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b(y)58 b(this)h(equation)f(leads)h(to)f(sev)-5 b(eral)59 b(w)-5 b(ell-kno)g(wn)59 b(phenomena.)726 10572 y(First,)f(for)f(con\034ning)g(p)5 b(oten)-5 b(tials)57 b Fs(V)37 b Ft(,)57 b(the)f(particle)h(will)g(come)g(to)f(a)h(stop)g (exp)5 b(onen)-5 b(tially)726 10771 y(fast)38 b(\(with)g(rate)1803 10697 y Fp(~)-71 b Fo(\015)p 1798 10733 77 7 v 1803 10829 a Fp(2)1933 10771 y Ft(if)44 b Fr(~)-89 b Fs(\015)47 b Ft(is)38 b(small)h(enough\))f(at)g(one)g(of)g(the)f(minima)j(of)d (the)h(p)5 b(oten)-5 b(tial.)68 b(If,)41 b(on)726 10971 y(the)f(other)g(hand,)45 b Fs(V)36 b Fr(\()p Fs(q)6 b Fr(\))46 b(=)g Fq(\000)p Fs(F)30 b Fq(\001)7 b Fs(q)f Ft(,)44 b(for)c(some)h Fs(F)69 b Fq(2)46 b Fn(R)4015 10910 y Fo(d)4101 10971 y Ft(,)e(the)c(particle)g(will)g(reac)-5 b(h)41 b(a)f(limiting)726 11170 y(sp)5 b(eed)58 b Fs(v)6 b Fr(\()p Fs(F)23 b Fr(\))50 b(=)1781 11104 y Fo(F)p 1781 11131 103 7 v 1799 11227 a Fp(~)-71 b Fo(\015)1961 11170 y Ft(whic)-5 b(h)58 b(is)g(prop)5 b(ortional)58 b(to)g(the)f(applied)h(\034eld.)82 b(This,)59 b(in)f(particular,)3549 11733 y(1)p eop %%Page: 2 2 2 1 bop 726 1471 a Ft(is)52 b(at)e(the)g(origin)g(of)h(Ohm's)g(la)-5 b(w.)73 b(Again,)51 b(the)f(approac)-5 b(h)52 b(is)f(exp)5 b(onen)-5 b(tial,)51 b(but)f(this)h(time)726 1671 y(with)60 b(rate)65 b Fr(~)-89 b Fs(\015)9 b Ft(.)87 b(In)60 b(particular,)h(if)e Fs(F)76 b Fr(=)54 b(0)p Ft(,)61 b(the)e(particle)g(comes)h(exp)5 b(onen)-5 b(tially)59 b(fast)h(to)f(a)726 1870 y(full)66 b(stop.)103 b(The)65 b(phenomenological)h(friction)f(force)f (summarizes)j(the)d(reaction)h(of)f(the)726 2069 y(en)-5 b(vironmen)g(t)55 b(of)f(the)f(particle)h(to)f(its)i(passage)g(and)f (the)g(energy)f(lost)h(b)-5 b(y)54 b(the)g(particle)f(is)726 2268 y(transferred)41 b(to)g(the)g(medium)h(surrounding)g(the)f (particle)g(b)-5 b(y)41 b(v)-9 b(arious)41 b(pro)5 b(cesses)42 b(\(suc)-5 b(h)42 b(as)726 2468 y(inelastic)60 b(collisions,)i(for)d (example\).)85 b(A)58 b(more)i(fundamen)-5 b(tal,)61 b(microscopic,)g(treatmen)-5 b(t)726 2667 y(of)44 b(the)f(phenomena)i (requires)f(therefore)f(considering)i(the)e(com)-5 b(bined)45 b(system)f(consisting)726 2866 y(of)83 b(the)f(particle)g(and)h(the)f (medium.)156 b(This)83 b(com)-5 b(bined)83 b(system)g(should)h(allo)-5 b(w)83 b(for)f(a)726 3065 y(Hamiltonian)56 b(treatmen)-5 b(t)55 b(in)g(whic)-5 b(h)56 b(the)f(total)g(energy)f(is)i(conserv)-5 b(ed.)976 3265 y(Our)71 b(goal)g(in)g(this)g(pap)5 b(er)71 b(is)g(to)g(presen)-5 b(t)71 b(and)h(study)f(a)f(Hamiltonian)h(mo)5 b(del)71 b(of)g(a)726 3464 y(system)46 b(comp)5 b(osed)46 b(of)f(a)h(particle)f(and)h(a)f(homogeneous)h(medium.)72 b(W)-14 b(e)45 b(sho)-5 b(w)46 b(rigorously)726 3663 y(that)51 b(it)f(has)h(the)f(b)5 b(eha)-5 b(viour)51 b(describ)5 b(ed)51 b(ab)5 b(o)-5 b(v)g(e)51 b(and)g(analyse)g(the)f (ph)-5 b(ysical)52 b(mec)-5 b(hanisms)726 3862 y(at)45 b(the)f(origin)h(of)f(the)g(observ)-5 b(ed)46 b(phenomena.)71 b(W)-14 b(e)44 b(sta)-5 b(y)44 b(within)h(the)f(con)-5 b(text)44 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b(the)g(other)f(sub)9 b(jected)50 b(to)h(a)f(time-indep)5 b(enden)-5 b(t)51 b(external)f(force)g Fs(F)69 b Fr(=)46 b Fq(\000r)p Fs(V)38 b Ft(.)72 b(W)-14 b(e)50 b(are)726 5656 y(mostly)e(in)-5 b(terested)48 b(in)g(the)f(more)g(di\036cult)h (case)g(where)f Fs(F)71 b Ft(is)48 b(constan)-5 b(t)48 b(\(so)f Fs(V)83 b Fr(=)46 b Fq(\000)p Fs(F)f Fq(\001)21 b Fs(q)6 b Ft(\),)726 5855 y(but)56 b(w)-5 b(e)55 b(will)h(also)g(deal) f(with)g(con\034ning)h(p)5 b(oten)-5 b(tials.)976 6054 y(More)55 b(precisely)-14 b(,)55 b(the)g(equations)h(of)f(motion)g(for) g(the)g(coupled)h(system)f(are:)2056 6356 y Fs(@)2153 6287 y Fp(2)2144 6397 y Fo(t)2228 6356 y Fs(\036)p Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))36 b Fq(\000)h Fs(c)3121 6287 y Fp(2)3195 6356 y Fr(\001)3333 6381 y Fo(y)3414 6356 y Fs(\036)p Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))45 b(=)h Fq(\000)p Fs(\032)p Fr(\()p Fs(x)36 b Fq(\000)h Fs(q)6 b Fr(\()p Fs(t)p Fr(\))p Fs(;)28 b(y)6 b Fr(\))788 b Ft(\(1.2\))1205 6696 y Fr(\177)-96 b Fs(q)6 b 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b(R)q Fr([)p Fs(f)18 b Fr(\()p Fs(v)6 b Fr(\)])47 b(=)f Fs(f)18 b Fr(\()p Fs(R)q(v)6 b Fr(\))p Fs(:)3549 11733 y Ft(5)p eop %%Page: 6 6 6 5 bop 726 1471 a Ft(No)-5 b(w,)79 b(if)74 b Fs(v)84 b Fr(=)78 b Fq(j)p Fs(v)6 b Fq(j)p Fs(e)1971 1496 y Fp(1)2047 1471 y Ft(,)79 b(one)74 b(\034nds,)80 b(after)73 b(a)i(few)e(c)-5 b(hanges)75 b(of)f(v)-9 b(ariables)75 b(\()p Fs(\025)i Fr(=)g Fq(j)p Fs(v)6 b Fq(j)p Fs(\034)94 b Ft(and)731 1627 y Fr(~)726 1671 y Fs(k)52 b Fr(=)1098 1605 y Fo(k)p 1059 1632 150 7 v 1059 1728 a Fi(j)p Fo(v)t Fi(j)1229 1671 y Ft(\):)791 2196 y Fs(f)18 b Fr(\()p Fs(v)6 b Fr(\))46 b(=)h Fq(\000j)p Fs(v)6 b Fq(j)1635 2127 y Fo(n)p Fi(\000)p Fp(2)1923 1970 y Fl(Z)2016 2347 y Fk(R)2093 2314 y Fj(d)2205 2196 y Fs(dx)2414 1970 y Fl(Z)2506 2347 y Fk(R)2583 2314 y Fj(n)2708 2196 y Fs(d)2798 2152 y Fr(~)2794 2196 y Fs(k)2914 1970 y Fl(Z)3080 2011 y Fp(+)p Fi(1)3006 2347 y Fp(0)3350 2196 y Fs(d\025)p Fq(r)p Fs(\032)3757 2221 y Fp(1)3832 2196 y Fr(\()p Fs(x)p Fr(\))p Fs(\032)4143 2221 y Fp(1)4216 2196 y Fr(\()p Fs(x)37 b Fr(+)g Fs(\025e)4753 2221 y Fp(1)4827 2196 y Fr(\))p Fq(j)15 b Fr(^)-98 b Fs(\032)5024 2221 y Fp(2)5098 2196 y Fr(\()p Fq(j)p Fs(v)6 b Fq(j)5345 2152 y Fr(~)5341 2196 y Fs(k)g Fr(\))p Fq(j)5544 2127 y Fp(2)5638 2083 y Fr(sin)q(\()p Fs(c\025)p Fq(j)6126 2040 y Fr(~)6122 2083 y Fs(k)f Fq(j)p Fr(\))p 5638 2158 686 7 v 5853 2332 a Fs(c)p Fq(j)5975 2288 y Fr(~)5971 2332 y Fs(k)h Fq(j)6344 2196 y Fs(:)726 2670 y Ft(The)66 b(rotational)f(in)-5 b(v)c(ariance)66 b(of)g Fs(\032)2935 2695 y Fp(1)3075 2670 y Ft(no)-5 b(w)66 b(implies)h(that)e Fs(f)4456 2695 y Fo(i)4511 2670 y Fr(\()p Fq(j)p Fs(v)6 b Fq(j)p Fs(e)4831 2695 y Fp(1)4906 2670 y Fr(\))63 b(=)h(0)i Ft(for)f Fs(i)f Fr(=)f(2)p Fs(;)28 b(:::;)g(d)p Ft(,)726 2869 y(so)60 b(that)f Fs(f)18 b Fr(\()p Fs(v)6 b Fr(\))59 b Ft(has)h(the)e(direction)h(of)g Fs(e)3227 2894 y Fp(1)3361 2869 y Ft(and)h(so)f(in)h(the)f(general)g(case)g(\()p Fs(v)g Fq(6)p Fr(=)52 b(0)p Ft(\))59 b(one)g(has)726 3069 y(indeed)2959 3268 y Fs(f)18 b Fr(\()p Fs(v)6 b Fr(\))46 b(=)g Fs(f)3576 3293 y Fo(r)3650 3268 y Fr(\()p Fq(j)p Fs(v)6 b Fq(j)p Fr(\))4024 3156 y Fs(v)p 3978 3230 179 7 v 3978 3382 a Fq(j)p Fs(v)g Fq(j)4176 3268 y Fs(:)726 3658 y Ft(W)-14 b(e)64 b(need)g(to)g(study)g(the)f (asymptotique)h(b)5 b(eha)-5 b(viour)64 b(of)g Fs(f)18 b Fr(\()p Fs(v)6 b Fr(\))63 b Ft(as)i Fq(j)p Fs(v)6 b Fq(j)65 b Ft(go)5 b(es)63 b(to)h Fr(0)g Ft(and)h(as)726 3857 y Fq(j)p Fs(v)6 b Fq(j)57 b Ft(go)5 b(es)55 b(to)g Fr(+)p Fq(1)p Ft(.)74 b(F)-14 b(or)56 b(that)e(purp)5 b(ose,)56 b(w)-5 b(e)56 b(write)2796 4296 y Fs(f)18 b Fr(\()p Fs(v)6 b Fr(\))46 b(=)3332 4070 y Fl(Z)3424 4448 y Fk(R)3501 4414 y Fj(n)3626 4296 y Fs(dk)61 b(f)18 b Fr(\()p Fs(v)6 b(;)28 b(k)5 b Fr(\))p Fs(;)726 4744 y Ft(with)56 b(\(after)e(some)i(manipulations\))1594 5141 y Fs(f)18 b Fr(\()p Fs(v)6 b(;)28 b(k)5 b Fr(\))166 b(=)g Fs(f)2616 5166 y Fo(r)2690 5141 y Fr(\()p Fq(j)p Fs(v)6 b Fq(j)p Fs(;)28 b Fq(j)p Fs(k)5 b Fq(j)p Fr(\))3321 5029 y Fs(v)p 3275 5103 V 3275 5255 a Fq(j)p Fs(v)h Fq(j)6114 5141 y Ft(\(2.5\))1353 5582 y Fs(f)1434 5607 y Fo(r)1508 5582 y Fr(\()p Fq(j)p Fs(v)g Fq(j)p Fs(;)28 b Fq(j)p Fs(k)5 b Fq(j)p Fr(\))167 b(=)f Fq(\000)2769 5470 y Fr(1)p 2684 5544 254 7 v 2684 5696 a Fq(j)p Fs(v)6 b Fq(j)2862 5648 y Fp(2)2957 5582 y Fq(j)15 b Fr(^)-98 b Fs(\032)3089 5607 y Fp(2)3164 5582 y Fr(\()p Fs(k)5 b Fr(\))p Fq(j)3431 5514 y Fp(2)3505 5582 y Fs(h)p Fr(\()3739 5470 y Fs(c)p 3686 5544 179 7 v 3686 5696 a Fq(j)p Fs(v)h Fq(j)3884 5582 y Fs(k)f Fr(\))2074 b Ft(\(2.6\))1769 6061 y Fs(h)p Fr(\()p Fs(\030)8 b Fr(\))164 b(=)2535 5835 y Fl(Z)2701 5876 y Fp(+)p Fi(1)2627 6212 y Fp(0)2971 6061 y Fs(d\025)3182 5835 y Fl(Z)3275 6212 y Fk(R)3352 6179 y Fj(d)3464 6061 y Fs(dx)56 b(@)3789 6086 y Fp(1)3863 6061 y Fs(\032)3949 6086 y Fp(1)4024 6061 y Fr(\()p Fs(x)p Fr(\))p Fs(\032)4335 6086 y Fp(1)4408 6061 y Fr(\()p Fs(x)4568 6086 y Fp(1)4679 6061 y Fr(+)37 b Fs(\025;)28 b(x)5111 6086 y Fi(?)5223 6061 y Fr(\))5308 5949 y(sin)g Fs(\025)p Fq(j)p 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Fp(2)3903 7400 y Fl(\022)4084 7522 y Fq(j)p Fs(v)h Fq(j)p 4045 7596 256 7 v 4045 7748 a Fs(c)p Fq(j)p Fs(k)f Fq(j)4321 7400 y Fl(\023)4443 7435 y Fo(`)4535 7634 y Fs(:)726 8088 y Ft(In)65 b(other)g(w)-5 b(ords,)68 b(for)c(\034xed)h Fs(k)5 b Ft(,)68 b Fs(f)18 b Fr(\()p Fs(v)6 b(;)28 b(k)5 b Fr(\))64 b Ft(v)-9 b(anishes)66 b(to)e(all)h(orders)g(in)g Fq(j)p Fs(v)6 b Fq(j)66 b Ft(as)f Fs(v)j Fq(!)62 b Fr(0)p Ft(.)102 b(So,)726 8288 y(as)70 b Fs(v)k Fq(!)68 b Fr(0)p Ft(,)73 b(the)68 b(force)g(on)g(the)h(particle)f(due)g(to)h(one)f(of)h (the)f(oscillators)h(of)f(frequency)726 8487 y Fs(!)6 b Fr(\()p Fs(k)f Fr(\))79 b(=)h Fs(c)p Fq(j)p Fs(k)5 b Fq(j)75 b Ft(presen)-5 b(t)76 b(at)e Fs(x)p Ft(,)80 b(decreases)75 b(faster)g(than)g(an)-5 b(y)75 b(p)5 b(o)-5 b(w)g(er)75 b(of)g Fq(j)p Fs(v)6 b Fq(j)76 b Ft(for)f(small)h Fs(v)726 8686 y Ft(\()p Fh(i.e.)146 b Ft(when)79 b Fq(j)p Fs(v)6 b Fq(j)87 b Fs(<<)f(c)p Fq(j)p Fs(k)5 b Fq(j)p Fs(R)2613 8711 y Fp(1)2688 8686 y Ft(\).)146 b(Roughly)79 b(sp)5 b(eaking,)86 b(the)79 b(coupling)h(of)f(the)f(particle)726 8885 y(to)65 b(suc)-5 b(h)65 b(an)g(oscillator)g(is)g(extremely)f(w)-5 b(eak)64 b(when)h Fq(j)p Fs(v)6 b Fq(j)65 b Ft(is)g(m)-5 b(uc)g(h)66 b(smaller)g(than)f Fs(c)p Fq(j)p Fs(k)5 b Fq(j)p Fs(R)6334 8910 y Fp(1)6409 8885 y Ft(.)726 9085 y(This)79 b(corresp)5 b(onds)78 b(to)g(a)f(w)-5 b(ell-kno)g(wn)79 b(piece)e(of)h(ph)-5 b(ysical)78 b(in)-5 b(tuition:)120 b(if)77 b(the)g(particle)726 9284 y(has)70 b(sp)5 b(eed)68 b Fs(v)6 b Ft(,)73 b(it)68 b(in)-5 b(teracts)69 b(during)g(a)f(time)h (of)f(order)4335 9216 y Fo(R)4435 9233 y Ff(1)p 4335 9246 165 7 v 4343 9341 a Fi(j)p Fo(v)t Fi(j)4588 9284 y Ft(with)g(an)-5 b(y)69 b(giv)-5 b(en)69 b(oscillator.)726 9501 y(F)-14 b(or)66 b(the)f(energy)g(transfer)g(b)5 b(et)-5 b(w)g(een)65 b(the)f(particle)h(and)h(the)f(oscillator)g(to)g (b)5 b(e)65 b(e\036cien)-5 b(t,)726 9700 y(this)57 b(in)-5 b(teraction)56 b(time)f(has)i(to)f(b)5 b(e)55 b(comparable)i(to)f(the)f (p)5 b(erio)g(d)56 b(of)f(the)h(oscillator)g(as)h(an)726 9900 y(explicit)51 b(computation)h(easily)f(con\034rms.)74 b(Indeed,)53 b(the)e(total)g(energy)g(transfer)g Fr(\001)p Fs(E)61 b Ft(to)51 b(a)726 10099 y(driv)-5 b(en)56 b(oscillator)g(of)f (frequency)f Fs(!)2868 10454 y Fr(\177)-93 b Fs(u)p Fr(\()p Fs(t)p Fr(\))36 b(+)h Fs(!)3454 10385 y Fp(2)3528 10454 y Fs(u)p Fr(\()p Fs(t)p Fr(\))46 b(=)g Fs(\033)6 b Fr(\()p Fs(t)p Fr(\))726 10808 y Ft(is)55 b(easily)f(computed)h(to)e(b)5 b(e)54 b Fr(\001)p Fs(E)i Fr(=)46 b Fs(\031)6 b Fq(j)j Fr(^)-92 b Fs(\033)6 b Fr(\()p Fs(!)g Fr(\))p Fq(j)3564 10748 y Fp(2)3638 10808 y Ft(.)73 b(Applying)54 b(this)h(to)f(\(1.4\))f (with)h Fs(q)6 b Fr(\()p Fs(t)p Fr(\))45 b(=)h Fs(v)6 b(t)p Ft(,)726 11007 y(one)65 b(\034nds)g Fr(\001)p Fs(E)70 b Fr(=)2053 10942 y Fo(\031)p 1987 10969 215 7 v 1987 11065 a Fi(j)p Fo(v)t Fi(j)2135 11031 y Ff(2)2221 11007 y Fq(j)15 b Fr(^)-98 b Fs(\032)2353 11032 y Fp(2)2428 11007 y Fr(\()p Fs(k)5 b Fr(\))p Fq(j)2695 10947 y Fp(2)2769 11007 y Fq(j)15 b Fr(^)-98 b Fs(\032)2901 11032 y Fp(1)2975 11007 y Fr(\()p Fs(c)p Fq(j)p Fs(k)5 b Fq(j)p Fs(=)p Fq(j)p Fs(v)h Fq(j)p Fs(;)28 b Fr(0\))p Fq(j)3824 10947 y Fp(2)3964 11007 y Ft(whic)-5 b(h)64 b(v)-9 b(anishes)65 b(again)g(to)e(all)h(orders)726 11235 y(in)56 b Fq(j)p Fs(v)6 b Fq(j)56 b Ft(as)g Fq(j)p Fs(v)6 b Fq(j)47 b(!)f Fr(0)p Ft(.)3549 11733 y(6)p eop %%Page: 7 7 7 6 bop 976 1471 a Ft(In)59 b(particular,)h(it)f(is)g(clear)g(from)g (this)h(observ)-9 b(ation)58 b(that,)i(when)f(coupling)h(the)e(par-)726 1671 y(ticle)64 b(to)g(a)h(family)f(of)h(oscillators,)i(all)e(of)f(the) g(same)h(\034xed)f(frequency)g(\(as)g(in)h(a)g(pin)-5 b(ball)726 1870 y(mac)g(hine)47 b(where)e(eac)-5 b(h)45 b(circular)g(obstacles)h(w)-5 b(ould)46 b(b)5 b(e)45 b(moun)-5 b(ted)46 b(on)g(a)f(spring\),)j(no)e(ohmic)726 2069 y(b)5 b(eha)-5 b(viour)49 b(can)f(b)5 b(e)48 b(exp)5 b(ected)47 b(since)i(the)f(friction)g(force)f(is)i(not)f(linear)h(in)g Fs(v)55 b Ft(at)47 b(small)j Fs(v)55 b Ft(in)726 2268 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5785 y Ft(.)726 5985 y(It)75 b(admits)h(a)f(unique)h(solution:)114 b Fs(t)79 b Fq(2)g Fn(R)115 b Fq(!)79 b Fs(Y)37 b Fr(\()p Fs(t)p Fr(\))79 b Fq(2)g(E)15 b Ft(.)133 b(Moreo)-5 b(v)g(er,)80 b(let)75 b(denotes)g(b)-5 b(y)726 6184 y Fs(W)906 6124 y Fo(t)883 6225 y Fp(0)1041 6184 y Fr(:)77 b Fs(Y)1260 6209 y Fp(0)1411 6184 y Fq(!)g Fs(Y)37 b Fr(\()p Fs(t)p Fr(\))73 b Ft(the)g(corresp)5 b(onding)74 b(con)-5 b(tin)g(uous)76 b(group,)i Fs(W)5008 6124 y Fo(t)4985 6225 y Fp(0)5140 6184 y Ft(turns)c(out)g(to)f(b)5 b(e)73 b(a)726 6383 y(linear)50 b(isometry)f(with)h(the)f(norm)h Fq(j)c(j)3074 6408 y Fi(E)3164 6383 y Ft(.)72 b(It)49 b(is)h(also)g(con)-5 b(tin)g(uous)51 b(on)e Fn(R)j Fq(\002)25 b(E)73 b Ft(\([LM],c)-5 b(hapter)726 6582 y(3\).)976 6782 y(No)g(w)82 b(de\034ne)i Fs(Z)12 b Fr(\()p Fs(t)p Fr(\))91 b(=)i Fs(W)2697 6711 y Fi(\000)p Fo(t)2674 6826 y Fp(0)2859 6782 y Fs(Y)37 b Fr(\()p Fs(t)p Fr(\))82 b Ft(or)i Fs(Y)37 b Fr(\()p Fs(t)p Fr(\))91 b(=)i Fs(W)4312 6721 y Fo(t)4289 6823 y 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Fs(Z)12 b Fr(\))46 b Fq(2)g(E)726 9701 y Ft(it)62 b(is)h(clear)f(that)1881 9659 y Fr(~)1844 9701 y Fs(G)g Ft(is)g(con)-5 b(tin)g(uous)64 b(on)f Fn(R)77 b Fq(\002)42 b(E)76 b Ft(and)63 b(lipsc)-5 b(hitz)63 b(on)g Fq(E)76 b Ft(b)5 b(ecause)62 b Fs(W)5985 9641 y Fo(t)5962 9742 y Fp(0)6106 9701 y Ft(is)h(an)726 9900 y(isometry)71 b(and)g Fs(G)1890 9925 y Fp(1)2035 9900 y Ft(is)h(lipsc)-5 b(hitz.)121 b(This)71 b(problem)h(satis\034es)g(all) f(the)f(conditions)i(of)e(the)726 10099 y(Cauc)-5 b(h)g(y-Lipsc)g(hitz) 60 b(theorem)d(\([ZQ)q(],c)-5 b(hapter)58 b(10\),)g(so)g(it)g(has)h(a)f (unique)g(solution)g(whic)-5 b(h)726 10299 y(is)75 b(de\034ned)f(on)g (an)g(op)5 b(en)74 b(in)-5 b(terv)c(al.)129 b(More)73 b(precisely)-14 b(,)79 b(there)73 b(exists)h(an)g(op)5 b(en)73 b(in)-5 b(terv)c(al)726 10498 y Fs(J)84 b Ft(suc)-5 b(h)69 b(that)f Fr(0)f Fq(2)g Fs(J)84 b Ft(and)68 b(there)f(exists)h(a) g(unique)g(function)g Fs(Z)79 b Fr(:)67 b Fs(t)g Fq(2)g Fs(J)83 b Fq(!)67 b Fs(Z)12 b Fr(\()p Fs(t)p Fr(\))67 b Fq(2)g(E)726 10697 y Ft(satisfying)56 b(\(3.6\).)73 b(Moreo)-5 b(v)g(er,)2751 10655 y Fr(~)2703 10697 y Fs(W)2883 10637 y Fo(t)2987 10697 y Fr(:)102 b Fs(Z)3248 10722 y Fp(0)3369 10697 y Fq(!)f Fs(Z)12 b Fr(\()p Fs(t)p Fr(\))55 b Ft(is)h(con)-5 b(tin)g(uous)56 b(on)g Fq(E)70 b Ft(for)55 b(ev)-5 b(ery)54 b Fs(t)46 b Fq(2)g Fs(J)726 10896 y Ft(and)56 b(so)g(w)-5 b(e)55 b(ha)-5 b(v)g(e)56 b(the)f(same)h(results) g(for)2264 11235 y Fs(W)2444 11166 y Fo(t)2548 11235 y Fr(:)46 b Fs(Y)2736 11260 y Fp(0)2857 11235 y Fq(2)g(E)61 b(!)46 b Fs(Y)37 b Fr(\()p Fs(t)p Fr(\))45 b(=)i Fs(W)4099 11166 y Fo(t)4076 11276 y Fp(0)4206 11193 y Fr(~)4157 11235 y Fs(W)4337 11166 y Fo(t)4395 11235 y Fs(Y)4491 11260 y Fp(0)4612 11235 y Fq(2)f(E)15 b Fs(:)3508 11733 y Ft(10)p eop %%Page: 11 11 11 10 bop 976 1471 a Ft(In)78 b(order)g(to)f(pro)-5 b(v)g(e)79 b(global)f(existence,)83 b(w)-5 b(e)78 b(no)-5 b(w)79 b(pro)-5 b(v)g(e)78 b(conserv)-9 b(ation)78 b(of)f(energy)-14 b(.)726 1671 y(W)g(e)76 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b(represen)-5 b(tation:)2228 2550 y Fs(Y)37 b Fr(\()p Fs(t)p Fr(\))46 b(=)g Fs(W)2952 2481 y Fo(t)2929 2591 y Fp(0)3010 2550 y Fs(Y)3106 2575 y Fp(0)3218 2550 y Fr(+)3384 2324 y Fl(Z)3550 2365 y Fo(t)3476 2701 y Fp(0)3636 2550 y Fs(ds)28 b(W)4008 2479 y Fo(t)p Fi(\000)p Fo(s)3985 2594 y Fp(0)4233 2550 y Fs(G)4364 2575 y Fp(1)4438 2550 y Fr(\()p Fs(Y)37 b Fr(\()p Fs(s)p Fr(\)\))p Fs(;)726 3008 y Ft(it)64 b(is)h(clear)e(that)g Fs(\036)p Fr(\()p Fs(t)p Fr(\))p Fs(;)28 b(\031)6 b Fr(\()p Fs(t)p Fr(\))62 b Ft(are)i(smo)5 b(oth)64 b(as)g(w)-5 b(ell)64 b(\(in)g Fs(x)g Ft(and)g Fs(y)6 b Ft(\).)100 b(Note)63 b(that)g Fs(\036)p Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))726 3207 y Ft(and)71 b Fs(\031)6 b Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))68 b Ft(are)i(also)g(smo)5 b(oth)70 b(in)g Fs(t)g Ft(\([LM],)j(c)-5 b(hapter)70 b(3\).)117 b(F)-14 b(or)70 b(suc)-5 b(h)71 b(initial)f(data)g(a)726 3407 y(simple)57 b(computation)e(then)g(yields:)3071 3737 y Fs(d)p 3041 3811 147 7 v 3041 3963 a(dt)3207 3849 y(H)13 b Fr(\()p Fs(Y)37 b Fr(\()p Fs(t)p Fr(\)\))45 b(=)i(0)p Fs(;)726 4270 y Ft(so)64 b(that,)h(for)d(smo)5 b(oth)64 b(initial)f(data,)i Fs(H)13 b Fr(\()p Fs(Y)37 b Fr(\()p Fs(t)p Fr(\)\))62 b Ft(is)i(a)f(constan)-5 b(t)63 b(for)g(all)g Fs(t)g Ft(in)h Fs(J)16 b Ft(.)97 b(W)-14 b(e)63 b(no)-5 b(w)726 4470 y(pro)g(v)g(e)58 b(that)e Fs(H)70 b Ft(is)58 b(con)-5 b(tin)g(uous)58 b(on)f Fq(E)15 b Ft(.)78 b(The)57 b(con)-5 b(tin)g(uit)g(y)57 b(of)g Fs(W)4704 4409 y Fo(t)4819 4470 y Ft(on)g Fq(E)71 b Ft(and)57 b(the)g(fact)f(that)726 4669 y(smo)5 b(oth)54 b(initial)f(data)g(are)f(dense)i(in)f Fq(E)67 b Ft(will)53 b(then)g(imply)h(the)e(result)h(for)g(all)g(initial)g(data.)726 4868 y(Since)75 b Fs(V)111 b Ft(is)74 b(con)-5 b(tin)g(uous,)81 b(it)74 b(only)g(remains)h(to)f(sho)-5 b(w)75 b(the)f(in)-5 b(teraction)73 b(term)h(in)h Fs(H)87 b Ft(is)726 5067 y(con)-5 b(tin)g(uous.)1462 5389 y Fq(j)1554 5256 y Fl(R)1633 5448 y Fk(R)1710 5415 y Fj(d)p Ff(+)p Fj(n)1981 5389 y Fs(dx)28 b(dy)34 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7611 y(2)2414 7570 y Ft(is)56 b(b)5 b(ounded)56 b(to)5 b(o.)976 7769 y(Finally)-14 b(,)88 b Fr(_)-78 b Fs(p)o Fr(\()p Fs(t)p Fr(\))55 b Ft(is)h(b)5 b(ounded)55 b(on)h Fs(J)16 b Ft(:)74 b(there)55 b(exists)g Fs(C)j(>)46 b Fr(0)55 b Ft(suc)-5 b(h)57 b(that)2703 8053 y Fq(8)p Fs(t)47 b Fq(2)e Fs(J)n(;)28 b(t)46 b(>)g Fr(0)92 b Fq(j)79 b Fr(_)-78 b Fs(p)o Fr(\()p Fs(t)p Fr(\))45 b Fq(j\024)i Fs(C)s(:)1552 b Ft(\(3.12\))726 8337 y(W)-14 b(e)45 b(ha)-5 b(v)g(e)46 b(supp)5 b(osed)46 b Fs(b)g Ft(to)f(b)5 b(e)45 b(\034nite,)i(so)f Fs(p)p Fr(\()p Fs(t)p Fr(\))e Ft(and)h Fs(q)6 b Fr(\()p Fs(t)p Fr(\))45 b Ft(are)g(also)h(b)5 b(ounded)46 b(for)f Fs(t)91 b(>)g Fr(0)p Fs(;)28 b(t)46 b Fq(2)726 8537 y Fs(J)n(:)976 8736 y Ft(By)54 b(energy)h(conserv)-9 b(ation)55 b Fr(\(3)p Fs(:)p Fr(3\))f Ft(implies:)1225 9126 y Fs(H)13 b Fr(\()p Fs(Y)1537 9151 y Fp(0)1612 9126 y Fr(\))45 b Fq(\025)1918 9014 y Fr(1)p 1918 9088 84 7 v 1918 9240 a(2)2020 9126 y Fs(p)p Fr(\()p Fs(t)p Fr(\))2294 9058 y Fp(2)2405 9126 y Fr(+)37 b Fs(V)f Fr(\()p Fs(q)6 b Fr(\()p Fs(t)p Fr(\)\))36 b(+)3326 9014 y Fs(c)3398 8954 y Fp(2)p 3326 9088 147 7 v 3357 9240 a Fr(4)3492 9126 y Fq(k)p Fs(\036)p Fr(\()p Fs(t)p Fr(\))p Fq(k)3947 9058 y Fp(2)4057 9126 y Fr(+)4243 9014 y(1)p 4243 9088 84 7 v 4243 9240 a(2)4346 9126 y Fq(k)p Fs(\031)6 b Fr(\()p Fs(t)p Fr(\))p Fq(k)4803 9058 y Fp(2)4803 9167 y(2)4913 9126 y Fr(+)5131 9014 y(1)p 5099 9088 147 7 v 5099 9240 a Fs(c)5171 9192 y Fp(2)5266 9126 y Fq(h)p Fs(\032)p Fr(;)28 b(\001)5629 9058 y Fi(\000)p Fp(1)5629 9167 y Fo(y)5806 9126 y Fs(\032)p Fq(i)726 9466 y Ft(and)55 b(so)g Fq(k)p Fs(\036)p Fr(\()p Fs(t)p Fr(\))p Fq(k)e Ft(and)i Fq(k)p Fs(\031)6 b Fr(\()p Fs(t)p Fr(\))p Fq(k)2538 9491 y Fp(2)2666 9466 y Ft(are)54 b(b)5 b(ounded)55 b(and)f(then)g Fq(j)p Fs(Y)38 b Fr(\()p Fs(t)p Fr(\))p Fq(j)4733 9491 y Fi(E)4876 9466 y Ft(to)5 b(o)54 b(whic)-5 b(h)55 b(is)f(a)h(con)-5 b(tra-)726 9666 y(diction)56 b(with)f(the)g(fact)f(that)h Fs(b)h Ft(is)f(\034nite.)976 9865 y(W)-14 b(e)79 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Fr(+)g Fs(v)5150 9784 y Fq(j)p Fs(z)7 b Fq(j)p 5150 9858 177 7 v 5203 10010 a Fs(c)5347 9662 y Fl(\023)2922 10434 y Fq(\000)p Fs(\032)3137 10459 y Fp(1)3239 10150 y Fl( )3371 10434 y Fs(x)36 b Fr(+)i Fs(v)3775 10322 y Fq(j)p Fs(z)7 b Fq(j)p 3775 10396 V 3827 10548 a Fs(c)4008 10434 y Fr(+)4175 10208 y Fl(Z)4341 10249 y Fo(t)4267 10585 y(t)p Fi(\000)4441 10527 y Fg(j)p Fj(z)s Fg(j)p 4441 10559 137 6 v 4482 10625 a Fj(c)4653 10390 y Fr(_)4633 10434 y Fs(h)p Fr(\()p Fs(s)p Fr(\))p Fs(ds)5101 10150 y Fl(!)5259 10250 y(i)5337 10434 y Fq(r)p Fs(\032)5561 10459 y Fp(1)5636 10434 y Fr(\()p Fs(x)p Fr(\))36 b(+)6107 10392 y(~)6063 10434 y Fs(A)p Fr(\()p Fs(t)p Fr(\))p Fs(:)3508 11733 y Ft(14)p eop %%Page: 15 15 15 14 bop 726 1471 a Ft(W)-14 b(e)55 b(can)h(no)-5 b(w)56 b(write)1807 1878 y Fs(\032)1893 1903 y Fp(1)1995 1595 y Fl( )2126 1878 y Fs(x)37 b Fr(+)g Fs(v)2530 1766 y Fq(j)p Fs(z)7 b Fq(j)p 2530 1840 177 7 v 2583 1992 a Fs(c)2764 1878 y Fr(+)2930 1652 y Fl(Z)3096 1694 y Fo(t)3022 2030 y(t)p Fi(\000)3196 1971 y Fg(j)p Fj(z)s Fg(j)p 3196 2004 137 6 v 3237 2070 a Fj(c)3408 1835 y Fr(_)3388 1878 y Fs(h)p Fr(\()p Fs(s)p Fr(\))p Fs(ds)3856 1595 y Fl(!)4024 1878 y Fq(\000)37 b Fs(\032)4276 1903 y Fp(1)4378 1644 y Fl(\022)4500 1878 y Fs(x)g Fr(+)g Fs(v)4904 1766 y Fq(j)p Fs(z)7 b Fq(j)p 4904 1840 177 7 v 4956 1992 a Fs(c)5101 1644 y Fl(\023)1511 2409 y Fr(=)1807 2183 y Fl(Z)1973 2224 y Fo(t)1899 2560 y(t)p Fi(\000)2073 2502 y Fg(j)p Fj(z)s Fg(j)p 2073 2534 137 6 v 2114 2600 a Fj(c)2285 2365 y Fr(_)2265 2409 y Fs(h)p Fr(\()p Fs(s)p Fr(\))35 b Fq(\001)j(r)p Fs(\032)2912 2434 y Fp(1)3014 2175 y Fl(\022)3136 2409 y Fs(x)f Fr(+)g Fs(v)3540 2297 y Fq(j)p Fs(z)7 b Fq(j)p 3540 2371 177 7 v 3593 2523 a Fs(c)3737 2175 y Fl(\023)3887 2409 y Fs(ds)2471 2923 y Fr(+)2620 2811 y(1)p 2620 2885 84 7 v 2620 3037 a(2)2723 2739 y Fl(D)2824 2923 y Ft(Hess)p Fs(\032)3239 2948 y Fp(1)3314 2923 y Fr(\()k(~)-94 b Fs(x)3474 2953 y Fo(t;)p Fi(j)p Fo(z)5 b Fi(j)3718 2923 y Fr(\))3811 2697 y Fl(Z)3976 2738 y Fo(t)3903 3074 y(t)p Fi(\000)4076 3016 y Fg(j)p Fj(z)s Fg(j)p 4076 3048 137 6 v 4118 3114 a Fj(c)4289 2879 y Fr(_)4269 2923 y Fs(h)o Fr(\()p Fs(s)p Fr(\))p Fs(ds)p Fr(;)4810 2697 y Fl(Z)4975 2738 y Fo(t)4902 3074 y(t)p Fi(\000)5076 3016 y Fg(j)p Fj(z)s Fg(j)p 5076 3048 V 5117 3114 a Fj(c)5288 2879 y Fr(_)5268 2923 y Fs(h)o Fr(\()p Fs(s)p Fr(\))p Fs(ds)5735 2739 y Fl(E)726 3399 y Ft(for)60 b(some)71 b Fr(~)-93 b Fs(x)1501 3429 y Fo(t;)p Fi(j)p Fo(z)5 b Fi(j)1800 3399 y Fq(2)53 b Fr([)p Fs(x)40 b Fr(+)g Fs(v)2420 3318 y Fi(j)p Fo(z)5 b Fi(j)p 2421 3360 148 7 v 2465 3456 a Fo(c)2588 3399 y Fr(;)28 b Fs(x)39 b Fr(+)i Fs(v)3072 3318 y Fi(j)p Fo(z)5 b Fi(j)p 3072 3360 V 3116 3456 a Fo(c)3279 3399 y Fr(+)3448 3265 y Fl(R)3559 3306 y Fo(t)3527 3458 y(t)p Fi(\000)3700 3399 y Fg(j)p Fj(z)s Fg(j)p 3700 3431 137 6 v 3742 3498 a Fj(c)3913 3355 y Fr(_)3893 3399 y Fs(h)o Fr(\()p Fs(s)p Fr(\))p Fs(ds)p Fr(])p Fs(:)60 b Ft(In)g(addition,)i(an)e(in)-5 b(tegration)726 3629 y(b)g(y)56 b(parts)g(yields:)1749 3808 y Fl(Z)1915 3849 y Fo(t)1841 4185 y(t)p Fi(\000)2015 4127 y Fg(j)p Fj(z)s Fg(j)p 2015 4159 V 2056 4225 a Fj(c)2227 3990 y Fr(_)2207 4034 y Fs(h)p Fr(\()p Fs(s)p Fr(\))p Fs(ds)45 b Fr(=)2915 3922 y Fq(j)p Fs(z)7 b Fq(j)p 2915 3996 177 7 v 2968 4148 a Fs(c)3132 3990 y Fr(_)3112 4034 y Fs(h)p Fr(\()p Fs(t)p Fr(\))36 b(+)3600 3808 y Fl(Z)3766 3849 y Fo(t)3692 4185 y(t)p Fi(\000)3866 4127 y Fg(j)p Fj(z)s Fg(j)p 3866 4159 137 6 v 3907 4225 a Fj(c)4030 4034 y Fr(\()p Fs(t)h Fq(\000)4378 3922 y(j)p Fs(z)7 b Fq(j)p 4378 3996 177 7 v 4430 4148 a Fs(c)4611 4034 y Fq(\000)37 b Fs(s)p Fr(\))4922 3990 y(\177)4920 4034 y Fs(h)o Fr(\()p Fs(s)p Fr(\))p Fs(ds:)726 4456 y Ft(As)56 b(a)f(result)728 4766 y Fr(\177)726 4810 y Fs(h)p Fr(\()p Fs(t)p Fr(\))165 b(=)h Fq(\000)1745 4697 y Fr(1)p 1621 4771 330 7 v 1621 4923 a(4)p Fs(\031)6 b(c)1877 4876 y Fp(3)1999 4584 y Fl(Z)47 b(Z)f(Z)2368 4961 y Fi(j)p Fo(z)5 b Fi(j\024)p Fo(ct)2764 4810 y Fs(dx)28 b(dy)35 b(dz)h(\032)3460 4835 y Fp(2)3534 4810 y Fr(\()p Fs(y)43 b Fq(\000)37 b Fs(z)7 b Fr(\))p Fs(\032)4124 4835 y Fp(2)4198 4810 y Fr(\()p Fs(y)f Fr(\))4415 4675 y Fl(\002)4504 4766 y Fr(_)4484 4810 y Fs(h)p Fr(\()p Fs(t)p Fr(\))36 b Fq(\001)h(r)p Fs(\032)5113 4835 y Fp(1)5187 4810 y Fr(\()p Fs(x)g Fr(+)g Fs(v)5656 4697 y Fq(j)p Fs(z)7 b Fq(j)p 5656 4771 177 7 v 5709 4923 a Fs(c)5853 4810 y Fr(\))5918 4625 y Fl(i)5996 4810 y Fq(r)p Fs(\032)6220 4835 y Fp(1)6294 4810 y Fr(\()p Fs(x)p Fr(\))1805 5293 y Fq(\000)2077 5181 y Fr(1)p 1954 5255 330 7 v 1954 5407 a(4)p Fs(\031)f(c)2210 5359 y Fp(2)2331 5067 y Fl(Z)47 b(Z)f(Z)2700 5444 y Fi(j)p Fo(z)5 b Fi(j\024)p Fo(ct)3096 5293 y Fs(dx)28 b(dy)35 b(dz)3725 5181 y(\032)3811 5206 y Fp(2)3886 5181 y Fr(\()p Fs(y)43 b Fq(\000)37 b Fs(z)7 b Fr(\))p Fs(\032)4476 5206 y Fp(2)4550 5181 y Fr(\()p Fs(y)f Fr(\))p 3725 5255 1042 7 v 4158 5407 a Fq(j)p Fs(z)h Fq(j)2469 5796 y(\002)2598 5612 y Fl(h\020)2803 5570 y(Z)2969 5611 y Fo(t)2895 5948 y(t)p Fi(\000)3069 5889 y Fg(j)p Fj(z)s Fg(j)p 3069 5921 137 6 v 3110 5988 a Fj(c)3234 5796 y Fr(\()p Fs(t)36 b Fq(\000)3581 5684 y(j)p Fs(z)7 b Fq(j)p 3581 5758 177 7 v 3633 5910 a Fs(c)3815 5796 y Fq(\000)37 b Fs(s)p Fr(\))4126 5753 y(\177)4124 5796 y Fs(h)o Fr(\()p Fs(s)p Fr(\))p Fs(ds)4591 5612 y Fl(\021)4726 5796 y Fq(\001)g(r)p Fs(\032)5033 5821 y Fp(1)5108 5796 y Fr(\()p Fs(x)f Fr(+)h Fs(v)5576 5684 y Fq(j)p Fs(z)7 b Fq(j)p 5576 5758 V 5629 5910 a Fs(c)5773 5796 y Fr(\))5838 5612 y Fl(i)5916 5796 y Fq(r)p Fs(\032)6140 5821 y Fp(1)6215 5796 y Fr(\()p Fs(x)p Fr(\))1805 6291 y Fq(\000)2077 6179 y Fr(1)p 1954 6253 330 7 v 1954 6405 a(4)p Fs(\031)f(c)2210 6357 y Fp(2)2331 6065 y Fl(Z)47 b(Z)f(Z)2700 6442 y Fi(j)p Fo(z)5 b Fi(j\024)p Fo(ct)3096 6291 y Fs(dx)28 b(dy)35 b(dz)3725 6179 y(\032)3811 6204 y Fp(2)3886 6179 y Fr(\()p Fs(y)43 b Fq(\000)37 b Fs(z)7 b Fr(\))p Fs(\032)4476 6204 y Fp(2)4550 6179 y Fr(\()p Fs(y)f Fr(\))p 3725 6253 1042 7 v 4158 6405 a Fq(j)p Fs(z)h Fq(j)2469 6794 y(\002)2618 6682 y Fr(1)p 2618 6756 84 7 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Fr(~)4112 11166 y Fs(A)o Fr(\()p Fs(s)p Fr(\))p Fq(j)p Fs(ds:)3508 11733 y Ft(16)p eop %%Page: 17 17 17 16 bop 726 1471 a Ft(If)47 b Fs(\036)983 1496 y Fp(0)1104 1471 y Ft(and)g Fs(\031)1513 1496 y Fp(0)1634 1471 y Ft(ha)-5 b(v)g(e)47 b(compact)f(supp)5 b(ort,)49 b(so)e(do)g Fs(A)p Fr(\()p Fs(s)p Fr(\))e Ft(and)4470 1429 y Fr(~)4426 1471 y Fs(A)p Fr(\()p Fs(s)p Fr(\))p Ft(.)70 b(One)47 b(should)h(remem)-5 b(b)5 b(er)726 1671 y(that)1134 1629 y Fr(~)1090 1671 y Fs(A)59 b Ft(di\033ers)h(from)f Fs(A)g Ft(b)-5 b(y)60 b(terms)f(whic)-5 b(h)60 b(ha)-5 b(v)g(e)60 b(compact)f(supp)5 b(ort.)86 b(Moreo)-5 b(v)g(er)60 b(one)f(of)726 1909 y(these)48 b(terms)f(is)h(b)5 b(ounded)48 b(b)-5 b(y)2698 1843 y Fp(1)p 2669 1870 125 7 v 2669 1966 a Fo(c)2728 1933 y Ff(2)2861 1909 y Ft(and)48 b(the)e(other)h(b)-5 b(y)4122 1828 y Fi(j)4177 1798 y Fp(_)4161 1828 y Fo(h)p Fp(\()p Fo(t)p Fp(\))p Fi(j)p 4122 1871 311 7 v 4215 1966 a Fo(c)4274 1933 y Ff(3)4452 1909 y Ft(.)72 b(Therefore,)48 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2930 V 5184 3014 a(c)5238 2990 y Ff(3)5322 2957 y Fp(\()p Fo(s)p Fi(\000)p Fo(t)p Fp(\))5651 3032 y Fs(ds)2937 3497 y Fq(\024)3252 3385 y Fs(Ac)3449 3325 y Fp(2)p Fi(\000)p Fo(")3726 3385 y Fr(+)37 b Fs(k)5 b(B)j Fr(\()p Fs(t)p Fr(\))p 3252 3459 1056 7 v 3732 3611 a Fs(\015)4327 3497 y Fr([)p Fs(e)4496 3368 y Fj(\015)p 4471 3395 119 6 v 4471 3480 a(c)4525 3456 y Ff(3)4609 3422 y Fo(\013)4741 3497 y Fq(\000)37 b Fr(1])2937 3853 y Fq(\024)166 b Fs(A)3357 3785 y Fi(0)3403 3853 y Fs(\013)q(c)3582 3785 y Fi(\000)p Fp(1)p Fi(\000)p Fo(")3964 3853 y Fr(+)37 b Fs(k)4221 3785 y Fi(0)4268 3853 y Fs(B)8 b Fr(\()p Fs(t)p Fr(\))p Fs(\013)q(c)4771 3785 y Fi(\000)p Fp(3)6031 3853 y Ft(\(4.10\))726 4219 y(where)60 b Fs(\013)h Ft(is)g(suc)-5 b(h)61 b(that)2338 4177 y Fr(~)2294 4219 y Fs(A)p Fr(\()p Fs(s)p Fr(\))53 b(=)h(0)60 b Ft(for)g Fs(s)54 b(>)h(\013)61 b Ft(\(note)e(that)g Fs(\013)d(<)e Fr(sup)q Fq(f)5248 4151 y Fp(2)p Fo(R)5414 4168 y Ff(2)p 5248 4181 232 7 v 5334 4276 a Fo(c)5499 4219 y Fs(;)5592 4151 y Fo(R)q Fp(+)p Fo(R)5895 4168 y Ff(2)p 5592 4181 368 7 v 5747 4276 a Fo(c)5980 4219 y Fq(g)p Ft(\))60 b(and)726 4418 y(pro)-5 b(vided)65 b Fs(c)e Ft(is)h(large)f(enough.)99 b(In)63 b(the)g(case)h(when)f Fs(\036)4212 4443 y Fp(0)4350 4418 y Ft(and)h Fs(\031)4776 4443 y Fp(0)4914 4418 y Ft(do)f(not)h(ha)-5 b(v)g(e)63 b(compact)726 4617 y(supp)5 b(ort,)49 b(w)-5 b(e)46 b(need)h(a)f (di\033eren)-5 b(t)46 b(b)5 b(ound)47 b(on)3537 4575 y Fr(~)3493 4617 y Fs(A)p Fr(\()p Fs(t)p Fr(\))e Ft(than)i(\(4.8\).)70 b(F)-14 b(or)46 b Fq(j)h Fs(y)52 b Fq(j\025)47 b Fs(R)5552 4642 y Fp(2)5626 4617 y Fs(;)56 b(\032)5814 4642 y Fp(2)5888 4617 y Fr(\()p Fs(y)6 b Fr(\))46 b(=)g(0)p Ft(,)726 4817 y(so)1063 5225 y Fs(A)p Fr(\()p Fs(t)p Fr(\))165 b(=)2049 5113 y(1)p 1858 5187 465 7 v 1858 5339 a(4)p Fs(\031)6 b(c)2114 5291 y Fp(2)2188 5339 y Fs(t)2248 5291 y Fp(2)2370 4999 y Fl(Z)2564 5225 y Fs(dx)2801 4999 y Fl(Z)2893 5376 y Fi(j)p Fo(y)t Fi(j\024)p Fo(R)3246 5393 y Ff(2)3348 5225 y Fs(dy)3577 4999 y Fl(Z)3669 5376 y Fo(S)3751 5393 y Fj(ct)3860 5376 y Fp(\()p Fo(y)t Fp(\))4071 5225 y Fs(d\033)34 b Fr([)p Fs(\036)4431 5250 y Fp(0)4505 5225 y Fr(\()p Fs(x;)28 b(\033)6 b Fr(\))36 b(+)h Fs(\033)43 b Fq(\001)37 b(r)5466 5250 y Fo(y)5546 5225 y Fs(\036)5645 5250 y Fp(0)5719 5225 y Fr(\()p Fs(x;)28 b(\033)6 b Fr(\))3498 5596 y(+)p Fs(t\031)3782 5621 y Fp(0)3857 5596 y Fr(\()p Fs(x;)28 b(\033)6 b Fr(\)])p Fs(\032)4389 5621 y Fp(2)4462 5596 y Fr(\()p Fs(y)g Fr(\))p Fq(r)p Fs(\032)4903 5621 y Fp(1)4977 5596 y Fr(\()p Fs(x)36 b Fq(\000)h Fs(q)6 b Fr(\()p Fs(t)p Fr(\)\))p Fs(:)726 5962 y Ft(If)61 b Fq(j)56 b Fs(y)63 b Fq(j\024)56 b Fs(R)1501 5987 y Fp(2)1575 5962 y Ft(,)63 b(then)e Fq(j)56 b Fs(\033)62 b Fq(j\025j)56 b Fs(ct)41 b Fq(\000)g Fs(R)3073 5987 y Fp(2)3203 5962 y Fq(j)p Ft(.)92 b(A)-5 b(ccording)61 b(to)g(\(4.1\),)h(and)f(the)g(h) -5 b(yp)5 b(othesis)62 b(on)726 6161 y Fs(\036)825 6186 y Fp(0)900 6161 y Ft(,)55 b Fs(\031)1096 6186 y Fp(0)2416 6360 y Fq(j)46 b Fs(\036)2607 6385 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Fp(2)p Fi(\000)p Fo(")3058 8686 y Fl(Z)3224 8728 y Fo(t)3150 9064 y Fp(0)3371 8800 y Fs(e)3494 8671 y Fj(\015)p 3468 8698 V 3468 8782 a(c)3522 8758 y Ff(3)3607 8725 y Fp(\()p Fo(s)p Fi(\000)p Fo(t)p Fp(\))p 3330 8874 648 7 v 3330 9026 a Fr(\(1)36 b(+)h Fs(cs)p Fr(\))3895 8978 y Fo(\027)3997 8912 y Fs(ds)2297 9417 y Fr(=)166 b Fs(Ac)2789 9349 y Fp(2)p Fi(\000)p Fo(")3058 9191 y Fl(Z)3244 9188 y Ff(1)p 3244 9206 57 6 v 3245 9272 a Fj(c)3150 9569 y Fp(0)3417 9305 y Fs(e)3540 9176 y Fj(\015)p 3515 9203 119 6 v 3515 9287 a(c)3569 9263 y Ff(3)3653 9230 y Fp(\()p Fo(s)p Fi(\000)p Fo(t)p Fp(\))p 3376 9379 648 7 v 3376 9531 a Fr(\(1)36 b(+)h Fs(cs)p Fr(\))3941 9483 y Fo(\027)4043 9417 y Fs(ds)g Fr(+)g Fs(Ac)4607 9349 y Fp(2)p Fi(\000)p Fo(")4875 9191 y Fl(Z)5042 9232 y Fo(t)4988 9524 y Ff(1)p 4988 9542 57 6 v 4989 9609 a Fj(c)5189 9305 y Fs(e)5312 9176 y Fj(\015)p 5286 9203 119 6 v 5286 9287 a(c)5340 9263 y Ff(3)5425 9230 y Fp(\()p Fo(s)p Fi(\000)p Fo(t)p Fp(\))p 5148 9379 648 7 v 5148 9531 a 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b(Finally)55 b(w)-5 b(e)56 b(ha)-5 b(v)g(e)56 b(for)f(all)g Fs(t)46 b Fq(\025)h Fr(0)1817 11094 y Fq(j)1883 11050 y Fr(_)1863 11094 y Fs(h)p Fr(\()p Fs(t)p Fr(\))p Fq(j)e(\024)i(j)2482 11050 y Fr(_)2462 11094 y Fs(h)p Fr(\(0\))p Fq(j)36 b Fr(+)h Fs(K)3160 11119 y Fp(3)3234 11094 y Fs(c)3306 11025 y Fp(1)p Fi(\000)p Fo(")3584 11094 y Fr(+)3770 10982 y Fs(K)3911 11007 y Fp(2)p 3770 11056 216 7 v 3794 11208 a Fs(\015)9 b(c)4005 11094 y(B)4139 11025 y Fp(2)4214 11094 y Fr(\()p Fs(t)p Fr(\))36 b(+)h Fs(k)4697 11025 y Fi(0)4745 11094 y Fs(B)8 b Fr(\()p Fs(t)p Fr(\))p Fs(c)5141 11025 y Fi(\000)p Fp(3)5318 11094 y Fs(:)3508 11733 y Ft(17)p eop %%Page: 18 18 18 17 bop 726 1471 a Ft(Since)56 b Fs(B)8 b Fr(\()p Fs(t)p Fr(\))55 b Ft(is)h(increasing,)g(w)-5 b(e)55 b(ha)-5 b(v)g(e,)56 b(for)f(all)h Fr(0)46 b Fq(\024)g Fs(t)g Fq(\024)g Fs(T)23 b Ft(,)1757 1921 y Fq(j)1823 1877 y Fr(_)1803 1921 y Fs(h)p Fr(\()p Fs(t)p Fr(\))p Fq(j)45 b(\024)i(j)2422 1877 y Fr(_)2402 1921 y Fs(h)o Fr(\(0\))p Fq(j)37 b Fr(+)g Fs(K)3100 1946 y Fp(3)3174 1921 y Fs(c)3246 1853 y Fp(1)p Fi(\000)p Fo(")3524 1921 y Fr(+)3710 1809 y Fs(K)3851 1834 y Fp(2)p 3710 1883 216 7 v 3734 2035 a Fs(\015)9 b(c)3945 1921 y(B)4079 1853 y Fp(2)4154 1921 y Fr(\()p Fs(T)23 b Fr(\))36 b(+)h Fs(k)4697 1853 y Fi(0)4745 1921 y Fs(B)8 b Fr(\()p Fs(T)23 b Fr(\))p Fs(c)5201 1853 y Fi(\000)p Fp(3)5378 1921 y Fs(:)726 2364 y Ft(So,)56 b(taking)e(the)g(suprem)-5 b(um)56 b(o)-5 b(v)g(er)55 b Fs(t)p Ft(,)g(w)-5 b(e)54 b(ha)-5 b(v)g(e)55 b(the)f(follo)-5 b(wing)55 b(inequalit)-5 b(y)54 b(for)h(all)f Fs(T)70 b Fq(\025)46 b Fr(0)p Ft(:)1754 2805 y Fs(B)8 b Fr(\()p Fs(T)23 b Fr(\))36 b Fq(\000)h Fs(k)2431 2737 y Fi(0)2479 2805 y Fs(B)8 b Fr(\()p Fs(T)23 b Fr(\))p Fs(c)2935 2737 y Fi(\000)p Fp(3)3158 2805 y Fq(\024)47 b(j)3400 2761 y Fr(_)3380 2805 y Fs(h)o Fr(\(0\))p Fq(j)37 b Fr(+)g Fs(K)4078 2830 y Fp(3)4152 2805 y Fs(c)4224 2737 y Fp(1)p Fi(\000)p Fo(")4502 2805 y Fr(+)4688 2693 y Fs(K)4829 2718 y Fp(2)p 4688 2767 V 4712 2919 a Fs(\015)9 b(c)4923 2805 y(B)5057 2737 y Fp(2)5132 2805 y Fr(\()p Fs(T)23 b Fr(\))p Fs(:)603 b Ft(\(4.12\))726 3289 y(Using)56 b(the)f(h)-5 b(yp)5 b(othesis)56 b(on)g Fs(F)78 b Ft(and)56 b Fs(H)3174 3314 y Fp(0)3248 3289 y Fr(\()p Fs(Y)3409 3314 y Fp(0)3484 3289 y Fr(\))p Ft(,)f(one)g(has)h Fq(j)4316 3245 y Fr(_)4296 3289 y Fs(h)p Fr(\(0\))p Fq(j)45 b Fs(<)h(K)12 b(c)5096 3228 y Fp(1)p Fi(\000)p Fo(")5337 3289 y Ft(,)56 b(so)1899 3739 y Fs(B)8 b Fr(\()p Fs(T)23 b Fr(\)\(1)36 b Fq(\000)h Fs(k)2724 3670 y Fi(0)2771 3739 y Fs(c)2843 3670 y Fi(\000)p Fp(3)3021 3739 y Fr(\))46 b Fq(\024)g Fr(\()p Fs(K)i Fr(+)37 b Fs(K)3868 3764 y Fp(3)3943 3739 y Fr(\))p Fs(c)4080 3670 y Fp(1)p Fi(\000)p Fo(")4357 3739 y Fr(+)4543 3626 y Fs(K)4684 3651 y Fp(2)p 4543 3701 V 4568 3853 a Fs(\015)9 b(c)4779 3739 y(B)4913 3670 y Fp(2)4987 3739 y Fr(\()p Fs(T)23 b Fr(\))p Fs(:)748 b Ft(\(4.13\))726 4192 y(An)56 b(easy)f(computation)g(and)h(the)f(con) -5 b(tin)g(uit)g(y)56 b(of)f Fs(B)8 b Fr(\()p Fs(t)p Fr(\))54 b Ft(tell)h(us)h(that:)2805 4557 y Fs(B)8 b Fr(\()p Fs(t)p Fr(\))46 b Fq(\024)g Fs(B)3476 4582 y Fi(\000)3920 4557 y Fq(8)p Fs(t)g Fq(\025)g Fr(0)726 4923 y Ft(or)2806 5122 y Fs(B)8 b Fr(\()p Fs(t)p Fr(\))45 b Fq(\025)i Fs(B)3477 5147 y Fp(+)3919 5122 y Fq(8)p Fs(t)f Fq(\025)h Fr(0)726 5421 y Ft(with)1428 5754 y Fs(B)1554 5779 y Fi(\006)1712 5754 y Fr(=)1973 5642 y Fs(\015)9 b(c)p 1907 5716 299 7 v 1907 5868 a Fr(2)p Fs(K)2131 5893 y Fp(2)2254 5470 y Fl( )2385 5754 y Fr(1)37 b Fq(\000)g Fs(k)2762 5685 y Fi(0)2809 5754 y Fs(c)2881 5685 y Fi(\000)p Fp(3)3096 5754 y Fq(\006)3262 5451 y Fl(s)p 3428 5451 2120 7 v 303 x Fr(\(1)g Fq(\000)g Fs(k)3870 5706 y Fi(0)3917 5754 y Fs(c)3989 5706 y Fi(\000)p Fp(3)4167 5754 y Fr(\))4232 5706 y Fp(2)4343 5754 y Fq(\000)4529 5642 y Fr(4)p Fs(K)4753 5667 y Fp(2)4828 5642 y Fr(\()p Fs(K)48 b Fr(+)37 b Fs(K)5389 5667 y Fp(3)5464 5642 y Fr(\))p 4529 5716 1000 7 v 4910 5868 a Fs(\015)9 b(c)5077 5820 y Fo(")5548 5470 y Fl(!)5707 5754 y Fs(:)726 6203 y Ft(W)-14 b(e)68 b(will)h(no)-5 b(w)68 b(tak)-5 b(e)68 b Fs(c)g Ft(large)g(enough)h(so)g(that)e(there)h(exists)g(t)-5 b(w)g(o)68 b(constan)-5 b(ts)69 b Fs(\014)77 b Ft(and)69 b Fs(\014)6409 6142 y Fi(0)726 6402 y Ft(suc)-5 b(h)57 b(that:)3130 6601 y Fs(B)3256 6626 y Fi(\000)3414 6601 y Fq(\024)47 b Fs(\014)9 b(c)3765 6533 y Fp(1)p Fi(\000)p Fo(")4005 6601 y Fs(;)2885 6900 y(B)3011 6925 y Fp(+)3167 6900 y Fq(\025)46 b Fs(\014)3445 6831 y Fi(0)3492 6900 y Fs(c)g(>)g(K)12 b(c)4010 6831 y Fp(1)p Fi(\000)p Fo(")4251 6900 y Fs(:)726 7199 y Ft(Note)55 b(no)-5 b(w)56 b(that)2534 7398 y Fs(B)8 b Fr(\(0\))46 b(=)p Fq(j)3169 7354 y Fr(_)3148 7398 y Fs(h)p Fr(\(0\))f Fq(j\024)i Fs(K)12 b(c)3949 7330 y Fp(1)p Fi(\000)p Fo(")4236 7398 y Fs(<)46 b(B)4537 7423 y Fp(+)726 7697 y Ft(and)56 b(so)g(w)-5 b(e)55 b(ha)-5 b(v)g(e)56 b(\034nally)g(the)f(follo)-5 b(wing)56 b(b)5 b(ound)55 b(for)4156 7653 y Fr(_)4136 7697 y Fs(h)o Fr(\()p Fs(t)p Fr(\))p Ft(:)2164 8062 y Fq(j)2230 8019 y Fr(_)2210 8062 y Fs(h)p Fr(\()p 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b(=)h Fq(\000)2894 9035 y Fs(\022)p 2862 9109 147 7 v 2862 9261 a(c)2934 9213 y Fp(3)3028 9147 y Fs(e)3105 9072 y Fi(\000)3259 9027 y Fj(\022)p 3229 9045 119 6 v 3229 9129 a(c)3283 9105 y Ff(3)3368 9072 y Fo(t)3427 9147 y Fs(g)6 b Fr(\()p Fs(t)p Fr(\))36 b(+)h Fs(e)3981 9072 y Fi(\000)4135 9027 y Fj(\022)p 4105 9045 V 4105 9129 a(c)4159 9105 y Ff(3)4243 9072 y Fo(t)4326 9147 y Fr(_)-70 b Fs(g)6 b Fr(\()p Fs(t)p Fr(\))45 b(=)i Fq(\000)4979 9035 y Fs(\022)p 4947 9109 147 7 v 4947 9261 a(c)5019 9213 y Fp(3)5133 9103 y Fr(_)5113 9147 y Fs(h)p Fr(\()p Fs(t)p Fr(\))36 b(+)h Fs(e)5678 9072 y Fi(\000)5831 9027 y Fj(\022)p 5802 9045 119 6 v 5802 9129 a(c)5856 9105 y Ff(3)5940 9072 y Fo(t)6023 9147 y Fr(_)-70 b Fs(g)6 b Fr(\()p Fs(t)p Fr(\))p Fs(;)3508 11733 y Ft(18)p eop %%Page: 19 19 19 18 bop 726 1471 a Ft(so)56 b(that)f Fr(\(4)p Fs(:)p Fr(7\))g Ft(b)5 b(ecomes)789 1908 y Fr(_)-70 b Fs(g)6 b Fr(\()p Fs(t)p Fr(\))165 b(=)i([)1599 1795 y Fs(\022)p 1567 1869 147 7 v 1567 2021 a(c)1639 1974 y Fp(3)1733 1908 y Fs(I)13 b(d)37 b Fr(+)g(\000)2212 1933 y Fo(v)2291 1908 y Fr(])g Fq(\001)g Fs(g)6 b Fr(\()p Fs(t)p Fr(\))1501 2325 y(+)1773 2212 y Fs(\022)p 1650 2287 330 7 v 1650 2439 a Fr(4)p Fs(\031)g(c)1906 2391 y Fp(5)2027 2099 y Fl(Z)47 b(Z)f(Z)2396 2476 y Fi(j)p Fo(z)5 b Fi(j\024)p Fo(ct)2793 2325 y Fs(dx)28 b(dy)34 b(dz)3422 2212 y(\032)3508 2237 y Fp(2)3582 2212 y Fr(\()p Fs(y)43 b Fq(\000)37 b Fs(z)7 b Fr(\))p Fs(\032)4172 2237 y Fp(2)4246 2212 y Fr(\()p Fs(y)f Fr(\))p 3422 2287 1042 7 v 3854 2439 a Fq(j)p Fs(z)h Fq(j)1833 2828 y(\002)1962 2644 y Fl(h\020)2167 2602 y(Z)2333 2643 y Fo(t)2259 2979 y(t)p Fi(\000)2433 2921 y Fg(j)p Fj(z)s Fg(j)p 2433 2953 137 6 v 2474 3019 a Fj(c)2598 2828 y Fr(\()p Fs(t)36 b Fq(\000)2945 2716 y(j)p Fs(z)7 b Fq(j)p 2945 2790 177 7 v 2998 2942 a Fs(c)3179 2828 y Fq(\000)37 b Fs(s)p Fr(\))p Fs(e)3614 2708 y Fj(\022)p 3584 2726 119 6 v 3584 2810 a(c)3638 2786 y Ff(3)3723 2753 y Fp(\()p Fo(t)p Fi(\000)p Fo(s)p Fp(\))4052 2828 y Fs(g)6 b Fr(\()p 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b(that)g(in)h(the)g(third)g(term)f(of)h(the)f(righ)-5 b(t-hand)61 b(side)g(w)-5 b(e)60 b(only)f(replaced)h(one)g(factor)747 5791 y Fr(_)726 5835 y Fs(h)p Fr(\()p Fs(s)p Fr(\))54 b Ft(b)-5 b(y)56 b Fs(e)1392 5760 y Fi(\000)1546 5715 y Fj(\022)p 1516 5733 V 1516 5817 a(c)1570 5793 y Ff(3)1655 5760 y Fo(s)1726 5835 y Fs(g)6 b Fr(\()p Fs(s)p Fr(\))p Ft(.)73 b(W)-14 b(e)55 b(will)g(use)h(\(4.14\))f(to)f(con)-5 b(trol)56 b(the)f(other)g(one.)74 b(W)-14 b(e)54 b(de\034ne)2897 6185 y Fs(M)18 b Fr(\()p Fs(t)p Fr(\))45 b(=)115 b(sup)3486 6324 y Fp(0)p Fi(\024)p Fo(s)p Fi(\024)p Fo(t)3901 6185 y Fq(j)p Fs(g)6 b Fr(\()p Fs(s)p Fr(\))p Fq(j)726 6644 y Ft(and)2887 6843 y Fs(N)18 b Fr(\()p Fs(t)p Fr(\))46 b(=)115 b(sup)3449 6983 y Fp(0)p Fi(\024)p Fo(s)p Fi(\024)p Fo(t)3864 6843 y Fq(j)24 b Fr(_)-70 b Fs(g)6 b Fr(\()p Fs(s)p Fr(\))p Fq(j)p Fs(:)726 7252 y Ft(W)-14 b(riting)72 b Fs(R)q Fr(\()p Fs(t)p Fr(\))g Ft(for)f(the)h(righ)-5 b(t-hand)73 b(side)f(of)g Fr(\(4)p Fs(:)p Fr(15\))f Ft(and)h(using)h 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216 7 v 3692 8478 a Fs(c)3764 8430 y Fp(4)3893 8364 y Fs(e)3970 8289 y Fp(2)p Fo(R)4136 8306 y Ff(2)4251 8244 y Fj(\022)p 4222 8262 119 6 v 4222 8346 a(c)4276 8322 y Ff(4)4369 8364 y Fs(N)g Fr(\()p Fs(t)p Fr(\))36 b(+)h Fs(e)5039 8244 y Fj(\022)p 5009 8262 V 5009 8346 a(c)5063 8322 y Ff(3)5148 8289 y Fo(t)5206 8364 y Fq(j)5296 8322 y Fr(~)5252 8364 y Fs(A)p Fr(\()p Fs(t)p Fr(\))p Fq(j)p Fs(:)726 8759 y Ft(So)56 b(w)-5 b(e)55 b(ha)-5 b(v)g(e,)56 b(for)f(all)h Fs(s)46 b Fq(\025)g Fr(0)p Ft(,)1648 9265 y Fq(j)24 b Fr(_)-70 b Fs(g)6 b Fr(\()p Fs(s)p Fr(\))p Fq(j)165 b(\024)2493 9081 y Fl(\020)2644 9153 y Fs(\022)p 2612 9227 147 7 v 2612 9379 a(c)2684 9331 y Fp(3)2816 9265 y Fq(\000)3027 9153 y Fs(\015)p 3002 9227 V 3002 9379 a(c)3074 9331 y Fp(3)3205 9265 y Fr(+)3391 9153 y Fs(K)3532 9178 y Fp(5)3606 9153 y Fs(e)3683 9078 y Fp(2)p Fo(\022)3837 9015 y Fj(R)3923 9040 y Ff(2)p 3837 9051 151 6 v 3852 9135 a Fj(c)3906 9111 y Ff(4)p 3391 9227 625 7 v 3547 9379 a Fs(c)3619 9331 y Fp(3+)p 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b(side)f(and)g(then)g(in)726 10276 y(the)g(left-hand)h (side)g(of)f(this)h(inequalit)-5 b(y)-14 b(,)55 b(w)-5 b(e)55 b(obtain)1307 10782 y Fs(N)18 b Fr(\()p Fs(t)p Fr(\)[1)37 b Fq(\000)2000 10670 y Fs(K)2141 10695 y Fp(6)p 2000 10744 216 7 v 2034 10896 a Fs(c)2106 10848 y Fp(4)2235 10782 y Fs(e)2312 10707 y Fp(2)p Fo(R)2478 10724 y Ff(2)2593 10662 y Fj(\022)p 2563 10680 119 6 v 2563 10764 a(c)2617 10740 y Ff(4)2710 10782 y Fr(])166 b Fq(\024)3218 10598 y Fl(\020)3369 10670 y Fs(\022)p 3337 10744 147 7 v 3337 10896 a(c)3409 10848 y Fp(3)3540 10782 y Fq(\000)3751 10670 y Fs(\015)p 3726 10744 V 3726 10896 a(c)3798 10848 y Fp(3)3929 10782 y Fr(+)4115 10670 y Fs(K)4256 10695 y Fp(5)4331 10670 y Fs(e)4408 10595 y Fp(2)p Fo(\022)4561 10532 y Fj(R)4647 10557 y Ff(2)p 4561 10568 151 6 v 4577 10652 a Fj(c)4631 10628 y Ff(4)p 4115 10744 625 7 v 4272 10896 a Fs(c)4344 10848 y Fp(3+)p Fo(")4796 10782 y Fr(+)5014 10670 y Fs(\022)p 4982 10744 147 7 v 4982 10896 a(c)5054 10848 y Fp(7)5148 10782 y Fs(K)5289 10807 y Fp(4)5364 10782 y Fs(e)5441 10707 y Fp(2)p Fo(\022)5594 10644 y Fj(R)5680 10669 y Ff(2)p 5594 10681 151 6 v 5610 10764 a Fj(c)5664 10740 y Ff(4)5773 10598 y Fl(\021)3882 11140 y Fq(\002)p Fs(M)18 b Fr(\()p Fs(t)p Fr(\))36 b(+)105 b(sup)4582 11280 y Fp(0)p Fi(\024)p Fo(s)p Fi(\024)p Fo(t)4969 11140 y Fr(\()p Fs(e)5160 11020 y Fj(\022)p 5130 11038 119 6 v 5130 11122 a(c)5184 11098 y Ff(3)5269 11065 y Fo(s)5340 11140 y Fq(j)5430 11098 y Fr(~)5386 11140 y Fs(A)p Fr(\()p Fs(s)p Fr(\))p Fq(j)p Fr(\))p Fs(:)155 b Ft(\(4.17\))3508 11733 y(19)p eop %%Page: 20 20 20 19 bop 726 1471 a Ft(W)-14 b(e)59 b(denote)g(b)-5 b(y)59 b Fs(k)1875 1496 y Fo(c)2002 1471 y Ft(the)g(in)-5 b(v)g(erse)60 b(of)f(the)f(factor)g(of)h Fs(N)18 b Fr(\()p Fs(t)p Fr(\))59 b Ft(in)g Fr(\(4)p Fs(:)p Fr(17\))p Fs(:)g Ft(Remark)g(that)f Fs(k)6205 1496 y Fo(c)6326 1471 y Fq(\030)726 1671 y Fr(1)37 b(+)1058 1605 y Fo(k)p 1032 1632 125 7 v 1032 1728 a(c)1091 1695 y Ff(4)1176 1671 y Fs(:)56 b Ft(Then,)f(using)i Fr(\(4)p Fs(:)p Fr(11\))p Ft(,)e(w)-5 b(e)55 b(ha)-5 b(v)g(e)726 2200 y Fs(N)18 b Fr(\()p Fs(t)p Fr(\))46 b Fq(\024)h Fs(k)1375 2225 y Fo(c)1443 2016 y Fl(\020)1594 2088 y Fs(\022)p 1562 2162 147 7 v 1562 2314 a(c)1634 2266 y Fp(3)1747 2200 y Fq(\000)1941 2088 y Fs(\015)p 1916 2162 V 1916 2314 a(c)1988 2266 y Fp(3)2102 2200 y Fr(+)2302 2088 y Fs(\022)p 2270 2162 V 2270 2314 a(c)2342 2266 y Fp(7)2436 2200 y Fs(K)2577 2225 y Fp(4)2652 2200 y Fs(e)2729 2125 y Fp(2)p Fo(\022)2882 2063 y Fj(R)2968 2088 y Ff(2)p 2882 2099 151 6 v 2898 2182 a Fj(c)2952 2158 y Ff(4)3080 2200 y Fr(+)3296 2088 y Fs(K)3437 2113 y Fp(5)p 3248 2162 312 7 v 3248 2314 a Fs(c)3320 2266 y Fp(3+)p Fo(")3579 2200 y Fs(e)3656 2125 y Fp(2)p Fo(\022)3810 2063 y Fj(R)3896 2088 y Ff(2)p 3810 2099 151 6 v 3825 2182 a Fj(c)3879 2158 y Ff(4)3988 2016 y Fl(\021)4087 2200 y Fs(M)18 b Fr(\()p Fs(t)p Fr(\))h(+)g Fs(k)4709 2225 y Fo(c)4778 2200 y Fs(Ac)4975 2132 y Fp(2)p Fi(\000)p Fo(")5311 2200 y Fr(sup)5243 2340 y Fp(0)p Fi(\024)p Fo(s)p Fi(\024)p Fo(t)5629 2200 y Fr([)5866 2088 y Fs(e)5993 1968 y Fj(\022)p 5963 1986 119 6 v 5963 2070 a(c)6017 2046 y Ff(3)6102 2013 y Fo(s)p 5695 2162 648 7 v 5695 2314 a Fr(\(1)37 b(+)g Fs(cs)p Fr(\))6261 2266 y Fo(\027)6363 2200 y Fr(])p Fs(:)726 2675 y Ft(A)55 b(simple)i(computation)e(sho)-5 b(ws)57 b(there)e(exists)g Fs(s)3830 2700 y Fi(\003)3952 2675 y Fs(>)46 b Fr(0)56 b Ft(so)g(that)1845 3180 y Fr(sup)1777 3320 y Fp(0)p Fi(\024)p Fo(s)p Fi(\024)p Fo(t)2163 3180 y Fr([)2400 3068 y Fs(e)2527 2948 y Fj(\022)p 2497 2966 119 6 v 2497 3050 a(c)2551 3026 y Ff(3)2636 2992 y Fo(s)p 2229 3142 648 7 v 2229 3294 a Fr(\(1)37 b(+)g Fs(cs)p Fr(\))2795 3246 y Fo(\027)2897 3180 y Fr(])166 b(=)3592 3068 y Fs(e)3719 2948 y Fj(\022)p 3689 2966 119 6 v 3689 3050 a(c)3743 3026 y Ff(3)3828 2992 y Fo(t)p 3424 3142 630 7 v 3424 3294 a Fr(\(1)37 b(+)g Fs(ct)p Fr(\))3972 3246 y Fo(\027)4406 3180 y Fq(8)p Fs(t)46 b Fq(\025)g Fs(t)4839 3205 y Fi(\003)4962 3180 y Fr(=)5157 3068 y Fs(s)5235 3093 y Fi(\003)p 5157 3142 155 7 v 5198 3294 a Fs(c)5331 3180 y(;)3109 3608 y Fr(=)166 b(1)830 b Fq(8)p Fs(t)47 b Fq(\024)f Fs(t)4751 3633 y Fi(\003)4873 3608 y Fr(=)5068 3496 y Fs(s)5146 3521 y Fi(\003)p 5068 3570 V 5110 3722 a Fs(c)5242 3608 y(:)726 4019 y Ft(No)-5 b(w)56 b(w)-5 b(e)55 b(ha)-5 b(v)g(e,)56 b(for)f(all)g Fr(0)47 b Fq(\024)f Fs(s)g Fq(\024)g Fs(t)p Ft(,)2444 4469 y Fq(j)p Fs(g)6 b Fr(\()p Fs(s)p Fr(\))p Fq(j)165 b(\024)i(j)p Fs(g)6 b Fr(\(0\))p Fq(j)36 b Fr(+)3882 4243 y Fl(Z)4048 4284 y Fo(s)3974 4620 y Fp(0)4147 4469 y Fq(j)24 b Fr(_)-70 b Fs(g)6 b Fr(\()p Fs(u)p Fr(\))p Fq(j)p Fs(du)2994 4921 y Fq(\024)167 b(j)p Fs(g)6 b Fr(\(0\))p Fq(j)36 b Fr(+)3882 4695 y Fl(Z)4048 4737 y Fo(s)3974 5073 y Fp(0)4147 4921 y Fs(N)18 b Fr(\()p Fs(u)p Fr(\))p Fs(du)2994 5396 y Fq(\024)167 b(j)p Fs(g)6 b Fr(\(0\))p Fq(j)36 b Fr(+)3882 5170 y Fl(Z)4048 5211 y Fo(t)3974 5547 y Fp(0)4134 5396 y Fs(N)18 b Fr(\()p Fs(u)p Fr(\))p Fs(du;)726 5844 y Ft(so)56 b(that)995 6283 y Fs(M)18 b Fr(\()p Fs(t)p Fr(\))166 b Fq(\024)g(j)p Fs(M)18 b Fr(\(0\))p Fq(j)37 b Fr(+)2512 6057 y Fl(Z)2678 6098 y Fo(t)2604 6434 y Fp(0)2764 6283 y Fs(N)18 b Fr(\()p Fs(u)p Fr(\))p Fs(du)1530 6757 y Fq(\024)166 b(j)p Fs(g)6 b Fr(\(0\))p Fq(j)36 b Fr(+)i Fs(k)2504 6782 y Fo(c)2572 6573 y Fl(\020)2723 6645 y Fs(\022)p 2691 6719 147 7 v 2691 6871 a(c)2763 6823 y Fp(3)2894 6757 y Fq(\000)3105 6645 y Fs(\015)p 3080 6719 V 3080 6871 a(c)3152 6823 y Fp(3)3283 6757 y Fr(+)3501 6645 y Fs(\022)p 3469 6719 V 3469 6871 a(c)3541 6823 y Fp(7)3635 6757 y Fs(K)3776 6782 y Fp(4)3851 6757 y Fs(e)3928 6682 y Fp(2)p Fo(\022)4081 6620 y Fj(R)4167 6645 y Ff(2)p 4081 6656 151 6 v 4097 6739 a Fj(c)4151 6715 y Ff(4)4296 6757 y Fr(+)4530 6645 y Fs(K)4671 6670 y Fp(5)p 4482 6719 312 7 v 4482 6871 a Fs(c)4554 6823 y Fp(3+)p Fo(")4813 6757 y Fs(e)4890 6682 y Fp(2)p Fo(\022)5044 6620 y Fj(R)5130 6645 y Ff(2)p 5044 6656 151 6 v 5059 6739 a Fj(c)5113 6715 y Ff(4)5222 6573 y Fl(\021)5349 6531 y(Z)5515 6572 y Fo(t)5441 6908 y Fp(0)5601 6757 y Fs(M)18 b Fr(\()p Fs(u)p Fr(\))p Fs(du)2489 7265 y Fr(+)p Fs(k)2704 7290 y Fo(c)2772 7265 y Fs(Ac)2969 7196 y Fp(2)p Fi(\000)p Fo(")3237 7039 y Fl(Z)3403 7080 y Fo(t)3330 7416 y Fp(0)3680 7152 y Fs(e)3807 7033 y Fj(\022)p 3778 7051 119 6 v 3778 7135 a(c)3832 7111 y Ff(3)3916 7077 y Fo(u)p 3509 7227 665 7 v 3509 7379 a Fr(\(1)37 b(+)g Fs(cu)p Fr(\))4092 7331 y Fo(\027)4194 7265 y Fs(du)g Fr(+)g Fs(k)4664 7290 y Fo(c)4732 7265 y Fs(Ac)4929 7196 y Fp(2)p Fi(\000)p Fo(")5170 7265 y Fs(t)5230 7290 y Fi(\003)5306 7265 y Fs(:)726 7727 y Ft(It)54 b(is)g(no)-5 b(w)54 b(easy)g(to)f(see)h(that)f(for)h(all)g Fs(\021)e(>)46 b Fr(0)p Ft(,)55 b(there)e(exists)g Fs(c)4530 7752 y Fp(0)4605 7727 y Fr(\()p Fs(\032;)28 b(K)s(;)g(F)5155 7752 y Fp(0)5228 7727 y Fs(;)g(\017;)g(\021)6 b Fr(\))54 b Ft(so)g(that,)f(for)726 7927 y(all)j Fs(c)46 b(>)g(c)1322 7952 y Fp(0)1452 7927 y Ft(one)55 b(has)1405 8362 y Fs(k)1491 8387 y Fo(c)1559 8177 y Fl(\020)1710 8249 y Fs(\022)p 1679 8324 147 7 v 1679 8476 a(c)1751 8428 y Fp(3)1882 8362 y Fq(\000)2093 8249 y Fs(\015)p 2068 8324 V 2068 8476 a(c)2140 8428 y Fp(3)2271 8362 y Fr(+)2489 8249 y Fs(\022)p 2457 8324 V 2457 8476 a(c)2529 8428 y Fp(7)2623 8362 y Fs(K)2764 8387 y Fp(4)2839 8362 y Fs(e)2916 8287 y Fp(2)p Fo(\022)3069 8224 y Fj(R)3155 8249 y Ff(2)p 3069 8260 151 6 v 3085 8344 a Fj(c)3139 8320 y Ff(4)3284 8362 y Fr(+)3518 8249 y Fs(K)3659 8274 y Fp(5)p 3470 8324 312 7 v 3470 8476 a Fs(c)3542 8428 y Fp(3+)p Fo(")3801 8362 y Fs(e)3878 8287 y Fp(2)p Fo(\022)4031 8224 y Fj(R)4117 8249 y Ff(2)p 4031 8260 151 6 v 4047 8344 a Fj(c)4101 8320 y Ff(4)4210 8177 y Fl(\021)4355 8362 y Fq(\024)46 b Fr(\()p Fs(\022)c Fq(\000)37 b Fs(\015)9 b Fr(\(1)36 b Fq(\000)h Fs(\021)6 b Fr(\)\))5595 8249 y(1)p 5564 8324 147 7 v 5564 8476 a Fs(c)5636 8428 y Fp(3)5730 8362 y Fs(:)726 8783 y Ft(Using)56 b(the)f(Gron)-5 b(w)g(all)57 b(lemma)f([ZQ],)g(w)-5 b(e)55 b(ha)-5 b(v)g(e)1194 9178 y Fq(j)p Fs(g)6 b Fr(\()p Fs(t)p Fr(\))p Fq(j)46 b(\024)g Fs(M)18 b Fr(\()p Fs(t)p Fr(\))165 b Fq(\024)h Fr(\()p Fq(j)p Fs(g)6 b Fr(\(0\))p Fq(j)36 b Fr(+)i Fs(k)3355 9203 y Fo(c)3423 9178 y Fs(Ac)3620 9109 y Fp(2)p Fi(\000)p Fo(")3860 9178 y Fs(t)3920 9203 y Fi(\003)3996 9178 y Fr(\))p Fs(e)4138 9103 y Fp(\()p Fo(\022)t Fi(\000)p Fo(\015)7 b Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\)\))4892 9058 y Fj(t)p 4856 9076 119 6 v 4856 9160 a(c)4910 9136 y Ff(3)2943 9563 y Fr(+)p Fs(k)3158 9588 y Fo(c)3227 9563 y Fs(Ac)3424 9494 y Fp(2)p Fi(\000)p Fo(")3664 9378 y Fl(\020)3791 9337 y(Z)3957 9378 y Fo(t)3883 9714 y Fp(0)4063 9450 y Fs(e)4186 9322 y Fj(\015)p 4160 9349 V 4160 9433 a(c)4214 9409 y Ff(3)4299 9375 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(s)p 4063 9525 654 7 v 4066 9677 a Fr(\(1)36 b(+)h Fs(cs)p Fr(\))4631 9629 y Fo(\027)4736 9563 y Fs(ds)4900 9378 y Fl(\021)5000 9563 y Fs(e)5077 9488 y Fp(\()p Fo(\022)t Fi(\000)p Fo(\015)7 b Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\)\))5830 9443 y Fj(t)p 5794 9461 119 6 v 5794 9545 a(c)5848 9521 y Ff(3)5941 9563 y Fs(:)44 b Ft(\(4.18\))3508 11733 y(20)p eop %%Page: 21 21 21 20 bop 726 1483 a Ft(Remem)-5 b(b)5 b(ering)57 b(that)2160 1439 y Fr(_)2140 1483 y Fs(h)p Fr(\()p Fs(t)p Fr(\))45 b(=)h Fs(e)2723 1408 y Fi(\000)2877 1363 y Fj(\022)p 2847 1381 119 6 v 2847 1465 a(c)2901 1441 y Ff(3)2986 1408 y Fo(t)3045 1483 y Fs(g)6 b Fr(\()p Fs(t)p Fr(\))p Ft(,)54 b(this)i(yields)726 1984 y Fq(j)792 1940 y Fr(_)772 1984 y Fs(h)p Fr(\()p Fs(t)p Fr(\))p Fq(j)166 b(\024)g Fr(\()p Fq(j)1696 1940 y Fr(_)1676 1984 y Fs(h)o Fr(\(0\))p Fq(j)36 b Fr(+)h Fs(k)2318 2009 y Fo(c)2386 1984 y Fs(Ac)2583 1915 y Fp(2)p Fi(\000)p Fo(")2824 1984 y Fs(t)2884 2009 y Fi(\003)2960 1984 y Fr(\))p Fs(e)3102 1909 y Fi(\000)3251 1855 y Fj(\015)p 3226 1882 V 3226 1966 a(c)3280 1942 y Ff(3)3364 1909 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(t)3806 1984 y Fr(+)g Fs(k)4058 2009 y Fo(c)4126 1984 y Fs(Ac)4323 1915 y Fp(2)p Fi(\000)p Fo(")4564 1800 y Fl(\020)4690 1758 y(Z)4856 1799 y Fo(t)4783 2135 y Fp(0)4962 1872 y Fs(e)5085 1743 y Fj(\015)p 5060 1770 V 5060 1854 a(c)5114 1830 y Ff(3)5198 1796 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(s)p 4962 1946 654 7 v 4965 2098 a Fr(\(1)g(+)g Fs(cs)p Fr(\))5531 2050 y Fo(\027)5636 1984 y Fs(ds)5800 1800 y Fl(\021)5899 1984 y Fs(e)5976 1909 y Fi(\000)6126 1855 y Fj(\015)p 6100 1882 119 6 v 6100 1966 a(c)6154 1942 y Ff(3)6239 1909 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(t)1270 2486 y Fq(\024)166 b Fr(\()p Fq(j)1696 2443 y Fr(_)1676 2486 y Fs(h)o Fr(\(0\))p Fq(j)36 b Fr(+)h Fs(k)2318 2511 y Fo(c)2386 2486 y Fs(Ac)2583 2418 y Fp(2)p Fi(\000)p Fo(")2824 2486 y Fs(t)2884 2511 y Fi(\003)2960 2486 y Fr(\))p Fs(e)3102 2411 y Fi(\000)3251 2358 y Fj(\015)p 3226 2385 V 3226 2469 a(c)3280 2445 y Ff(3)3364 2411 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(t)3806 2486 y Fr(+)g Fs(k)4058 2511 y Fo(c)4126 2486 y Fs(Ac)4323 2418 y Fp(2)p Fi(\000)p Fo(")4564 2302 y Fl(\020)4690 2260 y(Z)4881 2257 y Fj(t)p 4876 2275 57 6 v 4876 2341 a Ff(2)4783 2638 y Fp(0)5009 2374 y Fs(e)5131 2245 y Fj(\015)p 5106 2272 119 6 v 5106 2356 a(c)5160 2332 y Ff(3)5245 2299 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(s)p 5009 2448 654 7 v 5012 2600 a Fr(\(1)f(+)h Fs(cs)p Fr(\))5577 2552 y Fo(\027)5682 2486 y Fs(ds)5846 2302 y Fl(\021)5945 2486 y Fs(e)6022 2411 y Fi(\000)6172 2358 y Fj(\015)p 6146 2385 119 6 v 6146 2469 a(c)6200 2445 y Ff(3)6285 2411 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(t)3225 2985 y Fr(+)p Fs(k)3440 3010 y Fo(c)3508 2985 y Fs(Ac)3705 2917 y Fp(2)p Fi(\000)p Fo(")3945 2801 y Fl(\020)4072 2759 y(Z)4238 2800 y Fo(t)4189 3092 y Fj(t)p 4184 3110 57 6 v 4184 3177 a Ff(2)4344 2873 y Fs(e)4467 2744 y Fj(\015)p 4442 2771 119 6 v 4442 2855 a(c)4496 2831 y Ff(3)4580 2798 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(s)p 4344 2947 654 7 v 4347 3099 a Fr(\(1)g(+)g Fs(cs)p Fr(\))4913 3051 y Fo(\027)5017 2985 y Fs(ds)5181 2801 y Fl(\021)5281 2985 y Fs(e)5358 2910 y Fi(\000)5507 2856 y Fj(\015)p 5482 2883 119 6 v 5482 2967 a(c)5536 2943 y Ff(3)5621 2910 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(t)1270 3520 y Fq(\024)166 b Fr(\()p Fq(j)1696 3476 y Fr(_)1676 3520 y Fs(h)o Fr(\(0\))p Fq(j)36 b Fr(+)h Fs(k)2318 3545 y Fo(c)2386 3520 y Fs(Ac)2583 3451 y Fp(2)p Fi(\000)p Fo(")2824 3520 y Fs(t)2884 3545 y Fi(\003)2960 3520 y Fr(\))p Fs(e)3102 3445 y Fi(\000)3251 3391 y Fj(\015)p 3226 3418 V 3226 3502 a(c)3280 3478 y Ff(3)3364 3445 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(t)3806 3520 y Fr(+)g Fs(k)4058 3545 y Fo(c)4126 3520 y Fs(Ac)4323 3451 y Fp(2)p Fi(\000)p Fo(")4583 3408 y Fs(e)4660 3332 y Fi(\000)4810 3279 y Fj(\015)p 4784 3306 V 4784 3390 a(c)4838 3366 y Ff(3)4923 3332 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(t)p 4583 3482 745 7 v 4666 3560 a(\015)p 4642 3596 125 7 v 4642 3691 a(c)4701 3658 y Ff(3)4786 3634 y Fr(\(1)f Fq(\000)h Fs(\021)6 b Fr(\))5348 3520 y([)p Fs(e)5517 3391 y Fj(\015)p 5491 3418 119 6 v 5491 3502 a(c)5545 3478 y Ff(3)5630 3445 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))6001 3400 y Fj(t)p 5996 3418 57 6 v 5996 3485 a Ff(2)6118 3520 y Fq(\000)37 b Fr(1])3225 4014 y(+)p Fs(k)3440 4039 y Fo(c)3508 4014 y Fs(Ac)3705 3945 y Fp(2)p Fi(\000)p Fo(")3973 3788 y Fl(Z)4139 3829 y Fo(t)4090 4121 y Fj(t)p 4085 4139 V 4085 4205 a Ff(2)4487 3902 y Fs(ds)p 4245 3976 648 7 v 4245 4128 a Fr(\(1)g(+)g Fs(cs)p Fr(\))4811 4080 y Fo(\027)1270 4499 y Fq(\024)166 b Fr(\()p Fq(j)1696 4455 y Fr(_)1676 4499 y Fs(h)o Fr(\(0\))p Fq(j)36 b Fr(+)h Fs(k)2318 4524 y Fo(c)2386 4499 y Fs(Ac)2583 4431 y Fp(2)p Fi(\000)p Fo(")2824 4499 y Fs(t)2884 4524 y Fi(\003)2960 4499 y Fr(\))p Fs(e)3102 4424 y Fi(\000)3251 4370 y Fj(\015)p 3226 4397 119 6 v 3226 4481 a(c)3280 4457 y Ff(3)3364 4424 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(t)3806 4499 y Fr(+)4214 4387 y Fs(k)4300 4412 y Fo(c)p 3992 4461 599 7 v 3992 4613 a Fs(\015)9 b Fr(\(1)37 b Fq(\000)g Fs(\021)6 b Fr(\))4611 4499 y Fs(Ac)4808 4431 y Fp(5)p Fi(\000)p Fo(")5048 4499 y Fs(e)5125 4424 y Fi(\000)5275 4370 y Fj(\015)p 5249 4397 119 6 v 5249 4481 a(c)5303 4457 y Ff(3)5388 4424 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))5759 4379 y Fj(t)p 5754 4397 57 6 v 5754 4464 a Ff(2)3225 4969 y Fr(+)3755 4856 y Fs(k)3841 4881 y Fo(c)3909 4856 y Fs(Ac)4106 4796 y Fp(1)p Fi(\000)p Fo(")p 3374 4930 1354 7 v 3374 5087 a Fr(\()p Fs(\027)47 b Fq(\000)37 b Fr(1\)\(1)f(+)h Fs(c)4332 5022 y Fo(t)p 4324 5049 67 7 v 4324 5144 a Fp(2)4410 5087 y Fr(\))4475 5039 y Fo(\027)8 b Fi(\000)p Fp(1)4747 4969 y Fs(:)726 5441 y Ft(Consequen)-5 b(tly)3048 5597 y Fr(_)3028 5641 y Fs(h)p Fr(\()p Fs(t)p Fr(\))45 b(=)h Fs(O)5 b Fr(\()p Fs(t)3791 5572 y Fp(1)p Fi(\000)p Fo(\027)4043 5641 y Fr(\))p Fs(;)726 5940 y Ft(so)44 b(that)f(w)-5 b(e)43 b(can)g(conclude)h(that)e(there)h(exists) g Fs(q)3741 5965 y Fi(1)3882 5940 y Fr(\()p Fs(Y)4043 5965 y Fp(0)4118 5940 y Fs(;)28 b(K)s(;)g(F)5 b(;)28 b(")p Fr(\))44 b Fq(2)i Fn(R)5062 5879 y Fo(d)5192 5940 y Ft(with)d(the)g(prop)5 b(ert)-5 b(y)726 6139 y(that)2458 6338 y Fs(q)6 b Fr(\()p Fs(t)p Fr(\))46 b(=)g Fs(q)3023 6363 y Fi(1)3201 6338 y Fr(+)37 b Fs(v)6 b Fr(\()p Fq(\000)p Fs(F)23 b Fr(\))p Fs(t)36 b Fr(+)h Fs(O)5 b Fr(\()p 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Fp(5)p 4123 7250 312 7 v 4123 7402 a Fs(c)4195 7354 y Fp(3+)p Fo(")4454 7288 y Fs(e)4531 7213 y Fp(2)p Fo(\022)4685 7150 y Fj(R)4771 7175 y Ff(2)p 4685 7187 151 6 v 4700 7270 a Fj(c)4754 7246 y Ff(4)4863 7104 y Fl(\021)4962 7288 y Fs(M)18 b Fr(\()p Fs(t)p Fr(\))36 b(+)5582 7246 y(~)5534 7288 y Fs(N)726 7737 y Ft(where)1261 7695 y Fr(~)1213 7737 y Fs(N)80 b Ft(is)63 b(a)e(n)-5 b(um)g(b)5 b(er)63 b(whic)-5 b(h)63 b(dep)5 b(ends)62 b(on)g(ev)-5 b(erything)61 b(except)g(t,)i (yielding)f(in)g(stead)726 7937 y(of)56 b Fr(\(4)p Fs(:)p Fr(18\))1983 8166 y Fq(j)p Fs(g)6 b Fr(\()p Fs(t)p Fr(\))p Fq(j)46 b(\024)g(j)p Fs(g)6 b Fr(\(0\))p Fq(j)p Fs(e)3038 8090 y Fp(\()p Fo(\022)t Fi(\000)p Fo(\015)h Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\)\))3791 8046 y Fj(t)p 3755 8064 119 6 v 3755 8148 a(c)3809 8124 y Ff(3)3939 8166 y Fr(+)4153 8124 y(~)4105 8166 y Fs(N)19 b(e)4334 8090 y Fp(\()p Fo(\022)t Fi(\000)p Fo(\015)7 b Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\)\))5088 8046 y Fj(t)p 5052 8064 V 5052 8148 a(c)5106 8124 y Ff(3)726 8465 y Ft(and)56 b(hence)2454 8664 y Fq(j)2520 8620 y Fr(_)2500 8664 y Fs(h)p Fr(\()p Fs(t)p Fr(\))p Fq(j)46 b(\024)g Fr(\()p Fq(j)3184 8620 y Fr(_)3164 8664 y Fs(h)o Fr(\(0\))p Fq(j)36 b Fr(+)3769 8622 y(~)3720 8664 y Fs(N)19 b Fr(\))p Fs(e)4014 8589 y Fi(\000)4163 8535 y Fj(\015)p 4138 8562 V 4138 8646 a(c)4192 8622 y Ff(3)4276 8589 y Fp(\(1)p Fi(\000)p Fo(\021)t Fp(\))p Fo(t)4681 8664 y Fs(:)726 8963 y Ft(from)56 b(whic)-5 b(h)56 b(the)f(announced)h(b)5 b(eha)-5 b(viour)55 b(of)g Fs(q)6 b Fr(\()p Fs(t)p Fr(\))55 b Ft(follo)-5 b(ws)56 b(again.)p 6334 9162 7 113 v 6341 9057 100 7 v 6341 9162 V 6440 9162 7 113 v 726 9711 a Fu(5)264 b(The)87 b(con\034ned)h(case) 726 10075 y Ft(W)-14 b(e)52 b(turn)g(on)-5 b(to)53 b(the)f(case)g(of)g (a)g(con\034ning)h(p)5 b(oten)-5 b(tial.)72 b(W)-14 b(e)52 b(mak)-5 b(e)52 b(the)g(follo)-5 b(wing)53 b(assump-)726 10274 y(tions)j(on)g Fs(V)92 b Ft(and)56 b Fs(\032)1962 10299 y Fp(2)2036 10274 y Ft(:)1142 10606 y(\(C\))194 b Fr(lim)1446 10721 y Fi(j)p Fo(q)t Fi(j!)p Fp(+)p Fi(1)1984 10606 y Fq(j)p Fs(V)37 b Fr(\()p Fs(q)6 b Fr(\))p Fq(j)46 b Fr(=)g(+)p Fq(1)1142 11084 y Ft(\(W\))69 b Fr(^)-98 b Fs(\032)1583 11109 y Fp(2)1657 11084 y Fr(\()p Fs(k)5 b Fr(\))46 b Fq(6)p Fr(=)g(0)56 b Ft(for)f(all)g Fs(k)d Fq(2)46 b Fn(R)3140 11024 y Fp(3)3508 11733 y Ft(21)p eop %%Page: 22 22 22 21 bop 726 1471 a Ft(Let)59 b Fs(S)j Fr(=)53 b Fq(f)p Fs(q)1537 1411 y Fi(\003)1666 1471 y Fq(2)f Fn(R)1952 1411 y Fo(d)2038 1471 y Fq(jr)p Fs(V)38 b Fr(\()p Fs(q)2502 1411 y Fi(\003)2578 1471 y Fr(\))52 b(=)g(0)p Fq(g)60 b Ft(b)5 b(e)59 b(the)f(set)h(of)g(critical)g(p)5 b(oin)-5 b(ts)60 b(of)f Fs(V)37 b Ft(.)85 b(W)-14 b(e)59 b(supp)5 b(ose)726 1671 y(that)52 b Fs(S)61 b Ft(is)53 b(discrete.)73 b(F)-14 b(or)52 b(all)h Fs(q)f Fq(2)46 b Fn(R)3012 1610 y Fo(d)3098 1671 y Ft(,)53 b(w)-5 b(e)52 b(denote)g(b)-5 b(y)52 b Fs(\036)4296 1696 y Fo(q)4422 1671 y Ft(the)f(solution)i(of)f (\(1.2\).)72 b(There-)726 1870 y(fore,)55 b Fq(f)p Fr(\()p Fs(\036)1347 1895 y Fo(q)1420 1870 y Fs(;)28 b(q)6 b(;)28 b Fr(0)p Fs(;)g Fr(0\))p Fq(j)p Fs(q)52 b Fq(2)46 b Fs(S)10 b Fq(g)54 b Ft(is)i(the)f(set)g(of)g(the)g(equilibrium)h(p)5 b(oin)-5 b(ts)56 b(for)f(the)g(dynamics.)726 2148 y Fa(Theorem)65 b(3.)84 b Fh(Supp)-8 b(ose)66 b(that)i Fr(\()p Fs(H)2959 2173 y Fp(1)3033 2148 y Fr(\))p Fh(,)g Fr(\()p Fs(H)3420 2173 y Fp(2)3494 2148 y Fr(\))p Fh(,)h Fr(\()p Fs(C)12 b Fr(\))66 b Fh(and)h Fr(\()p Fs(W)23 b Fr(\))66 b Fh(ar)-8 b(e)67 b(satis\034e)-8 b(d)67 b(and)g Fs(n)60 b Fr(=)g(3)p Fh(.)726 2347 y(Denote)72 b(by)e Fs(Y)37 b Fr(\()p Fs(t)p Fr(\))65 b(=)h(\()p Fs(\036)p Fr(\()p Fs(t)p Fr(\);)28 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\);)28 b Fs(\031)6 b Fr(\()p Fs(t)p Fr(\);)28 b Fs(p)p Fr(\()p Fs(t)p Fr(\)\))64 b Fh(the)71 b(solution)f(of)g Fr(\(3)p Fs(:)p Fr(1\))p Fh(.)108 b(F)-13 b(or)70 b(al)8 b(l)72 b Fs(Y)5955 2372 y Fp(0)6095 2347 y Fq(2)65 b(D)5 b Fh(,)726 2546 y(ther)-8 b(e)60 b(exists)g Fs(q)1675 2486 y Fi(\003)1797 2546 y Fq(2)46 b Fs(S)69 b Fh(such)59 b(that:)3138 2844 y Fr(lim)3045 2944 y Fo(t)p Fi(!)p Fp(+)p Fi(1)3490 2844 y Fs(q)6 b Fr(\()p Fs(t)p Fr(\))45 b(=)h Fs(q)4060 2775 y Fi(\003)726 3203 y Fh(and)3152 3403 y Fr(lim)3059 3502 y Fo(t)p Fi(!)p Fp(+)p Fi(1)3534 3403 y Fr(_)-77 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))45 b(=)i(0)p Fs(:)726 3723 y Fh(If)72 b(mor)-8 b(e)g(over,)74 b Fs(q)1776 3662 y Fi(\003)1924 3723 y Fh(is)d(a)h(non-de)-8 b(gener)g(ate)73 b(minimum)e(for)h Fs(V)36 b Fh(,)75 b(then)d(ther)-8 b(e)72 b(exists)g(for)f(al)8 b(l)726 3922 y Fs(\021)72 b(<)65 b Fr(1)70 b Fh(a)g Fs(c)1454 3947 y Fp(0)1594 3922 y Fs(>)65 b Fr(0)70 b Fh(such)g(that)g(for)f(any) h Fs(c)65 b(>)g(c)3686 3947 y Fp(0)3831 3922 y Fh(and)70 b(for)f(al)8 b(l)71 b Fs(\036)4786 3947 y Fp(0)4931 3922 y Fh(and)f Fs(\031)5359 3947 y Fp(0)5503 3922 y Fh(with)g(c)-8 b(omp)g(act)726 4121 y(supp)g(ort,)59 b(we)h(have:)3033 4320 y Fs(q)6 b Fr(\()p Fs(t)p Fr(\))45 b(=)h Fs(q)3603 4252 y Fi(\003)3716 4320 y Fr(+)37 b Fs(")p Fr(\()p Fs(t)p Fr(\))726 4622 y Fh(with)60 b Fq(j)p Fs(")p Fr(\()p Fs(t)p Fr(\))p Fq(j)46 b(\024)g Fs(K)12 b(e)1917 4495 y Fg(\000)p Fj(\016)p 1917 4520 147 6 v 1932 4604 a(c)1986 4580 y Ff(3)2152 4622 y Fh(and)59 b Fs(\021)2582 4548 y Fo(\015)p 2583 4584 77 7 v 2588 4679 a Fp(2)2726 4622 y Fs(<)46 b(\016)52 b(<)3222 4548 y Fo(\015)p 3222 4584 V 3227 4679 a Fp(2)3319 4622 y Fh(.)76 b(A)60 b(similar)f(r)-8 b(esult)60 b(holds)f(for)90 b Fr(_)-77 b Fs(q)7 b Fr(\()p Fs(t)p Fr(\))p Fh(.)976 4900 y Ft(One)58 b(could)h(also,)i(as)e(in)g ([KKS1],)h(study)f(the)f(con)-5 b(v)g(ergence)59 b(of)f Fs(\036)p Fr(\()p Fs(t)p Fr(\))g Ft(to)g Fs(\036)5710 4925 y Fo(q)5775 4891 y Fg(\003)5853 4900 y Ft(,)h(but)g(w)-5 b(e)726 5099 y(shall)50 b(not)e(do)h(this)g(here.)71 b(Note)48 b(that)f(the)i(\034rst)f(part)h(of)f(the)g(theorem)g(do)5 b(es)49 b(not)f(require)g Fs(c)726 5298 y Ft(to)57 b(b)5 b(e)56 b(large.)77 b(In)56 b(fact,)g(using)i(the)e(linearization)g (metho)5 b(d)57 b(of)f([KKS1],)h(one)g(could)g(pro)-5 b(v)g(e)726 5498 y(the)49 b(con)-5 b(v)g(ergence)48 b(is)h(exp)5 b(onen)-5 b(tial)49 b(for)f(all)h Fs(c)f Ft(as)i(w)-5 b(ell.)72 b(In)48 b(this)h(w)-5 b(a)g(y)-14 b(,)51 b(w)-5 b(e)49 b(do)g(not,)g(ho)-5 b(w)g(ev)g(er,)726 5697 y(obtain)52 b(a)g(v)-5 b(ery)52 b(explicit)f(expression)h(for)g(the)f(exp)5 b(onen)-5 b(tial)52 b(rate)f(of)g(deca)-5 b(y)-14 b(.)73 b(Our)52 b(metho)5 b(d)726 5896 y(here)61 b(sho)-5 b(w)62 b(it)f(to)f(b)5 b(e)61 b(equal)g(to)2850 5822 y Fo(\015)p 2793 5858 191 7 v 2793 5953 a Fp(2)p Fo(c)2918 5920 y Ff(3)3003 5896 y Ft(,)i(con\034rming)f(the)e(solutions)i(in)g(this)f (mo)5 b(del)61 b(b)5 b(eha)-5 b(v)g(e)726 6095 y(v)g(ery)55 b(m)-5 b(uc)g(h)57 b(lik)-5 b(e)55 b(those)h(of)f(the)g (phenomenological)h(equation)f(\(1.1\).)726 6494 y Fa(Pro)5 b(of:)71 b Ft(In)46 b(order)h(to)e(pro)-5 b(v)g(e)47 b(the)e(con)-5 b(v)g(ergence)46 b(of)g Fs(q)6 b Fr(\()p Fs(t)p Fr(\))45 b Ft(and)77 b Fr(_)-77 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))p Ft(,)47 b(w)-5 b(e)46 b(follo)-5 b(w)47 b(the)e(metho)5 b(d)726 6693 y(of)56 b([KKS1)q(].)75 b(The)56 b(exp)5 b(onen)-5 b(tial)55 b(rate)g(is)h(then)g(obtained)g(b) -5 b(y)56 b(the)f(same)h(tec)-5 b(hniques)56 b(as)h(in)726 6892 y(the)e(case)h Fs(V)37 b Fr(\()p Fs(q)6 b Fr(\))45 b(=)h Fq(\000)p Fs(F)60 b Fq(\001)37 b Fs(q)6 b Ft(.)976 7092 y(Using)59 b(the)g(conserv)-9 b(ation)60 b(of)f(energy)g(and)h (the)f(h)-5 b(yp)5 b(othesis)60 b(on)g Fs(V)36 b Ft(,)61 b(one)f(kno)-5 b(ws)60 b(that)726 7291 y Fs(q)6 b Fr(\()p Fs(t)p Fr(\))p Ft(,)89 b Fr(_)-77 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))56 b Ft(and)71 b Fr(\177)-96 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))57 b Ft(are)g(b)5 b(ounded.)81 b(Let)56 b Fs(B)3531 7316 y Fo(R)3690 7291 y Fq(\032)49 b Fn(R)3991 7231 y Fp(3)4132 7291 y Ft(b)5 b(e)57 b(the)g(ball)h(of)f(radius)i Fs(R)f Ft(cen)-5 b(tered)726 7490 y(at)55 b Fr(0)p Ft(.)74 b(W)-14 b(e)55 b(de\034ne:)757 7853 y Fs(E)880 7878 y Fo(R)989 7853 y Fr(\()p Fs(t)p Fr(\))165 b(=)1659 7740 y Fs(p)1743 7680 y Fp(2)1817 7740 y Fr(\()p Fs(t)p Fr(\))p 1659 7814 348 7 v 1791 7966 a(2)2063 7853 y(+)37 b Fs(V)g Fr(\()p Fs(q)6 b Fr(\()p Fs(t)p Fr(\)\))35 b(+)2984 7740 y(1)p 2984 7814 84 7 v 2984 7966 a(2)3114 7627 y Fl(Z)3207 8004 y Fk(R)3284 7971 y Fj(d)3396 7853 y Fs(dx)3633 7627 y Fl(Z)3725 8004 y Fo(B)3825 8021 y Fj(R)3955 7853 y Fs(dy)g Fr(\()p Fs(c)4294 7784 y Fp(2)4368 7853 y Fq(jr)4552 7878 y Fo(y)4632 7853 y Fs(\036)p Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fq(j)5297 7784 y Fp(2)5408 7853 y Fr(+)37 b Fq(j)p Fs(\031)6 b Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fq(j)6287 7784 y Fp(2)6360 7853 y Fr(\))2801 8313 y(+)2958 8087 y Fl(Z)3050 8464 y Fk(R)3127 8431 y Fj(d)p Ff(+)p Fj(n)3398 8313 y Fs(dx)g(dy)35 b(\032)p Fr(\()p Fs(x)h Fq(\000)h Fs(q)6 b Fr(\()p Fs(t)p Fr(\))p Fs(;)28 b(y)6 b Fr(\))p Fs(\036)p Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fs(:)696 b Ft(\(5.1\))726 8714 y(W)-14 b(e)54 b(tak)-5 b(e)53 b Fs(R)47 b Fq(\025)g Fs(R)1844 8739 y Fp(2)1918 8714 y Ft(.)74 b(Using)54 b Fr(\(3)p Fs(:)p Fr(10\))p Ft(,)f(w)-5 b(e)54 b(can)g(write)f Fs(\036)h Ft(as)g 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b(;)28 b(t)p Fr(\))65 b Ft(are)g(di\033eren)-5 b(tiable.)104 b(Let)65 b(us)h(write)f Fs(n)p Fr(\()p Fs(y)6 b Fr(\))62 b(=)4387 9238 y Fo(y)p 4347 9274 151 7 v 4347 9369 a Fi(j)p Fo(y)t Fi(j)4583 9312 y Ft(and)k(let)f Fs(d\033)72 b Ft(b)5 b(e)65 b(the)g(surface)726 9540 y(area)56 b(elemen)-5 b(t)55 b(of)g Fs(@)9 b(B)2112 9565 y Fo(R)2221 9540 y Ft(.)74 b(Then)56 b(di\033eren)-5 b(tiating)55 b Fr(\(5)p Fs(:)p Fr(1\))p Ft(,)g(w)-5 b(e)55 b(ha)-5 b(v)g(e)776 9811 y Fs(d)p 746 9886 147 7 v 746 10038 a(dt)913 9924 y(E)1036 9949 y Fo(R)1144 9924 y Fr(\()p Fs(t)p Fr(\))165 b(=)1844 9811 y Fs(d)p 1814 9886 V 1814 10038 a(dt)1981 9739 y Fl(\020)2080 9924 y Fs(H)13 b Fr(\()p Fs(Y)37 b Fr(\()p Fs(t)p Fr(\)\))f Fq(\000)2906 9811 y Fr(1)p 2906 9886 84 7 v 2906 10038 a(2)3037 9698 y Fl(Z)3129 10075 y Fk(R)3206 10042 y Fj(d)3318 9924 y Fs(dx)3555 9698 y Fl(Z)3647 10075 y Fi(j)p Fo(y)t Fi(j)p Fo(>R)4038 9924 y Fs(dy)f Fr(\()p Fs(c)4377 9855 y Fp(2)4450 9924 y Fq(jr)4634 9949 y Fo(y)4715 9924 y Fs(\036)p Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fq(j)5380 9855 y Fp(2)5490 9924 y Fr(+)37 b Fq(j)p Fs(\031)6 b Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fq(j)6369 9855 y Fp(2)6443 9924 y Fr(\))6508 9739 y Fl(\021)1499 10396 y Fr(=)167 b Fs(c)1867 10328 y Fp(2)1969 10170 y Fl(Z)2061 10548 y Fk(R)2138 10515 y Fj(d)2250 10396 y Fs(dx)2487 10170 y Fl(Z)2579 10548 y Fo(@)7 b(B)2757 10565 y Fj(R)2888 10396 y Fs(d\033)f Fr(\()p Fs(y)g Fr(\))28 b Fs(n)p Fr(\()p Fs(y)6 b Fr(\))35 b Fq(\001)i(r)3893 10421 y Fo(y)3973 10396 y Fs(\036)p Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fs(\031)6 b Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))1499 10857 y(=)167 b Fs(c)1867 10788 y Fp(2)1969 10631 y Fl(Z)2061 11008 y Fk(R)2138 10975 y Fj(d)2250 10857 y Fs(dx)2487 10631 y Fl(Z)2579 11008 y Fo(@)7 b(B)2757 11025 y Fj(R)2888 10857 y Fs(d\033)f Fr(\()p Fs(y)g Fr(\))28 b Fs(n)p Fr(\()p Fs(y)6 b Fr(\))35 b Fq(\001)i Fr(\()p Fq(r)3958 10882 y Fo(y)4038 10857 y Fs(\036)4137 10788 y Fo(r)4210 10857 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fs(\031)4831 10788 y Fo(r)4904 10857 y Fr(\()p Fs(x;)g(y)6 b(;)28 b(t)p Fr(\))35 b(+)i Fq(r)5763 10882 y Fo(y)5844 10857 y Fs(\036)5943 10788 y Fo(r)6016 10857 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))1795 11235 y Fq(\002)p Fs(\031)2025 11166 y Fp(0)2099 11235 y Fr(\()p Fs(x;)g(y)6 b(;)28 b(t)p Fr(\))36 b(+)h Fq(r)2959 11260 y Fo(y)3039 11235 y Fs(\036)3138 11166 y Fp(0)3212 11235 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fs(\031)3833 11166 y Fo(r)3905 11235 y Fr(\()p Fs(x;)g(y)6 b(;)28 b(t)p Fr(\))36 b(+)h Fq(r)4765 11260 y Fo(y)4846 11235 y Fs(\036)4945 11166 y Fp(0)5019 11235 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fs(\031)5640 11166 y Fp(0)5713 11235 y Fr(\()p Fs(x;)g(y)6 b(;)28 b(t)p Fr(\)\))p Fs(:)3508 11733 y Ft(22)p eop %%Page: 23 23 23 22 bop 726 1471 a Ft(W)-14 b(e)57 b(b)5 b(ound)57 b(the)g(three)f(last)h(terms)g(b)-5 b(y)58 b(the)e(Y)-14 b(oung)57 b(inequalit)-5 b(y)-14 b(,)57 b(and)h(w)-5 b(e)57 b(then)g(in)-5 b(tegrate)726 1671 y(in)56 b Fs(t)p Ft(.)74 b(Hence,)55 b(for)g(all)g Fs(T)69 b(>)47 b Fr(0)p Ft(,)1188 2086 y Fs(E)1311 2111 y Fo(R)1419 2086 y Fr(\()p Fs(T)60 b Fr(+)1827 1973 y Fs(R)p 1827 2048 128 7 v 1855 2200 a(c)1974 2086 y Fr(\))37 b Fq(\000)g Fs(E)2365 2111 y Fo(R)2473 2086 y Fr(\()2558 1973 y Fs(R)h Fr(+)f Fs(R)3014 1998 y Fp(2)p 2558 2048 531 7 v 2787 2200 a Fs(c)3108 2086 y Fr(\))893 2530 y Fq(\024)166 b Fs(c)1260 2462 y Fp(2)1362 2304 y Fl(Z)1528 2345 y Fo(T)18 b Fp(+)p Fo(R)1474 2619 y Fj(R)p Ff(+)p Fj(R)1731 2644 y Ff(2)p 1474 2655 322 6 v 1608 2722 a Fj(c)1863 2530 y Fs(dt)2064 2304 y Fl(Z)2157 2682 y Fk(R)2234 2648 y Fj(d)2346 2530 y Fs(dx)2583 2304 y Fl(Z)2675 2682 y Fo(@)7 b(B)2853 2699 y Fj(R)2983 2530 y Fs(d\033)f Fr(\()p Fs(y)g Fr(\))3415 2346 y Fl(\020)3514 2530 y Fs(n)p Fr(\()p Fs(y)g Fr(\))36 b Fq(\001)h(r)4088 2555 y Fo(y)4168 2530 y Fs(\036)4267 2462 y Fo(r)4341 2530 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fs(\031)4962 2462 y Fo(r)5034 2530 y Fr(\()p Fs(x;)g(y)6 b(;)28 b(t)p Fr(\))1188 3036 y(+)1337 2923 y Fs(c)1409 2863 y Fp(2)p 1337 2997 147 7 v 1368 3149 a Fr(4)1503 3036 y(\()p Fq(jr)1752 3061 y Fo(y)1832 3036 y Fs(\036)1931 2967 y Fo(r)2004 3036 y Fr(\()p Fs(x;)g(y)6 b(;)28 b(t)p Fr(\))p Fq(j)2570 2967 y Fp(2)2681 3036 y Fr(+)37 b Fq(j)p Fs(\031)2994 2967 y Fo(r)3067 3036 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fq(j)3633 2967 y Fp(2)3707 3036 y Fr(\))36 b(+)h(2\()p Fq(jr)4306 3061 y Fo(y)4386 3036 y Fs(\036)4485 2967 y Fp(0)4560 3036 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fq(j)5126 2967 y Fp(2)5236 3036 y Fr(+)37 b Fq(j)p Fs(\031)5549 2967 y Fp(0)5623 3036 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fq(j)6189 2967 y Fp(2)6263 3036 y Fr(\))6328 2851 y Fl(\021)6427 3036 y Fs(:)726 3454 y Ft(W)-14 b(e)59 b(kno)-5 b(w)58 b(that)g Fs(E)1938 3479 y Fo(R)2047 3454 y Fr(\()2132 3386 y Fo(R)q Fp(+)p Fo(R)2435 3403 y Ff(2)p 2131 3416 368 7 v 2285 3511 a Fo(c)2518 3454 y Fr(\))51 b Fq(\024)h Fs(H)13 b Fr(\()p Fs(Y)37 b Fr(\()3249 3386 y Fo(R)q Fp(+)p Fo(R)3552 3403 y Ff(2)p 3249 3416 V 3403 3511 a Fo(c)3636 3454 y Fr(\)\))51 b(=)g Fs(H)13 b Fr(\()p Fs(Y)4309 3479 y Fp(0)4384 3454 y Fr(\))p Ft(,)59 b(and)g(the)f(h)-5 b(yp)5 b(othesis)60 b(on)e(the)726 3653 y(p)5 b(oten)-5 b(tial)55 b(and)h(\()p Fr(3)p Fs(:)p Fr(13)p Ft(\))f(tell)g(us)h(that:) 2091 4071 y Fs(E)2214 4096 y Fo(R)2322 4071 y Fr(\()p Fs(t)p Fr(\))165 b Fq(\025)h Fs(H)13 b Fr(\()p Fs(Y)38 b Fr(\()p Fs(t)p Fr(\)\))d Fq(\000)3798 3959 y Fr(1)p 3798 4033 84 7 v 3798 4185 a(2)3901 4071 y(\()p Fq(k)p Fs(\031)6 b Fq(k)4233 4003 y Fp(2)4233 4112 y(2)4344 4071 y Fr(+)37 b Fs(c)4582 4003 y Fp(2)4656 4071 y Fq(k)p Fs(\036)p Fq(k)4921 4003 y Fp(2)4996 4071 y Fr(\))2677 4471 y Fq(\025)166 b(\000)p Fs(H)13 b Fr(\()p Fs(Y)3413 4496 y Fp(0)3489 4471 y Fr(\))36 b(+)h(2)p Fs(V)3936 4496 y Fp(0)4047 4471 y Fr(+)4265 4358 y(2)p 4233 4433 147 7 v 4233 4585 a Fs(c)4305 4537 y Fp(2)4400 4471 y Fq(h)p Fs(\032)p Fr(;)28 b(\001)4763 4402 y Fi(\000)p Fp(1)4763 4512 y Fo(y)4940 4471 y Fs(\032)p Fq(i)726 4856 y Ft(where)56 b Fs(V)1304 4881 y Fp(0)1433 4856 y Ft(is)g(the)f(in\034m)-5 b(um)57 b(of)e Fs(V)37 b Ft(.)74 b(So)55 b(w)-5 b(e)56 b(ha)-5 b(v)g(e)2358 5264 y Fs(E)2481 5289 y Fo(R)2589 5264 y Fr(\()p Fs(T)60 b Fr(+)2997 5151 y Fs(R)p 2997 5226 128 7 v 3024 5378 a(c)3144 5264 y Fr(\))36 b Fq(\000)h Fs(E)3534 5289 y Fo(R)3643 5264 y Fr(\()3728 5151 y Fs(R)h Fr(+)f Fs(R)4184 5176 y Fp(2)p 3727 5226 531 7 v 3957 5378 a Fs(c)4278 5264 y Fr(\))46 b Fq(\025)g(\000)p Fs(C)726 5649 y Ft(where)56 b Fs(C)66 b Ft(is)56 b(a)g(constan)-5 b(t)55 b(not)g(dep)5 b(ending)56 b(on)g Fs(R)q(;)28 b(T)23 b Ft(.)74 b(Hence,)726 6125 y Fq(\000)p Fs(c)927 6056 y Fp(2)1030 5899 y Fl(Z)1196 5940 y Fo(T)18 b Fp(+)1414 5895 y Fj(R)p 1414 5913 86 6 v 1430 5980 a(c)1142 6214 y(R)p Ff(+)p Fj(R)1399 6239 y Ff(2)p 1142 6250 322 6 v 1276 6316 a Fj(c)1555 6125 y Fs(dt)1757 5899 y Fl(Z)1849 6276 y Fk(R)1926 6243 y Fj(d)2039 6125 y Fs(dx)2276 5899 y Fl(Z)2368 6276 y Fo(@)7 b(B)2546 6293 y Fj(R)2676 6125 y Fs(d\033)f Fr(\()p Fs(y)g Fr(\))3108 5940 y Fl(\020)3207 6125 y Fs(n)p Fr(\()p Fs(y)g Fr(\))p Fq(\001r)3708 6150 y Fo(y)3787 6125 y Fs(\036)3886 6056 y Fo(r)3960 6125 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fs(\031)4581 6056 y Fo(r)4653 6125 y Fr(\()p Fs(x;)g(y)6 b(;)28 b(t)p Fr(\)+)5322 6012 y(1)p 5322 6087 84 7 v 5322 6239 a(4)5424 6125 y(\()p Fq(jr)5673 6150 y Fo(y)5753 6125 y Fs(\036)5852 6056 y Fo(r)5925 6125 y Fq(j)5971 6056 y Fp(2)6046 6125 y Fr(+)p Fq(j)p Fs(\031)6322 6056 y Fo(r)6395 6125 y Fq(j)6441 6056 y Fp(2)6516 6125 y Fr(\))854 6900 y Fq(\024)46 b Fs(C)j Fr(+)37 b(2)p Fs(c)1518 6831 y Fp(2)1620 6674 y Fl(Z)1786 6715 y Fo(T)18 b Fp(+)2004 6670 y Fj(R)p 2004 6688 86 6 v 2020 6755 a(c)1732 6989 y(R)p Ff(+)p Fj(R)1989 7014 y Ff(2)p 1732 7025 322 6 v 1866 7091 a Fj(c)2145 6900 y Fs(dt)2347 6674 y Fl(Z)2439 7051 y Fk(R)2516 7018 y Fj(d)2629 6900 y Fs(dx)2866 6674 y Fl(Z)2958 7051 y Fo(@)7 b(B)3136 7068 y Fj(R)3266 6900 y Fs(d\033)f Fr(\()p Fs(y)g Fr(\)\()p Fq(jr)3919 6925 y Fo(y)3999 6900 y Fs(\036)4098 6831 y Fp(0)4172 6900 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fq(j)4738 6831 y Fp(2)4849 6900 y Fr(+)37 b Fq(j)p Fs(\031)5162 6831 y Fp(0)5236 6900 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fq(j)5802 6831 y Fp(2)5876 6900 y Fr(\))p Fs(:)127 b Ft(\(5.2\))726 7373 y(W)-14 b(e)55 b(\034rst)h(need)f(to)g(b)5 b(ound)56 b(the)f(righ)-5 b(t-hand)57 b(side.)74 b(This)56 b(is)g(Lemma)g Fr(3)p Fs(:)p Fr(3)f Ft(of)g([KKS1)q(]:)1246 7631 y Fl(Z)1412 7672 y Fo(T)18 b Fp(+)1630 7628 y Fj(R)p 1630 7646 86 6 v 1646 7712 a(c)1358 7946 y(R)p Ff(+)p Fj(R)1615 7971 y Ff(2)p 1358 7982 322 6 v 1492 8049 a Fj(c)1772 7857 y Fs(dt)1974 7631 y Fl(Z)2066 8009 y Fk(R)2143 7976 y Fj(d)2255 7857 y Fs(dx)2492 7631 y Fl(Z)2584 8009 y Fo(@)7 b(B)2762 8026 y Fj(R)2893 7857 y Fs(d\033)f Fr(\()p Fs(y)g Fr(\)\()p Fq(jr)3546 7882 y Fo(y)3625 7857 y Fs(\036)3724 7789 y Fp(0)3799 7857 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fq(j)4365 7789 y Fp(2)4475 7857 y Fr(+)37 b Fq(j)p Fs(\031)4788 7789 y Fp(0)4862 7857 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))p Fq(j)5428 7789 y Fp(2)5502 7857 y Fr(\))46 b Fq(\024)g Fs(I)5861 7882 y Fp(0)726 8320 y Ft(uniformly)56 b(in)g Fs(R)h Ft(and)e Fs(T)23 b Ft(.)976 8519 y(Then)55 b(using)h(\()p Fr(4)p Fs(:)p Fr(2)p Ft(\),)f(w)-5 b(e)56 b(ha)-5 b(v)g(e)1359 8959 y Fs(\036)1458 8890 y Fo(r)1532 8959 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))45 b(=)2351 8846 y Fq(\000)p Fr(1)p 2292 8920 330 7 v 2292 9073 a(4)p Fs(\031)6 b(c)2548 9025 y Fp(2)2670 8733 y Fl(Z)2762 9110 y Fi(j)p Fo(y)t Fi(\000)p Fo(z)f Fi(j\024)p Fo(ct)3333 8959 y Fs(dz)3599 8846 y(\032)3685 8871 y Fp(2)3759 8846 y Fr(\()p Fs(z)i Fr(\))p 3552 8920 468 7 v 3552 9073 a Fq(j)p Fs(y)43 b Fq(\000)37 b Fs(z)7 b Fq(j)4039 8959 y Fs(\032)4125 8984 y Fp(1)4227 8725 y Fl(\022)4349 8959 y Fs(x)37 b Fq(\000)g Fs(q)6 b Fr(\()p Fs(t)37 b Fq(\000)5075 8846 y(j)p Fs(y)43 b Fq(\000)37 b Fs(z)7 b Fq(j)p 5075 8920 V 5272 9073 a Fs(c)5562 8959 y Fr(\))5627 8725 y Fl(\023)5776 8959 y Fs(:)726 9444 y Ft(If)55 b Fq(j)p Fs(y)6 b Fq(j)47 b Fr(=)f Fs(R)q Ft(,)56 b(b)5 b(ecause)56 b Fs(\032)2223 9469 y Fp(2)2297 9444 y Fr(\()p Fs(z)7 b Fr(\))46 b(=)g(0)55 b Ft(for)g Fq(j)p Fs(z)7 b Fq(j)47 b(\025)g Fs(R)3649 9469 y Fp(2)3723 9444 y Ft(,)56 b(w)-5 b(e)55 b(ha)-5 b(v)g(e)56 b(for)f Fs(t)46 b(>)5007 9375 y Fo(R)q Fp(+)p Fo(R)5310 9392 y Ff(2)p 5007 9405 368 7 v 5161 9501 a Fo(c)1419 9899 y Fs(\036)1518 9830 y Fo(r)1592 9899 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))45 b(=)2411 9786 y Fq(\000)p Fr(1)p 2352 9860 330 7 v 2352 10012 a(4)p Fs(\031)6 b(c)2608 9965 y Fp(2)2730 9673 y Fl(Z)2822 10050 y Fi(j)p Fo(z)f Fi(j)p Fo()3574 10315 y Fo(R)q Fp(+)p Fo(R)3877 10332 y Ff(2)p 3574 10345 368 7 v 3728 10441 a Fo(c)887 10839 y Fs(\031)988 10770 y Fo(r)1061 10839 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))45 b(=)1841 10795 y(_)1801 10839 y Fs(\036)1900 10770 y Fo(r)1973 10839 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))46 b(=)2793 10726 y Fq(\000)p Fr(1)p 2734 10800 330 7 v 2734 10952 a(4)p Fs(\031)6 b(c)2990 10904 y Fp(2)3111 10613 y Fl(Z)3204 10990 y Fi(j)p Fo(z)f Fi(j)p Fo()h(Q)5613 4459 y Fp(0)5695 4434 y Fr(+)8 b Fs(R)5958 4459 y Fp(1)6079 4434 y Fr(=)6290 4392 y(~)6254 4434 y Fs(R)6380 4459 y Fp(1)726 4633 y Ft(w)-5 b(e)56 b(ha)-5 b(v)g(e)56 b Fs(\036)1453 4573 y Fo(r)1526 4633 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))45 b(=)h Fs(\031)2367 4573 y Fo(r)2441 4633 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))45 b(=)h(0)p Ft(.)74 b(So)55 b Fr(\(5)p Fs(:)p Fr(2\))g Ft(b)5 b(ecomes)1217 4904 y Fl(Z)1383 4945 y Fo(T)18 b Fp(+)1601 4900 y Fj(R)p 1601 4918 86 6 v 1617 4985 a(c)1329 5219 y(R)p Ff(+)p Fj(R)1586 5244 y Ff(2)p 1329 5255 322 6 v 1463 5321 a Fj(c)1743 5130 y Fs(dt)1945 4904 y Fl(Z)2037 5281 y Fi(j)p Fo(x)p Fi(j)p Fo()3292 7965 y Fr(~)3256 8007 y Fs(R)3382 8032 y Fp(1)3457 8007 y Ft(.)73 b(Hence)2471 8373 y Fr(lim)2378 8472 y Fo(t)p Fi(!)p Fp(+)p Fi(1)2822 8373 y Fq(r)p Fs(\032)3046 8398 y Fp(1)3121 8373 y Fr(\()p Fs(x)p Fr(\))36 b Fq(\001)68 b Fr(_)-77 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))45 b(=)h(0)166 b Fq(8)p Fs(x)47 b Fq(2)f Fn(R)4717 8304 y Fo(d)726 8810 y Ft(whic)-5 b(h)57 b(pro)-5 b(v)g(es)56 b(that)86 b Fr(_)-77 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))55 b Ft(tends)h(to)f(zero.)75 b(It)55 b(remains)i(to)e(sho)-5 b(w)56 b(that)f Fs(q)6 b Fr(\()p Fs(t)p Fr(\))55 b Ft(con)-5 b(v)g(erges)56 b(to)726 9009 y(some)g Fs(q)1222 8949 y Fi(\003)1354 9009 y Ft(whic)-5 b(h)56 b(satis\034es)h Fq(r)p Fs(V)37 b Fr(\()p Fs(q)2858 8949 y Fi(\003)2934 9009 y Fr(\))46 b(=)g(0)p Ft(.)74 b(Remem)-5 b(b)5 b(er)56 b(that)f Fs(\036)4715 9034 y Fo(q)4844 9009 y Ft(is)g(the)g(solution)i(of)e Fr(\(1)p Fs(:)p Fr(2\))726 9208 y Ft(corresp)5 b(onding)63 b(to)f Fs(q)6 b Fr(\()p Fs(t)p Fr(\))57 b(=)h Fs(q)6 b Ft(.)95 b(Let)61 b Fq(A)d Fr(=)g Fq(f)p Fs(Y)3606 9233 y Fo(q)3737 9208 y Fr(=)g(\()p Fs(\036)4088 9233 y Fo(q)4161 9208 y Fs(;)28 b(q)6 b(;)28 b Fr(0)p Fs(;)g Fr(0\))165 b Fs(q)64 b Fq(2)57 b Fn(R)5288 9148 y Fo(d)5374 9208 y Fs(;)f Fq(j)p Fs(q)6 b Fq(j)58 b(\024)g Fs(Q)6024 9233 y Fp(0)6099 9208 y Fq(g)p Ft(.)94 b Fq(A)726 9408 y Ft(is)57 b(compact)f(in)h Fq(E)15 b Ft(.)76 b(Finally)-14 b(,)57 b(w)-5 b(e)56 b(note)g(b)-5 b(y)57 b Fq(k)p Fs(:)p Fq(k)3636 9433 y Fo(R)3801 9408 y Ft(the)f Fs(L)4201 9347 y Fp(2)4332 9408 y Ft(norm)g(restricted)g(to)g(the)g(ball)g(of)726 9607 y(radius)h Fs(R)g Ft(and)e Fq(j)p Fs(Y)38 b Fq(j)1957 9547 y Fp(2)1957 9653 y Fi(E)10 b Fo(;R)2233 9607 y Fr(=)46 b Fq(kr)2629 9632 y Fo(y)2709 9607 y Fs(\036)p Fq(k)2891 9547 y Fp(2)2891 9653 y Fo(R)3037 9607 y Fr(+)37 b Fq(j)p Fs(q)6 b Fq(j)3375 9547 y Fp(2)3487 9607 y Fr(+)37 b Fq(k)p Fs(\031)6 b Fq(k)3920 9547 y Fp(2)3920 9653 y Fo(R)4065 9607 y Fr(+)37 b Fq(j)p Fs(p)p Fq(j)4407 9547 y Fp(2)4482 9607 y Ft(.)74 b(W)-14 b(e)54 b(\034rst)i(pro)-5 b(v)g(e)1435 10001 y Fr(inf)1366 10108 y Fo(Y)1444 10125 y Fj(q)1510 10108 y Fi(2A)1733 10001 y Fq(j)p Fs(Y)38 b Fr(\()p Fs(t)p Fr(\))e Fq(\000)h Fs(Y)2401 10026 y Fo(q)2475 10001 y Fq(j)2521 9933 y Fp(2)2521 10042 y Fi(E)10 b Fo(;R)6114 10001 y Ft(\(5.6\))1070 10368 y Fr(=)167 b Fq(j)p Fs(p)p Fr(\()p Fs(t)p Fr(\))p Fq(j)1732 10299 y Fp(2)1842 10368 y Fr(+)37 b Fq(k)p Fs(\031)6 b Fr(\()p Fs(t)p Fr(\))p Fq(k)2465 10299 y Fp(2)2465 10409 y Fo(R)2609 10368 y Fr(+)145 b(inf)2776 10483 y Fi(j)p Fo(q)t Fi(j\024)p Fo(Q)3128 10500 y Ff(0)3193 10368 y Fr(\()p Fq(kr)3479 10393 y Fo(y)3558 10368 y Fr(\()p Fs(\036)p Fr(\()p Fs(t)p Fr(\))36 b Fq(\000)h Fs(\036)4213 10393 y Fo(q)4286 10368 y Fr(\))p Fq(k)4434 10299 y Fp(2)4434 10409 y Fo(R)4580 10368 y Fr(+)g Fq(j)p Fs(q)6 b Fr(\()p Fs(t)p Fr(\))36 b Fq(\000)h Fs(q)6 b Fq(j)5390 10299 y Fp(2)5511 10368 y Fq(!)5677 10393 y Fo(t)p Fi(!)p Fp(+)p Fi(1)6148 10368 y Fr(0)p Fs(:)726 10835 y Ft(W)-14 b(e)56 b(kno)-5 b(w)56 b(that)f Fq(j)p Fs(p)p Fr(\()p Fs(t)p Fr(\))p Fq(j)46 b(!)i Fr(0)56 b Ft(as)g Fs(t)47 b Fq(!)g Fr(+)p Fq(1)p Ft(.)76 b(Then)56 b Fr(\(5)p Fs(:)p Fr(3\))g Ft(implies)h(that)e Fq(k)p Fs(\031)5453 10775 y Fo(r)5526 10835 y Fr(\()p Fs(t)p Fr(\))p Fq(k)5799 10860 y Fo(R)5954 10835 y Fq(!)48 b Fr(0)56 b Ft(as)726 11035 y Fs(t)49 b Fq(!)h Fr(+)p Fq(1)p Ft(.)79 b(The)57 b(b)5 b(ound)57 b(on)h Fq(k)p Fs(\031)2744 10974 y Fp(0)2818 11035 y Fr(\()p Fs(t)p Fr(\))p Fq(k)3091 11060 y Fo(R)3256 11035 y Ft(\(see)f(Lemma)g(3.3)g(of)g([KKS1)q(]\))f (then)h(sho)-5 b(ws)59 b(that)3508 11733 y(25)p eop %%Page: 26 26 26 25 bop 726 1471 a Ft(the)60 b(same)h(result)f(holds)h(for)f Fq(k)p Fs(\031)6 b Fr(\()p Fs(t)p Fr(\))p Fq(k)3060 1496 y Fo(R)3168 1471 y Ft(.)88 b(T)-14 b(o)60 b(estimate)f(the)h(in\034m)-5 b(um)62 b(o)-5 b(v)g(er)60 b Fs(q)66 b Ft(in)61 b Fr(\(5)p Fs(:)p Fr(6\))e Ft(w)-5 b(e)726 1671 y(tak)g(e)55 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))55 b Ft(for)g Fs(q)6 b Ft(.)74 b(Then)55 b(the)g(last)h(term)f(v)-9 b(anishes)56 b(and)f(w)-5 b(e)56 b(ha)-5 b(v)g(e)56 b(to)e(con)-5 b(trol)815 2097 y Fq(r)953 2122 y Fo(y)1033 2097 y Fr(\()p Fs(\036)1197 2029 y Fo(r)1270 2097 y Fr(\()p Fs(x;)28 b(y)6 b(;)28 b(t)p Fr(\))36 b Fq(\000)h Fs(\036)2091 2127 y Fo(q)t Fp(\()p Fo(t)p Fp(\))2318 2097 y Fr(\()p Fs(x;)28 b(y)6 b Fr(\)\))165 b(=)h Fq(r)3367 2122 y Fo(y)3447 1913 y Fl(h)3604 1985 y Fq(\000)p Fr(1)p 3545 2059 330 7 v 3545 2211 a(4)p Fs(\031)6 b(c)3801 2163 y Fp(2)3923 1871 y Fl(Z)4117 2097 y Fs(dz)4307 1985 y(\032)4393 2010 y Fp(2)4468 1985 y Fr(\()p Fs(y)43 b Fq(\000)37 b Fs(z)7 b Fr(\))p 4307 2059 665 7 v 4551 2211 a Fq(j)p Fs(z)g Fq(j)4992 1913 y Fl(\020)5091 2097 y Fs(\032)5177 2122 y Fp(1)5251 2097 y Fr(\()p Fs(x)37 b Fq(\000)g Fs(q)6 b Fr(\()p Fs(t)36 b Fq(\000)6041 1985 y(j)p Fs(z)7 b Fq(j)p 6041 2059 177 7 v 6094 2211 a Fs(c)6238 2097 y Fr(\)\))4889 2510 y Fq(\000)p Fs(\032)5104 2535 y Fp(1)5178 2510 y Fr(\()p Fs(x)37 b Fq(\000)g Fs(q)6 b Fr(\()p Fs(t)p Fr(\)\))5876 2326 y Fl(\021)o(i)726 2901 y Ft(for)71 b Fq(j)p Fs(y)6 b Fq(j)74 b(\024)f Fs(R)q Ft(,)i(the)c(term)g Fq(r)2552 2926 y Fo(y)2632 2901 y Fs(\036)2731 2841 y Fp(0)2877 2901 y Ft(b)5 b(eing)71 b(con)-5 b(trolled)72 b(using)g(once)f(again)g (Lemma)h(3.3)f(of)726 3133 y([KKS1)q(].)105 b(The)65 b(di\033erence)g Fs(\032)2593 3158 y Fp(1)2668 3133 y Fr(\()p Fs(x)43 b Fq(\000)h Fs(q)6 b Fr(\()p Fs(t)43 b Fq(\000)3485 3053 y Fi(j)p Fo(z)5 b Fi(j)p 3485 3095 148 7 v 3529 3191 a Fo(c)3652 3133 y Fr(\)\))43 b Fq(\000)h Fs(\032)4084 3158 y Fp(1)4158 3133 y Fr(\()p Fs(x)f Fq(\000)h Fs(q)6 b Fr(\()p Fs(t)p Fr(\)\))64 b Ft(can)i(b)5 b(e)65 b(written)g(using)726 3355 y(an)59 b(in)-5 b(tegral)59 b(dep)5 b(ending)59 b(only)f(on)89 b Fr(_)-76 b Fs(q)6 b Fr(\()p Fs(s)p Fr(\))57 b Ft(for)h Fs(s)51 b Fq(2)g Fr([)p Fs(t)39 b Fq(\000)4196 3287 y Fo(R)q Fp(+)p Fo(R)4499 3304 y Ff(2)p 4196 3317 368 7 v 4350 3412 a Fo(c)4584 3355 y Fs(;)28 b(t)p Fr(])58 b Ft(whic)-5 b(h)59 b(tends)g(to)f(zero)g (as)726 3554 y Fs(t)e Ft(go)5 b(es)55 b(to)g(in\034nit)-5 b(y)56 b(uniformly)f(in)h Fr(\()p Fs(x;)28 b(y)6 b Fr(\))45 b Fq(2)h Fs(B)3646 3579 y Fo(R)3755 3554 y Ft(.)74 b(All)55 b(this)h(pro)-5 b(v)g(es)56 b Fr(\(5)p Fs(:)p Fr(6\))p Ft(.)976 3753 y(Giv)-5 b(en)52 b(a)f(solution)i Fs(Y)37 b Fr(\()p Fs(t)p Fr(\))51 b Ft(of)g Fr(\(3)p Fs(:)p Fr(1\))p Ft(,)h(w)-5 b(e)52 b(call)g Fq(B)k Ft(the)c(set)f(of)h(all)4894 3711 y Fr(\026)4868 3753 y Fs(Y)84 b Fq(2)46 b(E)66 b Ft(suc)-5 b(h)53 b(that)e(there)726 3952 y(exists)65 b(some)g(sequence)g Fs(t)2377 3977 y Fo(n)2529 3952 y Fq(!)d Fr(+)p Fq(1)i Ft(with)h Fs(Y)37 b Fr(\()p Fs(t)3762 3977 y Fo(n)3852 3952 y Fr(\))61 b Fq(!)4231 3910 y Fr(\026)4206 3952 y Fs(Y)102 b Ft(in)65 b(the)f(norm)h Fq(j)p Fs(:)p Fq(j)5483 3977 y Fi(E)10 b Fo(;R)5778 3952 y Ft(for)65 b(all)g Fs(R)q Ft(.)726 4152 y(The)k(con)-5 b(tin)g(uit)g(y)69 b(of)f Fs(W)2260 4091 y Fo(t)2386 4152 y Ft(tells)h(us)g(that)e Fq(B)74 b Ft(is)69 b(an)f(in)-5 b(v)c(arian)k(t)69 b(set.)113 b(Then,)71 b Fr(\(5)p Fs(:)p Fr(6\))d Ft(tells)g(us)726 4351 y(that)e Fq(B)i(\032)c(A)p Ft(.)104 b(So,)69 b(for)2329 4309 y Fr(\026)2304 4351 y Fs(Y)101 b Fq(2)63 b(B)71 b Ft(there)65 b(exists)g(a)h Fs(C)4038 4291 y Fp(2)4178 4351 y Ft(curv)-5 b(e)65 b Fs(t)f Fq(!)76 b Fr(~)-96 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))63 b Fq(2)g Fn(R)5614 4291 y Fo(d)5766 4351 y Ft(suc)-5 b(h)67 b(that)726 4550 y Fs(W)906 4490 y Fo(t)990 4508 y Fr(\026)965 4550 y Fs(Y)90 b Fr(=)54 b Fs(Y)1440 4580 y Fp(~)-76 b Fo(q)t Fp(\()p Fo(t)p Fp(\))1657 4550 y Ft(.)87 b(But)59 b Fs(W)2304 4490 y Fo(t)2387 4508 y Fr(\026)2362 4550 y Fs(Y)97 b Ft(is)60 b(a)g(solution)g(of)g Fr(\(3)p Fs(:)p Fr(1\))e Ft(so)j(w)-5 b(e)59 b(m)-5 b(ust)61 b(ha)-5 b(v)g(e)5372 4511 y Fr(_)5367 4550 y(~)-95 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))52 b(=)h(0)p Ft(,)61 b(hence)739 4749 y Fr(~)-96 b Fs(q)7 b Fr(\()p Fs(t)p Fr(\))45 b(=)h Fs(q)1297 4689 y Fi(\003)1412 4749 y Ft(with)39 b Fq(r)p Fs(V)e Fr(\()p Fs(q)2191 4689 y Fi(\003)2267 4749 y Fr(\))46 b(=)g(0)39 b Ft(and)g Fs(q)3061 4689 y Fi(\003)3184 4749 y Fq(2)46 b Fs(S)10 b Ft(.)68 b(Therefore)4328 4707 y Fr(\026)4303 4749 y Fs(Y)83 b Fr(=)46 b Fs(Y)4753 4774 y Fo(q)4818 4741 y Fg(\003)4935 4749 y Ft(and)40 b Fq(B)51 b(\032)46 b(f)p Fs(Y)5756 4774 y Fo(q)5830 4749 y Fs(;)55 b(q)e Fq(2)45 b Fs(S)10 b Fq(g)p Ft(.)976 4949 y(W)-14 b(e)53 b(no)-5 b(w)53 b(pro)-5 b(v)g(e)54 b(that)f Fs(q)6 b Fr(\()p Fs(t)p Fr(\))45 b Fq(!)i Fs(q)3015 4888 y Fi(\003)3091 4949 y Ft(.)73 b(Supp)5 b(ose)54 b(there)f(exists)h Fs(R)q(;)28 b(\017)46 b(>)h Fr(0)53 b Ft(and)h(a)f(sequence)726 5148 y Fs(t)786 5173 y Fo(n)923 5148 y Fq(!)46 b Fr(+)p Fq(1)56 b Ft(suc)-5 b(h)56 b(that)2755 5479 y Fr(inf)2735 5586 y Fo(q)t Fi(2)p Fo(S)3005 5479 y Fq(j)p Fs(Y)37 b Fr(\()p Fs(t)3309 5504 y Fo(n)3399 5479 y Fr(\))f Fq(\000)h Fs(Y)3762 5504 y Fo(q)3836 5479 y Fq(j)3882 5504 y Fi(E)10 b Fo(;R)4158 5479 y Fq(\025)46 b Fs(\017:)1668 b Ft(\(5.7\))726 5910 y(But)81 b Fr(\(5)p Fs(:)p Fr(6\))f Ft(and)h(the)g(compactness)g(of)g Fq(A)f Ft(imply)i(that)e(there)g(exists)5414 5868 y Fr(\026)5389 5910 y Fs(Y)126 b Fq(2)88 b(A)81 b Ft(and)g(a)726 6109 y(subsequence)52 b Fs(t)1720 6049 y Fi(0)1720 6150 y Fo(n)1862 6109 y Ft(suc)-5 b(h)52 b(that)e Fs(Y)38 b Fr(\()p Fs(t)2846 6049 y Fi(0)2846 6150 y Fo(n)2935 6109 y Fr(\))46 b Fq(!)3283 6067 y Fr(\026)3258 6109 y Fs(Y)89 b Ft(in)51 b(the)g(norm)g Fq(j)p Fs(:)p Fq(j)4481 6134 y Fi(E)10 b Fo(;R)4763 6109 y Ft(for)51 b(all)g Fs(R)q Ft(,)h(where)5965 6067 y Fr(\026)5940 6109 y Fs(Y)83 b Fq(2)46 b(A)p Ft(.)726 6308 y(Then,)c(b)-5 b(y)38 b(de\034nition,) 2201 6266 y Fr(\026)2176 6308 y Fs(Y)83 b Fq(2)46 b(B)5 b Ft(.)68 b(But)37 b Fr(\(5)p Fs(:)p Fr(7\))h Ft(is)g(then)g(a)g(con)-5 b(tradiction)38 b(to)g Fq(B)51 b(\032)46 b(f)p Fs(Y)5756 6333 y Fo(q)5830 6308 y Fs(;)55 b(q)e Fq(2)45 b Fs(S)10 b Fq(g)p Ft(.)726 6507 y(So)2983 6707 y Fr(inf)2963 6814 y Fo(q)t Fi(2)p Fo(S)3233 6707 y Fq(j)p Fs(q)c Fr(\()p Fs(t)p Fr(\))36 b Fq(\000)h Fs(q)6 b Fq(j)46 b(!)h Fr(0)726 7073 y Ft(and)56 b(b)5 b(ecause)55 b Fs(S)65 b Ft(is)56 b(discrete,)f(there)g(exists)g Fs(q)3616 7013 y Fi(\003)3739 7073 y Fq(2)46 b Fs(S)64 b Ft(suc)-5 b(h)57 b(that)e Fs(q)6 b Fr(\()p Fs(t)p Fr(\))45 b Fq(!)h Fs(q)5404 7013 y Fi(\003)5480 7073 y Ft(.)976 7272 y(W)-14 b(e)67 b(no)-5 b(w)68 b(supp)5 b(ose)69 b(that)e Fs(q)2721 7212 y Fi(\003)2865 7272 y Ft(is)i(a)f(non-degenerate)f(minim)-5 b(um)70 b(for)e Fs(V)37 b Ft(.)111 b(Because)67 b(of)726 7472 y(the)57 b(translation)g(in)-5 b(v)c(ariance)57 b(of)g(the)g(in)-5 b(teraction)57 b(term,)g(w)-5 b(e)57 b(can)g(supp)5 b(ose)58 b(that)e Fs(q)6022 7411 y Fi(\003)6148 7472 y Fr(=)49 b(0)p Ft(.)726 7671 y(Hence,)55 b(the)g(same)h(computation)f(as)h(in)g (Section)f(4)g(leads)h(to:)811 8088 y Fr(\177)-95 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))165 b(=)h Fq(\000r)p Fs(V)38 b Fr(\()p Fs(q)6 b Fr(\()p Fs(t)p Fr(\)\))35 b Fq(\000)2577 7976 y Fs(\015)p 2552 8050 147 7 v 2552 8202 a(c)2624 8154 y Fp(3)2749 8088 y Fr(_)-77 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))36 b Fq(\000)3334 7976 y Fr(1)p 3210 8050 330 7 v 3210 8202 a(4)p Fs(\031)6 b(c)3466 8154 y Fp(2)3588 7862 y Fl(Z)46 b(Z)g(Z)3957 8240 y Fi(j)p Fo(z)5 b Fi(j\024)p Fo(ct)4353 8088 y Fs(dx)28 b(dy)35 b(dz)4982 7976 y(\032)5068 8001 y Fp(2)5142 7976 y Fr(\()p Fs(y)43 b Fq(\000)37 b Fs(z)7 b 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8999 y Fp(2)4607 8974 y Fr(\()p Fs(y)f Fr(\))p 3782 9048 1042 7 v 4215 9200 a Fq(j)p Fs(z)h Fq(j)2525 9589 y(\002)2674 9477 y Fr(1)p 2674 9551 84 7 v 2674 9703 a(2)2777 9405 y Fl(D)2879 9589 y Ft(Hess)p Fs(\032)3294 9614 y Fp(1)3368 9589 y Fr(\()k(~)-94 b Fs(x)3528 9619 y Fo(t;)p Fi(j)p Fo(z)5 b Fi(j)3773 9589 y Fr(\))3866 9363 y Fl(Z)4031 9404 y Fo(t)3957 9741 y(t)p Fi(\000)4131 9682 y Fg(j)p Fj(z)s Fg(j)p 4131 9714 137 6 v 4172 9781 a Fj(c)4354 9589 y Fr(_)-76 b Fs(q)6 b Fr(\()p Fs(s)p Fr(\))p Fs(ds)p Fr(;)4850 9363 y Fl(Z)5015 9404 y Fo(t)4941 9741 y(t)p Fi(\000)5115 9682 y Fg(j)p Fj(z)s Fg(j)p 5115 9714 V 5156 9781 a Fj(c)5338 9589 y Fr(_)-77 b Fs(q)6 b Fr(\()p Fs(s)p Fr(\))p Fs(ds)5759 9405 y Fl(E)5860 9589 y Fq(r)p Fs(\032)6084 9614 y Fp(1)6158 9589 y Fr(\()p Fs(x)p Fr(\))1861 10000 y(+)2034 9958 y(~)1990 10000 y Fs(A)p Fr(\()p Fs(t)p Fr(\))p Fs(:)3763 b Ft(\(5.8\))726 10331 y(Moreo)-5 b(v)g(er,)64 b Fq(r)p Fs(V)37 b Fr(\()p Fs(q)6 b Fr(\()p Fs(t)p Fr(\)\))56 b(=)h Fs(W)64 b Fq(\001)41 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))41 b(+)g Fs(f)18 b Fr(\()p Fs(q)6 b Fr(\()p Fs(t)p Fr(\)\))61 b Ft(where)g Fs(W)85 b Ft(is)62 b(the)g(Hessian)g(matrix)g (of)f Fs(V)726 10530 y Ft(at)i Fs(q)h Fr(=)59 b(0)k Ft(and)g Fs(f)18 b Fr(\()p Fs(q)6 b Fr(\))58 b(=)h Fs(o)p Fr(\()p Fq(j)p Fs(q)6 b Fq(j)p Fr(\))p Ft(.)96 b(Remark)63 b(that)3862 10488 y Fr(~)3818 10530 y Fs(A)p Fr(\()p Fs(t)p Fr(\))f Ft(has)h(compact)g(supp)5 b(ort.)96 b(W)-14 b(e)63 b(no)-5 b(w)726 10729 y(de\034ne)56 b Fs(Q)p Fr(\()p Fs(t)p Fr(\))46 b(=)g(\()p Fs(q)6 b Fr(\()p Fs(t)p Fr(\))p Fs(;)57 b Fr(_)-75 b Fs(q)t Fr(\()p Fs(t)p Fr(\)\))45 b Fq(2)h Fn(R)2815 10669 y Fp(2)p Fo(d)3023 10729 y Ft(and)3394 10687 y Fr(~)3346 10729 y Fs(W)78 b Ft(the)55 b Fr(2)p Fs(d)37 b Fq(\002)g Fr(2)p Fs(d)56 b Ft(matrix:)2769 11118 y Fr(~)2721 11160 y Fs(W)69 b Fr(=)3122 10926 y Fl(\022)3416 11059 y Fs(O)406 b(I)3327 11258 y Fq(\000)p Fs(W)189 b Fq(\000)3975 11184 y Fo(\015)p 3952 11220 125 7 v 3952 11315 a(c)4011 11282 y Ff(3)4096 11258 y Fs(I)4265 10926 y Fl(\023)4415 11160 y Fs(:)3508 11733 y Ft(26)p eop %%Page: 27 27 27 26 bop 726 1471 a Ft(Since)79 b Fr(0)g Ft(is)h(a)f(non-degenerate)f (minim)-5 b(um)81 b(for)e Fs(V)37 b Ft(,)85 b Fs(W)101 b Ft(is)80 b(a)f(diagonalizable)g(p)5 b(ositiv)-5 b(e)726 1671 y(de\034nite)47 b(matrix.)71 b(One)47 b(should)h(remark)f(that) 3730 1629 y Fr(~)3681 1671 y Fs(W)70 b Ft(is)48 b(diagonalizable)f(as)h (w)-5 b(ell)47 b(with)g(eigen-)726 1887 y(v)-9 b(alues)52 b Fs(\025)1314 1912 y Fo(k)1442 1887 y Fr(=)46 b Fq(\000)1823 1813 y Fo(\015)p 1766 1849 191 7 v 1766 1945 a Fp(2)p Fo(c)1891 1911 y Ff(3)2005 1887 y Fr(+)29 b Fs(i\013)2326 1912 y Fo(k)2408 1887 y Ft(,)53 b(and)e(so)h(for)f(all)h Fs(t)f Ft(in)h Fn(R)27 b Ft(,)61 b Fq(k)p Fs(e)4232 1798 y Fp(~)4193 1827 y Fo(W)18 b(t)4395 1887 y Fq(k)46 b Fr(=)g Fs(e)4776 1812 y Fi(\000)4954 1758 y Fj(\015)p 4900 1785 176 6 v 4900 1869 a Ff(2)p Fj(c)5011 1845 y Ff(3)5095 1812 y Fo(t)5154 1887 y Ft(.)72 b(W)-14 b(e)51 b(rewrite)g Fr(\(5)p 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Fp(~)2584 3142 y Fo(W)18 b(t)2785 3211 y Fs( )6 b Fr(\()p Fs(e)3079 3113 y Fp(~)3041 3142 y Fo(W)18 b(t)3242 3211 y Fs(X)13 b Fr(\()p Fs(t)p Fr(\))p Fs(;)3704 3169 y Fr(~)3657 3211 y Fs(W)22 b(e)3951 3113 y Fp(~)3913 3142 y Fo(W)c(t)4114 3211 y Fs(X)13 b Fr(\()p Fs(t)p Fr(\))36 b(+)h Fs(e)4773 3113 y Fp(~)4734 3142 y Fo(W)18 b(t)5002 3169 y Fr(_)4935 3211 y Fs(X)13 b Fr(\()p Fs(t)p Fr(\)\))965 3858 y Fq(j)1077 3816 y Fr(_)1011 3858 y Fs(X)g Fr(\()p Fs(t)p Fr(\))p Fq(j)165 b(\024)2001 3745 y Fr(1)p 1878 3819 330 7 v 1878 3971 a(4)p Fs(\031)6 b(c)2134 3923 y Fp(2)2255 3632 y Fl(Z)47 b(Z)f(Z)2624 4009 y Fi(j)p Fo(z)5 b Fi(j\024)p Fo(ct)3021 3858 y Fs(dx)28 b(dy)34 b(dz)3650 3745 y(\032)3736 3770 y Fp(2)3810 3745 y Fr(\()p Fs(y)43 b Fq(\000)37 b Fs(z)7 b Fr(\))p Fs(\032)4400 3770 y Fp(2)4474 3745 y Fr(\()p Fs(y)f Fr(\))p 3650 3819 1042 7 v 4082 3971 a Fq(j)p Fs(z)h Fq(j)4711 3673 y Fl(\020)4838 3632 y(Z)5004 3673 y Fo(t)4930 4009 y(t)p Fi(\000)5104 3951 y Fg(j)p Fj(z)s Fg(j)p 5104 3983 137 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b(in)f(Section)g(4,)h(w)-5 b(e)55 b(de\034ne)2873 7901 y Fs(M)18 b Fr(\()p Fs(t)p Fr(\))45 b(=)114 b(sup)3462 8041 y Fp(0)p Fi(\024)p Fo(s)p Fi(\024)p Fo(t)3876 7901 y Fq(j)p Fs(Y)38 b Fr(\()p Fs(s)p Fr(\))p Fq(j)726 8376 y Ft(and)2886 8575 y Fs(N)18 b Fr(\()p Fs(t)p Fr(\))46 b(=)115 b(sup)3448 8715 y Fp(0)p Fi(\024)p Fo(s)p Fi(\024)p Fo(t)3863 8575 y Fq(j)3952 8533 y Fr(_)3909 8575 y Fs(Y)37 b Fr(\()p Fs(s)p Fr(\))p Fq(j)726 8994 y Ft(Using)56 b Fr(\(5)p Fs(:)p Fr(9\))f Ft(and)h Fr(\(5)p Fs(:)p Fr(10\))p Ft(,)e(w)-5 b(e)56 b(ha)-5 b(v)g(e)55 b(for)h(all)f Fr(0)46 b Fq(\024)h Fs(s)e Fq(\024)i Fs(t)1284 9444 y Fq(j)1396 9402 y Fr(_)1330 9444 y Fs(X)12 b Fr(\()p Fs(s)p Fr(\))p Fq(j)165 b(\024)2215 9331 y Fs(K)2356 9356 y Fp(1)p 2215 9406 216 7 v 2249 9558 a Fs(c)2321 9510 y Fp(4)2450 9444 y Fs(e)2547 9306 y Fj(\015)6 b(R)2701 9331 y Ff(2)p 2547 9342 219 6 v 2597 9426 a Fj(c)2651 9402 y Ff(4)2794 9444 y Fq(k)2925 9402 y Fr(~)2877 9444 y Fs(W)22 b Fq(k)p Fs(M)c Fr(\()p Fs(t)p Fr(\))37 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(analyse)g(their)g(merits)h(and)f(limitations.)726 6709 y(W)-14 b(e)64 b(will)g(in)h(particular)f(argue)g(that,)i(unlik)-5 b(e)64 b(the)g(mo)5 b(del)64 b(w)-5 b(e)64 b(presen)-5 b(ted)65 b(here,)h(none)e(of)726 6908 y(the)79 b(mo)5 b(dels)79 b(of)g(the)g(heath)f(bath)h(and)h(of)e(its)h(coupling)h(to)e (the)h(particle)f(considered)726 7107 y(up)70 b(to)f(no)-5 b(w)70 b(can)f(correctly)f(describ)5 b(e)70 b(the)f(b)5 b(eha)-5 b(viour)69 b(of)g(a)h(particle)f(mo)-5 b(ving)70 b(through)726 7306 y(a)h(homogeneous)h(medium)f(in)g(the)f(presence)h (of)f(eac)-5 b(h)71 b(of)f(the)g(three)g(most)h(commonly)726 7506 y(studied)56 b(p)5 b(oten)-5 b(tials:)74 b Fs(V)83 b Fr(=)47 b(0)p Fs(;)28 b(V)82 b Fr(=)46 b Fq(\000)p Fs(F)60 b Fq(\001)37 b Fs(q)62 b Ft(or)55 b Fs(V)92 b Ft(con\034ning.)976 7705 y(With)62 b(an)h(ey)-5 b(e)62 b(to)-5 b(w)g(ards)63 b(p)5 b(ossible)64 b(generalizations)f(of)f(our)h (w)-5 b(ork,)64 b(w)-5 b(e)63 b(will)g(describ)5 b(e)726 7904 y(a)75 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Fr(2)p Fs(m)2858 9315 y Fo(\013)3009 9176 y Fr(+)3195 9063 y(1)p 3195 9137 84 7 v 3195 9290 a(2)3298 9176 y Fs(m)3444 9201 y Fo(\013)3539 9176 y Fs(!)3648 9107 y Fp(2)3642 9217 y Fo(\013)3737 9176 y Fs(q)3817 9107 y Fp(2)3811 9217 y Fo(\013)3905 8942 y Fl(\025)4030 9176 y Fr(+)4264 8968 y Fo(N)4203 9018 y Fl(X)4196 9370 y Fo(\013)p Fp(=1)4478 9176 y Fs(\033)4573 9201 y Fo(\013)4668 9176 y Fr(\()p Fs(q)6 b Fr(\))p Fs(q)4952 9201 y Fo(\013)5082 9176 y Fr(+)5317 8968 y Fo(N)5256 9018 y Fl(X)5249 9370 y Fo(\013)p Fp(=1)5531 9176 y Fs(W)5688 9201 y Fo(\013)5782 9176 y Fr(\()p Fs(q)g Fr(\))122 b Ft(\(6.1\))726 9639 y(with)2946 10025 y Fs(W)3103 10050 y Fo(\013)3197 10025 y Fr(\()p Fs(q)6 b Fr(\))46 b(=)3672 9913 y Fs(\033)3767 9938 y Fo(\013)3862 9913 y Fr(\()p Fs(q)6 b Fr(\))4072 9853 y Fp(2)p 3648 9987 522 7 v 3648 10139 a Fr(2)p Fs(m)3877 10164 y Fo(\013)3971 10139 y Fs(!)4080 10091 y Fp(2)4074 10180 y Fo(\013)4189 10025 y Fs(:)1879 b Ft(\(6.2\))726 10433 y(The)58 b(last)g(term,)g(whic)-5 b(h)58 b(is)h(just)e(an)h (additional)g(p)5 b(oten)-5 b(tial)58 b(for)f(the)g(particle,)h(is)h (referred)726 10633 y(to)c(as)h(the)f(\020coun)-5 b(terterm\021.)74 b(W)-14 b(riting)2171 11107 y Fs(V)2268 11132 y Fo(ef)13 b(f)2541 11107 y Fr(=)46 b Fs(V)74 b Fr(+)37 b Fs(W)-5 b(;)360 b(W)23 b Fr(\()p Fs(q)6 b Fr(\))45 b(=)4290 10900 y Fo(N)4229 10950 y Fl(X)4221 11301 y Fo(\013)p Fp(=1)4504 11107 y Fs(W)4661 11132 y Fo(\013)4755 11107 y Fr(\()p Fs(q)6 b Fr(\))p Fs(;)3508 11733 y Ft(28)p eop %%Page: 29 29 29 28 bop 726 1471 a Ft(one)53 b(can)f(think)g(of)g Fs(W)75 b Ft(as)53 b(a)f(p)5 b(oten)-5 b(tial)52 b(due)g(to)g(the)f(presence)i (of)f(the)f(bath,)i(but)g(whic)-5 b(h)52 b(is)726 1671 y(indep)5 b(enden)-5 b(t)61 b(of)g(the)f(state)g(of)g(the)g(latter.)89 b(W)-14 b(e'll)61 b(explain)f(its)h(role)f(b)5 b(elo)-5 b(w.)90 b(Of)61 b(course,)726 1870 y(for)k(an)-5 b(y)64 b(of)h(these)f(mo)5 b(dels)65 b(to)f(describ)5 b(e)65 b(dissipation,)j(the)c(limit)h Fs(N)80 b Fq(!)61 b Fr(+)p Fq(1)k 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y(friction.)976 3464 y(Let)54 b Fr(\()p Fs(A;)28 b(\026)p Fr(\))54 b Ft(b)5 b(e)55 b(a)g(measure)h(space)g(and)f(consider)h(the)f(Hamiltonian)1248 3936 y Fs(H)13 b Fr(\()p Fs(q)6 b(;)28 b(p;)g(\036;)g(\031)6 b Fr(\))164 b(=)2630 3823 y Fs(p)2714 3763 y Fp(2)p 2594 3897 229 7 v 2594 4049 a Fr(2)p Fs(m)2880 3936 y Fr(+)37 b Fs(V)g Fr(\()p Fs(q)6 b Fr(\))36 b(+)3592 3710 y Fl(Z)3684 4087 y Fo(A)3820 3936 y Fs(d\026)p Fr(\()p Fs(\013)q Fr(\))4271 3701 y Fl(\024)4378 3823 y Fq(j)p Fs(\031)4519 3848 y Fo(\013)4613 3823 y Fq(j)4659 3763 y Fp(2)p 4378 3897 357 7 v 4424 4049 a Fr(2)p Fs(\032)4593 4074 y Fo(\013)4791 3936 y Fr(+)4977 3823 y(1)p 4977 3897 84 7 v 4977 4049 a(2)5079 3936 y Fs(\032)5165 3961 y Fo(\013)5260 3936 y Fs(!)5369 3867 y Fp(2)5363 3977 y Fo(\013)5458 3936 y Fq(j)p Fs(\036)5603 3961 y Fo(\013)5698 3936 y Fq(j)5744 3867 y Fp(2)5818 3701 y Fl(\025)2906 4386 y Fr(+)3063 4160 y Fl(Z)3155 4537 y Fo(A)3291 4386 y Fs(d\026)p Fr(\()p Fs(\013)q Fr(\))p Fs(\033)3809 4411 y Fo(\013)3903 4386 y Fr(\()p Fs(q)6 b Fr(\))p Fs(\036)4212 4411 y Fo(\013)4343 4386 y Fr(+)4509 4160 y Fl(Z)4601 4537 y Fo(A)4737 4386 y Fs(d\026)p Fr(\()p Fs(\013)q Fr(\))p Fs(W)5317 4411 y Fo(\013)5411 4386 y Fr(\()p Fs(q)g Fr(\))493 b Ft(\(6.3\))726 4834 y(with)56 b(the)f(corresp)5 b(onding)56 b(second)f(order)h (equations)f(of)g(motion)2335 5199 y Fs(\032)2421 5224 y Fo(\013)2537 5155 y Fr(\177)2515 5199 y Fs(\036)2614 5224 y Fo(\013)2709 5199 y Fr(\()p Fs(t)p Fr(\))36 b(+)h Fs(\032)3187 5224 y Fo(\013)3281 5199 y Fs(!)3390 5131 y Fp(2)3384 5240 y Fo(\013)3479 5199 y Fs(\036)3578 5224 y Fo(\013)3673 5199 y Fr(\()p Fs(t)p Fr(\))45 b(=)h Fq(\000)p Fs(\033)4307 5224 y Fo(\013)4402 5199 y Fr(\()p Fs(q)6 b Fr(\()p Fs(t)p Fr(\)\))p Fs(;)726 5565 y Ft(and)2509 5764 y Fs(m)12 b Fr(\177)-95 b Fs(q)6 b Fr(\()p Fs(t)p Fr(\))45 b(=)h Fq(\000r)p Fs(V)3509 5789 y Fo(ef)13 b(f)3737 5764 y Fr(\()p Fs(q)6 b Fr(\()p Fs(t)p Fr(\)\))35 b(+)i Fs(f)18 b Fr(\()p Fs(t)p Fr(\))p Fs(;)726 6063 y Ft(where)2626 6305 y Fs(f)g Fr(\()p Fs(t)p Fr(\))45 b(=)h Fq(\000)3292 6079 y Fl(Z)3486 6305 y Fs(\033)3587 6237 y Fi(0)3581 6346 y Fo(\013)3675 6305 y Fr(\()p Fs(q)6 b Fr(\))p Fs(\036)3984 6330 y Fo(\013)4133 6305 y Fs(d\026)p Fr(\()p Fs(\013)q Fr(\))726 6679 y Ft(is)38 b(the)f(force)f(exerted)g(on)h(the)g (particle)g(b)-5 b(y)37 b(the)g(oscillator)g(bath.)68 b(Note)36 b(that)g(the)h(equations)726 6878 y(for)51 b(the)f(\020\034eld\021)65 b(v)-9 b(ariables)51 b Fs(\036)2519 6903 y Fo(\013)2664 6878 y Ft(are)f(linear,)i(re\035ecting)d(the)h (fact)g(that)g(the)g(Hamiltonian)g(is)726 7077 y(quadratic)e(in)g Fs(\036;)28 b(\031)6 b Ft(,)49 b(its)e(only)h(non-linearit)-5 b(y)49 b(b)5 b(eing)47 b(in)h Fs(q)6 b Ft(.)72 b(This)48 b(is)h(of)e(course)h(what)f(mak)-5 b(es)726 7277 y(this)56 b(t)-5 b(yp)5 b(e)55 b(of)g(mo)5 b(del)55 b(tractable.)976 7476 y(Needless)60 b(to)h(sa)-5 b(y)-14 b(,)62 b(taking)e(for)h Fs(A)f Ft(the)g(set)h(of)f Fs(N)79 b Ft(elemen)-5 b(ts)61 b(and)g(for)g Fs(\026)f 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8464 y Fp(2)4204 8439 y Fr(\()p Fs(k)5 b Fr(\))726 8804 y Ft(it)47 b(is)h(easy)f(to)g(see)g(that)f(the)h(mo)5 b(del)47 b(considered)h(in)f(this)h(pap)5 b(er)47 b(is)g(also)h(of)f (the)g(ab)5 b(o)-5 b(v)g(e)47 b(t)-5 b(yp)5 b(e)726 9004 y(\(with)55 b(the)g(coun)-5 b(terterm)55 b(iden)-5 b(tically)55 b(zero\).)976 9203 y(Not)41 b(all)h(c)-5 b(hoices)42 b(of)g(the)f(parameters)h(used)h(in)f(\(6.3\))f(to)h(describ)5 b(e)42 b(the)f(heath)h(bath)g(and)726 9402 y(its)j(coupling)g(to)f(the) g(particle)g(lead)g(to)g(ph)-5 b(ysically)45 b(reasonable)g(mo)5 b(dels.)71 b(W)-14 b(e)44 b(shall)h(deriv)-5 b(e)726 9601 y(a)55 b(certain)f(n)-5 b(um)g(b)5 b(er)55 b(of)f(conditions)h (for)f(that)g(purp)5 b(ose.)74 b(A)54 b(\034rst)h(ob)-5 b(vious)55 b(requiremen)-5 b(t)54 b(is)726 9801 y(that)65 b(one)h(and)g(the)f(same)h(c)-5 b(hoice)66 b(of)f(these)h(parameters)f (should)i(describ)5 b(e)66 b(the)f(system)726 10000 y(correctly)54 b(for)h(di\033eren)-5 b(t)56 b(c)-5 b(hoices)56 b(of)f Fs(V)36 b Ft(.)74 b(In)56 b(particular,)f(the)g(mo)5 b(del)55 b(should)i(mak)-5 b(e)55 b(sense)726 10199 y(when)74 b Fs(V)112 b Fr(=)77 b(0)p Ft(.)127 b(This)74 b(immediately)f(leads)h (to)f(the)f(follo)-5 b(wing)74 b(observ)-9 b(ations,)78 b(that)73 b(do)726 10398 y(not)66 b(seem)g(to)f(ha)-5 b(v)g(e)66 b(b)5 b(een)66 b(made)g(b)5 b(efore.)104 b(When)66 b(the)f(external)g(p)5 b(oten)-5 b(tial)65 b Fs(V)102 b Ft(v)-9 b(anishes)726 10598 y(iden)k(tically)-14 b(,)60 b(one)f(exp)5 b(ects)58 b(the)g(particle)g(to)g(b)5 b(e)59 b(able)g(to)f(\020sit)h(still\021)73 b(at)59 b(an)-5 b(y)59 b(p)5 b(oin)-5 b(t)59 b(in)g Fn(R)6322 10537 y Fo(d)6409 10598 y Ft(,)726 10797 y(in)65 b(the)e(surrounding)j(medium,) h(itself)c(at)h(rest,)i(without)d(exp)5 b(eriencing)63 b(an)-5 b(y)64 b(net)g(force.)726 10996 y(In)59 b(other)g(w)-5 b(ords,)61 b(the)d(Hamiltonian)h(m)-5 b(ust)60 b(in)f(that)f(case)h (allo)-5 b(w)59 b(for)g(a)f(stationary)h(state)3508 11733 y(29)p eop %%Page: 30 30 30 29 bop 726 1471 a Ft(with)64 b(the)g(particle)g(at)g(rest)g(an)-5 b(ywhere)64 b(in)g(space.)101 b(This)65 b(is)g(only)f(p)5 b(ossible,)67 b(as)e(is)g(easily)726 1671 y(c)-5 b(hec)g(k)g(ed,)56 b(if)f(one)h(c)-5 b(ho)5 b(oses)56 b(the)f(coun)-5 b(terterm)55 b Fs(W)78 b Ft(so)56 b(that:)1776 2133 y Fq(r)p Fs(W)23 b Fr(\()p Fs(q)6 b Fr(\))45 b(=)i Fq(r)p Fs(U)18 b Fr(\()p Fs(q)6 b Fr(\))p Fs(;)83 b Ft(where)55 b Fs(U)18 b Fr(\()p Fs(q)6 b Fr(\))46 b(=)4175 1907 y Fl(Z)4389 2021 y Fs(\033)4484 2046 y Fo(\013)4578 2021 y Fr(\()p Fs(q)6 b Fr(\))4788 1960 y Fp(2)p 4389 2095 474 7 v 4395 2247 a Fr(2)p Fs(\032)4564 2272 y Fo(\013)4658 2247 y Fs(!)4767 2199 y Fp(2)4761 2288 y Fo(\013)4937 2133 y Fs(d\026)p Fr(\()p Fs(\013)q Fr(\))p Fs(:)708 b Ft(\(6.4\))726 2585 y(This)40 b(\034rst)e (requiremen)-5 b(t)39 b(explains)f(therefore)g(v)-5 b(ery)38 b(simply)h(the)f(form)g(of)g(the)g(coun)-5 b(terterm)726 2784 y(in)54 b(\(6.2\))f(\(whic)-5 b(h)53 b(tends)h(to)f(b)5 b(e)53 b(motiv)-9 b(ated)52 b(in)i(a)f(di\033eren)-5 b(t)53 b(manner)h([CL])g([CT)o(])g([FLO1]\).)726 2983 y(One)c(notices)f(in)h(addition)g(immediately)f(that)g(the)g(coun)-5 b(terterm)49 b(in)h(\(6.2\))e(is)i(simply)g(the)726 3183 y(p)5 b(oten)-5 b(tial)61 b(energy)f(of)g(the)g(en)-5 b(vironmen)g(t)62 b(when)e(the)h(system)f(is)i(in)f(its)f(stationary)h (state)726 3382 y(with)56 b(the)f(particle)f(at)h(p)5 b(osition)56 b Fs(q)6 b Ft(!)74 b(W)-14 b(e)55 b(therefore)f(naturally) h(require)3162 3747 y Fs(U)18 b Fr(\()p Fs(q)6 b Fr(\))46 b Fs(<)g Fr(+)p Fq(1)2095 b Ft(\(6.5\))726 4113 y(and)56 b(note)f(that)g(with)g Fs(W)23 b Fr(\()p Fs(q)6 b Fr(\))45 b(=)h Fs(U)18 b Fr(\()p Fs(q)6 b Fr(\))p Ft(,)56 b(one)f(has)1102 4584 y Fs(H)13 b Fr(\()p Fs(q)6 b(;)28 b(p;)g(\036;)g(\031)6 b Fr(\))164 b(=)2483 4472 y Fs(p)2567 4412 y Fp(2)p 2448 4546 229 7 v 2448 4698 a Fr(2)p Fs(m)2734 4584 y Fr(+)37 b Fs(V)f Fr(\()p Fs(q)6 b Fr(\))37 b(+)2926 4829 y Fl(Z)3018 5206 y Fo(A)3154 5055 y Fs(d\026)p Fr(\()p Fs(\013)q Fr(\))3605 4771 y Fl(")3721 4943 y Fq(j)p Fs(\031)3862 4968 y Fo(\013)3956 4943 y Fq(j)4002 4883 y Fp(2)p 3721 5017 357 7 v 3767 5169 a Fr(2)p Fs(\032)3936 5194 y Fo(\013)4134 5055 y Fr(+)4320 4943 y(1)p 4320 5017 84 7 v 4320 5169 a(2)4423 5055 y Fs(\032)4509 5080 y Fo(\013)4603 5055 y Fs(!)4712 4987 y Fp(2)4706 5096 y Fo(\013)4829 4821 y Fl(\022)4951 5055 y Fs(\036)5050 5080 y Fo(\013)5182 5055 y Fr(+)5368 4943 y Fs(\033)5463 4968 y Fo(\013)5557 4943 y Fr(\()p Fs(q)6 b Fr(\))p 5368 5017 399 7 v 5378 5169 a Fs(\032)5464 5194 y Fo(\013)5558 5169 y Fs(!)5667 5121 y Fp(2)5661 5210 y Fo(\013)5786 4821 y Fl(\023)5909 4855 y Fp(2)5983 4771 y Fl(#)6114 5055 y Ft(\(6.6\))726 5560 y(so)56 b(that)f(the)g(resulting)h(Hamiltonian)f(is)h(no)-5 b(w)56 b(clearly)f(b)5 b(ounded)56 b(b)5 b(elo)-5 b(w.)976 5759 y(As)42 b(a)h(result,)i(it)e(is)g(clear)f(that)g(in)h (translationally)g(in)-5 b(v)c(arian)k(t)43 b(systems)g(\(homogeneous) 726 5958 y(medium\),)67 b(the)d(coun)-5 b(terterm)64 b(is)g(constan)-5 b(t)65 b(and)f(can)g(therefore)f(b)5 b(e)64 b(dropp)5 b(ed)64 b(from)g(the)726 6157 y(Hamiltonian.)76 b(Indeed,)57 b(in)f(that)f(case,)h(this)g(p)5 b(oten)-5 b(tial)56 b(energy)f(has)i(to)e(b)5 b(e)56 b(indep)5 b(enden)-5 b(t)726 6357 y(of)51 b Fs(q)58 b Ft(so)51 b(that)g(one)g(can)h(tak)-5 b(e)50 b Fs(W)74 b Ft(iden)-5 b(tically)51 b(equal)g(to)g(zero.)72 b(In)52 b(conclusion,)h(the)e (Hamil-)726 6556 y(tonian)46 b(\(6.3\))e(can)h(describ)5 b(e)45 b(the)g(coupling)h(of)e(a)h(particle)g(to)g(a)g(homogeneous)h (dissipativ)-5 b(e)726 6755 y(medium)57 b(only)e(if)2064 7120 y Fs(W)23 b Fr(\()p Fs(q)6 b Fr(\))45 b(=)i Fs(U)18 b Fr(\()p Fs(q)6 b Fr(\))p Fs(;)28 b(U)18 b Fr(\()p Fs(q)6 b Fr(\))45 b(=)h Fs(U)3764 7145 y Fi(\003)3887 7120 y Fs(<)g Fr(+)p Fq(1)p Fs(;)84 b Fq(8)p Fs(q)52 b Fq(2)46 b Fn(R)4985 7052 y Fo(d)5071 7120 y Fs(:)997 b Ft(\(6.7\))976 7486 y(The)76 b(only)h(mo)5 b(del)76 b(of)h(the)f(class)i (\(6.3\)-\(6.4\))d(that)h(seems)i(to)e(ha)-5 b(v)g(e)77 b(b)5 b(een)76 b(activ)-5 b(ely)726 7685 y(studied)72 b(in)g(the)e(literature)g(is)i(the)f(one)g(with)g(linear)g(coupling,)76 b Fh(i.e.)121 b Fs(\033)5431 7710 y Fo(\013)5525 7685 y Fr(\()p Fs(q)6 b Fr(\))72 b(=)g Fq(\000)p Fs(\025)6234 7710 y Fo(\013)6329 7685 y Fs(q)6 b Ft(.)726 7884 y(\(See)57 b([CL])g([CT])g([FLO1])g([FLO2])h([JP1])f(and)g(references)g (therein.\))78 b(The)57 b(Hamiltonian)726 8084 y(can)f(then)f(b)5 b(e)55 b(rewritten)1114 8514 y Fs(H)1252 8539 y Fo(C)8 b(L)1500 8514 y Fr(=)1731 8401 y Fs(p)1815 8341 y Fp(2)p 1696 8475 229 7 v 1696 8628 a Fr(2)p Fs(m)1981 8514 y Fr(+)37 b Fs(V)g Fr(\()p Fs(q)6 b Fr(\))36 b(+)2693 8288 y Fl(Z)2786 8665 y Fo(A)2921 8514 y Fs(d\026)p Fr(\()p Fs(\013)q Fr(\))3372 8280 y Fl(\024)3516 8401 y Fs(\031)3617 8341 y Fp(2)3611 8442 y Fo(\013)p 3479 8475 264 7 v 3479 8628 a Fr(2)p Fs(\032)3648 8653 y Fo(\013)3799 8514 y Fr(+)3985 8401 y(1)p 3985 8475 84 7 v 3985 8628 a(2)4088 8514 y Fs(\032)4174 8539 y Fo(\013)4269 8514 y Fs(!)4378 8445 y Fp(2)4372 8555 y Fo(\013)4467 8514 y Fr(\()p Fs(\036)4631 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9076 y([KKS1])149 b(K)-5 b(omec)g(h)46 b(A.,)h(Kunze)e(M,)h(Sp)5 b(ohn)45 b(H.,)i(Long-time)f(asymptotics)f (for)g(a)g(classical)1400 9275 y(particle)c(in)-5 b(teracting)41 b(with)g(a)h(scalar)g(w)-5 b(a)g(v)g(e)42 b(\034eld,)i(Comm.)f(P)-5 b(artial)41 b(Di\033eren)-5 b(tial)1400 9474 y(Equations)55 b Fa(22)p Ft(,)i(307-335)e(\(1997\).)726 9806 y([KKS2])149 b(K)-5 b(omec)g(h)55 b(A.,)f(Kunze)g(M,)h(Sp)5 b(ohn)54 b(H.,)g(E\033ectiv)-5 b(e)53 b(dynamics)i(for)f(a)g(mec)-5 b(hanical)1400 10005 y(particle)46 b(coupled)h(to)g(a)f(w)-5 b(a)g(v)g(e)48 b(\034eld,)h(Comm.)e(Math.)g(Ph)-5 b(ys.)48 b Fa(203)p Ft(,)i(1-19)d(\(1999\).)726 10338 y([KS])361 b(K)-5 b(omec)g(h)56 b(A.,)f(Sp)5 b(ohn)56 b(H.,)g(Soliton-lik)-5 b(e)56 b(asymptotics)f(for)h(a)f(classical)h(particle)1400 10537 y(in)-5 b(teracting)52 b(with)g(a)g(scalar)g(w)-5 b(a)g(v)g(e)53 b(\034eld,)g(Nonlinear)f(Anal.)g Fa(33)p Ft(,)i(13-24)f(\(1998\).)726 10869 y([LM])326 b(Lions)68 b(J.)h(L.,)i(Magenes)d(E.,)j(Probl\350mes)e(aux)f(limites)g(non)h (homog\350nes)f(\(v)-5 b(ol)1400 11068 y(1\),)55 b(Duno)5 b(d)55 b(\(1968\).)3508 11733 y(34)p eop %%Page: 35 35 35 34 bop 726 1471 a Ft([MS])338 b(M\366hrig)56 b(K.,)f(Smilansky)h (U.,)f(Nucl.)g(Ph)-5 b(ys.)57 b(A)p Fa(338)p Ft(,)f(227-268)g (\(1980\).)726 1803 y([R])460 b(Rudin)56 b(W.,)f(F)-14 b(unctional)56 b(analysis,)h(McGra)-5 b(w)56 b(Hill,)f(New-Y)-14 b(ork)54 b(\(1973\).)726 2136 y([ZQ])352 b(Zuily)61 b(C.,)h (Que\033elec)e(H.,)h(El\351men)-5 b(ts)61 b(d'analyse)g(p)5 b(our)61 b(l'agr\351gation,)h(Masson)1400 2335 y(\(1995\).)3508 11733 y(35)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0107170910679--