Generating Correct Mathematical Documents

1.  Mathematical Articles

• Contain ordinary text and mathematical content.

• Need realization on paper and on computer networks.

2.  A Segment of an Article

The gamma function may be defined in a suitable right-half plane by an integral, which essentially amounts to Fourier transform relative to the multiplicative group of postive real numbers of the reciprocal exponential function, and then extended to a meromorphic function in the complex plane with entire reciprocal. The following formula represents a variant of its Weierstrass product expansion: $${\int }_0^{\infty }{t}^x{e}^{-t}\frac{dt}{t}=\frac{1}{x}\prod _{k=1}^{\infty }\frac{{\left(1+\frac{1}{k}\right)}^x}{\left(1+\frac{x}{k}\right)}\text{.}$$ The product manifests simple poles at zero and each of the negative integers.

3.  Goals

From a Single Source:

1. A typeset version of high quality for paper preprints.

2. An online version for a web preprint.

3. A typeset version of high quality meeting the requirements of a journal chosen after the time of writing.

4.  Goals for Online Versions

1. catalogable.

2. universally accessible.

3. re-scalable and re-sizable.

4. searchable for mathematical content.

5. “clippable” for mathematical content.

5.  Mathematical Mainstream 1980-2000

Donald Knuth's TeX

LaTeX

Plain TeX

Other TeX Variants:

AMSTeX, ConTeXt, Omega, Texinfo

6.  How May One Provide Multiple Formattings?

Early Ideas:

1. Intuitive Authoring Systems: the WYSIWYG Idea

2. Write HTML and Translate From There.

3. Write LaTeX and Translate From There.

7.  WYSIWYG: The Good News

• Standard with Common “Word Processors”

• Available for LaTeX.

• Available for SGML Systems.

• Easy for Easy Tasks.

• OK for Very Simple Documents.

8.  WYSIWYG: The Bad News

• Hard for Hard Tasks.

• Slow for Those Who Write Frequently.

• Insufficiently Rich for Mathematics.

• Insufficiently Abstract.

• Inapplicable to Multiple Formattings.

• Some say WYSIAYG: “what you see is all you get”.

9.  WYSIWYG vs. Structured Markup

• Format-specific hanging indentation commands.

• Use of abstract list structures.

10.  Translating HTML

• Reliable

• But:

  1. No Math in HTML

  2. HTML Generally Less Rich Than LaTeX

  3. Special characters are translation headaches.
     Examples: # $ % & ~ _ ^ \ { } < >

11.  Translating LaTeX

• Almost Impossible

• Good Structure a Help

• May Require Human Intervention

• Need to Proof Read Twice

• htlatex in the TeXLive 6 distribution is remarkably good.

12.  XML

eXtensible Markup Language

• Data Under a Template for Translation.

• Enforced Separation of Content and Presentation.

• Universal Exchange.

• Originated by

  World Wide Web Consortium (W3C)

  Sun Microsystems

13.  XML

• Many Templates

• Synonym for XML Template:

  Document Type

• Two worlds

  1. Classical Documents:
     Examples: HTML, Docbook, TEI, …

  2. Electronic Data Interchange (EDI)
     Example: Graham William's TeX Catalogue found on CTAN
     help/Catalogue/catalogue.html

14.  The GELLMU Project

• Superseded my earlier ideas:

  • Strictly controlled dialect of LaTeX.

  • Adapting Texinfo (already suitable for multiple formattings).

• Relation to the Goals:

  • No present full realization of online version goals.

  • Proof that full realization of all goals and more is possible.

15.  GELLMU

• A markup interface for writing (SGML or) XML.

• LaTeX-like notation more succinct than that of XML.

• Extensible using GELLMU's \newcommand with arguments. (SGML has no analogue of macros with arguments.)

16.  Difference Between LaTeX Source and GELLMU Source

• Article prepared under a template for processing by latex, the Program.

• Article prepared under a template for processing by many programs.

It is a small step from LaTeX source to GELLMU source.

17.  The Idea of LaTeX-Like MarkUp

• Text + Commands

• A command is a function that operates on text.

• A command may take a non-negative number of arguments.

18.  Examples of LaTeX-Like Commands

Example of a command taking no argument:

\latex

LaTeX

Example of a command taking one argument:

this is \emph{emphasized} text

this is emphasized text

Example of a command (for math) taking two arguments:

\[ \frac{a z + b}{cz + d} \]

$$\frac{az+b}{cz+d}$$

19.  GELLMU Modes

1. Basic

2. Advanced

   1. Regular

   2. Other (less fully developed)

20.  Regular GELLMU: System Stages

1. LaTeX-Like Source.

2. Syntactic Translation to SGML.

3. Translation of SGML to Enriched XML.

4. Various Formattings of Enriched XML.

21.  Customizing

• Each stage presents opportunities for customizing.

• Each output format is the result of successive transforms.

• Additional intermediate transforms can be provided.

• These slides were prepared using a special formatting of regular GELLMU's article.

• The source markup for these slides is as readable as ordinary LaTeX source.

22.  The Syntatic Translator

23.  Syntactic Differences from LaTeX

• Command names (element names) may contain numbers.

• Example: \frac23 is a command name.

• Arguments must be delimited with braces or brackets.

• No white space between command name and first argument delimiter.

• No white space between delimiters of successive arguments.

• Bracketed arguments may not be optional.

24.  Use of GELLMU in Basic Mode for XHTML

Write:

the WWW \a[href="http://www.w3.org/"

]{Consortium} site

for generating the XML:

the WWW <a href="http://www.w3.org/"

>Consortium</a> site

to produce:

25.  \newcommand with XHTML

Definitions

\newcommand{\emph}[1]{\em{#1}}

\newcommand{\w3ref}[2][]{%

\a[href="http://www.w3.org/#1"]{#2}}

Invocations

Using GELLMU's \emph{newcommand}

one can reduce the markup required

for an anchor to \w3ref{W3C}'s

\w3ref{Math/}{MathML} site.

Rendering: Using GELLMU's newcommand one can reduce the markup required for an anchor to W3C's MathML site.

26.  Why is GELLMU's article “Didactic”?

• Intended as a first XML document type for LaTeX authors

• Sits in the middle between

  1. What LaTeX authors are accustomed to.

  2. What high end XML people think is needed.

• Room to adjust and expand.

27.  The Gamma Function: Its Weierstrass Product

$${\int }_0^{\infty }{t}^x{e}^{-t}\frac{dt}{t}=\frac{1}{x}\prod _{k=1}^{\infty }\frac{{\left(1+\frac{1}{k}\right)}^x}{\left(1+\frac{x}{k}\right)}$$

28.  Markup for the Gamma Identity

Regular GELLMU source for the identity:

\[ \int_{0}^{\infty}

      t^x e^{-t} \frac{d t}{t}

   \int:

   = \frac{1}{x}

   \prod_{k=1}^{\infty}

     \frac{

      \bal{1 + \frac{1}{k}}^x

     }{

      \bal{1 + \frac{x}{k}}

   } \prod: \]

29.  Gamma: Derived XML Markup

<displaymath>

<int>

 <msub>0</msub>

 <msup><infty/></msup>

 t<pow>x</pow> e<pow><minus/>t</pow>

 <frac>

  <numr>d t</numr>

  <denm>t</denm>

 </frac>

</int>

<equals/>

<frac><numr>1</numr><denm>x</denm></frac>

<prod>

 <msub>k<equals/>1</msub>

 <msup><infty/></msup>

 <frac>

  <numr>

   <bal>1<plus/>

    <frac>

     <numr>1</numr>

     <denm>k</denm>

    </frac>

   </bal><pow>x</pow>

  </numr>

  <denm>

   <bal>1 <plus/>

    <frac>

     <numr>x</numr>

     <denm>k</denm>

    </frac>

   </bal>

  </denm>

 </frac>

</prod>

</displaymath>

30.  Gamma: in MathML

<math

 xmlns="http://www.w3.org/1998/Math/MathML"

 class="display" mode="display">

<mrow>

 <mrow>

  <msubsup>

   <mo>&Integral;</mo>

   <mrow><mn>0</mn></mrow>

   <mi>&infin;</mi>

  </msubsup>

  <mrow>

   <msup>

    <mrow><mi>t</mi></mrow>

    <mrow><mi>x</mi></mrow>

   </msup>

   <mo> </mo>

  <msup>

   <mrow><mi>e</mi></mrow>

   <mrow><mi>-t</mi></mrow>

  </msup>

  <mo> </mo>

   <mfrac>

    <mrow><mi>dt</mi></mrow>

    <mi>t</mi>

   </mfrac>

  </mrow>

 </mrow>

 <mo>=</mo>

 <mrow>

  <mfrac>

   <mrow><mn>1</mn></mrow>

   <mi>x</mi>

  </mfrac>

 <mo> </mo>

  <msubsup>

   <mo>&Product;</mo>

   <mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow>

   <mi>&infin;</mi>

  </msubsup>

  <mrow>

   <mrow>

    <mfrac>

     <mrow>

      <msup>

       <mrow><mfenced>

        <mrow>

         <mn>1</mn><mo>+</mo>

         <mfrac><mn>1</mn><mi>k</mi></mfrac>

        </mrow>

       </mfenced></mrow>

       <mrow><mi>x</mi></mrow>

      </msup>

     </mrow>

     <mrow><mfenced>

      <mrow>

       <mn>1</mn><mo>+</mo>

       <mfrac><mi>x</mi><mi>k</mi></mfrac>

      </mrow>

     </mfenced></mrow>

    </mfrac>

   </mrow>

  </mrow>

 </mrow>

</mrow>

</math>

31.  Viewing MathML

Viewing support for MathML in web pages is not yet widely available. The above item can be rendered by:

• W3C's Amaya: wprod.html or wprod.xml.

• Mozilla's MathML development track: wprod.xml (only).

• With special plugin for MSIE: wprod.html (only).

32.  Generating MathML from article

• Ad hoc wprod.html was made from GELLMU source:
  wprod.glm.

• The short article form (slide 28) of GELLMU source above could be given automatic translation to MathML.

• An automatic translation should go through content MathML and from there to presentation MathML.

• An automatic translation would not be under the umbrella of general XML processing.

33.  Reliable Generation of MathML

Reliable translation will require:

A substantial non-XML, but XML-aware, parsing of all math zones in a GELLMU source document.

Occasional math parsing hints from authors in their markup.

Desirable, sometimes required:

1. Source markup labeling of math symbols.

2. Source markup typing of math symbols.

34.  MathML Generation Issues

• Will authors cooperate?

• Will standard web user agents cooperate?

35.  Two Final Notes

• For more information: http://www.albany.edu/~hammond/gellmu/

• GELLMU source for these slides is on the web:
  http://math.albany.edu:8000/~hammond/Presen/Correct/correct.glm