File is Postscript, 181159 bytes long (2518 lines, 12312 words). Produces 10 pages of text, including 1 figure. BODY %! 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y(Another)j(consequence)g(of)g(\(11\){\(13\))j(is)d(that)g(the)h(temp)q (eratures)e Fl(T)1482 2382 y Fp(\022)q(;)p Ff(e\013)1549 2375 y Fn(\()p Fl(N)5 b Fn(\))20 b(de\014ned)h(b)o(y)75 2435 y Fl(A)112 2442 y Fj(2)156 2435 y Fn(=)j(0)f(or)g(b)o(y)f Fk(h)p Fl(S)457 2417 y Fj(2)477 2435 y Fk(i)p Fl(=)p Fk(h)p Fl(R)576 2417 y Fj(2)598 2435 y Fk(i)j Fn(=)f(1)p Fl(=)p Fn(6)g(are)f Fm(not)k Fn(shifted)c(from)e Fl(T)1299 2442 y Fp(\022)1341 2435 y Fn(b)o(y)h(a)h(term) e(of)i(order)f Fl(N)1824 2417 y Fh(\000)p Fp(\036)75 2495 y Fn(\(con)o(trary)15 b(to)h(some)e(previous)i(predictions)e([29)q(]\):)20 b(the)15 b(sp)q(ecial)g(p)q(oin)o(t)h(in)f(the)g(Edw)o(ards)h(mo)q(del)75 2555 y(lies)k(at)i Fl(z)i Fn(=)f(0,)f(hence)f Fl(x)h Fn(=)g(0.)37 b(It)21 b(is)g(not)h(clear)f(to)g(me)f(whether)h(the)g(exp)q(onen)o(t)g Fl( )i Fn(\()p Fl(>)f(\036)p Fn(\))75 2615 y(go)o(v)o(erning)16 b(this)g(shift)g(is)g(univ)o(ersal)f([30)q(].)148 2676 y(Let)26 b(me)e(no)o(w)i(explain)f(wh)o(y)g(\(11\){\(13\))j(should)e(b)q(e)g(true.)49 b(The)26 b(corresp)q(ondence)g(b)q(e-)75 2736 y(t)o(w)o(een)c(crosso)o(v)o (er)h(scaling)f(functions)i(and)f(con)o(tin)o(uum)e(\014eld)i(theories)f(is)h (b)q(est)g(understo)q(o)q(d)963 2865 y(4)p eop %%Page: 5 5 bop 75 151 a Fn(in)14 b(a)g(Wilson-t)o(yp)q(e)g(renormalization-group)g (framew)o(ork.)19 b(Consider)14 b(a)h(Wilson-t)o(yp)q(e)f(R)o(G)g(map)75 212 y Fk(R)22 b Fn(acting)g(on)h(the)e(\(in\014nite-dimensional\))f(space)i (of)g(Hamiltonians)f(for)h(a)g(\014eld)g(theory)g(or)75 272 y(p)q(olymer)13 b(mo)q(del)f(with)i(some)g(\014xed)f(ultra)o(violet)g (cuto\013)i(\003)f(\(e.g.)f(on)i(the)f(lattice\).)19 b(Assume)13 b(that)75 332 y(there)i(exists)f(a)i(critical)e(\014xed)h(p)q(oin)o(t)g Fl(H)821 314 y Fh(\003)856 332 y Fn(with)g(stable)h(manifold)e Fk(M)1368 339 y Fp(s)1401 332 y Fn(and)i(unstable)f(manifold)75 392 y Fk(M)135 399 y Fp(u)157 392 y Fn(.)21 b(\(F)l(or)13 b(simplicit)o(y)d 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Fn(at)c(momen)o(ta)d Fk(j)p Fl(p)p Fk(j)g(\024)f Fn(\003)j(are)h Fm(e)n(qual)22 b Fn(to)16 b(the)g(correlation)75 813 y(functions)f(of)g Fl(H)j Fn(with)d(cuto\013)g(\003.)21 b(This)14 b(p)q(oin)o(t)h(of)g(view)f (has)h(b)q(een)g(emphasized)e(b)o(y)h(Wilson)g(and)75 874 y(Kogut)j([31])f (and)h(others)g([32,)f(33].)148 934 y(Supp)q(ose)g(that)f(the)f(unstable)h (manifold)e Fk(M)978 941 y Fp(u)1015 934 y Fn(is)i Fl(r)q Fn(-dimensional,)e (i.e.)19 b(there)14 b(are)h Fl(r)h Fn(relev)m(an)o(t)75 994 y(op)q(erators)f(with)f(exp)q(onen)o(ts)g Fl(\025)652 1001 y Fj(1)686 994 y Fk(\025)g Fl(\025)767 1001 y Fj(2)801 994 y Fk(\025)f Fl(:)8 b(:)g(:)13 b Fk(\025)h Fl(\025)1005 1001 y Fp(r)1038 994 y Fl(>)g Fn(0.)21 b(Then)14 b(there)f(is)h(an)g Fl(r)q Fn(-parameter)g(fam-)75 1054 y(ily)i(of)h(con)o(tin)o(uum)e(\014eld)i (theories.)23 b(Ho)o(w)o(ev)o(er,)15 b(this)i(family)e(is)i(mapp)q(ed)f(in)o (to)h(itself)f(b)o(y)h(spatial)75 1114 y(dilations,)22 b(so)g(there)f(is)g (really)f(only)h(an)h(\()p 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b(particular,)f(if)h Fk(C)j Fn(is)d(an)o(y)g Fl(r)q Fn(-dimensional)f(manifold)g(whic)o(h)75 1415 y(cuts)h(the)g(stable)h (manifold)d(transv)o(ersally)i([34],)g(then)g(there)g(is)g(a)g(one-to-one)i (corresp)q(ondence)75 1476 y(b)q(et)o(w)o(een)e(asymptotic)f(paths)i(of)g (approac)o(h)g(to)g Fk(M)1028 1483 y Fp(s)1061 1476 y Fn(within)f Fk(C)k Fn(and)d(con)o(tin)o(uum)d(\014eld)i(theories.)75 1536 y(The)j(crosso)o(v)o(er)g(scaling)g(functions)h(for)f(approac)o(h)h(to)g(the) f(critical)e(p)q(oin)o(t)j(from)e(within)h Fk(C)j Fn(are)75 1596 y(th)o(us)e(giv)o(en)f(precisely)f(\(with)h(resp)q(ect)h(to)g(the)g (nonlinear)f(scaling)h(\014elds\))f(b)o(y)h(the)f(con)o(tin)o(uum)75 1656 y(\014eld)i(theories.)34 b(Of)21 b(course,)g(the)g(nonlinear)f(scaling)h (\014elds)f(are)h(connected)f(with)g(the)h(usual)75 1716 y(ph)o(ysical)f (parameters)h(b)o(y)g(a)h(smo)q(oth)f(transformation;)j(this)d (transformation)g(giv)o(es)g(rise)g(to)75 1776 y(the)16 b(non)o(univ)o(ersal) 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b(in)o(terpretation)f(is)h(probably)h(not)f(new)h ([38],)75 813 y(but)14 b(I)f(ha)o(v)o(e)g(not)h(seen)g(it)f(an)o(ywhere)g(in) h(prin)o(t.)19 b(Of)14 b(course,)g(it)f(remains)f(to)i(w)o(ork)g(out)g(the)g (details)75 874 y(for)j(sp)q(eci\014c)e(R)o(G)h(maps)g Fk(R)p Fn(.)148 934 y(Returning)f(to)g(the)g(p)q(olymer)e(problem,)h(w)o(e)g(see)h (that)g(the)g(scaling)g(functions)g Fl(f)1637 941 y Fp(R)1665 934 y Fn(\()p Fl(x)p Fn(\),)g Fl(f)1784 941 y Fp(S)1809 934 y Fn(\()p Fl(x)p Fn(\))75 994 y(and)20 b Fl(f)197 1001 y Fp(A)226 994 y Fn(\()p Fl(x)p Fn(\))f(are)h(necessarily)f(nonanalytic)h(at)g Fl(x)g Fn(=)f(0)i(\(since)e(the)g(Edw)o(ards)i(mo)q(del)e(has)h(this)75 1054 y(prop)q(ert)o(y)f(at)g(least)g(for)h Fl(d)e(>)h Fn(2\).)30 b(The)19 b(scaling)g(functions)g(for)g Fl(x)g(<)f Fn(0)i(are)f(con)o(trolled) f(b)o(y)g(the)75 1114 y(crosso)o(v)o(er)g(to)g(a)g Fm(di\013er)n(ent)23 b Fn(\014xed)18 b(p)q(oin)o(t,)g(and)g(they)g(are)g(th)o(us)g(giv)o(en)f(b)o (y)g(a)h Fm(di\013er)n(ent)23 b Fn(manifold)75 1175 y(of)g(con)o(tin)o(uum)e (\014eld)h(theories.)40 b(Unfortunately)22 b(I)g(am)g(unable)g(to)h(sa)o(y)g (m)o(uc)o(h)d(ab)q(out)k(what)75 1235 y(these)d(theories)f(migh)o(t)f(b)q(e.) 36 b(Roughly)21 b(sp)q(eaking)g(one)h(w)o(ould)e(exp)q(ect)h(a)g Fl(')1517 1217 y Fj(6)1551 1235 y Fk(\000)14 b Fl(')1636 1217 y Fj(4)1677 1235 y Fn(theory)20 b(in)75 1295 y(the)c(symmetry)o(-breaking)e (region)i(\(or)g(in)g(p)q(olymer)f(language)i(an)f(Edw)o(ards)h(mo)q(del)e (with)h(t)o(w)o(o-)75 1355 y(b)q(o)q(dy)h(attraction)g(and)f(three-b)q(o)q (dy)h(repulsion\);)f(but)g(since)f(the)h Fl(')1334 1337 y Fj(6)1370 1355 y Fn(coupling)g(\(or)g(three-b)q(o)q(dy)75 1415 y(in)o(teraction\))d(is) g(a)i(dangerous)g(irrelev)m(an)o(t)d(v)m(ariable)i([39])f(in)h(dimension)e Fl(d)i Fk(\025)g Fn(3,)g(it)g(app)q(ears)h(that)75 1476 y(suc)o(h)j(con)o (tin)o(uum)e(theories)i(do)h(not)g(exist.)26 b(\(The)19 b(situation)f(is)g (analogous)j(to)d Fl(')1612 1457 y Fj(4)1650 1476 y Fn(theories)g(in)75 1536 y(the)h(brok)o(en)f(phase)i(in)f(dimension)e Fl(d)i Fk(\025)g Fn(4)g([40].\))29 b(P)o(ossibly)19 b(the)g(scaling)g(functions)g(are)g(then) 75 1596 y(giv)o(en)c(b)o(y)g(a)i(suitable)e(Landau)i(theory)l(.)k(The)16 b(scaling)g(functions)g(are)g(presumably)e(con)o(tin)o(uous)75 1656 y(at)j Fl(x)c Fn(=)h(0,)i(but)h(I)e(see)h(no)h(reason)g(for)g(all)e(of)i (their)e(deriv)m(ativ)o(es)g(to)i(b)q(e)f(con)o(tin)o(uous.)148 1716 y(In)11 b(dimension)e Fl(d)15 b(<)e Fn(3,)f(the)f(theta)g(p)q(oin)o(t)h (is)e(no)i(longer)f(Gaussian,)i(and)f(the)e(crosso)o(v)o(er)h(scaling)75 1776 y(functions)17 b(are)h(no)f(longer)h(giv)o(en)e(b)o(y)h(the)g(Edw)o (ards)h(mo)q(del.)23 b(Ho)o(w)o(ev)o(er,)15 b(one)i(still)g(exp)q(ects)f(the) 75 1837 y(scaling)j(functions)g(to)g(b)q(e)g(giv)o(en)f(b)o(y)g(t)o(w)o(o)h (con)o(tin)o(uum)e(\014eld)h(theories)h(|)g(one)g(for)g Fl(x)f(>)g Fn(0)h(and)75 1897 y(one)14 b(for)h Fl(x)e(<)h Fn(0)h(|)f(with)g (nonanalyticit)o(y)f(at)h Fl(x)g Fn(=)g(0.)20 b(It)14 b(w)o(ould)g(b)q(e)h (in)o(teresting)e(to)h(examine)e(the)75 1957 y(scaling)17 b(functions)h(in)f (exactly-soluble)f(mo)q(dels)h(suc)o(h)g(as)h(the)f(in)o(teracting)g (partially-directed)75 2017 y(self-a)o(v)o(oiding)e(w)o(alk)h([41)q(].)148 2077 y(The)h(analysis)g(giv)o(en)e(here)i(should)g(apply)f(also)h(to)g (magnetic)e(and)j(\015uid)e(tricritical)f(p)q(oin)o(ts)75 2138 y([42].)148 2198 y(I)e(wish)g(to)g(thank)g(Jim)e(Barrett,)i(Ric)o(hard)f (Brak,)h(Sergio)g(Caracciolo,)g(Rob)q(erto)h(F)l(ern\023)-24 b(andez,)75 2258 y(Mic)o(hael)11 b(Fisher,)h(T)l(on)o(y)h(Guttmann,)g(Ross)g (Hallett,)f(G)o(\023)-23 b(erard)13 b(Jannink,)g(An)o(tti)e(Kupiainen,)i(Bin) 75 2318 y(Li,)20 b(Neal)f(Madras,)i(Bernie)d(Nic)o(k)o(el,)g(Andrea)h(P)o (elissetto,)g(Stu)h(Whittington)g(and)h(esp)q(ecially)75 2378 y(Bertrand)c(Duplan)o(tier)g(for)h(v)o(ery)e(helpful)h(con)o(v)o(ersations)g (and/or)i(corresp)q(ondence.)25 b(This)18 b(re-)75 2439 y(searc)o(h)g(w)o(as) i(supp)q(orted)f(in)g(part)g(b)o(y)f(National)g(Science)g(F)l(oundation)h (gran)o(t)g(DMS{9200719,)75 2499 y(Departmen)o(t)11 b(of)i(Energy)f(con)o (tract)h(DE-F)o(G02-90ER40581)q(,)j(NA)l(TO)11 b(Collab)q(orativ)o(e)i (Researc)o(h)75 2559 y(Gran)o(t)18 b(CR)o(G)g(910251,)i(and)e(a)g(gran)o(t)g (from)f(the)g(New)g(Y)l(ork)g(Univ)o(ersit)o(y)e(Researc)o(h)i(Challenge)75 2619 y(F)l(und.)k(Ac)o(kno)o(wledgmen)o(t)13 b(is)j(also)g(made)f(to)h(the)g (donors)h(of)g(the)e(P)o(etroleum)f(Researc)o(h)h(F)l(und,)75 2679 y(administered)f(b)o(y)h(the)h(American)d(Chemical)h(So)q(ciet)o(y)l(,)g (for)i(partial)g(supp)q(ort)h(of)f(this)g(researc)o(h.)963 2865 y(6)p eop %%Page: 7 7 bop 75 160 a Fd(References)99 269 y Fn([1])24 b(P)l(.J.)19 b(Flory)l(,)g Fm(Principles)i(of)f(Polymer)g(Chemistry)j Fn(\(Cornell)c(Univ) o(ersit)o(y)d(Press,)k(Ithaca,)175 329 y(NY,)15 b(1953\).)99 431 y([2])24 b(H.)e(Y)l(amak)m(a)o(w)o(a,)i Fm(Mo)n(dern)f(The)n(ory)g(of)g (Polymer)h(Solutions)k Fn(\(Harp)q(er)23 b(and)h(Ro)o(w,)h(New)175 491 y(Y)l(ork,)16 b(1971\).)99 593 y([3])24 b(P)l(.G.)14 b(DeGennes,)h Fm(Sc)n(aling)i(Conc)n(epts)g(in)f(Polymer)g(Physics)j Fn(\(Cornell)14 b(Univ)o(ersit)o(y)e(Press,)175 653 y(Ithaca,)k(NY,)f(1979\).)99 755 y([4])24 b(K.F.)i(F)l(reed,)h Fm(R)n(enormalization)g(Gr)n(oup)e(The)n (ory)h(of)g(Macr)n(omole)n(cules)31 b Fn(\(Wiley)l(,)d(New)175 815 y(Y)l(ork,)16 b(1987\).)99 917 y([5])24 b(J.)d(des)f(Cloizeaux)h(and)g (G.)f(Jannink,)i Fm(Polymers)f(in)h(Solution:)32 b(Their)22 b(Mo)n(del)r(ling)h(and)175 977 y(Structur)n(e)e Fn(\(Oxford)16 b(Univ)o(ersit)o(y)e(Press,)i(New)g(Y)l(ork,)f(1990\).)99 1079 y([6])24 b(H.)16 b(F)l(ujita,)f Fm(Polymer)i(Solutions)22 b Fn(\(Elsevier,)14 b(Amsterdam,)f(1990\).)99 1180 y([7])24 b(Y)l(amak)m(a)o(w) o(a)13 b([2],)h(Sections)f(11)i(and)g(16)f(\(pp.)g(69{73)i(and)e(110{118\))j (and)e(parts)f(of)g(Sections)175 1241 y(15,)24 b(20b)f(and)g(21b)g(\(pp.)f (94{110,)j(153{164)g(and)e(167{169\).)h(See)d(also)i(DesCloizeaux{)175 1301 y(Jannink)17 b([5)o(],)f(Section)g(8.1)g(\(pp.)g(289{313\).)99 1402 y([8])24 b(Y)l(amak)m(a)o(w)o(a)36 b([2],)18 b(most)f(of)i(Section)f(15) h(\(pp.)f(94{110\))i(and)f(parts)g(of)f(Sections)g(20b)h(and)175 1463 y(21b)e(\(pp.)f(153{164)j(and)e(167{169\).)99 1564 y([9])24 b(C.)16 b(Dom)o(b)g(and)g(A.J.)f(Barrett,)h(P)o(olymer)e Fo(17)p Fn(,)i(179)h(\(1976\).)75 1666 y([10])24 b(B.G.)15 b(Nic)o(k)o(el,)e (Macromolecules)h Fo(24)p Fn(,)i(1358)i(\(1991\).)75 1768 y([11])24 b(V.)14 b(Privman,)g(P)l(.C.)g(Hohen)o(b)q(erg)h(and)h(A.)e(Aharon)o(y)l(,)h (in)f Fm(Phase)j(T)l(r)n(ansitions)f(and)g(Critic)n(al)175 1828 y(Phenomena)p Fn(,)g(V)l(ol.)e(14,)h(ed.)f(C.)g(Dom)o(b)g(and)h(J.L.)g (Leb)q(o)o(witz)f(\(Academic)e(Press,)j(London{)175 1888 y(San)i(Diego,)f (1991\).)75 1990 y([12])24 b(F)l(or)30 b(a)f(brief)f(review,)j(see)e(N.)f (Madras)i(and)g(G.)f(Slade,)i Fm(The)f(Self-A)o(voiding)i(Walk)175 2050 y Fn(\(Birkh\177)-24 b(auser,)15 b(Boston{Basel{Berlin,)g(1993\),)j (Section)d(10.1.)75 2152 y([13])24 b(M.)16 b(Muth)o(ukumar)e(and)j(B.G.)e (Nic)o(k)o(el,)e(J.)j(Chem.)f(Ph)o(ys.)g Fo(80)p Fn(,)h(5839)i(\(1984\).)75 2253 y([14])24 b(M.)16 b(Muth)o(ukumar)e(and)j(B.G.)e(Nic)o(k)o(el,)e(J.)j (Chem.)f(Ph)o(ys.)g Fo(86)p Fn(,)h(460)i(\(1987\).)75 2355 y([15])24 b(B.)16 b(Li,)f(N.)h(Madras)h(and)g(A.D.)e(Sok)m(al,)h(in)g (preparation.)75 2457 y([16])24 b(A.J.)15 b(Barrett)h(and)h(B.G.)e(Nic)o(k)o (el,)e(priv)m(ate)j(comm)o(unic)o(ation.)75 2558 y([17])24 b(Suc)o(h)17 b(an)h(analysis)g(is)f(p)q(erformed)f(in)h(DesCloizeaux-Jannink) h([5)o(],)f(Section)g(15.3)h(and)g(in)175 2619 y(F)l(reed)e([4],)f(Section)h (10.F.)963 2865 y(7)p eop %%Page: 8 8 bop 75 151 a Fn([18])24 b(H.)13 b(F)l(ujita)g(and)i(T.)e(Norisuy)o(e,)g (Macromolecules)e Fo(18)p Fn(,)j(1637)h(\(1985\);)g(K.)f(Hub)q(er)f(and)h (W.H.)175 212 y(Sto)q(c)o(kma)o(y)o(er,)c(Macromolecules)g Fo(20)p Fn(,)j(1400)h(\(1987\);)h(H.)c(F)l(ujita,)h(Macromolecules)e Fo(21)p Fn(,)j(179)175 272 y(\(1988\).)75 373 y([19])24 b(Hub)q(er)16 b(and)h(Sto)q(c)o(kma)o(y)o(er)d([18)q(])i(attributed)g(this)g(b)q(eha)o (vior)g(to)h(the)f(e\013ects)g(of)h(c)o(hain)f(sti\013-)175 434 y(ness.)e(In)f(fact,)h(as)g(p)q(oin)o(ted)g(out)g(b)o(y)f(Nic)o(k)o(el)e ([10],)i(c)o(hain)g(sti\013ness)i(is)e(quite)g(irrelev)m(an)o(t)f(here,)175 494 y(as)17 b(the)f(e\013ect)g(o)q(ccurs)h(also)f(for)h Fm(p)n(erfe)n(ctly)g (\015exible)22 b Fn(c)o(hains)16 b(\(self-a)o(v)o(oiding)f(w)o(alks\).)75 595 y([20])24 b(A.J.)15 b(Liu)i(and)f(M.E.)g(Fisher,)f(J.)h(Stat.)g(Ph)o(ys.) g Fo(58)p Fn(,)g(431)h(\(1990\).)75 697 y([21])24 b(B.)18 b(Duplan)o(tier,)g (J.)g(Chem.)f(Ph)o(ys.)h Fo(86)p Fn(,)h(4233)h(\(1987\);)h(Ph)o(ys.)d(Rev.)g Fo(A38)p Fn(,)h(3647)h(\(1988\);)175 757 y(and)d(priv)m(ate)f(comm)o(unic)o (ation.)75 859 y([22])24 b(M.J.)14 b(Stephen,)h(Ph)o(ys.)f(Lett.)h Fo(A53)p Fn(,)g(363)i(\(1975\);)f(A.L.)f(Lewis)g(and)h(F.W.)e(Adams,)g(Ph)o (ys.)175 919 y(Rev.)h Fo(B18)p Fn(,)h(5099)i(\(1978\);)f(B.)f(Duplan)o(tier,) f(Europh)o(ys.)h(Lett.)g Fo(1)p Fn(,)g(491)h(\(1986\).)75 1021 y([23])24 b(B.)16 b(Duplan)o(tier)f(and)i(H.)e(Saleur,)h(Ph)o(ys.)g(Rev.)f (Lett.)h Fo(59)p Fn(,)g(539)h(\(1987\).)75 1123 y([24])24 b(I)c(emphasize)e (that)j(the)e(issue)h(here)g(is)g Fm(not)25 b Fn(whether)20 b Fl(z)i Fn(is)e(small)e(or)j(large.)e(In)h(fact,)h(b)o(y)175 1183 y(applying)c(appropriate)h(resummation)d(metho)q(ds)i(to)g(the)g(small-) p Fl(z)h Fn(p)q(erturbation)g(expan-)175 1243 y(sion,)23 b(one)e(can)h (obtain)g(a)f(highly)g(accurate)g(description)g(of)h(the)f(con)o(tin)o(uum)e (Edw)o(ards)175 1303 y(mo)q(del)g(o)o(v)o(er)f(the)h(en)o(tire)g(range)h(0)f Fk(\024)h Fl(z)h Fk(\024)e(1)h Fn([14].)f(The)g(issue)g(is)h(that)g(the)f (con)o(tin)o(uum)175 1363 y(Edw)o(ards)i(mo)q(del)d(arises)h(as)h(a)g(w)o (eak-excluded-v)o(olume)c(limit)h(\()p Fl(\025)i Fk(!)g Fn(0,)h Fl(N)25 b Fk(!)19 b(1)g Fn(with)175 1424 y Fl(z)k Fk(\030)d Fl(\025N)352 1406 y Fj(2)p Fh(\000)p Fp(d=)p Fj(2)474 1424 y Fn(\014xed\))g(of)h(a)f(suitable)g(discrete-c)o(hain)f(mo)q(del,)h(and)h (is)f(relev)m(an)o(t)f Fm(only)25 b Fn(in)175 1484 y(this)14 b(limit)d(|)j(or,)g(what)h(turns)f(out)g(to)h(b)q(e)f(equiv)m(alen)o(t,)e (the)i(limit)d Fl(T)20 b Fk(!)14 b Fl(T)1548 1491 y Fp(\022)1567 1484 y Fn(,)g Fl(N)19 b Fk(!)13 b(1)h Fn(with)175 1544 y Fl(z)i Fk(\030)e Fn(\()p Fl(T)j Fk(\000)11 b Fl(T)411 1551 y Fp(\022)430 1544 y Fn(\))p Fl(N)493 1526 y Fj(2)p Fh(\000)p Fp(d=)p Fj(2)610 1544 y Fn(\014xed.)75 1646 y([25])24 b(Y)l(amak)m(a)o(w)o(a)d([2],)h(Section) g(40;)i(Y.)d(Miy)o(aki)f(and)j(H.)d(F)l(ujita,)j(Macromolecules)c Fo(14)p Fn(,)k(742)175 1706 y(\(1981\);)18 b(F)l(ujita)d([6],)h(Section)f (2.2.5.)75 1808 y([26])24 b(Ho)o(w)o(ev)o(er,)15 b(one)i(should)g Fm(not)22 b Fn(attempt)16 b(to)h(go)h(further)f(and)g(determine)d(the)j(non)o (univ)o(ersal)175 1868 y(scale)e(factors)h(in)f(terms)e(of)j(simple)d (microscopic)g(quan)o(tities)h(\(suc)o(h)h(as)h(the)f(\\binary)g(clus-)175 1928 y(ter)20 b(in)o(tegral"\).)g(These)g(factors)h(are)g(giv)o(en)e(b)o(y)h (complicated)f(man)o(y-b)q(o)q(dy)h(in)o(teractions)175 1988 y(in)o(v)o(olving)15 b Fm(al)r(l)23 b Fn(the)16 b(details)g(of)h(the)f(p)q (olymer)f(and)i(solv)o(en)o(t,)e(and)i(it)e(is)i(unlik)o(ely)d(that)j(they) 175 2048 y(can)g(b)q(e)f(explained)f(\(except)g(at)i(a)g(crude)e(appro)o (ximate)g(lev)o(el\))f(b)o(y)i Fm(any)21 b Fn(simple)14 b(theory)l(.)75 2150 y([27])24 b(F)l(or)15 b(a)g(critical)f(review)f(of)j(the)e(exp)q(erimen) o(tal)e(literature)i(on)h(the)g(static)f(scaling)h(b)q(eha)o(vior)175 2210 y(of)g(high-molecular-w)o(eigh)o(t)d(p)q(olymers)h(in)i(dilute)e (solution,)i(see)f(A.D.)f(Sok)m(al,)i(in)f(prepara-)175 2270 y(tion.)75 2372 y([28])24 b(H.R.)16 b(Haller,)g(C.)h(Destor)h(and)g(D.S.)f (Cannell,)f(Rev.)h(Sci.)f(Instrum.)f Fo(54)p Fn(,)i(973)i(\(1983\);)g(B.)175 2432 y(Ch)o(u,)i(R.)e(Xu,)h(T.)g(Maeda)g(and)g(H.S.)f(Dhadw)o(al,)i(Rev.)e (Sci.)f(Instrum.)h Fo(59)p Fn(,)h(716)h(\(1988\);)175 2493 y(K.B.)16 b(Stra)o(wbridge,)i(F.R.)e(Hallett)g(and)i(J.)f(W)l(atton,)i(Can.)e (J.)h(Appl.)e(Sp)q(ectroscop)o(y)h Fo(36)p Fn(,)175 2553 y(53)g(\(1991\).)75 2654 y([29])24 b(B.J.)15 b(Chera)o(yil,)g(J.F.)g(Douglas)j(and)e(K.F.)g(F)l (reed,)f(J.)g(Chem.)g(Ph)o(ys.)h Fo(83)p Fn(,)f(5293)j(\(1985\).)963 2865 y(8)p eop %%Page: 9 9 bop 75 153 a Fn([30])24 b(In)i(at)h(least)f(one)g(mo)q(del)f(the)h(shift)g (is)g(of)h(order)f Fl(N)1214 135 y Fh(\000)p Fj(\()p Fp(d=)p Fj(2)p Fh(\000)p Fj(1\))1396 153 y Fn(for)h(3)k Fl(<)g(d)g(<)g Fn(4,)d(and)175 213 y Fl(N)219 195 y Fh(\000)p Fj(1)p Fp(=)p Fj(2)302 213 y Fn(\(log)9 b Fl(N)c Fn(\))455 195 y Fh(\000)p Fj(7)p Fp(=)p Fj(11)572 213 y Fn(for)16 b Fl(d)f Fn(=)e(3)k([21].)75 315 y([31])24 b(K.G.)14 b(Wilson)g(and)h(J.)f(Kogut,)h(Ph)o(ys.)e(Rep)q(orts) i Fo(12C)p Fn(,)f(75)h(\(1974\),)h(see)e(esp)q(ecially)f(Section)175 375 y(12.2.)k(See)e(also)i(K.G.)f(Wilson,)g(Rev.)f(Mo)q(d.)h(Ph)o(ys.)g Fo(47)p Fn(,)g(773)h(\(1975\),)g(Section)f(IV.)75 477 y([32])24 b(J.)16 b(P)o(olc)o(hinski,)e(Nucl.)h(Ph)o(ys.)h Fo(B231)p Fn(,)f(269)j(\(1984\).)75 578 y([33])24 b(K.)11 b(Ga)o(w\030)-22 b(edzki)11 b(and)i(A.)d(Kupiainen,)i(in)f Fm(Critic)n(al)i(Phenomena,)j(R)n (andom)c(Systems,)i(Gauge)175 638 y(The)n(ories)k([L)n(es)g(Houches)i(1984],) f(Part)f(I)8 b Fn(,)17 b(ed.)g(K.)g(Osterw)o(alder)g(and)h(R.)f(Stora)i (\(North-)175 699 y(Holland,)d(Amsterdam,)d(1986\),)k(pp.)f(185{293.)75 800 y([34])24 b(In)16 b(the)g(language)h(of)g(Wilson)f(and)h(Kogut)g([31],)f Fk(C)j Fn(is)d(the)g(\\canonical)h(surface".)75 902 y([35])24 b(A.)16 b(Aharon)o(y)g(and)h(M.E.)e(Fisher,)g(Ph)o(ys.)h(Rev.)f Fo(B27)p Fn(,)h(4394)h(\(1983\).)75 1004 y([36])24 b(A.C.D.)e(v)m(an)i(En)o (ter,)g(R.)f(F)l(ern\023)-24 b(andez)23 b(and)h(A.D.)e(Sok)m(al,)k(Ph)o(ys.)c (Rev.)h(Lett.)g Fo(66)p Fn(,)h(3253)175 1064 y(\(1991\))18 b(and)f(J.)f(Stat.)g(Ph)o(ys.)g(\(to)g(b)q(e)h(published\).)75 1166 y([37])24 b(S.)e(W)l(ein)o(b)q(erg,)h(Ph)o(ys.)e(Rev.)g Fo(D8)p Fn(,)i(3497)h(\(1973\);)i(J.C.)21 b(Collins)h(and)h(A.J.)e (Macfarlane,)175 1226 y(Ph)o(ys.)16 b(Rev.)f Fo(D10)p Fn(,)h(1201)i (\(1974\);)f(S.W.)f(MacDo)o(w)o(ell,)e(Ph)o(ys.)i(Rev.)f Fo(D12)p Fn(,)h(1089)i(\(1975\).)75 1328 y([38])24 b(This)17 b(in)o(terpretation)f(w)o (as)h(w)o(ork)o(ed)f(out)h(in)f(1983)i(in)e(con)o(v)o(ersations)h(b)q(et)o(w) o(een)e(An)o(tti)h(Ku-)175 1388 y(piainen)e(and)h(the)f(author.)h(V)l(ery)e (similar)f(ideas)i(can)h(b)q(e)f(found)h(in)f(J.)g(Hughes)h(and)f(J.)g(Liu,) 175 1448 y(Nucl.)h(Ph)o(ys.)h Fo(B307)p Fn(,)f(183)j(\(1988\).)75 1550 y([39])24 b(M.E.)d(Fisher,)i(in)e Fm(R)n(enormalization)i(Gr)n(oup)e(in) i(Critic)n(al)g(Phenomena)h(and)f(Quantum)175 1610 y(Field)g(The)n(ory:)31 b(Pr)n(o)n(c)n(e)n(e)n(dings)21 b(of)h(a)g(Confer)n(enc)n(e)t Fn(,)h(ed.)e(J.D.)g(Gun)o(ton)h(and)g(M.S.)e(Green)175 1670 y(\(T)l(emple)14 b(Univ)o(ersit)o(y)l(,)f(Philadelphia,)i(1974\),)j(pp.)e (65{68.)75 1772 y([40])24 b(R.)16 b(F)l(ern\023)-24 b(andez,)15 b(J.)g(F)l(r\177)-24 b(ohlic)o(h)15 b(and)h(A.D.)f(Sok)m(al,)h Fm(R)n(andom)g(Walks,)i(Critic)n(al)f(Phenomena,)175 1832 y(and)23 b(T)l(riviality)g(in)f(Quantum)i(Field)f(The)n(ory)i Fn(\(Springer-V)l (erlag,)d(Berlin{Heidelb)q(erg{)175 1892 y(New)16 b(Y)l(ork,)f(1992\),)j (Section)d(15.3.)75 1994 y([41])24 b(R.)19 b(Zw)o(anzig)h(and)g(J.I.)f (Lauritzen,)h(Jr.,)f(J.)h(Chem.)d(Ph)o(ys.)j Fo(48)p Fn(,)g(3351)h(\(1968\);) h(J.I.)d(Lau-)175 2054 y(ritzen,)25 b(Jr.)e(and)i(R.)f(Zw)o(anzig,)h(J.)f (Chem.)e(Ph)o(ys.)h Fo(52)p Fn(,)j(3740)g(\(1970\);)j(R.)23 b(Brak,)i(A.J.)175 2114 y(Guttmann)19 b(and)i(S.G.)e(Whittington,)h(J.)f(Ph)o (ys.)h Fo(A25)p Fn(,)g(2437)h(\(1992\);)i(A.L.)c(Ow)o(czarek,)175 2174 y(T.)d(Prellb)q(erg)g(and)h(R.)f(Brak,)f(Univ)o(ersit)o(y)f(of)i(Melb)q (ourne)g(preprin)o(t)f(\(1992\).)75 2276 y([42])24 b(I.D.)11 b(La)o(wrie)h(and)h(S.)f(Sarbac)o(h,)h(in)e Fm(Phase)j(T)l(r)n(ansitions)f (and)h(Critic)n(al)g(Phenomena)t Fn(,)f(v)o(ol.)e(9,)175 2336 y(ed.)k(C.)h(Dom)o(b)e(and)i(J.L.)g(Leb)q(o)o(witz)f(\(Academic)e(Press,)j (London,)h(1984\);)g(C.M.)d(Knobler)175 2396 y(and)j(R.L.)f(Scott,)g(in)g Fm(ibid.)963 2865 y Fn(9)p eop %%Page: 10 10 bop 621 1445 a Fc(u)638 1481 y Fl(H)682 1463 y Fh(\003)863 1340 y Fb(:)367 1595 y(+)940 1375 y Fk(M)1000 1382 y Fp(u)1418 1271 y Fa(u)1382 1339 y Fl(H)1426 1321 y Fh(\003)1422 1351 y Fp(GS)p 264 1669 4 4 v 266 1667 V 269 1664 V 272 1662 V 272 1662 V 275 1660 V 278 1658 V 281 1656 V 281 1656 V 284 1653 V 287 1651 V 290 1649 V 290 1649 V 293 1647 V 296 1645 V 299 1643 V 299 1643 V 302 1641 V 305 1638 V 308 1636 V 308 1636 V 311 1634 V 314 1632 V 317 1630 V 317 1630 V 320 1628 V 323 1626 V 326 1624 V 326 1624 V 328 1622 V 331 1619 V 334 1617 V 334 1617 V 337 1615 V 340 1613 V 343 1611 V 343 1611 V 346 1609 V 349 1607 V 352 1605 V 352 1605 V 355 1603 V 358 1601 V 361 1599 V 361 1599 V 364 1597 V 367 1595 V 370 1593 V 370 1593 V 373 1591 V 376 1589 V 379 1587 V 379 1587 V 382 1585 V 385 1583 V 388 1581 V 388 1581 V 391 1579 V 393 1577 V 396 1575 V 396 1575 V 399 1573 V 402 1571 V 405 1570 V 405 1570 V 408 1568 V 411 1566 V 414 1564 V 414 1564 V 417 1562 V 420 1560 V 423 1558 V 423 1558 V 426 1556 V 429 1554 V 432 1553 V 432 1553 V 435 1551 V 438 1549 V 441 1547 V 441 1547 V 444 1545 V 447 1543 V 450 1541 V 450 1541 V 454 1539 V 458 1536 V 458 1536 V 463 1533 V 467 1531 V 467 1531 V 472 1528 V 476 1525 V 476 1525 V 481 1523 V 485 1520 V 485 1520 V 489 1517 V 494 1515 V 494 1515 V 498 1512 V 503 1510 V 503 1510 V 507 1507 V 512 1504 V 512 1504 V 516 1502 V 520 1499 V 520 1499 V 525 1497 V 529 1494 V 529 1494 V 534 1492 V 538 1489 V 538 1489 V 543 1487 V 547 1485 V 547 1485 V 551 1482 V 556 1480 V 556 1480 V 560 1477 V 565 1475 V 565 1475 V 569 1473 V 574 1470 V 574 1470 V 578 1468 V 582 1465 V 582 1465 V 587 1463 V 591 1461 V 591 1461 V 596 1459 V 600 1456 V 600 1456 V 605 1454 V 609 1452 V 609 1452 V 613 1450 V 618 1447 V 618 1447 V 622 1445 V 627 1443 V 627 1443 V 631 1441 V 636 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1334 V 906 1333 V 910 1331 V 910 1331 V 915 1330 V 919 1329 V 919 1329 V 923 1327 V 928 1326 V 928 1326 V 932 1325 V 937 1324 V 937 1324 V 941 1323 V 946 1321 V 946 1321 V 950 1320 V 954 1319 V 954 1319 V 959 1318 V 963 1317 V 963 1317 V 968 1316 V 972 1314 V 972 1314 V 977 1313 V 981 1312 V 981 1312 V 985 1311 V 990 1310 V 990 1310 V 994 1309 V 999 1308 V 999 1308 V 1003 1307 V 1008 1306 V 1008 1306 V 1012 1305 V 1016 1304 V 1016 1304 V 1021 1303 V 1025 1302 V 1025 1302 V 1030 1301 V 1034 1300 V 1034 1300 V 1039 1299 V 1043 1298 V 1043 1298 V 1047 1298 V 1052 1297 V 1052 1297 V 1056 1296 V 1061 1295 V 1061 1295 V 1065 1294 V 1070 1293 V 1070 1293 V 1074 1293 V 1078 1292 V 1078 1292 V 1083 1291 V 1087 1290 V 1087 1290 V 1092 1290 V 1096 1289 V 1096 1289 V 1101 1288 V 1105 1287 V 1105 1287 V 1110 1287 V 1114 1286 V 1114 1286 V 1118 1285 V 1123 1285 V 1123 1285 V 1127 1284 V 1132 1284 V 1132 1284 V 1136 1283 V 1141 1282 V 1141 1282 V 1145 1282 V 1149 1281 V 1149 1281 V 1154 1281 V 1158 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V 1406 1272 V 1406 1272 V 1411 1273 V 1415 1273 V 650 1711 4 9 v 652 1744 a Fb(6)501 1109 y(U)337 914 y Fk(M)397 921 y Fp(s)p 653 1802 4 4 v 653 1797 V 653 1793 V 653 1793 V 653 1788 V 653 1784 V 653 1784 V 653 1779 V 653 1775 V 653 1775 V 653 1771 V 653 1766 V 653 1766 V 653 1762 V 653 1757 V 653 1757 V 653 1753 V 653 1748 V 653 1748 V 652 1744 V 652 1740 V 652 1740 V 652 1735 V 652 1731 V 652 1731 V 652 1726 V 652 1722 V 652 1722 V 651 1717 V 651 1713 V 651 1713 V 651 1709 V 651 1704 V 651 1704 V 650 1700 V 650 1695 V 650 1695 V 650 1691 V 650 1686 V 650 1686 V 649 1682 V 649 1678 V 649 1678 V 649 1673 V 648 1669 V 648 1669 V 648 1664 V 648 1660 V 648 1660 V 647 1655 V 647 1651 V 647 1651 V 647 1647 V 646 1642 V 646 1642 V 646 1638 V 645 1633 V 645 1633 V 645 1629 V 644 1624 V 644 1624 V 644 1620 V 644 1616 V 644 1616 V 643 1611 V 643 1607 V 643 1607 V 642 1602 V 642 1598 V 642 1598 V 641 1593 V 641 1589 V 641 1589 V 640 1585 V 639 1580 V 639 1580 V 639 1576 V 638 1571 V 638 1571 V 638 1567 V 637 1562 V 637 1562 V 637 1558 V 636 1554 V 636 1554 V 635 1549 V 635 1545 V 635 1545 V 634 1540 V 633 1536 V 633 1536 V 633 1531 V 632 1527 V 632 1527 V 631 1523 V 631 1518 V 631 1518 V 630 1514 V 629 1509 V 629 1509 V 628 1505 V 628 1500 V 628 1500 V 627 1496 V 626 1492 V 626 1492 V 625 1487 V 625 1483 V 625 1483 V 624 1478 V 623 1474 V 623 1474 V 622 1469 V 621 1465 V 621 1465 V 620 1461 V 620 1456 V 620 1456 V 619 1452 V 618 1447 V 618 1447 V 617 1443 V 616 1438 V 616 1438 V 615 1434 V 614 1430 V 614 1430 V 613 1425 V 612 1421 V 612 1421 V 611 1416 V 610 1412 V 610 1412 V 609 1407 V 608 1403 V 608 1403 V 607 1399 V 606 1394 V 606 1394 V 605 1390 V 604 1385 V 604 1385 V 603 1381 V 602 1376 V 602 1376 V 601 1372 V 600 1368 V 600 1368 V 599 1363 V 598 1359 V 598 1359 V 597 1354 V 596 1350 V 596 1350 V 595 1345 V 593 1341 V 593 1341 V 592 1337 V 591 1332 V 591 1332 V 590 1328 V 589 1323 V 589 1323 V 588 1319 V 586 1314 V 586 1314 V 585 1310 V 584 1306 V 584 1306 V 583 1301 V 581 1297 V 581 1297 V 580 1292 V 579 1288 V 579 1288 V 578 1283 V 576 1279 V 576 1279 V 575 1275 V 574 1270 V 574 1270 V 572 1266 V 571 1261 V 571 1261 V 570 1257 V 568 1252 V 568 1252 V 567 1248 V 565 1244 V 565 1244 V 564 1239 V 563 1235 V 563 1235 V 561 1230 V 560 1226 V 560 1226 V 558 1221 V 557 1217 V 557 1217 V 555 1213 V 554 1208 V 554 1208 V 552 1204 V 551 1199 V 551 1199 V 549 1195 V 548 1190 V 548 1190 V 546 1186 V 545 1182 V 545 1182 V 543 1177 V 542 1173 V 542 1173 V 540 1168 V 538 1164 V 538 1164 V 537 1159 V 535 1155 V 535 1155 V 534 1151 V 532 1146 V 532 1146 V 530 1142 V 529 1137 V 529 1137 V 527 1133 V 525 1128 V 525 1128 V 524 1124 V 522 1120 V 522 1120 V 520 1115 V 519 1111 V 519 1111 V 517 1106 V 515 1102 V 515 1102 V 513 1097 V 512 1093 V 512 1093 V 510 1089 V 508 1084 V 508 1084 V 506 1080 V 504 1075 V 504 1075 V 503 1071 V 501 1066 V 501 1066 V 499 1062 V 497 1058 V 497 1058 V 495 1053 V 493 1049 V 493 1049 V 491 1044 V 490 1040 V 490 1040 V 488 1035 V 486 1031 V 486 1031 V 484 1027 V 482 1022 V 482 1022 V 480 1018 V 478 1013 V 478 1013 V 476 1009 V 474 1004 V 474 1004 V 472 1000 V 470 996 V 470 996 V 468 991 V 466 987 V 466 987 V 464 982 V 462 978 V 462 978 V 460 973 V 458 969 V 458 969 V 456 965 V 453 960 V 453 960 V 451 956 V 449 951 V 449 951 V 447 947 V 445 942 V 445 942 V 443 938 V 441 934 V 441 934 V 438 929 V 436 925 V 436 925 V 434 920 V 432 916 V 432 916 V 430 911 V 427 907 V 427 907 V 425 903 V 423 898 V 423 898 V 421 894 V 418 889 V 418 889 V 416 885 V 414 880 V 414 880 V 411 876 V 409 872 V 409 872 V 407 867 V 404 863 V 404 863 V 402 858 V 400 854 V 400 854 V 397 849 V 395 845 V 395 845 V 393 841 V 390 836 V 390 836 V 388 832 V 385 827 V 385 827 V 383 823 V 380 818 V 380 818 V 378 814 V 376 810 V 376 810 V 373 805 V 371 801 V 371 801 V 368 796 V 365 792 V 365 792 V 363 787 V 360 783 V 360 783 V 358 779 V 355 774 V 355 774 V 353 770 V 350 765 V 350 765 V 348 761 V 345 756 V 345 756 V 342 752 V 340 748 V 340 748 V 337 743 V 334 739 V 1701 879 a Fk(C)1153 813 y Fa(u)1136 778 y Fl(P)1631 826 y Fa(u)1613 791 y Fl(Q)p 398 1374 V 400 1372 V 402 1369 V 403 1367 V 405 1365 V 405 1365 V 407 1362 V 409 1360 V 411 1358 V 413 1355 V 413 1355 V 415 1353 V 417 1351 V 419 1348 V 421 1346 V 421 1346 V 423 1344 V 425 1341 V 426 1339 V 428 1337 V 428 1337 V 430 1334 V 432 1332 V 434 1330 V 436 1328 V 436 1328 V 439 1325 V 441 1322 V 444 1318 V 444 1318 V 446 1315 V 449 1312 V 451 1309 V 451 1309 V 454 1306 V 456 1304 V 459 1301 V 459 1301 V 462 1298 V 464 1295 V 467 1292 V 467 1292 V 469 1289 V 472 1286 V 474 1283 V 474 1283 V 477 1280 V 479 1277 V 482 1274 V 482 1274 V 485 1271 V 487 1268 V 490 1266 V 490 1266 V 492 1263 V 495 1260 V 497 1257 V 497 1257 V 500 1254 V 502 1251 V 505 1249 V 505 1249 V 508 1246 V 510 1243 V 513 1240 V 513 1240 V 515 1237 V 518 1235 V 520 1232 V 520 1232 V 523 1229 V 525 1226 V 528 1224 V 528 1224 V 530 1221 V 533 1218 V 536 1215 V 536 1215 V 538 1213 V 541 1210 V 543 1207 V 543 1207 V 546 1205 V 548 1202 V 551 1199 V 551 1199 V 553 1197 V 556 1194 V 559 1191 V 559 1191 V 561 1189 V 564 1186 V 566 1183 V 566 1183 V 569 1181 V 571 1178 V 574 1176 V 574 1176 V 576 1173 V 579 1171 V 582 1168 V 582 1168 V 584 1165 V 587 1163 V 589 1160 V 589 1160 V 592 1158 V 594 1155 V 597 1153 V 597 1153 V 599 1150 V 602 1148 V 604 1145 V 604 1145 V 607 1143 V 610 1140 V 612 1138 V 612 1138 V 615 1136 V 617 1133 V 620 1131 V 620 1131 V 622 1128 V 625 1126 V 627 1123 V 627 1123 V 630 1121 V 633 1119 V 635 1116 V 635 1116 V 638 1114 V 640 1112 V 643 1109 V 643 1109 V 645 1107 V 648 1105 V 650 1102 V 650 1102 V 653 1100 V 656 1098 V 658 1095 V 658 1095 V 661 1093 V 663 1091 V 666 1088 V 666 1088 V 668 1086 V 671 1084 V 674 1082 V 674 1082 V 676 1079 V 679 1077 V 681 1075 V 681 1075 V 684 1073 V 686 1071 V 689 1068 V 689 1068 V 691 1066 V 694 1064 V 697 1062 V 697 1062 V 699 1060 V 702 1057 V 704 1055 V 704 1055 V 707 1053 V 709 1051 V 712 1049 V 712 1049 V 714 1047 V 717 1045 V 720 1043 V 720 1043 V 722 1041 V 725 1038 V 727 1036 V 727 1036 V 730 1034 V 732 1032 V 735 1030 V 735 1030 V 738 1028 V 740 1026 V 743 1024 V 743 1024 V 745 1022 V 748 1020 V 750 1018 V 750 1018 V 753 1016 V 756 1014 V 758 1012 V 758 1012 V 761 1010 V 763 1008 V 766 1006 V 766 1006 V 768 1005 V 771 1003 V 774 1001 V 774 1001 V 776 999 V 779 997 V 781 995 V 781 995 V 784 993 V 786 991 V 789 989 V 789 989 V 792 988 V 794 986 V 797 984 V 797 984 V 801 981 V 804 978 V 804 978 V 808 976 V 812 973 V 812 973 V 816 970 V 820 968 V 820 968 V 824 965 V 828 962 V 828 962 V 832 960 V 835 957 V 835 957 V 839 955 V 843 952 V 843 952 V 847 950 V 851 947 V 851 947 V 855 945 V 859 942 V 859 942 V 863 940 V 866 938 V 866 938 V 870 935 V 874 933 V 874 933 V 878 930 V 882 928 V 882 928 V 886 926 V 890 924 V 890 924 V 894 921 V 898 919 V 898 919 V 902 917 V 905 915 V 905 915 V 909 912 V 913 910 V 913 910 V 917 908 V 921 906 V 921 906 V 925 904 V 929 902 V 929 902 V 933 900 V 937 898 V 937 898 V 941 896 V 945 893 V 945 893 V 948 891 V 952 890 V 952 890 V 956 888 V 960 886 V 960 886 V 964 884 V 968 882 V 968 882 V 972 880 V 976 878 V 976 878 V 980 876 V 984 874 V 984 874 V 988 873 V 992 871 V 992 871 V 996 869 V 999 867 V 999 867 V 1003 866 V 1007 864 V 1007 864 V 1011 862 V 1015 860 V 1015 860 V 1019 859 V 1023 857 V 1023 857 V 1027 856 V 1031 854 V 1031 854 V 1035 852 V 1039 851 V 1039 851 V 1043 849 V 1047 848 V 1047 848 V 1051 846 V 1055 845 V 1055 845 V 1059 843 V 1063 842 V 1063 842 V 1067 841 V 1071 839 V 1071 839 V 1075 838 V 1079 836 V 1079 836 V 1083 835 V 1087 834 V 1087 834 V 1091 832 V 1094 831 V 1094 831 V 1098 830 V 1102 829 V 1102 829 V 1106 827 V 1110 826 V 1110 826 V 1114 825 V 1118 824 V 1118 824 V 1122 823 V 1126 822 V 1126 822 V 1130 820 V 1134 819 V 1134 819 V 1138 818 V 1142 817 V 1142 817 V 1146 816 V 1150 815 V 1150 815 V 1154 814 V 1158 813 V 1158 813 V 1162 812 V 1166 811 V 1166 811 V 1171 810 V 1175 810 V 1175 810 V 1179 809 V 1183 808 V 1183 808 V 1187 807 V 1191 806 V 1191 806 V 1195 805 V 1199 805 V 1199 805 V 1203 804 V 1207 803 V 1207 803 V 1211 802 V 1215 802 V 1215 802 V 1219 801 V 1223 800 V 1223 800 V 1227 800 V 1231 799 V 1231 799 V 1235 798 V 1239 798 V 1239 798 V 1243 797 V 1247 797 V 1247 797 V 1251 796 V 1255 796 V 1255 796 V 1259 795 V 1264 795 V 1264 795 V 1268 794 V 1272 794 V 1272 794 V 1276 793 V 1280 793 V 1280 793 V 1284 793 V 1288 792 V 1288 792 V 1292 792 V 1296 792 V 1296 792 V 1300 791 V 1304 791 V 1304 791 V 1308 791 V 1313 791 V 1313 791 V 1317 790 V 1321 790 V 1321 790 V 1325 790 V 1329 790 V 1329 790 V 1333 790 V 1337 790 V 1337 790 V 1341 789 V 1345 789 V 1345 789 V 1349 789 V 1354 789 V 1354 789 V 1358 789 V 1362 789 V 1362 789 V 1366 789 V 1370 789 V 1370 789 V 1374 789 V 1378 789 V 1378 789 V 1383 790 V 1387 790 V 1387 790 V 1391 790 V 1395 790 V 1395 790 V 1399 790 V 1403 790 V 1403 790 V 1407 791 V 1412 791 V 1412 791 V 1416 791 V 1420 791 V 1420 791 V 1424 792 V 1428 792 V 1428 792 V 1432 792 V 1437 793 V 1437 793 V 1441 793 V 1445 793 V 1445 793 V 1449 794 V 1453 794 V 1453 794 V 1457 795 V 1462 795 V 1462 795 V 1466 795 V 1470 796 V 1470 796 V 1474 796 V 1478 797 V 1478 797 V 1483 798 V 1487 798 V 1487 798 V 1491 799 V 1495 799 V 1495 799 V 1499 800 V 1504 801 V 1504 801 V 1508 801 V 1512 802 V 1512 802 V 1516 803 V 1520 803 V 1520 803 V 1525 804 V 1529 805 V 1529 805 V 1533 806 V 1537 807 V 1537 807 V 1542 807 V 1546 808 V 1546 808 V 1550 809 V 1554 810 V 1554 810 V 1559 811 V 1563 812 V 1563 812 V 1567 813 V 1571 814 V 1571 814 V 1576 815 V 1580 816 V 1580 816 V 1584 817 V 1588 818 V 1588 818 V 1593 819 V 1597 820 V 1597 820 V 1601 821 V 1605 822 V 1605 822 V 1610 823 V 1614 824 V 1614 824 V 1618 826 V 1623 827 V 1623 827 V 1627 828 V 1631 829 V 1631 829 V 1635 831 V 1640 832 V 1640 832 V 1644 833 V 1648 834 V 1648 834 V 1653 836 V 1657 837 V 1657 837 V 1661 839 V 1666 840 V 1666 840 V 1670 841 V 1674 843 V 753 1017 a Fa(u)691 990 y Fl(H)731 997 y Fj(1)987 1114 y Fa(u)1005 1132 y Fl(H)1049 1109 y Ff(e\013)1045 1143 y Fj(1)836 1083 y Fb(j)p 751 1018 3 3 v 753 1019 V 754 1020 V 756 1021 V 757 1022 V 759 1023 V 761 1024 V 762 1026 V 764 1027 V 765 1028 V 767 1029 V 769 1030 V 769 1030 V 770 1031 V 772 1032 V 774 1033 V 775 1034 V 777 1035 V 779 1036 V 780 1037 V 782 1038 V 784 1039 V 785 1041 V 785 1041 V 787 1042 V 789 1042 V 790 1043 V 792 1044 V 794 1045 V 795 1046 V 797 1047 V 799 1048 V 800 1049 V 802 1050 V 802 1050 V 804 1051 V 805 1052 V 807 1053 V 809 1054 V 811 1055 V 812 1056 V 814 1057 V 816 1058 V 818 1059 V 818 1059 V 819 1059 V 821 1060 V 823 1061 V 824 1062 V 826 1063 V 828 1064 V 830 1064 V 831 1065 V 833 1066 V 833 1066 V 835 1067 V 837 1068 V 839 1069 V 841 1070 V 842 1070 V 844 1071 V 846 1072 V 848 1073 V 848 1073 V 850 1074 V 852 1075 V 854 1075 V 855 1076 V 857 1077 V 859 1078 V 861 1078 V 863 1079 V 863 1079 V 865 1080 V 866 1081 V 868 1081 V 870 1082 V 872 1083 V 874 1083 V 875 1084 V 877 1085 V 877 1085 V 879 1085 V 881 1086 V 882 1087 V 884 1087 V 886 1088 V 888 1089 V 890 1089 V 891 1090 V 891 1090 V 893 1090 V 895 1091 V 897 1092 V 898 1092 V 900 1093 V 902 1093 V 904 1094 V 905 1095 V 905 1095 V 907 1095 V 909 1096 V 911 1096 V 912 1097 V 914 1097 V 916 1098 V 917 1098 V 919 1099 V 919 1099 V 921 1099 V 923 1100 V 924 1100 V 926 1101 V 928 1101 V 929 1102 V 931 1102 V 933 1103 V 933 1103 V 934 1103 V 936 1104 V 938 1104 V 939 1104 V 941 1105 V 943 1105 V 945 1106 V 946 1106 V 946 1106 V 948 1107 V 950 1107 V 951 1107 V 953 1108 V 955 1108 V 956 1109 V 958 1109 V 960 1110 V 960 1110 V 961 1110 V 963 1110 V 965 1111 V 967 1111 V 969 1112 V 971 1112 V 973 1113 V 973 1113 V 975 1113 V 977 1114 V 979 1114 V 981 1115 V 984 1115 V 986 1115 V 664 1091 a Fa(u)602 1064 y Fl(H)642 1071 y Fj(2)1015 1207 y Fa(u)17 b Fl(H)1076 1184 y Ff(e\013)1072 1218 y Fj(2)783 1186 y Fb(q)p 662 1092 V 664 1093 V 665 1094 V 666 1096 V 668 1097 V 669 1098 V 670 1099 V 671 1101 V 673 1102 V 674 1103 V 674 1103 V 675 1105 V 677 1106 V 678 1107 V 679 1108 V 681 1110 V 682 1111 V 684 1112 V 685 1113 V 685 1113 V 686 1115 V 688 1116 V 689 1117 V 690 1118 V 691 1119 V 693 1120 V 694 1122 V 695 1123 V 695 1123 V 697 1124 V 698 1125 V 700 1126 V 701 1127 V 703 1129 V 704 1130 V 705 1131 V 705 1131 V 707 1132 V 708 1133 V 710 1134 V 711 1135 V 712 1136 V 714 1137 V 715 1138 V 715 1138 V 717 1139 V 718 1141 V 720 1142 V 721 1143 V 723 1144 V 725 1145 V 725 1145 V 726 1146 V 728 1147 V 730 1149 V 732 1150 V 734 1151 V 734 1151 V 735 1152 V 737 1153 V 739 1154 V 741 1155 V 743 1156 V 743 1156 V 744 1157 V 746 1158 V 748 1159 V 750 1160 V 751 1161 V 751 1161 V 753 1162 V 755 1163 V 756 1164 V 758 1165 V 760 1166 V 760 1166 V 762 1167 V 763 1168 V 765 1168 V 767 1169 V 768 1170 V 768 1170 V 770 1171 V 771 1171 V 773 1172 V 775 1173 V 776 1174 V 776 1174 V 778 1174 V 780 1175 V 781 1176 V 783 1176 V 784 1177 V 784 1177 V 786 1178 V 788 1178 V 789 1179 V 791 1179 V 792 1180 V 792 1180 V 794 1181 V 796 1181 V 798 1182 V 800 1183 V 800 1183 V 802 1184 V 804 1184 V 806 1185 V 808 1185 V 808 1185 V 810 1186 V 812 1187 V 814 1187 V 816 1188 V 816 1188 V 818 1188 V 820 1189 V 821 1189 V 823 1190 V 823 1190 V 825 1190 V 827 1191 V 829 1191 V 831 1192 V 831 1192 V 833 1192 V 835 1193 V 836 1193 V 838 1194 V 838 1194 V 840 1194 V 842 1194 V 844 1195 V 846 1195 V 846 1195 V 848 1195 V 849 1196 V 851 1196 V 853 1197 V 853 1197 V 855 1197 V 857 1197 V 859 1198 V 860 1198 V 860 1198 V 862 1198 V 864 1198 V 866 1199 V 868 1199 V 868 1199 V 869 1199 V 871 1200 V 873 1200 V 875 1200 V 875 1200 V 877 1200 V 878 1201 V 880 1201 V 882 1201 V 882 1201 V 884 1201 V 886 1202 V 887 1202 V 889 1202 V 889 1202 V 891 1202 V 894 1203 V 896 1203 V 896 1203 V 899 1203 V 901 1203 V 903 1204 V 903 1204 V 906 1204 V 908 1204 V 910 1204 V 910 1204 V 913 1204 V 915 1205 V 917 1205 V 917 1205 V 920 1205 V 922 1205 V 924 1205 V 924 1205 V 927 1205 V 929 1206 V 931 1206 V 931 1206 V 934 1206 V 936 1206 V 938 1206 V 938 1206 V 940 1206 V 943 1206 V 945 1207 V 945 1207 V 947 1207 V 950 1207 V 952 1207 V 952 1207 V 954 1207 V 956 1207 V 959 1207 V 959 1207 V 961 1207 V 963 1207 V 966 1207 V 966 1207 V 968 1208 V 970 1208 V 972 1208 V 972 1208 V 975 1208 V 977 1208 V 979 1208 V 979 1208 V 982 1208 V 984 1208 V 986 1208 V 986 1208 V 988 1208 V 991 1208 V 993 1208 V 993 1208 V 995 1208 V 997 1208 V 1000 1208 V 1000 1208 V 1002 1208 V 1004 1208 V 1006 1208 V 1006 1208 V 1009 1208 V 1011 1208 V 1013 1209 V 615 1136 a Fa(u)544 1118 y Fl(H)584 1125 y Fj(3)1025 1252 y Fa(u)1043 1269 y Fl(H)1087 1246 y Ff(e\013)1083 1280 y Fj(3)746 1259 y Fb(z)p 613 1137 V 615 1139 V 616 1140 V 617 1142 V 618 1144 V 619 1145 V 620 1147 V 622 1149 V 622 1149 V 623 1150 V 624 1152 V 625 1153 V 626 1155 V 627 1156 V 628 1158 V 629 1159 V 629 1159 V 630 1160 V 631 1162 V 632 1163 V 633 1164 V 634 1166 V 635 1167 V 636 1168 V 636 1168 V 638 1170 V 639 1172 V 641 1173 V 642 1175 V 643 1177 V 643 1177 V 645 1178 V 646 1180 V 647 1181 V 649 1183 V 650 1184 V 650 1184 V 651 1186 V 652 1187 V 654 1189 V 655 1190 V 656 1191 V 656 1191 V 657 1193 V 659 1194 V 660 1195 V 661 1196 V 662 1198 V 662 1198 V 663 1199 V 665 1200 V 666 1201 V 667 1202 V 668 1203 V 668 1203 V 669 1204 V 670 1206 V 671 1207 V 673 1208 V 674 1209 V 674 1209 V 675 1210 V 676 1211 V 678 1212 V 679 1214 V 679 1214 V 680 1215 V 682 1216 V 683 1217 V 684 1218 V 684 1218 V 686 1219 V 687 1220 V 688 1221 V 690 1222 V 690 1222 V 691 1223 V 692 1224 V 693 1225 V 695 1226 V 695 1226 V 696 1227 V 698 1228 V 700 1229 V 700 1229 V 701 1230 V 703 1231 V 705 1232 V 705 1232 V 707 1234 V 709 1235 V 709 1235 V 712 1237 V 714 1238 V 714 1238 V 716 1239 V 719 1240 V 719 1240 V 721 1242 V 723 1243 V 723 1243 V 726 1244 V 728 1245 V 728 1245 V 730 1246 V 732 1247 V 732 1247 V 735 1248 V 737 1249 V 737 1249 V 739 1249 V 741 1250 V 741 1250 V 743 1251 V 746 1252 V 746 1252 V 748 1253 V 750 1253 V 750 1253 V 752 1254 V 754 1254 V 754 1254 V 756 1255 V 758 1256 V 758 1256 V 760 1256 V 762 1257 V 762 1257 V 765 1257 V 767 1258 V 767 1258 V 769 1258 V 771 1259 V 771 1259 V 773 1259 V 775 1260 V 775 1260 V 777 1260 V 779 1260 V 779 1260 V 781 1261 V 783 1261 V 783 1261 V 785 1261 V 787 1262 V 787 1262 V 789 1262 V 791 1262 V 791 1262 V 793 1262 V 795 1263 V 795 1263 V 797 1263 V 799 1263 V 799 1263 V 801 1263 V 803 1264 V 803 1264 V 805 1264 V 807 1264 V 807 1264 V 809 1264 V 811 1264 V 811 1264 V 813 1264 V 815 1265 V 815 1265 V 817 1265 V 819 1265 V 819 1265 V 821 1265 V 823 1265 V 823 1265 V 825 1265 V 827 1265 V 827 1265 V 828 1265 V 830 1265 V 830 1265 V 832 1265 V 834 1265 V 834 1265 V 836 1265 V 838 1265 V 838 1265 V 840 1265 V 842 1265 V 842 1265 V 844 1265 V 846 1265 V 846 1265 V 848 1265 V 850 1265 V 850 1265 V 852 1265 V 853 1265 V 853 1265 V 855 1265 V 857 1265 V 857 1265 V 859 1265 V 861 1265 V 861 1265 V 863 1265 V 865 1265 V 865 1265 V 867 1265 V 869 1265 V 869 1265 V 871 1265 V 872 1265 V 872 1265 V 874 1265 V 876 1265 V 876 1265 V 878 1265 V 880 1265 V 880 1265 V 882 1264 V 884 1264 V 884 1264 V 886 1264 V 887 1264 V 887 1264 V 889 1264 V 891 1264 V 891 1264 V 893 1264 V 895 1264 V 895 1264 V 897 1264 V 899 1264 V 899 1264 V 901 1263 V 902 1263 V 902 1263 V 904 1263 V 906 1263 V 906 1263 V 908 1263 V 910 1263 V 910 1263 V 912 1263 V 914 1262 V 914 1262 V 915 1262 V 917 1262 V 917 1262 V 919 1262 V 921 1262 V 921 1262 V 923 1262 V 925 1262 V 925 1262 V 927 1261 V 928 1261 V 928 1261 V 930 1261 V 932 1261 V 932 1261 V 934 1261 V 936 1261 V 936 1261 V 938 1261 V 939 1260 V 939 1260 V 941 1260 V 943 1260 V 943 1260 V 945 1260 V 947 1260 V 947 1260 V 949 1260 V 951 1259 V 951 1259 V 952 1259 V 954 1259 V 954 1259 V 956 1259 V 958 1259 V 958 1259 V 960 1259 V 962 1258 V 962 1258 V 963 1258 V 965 1258 V 965 1258 V 967 1258 V 969 1258 V 969 1258 V 971 1258 V 973 1257 V 973 1257 V 974 1257 V 976 1257 V 976 1257 V 978 1257 V 980 1257 V 980 1257 V 982 1257 V 984 1256 V 984 1256 V 985 1256 V 987 1256 V 987 1256 V 989 1256 V 991 1256 V 991 1256 V 993 1256 V 994 1255 V 994 1255 V 996 1255 V 998 1255 V 998 1255 V 1000 1255 V 1002 1255 V 1002 1255 V 1004 1255 V 1005 1254 V 1005 1254 V 1007 1254 V 1009 1254 V 1009 1254 V 1011 1254 V 1013 1254 V 1013 1254 V 1015 1254 V 1016 1253 V 1016 1253 V 1018 1253 V 1020 1253 V 1020 1253 V 1022 1253 V 1024 1253 V 75 1955 a Fn(Figure)g(1:)23 b (Renormalization-group)17 b(\015o)o(w)g(in)g(the)g(neigh)o(b)q(orho)q(o)q(d)i (of)f(a)g(\014xed)e(p)q(oin)o(t)i Fl(H)1739 1937 y Fh(\003)1759 1955 y Fn(.)24 b Fk(M)1857 1962 y Fp(s)75 2016 y Fn(\(resp.)d Fk(M)276 2023 y Fp(u)298 2016 y Fn(\))g(is)g(the)g(stable)g(\(resp.)g (unstable\))g(manifold.)35 b(Case)22 b(Ia:)31 b(Mo)q(dels)21 b(in)g(the)g(go)q(o)q(d-)75 2076 y(solv)o(en)o(t)12 b(regime)f(ma)o(y)h(ha)o (v)o(e)g(correction-to-scaling)i(amplitudes)d(that)j(are)f(either)f(negativ)o (e)h(\()p Fl(P)7 b Fn(\))75 2136 y(or)17 b(p)q(ositiv)o(e)f(\()p Fl(Q)p Fn(\).)23 b(Case)18 b(I)q(I:)e(The)h(initial)e(Hamiltonians)h Fl(H)1198 2143 y Fp(n)1239 2136 y Fn(approac)o(h)h(the)g(stable)g(manifold,) 75 2196 y(while)k(the)h(lo)o(w-energy)g(e\013ectiv)o(e)e(Hamiltonians)h Fl(H)1092 2178 y Fp(ef)t(f)1088 2209 y(n)1176 2196 y Fk(\021)i(R)1280 2178 y Fp(n)1304 2196 y Fl(H)1344 2203 y Fp(n)1390 2196 y Fn(approac)o(h)g (the)f(unstable)75 2256 y(manifold.)951 2865 y(10)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ENDBODY