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(hniques)h(esp)s(ecially)d(designed)h(for)g(cylinder-lik)m(e)e(domains) h(also)456 2634 y(called)e(ducts)i(in)e(acoustics.)42 b(W)-8 b(e)24 b(ha)m(v)m(e)i(already)e(men)m(tioned)f([1,)i(15)o(])f (where)i(suc)m(h)456 2750 y(metho)s(ds)42 b(are)h(dev)m(elop)s(ed.)75 b(Both)43 b(of)f(these)i(pap)s(ers)f(stress)i(the)e(imp)s(ortance)456 2866 y(of)34 b(adv)-5 b(anced)36 b(metho)s(ds)g(sp)s(ecially)d (designed)j(for)f(acoustic)g(w)m(a)m(v)m(eguides.)54 b(It)35 b(is)456 2982 y(hardly)25 b(surpsising)g(that)g(carefully)f(p)s (erformed)h(n)m(umerical)f(analysis)h(is)g(equally)456 3099 y(imp)s(ortan)m(t)32 b(for)i(quan)m(tum)g(problems.)48 b(In)35 b([15)o(])f(the)h(authors)g(apply)f(a)g(second-)456 3215 y(order)43 b(\014nite)g(di\013erence)g(metho)s(d)g(and)g(implemen) m(t)e(an)i(iterativ)m(e)f(pro)s(cedure)456 3331 y(for)30 b(the)h(resulting)f(algebraic)f(system.)44 b(It)31 b(is)f(men)m(tioned) h(there)g(that)g(standard)456 3447 y(metho)s(ds)24 b(not)h(using)f(an)m (y)i(preconditioning)d(are)i(lik)m(ely)e(to)i(fail)d(esp)s(ecially)i (when)456 3563 y(a)32 b(large)f(w)m(a)m(v)m(e)k(n)m(um)m(b)s(er)e(is)f (in)m(v)m(olv)m(ed.)555 3680 y(The)40 b(n)m(umerical)e(approac)m(h)h (prop)s(osed)h(in)e([1])h(is)g(similar)d(in)i(spirit)g(to)h(that)456 3796 y(of)c(our)g(pap)s(er.)54 b(In)36 b(b)s(oth)f(cases)i(the)g (Helmholtz)c(equation)j(is)f(reduced)j(to)d(the)456 3912 y(so-called)30 b(coupled)j(mo)s(de)e(system)j(of)e(equations)g(via)g (the)g(separation)g(of)g(v)-5 b(ari-)456 4028 y(ables.)66 b(The)42 b(main)c(di\013erence)k(is)d(that)i(in)e([1])i(the)f(co)s (e\016cien)m(ts)i(of)e(the)h(ODE)456 4145 y(system)47 b(ha)m(v)m(e)g(to)e(b)s(e)i(computed)f(n)m(umerically)e(whereas)j(our)f (c)m(hoice)g(of)g(the)456 4261 y(F)-8 b(ourier)27 b(expansion)i (functions)g(allo)m(ws)f(us)h(to)g(\014nd)g(them)f(in)g(closed)h(form.) 41 b(The)456 4377 y(Diric)m(hlet)g(problem)i(studied)i(here)g(is)e (separable)i(as)f(opp)s(osed)h(to)e(the)i(m)m(uc)m(h)456 4493 y(less)36 b(straigh)m(tforw)m(ard)g(Robin)f(case.)56 b(The)37 b(examples)f(in)g(the)h(cited)f(pap)s(er)g(are)456 4610 y(related)d(to)g(higher)h(frequencies)h(while)e(w)m(e)i(concen)m (trate)g(on)f(the)g(lo)m(w)m(est)g(oscil-)456 4726 y(lation)f(mo)s(de)h (only)-8 b(.)51 b(On)35 b(the)g(other)h(hand,)g(the)f(transfer)h(metho) s(d)f(w)m(e)h(use)g(for)456 4842 y(the)k(\014nal)g(ODE)g(problem)f(is)g (able)h(to)g(handle)g(a)g(w)m(a)m(v)m(eguide)h(with)f(a)g(narro)m(w)456 4958 y(throat)32 b(|)g(a)g(situation)f(not)h(co)m(v)m(ered)j(in)d([1].) 555 5074 y(The)47 b(aim)d(of)i(this)g(pap)s(er)g(is)f(to)h(elab)s (orate)f(a)h(metho)s(d)f(suitable)g(for)h(p)s(er-)456 5191 y(turb)s(ed)d(cylinders)h(that)f(tak)m(es)h(accoun)m(t)h(of)d (their)h(geometry)-8 b(.)76 b(The)44 b(metho)s(d)p eop %%Page: 3 3 3 2 bop 1460 236 a Fm(PER)-6 b(TURBED)33 b(CYLINDERS)965 b(3)456 425 y Fs(based)32 b(on)e(the)i(F)-8 b(ourier)30 b(decomp)s(osition)f(in)h(one)h(direction)f(allo)m(ws)g(us)h(to)g (sepa-)456 541 y(rate)g(v)-5 b(ariables)29 b(in)i(the)g(Helmholtz)e (equation)i(explicitly)e(leading)h(to)h(a)f(system)456 658 y(of)38 b(ODEs.)62 b(This)39 b(is)f(done)h(for)g(a)f(fairly)f (general)h(geometry)h(in)f(the)h(next)h(sec-)456 774 y(tion.)65 b(In)41 b(section)f(3)g(w)m(e)h(discuss)g(di\013eren)m(t)g (b)s(oundary)f(conditions)g(in)m(v)m(olv)m(ed.)456 890 y(First)f(w)m(e)i(deal)f(with)g(standard)h(self-adjoin)m(t)e (conditions,)i(then)g(concen)m(trate)456 1006 y(on)29 b(non-compact)f(domains)h(and)g(de\014ne)i(resonances)g(b)m(y)f(a)f(sp) s(eci\014c)h(b)s(oundary)456 1123 y(condition)23 b(at)h(in\014nit)m(y) -8 b(.)40 b(What)25 b(is)f(often)h(called)e(the)i(radiation)e (condition)g(in)h(the)456 1239 y(literature)34 b(is)i(rewritten)g(in)f (terms)h(of)f(the)h(resulting)f(ODE)h(system.)54 b(W)-8 b(e)37 b(end)456 1355 y(up)26 b(with)f(a)h(non-self-adjoin)m(t)e(eigen) m(v)-5 b(alue)26 b(problem)e(on)i(a)g(\014nite)f(in)m(terv)-5 b(al)25 b(whose)456 1471 y(solution)30 b(appro)m(ximates)h(that)h(of)f (the)h(original)d(resonance)k(problem.)42 b(Finally)-8 b(,)456 1588 y(w)m(e)43 b(use)h(the)e(transfer)h(metho)s(d)f(of)g([2])g (to)h(\014nd)f(the)h(eigen)m(v)-5 b(alues)43 b(of)f(the)h(t)m(w)m(o)456 1704 y(problems.)h(Our)32 b(n)m(umerical)g(results)h(illustrate)e(the)i (closeness)i(of)d(eigen)m(v)-5 b(alues)456 1820 y(and)39 b(resonances)j(and)d(are)h(presen)m(ted)h(in)e(section)h(4)f(where)i(w) m(e)f(conclude)g(b)m(y)456 1936 y(discussing)32 b(the)h(rate)g(of)f (con)m(v)m(ergence.)555 2052 y(T)-8 b(o)23 b(b)s(e)g(able)g(to)f (compare)h(the)g(metho)s(d)f(of)h(the)g(pap)s(er)g(with)g(others)g(w)m (e)h(lo)s(ok)e(at)456 2169 y(the)i(\014nite)g(v)m(olume)f(metho)s(d)g (describ)s(ed,)k Fq(e.g.)40 b Fs(in)23 b([16)o(].)41 b(As)24 b(discussed)i(in)d(section)456 2285 y(4,)49 b(the)d(prop)s (osed)h(approac)m(h)g(tested)g(on)f(our)g(eigen)m(v)-5 b(alue)46 b(examples)g(pro)m(v)m(es)456 2401 y(to)e(b)s(e)h (signi\014can)m(tly)f(more)g(e\016cien)m(t)i(than)f(the)g(standard)h(t) m(w)m(o-dimensional)456 2517 y(pro)s(cedure.)790 2718 y(2.)55 b Fr(Sep)-7 b(ara)g(tion)38 b(of)g(v)-9 b(ariables)37 b(f)n(or)h(the)f(Lapla)n(cian)456 2893 y Fs(2.1.)55 b Fv(Change)48 b(of)e(v)-6 b(ariables.)49 b Fs(Consider)41 b(the)f(op)s(erator)g Fl(H)48 b Fs(:=)41 b Fp(\000)p Fs(\001)g(acting)456 3009 y(on)36 b Fp(L)664 2973 y Fo(2)703 3009 y Fs(\(\012\))h(sub)5 b(ject)38 b(to)e(Diric)m(hlet)e(b)s(oundary) j(conditions.)54 b(The)37 b(domain)e(\012)i(is)456 3125 y(de\014ned)d(as)1118 3283 y(\012)28 b(=)g Fp(f)p Fs(\()p Fl(\030)5 b(;)17 b(\021)t Fs(\))26 b(:)60 b Fl(a)28 b(<)g(\030)k(<)27 b(b;)50 b Fs(0)27 b Fl(<)h(\021)j(<)d(')p Fs(\()p Fl(\030)5 b Fs(\))p Fp(g)461 b Fs(\(2.1\))456 3447 y(in)40 b(the)h(Cartesian)f (co)s(ordinates)h(\()p Fl(\030)5 b(;)17 b(\021)t Fs(\).)66 b(The)42 b(p)s(ossibilit)m(y)d(of)h Fl(a)h Fs(and)g Fl(b)g Fs(b)s(eing)456 3563 y(in\014nite)27 b(is)h(not)h(excluded)h(here,)g (so)f(that)g(\012)g(is)f(not)g(necessarily)i(compact.)41 b(The)456 3680 y(function)29 b Fl(')p Fs(\()p Fl(\030)5 b Fs(\))30 b(is)f(assumed)i(to)f(b)s(e)h(smo)s(oth)e(and)h(satisfy)g Fl(')p Fs(\()p Fl(\030)5 b Fs(\))27 b Fl(>)h Fs(0)p Fl(;)47 b(\030)32 b Fp(2)c Fs([)p Fl(a;)17 b(b)p Fs(].)456 3796 y(T)-8 b(o)32 b(\014nd)h(the)g(sp)s(ectrum)g(of)f Fl(H)40 b Fs(w)m(e)34 b(solv)m(e)f(the)g(Helmholtz)e(equation)1174 3954 y Fp(\000)p Fs(\001)p Fl(f)11 b Fs(\()p Fl(\030)5 b(;)17 b(\021)t Fs(\))27 b(=)h Fl(\025f)11 b Fs(\()p Fl(\030)5 b(;)17 b(\021)t Fs(\))p Fl(;)210 b Fs(\()p Fl(\030)5 b(;)17 b(\021)t Fs(\))26 b Fp(2)i Fs(\012)519 b(\(2.2\))456 4112 y(with)32 b(the)h(b)s(oundary)g(conditions)1343 4270 y Fl(f)11 b Fs(\()p Fl(\030)5 b(;)17 b(\021)t Fs(\))43 b(=)h(0)17 b Fl(;)211 b Fs(\()p Fl(\030)5 b(;)17 b(\021)t Fs(\))26 b Fp(2)j Fl(@)5 b Fs(\012)688 b(\(2.3\))555 4429 y(The)34 b(c)m(hange)f(of)f(v)-5 b(ariables)1479 4587 y Fl(x)28 b Fs(=)f Fl(\030)5 b(;)211 b(y)31 b Fs(=)d Fl(\021)t(=')p Fs(\()p Fl(\030)5 b Fs(\))821 b(\(2.4\))456 4751 y(maps)37 b(the)g(p)s(erturb)s(ed)i(cylinder)e(\012)g(on)m(to)h (\012)2127 4766 y Fo(0)2219 4751 y Fs(=)52 b Fp(f)p Fs(\()p Fl(x;)17 b(y)t Fs(\))35 b(:)74 b Fl(a)36 b(<)f(x)i(<)e(b;)55 b Fs(0)36 b Fl(<)456 4867 y(y)d(<)d Fs(1)p Fp(g)j Fs(whic)m(h)i(is)f (either)f(a)h(rectangle)g(or)g(a)f(strip)h(\(in\014nite)f(or)h (semi-in\014nite\).)456 4983 y(W)-8 b(e)33 b(men)m(tion)e(that)h(a)h (similar)c(metho)s(d)j(has)h(b)s(een)g(used)h(b)m(y)f(Boriso)m(v)f(et)h (al.)42 b([7])456 5099 y(to)c(study)h(b)s(ound)g(states)g(asso)s (ciated)g(with)f(a)g(lo)s(cal)e(p)s(erturbation)i(of)g(a)g(strip)456 5216 y(or)i(la)m(y)m(er.)68 b(In)41 b(our)g(case)h(the)f(deformation)e (reduces)j(the)g(width)e(of)h(the)g(strip)p eop %%Page: 4 4 4 3 bop 456 236 a Fm(4)831 b(A.)33 b(ASLANY)-6 b(AN)30 b(AND)i(E.B.)j(D)n(A)-9 b(VIES)456 425 y Fs(lo)s(cally)h(,)36 b(and)i(there)g(are)g(no)g(b)s(ound)g(states.)60 b(The)38 b(transformation)e(\(2.4\))h(can)456 541 y(b)s(e)32 b(expressed)k(in)c (the)h(di\013eren)m(tial)d(form)i(as)682 791 y Fp(r)765 806 y Fk(\030)s(\021)868 791 y Fs(=)c Fl(W)14 b Fp(r)1161 806 y Fk(xy)1242 791 y Fl(;)212 b Fp(r)1564 806 y Fk(xy)1672 791 y Fs(=)1776 651 y Fj(\022)1921 686 y Fk(@)p 1901 702 81 4 v 1901 760 a(@)t(x)1921 806 y(@)p 1902 823 79 4 v 1902 880 a(@)t(y)2033 651 y Fj(\023)2123 791 y Fl(;)g(W)41 b Fs(=)2599 621 y Fj( )2719 725 y Fs(1)83 b Fp(\000)2938 682 y Fk(')2984 658 y Fi(0)3007 682 y Fk(y)p 2938 703 106 4 v 2968 760 a(')2719 855 y Fs(0)2935 816 y Fo(1)p 2929 832 46 4 v 2929 889 a Fk(')3095 621 y Fj(!)3191 791 y Fl(:)456 1041 y Fs(The)33 b(quadratic)f(form)g(corresp)s(onding)g (to)g Fl(H)40 b Fs(is)33 b(giv)m(en)f(b)m(y)1087 1252 y Fl(J)9 b Fs(\()p Fl(f)i Fs(\))44 b(=)1449 1116 y Fj(Z)1504 1342 y Fo(\012)1576 1252 y Fs([\()p Fp(r)1724 1267 y Fk(\030)s(\021)1800 1252 y Fl(f)5 b(;)17 b Fp(r)1980 1267 y Fk(\030)s(\021)2055 1252 y Fl(f)11 b Fs(\))22 b Fp(\000)h Fl(\025)p Fs(\()p Fl(f)5 b(;)17 b(f)11 b Fs(\)])16 b(d)p Fl(\030)5 b Fs(d)p Fl(\021)456 1467 y Fs(or,)32 b(equiv)-5 b(alen)m(tly)d(,)32 b(b)m(y)843 1678 y Fl(J)9 b Fs(\()p Fl(f)i Fs(\))44 b(=)1205 1543 y Fj(Z)1261 1768 y Fo(\012)1312 1777 y Fh(0)1367 1678 y Fs([\()p Fl(W)14 b Fp(r)1621 1693 y Fk(xy)1702 1678 y Fl(f)5 b(;)17 b(W)d Fp(r)1988 1693 y Fk(xy)2069 1678 y Fl(f)d Fs(\))22 b Fp(\000)g Fl(\025)p Fs(\()p Fl(f)5 b(;)17 b(f)11 b Fs(\)])16 b Fl(')p Fs(\()p Fl(x)p Fs(\)d)p Fl(x)p Fs(d)p Fl(y)t(:)456 1903 y Fs(This)32 b(can)h(b)s(e)g(rewritten) g(as)912 2094 y Fl(J)9 b Fs(\()p Fl(f)i Fs(\))28 b(=)1258 1959 y Fj(Z)1313 2184 y Fo(\012)1364 2193 y Fh(0)1420 2094 y Fs([\()p Fl(A)p Fp(r)1641 2109 y Fk(xy)1722 2094 y Fl(f)5 b(;)17 b Fp(r)1902 2109 y Fk(xy)1983 2094 y Fl(f)11 b Fs(\))22 b Fp(\000)h Fl(\025)p Fs(\()p Fl(f)5 b(;)17 b(f)11 b Fs(\)])16 b Fl(')p Fs(\()p Fl(x)p Fs(\)d)p Fl(x)p Fs(d)p Fl(y)t(;)273 b Fs(\(2.5\))456 2319 y(where)1158 2549 y Fl(A)45 b Fs(=)f Fl(W)1502 2508 y Fg(\003)1541 2549 y Fl(W)58 b Fs(=)1811 2379 y Fj( )2009 2473 y Fs(1)238 b Fp(\000)2383 2429 y Fk(')2429 2405 y Fi(0)2452 2429 y Fk(y)p 2383 2450 106 4 v 2413 2507 a(')1931 2623 y Fp(\000)2018 2579 y Fk(')2064 2556 y Fi(0)2087 2579 y Fk(y)p 2018 2600 V 2049 2657 a(')2227 2575 y Fo(\(1+\()p Fk(')2417 2552 y Fi(0)2441 2575 y Fk(y)r Fo(\))2505 2552 y Fh(2)2540 2575 y Fo(\))p 2227 2600 341 4 v 2357 2657 a Fk(')2403 2638 y Fh(2)2619 2379 y Fj(!)2715 2549 y Fl(:)502 b Fs(\(2.6\))456 2799 y(Hence)34 b(the)f(Helmholtz)d(equation) j(in)e(the)i(new)h(v)-5 b(ariables)31 b(tak)m(es)j(the)f(form)510 2958 y(\()p Fl(')17 b(f)677 2973 y Fk(x)721 2958 y Fs(\))759 2988 y Fk(x)841 2958 y Fp(\000)40 b Fs(\()o Fl(')1059 2918 y Fg(0)1083 2958 y Fl(y)19 b(f)1198 2973 y Fk(x)1242 2958 y Fs(\))1280 2988 y Fk(y)1360 2958 y Fp(\000)40 b Fs(\()p Fl(')1579 2918 y Fg(0)1602 2958 y Fl(y)20 b(f)1718 2973 y Fk(y)1759 2958 y Fs(\))1797 2991 y Fk(x)1857 2958 y Fs(+)1967 2878 y Fj(\000)2012 2958 y Fs(\(1)i(+)g(\()p Fl(')2321 2918 y Fg(0)2344 2958 y Fl(y)t Fs(\))2434 2918 y Fo(2)2473 2958 y Fs(\))17 b Fl(f)2576 2973 y Fk(y)2617 2958 y Fl(=')2730 2878 y Fj(\001)2775 2998 y Fk(y)2855 2958 y Fs(+)39 b Fl(\025'f)55 b Fs(=)44 b(0)p Fl(:)3244 3094 y Fs(\(2.7\))555 3258 y(Since)h(w)m(e)i(restrict)e(ourselv)m(es)h (to)f(the)h(Diric)m(hlet)c(case)47 b(no)e(c)m(hange)h(in)e(the)456 3374 y(b)s(oundary)e(conditions)g(is)g(required)g(here,)k(the)d (condition)e(\(2.3\))h(is)g(retained)456 3490 y(on)c Fl(@)5 b Fs(\012)723 3505 y Fo(0)764 3490 y Fs(.)61 b(Ho)m(w)m(ev)m (er,)43 b(in)37 b(a)i(generic)f(situation)f(one)i(can)g(still)d(use)j (\(2.4\),)h(\(2.7\))456 3607 y(pro)m(vided)45 b(ob)m(vious)g(c)m (hanges)i(are)e(made)g(to)f(the)i(original)c(b)s(oundary)j(condi-)456 3723 y(tions)f(where)j(necessary)-8 b(.)83 b(F)-8 b(or)44 b(instance,)49 b(instead)c(of)g(Neumann)g(b)s(oundary)456 3839 y(conditions)31 b(at)h Fl(\021)g Fs(=)27 b Fl(')p Fs(\()p Fl(\030)5 b Fs(\))32 b(one)h(w)m(ould)f(ha)m(v)m(e)1271 3924 y Fj(\000)1317 4005 y Fs(\(1)22 b(+)g Fl(')1588 3964 y Fg(0)p Fo(2)1646 4005 y Fs(\))p Fl(f)1732 4020 y Fk(y)1812 4005 y Fp(\000)39 b Fl('')2056 3964 y Fg(0)2096 4005 y Fl(f)2144 4020 y Fk(x)2188 3924 y Fj(\001)2234 3920 y(\014)2234 3980 y(\014)2267 4044 y Fk(y)r Fo(=1)2443 4005 y Fs(=)44 b(0)p Fl(;)456 4182 y Fs(while)31 b(Neumann)i(b)s (oundary)g(conditions)f(at)g Fl(x)c Fs(=)f Fl(a;)34 b(b)f Fs(w)m(ould)f(b)s(ecome)1552 4346 y(\()p Fl('f)1702 4361 y Fk(x)1768 4346 y Fp(\000)22 b Fl(')1931 4305 y Fg(0)1954 4346 y Fl(y)t(f)2054 4361 y Fk(y)2095 4346 y Fs(\))p Fp(j)2161 4378 y Fk(x)p Fo(=)p Fk(a;)11 b(b)555 4516 y Fs(Note)32 b(that)e(a)h(similar)d(transformation)h(can)j(b)s(e)f (also)f(done)i(for)e(a)h(more)g(gen-)456 4632 y(eral)g(domain)996 4796 y(\012)d(=)g Fp(f)p Fs(\()p Fl(\030)5 b(;)17 b(\021)t Fs(\))26 b(:)60 b Fl(a)28 b(<)f(\030)33 b(<)27 b(b;)50 b(')2124 4811 y Fo(1)2163 4796 y Fs(\()p Fl(\030)5 b Fs(\))27 b Fl(<)h(\021)j(<)d(')2665 4811 y Fo(2)2704 4796 y Fs(\()p Fl(\030)5 b Fs(\))p Fp(g)p Fl(:)456 4960 y Fs(The)33 b(c)m(hange)h(of)e(v)-5 b(ariables)1448 5161 y Fl(x)28 b Fs(=)g Fl(\030)5 b(;)211 b(y)31 b Fs(=)2139 5094 y Fl(\021)26 b Fp(\000)d Fl(')2377 5109 y Fo(1)p 2113 5138 329 4 v 2113 5229 a Fl(')2177 5244 y Fo(2)2239 5229 y Fp(\000)f Fl(')2402 5244 y Fo(1)p eop %%Page: 5 5 5 4 bop 1460 236 a Fm(PER)-6 b(TURBED)33 b(CYLINDERS)965 b(5)456 425 y Fs(leads)24 b(to)h(a)f(quadratic)h(form)e(of)h(t)m(yp)s (e)i(\(2.5\))e(whose)i(co)s(e\016cien)m(ts)g(are)f(not)f(quoted)456 541 y(here)33 b(for)f(brevit)m(y)-8 b(.)456 739 y(2.2.)55 b Fv(Discretisation)32 b(in)h(the)h Fl(y)t Fv(-direction.)46 b Fs(The)31 b(quadratic)e(form)f(\(2.5\))h(is)456 856 y(related)21 b(to)g(the)h(transformed)g(op)s(erator)f(on)h(the)g(w)m (eigh)m(ted)g(space)h Fp(L)2948 820 y Fo(2)2987 856 y Fs(\(\012)3095 871 y Fo(0)3135 856 y Fl(;)17 b(')p Fs(d)p Fl(x)22 b Fs(d)p Fl(y)t Fs(\).)456 972 y(Here)33 b(and)g(b)s(elo)m(w)f (w)m(e)i(use)f(the)g(notation)e Fp(r)d Fs(=)g Fp(r)2320 987 y Fk(xy)2401 972 y Fs(.)555 1088 y(T)-8 b(o)45 b(discretise)g(the)g (form)f Fl(J)9 b Fs(\()p Fl(f)i Fs(\))44 b(in)g(the)i Fl(y)t 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Fk(k)1856 3236 y Fs(=)27 b Fl(h)2015 3195 y Fg(0)2015 3261 y Fk(k)2058 3236 y Fl(;)212 b(h)2353 3195 y Fo(2)2353 3261 y Fk(k)2423 3236 y Fs(=)28 b Fl(\031)t(k)s(h)2696 3251 y Fk(k)456 3405 y Fs(\(throughout)k(the)h(pap)s(er)1445 3369 y Fg(0)1500 3405 y Fs(denotes)h(di\013eren)m(tiation)d(with)h(resp)s(ect)i(to)e Fl(x)p Fs(\).)555 3521 y(W)-8 b(e)27 b(notice)g(that)g(the)g(v)-5 b(ariables)25 b(are)i(separated)h(in)e(the)h(co)s(e\016cien)m(ts)h(of)f Fl(J)9 b Fs(\()p Fl(f)i Fs(\):)718 3690 y Fl(')p Fs(\()p Fl(x)p Fs(\))p Fl(A)986 3705 y Fk(ij)1047 3690 y Fs(\()p Fl(x;)17 b(y)t Fs(\))27 b(=)g Fl(B)1478 3705 y Fk(ij)1539 3690 y Fs(\()p Fl(x)p Fs(\))22 b(+)g Fl(C)1860 3705 y Fk(ij)1921 3690 y Fs(\()p Fl(y)t Fs(\))p Fl(D)2130 3705 y Fk(ij)2189 3690 y Fs(\()p Fl(x)p Fs(\))p Fl(;)212 b(i;)17 b(j)34 b Fs(=)27 b(1)p Fl(;)17 b Fs(2)p Fl(:)262 b Fs(\(2.9\))456 3865 y(The)27 b(en)m(tries)h Fl(A)1030 3880 y Fk(ij)1118 3865 y Fs(of)e(the)h(matrix)f Fl(A)h Fs(are)g(de\014ned)h(b)m(y)g 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Fl(\025')2747 4975 y Fj(Z)2846 5001 y Fo(1)2801 5201 y(0)2902 5111 y Fp(j)p Fl(f)11 b Fp(j)3017 5069 y Fo(2)3055 5111 y Fs(d)p Fl(y)3161 4940 y Fj(#)3235 5111 y Fs(d)p Fl(x)p eop %%Page: 6 6 6 5 bop 456 236 a Fm(6)831 b(A.)33 b(ASLANY)-6 b(AN)30 b(AND)i(E.B.)j(D)n(A)-9 b(VIES)1094 519 y Fs(=)1214 384 y Fj(Z)1314 410 y Fk(b)1270 609 y(a)1365 349 y Fj(")1488 395 y Fo(2)1434 425 y Fj(X)1423 635 y Fk(i;j)t Fo(=1)1622 519 y Fl(B)1696 534 y Fk(ij)1757 519 y Fl(E)1829 534 y Fk(ij)1912 519 y Fs(+)22 b Fl(D)2091 534 y Fk(ij)2151 519 y Fl(F)2214 534 y Fk(ij)2297 519 y Fp(\000)h Fl(\025'G)2595 349 y Fj(#)2669 519 y Fs(d)p Fl(x:)456 783 y Fs(As)33 b(w)m(e)g(substitute)g(the)g(expansion)g(\(2.8\))f(in)m(to)f(the)i(ab)s (o)m(v)m(e)g(in)m(tegral,)e(the)i(co)s(ef-)456 899 y(\014cien)m(ts)g Fl(E)856 914 y Fk(ij)917 899 y Fl(;)49 b(F)1056 914 y Fk(ij)1117 899 y Fl(;)g(G)32 b Fs(are)h(readily)e(computed)i(b)s(elo)m (w.)1016 1150 y Fl(E)1088 1165 y Fk(ij)1193 1150 y Fs(=)1313 1014 y Fj(Z)1413 1041 y 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b(is)g(of)g(main)f(in)m(terest)i(to)g(us,)g(let)f(us)h (consider)g(a)f(domain)f(\012)i(suc)m(h)h(that,)f(in)456 2442 y(notation)f(of)h(subsection)h(2.1)g Fl(x)28 b Fp(2)g Fs(\()p Fl(a;)17 b Fp(1)p Fs(\))32 b(and)1470 2602 y Fl(')p Fs(\()p Fl(x)p Fs(\))45 b Fp(\030)g Fs(1)p Fl(;)211 b(x)45 b Fp(\025)g Fl(X)822 b Fs(\(3.4\))456 2763 y(for)28 b(some)h Fl(X)35 b(>)28 b Fs(0.)42 b(Similar)25 b(assumptions)k(are)g (often)g(made)g(in)f(pap)s(ers)i(dealing)456 2879 y(with)35 b(scattering)g(problems,)h(for)f(instance)h(in)f([1,)g(15,)h(6)o(].)53 b(W)-8 b(e)36 b(tak)m(e)h(Diric)m(hlet)456 2995 y(b)s(oundary)29 b(conditions)f(on)g Fl(@)5 b Fs(\012)31 b(and)e(require)g(a)f (di\013eren)m(t)h(t)m(yp)s(e)h(of)e(condition)g(to)456 3111 y(b)s(e)j(satis\014ed)g(as)g Fp(j)p Fl(x)p Fp(j)c(!)h(1)p Fs(.)42 b(Namely)-8 b(,)30 b(for)h(a)f(giv)m(en)h Fl(\025)d Fp(2)g Ff(C)57 b Fs(there)31 b(alw)m(a)m(ys)h(exists)456 3228 y(a)g(unique)h(solution)e(of)h(\(2.2\))g(that)g(has)h(the)g(form) 456 3468 y Fl(f)11 b Fs(\()p Fl(x;)17 b(y)t Fs(\))26 b(=)i(\(exp)q(\()p Fp(\000)p Fl(t)1209 3483 y Fo(1)1249 3468 y Fl(x)p Fs(\))22 b(+)g Fl(s)1508 3483 y Fo(1)1564 3468 y Fs(exp)q(\()p Fl(t)1786 3483 y Fo(1)1826 3468 y Fl(x)p Fs(\)\))17 b Fl(g)2021 3483 y Fo(1)2060 3468 y Fs(\()p Fl(y)t Fs(\))k(+)2344 3343 y Fg(1)2307 3373 y Fj(X)2315 3586 y Fk(k)r Fo(=2)2468 3468 y Fl(s)2514 3483 y Fk(k)2556 3468 y Fl(g)2603 3483 y Fk(k)2646 3468 y Fs(\()p Fl(y)t Fs(\))c(exp)o(\()p Fl(t)3011 3483 y Fk(k)3054 3468 y Fl(x)p Fs(\))p Fl(;)50 b(x)28 b Fp(\025)g Fl(X)r(:)3244 3679 y Fs(\(3.5\))456 3839 y(Here)33 b(w)m(e)h(denote) 1226 3999 y Fl(t)1261 4014 y Fk(k)1331 3999 y Fs(=)28 b Fp(\000)1512 3909 y Fj(p)p 1612 3909 407 4 v 90 x Fs(\()p Fl(\031)t(k)s Fs(\))1801 3971 y Fo(2)1862 3999 y Fp(\000)23 b Fl(\025;)212 b Fs(Re)16 b Fl(t)2424 4014 y Fk(k)2495 3999 y Fl(<)27 b Fs(0;)456 4160 y Fl(g)503 4175 y Fk(k)580 4160 y Fs(are)36 b(the)g(same)f(as)g(in)g(subsection)h(2.2.)52 b(The)36 b(co)s(e\016cien)m(ts)h Fl(s)2828 4175 y 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arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /MFshow {{dup dup 0 get findfont exch 1 get scalefont setfont [ currentpoint ] exch dup 2 get 0 exch rmoveto dup dup 5 get exch 4 get {show} {stringwidth pop 0 rmoveto}ifelse dup 3 get {2 get neg 0 exch rmoveto pop} {pop aload pop moveto}ifelse} forall} bind def /MFwidth {0 exch {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont 5 get stringwidth pop add} {pop} ifelse} forall} bind def /MLshow { currentpoint stroke M 0 exch R MFshow } bind def /MRshow { currentpoint stroke M exch dup MFwidth neg 3 -1 roll R MFshow } def /MCshow { currentpoint stroke M exch dup MFwidth -2 div 3 -1 roll R MFshow } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 250 scalefont setfont LTb 3450 375 M 0 63 V 0 4352 R 0 -63 V LTa 63 375 M 6774 0 V LTa 3450 375 M 0 4415 V LTb 3992 1663 M [ [(Symbol) 250.0 0.0 true true (W)] ] -83.3 MCshow 402 4422 M [ [(Helvetica) 250.0 0.0 true true (1\))] ] -83.3 MCshow 1282 4422 M [ [(Symbol) 250.0 0.0 true true (a)] [(Helvetica) 250.0 0.0 true true ( = 0.9)] ] -83.3 MCshow 1.000 UL LT0 63 375 M 68 0 V 69 0 V 68 0 V 69 0 V 68 0 V 69 0 V 68 0 V 68 0 V 69 0 V 68 0 V 69 0 V 68 0 V 69 0 V 68 0 V 68 0 V 69 0 V 68 0 V 69 0 V 68 0 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def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C3 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill vpt 0 360 arc closepath } bind def /C4 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /MFshow {{dup dup 0 get findfont exch 1 get scalefont setfont [ currentpoint ] exch dup 2 get 0 exch rmoveto dup dup 5 get exch 4 get {show} {stringwidth pop 0 rmoveto}ifelse dup 3 get {2 get neg 0 exch rmoveto pop} {pop aload pop moveto}ifelse} forall} bind def /MFwidth {0 exch {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont 5 get stringwidth pop add} {pop} ifelse} forall} bind def /MLshow { currentpoint stroke M 0 exch R MFshow } bind def /MRshow { currentpoint stroke M exch dup MFwidth neg 3 -1 roll R MFshow } def /MCshow { currentpoint stroke M exch dup MFwidth -2 div 3 -1 roll R MFshow } def end %%EndProlog gnudict begin gsave 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sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /MFshow {{dup dup 0 get findfont exch 1 get scalefont setfont [ currentpoint ] exch dup 2 get 0 exch rmoveto dup dup 5 get exch 4 get {show} {stringwidth pop 0 rmoveto}ifelse dup 3 get {2 get neg 0 exch rmoveto pop} {pop aload pop moveto}ifelse} forall} bind def /MFwidth {0 exch {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont 5 get stringwidth pop add} {pop} ifelse} forall} bind def /MLshow { currentpoint stroke M 0 exch R MFshow } bind def /MRshow { currentpoint stroke M exch dup MFwidth neg 3 -1 roll R MFshow } def /MCshow { currentpoint stroke M exch dup MFwidth -2 div 3 -1 roll R MFshow } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 200 scalefont setfont LTb 903 434 M 63 0 V 5931 0 R -63 0 V stroke 783 434 M [ [(Helvetica) 200.0 0.0 true true (4.3)] ] -66.7 MRshow 903 1101 M 63 0 V 5931 0 R -63 0 V stroke 783 1101 M [ [(Helvetica) 200.0 0.0 true true (4.35)] ] -66.7 MRshow 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Fs(Eigen)m(v)-5 b(alues)33 b(of)f(problem)f Fv(1)i Fs(for)f(a)g(range)h(of)f Fl(\013)1327 3208 y @beginspecial -40 @hoffset -25 @voffset 50 @hscale 50 @vscale @setspecial %%BeginDocument: re_im.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: re_im.eps %%Creator: gnuplot %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 120 dict def gnudict begin /Color false def /Solid true def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -66 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M 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closepath fill vpt 0 360 arc closepath } bind def /C5 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc 2 copy moveto 2 copy vpt 180 270 arc closepath fill vpt 0 360 arc } bind def /C6 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 270 arc closepath fill vpt 0 360 arc closepath } bind def /C7 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 270 arc closepath fill vpt 0 360 arc closepath } bind def /C8 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C9 { BL [] 0 setdash 2 copy moveto 2 copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub exch vpt sub exch vpt2 Square } bind def /S0 { BL [] 0 setdash 2 copy moveto 0 vpt rlineto BL Bsquare } bind def /S1 { BL [] 0 setdash 2 copy vpt Square fill Bsquare } bind def /S2 { BL [] 0 setdash 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S3 { BL [] 0 setdash 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S4 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S5 { BL [] 0 setdash 2 copy 2 copy vpt Square fill exch vpt sub exch vpt sub vpt Square fill Bsquare } bind def /S6 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill Bsquare } bind def /S7 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt vpt2 Rec fill 2 copy vpt Square fill Bsquare } bind def /S8 { BL [] 0 setdash 2 copy vpt sub vpt Square fill Bsquare } bind def /S9 { BL [] 0 setdash 2 copy vpt sub vpt vpt2 Rec fill Bsquare } bind def /S10 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S11 { BL [] 0 setdash 2 copy vpt sub vpt Square fill 2 copy exch vpt sub exch vpt2 vpt Rec fill Bsquare } bind def /S12 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill Bsquare } bind def /S13 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy vpt Square fill Bsquare } bind def /S14 { BL [] 0 setdash 2 copy exch vpt sub exch vpt sub vpt2 vpt Rec fill 2 copy exch vpt sub exch vpt Square fill Bsquare } bind def /S15 { BL [] 0 setdash 2 copy Bsquare fill Bsquare } bind def /D0 { gsave translate 45 rotate 0 0 S0 stroke grestore } bind def /D1 { gsave translate 45 rotate 0 0 S1 stroke grestore } bind def /D2 { gsave translate 45 rotate 0 0 S2 stroke grestore } bind def /D3 { gsave translate 45 rotate 0 0 S3 stroke grestore } bind def /D4 { gsave translate 45 rotate 0 0 S4 stroke grestore } bind def /D5 { gsave translate 45 rotate 0 0 S5 stroke grestore } bind def /D6 { gsave translate 45 rotate 0 0 S6 stroke grestore } bind def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke } def /PentE { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore } def /CircE { stroke [] 0 setdash hpt 0 360 arc stroke } def /BoxFill { gsave Rec 1 setgray fill grestore } def /MFshow {{dup dup 0 get findfont exch 1 get scalefont setfont [ currentpoint ] exch dup 2 get 0 exch rmoveto dup dup 5 get exch 4 get {show} {stringwidth pop 0 rmoveto}ifelse dup 3 get {2 get neg 0 exch rmoveto pop} {pop aload pop moveto}ifelse} forall} bind def /MFwidth {0 exch {dup 3 get{dup dup 0 get findfont exch 1 get scalefont setfont 5 get stringwidth pop add} {pop} ifelse} forall} bind def /MLshow { currentpoint stroke M 0 exch R MFshow } bind def /MRshow { currentpoint stroke M exch dup MFwidth neg 3 -1 roll R MFshow } def /MCshow { currentpoint stroke M exch dup MFwidth -2 div 3 -1 roll R MFshow } def end %%EndProlog gnudict begin gsave 50 50 translate 0.050 0.050 scale 0 setgray newpath (Helvetica) findfont 200 scalefont setfont LTb 903 300 M 63 0 V 5931 0 R -63 0 V stroke 783 300 M [ [(Helvetica) 200.0 0.0 true true (0)] ] -66.7 MRshow 903 1057 M 63 0 V 5931 0 R -63 0 V stroke 783 1057 M [ [(Helvetica) 200.0 0.0 true true (0.01)] ] -66.7 MRshow 903 1813 M 63 0 V 5931 0 R -63 0 V stroke 783 1813 M [ [(Helvetica) 200.0 0.0 true true (0.02)] ] -66.7 MRshow 903 2570 M 63 0 V 5931 0 R -63 0 V 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copy vpt 270 450 arc closepath fill vpt 0 360 arc closepath } bind def /C10 { BL [] 0 setdash 2 copy 2 copy moveto vpt 270 360 arc closepath fill 2 copy moveto 2 copy vpt 90 180 arc closepath fill vpt 0 360 arc closepath } bind def /C11 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 180 arc closepath fill 2 copy moveto 2 copy vpt 270 360 arc closepath fill vpt 0 360 arc closepath } bind def /C12 { BL [] 0 setdash 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C13 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill 2 copy moveto 2 copy vpt 180 360 arc closepath fill vpt 0 360 arc closepath } bind def /C14 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 360 arc closepath fill vpt 0 360 arc } bind def /C15 { BL [] 0 setdash 2 copy vpt 0 360 arc closepath fill vpt 0 360 arc closepath } bind def /Rec { newpath 4 2 roll moveto 1 index 0 rlineto 0 exch rlineto neg 0 rlineto closepath } bind def /Square { dup Rec } bind def /Bsquare { vpt sub 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def /D7 { gsave translate 45 rotate 0 0 S7 stroke grestore } bind def /D8 { gsave translate 45 rotate 0 0 S8 stroke grestore } bind def /D9 { gsave translate 45 rotate 0 0 S9 stroke grestore } bind def /D10 { gsave translate 45 rotate 0 0 S10 stroke grestore } bind def /D11 { gsave translate 45 rotate 0 0 S11 stroke grestore } bind def /D12 { gsave translate 45 rotate 0 0 S12 stroke grestore } bind def /D13 { gsave translate 45 rotate 0 0 S13 stroke grestore } bind def /D14 { gsave translate 45 rotate 0 0 S14 stroke grestore } bind def /D15 { gsave translate 45 rotate 0 0 S15 stroke grestore } bind def /DiaE { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke } def /BoxE { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke } def /TriUE { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke } def /TriDE { stroke [] 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499 319 V 200 107 V 200 138 V 599 372 V 903 1000 Pls 1902 1663 Pls 2901 2310 Pls 3400 2626 Pls 3900 2936 Pls 4400 3246 Pls 4899 3546 Pls 5399 3837 Pls 5898 4156 Pls 6098 4263 Pls 6298 4401 Pls 6897 4773 Pls stroke grestore end showpage %%Trailer %%DocumentFonts: Symbol, Helvetica %%EndDocument @endspecial 32 w @beginspecial 110 @hoffset -20 @voffset 50 @hscale 50 @vscale @setspecial %%BeginDocument: im.eps %!PS-Adobe-2.0 EPSF-2.0 %%Title: im.eps %%Creator: gnuplot %%DocumentFonts: (atend) %%BoundingBox: 50 50 410 302 %%Orientation: Portrait %%EndComments /gnudict 120 dict def gnudict begin /Color false def /Solid true def /gnulinewidth 5.000 def /userlinewidth gnulinewidth def /vshift -66 def /dl {10 mul} def /hpt_ 31.5 def /vpt_ 31.5 def /hpt hpt_ def /vpt vpt_ def /M {moveto} bind def /L {lineto} bind def /R {rmoveto} bind def /V {rlineto} bind def /vpt2 vpt 2 mul def /hpt2 hpt 2 mul def /Lshow { currentpoint stroke M 0 vshift R show } def /Rshow { currentpoint stroke M dup stringwidth pop neg vshift R show } def /Cshow { currentpoint stroke M dup stringwidth pop -2 div vshift R show } def /UP { dup vpt_ mul /vpt exch def hpt_ mul /hpt exch def /hpt2 hpt 2 mul def /vpt2 vpt 2 mul def } def /DL { Color {setrgbcolor Solid {pop []} if 0 setdash } {pop pop pop Solid {pop []} if 0 setdash} ifelse } def /BL { stroke gnulinewidth 2 mul setlinewidth } def /AL { stroke gnulinewidth 2 div setlinewidth } def /UL { gnulinewidth mul /userlinewidth exch def } def /PL { stroke userlinewidth setlinewidth } def /LTb { BL [] 0 0 0 DL } def /LTa { AL [1 dl 2 dl] 0 setdash 0 0 0 setrgbcolor } def /LT0 { PL [] 0 1 0 DL } def /LT1 { PL [4 dl 2 dl] 0 0 1 DL } def /LT2 { PL [2 dl 3 dl] 1 0 0 DL } def /LT3 { PL [1 dl 1.5 dl] 1 0 1 DL } def /LT4 { PL [5 dl 2 dl 1 dl 2 dl] 0 1 1 DL } def /LT5 { PL [4 dl 3 dl 1 dl 3 dl] 1 1 0 DL } def /LT6 { PL [2 dl 2 dl 2 dl 4 dl] 0 0 0 DL } def /LT7 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 1 0.3 0 DL } def /LT8 { PL [2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 2 dl 4 dl] 0.5 0.5 0.5 DL } def /Pnt { stroke [] 0 setdash gsave 1 setlinecap M 0 0 V stroke grestore } def /Dia { stroke [] 0 setdash 2 copy vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath stroke Pnt } def /Pls { stroke [] 0 setdash vpt sub M 0 vpt2 V currentpoint stroke M hpt neg vpt neg R hpt2 0 V stroke } def /Box { stroke [] 0 setdash 2 copy exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath stroke Pnt } def /Crs { stroke [] 0 setdash exch hpt sub exch vpt add M hpt2 vpt2 neg V currentpoint stroke M hpt2 neg 0 R hpt2 vpt2 V stroke } def /TriU { stroke [] 0 setdash 2 copy vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath stroke Pnt } def /Star { 2 copy Pls Crs } def /BoxF { stroke [] 0 setdash exch hpt sub exch vpt add M 0 vpt2 neg V hpt2 0 V 0 vpt2 V hpt2 neg 0 V closepath fill } def /TriUF { stroke [] 0 setdash vpt 1.12 mul add M hpt neg vpt -1.62 mul V hpt 2 mul 0 V hpt neg vpt 1.62 mul V closepath fill } def /TriD { stroke [] 0 setdash 2 copy vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath stroke Pnt } def /TriDF { stroke [] 0 setdash vpt 1.12 mul sub M hpt neg vpt 1.62 mul V hpt 2 mul 0 V hpt neg vpt -1.62 mul V closepath fill} def /DiaF { stroke [] 0 setdash vpt add M hpt neg vpt neg V hpt vpt neg V hpt vpt V hpt neg vpt V closepath fill } def /Pent { stroke [] 0 setdash 2 copy gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath stroke grestore Pnt } def /PentF { stroke [] 0 setdash gsave translate 0 hpt M 4 {72 rotate 0 hpt L} repeat closepath fill grestore } def /Circle { stroke [] 0 setdash 2 copy hpt 0 360 arc stroke Pnt } def /CircleF { stroke [] 0 setdash hpt 0 360 arc fill } def /C0 { BL [] 0 setdash 2 copy moveto vpt 90 450 arc } bind def /C1 { BL [] 0 setdash 2 copy moveto 2 copy vpt 0 90 arc closepath fill vpt 0 360 arc closepath } bind def /C2 { BL [] 0 setdash 2 copy moveto 2 copy vpt 90 180 arc 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b(Abrahamsson)e(and)i (H.-O.)f(Kreiss.)g(Numerical)g(solution)g(of)h(the)g(coupled)f(mo)r(de) 626 5016 y(equations)27 b(in)h(duct)g(acoustics.)f Fn(J.)i(c)l(omput.)h (Phys.)p Fu(,)f(111:1{14,)c(1994.)497 5116 y([2])41 b(A.)36 b(A.)f(Abramo)n(v.)f(A)i(v)n(ersion)e(of)h(the)g(piv)n(otal)g (condensation)f(metho)r(d.)i Fn(Zh.)h(vychisl.)626 5216 y(Mat.)31 b(mat.)f(Fiz.)h(\(R)n(ussian\))p Fu(,)c(1:349{351,)d(1961.)p eop %%Page: 18 18 18 17 bop 456 236 a Fm(18)793 b(A.)33 b(ASLANY)-6 b(AN)30 b(AND)i(E.B.)j(D)n(A)-9 b(VIES)497 425 y Fu([3])41 b(A.)31 b(A.)f(Abramo)n(v.)f(A)i(metho)r(d)g(of)f(\014nding)g(the)h(eigen)n(v) -5 b(alues)29 b(and)h(eigenfunctions)g(of)g(a)626 525 y(self-conjugate)23 b(di\013eren)n(tial)h(problem.)f Fn(Comp.)28 b(Maths)f(Math.)h(Phys.)p Fu(,)e(31:27{36,)21 b(1991.)497 624 y([4])41 b(A.)30 b(A.)f(Abramo)n(v)f(and)g(L.)h(F.)h(Y) -7 b(ukhno.)29 b(Determining)g(the)g(n)n(um)n(b)r(er)g(of)g(eigen)n(v) -5 b(alues)28 b(of)626 724 y(a)f(sp)r(ectral)g(problem.)h Fn(Comp.)j(Maths)f(Math.)i(Phys.)p Fu(,)d(34:671{677,)23 b(1994.)497 824 y([5])41 b(S.)26 b(Agmon.)f(A)h(p)r(erturbation)f (theory)g(of)g(resonances.)f Fn(Communs)j(on)h(Pur)l(e)g(and)g(Appl.) 626 923 y(Maths)p Fu(,)h(LI:1255{1309,)23 b(1998.)497 1023 y([6])41 b(A.)31 b(Aslan)n(y)n(an,)e(L.)h(P)n(arno)n(vski,)e(and)i (D.)g(V)-7 b(assiliev.)29 b(Complex)h(resonances)e(in)i(acoustic)626 1123 y(w)n(a)n(v)n(eguides.)c Fn(Q.)k(Jl)f(Me)l(ch.)i(appl.)h(Math.)p Fu(,)d(53:429{447,)23 b(2000.)497 1224 y([7])41 b(D.)28 b(Boriso)n(v,)d(P)-7 b(.)26 b(Exner,)h(R.)g(Gadylshin,)g(and)g(D.)g (Krej)n(\024)-39 b(ci)l(\024)i(r)2477 1205 y(\023)2486 1224 y(ik.)24 b(Bound)j(states)f(in)i(w)n(eakly)626 1324 y(deformed)g(strips)f(and)g(la)n(y)n(ers.)f Fn(\(to)j(app)l(e)l(ar\))p Fu(,)h(2000.)497 1424 y([8])41 b(H.)31 b(L.)f(Cycon,)g(R.)g(G.)g(F)-7 b(ro)r(ese,)30 b(W.)g(Kirsc)n(h,)g(and)f(B.)h(Simon.)g Fn(Schr\177)-42 b(odinger)34 b(op)l(er)l(ators;)626 1523 y(with)c(applic)l(ation)h(to)e(quantum)e(me)l(chanics)j(and)f(glob)l (al)h(ge)l(ometry)p Fu(.)e(T)-7 b(exts)26 b(and)g(Mono-)626 1623 y(graphs)g(in)i(Ph)n(ysics.)f(Springer{V)-7 b(erlag,)25 b(Berlin,)i(1987.)497 1723 y([9])41 b(E.)29 b(B.)h(Da)n(vies)e(and)i (L.)f(P)n(arno)n(vski.)e(T)-7 b(rapp)r(ed)29 b(mo)r(des)g(in)h (acoustic)f(w)n(a)n(v)n(eguides.)e Fn(Q.)32 b(Jl)626 1822 y(Me)l(ch.)f(appl.)h(Math.)p Fu(,)d(51:477{492,)24 b(1998.)456 1922 y([10])40 b(D.)e(V.)g(Ev)-5 b(ans,)39 b(M.)e(Levitin,)j(and)d(D.)h(V)-7 b(assiliev.)37 b(Existence)f (theorems)h(for)g(trapp)r(ed)626 2021 y(mo)r(des.)28 b Fn(J.)i(Fluid)g(Me)l(ch.)p Fu(,)f(261:21{31,)24 b(1994.)456 2123 y([11])40 b(P)-7 b(.)21 b(Exner)e(and)i(D.)g(Krej)n(\024)-39 b(ci)l(\024)i(r)1464 2103 y(\023)1473 2123 y(ik.)18 b(Quan)n(tum)i(w)n (a)n(v)n(eguides)f(with)i(a)f(lateral)g(semitransparen)n(t)626 2223 y(barrier:)35 b(sp)r(ectral)27 b(and)h(scattering)e(prop)r (erties.)h Fn(J.)j(Phys.)h(A)p Fu(,)c(32:4475{4494,)c(1999.)456 2325 y([12])40 b(P)-7 b(.)40 b(Exner)f(and)h(D.)g(Krej)n(\024)-39 b(ci)l(\024)i(r)1541 2305 y(\023)1550 2325 y(ik.)37 b(W)-7 b(a)n(v)n(eguides)39 b(coupled)g(through)h(a)f(semitransparen)n(t)626 2424 y(barrier:)c(a)28 b(Birman{Sc)n(h)n(winger)d(analysis.)h Fn(\(to)k(app)l(e)l(ar\))p Fu(,)f(2000.)456 2526 y([13])40 b(P)-7 b(.)34 b(Exner,)h(P)-7 b(.)1117 2505 y(\024)1115 2526 y(Seba,)35 b(M.)f(T)-7 b(ater,)35 b(and)f(D.)h(V)-7 b(an)n(\024)-39 b(ek.)33 b(Bound)h(states)f(and)h(scattering)f(in)626 2626 y(quan)n(tum)d(w)n(a)n(v)n(eguides)d(coupled)i(laterally)f (through)g(a)h(b)r(oundary)g(windo)n(w.)g Fn(J.)i(Math.)626 2726 y(Phys.)p Fu(,)e(37:4867{4887,)23 b(1996.)456 2825 y([14])40 b(H.)23 b(D.)g(Maniar)e(and)h(J.)g(N.)g(Newman.)h(W)-7 b(a)n(v)n(e)21 b(di\013raction)h(b)n(y)g(a)g(long)f(arra)n(y)f(of)i (cylinders.)626 2925 y Fn(J.)30 b(Fluid)h(Me)l(ch.)p Fu(,)e(339:309{330,)23 b(1997.)456 3025 y([15])40 b(K.)e(Otto)g(and)g (E.)f(Larsson.)f(Iterativ)n(e)h(solution)h(of)g(the)g(Helmholtz)h (equation)e(b)n(y)h(a)626 3124 y(second-order)25 b(metho)r(d.)k Fn(SIAM)g(J.)h(Matrix)g(A)n(nal.)g(Appl.)p Fu(,)f(21:209{229,)24 b(1999.)456 3224 y([16])40 b(R.)28 b(S.)g(V)-7 b(arga.)26 b Fn(Matrix)k(Iter)l(ative)g(A)n(nalysis)p Fu(.)f(Pren)n(tice-Hall,)d (Englew)n(o)r(o)r(d)g(Cli\013s,)i(1962.)456 3326 y([17])40 b(P)-7 b(.)731 3305 y(\024)728 3326 y(Seba,)30 b(I.)g(Rotter,)f(M.)h(M) r(\177)-44 b(uller,)30 b(E.)f(P)n(ersson,)f(and)h(K.)h(Pic)n(h)n(ugin.) e(Collectiv)n(e)h(mo)r(des)626 3426 y(in)f(an)f(op)r(en)h(micro)n(w)n (a)n(v)n(e)d(billiard.)i Fn(Physic)l(al)32 b(R)l(eview)e(E)p Fu(,)e(61:66{70,)d(2000.)555 3613 y Ft(Dep)-6 b(ar)g(tment)36 b(of)d(Ma)-6 b(thema)g(tics,)36 b(King's)d(College)h(London,)e(Strand,) i(Lon-)456 3712 y(don)c(W)n(C2R)i(2LS,)e(UK)555 3812 y Fn(E-mail)h(addr)l(ess)7 b Fu(:)38 b Fa(aslanyan@mth.kcl)o(.a)o(c.u)o (k,)f(e.brian.davies@k)o(cl)o(.ac)o(.u)o(k)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0012190524211--