Content-Type: multipart/mixed; boundary="-------------0305211037806" This is a multi-part message in MIME format. ---------------0305211037806 Content-Type: text/plain; name="03-230.comments" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-230.comments" Department of Mathematics, Imperial College, London colin.cotter@imperial.ac.uk, s.reich@imperial.ac.uk ---------------0305211037806 Content-Type: text/plain; name="03-230.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="03-230.keywords" Markov processes, Computational methods in fluid dynamics, Computer simulation of molecular and particle dynamics ---------------0305211037806 Content-Type: application/postscript; name="EDPD.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="EDPD.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86 Copyright 1999 Radical Eye Software %%Title: DPD_epl1.dvi %%Pages: 7 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Helvetica %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o DPD.ps DPD_epl1 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2003.05.21:1444 %%BeginProcSet: texc.pro %! 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Fp(k)1817 560 y FC(\()p Fq(r)p FC(\))14 b Fr(dt)20 b FC(+)e Fr(\033)s(dW)2281 572 y Fp(k)2322 468 y Fo(i)2375 560 y Fr(;)584 b FC(\(4\))-282 839 y(where)27 b Fq(r)h FC(is)f(the)h(collection)f(of)h(the)g Fr(N)36 b FC(particle)27 b(p)r(osition)h(v)n(ectors)e Fq(r)1899 851 y Fp(i)1927 839 y FC(,)i Fq(p)f FC(is)h(the)g(asso)r(ciated)e(momen)n(tum)-282 939 y(v)n(ector,)g Fq(M)h FC(is)h(the)g(diagonal)e(mass)h(matrix,)g Fq(v)e FC(=)d Fq(M)1417 909 y Fx(\000)p Fy(1)1506 939 y Fq(p)p FC(,)28 b Fr(V)19 b FC(\()p Fq(r)p FC(\))28 b(is)g(the)g(p)r(oten)n(tial)f(energy)-7 b(,)1046 1122 y(_)1036 1144 y Fr(h)1084 1156 y Fp(k)1124 1144 y FC(\()p Fq(r)p FC(\))24 b(=)f Fn(r)1408 1156 y Fh(r)1443 1144 y Fr(h)1491 1156 y Fp(k)1532 1144 y FC(\()p Fq(r)p FC(\))c Fn(\001)g Fq(v)q Fr(;)-282 1349 y FC(and)32 b(the)h(functions)g Fr(h)443 1361 y Fp(k)483 1349 y FC(\()p Fq(r)p FC(\))h(can)e(again)f(b) r(e)i(c)n(hosen)e(quite)i(arbitrarily)-7 b(.)50 b(It)32 b(is)h(essen)n(tial,)g(ho)n(w)n(ev)n(er,)e(that)-282 1448 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FC(and)522 2509 y Fr(h)570 2521 y Fp(i)598 2509 y FC(\()p Fq(r)p FC(\))24 b(=)e Fr(x)859 2521 y Fp(i)887 2509 y Fr(;)180 b(h)1138 2521 y Fp(i)p Fy(+)p Fp(N)1276 2509 y FC(\()p Fq(r)p FC(\))24 b(=)e Fr(y)1531 2521 y Fp(i)1559 2509 y Fr(;)180 b(h)1810 2521 y Fp(i)p Fy(+2)p Fp(N)1980 2509 y FC(\()p Fq(r)p FC(\))24 b(=)f Fr(z)2234 2521 y Fp(i)2261 2509 y Fr(;)-282 2692 y(i)f FC(=)h(1)p Fr(;)14 b(:)g(:)g(:)f(;)h(N)9 b FC(,)28 b(in)f(\(4\),)h(whic)n(h)g(leads)f(to)g(the)h(standard)f (Langevin)f(mo)r(del)784 2896 y Fr(d)p Fq(r)50 b FC(=)f Fq(M)1121 2862 y Fx(\000)p Fy(1)1210 2896 y Fq(p)14 b Fr(dt;)1609 b FC(\(6\))770 3021 y Fr(d)p Fq(p)50 b FC(=)f Fn(\000)14 b FC([)p Fn(r)1201 3033 y Fh(r)1236 3021 y Fr(V)19 b FC(\()p Fq(r)p FC(\))h(+)e Fr(\015)5 b Fq(v)q FC(])14 b Fr(dt)k FC(+)g Fr(\033)s(d)p Fq(W)q Fr(;)947 b FC(\(7\))-282 3226 y(where)24 b Fq(W)q FC(\()p Fr(t)p FC(\))i(is)f(no)n(w)g(a)f Fr(N)9 b FC(-v)n(ector)24 b(of)h(indep)r (enden)n(t)h(random)e(Wiener)h(pro)r(cesses.)35 b(Y)-7 b(et)25 b(another)g(v)-5 b(arian)n(t)-282 3326 y(of)24 b(\(4\))g(is)f(obtained)h(in)g(the)h(con)n(text)e(of)h(the)g (Hamiltonian)g(particle)f(mesh)h(\(HPM\))h(metho)r(d)f([5,)7 b(6],)25 b(whic)n(h)-282 3425 y(itself)35 b(is)g(an)f(application)g(of) h(the)g(classical)e(particle-in-cell)h(\(PIC\))h(or)e(particle-mesh)h (metho)r(ds)h([1,)15 b(7])-282 3525 y(to)29 b(geoph)n(ysical)f(\015uid) h(dynamics)g(\(GFD\))i([17].)42 b(Here)29 b(the)h(functions)g Fr(h)2044 3537 y Fp(k)2084 3525 y FC(\()p Fq(r)p FC(\))h(w)n(ould)e (refer)f(to)i(some)e(cell)-282 3624 y(a)n(v)n(eraged)21 b(quan)n(tit)n(y)i(and)h Fr(k)i FC(w)n(ould)e(b)r(e)g(its)g(cell)g (index.)36 b(F)-7 b(or)23 b(example,)h(the)g(\015uid)g(densit)n(y)g(at) g(a)f(grid)g(p)r(oin)n(t)-282 3724 y Fq(x)-232 3736 y Fp(k)-163 3724 y FC(can)k(b)r(e)h(appro)n(ximated)e(b)n(y)909 3934 y Fr(h)957 3946 y Fp(k)998 3934 y FC(\()p Fq(r)p FC(\))e(=)1243 3830 y Fp(N)1212 3855 y Fo(X)1219 4032 y Fp(i)p Fy(=1)1346 3934 y Fr(m)1419 3946 y Fp(i)1447 3934 y Fr( )s 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Fr(i)i FC(and)f Fr(j)5 b FC(,)28 b(then)1095 4908 y Fr(h)1143 4920 y Fp(k)1184 4908 y FC(\()p Fq(r)p FC(\))c(=)f Fn(j)p Fq(r)1461 4920 y Fp(i)1507 4908 y Fn(\000)18 b Fq(r)1629 4920 y Fp(j)1665 4908 y Fn(j)p Fr(:)p eop %%Page: 3 3 3 2 bop -125 -141 a Fs(C.)28 b(J.)h(Cotter)d Fv(and)i Fs(S.)f(Reich)p FC(:)g Fz(An)k(extended)h(dissip)-6 b(a)g(tive)33 b(p)-6 b(ar)g(ticle)32 b(d)n(ynamics)g(model)210 b Fq(3)0 149 y Fw(Gener)l(alize)l(d)23 b(L)l(angevin)g(Dynamics.)37 b({)83 b Fz(Esp)-6 b(anol)19 b FC([4])g(ga)n(v)n(e)e(a)i (justi\014cation)h(of)f(the)g(DPD)h(equations)-125 249 y(\(1\)-\(2\))k(using)f(the)h(Mori-Zw)n(anzig)d(pro)5 b(jection)22 b(op)r(erator)g(calculus)h([13,)7 b(19])21 b(under)j(the)f(assumption)g(of)h(a)-125 349 y(time-scale)i(separation) e(b)r(et)n(w)n(een)i(the)g(memory)g(k)n(ernel)f(and)h(the)g(dynamic)g (v)-5 b(ariables)24 b(of)i(in)n(terest.)36 b(This)-125 448 y(assumption)d(allo)n(ws)e(one)h(to)h(replace)e(the)i(memory)f(k)n (ernel)g(b)n(y)g(a)g(Dirac)g Fr(\016)s 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Fo(i)2653 1055 y Fr(;)464 b FC(\(9\))-125 1336 y(where)27 b Fn(K)178 1348 y Fp(k)219 1336 y FC(\()p Fr(\034)9 b FC(\))29 b(is)e(the)h(memory)f(k)n(ernel)f (and)i Fr(U)1367 1348 y Fp(k)1407 1336 y FC(\()p Fr(t)p FC(\))g(is)f(a)g(stationary)f(zero-mean)g(Gaussian)g(pro)r(cess.)36 b(Its)-125 1436 y(auto-co)n(v)-5 b(ariance)25 b(function)j(satis\014es) f(the)h(\015uctuation)g(dissipation)f(relation)997 1637 y Ff(E)8 b FC([)p Fr(U)1126 1649 y Fp(k)1173 1637 y FC(\()p Fr(t)p FC(\))p Fr(U)1324 1649 y Fp(k)1365 1637 y FC(\()p Fr(s)p FC(\)])24 b(=)e Fr(k)1645 1649 y Fp(B)1703 1637 y Fr(T)12 b Fn(K)1827 1649 y Fp(k)1867 1637 y FC(\()p Fr(t)18 b Fn(\000)g Fr(s)p FC(\))p Fr(:)-125 1839 y FC(It)k(is)f(often) g(reasonable)f(to)h(assume)f(that)h(the)h(unresolv)n(ed)e(pro)r(cess)g (exhibits)h(an)g(exp)r(onen)n(tially)g(deca)n(ying)-125 1939 y(co)n(v)-5 b(ariance)26 b(in)i(whic)n(h)f(case)g(the)h(asso)r (ciated)e(memory)h(k)n(ernel)g(can)g(b)r(e)h(appro)n(ximated)e(b)n(y) 1217 2140 y Fn(K)1280 2152 y Fp(k)1321 2140 y FC(\()p Fr(\034)9 b FC(\))24 b(=)f Fr(\025)1590 2152 y Fp(k)1631 2140 y Fr(e)1670 2106 y Fx(\000)p Fp(\013)1765 2115 y Fe(k)1801 2106 y Fx(j)p Fp(\034)7 b Fx(j)1882 2140 y Fr(;)-125 2342 y FC(with)23 b Fr(\013)112 2354 y Fp(k)176 2342 y FC(the)g(deca)n(y)e(rate)h(and)g Fr(\025)909 2354 y Fp(k)973 2342 y FC(a)g(scaling)f(\(coupling\))i(parameter.)33 b(If)23 b(this)g(is)f(the)h(case,)g(then)g Fr(U)3010 2354 y Fp(k)3050 2342 y FC(\()p Fr(t)p FC(\))g(is,)-125 2442 y(in)k(fact,)g(an)f(Ornstein-Uhlen)n(b)r(ec)n(k)g(\(OU\))h(pro)r (cess)e(and)h(the)h(in)n(tegro-di\013eren)n(tial)d(equation)i (\(8\)-\(9\))h(can)-125 2541 y(b)r(e)i(reform)n(ulated)d(as)h(an)h (extended)g(Mark)n(o)n(vian)d(SDE)j([10,)11 b(16].)36 b(These)28 b(t)n(w)n(o)f(references)f(also)h(pro)n(vide)g(a)-125 2641 y(deriv)-5 b(ation)27 b(of)h(suc)n(h)f(SDEs)h(from)f(large)f (Hamiltonian)i(systems)f(sub)5 b(ject)27 b(to)h(random)e(initial)i (data.)0 2750 y(Let)h(us)g(assume,)f(for)g(simplicit)n(y)-7 b(,)30 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4897 y Fr(h)p FC(\()p Fg(r)s FC(\()p Fr(t)1004 4863 y Fx(0)1027 4897 y FC(\)\))c Fr(dt)1178 4863 y Fx(0)1202 4897 y Fr(;)180 b(U)1462 4909 y Fp(k)1526 4897 y FC(=)22 b Fr(e)1652 4863 y Fx(\000)p Fp(\013t)1776 4897 y Fr(s)1815 4909 y Fp(k)1856 4897 y FC(\(0\))d(+)2064 4820 y Fo(p)p 2147 4820 V 77 x FC(2)p Fr(k)2232 4909 y Fp(B)2288 4897 y Fr(T)12 b(\025\013)2464 4784 y Fo(Z)2547 4805 y Fp(t)2510 4973 y Fy(0)2590 4897 y Fr(e)2629 4863 y Fp(\013)p Fy(\()p Fp(t)2723 4838 y Fd(0)2746 4863 y Fx(\000)p Fp(t)p Fy(\))2866 4897 y Fr(dW)2987 4909 y Fp(k)3029 4897 y FC(\()p Fr(t)3091 4863 y Fx(0)3115 4897 y FC(\))p eop %%Page: 4 4 4 3 bop -282 -141 a Fq(4)2512 b Fy(EUR)n(OPHYSICS)21 b(LETTERS)-282 149 y FC(where)h Fr(s)-8 161 y Fp(k)33 149 y FC(\(0\))h(is)g(a)g(constan)n(t)f(randomly)g(sampled)g(from)h (the)g Fn(N)12 b FC(\(0)p Fr(;)i(\025k)1933 161 y Fp(B)1991 149 y Fr(T)e FC(\))23 b(distribution,)h(indep)r(enden)n(tly)-282 249 y(of)j Fr(dW)-67 261 y Fp(k)-25 249 y FC(.)37 b(This)27 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1 5 0 6 1 5 0 5 1 6 0 5 1 5 0 6 1 5 0 5 0 6 1 5 0 6 0 5 1 5 0 6 1 5 0 5 1 6 0 5 1 5 0 6 1 5 0 5 0 6 1 5 1 6 0 5 0 5 1 6 0 5 1 5 0 6 1 5 0 5 1 6 0 5 0 5 0 6 1 5 0 5 0 6 1 5 0 6 1 5 0 5 1 6 0 5 0 5 0 6 1 5 0 5 1 6 1 5 1 5 0 6 0 5 1 6 0 5 1 5 0 6 1 5 0 5 1 1483 1319 100 MP stroke 6 0 5 0 5 1 6 1 5 0 5 1 6 0 5 0 5 1 6 0 5 1 6 0 5 0 5 0 6 0 5 1 5 0 6 0 5 0 5 1 6 0 5 0 5 0 6 0 5 0 6 0 5 0 5 0 6 1 5 0 5 0 6 0 5 0 5 1 6 0 5 0 5 0 6 0 5 0 6 0 5 0 5 0 6 0 5 0 5 0 6 0 5 0 5 0 6 0 5 0 5 0 6 -1 5 0 5 0 6 0 5 0 6 -1 5 0 5 0 6 0 5 0 5 0 6 -1 5 0 5 0 6 0 5 -1 5 0 6 0 5 -1 6 0 5 0 5 0 6 -1 5 0 5 -1 6 0 5 0 5 0 6 -1 5 0 5 0 6 -1 5 0 5 0 6 0 5 -1 6 0 5 0 5 -1 6 0 5 0 5 -1 6 0 5 0 5 -1 6 0 5 0 5 -1 953 1324 100 MP stroke 6 0 5 0 6 -1 5 0 5 0 6 -1 5 0 5 0 6 -1 5 -1 899 1328 11 MP stroke 5 3 6 3 5 4 5 3 6 3 5 5 5 2 6 4 5 3 6 4 5 2 5 3 6 3 5 5 5 3 6 3 5 3 5 3 6 4 5 4 5 3 6 5 5 3 6 5 5 3 5 3 6 5 5 4 5 4 6 5 5 3 5 4 6 5 5 4 5 5 6 4 5 5 5 5 6 4 5 3 6 4 5 4 5 5 6 4 5 5 5 3 6 3 5 4 5 4 6 4 5 2 5 3 6 3 5 4 6 2 5 4 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5 4 5 4 6 3 5 4 6 4 5 2 5 3 6 4 5 4 5 3 6 3 5 3 5 4 6 3 5 4 5 3 6 2 5 3 5 3 6 3 5 3 6 4 5 5 5 3 6 3 5 3 5 4 6 3 5 4 5 3 6 3 5 2 5 4 6 2 5 4 6 4 5 3 5 3 6 4 5 3 5 3 6 3 5 3 5 4 6 3 5 4 5 4 6 4 5 3 5 2 6 3 5 2 6 3 5 3 5 3 6 4 5 3 5 3 6 4 5 3 5 2 6 3 5 3 5 3 6 3 5 4 6 2 5 4 3073 2084 100 MP stroke 5 2 6 3 5 3 5 4 6 3 5 3 5 4 6 3 5 3 5 3 6 3 5 2 5 3 6 4 5 3 6 3 5 4 5 3 6 2 5 3 5 2 6 3 5 4 5 2 6 4 5 3 5 3 6 4 5 3 6 3 5 2 5 3 6 3 5 3 5 2 6 3 5 3 5 3 6 3 5 3 5 3 6 3 5 3 5 4 6 3 5 3 6 3 5 5 5 4 6 2 5 3 5 3 6 3 5 4 5 4 6 3 5 2 5 3 6 2 5 4 6 2 5 4 5 3 6 2 5 3 5 3 6 3 5 4 5 4 6 4 5 1 5 3 6 4 5 3 5 4 6 4 5 3 6 3 5 4 5 3 6 3 5 3 5 3 6 4 5 3 5 2 6 1 5 3 5 3 6 3 5 4 6 3 5 4 5 3 6 4 5 3 5 3 6 4 5 2 2543 1778 100 MP stroke 5 3 6 3 5 3 5 3 6 2 5 2 5 5 6 3 5 2 6 3 5 4 5 2 6 3 5 4 5 3 6 4 5 2 5 3 6 3 5 3 5 4 6 3 5 3 6 4 5 3 5 2 6 3 5 3 5 3 6 2 5 2 5 3 6 3 5 3 5 3 6 3 5 4 6 4 5 3 5 3 6 3 5 3 5 4 6 2 5 3 5 3 6 3 5 3 5 2 6 3 5 2 5 4 6 3 5 2 6 4 5 3 5 3 6 2 5 3 5 2 6 4 5 3 5 3 6 3 5 3 5 3 6 2 5 3 6 3 5 2 5 3 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432 y Fr(s)2133 444 y Fp(l)2158 265 y Fo(3)2158 414 y(5)2227 432 y Fr(dt;)-125 714 y FC(and)28 b(equation)f(\(12\))g(to)653 978 y Fr(ds)735 990 y Fp(k)799 978 y FC(=)c Fn(\000)p Fr(\013s)1044 990 y Fp(k)1098 978 y Fr(t)c FC(+)f Fr(\025)1322 874 y Fp(K)1292 899 y Fo(X)1299 1078 y Fp(l)p Fy(=1)1426 978 y Fr(a)1470 990 y Fp(k)q(l)1542 956 y FC(_)1532 978 y Fr(h)1580 990 y Fp(l)1605 978 y FC(\()p Fq(r)p FC(\))c Fr(dt)20 b FC(+)1898 901 y Fo(p)p 1981 901 304 4 v 77 x FC(2)p Fr(k)2066 990 y Fp(B)2123 978 y Fr(T)12 b(\025\013)h(dB)2404 990 y Fp(k)2445 978 y Fr(;)-125 1235 y FC(where)40 b(the)g(w)n(eigh)n(ts)f Fr(a)635 1247 y Fp(k)q(l)737 1235 y FC(mo)r(del)g(the)i(desired)e (spatial)g(correlation)f(k)n(ernel.)72 b(This)40 b(approac)n(h)e(seems) -125 1335 y(particularly)27 b(suited)h(for)f(particle-mesh)f(metho)r (ds)i(suc)n(h)f(as)g(the)h(HPM)g(metho)r(d)g(describ)r(ed)f(in)h Fn(x)p FC(4.)-125 1584 y Fv(REFERENCES)-86 1775 y([1])67 b(C.K.)34 b(Birdsall)i(and)d(A.B.)h(Langdon.)57 b Fa(Plasma)35 b(Physics)g(via)g(Computer)h(Simulations)p Fv(.)57 b(McGra)n(w-Hill,)61 1866 y(New)26 b(York,)g(1981.)-86 1957 y([2])67 b(C.)24 b(Cotter)f(and)f(S.)h(Reic)n(h.)29 b(A)22 b(sub-grid)g(mo)r(del)h(for)g (the)f(Hamiltonian)h(particle-mesh)f(metho)r(d.)28 b(T)-6 b(ec)n(hnical)61 2049 y(rep)r(ort,)27 b(in)e(preparation.)-86 2140 y([3])67 b(H.)18 b(Dym)d(and)j(H.P)-6 b(.)17 b(McKean.)22 b Fa(Gaussian)f(pr)l(o)l(c)l(esses,)j(function)d(the)l(ory,)h(and)f (the)g(inverse)g(sp)l(e)l(ctr)l(al)h(pr)l(oblem)p Fv(.)61 2231 y(Academic)j(Press,)i(New)f(Y)-6 b(ork)25 b(-)g(London,)h(1976.) -86 2323 y([4])67 b(P)-6 b(.)26 b(Espanol.)36 b Fa(Physic)l(al)27 b(R)l(eview)h(E)p Fv(,)e(52:1734{1742,)31 b(1995.)-86 2414 y([5])67 b(J.)33 b(F)-6 b(rank,)34 b(G.)f(Gott)n(w)n(ald,)i(and)d (S.)g(Reic)n(h.)54 b(In)32 b(M.)g(Grieb)r(el)i(and)e(M.A.)h(Sc)n(h)n(w) n(eitzer,)h(editors,)h Fa(Meshfr)l(e)l(e)61 2505 y(Metho)l(ds)30 b(for)f(Partial)g(Di\013er)l(ential)g(Equations,)g(L)l(e)l(ctur)l(e)i 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Fv(,)e(2\(4\):533{562,)k(2002.)-125 3236 y([11])68 b(O.)29 b(Ma)h(and)f(J.A.)g(Izaguirre.)45 b(Targeted)30 b(molli\014ed)f(impulse)f(-)h(a)g(m)n(ulti-scale)g(sto)r (c)n(hastic)h(in)n(tegrator)g(for)61 3327 y(long)d(molecular)f (dynamics)e(sim)n(ulations.)35 b(T)-6 b(ec)n(hnical)27 b(rep)r(ort,)f(2002.)-125 3418 y([12])68 b(H.)26 b(Mori.)35 b Fa(Pr)l(o)l(g.)28 b(The)l(or.)g(Phys.)p Fv(,)f(34:399{416,)i(1965.) -125 3510 y([13])68 b(H.)26 b(Mori.)35 b Fa(Pr)l(o)l(g.)28 b(The)l(or.)g(Phys.)p Fv(,)f(33:423{450,)i(1965.)-125 3601 y([14])68 b(B.)27 b(\037ksendal.)34 b Fa(Sto)l(chastic)29 b(Di\013er)l(ential)f(Equations)p Fv(.)36 b(Springer-V)-6 b(erlag,)26 b(2000.)-125 3692 y([15])68 b(G.)28 b(P)n(a)n(vliotis)g (and)e(A.)g(Stuart.)37 b(White)27 b(noise)g(limit)g(for)g(inertial)h (particles)g(in)f(a)g(random)e(\014eld.)38 b Fa(SIAM)27 b(J.)61 3784 y(of)h(Multisc)l(ale)f(Mo)l(del)t(ling)g(and)h(Simulation) p Fv(,)e(2003.)-125 3875 y([16])68 b(L.)26 b(Rey-Bellet)g(and)f(L.E.)i (Thomas.)34 b Fa(Comm.)27 b(Math.)g(Phys.)p Fv(,)g(225:305{329,)j (2002.)-125 3966 y([17])68 b(R.)26 b(Salmon.)34 b Fa(Ge)l(ophysic)l(al) 29 b(Fluid)e(Dynamics)p Fv(.)34 b(Oxford)26 b(Univ)n(ersit)n(y)f (Press,)i(1999.)-125 4058 y([18])68 b(J.M.)27 b(Sanz{Serna)f(and)g (M.P.)g(Calv)n(o.)36 b Fa(Numeric)l(al)27 b(Hamiltonian)h(Pr)l(oblems)p Fv(.)35 b(Chapman)25 b(&)g(Hall,)i(1994.)-125 4149 y([19])68 b(R.)26 b(Zw)n(anzig.)35 b Fa(J.)28 b(Stat.)g(Phys.)p Fv(,)e(9:215{220,)k(1973.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0305211037806--