AISC Meets Natural Typography James H. Davenport Department of - - PowerPoint PPT Presentation

AISC Meets Natural Typography

James H. Davenport Department of Computer Science University of Bath Bath BA2 7AY England J.H.Davenport@bath.ac.uk July 31, 2008

Notation exists to be abused . . . the abuses of language without which any mathematical text threatens to be- come pedantic and even unreadable. (Bourbaki) ❜✉t s♦♠❡ ❛❜✉s❡ ✐s ♠♦r❡ ❤❛r♠❢✉❧ t❤❛♥ ♦t❤❡rs✳

1

Notation exists to be abused . . . the abuses of language without which any mathematical text threatens to be- come pedantic and even unreadable. (Bourbaki) but some abuse is more harmful than others.

2

The trivial differences Intervals The “Anglo-saxon” way (0, 1] and the “French” way ]0, 1].

■♥✈❡rs❡ ❢✉♥❝t✐♦♥s ■s ❛r❝s✐♥ t❤❡ ✭❛❄✱ ✇❤✐❝❤❄❄✮

s✐♥❣❧❡✲✈❛❧✉❡❞ ✐♥✈❡rs❡ ❛♥❞ ❆r❝s✐♥ t❤❡ ♠✉❧t✐✲ ✈❛❧✉❡❞ ✭❆♥❣❧♦✲❙❛①♦♥✮✱ ♦r t❤❡ ❝♦♥✈❡rs❡ ✭❋r❡♥❝❤✮❄ ✐ ♦r ❥ ■♥ ♣r❛❝t✐❝❡✱ t❤✐s ❝❛✉s❡s ❧✐tt❧❡ ❝♦♥❢✉s✐♦♥ ❢♦r ❡①♣❡rts✱ s♦♠❡ ❢♦r st✉❞❡♥ts✳

3

The trivial differences Intervals The “Anglo-saxon” way (0, 1] and the “French” way ]0, 1].

Inverse functions Is arcsin the (a?, which??)

single-valued inverse and Arcsin the multi- valued (Anglo-Saxon), or the converse (French)? ✐ ♦r ❥ ■♥ ♣r❛❝t✐❝❡✱ t❤✐s ❝❛✉s❡s ❧✐tt❧❡ ❝♦♥❢✉s✐♦♥ ❢♦r ❡①♣❡rts✱ s♦♠❡ ❢♦r st✉❞❡♥ts✳

4

The trivial differences Intervals The “Anglo-saxon” way (0, 1] and the “French” way ]0, 1].

Inverse functions Is arcsin the (a?, which??)

single-valued inverse and Arcsin the multi- valued (Anglo-Saxon), or the converse (French)? i or j In practice, this causes little confusion for experts, some for students.

5

metric tensor Is the metric tensor for flat Min- kowski space

     

−1 1 1 1

     

• r its nega-

tive

     

1 −1 −1 −1

     

? Is the temporal variable the last, rather than the first, co-

• rdinate, giving

     

1 1 1 −1

     

, or its negative?

6

Binomial coefficients

n

k

• , Ck

n, Cn k

?

✵ ✷ N❄ ❑♥♦✇❧❡❞❣❡ ♦❢ ❧✐♥❣✉✐st✐❝ ❝♦♥t❡①t ♠❛② ❤❡❧♣ ❞❡❝✐❞❡ t❤✐s q✉❡st✐♦♥✱ ❜✉t ✐s ❢❛r ❢r♦♠ ❝❡rt❛✐♥✳

✏ ✑

❈❧❡❛r❧② ✭✶✮✭♣✶✮❂✷✿ ✐t✬s ❛ q✉❛❞r❛t✐❝ r❡s✐❞✉❡ s②♠❜♦❧✦ ❭❧❡❢t✭❭❢r❛❝④✲✶⑥④♣⑥❭r✐❣❤t✮ ✐s✱ ❛❧❛s✱ ❤♦✇ ♣r♦❢❡s✲ s✐♦♥❛❧ t②♣❡s❡tt❡rs ❡♥❝♦❞❡ ✐t✳

7

Binomial coefficients

n

k

• , Ck

n, Cn k

?

0 ∈ N? Knowledge of linguistic context may help decide this question, but is far from certain.

✏ ✑

❈❧❡❛r❧② ✭✶✮✭♣✶✮❂✷✿ ✐t✬s ❛ q✉❛❞r❛t✐❝ r❡s✐❞✉❡ s②♠❜♦❧✦ ❭❧❡❢t✭❭❢r❛❝④✲✶⑥④♣⑥❭r✐❣❤t✮ ✐s✱ ❛❧❛s✱ ❤♦✇ ♣r♦❢❡s✲ s✐♦♥❛❧ t②♣❡s❡tt❡rs ❡♥❝♦❞❡ ✐t✳

8

Binomial coefficients

n

k

• , Ck

n, Cn k

?

0 ∈ N? Knowledge of linguistic context may help decide this question, but is far from certain.

−1

p

• Clearly (−1)(p−1)/2: it’s a quadratic residue

symbol! ❭❧❡❢t✭❭❢r❛❝④✲✶⑥④♣⑥❭r✐❣❤t✮ ✐s✱ ❛❧❛s✱ ❤♦✇ ♣r♦❢❡s✲ s✐♦♥❛❧ t②♣❡s❡tt❡rs ❡♥❝♦❞❡ ✐t✳

9

Binomial coefficients

n

k

• , Ck

n, Cn k

?

0 ∈ N? Knowledge of linguistic context may help decide this question, but is far from certain.

−1

p

• Clearly (−1)(p−1)/2: it’s a quadratic residue

symbol! \left(\frac{-1}{p}\right) is, alas, how profes- sional typesetters encode it.

10

“this has the usual mathematical meaning” (1) Mathematics a1 ∪ a2 ∪ a3 L

A

T E X a_1 \cup a_2 \cup a_3 OpenMath <OMS name="union" cd="set1"/> MathML <apply> <union/> <i>a1</i>...</apply>

11

“this has the usual mathematical meaning” (2) Mathematics {a1, a2, a3} L

A

T E X \bigcup \{a_1,a_2,a_3\} OpenMath <OMS name="big union" cd="set3"/>

• r <OMS name="apply to list" cd="fns2"/>

MathML <apply> <union/> <bvar>i</bvar> <domain ...> <set> <i> a1 </i>...</set>

12

“this has the usual mathematical meaning” (3) Mathematics 3

i=1 ai

L

A

T E X \bigcup_{i=1}^3 a_i OpenMath big union on make list MathML <apply> <union/> <bvar>i</bvar> <lowlimit>...

13

Pq and friends (1) Pq✭✉✮ ❂

❩ ✉

✵ ♣q✷✭t✮❞t

✭✶✻✿✷✺✿✶✮ ✭✇❤❡r❡ ♣q✷✭t✮ ♠❡❛♥s ♣q✭t✮✷✱ ❛♥❞ ♥♦t ♣ ✁ q✷✮ ❙❤♦rt ❢♦r ✶✷ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ❙♥✭✉✮ ❂

❩ ✉

✵ s♥✷✭t✮❞t❀

s✐♥❝❡ ♣❀ q❀ ✷ ❢s❀ ❝❀ ♥❀ ❞❣✳ ❇✉t ✇❤❡♥ q ❂ s Pq✭✉✮ ❂

❩ ✉

✵

✒

♣q✷✭t✮

✓

❞t ✿

14

Pq and friends (1) Pq(u) =

u

0 pq2(t)dt

(16.25.1) ✭✇❤❡r❡ ♣q✷✭t✮ ♠❡❛♥s ♣q✭t✮✷✱ ❛♥❞ ♥♦t ♣ ✁ q✷✮ ❙❤♦rt ❢♦r ✶✷ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ❙♥✭✉✮ ❂

❩ ✉

✵ s♥✷✭t✮❞t❀

s✐♥❝❡ ♣❀ q❀ ✷ ❢s❀ ❝❀ ♥❀ ❞❣✳ ❇✉t ✇❤❡♥ q ❂ s Pq✭✉✮ ❂

❩ ✉

✵

✒

♣q✷✭t✮

✓

❞t ✿

15

Pq and friends (1) Pq(u) =

u

0 pq2(t)dt

(16.25.1) (where pq2(t) means pq(t)2, and not p · q2) ❙❤♦rt ❢♦r ✶✷ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❢♦r♠ ❙♥✭✉✮ ❂

❩ ✉

✵ s♥✷✭t✮❞t❀

s✐♥❝❡ ♣❀ q❀ ✷ ❢s❀ ❝❀ ♥❀ ❞❣✳ ❇✉t ✇❤❡♥ q ❂ s Pq✭✉✮ ❂

❩ ✉

✵

✒

♣q✷✭t✮

✓

❞t ✿

16

Pq and friends (1) Pq(u) =

u

0 pq2(t)dt

(16.25.1) (where pq2(t) means pq(t)2, and not p · q2) Short for 12 equations of the form Sn(u) =

u

0 sn2(t)dt,

since p, q, ∈ {s, c, n, d}. ❇✉t ✇❤❡♥ q ❂ s Pq✭✉✮ ❂

❩ ✉

✵

✒

♣q✷✭t✮

✓

❞t ✿

17

Pq and friends (1) Pq(u) =

u

0 pq2(t)dt

(16.25.1) (where pq2(t) means pq(t)2, and not p · q2) Short for 12 equations of the form Sn(u) =

u

0 sn2(t)dt,

since p, q, ∈ {s, c, n, d}. But when q = s Pq(u) =

u

• pq2(t) − 1

t2

• dt − 1

u.

18

Pq and friends (2) pq(u) = pr(u) qr(u) (16.3.4) ✭❡①❝❡♣t t❤❛t ❤❡r❡ t❤❡r❡ ✐s ♥♦ ❞✐st✐♥❝t♥❡ss ❛s✲ s✉♠♣t✐♦♥✱ ❜✉t ♣♣ ✐s t♦ ❜❡ t❛❦❡♥ ❛s t❤❡ ❝♦♥st❛♥t ❢✉♥❝t✐♦♥ ✶✮✳

19

Pq and friends (2) pq(u) = pr(u) qr(u) (16.3.4) (except that here there is no distinctness as- sumption, but pp is to be taken as the constant function 1).

20

Juxtaposition (1) ▼✉❧t✐♣❧✐❝❛t✐♦♥ ❚❤✐s ✐s ❡♥❝♦❞❡❞ ❛s ✫■♥✈✐s✐❜❧❡❚✐♠❡s❀ ✐♥ ▼❛t❤▼▲✳ ❚❤✐s ♦♥❧② ❛♣♣❧✐❡s t♦ ✐t❛❧✐❝ ❧❡t✲ t❡rs✿ ❥✉①t❛♣♦s❡❞ r♦♠❛♥ ❧❡tt❡rs ❝♦♥st✐t✉t❡ ❛ s✐♥❣❧❡ ❧❡①❡♠❡✱ ❛s ✐♥ s✐♥ ♦r ♣q✳ ✭❋✉♥❝t✐♦♥✮ ❆♣♣❧✐❝❛t✐♦♥ s✐♥ ① ♦t❤❡r✇✐s❡ s✐♥✭①✮✳ s✐♥✭① ✰ ②✮ ✐s ❞✐☛❡r❡♥t ❢r♦♠ ✷✭① ✰ ②✮✱ ❛♥❞ ❢✭①✰②✮ ✐s ❄❄ ❚❤✐s ✐s ❡♥❝♦❞❡❞ ❛s ✫❆♣♣❧②❋✉♥❝t✐♦♥❀✳ ❆❞❞✐t✐♦♥ ✹ ❝♦✉❧❞ ♦t❤❡r✇✐s❡ ❜❡ r❡♥❞❡r❡❞ ❛s ✹✰ ✳ ❚❤✐s ✐s ✭♥♦✇✮ ❡♥❝♦❞❡❞ ❛s ✫■♥✈✐s✐❜❧❡P❧✉s❀✳

21

Juxtaposition (1) Multiplication This is encoded as &InvisibleTimes; in MathML. ❚❤✐s ♦♥❧② ❛♣♣❧✐❡s t♦ ✐t❛❧✐❝ ❧❡t✲ t❡rs✿ ❥✉①t❛♣♦s❡❞ r♦♠❛♥ ❧❡tt❡rs ❝♦♥st✐t✉t❡ ❛ s✐♥❣❧❡ ❧❡①❡♠❡✱ ❛s ✐♥ s✐♥ ♦r ♣q✳ ✭❋✉♥❝t✐♦♥✮ ❆♣♣❧✐❝❛t✐♦♥ s✐♥ ① ♦t❤❡r✇✐s❡ s✐♥✭①✮✳ s✐♥✭① ✰ ②✮ ✐s ❞✐☛❡r❡♥t ❢r♦♠ ✷✭① ✰ ②✮✱ ❛♥❞ ❢✭①✰②✮ ✐s ❄❄ ❚❤✐s ✐s ❡♥❝♦❞❡❞ ❛s ✫❆♣♣❧②❋✉♥❝t✐♦♥❀✳ ❆❞❞✐t✐♦♥ ✹ ❝♦✉❧❞ ♦t❤❡r✇✐s❡ ❜❡ r❡♥❞❡r❡❞ ❛s ✹✰ ✳ ❚❤✐s ✐s ✭♥♦✇✮ ❡♥❝♦❞❡❞ ❛s ✫■♥✈✐s✐❜❧❡P❧✉s❀✳

22

Juxtaposition (1) Multiplication This is encoded as &InvisibleTimes; in MathML. This only applies to italic let- ters: juxtaposed roman letters constitute a single lexeme, as in sin or pq. ✭❋✉♥❝t✐♦♥✮ ❆♣♣❧✐❝❛t✐♦♥ s✐♥ ① ♦t❤❡r✇✐s❡ s✐♥✭①✮✳ s✐♥✭① ✰ ②✮ ✐s ❞✐☛❡r❡♥t ❢r♦♠ ✷✭① ✰ ②✮✱ ❛♥❞ ❢✭①✰②✮ ✐s ❄❄ ❚❤✐s ✐s ❡♥❝♦❞❡❞ ❛s ✫❆♣♣❧②❋✉♥❝t✐♦♥❀✳ ❆❞❞✐t✐♦♥ ✹ ❝♦✉❧❞ ♦t❤❡r✇✐s❡ ❜❡ r❡♥❞❡r❡❞ ❛s ✹✰ ✳ ❚❤✐s ✐s ✭♥♦✇✮ ❡♥❝♦❞❡❞ ❛s ✫■♥✈✐s✐❜❧❡P❧✉s❀✳

23

Juxtaposition (1) Multiplication This is encoded as &InvisibleTimes; in MathML. This only applies to italic let- ters: juxtaposed roman letters constitute a single lexeme, as in sin or pq. (Function) Application sin x otherwise sin(x). s✐♥✭① ✰ ②✮ ✐s ❞✐☛❡r❡♥t ❢r♦♠ ✷✭① ✰ ②✮✱ ❛♥❞ ❢✭①✰②✮ ✐s ❄❄ ❚❤✐s ✐s ❡♥❝♦❞❡❞ ❛s ✫❆♣♣❧②❋✉♥❝t✐♦♥❀✳ ❆❞❞✐t✐♦♥ ✹ ❝♦✉❧❞ ♦t❤❡r✇✐s❡ ❜❡ r❡♥❞❡r❡❞ ❛s ✹✰ ✳ ❚❤✐s ✐s ✭♥♦✇✮ ❡♥❝♦❞❡❞ ❛s ✫■♥✈✐s✐❜❧❡P❧✉s❀✳

24

Juxtaposition (1) Multiplication This is encoded as &InvisibleTimes; in MathML. This only applies to italic let- ters: juxtaposed roman letters constitute a single lexeme, as in sin or pq. (Function) Application sin x otherwise sin(x). sin(x + y) is different from 2(x + y), and f(x+y) is ?? This is encoded as &ApplyFunction;. ❆❞❞✐t✐♦♥ ✹ ❝♦✉❧❞ ♦t❤❡r✇✐s❡ ❜❡ r❡♥❞❡r❡❞ ❛s ✹✰ ✳ ❚❤✐s ✐s ✭♥♦✇✮ ❡♥❝♦❞❡❞ ❛s ✫■♥✈✐s✐❜❧❡P❧✉s❀✳

25

Juxtaposition (1) Multiplication This is encoded as &InvisibleTimes; in MathML. This only applies to italic let- ters: juxtaposed roman letters constitute a single lexeme, as in sin or pq. (Function) Application sin x otherwise sin(x). sin(x + y) is different from 2(x + y), and f(x+y) is ?? This is encoded as &ApplyFunction;. Addition 41

2 could otherwise be rendered as

4+1

• 2. This is (now) encoded as &InvisiblePlus;.

26

Juxtaposition (2) Summation aibi could otherwise be rendered as

i aibi. It has no MathML counterpart.

❙✉❜t❧❡ ✈❛r✐❛♥ts ♦✈❡r ✉♣♣❡r✴❧♦✇❡r ❛♥❞ r♦✲ ♠❛♥✴❣r❡❡❦ ✐♥❞✐❝❡s✳ ❈♦♥❝❛t❡♥❛t✐♦♥ ♠✶✷ ❝♦✉❧❞ ❜❡ r❡♥❞❡r❡❞ ❛s ♠✶❀✷✳ ❚❤✐s ✐s ❡♥❝♦❞❡❞ ❛s ✫■♥✈✐s✐❜❧❡❈♦♠♠❛❀✳ ❊✈❡♥ ✇✐t❤♦✉t t❤✐s✱ ▼❛t❤▼▲ ✐s ❧❡ss ❛♠❜✐❣✉♦✉s t❤❛♥ ♦r❞✐♥❛r② ♥♦t❛t✐♦♥✿ ♠✶✷ ♠✐❣❤t ❡q✉❛❧❧② ❜❡ t❤❡ t✇❡❧t❤ ✐t❡♠ ♦❢ ❛ ✈❡❝t♦r✱ ❜✉t ▼❛t❤▼▲ ✇♦✉❧❞ ❞✐st✐♥❣✉✐s❤ t❤❡ ❢♦❧❧♦✇✐♥❣✳

27

Juxtaposition (2) Summation aibi could otherwise be rendered as

i aibi. It has no MathML counterpart.

Subtle variants over upper/lower and ro- man/greek indices. ❈♦♥❝❛t❡♥❛t✐♦♥ ♠✶✷ ❝♦✉❧❞ ❜❡ r❡♥❞❡r❡❞ ❛s ♠✶❀✷✳ ❚❤✐s ✐s ❡♥❝♦❞❡❞ ❛s ✫■♥✈✐s✐❜❧❡❈♦♠♠❛❀✳ ❊✈❡♥ ✇✐t❤♦✉t t❤✐s✱ ▼❛t❤▼▲ ✐s ❧❡ss ❛♠❜✐❣✉♦✉s t❤❛♥ ♦r❞✐♥❛r② ♥♦t❛t✐♦♥✿ ♠✶✷ ♠✐❣❤t ❡q✉❛❧❧② ❜❡ t❤❡ t✇❡❧t❤ ✐t❡♠ ♦❢ ❛ ✈❡❝t♦r✱ ❜✉t ▼❛t❤▼▲ ✇♦✉❧❞ ❞✐st✐♥❣✉✐s❤ t❤❡ ❢♦❧❧♦✇✐♥❣✳

28

Juxtaposition (2) Summation aibi could otherwise be rendered as

i aibi. It has no MathML counterpart.

Subtle variants over upper/lower and ro- man/greek indices. Concatenation m12 could be rendered as m1,2. This is encoded as &InvisibleComma;. Even without this, MathML is less ambiguous than ordinary notation: m12 might equally be the twelth item of a vector, but MathML would distinguish the following.

29

<msub> <mi> m </mi> <mrow> <mn> 1 </mn> <mn> 2 </mn> </mrow> </msub>

30

<msub> <mi> m </mi> <mrow> <mn> 12 </mn> </mrow> </msub> (of course, the <mrow> is redundant in this case).

31

An awful example. Then the functor T → {generically smooth T- morphisms T×SC′ → T ×SC} from ((S-schemes)) to ((sets)) is ❚❤❡♥ t❤❡ ❢✉♥❝t♦r ✩❚❭♠❛♣st♦❭④✩❣❡♥❡r✐❝❛❧❧② s♠♦♦t❤ ✩❚✩✲♠♦r♣❤✐s♠s ✩❚❭t✐♠❡s❴❙❭❈❛❧ ❈✬❭t♦ ❚❭t✐♠❡s❴❙❭❈❛❧ ❈❭⑥✩ ❢r♦♠ ✩✭✭❙✩✲s❝❤❡♠❡s✮✮ t♦ ✭✭s❡ts✮✮ ✐s

32

An awful example. Then the functor T → {generically smooth T- morphisms T×SC′ → T ×SC} from ((S-schemes)) to ((sets)) is Then the functor $T\mapsto\{$generically smooth $T$-morphisms $T\times_S\Cal C’\to T\times_S\Cal C\}$ from $((S$-schemes)) to ((sets)) is

33

The meanings of ± Arcsinh z1 ± Arcsinh z2 = Arcsinh

• z1
• 1 − z2

2 ± z2

• 1

means Arcsinh z1 + Arcsinh z2 ⊂ Arcsinh

• z1
• 1 − z2

2 + z2

• 1

∪ Arcsinh

• z1
• 1 − z2

2 − z2

• 1

and the fact that the same equation holds for Arcsinh z1 − Arcsinh z2.

34

Ways forward? ✎ ❋✉❧❧② s❡♠❛♥t✐❝ ♠❛r❦✉♣ ❉♦♥✬t ❤♦❧❞ ②♦✉r ❜r❡❛t❤ ✎ ❉♦ ♥♦t❤✐♥❣ ◗✉❛❧✐t② ♦❢ ▲

❆

❚ ❊ ❳ ✐s ✐♠♣r♦✈✐♥❣ ✎ ❍❡❧♣ ✐t ❛❧♦♥❣ ❙❧✐❣❤t❧② ♠♦r❡ s❡♠❛♥t✐❝ ▲

❆

❚ ❊ ❳

35

Ways forward?

• Fully semantic markup

❉♦♥✬t ❤♦❧❞ ②♦✉r ❜r❡❛t❤ ✎ ❉♦ ♥♦t❤✐♥❣ ◗✉❛❧✐t② ♦❢ ▲

❆

❚ ❊ ❳ ✐s ✐♠♣r♦✈✐♥❣ ✎ ❍❡❧♣ ✐t ❛❧♦♥❣ ❙❧✐❣❤t❧② ♠♦r❡ s❡♠❛♥t✐❝ ▲

❆

❚ ❊ ❳

36

Ways forward?

• Fully semantic markup

Don’t hold your breath ✎ ❉♦ ♥♦t❤✐♥❣ ◗✉❛❧✐t② ♦❢ ▲

❆

❚ ❊ ❳ ✐s ✐♠♣r♦✈✐♥❣ ✎ ❍❡❧♣ ✐t ❛❧♦♥❣ ❙❧✐❣❤t❧② ♠♦r❡ s❡♠❛♥t✐❝ ▲

❆

❚ ❊ ❳

37

Ways forward?

• Fully semantic markup

Don’t hold your breath

• Do nothing

◗✉❛❧✐t② ♦❢ ▲

❆

❚ ❊ ❳ ✐s ✐♠♣r♦✈✐♥❣ ✎ ❍❡❧♣ ✐t ❛❧♦♥❣ ❙❧✐❣❤t❧② ♠♦r❡ s❡♠❛♥t✐❝ ▲

❆

❚ ❊ ❳

38

Ways forward?

• Fully semantic markup

Don’t hold your breath

• Do nothing

Quality of L

A

T E X is improving ✎ ❍❡❧♣ ✐t ❛❧♦♥❣ ❙❧✐❣❤t❧② ♠♦r❡ s❡♠❛♥t✐❝ ▲

❆

❚ ❊ ❳

39

Ways forward?

• Fully semantic markup

Don’t hold your breath

• Do nothing

Quality of L

A

T E X is improving

• Help it along

❙❧✐❣❤t❧② ♠♦r❡ s❡♠❛♥t✐❝ ▲

❆

❚ ❊ ❳

40

Ways forward?

• Fully semantic markup

Don’t hold your breath

• Do nothing

Quality of L

A

T E X is improving

• Help it along

Slightly more semantic L

A

T E X

41

Example: fractions and QR symbols (\displaystyle and \textstyle) ❭❢r❛❝t✐♦♥④❛⑥④❜⑥ ❛❂❜ ✭♦r ♣♦ss✐❜❧② ✭❛❂❜✮✮ ❭qr④❛⑥④❜⑥

✒ ✓

✭❛❥❜✮

42

Example: fractions and QR symbols (\displaystyle and \textstyle) \fraction{a}{b} ❛❂❜ ✭♦r ♣♦ss✐❜❧② ✭❛❂❜✮✮ ❭qr④❛⑥④❜⑥

✒ ✓

✭❛❥❜✮

43

Example: fractions and QR symbols (\displaystyle and \textstyle) \fraction{a}{b} a b a/b (or possibly (a/b)) ❭qr④❛⑥④❜⑥

✒ ✓

✭❛❥❜✮

44

Example: fractions and QR symbols (\displaystyle and \textstyle) \fraction{a}{b} a b a/b (or possibly (a/b)) \qr{a}{b}

a

b

• ✭❛❥❜✮

45

Example: fractions and QR symbols (\displaystyle and textstyle) \fraction{a}{b} a b a/b (or possibly (a/b)) \qr{a}{b}

a

b

• (a|b)

46

The DML trichotomy ❘❡✈✐s❡❞

• 1. retro-digital, i.e. scanned.
• 2. retro-born-digital, i.e. reconstruction from

a .pdf or .ps.

• 3. born-digital, i.e.

not just the pixels, but the whole workflow. ✹✳ ❜♦r♥✲✐♥t❡❧❧✐❣❡♥t✲❞✐❣✐t❛❧✱ ✇✐t❤ s❡♠❛♥t✐❝s r❡✲ ❝♦✈❡r❛❜❧❡ ❢r♦♠ t❤❡ ♠❛r❦✉♣✳

47

The DML trichotomy Revised

• 1. retro-digital, i.e. scanned.
• 2. retro-born-digital, i.e. reconstruction from

a .pdf or .ps.

• 3. born-digital, i.e.

not just the pixels, but the whole workflow.

• 4. born-intelligent-digital, with semantics re-

coverable from the markup.

48