Display Math in Formula Where is it and where can it go Or Is T - - PowerPoint PPT Presentation

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Display Math in Formula Where is it and where can it go Or Is T - - PowerPoint PPT Presentation

Aug 14, 2023 118 likes •575 views

Display Math in Formula Where is it and where can it go Or Is T EX really good at math? The present Basics Equation Numbering (1) \placeformula \startformula ... \stopformula + + = Location of equation number (2)

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Display Math in Formula Where is it and where can it go

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Or Is T EX really good at math?

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The present Basics

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Equation Numbering

+ + = (1) \placeformula \startformula ... \stopformula

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Location of equation number

(2) + + = \setupformulas[location=left]

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Conversion of equation numbers

+ + + + = (C) \setupformulas[conversion=Characters]

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Formatting of equation numbers

+ + + = (4) \setupformulas[numberstyle=bold]

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Formatting of equation numbers

+ + + + = [5] \setupformulas[left={[},right={]}]

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Placement of fomrula

+ + = (6) \setupformulas[align=left]

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Placement of fomrula

+ + + = (7) \setupformulas[align=right]

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The present AMSTeX features

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Gather

= + + + + = + + + \startformula \startalign[ n=1 ] \NC ... \NR[+] \NC ... \NR[+] \stopalign \stopformula

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Left gather

= + = + + + + \startformula \startalign[n=1, align=left ] \NC ... \NR[+] \NC ... \NR[+] \stopalign \stopformula

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Right gather

= + + + + + = + + + + + \startformula \startalign[n=1, align=right ] \NC ... \NR[+] \NC ... \NR[+] \stopalign \stopformula

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Align

= + + + = + + \startformula \startalign[ n=2 ] \NC ... \NC ... \NR[+] \NC ... \NC ... \NR[+] \stopalign \stopformula

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Alignat

= = = + = \startformula \startalign[n=2, m=2, distance=2em ] \NC ... \NC ... \NC ... \NC ... \NR[+] \NC ... \NC ... \NC ... \NC ... \NR[+] \stopalign \stopformula

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Flaign

= + + = + + = + = + + \startformula \startalign[n=2, m=2, distance=1em plus 1fill ] \NC ... \NC ... \NC ... \NC ... \NR[+] \NC ... \NC ... \NC ... \NC ... \NR[+] \stopalign \stopformula

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Intertext

= + + + + = + + \startformula \startalign \NC ... \NC ... \NR[+] \intertext{...} \NC ... \NC ... \NR[+] \stopalign \stopformula

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Multi-column numbered equations

= + (8) = + (9) = (10) = + (11) \placeformula \startformulas \startformula \startalign ... \stopalign \stopformula \startformula \startalign ... \stopalign \stopformula \stopformulas

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Subformulas

= + + (12a) = + + + (12b) \startsubformulas \startformula \startalign \NC ... \NC ... \NR[+] \NC ... \NC ... \NR[+] \stopalign \stopformula \stopsubformulas

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The present Subexpressions

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Matrix

(13) \startformula \startmathmatrix[n=3] \NC ... \NC ... \NC ... \NR \NC ... \NC ... \NC ... \NR \NC ... \NC ... \NC ... \NR \stopmathmatrix \stopformula

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Matrix — parenthesis

        \startformula \startmathmatrix[n=3, left={\left(\,}, right={\,\right)}] \NC ... \NC ... \NC ... \NR \NC ... \NC ... \NC ... \NR \NC ... \NC ... \NC ... \NR \stopmathmatrix \stopformula

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Defining matrices

\definemathmatrix[pmatrix][left={\left(\,}, right={\,\right)}]

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Aligning matrices

                      \startformula \startmathmatrix[ location=low ] ... \stopmathmatrix \startmathmatrix[ location=middle ] ... \stopmathmatrix \startmathmatrix[ location=high ] ... \stopmathmatrix \stopformula

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Cases

=    + + , , \startformula ... = \startcases \NC ... \NC ... \NR \NC ... \NC ... \NR \stopcases \stopformula

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Substacks

  • +

+ + \startformula \sum_{\startsubstack \NC ... \NR \NC ... \NR \stopsubstack ... \stopformula

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Missing features

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Simple yet have no support

  • Arbitrary tag's as equation numbers

E = mc2 (Einstein's Formula) Need to come up with a consistent user interface

  • gathered, aligned, etc.

Is really simple to code from scratch, hard to reuse parts of mathalign

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Not so simple and still have no support

  • Complete support of multline

If you do not care about location of equation numbers, support is easy.

  • Proper support for equation numbers in multiline equations

Need a two pass algorithm, current support is only a one pass algorithm.

  • Location of equation numbers — ctags, tbtags

Currently ConT EXt does not even attempt to do this

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  • Proper support for split

Easy once location of equation number is done.

  • Correct support for align when there are multiple columns

Find the size of all columns and split the remaining space equally between them

  • Controlling page break between equations

Something more fine tuned than the current all or none approach

  • Using \shortdisplayskip

Can be done, (breqn does it), but I don't really understand T EX that well.

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Example

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   



First draft – August , 

4b1 t (xt, y2 t , u2 t , s2 t−1)

= Pr

  • Xt = xt, Y2

t = y2 t , U2 t = u2 t , S2 t−1 = s2 t−1

  • Y1,t = y1,t, U1,t = u1,t; 4ϕt−1

= Pr

  • U2

t = u2 t

  • Xt = xt, Y2

t = y2 t , S2 t−1 = s2 t−1Y1,t = y1,t, U1,t = u1,t; 3ϕt−1, g2 t

  • × Pr
  • Xt = xt, Y2

t = y2 t , S2 t−1 = s2 t−1

  • Y1,t = y1,t, U1,t = u1,t; 3ϕt−1, g2

t

  • (c)

= I

  • u2

t = g2 t (y2 t , s2 t−1)

  • × Pr
  • Xt = xt, Y2

t = y2 t , S2 t−1 = s2 t−1

  • Y1,t = y1,t, U1,t = u1,t; 3ϕt−1

= I

  • u2

t = g2 t (y2 t , s2 t−1)

3b1

t (xt, y2 t , s2 t−1)

=: 3F1(3b1

t , g2 t )(xt, y2 t , s2 t−1)

(.) where (c) follows from the sequential order in which the system variables are generated. . Consider 4o1

t = (y1,t, u1,t) ∈ (Y1,t × U1,t), y1 t+1 ∈ Y1, xt+1 ∈ X, s2 t ∈ S2, and 1ϕt = (4ϕt−1, l2 t ). Then a component (xt+1, s2 t ) of a realization 1b1 t+1 of 1B1 t+1 is

given by

1b1 t+1 = Pr

  • Xt+1 = xt+1, S2

t = s2 t

  • Y1

t+1 = y1 t+1, 4O1 t = 4o1 t ; 1ϕt

= Pr

  • Xt+1 = xt+1, S2

t = s2 t , Y1 t+1 = y1 t+1

  • 4O1

t = 4o1 t ; 1ϕt

Pr

  • Y1

t+1 = y1 t+1

  • 4O1

t = 4o1 t ; 1ϕt

(.) Now, Pr

  • Xt+1 = xt+1, S2

t = s2 t , Y1 t+1 = y1 t+1

  • 4O1

t = 4o1 t ; 1ϕt

= Pr

  • Y1

t+1 = y1 t+1

  • Xt+1 = xt+1, S2

t = s2 t , 4O1 t = 4o1 t ; 1ϕt

× Pr

  • Xt+1 = xt+1, S2

t = s2 t

  • 4O1

t = 4o1 t ; 1ϕt

= PN1

  • n1

t ∈ N1 : y1 t+1 = h1 t (xt+1, n1 t+1)

  • × Pr
  • Xt+1 = xt+1, S2

t = s2 t

  • 4O1

t = 4o1 t ; 1ϕt

(.) Further,

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   



First draft – August , 

Pr

  • Xt+1 = xt+1, S2

t = s2 t

  • 4O1

t = 4o1 t ; 1ϕt

=

  • xt∈X,y2

t ∈Y2

u2

t ∈U2,s2 t−1∈S2

Pr

  • Xt+1 = xt+1, Xt = xt, Y2

t = y2 t , U2 t = u2 t ,

S2

t−1 = s2 t−1, S2 t = s2 t | 4O1 t = 4o1 t ; 1ϕt

=

  • xt∈X,y2

t ∈Y2

u2

t ∈U2,s2 t−1∈S2

Pr

  • Xt+1 = xt+1 | Xt = xt, Y2

t = y2 t , U2 t = u2 t ,

S2

t−1 = s2 t−1, S2 t = s2 t , 4O1 t = 4o1 t ; 1ϕt

× Pr

  • S2

t = s2 t | Xt = xt, Y2 t = y2 t , U2 t = u2 t ,

S2

t−1 = s2 t−1, 4O1 t = 4o1 t ; 4ϕt−1, l2 t

  • × Pr
  • Xt = xt, Y2

t = y2 t , U2 t = u2 t , S2 t−1 = s2 t−1, 4O1 t = 4o1 t ; 4ϕt−1, l2 t

  • (d)

=

  • xt∈X,y2

t ∈Y2

u2

t ∈U2,s2 t−1∈S2

PW

  • wt ∈ W : xt+1 = f(xt, u1

t , u2 t , wt)

  • I
  • s2

t = l2 t (y2 t , u2 t , s1 t−1)

  • × Pr
  • Xt = xt, Y2

t = y2 t , U2 t = u2 t , S2 t−1 = s2 t−1, 4O1 t = 4o1 t ; 4ϕt−1

=

  • xt∈X,y2

t ∈Y2

u2

t ∈U2,s2 t−1∈S2

PW

  • wt ∈ W : xt+1 = f(xt, u1

t , u2 t , wt)

  • I
  • s2

t = l2 t (y2 t , u2 t , s1 t−1)

  • × 4b1

t (xt, y2 t , u2 t , s2 t−1).

(.) where (d) follows from the sequential order in which the system variables are

  • generated. Combining (.)–(.) we get

1b1 t+1(xt+1, s2 t ) =: 4F1(4b1 t , l2 t , y1 t+1, u1 t )(xt+1, s2 t )

(.) where 4F1 is given by (.)–(.).

  • Structural properties

In this section, we provide structural/qualitative properties of optimal control laws

  • f agent  that are true for every arbitrary but fixed control and state-update strate-

gies of agent . These properties are subsequently used to convert the model of variation  into a model similar to variation . Theorem . (Structure of optimal control laws of agent ). Consider variation 

  • f the model of Problem .. For any arbitrary but fixed control and state-update strategies
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The Future?

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What is wrong with current math support in T EX?

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Separation of content from presentation

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Display math is becoming write

  • nce format
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Automatic line breaks

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Can luaT EX help?

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Allow non-T EXperts to experiment with line breaking algorithms

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What is needed for automatic line breaks

\startdisplay ......... \stopdisplay 1. Dimensions (height, width, depth) of each “character”

  • 2. Mathcode of each character (relation, operator, delimiter,

punctuation, etc)

  • 3. What else . . .
  • 4. Can T

EX know this (commands like \over etc) . . .

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Easy to convert from display to inline

\startinline ......... \stopinline

  • T

EX should do the right thing for both display and inline modes

  • nath does that to a large extent.
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Thank you