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Fq(:)1141 3447 y Fm(@)1197 3406 y Fi(2)1192 3472 y Fl(t)1233 3447 y Fm(~)-45 b(v)t Fn(\()p Fm(t;)15 b(~)-47 b(x)p Fn(\))28 b(=)f Fk(\000)1756 3380 y Fn(1)p 1716 3424 128 4 v 1716 3515 a Fm(N)1804 3487 y Fi(2)1854 3447 y Fk(r)1937 3406 y Fi(2)1973 3447 y Fm(~)-45 b(v)t Fn(\()p Fm(t;)15 b(~)-47 b(x)p Fn(\))964 b Fq(\(1\))83 3654 y(\(note)44 b(the)g(unusual)g(sign)f(in)h (front)f(of)h Fk(r)1654 3618 y Fi(2)1693 3654 y Fq(\).)91 b(That)45 b(is,)j(the)c(v)n(ectors)h(look)e(lik)o(e)i(long)83 3774 y(w)o(a)n(v)n (elength)36 b(w)o(a)n(v)n(es)e(with,)j(say)-7 b(,)37 b Fm(n)e Fq(nodes)f(in)g(the)h Fm(x)g Fq(direction)g(and)g Fm(m)f Fq(nodes)g(in)h(the) g Fm(y)83 3895 y Fq(direction,)27 b(and)f(the)g(corresponding)g(L)-6 b(yapuno)n(v)27 b(e)n(xponent)g(is)e(proportional)i(to)1188 4203 y Fk(\006)1296 4136 y Fn(1)p 1275 4180 89 4 v 1275 4271 a Fm(N)1374 4026 y Fh(v)1374 4072 y(u)1374 4122 y(u)1374 4172 y(t)p 1462 4026 702 4 v -90 x(\022)1545 4136 y Fm(m)p 1533 4180 111 4 v 1533 4271 a(L)1599 4286 y Fl(x)1653 4082 y Fh(\023)1714 4105 y Fi(2)1776 4203 y Fn(+)1874 4082 y Fh(\022)1970 4136 y Fm(n)p 1945 4180 108 4 v 1945 4271 a(L)2011 4286 y Fl(y)2063 4082 y Fh(\023)2124 4105 y Fi(2)2189 4203 y Fm(:)83 4492 y Fq(Note)38 b(that)g(the)g(translation)h(modes\320constant)f(L)-6 b(yapuno)n(v)39 b(v)n(ectors)g(with)e(zero)h(e)n(xpo-)83 4612 y(nents,)49 b(which)c(are)f(tri)m(vially)j(present)d(in)g(an)n(y)h(system)g (with)f(translation)h(in)l(v)m(ariance,)83 4733 y(correspond)28 b(to)g(the)h(special)g(case)f Fm(m)j Fn(=)g Fm(n)g Fn(=)g(0)p Fq(.)44 b(This)28 b(phenomenology)h(w)o(as)e(observ)n(ed)83 4853 y(in)c(simulations)g(with)g(widely)g(v)m(arying)g(parameters,)i(such)d (as)h(aspect)g(ratio,)h(density)-7 b(,)25 b(and)83 4973 y(the)h(shape)g(of)f (the)h(particles)h([4,)f(5].)229 5094 y(The)g(tangent)h(\257o)m(w)e(of)h(the) g(molecular)h(dynamics)f(system)g(can)g(be)g(written)g(as)1193 5353 y Fm(@)1244 5368 y Fl(t)1290 5232 y Fh(\022)1368 5293 y Fm(\016)r(~)-47 b(x)1371 5413 y(\016)t(~)e(p)1487 5232 y Fh(\023)1576 5353 y Fn(=)28 b Fm(G)p Fn(\()p Fm(t)p Fn(\))1885 5232 y Fh(\022)1962 5293 y Fm(\016)r(~)-47 b(x)1965 5413 y(\016)t(~)e(p)2081 5232 y Fh(\023)2184 5353 y Fm(;)991 b Fq(\(2\))1676 5637 y(3)p eop %%Page: 4 4 bop 432 573 a Fq(where)27 b(the)h(components)g Fm(\016)r(~)-47 b(x;)34 b(\016)t(~)-49 b(p)27 b Fq(are)g(column)i(v)n(ectors)f(with)f Fm(N)38 b Fq(entries)28 b(each)g(of)f(which)432 693 y(is)h(a)h(v)n(ector)h (in)f Fj(R)1076 657 y Fi(2)1115 693 y Fq(.)46 b(The)29 b(quantity)h Fm(G)p Fn(\()p Fm(t)p Fn(\))e Fq(is)h(the)g(action)h(on)e(the)h(tangent)h (space)f(induced)432 814 y(by)35 b(the)h(\257o)m(w)f Fn(\010\()p Fm(t)p Fn(\))h Fq(of)f(the)h(dynamical)h(system:)57 b(If)35 b Fm( )2434 829 y Fi(0)2509 814 y Fq(is)g(the)h(instantaneous)g(state)h(of) 432 934 y(the)32 b(system)h(then)f Fm(G)p Fn(\()p Fm(t)p Fn(\))g Fq(is)h(gi)m(v)n(en)g(by)f Fm(G)p Fn(\()p Fm(t)p Fn(\))p Fm(f)47 b Fn(=)37 b Fm(D)s Fn(\010\()p Fm(t)p Fn(\))2481 949 y Fl( )2527 958 y Fg(0)2566 934 y Fm(f)11 b Fq(,)34 b(where)e Fn(\010\()p Fm(t)p Fn(\)\()p Fm( )3247 949 y Fi(0)3313 934 y Fn(+)26 b Fm("f)11 b Fn(\))36 b(=)432 1054 y(\010\()p Fm(t)p Fn(\))p Fm( )676 1069 y Fi(0)737 1054 y Fn(+)20 b Fm("D)s Fn(\010\()p Fm(t)p Fn(\))1144 1069 y Fl( )1190 1078 y Fg(0)1229 1054 y Fm(f)32 b Fn(+)20 b Fm(O)s Fn(\()p Fm(")1567 1018 y Fi(2)1606 1054 y Fn(\))p Fq(.)37 b(The)25 b(e)m(v)n(olution)i(operator)e(of)g(eq.)g (\(2\))g(may)h(be)f(written)432 1175 y(formally)h(as)1609 1343 y Fm(U)39 b Fn(=)27 b Ff(T)16 b Fn(exp)2066 1226 y Fh(Z)2166 1343 y Fm(G)p Fn(\()p Fm(t)p Fn(\))p Fm(dt)25 b(:)1059 b Fq(\(3\))432 1565 y(W)-8 b(e)24 b(proceed)h(to)f(construct)g(a)h(simpli\256ed)f(model)h (of)f(the)g(tangent)h(dynamics,)h(by)e(making)432 1685 y(a)h(series)h(of)g (modi\256cations)g(and)g(assumptions)g(about)g(the)g(nature)g(of)f Fm(G)h Fq(and)g Fm(U)10 b Fq(.)578 1812 y(W)-8 b(e)27 b(\256rst)f(replace)j (the)e(hard-core)g(interaction)i(with)e(a)g(short)g(range)g(`soft')f (potential.)432 1932 y(In)f(that)i(case)f Fm(G)f Fq(will)h(ha)n(v)n(e)h(a)e (block)h(structure)h(of)e(the)h(form)1611 2197 y Fm(G)p Fn(\()p Fm(t)p Fn(\))i(=)1930 2076 y Fh(\022)2076 2137 y Fn(0)169 b(1)2008 2257 y Fm(A)p Fn(\()p Fm(t)p Fn(\))102 b(0)2360 2076 y Fh(\023)2463 2197 y Fm(;)1061 b Fq(\(4\))432 2479 y(where)30 b(the)i(symmetric)g Fm(N)j Fk(\002)25 b Fm(N)42 b Fq(matrix)31 b Fm(A)g Fq(depends)g(on)f(the)i (instantaneous)f(positions)432 2599 y(of)h(the)g(particles)h(and)g(couples)f (only)g(nearest)h(neighbors.)56 b(Since)33 b(the)f(interactions)i(are)432 2719 y(purely)e(repulsi)m(v)n(e,)k(the)c(\257o)m(w)g(is)g(h)o(yperbolic,)k (which)c(implies)h(that)g Fm(A)f Fq(should)g(be)g(tak)o(en)432 2840 y(positi)m(v)n(e.)578 2967 y(At)g(this)g(stage)h(one)f(may)h(note)f (that)h(if)g(the)f(matrix)i Fm(A)p Fn(\()p Fm(t)p Fn(\))e Fq(in)g(eq.)h (\(4\))f(were)g(replaced)432 3087 y(by)h(the)g(ne)n(g)o(ati)m(v)n(e)j(of)d (the)h(discrete)g(Laplacian,)j(the)d(L)-6 b(yapuno)n(v)34 b(spectrum)g(of)f Fm(G)g Fq(w)o(ould)432 3208 y(be)i(precisely)h(that)g(described)f(in)g(eq.)h (\(1\).)66 b(Ho)m(we)m(v)n(er)l(,)38 b(the)e(matrices)g Fm(A)p Fn(\()p Fm(t)p Fn(\))f Fq(are)h(in)f(f)o(act)432 3328 y(generated)i(by)e (chaotic)i(dynamics,)j(and)c(therefore)h(\257uctuate)f(rapidly)-7 b(.)69 b(Furthermore,)432 3449 y(the)32 b(particles)i(in)e(the)h(g)o(as)f (rearrange)h(in)f(time,)j(so)d(that)h(the)g(positions)f(of)g(the)g(non-zero) 432 3569 y(elements)39 b(in)g(the)g(matrix)g(also)g(e)m(v)n(olv)n(e.)77 b(In)38 b(our)g(study)h(we)f(concentrate)i(on)e(the)h(\256rst)432 3689 y(feature.)54 b(That)31 b(is,)i(we)e(sho)m(w)f(ho)m(w)h(tangent)h (dynamics)f(of)g(the)g(type)h(\(4\))e(result)h(in)h(slo)m(w)432 3810 y(h)o(ydrodynamic)e(modes)f(in)h(spite)f(of)g(the)h(\257uctuations)g(in) f Fm(A)p Fn(\()p Fm(t)p Fn(\))p Fq(;)i(the)f(ef)m(fects)g(of)f(particle)432 3930 y(rearrangement)45 b(may)f(in)f(principle)i(dealt)f(with)g(similarly)-7 b(,)50 b(b)n(ut)43 b(need)h(to)g(be)f(studied)432 4050 y(further)-6 b(.)578 4178 y(The)36 b(abo)n(v)n(e)g(discussion)f(allo)m(ws)h(us)f(to)h (conclude)g(that)g(the)g(matrices)g Fm(A)p Fn(\()p Fm(t)p Fn(\))g Fq(should)432 4298 y(ha)n(v)n(e)g(non-zero)g(elements)h(only)f(at)g(those)g (positions)g(which)g(are)g(nonzero)g(in)f(the)i(dis-)432 4418 y(crete)28 b(Laplacian.)45 b(Furthermore,)29 b(momentum)g(conserv)m(ation)g (implies)f(that)g(the)g(sum)g(of)432 4539 y(elements)33 b(in)g(an)n(y)g(ro)m (w)f(and)g(column)h(of)f Fm(A)g Fq(must)h(v)m(anish.)58 b(This)32 b(speci\256es)g(completely)432 4659 y(the)g(matrix)h(structure)g(of)f Fm(A)p Fq(,)i(and)e(it)h(remains)g(to)f(model)h(the)g(time)g(dependence)g(of) f(the)432 4779 y(of)m(f-diagonal)21 b(non-zero)f(elements)h(of)f Fm(A)p Fq(.)35 b(F)n(or)20 b(this)g(we)f(in)l(v)n(ok)o(e)h(the)h(h)o (ypothesis)f(of)g(strong)432 4900 y(chaos)d([1,)h(3]:)33 b(The)18 b(elements)h(of)f Fm(A)f Fq(may)h(be)g(treated)h(as)f(independent)g(random)g (processes,)432 5020 y(with)28 b(a)g(correlation)i(time)f Fm(\034)40 b Fq(which)28 b(is)g(short)g(with)g(respect)h(to)g(other)f(time)h(scales)g (of)f(the)432 5141 y(system.)50 b(It)30 b(is)g(commonly)g(found)g(that)h (this)f(approximation)h(yields)f(results)g(which)g(are)432 5261 y(in)c(good)f(qualitati)m(v)n(e)k(agreement)e(with)f(those)f(of)h(the)g (actual)h(tangent)g(\257o)m(w)-7 b(.)578 5388 y(W)l(ith)30 b(this)f(in)h(mind)f(we)g(model)h(the)g(e)m(v)n(olution)h(operator)e Fm(U)40 b Fq(by)29 b(a)h(product)f(of)g(inde-)2025 5637 y(4)p eop %%Page: 5 5 bop 83 573 a Fq(pendent)26 b(random)g(matrices)h Fm(S)1199 588 y Fl(n)1457 699 y Fm(U)38 b Fn(=)1665 616 y Fh(Y)1696 790 y Fl(n)1787 699 y Fm(S)1847 714 y Fl(n)1920 699 y Fm(;)1255 b Fq(\(5\))83 936 y(where)1148 1071 y Fm(S)1208 1086 y Fl(n)1282 1071 y Fk(\030)28 b Ff(T)17 b Fn(exp)1636 954 y Fh(Z)1719 980 y Fl(n\034)1682 1143 y Fi(\()p Fl(n)p Fe(\000)p Fi(1\))p Fl(\034)1930 1071 y Fm(G)p Fn(\()p Fm(t)p Fn(\))p Fm(dt)25 b(:)946 b Fq(\(6\))83 1297 y(During)37 b(the)g(time)h(interv)m(al)h(of)d(length)i Fm(\034)11 b Fq(,)40 b Fm(A)p Fq(,)g(and)d(therefore)h Fm(G)p Fq(,)i(may)d(be)g(considered)83 1462 y(constant,)i(so)34 b(the)i(simplest)g (model)g(for)f Fm(S)41 b Fq(w)o(ould)35 b(be)g Fm(S)46 b Fn(=)2334 1341 y Fh(\022)2450 1402 y Fn(1)141 b Fm(\034)2411 1522 y(\034)11 b(A)105 b Fn(1)2710 1341 y Fh(\023)2771 1462 y Fq(.)66 b(Ho)m(we)m(v)n(er)36 b(it)83 1624 y(is)30 b(more)g(con)l(v)n(enient)h(to)f(correct)g(this)g(form)g (by)f(a)h(second)f(order)h(term)g(in)g Fm(\034)41 b Fq(in)30 b(order)g(to)83 1745 y(preserv)n(e)22 b(the)g(symplectic)h(property)e(which)h (holds)f(for)g Fm(U)10 b Fq(.)36 b(W)-8 b(e)22 b(thus)f(arri)m(v)n(e)i(at)f (our)f(model)83 1865 y([2,)26 b(3]:)1209 2030 y Fm(S)34 b Fn(=)1406 1909 y Fh(\022)1523 1970 y Fn(1)281 b Fm(\034)1484 2090 y(\034)11 b(A)103 b Fn(1)22 b(+)g Fm(\034)1935 2054 y Fi(2)1975 2090 y Fm(A)2064 1909 y Fh(\023)2167 2030 y Fm(:)1008 b Fq(\(7\))83 2252 y(The)20 b(matrices)g Fm(A)f Fq(are)h(independent,)i(and)d(their)h(of)m (f-diagonal)h(elements)f(are)g(independent)83 2373 y(and)d(identically)j (distrib)n(uted.)36 b(The)18 b(actual)g(probability)h(distrib)n(ution)f(of)g (the)f(of)m(f-diagonal)83 2493 y(elements)22 b(can)e(be)h(chosen)f (arbitrarily)-7 b(,)24 b(subject)d(to)g(the)g(constraint)g(of)f(uniform)h(h)o (yperbol-)83 2614 y(icity)-7 b(,)25 b(namely)e(that)f(the)g(support)g(of)g (the)g(distrib)n(ution)h(is)f(strictly)h(ne)n(g)o(ati)m(v)n(e,)j(and)c (bounded)83 2734 y(a)n(w)o(ay)k(from)g(zero.)229 2856 y(The)36 b(model)g(as)f(de\256ned)g(abo)n(v)n(e)i(mak)o(es)e(sense)g(in)h(an)n(y)f (space)h(dimension,)j(b)n(ut)c(for)83 2977 y(the)27 b(sak)o(e)g(of)g (simplicity)h(we)f(study)f(it)i(in)f(one)g(dimension.)41 b(There)28 b(it)f(bears)g(similarity)i(to)83 3097 y(the)k(tangent)g(dynamics)g(of)f(an)g (anharmonic)i(chain.)58 b(Ho)m(we)m(v)n(er)l(,)35 b(in)d(the)h(latter)h(case) e(the)83 3217 y(matrices)g Fm(S)j Fq(w)o(ould)30 b(be)h(elliptic)h(rather)f (than)g(h)o(yperbolic.)52 b(As)30 b(we)g(sho)m(w)g(belo)m(w)g(this)h(is)83 3338 y(an)j(essential)g(ingredient)h(in)f(the)g(mechanism)h(for)e(h)o (ydrodynamic)i(modes,)h(which)d(are)83 3458 y(not)g(present)g(in)g(the)g(L)-6 b(yapuno)n(v)34 b(spectra)g(of)e(anharmonic)i(chains)f([8].)59 b(Unfortunately)-7 b(,)83 3579 y(we)35 b(ha)n(v)n(e)h(not)f(been)g(able)h(to) f(\256nd)g(a)g(model)h(dynamics)g(in)f(real)h(space)f(whose)g(tangent)83 3699 y(space)28 b(dynamics)h(w)o(ould)e(resemble)i(that)g(generated)g(by)e (the)i(matrices)g(of)e(type)i(\(7\).)43 b(On)83 3819 y(the)26 b(other)g(hand)f(our)g(results)h(do)f(not)g(use)h(e)n(xplicitly)h(the)f (dimensionality)i(of)d(the)h(system)83 3940 y(and)g(seem)g(to)g(be)f (generalizable)j(to)e(higher)g(dimensions.)229 4062 y(Since)f(the)h(indi)m (vidual)g(matrices)g(are)f(symplectic,)i(the)e(L)-6 b(yapuno)n(v)26 b(e)n(xponents)g(of)e(\(5\))83 4182 y(come)d(in)g(pairs)g(of)g(equal)g (absolute)h(v)m(alue)g(and)f(opposite)g(signs.)35 b(T)l(ranslation)22 b(and)f(Galilei)83 4303 y(in)l(v)m(ariance)28 b(imply)f(the)g(e)n(xistence)g (of)f(tw)o(o)g(v)m(anishing)h(e)n(xponents.)40 b(W)-8 b(e)26 b(concentrate)i(our)83 4423 y(attention)f(on)e(the)h(L)-6 b(yapuno)n(v)27 b(e)n(xponents)f Fm(\025)1668 4438 y Fl(N)7 b Fe(\000)p Fi(1)1851 4423 y Fq(and)25 b Fm(\025)2080 4438 y Fl(N)7 b Fe(\000)p Fi(2)2263 4423 y Fq(of)25 b(smallest)i(positi)m(v)n(e)g(v)m(alue,)83 4544 y(and)f(the)g(corresponding)g(L)-6 b(yapuno)n(v)26 b(v)n(ectors)h Fm(v)1812 4559 y Fl(N)7 b Fe(\000)p Fi(1)1995 4544 y Fq(and)26 b Fm(v)2215 4559 y Fl(N)7 b Fe(\000)p Fi(2)2372 4544 y Fq(.)229 4666 y(Before)23 b(we)f(go)g(on,)h(it)g(is)f(necessary)h(to)f(mak)o(e)h (precise)g(what)f(we)g(mean)h(by)g(L)-6 b(yapuno)n(v)83 4786 y(v)n(ectors.)38 b(As)23 b(is)h(well-kno)m(wn,)h(essentially)g(all)g (numerical)h(methods)e(for)g(calculating)i(the)83 4907 y(tangent)k(space)g (dynamics)g(of)g(the)g(the)f(L)-6 b(yapuno)n(v)31 b(v)n(ectors)f Fm(v)2315 4922 y Fl(n)2392 4907 y Fq(are)f(de\256ned)h(as)f(follo)m(ws:)83 5027 y(One)19 b(starts)g(with)g(an)g(orthogonal)g(matrix)h Fm(Q)p Fq(,)h(mutiplies)f(it)g(from)e(the)i(left)f(with)g(the)h(tangent)83 5147 y(matrix)30 b(\(in)f(the)h(present)f(case)h Fm(S)6 b Fq(\))29 b(and)g(decomposes)g(the)h(result)f(as)g Fm(S)6 b(Q)33 b Fn(=)f Fm(Q)2918 5111 y Fe(0)2941 5147 y Fm(R)q Fq(,)f(where)83 5268 y Fm(Q)160 5232 y Fe(0)214 5268 y Fq(is)f(orthogonal)h(and)f Fm(R)h Fq(is)g(upper)f(triangular)-6 b(.)53 b(This)30 b(procedur)h(is)f (iterated)i(to)f(yield)g(a)83 5388 y(sequence)i(of)e Fm(Q)682 5403 y Fl(t)712 5388 y Fq(.)57 b(The)32 b(columns)h(of)f(the)g(orthogonal)h (matrices)g Fm(Q)2563 5403 y Fl(t)2625 5388 y Fq(are)f(what)g(we)g(will)1676 5637 y(5)p eop %%Page: 6 6 bop 432 573 a Fq(call)27 b(the)f(L)-6 b(yapuno)n(v)27 b(v)n(ectors.)39 b(The)26 b(reader)g(should)g(note)g(that)h(these)f(v)n(ectors)h(are)f(not)g (the)432 693 y(ones)f(whose)g(e)n(xistence)i(is)f(pro)n(v)n(ed)g(in)g(the)g (multiplicati)m(v)n(e)k(er)n(godic)d(theorem.)578 818 y(W)-8 b(e)24 b(can)g(no)m(w)f(state)h(the)g(main)h(result)f(of)f(this)h(paper)-6 b(.)37 b Fp(Existence)24 b(of)f(h)n(ydr)n(odynamic)432 938 y(L)-6 b(yapuno)o(v)39 b(modes)p Fq(:)66 b Fo(As)40 b(the)h(size)f Fm(N)50 b Fo(of)41 b(the)f(matrices)h(tends)f(to)g(in\256nity)i(the)e(e)n (xpo-)432 1058 y(nents)33 b Fm(\025)737 1073 y Fl(N)7 b Fe(\000)p Fi(1)926 1058 y Fo(and)33 b Fm(\025)1169 1073 y Fl(N)7 b Fe(\000)p Fi(2)1359 1058 y Fo(as)32 b(well)i(as)e(the)h(vector)o(s)g Fm(v)2338 1073 y Fl(N)7 b Fe(\000)p Fi(1)2528 1058 y Fo(and)33 b Fm(v)2761 1073 y Fl(N)7 b Fe(\000)p Fi(2)2951 1058 y Fo(ar)l(e)33 b(asymptotic)g(to)432 1179 y(the)h(e)n(xponents)g(and)f(vector)o(s)h(that)g (would)g(be)f(obtained)i(if)f(the)g(matrices)f Fm(A)h Fo(wher)l(e)f(r)l(e-) 432 1299 y(placed)23 b(e)n(verywher)l(e)g(by)g(the)f(ne)l(gative)i(of)e(the)h (discr)l(ete)g(Laplacian)g(\(pr)-5 b(operly)22 b(r)l(escaled\).)432 1420 y Fq(The)38 b(statement)h(holds)f(for)g(the)g(L)-6 b(yapuno)n(v)39 b(v)n(ectors,)j(which)c(are)g(random)g(objects,)k(in)432 1540 y(probability)-7 b(.)578 1664 y(The)43 b(statement)g(is)g(spelled)f(out)h (only)f(for)g(tw)o(o)g(L)-6 b(yapuno)n(v)43 b(modes,)k(which)c(ha)n(v)n(e)432 1785 y(nearly)22 b(equal)h(e)n(xponents,)h(and)d(where)h(the)g(de)m(viation)i (from)e(h)o(ydrodynamic)h(beha)n(vior)g(is)432 1905 y(the)28 b(smallest.)44 b(Ho)m(we)m(v)n(er)l(,)29 b(as)f(will)g(become)g(apparent)h (from)e(the)h(ar)n(guments)h(belo)m(w)-7 b(,)29 b(the)432 2025 y(result)h(can)h(be)f(e)n(xtended)i(to)e(a)g(number)h(of)f(L)-6 b(yapuno)n(v)31 b(modes)g(near)f(the)h(middle)g(of)f(the)432 2146 y(spectrum)25 b(which)f(is)g(proportional)h(to)1826 2062 y Fk(p)p 1909 2062 89 4 v 84 x Fm(N)11 b Fq(.)37 b(Our)23 b(numerical)j (studies)e(also)h(indicate)h(that)432 2266 y(this)g(is)f(in)h(f)o(act)g (true.)578 2391 y(As)21 b(already)i(e)n(xplained,)h(the)e(basic)h(reason)e (for)h(the)g(e)n(xistence)h(of)f(the)g(h)o(ydrodynamic)432 2511 y(modes)33 b(is)h(translation)h(in)l(v)m(ariance.)63 b(Ho)m(we)m(v)n(er) l(,)37 b(this)d(general)g(observ)m(ation)h(is)e(not)h(suf-)432 2631 y(\256cient,)h(and)e(the)g(actual)h(proof)f(is)f(not)h(tri)m(vial.)60 b(It)33 b(depends)g(on)f(the)h(random)g(nature)h(of)432 2752 y(successi)m(v)n(e)41 b(matrices,)k Fo(i.e)n(.)p Fq(,)h(on)40 b(strong)g(chaos.)81 b(Our)39 b(strate)n(gy)i(will)g(be)f(to)g(sho)m(w)g(the) 432 2872 y(e)n(xistence)23 b(of)f(h)o(ydrodynamic)h(modes)f(\256rst)g(in)g (the)g(spectrum)h(of)f(a)g Fo(single)g Fq(matrix)h(of)f(type)432 2993 y Fm(A)p Fq(,)k(i.e.,)h(a)e(ne)n(g)o(ati)m(v)n(e)k(random)d(Laplacian)h (\(sec.)f(3\);)f(then)h(this)g(will)g(be)f(used)g(to)h(sho)m(w)f(that)432 3113 y(such)g(modes)i(e)n(xist)f(in)h(the)f(L)-6 b(yapuno)n(v)27 b(spectrum)g(of)f(non-symplectic)h(products)f(of)g(type)432 3167 y Fh(Q)510 3233 y Fn(\(1)e(+)h Fm(A)795 3248 y Fl(n)842 3233 y Fn(\))30 b Fq(in)h(sec.)f(4,)i(which)f(in)f(turn)g(will)h(be)f(used)h (to)f(sho)m(w)g(the)g(same)h(property)g(for)432 3354 y(symplectic)c(products) f(in)f(sec.)i(5.)432 3723 y Fs(3)147 b(Spectral)37 b(pr)m(operties)g(of)g(a)g (single)g(matrix)432 3957 y Fq(The)26 b(matrix)g Fm(A)g Fq(de\256ned)g(in)g (section)g(2)f(tak)o(es)h(in)g(one)g(dimension)g(the)g(e)n(xplicit)i(form)788 4373 y Fm(A)g Fn(=)993 4103 y Fh(0)993 4249 y(B)993 4299 y(B)993 4349 y(B)993 4398 y(B)993 4452 y(@)1082 4178 y Fm(a)1133 4193 y Fi(1)1195 4178 y Fn(+)22 b Fm(a)1344 4193 y Fi(2)1552 4178 y Fk(\000)p Fm(a)1680 4193 y Fi(2)1948 4178 y Fn(0)162 b(0)101 b Fk(\001)17 b(\001)g(\001)174 b Fn(0)256 b Fk(\000)p Fm(a)3034 4193 y Fi(1)1148 4299 y Fk(\000)p Fm(a)1276 4314 y Fi(2)1485 4299 y Fm(a)1536 4314 y Fi(2)1598 4299 y Fn(+)22 b Fm(a)1747 4314 y Fi(3)1889 4299 y Fk(\000)p Fm(a)2017 4314 y Fi(3)2159 4299 y Fn(0)101 b Fk(\001)17 b(\001)g(\001)174 b Fn(0)316 b(0)1220 4382 y Fq(.)1220 4415 y(.)1220 4448 y(.)2977 4382 y(.)2977 4415 y(.)2977 4448 y(.)1148 4569 y Fk(\000)p Fm(a)1276 4584 y Fi(1)1611 4569 y Fn(0)288 b(0)162 b(0)101 b Fk(\001)17 b(\001)g(\001)101 b(\000)p Fm(a)2656 4584 y Fl(N)2825 4569 y Fm(a)2876 4584 y Fl(N)2966 4569 y Fn(+)22 b Fm(a)3115 4584 y Fi(1)3171 4103 y Fh(1)3171 4249 y(C)3171 4299 y(C)3171 4349 y(C)3171 4398 y(C)3171 4452 y(A)3286 4373 y Fm(;)238 b Fq(\(8\))432 4786 y(where)30 b(the)h Fm(a)917 4801 y Fl(n)994 4786 y Fq(are)g(positi)m(v)n(e)h (identically)g(distrib)n(uted)g(independent)f(random)g(numbers.)432 4907 y(The)42 b(distrib)n(ution)h(of)f(the)g Fm(a)1486 4922 y Fl(n)1575 4907 y Fq(is)g(arbitrary)-7 b(,)48 b(subject)43 b(to)f(the)g(condition)h Fn(0)50 b Fm(<)g(a)3422 4922 y Fi(min)3594 4907 y Fm(<)432 5027 y(a)e(<)g(a)706 5042 y Fi(max)883 5027 y Fq(with)41 b Fm(a)1156 5042 y Fi(min)1326 5027 y Fm(<)48 b(a)1501 5042 y Fi(max)1637 5027 y Fq(,)c(and)d(normalized)i(such)d(that)i Fk(h)p Fm(a)2905 4991 y Fe(\000)p Fi(1)2999 5027 y Fk(i)48 b Fn(=)g(1)p Fq(,)d(for)40 b(later)432 5147 y(con)l(v)n(enience.)48 b(The)29 b Fk(h\001i)f Fq(al)o(w)o(ays)g(denote)h(e)n(xpectation)i(with)e (respect)g(to)g(the)g(probability)432 5268 y(distrib)n(ution)20 b(of)e(the)h Fm(a)p Fq(.)36 b(W)-8 b(e)19 b(are)g(not)g(going)g(to)g(assume)g (that)g(the)h(width)e(of)h(the)g(distrib)n(ution)432 5388 y(is)25 b(small.)2025 5637 y(6)p eop %%Page: 7 7 bop 229 573 a Fq(The)26 b(matrix)h Fm(A)f Fq(may)g(be)f(written)i(as)e(a)h (product)1464 783 y Fm(A)i Fn(=)g Fk(\000)p Fm(@)p 1746 796 57 4 v 6 w Fk(A)p 1883 704 V Fm(@)1268 b Fq(\(9\))83 993 y(where)26 b Fm(@)p 358 1006 V 31 w Fq(and)p 613 914 V 26 w Fm(@)31 b Fq(are)26 b(the)g(discrete)h(deri)m(v)m(ati)m(v)n(es)h(whose)d(action)i(on)f (a)f(v)n(ector)i Fm(v)32 b Fk(2)c Fj(R)3096 1008 y Fl(N)3188 993 y Fq(is)777 1204 y Fn(\()p Fm(@)p 815 1217 V 6 w(v)t Fn(\))961 1219 y Fl(n)1035 1204 y Fn(=)g Fm(v)1186 1219 y Fl(n)1255 1204 y Fk(\000)23 b Fm(v)1402 1219 y Fl(n)p Fe(\000)p Fi(1)1565 1204 y Fm(;)220 b Fn(\()p 1850 1124 V Fm(@)6 b(v)t Fn(\))1996 1219 y Fl(n)2070 1204 y Fn(=)28 b Fm(v)2221 1219 y Fl(n)p Fi(+1)2380 1204 y Fk(\000)23 b Fm(v)2527 1219 y Fl(n)2600 1204 y Fm(;)524 b Fq(\(10\))83 1414 y(and)21 b Fk(A)g Fq(is)g(a)h(diagonal)g(matrix)h(with)e (diagonal)h(elements)h Fm(a)2177 1429 y Fl(n)2224 1414 y Fq(.)36 b(\(The)22 b(indices)g(are)f(e)n(xtended)83 1534 y(periodically)39 b(so)e(that)g Fm(a)975 1549 y Fl(N)7 b Fi(+1)1176 1534 y Fk(\021)44 b Fm(a)1348 1549 y Fi(1)1387 1534 y Fq(\).)72 b(If)37 b(all)h(the)f Fm(a)1974 1549 y Fl(n)2058 1534 y Fq(were)g(equal)h(to)f(one,)j Fk(\000)p Fm(A)e Fq(w)o(ould)83 1655 y(reduce)26 b(to)g(the)g(discrete)h (Laplacian)g(matrix)g Fm(@)1753 1618 y Fi(2)1821 1655 y Fk(\021)h Fm(@)p 1926 1668 V 1983 1575 V 6 w(@)6 b Fq(.)229 1775 y(W)-8 b(e)26 b(de\256ne)g(the)g(F)n(ourier)g(transform)g(matrix)h Fm(F)39 b Fq(with)26 b(elements)1318 2006 y Fm(F)1381 2021 y Fl(k)r(n)1495 2006 y Fn(=)1670 1939 y(1)p 1608 1983 172 4 v 1608 2001 a Fk(p)p 1691 2001 89 4 v 84 x Fm(N)1790 2006 y(e)1835 1965 y Fl(i)1869 1938 y Fg(2)p Fd(\031)p 1869 1950 69 4 v 1876 1991 a(N)1947 1965 y Fl(k)r(n)2058 2006 y Fm(;)1066 b Fq(\(11\))83 2286 y(which)38 b(is)h(a)f(unitary)h(transformation)h(taking)f Fj(R)1903 2250 y Fl(N)2008 2286 y Fq(to)2141 2261 y Fn(~)2125 2286 y Fj(C)2206 2250 y Fl(N)2273 2286 y Fq(,)j(the)d(subset)f(of)g Fj(C)3000 2301 y Fl(N)3106 2286 y Fq(\(with)83 2406 y(standard)29 b(basis)f(v)n(ectors)i Fm(e)1056 2421 y Fl(n)1103 2406 y Fq(\))e(consisting)h (of)g(v)n(ectors)k Fn(~)-53 b Fm(v)32 b Fq(for)d(which)f Fm(v)2593 2421 y Fe(\000)p Fl(k)2723 2406 y Fn(=)j Fm(v)2881 2370 y Fe(\003)2877 2431 y Fl(k)2921 2406 y Fq(,)e(which)g(is)83 2527 y(an)f Fm(N)38 b Fq(dimensional)29 b(v)n(ector)g(space)f(o)n(v)n(er)h Fj(R)p Fq(.)44 b(The)28 b(components)g(of)g Fm(A)g Fq(in)g(the)g(ne)m(w)g(basis)83 2647 y(are)887 2742 y Fn(~)861 2768 y Fm(A)934 2783 y Fl(k)r(l)1026 2768 y Fk(\021)g Fn(\()p Fm(F)14 b(AF)1396 2726 y Fe(y)1431 2768 y Fn(\))1469 2783 y Fl(k)r(l)1562 2768 y Fn(=)27 b Fm(\026)1724 2726 y Fe(\003)1724 2792 y Fl(k)1766 2768 y Fn(\()p Fk(h)p Fm(a)p Fk(i)17 b Fm(\016)1993 2783 y Fl(k)r(;l)2099 2768 y Fn(+)23 b(~)-50 b Fm(a)2248 2783 y Fl(k)r Fe(\000)p Fl(l)2368 2768 y Fn(\))p Fm(\026)2465 2783 y Fl(l)2516 2768 y Fm(;)608 b Fq(\(12\))83 2953 y(where)38 b Fm(\026)429 2968 y Fl(k)515 2953 y Fn(=)44 b(1)29 b Fk(\000)g Fn(exp)984 2857 y Fh(\020)1034 2953 y Fk(\000)1121 2914 y Fi(2)p Fl(\031)r(i)p 1121 2930 103 4 v 1141 2988 a(N)1234 2953 y Fm(k)1288 2857 y Fh(\021)1337 2953 y Fq(,)41 b(and)f Fn(~)-51 b Fm(a)38 b Fq(is)g(related)h(to)f(the)g(F)n (ourier)g(transform)g(of)g Fm(a)83 3093 y Fq(considered)26 b(as)g(a)f(v)n(ector)i(in)f Fj(R)1199 3057 y Fl(N)1291 3093 y Fq(by)1232 3303 y Fn(~)-50 b Fm(a)27 b Fn(=)h Fm(N)1501 3262 y Fe(\000)p Fi(1)p Fl(=)p Fi(2)1666 3303 y Fm(F)14 b Fn(\()-6 b Fm(~)-43 b(a)22 b Fk(\000)h(h)-7 b Fm(~)-42 b(a)p Fk(i)p Fn(\))25 b Fm(:)978 b Fq(\(13\))83 3513 y(The)43 b(random)h(v)m(ariables)i Fn(~)-51 b Fm(a)1098 3528 y Fl(k)1184 3513 y Fq(are)44 b(centered,)49 b(and)43 b(as)g(sums)f(of)h(independent)h(random)83 3634 y(numbers)26 b(their)g(`single-point')h(distrib)n(ution)f(is)g(nearly)g(Gaussian)f(with)h (v)m(ariance)1247 3816 y Fh(D)1298 3913 y Fk(j)q Fn(~)-50 b Fm(a)p Fk(j)1405 3872 y Fi(2)1444 3816 y Fh(E)1522 3913 y Fn(=)1636 3845 y Fk(h)p Fm(a)1726 3809 y Fi(2)1765 3845 y Fk(i)22 b(\000)h(h)p Fm(a)p Fk(i)2055 3797 y Fi(2)p 1636 3889 459 4 v 1821 3981 a Fm(N)2129 3913 y(;)995 b Fq(\(14\))83 4165 y(so)25 b(that)i(the)n(y)f(are)g (typically)i(small,)f(of)e(order)h Fk(O)s Fn(\()p Fm(N)1941 4129 y Fe(\000)p Fi(1)p Fl(=)p Fi(2)2106 4165 y Fn(\))p Fq(.)38 b(The)26 b(joint)g(distrib)n(ution)h(is)f(not)83 4286 y(Gaussian.)229 4406 y(Note)16 b(that)g Fm(\026)670 4421 y Fi(0)739 4406 y Fn(=)28 b(0)p Fq(,)18 b(so)e(that)g(ro)m(w)g Fn(0)g Fq(and)g(column)g Fn(0)g Fq(of)2128 4381 y Fn(~)2097 4406 y Fm(A)21 b Fq(are)16 b(zero,)21 b(with)16 b(the)g(translation)83 4526 y(v)n(ector)27 b Fm(e)407 4541 y Fi(0)472 4526 y Fq(being)f(tri)m(vially)i(a)e(zero)g(eigen) l(v)n(ector)-6 b(.)39 b(W)-8 b(e)26 b(de\256ne)g(the)g(slo)m(w)f(subspace) 1045 4737 y Fm(V)1102 4752 y Fi(s)1161 4737 y Fn(=)j(Span\()p Fk(f)p Fm(e)1609 4752 y Fi(0)1648 4737 y Fm(;)17 b(e)1737 4752 y Fi(1)1777 4737 y Fm(;)g(e)1866 4752 y Fe(\000)p Fi(1)1960 4737 y Fk(g)p Fn(\))22 b Fk(\\)2175 4711 y Fn(~)2159 4737 y Fj(C)2240 4695 y Fl(N)2332 4737 y Fm(;)792 b Fq(\(15\))83 4947 y(and)36 b(its)g(orthogonal)g(complement,)k(the)c(f)o(ast)g(subspace)g Fm(V)2245 4962 y Fi(f)2275 4947 y Fq(.)68 b(W)-8 b(e)35 b(will)h(consider)g (often)83 5067 y(belo)m(w)c(the)h(block)f(decomposition)h(of)1551 5042 y Fn(~)1525 5067 y Fm(A)f Fq(and)g(other)g(matrices)h(into)g(the)f(f)o (ast)g(and)g(slo)m(w)83 5188 y(subspaces,)26 b Fo(e)n(.g)n(.)p Fq(,)1330 5328 y Fn(~)1304 5353 y Fm(A)i Fn(=)1508 5232 y Fh(\022)1587 5293 y Fm(A)1660 5308 y Fi(f)5 b(f)1821 5293 y Fm(A)1894 5308 y Fi(f)t(s)1586 5413 y Fm(A)1659 5428 y Fi(sf)1820 5413 y Fm(A)1893 5428 y Fi(ss)1970 5232 y Fh(\023)2073 5353 y Fm(:)1051 b Fq(\(16\))1676 5637 y(7)p eop %%Page: 8 8 bop 432 573 a Fq(Note)25 b(that)i Fm(V)891 588 y Fi(f)947 573 y Fq(contains)f(slo)m(w)g(as)f(well)i(as)e(f)o(ast)h(modes.)578 693 y(The)f(block)g Fm(A)1085 708 y Fi(ss)1170 693 y Fq(has)f(small)i(norm)f (of)f(order)h Fk(O)s Fn(\()p Fm(N)2373 657 y Fe(\000)p Fi(2)2468 693 y Fn(\))p Fq(,)g(and)g(the)g(of)m(f-diagonal)h(blocks)432 814 y(ha)n(v)n(e)37 b(norm)g(of)f(order)h Fk(O)s Fn(\()p Fm(N)1490 777 y Fe(\000)p Fi(1)1585 814 y Fn(\))p Fq(.)70 b(Ho)m(we)m(v)n(er)l(,)40 b(there)e(are)f(more)g(speci\256c)f(properties)i(of)432 934 y Fm(A)g Fq(which)g(are)g(needed)g(to)g(establish)h(the)f(e)n(xistence)i(of)e (h)o(ydrodynamic)h(eigenmodes.)432 1054 y(Consider)g(the)i(eigen)l(v)m(alue)g (problem)f Fm(Av)51 b Fn(=)c Fm(\025v)t Fq(.)79 b(Letting)41 b Fm(v)50 b Fn(=)d Fm(@)p 2917 1067 57 4 v 6 w(u)p Fq(,)c(and)d(using)f(the) 432 1175 y(representation)27 b(\(9\))e(gi)m(v)n(es)i(an)f(equation)g(for)g Fm(u)1701 1387 y Fk(\000)p Fm(@)1834 1346 y Fi(2)1875 1387 y Fm(u)h Fn(=)g Fm(\025)p Fk(A)2198 1346 y Fe(\000)p Fi(1)2292 1387 y Fm(u)e(:)1100 b Fq(\(17\))578 1599 y(It)26 b(is)f(con)l(v)n(enient)j (to)d(proceed)i(by)e(writing)h(eq.)g(\(17\))g(in)f(F)n(ourier)i(component)f (form)1446 1811 y Fn(\()p Fk(j)p Fm(\026)1571 1826 y Fl(k)1613 1811 y Fk(j)1641 1770 y Fi(2)1702 1811 y Fk(\000)d Fm(\025)p Fn(\))6 b(~)-55 b Fm(u)1953 1826 y Fl(k)2023 1811 y Fn(=)27 b Fm(\025)2200 1728 y Fh(X)2243 1903 y Fl(q)2333 1785 y Fn(~)2336 1811 y Fm(b)2377 1826 y Fl(k)r Fe(\000)p Fl(q)2515 1811 y Fn(~)-55 b Fm(u)2565 1826 y Fl(q)2628 1811 y Fm(;)845 b Fq(\(18\))432 2116 y(with)21 b(the)775 2090 y Fn(~)779 2116 y Fm(b)820 2131 y Fl(k)884 2116 y Fq(bearing)g(a)g(relation)h(to)f Fm(a)1759 2080 y Fe(\000)p Fi(1)1759 2141 y Fl(n)1874 2116 y Fq(analogous)g(to)g(that)h (between)g Fn(~)-50 b Fm(a)2993 2131 y Fl(k)3056 2116 y Fq(and)21 b Fm(a)3275 2131 y Fl(n)3322 2116 y Fq(,)h(namely)1530 2346 y Fn(~)1533 2373 y Fm(b)1574 2388 y Fl(k)1645 2373 y Fn(=)28 b Fm(N)1837 2331 y Fe(\000)p Fi(1)p Fl(=)p Fi(2)2002 2373 y Fm(F)2065 2388 y Fl(k)r(n)2151 2373 y Fn(\()2223 2305 y(1)p 2199 2349 99 4 v 2199 2441 a Fm(a)2250 2456 y Fl(n)2329 2373 y Fk(\000)23 b Fn(1\))i Fm(:)932 b Fq(\(19\))432 2655 y(Since)695 2559 y Fh(D)774 2616 y Fi(1)p 756 2632 73 4 v 756 2690 a Fl(a)793 2671 y Fg(2)838 2559 y Fh(E)932 2655 y Fm(<)43 b Fn(1)p Fm(=a)1200 2619 y Fi(2)1200 2680 y(min)1359 2655 y Fq(is)37 b Fk(O)s Fn(\(1\))g Fq(we)g(\256nd)g(that)h(the)2408 2629 y Fn(~)2411 2655 y Fm(b)2452 2670 y Fl(k)2532 2655 y Fq(are)g Fk(O)s Fn(\()p Fm(N)2902 2619 y Fe(\000)p Fi(1)p Fl(=)p Fi(2)3067 2655 y Fn(\))f Fq(for)g(the)h(same)432 2776 y(reason)25 b(that)i(the)g Fn(~)-50 b Fm(a)1103 2791 y Fl(k)1171 2776 y Fq(are.)578 2896 y(W)-8 b(e)36 b(claim)i(that)f Fo(given)f(a)g(\256xed)h Fm(m)p Fo(,)i(and)d(for)g Fm(N)52 b Fk(!)41 b(1)36 b Fo(the)g(system)g(\(18\))g(has)f(two)432 3017 y(linearly)26 b(independent)h(solutions)f Fm(u)1748 2980 y Fi(\()p Fe(\006)p Fl(m)p Fi(\))1924 3017 y Fm(;)33 b(\025)2041 3032 y Fe(\006)p Fl(m)2188 3017 y Fo(suc)n(h)26 b(that)1471 3203 y Fn(1)p 1352 3247 288 4 v 1352 3358 a Fk(j)p Fm(u)1436 3307 y Fi(\()p Fe(\006)p Fl(m)p Fi(\))1436 3365 y Fl(m)1611 3358 y Fk(j)1703 3187 y Fh(X)1666 3375 y Fe(j)p Fl(k)r Fe(j6)p Fi(=)p Fl(m)1877 3270 y Fk(j)p Fm(u)1961 3219 y Fi(\()p Fe(\006)p Fl(m)p Fi(\))1961 3295 y Fl(k)2137 3270 y Fk(j)h Fn(=)h Fk(O)s Fn(\()p Fm(N)2504 3229 y Fe(\000)p Fi(1)p Fl(=)p Fi(2)2669 3270 y Fn(\))d Fm(;)741 b Fq(\(20\))432 3569 y Fo(and)1520 3579 y Fh(\014)1520 3629 y(\014)1520 3679 y(\014)1520 3729 y(\014)1520 3778 y(\014)1579 3661 y Fm(\025)1636 3676 y Fe(\006)p Fl(m)p 1558 3705 221 4 v 1558 3797 a Fk(j)p Fm(\026)1645 3812 y Fl(m)1711 3797 y Fk(j)1739 3768 y Fi(2)1810 3729 y Fk(\000)23 b Fn(1)1959 3579 y Fh(\014)1959 3629 y(\014)1959 3679 y(\014)1959 3729 y(\014)1959 3778 y(\014)2014 3729 y Fn(=)k Fk(O)s Fn(\()p Fm(N)2325 3687 y Fe(\000)p Fi(1)p Fl(=)p Fi(2)2491 3729 y Fn(\))e Fm(:)919 b Fq(\(21\))432 3956 y(W)-8 b(e)29 b(justify)h(the)g(claim)g(by)g (sho)m(wing)f(that)h(eqs.)f(\(20\))g(and)h(\(21\))f(are)h(consistent)g(with)f (the)432 4076 y(eigen)l(v)m(alue)e(equation)g(\(18\).)37 b(F)n(or)26 b(this)f(we)h(re)m(write)h(\(18\))e(as)1394 4335 y Fn(~)-55 b Fm(u)1444 4284 y Fi(\()p Fl(m)p Fi(\))1444 4360 y Fl(k)1593 4335 y Fn(=)1865 4268 y Fm(\025)1922 4283 y Fl(m)p 1706 4312 442 4 v 1706 4403 a Fk(j)p Fm(\026)1793 4418 y Fl(k)1835 4403 y Fk(j)1863 4375 y Fi(2)1925 4403 y Fk(\000)22 b Fm(\025)2081 4418 y Fl(m)2174 4252 y Fh(X)2217 4426 y Fl(q)2307 4309 y Fn(~)2311 4335 y Fm(b)2352 4350 y Fl(k)r Fe(\000)p Fl(q)2490 4335 y Fn(~)-55 b Fm(u)2540 4294 y Fi(\()p Fl(m)p Fi(\))2540 4360 y Fl(q)2686 4335 y Fm(:)787 b Fq(\(22\))432 4616 y(W)-8 b(e)17 b(assume)g(that)g (\(20\26121\))g(hold;)i(this)e(implies)g(that)g(the)g(sum)g(o)n(v)n(er)g Fm(q)22 b Fq(in)17 b(\(22\))g(is)g(dominated)432 4737 y(by)25 b(the)h(tw)o(o)g(terms)g(with)f Fm(q)32 b Fn(=)c Fk(\006)p Fm(m)p Fq(,)e(that)h(is,)678 5005 y Fn(~)-55 b Fm(u)728 4954 y Fi(\()p Fl(m)p Fi(\))728 5030 y Fl(k)876 5005 y Fn(=)1149 4938 y Fk(j)p Fm(\026)1236 4953 y Fl(m)1302 4938 y Fk(j)1330 4902 y Fi(2)p 990 4982 538 4 v 990 5074 a Fk(j)p Fm(\026)1077 5089 y Fl(k)1119 5074 y Fk(j)1147 5045 y Fi(2)1208 5074 y Fk(\000)23 b(j)p Fm(\026)1395 5089 y Fl(m)1461 5074 y Fk(j)1489 5045 y Fi(2)1538 5005 y Fn(\()p Fm(b)1617 5020 y Fl(k)r Fe(\000)p Fl(m)1783 5005 y Fn(~)-55 b Fm(u)1833 4964 y Fi(\()p Fl(m)p Fi(\))1833 5030 y Fl(m)1976 5005 y Fn(+)22 b Fm(b)2115 5020 y Fl(k)r Fi(+)p Fl(m)2282 5005 y Fn(~)-55 b Fm(u)2332 4954 y Fi(\()p Fl(m)p Fi(\))2332 5024 y Fe(\000)p Fl(m)2453 5005 y Fn(\))25 b Fm(;)221 b Fq(for)25 b Fk(j)p Fm(k)s Fk(j)i(6)p Fn(=)h Fm(m)p Fq(.)241 b(\(23\))432 5268 y(On)33 b(substituting)h(this)g(e)n (xpression)g(in)g(the)g(left-hand-side)h(of)e(\(20\))h(the)g(sum)f(o)n(v)n (er)i Fm(k)i Fq(is)432 5388 y(observ)n(ed)c(to)g(be)f(local,)k(in)d(the)g (sense)g(that)g(it)g(is)g(dominated)h(by)e(terms)i(with)e Fk(j)p Fm(k)s Fk(j)37 b(\030)h Fm(m)p Fq(,)2025 5637 y(8)p eop %%Page: 9 9 bop 83 589 a Fq(where)32 b Fk(j)p Fm(\026)451 604 y Fl(k)493 589 y Fk(j)521 553 y Fi(2)596 589 y Fk(\030)709 493 y Fh(\020)769 550 y Fi(2)p Fl(\031)r(k)p 769 566 117 4 v 796 624 a(N)895 493 y Fh(\021)945 516 y Fi(2)984 589 y Fq(.)56 b(Since)1320 563 y Fn(~)1323 589 y Fm(b)1364 604 y Fl(k)1439 589 y Fq(is)32 b Fk(O)s Fn(\()p Fm(N)1747 553 y Fe(\000)p Fi(1)p Fl(=)p Fi(2)1912 589 y Fn(\))p Fq(,)i(assumption)e(\(20\))f(is)h(v)n(eri\256ed.)56 b(On)83 710 y(the)26 b(other)g(hand,)g(using)g(\(20\))f(in)h(\(22\))f(for)h Fm(k)k Fn(=)e Fm(m)e Fq(gi)m(v)n(es)898 966 y Fn(~)-55 b Fm(u)948 925 y Fi(\()p Fl(m)p Fi(\))948 991 y Fl(m)1096 966 y Fn(=)1381 899 y Fm(\025)1438 914 y Fl(m)p 1210 943 466 4 v 1210 1034 a Fk(j)p Fm(\026)1297 1049 y Fl(m)1363 1034 y Fk(j)1391 1006 y Fi(2)1452 1034 y Fk(\000)23 b Fm(\025)1609 1049 y Fl(m)1685 966 y Fn(\()p Fm(b)1764 981 y Fi(0)1810 966 y Fn(~)-55 b Fm(u)1860 925 y Fi(\()p Fl(m)p Fi(\))1860 991 y Fl(m)2003 966 y Fn(+)22 b Fm(b)2142 981 y Fi(2)p Fl(m)2251 966 y Fn(~)-55 b Fm(u)2301 915 y Fi(\()p Fl(m)p Fi(\))2301 985 y Fe(\000)p Fl(m)2422 966 y Fn(\))25 b Fm(:)639 b Fq(\(24\))83 1249 y(Since)41 b Fm(b)391 1264 y Fi(0)471 1249 y Fq(and)654 1223 y Fn(~)658 1249 y Fm(b)699 1264 y Fi(2)p Fl(m)841 1249 y Fq(are)g Fm(O)s Fn(\()p Fm(N)1210 1213 y Fe(\000)p Fi(1)p Fl(=)p Fi(2)1374 1249 y Fn(\))f Fq(it)h(follo)m(ws)f (that)h Fm(\025)2142 1264 y Fl(m)2209 1249 y Fm(=)p Fn(\()p Fk(j)p Fm(\026)2383 1264 y Fl(m)2448 1249 y Fk(j)2476 1213 y Fi(2)2538 1249 y Fk(\000)22 b Fm(\025)2694 1264 y Fl(m)2761 1249 y Fn(\))47 b(=)g Fm(O)s Fn(\()p Fm(N)3173 1213 y Fi(1)p Fl(=)p Fi(2)3283 1249 y Fn(\))83 1370 y Fq(v)n(erifying)27 b(\(21\),)f(which)f(sho)m(ws)g(that)i(\(20\26121\))e(are)h(indeed)g (consistent)g(with)g(\(18\).)229 1490 y(In)g(terms)g(of)f(the)h(original)h(v) m(ariables)g Fm(v)t Fq(,)f(eq.)g(\(23\))f(reads)334 1744 y 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2926 y Fe(\000)p Fi(2)1656 2963 y Fn(\))p Fq(.)229 3083 y(This)32 b(is)f(established)i(easily)f(by)f(noting)h(that)g(the)g(sharp)f(cutof)m(f)h (on)g(the)f(probability)83 3203 y(distrib)n(ution)c(of)e(the)h Fm(a)g Fq(implies)h(that)f(e)m(v)n(ery)h(realization)h Fm(A)d Fq(satis\256es)h(the)g(bounds)1154 3413 y Fk(\000)p Fm(a)1282 3428 y Fi(min)1405 3413 y Fm(@)1461 3372 y Fi(2)1529 3413 y Fm(<)h(A)h(<)g Fk(\000)p Fm(a)1965 3428 y Fi(max)2101 3413 y Fm(@)2157 3372 y Fi(2)2223 3413 y Fm(;)901 b Fq(\(26\))83 3623 y(and)39 b(then)f(by)h(the)g(minimax)h(principle)f(it)g(follo)m(ws)g (that)g(the)g Fm(p)p Fq(th)g(eigen)l(v)m(alue)h(of)e Fm(A)h Fq(is)83 3743 y(lar)n(ger)24 b(than)g Fm(a)592 3758 y Fi(min)737 3743 y Fq(times)h(the)f Fm(p)p Fq(th)f(eigen)l(v)m(alue)j(of)d Fk(\000)p Fm(@)1981 3707 y Fi(2)2045 3743 y Fq(\(sorting)g(the)h(eigen)l(v)m (alues)h(of)f(both)83 3863 y(matrices)j(in)f(increasing)g(order\).)229 3984 y(The)35 b(results)g(of)f(this)h(section)g(can)g(be)g(summarized)h 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Fq(,)c(and)e(the)g(diagonal)83 4977 y(blocks)26 b(obe)n(ying)1187 5098 y Fm(D)1268 5113 y Fi(f)1327 5098 y Fm(>)h(\025)h(>)g(\013)q(\025)f(>)g (D)1950 5113 y Fi(s)2010 5098 y Fk(\025)h Fn(0)e Fm(;)934 b Fq(\(28\))83 5268 y(and)26 b(furthermore)1434 5388 y Fk(k)p Fm(A)1557 5403 y Fi(ss)1617 5388 y Fk(k)h Fm(<)h(\013)q(\025)d(:)1181 b Fq(\(29\))1676 5637 y(9)p eop %%Page: 10 10 bop 578 573 a Fq(The)32 b(orders)f(of)h(magnitude)h(for)e Fm(\025)h Fq(and)g Fm(")f Fq(are)h Fm(")j Fn(=)h Fk(O)s Fn(\()p Fm(N)2685 537 y Fe(\000)p Fi(1)p Fl(=)p Fi(2)2850 573 y Fn(\))c Fq(and)g Fm(\025)k Fn(=)f Fk(O)s Fn(\()p Fm(N)3511 537 y Fe(\000)p Fi(2)3606 573 y Fn(\))p Fq(,)432 693 y(whereas)17 b Fm(\013)29 b Fk(\030)f Fn(1)p Fm(=)p Fn(4)p Fq(.)34 b(Ho)m(we)m(v)n(er)l(,)21 b(to)d(k)o(eep)g(the)g (discussion)g(reasonably)g(general)h(we)e(are)h(not)432 814 y(going)23 b(to)h(use)f(these)h(speci\256c)g(v)m(alues)g(in)g(our)f(ar)n (guments)i(belo)m(w)-7 b(.)37 b(Rather)l(,)25 b(we)f(will)g(mak)o(e)432 934 y(statements)j(re)n(g)o(arding)g(arbitrary)f(matrices)h(which)f(satisfy)g (the)g(conditions)g(\(27\26129\).)578 1054 y(Although)31 b(this)h(will)f(not) g(be)h(used)f(belo)m(w)-7 b(,)33 b(it)f(is)f(rele)m(v)m(ant)i(to)f(note)f (that)h(if)f(we)g(let)h Fm(m)432 1175 y Fq(v)m(ary)-7 b(,)35 b(the)e(small)h(parameter)g(in)f(\(20\))f(and)h(\(21\))f(becomes)h Fm(m=)-5 b(N)2816 1139 y Fi(1)p Fl(=)p Fi(2)2927 1175 y Fq(.)58 b(This)33 b(means)g(that)432 1295 y(we)26 b(can)i(e)n(xpect)g(a)f(number)h (of)e(h)o(ydrodynamic)j(eigenmodes)f(which)f(is)f(proportional)i(to)432 1332 y Fk(p)p 515 1332 89 4 v 84 x Fm(N)10 b Fq(.)61 b(Another)34 b(w)o(ay)f(to)g(see)h(this)g(is)f(related)i(to)e(the)h(study)g(of)f(the)h (vibrations)g(of)f(one-)432 1536 y(dimensional)25 b(disordered)f(lattices)i (which)e(are)g(modeled)h(precisely)f(by)g(the)h(eigenmodes)432 1656 y(of)42 b(matrices)i(of)f(type)g(\(8\).)88 b(There)44 b(it)f(is)g(kno)m(wn)f(that)i(the)f(localization)i(length)f Fm(\030)j Fq(is)432 1777 y(proportional)38 b(to)g Fm(\025)1147 1741 y Fe(\000)p Fi(1)1147 1801 y Fl(m)1241 1777 y Fq(.)74 b(Since)38 b Fm(\025)1661 1792 y Fl(m)1771 1777 y Fk(\030)45 b Fn(\(2)p Fm(\031)t(=)-5 b(N)9 b Fn(\))2208 1741 y Fi(2)2285 1777 y Fq(the)38 b(localization)i(length)f(will)f(reach)432 1897 y Fm(N)45 b Fq(when)35 b Fm(m)40 b Fn(=)g Fk(O)s Fn(\()p Fm(N)1260 1861 y Fi(1)p Fl(=)p Fi(2)1371 1897 y Fn(\))p Fq(.)65 b(Thus,)37 b(ag)o(ain,)i(we)34 b(only)h(e)n(xpect)i(w)o(a)n(v)n(e-lik)o(e)f (modes)f(when)432 2017 y Fm(m)28 b(<)f Fk(O)s Fn(\()p Fm(N)856 1981 y Fi(1)p Fl(=)p Fi(2)967 2017 y Fn(\))p Fq(.)432 2356 y Fs(4)147 b(Pr)m(oducts)37 b(of)g(matrices)h(of)f(the)f(f)l(orm)i Fc(1)31 b(+)i Fb(\034)16 b(A)432 2582 y Fq(In)27 b(this)g(section)g(we)g(use) g(the)g(properties)h(deri)m(v)n(ed)h(in)e(section)h(3)e(to)i(deri)m(v)n(e)g (the)g(e)n(xistence)432 2703 y(of)23 b(h)o(ydrodynamic)i(modes)e(in)h(the)g (L)-6 b(yapuno)n(v)25 b(spectrum)f(of)f(the)h(product)3110 2636 y Fh(Q)3188 2723 y Fl(n)3235 2703 y Fn(\(1)17 b(+)h Fm(\034)11 b(A)3559 2718 y Fl(n)3606 2703 y Fn(\))p Fq(,)432 2823 y(where)20 b(the)g(matrices)h Fm(A)1283 2838 y Fl(n)1350 2823 y Fq(are)g(independent)g (realizations)h(of)e(the)g(random)g(matrix)h(de\256ned)432 2943 y(in)33 b(eq.)g(\(8\).)58 b(Beside)33 b(pro)n(viding)g(a)g(step)g(to)m (w)o(ards)f(pro)n(ving)i(the)f(e)n(xistence)h(of)e(h)o(ydrody-)432 3064 y(namic)38 b(modes)f(in)g(symplectic)i(products,)h(such)d(a)g(product)g (may)h(be)f(re)n(g)o(arded)h(as)f(the)432 3184 y(discrete)26 b(approximation)i(to)d(a)h(continuous)g(tangent)h(\257o)m(w)e(gi)m(v)n(en)i (by)1611 3405 y Fm(U)38 b Fn(=)28 b Ff(T)17 b Fn(exp)2068 3287 y Fh(Z)2167 3405 y Fm(A)p Fn(\()p Fm(t)p Fn(\))p Fm(dt)26 b(:)1010 b Fq(\(30\))432 3622 y([Compare)33 b(eqs.)g(\(3\))g(and)g(\(4\).])58 b(Although)33 b(this)g(does)g(not)g(correspond)f(to)h(the)g(tangent)432 3742 y(\257o)m(w)44 b(of)h(a)g(mechanical)i(system,)k(it)45 b(is)g(nonetheless)h(the)f(simplest)h(e)n(xample)g(where)432 3862 y(h)o(ydrodynamic)36 b(L)-6 b(yapuno)n(v)37 b(modes)e(can)h(be)f(e)n (xpected.)68 b(F)n(or)35 b(con)l(v)n(enience)i(of)e(further)432 3983 y(analysis)30 b(we)g(absorb)g Fm(\034)42 b Fq(into)31 b(the)g(de\256nition)g(of)f Fm(A)g Fq(and)h(change)g(to)f(F)n(ourier)h(basis) f(once)432 4103 y(and)25 b(for)h(all,)h(so)e(the)h(problem)g(becomes)h(that)f (of)f(a)h(product)1789 4213 y Fh(Y)1820 4388 y Fl(n)1895 4296 y Fn(\(1)c(+)2128 4271 y(~)2102 4296 y Fm(A)2175 4311 y Fl(n)2222 4296 y Fn(\))j Fm(:)1188 b Fq(\(31\))578 4545 y(Since)34 b(the)g(L)-6 b(yapuno)n(v)35 b(e)n(xponents)g(of)e(the)h(slo)m(w)g(part)g(are)g(e)n (xpected)i(to)e(be)f(smaller)432 4666 y(than)h(the)g(rest)f(we)g(aim)i(at)f (sho)m(wing)f(that)h(the)g(\256rst)f Fm(N)k Fk(\000)27 b Fn(3)33 b Fq(L)-6 b(yapuno)n(v)35 b(v)n(ectors)f(span)f(a)432 4786 y(subspace)17 b Fm(L)883 4801 y Fi(f)932 4786 y Fq(\(of)1084 4761 y Fn(~)1068 4786 y Fj(C)1149 4801 y Fl(N)1216 4786 y Fq(\))h(which)f(is) h(almost)g(orthogonal)g(to)g Fm(V)2522 4801 y Fi(s)2571 4786 y Fq(in)g(the)g(sense)f(that)h(for)g(an)n(y)g(tw)o(o)432 4907 y(unit)30 b(v)n(ectors)h Fm(u)999 4922 y Fi(f)1063 4907 y Fk(2)j Fm(L)1229 4922 y Fi(f)1291 4907 y Fq(and)c Fm(v)1515 4922 y Fi(s)1581 4907 y Fk(2)k Fm(V)1738 4922 y Fi(s)1800 4907 y Fq(one)c(has)g Fk(j)p Fm(u)2227 4922 y Fi(f)2281 4907 y Fk(\001)25 b Fm(v)2381 4922 y Fi(s)2413 4907 y Fk(j)33 b(\034)g Fn(1)p Fq(.)51 b(Although)30 b(the)h(subspace)432 5027 y Fm(L)498 5042 y Fi(f)565 5027 y Fq(changes)37 b(after)g(each)f(step,)k(we)c(sho)m(w)g(belo)m(w)g(that)h(the)g (`almost)g(orthogonality')h(is)432 5147 y(propag)o(ated)27 b(from)e(step)h(to)g(step.)578 5268 y(T)-8 b(o)35 b(sho)m(w)f(this,)k(we)d (propose)f(the)i(follo)m(wing)f(scheme.)66 b(T)-8 b(ak)o(e)36 b(an)f(arbitrary)g(v)n(ector)432 5388 y Fm(u)e Fk(2)g Fm(L)686 5403 y Fi(f)718 5388 y Fq(,)e(whose)e(components)h(in)g Fm(V)1752 5403 y Fi(f)1813 5388 y Fq(and)g Fm(V)2047 5403 y Fi(s)2108 5388 y Fq(are)g Fm(u)2318 5403 y Fi(f)2379 5388 y Fq(and)g Fm(u)2612 5403 y Fi(s)2673 5388 y Fq(respecti)m(v)n(ely)-7 b(,)34 b(normalized)2000 5637 y(10)p eop %%Page: 11 11 bop 83 573 a Fq(so)29 b(that)g Fk(k)p Fm(u)490 588 y Fi(f)521 573 y Fk(k)j Fn(=)g(1)p Fq(,)e(and)f(assume)g(that)h Fm(u)1558 588 y Fi(s)1618 573 y Fq(is)f(small.)48 b(The)29 b(action)h(of)f Fn(1)24 b(+)g Fm(A)29 b Fq(generates)h(a)83 693 y(ne)m(w)c(normalized)h(v)n (ector)g Fm(u)1095 657 y Fe(0)1143 693 y Fq(by)1282 1071 y Fm(u)1338 1030 y Fe(0)1388 1071 y Fn(=)1594 1003 y(\(1)22 b(+)g Fm(A)p Fn(\))p Fm(u)p 1502 1048 559 4 v 1502 1139 a Fk(k)p Fn([\(1)f(+)h Fm(A)p Fn(\))p Fm(u)p Fn(])1979 1154 y Fi(f)2010 1139 y Fk(k)2095 1071 y Fm(;)1029 b Fq(\(32\))83 1395 y(where)26 b Fn([)p Fk(\001)p Fn(])440 1410 y Fi(f)498 1395 y Fq(is)g(the)h(projection)h (onto)e(the)h Fn(f)7 b Fq(-component)27 b(W)-8 b(e)27 b(w)o(ould)f(lik)o(e)h (to)g(sho)m(w)f(that)h Fm(u)3289 1410 y Fi(s)83 1515 y Fq(remains)f(small)h (after)f(repeated)h(iteration)g(of)f(this)f(process.)229 1662 y(The)d(block)f(diagonalization)j(\(27\))c(sho)m(ws)g(that)i(the)g(subspaces) e Fm(V)2574 1677 y Fi(f)2626 1662 y Fq(and)h Fm(V)2851 1677 y Fi(s)2904 1662 y Fq(are)g(indeed)83 1782 y(almost)34 b(in)l(v)m(ariant)h (under)e(the)g(transformation)1858 1757 y Fn(~)1832 1782 y Fm(A)p Fq(.)60 b(Ho)m(we)m(v)n(er)l(,)36 b(in)d(trying)h(to)f(apply)h(this)83 1903 y(f)o(act)29 b(to)f(the)h(L)-6 b(yapuno)n(v)29 b(v)n(ectors)g(of)f(the)h (product)f(\(31\))g(we)g(immediately)j(encounter)e(the)83 2023 y(danger)23 b(that)h(the)f(small)g(perturbations)h(may)f(accumulate.)39 b(The)23 b(basic)g(problem)g(is)g(that)h(a)83 2144 y(v)n(ector)i(in)e Fm(V)521 2159 y Fi(s)577 2144 y Fq(is)h(contracted)h(with)e(respect)h(to)g (the)g(`slo)m(west')g(direction)g(in)g Fm(V)2833 2159 y Fi(f)2888 2144 y Fq(by)g(a)f(f)o(actor)83 2264 y(of)29 b(only)g Fn(1)24 b Fk(\000)g(O)s Fn(\()p Fm(\025)p Fn(\))29 b Fq(\(as)g(can)h(be)f(seen)g (from)g(the)h(bounds)e(on)h Fm(D)2395 2279 y Fi(f)2455 2264 y Fq(and)g Fm(D)2712 2279 y Fi(s)2744 2264 y Fq(\),)h(whereas)f(the)83 2384 y(perturbations)e(which)f(tilt)h(a)f(v)n(ector)h(in)f Fm(L)1582 2399 y Fi(f)1639 2384 y Fq(with)g(respect)h(to)f Fm(V)2323 2399 y Fi(f)2379 2384 y Fq(are)g(of)g(order)g Fm(")p Fq(,)g(which)g(is)83 2505 y(the)j(typical)i(size)e(of)g(the)h(of)m (f-diagonal)g(blocks)f([cf.)g(eqs.)h(\(27\)\261\(29\)])e(and)h(since)h(we)e (are)83 2625 y(interested)j(in)f(the)g(case)g Fm(\025)k Fk(\034)e Fm(")e Fq(this)g(contraction)h(is)f(not)g(strong)f(enough)h(to)g(o)n(v)n (ercome)83 2745 y(the)c(perturbation.)229 2893 y(This)35 b(order)g(of)g (magnitude)h(ar)n(gument)g(can)f(be)g(made)h(e)n(xplicit)h(by)e(constructing) h(a)83 3013 y(series)19 b(of)f(matrices)i(with)f(the)g(properties)h(gi)m(v)n (en)g(by)e(eqs.)h(\(27\)\261\(29\),)i(which)d(tak)o(e)i(a)f(v)n(ector)83 3133 y(in)29 b Fm(V)248 3148 y Fi(f)307 3133 y Fq(and)f(rotate)i(it)f(such)f (that)h(the)g(outcome)h(is)e(a)g(v)n(ector)i(which)f(has)f(an)g(angle)h(with) g Fm(V)3290 3148 y Fi(f)83 3254 y Fq(of)c(order)h(1.)37 b(This)26 b(counter)n(-e)n(xample)i(is)e(gi)m(v)n(en)h(in)e(appendix)i(A.)229 3401 y(An)36 b(essential)h(ingredient)g(in)g(the)f(construction)h(of)f(this)g (counter)n(-e)n(xample)j(is)d(that)83 3521 y(the)h(matrix)572 3496 y Fn(~)546 3521 y Fm(A)g Fq(has)f(to)h(be)f(chosen)h(speci\256cally)g (gi)m(v)n(en)h Fm(u)e Fq(which)g(in)h(turn)g(depends)f(on)83 3642 y(former)46 b(realizations,)53 b(in)47 b(violation)g(of)f(the)g (independence)h(assumption.)99 b(In)45 b(other)83 3762 y(w)o(ords,)23 b(although)g(such)f(a)h(`bad')g(sequence)g(is)f(possible)h(one)g(naturally)g (e)n(xpects)h(that)g(this)83 3882 y(is)d(an)h(e)m(v)n(ent)h(with)f(v)n(ery)g (lo)m(w)g(probability)-7 b(.)37 b(T)-8 b(ypically)24 b(the)e(perturbations)g (to)g Fm(u)2872 3897 y Fi(s)2925 3882 y Fq(generated)83 4003 y(by)31 b(the)g(of)m(f-diagonal)h(part)f(of)g(the)g(matrices)h Fm(R)g Fq(do)f(not)g(ha)n(v)n(e)g(the)h(same)f(direction,)j(and)83 4123 y(should)h(serv)n(e)h(to)g(cancel)h(one)f(another)-6 b(.)68 b(Therefore)36 b(the)g(statement)h(one)f(can)g(hope)g(to)83 4243 y(sho)m(w)31 b(is)g(that)h Fo(in)f(the)h(sequence)g(g)o(ener)n(ated)g (by)g(iter)n(ation)g(of)g(eq.)g(\(32\),)h(the)e(pr)-5 b(obability)83 4364 y(that)40 b Fk(k)p Fm(u)387 4379 y Fi(s)419 4364 y Fk(k)46 b Fm(>)h(C)7 b(")39 b Fo(for)h(some)f(\256xed)h Fm(C)47 b Fo(is)39 b(very)i(small)p Fq(,)i(as)d(w)o(as)f(sho)m(wn)g(for)h(a)g(similar)83 4484 y(e)n(xample)28 b(in)f([9].)40 b(Here)26 b(we)g(will)h(only)g(pro)n(v)n (e)g(the)g(weak)o(er)g(statement)h(that)f(the)g(v)m(ariance)83 4605 y Fk(hk)p Fm(u)228 4620 y Fi(s)259 4605 y Fk(k)309 4568 y Fi(2)348 4605 y Fk(i)j Fq(is)g Fk(O)s Fn(\()p Fm(")681 4568 y Fi(2)720 4605 y Fn(\))p Fq(,)i(and)e(tak)o(e)h(that)g(as)f(an)g(indication) i(that)f(the)f(probabilistic)i(statement)83 4725 y(is)27 b(correct,)i(since)e (the)g(beha)n(vior)h(of)f(higher)g(moments)h(can)f(be)g(treated)h(in)f(an)g (analogous)83 4845 y(manner)-6 b(.)229 4992 y(T)e(o)26 b(pro)n(v)n(e)h(this)e (statement)j(we)d(look)h(at)g(the)g Fm(s)p Fq(-component)g(of)g(eq.)g (\(32\),)1106 5345 y Fm(u)1162 5304 y Fe(0)1162 5369 y Fi(s)1221 5345 y Fn(=)1383 5277 y Fm(A)1456 5292 y Fi(sf)1515 5277 y Fm(u)1571 5292 y Fi(f)1624 5277 y Fn(+)c(\(1)g(+)g Fm(A)2002 5292 y Fi(ss)2062 5277 y Fn(\))p Fm(u)2156 5292 y Fi(s)p 1334 5321 902 4 v 1334 5413 a Fk(k)p Fn(\(1)g(+)g Fm(A)1664 5428 y Fi(f)5 b(f)1722 5413 y Fn(\))p Fm(u)1816 5428 y Fi(f)1868 5413 y Fn(+)22 b Fm(A)2039 5428 y Fi(f)5 b(s)2098 5413 y Fm(u)2154 5428 y Fi(s)2186 5413 y Fk(k)2271 5345 y Fm(:)853 b Fq(\(33\))1651 5637 y(11)p eop %%Page: 12 12 bop 432 573 a Fq(The)26 b(quantity)g Fk(hk)p Fm(u)1120 537 y Fe(0)1120 597 y Fi(s)1152 573 y Fk(k)1202 537 y Fi(2)1241 573 y Fk(i)f Fq(is)h(a)f(sum)h(of)f(three)h(terms)g Fm(E)2328 588 y Fi(1)2390 573 y Fn(+)c Fm(E)2560 588 y Fi(2)2622 573 y Fn(+)g Fm(E)2792 588 y Fi(3)2832 573 y Fq(:)444 858 y Fm(E)516 873 y Fi(1)583 858 y Fn(=)687 712 y Fh(*)764 791 y Fk(k)p Fm(A)887 806 y Fi(s)o(f)945 791 y Fm(u)1001 806 y Fi(f)1032 791 y Fk(k)1082 755 y Fi(2)p 764 835 358 4 v 902 927 a Fm(`)943 898 y Fi(2)1131 712 y Fh(+)1215 858 y Fm(;)42 b(E)1356 873 y Fi(2)1423 858 y Fn(=)28 b(2)1593 712 y Fh(*)1669 791 y Fm(A)1742 806 y Fi(sf)1801 791 y Fm(u)1857 806 y Fi(f)1909 791 y Fk(\001)22 b Fn(\(1)g(+)g Fm(A)2239 806 y Fi(ss)2299 791 y Fn(\))p Fm(u)2393 806 y Fi(s)p 1669 835 756 4 v 2007 927 a Fm(`)2048 898 y Fi(2)2435 712 y Fh(+)2518 858 y Fm(;)42 b(E)2659 873 y Fi(3)2727 858 y Fn(=)2830 712 y Fh(*)2907 791 y Fk(k)p Fn(\(1)22 b(+)g Fm(A)3237 806 y Fi(ss)3297 791 y Fn(\))p Fm(u)3391 806 y Fi(s)3422 791 y Fk(k)3472 755 y Fi(2)p 2907 835 605 4 v 3169 927 a Fm(`)3210 898 y Fi(2)3521 712 y Fh(+)3630 858 y Fm(;)3500 1035 y Fq(\(34\))432 1155 y(where)j Fm(`)j Fn(=)f Fk(k)p Fn(\(1)22 b(+)g Fm(A)1208 1170 y Fi(f)5 b(f)1266 1155 y Fn(\))p Fm(u)1360 1170 y Fi(f)1413 1155 y Fn(+)22 b Fm(A)1584 1170 y Fi(f)5 b(s)1643 1155 y Fm(u)1699 1170 y Fi(s)1730 1155 y Fk(k)p Fq(.)578 1277 y(T)-8 b(o)24 b(bound)g(these)h(terms)g(we)f(\256rst)g(need)h(a)f(lo)m(wer)h(bound)f(on)g (the)h(denominator)h Fm(`)p Fq(.)37 b(Let)432 1397 y Fm(v)479 1412 y Fi(f)538 1397 y Fn(=)27 b Fm(R)716 1361 y Fl(T)715 1422 y Fi(f)5 b(f)773 1397 y Fm(u)829 1412 y Fi(f)882 1397 y Fn(+)22 b Fm(R)1055 1361 y Fl(T)1054 1422 y Fi(f)5 b(s)1113 1397 y Fm(u)1169 1412 y Fi(s)1200 1397 y Fq(,)26 b(and)g(de\256ne)g Fm(d)f Fq(by)1699 1620 y Fk(k)p Fm(D)1830 1635 y Fi(f)1861 1620 y Fm(v)1908 1635 y Fi(f)1939 1620 y Fk(k)i(\021)h Fm(d)p Fk(k)p Fm(v)2269 1635 y Fi(f)2300 1620 y Fk(k)d Fm(:)1098 b Fq(\(35\))432 1844 y(Note)20 b(that)h Fm(d)f Fq(can)h(v)m(ary)g(widely)f (between)h Fk(O)s Fn(\(1\))f Fq(v)m(alues)h(and)g Fk(O)s Fn(\()p Fm(\025)p Fn(\))p Fq(.)35 b(But,)22 b(using)e(the)h(lo)m(wer)432 1964 y(bound)k(on)h Fm(D)921 1979 y Fi(f)977 1964 y Fq(of)f(\(28\),)h(we)g (see)f(that)544 2187 y Fk(k)p Fn(\(1)c(+)h Fm(D)881 2202 y Fi(f)912 2187 y Fn(\))p Fm(v)997 2202 y Fi(f)1028 2187 y Fk(k)1078 2146 y Fi(2)1145 2187 y Fn(=)28 b Fk(k)p Fm(v)1346 2202 y Fi(f)1377 2187 y Fk(k)1427 2146 y Fi(2)1488 2187 y Fn(+)22 b(2)p Fm(v)1682 2202 y Fi(f)1735 2187 y Fk(\001)g Fm(D)1866 2202 y Fi(f)1897 2187 y Fm(v)1944 2202 y Fi(f)1997 2187 y Fn(+)g Fk(k)p Fm(D)2226 2202 y Fi(f)2257 2187 y Fm(v)2304 2202 y Fi(f)2335 2187 y Fk(k)2385 2146 y Fi(2)2452 2187 y Fk(\025)28 b Fn(\(1)22 b(+)g(2)p Fm(\025)g Fn(+)g Fm(d)3041 2146 y Fi(2)3080 2187 y Fn(\))p Fk(k)p Fm(v)3215 2202 y Fi(f)3246 2187 y Fk(k)3296 2146 y Fi(2)3361 2187 y Fm(:)112 b Fq(\(36\))432 2411 y(Expanding)26 b Fm(`)f Fq(as)1382 2534 y Fm(`)j Fn(=)f Fk(k)p Fm(R)1678 2549 y Fi(f)5 b(f)1736 2534 y Fn(\(1)22 b(+)g Fm(D)2024 2549 y Fi(f)2055 2534 y Fn(\))p Fm(v)2140 2549 y Fi(f)2193 2534 y Fn(+)g Fm(R)2365 2549 y Fi(f)5 b(s)2424 2534 y Fm(D)2505 2549 y Fi(s)2537 2534 y Fm(v)2584 2549 y Fi(s)2617 2534 y Fk(k)25 b Fm(;)781 b Fq(\(37\))432 2712 y(and)24 b(using)h(the)g(estimates)h Fk(k)p Fm(R)1526 2727 y Fi(f)t(s)1584 2712 y Fm(D)1665 2727 y Fi(s)1697 2712 y Fk(k)i Fn(=)f Fk(O)s Fn(\()p Fm("\025)p Fn(\))d Fq(and)h Fk(k)p Fn(1)19 b Fk(\000)i Fm(R)2625 2727 y Fi(f)5 b(f)2683 2712 y Fk(k)27 b Fn(=)h Fk(O)s Fn(\()p Fm(")3030 2676 y Fi(2)3069 2712 y Fn(\))c Fq(\(cf.)h(eq.)g(\(28\)\),)432 2833 y(we)g(get)h(from)g (\(36\))f(the)h(desired)g(lo)m(wer)g(bound)g(on)f Fm(`)p Fq(:)1305 3056 y Fm(`)1346 3015 y Fi(2)1413 3056 y Fm(>)i Fn(\(1)22 b(+)g(2)p Fm(\025)g Fn(+)g Fm(d)2000 3015 y Fi(2)2039 3056 y Fn(\))p Fk(k)p Fm(v)2174 3071 y Fi(f)2205 3056 y Fk(k)2255 3015 y Fi(2)2294 3056 y Fn(\(1)g Fk(\000)h(O)s Fn(\()p Fm(")p Fn(\)\))i Fm(:)703 b Fq(\(38\))578 3280 y(W)-8 b(e)26 b(can)g(no)m(w)f(bound)h Fm(E)1455 3295 y Fi(1)1494 3280 y Fq(,)g Fm(E)1617 3295 y Fi(2)1683 3280 y Fq(and)f Fm(E)1927 3295 y Fi(3)1967 3280 y Fq(.)37 b(First,)26 b(we)g(ha)n(v)n(e)1005 3503 y Fk(k)p Fm(A)1128 3518 y Fi(sf)1187 3503 y Fm(u)1243 3518 y Fi(f)1273 3503 y Fk(k)i Fn(=)f Fk(k)p Fm(R)1578 3518 y Fi(sf)1637 3503 y Fm(D)1718 3518 y Fi(f)1749 3503 y Fm(v)1796 3518 y Fi(f)1827 3503 y Fk(k)22 b Fn(+)g Fk(O)s Fn(\()p Fm(\025")p Fn(\))27 b Fm(<)h("d)p Fk(k)p Fm(v)2583 3518 y Fi(f)2613 3503 y Fk(k)22 b Fn(+)g Fk(O)s Fn(\()p Fm(\025")p Fn(\))j Fm(:)404 b Fq(\(39\))432 3727 y(Thus,)26 b(ne)n(glecting)h(higher)f (order)g(corrections)g(in)g Fm(")p Fq(,)g(we)f(get)1651 4006 y Fm(E)1723 4021 y Fi(1)1790 4006 y Fm(<)2058 3939 y(")2104 3903 y Fi(2)2143 3939 y Fm(d)2194 3903 y Fi(2)p 1903 3983 485 4 v 1903 4075 a Fn(1)d(+)g(2)p Fm(\025)g Fn(+)g Fm(d)2349 4046 y Fi(2)2424 4006 y Fm(:)1049 b Fq(\(40\))578 4263 y(The)27 b(bound)g(on)g(the)h(term)f Fm(E)1611 4278 y Fi(2)1678 4263 y Fq(mak)o(es)g(essential)h(use)f(of)g(the)h(translation)g(in)l(v)m(ariance.) 432 4384 y(F)n(or)d(this,)i(we)e(note)h(that)1941 4478 y Fm(A)2014 4493 y Fi(f)5 b(s)2073 4478 y Fm(u)2129 4493 y Fi(s)p 1750 4522 602 4 v 1750 4613 a Fk(k)p Fn(\(1)22 b(+)g Fm(A)2080 4628 y Fi(f)5 b(f)2138 4613 y Fn(\))p Fm(u)2232 4628 y Fi(f)2262 4613 y Fk(k)2312 4584 y Fi(2)3500 4545 y Fq(\(41\))432 4786 y(transforms)34 b(as)g(a)g(v)n(ector)l(,)k(that)d(is,)i(its)d Fm(k)s Fq(th)g(component)h(is)f(multiplied)j(by)d Fn(exp)q(\()p Fm(i)3434 4747 y Fi(2)p Fl(\031)p 3434 4763 79 4 v 3442 4821 a(N)3522 4786 y Fm(k)s(x)p Fn(\))432 4907 y Fq(under)25 b(a)g(relabeling)i (of)e(the)g(coordinates)i Fm(n)g Fk(!)h Fm(n)21 b Fn(+)g Fm(x)p Fq(.)38 b(Therefore,)26 b(because)g(of)f(transla-)432 5027 y(tion)k(in)l(v)m(ariance,)i(the)e(e)n(xpectation)i(v)m(alue)f(of)e(\(41\))h (must)f(remain)i(in)l(v)m(ariant)g(under)f(such)432 5147 y(transformation,)h (which)e(means)h(it)f(must)h(v)m(anish.)45 b(Since)28 b(the)h(denominator)g (in)g Fm(E)3426 5162 y Fi(2)3493 5147 y Fq(is)f Fm(`)3630 5111 y Fi(2)432 5268 y Fq(\(which)23 b(also)g(depends)g(on)g Fm(u)1463 5283 y Fi(s)1494 5268 y Fq(\))g(and)g(not)g Fk(k)p Fn(\(1)16 b(+)g Fm(A)2192 5283 y Fi(f)6 b(f)2251 5268 y Fn(\))p Fm(u)2345 5283 y Fi(f)2375 5268 y Fk(k)2425 5232 y Fi(2)2464 5268 y Fq(,)24 b(we)f(need)g(some)h(gymnastics)f(to)432 5388 y(e)n(xhibit)k(the)f(v)m (anishing)g(term.)39 b(In)25 b(order)h(to)f(see)h(this)g(we)f(write)2000 5637 y(12)p eop %%Page: 13 13 bop 355 779 a Fm(E)427 794 y Fi(2)494 779 y Fn(=)28 b(2)647 658 y Fh(\034)831 712 y Fm(A)904 727 y Fi(sf)963 712 y Fm(u)1019 727 y Fi(f)1071 712 y Fk(\001)22 b Fm(u)1177 727 y Fi(s)p 719 756 602 4 v 719 848 a Fk(k)p Fn(\(1)g(+)g Fm(A)1049 863 y Fi(f)5 b(f)1107 848 y Fn(\))p Fm(u)1201 863 y Fi(f)1231 848 y Fk(k)1281 819 y Fi(2)1352 779 y Fn(+)1506 712 y Fm(A)1579 727 y Fi(sf)1638 712 y Fm(u)1694 727 y Fi(f)1746 712 y Fk(\001)22 b Fm(A)1869 727 y Fi(ss)1929 712 y Fm(u)1985 727 y Fi(s)p 1460 756 V 1460 848 a Fk(k)p Fn(\(1)g(+)g Fm(A)1790 863 y Fi(f)5 b(f)1848 848 y Fn(\))p Fm(u)1942 863 y Fi(f)1972 848 y Fk(k)2022 819 y Fi(2)559 1019 y Fk(\000)646 952 y Fm(A)719 967 y Fi(sf)778 952 y Fm(u)834 967 y Fi(f)887 952 y Fk(\001)22 b Fn(\(1)g(+)g Fm(A)1217 967 y Fi(ss)1277 952 y Fn(\))p Fm(u)1371 967 y Fi(s)1419 952 y Fn([2\(1)g(+)g Fm(A)1775 967 y Fi(f)t(f)1832 952 y Fn(\))p Fm(u)1926 967 y Fi(f)1979 952 y Fk(\001)g Fm(A)2102 967 y Fi(f)5 b(s)2161 952 y Fm(u)2217 967 y Fi(s)2270 952 y Fn(+)22 b Fk(k)p Fm(A)2491 967 y Fi(f)5 b(s)2550 952 y Fm(u)2606 967 y Fi(s)2638 952 y Fk(k)2688 915 y Fi(2)2727 952 y Fn(])p 646 996 2108 4 v 1359 1087 a Fk(k)p Fn(\(1)22 b(+)g Fm(A)1689 1102 y Fi(f)5 b(f)1747 1087 y Fn(\))p Fm(u)1841 1102 y Fi(f)1871 1087 y Fk(k)1921 1058 y Fi(2)1961 1087 y Fm(`)2002 1058 y Fi(2)2764 898 y Fh(\035)2852 1019 y Fm(:)3151 903 y Fq(\(42\))83 1252 y(The)34 b(\256rst)g(term)h(in)f (\(42\))g(v)m(anishes)g(because)h(of)e(translation)j(in)l(v)m(ariance,)i(as)c (e)n(xplained)83 1372 y(before.)k(The)26 b(second)f(term)i(is)e(bounded)h(by) 1454 1555 y Fn(2)p Fm(\013)q(\025")17 b Fk(h)o(k)p Fm(u)1830 1570 y Fi(s)1861 1555 y Fk(ki)1201 b Fq(\(43\))83 1738 y(and)26 b(the)g(dominant)h(part)f(of)f(the)h(third)g(is)671 1978 y Fn(4)730 1910 y(\()p Fm(A)841 1925 y Fi(sf)900 1910 y Fm(u)956 1925 y Fi(f)1009 1910 y Fk(\001)c Fm(u)1115 1925 y Fi(s)1146 1910 y Fn(\)\(\(1)g(+)g Fm(A)1502 1925 y Fi(f)5 b(f)1560 1910 y Fn(\))p Fm(u)1654 1925 y Fi(f)1706 1910 y Fk(\001)22 b Fm(A)1829 1925 y Fi(f)5 b(s)1888 1910 y Fm(u)1944 1925 y Fi(s)1976 1910 y Fn(\))p 730 1954 1284 4 v 1031 2046 a Fk(k)p Fn(\(1)22 b(+)g Fm(A)1361 2061 y Fi(f)5 b(f)1419 2046 y Fn(\))p Fm(u)1513 2061 y Fi(f)1543 2046 y Fk(k)1593 2017 y Fi(2)1632 2046 y Fm(`)1673 2017 y Fi(2)2051 1978 y Fm(<)2165 1910 y Fn(4)p Fm(d")2311 1874 y Fi(2)2366 1910 y Fk(hk)p Fm(u)2511 1925 y Fi(s)2542 1910 y Fk(k)2592 1874 y Fi(2)2631 1910 y Fk(i)p 2165 1954 506 4 v 2175 2046 a Fn(1)22 b(+)g(2)p Fm(\025)g Fn(+)g Fm(d)2621 2017 y Fi(2)2705 1978 y Fm(:)419 b Fq(\(44\))83 2210 y(The)26 b(last)g(term)h(is)e(bounded)h(by)1121 2440 y Fm(E)1193 2455 y Fi(3)1261 2440 y Fm(<)1448 2373 y Fn(1)c(+)g(2)p Fm(\013)q(\025)p 1374 2417 485 4 v 1374 2509 a Fn(1)g(+)g(2)p Fm(\025)g Fn(+)g Fm(d)1820 2480 y Fi(2)1885 2344 y Fh(D)1936 2440 y Fk(k)p Fm(u)2042 2455 y Fi(s)2074 2440 y Fk(k)2124 2399 y Fi(2)2163 2344 y Fh(E)2256 2440 y Fm(:)83 2656 y Fq(Collecting)27 b(the)f(bounds)g(yields)602 2799 y Fh(D)653 2895 y Fk(k)p Fm(u)759 2854 y Fe(0)759 2920 y Fi(s)790 2895 y Fk(k)840 2854 y Fi(2)879 2799 y Fh(E)958 2895 y Fm(<)1071 2828 y(d)1122 2792 y Fi(2)1161 2828 y Fm(")1207 2792 y Fi(2)1269 2828 y Fn(+)c(\(1)f(+)h(2)p Fm(\013)q(\025)g Fn(+)g(4)p Fm(d")2008 2792 y Fi(2)2047 2828 y Fn(\))17 b Fk(h)o(k)p Fm(u)2246 2843 y Fi(s)2277 2828 y Fk(k)2327 2792 y Fi(2)2366 2828 y Fk(i)p 1071 2872 1334 4 v 1496 2964 a Fn(1)22 b(+)g(2)p Fm(\025)g Fn(+)g Fm(d)1942 2935 y Fi(2)2437 2895 y Fn(+)g(2)p Fm(\013)q(\025")j(:)349 b Fq(\(45\))83 3117 y(It)17 b(appears)g(from)g (\(45\))g(that)g(although)g(lar)n(ge)g(perturbations)g(are)g(possible)g(when) g Fm(d)g Fq(is)g Fk(O)s Fn(\(1\))p Fq(,)83 3237 y(the)26 b(contraction)h (rate)g(increases)f(precisely)h(enough)e(to)h(compensate)h(this)f(contrib)n (ution.)83 3358 y(Thus)f(if)h Fk(hk)p Fm(u)545 3373 y Fi(s)576 3358 y Fk(k)626 3322 y Fi(2)666 3358 y Fk(i)f Fq(is)g Fk(O)s Fn(\()p Fm(")989 3322 y Fi(2)1028 3358 y Fn(\))h Fq(to)g(start)g(with,)g(it)g (will)g(stay)g(so)f(inde\256nitely)-7 b(.)229 3478 y(In)28 b(summary)-7 b(,)29 b(assuming)e(that)i(the)f(v)m(ariance)h(is)e(indeed)h(a)g (measure)g(of)f(typical)i(\257uc-)83 3599 y(tuations,)j(we)d(ha)n(v)n(e)h (sho)m(wn)f(that,)j(for)d Fm(N)43 b Fk(\035)32 b Fn(1)p Fq(,)f(the)f (subspace)f Fm(L)2453 3614 y Fi(f)2514 3599 y Fq(spanned)g(by)g(the)h (\256rst)83 3719 y Fm(N)20 b Fk(\000)10 b Fn(3)20 b Fq(L)-6 b(yapuno)n(v)21 b(v)n(ectors)g(of)e(the)i(product)f(\(31\))g(is,)h(with)f(v)n (ery)h(high)f(probability)-7 b(,)23 b(almost)83 3839 y(orthogonal)33 b(to)g Fm(V)726 3854 y Fi(s)757 3839 y Fq(.)58 b(This)33 b(implies)g(that)g (the)g(last)g(three)g(L)-6 b(yapuno)n(v)34 b(v)n(ectors)f(\(including)83 3960 y(the)28 b(translation\))i(remain)e(approximately)i(in)e Fm(V)1819 3975 y Fi(s)1851 3960 y Fq(.)44 b(This)28 b(means)h(by)e (de\256nition)i(that)g(the)n(y)83 4080 y(are)d(h)o(ydrodynamic,)h(in)e(the)h (sense)e(that)i(the)n(y)g(are)g(well)g(approximated)g(by)f(eigen)l(v)n (ectors)83 4200 y(of)g(the)h(discrete)g(Laplacian.)39 b(Since)26 b(the)f(action)i(of)1960 4175 y Fn(~)1934 4200 y Fm(A)f Fq(on)f Fm(V)2217 4215 y Fi(s)2274 4200 y Fq(has)g(tw)o(o)g(eigen)l(v)m(alues)h (close)83 4321 y(to)33 b Fn(\(2)p Fm(\031)t(=)-5 b(N)10 b Fn(\))511 4285 y Fi(2)582 4321 y Fq(as)32 b(sho)m(wn)g(in)g(the)h(pre)m(vious)g (section,)i(it)e(follo)m(ws)f(as)g(a)h(corollary)g(that)g(the)83 4441 y(tw)o(o)27 b(smallest)i(non-tri)m(vial)g(L)-6 b(yapuno)n(v)28 b(e)n(xponents)h(ha)n(v)n(e)f(approximately)h(this)f(v)m(alue,)h(so)83 4562 y(that)d(the)n(y)h(are)f(also)g(h)o(ydrodynamic.)39 b(This)25 b(completes)i(the)f(demonstration.)83 4899 y Fs(5)147 b(Pr)m(oducts)37 b(of)g(symplectic)g(matrices)83 5125 y Fq(W)-8 b(e)26 b(no)m(w)f(turn)h(to)g (products)g(of)f(matrices)i(of)e(the)h(form)1209 5353 y Fm(S)34 b Fn(=)1406 5232 y Fh(\022)1523 5293 y Fn(1)281 b Fm(\034)1484 5413 y(\034)11 b(A)103 b Fn(1)22 b(+)g Fm(\034)1935 5377 y Fi(2)1975 5413 y Fm(A)2064 5232 y Fh(\023)2167 5353 y Fm(:)957 b Fq(\(46\))1651 5637 y(13)p eop %%Page: 14 14 bop 432 573 a Fq(W)-8 b(e)43 b(disre)n(g)o(ard)i(the)f(tw)o(o)f(translation)h (modes)g(in)f Fm(S)49 b Fq(for)43 b(con)l(v)n(enience)j(and)d(vie)m(w)h(the) 432 693 y(matrices)23 b Fm(S)k Fq(as)21 b Fn(\(2)p Fm(N)j Fk(\000)13 b Fn(2\))g Fk(\002)g Fn(\(2)p Fm(N)24 b Fk(\000)13 b Fn(2\))22 b Fq(matrices.)38 b(Let)22 b(us)f(recall)i(that)f(since)g(the)g(matrices)432 814 y Fm(S)i Fq(are)18 b(symplectic)i(and)e(h)o(yperbolic,)k(the)c(L)-6 b(yapuno)n(v)20 b(e)n(xponents)f(are)f(non-zero)g(and)h(come)432 934 y(in)27 b(pairs)h(of)f(opposite)h(signs.)42 b(W)-8 b(e)28 b(concentrate)h(on)e(modes)h(number)g Fm(N)33 b Fk(\000)24 b Fn(2)j Fq(and)h Fm(N)33 b Fk(\000)24 b Fn(1)432 1054 y Fq(which)h(are)h (the)g(smallest)h(positi)m(v)n(e)g(ones.)578 1175 y(W)-8 b(e)30 b(reduce)h(the)f(problem)g(to)g(an)g(equi)m(v)m(alent)j(one)d(to)g(which)g (the)g(results)g(of)g(section)432 1295 y(4)i(can)g(be)h(applied)g(directly)-7 b(.)59 b(W)-8 b(e)33 b(denote)f(by)h Fm(L)2206 1310 y Fi(+)2297 1295 y Fq(the)g(subspace)f(spanned)g(by)g(the)h(\256rst)432 1416 y Fm(N)h Fk(\000)24 b Fn(1)29 b Fq(L)-6 b(yapuno)n(v)30 b(v)n(ectors.)48 b(It)28 b(is)h(spanned)g(by)g(a)f(set)h(of)g Fm(N)34 b Fk(\000)25 b Fn(1)j Fq(independent)i(v)n(ectors,)432 1536 y(which)h(we)g(display)h(in)g(the)g(form)f(of)h(a)f Fn(\()p Fm(N)36 b Fk(\000)26 b Fn(1\))f Fk(\002)g Fn(\(2)p Fm(N)36 b Fk(\000)26 b Fn(2\))31 b Fq(matrix)h Fk(V)8 b Fq(.)55 b(The)32 b Fm(N)k Fk(\000)26 b Fn(1)432 1656 y Fq(v)n(ectors)g(can)g(al)o(w)o(ays)g (be)f(chosen)h(in)g(such)f(a)h(w)o(ay)f(that)i Fk(V)34 b Fq(is)25 b(of)g(the)i(normal)f(form)1799 1898 y Fk(V)36 b Fn(=)1999 1777 y Fh(\022)2088 1838 y Fj(1)2077 1958 y Fm(V)2172 1777 y Fh(\023)2275 1898 y Fm(;)1198 b Fq(\(47\))432 2142 y(where)24 b(both)g(blocks)g(are)h Fn(\()p Fm(N)k Fk(\000)19 b Fn(1\))f Fk(\002)i Fn(\()p Fm(N)29 b Fk(\000)19 b Fn(1\))p Fq(.)37 b(Acting)24 b(on)g Fk(V)32 b Fq(with)24 b Fm(S)30 b Fq(gi)m(v)n(es)25 b(a)f(spanning)432 2263 y(set)h(of)h(the)g(image)h(subspace)e Fm(L)1565 2226 y Fe(0)1565 2287 y Fi(+)1625 2263 y Fq(,)1437 2526 y Fm(S)6 b Fk(V)36 b Fn(=)1703 2405 y Fh(\022)1999 2466 y Fn(1)22 b(+)g Fm(\034)11 b(V)1781 2586 y(\034)g(A)23 b Fn(+)f(\(1)g(+)g Fm(\034)2288 2550 y Fi(2)2353 2561 y Fn(~)2328 2586 y Fm(A)p Fn(\))p Fm(V)2534 2405 y Fh(\023)2637 2526 y Fm(;)836 b Fq(\(48\))432 2798 y(and)25 b(changing)i(basis)e(to)h(normal)g(form)g(gi)m(v)n(es)h Fk(V)2182 2762 y Fe(0)2233 2798 y Fn(=)2336 2677 y Fh(\022)2437 2738 y Fj(1)2414 2858 y Fm(V)2493 2822 y Fe(0)2533 2677 y Fh(\023)2619 2798 y Fq(where)1624 3103 y Fm(V)1703 3062 y Fe(0)1754 3103 y Fn(=)g Fm(\034)11 b(A)23 b Fn(+)2225 3036 y Fm(V)p 2114 3080 301 4 v 2114 3172 a Fn(1)f(+)g Fm(\034)11 b(V)2450 3103 y(:)1023 b Fq(\(49\))432 3332 y(A)23 b(con)l(v)n(enient)j(property)e(of)g(this)g (matrix)h(dynamical)h(system)e(is)g(that)h(if)f Fm(V)45 b Fq(is)24 b(symmetric)432 3453 y(to)h(be)n(gin)i(with,)f(it)g(stays)g(so)f(as)g(a)h (consequence)g(of)f(the)i(symplectic)g(property)f(of)f Fm(S)31 b Fq([10].)578 3573 y(By)h(de\256nition,)j(an)n(y)e(v)n(ector)g Fm(v)40 b Fk(2)d Fm(L)1898 3588 y Fi(+)1989 3573 y Fq(has)32 b(a)g(block)h(representation)g Fm(v)40 b Fn(=)d(\()16 b Fm(u;)h(V)k(u)16 b Fn(\))p Fq(.)432 3693 y(From)25 b(eq.)h(\(48\))g(it)g(follo)m(ws)g(that)g (its)g(image)h(is)e Fn(\()17 b Fm(u)2211 3657 y Fe(0)2234 3693 y Fm(;)g(V)2356 3657 y Fe(0)2379 3693 y Fm(u)2435 3657 y Fe(0)2474 3693 y Fn(\))p Fq(,)26 b(where)1703 3890 y Fm(u)1759 3849 y Fe(0)1810 3890 y Fn(=)h(\(1)22 b(+)g Fm(\034)11 b(V)22 b Fn(\))p Fm(u)j(:)1102 b Fq(\(50\))432 4087 y(Hence,)25 b(the)g(\256rst)f Fm(N)30 b Fk(\000)19 b Fn(1)24 b Fq(L)-6 b(yapuno)n(v)26 b(modes)e(of)g(the)h (products)g(of)f(the)g Fm(S)30 b Fq(are)25 b(the)g(same)g(as)432 4207 y(those)g(of)g(the)g(product)1268 4140 y Fh(Q)1346 4228 y Fl(n)1393 4207 y Fn(\(1)c(+)g Fm(\034)11 b(V)1708 4222 y Fl(n)1755 4207 y Fn(\))25 b Fq(where)g(the)h(matrices)g Fm(V)2669 4222 y Fl(n)2741 4207 y Fq(are)f(e)m(v)n(olving)i(according)432 4327 y(to)f(eq.)g(\(49\):)37 b Fm(V)976 4342 y Fl(n)p Fi(+1)1140 4327 y Fn(=)28 b Fm(\034)11 b(A)1370 4342 y Fl(n)1440 4327 y Fn(+)22 b Fm(V)1595 4342 y Fl(n)1642 4327 y Fm(=)p Fn(\(1)f(+)h Fm(\034)11 b(V)2007 4342 y Fl(n)2055 4327 y Fn(\))p Fq(.)578 4448 y(In)38 b(vie)m(w)g(of)g(this)g(equi)m(v)m(alence,)44 b(it)38 b(suf)m(\256ces)g(to)g(sho)m(w)f(that)i(the)f(matrix)h Fm(V)59 b Fq(has)38 b(the)432 4568 y(properties)26 b(formulated)g(in)g(eqs.)f (\(27\26129\))g(and)g(to)h(apply)f(the)h(results)f(of)g(section)h(4.)38 b(First)432 4688 y(note)26 b(if)h Fm(A)795 4703 y Fl(n)870 4688 y Fn(=)h Fk(\000)p Fm(@)1107 4652 y Fi(2)1174 4688 y Fq(\(minus)e(the)h (discrete)g(Laplacian\))h(for)e(all)h Fm(n)p Fq(,)f(then)h(the)g Fm(V)3231 4703 y Fl(n)3303 4688 y Fq(con)l(v)n(er)n(ge)432 4809 y(to)f Fm(f)11 b Fn(\()p Fk(\000)p Fm(@)767 4773 y Fi(2)807 4809 y Fn(\))p Fq(,)26 b(where)1491 5012 y Fm(f)11 b Fn(\()p Fm(x)p Fn(\))28 b(=)1822 4944 y Fm(\034)11 b(x)p 1822 4988 109 4 v 1852 5080 a Fn(2)1963 5012 y(+)2061 4849 y Fh(s)p 2144 4849 467 4 v 163 x Fm(x)23 b Fn(+)2320 4890 y Fh(\022)2391 4944 y Fm(\034)11 b(x)p 2391 4988 109 4 v 2421 5080 a Fn(2)2510 4890 y Fh(\023)2571 4913 y Fi(2)432 5243 y Fq(is)39 b(the)h(lar)n(ger)g(root) f(of)h(the)f(quadratic)i(equation)f Fm(f)11 b Fn(\()p Fm(x)p Fn(\))47 b(=)f Fm(\034)11 b(x)30 b Fn(+)2955 5196 y Fl(f)7 b Fi(\()p Fl(x)p Fi(\))p 2891 5220 265 4 v 2891 5277 a(1+)p Fl(\034)h(f)f Fi(\()p Fl(x)p Fi(\))3166 5243 y Fq(.)78 b(F)n(or)40 b(small)432 5388 y Fm(x)30 b(>)g Fn(0)c Fq(this)i(is)f(close)g(to)h Fm(x)1366 5352 y Fi(1)p Fl(=)p Fi(2)1476 5388 y Fq(,)g(and)f(therefore)h(we)f (assume)g(that)h Fm(V)49 b Fq(has)27 b(a)g(representation)2000 5637 y(14)p eop %%Page: 15 15 bop 83 573 a Fq(of)31 b(the)i(type)f(gi)m(v)n(en)h(by)e(eqs.)h(\(27\26129\),) h(with)f Fm(")j Fn(=)h Fk(O)s Fn(\()p Fm(N)2138 537 y Fe(\000)p Fi(1)p Fl(=)p Fi(2)2304 573 y Fn(\))31 b Fq(as)g(before)h(and)g Fm(\025)f Fq(is)h(no)m(w)83 693 y Fm(f)11 b Fn(\(4)p Fm(\031)t(=)-5 b(N)420 657 y Fi(2)459 693 y Fn(\))27 b(=)h Fk(O)s Fn(\()p Fm(N)836 657 y Fe(\000)p Fi(1)931 693 y Fn(\))p Fq(.)37 b(The)26 b(aim)g(is)g(to)g(sho)m(w)f(that)h(this)g(property)g(is)g(carried)g(on)g(to)f Fm(V)3265 657 y Fe(0)3288 693 y Fq(.)229 816 y(In)h(order)f(to)h(a)n(v)n(oid) g(the)g(necessity)g(of)f(presenting)i(e)m(v)n(en)f(more)g(technical)i (details,)f(we)83 937 y(present)38 b(the)h(ar)n(gument)g(for)f(the)h(case)f (where)g(the)h(slo)m(w)f(subspace)g(contains)h(a)f(single)83 1057 y(mode,)29 b(rather)f(than)h(a)e(pair)h(of)g(nearly)g(de)n(generate)i (modes.)44 b(Since)28 b Fm(V)2620 1021 y Fe(0)2671 1057 y Fq(is)g(symmetric)h (its)83 1177 y(smallest)e(eigen)l(v)m(alue)g(is)f(gi)m(v)n(en)h(by)1018 1451 y Fm(\025)1075 1466 y Fl(V)1131 1447 y Fa(0)1185 1451 y Fn(=)48 b(min)1289 1514 y Fe(k)p Fl(u)p Fe(k)p Fi(=1)1507 1451 y Fm(u)22 b Fk(\001)g Fn(\()p Fm(\034)11 b(A)22 b Fn(+)2041 1383 y Fm(V)p 1929 1427 301 4 v 1929 1519 a Fn(1)g(+)g Fm(\034)11 b(V)2240 1451 y Fn(\))p Fm(u)25 b(:)765 b Fq(\(51\))83 1723 y(It)26 b(follo)m(ws)h(from)f(our)g(assumptions)g(that)h(there)g(e)n(xist)g (\(normalized\))h(eigen)l(v)n(ectors)f(of)f Fm(V)83 1844 y Fq(and)g Fm(A)p Fq(:)569 1972 y Fm(Ae)687 1987 y Fl(A)772 1972 y Fn(=)h Fm(\025)932 1987 y Fi(1)972 1972 y Fn(\(1)21 b(+)h Fm(c)1220 1987 y Fl(A)1277 1972 y Fm(")p Fn(\))p Fm(e)1406 1987 y Fl(A)1489 1972 y Fm(;)220 b(V)22 b(e)1860 1987 y Fl(V)1949 1972 y Fn(=)27 b Fm(f)11 b Fn(\()p Fm(\025)2206 1987 y Fi(1)2245 1972 y Fn(\)\(1)22 b(+)g Fm(c)2532 1987 y Fl(V)2593 1972 y Fm(")p Fn(\))p Fm(e)2722 1987 y Fl(V)2808 1972 y Fm(;)316 b Fq(\(52\))83 2155 y(where)21 b Fm(\025)410 2170 y Fi(1)469 2155 y Fq(is)g(the)g(smallest)g(positi)m(v)n(e)i(eigen)l(v)m(alue)f(of)e Fk(\000)p Fm(@)2090 2119 y Fi(2)2152 2155 y Fq(and)g Fm(e)2364 2170 y Fl(A)2442 2155 y Fq(and)h Fm(e)2655 2170 y Fl(V)2736 2155 y Fq(are)g(close)g(to)g Fm(e)3256 2170 y Fi(1)3295 2155 y Fq(,)83 2276 y(the)i(corresponding)f(eigen)l(v)n(ector)-6 b(.)38 b(The)23 b(v)m(ariational)h(principle)g(then)e(gi)m(v)n(es)h (immediately)83 2396 y(a)j(lo)m(wer)g(bound)f(on)h Fm(\025)874 2411 y Fl(V)930 2392 y Fa(0)957 2396 y Fq(,)1217 2524 y Fm(\025)1274 2539 y Fl(V)1330 2520 y Fa(0)1384 2524 y Fm(>)i(f)11 b Fn(\()p Fm(\025)1642 2539 y Fi(1)1681 2524 y Fn(\)\(1)22 b(+)g Fm("c)2014 2539 y Fl(V)2070 2520 y Fa(0)2097 2524 y Fn(\))j Fm(;)964 b Fq(\(53\))83 2707 y(for)24 b(some)g Fm(c)507 2722 y Fl(V)563 2703 y Fa(0)614 2707 y Fq(between)g Fm(c)1019 2722 y Fl(A)1100 2707 y Fq(and)g Fm(c)1313 2722 y Fl(V)1373 2707 y Fq(.)37 b(T)-8 b(o)24 b(get)g(an)g(upper)g(bound)g(on)f Fm(\025)2549 2722 y Fl(V)2606 2703 y Fa(0)2656 2707 y Fq(recall)i(that)g(it)f(w)o(as)83 2828 y(sho)m(wn)30 b(in)g(sec.)h(3)f(eq.)h(\(25\))f(that)i(the)e Fm(k)k Fq(component)d(of)f Fm(e)2171 2843 y Fl(A)2258 2828 y Fq(is)h(of)f(order)g Fm("k)2817 2792 y Fe(\000)p Fi(1)2911 2828 y Fq(,)i(and)e(note)83 2948 y(that)c(the)h(bound)e(\(26\))g(implies)1017 3176 y Fm(f)11 b Fn(\()p Fk(\000)p Fm(a)1242 3191 y Fi(min)1364 3176 y Fm(@)1420 3135 y Fi(2)1460 3176 y Fn(\))28 b Fm(<)g(V)49 b(<)27 b(f)11 b Fn(\()p Fk(\000)p Fm(a)2064 3191 y Fi(max)2200 3176 y Fm(@)2256 3135 y Fi(2)2296 3176 y Fn(\))26 b Fm(:)764 b Fq(\(54\))83 3404 y(W)-8 b(e)26 b(no)m(w)f(use)h Fm(u)h Fn(=)h Fm(e)834 3419 y Fl(A)916 3404 y Fq(in)e(eq.)g(\(51\))g(and)f(get)504 3631 y Fm(\025)561 3646 y Fl(V)618 3627 y Fa(0)672 3631 y Fm(<)i(\025)832 3646 y Fi(1)872 3631 y Fn(\(1)22 b(+)g Fm(c)1121 3646 y Fl(A)1178 3631 y Fm(")p Fn(\))f(+)h Fm(e)1426 3646 y Fl(A)1506 3631 y Fk(\001)f Fm(f)11 b Fn(\()p Fk(\000)p Fm(a)1780 3646 y Fi(max)1916 3631 y Fm(@)1972 3590 y Fi(2)2013 3631 y Fn(\))p Fm(e)2096 3646 y Fl(A)2180 3631 y Fm(<)28 b(f)11 b Fn(\()p Fm(\025)2438 3646 y Fi(1)2477 3631 y Fn(\)\(1)22 b(+)i(\026)-51 b Fm(c")p Fn(\))25 b Fm(;)251 b Fq(\(55\))83 3859 y(for)36 b(some)g(constant)j Fn(\026)-51 b Fm(c)p Fq(.)69 b(Eqs.)36 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/VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 2 string 0 1 255{IE S dup 360 add 36 4 index cvrs cvn put}for pop 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V{}B /RV statusdict begin /product where{pop false[ (Display)(NeXT)(LaserWriter 16/600)]{dup length product length le{dup length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end /TeXBase1Encoding [ /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef /Zcaron /zcaron /caron /dotlessi /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there down to here. /grave /quotesingle /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends /.notdef /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def TeXDict begin /rf{findfont dup length 1 add dict begin{1 index /FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type /integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics /Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont /setfont cvx]cvx def}def /ObliqueSlant {dup sin S cos div neg}B /SlantFont{4 index mul add}def /ExtendFont{3 -1 roll mul exch}def /ReEncodeFont{/Encoding exch def}def end TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale true def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end TeXDict begin @defspecial /magscale false def @fedspecial end TeXDict begin 39158280 55380996 1200 600 600 (showfig.dvi) @start /Fa 139[22 3[40 40 40 4[22 5[35 1[35 38[22 58[{ TeXBase1Encoding ReEncodeFont}8 79.7011 /Times-Roman rf /Fb 5 106 df 46 D<141E147EEB01FE130713FFB5FCA3130F1200B3B3A3007FB612C0A4223479B331> 49 D70 D103 D105 D E /Fc 2 108 df<90260FFFC0903803FFFE496D5BA29026003FF09038003FC0F01F00027F 151E810273151C8102F1153C6F143814E08101016D1478701370ECC03F820103011F14F0 705BEC800F8201070107130104F85BEC000316FC491503DB01FE5B130EED00FF011E1507 DC7F87C7FC131CEE3FC7013C15CFEE1FEE1338EE0FFE1378705A1370160313F0705A1201 D807F81400B512C017705C3F307CAF3F>78 D107 D E /Fd 5 119 df<0203B612F0021F15F8147F49B7FC010716F04916E090271FF80FF8C7FC 90393FC003FC90387F800101FEC77E485A4848147E0007157F5B485AA2485AA2123F5B5E 007F5D90C8FCA21501485D5AA24B5AA24B5AA2007E4A5A5E4B5A003E4AC8FC003F147E6C 5C6D485A000F495A3907C007C02601F03FC9FC38007FFCEB1FE0352C7DAA39>27 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6C7EA24B6C7EA29238F003FF15E002016D7F15C002036D7F5D02076E7E92C7FC4A6E7E14 0E021E6E7E141C023C6E7E143802786E7E147002F06E7E5C01016F7F5C01036F7F5C0107 707E91C9FC49707E130E011E707E131C013C707E13380178707E137001F0707E5B000171 13805B00037113C05B0007F07FE090CBFC48F03FF0120E001EF01FF8001FBAFC4819FCA2 4819FEA2BCFCA348477BC653>1 D<007FBA1280BB12C0A26C1980CEFCB0007FBA1280BB 12C0A26C198042187BA44D>61 D E /Ff 1 1 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D E /Fg 137[50 1[28 39 33 1[50 50 50 78 3[28 1[50 33 44 50 44 1[44 27[61 72 10[50 50 50 50 50 50 50 50 50 50 3[25 44[{TeXBase1Encoding ReEncodeFont}28 99.6264 /Times-Roman rf end TeXDict begin 1 0 bop 94 4258 a @beginspecial @setspecial /X-CENTER-PLOT 0 def /Y-CENTER-PLOT 0 def /page-height 500 def /page-width 500 def 538.583 page-width gt not { % Rotation required 90 rotate 0 page-width neg translate % Handle centering Y-CENTER-PLOT 1 eq { % Center in y page-height 538.583 sub 2 div } { % Don't center in y 0 } ifelse X-CENTER-PLOT 1 eq { % Center in x page-width 529.455 sub 2 div } { % Don't center in x 0 } ifelse translate } { % No rotation - just handle centering X-CENTER-PLOT 1 eq { % Center in x page-width 538.583 sub 2 div } { % Don't center in x 0 } ifelse Y-CENTER-PLOT 1 eq { % Center in y page-height 529.455 sub 2 div } { % Don't center in y 0 } ifelse translate } ifelse 0.048 0.048 scale /choose-font % stack: fontsize fontname => --- { findfont exch scalefont setfont newpath 0 0 moveto (0) true charpath flattenpath pathbbox /top exch def pop /bottom exch def pop bottom top bottom top add 2 div /center-font-val exch def /upper-font-val exch def /lower-font-val exch def } def /center-x-just % stack: (string) x y => (string) newx y { exch 2 index stringwidth pop 2 div sub exch } def /left-x-just % stack: (string) x y => (string) newx y { } def /right-x-just % stack: (string) x y => (string) newx y { exch 2 index stringwidth pop sub exch } def /center-y-just % stack: (string) x y => (string) x newy { center-font-val sub } def /lower-y-just % stack: (string) x y => (string) x newy { lower-font-val sub } def /upper-y-just % stack: (string) x y => (string) x newy { upper-font-val sub } def /just-string % stack: (string) x y just => --- { dup 0 eq { pop center-x-just center-y-just } if dup 1 eq { pop left-x-just center-y-just } if dup 2 eq { pop left-x-just upper-y-just } if dup 3 eq { pop center-x-just upper-y-just } if dup 4 eq { pop right-x-just upper-y-just } if dup 5 eq { pop right-x-just center-y-just } if dup 6 eq { pop right-x-just lower-y-just } if dup 7 eq { pop center-x-just lower-y-just } if dup 8 eq { pop left-x-just lower-y-just } if moveto show } def /mark0 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto size size add 0 rlineto 0 size size add rlineto 0 size size add sub 0 rlineto closepath fill} def /mark1 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto size size add 0 rlineto 0 size size add rlineto 0 size size add sub 0 rlineto closepath stroke} def /mark2 {/size exch def /y exch def /x exch def newpath x y moveto x y size 0 360 arc stroke} def /mark3 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto x size add y size add lineto x size sub y size add moveto x size add y size sub lineto stroke} def /mark4 {/size exch def /y exch def /x exch def newpath x size sub y moveto x y size add lineto x size add y lineto x y size sub lineto closepath stroke} def /mark5 {/size exch def /y exch def /x exch def x y size mark1 newpath x size sub y moveto size size add 0 rlineto stroke} def /mark6 {/size exch def /y exch def /x exch def newpath x y moveto x y size 0 360 arc fill} def /mark7 {/size exch def /y exch def /x exch def newpath x y moveto x size sub y size sub lineto x size add y size sub lineto closepath fill newpath x y moveto x size add y size add lineto x size sub y size add lineto closepath fill} def 0 setgray 300 /Times-Bold choose-font 200 /Times-Roman choose-font 14 setlinewidth [] 0 setdash /pos 7984 def /diff 355 def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 7566 7984 24 mark0 14 setlinewidth [25] 0 setdash /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.125 0 setgray 7566 pos 24 mark1 14 setlinewidth [50] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.25 0 setgray 7566 pos 24 mark2 14 setlinewidth [75] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.375 0 setgray 7566 pos 24 mark3 14 setlinewidth [100] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.5 0 setgray 7566 pos 24 mark4 14 setlinewidth [125] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.625 0 setgray 7566 pos 24 mark5 0 0 setgray [] 0 setdash 8 setlinewidth newpath 950 724 moveto 7476 724 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 1160 moveto 7476 1160 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 1595 moveto 7476 1595 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 2031 moveto 7476 2031 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 2466 moveto 7476 2466 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 2901 moveto 7476 2901 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 3337 moveto 7476 3337 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 3772 moveto 7476 3772 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 4208 moveto 7476 4208 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 4643 moveto 7476 4643 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 5079 moveto 7476 5079 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 5514 moveto 7476 5514 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 5950 moveto 7476 5950 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 6385 moveto 7476 6385 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 6821 moveto 7476 6821 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 7256 moveto 7476 7256 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 7691 moveto 7476 7691 lineto stroke [] 0 setdash 8 setlinewidth newpath 1186 7954 moveto 1186 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 2397 7954 moveto 2397 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 3608 7954 moveto 3608 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 4818 7954 moveto 4818 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 6029 7954 moveto 6029 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 7240 7954 moveto 7240 380 lineto stroke 8 setlinewidth 1337 772 24 mark0 1489 813 24 mark0 1640 849 24 mark0 1791 915 24 mark0 1943 972 24 mark0 2094 1008 24 mark0 2246 1084 24 mark0 2397 1127 24 mark0 2548 1149 24 mark0 2700 1204 24 mark0 2851 1273 24 mark0 3002 1325 24 mark0 3154 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get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 2 string 0 1 255{IE S dup 360 add 36 4 index cvrs cvn put}for pop 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V{}B /RV statusdict begin /product where{pop false[ (Display)(NeXT)(LaserWriter 16/600)]{dup length product length le{dup length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end /TeXBase1Encoding [ /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef /Zcaron /zcaron /caron /dotlessi /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there down to here. /grave /quotesingle /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends /.notdef /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def TeXDict begin /rf{findfont dup length 1 add dict begin{1 index /FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type /integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics /Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont /setfont cvx]cvx def}def /ObliqueSlant {dup sin S cos div neg}B /SlantFont{4 index mul add}def /ExtendFont{3 -1 roll mul exch}def /ReEncodeFont{/Encoding exch def}def end TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale true def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end TeXDict begin @defspecial /magscale false def @fedspecial end 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/page-height 500 def /page-width 500 def 538.583 page-width gt not { % Rotation required 90 rotate 0 page-width neg translate % Handle centering Y-CENTER-PLOT 1 eq { % Center in y page-height 538.583 sub 2 div } { % Don't center in y 0 } ifelse X-CENTER-PLOT 1 eq { % Center in x page-width 529.455 sub 2 div } { % Don't center in x 0 } ifelse translate } { % No rotation - just handle centering X-CENTER-PLOT 1 eq { % Center in x page-width 538.583 sub 2 div } { % Don't center in x 0 } ifelse Y-CENTER-PLOT 1 eq { % Center in y page-height 529.455 sub 2 div } { % Don't center in y 0 } ifelse translate } ifelse 0.048 0.048 scale /choose-font % stack: fontsize fontname => --- { findfont exch scalefont setfont newpath 0 0 moveto (0) true charpath flattenpath pathbbox /top exch def pop /bottom exch def pop bottom top bottom top add 2 div /center-font-val exch def /upper-font-val exch def /lower-font-val exch def } def /center-x-just % stack: (string) x y => (string) newx y { exch 2 index stringwidth pop 2 div sub exch } def /left-x-just % stack: (string) x y => (string) newx y { } def /right-x-just % stack: (string) x y => (string) newx y { exch 2 index stringwidth pop sub exch } def /center-y-just % stack: (string) x y => (string) x newy { center-font-val sub } def /lower-y-just % stack: (string) x y => (string) x newy { lower-font-val sub } def /upper-y-just % stack: (string) x y => (string) x newy { upper-font-val sub } def /just-string % stack: (string) x y just => --- { dup 0 eq { pop center-x-just center-y-just } if dup 1 eq { pop left-x-just center-y-just } if dup 2 eq { pop left-x-just upper-y-just } if dup 3 eq { pop center-x-just upper-y-just } if dup 4 eq { pop right-x-just upper-y-just } if dup 5 eq { pop right-x-just center-y-just } if dup 6 eq { pop right-x-just lower-y-just } if dup 7 eq { pop center-x-just lower-y-just } if dup 8 eq { pop left-x-just lower-y-just } if moveto show } def /mark0 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto size size add 0 rlineto 0 size size add rlineto 0 size size add sub 0 rlineto closepath fill} def /mark1 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto size size add 0 rlineto 0 size size add rlineto 0 size size add sub 0 rlineto closepath stroke} def /mark2 {/size exch def /y exch def /x exch def newpath x y moveto x y size 0 360 arc stroke} def /mark3 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto x size add y size add lineto x size sub y size add moveto x size add y size sub lineto stroke} def /mark4 {/size exch def /y exch def /x exch def newpath x size sub y moveto x y size add lineto x size add y lineto x y size sub lineto closepath stroke} def /mark5 {/size exch def /y exch def /x exch def x y size mark1 newpath x size sub y moveto size size add 0 rlineto stroke} def /mark6 {/size exch def /y exch def /x exch def newpath x y moveto x y size 0 360 arc fill} def /mark7 {/size exch def /y exch def /x exch def newpath x y moveto x size sub y size sub lineto x size add y size sub lineto closepath fill newpath x y moveto x size add y size add lineto x size sub y size add lineto closepath fill} def 0 setgray 300 /Times-Bold choose-font 200 /Times-Roman choose-font 14 setlinewidth [] 0 setdash /pos 7984 def /diff 355 def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 7566 7984 24 mark0 14 setlinewidth [25] 0 setdash /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.125 0 setgray 7566 pos 24 mark1 14 setlinewidth [50] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.25 0 setgray 7566 pos 24 mark2 14 setlinewidth [75] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.375 0 setgray 7566 pos 24 mark3 14 setlinewidth [100] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.5 0 setgray 7566 pos 24 mark4 14 setlinewidth [125] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.625 0 setgray 7566 pos 24 mark5 0 0 setgray [] 0 setdash 8 setlinewidth newpath 950 601 moveto 7476 601 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 981 moveto 7476 981 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 1360 moveto 7476 1360 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 1740 moveto 7476 1740 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 2120 moveto 7476 2120 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 2499 moveto 7476 2499 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 2879 moveto 7476 2879 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 3259 moveto 7476 3259 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 3638 moveto 7476 3638 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 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/bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 2 string 0 1 255{IE S dup 360 add 36 4 index cvrs cvn put}for pop 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V{}B /RV statusdict begin /product where{pop false[ (Display)(NeXT)(LaserWriter 16/600)]{dup length product length le{dup length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end /TeXBase1Encoding [ /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef /Zcaron /zcaron /caron /dotlessi /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there down to here. /grave /quotesingle /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends /.notdef /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def TeXDict begin /rf{findfont dup length 1 add dict begin{1 index /FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type /integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics /Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont /setfont cvx]cvx def}def /ObliqueSlant {dup sin S cos div neg}B /SlantFont{4 index mul add}def /ExtendFont{3 -1 roll mul exch}def /ReEncodeFont{/Encoding exch def}def end TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale true def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end TeXDict begin @defspecial /magscale false def @fedspecial end TeXDict begin 39158280 55380996 1200 600 600 (showfig.dvi) @start /Fa 139[22 3[40 40 40 4[22 5[35 1[35 38[22 58[{ TeXBase1Encoding ReEncodeFont}8 79.7011 /Times-Roman rf /Fb 5 106 df 46 D<141E147EEB01FE130713FFB5FCA3130F1200B3B3A3007FB612C0A4223479B331> 49 D70 D103 D105 D E /Fc 2 108 df<90260FFFC0903803FFFE496D5BA29026003FF09038003FC0F01F00027F 151E810273151C8102F1153C6F143814E08101016D1478701370ECC03F820103011F14F0 705BEC800F8201070107130104F85BEC000316FC491503DB01FE5B130EED00FF011E1507 DC7F87C7FC131CEE3FC7013C15CFEE1FEE1338EE0FFE1378705A1370160313F0705A1201 D807F81400B512C017705C3F307CAF3F>78 D107 D E /Fd 5 119 df<0203B612F0021F15F8147F49B7FC010716F04916E090271FF80FF8C7FC 90393FC003FC90387F800101FEC77E485A4848147E0007157F5B485AA2485AA2123F5B5E 007F5D90C8FCA21501485D5AA24B5AA24B5AA2007E4A5A5E4B5A003E4AC8FC003F147E6C 5C6D485A000F495A3907C007C02601F03FC9FC38007FFCEB1FE0352C7DAA39>27 D<121FEA3F80EA7FC0EAFFE0A5EA7FC0EA3F80EA1F000B0B788A1C>58 D<121FEA3F80EA7FC0EAFFE0A313F0A2127FEA3FB0EA1F301200A413701360A213E013C0 A2120113801203EA07001206120E5A5A12300C1E788A1C>I<91267FFFF00203B512F891 B56C5CA29126003FFC9139001FFE00F207F0634B6C6F5A1577DB73FF5E1A0703F17F03E1 94C7FCA203E06D5C14014B6C6C140EA2706C141E1403DB801F151C831A3C02076D7E0300 1638707E1A784A1303020E6E1470A2706D13F0141E021C6D6D5BA2057F1301023C15E002 385FEF3FF019030278EC1FF802705F170FF0FC0714F04ADA07FE90C8FCA2716C5A13014A 6E138EA2199E01036F13DE4A16DCF07FFCA20107163F91C95BA2181F5B010E705A131E18 07133F49705A3803FFC0B6FC18014A5E55447DC350>78 D<013F1407D9FFC0EB1F802601 C3E0EB3FC0380381F0D80701147F000E7F001E153F001C151FD83C03140F12381607EA70 075CA2D8F00FEC038000E05BEA001F4A13071700133F5C5E017F140E91C7FCA2161E4914 1C5B5EA2167848481470A25E12004B5A15036D5C4BC7FC017E130E6D5B6D6C5A90380FC0 F0903803FFC0010090C8FC2A2D7DAB30>118 D E /Fe 2 62 df<163CA2167EA216FFA24B7FA24B7FA24B6C7EA292380E3FF0A24B6C7EA24B 6C7EA24B6C7EA29238F003FF15E002016D7F15C002036D7F5D02076E7E92C7FC4A6E7E14 0E021E6E7E141C023C6E7E143802786E7E147002F06E7E5C01016F7F5C01036F7F5C0107 707E91C9FC49707E130E011E707E131C013C707E13380178707E137001F0707E5B000171 13805B00037113C05B0007F07FE090CBFC48F03FF0120E001EF01FF8001FBAFC4819FCA2 4819FEA2BCFCA348477BC653>1 D<007FBA1280BB12C0A26C1980CEFCB0007FBA1280BB 12C0A26C198042187BA44D>61 D E /Ff 1 1 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D E /Fg 137[50 1[28 39 33 1[50 50 50 78 3[28 1[50 33 44 50 44 1[44 27[61 72 11[50 1[50 50 50 50 50 50 50 3[25 44[{ TeXBase1Encoding ReEncodeFont}26 99.6264 /Times-Roman rf end TeXDict begin 1 0 bop 94 4258 a @beginspecial @setspecial /X-CENTER-PLOT 0 def /Y-CENTER-PLOT 0 def /page-height 500 def /page-width 500 def 538.583 page-width gt not { % Rotation required 90 rotate 0 page-width neg translate % Handle centering Y-CENTER-PLOT 1 eq { % Center in y page-height 538.583 sub 2 div } { % Don't center in y 0 } ifelse X-CENTER-PLOT 1 eq { % Center in x page-width 529.455 sub 2 div } { % Don't center in x 0 } ifelse translate } { % No rotation - just handle centering X-CENTER-PLOT 1 eq { % Center in x page-width 538.583 sub 2 div } { % Don't center in x 0 } ifelse Y-CENTER-PLOT 1 eq { % Center in y page-height 529.455 sub 2 div } { % Don't center in y 0 } ifelse translate } ifelse 0.048 0.048 scale /choose-font % stack: fontsize fontname => --- { findfont exch scalefont setfont newpath 0 0 moveto (0) true charpath flattenpath pathbbox /top exch def pop /bottom exch def pop bottom top bottom top add 2 div /center-font-val exch def /upper-font-val exch def /lower-font-val exch def } def /center-x-just % stack: (string) x y => (string) newx y { exch 2 index stringwidth pop 2 div sub exch } def /left-x-just % stack: (string) x y => (string) newx y { } def /right-x-just % stack: (string) x y => (string) newx y { exch 2 index stringwidth pop sub exch } def /center-y-just % stack: (string) x y => (string) x newy { center-font-val sub } def /lower-y-just % stack: (string) x y => (string) x newy { lower-font-val sub } def /upper-y-just % stack: (string) x y => (string) x newy { upper-font-val sub } def /just-string % stack: (string) x y just => --- { dup 0 eq { pop center-x-just center-y-just } if dup 1 eq { pop left-x-just center-y-just } if dup 2 eq { pop left-x-just upper-y-just } if dup 3 eq { pop center-x-just upper-y-just } if dup 4 eq { pop right-x-just upper-y-just } if dup 5 eq { pop right-x-just center-y-just } if dup 6 eq { pop right-x-just lower-y-just } if dup 7 eq { pop center-x-just lower-y-just } if dup 8 eq { pop left-x-just lower-y-just } if moveto show } def /mark0 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto size size add 0 rlineto 0 size size add rlineto 0 size size add sub 0 rlineto closepath fill} def /mark1 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto size size add 0 rlineto 0 size size add rlineto 0 size size add sub 0 rlineto closepath stroke} def /mark2 {/size exch def /y exch def /x exch def newpath x y moveto x y size 0 360 arc stroke} def /mark3 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto x size add y size add lineto x size sub y size add moveto x size add y size sub lineto stroke} def /mark4 {/size exch def /y exch def /x exch def newpath x size sub y moveto x y size add lineto x size add y lineto x y size sub lineto closepath stroke} def /mark5 {/size exch def /y exch def /x exch def x y size mark1 newpath x size sub y moveto size size add 0 rlineto stroke} def /mark6 {/size exch def /y exch def /x exch def newpath x y moveto x y size 0 360 arc fill} def /mark7 {/size exch def /y exch def /x exch def newpath x y moveto x size sub y size sub lineto x size add y size sub lineto closepath fill newpath x y moveto x size add y size add lineto x size sub y size add lineto closepath fill} def 0 setgray 300 /Times-Bold choose-font 200 /Times-Roman choose-font 14 setlinewidth [] 0 setdash /pos 7984 def /diff 355 def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 7566 7984 24 mark0 14 setlinewidth [25] 0 setdash /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.125 0 setgray 7566 pos 24 mark1 14 setlinewidth [50] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.25 0 setgray 7566 pos 24 mark2 14 setlinewidth [75] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.375 0 setgray 7566 pos 24 mark3 14 setlinewidth [100] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.5 0 setgray 7566 pos 24 mark4 14 setlinewidth [125] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.625 0 setgray 7566 pos 24 mark5 0 0 setgray [] 0 setdash 8 setlinewidth newpath 950 724 moveto 7476 724 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 1158 moveto 7476 1158 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 1592 moveto 7476 1592 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 2027 moveto 7476 2027 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 2461 moveto 7476 2461 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 2895 moveto 7476 2895 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 3329 moveto 7476 3329 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 3763 moveto 7476 3763 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 4197 moveto 7476 4197 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 4631 moveto 7476 4631 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 5065 moveto 7476 5065 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 5499 moveto 7476 5499 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 5933 moveto 7476 5933 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 6368 moveto 7476 6368 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 6802 moveto 7476 6802 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 7236 moveto 7476 7236 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 7670 moveto 7476 7670 lineto stroke [] 0 setdash 8 setlinewidth newpath 1126 7954 moveto 1126 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 2336 7954 moveto 2336 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 3547 7954 moveto 3547 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 4758 7954 moveto 4758 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 5969 7954 moveto 5969 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 7179 7954 moveto 7179 380 lineto 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Fg(60)i(0)p Fd(:)p Fg(80)d(1)p Fd(:)p Fg(00)94 4666 y Fb()33 b()p Fa()c()p eop end userdict /end-hook known{end-hook}if %%EndDocument @endspecial 204 x Fq(Figure)23 b(3:)35 b(Same)23 b(as)g(\256g.)f(1)h(for)f (the)h(product)1694 4993 y Fh(Q)1773 5080 y Fl(n)1820 5060 y Fn(\(1)16 b(+)g Fm(A)2088 5075 y Fl(n)2134 5060 y Fn(\))p Fq(,)23 b(e)n(xcept)h(that)g(the)f(v)n(ertical)i(axis)83 5180 y(measures)h Fm(r)532 5195 y Fl(k)575 5180 y Fm(=\033)s Fq(.)1651 5637 y(21)p eop %%Page: 22 22 bop 432 4856 a @beginspecial 100 @llx 200 @lly 604 @urx 704 @ury 5040 @rwi @setspecial %%BeginDocument: fig4.ps /TeXDict 300 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 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roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]/id ch-image N /rw ch-width 7 add 8 idiv string N /rc 0 N /gp 0 N /cp 0 N{rc 0 ne{rc 1 sub /rc X rw}{G}ifelse}imagemask restore}B /G{{id gp get /gp gp 1 add N dup 18 mod S 18 idiv pl S get exec}loop}B /adv{cp add /cp X}B /chg{rw cp id gp 4 index getinterval putinterval dup gp add /gp X adv}B /nd{/cp 0 N rw exit}B /lsh{rw cp 2 copy get dup 0 eq{pop 1}{ dup 255 eq{pop 254}{dup dup add 255 and S 1 and or}ifelse}ifelse put 1 adv}B /rsh{rw cp 2 copy get dup 0 eq{pop 128}{dup 255 eq{pop 127}{dup 2 idiv S 128 and or}ifelse}ifelse put 1 adv}B /clr{rw cp 2 index string putinterval adv}B /set{rw cp fillstr 0 4 index getinterval putinterval adv}B /fillstr 18 string 0 1 17{2 copy 255 put pop}for N /pl[{adv 1 chg} {adv 1 chg nd}{1 add chg}{1 add chg nd}{adv lsh}{adv lsh nd}{adv rsh}{ adv rsh nd}{1 add adv}{/rc X nd}{1 add set}{1 add clr}{adv 2 chg}{adv 2 chg nd}{pop nd}]dup{bind pop}forall N /D{/cc X dup type /stringtype ne{] }if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 2 index S get sf div put}if put /ctr ctr 1 add N}B /I{ cc 1 add D}B /bop{userdict /bop-hook known{bop-hook}if /SI save N @rigin 0 0 moveto /V matrix currentmatrix dup 1 get dup mul exch 0 get dup mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N /eop{SI restore userdict /eop-hook known{eop-hook}if showpage}N /@start{userdict /start-hook known{start-hook}if pop /VResolution X /Resolution X 1000 div /DVImag X /IE 256 array N 2 string 0 1 255{IE S dup 360 add 36 4 index cvrs cvn put}for pop 65781.76 div /vsize X 65781.76 div /hsize X}N /p{show}N /RMat[1 0 0 -1 0 0]N /BDot 260 string N /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V{}B /RV statusdict begin /product where{pop false[ (Display)(NeXT)(LaserWriter 16/600)]{dup length product length le{dup length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse} forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /delta X 0 rmoveto}B /M{S p delta add tail}B /b{S p tail}B /c{-4 M} B /d{-3 M}B /e{-2 M}B /f{-1 M}B /g{0 M}B /h{1 M}B /i{2 M}B /j{3 M}B /k{ 4 M}B /w{0 rmoveto}B /l{p -4 w}B /m{p -3 w}B /n{p -2 w}B /o{p -1 w}B /q{ p 1 w}B /r{p 2 w}B /s{p 3 w}B /t{p 4 w}B /x{0 S rmoveto}B /y{3 2 roll p a}B /bos{/SS save N}B /eos{SS restore}B end /TeXBase1Encoding [ /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef /Zcaron /zcaron /caron /dotlessi /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there down to here. /grave /quotesingle /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends /.notdef /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def TeXDict begin /rf{findfont dup length 1 add dict begin{1 index /FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type /integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics /Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont /setfont cvx]cvx def}def /ObliqueSlant {dup sin S cos div neg}B /SlantFont{4 index mul add}def /ExtendFont{3 -1 roll mul exch}def /ReEncodeFont{/Encoding exch def}def end TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale true def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end TeXDict begin @defspecial /magscale false def @fedspecial end TeXDict begin 39158280 55380996 1200 600 600 (showfig.dvi) @start /Fa 139[22 3[40 40 40 4[22 5[35 1[35 38[22 58[{ TeXBase1Encoding ReEncodeFont}8 79.7011 /Times-Roman rf /Fb 5 106 df 46 D<141E147EEB01FE130713FFB5FCA3130F1200B3B3A3007FB612C0A4223479B331> 49 D70 D103 D105 D E /Fc 205[25 50[{TeXBase1Encoding ReEncodeFont}1 49.8132 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393FC003FC90387F800101FEC77E485A4848147E0007157F5B485AA2485AA2123F5B5E00 7F5D90C8FCA21501485D5AA24B5AA24B5AA2007E4A5A5E4B5A003E4AC8FC003F147E6C5C 6D485A000F495A3907C007C02601F03FC9FC38007FFCEB1FE0352C7DAA39>27 D<121FEA3F80EA7FC0EAFFE0A5EA7FC0EA3F80EA1F000B0B788A1C>58 D<121FEA3F80EA7FC0EAFFE0A313F0A2127FEA3FB0EA1F301200A413701360A213E013C0 A2120113801203EA07001206120E5A5A12300C1E788A1C>I<91267FFFF00203B512F891 B56C5CA29126003FFC9139001FFE00F207F0634B6C6F5A1577DB73FF5E1A0703F17F03E1 94C7FCA203E06D5C14014B6C6C140EA2706C141E1403DB801F151C831A3C02076D7E0300 1638707E1A784A1303020E6E1470A2706D13F0141E021C6D6D5BA2057F1301023C15E002 385FEF3FF019030278EC1FF802705F170FF0FC0714F04ADA07FE90C8FCA2716C5A13014A 6E138EA2199E01036F13DE4A16DCF07FFCA20107163F91C95BA2181F5B010E705A131E18 07133F49705A3803FFC0B6FC18014A5E55447DC350>78 D E /Fh 2 62 df<163CA2167EA216FFA24B7FA24B7FA24B6C7EA292380E3FF0A24B6C7EA24B 6C7EA24B6C7EA29238F003FF15E002016D7F15C002036D7F5D02076E7E92C7FC4A6E7E14 0E021E6E7E141C023C6E7E143802786E7E147002F06E7E5C01016F7F5C01036F7F5C0107 707E91C9FC49707E130E011E707E131C013C707E13380178707E137001F0707E5B000171 13805B00037113C05B0007F07FE090CBFC48F03FF0120E001EF01FF8001FBAFC4819FCA2 4819FEA2BCFCA348477BC653>1 D<007FBA1280BB12C0A26C1980CEFCB0007FBA1280BB 12C0A26C198042187BA44D>61 D E /Fi 1 1 df<007FB912E0BA12F0A26C18E03C04789A4D>0 D E /Fj 135[50 3[28 39 33 1[50 50 50 4[28 2[33 44 1[44 1[44 20[61 7[72 11[50 1[50 50 50 1[50 50 50 1[25 1[25 44[{ TeXBase1Encoding ReEncodeFont}23 99.6264 /Times-Roman rf end TeXDict begin 1 0 bop 94 4258 a @beginspecial @setspecial /X-CENTER-PLOT 0 def /Y-CENTER-PLOT 0 def /page-height 500 def /page-width 500 def 538.583 page-width gt not { % Rotation required 90 rotate 0 page-width neg translate % Handle centering Y-CENTER-PLOT 1 eq { % Center in y page-height 538.583 sub 2 div } { % Don't center in y 0 } ifelse X-CENTER-PLOT 1 eq { % Center in x page-width 529.455 sub 2 div } { % Don't center in x 0 } ifelse translate } { % No rotation - just handle centering X-CENTER-PLOT 1 eq { % Center in x page-width 538.583 sub 2 div } { % Don't center in x 0 } ifelse Y-CENTER-PLOT 1 eq { % Center in y page-height 529.455 sub 2 div } { % Don't center in y 0 } ifelse translate } ifelse 0.048 0.048 scale /choose-font % stack: fontsize fontname => --- { findfont exch scalefont setfont newpath 0 0 moveto (0) true charpath flattenpath pathbbox /top exch def pop /bottom exch def pop bottom top bottom top add 2 div /center-font-val exch def /upper-font-val exch def /lower-font-val exch def } def /center-x-just % stack: (string) x y => (string) newx y { exch 2 index stringwidth pop 2 div sub exch } def /left-x-just % stack: (string) x y => (string) newx y { } def /right-x-just % stack: (string) x y => (string) newx y { exch 2 index stringwidth pop sub exch } def /center-y-just % stack: (string) x y => (string) x newy { center-font-val sub } def /lower-y-just % stack: (string) x y => (string) x newy { lower-font-val sub } def /upper-y-just % stack: (string) x y => (string) x newy { upper-font-val sub } def /just-string % stack: (string) x y just => --- { dup 0 eq { pop center-x-just center-y-just } if dup 1 eq { pop left-x-just center-y-just } if dup 2 eq { pop left-x-just upper-y-just } if dup 3 eq { pop center-x-just upper-y-just } if dup 4 eq { pop right-x-just upper-y-just } if dup 5 eq { pop right-x-just center-y-just } if dup 6 eq { pop right-x-just lower-y-just } if dup 7 eq { pop center-x-just lower-y-just } if dup 8 eq { pop left-x-just lower-y-just } if moveto show } def /mark0 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto size size add 0 rlineto 0 size size add rlineto 0 size size add sub 0 rlineto closepath fill} def /mark1 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto size size add 0 rlineto 0 size size add rlineto 0 size size add sub 0 rlineto closepath stroke} def /mark2 {/size exch def /y exch def /x exch def newpath x y moveto x y size 0 360 arc stroke} def /mark3 {/size exch def /y exch def /x exch def newpath x size sub y size sub moveto x size add y size add lineto x size sub y size add moveto x size add y size sub lineto stroke} def /mark4 {/size exch def /y exch def /x exch def newpath x size sub y moveto x y size add lineto x size add y lineto x y size sub lineto closepath stroke} def /mark5 {/size exch def /y exch def /x exch def x y size mark1 newpath x size sub y moveto size size add 0 rlineto stroke} def /mark6 {/size exch def /y exch def /x exch def newpath x y moveto x y size 0 360 arc fill} def /mark7 {/size exch def /y exch def /x exch def newpath x y moveto x size sub y size sub lineto x size add y size sub lineto closepath fill newpath x y moveto x size add y size add lineto x size sub y size add lineto closepath fill} def 0 setgray 300 /Times-Bold choose-font 200 /Times-Roman choose-font 14 setlinewidth [] 0 setdash /pos 7984 def /diff 355 def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 7566 7984 24 mark0 14 setlinewidth [25] 0 setdash /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.125 0 setgray 7566 pos 24 mark1 14 setlinewidth [50] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.25 0 setgray 7566 pos 24 mark2 14 setlinewidth [75] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.375 0 setgray 7566 pos 24 mark3 14 setlinewidth [100] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.5 0 setgray 7566 pos 24 mark4 14 setlinewidth [125] 0 setdash 0 0 setgray /pos pos diff sub def newpath 7566 pos moveto 8886 pos lineto stroke [] 0 setdash 8 setlinewidth 0.625 0 setgray 7566 pos 24 mark5 0 0 setgray [] 0 setdash 8 setlinewidth newpath 950 442 moveto 7476 442 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 1035 moveto 7476 1035 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 1627 moveto 7476 1627 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 2219 moveto 7476 2219 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 2811 moveto 7476 2811 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 3403 moveto 7476 3403 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 3995 moveto 7476 3995 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 4587 moveto 7476 4587 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 5180 moveto 7476 5180 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 5772 moveto 7476 5772 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 6364 moveto 7476 6364 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 6956 moveto 7476 6956 lineto stroke [] 0 setdash 2 setlinewidth newpath 950 7548 moveto 7476 7548 lineto stroke [] 0 setdash 8 setlinewidth newpath 1154 7954 moveto 1154 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 2390 7954 moveto 2390 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 3626 7954 moveto 3626 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 4862 7954 moveto 4862 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 6098 7954 moveto 6098 380 lineto stroke [] 0 setdash 2 setlinewidth newpath 7334 7954 moveto 7334 380 lineto stroke 8 setlinewidth 1386 1144 24 mark0 1695 1819 24 mark0 2004 2447 24 mark0 2313 3017 24 mark0 2622 3529 24 mark0 2931 3985 24 mark0 3240 4391 24 mark0 3549 4755 24 mark0 3858 5083 24 mark0 4167 5381 24 mark0 4476 5652 24 mark0 4785 5902 24 mark0 5094 6132 24 mark0 5403 6346 24 mark0 5712 6542 24 mark0 6021 6724 24 mark0 6330 6889 24 mark0 6639 7040 24 mark0 6948 7178 24 mark0 14 setlinewidth [] 0 setdash newpath 1386 1144 moveto 1695 1819 lineto 2004 2447 lineto 2313 3017 lineto 2622 3529 lineto 2931 3985 lineto 3240 4391 lineto 3549 4755 lineto 3858 5083 lineto 4167 5381 lineto 4476 5652 lineto 4785 5902 lineto 5094 6132 lineto 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y(1)p Fg(:)p Fj(80)321 1705 y(2)p Fg(:)p Fj(00)321 1469 y(2)p Fg(:)p Fj(20)321 1228 y(2)p Fg(:)p Fj(40)535 4210 y(0)p Fg(:)p Fj(00)306 b(0)p Fg(:)p Fj(20)g(0)p Fg(:)p Fj(40)e(0)p Fg(:)p Fj(60)i(0)p Fg(:)p Fj(80)g(1)p Fg(:)p Fj(00)94 4666 y Fb()33 b()p Fa()c()p eop end userdict /end-hook known{end-hook}if %%EndDocument @endspecial 204 x Fq(Figure)22 b(4:)36 b(Same)23 b(as)g(\256g.)f(2)h(for)f (the)h(product)2043 4993 y Fh(Q)2122 5080 y Fl(n)2168 5060 y Fn(\(1)16 b(+)g Fm(A)2436 5075 y Fl(n)2482 5060 y Fn(\))p Fq(,)24 b(e)n(xcept)g(that)f(the)h(v)n(ertical)g(axis)432 5180 y(measures)837 5096 y Fk(p)p 920 5096 47 4 v 84 x Fm(\016)967 5195 y Fl(k)1009 5180 y Fm(=\033)t Fq(.)2000 5637 y(22)p eop %%Page: 23 23 bop 83 573 a Fq(The)26 b(image)h(of)e Fm(u)g Fq(is)650 832 y Fn(\(1)c(+)882 807 y(~)857 832 y Fm(A)p Fn(\))985 711 y Fh(\022)1062 772 y Fm(u)1118 787 y Fi(f)1062 892 y Fm(u)1118 907 y Fi(s)1166 711 y Fh(\023)1255 832 y Fn(=)1358 711 y Fh(\022)1436 772 y Fn(\(1)h(+)g Fm(\025)g Fn(+)g Fm(")1866 736 y Fi(2)1905 772 y Fn(\(1)g Fk(\000)h Fm(u)2170 787 y Fi(s)2201 772 y Fn(\)\))p Fm(u)2333 787 y Fi(f)2386 772 y Fn(+)f Fm("v)2577 787 y Fi(f)1590 892 y Fn(\(1)g(+)g Fm(\013)q(\025)f Fk(\000)i Fm(")2084 856 y Fi(2)2123 892 y Fn(\))p Fm(u)2217 907 y Fi(s)2271 892 y Fn(+)f Fm(")2415 856 y Fi(2)2624 711 y Fh(\023)2727 832 y Fm(:)397 b Fq(\(59\))83 1089 y(After)26 b(normalizing)h(the)f Fm(f)36 b Fq(component)27 b(to)f(1,)g(the)g Fm(s)f Fq(component)i(becomes)786 1313 y Fm(u)842 1272 y Fe(0)842 1337 y Fi(s)901 1313 y Fk(\030)1006 1216 y Fh(h)1045 1313 y Fn(1)22 b(+)g(\()p Fm(\013)h Fk(\000)g Fn(1\))p Fm(\025)e Fn(+)h Fm(")1746 1272 y Fi(2)1785 1313 y Fn(\(3)p Fm(=)p Fn(2)g Fk(\000)h Fm(u)2148 1328 y Fi(s)2179 1313 y Fn(\))2217 1216 y Fh(i)2273 1313 y Fm(u)2329 1328 y Fi(s)2382 1313 y Fn(+)f Fm(")2526 1272 y Fi(2)2591 1313 y Fm(:)533 b Fq(\(60\))83 1556 y(Evidently)-7 b(,)30 b(e)m(v)n(en)f(if)f Fm(u)888 1571 y Fi(s)950 1556 y Fn(=)i(0)d Fq(initially)-7 b(,)31 b(by)d(choosing)2047 1531 y Fn(~)2021 1556 y Fm(A)g Fq(as)f(abo)n(v)n(e,)j Fm(u)2584 1571 y Fi(s)2643 1556 y Fq(can)e(be)g (increased)83 1676 y(to)e(an)g Fk(O)s Fn(\(1\))f Fq(v)m(alue)i(\(as)e Fm(";)33 b(\025)27 b Fk(!)h Fn(0)p Fq(\))d(if)h Fm(\025)h Fn(=)h Fk(O)s Fn(\()p Fm(")1798 1640 y Fi(2)1837 1676 y Fn(\))p Fq(.)229 1797 y Fp(Ackno)o(wledgments)p Fq(:)42 b(W)-8 b(e)29 b(ha)n(v)n(e)h (pro\256ted)e(from)h(v)n(ery)g(useful)g(discussions)f(with)h(S.)83 1917 y(Lepri,)c(C.)e(Li)m(v)n(erani,)k(Z.)c(Olami,)i(A.)e(Politi,)i(and)f (L.-S.)f(Y)-11 b(oung.)36 b(But)24 b(we)e(are)i(particularly)83 2038 y(indebted)36 b(to)f(H.)g(Posch)g(for)f(ha)n(ving)i(pro)n(vided)f(and)g (discussed)g(with)g(us)g(his)f(beautiful)83 2158 y(numerical)19 b(e)n(xperiments.)37 b(This)18 b(w)o(ork)f(w)o(as)g(partially)j(supported)d (by)h(the)g(F)n(onds)g(National)83 2278 y(Suisse,)28 b(and)f(part)h(of)g(it)g (w)o(as)e(done)i(in)g(the)g(pleasant)g(atmosphere)g(of)g(the)g(ESI)f(in)h(V) -6 b(ienna)83 2399 y(and)26 b(at)g(the)g(Istituto)g(Nazionale)h(di)f(Ottica)h (in)e(Florence.)83 2741 y Fs(Refer)m(ences)134 2967 y Fq([1])49 b(P)n(aladin,)27 b(G.)f(and)f(V)-8 b(ulpiani,)27 b(A.:)38 b(J.)25 b(Ph)o(ys.)h(A)p Fp(19)p Fq(,)g(1881)f(\(1986\).)134 3169 y([2])49 b(Li)m(vi,)44 b(R.,)g(Politi,)h(A.,)e(Ruf)m(fo,)h(S.,)f(and)d(V)-8 b(ulpiani,)45 b(A.:)65 b(J.)39 b(Stat.)i(Ph)o(ys.)f Fp(46)p Fq(,)j(197)302 3289 y(\(1987\).)134 3491 y([3])49 b(Eckmann)26 b(J.-P)-11 b(.)26 b(and)f(W)-8 b(ayne,)27 b(C.E.:)38 b(J.)26 b(Stat.)h(Ph)o(ys.)e Fp(50)p Fq(,)h(853)g(\(1988\).)134 3693 y([4])49 b(Dellago,)h(Ch.,)g(Posch,)e(H.A.,)i(and)44 b(Hoo)n(v)n(er)l(,)50 b(W)-9 b(.G.:)75 b(Ph)o(ys.)45 b(Re)m(v)-7 b(.)45 b(E)p Fp(53)p Fq(,)50 b(1485)302 3813 y(\(1996\).)134 4015 y([5])f(Posch,)44 b(H.A.)d(and)g(Hirschl,)j(R.:)68 b(Simulation)42 b(of)f(billiards)h(and)e(of) h(hard-body)302 4135 y(\257uids,)25 b(preprint)i(\(1999\).)134 4337 y([6])49 b(F)n(orster)l(,)31 b(D.:)45 b Fo(Hydr)-5 b(odynamic)30 b(Fluctuations,)i(Br)-5 b(ok)o(en)30 b(Symmetry)-6 b(,)31 b(and)f(Corr)l (ela-)302 4457 y(tion)c(Fluctuations)p Fq(,)h(Benjamin)g(Cummings,)g(London,) f(\(1983\).)134 4659 y([7])49 b(Lepri,)26 b(S.,)h(Li)m(vi,)g(R.,)f(and)g (Politi,)h(A.:)37 b(Europh)o(ys.)27 b(Lett.)g Fp(43)p Fq(,)f(271)f(\(1998\).) 134 4861 y([8])49 b(Politi,)27 b(A.:)37 b(pri)m(v)m(ate)27 b(communication.)134 5063 y([9])49 b(Y)-11 b(oung,)25 b(L.S.:)38 b(Er)n(g.)26 b(Th.)h(Dyn.)e(Syst.)h Fp(6)p Fq(,)g(627)f(\(1986\).)83 5264 y([10])49 b(P)n(arto)n(vi,)27 b(M.H.:)38 b(Ph)o(ys.)25 b(Re)m(v)-7 b(.)27 b(Lett.)h Fp(82)p Fq(,)e(3424)f(\(1999\).)1651 5637 y(23)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------9908181016586--