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FC(Here)g(the)g Fw(split)e FC(en)o(vironmen)o(t)g (gets)i(a)f(n)o(um)o(b)q(er)f(from)g(the)i(outer)g Fw(gather)d FC(en)o(vironmen)o(t;)258 1012 y(n)o(um)o(b)q(ers)k(for)h(individual)e (lines)i(of)f(the)h Fw(align*)f FC(are)h(suppressed)j(b)q(ecause)e(of)e (the)i(star.)258 1102 y Fw(\\begin{gather})258 1151 y(\\begin{split})k (\\varphi\(x,z\))258 1201 y(&=z-\\gamma_{10}x)o(-\\gam)o(ma_{m)o(n}x^m) o(z^n\\\\)258 1251 y(&=z-Mr^{-1}x-Mr^)o({-\(m+)o(n\)}x^)o(mz^n)258 1301 y(\\end{split}\\\\[6p)o(t])258 1351 y(\\begin{align*})258 1401 y(\\zeta^0)h(&=\(\\xi^0\)^2,\\\\)258 1450 y(\\zeta^1)g (&=\\xi^0\\xi^1,\\\\)258 1500 y(\\zeta^2)g(&=\(\\xi^1\)^2,)258 1550 y(\\end{align*})258 1600 y(\\end{gather})320 1690 y FC(The)14 b Fw(*)p FC(-ed)g(form)e(of)i Fw(gather)e FC(with)i(the)g(non-)p Fw(*)p FC(-ed)g(form)e(of)h Fw(align)p FC(.)610 1778 y Fy(')p FC(\()p Fy(x;)7 b(z)r FC(\))12 b(=)g Fy(z)f FB(\000)e Fy(\015)881 1784 y FA(10)917 1778 y Fy(x)g FB(\000)h Fy(\015)1013 1784 y Fx(mn)1065 1778 y Fy(x)1089 1761 y Fx(m)1120 1778 y Fy(z)1141 1761 y Fx(n)745 1849 y FC(=)i Fy(z)f FB(\000)e Fy(M)c(r)925 1832 y Fu(\000)p FA(1)970 1849 y Fy(x)k FB(\000)g Fy(M)c(r)1109 1832 y Fu(\000)p FA(\()p Fx(m)p FA(+)p Fx(n)p FA(\))1238 1849 y Fy(x)1262 1832 y Fx(m)1293 1849 y Fy(z)1314 1832 y Fx(n)872 1941 y Fy(\020)893 1924 y FA(0)924 1941 y FC(=)11 b(\()p Fy(\030)1003 1924 y FA(0)1022 1941 y FC(\))1038 1924 y FA(2)1057 1941 y Fy(;)547 b FC(\(79\))872 2008 y Fy(\020)893 1991 y FA(1)924 2008 y FC(=)11 b Fy(\030)987 1991 y FA(0)1006 2008 y Fy(\030)1026 1991 y FA(1)1045 2008 y Fy(;)559 b FC(\(80\))872 2076 y Fy(\020)893 2059 y FA(2)924 2076 y FC(=)11 b(\()p Fy(\030)1003 2059 y FA(1)1022 2076 y FC(\))1038 2059 y FA(2)1057 2076 y Fy(;)547 b FC(\(81\))258 2164 y(Some)12 b(text)j(after)f(to)g(test)h(the)f(b)q (elo)o(w-displa)o(y)f(spacing.)258 2254 y Fw(\\begin{gather*})258 2304 y(\\begin{split})19 b(\\varphi\(x,z\))258 2354 y (&=z-\\gamma_{10}x)o(-\\gam)o(ma_{m)o(n}x^m)o(z^n\\\\)258 2403 y(&=z-Mr^{-1}x-Mr^)o({-\(m+)o(n\)}x^)o(mz^n)258 2453 y(\\end{split}\\\\[6p)o(t])258 2503 y(\\begin{align})g (\\zeta^0&=\(\\xi^0\)^)o(2,\\\\)p 1690 2503 V eop %%Page: 36 6 36 5 bop 257 116 a Ft(Sample)13 b(pap)q(er)h(for)g(the)g Fw(amsmath)f Ft(pac)o(k)n(age)686 b FC(36)p 257 2503 1 2284 v 258 262 a Fh(A.4)56 b(Align)258 338 y FC(Num)o(b)q(ered)14 b(v)o(ersion:)546 424 y Fy(\015)567 430 y Fx(x)589 424 y FC(\()p Fy(t)p FC(\))d(=)h(\(cos)c Fy(tu)h FC(+)g(sin)e Fy(tx;)g(v)q FC(\))p Fy(;)592 b FC(\(72\))547 486 y Fy(\015)568 492 y Fx(y)589 486 y FC(\()p Fy(t)p FC(\))11 b(=)h(\()p Fy(u;)7 b FC(cos)g Fy(tv)k FC(+)e(sin)e Fy(ty)q FC(\))p Fy(;)595 b FC(\(73\))548 578 y Fy(\015)569 584 y Fx(z)589 578 y FC(\()p Fy(t)p FC(\))11 b(=)691 519 y Fv(\022)722 578 y FC(cos)c Fy(tu)i FC(+)879 550 y Fy(\013)p 879 568 27 2 v 880 606 a(\014)917 578 y FC(sin)e Fy(tv)q(;)g FB(\000)1068 550 y Fy(\014)p 1067 568 V 1067 606 a(\013)1106 578 y FC(sin)g Fy(tu)i FC(+)g(cos)f Fy(tv)1352 519 y Fv(\023)1390 578 y Fy(:)214 b FC(\(74\))258 686 y(Some)12 b(text)j(after)f(to)g(test)h(the)f(b)q(elo)o(w-displa)o(y)f(spacing.) 258 772 y Fw(\\begin{align})258 822 y(\\gamma_x\(t\)&=\(\\c)o(os)19 b(tu+\\sin)h(tx,v\),\\\\)258 872 y(\\gamma_y\(t\)&=\(u,)o(\\cos)e (tv+\\sin)j(ty\),\\\\)258 922 y(\\gamma_z\(t\)&=\\le)o(ft\(\\c)o(os)e (tu+\\frac\\alpha\\be)o(ta\\s)o(in)g(tv,)301 972 y (-\\frac\\beta\\alpha\\s)o(in)g(tu+\\cos)h(tv\\right\).)258 1021 y(\\end{align})320 1107 y FC(Unn)o(um)o(b)q(ered)14 b(v)o(ersion:)546 1193 y Fy(\015)567 1199 y Fx(x)589 1193 y FC(\()p Fy(t)p FC(\))d(=)h(\(cos)c Fy(tu)h FC(+)g(sin)e Fy(tx;)g(v)q FC(\))p Fy(;)547 1255 y(\015)568 1261 y Fx(y)589 1255 y FC(\()p Fy(t)p FC(\))k(=)h(\()p Fy(u;)7 b FC(cos)g Fy(tv)k FC(+)e(sin)e Fy(ty)q FC(\))p Fy(;)548 1347 y(\015)569 1353 y Fx(z)589 1347 y FC(\()p Fy(t)p FC(\))k(=)691 1289 y Fv(\022)722 1347 y FC(cos)c Fy(tu)i FC(+)879 1319 y Fy(\013)p 879 1338 V 880 1376 a(\014)917 1347 y FC(sin)e Fy(tv)q(;)g FB(\000)1068 1319 y Fy(\014)p 1067 1338 V 1067 1376 a(\013)1106 1347 y FC(sin)g Fy(tu)i FC(+)g(cos)f Fy(tv)1352 1289 y Fv(\023)1390 1347 y Fy(:)258 1456 y FC(Some)k(text)j(after)f(to)g(test)h(the)f(b)q(elo)o(w-displa)o (y)f(spacing.)258 1542 y Fw(\\begin{align*})258 1591 y(\\gamma_x\(t\)&=\(\\c)o(os)19 b(tu+\\sin)h(tx,v\),\\\\)258 1641 y(\\gamma_y\(t\)&=\(u,)o(\\cos)e(tv+\\sin)j(ty\),\\\\)258 1691 y(\\gamma_z\(t\)&=\\le)o(ft\(\\c)o(os)e(tu+\\frac\\alpha\\be)o (ta\\s)o(in)g(tv,)301 1741 y(-\\frac\\beta\\alpha\\s)o(in)g(tu+\\cos)h (tv\\right\).)258 1791 y(\\end{align*})320 1876 y FC(A)14 b(v)n(ariation:)656 1962 y Fy(x)e FC(=)f Fy(y)399 b FC(b)o(y)13 b(\(84\))331 b(\(75\))645 2025 y Fy(x)669 2007 y Fu(0)692 2025 y FC(=)11 b Fy(y)756 2007 y Fu(0)1154 2025 y FC(b)o(y)i(\(85\))331 b(\(76\))570 2087 y Fy(x)9 b FC(+)h Fy(x)669 2070 y Fu(0)692 2087 y FC(=)h Fy(y)g FC(+)f Fy(y)829 2070 y Fu(0)1154 2087 y FC(b)o(y)j(Axiom)f(1.)238 b(\(77\))258 2173 y(Some)12 b(text)j(after)f(to)g(test)h(the)f(b)q(elo)o(w-displa)o(y)f(spacing.) 258 2259 y Fw(\\begin{align})258 2308 y(x&)21 b(=y)h(&&)f(\\text)g({by) g(\(\\ref{eq:C}\)}\\\\)258 2358 y(x'&)g(=)h(y')f(&&)g(\\text)g({by)g (\(\\ref{eq:D}\)}\\\\)258 2408 y(x+x')g(&)g(=)h(y+y')f(&&)g(\\text)g ({by)g(Axiom)g(1.})258 2458 y(\\end{align})p 1690 2503 1 2284 v eop %%Page: 35 7 35 6 bop 257 116 a Ft(Sample)13 b(pap)q(er)h(for)g(the)g Fw(amsmath)f Ft(pac)o(k)n(age)686 b FC(35)p 257 2503 1 2284 v 258 262 a Fh(A.3)56 b(Gather)258 338 y FC(Num)o(b)q(ered)14 b(v)o(ersion)g(with)f Fw(\\notag)g FC(on)h(the)g(second)h(line:)684 429 y Fy(D)q FC(\()p Fy(a;)7 b(r)q FC(\))12 b FB(\021)g(f)p Fy(z)h FB(2)f Fz(C)5 b FC(:)20 b FB(j)o Fy(z)12 b FB(\000)d Fy(a)p FB(j)i Fy(<)h(r)q FB(g)p Fy(;)359 b FC(\(69\))582 492 y(seg)q(\()p Fy(a;)7 b(r)q FC(\))12 b FB(\021)f(f)p Fy(z)j FB(2)d Fz(C)5 b FC(:)13 b FB(=)p Fy(z)h FC(=)e FB(=)p Fy(a;)20 b FB(j)p Fy(z)11 b FB(\000)e Fy(a)p FB(j)i Fy(<)h(r)q FB(g)p Fy(;)487 554 y(c)p FC(\()p Fy(e;)7 b(\022)q(;)g(r)q FC(\))k FB(\021)h(f)p FC(\()p Fy(x;)7 b(y)q FC(\))k FB(2)h Fz(C)5 b FC(:)20 b FB(j)p Fy(x)9 b FB(\000)g Fy(e)p FB(j)i Fy(<)h(y)d FC(tan)e Fy(\022)q(;)21 b FC(0)11 b Fy(<)h(y)h(<)f(r)q FB(g)p Fy(;)161 b FC(\(70\))744 629 y Fy(C)s FC(\()p Fy(E)r(;)7 b(\022)q(;)g(r)q FC(\))k FB(\021)983 590 y Fv([)974 679 y Fx(e)p Fu(2)p Fx(E)1045 629 y Fy(c)p FC(\()p Fy(e;)c(\022)q(;)g(r)q FC(\))p Fy(:)412 b FC(\(71\))258 760 y Fw(\\begin{gather})258 809 y (D\(a,r\)\\equiv\\{z\\)o(in\\ma)o(thbf{)o(C}\\co)o(lon)19 b(\\abs{z-a})g(\013)p FB(g)f FC(=)h(meas)997 428 y Fx(n)1019 422 y FB(f)p Fy(x)f FB(2)h Fy(R)1147 405 y Fx(n)1174 422 y FC(:)20 b FB(j)o Fy(f)t FC(\()p Fy(x)p FC(\))p FB(j)12 b(\025)g Fy(\013)p FB(g)41 b(8)p Fy(\013)11 b(>)h FC(0)p Fy(:)257 506 y Fw(\\[\\meas_1\\{u\\in)19 b(R_+^1\\colon)h(f^*\(u\)>\\alpha\\})257 556 y(=\\meas_n\\{x\\in)f (R^n\\colon)h(\\abs{f\(x\)}\\geq\\alp)o(ha\\})257 606 y(\\quad)h(\\forall\\alpha>0.\\])257 683 y(\\esssup)13 b FC(and)g Fw(\\meas)g FC(w)o(ould)g(b)q(e)i(de\014ned)g(in)e(the)i(do) q(cumen)o(t)e(pream)o(ble)g(as)257 760 y Fw(\\DeclareMathOperato)o (r*{\\e)o(sssup)o(}{ess)o(\\,sup)o(})257 810 y(\\DeclareMathOperato)o (r{\\me)o(as}{m)o(eas})320 887 y FC(The)e(follo)o(wing)d(sp)q(ecial)k (op)q(erator)f(names)f(are)i(prede\014ned)h(in)d(the)i Fw(amsmath)d FC(pac)o(k)n(age:)257 937 y Fw(\\varlimsup)p FC(,)17 b Fw(\\varliminf)p FC(,)g Fw(\\varinjlim)p FC(,)g(and)h Fw(\\varprojlim)p FC(.)28 b(Here's)20 b(what)e(they)257 987 y(lo)q(ok)13 b(lik)o(e)g(in)h(use:)p 722 1037 58 2 v 722 1070 a(lim)786 1076 y Fx(n)p Fu(!1)875 1070 y FB(Q)p FC(\()p Fy(u)949 1076 y Fx(n)971 1070 y Fy(;)7 b(u)1014 1076 y Fx(n)1045 1070 y FB(\000)j Fy(u)1111 1053 y FA(#)1140 1070 y FC(\))i FB(\024)f FC(0)384 b(\(48\))722 1133 y(lim)p 722 1140 V 786 1139 a Fx(n)p Fu(!1)882 1133 y FB(j)o Fy(a)915 1139 y Fx(n)p FA(+1)980 1133 y FB(j)6 b Fy(=)h FB(j)p Fy(a)1060 1139 y Fx(n)1082 1133 y FB(j)k FC(=)h(0)446 b(\(49\))722 1202 y(lim)722 1221 y FB(\000)-16 b(!)779 1202 y FC(\()p Fy(m)831 1184 y Fx(\025)831 1212 y(i)854 1202 y FB(\001)p FC(\))882 1184 y Fu(\003)912 1202 y FB(\024)12 b FC(0)639 b(\(50\))722 1268 y(lim)722 1288 y FB( )-16 b(\000)786 1275 y Fx(p)p Fu(2)p Fx(S)r FA(\()p Fx(A)p FA(\))901 1268 y Fy(A)932 1274 y Fx(p)962 1268 y FB(\024)12 b FC(0)589 b(\(51\))257 1352 y Fw(\\begin{align})257 1402 y(&\\varlimsup_{n\\righ)o(tarro)o(w\\inf)o(ty})410 1452 y(\\mathcal{Q}\(u_n,u)o(_n-u^)o({\\#}\))o(\\le0\\)o(\\)257 1502 y(&\\varliminf_{n\\righ)o(tarro)o(w\\inf)o(ty})301 1551 y(\\left\\lvert)19 b(a_{n+1}\\right\\rvert/)o(\\left)o(\\lve)o(rt) g(a_n\\right\\rvert=0)o(\\\\)257 1601 y(&\\varinjlim)h (\(m_i^\\lambda\\cdot)o(\)^*\\l)o(e0\\\\)257 1651 y (&\\varprojlim_{p\\in)e(S\(A\)}A_p\\le0)257 1701 y(\\end{align})257 1816 y Fh(9.12)56 b Fd(\\mod)17 b Fh(and)i(its)g(relativ)n(es)257 1892 y FC(The)13 b(commands)c Fw(\\mod)i FC(and)h Fw(\\pod)f FC(are)h(v)n(arian)o(ts)f(of)g Fw(\\pmod)g FC(preferred)j(b)o(y)e(some) f(authors;)257 1942 y Fw(\\mod)20 b FC(omits)f(the)j(paren)o(theses,)i (whereas)e Fw(\\pod)e FC(omits)f(the)j(`mo)q(d')c(and)j(retains)g(the) 257 1992 y(paren)o(theses.)g(Examples:)776 2076 y Fy(x)11 b FB(\021)h Fy(y)f FC(+)f(1)41 b(\(mo)q(d)13 b Fy(m)1136 2059 y FA(2)1155 2076 y FC(\))445 b(\(52\))776 2143 y Fy(x)11 b FB(\021)h Fy(y)f FC(+)f(1)41 b(mo)q(d)12 b Fy(m)1119 2126 y FA(2)1616 2143 y FC(\(53\))776 2211 y Fy(x)f FB(\021)h Fy(y)f FC(+)f(1)41 b(\()p Fy(m)1042 2194 y FA(2)1061 2211 y FC(\))539 b(\(54\))257 2295 y Fw(\\begin{align})257 2344 y(x&\\equiv)21 b(y+1\\pmod{m^2}\\\\)257 2394 y(x&\\equiv)g(y+1\\mod{m^2}\\\\)257 2444 y(x&\\equiv)g (y+1\\pod{m^2})257 2494 y(\\end{align})p eop %%Page: 22 20 22 19 bop 257 116 a Ft(Sample)13 b(pap)q(er)h(for)g(the)g Fw(amsmath)f Ft(pac)o(k)n(age)686 b FC(22)257 262 y Fh(9.8)56 b(Extensible)16 b(arro)n(ws)257 338 y Fw(\\xleftarrow)d FC(and)i Fw(\\xrightarrow)d FC(pro)q(duce)k(arro)o(ws)f(that)g(extend)h (automatically)c(to)257 388 y(accommo)q(date)i(un)o(usually)i(wide)g (subscripts)i(or)e(sup)q(erscripts.)27 b(The)17 b(text)g(of)e(the)i (sub-)257 438 y(script)g(or)f(sup)q(erscript)j(are)d(giv)o(en)g(as)g (an)g(optional)f(resp.)i(mandatory)d(argumen)o(t:)22 b(Ex-)257 488 y(ample:)687 585 y(0)738 561 y Fx(\013)719 585 y FB( )-14 b(\000)740 612 y Fx(\020)790 585 y Fy(F)14 b FB(\002)c(4)p FC([)p Fy(n)e FB(\000)i FC(1])1055 557 y Fx(@)1073 561 y Fs(0)1089 557 y Fx(\013)p FA(\()p Fx(b)p FA(\))1041 585 y FB(\000)-14 b(\000)-9 b(\000)g(\000)c(!)11 b Fy(E)1210 567 y Fx(@)1228 571 y Fs(0)1244 567 y Fx(b)257 686 y Fw(\\[0)22 b(\\xleftarrow[\\ze)o(ta]{\\)o(alpha)o(})d (F\\times\\triangle[)o(n-1])301 736 y(\\xrightarrow{\\par)o(tial_)o (0\\alp)o(ha\(b\))o(})g(E^{\\partial_0b}\\])257 850 y Fh(9.9)56 b Fd(\\overset)p Fh(,)15 b Fd(\\underset)p Fh(,)g(and)k Fd(\\sideset)257 927 y FC(Examples:)844 989 y Fu(\003)834 1025 y Fy(X)87 b(X)965 1050 y Fu(\003)1085 989 y Fx(a)1075 1025 y Fy(X)1087 1052 y Fx(b)257 1121 y Fw(\\[\\overset{*}{X}\\qq)o(uad\\u)o(nders)o(et{*})o({X}\\q)o(quad) 257 1171 y(\\overset{a}{\\unders)o(et{b})o({X}}\\)o(])320 1246 y FC(The)13 b(command)e Fw(\\sideset)g FC(is)i(for)g(a)g(rather)h (sp)q(ecial)f(purp)q(ose:)19 b(putting)13 b(sym)o(b)q(ols)f(at)257 1296 y(the)k(subscript)g(and)f(sup)q(erscript)h(corners)h(of)d(a)h (large)f(op)q(erator)h(sym)o(b)q(ol)e(suc)o(h)j(as)1594 1265 y Fv(P)1652 1296 y FC(or)257 1315 y Fv(Q)297 1346 y FC(,)d(without)g(a\013ecting)i(the)f(placemen)o(t)f(of)h(limits.)h (Examples:)783 1408 y Fu(\003)783 1466 y(\003)803 1402 y Fv(Y)856 1408 y Fu(\003)856 1466 y(\003)820 1493 y Fx(k)985 1402 y Fv(X)1045 1408 y Fu(0)960 1491 y FA(0)p Fu(\024)p Fx(i)p Fu(\024)p Fx(m)1076 1441 y Fy(E)1107 1447 y Fx(i)1121 1441 y Fy(\014)r(x)257 1565 y Fw (\\[\\sideset{_*^*}{_*)o(^*}\\p)o(rod_k)o(\\qqua)o(d)257 1615 y(\\sideset{}{'}\\sum_{)o(0\\le)j(i\\le)j(m})h(E_i\\beta)e(x)257 1665 y(\\])257 1779 y Fh(9.10)56 b(The)18 b Fd(\\text)f Fh(command)257 1856 y FC(The)e(main)c(use)k(of)e(the)i(command)c Fw(\\text)i 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y(the)j(same)e(t)o(yp)q(ographical)g(treatmen)o(t.)18 b(Examples:)735 2491 y FB(k)p Fy(f)t FB(k)801 2503 y Fu(1)848 2491 y FC(=)12 b(ess)7 b(sup)1012 2501 y Fx(x)p Fu(2)p Fx(R)1078 2493 y Fq(n)1108 2491 y FB(j)p Fy(f)t FC(\()p Fy(x)p FC(\))p FB(j)p eop %%Page: 21 21 21 20 bop 257 116 a Ft(Sample)13 b(pap)q(er)h(for)g(the)g Fw(amsmath)f Ft(pac)o(k)n(age)686 b FC(21)257 262 y Fh(9.4)56 b(Accen)n(ts)19 b(in)f(math)257 338 y FC(Double)c(accen)o(ts:)630 414 y(^)630 425 y(^)619 436 y Fy(H)707 417 y FC(\024)707 425 y(\024)698 436 y Fy(C)780 415 y FC(~)780 425 y(~)772 436 y Fy(T)855 414 y FC(\023)855 425 y(\023)844 436 y Fy(A)926 414 y FC(\022)926 425 y(\022)916 436 y Fy(G)1005 415 y FC(_)1005 425 y(_)991 436 y Fy(D)1077 415 y FC(\177)1077 425 y(\177)1068 436 y Fy(D)1154 414 y FC(\025)1154 425 y(\025)1145 436 y Fy(B)1230 420 y FC(\026)1230 425 y(\026)1220 436 y Fy(B)1298 413 y(~)1298 425 y(~)1295 436 y(V)257 527 y Fw(\\[\\Hat{\\Hat{H}}\\qua)o(d\\Che)o(ck{\\C)o(heck{)o(C}}\\q)o (uad)257 577 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Fv(\014)1058 1930 y(\014)1058 1955 y(\014)1081 1942 y(e)1072 1952 y Fy(D)r(u)1132 1905 y Fv(\014)1132 1930 y(\014)1132 1955 y(\014)1153 1952 y FC(\([)o Fy(t;)e(s)p FC([\)\))i FB(\000)h Fy(f)t FC(\()s(^)-24 b Fy(u)p FC(\()p Fy(t)p FC(\)\))p 745 2029 711 2 v 999 2034 a Fv(\014)999 2059 y(\014)999 2084 y(\014)1022 2071 y(e)1013 2082 y Fy(D)r(u)1073 2034 y Fv(\014)1073 2059 y(\014)1073 2084 y(\014)1093 2082 y FC(\([)p Fy(t;)7 b(s)p FC([)o(\))p eop %%Page: 13 29 13 28 bop 257 116 a Ft(Sample)13 b(pap)q(er)h(for)g(the)g Fw(amsmath)f Ft(pac)o(k)n(age)686 b FC(13)257 262 y(The)15 b(functions)f Fy(v)542 268 y Fx(h)575 262 y FC(=)e Fy(f)t FC(\()p Fy(u)683 268 y Fx(h)705 262 y FC(\))i(are)g(lo)q(cally)f(Lipsc) o(hitz)h(con)o(tin)o(uous)g(in)f(\012,)g(and)h(the)g(de\014ni-)257 311 y(tion)h(of)f(di\013eren)o(tial)h(implies)d(that)j FB(jr)p Fy(v)900 317 y Fx(h)921 311 y FB(j)e(\024)g Fy(K)d FB(jr)p Fy(u)1107 317 y Fx(h)1128 311 y FB(j)k FC(almost)f(ev)o (erywhere)k(in)d(\012.)21 b(The)257 361 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1654 y Fx(u)732 1648 y FC(\))14 b(satisfying)f(\(14\),)h(\(15\))o(,)g(then)401 1737 y(\()p Fy(v)438 1720 y FA(+)466 1737 y FC(\()p Fy(x)p FC(\))c FB(\000)f Fy(v)594 1720 y Fu(\000)623 1737 y FC(\()p Fy(x)p FC(\)\))g FB(\012)h Fy(\027)767 1743 y Fx(v)797 1737 y FC(=)i(\()p Fy(f)t FC(\()p Fy(u)921 1720 y FA(+)950 1737 y FC(\()p Fy(x)p FC(\)\))d FB(\000)h Fy(f)t FC(\()p Fy(u)1137 1720 y Fu(\000)1165 1737 y FC(\()p Fy(x)p FC(\)\)\))g FB(\012)f Fy(\027)1325 1743 y Fx(u)1388 1737 y FC(if)k Fy(x)e FB(2)h Fy(S)1526 1743 y Fx(v)257 1827 y FC(and)i Fy(f)t FC(\()p Fy(u)402 1812 y Fu(\000)431 1827 y FC(\()p Fy(x)p FC(\)\))d(=)h Fy(f)t FC(\()p Fy(u)622 1812 y FA(+)651 1827 y FC(\()p Fy(x)p FC(\)\))i(if)f Fy(x)e FB(2)g Fy(S)874 1833 y Fx(u)896 1827 y FB(n)p Fy(S)942 1833 y Fx(v)962 1827 y FC(.)18 b(Hence,)d(b)o(y)f(\(1.8\))f(w) o(e)h(get)286 1937 y Fy(J)t(v)q FC(\()p Fy(B)r FC(\))f(=)456 1881 y Fv(Z)479 1975 y Fx(B)q Fu(\\)p Fx(S)547 1979 y Fq(v)568 1937 y FC(\()p Fy(v)605 1920 y FA(+)643 1937 y FB(\000)c Fy(v)705 1920 y Fu(\000)734 1937 y FC(\))g FB(\012)h Fy(\027)822 1943 y Fx(v)848 1937 y Fy(d)p FB(H)905 1943 y Fx(n)p Fu(\000)p FA(1)981 1937 y FC(=)1025 1881 y Fv(Z)1048 1975 y Fx(B)q Fu(\\)p Fx(S)1116 1979 y Fq(v)1137 1937 y FC(\()p Fy(f)t FC(\()p Fy(u)1217 1920 y FA(+)1245 1937 y FC(\))g FB(\000)f Fy(f)t FC(\()p Fy(u)1376 1920 y Fu(\000)1405 1937 y FC(\)\))h FB(\012)f Fy(\027)1509 1943 y Fx(u)1537 1937 y Fy(d)p FB(H)1594 1943 y Fx(n)p Fu(\000)p FA(1)981 2053 y FC(=)1025 1996 y Fv(Z)1048 2090 y Fx(B)q Fu(\\)p Fx(S)1116 2094 y Fq(u)1139 2053 y FC(\()p Fy(f)t FC(\()p Fy(u)1219 2035 y FA(+)1247 2053 y FC(\))h FB(\000)f Fy(f)t FC(\()p Fy(u)1378 2035 y Fu(\000)1407 2053 y FC(\)\))g FB(\012)h Fy(\027)1511 2059 y Fx(u)1539 2053 y Fy(d)p FB(H)1596 2059 y Fx(n)p Fu(\000)p FA(1)257 2221 y FC(and)k(Lemma)d(6.1)i(is)h(pro)o(v)o(ed.)p 1659 2221 2 29 v 1661 2195 25 2 v 1661 2221 V 1686 2221 2 29 v 320 2304 a(T)m(o)k(pro)o(v)o(e)i(\(31\),)h(it)e(is)g(not)h (restrictiv)o(e)h(to)e(assume)g(that)h Fy(k)i FC(=)g(1.)34 b(Moreo)o(v)o(er,)22 b(to)257 2354 y(simplify)10 b(our)j(notation,)f (from)g(no)o(w)g(on)h(w)o(e)g(shall)f(assume)h(that)g(\012)e(=)h Fz(R)1412 2339 y Fx(n)1434 2354 y FC(.)18 b(The)13 b(pro)q(of)g(of)257 2403 y(\(31\))j(is)g(divided)f(in)o(to)g(t)o(w)o(o)g(steps.)26 b(In)15 b(the)i(\014rst)g(step)f(w)o(e)g(pro)o(v)o(e)g(the)h(statemen)o (t)e(in)h(the)257 2453 y(one-dimensional)10 b(case)i(\()p Fy(n)g FC(=)g(1\),)f(using)h(Theorem)f(5.2.)16 b(In)c(the)g(second)h (step)f(w)o(e)g(ac)o(hiev)o(e)257 2503 y(the)j(general)f(result)h (using)e(Theorem)h(7.1.)p eop %%Page: 12 30 12 29 bop 257 116 a Ft(Sample)13 b(pap)q(er)h(for)g(the)g Fw(amsmath)f Ft(pac)o(k)n(age)686 b FC(12)257 262 y Fz(Theorem)15 b(7.1.)21 b Fr(L)n(et)15 b FC(\012)c FB(\032)h Fz(R)748 246 y Fx(n)785 262 y Fr(b)n(e)j(an)h(op)n(en)f(set,)g(let)f Fy(u)d FB(2)h Fy(B)r(V)e FC(\(\012;)d Fz(R)1372 246 y Fx(m)1402 262 y FC(\))p Fr(,)15 b(and)h(let)407 372 y Fy(T)437 355 y 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FC(for)i(an)o(y)f(function)h Fy(g)6 b FC(:)20 b(]0)p Fy(;)7 b FC(+)p FB(1)p FC([)j FB(!)h Fz(R)p FC(.)257 1466 y Fr(R)n(emark)k(7.2.)21 b FC(Let)d Fy(g)6 b FC(:)14 b Fz(R)666 1451 y Fx(n)706 1466 y FB(!)j Fz(R)h FC(b)q(e)g(a)f(Lipsc)o (hitz)h(con)o(tin)o(uous)f(function)h(and)f(assume)257 1515 y(that)738 1614 y Fy(L)p FC(\()p Fy(z)r FC(\))c(=)29 b(lim)875 1643 y Fx(h)p Fu(!)p FA(0)944 1635 y Fs(+)980 1586 y Fy(g)q FC(\()p Fy(hz)r FC(\))10 b FB(\000)f Fy(g)q FC(\(0\))p 980 1605 224 2 v 1080 1643 a Fy(h)257 1716 y FC(exists)15 b(for)f(ev)o(ery)g Fy(z)g FB(2)d Fz(Q)653 1701 y Fx(n)690 1716 y FC(and)i(that)h Fy(L)g FC(is)g(a)g(linear)f (function)h(of)f Fy(z)r FC(.)18 b(Then)d Fy(g)g FC(is)f(di\013eren-)257 1766 y(tiable)g(at)f(0.)257 1830 y Fr(R)n(emark)i(7.3.)21 b FC(Let)15 b Fy(A)5 b FC(:)14 b Fz(R)673 1815 y Fx(n)708 1830 y FB(!)e Fz(R)798 1815 y Fx(m)844 1830 y FC(b)q(e)k(a)e(linear)g (function,)h(and)f(let)h Fy(f)9 b FC(:)14 b Fz(R)1462 1815 y Fx(m)1506 1830 y FB(!)f Fz(R)h 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b(If)16 b Fy(H)j FC(is)e(a)f(separator)h(then)g Fy(r)q FC(\()p FB(A)1513 2256 y Fu(0)1525 2271 y FC(\))e Fy(<)h(r)q FC(\()p FB(A)p FC(\))257 2321 y(and)e(there)h(is)f(no)g(con)o (tribution)f(from)f Fy(\031)q FC(\()p FB(A)952 2306 y Fu(0)964 2321 y Fy(;)7 b(t)p FC(\).)p 1659 2321 2 29 v 1661 2295 25 2 v 1661 2321 V 1686 2321 2 29 v 320 2403 a(The)13 b(P)o(oincar)o(\023)-20 b(e)14 b(p)q(olynomial)c(of)j(an)g (arrangemen)o(t)f(will)g(app)q(ear)i(rep)q(eatedly)g(in)f(these)257 2453 y(notes.)19 b(It)12 b(will)f(b)q(e)i(sho)o(wn)g(to)f(equal)g(the)h (P)o(oincar)o(\023)-20 b(e)13 b(p)q(olynomial)c(of)j(the)h(graded)g (algebras)257 2503 y(whic)o(h)19 b(w)o(e)h(are)f(going)f(to)h(asso)q (ciate)g(with)g FB(A)p FC(.)34 b(It)19 b(is)g(also)f(the)i(P)o(oincar)o (\023)-20 b(e)19 b(p)q(olynomial)p eop %%Page: 8 34 8 33 bop 257 116 a Ft(Sample)13 b(pap)q(er)h(for)g(the)g Fw(amsmath)f Ft(pac)o(k)n(age)707 b FC(8)320 262 y(Supp)q(ose)18 b(that)f Fy(\013)g 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Fx(B)q FA(\()p Fx(X)r FA(\))1071 1569 y FC(.)k(In)13 b(particular)h FB(j)p Fy(Q)1389 1575 y Fx(X)1420 1569 y FB(j)d FC(=)h Fy(w)1518 1554 y Fx(b)p FA(\()p Fx(X)r FA(\))1590 1569 y FC(.)320 1619 y(Next)21 b(note)g(that)f Fy(b)p FC(\()p Fy(X)s FC(\))k(=)f(dim)5 b Fy(X)s FC(.)39 b(W)m(e)20 b(see)h(this)g(b)o(y)f(c)o(ho)q(osing)h(a)f(basis)h(for)f Fy(X)257 1669 y FC(consisting)14 b(of)g(v)o(ectors)h Fy(v)659 1654 y Fx(k)693 1669 y FC(de\014ned)g(b)o(y)783 1795 y Fy(v)804 1778 y Fx(k)803 1806 y(i)837 1795 y FC(=)881 1724 y Fv(\()914 1767 y FC(1)41 b(if)13 b Fy(i)f FB(2)f FC(\003)1108 1773 y Fx(k)1129 1767 y Fy(;)914 1827 y FC(0)41 b(otherwise.)257 1922 y Fw(\\[v^{k}_{i}=)257 1971 y(\\begin{cases})19 b(1)j(&)g(\\text{if)e($i)h(\\in)g (\\Lambda_{k}$},\\\\)257 2021 y(0)h(&\\text{otherwise.)o(})d (\\end{cases})257 2071 y(\\])257 2154 y Fz(Lemma)d(6.2.)21 b Fr(L)n(et)14 b FB(A)h Fr(b)n(e)g(an)h(arr)n(angement.)i(Then)700 2251 y Fy(\037)p FC(\()p FB(A)p Fy(;)7 b(t)p FC(\))k(=)888 2212 y Fv(X)880 2301 y Fu(B)q(\022A)955 2251 y FC(\()p FB(\000)p FC(1\))1040 2234 y Fu(j)q(B)q(j)1085 2251 y Fy(t)1100 2234 y FA(dim)c Fx(T)t FA(\()p Fu(B)q FA(\))1235 2251 y Fy(:)320 2388 y FC(In)17 b(order)g(to)g(compute)g Fy(R)744 2372 y Fu(0)o(0)782 2388 y FC(recall)g(the)g(de\014nition)g (of)f Fy(S)r FC(\()p Fy(X)q(;)7 b(Y)j FC(\))17 b(from)f(Lemma)e(3.1.) 257 2437 y(Since)h Fy(H)g FB(2)d(B)q FC(,)i FB(A)543 2443 y Fx(H)587 2437 y FB(\022)e(B)q FC(.)19 b(Th)o(us)c(if)e Fy(T)6 b FC(\()p FB(B)q FC(\))13 b(=)g Fy(Y)23 b FC(then)15 b FB(B)e(2)f Fy(S)r FC(\()p Fy(H)q(;)7 b(Y)i FC(\).)20 b(Let)14 b Fy(L)1485 2422 y Fu(00)1519 2437 y FC(=)e Fy(L)p FC(\()p FB(A)1640 2422 y Fu(00)1662 2437 y FC(\).)p eop %%Page: 7 35 7 34 bop 257 116 a Ft(Sample)13 b(pap)q(er)h(for)g(the)g Fw(amsmath)f Ft(pac)o(k)n(age)707 b FC(7)257 262 y(The)13 b(function)e Fy(H)k FC(is)d Fr(B-di\013er)n(entiable)h(in)g(set)g Fy(S)i FC(if)c(it)g(is)h(B-di\013eren)o(tiable)g(at)g(ev)o(ery)h(p)q (oin)o(t)257 311 y(in)h Fy(S)r FC(.)19 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Fy(l)q FC(\))841 1478 y Fx(p)p Fu(\000)p FA(2)966 1456 y Fv(X)910 1545 y Fx(l)920 1549 y Fs(1)937 1545 y FA(+)p Fu(\001\001\001)p FA(+)p Fx(l)1027 1549 y Fq(p)1045 1545 y FA(=)p Fx(l)1107 1441 y(p)1089 1456 y Fv(Y)1088 1544 y Fx(i)p FA(=1)1149 1437 y Fv(\022)1180 1467 y Fy(n)1205 1473 y Fx(i)1186 1524 y Fy(l)1198 1530 y Fx(i)1218 1437 y Fv(\023)517 1634 y FB(\001)g FC([\()p Fy(n)g FB(\000)g Fy(l)q FC(\))h FB(\000)f FC(\()p Fy(n)762 1640 y Fx(i)785 1634 y FB(\000)h Fy(l)839 1640 y Fx(i)853 1634 y FC(\)])881 1617 y Fx(n)902 1621 y Fq(i)915 1617 y Fu(\000)p Fx(l)951 1621 y Fq(i)973 1576 y Fv(\022)1004 1634 y FC(1)f FB(\000)1086 1606 y Fy(l)1098 1612 y Fx(p)p 1080 1625 45 2 v 1080 1663 a Fy(n)1105 1669 y Fx(p)1129 1576 y Fv(\023)1167 1634 y FC([\()p Fy(n)f FB(\000)i Fy(l)q FC(\))g FB(\000)f FC(\()p Fy(n)1391 1640 y Fx(p)1420 1634 y FB(\000)g Fy(l)1473 1640 y Fx(p)1493 1634 y FC(\)])p Fy(:)1616 1558 y FC(\(22\))320 1749 y(The)14 b(en)o(umeration)g(of)f Fy(H)727 1755 y Fx(c)758 1749 y FC(in)h(a)g Fy(K)877 1755 y Fx(n)898 1759 y Fs(1)914 1755 y Fu(\001\001\001)o Fx(n)964 1759 y Fq(p)998 1749 y FC(graph)g(can)g(also)g(b)q(e)h(carried)g(out)f(b)o (y)g(The-)257 1799 y(orem)38 b(7.2)18 b(or)39 b(7.3)18 b(together)j(with)e(the)h(algebraic)e(metho)q(d)h(of)j(\(2\).)34 b(Some)19 b(elegan)o(t)257 1849 y(represen)o(tations)d(ma)o(y)c(b)q(e)j (obtained.)j(F)m(or)c(example,)e Fy(H)1163 1855 y Fx(c)1194 1849 y FC(in)i(a)f Fy(K)1312 1855 y Fx(n)1333 1859 y Fs(1)1349 1855 y Fx(n)1370 1859 y Fs(2)1386 1855 y Fx(n)1407 1859 y Fs(3)1439 1849 y FC(graph)h(ma)o(y)e(b)q(e)257 1899 y(written)435 2004 y Fy(H)470 2010 y Fx(c)498 2004 y FC(=)561 1976 y Fy(n)586 1982 y FA(1)605 1976 y FC(!)7 b Fy(n)649 1982 y FA(2)667 1976 y FC(!)g Fy(n)711 1982 y FA(3)729 1976 y FC(!)p 535 1995 233 2 v 535 2033 a Fy(n)560 2039 y FA(1)588 2033 y FC(+)i Fy(n)654 2039 y FA(2)682 2033 y FC(+)g Fy(n)748 2039 y FA(3)779 1965 y Fv(X)803 2053 y Fx(i)846 1946 y Fv(\024\022)898 1976 y Fy(n)923 1982 y FA(1)913 2033 y Fy(i)942 1946 y Fv(\023)o(\022)1083 1976 y Fy(n)1108 1982 y FA(2)1003 2033 y Fy(n)1028 2039 y FA(3)1056 2033 y FB(\000)g Fy(n)1122 2039 y FA(1)1150 2033 y FC(+)g Fy(i)1206 1946 y Fv(\023)o(\022)1347 1976 y Fy(n)1372 1982 y FA(3)1267 2033 y Fy(n)1292 2039 y FA(3)1320 2033 y FB(\000)g Fy(n)1386 2039 y FA(2)1414 2033 y FC(+)g Fy(i)1470 1946 y Fv(\023)539 2134 y FC(+)586 2076 y Fv(\022)616 2106 y Fy(n)641 2112 y FA(1)669 2106 y FB(\000)g FC(1)667 2163 y Fy(i)731 2076 y Fv(\023\022)836 2106 y Fy(n)861 2112 y FA(2)889 2106 y FB(\000)h FC(1)792 2163 y Fy(n)817 2169 y FA(3)845 2163 y FB(\000)g Fy(n)912 2169 y FA(1)939 2163 y FC(+)g Fy(i)995 2076 y Fv(\023\022)1100 2106 y Fy(n)1125 2112 y FA(3)1153 2106 y FB(\000)f FC(1)1056 2163 y Fy(n)1081 2169 y FA(3)1109 2163 y FB(\000)h Fy(n)1176 2169 y FA(2)1203 2163 y FC(+)g Fy(i)1259 2076 y Fv(\023\025)1319 2134 y Fy(:)1616 2069 y FC(\(23\))257 2282 y FD(5)67 b(Secret)23 b(Key)f(Exc)n(hanges)257 2373 y FC(Mo)q(dern)c(cryptograph) o(y)f(is)g(fundamen)o(tally)d(concerned)19 b(with)e(the)g(problem)f(of) g(secure)257 2423 y(priv)n(ate)f(comm)o(unicatio)o(n.)j(A)d(Secret)i (Key)e(Exc)o(hange)h(is)e(a)h(proto)q(col)g(where)h(Alice)f(and)257 2473 y(Bob,)20 b(ha)o(ving)d(no)i(secret)h(information)c(in)i(common)e (to)i(start,)i(are)f(able)f(to)h(agree)g(on)p eop %%Page: 4 38 4 37 bop 257 116 a Ft(Sample)13 b(pap)q(er)h(for)g(the)g Fw(amsmath)f Ft(pac)o(k)n(age)707 b FC(4)320 262 y(Set)593 350 y Fy(D)q FC(\()p Fy(t)659 356 y FA(1)678 350 y Fy(;)7 b(:)g(:)g(:)e(;)i(t)786 356 y Fx(n)808 350 y FC(\))12 b(=)892 322 y Fy(\016)p 884 340 36 2 v 884 378 a(\016)r(t)936 350 y FC(det)c Fz(K)p FC(\()p Fy(t;)f(t)1103 356 y FA(1)1121 350 y Fy(;)g(:)g(:)g(:)e(;)i(t)1229 356 y Fx(n)1251 350 y FC(\))p FB(j)1279 362 y Fx(t)p FA(=1)1343 350 y Fy(:)257 439 y FC(Then)527 516 y Fy(D)q FC(\()p Fy(t)593 522 y FA(1)612 516 y Fy(;)g(:)g(:)g(:)e(;)i(t)720 522 y Fx(n)742 516 y FC(\))12 b(=)814 476 y Fv(X)816 565 y Fx(i)p Fu(2)p Fp(n)880 516 y Fy(D)914 522 y Fx(i)936 516 y FC(det)7 b Fz(K)p FC(\()p Fy(t)12 b FC(=)g(1)p Fy(;)7 b(t)1179 522 y FA(1)1197 516 y Fy(;)g(:)g(:)g(:)e(;)i(t)1305 522 y Fx(n)1327 516 y FC(;)g Fy(i)p FB(j)p Fy(i)p FC(\))p Fy(;)202 b FC(\(13\))257 634 y(where)11 b Fz(K)p FC(\()p Fy(t)h FC(=)g(1)p Fy(;)7 b(t)552 640 y FA(1)570 634 y Fy(;)g(:)g(:)g(:)t(;)g(t)677 640 y Fx(n)699 634 y FC(;)g Fy(i)p FB(j)p Fy(i)p FC(\))i(is)g(the)h Fy(i)p FC(th)g(principal)e (submatrix)g(of)h Fz(K)p FC(\()p Fy(t)j FC(=)g(1)p Fy(;)7 b(t)1533 640 y FA(1)1551 634 y Fy(;)g(:)g(:)g(:)e(;)i(t)1659 640 y Fx(n)1681 634 y FC(\).)320 684 y(Theorem)27 b(3.3)13 b(leads)h(to)312 761 y(det)8 b Fz(K)p FC(\()p Fy(t)445 767 y FA(1)464 761 y Fy(;)f(t)498 767 y FA(1)516 761 y Fy(;)g(:)g(:)g(:)e(;)i(t)624 767 y Fx(n)646 761 y FC(\))12 b(=)718 721 y Fv(X)717 810 y Fx(I)r Fu(2)p Fp(n)778 761 y FC(\()p FB(\000)p FC(1\))863 744 y Fu(j)p Fx(I)r Fu(j)902 761 y Fy(t)917 744 y Fx(n)p Fu(\000j)o Fx(I)r Fu(j)1009 721 y Fv(Y)1010 810 y Fx(i)p Fu(2)p Fx(I)1069 761 y Fy(t)1084 767 y Fx(i)1105 721 y Fv(Y)1105 810 y Fx(j)r Fu(2)p Fx(I)1159 761 y FC(\()p Fy(D)1209 767 y Fx(j)1237 761 y FC(+)d Fy(\025)1302 767 y Fx(j)1320 761 y Fy(t)1335 767 y Fx(j)1352 761 y FC(\))e(det)h Fz(A)1476 744 y FA(\()p Fx(\025t)p FA(\))1536 761 y FC(\()p 1552 727 22 2 v Fy(I)t FB(j)p 1586 727 V Fy(I)s FC(\))p Fy(:)1616 859 y FC(\(14\))257 934 y(Note)15 b(that)302 1011 y(det)7 b Fz(K)p FC(\()p Fy(t)13 b FC(=)e(1)p Fy(;)c(t)545 1017 y FA(1)563 1011 y Fy(;)g(:)g(:)g(:)e(;)i(t)671 1017 y Fx(n)693 1011 y FC(\))12 b(=)765 972 y Fv(X)764 1061 y Fx(I)r Fu(2)p Fp(n)825 1011 y FC(\()p FB(\000)p FC(1\))910 994 y Fu(j)p Fx(I)r Fu(j)956 972 y Fv(Y)957 1061 y Fx(i)p Fu(2)p Fx(I)1016 1011 y Fy(t)1031 1017 y Fx(i)1052 972 y Fv(Y)1051 1061 y Fx(j)r Fu(2)p Fx(I)1106 1011 y FC(\()p Fy(D)1156 1017 y Fx(j)1183 1011 y FC(+)e Fy(\025)1249 1017 y Fx(j)1267 1011 y Fy(t)1282 1017 y Fx(j)1299 1011 y FC(\))d(det)h Fz(A)1423 994 y FA(\()p Fx(\025)p FA(\))1471 1011 y FC(\()p 1487 978 V Fy(I)s FB(j)p 1520 978 V Fy(I)s FC(\))k(=)g(0)p Fy(:)1616 1110 y FC(\(15\))320 1185 y(Let)i Fy(t)409 1191 y Fx(i)434 1185 y FC(=)h(^)-24 b Fy(x)502 1191 y Fx(i)516 1185 y Fy(;)7 b(i)k FC(=)h(1)p Fy(;)7 b(:)g(:)g(:)t(;)g(n)p FC(.)18 b(Lemma)25 b(3.1)13 b(yields)299 1239 y Fv(\022)330 1258 y(X)335 1346 y Fx(i)p Fu(2)p Fp(n)397 1297 y Fy(a)419 1303 y Fx(l)429 1307 y Fq(i)445 1297 y Fy(x)469 1303 y Fx(i)482 1239 y Fv(\023)520 1297 y FC(det)7 b Fz(K)p FC(\()p Fy(t)12 b FC(=)g(1)p Fy(;)7 b(x)772 1303 y FA(1)790 1297 y Fy(;)g(:)g(:)g(:)e(;)i(x)907 1303 y Fx(n)928 1297 y FC(;)g Fy(l)q FB(j)p Fy(l)q FC(\))400 1429 y(=)444 1371 y Fv(\022)478 1390 y(Y)480 1478 y Fx(i)p Fu(2)p Fp(n)545 1429 y FC(^)-24 b Fy(x)566 1435 y Fx(i)579 1371 y Fv(\023)654 1390 y(X)617 1481 y Fx(I)r Fu(\022)p Fp(n)p Fu(\000f)p Fx(l)p Fu(g)751 1429 y FC(\()p FB(\000)p FC(1\))836 1412 y Fu(j)q Fx(I)r Fu(j)882 1429 y FC(p)q(er)8 b Fz(A)984 1412 y FA(\()p Fx(\025)p FA(\))1032 1429 y FC(\()p Fy(I)s FB(j)p Fy(I)s FC(\))f(det)h Fz(A)1226 1412 y FA(\()p Fx(\025)p FA(\))1274 1429 y FC(\()p 1290 1396 V Fy(I)13 b FB([)c(f)p Fy(l)q FB(gj)p 1425 1396 V Fy(I)j FB([)d(f)p Fy(l)q FB(g)p FC(\))p Fy(:)41 b FC(\(16\))257 1553 y Fw(\\begin{multline})257 1602 y(\\biggl\(\\sum_{\\,i\\in)o(\\math)o (bf{n})o(}a_{l)18 b(_i}x_i\\biggr\))257 1652 y(\\det\\mathbf{K}\(t=1,)o (x_1,\\)o(dots,)o(x_n;l)g(|l)k(\)\\\\)257 1702 y (=\\biggl\(\\prod_{\\,i\\)o(in\\ma)o(thbf{)o(n}}\\h)o(at)d (x_i\\biggr\))257 1752 y(\\sum_{I\\subseteq\\ma)o(thbf{)o(n}-\\{)o(l)g (\\}})257 1802 y(\(-1\)^{\\envert{I}}\\p)o(er\\ma)o(thbf{)o(A}^{\()o (\\lamb)o(da\)})o(\(I|I\))257 1852 y(\\det\\mathbf{A}^{\(\\l)o(ambda)o (\)})257 1901 y(\(\\overline)h(I\\cup\\{l)g(\\}|\\overline)g (I\\cup\\{l)g(\\}\).)257 1951 y(\\label{sum-ali})257 2001 y(\\end{multline})320 2073 y FC(By)14 b(\(3\),)g(\(6\),)f(and)h (\(7\),)f(w)o(e)h(ha)o(v)o(e)257 2144 y Fz(Prop)q(osition)e(3.5.)777 2252 y Fy(H)812 2258 y Fx(c)840 2252 y FC(=)901 2224 y(1)p 889 2243 46 2 v 889 2281 a(2)p Fy(n)966 2201 y Fx(n)946 2213 y Fv(X)950 2302 y Fx(l)p FA(=0)1006 2252 y FC(\()p FB(\000)p FC(1\))1091 2235 y Fx(l)1104 2252 y Fy(D)1138 2258 y Fx(l)1151 2252 y Fy(;)453 b FC(\(17\))257 2367 y Fr(wher)n(e)528 2444 y Fy(D)562 2450 y Fx(l)587 2444 y FC(=)638 2404 y Fv(X)631 2494 y Fx(I)646 2498 y Fq(l)658 2494 y Fu(\022)p Fp(n)711 2444 y Fy(D)q FC(\()p Fy(t)777 2450 y FA(1)797 2444 y Fy(;)7 b(:)g(:)g(:)t(;)g(t)904 2450 y Fx(n)926 2444 y FC(\)2)p FB(j)975 2466 y Fx(t)988 2470 y Fq(i)1001 2466 y FA(=)1026 2434 y Fo(n)1054 2450 y FA(0)p Fx(;)13 b Fn(if)f Fx(i)p Fu(2)p Fx(I)1177 2454 y Fq(l)1054 2482 y FA(1)p Fx(;)f Fn(otherwise)1253 2466 y Fx(;)18 b(i)p FA(=1)p Fx(;:::;n)1407 2444 y Fy(:)197 b FC(\(18\))p eop %%Page: 3 39 3 38 bop 257 116 a Ft(Sample)13 b(pap)q(er)h(for)g(the)g Fw(amsmath)f Ft(pac)o(k)n(age)707 b FC(3)320 262 y(Note)19 b(that)h(all)e(basic)h(prop)q(erties)i(of)d(determinan)o(ts)h(are)h (direct)g(consequences)i(of)257 311 y(Lemma)j(3.2.)17 b(W)m(rite)649 363 y Fv(X)650 452 y Fx(j)r Fu(2)p Fp(n)716 403 y Fy(b)734 409 y Fx(ij)766 403 y FC(^)-24 b Fy(y)783 409 y Fx(j)812 403 y FC(=)856 363 y Fv(X)857 452 y Fx(j)r Fu(2)p Fp(n)923 403 y Fy(b)941 381 y FA(\()p Fx(\025)p FA(\))941 414 y Fx(ij)992 403 y FC(^)g Fy(y)1009 409 y Fx(j)1036 403 y FC(+)9 b(\()p Fy(b)1111 409 y Fx(ii)1146 403 y FB(\000)h Fy(\025)1212 409 y Fx(i)1226 403 y FC(\))s(^)-24 b Fy(y)1262 409 y Fx(i)1279 403 y FC(^)g Fy(y)340 b FC(\(7\))257 536 y(where)701 627 y Fy(b)719 606 y FA(\()p Fx(\025)p FA(\))719 639 y Fx(ii)778 627 y FC(=)12 b Fy(\025)846 633 y Fx(i)860 627 y Fy(;)48 b(b)938 606 y FA(\()p Fx(\025)p FA(\))938 639 y Fx(ij)997 627 y FC(=)12 b Fy(b)1059 633 y Fx(ij)1088 627 y Fy(;)48 b(i)11 b FB(6)p FC(=)h Fy(j:)390 b FC(\(8\))257 733 y(Let)17 b Fz(B)368 718 y FA(\()p Fx(\025)p FA(\))431 733 y FC(=)e(\()p Fy(b)512 712 y FA(\()p Fx(\025)p FA(\))512 745 y Fx(ij)560 733 y FC(\).)25 b(By)17 b(\(6\))f(and)g(\(7\),)g(it)g(is)g(straigh)o(tforw)o(ard)f(to)h (sho)o(w)g(the)h(follo)o(wing)257 783 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