compare {mathematica, latex, sympy, mathml} for storing expressions a database

[bhpayne]

Produce a report comparing formats

what are we comparing?

• presentation mathml, content mathml, mathematica, latex, abstract syntax tree

what is the evaluation?

• can it describe all the notation necessary for physics
• how easy is the input method? (character count)
• can it be transformed to other representations

In general, we need to evaluate

• ability to input
• ability to transform between representations
• ability to audit correctness

Not a problem:

• storage

consumer cares about:

• speed
• ease of use
• presentation
• accessibility - multi-device, multi-OS
• easy setup

To start, evaluate:

• character count
• rendering

[bhpayne]

A set of test cases for flexing the capability of syntax, provided in Latex

1. polynomial
   Latex:

a x^2 + b x + c = 0

2. Stoke's theorem
   Latex:

 \int \int_{\sum} \vec{\nabla} \times \vec{F} \dot d\sum = \oint_{\partial \sum} \vec{F}\dot d\vec{r}

Source: https://en.wikipedia.org/wiki/Stokes'_theorem

3. Tensor analysis
   Latex:

Y^i(X_j) = \Delta^i_{\ j}

Einstein notation: contravariant = superscript, covariant = subscript

4a) creation operator
Latex:

 \hat{a}^+ |n\rangle = \sqrt{n+1} |n+1\rangle

https://en.wikipedia.org/wiki/Creation_and_annihilation_operators

4b) uncertainty principle
Latex:

\sigma_x \sigma_p \geq \frac{\hbar}{2}

4c) Lüders projection
Latex:

 |\psi\rangle \rightarrow \sum_n  |c_n|^2 P_n,\; P_n = \sum_i |\psi_{ni}\rangle \lang \psi_{ni}|

wikipedia:

 <math> |\psi\rang \rightarrow \sum_n |c_n|^2 P_n,\; P_n = \sum_i |\psi_{ni}\rang \lang \psi_{ni}|</math>

https://en.wikipedia.org/wiki/Measurement_in_quantum_mechanics

https://www.physicsforums.com/threads/formulae-of-various-topics.102736/

[bhpayne]

Content MathML for 1:

<math xmlns="http://www.w3.org/1998/Math/MathML">
<apply>
<eq/>
<apply>
  <plus/>
    <apply><ci>a</ci> <power/><ci> x </ci><ci> 2 </ci></apply>
    <apply><ci>b</ci> <ci> x </ci>                    </apply>
    <apply><ci>c</ci>                                 </apply>
</apply>
<apply>
  <cn>0</cn>
</apply>
</apply>
</math>

http://www.w3.org/TR/REC-MathML/chapter2.html

[bhpayne]

Presentation MathML for 1:

<mrow>
<mrow>
 <mn>a</mn>
 <msup>
  <mi>x</mi>
  <mn>2</mn>
</msup>
<mo>+</mo>
<mrow>
  <mn>b</mn>
  <mo>&InvisibleTimes;</mo>
  <mi>x</mi>
</mrow>
<mo>+</mo>
<mn>c</mn>
 </mrow>
<mo>=</mo>
<mn>0</mn>
</mrow>

Stokes' Theorem for 2

<mo>&#x222B;</mo>
<msub>
    <mo>&#x222B;</mo>
    <mo>&#x3A3;</mo>
</msub>
<mover>
  <mo>&#x2207;</mo>
  <mo>&rarr;</mo>
</mover>
<mo>&#x2A2F;</mo>
<mover>
<mi>F</mi>
<mo>&rarr;</mo>
 </mover>
<mi>&#xB7;</mi>
<mi>d</mi>
<mo>&#x3A3;</mo>
<mo>=</mo>
<msub>
    <mo>&#x222E;</mo>
    <mrow>
        <mo>&#x2202;</mo>
        <mo>&#x3A3;</mo>
    </mrow>
</msub>
 <mover>
<mi>F</mi>
<mo>&rarr;</mo>
 </mover>
<mi>&#xB7;</mi>
<mi>d</mi>   
<mover>
<mi>r</mi>
<mo>&rarr;</mo>
</mover>

Tensor notation for number 3 in MathML

<msup>
    <mi>Y</mi>
    <mi>i</mi>
</msup>
<mo maxsize='1'>(</mo>
<msub>
    <mi>X</mi>
    <mi>j</mi>
</msub>
<mo maxsize='1'>)</mo>
<mo>=</mo>
<msubsup>
    <mtext>&#x394;</mtext>
    <mrow>
        <mspace width='1em'></mspace>
        <mi>j</mi>
    </mrow>
    <mi>i</mi>
</msubsup>

4a, creation operator

<msup>
<mover>
    <mtext>a</mtext>
<mo>&#x5E;</mo>
</mover>
    <mo>&#x2020;</mo>
</msup>
<mo maxsize='1'>|</mo>
<mi>n</mi>
<mo maxsize='1'>&#x232A;</mo>
<mo>=</mo>
<msqrt>
    <mi>n</mi>
    <mo>+</mo>
    <mn>1</mn>
</msqrt>
<mo maxsize='1'>|</mo>
<mi>n</mi>
<mo>+</mo>
<mn>1</mn>
<mo maxsize='1'>&#x232A;</mo>

quantum mechanics in Presentation MathML

<mfenced open="&#x2329;" close="&#x232A;" separators="|">
<mi>p</mi>
<mo>a</mo>
<mi>p+</mi>
</mfenced>

Presentation MathML for 4b) uncertainty principle

<msub>
    <mtext>&#x3C3;</mtext>
    <mi>x</mi>
</msub>
<msub>
    <mtext>&#x3C3;</mtext>
    <mi>p</mi>
</msub>
<mo>&#x2265;</mo>
<mfrac>
    <mtext>&#x210F;</mtext>
    <mn>2</mn>
</mfrac>

http://www.fmath.info/java/latex-mathml-converter/

Following works in Firefox but not Chrome to render the presentation MathML
http://www.tutorialspoint.com/cgi-bin/practice.cgi?file=html5-21.htm
http://www-archive.mozilla.org/projects/mathml/demo/tester.html

Following works in all browsers and uses MathJax
http://www.tutorialspoint.com/try_mathml_online.php

Following works in all browsers and generates GIF
http://www.mathmlcentral.com/Tools/FromMathML.jsp

MathML examples
http://www.tutorialspoint.com/html5/html5_mathml.htm

[bhpayne]

Correction: MathML = abstract syntax tree

[bhpayne]

Citation sources

AUTHORING XML DOCUMENTS WITH XHTML AND MATHML SUPPORT
http://www.sjsu.edu/cs/programs/mscs/research/projects/wu_xiaoheng.pdf

MathML
http://www.pragma-ade.com/general/manuals/mmlprime.pdf

Leslie Lamport
LaTeX: A Document Preparation System
http://www.amazon.com/LaTeX-Document-Preparation-Addison-Wesley-Techniques/dp/0201529831

Donald E Knuth
The TeXbook
http://www.amazon.com/TeXbook-Computers-Typesetting-Donald-Knuth/dp/0201134489

https://en.wikipedia.org/wiki/Wikipedia:Rendering_math

[bhpayne]

Mathematica to MathML

http://www.mathmlcentral.com/Tools/ToMathML.jsp

[bhpayne]

Unicode is needed to support Dirac notation in MathML

http://www.w3.org/TR/MathML2/bycodes.html
http://www.charbase.com/2329-unicode-left-pointing-angle-bracket
http://www.robinlionheart.com/stds/html4/entities-mathml/
http://access2science.com/mathml/MathMLUnicodeSingle.html

[bhpayne]

MathML to Latex

http://tex.stackexchange.com/questions/85639/i-want-to-convert-mathml-to-latex

[bhpayne]

http://tex.stackexchange.com/questions/57717/relationship-between-mathml-and-tex

[bhpayne]

Report started in doc/syntax_comparison/syntax_comparison.pdf

[bhpayne]

http://www.unicode.org/notes/tn28/UTN28-PlainTextMath-v3.1.pdf
Subscripts and superscripts are supported.
The Dirac notation is used as an example in the PDF.

https://www.johndcook.com/blog/2013/02/18/unicode-to-latex/

[bhpayne]

Sticking with latex for now

[bhpayne]

See also "Towards semantic mathematical editing"
https://www.texmacs.org/joris/semedit/semedit.html