1 Text Tests

Georgia: 0123456789, $20, €30, £60. Naïve Æsop’s Œuvres in français weremy first reading. Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Utpurus elit, vestibulum ut, placerat ac, adipiscing vitae, felis. Curabitur dictumgravida mauris. Nam arcu libero, nonummy eget, consectetuer id, vulputate a,magna. Donec vehicula augue eu neque. Pellentesque habitant morbi tristiquesenectus et netus et malesuada fames ac turpis egestas. Mauris ut leo. Crasviverra metus rhoncus sem. Nulla et lectus vestibulum urna fringilla ultrices.Phasellus eu tellus sit amet tortor gravida placerat. Integer sapien est, iaculisin, pretium quis, viverra ac, nunc. Praesent eget sem vel leo ultrices bibendum.Aenean faucibus. Morbi dolor nulla, malesuada eu, pulvinar at, mollis ac, nulla.Curabitur auctor semper nulla. Donec varius orci eget risus. Duis nibh mi,congue eu, accumsan eleifend, sagittis quis, diam. Duis eget orci sit amet orcidignissim rutrum.Georgia Italic: 0123456789, $20, €30, £60. Naïve Æsop’s Œuvres in

français were my first reading. Nam dui ligula, fringilla a, euismod sodales,sollicitudin vel, wisi. Morbi auctor lorem non justo. Nam lacus libero, pretiumat, lobortis vitae, ultricies et, tellus. Donec aliquet, tortor sed accumsan biben-dum, erat ligula aliquet magna, vitae ornare odio metus a mi. Morbi ac orciet nisl hendrerit mollis. Suspendisse ut massa. Cras nec ante. Pellentesque anulla. Cum sociis natoque penatibus et magnis dis parturient montes, nasce-tur ridiculus mus. Aliquam tincidunt urna. Nulla ullamcorper vestibulumturpis. Pellentesque cursus luctus mauris.Georgia Bold: 0123456789, $20, €30, £60. Naïve Æsop’s Œu-

vres in français were my first reading. Nulla malesuada porttitordiam. Donec felis erat, congue non, volutpat at, tincidunt tristique,libero. Vivamus viverra fermentum felis. Donec nonummy pellen-tesque ante. Phasellus adipiscing semper elit. Proin fermentummassa ac quam. Sed diam turpis, molestie vitae, placerat a, mo-lestie nec, leo. Maecenas lacinia. Nam ipsum ligula, eleifend at, ac-cumsan nec, suscipit a, ipsum. Morbi blandit ligula feugiat magna.Nunc eleifend consequat lorem. Sed lacinia nulla vitae enim. Pel-lentesque tincidunt purus vel magna. Integer non enim. Praesenteuismod nunc eu purus. Donec bibendum quam in tellus. Nullamcursus pulvinar lectus. Donec et mi. Nam vulputate metus eu enim.Vestibulum pellentesque felis eu massa.Georgia Bold Italic: 0123456789, $20, €30, £60. Naïve Æsop’s

Œuvres in français were my first reading. Quisque ullamcorperplacerat ipsum. Cras nibh. Morbi vel justo vitae lacus tinciduntultrices. Lorem ipsum dolor sit amet, consectetuer adipiscing elit.In hac habitasse platea dictumst. Integer tempus convallis augue.Etiam facilisis. Nunc elementum fermentum wisi. Aenean plac-erat. Ut imperdiet, enim sed gravida sollicitudin, felis odio plac-erat quam, ac pulvinar elit purus eget enim. Nunc vitae tortor.Proin tempus nibh sit amet nisl. Vivamus quis tortor vitae risus

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porta vehicula.Franklin Gothic Book: 0123456789, $20, €30, £60. Naïve Æsop’s Œuvres

in français were my first reading. Fusce mauris. Vestibulum luctus nibh at lectus.Sed bibendum, nulla a faucibus semper, leo velit ultricies tellus, ac venenatis arcuwisi vel nisl. Vestibulum diam. Aliquam pellentesque, augue quis sagittis posuere,turpis lacus congue quam, in hendrerit risus eros eget felis. Maecenas eget eratin sapien mattis porttitor. Vestibulum porttitor. Nulla facilisi. Sed a turpis eu lacuscommodo facilisis. Morbi fringilla, wisi in dignissim interdum, justo lectus sagittisdui, et vehicula libero dui cursus dui. Mauris tempor ligula sed lacus. Duis cursusenim ut augue. Cras ac magna. Cras nulla. Nulla egestas. Curabitur a leo. Quisqueegestas wisi eget nunc. Nam feugiat lacus vel est. Curabitur consectetuer.Franklin Gothic Book Italic: 0123456789, $20, €30, £60. Naïve Æsop’s Œu-

vres in français were my first reading. Suspendisse vel felis. Ut lorem lorem, in-terdum eu, tincidunt sit amet, laoreet vitae, arcu. Aenean faucibus pede eu ante.Praesent enim elit, rutrum at, molestie non, nonummy vel, nisl. Ut lectus eros,malesuada sit amet, fermentum eu, sodales cursus, magna. Donec eu purus.Quisque vehicula, urna sed ultricies auctor, pede lorem egestas dui, et convalliselit erat sed nulla. Donec luctus. Curabitur et nunc. Aliquam dolor odio, commodopretium, ultricies non, pharetra in, velit. Integer arcu est, nonummy in, fermentumfaucibus, egestas vel, odio.Franklin Gothic Medium: 0123456789, $20, €30, £60. Naïve Æsop’s Œuvres

in français were my first reading. Sed commodo posuere pede. Mauris ut est. Utquis purus. Sed ac odio. Sed vehicula hendrerit sem. Duis non odio. Morbi ut dui.Sed accumsan risus eget odio. In hac habitasse platea dictumst. Pellentesquenon elit. Fusce sed justo eu urna porta tincidunt. Mauris felis odio, sollicitudin sed,volutpat a, ornare ac, erat. Morbi quis dolor. Donec pellentesque, erat ac sagittissemper, nunc dui lobortis purus, quis congue purus metus ultricies tellus. Proinet quam. Class aptent taciti sociosqu ad litora torquent per conubia nostra, perinceptos hymenaeos. Praesent sapien turpis, fermentum vel, eleifend faucibus,vehicula eu, lacus.Franklin Gothic Medum Italic: 0123456789, $20, €30, £60. Naïve Æsop’s

Œuvres in français were my first reading. Pellentesque habitant morbi tristiquesenectus et netus et malesuada fames ac turpis egestas. Donec odio elit, dic-tum in, hendrerit sit amet, egestas sed, leo. Praesent feugiat sapien aliquet odio.Integer vitae justo. Aliquam vestibulum fringilla lorem. Sed neque lectus, con-sectetuer at, consectetuer sed, eleifend ac, lectus. Nulla facilisi. Pellentesqueeget lectus. Proin eu metus. Sed porttitor. In hac habitasse platea dictumst. Sus-pendisse eu lectus. Ut mi mi, lacinia sit amet, placerat et, mollis vitae, dui. Sedante tellus, tristique ut, iaculis eu, malesuada ac, dui. Mauris nibh leo, facilisisnon, adipiscing quis, ultrices a, dui.Franklin Gothic Medium Condensed: 0123456789, $20, €30, £60. Naïve Æsop’s Œuvres

in français were my first reading. Morbi luctus, wisi viverra faucibus pretium, nibh est plac-erat odio, nec commodo wisi enim eget quam. Quisque libero justo, consectetuer a, feugiatvitae, porttitor eu, libero. Suspendisse sed mauris vitae elit sollicitudin malesuada. Maece-nas ultricies eros sit amet ante. Ut venenatis velit. Maecenas sed mi eget dui varius euismod.Phasellus aliquet volutpat odio. Vestibulum ante ipsum primis in faucibus orci luctus et ultrices

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posuere cubilia Curae; Pellentesque sit amet pede ac sem eleifend consectetuer. Nullam ele-mentum, urna vel imperdiet sodales, elit ipsum pharetra ligula, ac pretium ante justo a nulla.Curabitur tristique arcu eu metus. Vestibulum lectus. Proin mauris. Proin eu nunc eu urna hen-drerit faucibus. Aliquam auctor, pede consequat laoreet varius, eros tellus scelerisque quam,pellentesque hendrerit ipsum dolor sed augue. Nulla nec lacus.Franklin Gothic Demibold: 0123456789, $20, €30, £60. Naïve Æsop’s Œu-

vres in français were my first reading. Suspendisse vitae elit. Aliquam arcuneque, ornare in, ullamcorper quis, commodo eu, libero. Fusce sagittis erat aterat tristique mollis. Maecenas sapien libero, molestie et, lobortis in, sodaleseget, dui. Morbi ultrices rutrum lorem. Nam elementum ullamcorper leo. Morbidui. Aliquam sagittis. Nunc placerat. Pellentesque tristique sodales est. Mae-cenas imperdiet lacinia velit. Cras non urna. Morbi eros pede, suscipit ac, variusvel, egestas non, eros. Praesent malesuada, diam id pretium elementum, erossem dictum tortor, vel consectetuer odio sem sed wisi.Franklin Gothic Demibold Italic: 0123456789, $20, €30, £60. Naïve Æsop’s

Œuvres in français were my first reading. Sed feugiat. Cum sociis natoque pe-natibus et magnis dis parturient montes, nascetur ridiculus mus. Ut pellen-tesque augue sed urna. Vestibulum diam eros, fringilla et, consectetuer eu,nonummy id, sapien. Nullam at lectus. In sagittis ultrices mauris. Curabiturmalesuada erat sit amet massa. Fusce blandit. Aliquam erat volutpat. Aliquameuismod. Aenean vel lectus. Nunc imperdiet justo nec dolor.Franklin Gothic Demibold Condensed: 0123456789, $20, €30, £60. Naïve Æsop’s Œu-

vres in français were my first reading. Etiam euismod. Fusce facilisis lacinia dui. Sus-pendisse potenti. In mi erat, cursus id, nonummy sed, ullamcorper eget, sapien. Prae-sent pretium, magna in eleifend egestas, pede pede pretium lorem, quis consectetuer tortorsapien facilisis magna. Mauris quis magna varius nulla scelerisque imperdiet. Aliquam nonquam. Aliquam porttitor quam a lacus. Praesent vel arcu ut tortor cursus volutpat. In vi-tae pede quis diam bibendum placerat. Fusce elementum convallis neque. Sed dolor orci,scelerisque ac, dapibus nec, ultricies ut, mi. Duis nec dui quis leo sagittis commodo.Franklin Gothic Heavy: 0123456789, $20, €30, £60. Naïve Æsop’s Œu-

vres in français weremy first reading. Aliquam lectus. Vivamus leo. Quisqueornare tellus ullamcorper nulla. Mauris porttitor pharetra tortor. Sed fringillajusto sed mauris. Mauris tellus. Sed non leo. Nullam elementum, magnain cursus sodales, augue est scelerisque sapien, venenatis congue nullaarcu et pede. Ut suscipit enim vel sapien. Donec congue. Maecenas urnami, suscipit in, placerat ut, vestibulum ut, massa. Fusce ultrices nulla etnisl.Franklin Gothic Heavy Italic: 0123456789, $20, €30, £60. Naïve Æsop’s

Œuvres in français were my first reading. Etiam ac leo a risus tristiquenonummy. Donec dignissim tincidunt nulla. Vestibulum rhoncus molestieodio. Sed lobortis, justo et pretium lobortis, mauris turpis condimentumaugue, nec ultricies nibh arcu pretium enim. Nunc purus neque, placeratid, imperdiet sed, pellentesque nec, nisl. Vestibulum imperdiet neque nonsem accumsan laoreet. In hac habitasse platea dictumst. Etiam condimen-tum facilisis libero. Suspendisse in elit quis nisl aliquam dapibus. Pellen-tesque auctor sapien. Sed egestas sapien nec lectus. Pellentesque vel dui

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vel neque bibendum viverra. Aliquam porttitor nisl nec pede. Proin mattislibero vel turpis. Donec rutrum mauris et libero. Proin euismod porta felis.Nam lobortis, metus quis elementum commodo, nunc lectus elementummauris, eget vulputate ligula tellus eu neque. Vivamus eu dolor.

2 Math Tests

Math test are taken from[1]. Note that we do not have \jmath, so we took onefrom CM.

2.1 Math Alphabets

Math Italic (\mathnormal)

, , , , , , , , , ,

A, B, C,D, E, F, G,H, I, J, K, L,M,N,O, P,Q,R, S, T,U, V,W,X, Y , Z,

a, b, c, d, e, f , g, h, i, j, k, l,m, n, o, p, q, r, s, t, u, v,w, x, y, z, ı, ,

A, B, Γ, ∆, E, Z,H,Θ, I, K, Λ,M,N,Ξ,O,Π, P, Σ, T, Υ ,Φ,X,Ψ,Ω,∇,α, β, γ, δ, ε, ζ , η, θ, ι, κ, λ, µ, ν, ξ, o, π, ρ, σ, τ, υ, φ, χ, ψ,ω, ε,ϑ, $, %, ς,ϕ,

Math Roman (\mathrm)

0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F,G,H, I, J,K,L,M,N,O,P,Q,R,S,T,U,V,W,X,Y,Z,

a,b, c,d, e, f, g,h, i, j, k, l,m,n, o,p,q, r, s, t,u, v,w, x, y, z, ı, ,

A,B,Γ,∆,E,Z,H,Θ, I,K,Λ,M,N,Ξ,O,Π,P,Σ,T,Υ,Φ,X,Ψ,Ω,∇,α, β, γ, δ, ε, ζ , η, θ, ι, κ, λ, µ, ν, ξ, o, π, ρ, σ, τ, υ, φ, χ, ψ,ω, ε,ϑ, $, %, ς,ϕ,

Math Bold (\mathbf)

,,,,,,,,,,

A,B,C,D,E, F,G,H, I, J,K, L,M,N,O,P,Q,R,S, T,U,V,W,X, Y,Z,a, b, c, d, e, f, g, h, i, j, k, l,m, n, o, p, q, r, s, t, u, v,w, x, y, z, ı, ,A,B, Γ,∆,E,Z,H,Θ, I,K,Λ,M,N, Ξ,O,Π,P, Σ, T, Υ,Φ,X,Ψ,Ω,∇,

Math Sans Serif (\mathsf)

, , , , , , , , , ,

A,B,C,D, E, F,G,H, I, J,K, L,M,N,O,P,Q,R,S, T,U, V,W, X, Y, Z,a, b, c, d, e, f, g, h, i, j, k, l,m, n, o, p, q, r, s, t, u, v,w, x, y, z, ı, ,A,B, Γ,∆, E, Z,H,Θ, I,K,Λ,M,N, Ξ,O,Π,P, Σ, T, Υ,Φ, X,Ψ,Ω,∇,

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2.2 Character Sidebearings

|A| + |B| + |C| + |D| + |E| + |F| + |G| + |H| + |I| + |J| + |K| + |L| + |M|+

|N| + |O| + |P| + |Q| + |R| + |S| + |T| + |U| + |V | + |W | + |X| + |Y | + |Z|+

|a| + |b| + |c| + |d| + |e| + |f | + |g| + |h| + |i| + |j| + |k| + |l| + |m|+

|n| + |o| + |p| + |q| + |r| + |s| + |t| + |u| + |v| + |w| + |x| + |y| + |z| + |ı| + ||+

|A| + |B| + |Γ| + |∆| + |E| + |Z| + |H| + |Θ| + |I| + |K| + |Λ| + |M|+

|N| + |Ξ| + |O| + |Π| + |P| + |Σ| + |T| + |Υ| + |Φ| + |X| + |Ψ| + |Ω| + |∇|+|α| + |β| + |γ| + |δ| + |ε| + |ζ | + |η| + |θ| + |ι| + |κ| + |λ| + |µ|+

|ν| + |ξ| + |o| + |π| + |ρ| + |σ| + |τ| + |υ| + |φ| + |χ| + |ψ| + |ω|+

|ε| + |ϑ| + |$| + |%| + |ς| + |ϕ|+

|A| + |B| + |C| + |D| + |E| + |F| + |G| + |H| + |I| + |J| + |K| + |L| + |M|+

|N| + |O| + |P| + |Q| + |R| + |S| + |T| + |U| + |V| + |W| + |X| + |Y| + |Z|+

|a| + |b| + |c| + |d| + |e| + |f| + |g| + |h| + |i| + |j| + |k| + |l| + |m|+

|n| + |o| + |p| + |q| + |r| + |s| + |t| + |u| + |v| + |w| + |x| + |y| + |z| + |ı| + ||+

|A| + |B| + |Γ| + |∆| + |E| + |Z| + |H| + |Θ| + |I| + |K| + |Λ| + |M|+

|N| + |Ξ| + |O| + |Π| + |P| + |Σ| + |T| + |Υ| + |Φ| + |X| + |Ψ| + |Ω| + |∇|+

|A| + |B| + |C| + |D| + |E| + |F| + |G| + |H| + |I| + |J| + |K| + |L| + |M|+

|N| + |O| + |P| + |Q| + |R| + |S| + |T| + |U| + |V| + |W| + |X| + |Y| + |Z|+

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2.3 Superscript positioning

A2 + B2 + C2 +D2 + E2 + F2 + G2 +H2 + I2 + J2 + K2 + L2 +M2+

N2 + O2 + P2 + Q2 + R2 + S2 + T2 +U2 + V2 +W2 + X2 + Y 2 + Z2+

a2 + b2 + c2 + d2 + e2 + f 2 + g2 + h2 + i2 + j2 + k2 + l2 +m2+

n2 + o2 + p2 + q2 + r2 + s2 + t2 + u2 + v2 +w2 + x2 + y2 + z2 + ı2 + 2+

A2 + B2 + Γ2 + ∆2 + E2 + Z2 +H2 + Θ2 + I2 + K2 + Λ2 +M2+

N2 + Ξ2 + O2 + Π2 + P2 + Σ2 + T2 + Υ2 + Φ2 + X2 +Ψ2 + Ω2 +∇2+α2 + β2 + γ2 + δ2 + ε2 + ζ2 + η2 + θ2 + ι2 + κ2 + λ2 + µ2+

ν2 + ξ2 + o2 + π2 + ρ2 + σ2 + τ2 + υ2 + φ2 + χ2 + ψ2 + ω2+

ε2 + ϑ2 + $2 + %2 + ς2 + ϕ2+

A2 + B2 + C2 + D2 + E2 + F2 + G2 + H2 + I2 + J2 + K2 + L2 +M2+

N2 + O2 + P2 + Q2 + R2 + S2 + T2 + U2 + V2 +W2 + X2 + Y2 + Z2+

a2 + b2 + c2 + d2 + e2 + f2 + g2 + h2 + i2 + j2 + k2 + l2 +m2+

n2 + o2 + p2 + q2 + r2 + s2 + t2 + u2 + v2 + w2 + x2 + y2 + z2 + ı2 + 2+

A2 + B2 + Γ2 + ∆2 + E2 + Z2 + H2 + Θ2 + I2 + K2 + Λ2 +M2+

N2 + Ξ2 + O2 + Π2 + P2 + Σ2 + T2 + Υ2 + Φ2 + X2 +Ψ2 + Ω2 +∇2+

A2 + B2 + C2 +D2 + E2 + F2 + G2 +H2 + I2 + J2 + K2 + L2 +M2+

N2 +O2 + P2 +Q2 + R2 + S2 + T2 +U2 + V2 +W2 + X2 + Y2 + Z2+

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2.4 Subscript positioning

Ai + Bi + Ci +Di + Ei + Fi + Gi +Hi + Ii + Ji + Ki + Li +Mi+

Ni + Oi + Pi + Qi + Ri + Si + Ti +Ui + Vi +Wi + Xi + Yi + Zi+

ai + bi + ci + di + ei + fi + gi + hi + ii + ji + ki + li +mi+

ni + oi + pi + qi + ri + si + ti + ui + vi +wi + xi + yi + zi + ıi + i+

Ai + Bi + Γi + ∆i + Ei + Zi +Hi + Θi + Ii + Ki + Λi +Mi+

Ni + Ξi + Oi +Πi + Pi + Σi + Ti + Υi +Φi + Xi +Ψi +Ωi +∇i+αi + βi + γi + δi + εi + ζi + ηi + θi + ιi + κi + λi + µi+

νi + ξi + oi + πi + ρi + σi + τi + υi + φi + χi + ψi + ωi+

εi + ϑi + $i + %i + ςi + ϕi+

Ai + Bi + Ci + Di + Ei + Fi + Gi + Hi + Ii + Ji + Ki + Li +Mi+

Ni + Oi + Pi + Qi + Ri + Si + Ti + Ui + Vi +Wi + Xi + Yi + Zi+

ai + bi + ci + di + ei + fi + gi + hi + ii + ji + ki + li +mi+

ni + oi + pi + qi + ri + si + ti + ui + vi + wi + xi + yi + zi + ıi + i+

Ai + Bi + Γi + ∆i + Ei + Zi + Hi + Θi + Ii + Ki + Λi +Mi+

Ni + Ξi + Oi + Πi + Pi + Σi + Ti + Υi + Φi + Xi +Ψi + Ωi +∇i+

Ai + Bi + Ci +Di + Ei + Fi + Gi +Hi + Ii + Ji + Ki + Li +Mi+

Ni +Oi + Pi +Qi + Ri + Si + Ti +Ui + Vi +Wi + Xi + Yi + Zi+

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2.5 Accent positioning

A + B + C + D + E + F + G + H + I + J + K + L + M+

N + O + P + Q + R + S + T + U + V + W + X + Y + Z+

a + b + c + d + e + f + g + h + i + j + k + l + m+

n + o + p + q + r + s + t + u + v + w + x + y + z + ı + +

A + B + Γ + ∆ + E + Z + H + Θ + I + K + Λ + M+

N + Ξ + O + Π + P + Σ + T + Υ + Φ + X + Ψ + Ω + ∇+α + β + γ + δ + ε + ζ + η + θ + ι + κ + λ + µ+

ν + ξ + o + π + ρ + σ + τ + υ + φ + χ + ψ + ω+

ε + ϑ + $ + % + ς + ϕ+

A + B + C + D + E + F + G + H + I + J + K + L + M+

N + O + P + Q + R + S + T + U + V + W + X + Y + Z+

a + b + c + d + e + f + g + h + i + j + k + l + m+

n + o + p + q + r + s + t + u + v + w + x + y + z + ı + +

A + B + Γ + ∆ + E + Z + H + Θ + I + K + Λ + M+

N + Ξ + O + Π + P + Σ + T + Υ + Φ + X + Ψ + Ω + ∇+

A + B + C + D + E + F + G + H + I + J + K + L + M+

N + O + P + Q + R + S + T + U + V + W + X + Y + Z+

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2.6 Differentials

dA + dB + dC + dD + dE + dF + dG + dH + dI + dJ + dK + dL + dM+

dN + dO + dP + dQ + dR + dS + dT + dU + dV + dW + dX + dY + dZ+

da + db + dc + dd + de + df + dg + dh + di + dj + dk + dl + dm+

dn + do + dp + dq + dr + ds + dt + du + dv + dw + dx + dy + dz + dı + d+

dA + dB + dΓ + d∆ + dE + dZ + dH + dΘ + dI + dK + dΛ + dM+

dN + dΞ + dO + dΠ + dP + dΣ + dT + dΥ + dΦ + dX + dΨ+ dΩ + d∇+dα + dβ + dγ + dδ + dε + dζ + dη + dθ + dι + dκ + dλ + dµ+

dν + dξ + do + dπ + dρ + dσ + dτ + dυ + dφ + dχ + dψ + dω+

dε + dϑ + d$ + d% + dς + dϕ+

dA + dB + dΓ + d∆ + dE + dZ + dH + dΘ + dI + dK + dΛ + dM+

dN + dΞ + dO + dΠ + dP + dΣ + dT + dΥ + dΦ + dX + dΨ+ dΩ + d∇+

dA + dB + dC + dD + dE + dF + dG + dH + dI + dJ + dK + dL + dM+

dN + dO + dP + dQ + dR + dS + dT + dU + dV + dW + dX + dY + dZ+

da + db + dc + dd + de + df + dg + dh + di + dj + dk + dl + dm+

dn + do + dp + dq + dr + ds + dt + du + dv + dw + dx + dy + dz + dı + d+

dA + dB + dΓ + d∆ + dE + dZ + dH + dΘ + dI + dK + dΛ + dM+

dN + dΞ + dO + dΠ + dP + dΣ + dT + dΥ + dΦ + dX + dΨ + dΩ + d∇+dα + dβ + dγ + dδ + dε + dζ + dη + dθ + dι + dκ + dλ + dµ+

dν + dξ + do + dπ + dρ + dσ + dτ + dυ + dφ + dχ + dψ + dω+

dε + dϑ + d$ + d% + dς + dϕ+

dA + dB + dΓ + d∆ + dE + dZ + dH + dΘ + dI + dK + dΛ + dM+

dN + dΞ + dO + dΠ + dP + dΣ + dT + dΥ + dΦ + dX + dΨ + dΩ + d∇+

∂A + ∂B + ∂C + ∂D + ∂E + ∂F + ∂G + ∂H + ∂I + ∂J + ∂K + ∂L + ∂M+

∂N + ∂O + ∂P + ∂Q + ∂R + ∂S + ∂T + ∂U + ∂V + ∂W + ∂X + ∂Y + ∂Z+

∂a + ∂b + ∂c + ∂d + ∂e + ∂f + ∂g + ∂h + ∂i + ∂j + ∂k + ∂l + ∂m+

∂n + ∂o + ∂p + ∂q + ∂r + ∂s + ∂t + ∂u + ∂v + ∂w + ∂x + ∂y + ∂z + ∂ı + ∂+

∂A + ∂B + ∂Γ + ∂∆ + ∂E + ∂Z + ∂H + ∂Θ + ∂I + ∂K + ∂Λ + ∂M+

∂N + ∂Ξ + ∂O + ∂Π + ∂P + ∂Σ + ∂T + ∂Υ + ∂Φ + ∂X + ∂Ψ+ ∂Ω + ∂∇+∂α + ∂β + ∂γ + ∂δ + ∂ε + ∂ζ + ∂η + ∂θ + ∂ι + ∂κ + ∂λ + ∂µ+

∂ν + ∂ξ + ∂o + ∂π + ∂ρ + ∂σ + ∂τ + ∂υ + ∂φ + ∂χ + ∂ψ + ∂ω+

∂ε + ∂ϑ + ∂$ + ∂% + ∂ς + ∂ϕ+

∂A + ∂B + ∂Γ + ∂∆ + ∂E + ∂Z + ∂H + ∂Θ + ∂I + ∂K + ∂Λ + ∂M+

∂N + ∂Ξ + ∂O + ∂Π + ∂P + ∂Σ + ∂T + ∂Υ + ∂Φ + ∂X + ∂Ψ+ ∂Ω + ∂∇+

9

2.7 Slash kerning

1/A + 1/B + 1/C + 1/D + 1/E + 1/F + 1/G + 1/H + 1/I + 1/J + 1/K + 1/L + 1/M+1/N + 1/O + 1/P + 1/Q + 1/R + 1/S + 1/T + 1/U + 1/V + 1/W + 1/X + 1/Y + 1/Z+1/a + 1/b + 1/c + 1/d + 1/e + 1/f + 1/g + 1/h + 1/i + 1/j + 1/k + 1/l + 1/m+1/n + 1/o + 1/p + 1/q + 1/r + 1/s + 1/t + 1/u + 1/v + 1/w + 1/x + 1/y + 1/z + 1/ı + 1/+1/A + 1/B + 1/Γ + 1/∆ + 1/E + 1/Z + 1/H + 1/Θ + 1/I + 1/K + 1/Λ + 1/M+1/N + 1/Ξ + 1/O + 1/Π + 1/P + 1/Σ + 1/T + 1/Υ + 1/Φ + 1/X + 1/Ψ+ 1/Ω + 1/∇+1/α + 1/β + 1/γ + 1/δ + 1/ε + 1/ζ + 1/η + 1/θ + 1/ι + 1/κ + 1/λ + 1/µ+1/ν + 1/ξ + 1/o + 1/π + 1/ρ + 1/σ + 1/τ + 1/υ + 1/φ + 1/χ + 1/ψ + 1/ω+1/ε + 1/ϑ + 1/$ + 1/% + 1/ς + 1/ϕ+

A/2 + B/2 + C/2 +D/2 + E/2 + F/2 + G/2 +H/2 + I/2 + J/2 + K/2 + L/2 +M/2+N/2 + O/2 + P/2 + Q/2 + R/2 + S/2 + T/2 +U/2 + V/2 +W/2 + X/2 + Y/2 + Z/2+a/2 + b/2 + c/2 + d/2 + e/2 + f /2 + g/2 + h/2 + i/2 + j/2 + k/2 + l/2 +m/2+n/2 + o/2 + p/2 + q/2 + r/2 + s/2 + t/2 + u/2 + v/2 +w/2 + x/2 + y/2 + z/2 + ı/2 + /2+A/2 + B/2 + Γ/2 + ∆/2 + E/2 + Z/2 +H/2 + Θ/2 + I/2 + K/2 + Λ/2 +M/2+N/2 + Ξ/2 + O/2 + Π/2 + P/2 + Σ/2 + T/2 + Υ/2 + Φ/2 + X/2 +Ψ/2 + Ω/2 +∇/2+α/2 + β/2 + γ/2 + δ/2 + ε/2 + ζ/2 + η/2 + θ/2 + ι/2 + κ/2 + λ/2 + µ/2+ν/2 + ξ/2 + o/2 + π/2 + ρ/2 + σ/2 + τ/2 + υ/2 + φ/2 + χ/2 + ψ/2 + ω/2+ε/2 + ϑ/2 + $/2 + %/2 + ς/2 + ϕ/2+

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2.8 Big operatorsn∑i=1

xnn∏i=1

xnn∐i=1

xn∫ n

i=1xn

∮ n

i=1xn

n⊗i=1

xnn⊕i=1

xnn⊙i=1

xnn∧i=1

xnn∨i=1

xnn⊎i=1

xnn⋃i=1

xnn⋂i=1

xnn⊔i=1

xn

2.9 Radicals

√x + y

√x2 + y2

√x2i + y

2j

√(cos x2

) √(sin x2

)√√√√√√√√√√√√√√√√√√√

x + y

2.10 Over- and underbraces︷︸︸︷x

︷ ︸︸ ︷x + y

︷ ︸︸ ︷x2 + y2

︷ ︸︸ ︷x2i + y

2j x︸︷︷︸ x + y︸ ︷︷ ︸ xi + yj︸ ︷︷ ︸ x2i + y

2j︸ ︷︷ ︸

2.11 Normal and wide accents

x x ~x x x xx x x xx xxx x x xx xxx

2.12 Long arrows

←−→ ↔ ←− −→ ←→ ⇐=⇒ ⇔ ⇐= =⇒ ⇐⇒

2.13 Left and right delimters

−(f ) − −[f ] − −bf c − −df e − −〈f 〉 − −f −

− (f ) − −[f]− − bf c − − df e − − 〈f 〉 − − f −

− )f ( − −]f[− − /f / − −\f \ − − /f \ − − \f /−

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2.14 Big-g-g delimters

−

⌊⌊⌊⌊b−c⌋⌋⌋⌋

− −

(((((−)))))

−

−

⌈⌈⌈⌈

d−e⌉⌉⌉⌉

− −

−

−

−

[[[[[−]]]]]

− −

−

−

−

⟨⟨⟨⟨⟨⟨⟨〈−〉⟩⟩⟩⟩⟩⟩⟩

− −

−

−

−

xxxxxxx↑−↓yyy

yyyy − −

~wwwwwww~wwwwww~wwwww~wwww~www~ww~w⇑−⇓wwwwww

wwwwwwwwwwwwwwwwwwwwww−

2.15 Miscellanneous formulae

Taken from [2]

hν = E

Let A = (aij) be the adjacency matrix of graph G. The corresponding Kirchhoffmatrix K = (kij) is obtained from A by replacing in −A each diagonal entry bythe degree of its corresponding vertex; i.e., the ith diagonal entry is identifiedwith the degree of the ith vertex. It is well known that

detK(i|i) = the number of spanning trees of G, i = 1, . . . , n (1)

where K(i|i) is the ith principal submatrix of K.Let Ci(j) be the set of graphs obtained from G by attaching edge (vivj) to eachspanning tree of G. Denote by Ci =

⋃j Ci(j). It is obvious that the collec-

tion of Hamiltonian cycles is a subset of Ci. Note that the cardinality of Ci iskii detK(i|i). Let X = x1, . . . , xn. Define multiplication for the elements of Xby

xixj = xjxi, x2i = 0, i, j = 1, . . . , n. (2)

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Let kij = kijxj and kij = −∑

j 6=i kij. Then the number of Hamiltonian cyclesHc isgiven by the relation( n∏

j=1

xj

)Hc =

12kij det K(i|i), i = 1, . . . , n. (3)

The task here is to express (3) in a form free of any xi, i = 1, . . . , n. The resultalso leads to the resolution of enumeration of Hamiltonian paths in a graph.It is well known that the enumeration of Hamiltonian cycles and paths in acomplete graph Kn and in a complete bipartite graph Kn1n2 can only be foundfrom first combinatorial principles. One wonders if there exists a formulawhich can be used very efficiently to produce Kn and Kn1n2 . Recently, using La-grangian methods, Goulden and Jackson have shown thatHc can be expressedin terms of the determinant and permanent of the adjacency matrix. However,the formula of Goulden and Jackson determines neither Kn nor Kn1n2 effec-tively. In this paper, using an algebraic method, we parametrize the adjacencymatrix. The resulting formula also involves the determinant and permanent,but it can easily be applied to Kn and Kn1n2 . In addition, we eliminate the per-manent fromHc and show thatHc can be represented by a determinantal func-tion of multivariables, each variable with domain 0,1. Furthermore, we showthat Hc can be written by number of spanning trees of subgraphs. Finally, weapply the formulas to a complete multigraph Kn1...np .The conditions aij = aji, i, j = 1, . . . , n, are not required in this paper. All for-mulas can be extended to a digraph simply by multiplyingHc by 2.The boundedness, property of Φ0, then yields∫

D

∣∣∂u∣∣2 eα|z|2 ≥ c6α∫D|u|2 eα|z|

2+ c7δ−2

∫A|u|2 eα|z|

2.

Let B(X) be the set of blocks of ΛX and let b(X) = |B(X)|. If φ ∈ QX then φ isconstant on the blocks of ΛX .

PX = φ ∈M | Λφ = ΛX, QX = φ ∈M | Λφ ≥ ΛX. (4)

If Λφ ≥ ΛX then Λφ = ΛY for some Y ≥ X so that

QX =⋃Y≥X

PY .

Thus by Möbius inversion

|PY | =∑X≥Y

µ(Y,X) |QX| .

Thus there is a bijection from QX toWB(X). In particular |QX| = wb(X).

13

W (Φ) =

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥

ϕ

(ϕ1, ε1)0 . . . 0

ϕkn2(ϕ2, ε1)

ϕ

(ϕ2, ε2). . . 0

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

ϕkn1(ϕn, ε1)

ϕkn2(ϕn, ε2)

. . .ϕknn−1(ϕn, εn−1)

ϕ

(ϕn, εn)

∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥∥References

[1] Walter Schmidt. Using Common PostScript Fonts With LATEX. PSNFSSVersion 9.2, September 2004. http://ctan.tug.org/tex-archive/macros/latex/required/psnfss.

[2] Michael Downes and Barbara Beeton. The amsart, amsproc, andamsbook document classes. American Mathematical Society, August2004. http://www.ctan.org/tex-archive/macros/latex/required/amslatex/classes.

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