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b Fq(x)g Fs(and)e(time)h Fq(t)p Fs(,)h(so)f(that)1449 5369 y Fq(@)1497 5383 y Fp(t)1527 5369 y Fq(h)p Fs(\()p Fq(t;)15 b(x)p Fs(\))26 b(=)f Fq(\016)s Fs(\()p Fq(X)2049 5383 y Fp(t)2100 5369 y Fl(\000)20 b Fq(x)p Fs(\))p Fq(:)1854 5652 y Fs(3)p eop %%Page: 4 4 4 3 bop 382 548 a Fs(F)-8 b(or)32 b(details)f(concerning)g(the)g (construction)g(and)g(primary)f(prop)s(erties)g(of)h(this)g(pro-)382 687 y(cess)h(and)f(a)h(rigorous)f(v)m(ersion)g(of)h(these)g(equations,) g(see)g([17)r(].)44 b(Let)33 b(us)e(just)g(empha-)382 826 y(size)g(a)g(couple)f(of)h(features:)42 b(The)31 b(pro)s(cess)f(\()p Fq(X)2011 840 y Fp(t)2041 826 y Fq(;)15 b(t)27 b Fl(\025)e Fs(0\))32 b(is)e(a)h(random)f(pro)s(cess,)h(ev)m(en) 382 965 y(though)21 b(the)h(previous)e(`di\013eren)m(tial)h(equations') h(lo)s(ok)f(v)m(ery)h(deterministic.)36 b(One)21 b(rea-)382 1103 y(son)i(is)f(that)h(\(in)f(the)h(stationary)h(regime\),)h(the)e (function)e Fq(x)26 b Fl(7!)f Fq(h)p Fs(\()p Fq(t;)15 b(x)p Fs(\))24 b(is)e(not)h(regular;)382 1242 y(in)30 b(fact,)k(it)d(is)g(a)h(Bro)m(wnian)f(motion)h(in)e(the)i(space)h(v)-5 b(ariable)30 b(\(for)i(\014xed)f Fq(t)p Fs(\).)46 b(Second,)382 1381 y Fq(X)457 1395 y Fp(t)517 1381 y Fs(is)28 b(not)i(a)h(usual)d (sto)s(c)m(hastic)j(pro)s(cess)e(\(it)h(is)e(not)i(solution)f(of)h(a)g (sto)s(c)m(hastic)h(di\013er-)382 1520 y(en)m(tial)j(equation)g(for)g (instance\),)i(it)d(has)h(the)h(2)p Fq(=)p Fs(3)h(scaling:)47 b(\()p Fq(\013)2600 1487 y Fk(\000)p Fn(2)p Fp(=)p Fn(3)2766 1520 y Fq(X)2841 1534 y Fp(\013t)2916 1520 y Fq(;)15 b(t)32 b Fl(\025)g Fs(0\))j(has)382 1659 y(the)c(same)g(la)m(w)g(as)g (\()p Fq(X)1154 1673 y Fp(t)1184 1659 y Fq(;)15 b(t)27 b Fl(\025)f Fs(0\).)43 b(In)30 b(particular,)g(\()p Fq(\013)2182 1626 y Fk(\000)p Fn(1)p Fp(=)p Fn(3)2347 1659 y Fq(h)p Fs(\()p Fq(\013)2492 1626 y Fn(2)p Fp(=)p Fn(3)2604 1659 y Fq(x;)15 b(\013t)p Fs(\))p Fq(;)g(t)27 b Fl(\025)f Fs(0)p Fq(;)15 b(x)27 b Fl(2)f Fo(R)r Fs(\))382 1798 y(has)36 b(the)g(same)g(la)m(w)f(as)i(\()p Fq(h)p Fs(\()p Fq(x;)15 b(t)p Fs(\))p Fq(;)g(t)36 b Fl(\025)e Fs(0)p Fq(;)15 b(x)35 b Fl(2)f Fo(R)s Fs(\))42 b(so)36 b(that)g Fq(h)h Fs(has)e(the)h(same)h(scaling)382 1937 y(prop)s(ert)m(y)30 b(as)g(the)h(asymptotic)f(scaling)g(conjectured)h(for)f(the)g(KPZ)g (equation.)523 2076 y(The)j(pro)s(cess)f(\()p Fq(X)1142 2090 y Fp(t)1173 2076 y Fq(;)15 b(t)30 b Fl(\025)f Fs(0\))34 b(can)g(b)s(e)e(view)m(ed)h(as)g(the)h(scaling)e(limit)f(of)i(a)h (discrete)382 2214 y(negativ)m(ely)44 b(reinforced)e(\(i.e.)81 b(self-rep)s(ellen)m(t\))42 b(random)h(w)m(alk)g(\()p Fq(S)2739 2228 y Fp(n)2786 2214 y Fq(;)15 b(n)47 b Fl(\025)g Fs(0\))e(on)e Fo(Z)382 2353 y Fs(called)21 b(the)g(`true)h(self-a)m(v)m (oiding)e(w)m(alk')i(in)e(the)i(ph)m(ysics)e(literature.)37 b(This)19 b(is)i(a)h(nearest-)382 2492 y(neigh)m(b)s(our)28 b(w)m(alk)i(on)g Fo(Z)25 b Fs(that)31 b(decides)e(at)i(eac)m(h)g(step)f (to)h(jump)e(to)i(the)f(left)g(or)g(to)g(the)382 2631 y(righ)m(t)c(according)g(to)h(a)g(probabilit)m(y)d(dep)s(ending)f(on)k 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3892 y Fs(\()p Fq(!)s Fs(\)\))897 4154 y(=)1110 4068 y Fm(X)1051 4259 y Fp(!)1095 4269 y Ff(j)1127 4259 y Fp(;!)1191 4269 y Ff(j)s Fd(+1)1316 4154 y Fq(n)1371 4168 y Fp(j)t Fn(+1)1497 4154 y Fq(r)s Fs(\()p Fq(z)1618 4168 y Fp(j)t Fn(+1)1745 4154 y Fs(\))p Fq(\026)1835 4169 y Fp(s)1868 4179 y Ff(j)1900 4169 y Fp(;\025;\022)2019 4154 y Fs(\()p Fq(!)2111 4168 y Fp(j)2148 4154 y Fs(\))p Fq(\026)2238 4169 y Fp(s)2271 4179 y Ff(j)s Fd(+1)2380 4169 y Fp(;\025;\022)2500 4154 y Fs(\()p Fq(!)2592 4168 y Fp(j)t Fn(+1)2718 4154 y Fs(\))p Fq(f)g Fs(\()p Fq(!)s Fs(\))382 4431 y(and)33 b(a)i(similar)c(den)m(tit)m(y)k(holds)d (for)i(the)g(jumps)f(to)i(the)f(left.)52 b(It)34 b(follo)m(ws)f(that)i (giv)m(en)382 4570 y(the)k(lo)s(cal)f(parities)f Fq(n)1157 4584 y Fp(j)1219 4570 y Fs(+)25 b Fq(z)1357 4584 y Fp(j)1433 4570 y Fs(=)39 b Fq(s)1586 4584 y Fp(j)1652 4570 y Fs(mo)s(d)30 b(2,)41 b(the)e(probabilit)m(y)d(measures)i Fq(\026)3059 4585 y Fg(s)p Fp(;\025;\022)3249 4570 y Fs(are)382 4709 y(stationary)e(for)f(the)h(dynamics.)55 b(These)35 b(are)h(the)g Fj(e)-5 b(quilibrium)37 b(Gibbs)h(me)-5 b(asur)g(es)44 b Fs(of)382 4848 y(our)e(system.)78 b(F)-8 b(or)43 b(a)g(similar)d (computation)i(in)g(the)h(con)m(text)h(of)f(a)g(simpler)d(one-)382 4986 y(comp)s(onen)m(t)31 b(domain)e(gro)m(wth)i(mo)s(del)e(see)i(also) f([1)q(].)523 5125 y(In)m(v)-5 b(ariance)39 b(under)e(spatial)h (translations)f(is)h(unfortunately)f(lost)i(in)f(this)f(v)m(ery)382 5264 y(general)42 b(setup.)74 b(In)40 b(order)i(to)g(imp)s(ose)e(it,)45 b(w)m(e)d(restrict)f(ourselv)m(es)g(to)i(one)f(of)f(the)382 5403 y(follo)m(wing)29 b(t)m(w)m(o)i(c)m(hoices:)42 b(either)30 b Ft(s)25 b Fs(=)g Ft(0)31 b Fs(or)f Ft(s)c Fs(=)f Ft(1)p Fs(.)1831 5652 y(10)p eop %%Page: 11 11 11 10 bop 382 548 a Fi(3.3)112 b(The)38 b(h)m(ydro)s(dynamic)f (equations)382 746 y Fs(F)-8 b(or)26 b(the)g(rest)g(of)g(this)e (section)i(w)m(e)g(\014x)f(either)g Ft(s)h Fs(=)f Ft(0)h Fs(or)f Ft(s)h Fs(=)f Ft(1)h 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b(u)p Fs(\))123 b(and)d Fq(\022)s Fs(\()p Fq(\032;)15 b Fl(\000)p Fq(u)p Fs(\))26 b(=)f(1)p Fq(=\022)s Fs(\()p Fq(\032;)15 b(u)p Fs(\))p Fq(:)371 b Fs(\(13\))523 3277 y(In)31 b(order)f(to)i(guess)g (the)f(system)g(of)h(h)m(ydro)s(dynamic)d(equations)i(w)m(e)h(ha)m(v)m (e)g(to)g(see)382 3416 y(\014rst)37 b(ho)m(w)h(the)g(in\014nitisimal)33 b(generator)40 b(acts)f(on)e(the)i(conserv)m(ed)f(quan)m(tities.)63 b(An)382 3555 y(easy)31 b(computation)f(sho)m(ws:)670 3749 y Fq(Ln)787 3763 y Fp(j)848 3749 y Fs(=)944 3675 y Fm(\000)986 3749 y Fq(n)1041 3763 y Fp(j)t Fk(\000)p Fn(1)1167 3749 y Fq(r)s Fs(\()p Fq(z)1288 3763 y Fp(j)t Fk(\000)p Fn(1)1415 3749 y Fs(\))20 b Fl(\000)g Fq(n)1616 3763 y Fp(j)1652 3749 y Fq(r)s Fs(\()p Fl(\000)p Fq(z)1844 3763 y Fp(j)1881 3749 y Fs(\))1916 3675 y Fm(\001)1978 3749 y Fl(\000)2069 3675 y Fm(\000)2111 3749 y Fq(n)2166 3763 y Fp(j)2202 3749 y Fq(r)s Fs(\()p Fq(z)2323 3763 y Fp(j)2359 3749 y Fs(\))h Fl(\000)f Fq(n)2561 3763 y Fp(j)t Fn(+1)2687 3749 y Fq(r)s 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5367 y(@)1119 5381 y Fp(t)1149 5367 y Fq(u)20 b Fs(+)g Fq(@)1360 5381 y Fp(x)1404 5293 y Fm(\000)1446 5367 y Fq(\025)p Fs(\()p Fq(\032;)15 b(u)p Fs(\)\()p Fq(\022)s Fs(\()p Fq(\032;)g(u)p Fs(\))22 b(+)e Fq(\022)s Fs(\()p Fq(\032;)15 b(u)p Fs(\))2366 5334 y Fk(\000)p Fn(1)2461 5367 y Fs(\))2496 5293 y Fm(\001)2563 5367 y Fs(=)25 b(0)p Fq(:)3209 5289 y Fs(\(14\))1831 5652 y(11)p eop %%Page: 12 12 12 11 bop 523 548 a Fs(Under)39 b(gro)m(wth)h(conditions)e(on)h(the)h (rate)g(function)e Fq(r)s Fs(\()p Fq(z)t Fs(\),)k(as)e Fq(z)45 b Fl(!)40 b(1)p Fs(,)i(Y)-8 b(au's)382 687 y(`relativ)m(e)32 b(en)m(trop)m(y)h(metho)s(d')f(\(see)h(e.g.)47 b([18)q(],)33 b([5)q(],)g([9)q(]\))f(in)f(principle)e(can)j(b)s(e)g(applied)382 826 y(to)g(our)f(system)h(of)f(in)m(teracting)h(particles,)f(resulting) e(in)i(the)g(v)-5 b(alidit)m(y)30 b(of)i(the)f(ab)s(o)m(v)m(e)382 965 y(system)42 b(of)h(PDEs)f(in)f(the)i(h)m(ydro)s(dynamic)d(limit,)j Fj(as)h(long)h(as)f(the)g(solutions)h(ar)-5 b(e)382 1103 y(smo)g(oth)p Fs(.)523 1242 y(F)d(rom)34 b(the)g(system)f(\(14\))i(w)m (e)f(can)g(deriv)m(e)f(the)h(system)f(\(7\))i(b)m(y)e(taking)h(a)g (second)382 1381 y(limit:)j(W)-8 b(e)29 b(replace)f Fq(\032)p Fs(\()p Fq(t;)15 b(x)p Fs(\))29 b(b)m(y)e Fq(\013)1545 1348 y Fn(2)p Fp(=)p Fn(3)1656 1381 y Fq(\032)p Fs(\()p Fq(\013t;)15 b(\013)1927 1348 y Fn(2)p Fp(=)p Fn(3)2038 1381 y Fq(x)p Fs(\))29 b(and)e Fq(u)p Fs(\()p Fq(t;)15 b(x)p Fs(\))29 b(b)m(y)e Fq(\013)2785 1348 y Fn(1)p Fp(=)p Fn(3)2896 1381 y Fq(u)p Fs(\()p Fq(\013t;)15 b(\013)3172 1348 y Fn(2)p Fp(=)p Fn(3)3283 1381 y Fq(x)p Fs(\))382 1520 y(W)-8 b(e)31 b(note)g(that)g(for)g(small)d(v)-5 b(alues)30 b(of)h(the)f(v)-5 b(ariables)29 b Fq(\032)i Fs(and)e Fq(u)p Fs(,)949 1713 y Fq(\025)p Fs(\()p Fq(\032;)15 b(u)p Fs(\))27 b(=)e Fq(\032)20 b Fs(+)g Fq(o)p Fs(\()p Fq(\032)p Fs(\))p Fq(;)198 b(\022)s Fs(\()p Fq(\032;)15 b(u)p Fs(\))25 b(=)g(1)c(+)f Fq(cu)g Fs(+)g Fq(o)p Fs(\()p Fq(u)p Fs(\))p Fq(:)382 1906 y Fs(where)1234 2145 y Fq(c)26 b Fs(=)1395 1989 y Fm( )1487 2083 y Fq(@)1540 2050 y Fn(2)1579 2083 y Fq(Z)p 1487 2124 162 4 v 1499 2207 a(@)5 b(\022)1598 2181 y Fn(2)1658 2013 y Fm(\014)1658 2067 y(\014)1658 2122 y(\014)1658 2176 y(\014)1688 2235 y Fp(\025)p Fn(=0)p Fp(;\022)r Fn(=1)1968 1989 y Fm(!)2040 2012 y Fk(\000)p Fn(1)2160 2145 y Fl(2)25 b Fs(\(0)p Fq(;)15 b Fl(1)p Fs(\))p Fq(:)382 2389 y Fs(Letting)35 b(no)m(w)f Fq(\013)f Fl(!)f Fs(0,)k(w)m(e)f(arriv)m(e)f(at)i(\(7\).)54 b(W)-8 b(e)36 b(should)c(emphasize)i(here)h(that)g(this)382 2528 y(scaling)i(limit)f(do)s(es)i(not)h(dep)s(end)d(m)m(uc)m(h)i(on)h (the)f(details)f(of)i(microscopic)e(system.)382 2667 y(Also,)30 b(from)g(an)m(y)h(conserv)-5 b(ation)30 b(la)m(w)g(of)h(the) g(form)1389 2751 y Fm(\()1503 2829 y Fq(@)1551 2843 y Fp(t)1581 2829 y Fq(\032)21 b Fs(+)f Fq(@)1788 2843 y Fp(x)1832 2829 y Fq(J)9 b Fs(\()p Fq(\032;)15 b(u)p Fs(\))26 b(=)f(0)1503 2983 y Fq(@)1551 2997 y Fp(t)1581 2983 y Fq(u)c Fs(+)f Fq(@)1793 2997 y Fp(x)1837 2983 y Fq(K)7 b Fs(\()p Fq(\032;)15 b(u)p 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(and)e(righ)m(t)h(\(column\))f(eigen)m(v)m(ectors)j(will)40 b(b)s(e)382 687 y(denoted)30 b(b)m(y)h Fq(l)h Fs(and)e Fq(r)s Fs(,)g(resp)s(ectiv)m(ely)-8 b(,)30 b Fq(m)g Fs(and)g Fq(s)p Fs(.)40 b(That)30 b(is:)965 916 y Fq(l)22 b Fl(\001)e(r)p Fq(J)34 b Fs(=)25 b Fq(\025l)r(;)869 b Fl(r)p Fq(J)29 b Fl(\001)20 b Fq(r)28 b Fs(=)d Fq(\025r)m(;)437 b Fs(\(20\))965 1080 y Fq(m)20 b Fl(\001)g(r)p Fq(J)34 b Fs(=)25 b Fq(\026m;)775 b Fl(r)p Fq(J)29 b Fl(\001)20 b Fq(s)25 b Fs(=)g Fq(\026s:)421 b Fs(\(21\))523 1309 y(F)-8 b(or)31 b(our)f(system)h(w)m(e)f(\014nd:) 1521 1561 y Fl(r)p Fq(J)k Fs(=)1777 1433 y Fm(\022)1886 1504 y Fq(u)83 b(\032)1889 1617 y Fs(1)k(0)2109 1433 y Fm(\023)2191 1561 y Fq(;)382 1813 y Fs(and)712 2042 y Fq(\025)26 b Fs(=)f(+)968 1981 y(1)p 968 2021 46 4 v 968 2105 a(2)1023 2042 y(\()1058 1957 y Fm(p)p 1149 1957 296 4 v 85 x Fq(u)1201 2016 y Fn(2)1261 2042 y Fs(+)19 b(4)p Fq(\032)i Fs(+)f Fq(u)p Fs(\))p Fq(;)277 b(l)28 b Fs(=)d(\()p Fq(\025;)15 b(\032)p Fs(\))p Fq(;)241 b(r)27 b Fs(=)e(\()p Fq(\025;)15 b Fs(1\))2943 2005 y Fp(T)3000 2042 y Fq(;)184 b Fs(\(22\))712 2258 y Fq(\026)25 b Fs(=)g Fl(\000)969 2196 y Fs(1)p 969 2237 46 4 v 969 2320 a(2)1024 2258 y(\()1059 2173 y Fm(p)p 1151 2173 296 4 v 1151 2258 a Fq(u)1203 2232 y Fn(2)1262 2258 y Fs(+)20 b(4)p Fq(\032)h Fl(\000)f Fq(u)p Fs(\))p Fq(;)226 b(m)25 b Fs(=)g(\()p Fq(\026;)15 b(\032)p Fs(\))p Fq(;)242 b(s)25 b Fs(=)g(\()p Fq(\026;)15 b Fs(1\))2949 2220 y Fp(T)3005 2258 y Fq(:)179 b Fs(\(23\))382 2487 y(Note)32 b(that)f Fq(l)22 b Fl(\001)e Fq(s)25 b Fs(=)g Fq(m)20 b Fl(\001)g Fq(r)28 b Fs(=)d(0,)31 b(as)g(it)f(should)e(b)s(e.)523 2626 y(W)-8 b(e)32 b(conclude)d(that)i (for)f(our)g(system,)1113 2855 y Fl(D)1183 2870 y Fn(h)n(yp)1326 2855 y Fs(=)25 b Fl(f)p Fs(\()p Fq(\032;)15 b(u)p Fs(\))27 b Fl(2)e Fo(R)k Fl(\002)20 b Fo(R)34 b Fs(:)25 b Fq(u)2159 2818 y Fn(2)2219 2855 y Fs(+)20 b(4)p Fq(\032)25 b(>)g Fs(0)p Fl(g)p Fq(:)382 3084 y Fs(Note)32 b(that)f(in)e(the)h(ph)m (ysically)e(relev)-5 b(an)m(t)31 b(domain)e(with)g(non-negativ)m(e)j (densities)1235 3314 y Fl(D)1305 3329 y Fn(ph)1412 3314 y Fs(:=)26 b Fl(f)p Fs(\()p Fq(\032;)15 b(u)p Fs(\))27 b Fl(2)d Fo(R)29 b Fl(\002)20 b Fo(R)34 b Fs(:)26 b Fq(\032)f Fl(\025)g Fs(0)p Fl(g)p Fq(;)382 3543 y Fs(there)34 b(is)f(one)h (single)f(p)s(oin)m(t)f(where)i(strict)g(h)m(yp)s(erb)s(olicit)m(y)d (is)i(lost,)i(namely)e(\()p Fq(\032;)15 b(u)p Fs(\))32 b(=)382 3682 y(\(0)p Fq(;)15 b Fs(0\).)41 b(On)24 b(the)i(other)f (hand,)h(w)m(e)g(found)e(that)h(the)h(system)f(is)g(still)e(h)m(yp)s (erb)s(olic)f(in)i(the)382 3821 y(ph)m(ysically)34 b(meaningless)g (domain)h Fl(D)1715 3836 y Fn(h)n(yp)1856 3821 y Fl(n)25 b(D)1996 3836 y Fn(ph)2112 3821 y Fs(=)34 b Fl(f)p Fs(\()p Fq(\032;)15 b(u)p Fs(\))36 b Fl(2)e(D)2672 3836 y Fn(h)n(yp)2824 3821 y Fs(:)h Fq(\032)f(<)g Fs(0)p Fl(g)h(6)p Fs(=)f Fl(;)p Fs(.)382 3959 y(A)m(t)28 b(the)f(momen)m(t)h(nothing)e(seems)h (to)h(prev)m(en)m(t)g(solutions)d(to)j(\015o)m(w)f(in)m(to)h(this)e (domain.)382 4098 y(Later)36 b(w)m(e)g(shall)e(see)i(that)g(Lax's)g (maxim)m(um)e(principle)e(\(v)-5 b(alid)34 b(for)i(stable)f(en)m(trop)m (y)382 4237 y(solutions\))29 b(tak)m(es)j(care)f(of)g(this)e(problem.) 382 4480 y Fi(4.3)112 b(Riemann)37 b(in)m(v)-6 b(arian)m(ts,)36 b(c)m(haracteristics)382 4678 y Fs(In)48 b(the)i(generic)f(t)m(w)m (o-comp)s(onen)m(t)i(case,)56 b(w)m(e)49 b(are)h(lo)s(oking)e(for)h (scalar)g(functions)382 4817 y Fl(D)452 4832 y Fn(h)n(yp)614 4817 y Fl(3)43 b Fq(v)48 b Fl(7!)43 b Fq(w)r Fs(\()p Fq(v)s Fs(\))j Fl(2)d Fo(R)51 b Fs(and)41 b(space-time)h(tra)5 b(jectories)42 b Fo(R)2584 4831 y Fn(+)2693 4817 y Fl(3)h Fq(t)h Fl(7!)g Fq(\030)t Fs(\()p Fq(t)p Fs(\))g Fl(2)g Fo(R)382 4956 y Fs(suc)m(h)35 b(that)h(for)g(smo)s(oth)f(solutions)f (of)i(\(15\))h(\(or,)g(equiv)-5 b(alen)m(tly)d(,)36 b(of)g(\(17\)\))h Fq(w)h Fs(is)d(con-)382 5095 y(serv)m(ed)30 b(along)h(the)f(tra)5 b(jectory)32 b Fq(\030)t Fs(\()p Fq(t)p Fs(\),)f(i.e.)1530 5279 y Fq(d)p 1513 5320 81 4 v 1513 5403 a(dt)1603 5341 y(w)1670 5267 y Fm(\000)1712 5341 y Fq(v)s Fs(\()p Fq(t;)15 b(\030)t Fs(\()p Fq(t)p Fs(\))2014 5267 y Fm(\001)2083 5341 y Fs(=)25 b(0)p Fq(:)1831 5652 y Fs(14)p eop %%Page: 15 15 15 14 bop 382 548 a Fs(Using)29 b(\(17\))j(w)m(e)f(\014nd:)1438 738 y Fq(d\030)p 1438 779 92 4 v 1444 862 a(dt)1564 800 y Fs(=)1670 738 y(\()p Fl(r)p Fq(w)23 b Fl(\001)d(r)p Fq(J)9 b Fs(\))21 b Fl(\001)f Fq(@)2198 752 y Fp(x)2242 738 y Fq(v)p 1670 779 620 4 v 1806 862 a Fl(r)p Fq(w)i Fl(\001)f Fq(@)2063 876 y Fp(x)2107 862 y Fq(v)2300 800 y(:)884 b Fs(\(24\))523 1044 y(In)30 b(order)f(to)j(solv)m(e)e(\(24\),) i Fl(r)p Fq(w)h Fs(m)m(ust)d(b)s(e)g(a)g(left)g(eigen)m(v)m(ector)j(of) d(the)h(matrix)f Fl(r)p Fq(J)9 b Fs(.)382 1183 y(It)29 b(follo)m(ws)f(that)h(this)e(relation)h(admits)g(t)m(w)m(o)i (solutions:)38 b(one)29 b(for)g(eac)m(h)h(eigen)m(v)-5 b(alue)28 b(of)382 1322 y Fl(r)p Fq(J)9 b Fs(.)40 b(W)-8 b(e)29 b(denote)f(the)g(t)m(w)m(o)h(solutions)d(b)m(y)i Fq(w)j Fs(\(corresp)s(onding)26 b(to)i(the)g(eigen)m(v)-5 b(alue)28 b Fq(\025)p Fs(\),)382 1460 y(resp)s(ectiv)m(ely)-8 b(,)35 b(b)m(y)f Fq(z)39 b Fs(\(corresp)s(onding)32 b(to)j(the)g(eigen) m(v)-5 b(alue)34 b Fq(\026)p Fs(\).)52 b(The)34 b(gradien)m(ts)g Fl(r)p Fq(w)r Fs(,)382 1599 y(resp)s(ectiv)m(ely)28 b Fl(r)p Fq(z)t Fs(,)i(are)f(parallel)e(to)j(the)f(ro)m(w)h(v)m(ectors)g Fq(l)r Fs(,)f(resp)s(ectiv)m(ely)g Fq(m)p Fs(,)g(de\014ned)f(in)382 1738 y(\(20\),)k(resp)s(ectiv)m(ely)e(\(21\).)42 b(In)30 b(other)g(w)m(ords,)1446 1985 y Fl(r)p Fq(w)23 b Fl(\001)d Fq(s)25 b Fs(=)g(0)p Fq(;)1991 1924 y(d\030)p 1991 1964 92 4 v 1997 2048 a(dt)2117 1985 y Fs(=)g Fq(\025;)1467 2205 y Fl(r)p Fq(z)f Fl(\001)c Fq(r)28 b Fs(=)d(0)p Fq(;)1991 2144 y(d\030)p 1991 2184 V 1997 2268 a(dt)2117 2205 y Fs(=)g Fq(\026:)382 2436 y Fs(These)20 b(equations,)i(of)f(course,)i (do)d(not)h(determine)e(uniquely)f(the)i(functions)f Fq(w)r Fs(\()p Fq(v)s Fs(\))j(and)382 2574 y Fq(z)t Fs(\()p Fq(v)s Fs(\).)48 b(Giv)m(en)32 b(t)m(w)m(o)i(smo)s(oth,)f(monotone)g (maps)f Fq(f)5 b(;)15 b(g)32 b Fs(:)c Fo(R)38 b Fl(!)28 b Fo(R)s Fs(,)39 b(the)32 b(transformation)401 2713 y(^)-64 b Fq(w)37 b Fs(:=)d Fq(f)10 b Fs(\()p Fq(w)r Fs(\),)44 b(^)-51 b Fq(z)39 b Fs(:=)34 b Fq(g)s Fs(\()p Fq(z)t Fs(\))k(lea)m(v)m(es)f(the)f(ab)s(o)m(v)m(e)h(equations)e(in)m(v)-5 b(arian)m(t.)57 b(The)35 b(functions)382 2852 y Fq(w)f Fs(and)c Fq(z)35 b Fs(are)d(called)e(the)i Fj(R)n(iemann)i(invariants)p Fs(,)f(or)e Fj(char)-5 b(acteristic)35 b(c)-5 b(o)g(or)g(dinates)34 b Fs(of)382 2991 y(the)c(problem.)523 3130 y(In)d(our)h(case)h(the)g (most)f(con)m(v)m(enien)m(t)i(c)m(hoice)f(of)f(the)g(Riemann)f(in)m(v) -5 b(arian)m(ts)28 b Fq(w)i Fs(and)382 3269 y Fq(z)k Fs(is)c(the)g(follo)m(wing:)40 b(let)1024 3499 y Fl(D)1094 3513 y Fp(w)1176 3499 y Fs(:=)25 b Fl(f)p Fs(\()p Fq(\032;)15 b(u)p Fs(\))27 b Fl(2)e(D)1734 3514 y Fn(h)n(yp)1877 3499 y Fs(:)1927 3414 y Fm(p)p 2018 3414 296 4 v 85 x Fq(u)2070 3473 y Fn(2)2130 3499 y Fs(+)20 b(4)p Fq(\032)g Fl(\000)g Fq(u)26 b Fl(\025)f Fs(0)p Fl(g)p Fq(;)1024 3663 y Fl(D)1094 3677 y Fp(z)1159 3663 y Fs(:=)g Fl(f)p Fs(\()p Fq(\032;)15 b(u)p Fs(\))27 b Fl(2)e(D)1717 3678 y Fn(h)n(yp)1860 3663 y Fs(:)1910 3578 y Fm(p)p 2001 3578 V 85 x Fq(u)2053 3637 y Fn(2)2113 3663 y Fs(+)20 b(4)p Fq(\032)h Fs(+)f Fq(u)25 b Fl(\025)g Fs(0)p Fl(g)p Fq(;)382 3893 y Fs(and)30 b(de\014ne)f Fq(w)f Fs(:)e Fl(D)1035 3907 y Fp(w)1117 3893 y Fl(!)f Fo(R)r Fs(,)37 b Fq(z)29 b Fs(:)d Fl(D)1546 3907 y Fp(z)1611 3893 y Fl(!)f Fo(R)39 b Fs(b)m(y)30 b(the)h(form)m(ulas:)935 4140 y Fq(w)r Fs(\()p Fq(\032;)15 b(u)p Fs(\))26 b(=)f Fl(\000)1404 4018 y Fm(q)p 1495 4018 550 4 v 37 x(p)p 1586 4055 296 4 v 85 x Fq(u)1638 4114 y Fn(2)1698 4140 y Fs(+)20 b(4)p Fq(\032)g Fl(\000)g Fq(u)2059 4039 y Fm(\020)2114 4055 y(p)p 2205 4055 V 85 x Fq(u)2257 4114 y Fn(2)2316 4140 y Fs(+)g(4)p Fq(\032)h Fs(+)f(2)p Fq(u)2708 4039 y Fm(\021)2778 4140 y Fq(;)935 4363 y(z)t Fs(\()p Fq(\032;)15 b(u)p Fs(\))26 b(=)f Fl(\000)1383 4240 y Fm(q)p 1474 4240 550 4 v 38 x(p)p 1565 4278 296 4 v 85 x Fq(u)1617 4336 y Fn(2)1676 4363 y Fs(+)20 b(4)p Fq(\032)h Fs(+)f Fq(u)2038 4262 y Fm(\020)2092 4278 y(p)p 2183 4278 V 85 x Fq(u)2235 4336 y Fn(2)2295 4363 y Fs(+)g(4)p Fq(\032)h Fl(\000)e Fs(2)p Fq(u)2686 4262 y Fm(\021)2756 4363 y Fq(:)523 4593 y Fs(Note)32 b(that)f Fl(D)1010 4607 y Fp(w)1087 4593 y Fl(\\)20 b(D)1238 4607 y Fp(z)1304 4593 y Fs(=)25 b Fl(D)1470 4608 y Fn(ph)1553 4593 y Fs(,)31 b(so)g(that)g(b)s(oth)f(Riemann)f(in)m(v)-5 b(arian)m(ts)30 b(are)h(de\014ned)382 4732 y(in)e(the)i(ph)m(ysically)d(relev)-5 b(an)m(t)30 b(sub)s(domain.)523 4871 y(It)42 b(is)f(straigh)m(tforw)m (ard)h(to)g(c)m(hec)m(k)i(that)e(b)s(oth)f(Riemann)g(in)m(v)-5 b(arian)m(ts)41 b Fq(w)j Fs(and)d Fq(z)382 5010 y Fs(de\014ned)e(ab)s (o)m(v)m(e)j(are)f Fj(c)-5 b(onvex)52 b Fs(functions)39 b(of)h(the)h(v)-5 b(ariables)39 b(\()p Fq(\032;)15 b(u)p Fs(\).)72 b(This)39 b(fact)i(will)382 5148 y(ha)m(v)m(e)31 b(crucial)f(imp)s(ortance)f(in)g(later)i(analysis.)1831 5652 y(15)p eop %%Page: 16 16 16 15 bop 457 2738 a currentpoint currentpoint translate 0.70546 0.70546 scale neg exch neg exch translate 457 2738 a @beginspecial 63 @llx 201 @lly 546 @urx 589 @ury 4830 @rwi @setspecial %%BeginDocument: zw.eps % MathWorks dictionary /MathWorks 160 dict begin % definition operators /bdef {bind def} bind def /ldef {load def} bind def /xdef {exch def} bdef /xstore {exch store} bdef % operator abbreviations /c /clip ldef /cc /concat ldef /cp /closepath ldef /gr /grestore ldef /gs /gsave ldef /mt /moveto ldef /np /newpath ldef /cm /currentmatrix ldef /sm /setmatrix ldef /rc {rectclip} bdef /rf {rectfill} bdef /rm /rmoveto ldef /rl /rlineto ldef /s /show ldef /sc {setcmykcolor} bdef /sr /setrgbcolor ldef /sg /setgray ldef /w /setlinewidth ldef /j /setlinejoin ldef /cap /setlinecap ldef % page state control /pgsv () def /bpage {/pgsv save def} bdef /epage {pgsv restore} bdef /bplot /gsave ldef /eplot {stroke grestore} bdef % orientation switch /portraitMode 0 def /landscapeMode 1 def % coordinate system mappings /dpi2point 0 def % font control /FontSize 0 def /FMS { /FontSize xstore %save size off stack findfont [FontSize 0 0 FontSize neg 0 0] makefont setfont }bdef /reencode { exch dup where {pop load} {pop StandardEncoding} ifelse exch dup 3 1 roll findfont dup length dict begin { 1 index /FID ne {def}{pop pop} ifelse } forall /Encoding exch def currentdict end definefont pop } bdef /isroman { findfont /CharStrings get /Agrave known } bdef /FMSR { 3 1 roll 1 index dup isroman {reencode} {pop pop} ifelse exch FMS } bdef /csm { 1 dpi2point div -1 dpi2point div scale neg translate landscapeMode eq {90 rotate} if } bdef % line types: solid, dotted, dashed, dotdash /SO { [] 0 setdash } bdef /DO { [.5 dpi2point mul 4 dpi2point mul] 0 setdash } bdef /DA { [6 dpi2point mul] 0 setdash } bdef /DD { [.5 dpi2point mul 4 dpi2point mul 6 dpi2point mul 4 dpi2point mul] 0 setdash } bdef % macros for lines and objects /L { lineto stroke } bdef /MP { 3 1 roll moveto 1 sub {rlineto} repeat } bdef /AP { {rlineto} repeat } bdef /PP { closepath eofill } bdef /DP { closepath stroke } bdef /MR { 4 -2 roll moveto dup 0 exch rlineto exch 0 rlineto neg 0 exch rlineto closepath } bdef /FR { MR stroke } bdef /PR { MR fill } bdef /L1i { { currentfile picstr readhexstring pop } image } bdef /tMatrix matrix def /MakeOval { newpath tMatrix currentmatrix pop translate scale 0 0 1 0 360 arc tMatrix setmatrix } bdef /FO { MakeOval stroke } bdef /PO { MakeOval fill } bdef /PD { currentlinecap 1 setlinecap 3 1 roll 2 copy moveto lineto stroke setlinecap } bdef /FA { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arc tMatrix setmatrix stroke } bdef /PA { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arc closepath tMatrix setmatrix fill } bdef /FAn { newpath tMatrix currentmatrix pop translate scale 0 0 1 5 -2 roll arcn tMatrix setmatrix stroke } bdef /PAn { newpath tMatrix currentmatrix pop translate 0 0 moveto scale 0 0 1 5 -2 roll arcn closepath tMatrix setmatrix fill } bdef /vradius 0 def /hradius 0 def /lry 0 def /lrx 0 def /uly 0 def /ulx 0 def /rad 0 def /MRR { /vradius xdef /hradius xdef /lry xdef /lrx xdef /uly xdef /ulx xdef newpath tMatrix currentmatrix pop ulx hradius add uly vradius add translate hradius vradius scale 0 0 1 180 270 arc tMatrix setmatrix lrx hradius sub uly vradius add translate hradius vradius scale 0 0 1 270 360 arc tMatrix setmatrix lrx hradius sub lry vradius sub translate hradius vradius scale 0 0 1 0 90 arc tMatrix setmatrix ulx hradius add lry vradius sub translate hradius vradius scale 0 0 1 90 180 arc tMatrix setmatrix closepath } bdef /FRR { MRR stroke } bdef /PRR { MRR fill } bdef /MlrRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lry uly sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 90 270 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 270 90 arc tMatrix setmatrix closepath } bdef /FlrRR { MlrRR stroke } bdef /PlrRR { MlrRR fill } bdef /MtbRR { /lry xdef /lrx xdef /uly xdef /ulx xdef /rad lrx ulx sub 2 div def newpath tMatrix currentmatrix pop ulx rad add uly rad add translate rad rad scale 0 0 1 180 360 arc tMatrix setmatrix lrx rad sub lry rad sub translate rad rad scale 0 0 1 0 180 arc tMatrix setmatrix closepath } bdef /FtbRR { MtbRR stroke } bdef /PtbRR { MtbRR fill } bdef currentdict end def MathWorks begin 0 cap end MathWorks begin bpage bplot /dpi2point 12 def portraitMode 0216 7344 csm 547 273 5794 4658 MR c np 92 dict begin %Colortable dictionary /c0 { 0 0 0 sr} bdef /c1 { 1 1 1 sr} bdef /c2 { 1 0 0 sr} bdef /c3 { 0 1 0 sr} bdef /c4 { 0 0 1 sr} bdef /c5 { 1 1 0 sr} bdef /c6 { 1 0 1 sr} bdef /c7 { 0 1 1 sr} bdef c0 1 j 1 sg 0 0 6918 5185 PR 6 w 0 4224 5360 0 0 -4224 899 4613 4 MP PP -5360 0 0 4224 5360 0 0 -4224 899 4613 5 MP stroke 4 w DO SO 6 w 0 sg 899 4613 mt 6259 4613 L 899 389 mt 6259 389 L 899 4613 mt 899 389 L 6259 4613 mt 6259 389 L 899 4613 mt 6259 4613 L 899 4613 mt 899 389 L 899 4613 mt 899 4559 L 899 389 mt 899 443 L /Helvetica /ISOLatin1Encoding 120 FMSR 796 4759 mt (-4) s 1569 4613 mt 1569 4559 L 1569 389 mt 1569 443 L 1466 4759 mt (-3) s 2239 4613 mt 2239 4559 L 2239 389 mt 2239 443 L 2136 4759 mt (-2) s 2909 4613 mt 2909 4559 L 2909 389 mt 2909 443 L 2806 4759 mt (-1) s 3579 4613 mt 3579 4559 L 3579 389 mt 3579 443 L 3546 4759 mt (0) s 4249 4613 mt 4249 4559 L 4249 389 mt 4249 443 L 4216 4759 mt (1) s 4919 4613 mt 4919 4559 L 4919 389 mt 4919 443 L 4886 4759 mt (2) s 5589 4613 mt 5589 4559 L 5589 389 mt 5589 443 L 5556 4759 mt (3) s 6259 4613 mt 6259 4559 L 6259 389 mt 6259 443 L 6226 4759 mt (4) s 899 4613 mt 953 4613 L 6259 4613 mt 6205 4613 L 728 4657 mt (-4) s 899 4085 mt 953 4085 L 6259 4085 mt 6205 4085 L 728 4129 mt (-3) s 899 3557 mt 953 3557 L 6259 3557 mt 6205 3557 L 728 3601 mt (-2) s 899 3029 mt 953 3029 L 6259 3029 mt 6205 3029 L 728 3073 mt (-1) s 899 2501 mt 953 2501 L 6259 2501 mt 6205 2501 L 798 2545 mt (0) s 899 1973 mt 953 1973 L 6259 1973 mt 6205 1973 L 798 2017 mt (1) s 899 1445 mt 953 1445 L 6259 1445 mt 6205 1445 L 798 1489 mt (2) s 899 917 mt 953 917 L 6259 917 mt 6205 917 L 798 961 mt (3) s 899 389 mt 953 389 L 6259 389 mt 6205 389 L 798 433 mt (4) s 899 4613 mt 6259 4613 L 899 389 mt 6259 389 L 899 4613 mt 899 389 L 6259 4613 mt 6259 389 L gs 899 389 5361 4225 MR c np 2564 389 mt 2574 413 L 2574 413 mt 2587 442 L 2587 442 mt 2611 495 L 2611 495 mt 2635 547 L 2635 547 mt 2641 561 L 2641 561 mt 2659 600 L 2659 600 mt 2684 653 L 2684 653 mt 2708 702 L 2708 702 mt 2710 706 L 2710 706 mt 2736 759 L 2736 759 mt 2762 811 L 2762 811 mt 2775 836 L 2775 836 mt 2790 864 L 2790 864 mt 2817 917 L 2817 917 mt 2842 963 L 2842 963 mt 2845 970 L 2845 970 mt 2875 1023 L 2875 1023 mt 2904 1075 L 2904 1075 mt 2909 1084 L 2909 1084 mt 2935 1128 L 2935 1128 mt 2966 1181 L 2966 1181 mt 2976 1197 L 2976 1197 mt 2999 1234 L 2999 1234 mt 3004 1243 L 3092 1378 mt 3102 1392 L 3102 1392 mt 3110 1404 L 3110 1404 mt 3139 1445 L 3139 1445 mt 3177 1497 L 3177 1497 mt 3177 1498 L 3177 1498 mt 3217 1551 L 3217 1551 mt 3244 1585 L 3244 1585 mt 3259 1603 L 3259 1603 mt 3303 1656 L 3303 1656 mt 3311 1665 L 3311 1665 mt 3349 1709 L 3349 1709 mt 3378 1740 L 3378 1740 mt 3398 1762 L 3398 1762 mt 3445 1809 L 3445 1809 mt 3451 1815 L 3451 1815 mt 3507 1867 L 3507 1867 mt 3512 1872 L 3512 1872 mt 3567 1920 L 3567 1920 mt 3579 1930 L 3579 1930 mt 3634 1973 L 3634 1973 mt 3646 1982 L 3646 1982 mt 3708 2026 L 3708 2026 mt 3713 2029 L 3713 2029 mt 3780 2072 L 3780 2072 mt 3791 2079 L 3791 2079 mt 3847 2110 L 3847 2110 mt 3888 2131 L 3888 2131 mt 3914 2144 L 3914 2144 mt 3981 2174 L 3981 2174 mt 4004 2184 L 4004 2184 mt 4048 2201 L 4048 2201 mt 4115 2225 L 4115 2225 mt 4151 2237 L 4151 2237 mt 4182 2246 L 4182 2246 mt 4249 2265 L 4249 2265 mt 4316 2282 L 4316 2282 mt 4350 2290 L 4350 2290 mt 4383 2297 L 4383 2297 mt 4450 2309 L 4450 2309 mt 4517 2321 L 4517 2321 mt 4584 2332 L 4584 2332 mt 4651 2342 L 4651 2342 mt 4657 2343 L 4657 2343 mt 4718 2350 L 4718 2350 mt 4785 2357 L 4785 2357 mt 4852 2364 L 4852 2364 mt 4919 2370 L 4919 2370 mt 4986 2376 L 5147 2389 mt 5187 2392 L 5187 2392 mt 5236 2395 L 5236 2395 mt 5254 2396 L 5254 2396 mt 5321 2400 L 5321 2400 mt 5388 2403 L 5388 2403 mt 5455 2406 L 5455 2406 mt 5522 2409 L 5522 2409 mt 5589 2412 L 5589 2412 mt 5656 2414 L 5656 2414 mt 5723 2417 L 5723 2417 mt 5790 2419 L 5790 2419 mt 5857 2421 L 5857 2421 mt 5924 2424 L 5924 2424 mt 5991 2426 L 5991 2426 mt 6058 2428 L 6058 2428 mt 6125 2430 L 6125 2430 mt 6192 2432 L 6192 2432 mt 6259 2433 L 2032 389 mt 2038 406 L 2038 406 mt 2051 442 L 2051 442 mt 2063 473 L 2091 548 mt 2105 584 L 2105 584 mt 2111 600 L 2111 600 mt 2132 653 L 2132 653 mt 2153 706 L 2153 706 mt 2172 755 L 2172 755 mt 2174 759 L 2174 759 mt 2195 811 L 2195 811 mt 2217 864 L 2217 864 mt 2239 917 L 2239 917 mt 2239 917 L 2239 917 mt 2262 970 L 2262 970 mt 2285 1023 L 2285 1023 mt 2306 1071 L 2306 1071 mt 2308 1075 L 2308 1075 mt 2332 1128 L 2332 1128 mt 2356 1181 L 2356 1181 mt 2373 1218 L 2373 1218 mt 2381 1234 L 2381 1234 mt 2406 1287 L 2406 1287 mt 2432 1339 L 2432 1339 mt 2440 1357 L 2440 1357 mt 2458 1392 L 2458 1392 mt 2485 1445 L 2485 1445 mt 2507 1487 L 2507 1487 mt 2513 1498 L 2513 1498 mt 2541 1551 L 2541 1551 mt 2570 1603 L 2570 1603 mt 2574 1610 L 2574 1610 mt 2601 1656 L 2601 1656 mt 2632 1709 L 2632 1709 mt 2641 1725 L 2641 1725 mt 2664 1762 L 2664 1762 mt 2697 1815 L 2697 1815 mt 2708 1832 L 2708 1832 mt 2732 1867 L 2732 1867 mt 2768 1920 L 2768 1920 mt 2775 1931 L 2775 1931 mt 2806 1973 L 2806 1973 mt 2842 2022 L 2842 2022 mt 2845 2026 L 2845 2026 mt 2887 2079 L 2887 2079 mt 2909 2105 L 2909 2105 mt 2932 2131 L 2932 2131 mt 2976 2180 L 2976 2180 mt 2980 2184 L 2980 2184 mt 3032 2237 L 3032 2237 mt 3043 2247 L 3043 2247 mt 3090 2290 L 3090 2290 mt 3110 2307 L 3110 2307 mt 3156 2343 L 3156 2343 mt 3177 2358 L 3177 2358 mt 3233 2395 L 3233 2395 mt 3244 2402 L 3244 2402 mt 3311 2437 L 3311 2437 mt 3335 2448 L 3335 2448 mt 3336 2449 L 3413 2474 mt 3445 2483 L 3445 2483 mt 3512 2495 L 3512 2495 mt 3579 2501 L 3579 2501 mt 3646 2501 L 3646 2501 mt 3713 2501 L 3713 2501 mt 3780 2501 L 3780 2501 mt 3847 2501 L 3847 2501 mt 3914 2501 L 3914 2501 mt 3981 2501 L 3981 2501 mt 4048 2501 L 4048 2501 mt 4115 2501 L 4115 2501 mt 4182 2501 L 4182 2501 mt 4249 2501 L 4249 2501 mt 4316 2501 L 4316 2501 mt 4383 2501 L 4383 2501 mt 4450 2501 L 4450 2501 mt 4517 2501 L 4517 2501 mt 4584 2501 L 4584 2501 mt 4651 2501 L 4651 2501 mt 4718 2501 L 4718 2501 mt 4785 2501 L 4785 2501 mt 4852 2501 L 4852 2501 mt 4919 2501 L 4919 2501 mt 4986 2501 L 4986 2501 mt 5053 2501 L 5053 2501 mt 5120 2501 L 5120 2501 mt 5187 2501 L 5187 2501 mt 5254 2501 L 5254 2501 mt 5321 2501 L 5321 2501 mt 5388 2501 L 5388 2501 mt 5455 2501 L 5455 2501 mt 5522 2501 L 5522 2501 mt 5589 2501 L 5589 2501 mt 5656 2501 L 5656 2501 mt 5723 2501 L 5723 2501 mt 5731 2501 L 5812 2501 mt 5857 2501 L 5857 2501 mt 5924 2501 L 5924 2501 mt 5991 2501 L 5991 2501 mt 6058 2501 L 6058 2501 mt 6125 2501 L 6125 2501 mt 6192 2501 L 6192 2501 mt 6259 2501 L 4503 4613 mt 4470 4560 L 4470 4560 mt 4450 4530 L 4450 4530 mt 4435 4507 L 4435 4507 mt 4401 4455 L 4401 4455 mt 4383 4428 L 4383 4428 mt 4376 4418 L 4331 4351 mt 4330 4349 L 4330 4349 mt 4316 4328 L 4316 4328 mt 4294 4296 L 4294 4296 mt 4257 4243 L 4257 4243 mt 4249 4232 L 4249 4232 mt 4220 4191 L 4220 4191 mt 4182 4138 L 4182 4138 mt 4182 4137 L 4182 4137 mt 4144 4085 L 4144 4085 mt 4115 4046 L 4115 4046 mt 4105 4032 L 4105 4032 mt 4065 3979 L 4065 3979 mt 4048 3957 L 4048 3957 mt 4025 3927 L 4025 3927 mt 3984 3874 L 3984 3874 mt 3981 3870 L 3981 3870 mt 3942 3821 L 3942 3821 mt 3914 3787 L 3914 3787 mt 3899 3768 L 3899 3768 mt 3855 3715 L 3855 3715 mt 3847 3706 L 3847 3706 mt 3811 3663 L 3811 3663 mt 3780 3627 L 3780 3627 mt 3765 3610 L 3765 3610 mt 3718 3557 L 3718 3557 mt 3713 3551 L 3713 3551 mt 3670 3504 L 3670 3504 mt 3646 3478 L 3646 3478 mt 3621 3451 L 3621 3451 mt 3579 3408 L 3579 3408 mt 3570 3399 L 3570 3399 mt 3518 3346 L 3518 3346 mt 3512 3340 L 3512 3340 mt 3464 3293 L 3464 3293 mt 3445 3275 L 3445 3275 mt 3409 3240 L 3409 3240 mt 3378 3212 L 3378 3212 mt 3351 3187 L 3351 3187 mt 3311 3152 L 3311 3152 mt 3291 3135 L 3291 3135 mt 3244 3095 L 3244 3095 mt 3229 3082 L 3229 3082 mt 3177 3040 L 3177 3040 mt 3163 3029 L 3163 3029 mt 3110 2988 L 3110 2988 mt 3094 2976 L 3094 2976 mt 3043 2939 L 3043 2939 mt 3022 2923 L 3022 2923 mt 2976 2892 L 2976 2892 mt 2944 2871 L 2944 2871 mt 2909 2848 L 2909 2848 mt 2861 2818 L 2861 2818 mt 2842 2807 L 2842 2807 mt 2775 2768 L 2775 2768 mt 2771 2765 L 2771 2765 mt 2708 2731 L 2708 2731 mt 2704 2729 L 2639 2682 mt 2614 2659 L 2614 2659 mt 2574 2620 L 2574 2620 mt 2562 2607 L 2562 2607 mt 2515 2554 L 2515 2554 mt 2507 2544 L 2507 2544 mt 2473 2501 L 2473 2501 mt 2440 2457 L 2440 2457 mt 2434 2448 L 2434 2448 mt 2397 2395 L 2397 2395 mt 2373 2360 L 2373 2360 mt 2362 2343 L 2362 2343 mt 2329 2290 L 2329 2290 mt 2306 2252 L 2306 2252 mt 2297 2237 L 2297 2237 mt 2266 2184 L 2266 2184 mt 2239 2136 L 2239 2136 mt 2237 2131 L 2237 2131 mt 2208 2079 L 2208 2079 mt 2180 2026 L 2180 2026 mt 2172 2010 L 2172 2010 mt 2154 1973 L 2154 1973 mt 2127 1920 L 2127 1920 mt 2105 1874 L 2105 1874 mt 2102 1867 L 2102 1867 mt 2077 1815 L 2077 1815 mt 2053 1762 L 2053 1762 mt 2038 1730 L 2038 1730 mt 2029 1709 L 2029 1709 mt 2005 1656 L 2005 1656 mt 1982 1603 L 1982 1603 mt 1971 1577 L 1971 1577 mt 1960 1551 L 1960 1551 mt 1938 1498 L 1938 1498 mt 1916 1445 L 1916 1445 mt 1904 1415 L 1904 1415 mt 1895 1392 L 1895 1392 mt 1874 1339 L 1874 1339 mt 1854 1287 L 1854 1287 mt 1837 1244 L 1837 1244 mt 1833 1234 L 1833 1234 mt 1813 1181 L 1813 1181 mt 1794 1128 L 1794 1128 mt 1774 1075 L 1774 1075 mt 1770 1064 L 1770 1064 mt 1755 1023 L 1755 1023 mt 1736 970 L 1736 970 mt 1717 917 L 1717 917 mt 1703 876 L 1703 876 mt 1699 864 L 1699 864 mt 1681 811 L 1681 811 mt 1663 759 L 1663 759 mt 1645 706 L 1645 706 mt 1636 679 L 1636 679 mt 1627 653 L 1627 653 mt 1617 621 L 1592 545 mt 1576 495 L 1576 495 mt 1569 474 L 1569 474 mt 1559 442 L 1559 442 mt 1542 389 L 3472 4613 mt 3445 4571 L 3445 4571 mt 3444 4569 L 3400 4501 mt 3378 4468 L 3378 4468 mt 3369 4455 L 3369 4455 mt 3334 4402 L 3334 4402 mt 3311 4367 L 3311 4367 mt 3299 4349 L 3299 4349 mt 3263 4296 L 3263 4296 mt 3244 4269 L 3244 4269 mt 3226 4243 L 3226 4243 mt 3189 4191 L 3189 4191 mt 3177 4174 L 3177 4174 mt 3151 4138 L 3151 4138 mt 3113 4085 L 3113 4085 mt 3110 4081 L 3110 4081 mt 3074 4032 L 3074 4032 mt 3043 3991 L 3043 3991 mt 3034 3979 L 3034 3979 mt 2994 3927 L 2994 3927 mt 2976 3904 L 2976 3904 mt 2952 3874 L 2952 3874 mt 2910 3821 L 2910 3821 mt 2909 3819 L 2909 3819 mt 2868 3768 L 2868 3768 mt 2842 3737 L 2842 3737 mt 2824 3715 L 2824 3715 mt 2779 3663 L 2779 3663 mt 2775 3658 L 2775 3658 mt 2734 3610 L 2734 3610 mt 2708 3581 L 2708 3581 mt 2687 3557 L 2687 3557 mt 2641 3507 L 2641 3507 mt 2639 3504 L 2639 3504 mt 2590 3451 L 2590 3451 mt 2574 3435 L 2574 3435 mt 2539 3399 L 2539 3399 mt 2507 3366 L 2507 3366 mt 2487 3346 L 2487 3346 mt 2440 3300 L 2440 3300 mt 2433 3293 L 2433 3293 mt 2377 3240 L 2377 3240 mt 2373 3236 L 2373 3236 mt 2320 3187 L 2320 3187 mt 2306 3175 L 2306 3175 mt 2260 3135 L 2260 3135 mt 2239 3117 L 2239 3117 mt 2197 3082 L 2197 3082 mt 2172 3059 L 2172 3059 mt 2144 3029 L 2144 3029 mt 2105 2982 L 2105 2982 mt 2101 2976 L 2101 2976 mt 2063 2923 L 2063 2923 mt 2038 2887 L 2038 2887 mt 2027 2871 L 2027 2871 mt 1994 2818 L 1994 2818 mt 1971 2780 L 1971 2780 mt 1962 2765 L 1962 2765 mt 1935 2716 L 1896 2645 mt 1876 2607 L 1876 2607 mt 1849 2554 L 1849 2554 mt 1837 2529 L 1837 2529 mt 1823 2501 L 1823 2501 mt 1798 2448 L 1798 2448 mt 1773 2395 L 1773 2395 mt 1770 2388 L 1770 2388 mt 1750 2343 L 1750 2343 mt 1726 2290 L 1726 2290 mt 1703 2237 L 1703 2237 mt 1703 2236 L 1703 2236 mt 1681 2184 L 1681 2184 mt 1659 2131 L 1659 2131 mt 1638 2079 L 1638 2079 mt 1636 2075 L 1636 2075 mt 1617 2026 L 1617 2026 mt 1596 1973 L 1596 1973 mt 1576 1920 L 1576 1920 mt 1569 1903 L 1569 1903 mt 1556 1867 L 1556 1867 mt 1536 1815 L 1536 1815 mt 1516 1762 L 1516 1762 mt 1502 1722 L 1502 1722 mt 1497 1709 L 1497 1709 mt 1479 1656 L 1479 1656 mt 1460 1603 L 1460 1603 mt 1442 1551 L 1442 1551 mt 1435 1532 L 1435 1532 mt 1424 1498 L 1424 1498 mt 1406 1445 L 1406 1445 mt 1388 1392 L 1388 1392 mt 1371 1339 L 1371 1339 mt 1368 1332 L 1368 1332 mt 1353 1287 L 1353 1287 mt 1336 1234 L 1336 1234 mt 1319 1181 L 1319 1181 mt 1303 1128 L 1303 1128 mt 1301 1123 L 1301 1123 mt 1286 1075 L 1286 1075 mt 1270 1023 L 1270 1023 mt 1254 970 L 1254 970 mt 1238 917 L 1238 917 mt 1234 905 L 1234 905 mt 1222 864 L 1222 864 mt 1206 811 L 1206 811 mt 1191 759 L 1191 759 mt 1175 706 L 1175 706 mt 1167 677 L 1167 677 mt 1160 653 L 1160 653 mt 1145 600 L 1145 600 mt 1130 547 L 1130 547 mt 1115 495 L 1115 495 mt 1100 442 L 1100 442 mt 1100 441 L 1100 441 mt 1086 389 L 2607 4613 mt 2574 4562 L 2574 4562 mt 2573 4560 L 2573 4560 mt 2539 4507 L 2539 4507 mt 2507 4459 L 2507 4459 mt 2504 4455 L 2504 4455 mt 2483 4422 L 2438 4355 mt 2434 4349 L 2434 4349 mt 2397 4296 L 2397 4296 mt 2373 4261 L 2373 4261 mt 2361 4243 L 2361 4243 mt 2324 4191 L 2324 4191 mt 2306 4166 L 2306 4166 mt 2286 4138 L 2286 4138 mt 2248 4085 L 2248 4085 mt 2239 4073 L 2239 4073 mt 2209 4032 L 2209 4032 mt 2172 3984 L 2172 3984 mt 2169 3979 L 2169 3979 mt 2129 3927 L 2129 3927 mt 2105 3897 L 2105 3897 mt 2087 3874 L 2087 3874 mt 2045 3821 L 2045 3821 mt 2038 3812 L 2038 3812 mt 2003 3768 L 2003 3768 mt 1971 3730 L 1971 3730 mt 1959 3715 L 1959 3715 mt 1914 3663 L 1914 3663 mt 1904 3651 L 1904 3651 mt 1869 3610 L 1869 3610 mt 1837 3574 L 1837 3574 mt 1822 3557 L 1822 3557 mt 1774 3504 L 1774 3504 mt 1770 3500 L 1770 3500 mt 1729 3451 L 1729 3451 mt 1703 3415 L 1703 3415 mt 1692 3399 L 1692 3399 mt 1659 3346 L 1659 3346 mt 1636 3307 L 1636 3307 mt 1628 3293 L 1628 3293 mt 1599 3240 L 1599 3240 mt 1571 3187 L 1571 3187 mt 1569 3183 L 1569 3183 mt 1545 3135 L 1545 3135 mt 1519 3082 L 1519 3082 mt 1502 3045 L 1502 3045 mt 1494 3029 L 1494 3029 mt 1470 2976 L 1470 2976 mt 1447 2923 L 1447 2923 mt 1435 2896 L 1435 2896 mt 1424 2871 L 1424 2871 mt 1402 2818 L 1402 2818 mt 1380 2765 L 1380 2765 mt 1368 2735 L 1368 2735 mt 1359 2712 L 1359 2712 mt 1338 2659 L 1338 2659 mt 1318 2607 L 1318 2607 mt 1301 2562 L 1301 2562 mt 1298 2554 L 1298 2554 mt 1278 2501 L 1278 2501 mt 1259 2448 L 1259 2448 mt 1240 2395 L 1240 2395 mt 1239 2393 L 1212 2317 mt 1203 2290 L 1203 2290 mt 1184 2237 L 1184 2237 mt 1167 2186 L 1167 2186 mt 1166 2184 L 1166 2184 mt 1149 2131 L 1149 2131 mt 1131 2079 L 1131 2079 mt 1114 2026 L 1114 2026 mt 1100 1982 L 1100 1982 mt 1097 1973 L 1097 1973 mt 1080 1920 L 1080 1920 mt 1064 1867 L 1064 1867 mt 1047 1815 L 1047 1815 mt 1033 1769 L 1033 1769 mt 1031 1762 L 1031 1762 mt 1015 1709 L 1015 1709 mt 999 1656 L 999 1656 mt 983 1603 L 983 1603 mt 967 1551 L 967 1551 mt 966 1546 L 966 1546 mt 952 1498 L 952 1498 mt 937 1445 L 937 1445 mt 922 1392 L 922 1392 mt 906 1339 L 906 1339 mt 899 1313 L gr 3545 4902 mt (u) s /Symbol /ISOLatin1Encoding 120 FMSR 673 2535 mt -90 rotate (r) s 90 rotate gs 899 389 5361 4225 MR c np /Helvetica /ISOLatin1Encoding 120 FMSR 2980 1353 mt (-3) s 4998 2425 mt (-3) s 2044 552 mt (0) s 3342 2504 mt (0) s 5739 2543 mt (0) s 4321 4426 mt (3) s 2638 2748 mt (3) s 1571 625 mt (3) s 3389 4577 mt (6) s 1882 2723 mt (6) s 2427 4431 mt (9) s 1192 2397 mt (9) s 0.333333 sg 7 -8 7 -8 6 -8 7 -8 7 -8 7 -8 6 -8 7 -9 7 -8 6 -8 7 -8 7 -8 6 -9 7 -8 7 -8 7 -9 6 -8 7 -9 7 -8 6 -9 7 -8 7 -9 6 -8 7 -9 7 -8 7 -9 6 -9 7 -8 7 -9 6 -9 7 -9 7 -9 6 -8 7 -9 7 -9 7 -9 6 -9 7 -9 7 -9 6 -9 7 -9 7 -9 7 -9 6 -9 7 -9 7 -10 6 -9 7 -9 7 -9 6 -10 7 -9 7 -9 6 -10 7 -9 7 -9 7 -10 6 -9 7 -10 7 -9 6 -10 7 -9 7 -10 6 -10 7 -9 7 -10 7 -10 6 -9 7 -10 7 -10 6 -10 7 -10 7 -9 6 -10 7 -10 7 -10 7 -10 6 -10 7 -10 7 -10 6 -10 7 -10 7 -10 7 -11 6 -10 7 -10 7 -10 6 -10 7 -11 7 -10 6 -10 7 -11 7 -10 7 -11 6 -10 7 -10 7 -11 6 -10 7 -11 0 0 899 4613 100 MP 7 -5 7 -5 6 -6 7 -5 7 -6 7 -5 6 -6 7 -5 7 -6 6 -5 7 -6 7 -6 6 -5 7 -6 7 -6 7 -5 6 -6 7 -6 7 -6 6 -6 7 -5 7 -6 6 -6 7 -6 7 -6 7 -6 6 -6 7 -6 7 -6 6 -6 7 -7 7 -6 6 -6 7 -6 7 -6 7 -7 6 -6 7 -6 7 -7 6 -6 7 -6 7 -7 7 -6 6 -7 7 -6 7 -7 6 -6 7 -7 7 -7 6 -6 7 -7 7 -7 6 -6 7 -7 7 -7 7 -7 6 -6 7 -7 7 -7 6 -7 7 -7 7 -7 6 -7 7 -7 7 -7 7 -7 6 -7 7 -7 7 -7 6 -8 7 -7 7 -7 6 -7 7 -8 7 -7 7 -7 6 -8 7 -7 7 -7 6 -8 7 -7 7 -8 6 -7 7 -8 7 -7 7 -8 6 -7 7 -8 7 -8 6 -7 7 -8 7 -8 7 -8 6 -8 7 -7 7 -8 6 -8 7 -8 7 -8 6 -8 100 AP 7 -3 7 -3 6 -3 7 -2 7 -3 7 -3 6 -3 7 -3 7 -3 6 -3 7 -3 7 -3 6 -3 7 -3 7 -3 7 -3 6 -3 7 -3 7 -3 6 -3 7 -4 7 -3 6 -3 7 -4 7 -3 7 -3 6 -4 7 -3 7 -4 6 -3 7 -3 7 -4 6 -4 7 -3 7 -4 7 -3 6 -4 7 -4 7 -3 6 -4 7 -4 7 -4 6 -4 7 -3 7 -4 7 -4 6 -4 7 -4 7 -4 6 -4 7 -4 7 -4 6 -4 7 -4 7 -4 7 -5 6 -4 7 -4 7 -4 6 -4 7 -5 7 -4 6 -4 7 -5 7 -4 7 -5 6 -4 7 -4 7 -5 6 -5 7 -4 7 -5 6 -4 7 -5 7 -5 7 -4 6 -5 7 -5 7 -4 6 -5 7 -5 7 -5 6 -5 7 -5 7 -5 7 -5 6 -5 7 -5 7 -5 6 -5 7 -5 7 -5 6 -5 7 -5 7 -5 7 -5 6 -6 7 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