Reconciling unicode-mathand LATEX 2εWill Robertson

July 20, 2015

What’s unicode-math?
\usepackage{unicode-math}\setmathfont{Cambria Math}\setmathfont{Cambria Math}[
range={\mathrel}, Colour=ForestGreen]\setmathfont{Cambria Math}[
range={\mathopen,\mathclose}, Colour=blue]\setmathfont{Cambria Math}[
range={\mathop,\mathscr}, Colour=red]\[ F(s)=\mathscr{L}\,\bigl\{f(t)\bigr\}= \int_0^∞ \mathup e^{-st}f(t)\, \mathup d t \]𝐹(𝑠) = ℒ 𝑓(𝑡) =e𝑓(𝑡) d𝑡

Introduction
LATEX defaults

Input:\[ fin = \mathit{fin} \]Output:f in = finIntroduction
With unicode-math

Input:\[ fin = \mathit{fin} \]Output:f in = f in

Introduction
Similarly:

Input:\[ fin = \mathbf{fin} \]Default:
f in = finWith unicode-math:f in = f inSomething needs ﬁxing!

Contents

Overview of Unicode Mathematics

Historical and current examples
Sidenote: symbols

TEX’s methods

Unicode mathematics▶

I hope you know what Unicode is▶ Thousands of mathematical glyphs (≈ 2500)▶ Standard LATEX names thanks to Barbara Beeton▶ As a non-mathematician, very fun to scroll throughseemingly endless tables like:
"2A00

"2A01

"2A02

"2A03

"2A04

"2A05

"2A06

"2A07

⨀ ⨁ ⨂ ⨃ ⨄ ⨅ ⨆ ⨇
"2A59

"2A60

"2A61

"2A62

"2A63

"2A64

"2A65

"2A66

⩙

⩠

⩡

⩢

⩣ ⩤ ⩥ ⩦

Ӆฬ

ExtracttheиККψϯ from଒ӆ symbols≙̥ table
อ
иККψπ

଑

иККψΞ

ଓ
ଠ
ଡ
ଢ
ଣ
ତ
ଃ
଄
ଈ
ଉ
଩
ପ
ଯ
ର
଱

иККψΪ

иККψΫ
иККψβ
иККψζ
иККψυ
иККЏи
иККЏϻ
иККЏК
иККЏЗ
иККЏϋ
иККЏψ
иККЏЏ
иККЏЌ
иККЏπ

Ӈ
ӈ
Ӊ
ӊ
Ӌ
ӌ
Ӎ
ӎ
ӏ
Ӑ
ӑ
Ӓ
ӓ
Ӕ
ӕ

≘

≚
≛
≜
≝
≞
≟
≠
≡
≢
≣
≤
≥
≦
≧
≨

̤

̦
̧
̨
̩
̪
̫
̬
̭
̮
̯
̰
̱
̲
̳
̴

ฮ
฻
฼
฽
฾
฿
ฤ
ล
ฦ
ว
ไ
ๅ
๊
๋
์

$`+2[$r2/;2[$p222[$bi`2[$i`BM;H2[UV$2[/27$K2b2[

Ł
ł
Ń
ń
Ņ
ņ
Ň
ň
ŉ

$[m2bi2[$M2UTV$2[mBpUTV$M2[mBp$1[mBp$H2[UTV$;2[UTV$H2[[UV$;2[[UV$HM2[[UVSymbols▶

Access to all these symbols is nice
▶ All the symbols stuﬀ is ﬁne, ± a few naming issues▶ Standardisation should be considered more rigorously

The ‘controversial’ stuﬀ▶

Alphabetic symbols are encoded individually:"00057"1D44A"1D416"1D51A"1D4E6"1D54E"1D5B6

W 𝑊 𝐖 𝔚 𝓦 𝕎 𝖶▶

Strong rationale: diﬀerent letter shapes have diﬀerent
meanings in mathematics, and therefore should be
distinguishable in plain text
▶ I.e., symbols are expressible without changing fonts▶ How does this translate to LATEX?
▶ (This is where I went wrong.)Alphabetic
ЛЅЦ
Ksymbols
t
+



𝐴
𝐵
𝐶
𝐷
𝐸
𝐹
𝐺
𝐻
𝐼
𝐽
𝐾
𝐿

ۭ
‫ۮ‬
‫ۯ‬
۰
۱
۲
۳
۴
۵
۶
۷
۸

ϻβϋЗϋ
ϻβϋЗψ
ϻβϋЗЏ
ϻβϋЗЌ
ϻβϋЗπ
ϻβϋЗϯ
ϻβϋЗΞ
ϻβϋЗΪ
ϻβϋЗΫ
ϻβϋЗβ
ϻβϋЗζ
ϻβϋЗυ

Ӷ
ӷ
Ӹ
ӹ
Ӻ
ӻ
Ӽ
ӽ
Ӿ
ӿ
Ԁ
ԁ

ন
঩
প
ফ
ব
ভ
ম
য
র
঱
ল
঳

T

ƻ
Ƽ
ƽ
ƾ
ƿ
ǀ
ǁ
ǂ
ǃ
Ǆ
ǅ
ǆ

2

K,aR

/

$KBi

L

$KBi"
$KBi*

L

$KBi.
$KBi1

L

$KBi6
$KBi:

L

$KBi>

L

$KBiA
$KBiC

L

$KBiE
$KBiG

L

L

L

L

L

L

ϻβЏυβᅧ౗𝛽ϻβЏυζᅨ symbols
ౘ𝛾Alphabetic
ϻβЏυυ
ϻβЌии
ϻβЌиϻ
ϻβЌиК
ϻβЌиЗ
ϻβЌиϋ
ϻβЌиψ
ϻβЌиЏ
ϻβЌиЌ
ϻβЌиπ
ϻβЌиϯ
ϻβЌиΞ
ϻβЌиΪ
ϻβЌиΫ

ᅩ
ᅾ
ᅫ
ᅬ
ᅭ
ᅮ
ᅯ
ᅱ
ᅲ
ᅳ
ᅴ
ᇢ
ᅵ
ᅶ

ౙ
ౚ
౛
౜
ౝ
౞
౟
ౠ
ౡ
ౢ
ౣ
౤
౥
౦

𝛿
𝜀
𝜁
𝜂
𝜃
𝜄
𝜅
𝜆
𝜇
𝜈
𝜉
𝜊
𝜋
𝜌

জ
ঝ
ঞ
ট
ঠ
ড
ঢ
ণ
ত
থ
দ
ধ
ন
঩
প
ফ

‫܌‬
‫ݔ‬
‫ܤ‬
ܰ
࡜
݈
ࠬ
‫ݠ‬
‫ݬ‬
‫ݸ‬
‫ބ‬
‫ސ‬
ࡐ
ި
ߌ
߰

$KBi#2i
$KBi;KK$KBi/2Hi
$KBi2TbBHQM$KBix2i
$KBi2i
$KBii?2i
$KBiBQi
$KBiFTT$KBiHK#/$KBiKm$KBiMm$KBitB$KBiQKB+`QM$KBiTB$KBi`?QϻβψЗπ

ච
ඡ
ඣ
ඤ
ඥ
ඦ
ඨ
ඩ
ඪ
ණ
ඬ
ථ
඲
ඳ

છ
જ
ઝ
ઞ
ટ
ઠ
ડ
ઢ
ણ
ત
થ
દ
ધ
ન

𝔸
𝔹
𝔻
𝔼
𝔽
𝔾
𝕀
𝕁
𝕂
𝕃
𝕄
𝕆
𝕊
𝕋

Alphabetic symbols
ϻβψЗϯ
ϻβψЗΪ

ϻβψЗΫ
ϻβψЗβ
ϻβψЗζ
ϻβψϋи
ϻβψϋϻ
ϻβψϋК
ϻβψϋЗ
ϻβψϋϋ
ϻβψϋЏ
ϻβψϋΞ
ϻβψϋΪ

ߠ
ߡ
ߢ
ߣ
ߤ
ߥ
ߦ
ߧ
ߨ
ߩ
ߪ
߫
߬
߭

৵
৶
৸
৹
৺
৻
৽
৾
৿
਀
ਁ
ਃ
ਇ
ਈ

$"##

L

$"##"
$"##.

L

$"##1
$"##6

L

$"##:
$"##A

L

$"##C
$"##E

L

$"##G

L

$"##J
$"##P

L

$"##a
$"##h

L

L

L

L

L

L

L

What does LATEX do with maths alphabets?
Re =ρv cos(θ)
for 0 ≤ θ < π/2L
H ∼ Hom(Z)k = [kx , ky , kz ]Tk ∈ R3What are the diﬀerent fonts here?
▶ ρv etc — \mathnormal▶ Re — \mathit▶ cos — \mathrm (actually \operator@font)▶ ‘for …’ — \textrm▶ (·)T or (·)T — your choice▶ Hom — \mathbf▶ v — \mathbf — but should it be?
▶ R — \mathbb

What does LATEX do with maths alphabets?
Re =ρv cos(θ)
for 0 ≤ θ < π/2L
H ∼ Hom(Z)k = [kx , ky , kz ]Tk ∈ R3What are the diﬀerent fonts here?
▶ ρv etc — \mathnormal▶ Re — \mathit▶ cos — \mathrm (actually \operator@font)▶ ‘for …’ — \textrm▶ (·)T or (·)T — your choice▶ Hom — \mathbf▶ v — \mathbf — but should it be?
▶ R — \mathbb

What does LATEX do with maths alphabets?
Re =ρv cos(θ)
for 0 ≤ θ < π/2L
H ∼ Hom(Z)k = [kx , ky , kz ]Tk ∈ R3What are the diﬀerent fonts here?
▶ ρv etc — \mathnormal▶ Re — \mathit▶ cos — \mathrm (actually \operator@font)▶ ‘for …’ — \textrm▶ (·)T or (·)T — your choice▶ Hom — \mathbf▶ v — \mathbf — but should it be?
▶ R — \mathbb

What does LATEX do with maths alphabets?
Re =ρv cos(θ)
for 0 ≤ θ < π/2L
H ∼ Hom(Z)k = [kx , ky , kz ]Tk ∈ R3What are the diﬀerent fonts here?
▶ ρv etc — \mathnormal▶ Re — \mathit▶ cos — \mathrm (actually \operator@font)▶ ‘for …’ — \textrm▶ (·)T or (·)T — your choice▶ Hom — \mathbf▶ v — \mathbf — but should it be?
▶ R — \mathbb

What does LATEX do with maths alphabets?
Re =ρv cos(θ)
for 0 ≤ θ < π/2L
H ∼ Hom(Z)k = [kx , ky , kz ]Tk ∈ R3What are the diﬀerent fonts here?
▶ ρv etc — \mathnormal▶ Re — \mathit▶ cos — \mathrm (actually \operator@font)▶ ‘for …’ — \textrm▶ (·)T or (·)T — your choice▶ Hom — \mathbf▶ v — \mathbf — but should it be?
▶ R — \mathbb

What does LATEX do with maths alphabets?
Re =ρv cos(θ)
for 0 ≤ θ < π/2L
H ∼ Hom(Z)k = [kx , ky , kz ]Tk ∈ R3What are the diﬀerent fonts here?
▶ ρv etc — \mathnormal▶ Re — \mathit▶ cos — \mathrm (actually \operator@font)▶ ‘for …’ — \textrm▶ (·)T or (·)T — your choice▶ Hom — \mathbf▶ v — \mathbf — but should it be?
▶ R — \mathbb

What does LATEX do with maths alphabets?
Re =ρv cos(θ)
for 0 ≤ θ < π/2L
H ∼ Hom(Z)k = [kx , ky , kz ]Tk ∈ R3What are the diﬀerent fonts here?
▶ ρv etc — \mathnormal▶ Re — \mathit▶ cos — \mathrm (actually \operator@font)▶ ‘for …’ — \textrm▶ (·)T or (·)T — your choice▶ Hom — \mathbf▶ v — \mathbf — but should it be?
▶ R — \mathbb

What does LATEX do with maths alphabets?
Re =ρv cos(θ)
for 0 ≤ θ < π/2L
H ∼ Hom(Z)k = [kx , ky , kz ]Tk ∈ R3What are the diﬀerent fonts here?
▶ ρv etc — \mathnormal▶ Re — \mathit▶ cos — \mathrm (actually \operator@font)▶ ‘for …’ — \textrm▶ (·)T or (·)T — your choice▶ Hom — \mathbf▶ v — \mathbf — but should it be?
▶ R — \mathbb

Maths alphabet fonts▶

Many many alphabet styles used in mathematics▶ T X achieved this by switching fonts
E▶ In Unicode mathematics, these are all in the one font
▶ Or are they?

What’s missing?
▶

Unicode mathematics has a few potential gaps
▶ To date, no provision for ‘text spacing’ (v maths)▶ Although it’s up to the font designer, symbols in UM
fonts are kerned as single-letter variables.
▶ Ramiﬁcations:
Upright roman not spaced for text (sin x)
Italic roman not spaced for text (Re)▶ Bold roman not spaced for text (Hom)▶
▶

What’s missing?
▶

Unicode mathematics has a few potential gaps
▶ To date, no provision for ‘text spacing’ (v maths)▶ Although it’s up to the font designer, symbols in UM
fonts are kerned as single-letter variables.
▶ Ramiﬁcations:
Upright roman not spaced for text (sin x)
Italic roman not spaced for text (Re)▶ Bold roman not spaced for text (Hom)▶
▶

Unfortunately, unicode-math has been calling these\mathrm, \mathit, and \mathbf for many years.

This gives a good excuse for a talk▶

Historical/interesting examples of mathematics▶ How unicode-math works, T X-nicallyE▶ Fixing the alphabet problem

Overview of Unicode Mathematics

Historical and current examples
Sidenote: symbols

TEX’s methods

\mathrm with maths spacing
Historical example

Osborne Reynolds. “An Experimental Investigation of the Circumstances Which
Determine Whether the Motion of Water Shall Be Direct or Sinuous, and of the Law ofResistance in Parallel Channels”. In: Phil. Trans. R. Soc. Lond. 174 (1883), pp. 935–982.
doi: 10.1098/rstl.1883.0029

integrable in time. With the Cauchy–Schwartz inequality, from (7.12) we\mathrmwithintegrable.
maths To
spacingU(t)is also squarecomplete the proof of regulation ofModernexampleBarbalat’s lemma, all that remains to show is that dU(t)2 /dt is uniformly b
Toward this end, we calculate
#
$
d
D̂(t)
2
AD̂(t)˙U(t) = 2U(t)K e
G2 (t) ,Ẋ(t) + D̂(t)G1(t) +
dt
D

where
% 1 & Adaptive, and PDE Systems.
Miroslav Krstić. Delay Compensation for Nonlinear,'AD̂(t)
Birkhäuser Boston,
2009.
doi:
10.1007/978-0-8176-4877-0
I + AD̂(t)(1 − y) g(y,t)dy ,X(t) +
G (t) = Ae
10AD̂(t)% 1

u(0,t) +AD̂(t)g(y,t)dy ,G (t) = BU(t) − Be
Also note2
0
d
d
vs
and
dt
dt
g(y,t) = eAD̂(t)(1−y) Bu(y,t) .

˙ is uniformly bounded over t ≥ 0 according to (7.13)–(7.15)
The signal D̂(t)˙given bye and e⎧⎨ 0,Proj[0,D̄] {τD } = τD 0,⎩
1,

for the scalar (delay case) and by⎧⎪⎨ I,
ProjΠ {τθ } = τθT⎪θP
⎩ I − ∇θ̂ P∇,∇ PT ∇ Pθ̂θD̂ = 0 and τD < 0,D̂ = D̄ and τD > 0,otherwiseθ̂ ∈ Π̊ or ∇θ̂ P T τ ≤ 0,
θ̂ ∈ ∂ Π and ∇θ̂ P T τ > 0,

(9

(9

for the vector (plant parameter) case.
The transformed state of the actuator isw(x,t) = e(x,t) − D̂(t)− K(θ̂ )e" x

K(θ̂ )eA(θ̂ )D̂(t)(x−y) B(θ̂ )e(y,t)dy0
A(θ̂ )D̂(t)x(9

X̃(t),and the constant b is chosen such thatb ≥ 4 sup |PB|2 (θ̂ )
θ̂ ∈Π

D̄
.
λ

(9

\mathrm
withof text
3.1. Effects
size spacing
As indicated above in Section 2.1, D’Arcy Thompson
used the Froude number to compare the walking speedsof different sized characters in Gulliver’s Travels. This isclearly one of the major benefits of the Froude number, withthe primary application being in the study of children’s gait
[1,2,32,41–44]. Alexander [32] showed that when dimensionless stride length was plotted as a function of dimensionless speed β whereβ = v/(gL)1/2 = (Fr)1/2(3)

then data for children aged over 4 years were the sameas adults. He used this empirical relationship to predict
the walking speeds for two small hominids (with estimatedChristopher
Mark
J. O’Malley.
“Froudethe contribution
heightsL. Vaughanof 1.19and
and
1.39m) wholeftandtheirfootprintsofinnaval
volarchitecture to our understanding of bipedal locomotion”. In: Gait & Posture 21.3
canic
ash
at
Laetoliin
East
Africa3.7
million
years
ago
[1].
(2005), pp. 350–362. doi: 10.1016/j.gaitpost.2004.01.011
Alexander estimated the speeds to be 0.64 and 0.75 m/s, re-

ph
ch
to
lo
le
op
m
an
th
th

β(

w
fo

3.

same phase relationship; (2) corresponding feet have equal\mathrmwith(%text
spacing
duty factorsof cyclein ground contact); (3) relative (i.e.

dimensionless) stride lengths are equal; (4) forces on corre…sometimes
sponding feet are equal multiples of body weight; and (5)power outputs are proportional to body weight times speed.
He hypothesised, and provided the necessary experimental
evidence to demonstrate, that animals meet these five criteria when they travel at speeds that translate to equal values of Fr [31]. Evidence in support of criterion 3 has beenpresented in Fig. 7a, while the data for criterion 2 may beseen in Fig. 7b. At Fr values below 2, the phase differences
lie between 0.4 and 0.5, and the animals utilise symmetrical gaits such as walking, trotting and pacing. There is anabrupt transition at Fr values between 2 and 3, and above
Christopher
L. Vaughan use
and Mark
J. O’Malley. “Froude
the contributionof naval
3 the animals
asymmetricalgaitsandsuch
as cantering
andarchitecture to our understanding of bipedal locomotion”. In: Gait & Posture 21.3
galloping.
Although
Alexander
developedthedynamic
sim(2005), pp. 350–362. doi: 10.1016/j.gaitpost.2004.01.011
ilarity hypothesis for quadrupedal animals [31], it may also
be applied to bipedal gait [33].

ila
in
Py
er
de
je
en
(F
al

fre
de
H
of
w
in
Va
ta
is

U cos nt.
Dy ¼ vð0; 0Þdt þ ' ' '10 \mathrm
The velocity
profilein aspacing
fluid between two parallel surfaces due to
with
text
oscillating
gradientshown
in Eq.
bewill be disAs pressure
well as moving
bodily,wasthe £uid
element
will (7.6)
rotatetoand00torted as indicated by(the corners, which are labeled A0 ; B;
C
and D0 to) ,!
p
ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ"#
represent the element at time cosht ¼ dt. The
rotationof the yside CD to its newð1 þ
iÞ ðn=2nPx0
is indicatedis positive
position
C 0 DRe
paﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
ﬃ # when
1by$the angle" da,whereeint measured
uðy; tÞ ¼
irn
counterclockwise. Similarly, therotationBC a
to its new positioncoshð1 þofiÞthe side
ðn=2n
B0 C 0 is indicated by the angle db, where b is positive when measured clockwise. Expressions for da and db in terms of the velocity components may beobtained as follows:
From the geometry of the element as it appears at time t ¼ dt,$
y component of D0 C 0
da ¼ tan
x component of D0 C 0% 1&½vð2dx; (12dyÞ dt þ ' ' '% ( ½vð(12dx; (12dyÞ dt þ ' ' '%
¼ tan(1dx þ ' ' '(1#where v is evaluated ¢rst at the point D, whose coordinates are ð12dx; (12dyÞ,I. G.andCurrie.
Fundamental
Fluids.
Thirdcoordinates
Edition. Marcelare
Dekker,secondlyat Mechanics
the pointof C,
whoseð(122003
dx; (12dyÞ. The
0 0x component of the side D C will be only slightly di¡erent from dx, andWhere is \mathbf used?

Horace Lamb. Hydrodynamics. Fourth Edition. Cambridge University Press, 1916

of an equivalent surface current density. In the expressions to follow, themagnetisation of each magnet has been assumed to be homogeneous andconstant, which is usually a reasonable assumption for modern rare earthmagnetic material; hence terms involving r · M and r ⇥ M equate to zeroand have been omitted from Eqs 2.11 and 2.12.
In the first step, the integration takes place over the surface of the firstmagnet S1 , which is written for the charge model as

Where is \mathbf used?

B1 (x2 ) =

µ0
4p

I ⇥

S1

M1 · n̂s1

⇤ x2

| x2

x1
x1 | 3ds1 ,(2.11)

and for the current model as
B1 (x2 ) =

µ0
4p

I h

S1

i
x2
M1 ⇥ n̂s01 ⇥| x2

x1
x1 | 3ds01 ,(2.12)

where n̂ is the normal vector from the differential surface of integration ds.
In the second step, the integration of the function of the magnetic field of
the first magnet takes place over the surface of the second magnet S2 , andthe integral for the charge model is

William S P Robertson. “Modelling and design of magnetic levitation systems forI ⇥vibration isolation”. PhD thesis. The
University ⇤of Adelaide, 2013

F=

S2

M2 · n̂s2 B1 (x2 ) ds2 ,and for the current model isI hi(2.13)

Where is \mathbf used?

Saunders Mac Lane. Categories for the Working Mathematician. 2nd ed. Springer, 1998

Overview of Unicode Mathematics

Historical and current examples
Sidenote: symbols

TEX’s methods

Miroslav Krstić, Ioannis Kanellakopoulos, and Petar Kokotović. Nonlinear and Adaptive
Control Design. Ed. by Simon Haykin. John Wiley and Sons, 1995. isbn: 0-471-12732-9

Linearizable Feedforward Systems of Type I
Consider the class of strict-feedforward systems given byn−1ẋ1 = x2 + ∑ π j (x j )x j+1 + πn(xn )u ,j=2

ẋi = xi+1 ,ẋn = u ,i = 2, . . . , n − 1 ,where π j (0) = 0. Any system in this class is DECI.
Theorem G.5. The diffeomorphic transformationy 1 = x1 − ∑% xj

y i = xi ,i = 2, . . . , n ,

n

j=2 0

π j (s)ds ,converts the strict-feedforward system (G.34)–(G.36) into the chain of inte(G.8)–(G.9). The feedback law
3 With possibly nonzero initial conditions.

@�det g�= �det g�(g −1 )µ⌫ ,@gµ⌫@(g −1 )
= −(g −1 )µ (g −1 ) ⌫ ,gµ⌫✓2(2) = �det F��det g�−1 ,(g −1 ) Fµ �F ⌫ = ✓(2) gµ⌫ ,@✓(1)= −g F #µ F #⌫ ,
@gµ⌫
@✓(2)
1
= − ✓(2) (g −1 )µ⌫ ,
@gµ⌫
2
@✓(1)
#µ⌫
= 2F
Jared Speck. “The global stability of the
Minkowski, spacetime solution to the@Fµ⌫
Einstein-nonlinear system in wave
coordinates”. In: Analysis & PDE 7.4 (2014),
pp. 771–901. doi: 10.2140/apde.2014.7.771@✓(2) � #µ⌫= F,
Courtesy http://math.stackexchange.com/a/1250067/2961
@Fµ⌫
#µ⌫

Dr. Seuss. On Beyond Zebra! Random House, 1955
Courtesy http://math.stackexchange.com/a/1197641

Overview of Unicode Mathematics

Historical and current examples
Sidenote: symbols

TEX’s methods

Setting up maths in plain (Xe/Lua)TEX
Deﬁne some fonts:\font\Urm = "[texgyrepagella-regular.otf]:color=AA00AA"\font\Ubf = "[texgyrepagella-bold.otf]:color=0000FF"\font\Uit = "[texgyrepagella-italic.otf] :color=00AA00"\font\Umm = "[texgyrepagella-math.otf]:color=FF0000"

Which produces: rm bf it mm(Remember: red is the maths font in particular)

Now set up some maths families\newfam\Urmfam\newfam\Ubffam\newfam\Uitfam\textfont\Urmfam\Urm\textfont\Ubffam\Ubf\textfont\Uitfam\Uit\textfont1\mm(\textfont1 just means default symbols)

Switching to these fonts?
$$
a + b + c \quad α + β + γ$$$${\fam\Urmfam abcαβγ} \quad{\fam\Uitfam abcαβγ} \quad{\fam\Ubffam abcαβγ}
$$

a+b+c
abc

+ +

abcαβγ abc(What’s going on with symbols?)

Remap to Unicode\Umathcode`\a = 7 1 "1D44E\relax\Umathcode`\b = 7 1 "1D44F\relax\Umathcode`\c = 7 1 "1D450\relax\Umathcode`\α = 7 1 "1D6FC\relax\Umathcode`\β = 7 1 "1D6FD\relax\Umathcode`\γ = 7 1 "1D6FE\relax

𝑎+𝑏+𝑐 𝛼+𝛽+𝛾("1D44E is Unicode Plane 1 mathematical small a)

Remap to elsewhere in Unicode\Umathcode`\a = 7 1 "1D482\relax\Umathcode`\b = 7 1 "1D483\relax\Umathcode`\c = 7 1 "1D484\relax\Umathcode`\α = 7 1 "1D736\relax\Umathcode`\β = 7 1 "1D737\relax\Umathcode`\γ = 7 1 "1D738\relax

𝒂+𝒃+𝒄 𝜶+𝜷+𝜸("1D44E is Unicode Plane 1 mathematical small a)

Rundown so far▶

Maths is not like text
▶ In text, an input char, say ‘ascii a’ (U+97) is represented
at the same code-point in the current font
▶ In maths, an input char is (luckily!) mapped towhatever code-point we like!
▶ In this case, input char U+97 → font glyph U+1D44E

But what about alphabets?
Traditional TEXThis is very diﬀerent than traditional TEX maths where wehave a number of maths fonts with diﬀerent shapes in ascii
slots:▶ e.g., input char A65 → A65 maths font (\mathnormal)▶ e.g., input char A65 → A65 bold font (\mathbf)▶ e.g., input char A65 → A65 cal font (\mathcal)

But what about alphabets?
Unicode mathematics

Now we need to map within the same font:▶ e.g., input char A65 → A1D434 maths font (\mathnormal)▶ e.g., input char A65 → A1D400 maths font (\mathbf)▶ e.g., input char A65 → A1D49C maths font (\mathcal)

But what about alphabets?
Unicode mathematics

The naive approach (up until now):▶ Ignore classical T X maths; Unicode is the future!
E▶ \mathcal maps to calligraphic range.
▶ \mathfrak maps to fraktur range.
▶ \mathbf maps to bold upright symbols. (Maybe okay?)▶ \mathit maps to math italic symbols. (Not okay!)
Impedance mismatch between what TEX users use andwhat Unicode mathematics provides.

Kerning

The big problem here:▶ \mathbf traditionally uses a text font (why not?)▶ \mathit traditionally uses a text font (needed!)▶ \mathtt traditionally uses a text font▶ \mathsf traditionally uses a text fontWhere is \mathbf used?

▶

It’s not just one thing!
▶ Single symbols with math-like spacing▶ Multi-letter names with text-like spacing
Unicode only provides for the ﬁrst.
LATEX sort of only provides the second.

What about \mathit?

LATEX deﬁnes:▶

\mathnormal — italic symbols with maths spacing▶

\textit — italic letters with text spacing and text

behaviour▶ \mathit — italic w. text spacing and maths behaviour
These could all be separate fonts. Unicode only provides
for the \mathnormal case.

Similarly \mathrm

LATEX deﬁnes upright roman letters:▶

\textrm — with text spacing and text behaviour▶

\mathrm — with text spacing and maths behaviour

These could all be separate fonts. Unicode only provides
for ???.
Up to the font designer.

This is LATEX 2ε+amsmath▶

\mathnormal▶

\mathit▶

\mathrm▶

\mathbf▶

\mathsf▶

\mathtt▶

\mathcal▶

\mathfrak\boldsymbol / bm allowed for bold in many cases too.

Categories of alphabets in Unicode

Unambiguous:▶ bb, bbit, scr, bfscr, cal, bfcal, frak, bﬀrak
Ambiguous:▶ rm, it, tt, bfup, bﬁt, sfup, sﬁt, bfsfup, bfsﬁt, bfsf
I called these \mathrm, \mathit, etc., and that was …

…a poor decision

As discussed:▶ ‘\mathit’ ALWAYS (in practice) used for Re, Fr, …
So unicode-math’s deﬁnition for \mathit is no good.
▶ ‘bf’ might be for a × b or Hom(·)
So UM’s deﬁnition for \mathbf (and \mathrm,\mathsf, \mathtt) potentially wrong.
▶ So what to do?

Switching to symbol ranges▶

ALL mapping ranges are deﬁned with new commands\symbb, \symcal, \symbf, etc. etc.
▶ ALL traditional LATEX font switches are given alias
names \mathtextrm, \mathtextit, \mathtextbf, etc.
▶ Unambiguous ranges: \mathfrak := \symfrak▶ By default: \mathbf := \mathtextbf(i.e., as per LATEX 2ε )▶ Package options (mathbf=sym) to switch

Tell me about these symbol ranges▶

\symliteral switches to ‘literal’ input syntax.

▶

\symbf follows TEX or ‘ISO’ conventions with uprightlatin and italic greek letters.
▶ \symsf switches to upright or italic sans serif according
to a package option.

Tell me about these symbol ranges
Continued▶

\symbfit, \symbfup, \symsfit, \symsfup can be used

where needed.
▶ Why is \symtt not aliased to \mathtt?
Few slots; if you want your code font and yourtypewriter maths font to match, you’re better oﬀ
sticking with traditional \mathtt.
▶ There is no contextual math style yet;\symbf{\symsf{X}} does NOT produce \symbfsf{X}.
It probably should.

Undoing the mapping for\DeclareMathAlphabet▶

Latest version of unicode-math ‘ﬁxes’\DeclareMathAlphabet.
▶ This is what allows the distinction between \mathXYZ
and \symXYZ.
▶ Necessary for using character shapes outside ofUnicode.
▶ Requires some care hacking the NFSS.
▶ ‘User-level’ interface \setmathfontface.

\documentclass{article}\usepackage{unicode-math}\setmathfont{texgyrepagella-math.otf}[Scale=0.85]\setmathfontface\mathchor{texgyrechorus-mediumitalic.otf}\setoperatorfont\mathchor\begin{document}\[(\sin x)^2 + (\cos x)^2 = 1\]\end{document}cCN Ǫ ӝ,Rc Ǫ ӝ  ∼ F ⊗ G . (This holds moreherent sheaf. Describe an isomorphism Hom (F , G ) =generally if G is an O-module, but we won’t use that, so you may as well prove
the simpler result given in this exercise.)
13.7.C. E XERCISE . Suppose(13.7.1.1)

0→F →G →H →0is an exact sequence of quasicoherent sheaves on a scheme X, where H is a locally
free quasicoherent sheaf, and suppose E is a quasicoherent sheaf. By left-exactnessof Hom (Exercise 2.5.H),
0 → Hom (H , E ) → Hom (G , E ) → Hom (F , E ) → 0

is exact except possibly on the right. Show that it is also exact on the right. (Hint:⊕n , so!
this is local, so you can assume that X is affine, say Spec A, and H = A(13.7.1.1) can be written as 0 → M → N → A⊕n → 0. Show that this exact
sequence splits, so we can write N = M ⊕ A⊕n in a way that respects the exact
sequence.) In particular, if F , G , H , and OX are all coherent, and H is locally
free, then we have an exact sequence of coherent sheaves
0 → H ∨ → G ∨ → F ∨ → 0.

13.7.D. E XERCISE ( THE SUPPORT OF A FINITE TYPE QUASICOHERENT SHEAF IS
CLOSED ).
Suppose F is a sheaf of abelian groups. Recall Definition 2.4.2 ofRavi
Vakil. Foundations
Geometry.
Notes. 2.6.F(b))
Stanford of
University,
2015.
the support
of a sectionofsAlgebraicof F , anddefinitionLecture(cf. Exercise
the supportofurl:
http://math.stanford.edu/~vakil/216blog/
F . (Support is a stalk-local notion, and hence behaves well with respect to restriction to open sets, or to stalks. Warning: Support is where the germ(s) are nonzero,not where the value(s) are nonzero.) Show that the support of a finite type quasicoherent sheaf on a scheme X is a closed subset. (Hint: Reduce to the case X

From \mathnormal to \mathXYZAs discussed above, we have default mapping set up▶ e.g., input char A65 → A1D434 maths fontWhen we switch to \mathXYZ (such as \mathit), we needto undo this mapping so we can revert to▶ e.g., input \mathbf{A65 } → A65 bold maths font

Inside the NFSS
Brieﬂy\cs_set:Npn \use@mathgroup #1 #2{\mode_if_math:T{\math@bgroup\cs_if_eq:cNF {M@\f@encoding} #1 {#1}\__um_switchto_literal:\mathgroup #2 \relax\math@egroup}}

Downside to this approach▶

All this symbol remapping doesn’t come for free.
▶ The literals mapping is probably not yet complete, but
it current reads:\Umathcode 97=7\symoperators 97\scan_stop: \Umathcode 98=7\symoperators 98\sc
an_stop: \Umathcode 99=7\symoperators 99\scan_stop: \Umathcode 100=7\symoperato
rs 100\scan_stop: \Umathcode 101=7\symoperators 101\scan_stop: \Umathcode 102=7\symoperators 102\scan_stop: \Umathcode 103=7\symoperators 103\scan_stop: \Umat
hcode 104=7\symoperators 104\scan_stop: \Umathcode 105=7\symoperators 105\scan_stop: \Umathcode 106=7\symoperators 106\scan_stop: \Umathcode 107=7\symoperators 107\scan_stop: \Umathcode 108=7\symoperators 108\scan_stop: \Umathcode 109=7\
symoperators 109\scan_stop: \Umathcode 110=7\symoperators 110\scan_stop: \Umathcode 111=7\symoperators 111\scan_stop: \Umathcode 112=7\symoperators 112\scan_s
top: \Umathcode 113=7\symoperators 113\scan_stop: \Umathcode 114=7\symoperators
114\scan_stop: \Umathcode 115=7\symoperators 115\scan_stop: \Umathcode 116=7\symoperators 116\scan_stop: \Umathcode 117=7\symoperators 117\scan_stop: \Umathc
ode 118=7\symoperators 118\scan_stop: \Umathcode 119=7\symoperators 119\scan_st
op: \Umathcode 120=7\symoperators 120\scan_stop: \Umathcode 121=7\symoperators
121\scan_stop: \Umathcode 122=7\symoperators 122\scan_stop: \Umathcode 104=7\symoperators 104\scan_stop: \Umathcode 119886=7\symoperators 119886\scan_stop: \U
mathcode 119887=7\symoperators 119887\scan_stop: \Umathcode 119888=7\symoperato
rs 119888\scan_stop: \Umathcode 119889=7\symoperators 119889\scan_stop: \Umathc
ode 119890=7\symoperators 119890\scan_stop: \Umathcode 119891=7\symoperators 119891\scan_stop: \Umathcode 119892=7\symoperators 119892\scan_stop: \Umathcode 1
19893=7\symoperators 119893\scan_stop: \Umathcode 119894=7\symoperators 119894\
scan_stop: \Umathcode 119895=7\symoperators 119895\scan_stop: \Umathcode 119896=7\symoperators 119896\scan_stop: \Umathcode 119897=7\symoperators 119897\scan_stop: \Umathcode 119898=7\symoperators 119898\scan_stop: \Umathcode 119899=7\symoperators 119899\scan_stop: \Umathcode 119900=7\symoperators 119900\scan_stop:\Umathcode 119901=7\symoperators 119901\scan_stop: \Umathcode 119902=7\symoperators 119902\scan_stop: \Umathcode 119903=7\symoperators 119903\scan_stop: \Umathcode 119904=7\symoperators 119904\scan_stop: \Umathcode 119905=7\symoperators
119905\scan_stop: \Umathcode 119906=7\symoperators 119906\scan_stop: \Umathcod
e 119907=7\symoperators 119907\scan_stop: \Umathcode 119908=7\symoperators 1199
08\scan_stop: \Umathcode 119909=7\symoperators 119909\scan_stop: \Umathcode 119
910=7\symoperators 119910\scan_stop: \Umathcode 119911=7\symoperators 119911\sc
an_stop: \Umathcode 8462=7\symoperators 8462\scan_stop: \Umathcode 65=7\symoperators 65\scan_stop: \Umathcode 66=7\symoperators 66\scan_stop: \Umathcode 67=7\
symoperators 67\scan_stop: \Umathcode 68=7\symoperators 68\scan_stop: \Umathcod
e 69=7\symoperators 69\scan_stop: \Umathcode 70=7\symoperators 70\scan_stop: \U
mathcode 71=7\symoperators 71\scan_stop: \Umathcode 72=7\symoperators 72\scan_s
top: \Umathcode 73=7\symoperators 73\scan_stop: \Umathcode 74=7\symoperators 74\scan_stop: \Umathcode 75=7\symoperators 75\scan_stop: \Umathcode 76=7\symopera
tors 76\scan_stop: \Umathcode 77=7\symoperators 77\scan_stop: \Umathcode 78=7\symoperators 78\scan_stop: \Umathcode 79=7\symoperators 79\scan_stop: \Umathcode
80=7\symoperators 80\scan_stop: \Umathcode 81=7\symoperators 81\scan_stop: \Umathcode 82=7\symoperators 82\scan_stop: \Umathcode 83=7\symoperators 83\scan_st
op: \Umathcode 84=7\symoperators 84\scan_stop: \Umathcode 85=7\symoperators 85\
scan_stop: \Umathcode 86=7\symoperators 86\scan_stop: \Umathcode 87=7\symoperat▶

ors 87\scan_stop: \Umathcode 88=7\symoperators 88\scan_stop: \Umathcode 89=7\symoperators 89\scan_stop: \Umathcode 90=7\symoperators 90\scan_stop: \Umathcode
119860=7\symoperators 119860\scan_stop: \Umathcode 119861=7\symoperators 119861\scan_stop: \Umathcode 119862=7\symoperators 119862\scan_stop: \Umathcode 11986
3=7\symoperators 119863\scan_stop: \Umathcode 119864=7\symoperators 119864\scan_stop: \Umathcode 119865=7\symoperators 119865\scan_stop: \Umathcode 119866=7\symoperators 119866\scan_stop: \Umathcode 119867=7\symoperators 119867\scan_stop: \Umathcode 119868=7\symoperators 119868\scan_stop: \Umathcode 119869=7\symope
rators 119869\scan_stop: \Umathcode 119870=7\symoperators 119870\scan_stop: \Umathcode 119871=7\symoperators 119871\scan_stop: \Umathcode 119872=7\symoperators 119872\scan_stop: \Umathcode 119873=7\symoperators 119873\scan_stop: \Umathco
de 119874=7\symoperators 119874\scan_stop: \Umathcode 119875=7\symoperators 119875\scan_stop: \Umathcode 119876=7\symoperators 119876\scan_stop: \Umathcode 119877=7\symoperators 119877\scan_stop: \Umathcode 119878=7\symoperators 119878\s
can_stop: \Umathcode 119879=7\symoperators 119879\scan_stop: \Umathcode 119880=
7\symoperators 119880\scan_stop: \Umathcode 119881=7\symoperators 119881\scan_s
top: \Umathcode 119882=7\symoperators 119882\scan_stop: \Umathcode 119883=7\symoperators 119883\scan_stop: \Umathcode 119884=7\symoperators 119884\scan_stop:\Umathcode 119885=7\symoperators 119885\scan_stop: \Umathcode 945=7\symoperators 945\scan_stop: \Umathcode 946=7\symoperators 946\scan_stop: \Umathcode 947=7\
symoperators 947\scan_stop: \Umathcode 948=7\symoperators 948\scan_stop: \Umathcode 949=7\symoperators 949\scan_stop: \Umathcode 950=7\symoperators 950\scan_s
top: \Umathcode 951=7\symoperators 951\scan_stop: \Umathcode 952=7\symoperators
952\scan_stop: \Umathcode 953=7\symoperators 953\scan_stop: \Umathcode 954=7\symoperators 954\scan_stop: \Umathcode 955=7\symoperators 955\scan_stop: \Umathc
ode 956=7\symoperators 956\scan_stop: \Umathcode 957=7\symoperators 957\scan_st
op: \Umathcode 958=7\symoperators 958\scan_stop: \Umathcode 959=7\symoperators
959\scan_stop: \Umathcode 960=7\symoperators 960\scan_stop: \Umathcode 961=7\symoperators 961\scan_stop: \Umathcode 962=7\symoperators 962\scan_stop: \Umathco
de 963=7\symoperators 963\scan_stop: \Umathcode 964=7\symoperators 964\scan_sto
p: \Umathcode 965=7\symoperators 965\scan_stop: \Umathcode 966=7\symoperators 9
66\scan_stop: \Umathcode 967=7\symoperators 967\scan_stop: \Umathcode 968=7\symoperators 968\scan_stop: \Umathcode 969=7\symoperators 969\scan_stop: \Umathcod
e 1013=7\symoperators 1013\scan_stop: \Umathcode 977=7\symoperators 977\scan_st
op: \Umathcode 1008=7\symoperators 1008\scan_stop: \Umathcode 981=7\symoperators 981\scan_stop: \Umathcode 1009=7\symoperators 1009\scan_stop: \Umathcode 982=
7\symoperators 982\scan_stop: \Umathcode 120572=7\symoperators 120572\scan_stop: \Umathcode 120573=7\symoperators 120573\scan_stop: \Umathcode 120574=7\symope
rators 120574\scan_stop: \Umathcode 120575=7\symoperators 120575\scan_stop: \Umathcode 120576=7\symoperators 120576\scan_stop: \Umathcode 120577=7\symoperators 120577\scan_stop: \Umathcode 120578=7\symoperators 120578\scan_stop: \Umathco
de 120579=7\symoperators 120579\scan_stop: \Umathcode 120580=7\symoperators 120
580\scan_stop: \Umathcode 120581=7\symoperators 120581\scan_stop: \Umathcode 120582=7\symoperators 120582\scan_stop: \Umathcode 120583=7\symoperators 120583\s

can_stop: \Umathcode 120584=7\symoperators 120584\scan_stop: \Umathcode 120585=
7\symoperators 120585\scan_stop: \Umathcode 120586=7\symoperators 120586\scan_s
top: \Umathcode 120587=7\symoperators 120587\scan_stop: \Umathcode 120588=7\symoperators 120588\scan_stop: \Umathcode 120589=7\symoperators 120589\scan_stop:\Umathcode 120590=7\symoperators 120590\scan_stop: \Umathcode 120591=7\symopera
tors 120591\scan_stop: \Umathcode 120592=7\symoperators 120592\scan_stop: \Umat
hcode 120593=7\symoperators 120593\scan_stop: \Umathcode 120594=7\symoperators
120594\scan_stop: \Umathcode 120595=7\symoperators 120595\scan_stop: \Umathcode
120596=7\symoperators 120596\scan_stop: \Umathcode 120598=7\symoperators 12059
8\scan_stop: \Umathcode 120599=7\symoperators 120599\scan_stop: \Umathcode 1206
00=7\symoperators 120600\scan_stop: \Umathcode 120601=7\symoperators 120601\scan_stop: \Umathcode 120602=7\symoperators 120602\scan_stop: \Umathcode 120603=7\
symoperators 120603\scan_stop: \Umathcode 913=7\symoperators 913\scan_stop: \Umathcode 914=7\symoperators 914\scan_stop: \Umathcode 915=7\symoperators 915\scan_stop: \Umathcode 916=7\symoperators 916\scan_stop: \Umathcode 917=7\symoperators 917\scan_stop: \Umathcode 918=7\symoperators 918\scan_stop: \Umathcode 919=
7\symoperators 919\scan_stop: \Umathcode 920=7\symoperators 920\scan_stop: \Umathcode 921=7\symoperators 921\scan_stop: \Umathcode 922=7\symoperators 922\scan_stop: \Umathcode 923=7\symoperators 923\scan_stop: \Umathcode 924=7\symoperators 924\scan_stop: \Umathcode 925=7\symoperators 925\scan_stop: \Umathcode 926=7\symoperators 926\scan_stop: \Umathcode 927=7\symoperators 927\scan_stop: \Umat
hcode 928=7\symoperators 928\scan_stop: \Umathcode 929=7\symoperators 929\scan_stop: \Umathcode 930=7\symoperators 930\scan_stop: \Umathcode 931=7\symoperators 931\scan_stop: \Umathcode 932=7\symoperators 932\scan_stop: \Umathcode 933=7\
symoperators 933\scan_stop: \Umathcode 934=7\symoperators 934\scan_stop: \Umathcode 935=7\symoperators 935\scan_stop: \Umathcode 936=7\symoperators 936\scan_s
top: \Umathcode 937=7\symoperators 937\scan_stop: \Umathcode 1012=7\symoperators 1012\scan_stop: \Umathcode 120546=7\symoperators 120546\scan_stop: \Umathcode
120547=7\symoperators 120547\scan_stop: \Umathcode 120548=7\symoperators 12054
8\scan_stop: \Umathcode 120549=7\symoperators 120549\scan_stop: \Umathcode 1205
50=7\symoperators 120550\scan_stop: \Umathcode 120551=7\symoperators 120551\scan_stop: \Umathcode 120552=7\symoperators 120552\scan_stop: \Umathcode 120553=7\
symoperators 120553\scan_stop: \Umathcode 120554=7\symoperators 120554\scan_sto
p: \Umathcode 120555=7\symoperators 120555\scan_stop: \Umathcode 120556=7\symop
erators 120556\scan_stop: \Umathcode 120557=7\symoperators 120557\scan_stop: \U
mathcode 120558=7\symoperators 120558\scan_stop: \Umathcode 120559=7\symoperators 120559\scan_stop: \Umathcode 120560=7\symoperators 120560\scan_stop: \Umathc
ode 120561=7\symoperators 120561\scan_stop: \Umathcode 120562=7\symoperators 120562\scan_stop: \Umathcode 120563=7\symoperators 120563\scan_stop: \Umathcode 1
20564=7\symoperators 120564\scan_stop: \Umathcode 120565=7\symoperators 120565\
scan_stop: \Umathcode 120566=7\symoperators 120566\scan_stop: \Umathcode 120567=7\symoperators 120567\scan_stop: \Umathcode 120568=7\symoperators 120568\scan_stop: \Umathcode 120569=7\symoperators 120569\scan_stop: \Umathcode 120570=7\symoperators 120570\scan_stop: \Umathcode 120563=7\symoperators 120563\scan_stop:

This adds approx 100× overhead to the font switch.
(hundreds of microseconds per)έρος Β - Κεφάλαιο 2ο

SummaryEΠΑΝΑΛΗΨΗ – ΑΝΑΚΕΦΑΛΑΙΩΣΗ 2oυ KΕΦΑΛΑΙΟΥ
Τριγωνομετρικοί αριθμοί γωνίας ω με

!ω!1Σε ορθοκανονικό σύστημα αξόνων Οxy, αν είναι ω = x O ,και Μ(x, y) είναι ένα οποιοδήποτε σημείο της πλευράς Ο ,διαφορετικό από το Ο, τότε:
ρ

ΟΜ

2
!"
x2
και ημωyρ συνω

x
εφω
ρ

y
.
x

12 + 22 = !"5,
Π.χ. αν Μ(1, 2), τότε ρ = !"
2
2!"
5
,
ημω =
=
5
!"
5

y

Μ

y
ρ
ω
O

x

x

2
1
!"
5
συνω =, εφω = 1 = 2.
=
5
!"
5

• Τα πρόσημα των τριγωνομετρικών αριθμών
μιας γωνίας ω με 0 ! ω ! 180 φαίνονταιστον διπλανό πίνακα:ωημω+συreference.)νω(Thanks to Apostolos Syropoulis for the
ε ω

1
+

Οι παραπληρωματικές γωνίες έχουν το ίδιο ημίτονο και αντίθετους τους άλλους
τριγωνομετρικούς αριθμούς. Δηλαδή,
ημ 1
ω
ημω
συν 1
ω
συνω
εφ 1
ω
εφω

\setmainfont{Kurier-Regular.otf}\setmathfont{texgyrepagella-math.otf}[Scale=0.85]\[\mathrm{ημ}(ω) = \frac{y}{ρ}\]Ƌȇ ި 

ǫ߼

But probably \mathrm should now be called \mathup!
(My friend Nino says ‘rm’ means remove; cf. GNU/Linux.)As we’ll see, it’s convenient to write the periodized density as a convolution with a sum of shifted δ’s:

depend on the choice of Tq .

‘Sha’ρp (x) =

∞
!

k=−∞

ρ(x − pk) =∞!

δ(x − kp) ∗ ρ(x) =

" !
∞

#
δ(x − kp) ∗ ρ(x) .

7. Examples
and Experiments
k=−∞
k=−∞

section contains tables and numericalexamples that illustrate the main
NowThisintroduce∞!
themes of this paper. In SectionIII p7.1,
Table
1, which contains 16 exam(x) =we explainδ(x − kp),k=−∞ples of pairs f, g such that the strong Beilinson-Blochconjecture and Theorem 6.1

together
imply the existence of nontrivial elements of the Shafarevich-Tate groupsothat, simply,
III p ∗ ρ . the higher-weight modular symbol
p =outlinesof the motive attached to f . Section ρ7.2

computations
that
were
used inTable
1. Section
discusses
Table
III
show.
Bracewellcallsmaking
it the “shahfunction”,
after the7.3
Cyrillic
letter, andthis 2,has
p is the star of thecaughtIt’s also referredas the Dirac(with spacingp).
whichon.
summarizes
the toresultsof ancomb
extensive
computationof conjectural orders of

Shafarevich-Tategroups formodular motives of low level and weight. Section 7.4
Using
the convolution theorem,
we have
givesOsgood.
specific The
exampleswhich various
hypotheses fail.
NoteNotes.
that inStanford§7 “modular
Brad
Fourier inTransform
and its Applications.
LectureF ρ = F ρ · F IIIp .
University,
2008
symbol” hasa diﬀerent meaning fromp in §5, being
related to homology rather thancohomology.
For precise
definitions
see [SV].
What
is F IIIp ? That’sa really interestingquestion.

7.1. Visible X Table 1. Table 1 on page 11 lists sixteen pairs of newforms f
5.2
III Distribution
and g The(of equal
weights and levels) along with at least one prime q such that there
is a prime q | q with f ≡ g (mod q). In each case, ords=k/2 L(g, k/2) ≥ 2 whileWe
want
to develop
the propertiesof III
particularly
Fouriertransform.
fact, we met
distribution
L(f,
k/2)
̸= 0. The
notation
isp ,asfollows.itsThefirstcolumnIncontains
a thislabel
whoseearlier, in Chapter 1. Rather, we met its Fourier transform — it’s the continuous buzz signal, as we’ll
structureisdiscuss further, below.
[Level]k[Weight][GaloisOrbit]
As a “standard” we take the spacing p to be 1,!
so we sum over the integer points and defineThis label determines a newform ∞g =an q n , up to Galois
conjugacy. For example,∞!
!
127k4C
denotesa newform
third
orbit
of δnewforms
in Sin4 (Γ0 (127)).
Neil
Watkins,
WilliamDummigan,andδ(xMark
Stein.
“Constructing
elements
III(x) in= the− k)Galois
or III
=
k.
k=−∞
k=−∞
Shafarevich-Tategroupsmodularmotives”.
NumbertheoryThe Galois orbits
are ofordered
first
by theIn:degreeof Q(.
. . , andan , .algebraic. .), thengeometryby the303(2003)of absolute values |Tr(ap (g))| for p not dividing the level, with positive
sequence
trace being first in the event that the two absolute values are equal, and the first\setmathfont{texgyrepagella-math.otf}[Scale=0.85]\setmathfontface\mathcyr{Charter Roman}\[
ρ_p = \mathcyr{Ш}_p \ast ρ \qquad\dot{\mathcyr{у}} = A(x,t)\mathcyr{у} +F(x,u)θ
\qquad y = mx+c\]߼֡  Ш ֡ ຆ ߼уɥ  ƻ Ǫ Ǧ уǀ Ǫ ǧ ࠸ǫ  ǟǪ

Ǘ

New commands to diﬀerentiate maths and text
AMTmi +?`+i2`b, ǵ#+ 77BǶ
$i2ti`K$i2tiBi$i2ti#7$i2tib7$i2tiii

#+ {

#+ {

#+ {

#+ {

#+ 77B$Ki?`K$Ki?Bi$Ki?#7$Ki?b7$Ki?ii

#+{

#+{

#+{

abcffiabcffi$bvKmT$bvKBi$bvK#7$bvKb7$bvKii

BCDग२३४७ ७ ९ऴवशहह़

઼ઽાુુૄ

஌஍எ஑஑ஔ

ﬁn