Content-Type: multipart/mixed; boundary="-------------9901171941932" This is a multi-part message in MIME format. ---------------9901171941932 Content-Type: text/plain; name="99-17.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-17.comments" ympark@bubble.yonsei.ac.kr yoohj@gauss.kyungpook.ac.kr ---------------9901171941932 Content-Type: text/plain; name="99-17.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="99-17.keywords" Non-equilibrium dynamics, superstable interactions, Hamiltonian equations, tempered solution ---------------9901171941932 Content-Type: application/postscript; name="dyn4.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="dyn4.ps" %!PS-Adobe-2.0 %%Creator: dvips 5.58 Copyright 1986, 1994 Radical Eye Software %%Title: dyn4.dvi %%CreationDate: Fri Mar 20 09:40:24 1992 %%Pages: 31 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSCommandLine: C:\EMTEX\BIN\dvips32.EXE dyn4 -pj:C:\TMP\dv4.mfj %DVIPSParameters: dpi=600, compressed, comments removed %DVIPSSource: TeX output 1992.03.20:0939 %%BeginProcSet: texc.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N /IE 0 N /ctr 0 N /df-tail{ /nn 8 dict N nn begin /FontType 3 N /FontMatrix fntrx N /FontBBox FBB N string /base X array /BitMaps X /BuildChar{CharBuilder}N 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b Ft(\(1.3\))-246 4513 y(where)37 b Fk(A)32 b Fg(\025)h Ft(0)i(is)g(a)g(constan)m(t)h(app)s(earing)f(in)f(the)i (sup)s(erstable)g(condition)e(\(see)i(\(2.1\)\))f(and)g(mak)m(es)-246 4712 y Fk(H)8 b Ft(\()p Fk(!)t(;)17 b(\026;)g(\033)t Ft(\))31 b(nonnegativ)m(e)i(n)m(um)m(b)s(ers.)44 b(Let)33 b(us)g(de\014ne)p 918 4929 89 4 v 918 5009 a Fk(H)7 b Ft(\()p Fk(!)t Ft(\))27 b(:=)h(sup)1357 5089 y Fi(\026)1530 5009 y Ft(sup)1468 5095 y Fi(\033)r Fh(\025)p Fi(g)r Fs(\()p Fh(j)p Fi(\026)p Fh(j)p Fs(\))1755 5009 y Fk(\033)1814 4968 y Fh(\000)p Fi(d)1909 5009 y Fk(H)8 b Ft(\()p Fk(!)t(;)17 b(\026;)g(\033)t Ft(\))p Fk(:)961 b Ft(\(1.4\))p -246 5236 V -246 5316 a Fk(H)8 b Ft(\()p Fk(!)t Ft(\))29 b(is)g(called)f (the)j(logarithmic)26 b(\015uctuation)j(of)g(energy)i(of)e Fk(!)k Ft([F)-8 b(r1].)42 b(The)31 b(allo)m(w)m(ed)e(con\014guration) -246 5515 y(space)34 b(is)e(de\014ned)i(b)m(y)p 642 5733 71 4 v 642 5813 a(\012)28 b(:=)g Fg(f)p Fk(!)j Ft(:)61 b Fk(!)35 b Ft(is)d(lo)s(cally)e(\014nite)j(and)p 2110 5733 89 4 v 99 w Fk(H)7 b Ft(\()p Fk(!)t Ft(\))27 b Fk(<)h Fg(1g)p Fk(:)686 b Ft(\(1.5\))p eop %%Page: 4 4 4 3 bop -246 -9 a Ft(4)3285 b Fm(Bahn)33 b Fl(et)i(al)-246 279 y Ft(It)f(turns)g(out)g(that)p 504 199 71 4 v 33 w(\012)g(carries)f(a)h(large)e(class)i(of)f(probabilit)m(y)f(measures)i (including)e(Gibbs)h(random)-246 478 y(\014elds)g(with)f(in)m (teraction)g Fk(U)43 b Ft([DF,MPPu,CPY].)-99 678 y(W)-8 b(e)23 b(will)e(sho)m(w)j(that)f(in)g(dimensions)f(one)h(and)h(t)m(w)m (o,)h(under)f(prop)s(er)g(conditions)e(on)h(the)g(p)s(oten)m(tial,)-246 877 y(there)42 b(exists)f(a)g(temp)s(ered)g(solution)e(\(see)j(Section) f(2)g(for)f(the)h(de\014nition\))f Fk(!)2713 892 y Fi(t)2783 877 y Ft(to)g(\(1.1\))h(with)f(an)m(y)-246 1076 y(initial)d (con\014guration)j Fk(!)45 b Fg(2)p 857 996 V 42 w Ft(\012.)68 b(This)41 b(solution)e(can)i(b)s(e)g(obtained)f(as)h(the)g(limit)c(of)j (solutions)g(to)-246 1275 y(\014nite)g(subsystems.)68 b(F)-8 b(urthermore,)41 b(the)f(temp)s(ered)g(solution)f(is)g(unique)i (when)g(the)f(p)s(oten)m(tial)e Fk(U)-246 1475 y Ft(is)31 b(either)h(ev)m(erywhere)i(t)m(w)m(o-times)d(con)m(tin)m(uously)h (di\013eren)m(tiable)e(or)h(sub)s(exp)s(onen)m(tially)g(decreasing)-246 1674 y(\(see)36 b(Theorem)f(2.2\))f(when)i(it)e(is)g(singular)f(at)h (the)h(origin.)48 b(The)36 b(main)c(idea)j(used)g(in)f(this)h(pap)s(er) f(is)-246 1873 y(to)f(decomp)s(ose)g(the)g(p)s(oten)m(tial)d Fk(U)44 b Ft(in)m(to)31 b(t)m(w)m(o)j(parts)1099 2152 y Fk(U)10 b Ft(\()p Fk(x)p Ft(\))28 b(=)g Fk(U)1514 2111 y Fs(\(1\))1609 2152 y Ft(\()p Fk(x)p Ft(\))22 b(+)g Fk(U)1936 2111 y Fs(\(2\))2031 2152 y Ft(\()p Fk(x)p Ft(\))p Fk(;)-246 2431 y Ft(where)42 b Fk(U)120 2395 y Fs(\(1\))255 2431 y Ft(is)e(of)g(\014nite)g(range)h(and)f Fk(U)1284 2395 y Fs(\(2\))1420 2431 y Ft(is)g(a)g(smo)s(oth)f(p)s(oten) m(tial)g(whic)m(h)i(decreases)i(p)s(olynomi-)-246 2631 y(ally)-8 b(.)59 b(See)39 b(\(2.2\))e(and)h(\(2.6\).)60 b(F)-8 b(or)37 b Fk(U)1144 2595 y Fs(\(1\))1239 2631 y Ft(,)j(w)m(e)f(use)g(the)g(metho)s(d)e(dev)m(elop)s(ed)i(in)f([F)-8 b(r1])37 b(with)h(suitable)-246 2830 y(mo)s(di\014cations)31 b(and)h(for)g Fk(U)768 2794 y Fs(\(2\))863 2830 y Ft(,)h(w)m(e)h (utilize)c(the)j(deca)m(y)h(prop)s(erties)f(of)f Fk(U)2444 2794 y Fs(\(2\))2571 2830 y Ft(stated)i(in)d(\(2.6\).)-99 3029 y(Emplo)m(ying)h(the)i(metho)s(d)f(dev)m(elop)s(ed)i(in)e(section) h(6)g(and)f(using)h(an)g(appropriate)e(mo)s(di\014cation)-246 3229 y(of)e(the)h(metho)s(d)e(of)h([F)-8 b(r1,)30 b(Section)g(5],)g(it) g(ma)m(y)g(b)s(e)g(p)s(ossible)f(to)h(sho)m(w)h(the)g(\(an)m(ti\))e (self-adjoin)m(tness)h(of)-246 3428 y(the)36 b(corresp)s(onding)f (Liouville)e(op)s(erator)h([MPPu,)k(F)-8 b(r1].)51 b(Ho)m(w)m(ev)m(er) 38 b(w)m(e)e(do)f(not)g(pursue)i(the)f(pro)s(of)-246 3627 y(here)e(and)e(lea)m(v)m(e)i(it)d(to)h(further)h(study)-8 b(.)-99 3826 y(W)g(e)36 b(organize)f(this)h(pap)s(er)g(as)h(follo)m (ws.)53 b(In)36 b(section)g(2,)h(w)m(e)g(giv)m(e)f(necessary)j (notations,)d(imp)s(ose)-246 4026 y(conditions)e(to)g(the)h(p)s(oten)m (tial,)f(and)h(giv)m(e)f(main)f(results)i(of)f(this)h(pap)s(er.)49 b(In)35 b(section)g(3,)g(w)m(e)h(discuss)-246 4225 y(the)g(cuto\013)f (of)g(total)e(energy)-8 b(.)53 b(In)35 b(section)g(4,)h(w)m(e)g(presen) m(t)h(the)f(a)e(priori)g(b)s(ound)h(whic)m(h)h(is)e(a)h(core)h(in)-246 4424 y(pro)m(ving)30 b(the)g(results)g(of)g(this)f(pap)s(er.)43 b(W)-8 b(e)30 b(pro)m(v)m(e)h(the)g(existence)g(and)f(the)g(uniqueness) i(of)e(temp)s(ered)-246 4623 y(solution)h(in)h(section)h(5)f(and)h (section)g(6)f(resp)s(ectiv)m(ely)-8 b(.)-246 4953 y Fu(2)161 b(Notations)54 b(and)g(Main)g(Results)-246 5251 y Ft(In)33 b(this)g(section)g(w)m(e)h(giv)m(e)e(conditions)g(to)h(the)g (in)m(teraction)f(and)h(state)g(our)f(main)g(results.)44 b(As)34 b(men-)-246 5450 y(tioned)h(in)f(In)m(tro)s(duction,)i(w)m(e)g (will)d(consider)i(a)g(symmetric)f(pair)g(p)s(oten)m(tial)f Fk(U)43 b Ft(:)48 b Fj(R)2911 5414 y Fi(d)2990 5450 y Fg(!)31 b Ft(\()p Fg(\0001)p Fk(;)17 b Fg(1)p Ft(],)-246 5650 y Fk(d)36 b Ft(=)f(1)p Fk(;)17 b Ft(2,)38 b(allo)m(wing)d(the)j(p) s(ossibilit)m(y)d(of)i(b)s(eing)f(of)h(in\014nite)g(range)g(and)g(of)g (singularit)m(y)f(at)h(the)h(ori-)-246 5849 y(gin.)46 b(W)-8 b(e)35 b(need,)g(ho)m(w)m(ev)m(er,)h(some)e(regularit)m(y)e (conditions)h(on)h(the)g(p)s(oten)m(tial)e(in)h(order)h(to)f(sho)m(w)i (the)p eop %%Page: 5 5 5 4 bop -246 -9 a Fm(Dynamics)32 b(of)g(In\014nite)h(P)m(article)f (Systems)2123 b Ft(5)-246 279 y(existence)42 b(and)e(uniqueness)i(of)d (the)i(solutions)e(to)g(\(1.1\).)66 b(First,)41 b(w)m(e)g(need)g(sup)s (erstabilit)m(y)e(of)g(the)-246 478 y(in)m(teraction)25 b(to)g(con)m(trol)g(the)h(n)m(um)m(b)s(er)g(of)f(particles,)h(just)g (as)g(in)f(equilibrium)d(theory)27 b([Ru1,2].)41 b(There)-246 678 y(exist)33 b Fk(A)28 b Fg(\025)g Ft(0)k(and)h Fk(B)g(>)27 b Ft(0)33 b(suc)m(h)h(that)e(for)g(an)m(y)h(\014nite)g(collection)d Fk(q)2244 693 y Fs(1)2284 678 y Fk(;)17 b(q)2371 693 y Fs(2)2410 678 y Fk(;)g Fg(\001)g(\001)g(\001)e Fk(q)2630 693 y Fi(n)2710 678 y Ft(of)32 b(p)s(oin)m(ts)g(of)g Fj(R)3291 642 y Fi(d)3337 678 y Ft(,)881 868 y Fi(n)830 898 y Ff(X)838 1110 y Fi(k)r Fs(=1)991 898 y Ff(X)1000 1110 y Fi(j)t Fh(6)p Fs(=)p Fi(k)1151 992 y Fk(U)10 b Ft(\()p Fk(q)1308 1007 y Fi(k)1374 992 y Fg(\000)22 b Fk(q)1516 1007 y Fi(j)1553 992 y Ft(\))28 b Fg(\025)g(\000)p Fk(An)23 b Ft(+)f Fk(B)2152 898 y Ff(X)2149 1117 y Fi(i)p Fh(2)p Fd(Z)2270 1098 y Fe(d)2317 992 y Fk(n)2375 951 y Fs(2)2375 1017 y Fi(i)2414 992 y Fk(;)892 b Ft(\(2.1\))-246 1339 y(where)45 b Fj(Z)116 1303 y Fi(d)197 1339 y Ft(is)d(the)i Fk(d)p Ft(-dimensional)39 b(in)m(teger)k(lattice)f(and)h(for)g(eac)m(h) g(site)g Fk(i)j Fg(2)g Fj(Z)2795 1303 y Fi(d)2833 1339 y Ft(,)g Fk(n)2964 1354 y Fi(i)3035 1339 y Ft(denotes)e(the)-246 1538 y(n)m(um)m(b)s(er)g(of)f(particles)f(within)g(the)i(unit)f(b)s(o)m (x)g(with)g(cen)m(ter)i Fk(i)p Ft(.)76 b(This)43 b(condition)f(can)h(b) s(e)h(v)m(eri\014ed)-246 1737 y(under)31 b(natural)d(assumptions,)i (e.g.,)g(that)g Fk(U)40 b Ft(is)29 b(not)g(in)m(tegrable)f(near)i(zero) g(and)f(in)m(tegrable)g(outside)-246 1937 y(the)38 b(origin)c([Ru2].)57 b(W)-8 b(e)37 b(assume)h Fk(U)10 b Ft(\()p Fk(x)p Ft(\))38 b(is)e(con)m(tin)m(uously)h(di\013eren)m(tiable)f(outside)h(from)f(the) h(origin)-246 2136 y(and)c(supp)s(ose)h(that)e Fk(U)43 b Ft(is)33 b(decomp)s(osed)g(in)m(to)f(t)m(w)m(o)h(parts)g(of)f (singular)f(and)h(analytic)g(p)s(oten)m(tials;)1099 2450 y Fk(U)10 b Ft(\()p Fk(x)p Ft(\))28 b(=)g Fk(U)1514 2409 y Fs(\(1\))1609 2450 y Ft(\()p Fk(x)p Ft(\))22 b(+)g Fk(U)1936 2409 y Fs(\(2\))2031 2450 y Ft(\()p Fk(x)p Ft(\))p Fk(:)1144 b Ft(\(2.2\))-246 2765 y(F)-8 b(urthermore,)31 b(w)m(e)g(assume)g(that)f Fk(U)1106 2729 y Fs(\(1\))1228 2765 y Fg(\025)e Ft(0)i(for)g(all)e Fk(x)j Ft(and)f(there)h(exists)g(a) f(p)s(ositiv)m(e)g(n)m(um)m(b)s(er)g Fk(r)j Ft(suc)m(h)-246 2964 y(that)1123 3279 y Fk(U)1199 3238 y Fs(\(1\))1294 3279 y Ft(\()p Fk(x)p Ft(\))28 b(=)f(0)p Fk(;)115 b Ft(if)48 b Fg(j)p Fk(x)p Fg(j)28 b(\025)g Fk(r)m(:)1168 b Ft(\(2.3\))-246 3594 y(It)47 b(is)f(imp)s(ortan)m(t)f(that)h(singularit)m(y)f(of)h Fk(U)57 b Ft(can)47 b(not)g(b)s(e)f(to)s(o)g(strong.)86 b(Sometimes)45 b(the)i(v)m(elo)s(cit)m(y)-246 3793 y(dep)s(ends)35 b(on)d(the)h(strength)h(of)e(the)h(in)m(teraction)e(and)i(this)f(v)m (elo)s(cit)m(y)h(can)f(b)s(ecome)h(arbitrarily)d(large)-246 3992 y(in)38 b(the)h(extreme)g(case.)62 b(Th)m(us,)41 b(as)e(in)e([F)-8 b(r1],)40 b(w)m(e)f(assume)g(that)f(there)h(exist)g (p)s(ositiv)m(e)f(constan)m(ts)i Fk(a)-246 4192 y Ft(and)33 b Fk(b)g Ft(suc)m(h)h(that)936 4506 y Fg(j)p Fk(x)p Fg(j)17 b(j)g Ft(grad)32 b Fk(U)1407 4465 y Fs(\(1\))1502 4506 y Ft(\()p Fk(x)p Ft(\))p Fg(j)27 b(\024)h Fk(a)23 b Ft(+)f Fk(b)17 b(U)2099 4465 y Fs(\(1\))2194 4506 y Ft(\()p Fk(x)p Ft(\))p Fk(:)981 b Ft(\(2.4\))-246 4821 y(F)-8 b(rom)30 b(a)i(mathematical)c(p)s(oin)m(t)j(of)g(view,)h(\(2.4\))f (means)g(that)h(the)g(singularit)m(y)e(can)h(not)h(b)s(e)f(stronger) -246 5020 y(than)i(that)f(of)g Fg(j)p Fk(x)p Fg(j)415 4984 y Fh(\000)p Fi(b)504 5020 y Ft(.)44 b(Also)32 b(w)m(e)h(need)h(lo) s(cal)c(Lipsc)m(hitz)j(condition.)42 b(Let)33 b Fk(L)28 b(>)f Ft(1)33 b(and)f(supp)s(ose)i(that)268 5335 y Fg(j)17 b Ft(grad)32 b Fk(U)611 5294 y Fs(\(1\))706 5335 y Ft(\()p Fk(x)p Ft(\))23 b Fg(\000)f Ft(grad)33 b Fk(U)1258 5294 y Fs(\(1\))1353 5335 y Ft(\()p Fk(y)t Ft(\))p Fg(j)27 b(\024)h Fk(L)p Ft([)p Fk(L)23 b Ft(+)f Fk(U)1997 5294 y Fs(\(1\))2092 5335 y Ft(\()p Fk(x)p Ft(\))g(+)g Fk(U)2419 5294 y Fs(\(1\))2514 5335 y Ft(\()p Fk(y)t Ft(\)])2669 5294 y Fi(c)2736 5335 y Fg(j)p Fk(x)h Fg(\000)f Fk(y)t Fg(j)312 b Ft(\(2.5\))-246 5650 y(holds)33 b(for)g(all)e Fk(x;)17 b(y)31 b Fg(6)p Ft(=)e(0)j(with)h(some)g(constan)m(t)h Fk(c)28 b(>)h Ft(1.)44 b(\(2.5\))33 b(means)g(that)g(the)g(singularit)m (y)f(can)h(not)-246 5849 y(b)s(e)h(to)s(o)f(w)m(eak,)i(logarithmic)29 b(singularities)i(are)j(excluded.)47 b(Finally)31 b(w)m(e)j(assume)g (that)f(the)h(analytic)p eop %%Page: 6 6 6 5 bop -246 -9 a Ft(6)3285 b Fm(Bahn)33 b Fl(et)i(al)-246 279 y Ft(p)s(oten)m(tial)c Fk(U)242 243 y Fs(\(2\))337 279 y Ft(\()p Fk(x)p Ft(\))i(is)f(ev)m(erywhere)k(t)m(w)m(o)d(times)f (con)m(tin)m(uously)h(di\013eren)m(tiable)e(and)i(decreases)i(p)s(oly-) -246 478 y(nomially:)43 b(There)35 b(exist)f(a)g(decreasing)g(function) g Fk( )g Ft(:)c Fj(R)1868 442 y Fs(+)1963 478 y Fg(!)f Fj(R)2158 442 y Fs(+)2257 478 y Ft(and)34 b(constan)m(ts)h Fk(\015)g(>)30 b Ft(2)p Fk(d)22 b Ft(+)h(3)34 b(and)-246 678 y(0)28 b Fk(<)f(M)44 b Ft(suc)m(h)34 b(that)1024 877 y Fk( )t Ft(\()p Fk(u)p Ft(\))27 b Fg(\024)h Fk(M)10 b Ft(\(1)22 b(+)g Fk(u)p Ft(\))1760 836 y Fh(\000)p Fi(\015)1859 877 y Fk(;)114 b(u)28 b Fg(\025)g Ft(0)p Fk(;)-246 1106 y Ft(and)356 1350 y Fg(j)p Fk(U)460 1309 y Fs(\(2\))555 1350 y Ft(\()p Fk(x)p Ft(\))p Fg(j)22 b Ft(+)g Fg(j)17 b Ft(grad)32 b Fk(U)1177 1309 y Fs(\(2\))1272 1350 y Ft(\()p Fk(x)p Ft(\))p Fg(j)22 b Ft(+)1618 1225 y Fi(d)1564 1255 y Ff(X)1551 1467 y Fi(l)q(;k)r 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5106 y Fk(:)507 b Ft(\(3.3\))-246 5411 y(Notice)32 b(that)h Fg(j)p Fk(y)25 b Fg(\000)d Fk(x)p Fg(j)28 b(\025)g Ft(2)p Fk(\033)37 b Ft(for)32 b Fg(j)p Fk(y)t Fg(j)26 b(\024)i(j)p Fk(x)22 b Fg(\000)h Fk(y)t Fg(j)p Ft(.)42 b(One)33 b(needs)h(to)e(sho)m(w)i(that)f(for)f Fk(y)e Fg(2)e Fk(D)3018 5426 y Fs(1)412 5554 y Ff( )630 5657 y Ft(1)p 501 5701 307 4 v 501 5811 a(1)22 b(+)680 5763 y Fi(\025)p Fh(j)p Fi(y)r Fh(j)p 680 5788 118 4 v 717 5845 a Fi(\033)817 5554 y Ff(!)913 5584 y(\022)1226 5657 y Ft(1)p 996 5701 510 4 v 996 5793 a(1)g(+)g Fk(\025)p Fg(j)p Fk(x)g Fg(\000)h Fk(y)t Fg(j)1515 5584 y Ff(\023)1616 5724 y Fg(\024)1722 5554 y Ff( )1987 5657 y Ft(1)p 1810 5701 402 4 v 1810 5811 a(1)f(+)1989 5763 y Fi(\025)p Fh(j)p Fi(x)p Fh(\000)p Fi(y)r Fh(j)p 1989 5788 213 4 v 2074 5845 a Fi(\033)2222 5554 y Ff(!)2317 5584 y(\022)2542 5657 y Ft(1)p 2400 5701 333 4 v 2400 5793 a(1)g(+)g Fk(\025)p Fg(j)p Fk(y)t Fg(j)2743 5584 y Ff(\023)2833 5724 y Fk(;)p eop %%Page: 10 10 10 9 bop -246 -9 a Ft(10)3236 b Fm(Bahn)33 b Fl(et)i(al)-246 279 y Ft(whic)m(h)e(is)g(equiv)-5 b(alen)m(t)32 b(to)1040 411 y Fk(\033)26 b Ft(+)c Fk(\025)p Fg(j)p Fk(x)g Fg(\000)g Fk(y)t Fg(j)p 1040 455 520 4 v 1128 547 a Fk(\033)k Ft(+)c Fk(\025)p Fg(j)p Fk(y)t Fg(j)1597 478 y(\024)1712 411 y Ft(1)g(+)g Fk(\025)p Fg(j)p Fk(x)g Fg(\000)h Fk(y)t Fg(j)p 1712 455 510 4 v 1801 547 a Ft(1)f(+)g Fk(\025)p Fg(j)p Fk(y)t Fg(j)2232 478 y Fk(:)-246 729 y Ft(Since)45 b(\()p Fk(\033)26 b Ft(+)c Fk(\025)p Fg(j)p Fk(x)g Fg(\000)h Fk(y)t Fg(j)o Ft(\))p Fk(=)p Ft(\()p Fk(\033)i Ft(+)d Fk(\025)p Fg(j)p Fk(y)t Fg(j)o Ft(\))44 b(is)g(monotonic)f(decreasing)i (w.r.t.)79 b Fk(\033)49 b Ft(if)43 b Fg(j)p Fk(y)t Fg(j)j(\024)j(j)p Fk(x)30 b Fg(\000)h Fk(y)t Fg(j)p Ft(,)46 b(this)-246 929 y(pro)m(v)m(es)38 b(\(3.3\).)53 b(No)m(w)36 b(\(3.2\))g(follo)m(ws) e(from)h(the)h(metho)s(d)f(used)i(in)e([F)-8 b(r1,)37 b(page)e(545])h(and)g(\(f)7 b(\))35 b(follo)m(ws)-246 1128 y(from)d(\(3.2\))g(b)m(y)h(the)g(metho)s(d)f(same)h(as)f(that)h (giv)m(en)f(in)g([F)-8 b(r1].)p 2141 1128 60 60 v -99 1429 a(F)g(rom)32 b(no)m(w)j(on)f(w)m(e)h(will)d(\014x)j Fk(p)f Ft(suc)m(h)h(that)f Fk(\015)i(>)30 b Ft(2)p Fk(p)23 b Ft(+)g(1)30 b Fk(>)g Ft(2\()p Fk(d)22 b Ft(+)i(1\))e(+)i(1)33 b(in)h(the)g(de\014nition)f(of)h Fk(')-246 1629 y Ft(in)d(\(3.1\).)42 b(The)33 b(smo)s(oth)d(v)m(ersion)i(of)f(total)f(energy)i(corresp)s (onding)f(to)g Fk(H)8 b Ft(\()p Fk(!)t(;)17 b(\026;)g(\033)t Ft(\))29 b(giv)m(en)j(in)e(\(1.3\))h(is)-246 1828 y(de\014ned)j(as)862 2122 y Fk(W)14 b Ft(\()p Fk(!)t(;)j(\026;)g(\033)t Ft(\))25 b(=)1444 2027 y Ff(X)1455 2239 y Fi(k)r Fh(2)p Fi(I)1604 2122 y Fk(f)11 b Ft(\()p Fk(q)1744 2137 y Fi(k)1809 2122 y Fg(\000)23 b Fk(\026;)17 b(\033)t Ft(\))p Fk(W)2201 2137 y Fi(k)2243 2122 y Ft(\()p Fk(!)t Ft(\))p Fk(;)922 b Ft(\(3.4\))-246 2431 y(where)34 b Fk(q)79 2446 y Fi(k)150 2431 y Ft(=)27 b Fk(q)296 2446 y Fi(k)339 2431 y Ft(\()p Fk(!)t Ft(\))p Fk(;)17 b(p)573 2446 y Fi(k)642 2431 y Ft(=)28 b Fk(p)795 2446 y Fi(k)838 2431 y Ft(\()p Fk(!)t Ft(\),)j(and)845 2725 y Fk(W)937 2740 y Fi(k)979 2725 y Ft(\()p Fk(!)t Ft(\))c(=)h(2)p Fk(A)22 b Ft(+)g Fg(j)p Fk(p)1570 2740 y Fi(k)1612 2725 y Fg(j)1640 2684 y Fs(2)1701 2725 y Ft(+)1799 2630 y Ff(X)1808 2843 y Fi(j)t Fh(6)p Fs(=)p Fi(k)1960 2725 y Fk(U)10 b Ft(\()p Fk(q)2117 2740 y Fi(k)2183 2725 y Fg(\000)22 b Fk(q)2325 2740 y Fi(j)2362 2725 y Ft(\))p Fk(;)906 b Ft(\(3.5\))-246 3043 y(where)34 b Fk(A)f Ft(is)f(the)h(same)f(as)h(in)f(\(2.1\).)43 b(De\014ne)33 b(the)g(logarithmic)c(\015uctuation)j(of)g Fk(W)46 b Ft(b)m(y)p 897 3257 106 4 v 897 3337 a Fk(W)13 b Ft(\()p Fk(!)t Ft(\))27 b(=)h(sup)1326 3416 y Fi(\026)1516 3337 y Ft(sup)1437 3423 y Fi(\033)r Fh(\025)p Fs(2)p Fi(g)r Fs(\()p Fh(j)p Fi(\026)p Fh(j)p Fs(\))1759 3337 y Fk(\033)1818 3296 y Fh(\000)p Fi(d)1914 3337 y Fk(W)14 b Ft(\()p Fk(!)t(;)j(\026;)g (\033)t Ft(\))p Fk(:)939 b Ft(\(3.6\))-99 3640 y(Recall)28 b(that)h(for)g(an)m(y)h Fk(i)e Fg(2)g Fj(Z)951 3604 y Fi(d)1019 3640 y Ft(and)h Fk(!)i Fg(2)p 1391 3560 71 4 v 28 w Ft(\012)q(,)f Fk(n)1577 3655 y Fi(i)1605 3640 y Ft(\()p Fk(!)t Ft(\))f(is)g(the)h(n)m(um)m(b)s(er)g(of)f(particles)g (of)g Fk(!)k Ft(in)28 b(the)i(unit)-246 3840 y(b)s(o)m(x)j(with)f(cen)m (ter)i(at)f Fk(i)p Ft(.)43 b(The)34 b(sup)s(erstabilit)m(y)e(of)g Fk(U)43 b Ft(implies)30 b(the)j(follo)m(wing:)-246 4118 y Fp(Lemma)k(3.2.)50 b Fl(Ther)-5 b(e)34 b(exists)g Ft(0)28 b Fk(<)f(\025)h(<)f Ft(1)35 b Fl(such)g(that)835 4412 y Fk(W)14 b Ft(\()p Fk(!)t(;)j(\026;)g(\033)t Ft(\))26 b Fg(\025)1429 4344 y Fk(B)p 1429 4389 80 4 v 1444 4480 a Ft(4)1538 4317 y Ff(X)1535 4536 y Fi(i)p Fh(2)p Fd(Z)1656 4518 y Fe(d)1703 4412 y Fk(f)11 b Ft(\()p Fk(i)22 b Fg(\000)g Fk(\026;)17 b(\033)t Ft(\))33 b Fk(n)2245 4427 y Fi(i)2273 4412 y Ft(\()p Fk(!)t Ft(\))2414 4370 y Fs(2)-246 4723 y Fl(and)782 4855 y Fk(@)p 752 4899 116 4 v 752 4990 a(@)5 b(\033)878 4922 y(W)14 b Ft(\()p Fk(!)t(;)j(\026;)g(\033)t Ft(\))26 b Fg(\025)1472 4855 y Fk(B)p 1472 4899 80 4 v 1487 4990 a Ft(4)1581 4827 y Ff(X)1577 5047 y Fi(i)p Fh(2)p Fd(Z)1699 5028 y Fe(d)1745 4922 y Fk(f)1804 4881 y Fh(0)1827 4922 y Ft(\()p Fk(i)d Fg(\000)f Fk(\026;)17 b(\033)t Ft(\))33 b Fk(n)2311 4937 y Fi(i)2339 4922 y Ft(\()p Fk(!)t Ft(\))2480 4881 y Fs(2)2519 4922 y Fk(:)-246 5320 y Fp(Pro)s(of.)49 b Ft(Let)40 b Fk(R)g(>)f Ft(0)g(b)s(e)h(a)f (su\016cien)m(tly)h(large)e(natural)g(n)m(um)m(b)s(er)i(whic)m(h)g (will)d(b)s(e)j(\014xed)g(later)f(and)-246 5519 y(write)33 b(the)g(p)s(oten)m(tial)e Fk(U)10 b Ft(\()p Fk(x)p Ft(\))33 b(as)1004 5813 y Fk(U)10 b Ft(\()p Fk(x)p Ft(\))28 b(:=)g Fk(U)1446 5772 y Fs(\()p Fi(R;)p Fs(1\))1614 5813 y Ft(\()p Fk(x)p Ft(\))22 b(+)g Fk(U)1941 5772 y Fs(\()p Fi(R;)p Fs(2\))2110 5813 y Ft(\()p Fk(x)p Ft(\))p Fk(;)1065 b Ft(\(3.7\))p eop %%Page: 11 11 11 10 bop -246 -9 a Fm(Dynamics)32 b(of)g(In\014nite)h(P)m(article)f (Systems)2074 b Ft(11)-246 279 y(where)34 b Fk(U)112 243 y Fs(\()p Fi(R;)p Fs(1\))280 279 y Ft(\()p Fk(x)p Ft(\))28 b(:=)g Fk(U)10 b Ft(\()p Fk(x)p Ft(\)1)826 295 y Fs([0)p Fi(;R)p 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Fi(k)r Fh(2)p Fd(Z)1947 4264 y Fe(d)1993 4158 y Fk(n)2051 4173 y Fi(k)2116 4158 y Ft(+)2224 4091 y(3)p 2224 4135 49 4 v 2224 4227 a(4)2283 4158 y Fk(B)2390 4064 y Ff(X)2379 4283 y Fi(k)r Fh(2)p Fd(Z)2515 4264 y Fe(d)2561 4158 y Fk(n)2619 4117 y Fs(2)2619 4183 y Fi(k)2662 4158 y Fk(;)596 b Ft(\(3.10\))-246 4472 y(where)34 b Fk(A)28 b Fg(\025)g Ft(0)k(and)h Fk(B)g(>)27 b Ft(0)33 b(are)f(the)h(same)g(constan)m(ts)h(app)s(eared)f(in)e(\(2.1\).)-99 4672 y(In)37 b(order)g(to)g(con)m(trol)f Fk(W)848 4635 y Fs(\()p Fi(R;)p Fs(1\))1015 4672 y Ft(,)i(w)m(e)g(mo)s(dify)d(the)i (metho)s(d)g(used)h(in)e(the)h(pro)s(of)f(of)h([F)-8 b(r1,)37 b(Lemma)-246 4871 y(3.1].)44 b(In)m(tro)s(duce)509 5161 y(\003)577 5176 y Fi(u)650 5161 y Ft(:=)27 b Fg(f)p Fk(x)h Fg(2)g Fj(R)1073 5119 y Fi(d)1147 5161 y Ft(:)g Fk(u)1258 5119 y Fs(\()p Fi(i)p Fs(\))1368 5161 y Fg(\024)h Fk(x)1529 5119 y Fs(\()p Fi(i)p Fs(\))1640 5161 y Fk(<)e(u)1799 5119 y Fs(\()p Fi(i)p Fs(\))1904 5161 y Ft(+)22 b Fk(mR)q(;)115 b Ft(1)27 b 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5849 y Fg(\\)15 b Ft(\003)2881 5864 y Fi(u)2926 5849 y Ft(.)42 b(If)29 b Fk(\032)3139 5864 y Fi(u)3214 5849 y Ft(denotes)p eop %%Page: 12 12 12 11 bop -246 -9 a Ft(12)3236 b Fm(Bahn)33 b Fl(et)i(al)-246 279 y Ft(the)43 b(minim)m(um)c(of)j Fk(f)547 294 y Fi(k)634 279 y Ft(=)i Fk(f)11 b Ft(\()p Fk(q)894 294 y Fi(k)965 279 y Fg(\000)29 b Fk(\026;)17 b(\033)t Ft(\))42 b(for)f Fk(k)47 b Fg(2)e Fk(I)1723 294 y Fi(u)1768 279 y Ft(,)g(and)d Fk(\025mR)2256 194 y Fg(p)p 2339 194 51 4 v 85 x Fk(d)i(<)g(\017)p Ft(,)h(i.e.,)g(the)d(diameter)f(of)-246 478 y(\003)-178 493 y Fi(u)-97 478 y Ft(is)36 b(less)g(than)g Fk(\017=\025)p Ft(,)h(then)g(using)f(Lemma)f(3.1)g(\(c\))h(and)h Fk(U)2050 442 y Fs(\()p Fi(R;)p Fs(1\))2218 478 y Ft(\()p Fk(x)p Ft(\))d Fg(\025)g(\000)p Fk(M)10 b Ft(,)38 b(w)m(e)f(obtain)e(for)h (eac)m(h)-246 678 y([)p Fk(k)s(;)17 b(j)6 b Ft(])28 b Fg(2)g Fk(P)137 693 y Fi(u)182 678 y Ft(,)-25 942 y(\()p Fk(f)61 957 y Fi(k)126 942 y Ft(+)22 b Fk(f)272 957 y Fi(j)308 942 y Ft(\))p Fk(U)422 901 y Fs(\()p Fi(R;)p Fs(1\))590 942 y Ft(\()p Fk(q)671 957 y Fi(k)736 942 y Fg(\000)h Fk(q)879 957 y Fi(j)916 942 y Ft(\))83 b(=)h(2)p Fk(\032)1296 957 y Fi(u)1341 942 y Fk(U)1417 901 y Fs(\()p Fi(R;)p Fs(1\))1585 942 y Ft(\()p Fk(q)1666 957 y Fi(k)1731 942 y Fg(\000)23 b Fk(q)1874 957 y Fi(j)1911 942 y Ft(\))f(+)g(\()p Fk(f)2155 957 y Fi(k)2220 942 y Ft(+)g Fk(f)2366 957 y Fi(j)2424 942 y Fg(\000)h Ft(2)p Fk(\032)2623 957 y Fi(u)2668 942 y Ft(\))p Fk(U)2782 901 y Fs(\()p Fi(R;)p Fs(1\))2950 942 y Ft(\()p Fk(q)3031 957 y Fi(k)3096 942 y Fg(\000)g Fk(q)3239 957 y Fi(j)3276 942 y Ft(\))1037 1166 y Fg(\025)83 b Ft(2)p Fk(\032)1296 1181 y Fi(u)1341 1166 y Fk(U)1417 1125 y Fs(\()p Fi(R;)p Fs(1\))1585 1166 y Ft(\()p Fk(q)1666 1181 y Fi(k)1731 1166 y Fg(\000)23 b Fk(q)1874 1181 y Fi(j)1911 1166 y Ft(\))f Fg(\000)g Ft(2)p Fk(M)10 b(\032)2273 1181 y Fi(u)2336 1086 y Ff(\000)2382 1166 y Ft(\(1)21 b(+)h Fk(\017)p Ft(\))2665 1125 y Fs(\()p Fi(p)p Fs(+1\))2873 1166 y Fg(\000)g Ft(1)3021 1086 y Ff(\001)3083 1166 y Fk(;)-246 1431 y Ft(and)33 b(so)g(it)e(follo)m(ws)h (from)f(the)i(ab)s(o)m(v)m(e)g(b)s(ound,)g(\(3.10\))f(and)h(Lemma)e (3.1)h(\(c\))h(that)-20 1601 y Ff(X)-80 1817 y Fs([)p Fi(k)r(;j)t Fs(])p Fh(2)p Fi(P)144 1825 y Fe(u)183 1696 y Ft(\()p Fk(f)269 1711 y Fi(k)334 1696 y Ft(+)22 b Fk(f)480 1711 y Fi(j)516 1696 y Ft(\))p Fk(U)630 1655 y Fs(\()p Fi(R;)p Fs(1\))798 1696 y Ft(\()p Fk(q)879 1711 y Fi(k)944 1696 y Fg(\000)h Fk(q)1087 1711 y Fi(j)1124 1696 y Ft(\))-80 2008 y Fg(\025)29 b(\000)p Fk(A)206 1914 y Ff(X)193 2126 y Fi(k)r Fh(2)p Fd(Z)329 2134 y Fe(u)380 2008 y Fk(\032)430 2023 y Fi(u)475 2008 y Fk(n)533 2023 y Fi(k)598 2008 y Ft(+)706 1941 y(3)p 706 1985 49 4 v 706 2077 a(4)765 2008 y Fk(B)874 1914 y Ff(X)861 2126 y Fi(k)r Fh(2)p Fd(Z)997 2134 y Fe(u)1048 2008 y Fk(\032)1098 2023 y Fi(u)1144 2008 y Fk(n)1202 1967 y Fs(2)1202 2033 y Fi(k)1267 2008 y Fg(\000)22 b Ft(2)p Fk(M)1536 1928 y Ff(\000)1582 2008 y Ft(\(1)g(+)g Fk(\017)p Ft(\))1866 1967 y Fs(\()p Fi(p)p Fs(+1\))2073 2008 y Fg(\000)h Ft(1)2222 1928 y Ff(\001)2343 1914 y(X)2284 2130 y Fs([)p Fi(k)r(;j)t Fs(])p Fh(2)p Fi(P)2508 2138 y Fe(u)2563 2008 y Fk(\032)2613 2023 y Fi(u)-80 2321 y Fg(\025)29 b(\000)p Fk(A)206 2226 y Ff(X)193 2438 y Fi(k)r Fh(2)p Fd(Z)329 2446 y Fe(u)380 2321 y Fk(\032)430 2336 y Fi(u)475 2321 y Fk(n)533 2336 y Fi(k)598 2321 y Ft(+)706 2253 y(3)p 706 2298 V 706 2389 a(4)765 2321 y Fk(B)5 b Ft(\(1)22 b(+)g Fk(\017)p Ft(\))1128 2280 y Fh(\000)p Fs(\()p Fi(p)p Fs(+1\))1398 2226 y Ff(X)1385 2438 y Fi(k)r Fh(2)p Fd(Z)1521 2446 y Fe(u)1572 2321 y Fk(f)11 b Ft(\()p Fk(k)25 b Fg(\000)e Fk(\026;)17 b(\033)t Ft(\))p Fk(n)2103 2280 y Fs(2)2103 2346 y Fi(k)2167 2321 y Fg(\000)23 b Fk(M)10 b Ft(\(\(1)22 b(+)g Fk(\017)p Ft(\))2693 2280 y Fs(\()p Fi(p)p Fs(+1\))2900 2321 y Fg(\000)h Ft(1\))3162 2226 y Ff(X)3104 2442 y Fs([)p Fi(k)r(;j)t Fs(])p Fh(2)p Fi(P)3328 2450 y Fe(u)3366 2321 y Ft(\()p Fk(f)3452 2336 y Fi(k)3516 2321 y Ft(+)f Fk(f)3662 2336 y Fi(j)3699 2321 y Ft(\))p Fk(:)-99 2625 y Ft(No)m(w)40 b(let)e Fk(z)44 b Fg(2)c Ft(\003)540 2640 y Fs(0)606 2625 y Fg(\\)27 b Fk(R)34 b Fj(Z)876 2588 y Fi(d)953 2625 y Ft(b)s(e)40 b(giv)m(en.)63 b(Then)41 b Fg(f)p Ft(\003)1785 2640 y Fi(u)1869 2625 y Ft(:)e Fk(u)f Fg(2)i Fk(z)32 b Ft(+)26 b Fk(R)q(m)p Fj(Z)2543 2588 y Fi(d)2581 2625 y Fg(g)39 b Ft(is)g(a)g(partition)f(of)g Fj(R)3460 2588 y Fi(d)3507 2625 y Ft(.)-246 2824 y(Let)44 b Fk(Q)17 2839 y Fi(z)100 2824 y Ft(b)s(e)g(the)g(union)e(of)h(all)e Fk(P)1035 2839 y Fi(u)1124 2824 y Ft(suc)m(h)j(that)g Fk(u)h Fg(2)i Fk(z)34 b Ft(+)29 b Fk(R)q(m)p Fj(Z)2204 2788 y Fi(d)2243 2824 y Ft(.)75 b(By)44 b(the)g(ab)s(o)m(v)m(e)g (estimate)f(and)-246 3023 y Fk(U)-170 2987 y Fs(\()p Fi(R;)p Fs(1\))-2 3023 y Ft(\()p Fk(x)p Ft(\))28 b Fg(\025)h(\000)p Fk(M)10 b Ft(,)33 b(w)m(e)h(see)g(that)-246 3288 y Fk(W)-140 3247 y Fs(\()p Fi(R;)p Fs(1\))27 3288 y Ft(\()p Fk(!)t(;)17 b(\026:\033)t Ft(\))82 b Fg(\025)610 3220 y Ft(3)p 610 3265 V 610 3356 a(4)668 3288 y Fk(B)5 b Ft(\(1)22 b(+)g Fk(\017)p Ft(\))1031 3247 y Fh(\000)p Fs(\()p Fi(p)p Fs(+1\))1299 3193 y Ff(X)1288 3413 y Fi(k)r Fh(2)p Fd(Z)1424 3394 y Fe(d)1471 3288 y Fk(f)11 b Ft(\()p Fk(k)24 b Fg(\000)f Fk(\026;)17 b(\033)t Ft(\))p Fk(n)2001 3247 y Fs(2)2001 3312 y Fi(k)2065 3288 y Fg(\000)23 b Fk(M)10 b Ft(\(\(1)23 b(+)f Fk(\017)p Ft(\))2592 3247 y Fs(\()p Fi(p)p Fs(+1\))2799 3288 y Fg(\000)g Ft(1\))3064 3193 y Ff(X)3002 3409 y Fs([)p Fi(k)r(;j)t Fs(])p Fh(2)p Fi(Q)3237 3417 y Fe(z)3270 3288 y Ft(\()p Fk(f)3356 3303 y Fi(k)3421 3288 y Ft(+)g Fk(f)3567 3303 y Fi(j)3603 3288 y Ft(\))600 3573 y Fg(\000)p Fk(M)906 3479 y Ff(X)799 3695 y Fs([)p Fi(k)r(;j)t Fs(])p Fh(2)p Fi(P)10 b Fh(n)p Fi(Q)1124 3703 y Fe(z)1156 3573 y Ft(\()p Fk(f)1242 3588 y Fi(k)1307 3573 y Ft(+)22 b Fk(f)1453 3588 y Fi(j)1490 3573 y Ft(\))439 3886 y Fg(\025)610 3819 y Ft(3)p 610 3863 V 610 3954 a(4)668 3886 y Fk(B)5 b Ft(\(1)22 b(+)g Fk(\017)p Ft(\))1031 3845 y Fh(\000)p Fs(\()p Fi(p)p Fs(+1\))1299 3791 y Ff(X)1288 4011 y Fi(k)r Fh(2)p Fd(Z)1424 3992 y Fe(d)1471 3886 y Fk(f)11 b Ft(\()p Fk(k)24 b Fg(\000)f Fk(\026;)17 b(\033)t Ft(\))p Fk(n)2001 3845 y Fs(2)2001 3911 y Fi(k)2065 3886 y Fg(\000)23 b Fk(M)10 b Ft(\(\(1)23 b(+)f Fk(\017)p Ft(\))2592 3845 y Fs(\()p Fi(p)p Fs(+1\))2799 3886 y Fg(\000)g Ft(1\))3045 3791 y Ff(X)3002 4007 y Fs([)p Fi(k)r(;j)t Fs(])p Fh(2)p Fi(P)3233 3886 y Ft(\()p Fk(f)3319 3901 y Fi(k)3384 3886 y Ft(+)g Fk(f)3530 3901 y Fi(j)3566 3886 y Ft(\))600 4172 y Fg(\000)p Fk(M)906 4077 y Ff(X)799 4293 y Fs([)p Fi(k)r(;j)t Fs(])p Fh(2)p Fi(P)10 b Fh(n)p Fi(Q)1124 4301 y Fe(z)1156 4172 y Ft(\()p Fk(f)1242 4187 y Fi(k)1307 4172 y Ft(+)22 b Fk(f)1453 4187 y Fi(j)1490 4172 y Ft(\))p Fk(:)1837 b Ft(\(3.11\))-246 4462 y(W)-8 b(e)33 b(notice)g(that)528 4632 y Ff(X)484 4848 y Fs([)p Fi(k)r(;j)t Fs(])p Fh(2)p Fi(P)715 4727 y Ft(\()p Fk(f)801 4742 y Fi(k)866 4727 y Ft(+)22 b Fk(f)1012 4742 y Fi(j)1049 4727 y Ft(\))27 b Fg(\024)i Fk(c)1262 4742 y Fs(4)1345 4632 y Ff(X)1334 4852 y Fi(k)r Fh(2)p Fd(Z)1470 4833 y Fe(d)1716 4632 y Ff(X)1517 4852 y Fi(j)t Fh(2)p Fd(Z)1646 4833 y Fe(d)1677 4852 y Fs(;)p Fh(j)p Fi(k)r Fh(\000)p Fi(j)t Fh(j\024)p Fi(R)p Fs(+2)2076 4727 y Fk(n)2134 4742 y Fi(k)2194 4727 y Fk(n)2252 4742 y Fi(j)2305 4727 y Fk(f)11 b Ft(\()p Fk(k)25 b Fg(\000)e Fk(\026;)17 b(\033)t Ft(\))p Fk(:)-246 5038 y Ft(Since)33 b Fk(f)11 b Ft(\()p Fk(k)24 b Fg(\000)f Fk(\026;)17 b(\033)t Ft(\))27 b Fg(\024)h Fk(f)11 b Ft(\()p Fk(j)27 b Fg(\000)22 b Fk(\026;)17 b(\033)t Ft(\))33 b(\(1)21 b(+)h Fk(\025)p Ft(\()p Fk(R)h Ft(+)e(2\)\))1729 5002 y Fs(\()p Fi(p)p Fs(+1\))1946 5038 y Ft(if)31 b Fg(j)p Fk(k)25 b Fg(\000)d Fk(j)6 b Fg(j)27 b(\024)i Fk(R)22 b Ft(+)g(2,)32 b(b)m(y)i(Lemma)d(3.1)h(\(c\),)-246 5237 y(and)h(the)h(n)m(um)m(b)s(er)f(of)f Fk(j)39 b Ft(suc)m(h)c(that)d Fg(j)p Fk(k)26 b Fg(\000)c Fk(j)6 b Fg(j)28 b(\024)h Fk(R)23 b Ft(+)g(2)32 b(has)i(an)f(order)g(of)f(\()p Fk(R)23 b Ft(+)g(2\))2777 5201 y Fi(d)2817 5237 y Ft(,)33 b(w)m(e)h(use)g(Sc)m(h)m(w)m(arz)-246 5436 y(inequalit)m(y)e(to)g (obtain)495 5606 y Ff(X)451 5822 y Fs([)p Fi(k)r(;j)t Fs(])p Fh(2)p Fi(P)682 5701 y Ft(\()p Fk(f)768 5716 y Fi(k)833 5701 y Ft(+)22 b Fk(f)979 5716 y Fi(j)1016 5701 y Ft(\))27 b Fg(\024)h Fk(c)1228 5716 y Fs(5)1284 5701 y Ft(\(3)22 b(+)g Fk(R)q Ft(\))1604 5660 y Fi(p)p Fs(+1+)p Fi(d)1853 5606 y Ff(X)1842 5826 y Fi(k)r Fh(2)p Fd(Z)1978 5807 y Fe(d)2025 5701 y Fk(f)11 b Ft(\()p Fk(k)24 b Fg(\000)f Fk(\026;)17 b(\033)t Ft(\))33 b Fk(n)2588 5716 y Fi(k)2631 5701 y Ft(\()p Fk(!)t Ft(\))2772 5660 y Fs(2)2810 5701 y Fk(:)448 b Ft(\(3.12\))p eop %%Page: 13 13 13 12 bop -246 -9 a Fm(Dynamics)32 b(of)g(In\014nite)h(P)m(article)f (Systems)2074 b Ft(13)-246 279 y(Hence,)34 b(inserting)e(\(3.12\))g(in) m(to)g(\(3.11\),)f(w)m(e)j(ha)m(v)m(e)89 540 y Fk(W)195 498 y Fs(\()p Fi(R;)p Fs(1\))362 540 y Ft(\()p Fk(!)t(;)17 b(\026:\033)t Ft(\))89 764 y Fg(\025)194 623 y Ff(\022)277 696 y Ft(3)p 277 741 49 4 v 277 832 a(4)336 764 y Fk(B)5 b Ft(\(1)22 b(+)g Fk(\017)p Ft(\))699 723 y Fh(\000)p Fs(\()p Fi(p)p Fs(+1\))961 764 y Fg(\000)h Fk(M)10 b Ft(\(\(1)22 b(+)g Fk(\017)p Ft(\))1487 723 y Fs(\()p Fi(p)p Fs(+1\))1695 764 y Fg(\000)g Ft(1\))17 b Fk(c)1940 779 y Fs(5)1979 764 y Ft(\(3)22 b(+)g Fk(R)q Ft(\))2299 723 y Fi(p)p Fs(+1+)p Fi(d)2520 623 y Ff(\023)2621 669 y(X)2610 889 y Fi(k)r Fh(2)p Fd(Z)2746 870 y Fe(d)2792 764 y Fk(f)11 b Ft(\()p Fk(k)25 b Fg(\000)e Fk(\026;)17 b(\033)t Ft(\))p Fk(n)3323 723 y Fs(2)3323 788 y Fi(k)189 1041 y Fg(\000)p Fk(M)494 946 y Ff(X)387 1162 y Fs([)p Fi(k)r(;j)t Fs(])p Fh(2)p Fi(P)10 b Fh(n)p Fi(Q)712 1170 y Fe(z)745 1041 y Ft(\()p Fk(f)831 1056 y Fi(k)896 1041 y Ft(+)22 b Fk(f)1042 1056 y Fi(j)1079 1041 y Ft(\))2168 b(\(3.13\))-99 1340 y(W)-8 b(e)25 b(are)g(going)e(to)i(sum)f(the)i(b)s (oth)e(side)h(of)g(the)g(inequalit)m(y)f(\(3.13\))g(o)m(v)m(er)i Fk(z)32 b Fg(2)c Ft(\003)2772 1355 y Fs(0)2818 1340 y Fg(\\)6 b Fk(R)q Fj(Z)3034 1304 y Fi(d)3073 1340 y Ft(.)41 b(The)25 b(term)-246 1539 y(in)k(the)i(left)e(hand)h(side)g(and)g(the)g 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5190 y Fk(C)7 b Ft(\()p Fk(R)q(;)17 b(\017;)g(m)p Ft(\))28 b Fg(\025)1853 5123 y Ft(1)p 1853 5167 49 4 v 1853 5258 a(2)1912 5190 y Fk(B)5 b(:)1267 b Ft(\(3.16\))-246 5450 y(Finally)36 b(w)m(e)j(tak)m(e)g(and)f(\014x)g (0)f Fk(<)g(\025)g(<)f Ft(1)i(so)g(small)e(that)i Fk(\025mR)2108 5365 y Fg(p)p 2191 5365 51 4 v 85 x Fk(d)f(<)g(\017)p Ft(.)60 b(By)38 b(\(3.8\))g(-)f(\(3.9\),)i(\(3.15\)-)-246 5650 y(\(3.16\),)f(the)f(\014rst)h(statemen)m(t)f(in)f(the)i(Lemma)d (follo)m(ws)h(directly)-8 b(.)56 b(Using)37 b(Lemma)e(3.1)i(\(d\))g (instead)-246 5849 y(of)32 b(Lemma)g(3.1\(c\),)g(w)m(e)i(obtain)d(the)i (second)h(inequalit)m(y)e(in)f(the)i(same)g(w)m(a)m(y)-8 b(.)p 2708 5849 60 60 v eop %%Page: 14 14 14 13 bop -246 -9 a Ft(14)3236 b Fm(Bahn)33 b Fl(et)i(al)-99 279 y Ft(In)f(the)h(rest)g(of)f(this)g(pap)s(er)h Fk(\025)30 b(>)h Ft(0)j(will)e(b)s(e)j(\014xed)g(suc)m(h)h(that)f(Lemma)e(3.2.)h (holds.)48 b(Finally)-8 b(,)32 b(let)-246 478 y(us)i(remark)e(that)g Fk(H)40 b Ft(and)33 b Fk(W)46 b 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Fs(10)1587 5317 y Fk(\033)1646 5276 y Fi(d)1687 5317 y Ft(\()1762 5193 y Fh(1)1725 5222 y Ff(X)1730 5432 y Fi(n)p Fs(=1)1885 5317 y Fk(n)1943 5276 y Fh(\000)p Fs(\()p Fi(p)p Fh(\000)p Fi(d)p Fs(\))2184 5317 y Ft(\))p 2255 5237 89 4 v 33 w Fk(H)8 b Ft(\()p Fk(!)t Ft(\))p Fk(:)773 b Ft(\(3.22\))-246 5650 y(Dividing)31 b(b)s(oth)h(sides)i(of)e(\(3.22\))g(b)m(y)i Fk(\033)1205 5613 y Fi(d)1279 5650 y Ft(and)f(using)f(the)h(fact)g (that)g Fk(p)22 b Fg(\000)h Fk(d)28 b(>)g Ft(1,)33 b(w)m(e)h(get)f(the) g(desired)-246 5849 y(results.)p 156 5849 60 60 v eop %%Page: 16 16 16 15 bop -246 -9 a Ft(16)3236 b Fm(Bahn)33 b Fl(et)i(al)-246 279 y Fu(4)161 b(A)54 b(priori)h(Bound)-246 580 y Ft(Recall)39 b(that)g(a)h(con)m(tin)m(uous)h(tra)5 b(jectory)40 b Fk(!)1382 595 y Fi(t)1452 580 y Ft(:)g Fj(R)51 b Fg(!)p 1771 500 71 4 v 40 w Ft(\012)73 b(is)40 b(called)f(a)g(temp)s(ered)h (solution)f(to)g(\(1.1\))-246 779 y(with)34 b(initial)c (con\014guration)j Fk(!)k Ft(if)c Fk(!)1103 794 y Fs(0)1172 779 y Ft(=)c Fk(!)37 b Ft(and)d(the)h(comp)s(onen)m(ts)f Fk(q)2317 794 y Fi(k)2360 779 y Ft(\()p Fk(t)p Ft(\))c(=)f Fk(q)2649 794 y Fi(k)2692 779 y Ft(\()p Fk(w)2800 794 y Fi(t)2830 779 y Ft(\),)34 b Fk(p)2978 794 y Fi(k)3020 779 y Ft(\()p Fk(t)p Ft(\))c(=)g Fk(p)3316 794 y Fi(k)3359 779 y Ft(\()p Fk(w)3467 794 y Fi(t)3496 779 y Ft(\))-246 979 y(are)36 b(con)m(tin)m(uously)g(di\013eren)m(tiable)e(and)i (satisfy)f(\(1.1\))g(for)g(all)e Fk(k)j Fg(2)d Fk(I)44 b Ft(and)35 b Fk(t)e Fg(2)g Fj(R)5 b Ft(.)58 b(W)-8 b(e)36 b(assume)g(the)-246 1178 y(dimension)c Fk(d)27 b Fg(\024)h Ft(2.)43 b(The)34 b(follo)m(wing)c(is)i(a)g(lo)s(cal)e(v)m(ersion)k(of) e(the)h(la)m(w)f(of)g(energy)h(conserv)-5 b(ation.)-246 1469 y Fp(Prop)s(osition)36 b(4.1.)49 b Fl(Ther)-5 b(e)36 b(exists)f(a)h(c)-5 b(onstant)36 b Fk(K)i(>)30 b Ft(0)35 b Fl(such)h(that)h(along)e(any)i(temp)-5 b(er)g(e)g(d)35 b(solution)-246 1668 y Fk(!)-185 1683 y Fi(t)-120 1668 y Fl(to)g(\(1.1\))f(we)g(have)515 1904 y Fk(@)p 497 1949 93 4 v 497 2040 a(@)5 b(t)599 1972 y(W)14 b Ft(\()p Fk(!)804 1987 y Fi(t)833 1972 y Fk(;)j(\026;)g(\033)t Ft(\))27 b Fg(\024)h Fk(K)41 b(g)t Ft(\()p Fg(j)p Fk(\026)p Fg(j)20 b Ft(+)i Fk(\033)t Ft(\)[)p 1779 1892 106 4 v Fk(W)14 b Ft(\()p Fk(!)1984 1987 y Fi(t)2013 1972 y Ft(\)])2078 1931 y Fs(1)p Fi(=)p Fs(2)2227 1904 y Fk(@)p 2198 1949 116 4 v 2198 2040 a(@)5 b(\033)2324 1972 y(W)14 b Ft(\()p Fk(!)2529 1987 y Fi(t)2558 1972 y Fk(;)j(\026;)g(\033)t Ft(\))-246 2275 y Fl(for)35 b(al)5 b(l)34 b Fk(t)28 b Fg(2)g Fj(R)5 b Fk(;)18 b(\026)32 b Fg(2)d Fj(R)569 2239 y Fi(d)615 2275 y Fk(;)35 b Fl(and)f Fk(\033)e Fg(\025)c Ft(2)p Fl(.)-246 2684 y Fp(Pro)s(of.)49 b Ft(In)32 b(order)g(to)f(pro)m (v)m(e)i(the)f(prop)s(osition,)f(w)m(e)h(mo)s(dify)e(and)i(extend)h (the)f(metho)s(d)g(used)g(in)f(the)-246 2883 y(pro)s(of)26 b(of)g([F)-8 b(r1,)28 b(Prop)s(osition)d(4.1].)41 b(Recall)25 b(the)i(decomp)s(osition)e Fk(U)10 b Ft(\()p Fk(x)p Ft(\))29 b(=)e Fk(U)2574 2847 y Fs(\(1\))2669 2883 y Ft(\()p Fk(x)p Ft(\))10 b(+)g 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y Fs(1)196 4761 y Ft(,)h(it)g(is)g(not)h(hard)f(to)g(see)i(that)e Fg(j)p Fk(W)1460 4776 y Fi(k)1503 4761 y Fg(j)27 b(\024)h Fk(W)1755 4776 y Fi(k)1820 4761 y Ft(+)22 b Fk(M)2073 4686 y Ff(P)2178 4790 y Fi(j)t Fh(6)p Fs(=)p Fi(k)2308 4761 y Ft(\(1)g(+)g Fg(j)p Fk(q)2586 4776 y Fi(k)2651 4761 y Fg(\000)g Fk(q)2793 4776 y Fi(j)2830 4761 y Fg(j)p Ft(\))2896 4725 y Fh(\000)p Fi(\015)2995 4761 y Ft(.)43 b(Th)m(us)458 5064 y Fg(j)p Fk(D)567 5079 y Fs(1)606 5064 y Fg(j)27 b(\024)766 4970 y Ff(X)777 5182 y Fi(k)r Fh(2)p Fi(I)927 5064 y Fg(j)17 b Ft(grad)32 b Fk(f)1242 5079 y Fi(k)1285 5064 y Fg(j)g(j)p Fk(p)1422 5079 y Fi(k)1465 5064 y Fg(j)17 b Ft([)p Fk(W)1629 5079 y Fi(k)1693 5064 y Ft(+)22 b Fk(M)1946 4970 y Ff(X)1955 5182 y Fi(j)t Fh(6)p Fs(=)p Fi(k)2090 5064 y Ft(\(1)f(+)h Fg(j)p Fk(q)2367 5079 y Fi(k)2432 5064 y Fg(\000)h Fk(q)2575 5079 y Fi(j)2612 5064 y Fg(j)p Ft(\))2678 5023 y Fh(\000)p Fi(\015)2776 5064 y Ft(])p Fk(:)-246 5405 y Ft(Since)33 b Fg(j)p Fk(p)86 5420 y Fi(k)128 5405 y Fg(j)28 b(\024)g Fk(K)372 5420 y Fs(1)444 5405 y Fk(g)t Ft(\()p Fg(j)p Fk(q)604 5420 y Fi(k)646 5405 y Fg(j)p Ft(\))33 b([)p 772 5325 106 4 v Fk(W)14 b Ft(\()p Fk(!)977 5420 y Fi(t)1006 5405 y Ft(\)])1071 5369 y Fs(1)p Fi(=)p Fs(2)1181 5405 y Ft(,)33 b(w)m(e)g(use)h(Lemma)d(3.1)h(\(f)7 b(\))32 b(to)h(get)-35 5708 y Fg(j)p Fk(D)74 5723 y Fs(1)113 5708 y Fg(j)28 b(\024)g Ft(4)p Fk(K)406 5723 y Fs(1)478 5708 y Fk(g)t Ft(\()p Fg(j)p Fk(\026)p Fg(j)21 b Ft(+)h Fk(\033)t Ft(\)[)p 925 5628 V Fk(W)13 b Ft(\()p Fk(!)1129 5723 y Fi(t)1159 5708 y Ft(\)])1224 5667 y Fs(1)p Fi(=)p Fs(2)1350 5614 y Ff(X)1403 5826 y Fi(k)1511 5708 y Fk(f)1570 5667 y Fh(0)1559 5733 y Fi(k)1635 5708 y Ft([)p Fk(W)1754 5723 y Fi(k)1797 5708 y Ft(\()p Fk(!)1896 5723 y Fi(t)1925 5708 y Ft(\))22 b(+)g Fk(M)2238 5614 y Ff(X)2247 5826 y Fi(j)t Fh(6)p Fs(=)p Fi(k)2381 5708 y Ft(\(1)g(+)g Fg(j)p Fk(q)2659 5723 y Fi(k)2724 5708 y Fg(\000)h Fk(q)2867 5723 y Fi(j)2903 5708 y Fg(j)p Ft(\))2969 5667 y Fh(\000)p Fi(\015)3068 5708 y Ft(])p Fk(:)211 b Ft(\(4.2\))p eop %%Page: 17 17 17 16 bop -246 -9 a Fm(Dynamics)32 b(of)g(In\014nite)h(P)m(article)f (Systems)2074 b Ft(17)-246 279 y(By)38 b(Lemma)e(3.1)h(\(d\),)i Fk(f)682 243 y Fh(0)671 305 y Fi(k)750 279 y Fg(\024)d Ft(\(1)25 b(+)g Fk(\025)p Fg(j)p Fk(q)1204 294 y Fi(k)1272 279 y Fg(\000)h Fk(q)1418 294 y Fi(j)1455 279 y Fg(j)p Ft(\))1521 243 y Fi(p)p Fs(+1)1684 279 y Fk(f)1743 243 y Fh(0)1732 304 y Fi(j)1768 279 y Ft(.)58 b(Th)m(us)39 b(using)e(Sc)m(h)m(w)m(arz)j(inequalit)m(y)c(and)h(the)-246 478 y(fact)c(that)f Fk(\015)h(>)27 b Ft(2)p Fk(p)p Ft(,)33 b(w)m(e)g(get)450 680 y Ff(X)461 892 y Fi(k)r Fh(2)p Fi(I)610 775 y Fk(f)669 734 y Fh(0)658 799 y Fi(k)717 680 y Ff(X)726 892 y Fi(j)t Fh(6)p Fs(=)p Fi(k)861 775 y Ft(\(1)22 b(+)g Fg(j)p Fk(q)1139 790 y Fi(k)1204 775 y Fg(\000)h Fk(q)1347 790 y Fi(j)1383 775 y Fg(j)p Ft(\))1449 734 y Fh(\000)p Fi(\015)1576 775 y Fg(\024)28 b Fk(K)1764 790 y Fs(2)1831 680 y Ff(X)1820 900 y Fi(k)r Fh(2)p Fd(Z)1956 881 y Fe(d)2003 775 y Fk(f)2062 734 y Fh(0)2085 775 y Ft(\()p Fk(k)d Fg(\000)d Fk(\026;)17 b(\033)t Ft(\))33 b Fk(n)2589 790 y Fi(k)2632 775 y Ft(\()p Fk(!)t Ft(\))2773 734 y Fs(2)2812 775 y Fk(:)494 b Ft(\(4.3\))-246 1095 y(F)-8 b(rom)31 b(\(4.2\)-\(4.3\))g(and)i(Lemma)e(3.2,)i(one)f(obtains) 648 1391 y Fg(j)p Fk(D)757 1406 y Fs(1)796 1391 y Fg(j)27 b(\024)h Fk(K)1039 1406 y Fs(3)1095 1391 y Fk(g)t Ft(\()p Fg(j)p Fk(\026)p Fg(j)21 b Ft(+)h Fk(\033)t Ft(\))17 b([)p 1559 1311 106 4 v Fk(W)c Ft(\()p Fk(!)1763 1406 y Fi(t)1792 1391 y Ft(\)])1857 1350 y Fs(1)p Fi(=)p Fs(2)2040 1324 y Fk(@)p 2011 1368 116 4 v 2011 1460 a(@)5 b(\033)2136 1391 y(W)14 b Ft(\()p Fk(!)2341 1406 y Fi(t)2370 1391 y Fk(;)j(\026;)g(\033)t Ft(\))p Fk(:)692 b Ft(\(4.4\))-99 1688 y(F)-8 b(or)31 b Fk(D)156 1703 y Fs(3)196 1688 y Ft(,)h(since)h Fg(j)17 b Ft(grad)33 b Fk(U)838 1652 y Fs(\(2\))933 1688 y Ft(\()p Fk(q)1014 1703 y Fi(k)1079 1688 y Fg(\000)22 b Fk(q)1221 1703 y Fi(j)1258 1688 y Ft(\))p Fg(j)27 b(\024)h Fk(M)44 b Ft(\(1)22 b(+)g Fg(j)p Fk(q)1872 1703 y Fi(k)1937 1688 y Fg(\000)h Fk(q)2080 1703 y Fi(j)2116 1688 y Fg(j)p Ft(\))2182 1652 y Fh(\000)p Fi(\015)2281 1688 y Fk(;)33 b Ft(w)m(e)h(ha)m(v)m(e)-246 1984 y Fg(j)p Fk(D)-137 1999 y Fs(3)-98 1984 y Fg(j)28 b(\024)g Fk(K)146 1999 y Fs(4)219 1889 y Ff(X)271 2101 y Fi(k)379 1889 y Ff(X)388 2101 y Fi(j)t Fh(6)p Fs(=)p Fi(k)540 1984 y Fg(j)17 b Ft(grad)32 b Fk(f)11 b Ft(\()p Fk(z)949 1999 y Fi(k)r(j)1034 1984 y Fg(\000)f Fk(\026;)17 b(\033)t Ft(\))p Fg(j)g(j)p Fk(q)1437 1999 y Fi(k)1488 1984 y Fg(\000)10 b Fk(q)1618 1999 y Fi(j)1655 1984 y Fg(j)17 b Ft(\(1)10 b(+)g Fg(j)p Fk(q)1954 1999 y Fi(k)2005 1984 y Fg(\000)g Fk(q)2135 1999 y Fi(j)2172 1984 y Fg(j)p Ft(\))2238 1943 y Fh(\000)p Fi(\015)2370 1984 y Ft(\()p Fk(g)t Ft(\()p Fg(j)p Fk(q)2568 1999 y Fi(k)2610 1984 y Fg(j)p Ft(\))g(+)g Fk(g)t Ft(\()p Fg(j)p Fk(q)2932 1999 y Fi(j)2967 1984 y Fg(j)p Ft(\)\)[)p 3098 1904 106 4 v Fk(W)j Ft(\()p Fk(!)3302 1999 y Fi(t)3332 1984 y Ft(\)])3397 1943 y Fs(1)p Fi(=)p Fs(2)3507 1984 y Fk(;)-246 2304 y Ft(where)34 b Fk(z)81 2319 y Fi(k)r(j)189 2304 y Ft(is)e(a)g(some)h(p)s(oin)m(t)f(b)s(et)m(w)m(een)i Fk(q)1287 2319 y Fi(k)1363 2304 y Ft(and)e Fk(q)1595 2319 y Fi(j)1632 2304 y Ft(.)44 b(Note)32 b(that,)h(b)m(y)g(Lemma)e (3.1)i(\(f)7 b(\),)422 2601 y Fg(j)17 b Ft(grad)49 b Fk(f)11 b Ft(\()p Fk(z)848 2616 y Fi(k)r(j)945 2601 y Fg(\000)23 b Fk(\026;)17 b(\033)t Ft(\))p Fg(j)26 b(\024)1541 2533 y Ft(1)p 1415 2578 302 4 v 1415 2669 a Fk(g)t Ft(\()p Fg(j)p Fk(z)1577 2684 y Fi(k)r(j)1651 2669 y Fg(j)p Ft(\))1760 2601 y(4)33 b Fk(g)t Ft(\()p Fg(j)p Fk(\026)p Fg(j)20 b Ft(+)i Fk(\033)t Ft(\))17 b Fk(f)2337 2560 y Fh(0)2360 2601 y Ft(\()p Fk(z)2443 2616 y Fi(k)r(j)2540 2601 y Fg(\000)23 b Fk(\026;)17 b(\033)t Ft(\))p Fk(:)-246 2897 y Ft(W)-8 b(e)25 b(note)f(that)f Fk(g)t Ft(\()p Fk(x)t Ft(+)t Fk(y)t Ft(\))k Fg(\024)i Fk(g)t Ft(\()p Fk(x)p Ft(\))p Fk(g)t Ft(\()p Fk(y)t Ft(\))21 b(for)j(an)m(y)g Fk(x;)17 b(y)31 b Fg(\025)d Ft(0,)e(and)e(so)g Fk(g)t Ft(\()p Fg(j)p Fk(q)2310 2912 y Fi(k)2352 2897 y Fg(j)p Ft(\))t(+)t Fk(g)t Ft(\()p Fg(j)p Fk(g)2666 2912 y Fi(j)2701 2897 y Fg(j)p Ft(\))j Fg(\024)i Ft(2)p Fk(g)t Ft(\()p Fg(j)p Fk(z)3111 2912 y Fi(k)r(j)3185 2897 y Fg(j)p Ft(\))p Fk(g)t Ft(\()p Fg(j)p Fk(q)3411 2912 y Fi(k)3456 2897 y Fg(\000)-246 3096 y Fk(q)-203 3111 y Fi(j)-166 3096 y Fg(j)p Ft(\).)43 b(Using)32 b(Sc)m(h)m(w)m(arz)j(inequalit)m(y)-8 b(,)31 b(Lemma)h(3.1)g(\(d\))g(and)h(Lemma)e(3.2,)i(w)m(e)g(obtain)-15 3392 y Fg(j)p Fk(D)94 3407 y Fs(3)133 3392 y Fg(j)28 b(\024)g Fk(K)377 3407 y Fs(5)449 3392 y Fk(g)t Ft(\()p Fg(j)p Fk(\026)p Fg(j)21 b Ft(+)h Fk(\033)t Ft(\)[)p 896 3312 106 4 v Fk(W)14 b Ft(\()p Fk(!)1101 3407 y Fi(t)1130 3392 y Ft(\)])1195 3351 y Fs(1)p Fi(=)p Fs(2)18 3617 y Fg(\002)124 3522 y Ff(X)112 3742 y Fi(k)r Fh(2)p Fd(Z)248 3723 y Fe(d)303 3522 y Ff(X)295 3742 y Fi(j)t Fh(2)p Fd(Z)425 3723 y Fe(d)455 3617 y Ft([)p Fk(n)540 3632 y Fi(k)583 3617 y Ft(\()p Fk(!)t Ft(\))32 b Fk(n)814 3632 y Fi(j)851 3617 y Ft(\()p Fk(!)t Ft(\))h Fk(f)1084 3575 y Fh(0)1107 3617 y Ft(\()p Fk(l)1174 3632 y Fi(k)r(j)1271 3617 y Fg(\000)22 b Fk(\026;)17 b(\033)t Ft(\))p Fg(j)p Fk(k)25 b Fg(\000)d Fk(j)6 b Fg(j)p Ft(\(1)22 b(+)g Fg(j)p Fk(k)j Fg(\000)d Fk(j)6 b Fg(j)p Ft(\))2369 3575 y Fh(\000)p Fi(\015)2501 3617 y Ft(\(1)22 b(+)g(log\(1)g(+)g Fg(j)p Fk(k)j Fg(\000)e Fk(j)6 b Fg(j)p Ft(\)\)])-15 3889 y Fg(\024)28 b Fk(K)180 3847 y Fh(0)173 3913 y Fs(5)246 3889 y Fk(g)t Ft(\()p Fg(j)p Fk(\026)p Fg(j)20 b Ft(+)i Fk(\033)t Ft(\)[)p 692 3809 V Fk(W)14 b Ft(\()p Fk(!)897 3904 y Fi(t)926 3889 y Ft(\)])991 3847 y Fs(1)p Fi(=)p Fs(2)18 4113 y Fg(\002)124 4018 y Ff(X)112 4238 y Fi(k)r Fh(2)p Fd(Z)248 4219 y Fe(d)295 4113 y Fk(f)354 4072 y Fh(0)343 4137 y Fi(k)419 4113 y Fk(n)477 4072 y Fs(2)477 4137 y Fi(k)520 4113 y Ft(\()p Fk(!)t Ft(\))32 b([)728 4018 y Ff(X)720 4238 y Fi(j)t Fh(2)p Fd(Z)850 4219 y Fe(d)897 4113 y Fg(j)p Fk(k)25 b Fg(\000)d Fk(j)6 b Fg(j)17 b Ft(\(1)22 b(+)g Fg(j)p Fk(k)i Fg(\000)f Fk(j)6 b Fg(j)p Ft(\))1713 4072 y Fh(\000)p Fi(\015)1845 4113 y Ft(\(1)22 b(+)g(log\(1)g(+)g Fg(j)p Fk(k)j Fg(\000)d Fk(j)6 b Fg(j)p Ft(\))33 b(\(1)22 b(+)g Fk(\025)p Fg(j)p Fk(k)j Fg(\000)d Fk(j)6 b Fg(j)p Ft(\))3312 4072 y Fi(p)p Fs(+1)3442 4113 y Ft(])-15 4430 y Fg(\024)28 b Fk(K)180 4389 y Fh(0)q(0)173 4455 y Fs(5)256 4430 y Fk(g)t Ft(\()p Fg(j)p Fk(\026)p Fg(j)21 b Ft(+)h Fk(\033)t Ft(\))17 b([)p 720 4350 V Fk(W)c Ft(\()p Fk(!)924 4445 y Fi(t)953 4430 y Ft(\)])1018 4389 y Fs(1)p Fi(=)p Fs(2)1168 4363 y Fk(@)p 1138 4407 116 4 v 1138 4499 a(@)5 b(\033)1264 4430 y(W)14 b Ft(\()p Fk(!)1469 4445 y Fi(t)1498 4430 y Fk(;)j(\026;)g(\033)t Ft(\))p Fk(;)1564 b Ft(\(4.5\))-246 4727 y(where)45 b Fk(l)76 4742 y Fi(k)r(j)194 4727 y Ft(is)d(a)h(p)s(oin)m(t)f(b)s(et)m (w)m(een)j Fk(k)h Ft(and)d Fk(j)6 b Ft(.)75 b(Here)44 b(w)m(e)g(ha)m(v)m(e)g(used)g Fk(\015)50 b(>)c Ft(2)p Fk(p)29 b Ft(+)g(1)43 b(to)f(get)h(the)h(last)-246 4926 y(inequalit)m(y)-8 b(.)-99 5125 y(Finally)30 b(for)i Fk(D)463 5140 y Fs(2)502 5125 y Ft(,)h(w)m(e)g(use)h(\(2.4\))e(to)g (obtain)g(the)h(follo)m(wing)c(b)s(ound:)-80 5421 y Fg(j)p Fk(D)29 5436 y Fs(2)68 5421 y Fg(j)f(\024)g Fk(K)312 5436 y Fs(6)368 5327 y Ff(X)421 5539 y Fi(k)687 5327 y Ff(X)529 5543 y Fi(j)t Fh(6)p Fs(=)p Fi(k)r Fs(;)p Fh(j)p Fi(q)728 5555 y Fe(k)763 5543 y Fh(\000)p Fi(q)850 5553 y Fe(j)882 5543 y Fh(j\024)p Fi(r)1007 5421 y Fg(j)17 b Ft(grad)32 b Fk(f)11 b Ft(\()p Fk(z)1416 5436 y Fi(k)r(j)1513 5421 y Fg(\000)23 b Fk(\026;)17 b(\033)t Ft(\))p Fg(j)g Ft(\()p Fk(a)k Ft(+)h Fk(b)33 b(U)2216 5380 y Fs(\(1\))2311 5421 y Ft(\()p Fk(q)2392 5436 y Fi(k)2457 5421 y Fg(\000)23 b Fk(q)2600 5436 y Fi(j)2637 5421 y Ft(\)\))33 b Fk(g)t Ft(\()p Fg(j)p Fk(q)2906 5436 y Fi(k)2948 5421 y Fg(j)p Ft(\))g([)p 3074 5341 106 4 v Fk(W)13 b Ft(\()p Fk(!)3278 5436 y Fi(t)3308 5421 y Ft(\)])3373 5380 y Fs(1)p Fi(=)p Fs(2)-13 5748 y Fg(\024)28 b Fk(K)182 5707 y Fh(0)175 5772 y Fs(6)248 5748 y Fk(g)t Ft(\()p Fg(j)p Fk(\026)p Fg(j)20 b Ft(+)i Fk(\033)t Ft(\)[)p 694 5668 V Fk(W)14 b Ft(\()p Fk(!)899 5763 y Fi(t)928 5748 y Ft(\)])993 5707 y Fs(1)p Fi(=)p Fs(2)1153 5653 y Ff(X)1206 5865 y Fi(k)1472 5653 y Ff(X)1313 5869 y Fi(j)t Fh(6)p Fs(=)p Fi(k)r Fs(;)p Fh(j)p Fi(q)1512 5881 y Fe(k)1548 5869 y Fh(\000)p Fi(q)1635 5879 y Fe(j)1667 5869 y Fh(j\024)p Fi(r)1792 5748 y Fk(f)1851 5707 y Fh(0)1874 5748 y Ft(\()p Fk(z)1957 5763 y Fi(k)r(j)2054 5748 y Fg(\000)23 b Fk(\026;)17 b(\033)t Ft(\))32 b(\()p Fk(a)23 b Ft(+)f Fk(b)33 b(U)2746 5707 y Fs(\(1\))2841 5748 y Ft(\()p Fk(q)2922 5763 y Fi(k)2987 5748 y Fg(\000)23 b Fk(q)3130 5763 y Fi(j)3167 5748 y Fg(j)p Ft(\)\))3297 5680 y Fk(g)t Ft(\()p Fg(j)p Fk(q)3457 5695 y Fi(k)3499 5680 y Fg(j)p Ft(\))p 3281 5725 302 4 v 3281 5816 a Fk(g)t Ft(\()p Fg(j)p Fk(z)3443 5831 y Fi(k)r(j)3516 5816 y Fg(j)p Ft(\))3592 5748 y Fk(:)p eop %%Page: 18 18 18 17 bop -246 -9 a Ft(18)3236 b Fm(Bahn)33 b Fl(et)i(al)-246 279 y Ft(Notice)d(that)h(if)e Fg(j)p Fk(q)432 294 y Fi(k)497 279 y Fg(\000)23 b Fk(q)640 294 y Fi(j)676 279 y Fg(j)28 b Fk(<)f(r)s Ft(,)33 b Fk(g)t Ft(\()p Fg(j)p Fk(q)1102 294 y Fi(k)1143 279 y Fg(j)p Ft(\))28 b Fg(\024)g Fk(K)1425 294 y Fs(7)1497 279 y Fk(g)t Ft(\()p Fg(j)p Fk(z)1659 294 y Fi(k)r(j)1734 279 y Fg(j)p Ft(\).)43 b(By)33 b(Lemma)e(3.1)h (\(d\),)450 477 y Ff(X)503 690 y Fi(k)769 477 y Ff(X)610 693 y Fi(j)t Fh(6)p Fs(=)p Fi(k)r Fs(;)p Fh(j)p Fi(q)809 705 y Fe(k)845 693 y Fh(\000)p Fi(q)932 703 y Fe(j)964 693 y Fh(j\024)p Fi(r)1089 572 y Fk(f)1148 531 y Fh(0)1171 572 y Ft(\()p Fk(z)1254 587 y Fi(k)r(j)1351 572 y Fg(\000)23 b Fk(\026;)17 b(\033)t Ft(\))82 b Fg(\024)i Fk(K)1977 587 y Fs(8)2044 477 y Ff(X)2033 697 y Fi(k)r Fh(2)p Fd(Z)2169 678 y Fe(d)2270 477 y Ff(X)2215 693 y Fh(j)p Fi(j)t Fh(\000)p Fi(k)r Fh(j\024)p Fi(r)2486 572 y Fk(n)2544 587 y Fi(k)2620 572 y Fk(n)2678 587 y Fi(j)2748 572 y Fk(f)2807 531 y Fh(0)2796 597 y Fi(k)1733 861 y Fg(\024)g Fk(K)1984 820 y Fh(0)1977 885 y Fs(8)2044 766 y Ff(X)2033 986 y Fi(k)r Fh(2)p Fd(Z)2169 967 y Fe(d)2215 861 y Fk(f)2274 820 y Fh(0)2263 885 y 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Fk(U)10 b Ft(\()p Fk(q)1127 2402 y Fi(k)1193 2387 y Fg(\000)23 b Fk(q)1336 2402 y Fi(l)1362 2387 y Ft(\))83 b Fg(\025)g(\000)p Fk(A)1879 2292 y Ff(X)1810 2512 y Fi(l)q Fh(2)p Fs(\003)1928 2521 y Fc(2)1963 2512 y Fh(\\)p Fd(Z)2060 2493 y Fe(d)2107 2387 y Fk(n)2165 2402 y Fi(l)2191 2387 y Ft(\()p Fk(!)2290 2402 y Fs(0)2329 2387 y Ft(\))1483 2669 y Fg(\025)g(\000)p Fk(A)1950 2574 y Ff(X)1827 2794 y Fi(l)q Fh(2)p Fi(B)1949 2802 y Fe(\033)1991 2794 y Fs(\()p Fi(\026)p Fs(\))p Fh(\\)p Fd(Z)2186 2775 y Fe(d)2232 2669 y Fk(n)2290 2684 y Fi(l)2317 2669 y Ft(\()p Fk(!)2416 2684 y Fs(0)2455 2669 y Ft(\))2493 2628 y Fs(2)1483 2989 y Fg(\025)g(\000)1730 2922 y Ft(2)p Fk(A)p 1730 2966 122 4 v 1752 3058 a(B)1879 2989 y(H)8 b Ft(\()p Fk(!)2067 3004 y Fs(0)2106 2989 y Fk(;)17 b(\026;)g(\033)25 b Ft(+)d(1\))p Fk(:)788 b Ft(\(5.3\))-246 3285 y(Here)43 b(w)m(e)h(ha)m(v)m(e)f(used)h(\(3.18\))d(to)h(get)g(the)h(last)e (inequalit)m(y)-8 b(.)72 b(On)42 b(the)h(other)f(hand,)j(b)m(y)e(using) f(the)-246 3484 y(decomp)s(osition)31 b(of)h Fk(U)39 b Ft(=)27 b Fk(U)792 3448 y Fs(\(1\))909 3484 y Ft(+)22 b Fk(U)1083 3448 y Fs(\(2\))1211 3484 y Ft(in)31 b(\(2.2\),)i(let)f(us) h(put)247 3684 y Ff(X)234 3897 y Fi(k)r Fh(2)p Fs(\003)369 3906 y Fc(1)424 3684 y Ff(X)420 3897 y Fi(l)q Fh(2)p Fs(\003)538 3906 y Fc(2)589 3779 y Fk(U)10 b Ft(\()p Fk(q)746 3794 y Fi(k)812 3779 y Fg(\000)22 b Fk(q)954 3794 y Fi(l)981 3779 y Ft(\))27 b(=)1162 3684 y Ff(X)1150 3897 y Fi(k)r Fh(2)p Fs(\003)1285 3906 y Fc(1)1340 3684 y Ff(X)1335 3897 y Fi(l)q Fh(2)p Fs(\003)1453 3906 y Fc(2)1504 3779 y Fk(U)1580 3738 y Fs(\(1\))1675 3779 y Ft(\()p Fk(q)1756 3794 y Fi(k)1821 3779 y Fg(\000)c Fk(q)1964 3794 y Fi(l)1990 3779 y Ft(\))f(+)2161 3684 y Ff(X)2148 3897 y Fi(k)r Fh(2)p Fs(\003)2283 3906 y Fc(1)2338 3684 y Ff(X)2334 3897 y Fi(l)q Fh(2)p Fs(\003)2452 3906 y Fc(2)2503 3779 y Fk(U)2579 3738 y Fs(\(2\))2674 3779 y Ft(\()p Fk(q)2755 3794 y Fi(k)2820 3779 y Fg(\000)h Fk(q)2963 3794 y Fi(l)2989 3779 y Ft(\))p Fk(:)279 b 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b(\026;)g(\033)26 b Ft(+)c(1\))p Fk(:)789 b Ft(\(5.5\))-246 4959 y(Inserting)33 b(\(5.3\))f(and)h(\(5.5\))f(in)m(to)g(\(5.1\),)g(w)m(e)h(pro)m(v)m(ed)h (the)f(lemma.)p 2313 4959 60 60 v -99 5263 a(When)28 b Fk(q)223 5278 y Fi(k)266 5263 y Ft(\(0\))g(is)f(outside)h(of)g(the)g Fk(r)1158 5278 y Fi(n)1205 5263 y Ft(-ball,)e(de\014ne)j Fk(q)1774 5212 y Fs(\()p Fi(n)p Fs(\))1770 5290 y Fi(k)1876 5263 y Ft(\()p Fk(t)p Ft(\))f Fg(\021)g Fk(q)2163 5278 y Fi(k)2206 5263 y Ft(\(0\))f(and)h Fk(p)2592 5212 y Fs(\()p Fi(n)p Fs(\))2592 5290 y Fi(k)2694 5263 y Ft(\()p Fk(t)p Ft(\))g Fg(\021)g Ft(0)g(for)f(all)f Fk(t)i Fg(\025)g Ft(0.)-246 5542 y Fp(Lemma)37 b(5.2.)50 b Fl(F)-7 b(or)24 b(e)-5 b(ach)25 b Fk(k)31 b Fg(2)d Fk(I)8 b Fl(,)27 b(and)e Fk(T)41 b(>)28 b Ft(0)p Fl(,)f(the)e(se)-5 b(quenc)g(e)25 b(of)g(c)-5 b(ontinuous)26 b(functions)f Ft(\()p Fk(q)3154 5491 y Fs(\()p Fi(n)p Fs(\))3150 5570 y Fi(k)3255 5542 y Ft(\()p Fg(\001)p Ft(\)\))3397 5506 y Fh(1)3397 5566 y Fi(n)p Fs(=1)-246 5741 y Fl(and)35 b Ft(\()p Fk(p)31 5690 y Fs(\()p Fi(n)p Fs(\))31 5769 y Fi(k)132 5741 y Ft(\()p Fg(\001)p Ft(\)\))274 5705 y Fh(1)274 5766 y Fi(n)p Fs(=1)446 5741 y Fl(on)f Ft([0)p Fk(;)17 b(T)d Ft(])35 b Fl(ar)-5 b(e)34 b(e)-5 b(qui-c)g(ontinuous.)p eop %%Page: 21 21 21 20 bop -246 -9 a Fm(Dynamics)32 b(of)g(In\014nite)h(P)m(article)f (Systems)2074 b Ft(21)-246 279 y Fp(Pro)s(of.)49 b Ft(F)-8 b(or)41 b(an)m(y)i Fg(\000)p Fk(T)57 b Fg(\024)45 b Fk(t)832 294 y Fs(1)915 279 y Fg(\024)f Fk(t)1071 294 y Fs(2)1154 279 y Fg(\024)g Fk(T)14 b Ft(,)44 b(it)d(follo)m(ws)g(from)f(Lemma)h (5.1,)j(Prop)s(osition)c(4.2,)k(and)-246 478 y(Prop)s(osition)31 b(4.3)h(that)653 734 y Fg(j)p Fk(q)728 683 y Fs(\()p Fi(n)p Fs(\))724 762 y Fi(k)829 734 y Ft(\()p Fk(t)902 749 y Fs(2)942 734 y Ft(\))22 b Fg(\000)g Fk(q)1148 683 y Fs(\()p Fi(n)p Fs(\))1144 762 y Fi(k)1250 734 y Ft(\()p Fk(t)1323 749 y Fs(1)1363 734 y Ft(\))p Fg(j)82 b(\024)1672 599 y Ff(Z)1771 625 y Fi(t)1796 634 y Fc(2)1727 824 y Fi(t)1752 833 y Fc(1)1852 734 y Fg(j)p Fk(p)1929 683 y Fs(\()p Fi(n)p Fs(\))1929 762 y Fi(k)2031 734 y Ft(\()p Fk(s)p Ft(\))p Fg(j)17 b Fk(ds)1511 959 y Fg(\024)84 b Fk(c)1714 974 y Fs(12)1805 959 y Fk(g)t Ft(\()p Fg(j)p Fk(q)1965 974 y Fi(k)2007 959 y Ft(\()p Fk(!)2106 974 y Fs(0)2145 959 y Ft(\))p Fg(j)p Ft(\))33 b Fg(j)p Fk(t)2345 974 y Fs(2)2406 959 y Fg(\000)23 b Fk(t)2541 974 y Fs(1)2581 959 y Fg(j)p Fk(:)697 b Ft(\(5.6\))-99 1214 y(On)32 b(the)h(other)g (hand)400 1470 y Fg(j)p Fk(p)477 1419 y Fs(\()p Fi(n)p Fs(\))477 1498 y Fi(k)579 1470 y Ft(\()p Fk(t)652 1485 y Fs(2)691 1470 y Ft(\))23 b Fg(\000)f Fk(p)900 1419 y Fs(\()p Fi(n)p Fs(\))900 1498 y Fi(k)1002 1470 y Ft(\()p Fk(t)1075 1485 y Fs(1)1115 1470 y Ft(\))p Fg(j)27 b(\024)1313 1376 y Ff(X)1322 1588 y Fi(j)t Fh(6)p Fs(=)p Fi(k)1474 1335 y Ff(Z)1573 1361 y Fi(t)1598 1370 y Fc(2)1529 1560 y Fi(t)1554 1569 y Fc(1)1654 1470 y Fg(j)17 b Ft(grad)f Fk(U)10 b Ft(\()p Fk(q)2066 1419 y Fs(\()p Fi(n)p Fs(\))2062 1496 y Fi(j)2168 1470 y Ft(\()p Fk(s)p 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4471 y Ft(\()p Fk(!)-38 4486 y Fi(t)-9 4471 y Fk(;)p 35 4416 65 4 v 17 w(!)99 4486 y Fi(t)129 4471 y Fk(;)17 b(m)p Ft(\))42 b(satis\014es)h (\(6.1\)-\(6.3\))e(with)h(suitably)g(c)m(hosen)i(sequences)h(\()p Fk(a)2660 4486 y Fi(m)2727 4471 y Ft(\))2765 4486 y Fi(m)p Fh(\025)p Fs(0)2922 4471 y Ft(,)g(\()p Fk(b)3073 4486 y Fi(m)3140 4471 y Ft(\))3178 4486 y Fi(m)p Fh(\025)p Fs(0)3377 4471 y Ft(and)-246 4670 y(constan)m(ts)36 b Fk(L)254 4685 y Fs(1)294 4670 y Ft(\()p Fk(R)q Ft(\))p Fk(;)17 b(L)555 4685 y Fs(2)595 4670 y Ft(\()p Fk(R)q Ft(\).)49 b(Giv)m(en)34 b Fk(R)f Fg(\025)e Fk(r)26 b Ft(+)e(1)34 b(and)h Fk(T)45 b(>)31 b Ft(0,)j(the)h(r.h.s.)51 b(of)34 b(\(6.6\))g(is)g(an)g(absolutely)-246 4870 y(con)m(tin)m(uous) 41 b(function)e(of)g Fk(t)h Fg(2)h Ft([0)p Fk(;)17 b(T)d Ft(],)40 b(th)m(us)h(w)m(e)g(can)f(di\013eren)m(tiate)f(it)g(almost)f (ev)m(erywhere.)68 b(By)40 b(a)-246 5069 y(notational)33 b(con)m(v)m(enience,)39 b(let)c(us)h(drop)g(the)g(subscript)g Fk(R)h Ft(in)d Fk(f)2158 5084 y Fi(R)2216 5069 y Ft(\()p Fg(\001)p Ft(\);)j Fk(f)11 b Ft(\()p Fg(\001)p Ft(\))32 b Fg(\021)h Fk(f)2737 5084 y Fi(R)2795 5069 y Ft(\()p Fg(\001)p Ft(\))i(\(cf.)52 b Fk(f)11 b Ft(\()p Fk(x;)17 b(\033)t Ft(\))36 b(in)-246 5268 y(\(3.1\)\))d(and)h(denote)g Fk(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))29 b(:=)h Fk(f)11 b Ft(\()p Fg(j)p Fk(q)1259 5283 y Fi(k)1301 5268 y Fg(j)23 b(\000)g Ft(2)r(~)-51 b Fk(cR)q Ft(\(1)22 b(+)h Fk(mg)t Ft(\()p Fk(m)p Ft(\)\)\),)p 2260 5187 59 4 v 34 w Fk(f)10 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))30 b(:=)f Fk(f)11 b Ft(\()p Fg(j)p 2943 5213 47 4 v Fk(q)2989 5292 y Fi(k)3032 5268 y Fg(j)22 b(\000)i Ft(2)r(~)-51 b Fk(c)o(R)q Ft(\(1)23 b(+)-246 5467 y Fk(mg)t Ft(\()p Fk(m)p Ft(\)\)\).)43 b(By)34 b(di\013eren)m(tiating)c(the)j(righ)m(t)f (hand)h(side)f(of)h(\(6.6\),)f(w)m(e)h(ha)m(v)m(e)349 5713 y Fk(d)p 332 5758 86 4 v 332 5849 a(dt)428 5780 y Ft(\()p Fk(\016)509 5795 y Fi(R)566 5780 y Ft(\()p Fk(t;)17 b(m)p Ft(\)\))28 b(=)g Fk(A)1049 5739 y Fs(\(1\))1143 5780 y Ft(\()p Fk(t;)17 b(m)p Ft(\))22 b(+)g Fk(A)1576 5739 y Fs(\(2\))1671 5780 y Ft(\()p Fk(t;)17 b(m)p Ft(\))22 b(+)g Fk(A)2104 5739 y Fs(\(3\))2199 5780 y Ft(\()p Fk(t;)17 b(m)p Ft(\))22 b(+)g Fk(A)2632 5739 y Fs(\(4\))2726 5780 y Ft(\()p Fk(t;)17 b(m)p Ft(\))367 b(\(6.7\))p eop %%Page: 24 24 24 23 bop -246 -9 a Ft(24)3236 b Fm(Bahn)33 b Fl(et)i(al)-246 279 y Ft(where)193 554 y Fk(A)266 513 y Fs(\(1\))360 554 y Ft(\()p Fk(t;)17 b(m)p Ft(\))83 b(=)842 459 y Ff(X)854 671 y Fi(k)r Fh(2)p Fi(I)1003 443 y Ff(\020)1090 486 y Fk(d)p 1072 531 86 4 v 1072 622 a(dt)1168 554 y(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))1565 443 y Ff(\021)p 1624 473 59 4 v 111 x Fk(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))2021 443 y Ff(\020)2080 554 y Fg(j)p Fk(q)2151 569 y Fi(k)2216 554 y Fg(\000)p 2316 499 47 4 v 23 w Fk(q)2363 577 y Fi(k)2405 554 y Fg(j)2433 569 y Fs(1)2494 554 y Ft(+)22 b Fg(j)p Fk(p)2669 569 y Fi(k)2734 554 y Fg(\000)p 2834 499 49 4 v 23 w Fk(p)2882 577 y Fi(k)2925 554 y Fg(j)2953 569 y Fs(1)2992 443 y Ff(\021)3052 554 y Fk(;)254 b Ft(\(6.8\))193 855 y Fk(A)266 814 y Fs(\(2\))360 855 y Ft(\()p Fk(t;)17 b(m)p Ft(\))83 b(=)842 760 y Ff(X)854 973 y Fi(k)r Fh(2)p Fi(I)1003 855 y Fk(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))1400 745 y Ff(\020)1486 788 y Fk(d)p 1469 832 86 4 v 1469 923 a(dt)p 1565 774 59 4 v 1565 855 a(f)10 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))1961 745 y Ff(\021)q(\020)2080 855 y Fg(j)p Fk(q)2151 870 y Fi(k)2216 855 y Fg(\000)p 2316 800 47 4 v 23 w Fk(q)2363 879 y Fi(k)2405 855 y Fg(j)2433 870 y Fs(1)2494 855 y Ft(+)22 b Fg(j)p Fk(p)2669 870 y Fi(k)2734 855 y Fg(\000)p 2834 800 49 4 v 23 w Fk(p)2882 879 y Fi(k)2925 855 y Fg(j)2953 870 y Fs(1)2992 745 y Ff(\021)3052 855 y Fk(;)193 1157 y(A)266 1115 y Fs(\(3\))360 1157 y Ft(\()p Fk(t;)17 b(m)p Ft(\))83 b(=)842 1062 y Ff(X)854 1274 y Fi(k)r Fh(2)p Fi(I)1003 1157 y Fk(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))p 1400 1075 59 4 v Fk(f)10 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))1824 1089 y Fk(d)p 1806 1134 86 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Fs(2)1879 2222 y Fg(\024)28 b Fk(c)2026 2237 y Fs(16)2118 2222 y Fk(g)t Ft(\()p Fg(j)p Fk(q)2278 2237 y Fi(k)2319 2222 y Fg(j)p Ft(\))2385 2181 y Fi(d=)p Fs(2)3333 2222 y Ft(\(6.9\))-246 2496 y(holds)33 b(for)f(an)m(y)h Fk(k)e Fg(2)d Fk(I)8 b Ft(,)32 b(w)m(e)i(use)f (\(6.5\))f(to)h(see)g(that)603 2771 y Fg(j)658 2703 y Fk(d)p 641 2748 86 4 v 641 2839 a(dt)736 2771 y(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))p Fg(j)27 b(\024)h Fk(c)1335 2786 y Fs(17)1427 2771 y Fk(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)22 b Ft(+)g(1\))32 b(\()p Fk(g)t Ft(\()p Fk(R)q Ft(\))17 b Fk(g)t Ft(\()p Fk(m)p Ft(\)\))2532 2730 y Fi(d=)p Fs(2)2642 2771 y Fk(;)616 b Ft(\(6.10\))-246 3045 y(and)603 3320 y Fg(j)658 3252 y Fk(d)p 641 3297 V 641 3388 a(dt)p 736 3239 59 4 v 736 3320 a(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))p Fg(j)27 b(\024)h Fk(c)1335 3335 y Fs(17)p 1443 3239 V 1443 3320 a Fk(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)22 b Ft(+)g(1\))33 b(\()p Fk(g)t Ft(\()p Fk(R)q Ft(\))17 b Fk(g)t Ft(\()p Fk(m)p 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y Ft(\()p Fk(t;)17 b(m)22 b Ft(+)g(1\))p Fk(:)313 b Ft(\(6.13\))-99 4692 y(Finally)-8 b(,)30 b(let)h(us)j(consider)f(the) g(last)e(term)h Fk(A)1556 4656 y Fs(\(4\))1651 4692 y Ft(\()p Fk(t;)17 b(m)p Ft(\))32 b(in)g(\(6.8\).)43 b(Observ)m(e)35 b(that)230 4967 y Fg(j)302 4899 y Fk(d)p 284 4944 86 4 v 284 5035 a(dt)380 4967 y Fg(j)p Fk(p)457 4982 y Fi(k)521 4967 y Fg(\000)p 621 4912 49 4 v 23 w Fk(p)670 4990 y Fi(k)713 4967 y Fg(j)741 4982 y Fs(1)796 4967 y Fg(j)83 b(\024)1068 4872 y Ff(X)1082 5084 y Fi(l)q Fh(6)p Fs(=)p Fi(k)1228 4967 y Fg(j)33 b Ft(grad)g Fk(U)1588 4926 y Fs(\(1\))1683 4967 y Ft(\()p Fk(q)1764 4982 y Fi(l)1812 4967 y Fg(\000)23 b Fk(q)1955 4982 y Fi(k)1998 4967 y Ft(\))f Fg(\000)g Ft(grad)33 b Fk(U)2456 4926 y Fs(\(1\))2551 4967 y Ft(\()p 2589 4912 47 4 v Fk(q)2636 4990 y Fi(l)2684 4967 y Fg(\000)p 2784 4912 V 23 w Fk(q)2831 4990 y Fi(k)2873 4967 y Ft(\))p Fg(j)1068 5245 y Ft(+)1161 5150 y Ff(X)1174 5363 y Fi(l)q Fh(6)p Fs(=)p Fi(k)1320 5245 y Fg(j)g Ft(grad)g Fk(U)1680 5204 y Fs(\(2\))1775 5245 y Ft(\()p Fk(q)1856 5260 y Fi(l)1905 5245 y Fg(\000)22 b Fk(q)2047 5260 y Fi(k)2090 5245 y Ft(\))g Fg(\000)h Ft(grad)33 b Fk(U)2549 5204 y Fs(\(2\))2644 5245 y Ft(\()p 2682 5190 V Fk(q)2728 5269 y Fi(l)2777 5245 y Fg(\000)p 2876 5190 V 22 w Fk(q)2923 5269 y Fi(k)2966 5245 y Ft(\))p Fg(j)p Fk(:)226 b Ft(\(6.14\))-246 5538 y(Th)m(us)907 5813 y Fg(j)p Fk(A)1008 5772 y Fs(\(4\))1103 5813 y Ft(\()p Fk(t;)17 b(m)p Ft(\))p Fg(j)27 b(\024)h Fk(Q)1580 5828 y Fs(1)1620 5813 y Ft(\()p Fk(t;)17 b(m)p Ft(\))22 b(+)g Fk(Q)2057 5828 y Fs(2)2097 5813 y Ft(\()p Fk(t;)17 b(m)p Ft(\))p Fk(;)921 b Ft(\(6.15\))p eop %%Page: 25 25 25 24 bop -246 -9 a Fm(Dynamics)32 b(of)g(In\014nite)h(P)m(article)f (Systems)2074 b Ft(25)-246 279 y(where)86 581 y Fk(Q)163 596 y Fs(1)203 581 y Ft(\()p Fk(t;)17 b(m)p Ft(\))2842 b(\(6.16\))86 814 y(=)190 719 y Ff(X)201 931 y Fi(k)r Fh(2)p Fi(I)350 643 y Ff( )429 814 y Fk(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))p 826 732 59 4 v Fk(f)10 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))1239 719 y Ff(X)1253 931 y Fi(l)q Fh(6)p Fs(=)p Fi(k)1399 703 y Ff(\020)1459 814 y Fg(j)33 b Ft(grad)g Fk(U)1819 772 y Fs(\(1\))1914 814 y Ft(\()p Fk(q)1995 829 y Fi(l)2043 814 y Fg(\000)22 b Fk(q)2185 829 y Fi(k)2228 814 y Ft(\))g Fg(\000)h Ft(grad)33 b Fk(U)2687 772 y Fs(\(1\))2782 814 y Ft(\()p 2820 759 47 4 v Fk(q)2867 837 y Fi(l)2915 814 y Fg(\000)p 3014 759 V 22 w Fk(q)3061 837 y Fi(k)3104 814 y Ft(\))p Fg(j)3170 703 y Ff(\021)3229 643 y(!)3325 814 y Fk(;)86 1062 y(Q)163 1077 y Fs(2)203 1062 y Ft(\()p Fk(t;)17 b(m)p Ft(\))86 1294 y(=)190 1200 y Ff(X)201 1412 y Fi(k)r Fh(2)p Fi(I)350 1124 y Ff( )429 1294 y Fk(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))p 826 1213 59 4 v Fk(f)10 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))1239 1200 y Ff(X)1253 1412 y Fi(l)q Fh(6)p Fs(=)p Fi(k)1399 1184 y Ff(\020)1459 1294 y Fg(j)33 b Ft(grad)g Fk(U)1819 1253 y Fs(\(2\))1914 1294 y Ft(\()p Fk(q)1995 1309 y Fi(l)2043 1294 y Fg(\000)22 b Fk(q)2185 1309 y Fi(k)2228 1294 y Ft(\))g Fg(\000)h Ft(grad)33 b Fk(U)2687 1253 y Fs(\(2\))2782 1294 y Ft(\()p 2820 1240 47 4 v Fk(q)2867 1318 y Fi(l)2915 1294 y Fg(\000)p 3014 1240 V 22 w Fk(q)3061 1318 y Fi(k)3104 1294 y Ft(\))p Fg(j)3170 1184 y Ff(\021)3229 1124 y(!)3325 1294 y Fk(:)-99 1622 y Ft(In)g(order)f(to)g(con)m(trol)g(the)h(\014rst)g(term)f Fk(Q)1403 1637 y Fs(1)1443 1622 y Ft(\()p Fk(t;)17 b(m)p Ft(\),)33 b(w)m(e)h(use)f(\(2.5\))f(to)g(see)i(that)294 1829 y Ff(X)308 2041 y Fi(l)q Fh(6)p Fs(=)p Fi(k)454 1924 y Fg(j)f Ft(grad)g Fk(U)814 1882 y Fs(\(1\))909 1924 y Ft(\()p Fk(q)990 1939 y Fi(l)1038 1924 y Fg(\000)23 b Fk(q)1181 1939 y Fi(k)1224 1924 y Ft(\))f Fg(\000)g Ft(grad)33 b Fk(U)1682 1882 y Fs(\(1\))1777 1924 y Ft(\()p 1815 1869 V Fk(q)1862 1947 y Fi(l)1910 1924 y Fg(\000)p 2010 1869 V 23 w Fk(q)2057 1947 y Fi(k)2099 1924 y Ft(\))p Fg(j)294 2202 y(\024)28 b Fk(L)515 2107 y Ff(X)529 2319 y Fi(l)q Fh(6)p Fs(=)p Fi(k)659 2202 y Ft([)17 b Fk(L)22 b Ft(+)g Fk(U)965 2161 y Fs(\(1\))1060 2202 y Ft(\()p 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Fk(f)1432 4050 y Ff(\000)1477 4131 y Fg(j)p 1505 4076 V Fk(q)1551 4154 y Fi(k)1594 4131 y Fg(j)g(\000)h Ft(2)r(~)-51 b Fk(c)o(R)q Ft(\(1)22 b(+)g Fk(mg)t Ft(\()p Fk(m)p Ft(\)\))2451 4050 y Ff(\001)156 4400 y Fg(\002)250 4305 y Ff(X)264 4518 y Fi(l)q Fh(6)p Fs(=)p Fi(k)394 4400 y Ft([)p Fk(L)g Ft(+)g Fk(U)683 4359 y Fs(\(1\))778 4400 y Ft(\()p Fk(q)859 4415 y Fi(l)908 4400 y Fg(\000)g Fk(q)1050 4415 y Fi(k)1093 4400 y Ft(\))g(+)g Fk(U)1327 4359 y Fs(\(1\))1422 4400 y Ft(\()p 1460 4345 V Fk(q)1507 4424 y Fi(l)1555 4400 y Fg(\000)p 1655 4345 V 23 w Fk(q)1702 4424 y Fi(k)1744 4400 y Ft(\)])1809 4359 y Fi(c)1877 4400 y Fg(j)p Fk(q)1948 4415 y Fi(l)1996 4400 y Fg(\000)p 2096 4345 V 23 w Fk(q)2143 4424 y Fi(l)2169 4400 y Fg(j)2197 4415 y Fs(1)58 4678 y Fg(\024)163 4584 y Ff(X)183 4796 y Fi(l)q Fh(2)p Fi(I)324 4678 y Fk(f)383 4598 y Ff(\000)428 4678 y Fg(j)p Fk(q)499 4693 y Fi(l)525 4678 y Fg(j)g(\000)h Ft(2)r(~)-51 b Fk(c)o(R)q Ft(\(1)22 b(+)g(\()p Fk(m)h Ft(+)f(1\))p Fk(g)t Ft(\()p Fk(m)f Ft(+)h(1\)\))1796 4598 y Ff(\001)1875 4678 y Fk(f)1934 4598 y Ff(\000)1979 4678 y Fg(j)p 2007 4623 V Fk(q)2054 4702 y Fi(l)2080 4678 y Fg(j)f(\000)i Ft(2)r(~)-51 b Fk(cR)q Ft(\(1)22 b(+)g(\()p Fk(m)g Ft(+)g(1\))p Fk(g)t Ft(\()p Fk(m)g Ft(+)g(1\)\))3351 4598 y Ff(\001)156 4948 y Fg(\002j)p Fk(q)304 4963 y Fi(l)352 4948 y Fg(\000)p 452 4893 V 23 w Fk(q)499 4971 y Fi(l)525 4948 y Fg(j)553 4963 y Fs(1)642 4853 y Ff(X)656 5065 y Fi(k)r Fh(6)p Fs(=)p Fi(l)786 4948 y Ft([)p Fk(L)g Ft(+)g Fk(U)1075 4907 y Fs(\(1\))1170 4948 y Ft(\()p Fk(q)1251 4963 y Fi(l)1300 4948 y Fg(\000)g Fk(q)1442 4963 y Fi(k)1485 4948 y Ft(\))g(+)g Fk(U)1719 4907 y Fs(\(1\))1814 4948 y Ft(\()p 1852 4893 V Fk(q)1899 4971 y Fi(l)1947 4948 y Fg(\000)p 2047 4893 V 23 w Fk(q)2094 4971 y Fi(k)2136 4948 y Ft(\)])2201 4907 y Fi(c)58 5209 y Fg(\024)28 b Fk(c)205 5224 y Fs(20)313 5209 y Fk(g)t Ft(\()p Fk(R)q Ft(\))515 5168 y Fi(cd)619 5209 y Fk(g)t Ft(\()p Fk(m)p Ft(\))831 5168 y Fi(cd)935 5209 y Fk(\016)978 5224 y Fi(R)1035 5209 y Ft(\()p Fk(t;)17 b(m)23 b Ft(+)f(1\))p Fk(:)1813 b Ft(\(6.19\))-246 5511 y(Hence)34 b(b)m(y)g(\(6.16\)-\(6.19\),)c(w)m(e)k(obtain)e(that)800 5813 y Fk(Q)877 5828 y Fs(1)917 5813 y Ft(\()p Fk(t;)17 b(m)p Ft(\))28 b Fg(\024)g Fk(c)1332 5828 y Fs(21)1423 5813 y Fk(g)t Ft(\()p Fk(R)q Ft(\))1625 5772 y Fi(cd)1695 5813 y Fk(g)t Ft(\()p Fk(m)p Ft(\))1907 5772 y Fi(cd)1978 5813 y Fk(\016)2021 5828 y Fi(R)2079 5813 y Ft(\()p Fk(t;)17 b(m)22 b Ft(+)g(1\))797 b(\(6.20\))p eop %%Page: 26 26 26 25 bop -246 -9 a Ft(26)3236 b Fm(Bahn)33 b Fl(et)i(al)-99 279 y Ft(Let)d(us)i(estimate)d(the)i(second)h(term)e Fk(Q)1384 294 y Fs(2)1424 279 y Ft(\()p Fk(t;)17 b(m)p Ft(\))33 b(in)e(\(6.16\).)43 b(W)-8 b(e)33 b(use)h(\(2.6\))e(to)g (obtain)-80 448 y Ff(X)-65 660 y Fi(l)q Fh(6)p Fs(=)p Fi(k)81 543 y Fg(j)h Ft(grad)g Fk(U)441 502 y Fs(\(2\))536 543 y Ft(\()p Fk(q)617 558 y Fi(l)665 543 y Fg(\000)23 b Fk(q)808 558 y Fi(k)851 543 y Ft(\))f Fg(\000)g Ft(grad)33 b Fk(U)1309 502 y Fs(\(2\))1404 543 y Ft(\()p 1442 488 47 4 v Fk(q)1489 566 y Fi(l)1537 543 y Fg(\000)p 1637 488 V 23 w Fk(q)1683 566 y Fi(k)1726 543 y Ft(\))17 b Fg(j)-80 821 y(\024)26 727 y Ff(X)40 939 y Fi(l)q Fh(6)p Fs(=)p Fi(k)169 821 y Ft(\()p Fk( )t Ft(\()p Fg(j)p Fk(q)383 836 y Fi(l)431 821 y Fg(\000)23 b Fk(q)574 836 y Fi(k)617 821 y Fg(j)p Ft(\))f(+)g Fk( )t Ft(\()p Fg(j)p 936 766 V Fk(q)982 845 y Fi(l)1030 821 y Fg(\000)p 1130 766 V 23 w Fk(q)1177 845 y Fi(k)1219 821 y Fg(j)p Ft(\)\))33 b([)p Fg(j)p Fk(q)1454 836 y Fi(l)1502 821 y Fg(\000)p 1602 766 V 23 w Fk(q)1649 845 y Fi(l)1675 821 y Fg(j)1703 836 y Fs(1)1764 821 y Ft(+)22 b Fg(j)p Fk(q)1933 836 y Fi(k)1998 821 y Fg(\000)p 2097 766 V 22 w Fk(q)2144 845 y Fi(k)2187 821 y Fg(j)2215 836 y Fs(1)2254 821 y Ft(])-80 1109 y(=)24 999 y Ff(\020)100 1015 y(X)114 1227 y Fi(l)q Fh(6)p Fs(=)p Fi(k)261 1109 y Fk( )t Ft(\()p Fg(j)p Fk(q)437 1124 y Fi(l)485 1109 y Fg(\000)g Fk(q)627 1124 y Fi(k)670 1109 y Fg(j)p Ft(\))g(+)g Fk( )t Ft(\()p Fg(j)p 989 1054 V Fk(q)1035 1133 y Fi(l)1083 1109 y Fg(\000)p 1183 1054 V 23 w Fk(q)1230 1133 y Fi(k)1273 1109 y Fg(j)p Ft(\))1339 999 y Ff(\021)1398 1109 y Fg(j)p Fk(q)1469 1124 y Fi(k)1533 1109 y Fg(\000)p 1633 1054 V 23 w Fk(q)1680 1133 y Fi(k)1722 1109 y Fg(j)1750 1124 y Fs(1)1812 1109 y Ft(+)1910 1015 y Ff(X)1924 1227 y Fi(l)q Fh(6)p Fs(=)p Fi(k)2054 1109 y Ft(\()p Fk( )t Ft(\()p Fg(j)p Fk(q)2268 1124 y Fi(l)2316 1109 y Fg(\000)g Fk(q)2458 1124 y Fi(k)2501 1109 y Fg(j)p Ft(\))g(+)g Fk( )t Ft(\()p Fg(j)p 2820 1054 V Fk(q)2866 1133 y Fi(l)2914 1109 y Fg(\000)p 3014 1054 V 23 w Fk(q)3061 1133 y Fi(k)3104 1109 y Fg(j)p Ft(\)\))p Fg(j)p Fk(q)3279 1124 y Fi(l)3326 1109 y Fg(\000)p 3426 1054 V 23 w Fk(q)3473 1133 y Fi(l)3499 1109 y Fg(j)3527 1124 y Fs(1)-80 1358 y Fg(\021)29 b Fk(S)86 1373 y Fs(1)125 1358 y Ft(\()p Fk(t;)17 b(k)s Ft(\))22 b(+)g Fk(S)514 1373 y Fs(2)553 1358 y Ft(\()p Fk(t;)17 b(k)s Ft(\))p Fk(:)2527 b Ft(\(6.21\))-246 1621 y(Th)m(us,)35 b(b)m(y)e(the)g (de\014nition)f(of)g Fk(Q)954 1636 y Fs(2)993 1621 y Ft(\()p Fk(t;)17 b(m)p Ft(\))33 b(in)f(\(6.16\),)g(w)m(e)h(ha)m(v)m(e) 13 1885 y Fk(Q)90 1900 y Fs(2)130 1885 y Ft(\()p Fk(t;)17 b(m)p Ft(\))83 b Fg(\024)614 1791 y Ff(X)625 2003 y Fi(k)r Fh(2)p Fi(I)774 1885 y Fk(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))p 1171 1804 59 4 v Fk(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))p Fk(S)1628 1900 y Fs(1)1667 1885 y Ft(\()p Fk(t;)g(k)s Ft(\))22 b(+)1996 1791 y Ff(X)2007 2003 y Fi(k)r Fh(2)p Fi(I)2157 1885 y Fk(f)11 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))p 2554 1804 V Fk(f)10 b Ft(\()p Fk(t;)17 b(k)s(;)g(m)p Ft(\))g Fk(S)3027 1900 y Fs(2)3066 1885 y Ft(\()p Fk(t;)g(k)s Ft(\))453 2151 y Fg(\021)84 b Fk(Q)691 2100 y Fs(\(1\))691 2175 y(2)785 2151 y Ft(\()p Fk(t;)17 b(m)p Ft(\))23 b(+)f Fk(Q)1223 2100 y Fs(\(2\))1223 2175 y(2)1317 2151 y Ft(\()p Fk(t;)17 b(m)p Ft(\))p Fk(:)1701 b Ft(\(6.22\))-246 2415 y(In)33 b(order)g(to)f(con)m(trol)g Fk(Q)655 2364 y Fs(\(1\))655 2439 y(2)750 2415 y Ft(\()p Fk(t;)17 b(m)p Ft(\),)32 b(w)m(e)i(notice)e(that)h(b)m(y)g(Lemma)e(3.1)h(\(a\))h(and)f(\(2.6\),) g(the)h(b)s(ound)912 2678 y Fk( )t Ft(\()p Fg(j)p Fk(q)1088 2693 y Fi(l)1136 2678 y Fg(\000)23 b Fk(q)1279 2693 y Fi(k)1322 2678 y Fg(j)p Ft(\))k Fg(\024)h Fk(c)1562 2693 y Fs(22)1653 2678 y Fk(f)11 b Ft(\()p Fk(q)1793 2693 y Fi(l)1841 2678 y Fg(\000)23 b Fk(q)1984 2693 y Fi(k)2027 2678 y Fk(;)17 b(g)t Ft(\()p Fg(j)p Fk(q)2231 2693 y Fi(k)2273 2678 y Fg(j)p Ft(\)\))908 b(\(6.23\))-246 2942 y(holds.)44 b(W)-8 b(e)32 b(then)i(use)f(the)g(ab)s(o)m(v)m(e)h(b)s (ound,)f(Sc)m(h)m(w)m(arz)h(inequalit)m(y)e(and)g(Lemma)g(3.2)g(to)g (see)i(that)768 3111 y Ff(X)782 3323 y Fi(l)q Fh(6)p Fs(=)p Fi(k)928 3206 y Fk( )t Ft(\()p Fg(j)p Fk(q)1104 3221 y Fi(l)1152 3206 y Fg(\000)23 b Fk(q)1295 3221 y Fi(k)1338 3206 y Fg(j)p Ft(\))k Fg(\024)h Fk(c)1578 3221 y Fs(23)1670 3206 y Fk(W)14 b Ft(\()p Fk(!)1875 3221 y Fi(t)1904 3206 y Fk(;)j(q)1991 3221 y Fi(k)2033 3206 y Fk(;)g(g)t Ft(\()p Fg(j)p Fk(q)2237 3221 y Fi(k)2279 3206 y Fg(j)p Ft(\)\))2383 3165 y Fi(d=)p Fs(2)2493 3206 y Fk(:)765 b Ft(\(6.24\))-246 3488 y(Rep)s(eating)32 b(the)h(same)f(calculation)f(after)h(a)g(c)m(hange)i Fk(!)1820 3503 y Fi(t)1882 3488 y Ft(b)m(y)p 2017 3434 65 4 v 33 w Fk(!)2081 3503 y Fi(t)2111 3488 y Ft(,)f(w)m(e)g(conclude)g (that)814 3752 y Fk(S)874 3767 y Fs(1)914 3752 y Ft(\()p Fk(t;)17 b(k)s Ft(\))27 b Fg(\024)h Fk(c)1297 3767 y Fs(24)1389 3752 y Ft(\()p Fk(g)t Ft(\()p Fk(R)q Ft(\))17 b Fk(g)t Ft(\()p Fk(m)p Ft(\)\))1896 3711 y Fi(d=)p Fs(2)2038 3752 y Fg(j)p Fk(q)2109 3767 y Fi(k)2174 3752 y Fg(\000)p 2274 3697 47 4 v 23 w Fk(q)2320 3776 y Fi(k)2363 3752 y Fg(j)2391 3767 y Fs(1)2430 3752 y Fk(;)-246 4016 y Ft(and)33 b(so)737 4280 y Fk(Q)814 4229 y Fs(\(1\))814 4304 y(2)909 4280 y Ft(\()p Fk(t;)17 b(m)p Ft(\))28 b Fg(\024)g Fk(c)1324 4295 y Fs(24)1415 4280 y Ft(\()p Fk(g)t Ft(\()p Fk(R)q Ft(\))p Fk(g)t Ft(\()p Fk(m)p Ft(\)\))1905 4239 y Fs(1)p Fi(=)p Fs(2)2014 4280 y Fk(\016)2057 4295 y Fi(R)2115 4280 y Ft(\()p Fk(t;)17 b(m)22 b Ft(+)g(1\))p Fk(:)734 b Ft(\(6.25\))-99 4544 y(Next,)33 b(w)m(e)h(consider)f Fk(Q)767 4493 y Fs(\(2\))767 4568 y(2)861 4544 y Ft(\()p Fk(t;)17 b(m)p Ft(\).)44 b(W)-8 b(e)33 b(write)277 4807 y Fk(S)337 4822 y Fs(2)376 4807 y Ft(\()p Fk(t;)17 b(k)s Ft(\))84 b(=)829 4713 y Ff(X)843 4925 y Fi(l)q Fh(6)p Fs(=)p Fi(k)989 4807 y Fk( )t Ft(\()p Fg(j)p Fk(q)1165 4822 y Fi(l)1213 4807 y Fg(\000)23 b Fk(q)1356 4822 y Fi(k)1399 4807 y Fg(j)p Ft(\))33 b Fg(j)p Fk(q)1569 4822 y Fi(l)1617 4807 y Fg(\000)p 1716 4752 V 22 w Fk(q)1763 4831 y Fi(l)1789 4807 y Fg(j)1817 4822 y Fs(1)1879 4807 y Ft(+)1977 4713 y Ff(X)1991 4925 y Fi(l)q Fh(6)p Fs(=)p Fi(k)2137 4807 y Fk( )t Ft(\()p Fg(j)p 2270 4752 V Fk(q)2316 4831 y Fi(l)2365 4807 y Fg(\000)p 2464 4752 V 22 w Fk(q)2511 4831 y Fi(k)2554 4807 y Fg(j)p Ft(\))g Fg(j)p Fk(q)2724 4822 y Fi(l)2772 4807 y Fg(\000)p 2871 4752 V 22 w Fk(q)2918 4831 y Fi(l)2944 4807 y Fg(j)2972 4822 y Fs(1)668 5056 y Fg(\021)84 b Fk(S)889 5071 y Fs(2)p Fi(;)p Fs(1)983 5056 y Ft(\()p 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27 27 26 bop -246 -9 a Fm(Dynamics)32 b(of)g(In\014nite)h(P)m(article)f (Systems)2074 b Ft(27)-246 280 y(By)38 b(symmetry)-8 b(,)39 b(it)e(is)g(enough)h(to)f(con)m(trol)f(only)h Fk(Q)1692 229 y Fs(\(2)p Fi(;)p Fs(1\))1692 305 y(2)1842 280 y Ft(\()p Fk(t;)17 b(m)p Ft(\).)58 b(F)-8 b(or)37 b(that)g(purp)s(ose,)j(let)c(us)i(further)-246 480 y(decomp)s(ose)33 b Fk(S)305 495 y Fs(2)p Fi(;)p Fs(1)432 480 y Ft(as)g(follo)m(ws:)18 779 y Fk(S)78 794 y Fs(2)p Fi(;)p Fs(1)172 779 y Ft(\()p Fk(t;)17 b(k)s Ft(\))28 b(=)672 684 y Ff(X)512 900 y Fi(l)q Fh(6)p Fs(=)p Fi(k)r Fs(:)p Fh(j)p Fi(q)700 912 y Fe(l)722 900 y Fh(\000)p Fi(q)809 912 y Fe(k)847 900 y Fh(j\024)p Fi(R)992 779 y Fk( )t Ft(\()p Fg(j)p Fk(q)1168 794 y Fi(l)1216 779 y Fg(\000)22 b Fk(q)1358 794 y Fi(k)1401 779 y Fg(j)p Ft(\))33 b Fg(j)p Fk(q)1571 794 y Fi(l)1619 779 y Fg(\000)p 1719 724 47 4 v 23 w Fk(q)1765 802 y Fi(l)1791 779 y Fg(j)1819 794 y Fs(1)1881 779 y Ft(+)2097 684 y Ff(X)1979 900 y Fi(l)q Fs(:)11 b Fh(j)p Fi(q)2084 912 y 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Fs(2)1442 2176 y Ft(\()p Fk(t;)17 b(m)p Ft(\))p Fk(:)1576 b Ft(\(6.28\))-246 2475 y(By)39 b(Prop)s(osition)d(4.3,)i(it) f(is)g(not)h(hard)g(\(and)g(v)m(ery)h(imp)s(ortan)m(t\))d(to)h(see)i (that)f(if)e Fk(f)2864 2395 y Ff(\000)2910 2475 y Fg(j)p Fk(q)2981 2490 y Fi(k)3023 2475 y Fg(j)26 b(\000)g Ft(2)r(~)-51 b Fk(cR)q Ft(\(1)25 b(+)-246 2674 y Fk(mg)t Ft(\()p Fk(m)p Ft(\)\))89 2594 y Ff(\001)169 2674 y Fg(6)p Ft(=)34 b(0,)j Fk(f)451 2594 y Ff(\000)496 2674 y Fg(j)p 524 2620 V Fk(q)571 2698 y Fi(k)613 2674 y Fg(j)25 b(\000)g Ft(2)r(~)-51 b Fk(cR)q Ft(\(1)24 b(+)h Fk(mg)t Ft(\()p Fk(m)p Ft(\)\))1481 2594 y Ff(\001)1560 2674 y Fg(6)p Ft(=)34 b(0,)j(and)g Fg(j)p Fk(q)2048 2689 y Fi(l)2098 2674 y Fg(\000)25 b Fk(q)2243 2689 y Fi(k)2286 2674 y Fg(j)34 b(\024)h Fk(R)q Ft(,)i(then)g Fk(f)2884 2594 y Ff(\000)2929 2674 y Fg(j)p Fk(q)3000 2689 y Fi(l)3026 2674 y Fg(j)25 b(\000)g Ft(2)r(~)-51 b Fk(cR)q Ft(\(1)24 b(+)-246 2874 y(\()p Fk(m)c Ft(+)f(1\))p Fk(g)t Ft(\()p Fk(m)f Ft(+)h(1\)\))491 2793 y 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Fi(t)1321 5254 y Ft(\))33 b Fk( )t Ft(\()p Fg(j)p Fk(l)24 b Fg(\000)f Fk(q)1721 5269 y Fi(k)1764 5254 y Fg(j)p Ft(\))33 b Fk(g)t Ft(\()p Fg(j)p Fk(l)23 b Fg(\000)f Fk(q)2174 5269 y Fi(k)2217 5254 y Fg(j)p Ft(\))-39 5577 y Fg(\024)28 b Fk(c)108 5592 y Fs(27)200 5577 y Fk(g)t Ft(\()p Fk(R)q Ft(\))p Fk(g)t Ft(\()p Fk(m)p Ft(\))614 5466 y Ff(\020)784 5482 y(X)688 5698 y Fi(l)q Fs(:)p Fh(j)p Fi(l)q Fh(\000)p Fi(q)859 5710 y Fe(k)896 5698 y Fh(j\025)p Fi(R)1040 5577 y Fk( )t Ft(\()p Fg(j)p Fk(l)c Fg(\000)f Fk(q)1369 5592 y Fi(k)1412 5577 y Fg(j)p Ft(\))p Fk(g)t Ft(\()p Fg(j)p Fk(l)g Fg(\000)g Fk(q)1790 5592 y Fi(k)1832 5577 y Fg(j)p Ft(\))1898 5536 y Fs(2)1937 5466 y Ff(\021)1997 5489 y Fs(1)p Fi(=)p Fs(2)2107 5466 y Ff(\020)2186 5482 y(X)2183 5702 y Fi(l)q Fh(2)p Fd(Z)2302 5683 y Fe(d)2349 5577 y Fk(n)2407 5592 y Fi(l)2433 5577 y Ft(\()p Fk(!)2532 5592 y Fi(t)2562 5577 y Ft(\))2600 5536 y Fs(2)2639 5577 y Fk(f)11 b Ft(\()p Fk(l)24 b Fg(\000)f Fk(q)2932 5592 y Fi(k)2975 5577 y Fk(;)17 b(g)t Ft(\()p Fg(j)p Fk(q)3179 5592 y Fi(k)3220 5577 y Fg(j)p Ft(\)\))3324 5466 y Ff(\021)3383 5489 y Fs(1)p Fi(=)p Fs(2)-39 5849 y Fg(\024)28 b Fk(c)108 5808 y Fh(0)108 5873 y Fs(27)200 5849 y Fk(g)t Ft(\()p Fk(R)q Ft(\))p Fk(g)t Ft(\()p Fk(m)p Ft(\)[)p Fk( )t Ft(\()p Fk(R)q Ft(\))p Fk(R)934 5808 y Fi(d)p Fs(+)p Fi(\017)1054 5817 y Fc(1)1091 5849 y Ft(])1118 5808 y Fs(1)p Fi(=)p Fs(2)1228 5849 y Ft([)p Fk(g)t Ft(\()p Fk(R)q Ft(\))p Fk(g)t Ft(\()p Fk(m)p Ft(\)])1696 5808 y Fi(d=)p Fs(2)1806 5849 y Fk(;)p eop %%Page: 28 28 28 27 bop -246 -9 a Ft(28)3236 b Fm(Bahn)33 b Fl(et)i(al)-246 279 y Ft(where)i(0)32 b Fk(<)f(\017)266 294 y Fs(1)338 279 y Fk(<)h Ft(1)j(is)g(a)g(small)d(constan)m(t.)52 b(Substituting)35 b(the)g(ab)s(o)m(v)m(e)h(b)s(ound)g(in)m(to)e(the)i (expression)-246 478 y(of)c Fk(D)-54 493 y Fs(2)-14 478 y Ft(\()p Fk(t;)17 b(m)p Ft(\))33 b(in)e(\(6.28\),)h(w)m(e)i(conclude)f (that)126 739 y Fk(D)207 754 y Fs(2)246 739 y Ft(\()p Fk(t;)17 b(m)p Ft(\))83 b Fg(\024)h Fk(c)772 698 y Fh(0)772 764 y Fs(28)863 739 y Fk(g)t Ft(\()p Fk(R)q Ft(\))1065 698 y Fs(2)1104 739 y Fk(g)t Ft(\()p Fk(m)p Ft(\))1316 698 y Fs(2)1355 739 y Ft([)p Fk( )t Ft(\()p Fk(R)q Ft(\))p Fk(R)1675 698 y Fi(d)p Fs(+)p Fi(\017)1795 707 y Fc(1)1833 739 y Ft(])1860 698 y Fs(1)p Fi(=)p Fs(2)2369 645 y Ff(X)1987 864 y Fi(l)q Fh(2)p Fd(Z)2106 845 y Fe(d)2136 864 y Fs(:)p Fh(j)p Fi(l)q Fh(j\024)p Fs(2)r(~)-37 b Fi(c)o(R)p Fs(\(1+)p Fi(mg)r Fs(\()p Fi(m)p Fs(\)\)+3)p Fi(R)2912 739 y Fk(n)2970 754 y Fi(l)2996 739 y Ft(\()p Fk(!)3095 754 y Fi(t)3125 739 y Ft(\))569 1015 y Fg(\024)84 b Fk(c)772 1030 y Fs(28)863 1015 y Fk(g)t Ft(\()p Fk(R)q Ft(\))1065 974 y Fs(2)1104 1015 y Fk(g)t Ft(\()p Fk(m)p Ft(\))1316 974 y Fs(2)1355 1015 y Ft([)p Fk( )t Ft(\()p Fk(R)q Ft(\))p Fk(R)1675 974 y Fi(d)p Fs(+)p Fi(\017)1795 983 y Fc(1)1833 1015 y Ft(])1860 974 y Fs(1)p Fi(=)p Fs(2)1970 1015 y Ft([)p Fk(R)q(mg)t Ft(\()p Fk(m)p Ft(\)])2396 974 y Fi(d)2437 1015 y Fk(:)821 b Ft(\(6.30\))-246 1276 y(Here)38 b(w)m(e)h(ha)m(v)m(e) 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b(from)f(\(6.13\),)g(\(6.15\),)f(\(6.20\),)h(\(6.25\),)g (and)h(\(6.31\))f(w)m(e)h(conclude)g(that)73 2482 y Fg(j)129 2415 y Fk(d)p 111 2459 86 4 v 111 2550 a(dt)207 2482 y(\016)250 2497 y Fi(R)308 2482 y Ft(\()p Fk(t;)17 b(m)p Ft(\))p Fg(j)83 b(\024)p 819 2402 110 4 v 83 w Fk(C)889 2497 y Fs(1)928 2482 y Ft([)p Fk(g)t Ft(\()p Fk(R)q Ft(\))1157 2441 y Fi(cd)1228 2482 y Fk(g)t Ft(\()p Fk(m)p Ft(\))1440 2441 y Fi(cd)1532 2482 y Ft(+)22 b(\()p Fk(g)t Ft(\()p Fk(R)q Ft(\))p Fk(g)t Ft(\()p Fk(m)p Ft(\)\))2120 2441 y Fi(d=)p Fs(2)2230 2482 y Ft(])17 b Fk(\016)2317 2497 y Fi(R)2375 2482 y Ft(\()p Fk(t;)g(m)22 b Ft(+)g(1\))869 2706 y(+)p 945 2626 77 4 v Fk(C)1022 2721 y Fs(2)1078 2706 y Fk(g)t Ft(\()p Fk(R)q Ft(\))1280 2665 y Fs(2)1318 2706 y Fk(g)t Ft(\()p Fk(m)p Ft(\))1530 2665 y Fs(2)1586 2706 y Ft([)p Fk( )t Ft(\()p Fk(R)q Ft(\))p Fk(R)1906 2665 y Fi(d)p Fs(+)p Fi(\017)2026 2674 y Fc(1)2065 2706 y Ft(])2092 2665 y Fs(1)p Fi(=)p Fs(2)2218 2706 y Ft([)p Fk(R)q(mg)t Ft(\()p Fk(m)p Ft(\)])2644 2665 y Fi(d)659 2930 y Fg(\024)83 b Fk(C)889 2945 y Fs(1)928 2930 y Ft(\()p Fk(g)t Ft(\()p Fk(R)q Ft(\))1168 2889 y Fi(cd)1272 2930 y Fk(g)t Ft(\()p Fk(m)p Ft(\))1484 2889 y Fi(cd)1554 2930 y Ft(\))33 b Fk(\016)1668 2945 y Fi(R)1726 2930 y Ft(\()p Fk(t;)17 b(m)23 b Ft(+)f(1\))f(+)i Fk(C)2326 2945 y Fs(2)2381 2930 y Fk(m)2466 2889 y Fs(2)p Fi(d)2576 2930 y Ft([)p Fk( )t Ft(\()p Fk(R)q Ft(\)])2848 2889 y Fs(1)p Fi(=)p Fs(2)2958 2930 y Fk(R)3033 2889 y Fi(d)p Fs(+)3134 2862 y Fc(2)p 3134 2874 31 3 v 3134 2915 a(3)3174 2889 y Fi(d)3285 2930 y Ft(\(6.32\))-246 3191 y(where)p 36 3111 110 4 v 34 w Fk(C)106 3206 y Fs(1)145 3191 y Fk(;)17 b(C)259 3206 y Fs(1)298 3191 y Ft(,)p 358 3111 77 4 v 33 w Fk(C)435 3206 y Fs(2)507 3191 y Ft(and)32 b Fk(C)766 3206 y Fs(2)838 3191 y Ft(are)h(p)s(ositiv)m(e)f(constan)m (ts.)-99 3391 y(T)-8 b(o)35 b(use)h(the)f(iteration)e(metho)s(d,)i(w)m (e)i(see)f(that)e(for)h(all)e(0)f Fg(\024)g Fk(t)g Fg(\024)g Fk(T)14 b Ft(,)36 b(the)f(sequence)j Fk(\016)3101 3406 y Fi(R)3159 3391 y Ft(\()p Fk(t;)17 b(m)p Ft(\))32 b(:=)-246 3590 y Fk(d)-195 3605 y Fi(R)-137 3590 y Ft(\()p Fk(!)-38 3605 y Fi(t)-9 3590 y Fk(;)p 35 3535 65 4 v 17 w(!)99 3605 y Fi(t)129 3590 y Fk(;)17 b(m)p Ft(\))35 b(satis\014es)h(the)g (relation)d(\(6.1\))i(with)f Fk(L)1750 3605 y Fs(1)1822 3590 y Ft(=)f Fk(C)2001 3605 y Fs(1)2056 3590 y Fk(g)t Ft(\()p Fk(R)q Ft(\))2258 3554 y Fi(cd)2329 3590 y Ft(,)j Fk(L)2458 3605 y Fs(2)2530 3590 y Ft(=)c Fk(T)47 b(C)2812 3605 y Fs(2)2868 3590 y Ft([)p Fk( )t Ft(\()p Fk(R)q Ft(\)])3140 3554 y Fs(1)p Fi(=)p Fs(2)3250 3590 y Fk(R)3325 3553 y Fi(d)p Fs(+)3426 3526 y Fc(2)p 3426 3538 31 3 v 3426 3579 a(3)3466 3553 y Fi(d)3507 3590 y Ft(,)-246 3789 y Fk(a)-195 3804 y Fi(m)-87 3789 y Ft(=)41 b Fk(g)t Ft(\()p Fk(m)p Ft(\))242 3753 y Fi(cd)312 3789 y Ft(,)i(and)d Fk(b)620 3804 y Fi(m)728 3789 y Ft(=)h Fk(m)930 3753 y Fs(2)p Fi(d)1006 3789 y Ft(.)67 b(It)40 b(is)g(easy)h(to)f(c)m(hec)m (k)j(that)d(the)h(conditions)e(in)h(\(6.2\)-\(6.3\))f(are)-246 3988 y(ful\014lled.)j(Th)m(us)34 b(b)m(y)f(\(6.4\))f(w)m(e)i(ha)m(v)m (e)110 4249 y Fk(\016)153 4264 y Fi(R)211 4249 y Ft(\()p Fk(!)310 4264 y Fi(t)339 4249 y Fk(;)p 383 4194 65 4 v 17 w(!)447 4264 y Fi(t)477 4249 y Fk(;)17 b(m)p Ft(\))83 b Fg(\024)928 4125 y Fh(1)892 4155 y Ff(X)887 4364 y Fi(m)p Fs(=0)1057 4249 y Fk(\016)1100 4264 y Fi(R)1158 4249 y Ft(\()p Fk(!)1257 4264 y Fs(0)1296 4249 y Fk(;)p 1340 4194 V 17 w(!)1404 4264 y Fs(0)1443 4249 y Fk(;)17 b(m)p Ft(\))1652 4182 y(1)p 1620 4226 113 4 v 1620 4318 a Fk(m)p Ft(!)1742 4249 y(\()p Fk(C)1850 4264 y Fs(1)1906 4249 y Fk(g)t Ft(\()p Fk(R)q Ft(\))2108 4208 y Fi(cd)2178 4249 y Fk(t)p Ft(\))2251 4208 y Fi(m)2351 4249 y Fk(g)t Ft(\()p Fk(m)p Ft(\))2563 4208 y Fi(cdm)887 4570 y Ft(+)p Fk(C)1033 4585 y Fs(3)1089 4570 y Ft([)p Fk( )t Ft(\()p Fk(R)q Ft(\)])1361 4529 y Fs(1)p Fi(=)p Fs(2)1471 4570 y Fk(R)1546 4529 y Fi(d)p Fs(+)1647 4502 y Fc(2)p 1648 4514 31 3 v 1648 4555 a(3)1688 4529 y Fi(d)1786 4446 y Fh(1)1749 4475 y Ff(X)1745 4685 y Fi(m)p Fs(=0)1956 4503 y Ft(1)p 1924 4547 113 4 v 1924 4638 a Fk(m)p Ft(!)2047 4570 y(\()p Fk(C)2155 4585 y Fs(1)2227 4570 y Fk(g)t Ft(\()p Fk(R)q Ft(\))2429 4529 y Fi(cd)2499 4570 y Fk(t)p Ft(\))2572 4529 y Fi(m)2672 4570 y Fk(g)t Ft(\()p Fk(m)p Ft(\))2884 4529 y Fi(cdm)3017 4570 y Fk(m)3102 4529 y Fs(2)p Fi(d)3285 4570 y Ft(\(6.33\))-246 4839 y(where)38 b Fk(C)110 4854 y Fs(3)186 4839 y Ft(is)e(a)g(p)s(ositiv)m(e)g(constan) m(t.)56 b(Notice)37 b(that)f Fk(\016)1750 4854 y Fi(R)1808 4839 y Ft(\()p Fk(!)1907 4854 y Fs(0)1946 4839 y Fk(;)p 1990 4784 65 4 v 17 w(!)2054 4854 y Fs(0)2093 4839 y Fk(;)17 b(m)p Ft(\))35 b(=)f(0)i(for)h(all)d Fk(m)h Ft(=)f(0)p Fk(;)17 b Ft(1)p Fk(;)g Fg(\001)g(\001)g(\001)d Ft(.)55 b(By)-246 5039 y(using)36 b(H\177)-49 b(older's)36 b(inequalit)m(y)f (with)g(conjugate)h(pair)f Fk(p)e Ft(=)g(\(1)24 b(+)g Fk(\017)p Ft(\))37 b(and)f Fk(p)2487 5002 y Fh(0)2543 5039 y Ft(=)2662 4999 y Fs(1)p 2662 5016 36 4 v 2665 5073 a Fi(\017)2707 5039 y Fk(p)g Ft(with)g(small)d Fk(\017)h(>)f Ft(0,)-246 5238 y(w)m(e)h(obtain)229 5499 y Fk(\016)272 5514 y Fi(R)330 5499 y Ft(\()p Fk(!)429 5514 y Fi(t)458 5499 y Fk(;)p 502 5444 65 4 v 17 w(!)566 5514 y Fi(t)596 5499 y Fk(;)17 b(m)p Ft(\))27 b Fg(\024)167 b Fk(C)1104 5514 y Fs(3)1176 5499 y Ft([)p Fk( )t Ft(\()p Fk(R)q Ft(\)])1448 5458 y Fs(1)p Fi(=)p Fs(2)1558 5499 y Fk(R)1633 5458 y Fi(d)p Fs(+)1734 5430 y Fc(2)p 1735 5442 31 3 v 1735 5484 a(3)1775 5458 y Fi(d)1832 5418 y Ff(\000)1878 5499 y Ft(exp)q([\()p Fk(C)2162 5514 y Fs(1)2201 5499 y Fk(t)p Ft(\))2274 5458 y Fs(1+)p Fi(\017)2397 5499 y Fk(g)t Ft(\()p Fk(R)q Ft(\))2599 5458 y Fi(cd)p Fs(\(1+)p Fi(\017)p Fs(\))2843 5499 y Ft(])2870 5418 y Ff(\001)2962 5407 y Fc(1)p 2926 5419 104 3 v 2926 5460 a(1+)p Fe(\017)1034 5774 y Fg(\002)1128 5604 y Ff( )1248 5650 y Fh(1)1211 5679 y Ff(X)1207 5889 y Fi(m)p Fs(=0)1418 5707 y Ft(1)p 1386 5751 113 4 v 1386 5842 a Fk(m)p Ft(!)1508 5774 y([)p Fk(g)t Ft(\()p Fk(m)p Ft(\))1747 5733 y Fi(cdm)1880 5774 y Fk(m)1965 5733 y Fs(2)p Fi(d)2041 5774 y Ft(])2068 5733 y Fi(p)2104 5710 y Fb(0)2131 5604 y Ff(!)2231 5599 y Fc(1)p 2219 5611 55 3 v 2219 5656 a Fe(p)2251 5642 y Fb(0)2304 5774 y Fk(:)954 b Ft(\(6.34\))p eop %%Page: 29 29 29 28 bop -246 -9 a Fm(Dynamics)32 b(of)g(In\014nite)h(P)m(article)f (Systems)2074 b Ft(29)-246 279 y(Notice)36 b(that)g(the)g(last)f(term)g (con)m(v)m(erges)k(to)c(a)h(constan)m(t.)54 b(W)-8 b(e)37 b(c)m(hec)m(k)h(here)f(that)e(when)i Fk(R)g Ft(is)f(large,)-246 478 y Fk(g)t Ft(\()p Fk(R)q Ft(\))28 b(is)g(similar)d(to)j(\(log)17 b Fk(R)q Ft(\))803 442 y Fs(1)p Fi(=d)942 478 y Ft(\(see)29 b(\(1.2\)\).)42 b(On)28 b(the)h(other)g(hand,)g(when)h(the)f(p)s(oten)m (tial)d(is)i(singular)-246 678 y(at)39 b(the)g(origin,)f(then)h Fk(c)f(>)g Ft(1)h(\(see)h(\(2.5\)\).)61 b(Th)m(us)40 b(the)f(exp)s(onen)m(tial)f(term)h(in)f(the)h(r.h.s.)62 b(of)39 b(\(6.34\))-246 877 y(increases)47 b(faster)e(than)h(an)m(y)g (p)s(olynomial)41 b(order,)49 b(but)d(slo)m(w)m(er)g(than)f(an)m(y)h (exp)s(onen)m(tial)f(order)h(as)-246 1076 y Fk(R)f Fg(!)e(1)p Ft(.)71 b(But,)44 b(since)e(w)m(e)h(are)f(assuming)f(that)h Fg(j)p Fk(U)1777 1040 y Fs(\(2\))1871 1076 y Ft(\()p Fk(x)p Ft(\))p Fg(j)i(\024)g Fk( )t Ft(\()p Fg(j)p Fk(x)p Fg(j)p Ft(\))f Fg(\024)h Ft(exp)q([)p Fg(\000)p Ft(\(log)17 b Fg(j)p Fk(x)p Fg(j)p Ft(\))3196 1040 y Fi(c)p Fs(+)p Fi(\017)3311 1049 y Fc(0)3348 1076 y Ft(])42 b(for)-246 1275 y(some)f Fk(\017)46 1290 y Fs(0)126 1275 y Fk(>)g Ft(0)g(and)f(since)h(the)g(function)f Fk(R)i Fg(!)f Fk(\016)1643 1290 y Fi(R)1700 1275 y Ft(\()p Fk(!)1799 1290 y Fi(t)1829 1275 y Fk(;)p 1873 1221 65 4 v 17 w(!)1937 1290 y Fi(t)1966 1275 y Fk(;)17 b(m)p Ft(\))41 b(is)f(increasing)f(as)i Fk(R)g Ft(increases,)j(b)m(y)-246 1475 y(taking)33 b Fk(\017)g Ft(suc)m(h)h(that)f Fk(c)p Ft(\(1)23 b(+)f Fk(\017)p Ft(\))29 b Fk(<)f(c)22 b Ft(+)h Fk(\017)1222 1490 y Fs(0)1295 1475 y Ft(and)33 b(letting)e Fk(R)j Ft(go)f(to)g(in\014nit)m(y)-8 b(,)32 b(w)m(e)i(conclude)g(from)e (\(6.34\))-246 1674 y(that)798 1949 y Fk(\016)841 1964 y Fi(R)894 1973 y Fc(0)933 1949 y Ft(\()p Fk(!)1032 1964 y Fi(t)1062 1949 y Fk(;)p 1106 1895 V 17 w(!)1170 1964 y Fi(t)1199 1949 y Fk(;)17 b Ft(0\))28 b(=)f(0)p Fk(;)114 b Ft(for)32 b(all)96 b Fk(R)2075 1964 y Fs(0)2142 1949 y Fg(\025)28 b Fk(r)d Ft(+)d(1)p Fk(:)795 b Ft(\(6.35\))-246 2225 y(Since)33 b Fk(R)83 2240 y Fs(0)150 2225 y Fg(\025)c Fk(r)24 b Ft(+)e(1)33 b(can)g(b)s(e)f(tak)m(en)i(arbitrarily)c(large,)h (\(6.35\))h(concludes)i(the)f(uniqueness.)p 3234 2225 60 60 v -99 2424 a(When)40 b(the)g(p)s(oten)m(tial)e(is)h(not)h (singular)e(at)h(the)h(origin,)f(the)h(pro)s(of)f(of)g(uniqueness)j (under)e(the)-246 2623 y(condition)29 b(of)h(p)s(olynomial)d (decreasing)j(rate)h(follo)m(ws)e(as)h(a)g(simple)f(corollary)f(to)i (the)h(metho)s(d)f(giv)m(en)-246 2823 y(in)i(the)h(ab)s(o)m(v)m(e.)-246 3069 y Fp(Pro)s(of)j(of)h(Theorem)f(2.3.)44 b Ft(Since)32 b(the)g(p)s(oten)m(tial)e Fk(U)10 b Ft(\()p Fk(x)p Ft(\))32 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Fs(2)2538 3942 y Fk(R)2613 3901 y Fi(d)p Fs(+)2714 3874 y Fc(2)p 2714 3886 31 3 v 2714 3927 a(3)2754 3901 y Fi(d)2795 3942 y Fk(m)2880 3901 y Fs(2)p Fi(d)2956 3942 y Fk(:)302 b Ft(\(6.36\))-246 4218 y(That)26 b(is,)h(w)m(e)g(ma)m(y)f(replace)g(the)g(constan)m(t)h Fk(cd)e Ft(b)m(y)i Fk(d=)p Ft(2)e(in)g(\(6.34\).)40 b(Notice)26 b(again)e(that)i Fk(g)t Ft(\()p Fk(R)q Ft(\))f(is)g(similar)-246 4417 y(to)40 b(\(log)17 b Fk(R)q Ft(\))175 4381 y Fs(1)p Fi(=d)326 4417 y Ft(for)40 b(large)g(v)-5 b(alues)40 b(of)g Fk(R)q Ft(.)68 b(Since)40 b(exp)q([\()p Fk(C)1863 4432 y Fs(1)1902 4417 y Fk(T)14 b Ft(\))2011 4381 y Fs(1+)p Fi(\017)2134 4417 y Ft(\(log)i Fk(R)q Ft(\))2437 4352 y Fc(1+)p Fe(\017)p 2437 4365 104 3 v 2474 4406 a Fc(2)2555 4417 y Ft(])40 b(increases)i(slo)m(w)m(er)f(than)-246 4616 y(\(1)t(+)t Fk(R)q Ft(\))38 4580 y Fi(\013)111 4616 y Ft(for)24 b(an)m(y)g Fk(\013)k(>)g Ft(0)23 b(as)h Fk(R)29 b Fg(!)e(1)p Ft(,)f(w)m(e)e(see)h(that)f(the)g(uniqueness)i(holds)d (with)g Fk( )t Ft(\()p Fk(R)q Ft(\))28 b(=)g(\(1)t(+)t Fk(R)q Ft(\))3408 4580 y Fh(\000)p Fi(\015)3507 4616 y Ft(,)-246 4816 y(the)33 b(appropriate)f(rate)h(used)g(to)g(deriv)m(e) g(the)g(a)f(priori)f(b)s(ound.)p 2131 4816 60 60 v -246 5251 a Fp(A)m(CKNO)m(WLEDGMENTS.)-99 5450 y Ft(This)44 b(w)m(ork)i(w)m(as)g(supp)s(orted)f(in)f(part)g(b)m(y)i(Korea)e (Researc)m(h)i(F)-8 b(oundation)44 b(\(98-015-D00043\).)-246 5650 y(H.)k(J.)f(Y)-8 b(o)s(o)46 b(w)m(as)i(supp)s(orted)g(b)m(y)g (gran)m(t)f(of)g(P)m(ost-Do)s(c.)87 b(Program)46 b(from)g(Kyungp)s(o)s (ok)h(National)-246 5849 y(Univ)m(ersit)m(y)34 b(\(1998\).)p eop %%Page: 30 30 30 29 bop -246 -9 a Ft(30)3236 b Fm(Bahn)33 b Fl(et)i(al)-99 279 y Fa(References)-246 551 y Fp([AKR1])48 b Ft(S.)39 b(Alb)s(ev)m(erio,)h(Y)-8 b(u.)62 b(G.)38 b(Kondratiev)h(and)f(M.)i (R\177)-49 b(oc)m(kner,)41 b(Analysis)d(and)h(geometry)g(on)-2 750 y(con\014guration)32 b(spaces,)i Fl(J.)h(F)-7 b(unct.)44 b(A)n(nal.)f Fp(154)p Ft(,)33 b(444-500)d(\(1998\).)-246 1029 y Fp([AKR2])48 b Ft(S.)39 b(Alb)s(ev)m(erio,)h(Y)-8 b(u.)62 b(G.)38 b(Kondratiev)h(and)f(M.)i(R\177)-49 b(oc)m(kner,)41 b(Analysis)d(and)h(geometry)g(on)-2 1228 y(con\014guration)32 b(spaces:)45 b(Gibbsian)31 b(case,)j Fl(J.)h(F)-7 b(unct.)44 b(A)n(nal.)e Fp(157)p Ft(,)33 b(242-271)e(\(1998\).)-246 1507 y Fp([CFS])50 b Ft(I.P)-8 b(.)42 b(Cornfeld,)i(S.)e(V.)g(F)-8 b(omin)40 b(and)h(Y)-8 b(a.)71 b(G.)41 b(Sinai,)i Fl(Er)-5 b(go)g(dic)42 b(the)-5 b(ory)p Ft(,)45 b(\(Springer-V)-8 b(erlag,)-2 1706 y(New)34 b(Y)-8 b(ork-Heidelb)s(erg-Berlin,)29 b(1982\).)-246 1985 y Fp([CPY])49 b Ft(V.)39 b(Choi,)i(Y.M.)f(P)m(ark)h (and)e(H.J.)h(Y)-8 b(o)s(o,)41 b(Diric)m(hlet)c(forms)i(and)h(Diric)m (hlet)d(Op)s(erators)i(for)-2 2184 y(in\014nite)d(particle)f(systems:) 53 b(Essen)m(tial)37 b(self-adjoin)m(tness,)g Fl(J.)i(Math.)56 b(Phys.)f Fp(39)p Ft(,)38 b(6509-6536)-2 2383 y(\(1998\))-246 2662 y Fp([DF])50 b Ft(R.L.)29 b(Dobrushin)g(and)h(J.)f(F)-8 b(ritz,)29 b(Non-equilibrium)e(dynamics)i(of)g(one-dimensional)e (in\014nite)-2 2861 y(particle)34 b(systems)i(with)f(a)g(hard-core)g (in)m(teraction,)f Fl(Commun.)51 b(Math.)g(Phys.)g Fp(55)p Ft(,)36 b(275-292)-2 3060 y(\(1977\).)-246 3339 y Fp([FD])50 b Ft(J.)29 b(F)-8 b(ritz)27 b(and)i(R.L.)g(Dobrushin,)h (Non-equilibrium)25 b(dynamics)k(of)f(t)m(w)m(o-dimensional)f (in\014nite)-2 3538 y(particle)44 b(systems)j(with)d(a)h(singular)f(in) m(teraction,)j Fl(Commun.)78 b(Math.)i(Phys.)h Fp(57)p Ft(,)48 b(67-81)-2 3738 y(\(1977\).)-246 4016 y Fp([F)-9 b(r1]])49 b Ft(J.)38 b(F)-8 b(ritz,)37 b(Some)g(remarks)g(on)g (non-equilibrium)d(dynamics)j(of)f(in\014nite)g(particle)g(systems,)-2 4216 y Fl(J.)f(Statist.)45 b(Phys.)f Fp(34)p Ft(,)32 b(539-555)f(\(1984\).)-246 4494 y Fp([F)-9 b(r2])49 b Ft(J.)40 b(F)-8 b(ritz,)40 b(Gradien)m(t)e(dynamics)h(of)g(in\014nite)f (p)s(oin)m(t)h(systems,)j Fl(A)n(nnal.)62 b(of)41 b(Pr)-5 b(ob)g(ab.)63 b Fp(15)39 b Ft(\(2\),)-2 4694 y(478-514)31 b(\(1987\).)-246 4972 y Fp([Lanf1])51 b Ft(O.E.)24 b(Lanford)f(I)s(I)s (I,)h(The)g(classical)f(mec)m(hanics)h(of)f(one-dimensional)e(systems)k (of)e(in\014nitely)-2 5172 y(man)m(y)33 b(particles,)f Fl(Commun.)43 b(Math.)i(Phys.)f Fp(9)p Ft(,)32 b(169-191)f(\(1968\).) -246 5450 y Fp([Lanf2])51 b Ft(O.E.)33 b(Lanford)f(I)s(I)s(I,)h(Time)f (ev)m(olution)g(of)h(large)f(classical)f(systems.)46 b(In:)e Fl(Dynamic)-5 b(al)34 b(Sys-)-2 5650 y(tems.)79 b(The)-5 b(ory)46 b(and)g(Applic)-5 b(ations)p Ft(,)47 b(Lecture)f(Notes)g(in)e(Ph)m(ysics.)83 b Fp(38)45 b Ft(1-111,)i(\(Springer,)-2 5849 y(Berlin,)32 b(1975\).)p eop %%Page: 31 31 31 30 bop -246 -9 a Fm(Dynamics)32 b(of)g(In\014nite)h(P)m(article)f (Systems)2074 b Ft(31)-246 279 y Fp([Lang])51 b Ft(R.)21 b(Lang,)j(Unendlic)m(h-dimensionle)19 b(Wienerprozesse)24 b(mit)c(W)-8 b(ec)m(hselwirkung.)41 b(I,)22 b Fl(Z.)j(Wahrsch.)-2 478 y(V)-7 b(erw.)45 b(Gebiete)32 b Fp(38)p Ft(,)h(55-72)e(\(1977\).) -246 761 y Fp([MPPr])49 b Ft(C.)38 b(Marc)m(hioro,)g(A.)g(P)m (ellegrinotti)d(and)j(E.)g(Presutti,)i(Existence)f(of)e(time)f(ev)m (olution)h(of)-2 960 y Fk(\027)6 b Ft(-dimensional)30 b(statistical)h(mec)m(hanics,)i Fl(Commun.)44 b(Math.)h(Phys.)e Fp(40)p Ft(,)33 b(175-185)d(\(1975\))-246 1242 y Fp([MPPu])49 b Ft(C.)23 b(Marc)m(hioro,)i(A.)e(P)m(ellegrinotti)d(and)i(M.)i (Pulviren)m(ti,)f(Selfadjoin)m(tness)g(of)g(the)g(Liouville)-2 1441 y(op)s(erator)32 b(for)g(in\014nite)g(classicl)f(systems,)j Fl(Commun.)44 b(Math.)h(Phys.)e Fp(58)p Ft(,)33 b(113-129)e(\(1978\).) -246 1724 y Fp([Osa])50 b Ft(H.)37 b(Osada,)h(Diric)m(hlet)c(form)i (approac)m(h)h(to)f(in\014nite-dimensional)d(Wiener)j(pro)s(cesses)j (with)-2 1923 y(singular)31 b(in)m(teraction,)h Fl(Commun.)44 b(Math.)h(Phys.)e Fp(176)p Ft(,)33 b(117-131)d(\(1996\).)-246 2205 y Fp([Ru1])49 b Ft(D.)32 b(Ruelle,)g Fl(Statistic)-5 b(al)35 b(Me)-5 b(chanics,)34 b(R)n(igor)-5 b(ous)34 b(R)-5 b(esults)p Ft(,)32 b(\(Benjamin,)g(New)h(Y)-8 b(ork,)33 b(1969\).)-246 2488 y Fp([Ru2])49 b Ft(D.)30 b(Ruelle,)f(Sup)s(erstable)h(in)m(teractions)f(in)h(classical)e(mec)m (hanics,)j Fl(Commun.)43 b(Math.)h(Phys.)-2 2687 y Fp(18)p Ft(,)33 b(127-159)e(\(1970\).)-246 2969 y Fp([S-S])50 b Ft(Ra.)42 b(Siegm)m(und-Sc)m(h)m(ultze,)31 b(On)f(non-equilibrium)d (dynamics)j(of)f(m)m(ultidimensional)d(in\014nite)-2 3168 y(particle)k(systems)i(in)e(translation)f(in)m(v)-5 b(arian)m(t)30 b(case,)i Fl(Commun.)43 b(Math.)i(Phys.)d Fp(100)p Ft(,)32 b(245-265)-2 3368 y(\(1985\).)-246 3650 y Fp([Si1])49 b Ft(Y)-8 b(a.)60 b(G.)37 b(Sinai,)h(Construction)h(of)e (dynamics)h(in)g(one-dimensional)d(systems)40 b(of)d(Statistical)-2 3849 y(mec)m(hanics,)c Fl(The)-5 b(or.)44 b(Math.)h(Phys.)f Fp(11)p Ft(,)32 b(248-258)f(\(1972\).)-246 4131 y Fp([Si2])49 b Ft(Y)-8 b(a.)42 b(G.)31 b(Sinai,)e(Construction)i(of)f(cluster)h (dynamics)g(for)f(dynamical)f(systems)j(of)e(statistical)-2 4331 y(mec)m(hanics,)j(V)-8 b(estinik)32 b(Mosk)m(o)m(w)j(Univ.)43 b Fl(Ser.)i(Mat.)g(Me)-5 b(ch.)43 b Fp(29)33 b Ft(152-158)d(\(1974\).) -246 4613 y Fp([Sp])50 b Ft(H.)44 b(Sp)s(ohn,)j(Equilibrium)41 b(\015uctuations)k(for)e(in)m(teracting)g(Bro)m(wnian)h(particles,)i Fl(Commun.)-2 4812 y(Math.)f(Phys.)f Fp(103)p Ft(,)32 b(1-33)g(\(1985\).)-246 5094 y Fp([Y)-9 b(os])49 b Ft(M.W.)31 b(Y)-8 b(oshida,)30 b(Construction)h(of)f(in\014nite)f(dimensional)f (in)m(teracting)h(di\013usion)h(pro)s(cesses)-2 5294 y(through)j(Diric)m(hlet)d(forms,)i Fl(Pr)-5 b(ob.)45 b(Th.)f(R)-5 b(el.)44 b(Fields)32 b Fp(106)p Ft(,)h(265-297)d (\(1996\).)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9901171941932--