq(\)\))24 b Fj(\024)e(W)7 b Fq(\()p Fk(f)i Fq(\(0\);)14 b Fk(R)q Fq(\()p Fk(t;)g Fq(0\)\))19 b Fj(\000)f Fk(C)1982 3700 y Fi(9)2033 3575 y Fg(Z)2116 3596 y Fe(s)2079 3764 y Fi(0)2151 3688 y Fk(d\034)24 b Fj(W)7 b Fq(\()p Fk(f)i Fq(\()p Fk(\034)g Fq(\);)14 b Fk(R)q Fq(\()p Fk(t;)g(\034)9 b Fq(\)\))14 b Fk(@)2901 3700 y Fe(\034)2958 3688 y Fq(log)h Fk(R)q Fq(\()p Fk(t;)f(\034)9 b Fq(\))p Fk(;)515 3890 y Fq(from)27 b(whic)n(h,)h(recalling)f(\(3.5\))g(and)h(\(A.2\),)719 4075 y Fj(W)7 b Fq(\()p Fk(f)i Fq(\()p Fk(t)p Fq(\);)14 b Fk(R)q Fq(\()p Fk(t)p Fq(\)\))84 b(=)f Fj(W)7 b Fq(\()p Fk(f)i Fq(\()p Fk(t)p Fq(\);)14 b Fk(R)q Fq(\()p Fk(t;)g(t)p Fq(\)\))51 b Fj(\024)f(W)7 b Fq(\()p Fk(f)i Fq(\(0\);)14 b Fk(R)q Fq(\()p Fk(t;)g Fq(0\)\))g Fk(e)2804 4035 y Fe(C)2852 4043 y Fc(9)2895 4035 y Fi(log)3002 4006 y Fa(R)p Fc(\()p Fa(t;)p Fc(0\))p 3002 4022 159 3 v 3004 4056 a Fa(R)p Fc(\()p Fa(t;t)p Fc(\))1295 4200 y Fj(\024)83 b Fq(2)1485 4166 y Fe(C)1533 4174 y Fc(9)1582 4200 y Fj(W)7 b Fq(\()p Fk(f)1744 4212 y Fi(0)1781 4200 y Fq(;)14 b(2)p Fk(R)q Fq(\()p Fk(t)p Fq(\)\))37 b Fj(\024)f Fq(2)2230 4166 y Fe(C)2278 4174 y Fc(9)2310 4166 y Fi(+1)2412 4200 y Fk(Q)2478 4212 y Fi(0)2529 4200 y Fk(R)q Fq(\()p Fk(t)p Fq(\))p Fk(:)515 4355 y Fq(Lemma)27 b(3.1)g(is)g(th)n(us)h(pro)n(v)n (ed.)p 3331 4310 49 4 v 3331 4359 4 50 v 3376 4359 V 3331 4362 49 4 v 515 4625 a Fp(References)556 4807 y Fq([1])41 b(W.)31 b(Braun)f(and)g(K.)h(Hepp,)h(The)e(Vlaso)n(v)f (dynamics)h(and)h(its)f(\015uctuations)h(in)g(the)685 4907 y(1)p Fk(=)-5 b(N)37 b Fq(limit)30 b(of)f(in)n(teracting)e (classical)h(particles,)g Fm(Comm.)k(Math.)g(Phys.)e Fr(56)e Fq(\(1977\))685 5006 y(101{113.)1905 5255 y(13)p eop %%Page: 14 14 14 13 bop 556 523 a Fq([2])41 b(P)-7 b(.)41 b(Butt\022)-42 b(a,)44 b(E.)c(Caglioti)f(and)i(C.)f(Marc)n(hioro,)i(On)e(the)h(long)f (time)h(b)r(eha)n(vior)e(of)685 623 y(in\014nitely)c(extended)f (systems)f(of)h(particles)f(in)n(teracting)g(via)g(Kac)g(p)r(oten)n (tials,)i Fm(J.)685 722 y(Stat.)30 b(Phys.)f Fr(108)e Fq(\(2002\))f(317{339.)556 888 y([3])41 b(P)-7 b(.)21 b(Butt\022)-42 b(a,)22 b(E.)e(Caglioti)g(and)g(C.)h(Marc)n(hioro,)f(On) g(the)h(motion)g(of)f(a)h(c)n(harged)e(particle)685 988 y(in)n(teracting)35 b(with)g(an)g(in\014nitely)h(extended)f(system,)i Fm(Comm.)g(Math.)h(Phys.)f Fr(233)685 1088 y Fq(\(2003\))27 b(545{569.)556 1254 y([4])41 b(P)-7 b(.)37 b(Butt\022)-42 b(a,)39 b(E.)d(Caglioti)g(and)g(C.)h(Marc)n(hioro,)g(On)f(the)h (violation)f(of)g(Ohm's)h(la)n(w)685 1353 y(for)31 b(b)r(ounded)g(in)n (teractions:)43 b(a)31 b(one)f(dimensional)h(system,)g Fm(Comm.)j(Math.)h(Phys.)685 1453 y Fq(\(2004\),)27 b(to)g(app)r(ear.) 556 1619 y([5])41 b(E.)24 b(Caglioti,)h(S.)f(Caprino,)g(C.)h(Marc)n (hioro)c(and)k(M.)f(Pulviren)n(ti,)h(The)f(Vlaso)n(v)f(equa-)685 1719 y(tion)28 b(with)g(in\014nite)h(mass,)d Fm(A)n(r)l(ch.)31 b(R)l(ational)f(Me)l(ch.)h(A)n(nal.)d Fr(159)f Fq(\(2001\))f(85{108.) 556 1885 y([6])41 b(S.)21 b(Caprino,)f(A)h(Vlaso)n(v-P)n(oisson)16 b(plasma)j(mo)r(del)h(with)h(non-)p Fk(L)2669 1897 y Fi(1)2725 1885 y Fq(data,)h Fm(Math.)i(Meth-)685 1984 y(o)l(ds)31 b(Appl.)g(Sci.)d Fq(\(2004\),)e(to)i(app)r(ear.)556 2150 y([7])41 b(R.L.)48 b(Dobrushin,)53 b(Vlaso)n(v)46 b(equations,)52 b Fm(Soviet)d(Journ.)f(F)-6 b(unctional)48 b(A)n(nal.)h Fr(13)685 2250 y Fq(\(1979\))27 b(115{123.)556 2416 y([8])41 b(L.D.)34 b(Landau)f(and)g(E.M.)h(Lifshitz,)h Fm(Physic)l(al)i(Kinetics)p Fq(,)f(Course)c(of)h(Theoretical)685 2516 y(Ph)n(ysics,)27 b(V)-7 b(ol.)27 b(10)g(\(P)n(ergamon)e(Press,)i (1981\).)556 2682 y([9])41 b(H.)22 b(Neunzert,)h(An)f(in)n(tro)r (duction)f(to)g(the)h(nonlinear)e(Boltzmann-Vlaso)n(v)g(equation,)685 2781 y(Kinetic)27 b(theories)e(and)h(the)g(Boltzmann)g(equation)g (\(Mon)n(tecatini,)g(1981\),)f(60{110.)685 2881 y Fm(L)l(e)l(ctur)l(e)k (Notes)h(in)g(Math.)f Fr(1048)p Fq(,)d(Springer,)h(Berlin,)g(1984.)515 3047 y([10])40 b(K.)27 b(Pfa\013elmoser,)f(Global)h(classical)f (solutions)g(of)i(the)f(Vlaso)n(v-P)n(oisson)d(system)j(in)685 3147 y(three)k(dimensions)f(for)g(general)g(initial)g(data,)h Fm(Journ.)i(Di\013.)g(Eq.)e Fr(95)g Fq(\(1992\))e(281{)685 3246 y(303.)515 3412 y([11])40 b(J.)23 b(Shae\013er,)g(Global)g (existence)f(of)h(smo)r(oth)g(solutions)f(to)h(the)g(Vlaso)n(v-P)n (oisson)c(sys-)685 3512 y(tem)31 b(in)e(three)h(dimensions,)g Fm(Comm.)j(Partial)g(Di\013er)l(ential)f(Equations)e Fr(16)f Fq(\(1991\))685 3611 y(1313{1335.)515 3778 y([12])40 b(H.)e(Sp)r(ohn,)i(On)e(the)f(Vlaso)n(v)f(hierarc)n(h)n(y)-7 b(,)38 b Fm(Math.)i(Metho)l(ds)g(Appl.)g(Sci.)e Fr(3)f Fq(\(1981\))685 3877 y(445{455.)515 4043 y([13])j(S.)47 b(W)-7 b(ollman,)52 b(Global-in-time)46 b(solutions)g(to)g(the)h (three-dimensional)f(Vlaso)n(v-)685 4143 y(P)n(oisson)26 b(system,)h Fm(Journ.)j(Math.)h(A)n(nal.)f(Appl.)f Fr(176)e Fq(\(1993\))f(76{91.)1905 5255 y(14)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0405271430842--