Arun Ram: MathML for math research papers

MathML for mathematics research articles

Arun Ram
Department of Mathematics
University of Wisconsin, Madison
Madison, WI 53706 USA

Last updates: 3 June 2007

Introduction

Given the ubiquity with which the internet has taken over our lives and the extent to which society is using web pages written in HTML it is, perhaps, a bit surprising that research mathematicians still use TeX to distribute their results. Current web browsers and authoring tools are much more friendly towards formatting, graphics, animation, colors and cross referencing. The possibilities are endless, and it is a pity that we don't currently have them available in our standard mathematics presentations. For example,

color,
animated graphics,
proofs that appear only when you want (click on) them,
point and click jumping to references and previous equations,

are only a few of the possible tools that mathematicians might like. Perhaps it is time that we, as a community, make the switch and stop using TeX in favor of web markup.

In fact, some TeX addons (HyperTeX, Beamer), attempt to incorporate some weblike features. In 1998, MathML version 2 was released by the World Wide Web Consortium, the standards organisation for Web markup. So what is holding up our conversion?

Because mathematicians have hardly noticed the existence of web markup possibilities for mathematics, software for MathML has developed extremely slowly. Only recently have some available browsers really started to process MathML with good looking results.

Mozilla Firefox http://www.mozilla.org

What about authoring tools? A few hours of playing with Dreamweaver shows how easy typing into web pages can be. Unfortunately, Dreamweaver doesn't handle MathML. But

Amaya http://www.w3.org/Amaya

does and it is because of Mozilla and Amaya that I think that use of MathML is a realistic possiblity for research mathematicians. Though Amaya crashes way too often to give it any more than beta-beta-beta status it is on the right track and a good programmer might take this template and make it into a wonderful MathML authoring tool in a month or two. I think that the mathematical community will jump to web presentation as soon as the rendering and authoring tools become polished. Greater usage of these tools by mathematicians would quickly push the currently available browsers to render mathematical markup even better.

This article has two goals:

To illustrate that MathML is capable of satisfying the research mathematician's typsetting and presentation needs, and
To provide a "getting started" guide for using MathML for writing mathematics papers.

Production of this article was partially supported by research and teaching intiatives supported by the National Science Foundation under Grant No. DMS-0353038. I thank National Science Foundation for continuing support of my research and teaching. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation. I thank Rahbar Virk for his help while learning MathML and Amaya.

Background information

The first and most important step is to get a browser that can display MathML. I recommend Mozilla Firefox, or Camino if you use a Mac. Download the necessary fonts from MathML at MIT or the appropriate font installer in the right hand column of the fonts for MathML enabled Mozilla (very soon this step will be replaced with the STIX fonts). Test your browser with the MathML torture test. Other browser options are:

Amaya
Internet Explorer with the Design Science MathPlayer plugin

Opera does not advertise any support of MathML.

MathML is an add-on to HTML, the base language for writing webpages. All the basic formatting is done with HTML (titles, paragraphs, links,... ) and so only the formatting of mathematical formulas needs to be done with MathML commands. The result is that MathML has fewer commands than TeX. It is not necessary to learn HTML and MathML separately. After reading this article, start by modifying the template for an article at http://www.math.wisc.edu/~ram/MathGlossary/template.xml (also see Appendix 2). If there is something you want to typeset which is not found in the template then look at the following useful pages, full of examples:

Notes by Arun Ram in MathML http://www.math.wisc.edu/~ram/MathGlossary
Mozilla MathML Demos page http://www.mozilla.org/projects/mathml/demo/
The W3C MathML test suite http://www.w3.org/Math/testsuite/,

MathML comes in two flavours: presentation markup and content markup. At the moment, most browsers do not render content markup (see MathML for Mozilla and Amaya overview). It seems that presentation markup is sufficient for mathematical typesetting needs and this article treats only presentation markup.

Style sheets for formatting are standard in TeX (e.g. amssym). The analogue for HTML is CSS (Cascading Style Sheets). There is no need to learn CSS separately, the necessary commands to get started are in the header of the template for a MathML typeset research article at http://www.math.wisc.edu/~ram/MathGlossary/template.xml. Appendix 1 contains a list of the most useful HTML commands and (all) MathML commands. The authoritative sources for commands and syntax are the W3C (World Wide Web Consortium) recommendations:

HTML (Hypertext Markup Language) http://www.w3.org/TR/html401/
CSS (Cascading Style Sheets) http://www.w3.org/TR/CSS21/
XHTML (Extensible hypertext markup language) http://www.w3.org/TR/xhtml1/
XML (Extensible markup language) http://www.w3.org/TR/xml/
XSL (Extensible stylesheet language) http://www.w3.org/TR/xsl/
MathML (Mathematics markup language) http://www.w3.org/TR/MathML2/
SVG (Scalable vector graphics) http://www.w3.org/TR/SVG/

I have found the W3C specifications more useful than books from the library or the bookstore. If you do go read the MathML specification, read Chapter 3 first.

Formatting an article with HTML and MathML

The following gives some basics for creating pages with HTML and MathML. The code for most of the examples in this article is included in the template article at http://www.math.wisc.edu/~ram/MathGlossary/template.xml. If you are viewing this article on the web use View Source in the View menu of your browser to see the code or download this page and edit it in your favourite editor.

Though MathML is simpler to learn than TeX, it has one great annoyance (which is, very likely, the reason it hasn't caught on more quickly): nearly every printing character is supposed to put inside <mi> </mi> or <mo> </mo> or <mn> </mn> (for "math identifier", "math operator" and "math number"). This means that the equation $x^{2} + y^{2} = 1$ is typeset with the code

<math xmlns="http://www.w3.org/1998/Math/MathML">
<msup><mi>x</mi><mn>2</mn></msup>
<mo>+</mo>
<msup><mi>y</mi><mn>2</mn></msup>
<mo>=</mo><mn>1</mn>
</math>

The consequence is that it is tedious to type in MathML by hand and some software that helps is desirable. I recommend use of Amaya but, as you use it, keep in mind that it is still in beta-version (see the suggestions for using Amaya below). Hopefully, Amaya will be improved quickly.

Features from HTML and CSS

For displayed equations the simplest option is to use <blockquote> put equation here </blockquote>. Alternatively, the header in the template article at http://www.math.wisc.edu/~ram/MathGlossary/template.xml (see also Appendix 2) includes the CSS lines table.dispeq{ width:100%; } td.eq{ text-align:center } td.eqno.after { content: "("counter(section)"."counter(eqn)")"; counter-increment: eqn; font-style: normal; } With these header lines displayed equations are inserted with <table class="dispeq"> <a name="eqname"></a> <tr> <td class="eq"> Insert equation here </td> <td class="eqno"></td> </tr> </table> where the line <td class="eqno"></td> is needed only for a numbered equation and the <a name="eqname"></a> is only needed to refer back to this equation later in the text.
The header material in the template article at http://www.math.wisc.edu/~ram/MathGlossary/template.xml (see also Appendix 2) includes CSS command (inside the <style> ... </style> tags) which provide formatting and automatic numbering for title, author, sections, subsections, equation numbers, theorems, propositions and lemmas. With these lines in the header, the title and author are put in with
```
<h2 class="title">My title</h2>

<p class="author">
My Name <br />
Department of My Stuff <br />
University of My Place <br />
City, ST 99999 SUE 
</p>
```
the section and subsection headers are put in with
<h2 class="section">A section</h2> and <h3 class="subsection">A subsection</h3>
respectively, and a theorem with a link back to equation (2.1) is coded with <div class="theorem"> My theorem with a link back to equation (<a href="#firsteqn">2.1</a>). </div>
A standard bibliographic entry might look something like
[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575.
Dressed up a bit, it has links to the authors' web pages, a link to the Math Review, a link to the journal, link to the arXiv version, and an anchor so that the reader can click on [BG] wherever it happens to be in the text and immediately jump to this reference entry.
[BG] A. Braverman and D. Gaitsgory, Crystals via the affine Grassmanian, Duke Math. J. 107 no. 3, (2001), 561-575; arXiv:math/9909077v2, MR1828302 (2002e:20083)
Because most web pages are continuous media, not paged media, the conventions for footnotes and acknowledgments may differ. A reasonable convention would be to place acknowledgments at the end of the introduction. Footnotes at the end of the document with anchors will have easy point and click link access. For information on including specific formatting rules for the printed version see Section 7 of the CSS Specification. Page breaks are discussed in Section 13.

A link to the Mozilla MathML torture test is coded with


 <a href="http://www.mozilla.org/projects/mathml/demo/texvsmml.xhtml">
Mozilla MathML torture test</a>.

The graphic in the line

span {v_{1}, \dots, v_{k}} \overset{μ}{\mapsto} Conv (λ^{(1)}, \dots, λ^{(k)}) =

is coded with

<img xmlns="http://www.w3.org/1999/xhtml" 
align="middle" alt="convex hull" src="convexhull.png" />

Nearly any graphic you make with your favourite picture drawing program can be included in this fashion (in this example, the .png file containing the graphic was made with the picture drawing program Fireworks).

In the example in (3.1), the 'Conv' is coded with <mi title="convex hull">Conv</mi> which has the effect that a tootip containing the words "convex hull" appears on mouse over.

MathML Features

There are plenty of math symbols available in MathML.

$∣ ‖ † ‡ ℏ ℧ ℵ \leftarrow ↑ ↓ \to \leftrightarrow ↕ \Leftarrow ⇑ ⇓ \Rightarrow \Leftrightarrow ⇕ ↖ ↗ ↙ ↘ \mapsto ↤ ↥ ↧ ↩ ↪ ⇄ ⇆ ⇅ ⇵ ⇤ ⇥ \forall \partial \exists ∄ \in \notin ∋ ∌ \prod ∐ \sum \pm \mp \propto \infty \land \lor \cup \cap \int \iint \oint ∴ \sim ≀ ≃ ≄ ≅ ≇ \approx ≉ \approx ≉ \neq \equiv < > \leq \geq ≪ ≫ ≮ ≯ ⩽̸ ⩾̸ ≺ ≻ ≼ ≽ \subset \supset ⊄ ⊅ \subseteq \supseteq ⊈ ⊉ ⊊ ⊋ ⊎ ⊏ ⊐ ⊑ ⊒ ⊔ ⊓ \oplus ⊖ \times \otimes ⊢ ⊣ ⊤ ⊥ ⊨ ⊲ ⊳ ⊴ ⊵ ⋄ \cdot ☆ ⋈ ⋉ ⋊ \dots ⋮ \dots ⋰ ⋱ ⌈ C^{'} ⌉ \subseteq ⌊ C^{''} ⌋ ⟨ ⟩ ♭ ♮ ♯ ♠ ♣$

The Mozilla MathML character table is extremely helpful, both for finding available characters and for determining character names. The following links will help for navigating these tables:

Logic Arrows Operators and Relations

Greek letters are coded with α β ... Φ ∇,
$Γ Λ Σ Ψ Δ Ξ ϒ Ω Θ Π Φ \nabla α β γ δ ε ϵ \in ζ η θ ϑ ι κ λ μ ν ξ o π ϖ ρ ϱ σ ς τ υ φ ϕ χ ψ ω$
blackboard bold characters with &Aopf; &Bopf; ... &yopf; &zopf;,
$𝔸 𝔹 ℂ 𝔻 𝔼 𝔽 𝔾 ℍ 𝕀 𝕁 𝕂 𝕃 𝕄 ℕ 𝕆 ℙ ℚ ℝ 𝕊 𝕋 𝕌 𝕍 𝕎 𝕏 𝕐 ℤ 𝕒 𝕓 𝕔 𝕕 𝕖 𝕗 𝕘 𝕙 𝕚 𝕛 𝕜 𝕝 𝕞 𝕟 𝕠 𝕡 𝕢 𝕣 𝕤 𝕥 𝕦 𝕧 𝕨 𝕩 𝕪 𝕫$
calligraphic characters with &Ascr; &Bscr; ... &yscr; &zscr;,
$𝒜 ℬ 𝒞 𝒟 ℰ ℱ 𝒢 ℋ ℐ 𝒥 𝒦 ℒ ℳ 𝒩 𝒪 𝒫 𝒬 ℛ 𝒮 𝒯 𝒰 𝒱 𝒲 𝒳 𝒴 𝒵$
$𝒶 𝒷 𝒸 𝒹 ℯ 𝒻 ℊ 𝒽 𝒾 𝒿 𝓀 𝓁 𝓂 𝓃 ℴ 𝓅 𝓆 𝓇 𝓈 𝓉 𝓊 𝓋 𝓌 𝓍 𝓎 𝓏$
fraktur characters with &Afr; &Bfr; ... &yfr; &zfr;, and

$𝔄 𝔅 ℭ 𝔇 𝔈 𝔉 𝔊 ℌ ℑ 𝔍 𝔎 𝔏 𝔐 𝔑 𝔒 𝔓 𝔔 ℜ 𝔖 𝔗 𝔘 𝔙 𝔚 𝔛 𝔜 ℨ 𝔞 𝔟 𝔠 𝔡 𝔢 𝔣 𝔤 𝔥 𝔦 𝔧 𝔨 𝔩 𝔪 𝔫 𝔬 𝔭 𝔮 𝔯 𝔰 𝔱 𝔲 𝔳 𝔴 𝔵 𝔶 𝔷$

cyrllic characters with &Acy; &Bcy; ... &ycy; &dzcy;.
$Ё Ђ Ѓ Є Ѕ І Ї Ј Љ Њ Ћ Ќ Ў Џ А Б В Г Д Е Ж З И Й К Л М Н О П Р С Т У Ф Х Ц Ч Ш Щ Ъ Ы Ь Э Ю Я$
$а б в г д е ж з и й к л м н о п р с т у ф х ц ч ш щ ъ ы ь э ю я ё ђ ѓ є ѕ і ї ј љ њ ћ ѓ ќ ў џ$

Math formulas and notation are typeset inside
<math xmlns="http://www.w3.org/1998/Math/MathML"> math stuff here </math>.
This is analogous to $ in TeX.

For superscripts use <msup> and for subscripts use <msub>. For multisubscripts and superscripts and prescripts use <mmultiscripts>.

b_{n}, x^{2}, R_{i}^{j}_{k}_{l}, and {}_{ℓ}^{m}_{n}_{p}R_{i}^{j}_{k}_{l} .

Sums (∑), products (∏), integrals (∫) and other similar "large" operators should resize appropriately if they are entered in the format <mo>∑</mo>. Similarly, parentheses, braces and brackets should resize appropriately if they are entered as <mo>(</mo> ... <mo>)</mo>.

G = \underset{w \in \tilde{W}}{⊔} I w I, (\sum_{i = 1}^{\infty} ⟨ \frac{m}{n} ⟩) ⟨ \prod_{i = 1}^{\infty} {\frac{m}{n}} ⟩ {\int_{- \infty}^{G} (\frac{m}{n})}

If parentheses don't automatically size correctly enclose the content inside <mrow> ... </mrow>. Parentheses are not required to be paired so a "cases" is easily accomplished with no extra effort.

T_{w} T_{s_{i}} = {\begin{matrix} T_{w s_{i}}, & if w s_{i} > w, \\ 0, & otherwise. \end{matrix}

Overbars, wide tildes, wide hats, vector symbols, overbraces, and underbraces should be done with <mover> and <munder>.

\overset{⏞}{x + y + z} \div \tilde{x} \otimes \hat{x} ⋉ \overline{x} \pm \dot{x} \circ \ddot{x} < \overset{˚}{x} \in \overset{\leftarrow}{x} \neq \vec{x} \land \overset{\leftrightarrow}{x}

The arrows stretch well in Mozilla and the &Tilde;, &OverBar;, &Hat; are also supposed to stretch, but the only one that does is &Hat;.

\tilde{xxx} \overset{goes to}{\to} \vec{vector} and \overline{xxx} \tilde{\to} \hat{xxxx} \overset{ϕ}{\leftarrow} {\underset{\underset{k + ℓ elements}{⏟}}{\overset{\overset{k a ’s}{⏞}}{a, ..., a}, \overset{\overset{ℓ b ’s}{⏞}}{b, ..., b}}}

where the last example is stolen from the Mozilla MathML torture test.

For fractions use

<mfrac linethickness="1"> <mrow> numerator </mrow> <mrow> denominator </mrow></mfrac>,

where the linethickness="1" should be left out for ordinary fractions and set to 0, instead of 1, for binomial coefficients.

\sum_{\binom{0 \leq i \leq m}{0 < j < n}} (\binom{n}{k / 2}) (\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}) \frac{x + \frac{y}{2}}{k + \frac{1}{x}} \frac{x + \frac{y}{2}}{k + \frac{1}{x}}

are examples stolen from the Mozilla MathML torture test. Resizing in fractions done with the displaystyle and scriptlevel attributes of the <mstyle> tag.

Matrices, math tables and aligned equations are done with <mtable>. The equation alignment in the second example below is done with <mtable columnalign="right center left">.

(\begin{matrix} (\begin{matrix} ⋱ & b \\ ⋰ & d \end{matrix}) & (\begin{matrix} \dots & ⋮ \\ g & h \end{matrix}) \\ 0 & (\begin{matrix} i & j \\ k & l \end{matrix}) \end{matrix}) \begin{array}{rcl} h_{λ^{\lor}} (f) h_{μ^{\lor}} (g) & = & h_{μ^{\lor}} (f^{- 1}) h_{λ^{\lor} + μ^{\lor}} (f) h_{λ^{\lor} + μ^{\lor}} (g) h_{λ^{\lor}} (g^{- 1}) \\ = & h_{μ^{\lor}} (f^{- 1}) h_{λ^{\lor} + μ^{\lor}} (f g) h_{λ^{\lor}} g^{- 1}) \\ = & h_{μ^{\lor}} (f^{- 1}) h_{μ^{\lor}} (f g) h_{λ^{\lor}} (f g) h_{λ^{\lor}} (g^{- 1}) \\ = & h_{μ^{\lor}} (g) h_{λ^{\lor}} (f) . \end{array}

The malignmark, maligngroup and groupalign are not yet implemented in Mozilla. Check the page status report of MathML tags in Mozilla for the latest updates.

Resizing is best done by enclosing the content in an <mstyle scriptlevel="-1"> ... </mstyle> where the -1 is changed to -2 or +1 or +2, as appropriate.

Positioning (up/down/left/right) of an object is controlled with <mpadded>. This doesn't render exactly right in Mozilla as

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mpadded width="+1em"><mi>&alpha;</mi></mpadded>
<mpadded height="-2em"><mi>&beta;</mi></mpadded>
<mpadded depth="+4em"><mi>&gamma;</mi></mpadded>
<mpadded lspace="+1em"><mi>&delta;</mi></mpadded>
</math>

renders as

α β γ δ

and the example, from 3.3.6.5 of the MathML specification, of a (strongly discouraged) overstruck C to try to make a sign for the complex numbers renders as

C |

For text inside math use <mtext>   text here   </mtext>. The   puts a space.
For inserting space inside math mode use <mspace width="2em"/>, where the 2 can be adjusted as desired. Other options are the space characters
 , &ensp;, &emsp;, &emsp13;, &emsp14;, &numsp;, &puncsp;,  , &hairsp;,
illustrated by
$nbsp ensp emsp emsp13 emsp14 numsp puncsp thinsp hairsp$ , respectively.
Phantom content is inserted with <mphantom> ... </mphantom> (analogous to the command \phantom in TeX).
Comments are inserted as in HTML, with .
Boxes can be placed around content using <mtable frame="solid">, color with <mstyle color="your color"> your colored content </mstyle>, and a colored background is accomplished with <mstyle background="yellow"> stuff </mstyle>.

$\begin{matrix} x^{2} + y^{2} = 1 \end{matrix} x^{3} + y^{3} = 2 x + 7 + 3 + z + q + q$

where the last example is Example miScolorname15 from the MathML testsuite. The MathML tag <menclose> is not yet implemented in Mozilla. Check the page status report of MathML tags in Mozilla for the latest updates.
The <maction> tag provides a nice option of "toggling" content, by clicking on it $3 + 2 = 32 fooled you fooled you + 1$ (try it, click on it).

Using Amaya for MathML authoring

Amaya is very useful for putting in all those <mi>, <mo>, setting up subscripts, superscripts, matrices and fractions. Amaya, at the moment, is painful for entering math symbols and it has the annoying problem that, even if you get it entered right, when you save your document, Amaya will automatically change character entities that have been entered manually. For example &langle; will change into some Chinese character.

Amaya crashes often. Until Amaya gets more robust the best option is to use Amaya in tandem with your favourite editor. Do the layout of the text and the equations with Amaya. Type a few paragraphs into Amaya then copy the code into your favourite editor and fix up the character names. Use a slightly different name for Amaya file and your document file.

Always start a new document by opening an existing template.
Double click the lower edge of the display window (on mouse over it turns pink) to display in split mode and see the code that Amaya produces.
Type into the display window to create text. I have found Amaya most useful when I don't use the math tool palette.
Use the keyboard shortcuts to start math tags and use the arrow keys to navigate the boxes Amaya produces for matrices, superscripts and subscripts.
Use the right arrow key to move the cursor out of math mode.
Save often.
Select Display all shortcuts in the Preferences > General dialog and restart Amaya.
If you really want to start a blank document with Amaya do New XHTMLdocument from the File Menu, make sure that the Documentprofile is XHTML1.1 and that the filename ends in .xml and type math into it. Amaya will automatically put in the correct DTD (Document Type Definition).

Amaya MathML keyboard shortcuts for Mac OS X

Command	Tag	Shortcut
New Formula	<math>	Ctrl-m Ctrl-m
Subscript	<msub>	Ctrl-m Ctrl-v
Superscript	<msup>	Ctrl-m Ctrl-6
Subscript and Superscript	<msubsup>	Ctrl-m Ctrl-b
Multiscripts	<mmultiscripts>	Ctrl-m Ctrl-s
Under	<munder>	Ctrl-m Ctrl-u
Over	<mover>	Ctrl-m Ctrl-o
Under and over	<munderover>	Ctrl-m Ctrl-k
Fraction	<mfrac>	Ctrl-m Ctrl-f
New Matrix	<mtable>	Ctrl-m Ctrl-h
Root	<mroot>	Ctrl-m Ctrl-r
Square root	<msqrt>	Ctrl-m Ctrl-q

Problems with Amaya

The most annoying problem is that Amaya automatically changes character entities that have been entered manually. For example &langle; will change into some Chinese character.
Amaya needs an efficient way to enter characters.
Amaya tends to crash when it gets confused.
The rows and columns are reversed in the matrix dialog box.
Looking at example pages shows that Mozilla is superior for MathML rendering.

Tips

The following useful information and example pages are provided by Mozilla and W3C:
If the usual minus sign doesn't show up properly hold down the Option key and type the usual minus sign to get a minus sign that displays properly (this may be Mac specific).
A math character will slant less if put in an <mo> instead of an <mi>.
Since HTML prefers that the user control spacing from style files the best hack for putting spaces in seems to be to go into math mode and put in mathspaces. Using spacing to tweak the look of the output is strongly discouraged as the results are usually renderer dependent. The ultimate tweaking is shown at the MathML test suite torture test complex 3 .
CSS rules in a document are included inside <style type="text/css"> ... </style> inside the <head> ... </head>. Alternatively, they can all be put into a style sheet, and then loaded with @import "mathsty.css";
Inserting media specific (for example, for printers) CSS formatting rules is explained in Section 7 of the CSS Specification. Page breaks are discussed in Section 13 of the CSS specification.
When printing from Mozilla be sure that "Shrink to fit page width" is not selected. Sometimes the shrunk MathML does not print correctly.
The details for automatic numbering are explained in Section 12.4 and 12.2 of the CSS specification. Alternatively, automatic numbering and referencing ought to be doable with fairly simple javascript.
The use of namespaces makes it possible to put XHTML commands (for example, links and graphics) inside MathML.
Using UTF-8 character encoding is always the best. For ordinary html pages use the XHTML Transitional DTD.

The page Authoring MathML for Mozilla suggests replacing the usual DTD with

<?xml version="1.0"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN" 
               "http://www.w3.org/TR/MathML2/dtd/xhtml-math11-f.dtd" [
  <!ENTITY mathml "http://www.w3.org/1998/Math/MathML"> 
]>
<html xmlns="http://www.w3.org/1999/xhtml">

This has the effect that the usual entrance into math mode

<math xlmns="http://www.w3.org/1998/Math/MathML">

only needs to be coded with <math xlmns="&mathml;">

<mi>ij</mi> renders as $ij$ , <mi>i</mi><mi>j</mi> renders as $i j$ , and <mi> <mi>i</mi><mi>j</mi> </mi> renders as $ij$ .
Doing graphics with SVG (Scalable vector graphics) should obviate the need for external graphics files and all browsers that handle MathML also render SVG. Inkscape is a drawing program that will output graphics in SVG (Scalable Vector Graphics). It should be possible to include formulas in MathML inside an SVG picture with the use of the foreignObject element of SVG. There is also a combined XHTML1.1+MathML2.0+SVG1.1 document type definition. However, I have been unable to determine whether the foreignObject tag is implemented in Inkscape, and I have not yet succeeded in getting mixed markup to render properly in Mozilla (it renders fine in Amaya).
Other media types, such as animations, video and sound, can be included. For these, it is most efficient to produce the appropriate code using an application like Dreamweaver (or copy it from some other web page) and then copy it into your MathML document.
Parentheses are supposed to resize appropriately if they are put in an <mo> tag. See 3.2.5.8 of the MathML specification.

Problems with MathML

There seems to be no character for an upwards pointing &sube;
<mfenced> is redundant since it is equivalent to putting the parentheses inside <mo> tags. Perhaps <mfenced> should be deprecated.
What is the future of the relationship between MathML, CSS, and SVG? Will MathML 3.0 have good options for putting XHTML commands (for example, for links and graphics) inside MathML math mode? See the discussion at 7.1.4 of the MathML specification.

Problems with Mozilla

The tags <mglyph>, <menclose>, <mlabeledtr>, <malignmark>, and <maligngroup>, are not yet implemented. Check the page status report of MathML tags in Mozilla for the latest updates.

Unfortunately, the placement of the limits on large sums and products is controlled only by setting displaystyle="true" and getting scriptlevel="-1". Setting accent="true" and/or accentunder="false" only seems to affect the size of the limit, not its placement.
The malignmark, maligngroup and groupalign are not yet implemented in Mozilla. They can be tested with the maligngroup1 example of the MathML test suite and the example given in 3.5.5.8 of the MathML specification.
Except for the "toggle" the attributes of <maction> do not render in Mozilla (see examples mactionBstatus1, mactionBhigh1, and mactionBtooltip1, of the MathML test suite). The tooltip in the Mozilla demo is done with the title attribute from HTML.
This may be Mac specific: The usual minus sign from the keyboard doesn't render at all; hold down the Option key and type the usual minus sign to get a minus sign that renders properly.
The arrows stretch well in Mozilla and the &Tilde;, &OverBar;, &Hat; are also supposed to stretch, but the only one that does is &Hat;.

$\tilde{xxx} \overset{goes to}{\to} \overline{xxx} \vec{vector} \overline{xxx} \overset{}{\to} \hat{xxx} \overset{goes to}{\leftarrow} \hat{xxx}$

The height and depth attributes of <mpadded> do not render properly. Also, negative width values in <mspace> do not render properly.

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mpadded width="+1em"><mi>&alpha;</mi></mpadded>
<mpadded height="-2em"><mi>&beta;</mi></mpadded>
<mpadded depth="+4em"><mi>&gamma;</mi></mpadded>
<mpadded lspace="+1em"><mi>&delta;</mi></mpadded>
</math>

renders as

α β γ δ

and the example, from 3.3.6.5 of the MathML specification,

<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
  <mi> C </mi>
  <mpadded width="0em">
    <mspace width="-0.3em"/>
    <mtext> | </mtext>
  </mpadded>
</mrow>
</math>

of a (strongly discouraged) overstruck C to try to make a sign for the complex numbers renders as

C |

The Greek letters shouldn't be slanted. Currently, in order to make them not slanted they need to be put inside an <mo> tag, which is undesirable since they function as characters not as operators.
&sqcup; and other similar operators don't look right when they resize. The naive coding produces
$G = \underset{w \in \tilde{W}}{⊔} I w I but the tweaked version G = \underset{w \in \tilde{W}}{⊔} I w I$ looks much better.
The renderer ought to handle spacing for text inside math better so that
$S_{p} = {paths 1 \overset{c_{1}}{\to} \dots \overset{c_{ℓ}}{\to} w whose folding ends in p} .$

doesn't need the tweak in it to have the space correct around the text.
The &compfn; renders properly on the Powerbook, but not on the MacBook Pro: $f \circ g$ .
Some other characters with problems: The printed $\leq$ is ugly and doesn't match $<$ , &varr; inside an <mo> renders as $↕$ , &vee; won't resize to a smaller size to put in a subscript $h_{μ^{\lor}}$ , ...

HTML and MathML commands

Basic HTML

<em> italics <em>, <strong> bold </strong>, <dfn> term being defined </dfn>
<h2> heading size 2 </h2>, <h3> heading size 3 </h3>, etc.
<p> paragraph here </p>, <br /> for a line break,   for a space, <pre>
```
preformatted
  text
```
</pre>, and <hr /> for an hrule
Make a bulleted list (unordered list) with
<ul> <li> list item </li> <li> list item </li> </ul>
list item

list item

Make a numbered list (ordered list) with

<ol> <li> list item one </li> <li> list item two </li> </ol>
	list item list item

Make a definition list with

<dl> <dt> definition term one</dt> <dd> definition description of term one</dd> <dt> definition term two </dt> <dd> definition description of term two</dd> </dl>
	definition term one definition description of term one definition term two definition description of term two

Make a table with:

<table>
<thead> 
     <tr> header </tr>
</thead>
<tfoot>
     <tr> footer </tr>
</tfoot>
<tbody>
     <tr> 
         <td>first</td>
         <td>second</td>
         <td>third</td>
     </tr>
     <tr> 
         <td>fourth</td>
         <td>fifth</td>
         <td>sixth</td>
     </tr>
</tbody>
<tbody>
     <tr> 
         <td>First</td>
         <td>Second</td>
         <td>Third</td>
     </tr>
     <tr> 
         <td>Fourth</td>
         <td>Fifth</td>
         <td>Sixth</td>
     </tr>
</tbody>
</table>

header footer

first	second	third
fourth	fifth	sixth
First	Second	Third
Fourth	Fifth	Sixth

<div class="mytype" name="myname"> ... </div> is a way to group a set of lines and attach names, classes, special formats and other attributes to this collection of lines.

MathML Presentation Markup tags

element
1. attribute = "value"
the nine common attributes are
1. mathbackground="#rgb | #rrggbb | html-color name"
2. mathcolor="#rgb | #rrggbb | html-color name"
3. mathsize="small | normal | big | number | v-unit"
4. mathvariant="number | bold | italic | script | bold-fraktur | double-struck | bold-script | fraktur | sans-serif | bold-sans-serif | sans-serif-italic | sans-serif-bold-italic | monospace"
5. fontsize="number v-unit"
6. fontweight=" normal | bold"
7. fontstyle="normal | italic"
8. fontfamily="string | css-fontfamily"
9. color="#rgb | #rrggbb | html-color name"
all MathML presentation elements accept the following attributes
1. class
2. style
3. id
4. xlink:href:
5. other:
<mn> number
1. the nine common attributes
<mo> operator
1. the nine common attributes
2. form="prefix | infix | postfix"
3. lspace="number h-unit | namedspace"
4. rspace="number h-unit | namedspace"
5. stretchy="true | false"
6. symmetric="true | false"
7. maxsize="number[v-unit | h-unit] | namedspace"
8. minsize="number | namedspace"
9. largeop="true | false"
10. movablelimits="true | false"
11. fence="true | false"
12. separator="true | false"
13. accent="true | false"
<mi> identifier
1. the nine common attributes
<mtext> text
1. the nine common attributes
<mspace> blank space
1. the nine common attributes and
2. width="number h-unit | namedspace"
3. height="number v-unit"
4. depth="number v-unit"
<ms> string
1. the nine common attributes
2. lquote="character" default is &quot
3. rquote="character" default is &quot
<mglyph> character glyph
1. the nine common attributes and
2. fontfamily="string | css-fontfamily "
3. index=" integer "
4. alt="string"
<mrow> horizontal row
<mfrac> fraction
1. linethickness="number [v-unit] | thin | medium | thick"
2. numalign=" left | center | right"
3. denomalign="(left | center | right)+"
4. bevelled="true | false"
<msqrt> square root
<mroot> root
<mstyle> apply style
1. all attributes of all presentation elements except for those that have a required value
2. scriptlevel="[+|-] unsigned integer"
3. displaystyle="true | false"
4. scriptsizemultiplier="number" default is 0.71
5. scriptminsize="number v-unit" default is 8pt
6. color="#rgb | #rrggbb | transparent | html-color-name"
7. background="#rgb | #rrggbb | transparent | html-color-name" default is transparent
8. veryverythinmathspace="number h-unit" default is 1/18 em
9. verythinmathspace="number h-unit"default is 1/9 em
10. thinmathspace="number h-unit"default is 1/6 em
11. mediummathspace="number h-unit" default is 2/9 em
12. thickmathspace="number h-unit"default is 5/18 em
13. verythickmathspace="number h-unit" default is 1/3 em
14. veryverythickmathspace="number h-unit" default is 7/18 em
<mspace> blank space
1. the nine common attributes
2. width="number h-unit | namedspace"
3. height="number v-unit"
4. depth="number v-unit"
<mphantom> make content invisible
<mfenced> add fences around content
1. open="string" default is (
2. close="string" default is )
3. separators="character*" default is ,
<menclose> enclose content in a stretchy symbol
1. notation="longdiv | actuarial | radical" default is radical
<merror> error messages
<msub> subscript
1. subscriptshift="number v-unit"
<msup> superscript
1. superscriptshift="number v-unit"
<msubsup> subscript and superscript
1. subscriptshift="number v-unit"
2. superscriptshift="number v-unit"
<munder> underbrace
1. accentunder="true | false"
<mover> overbrace
1. accent="true | false"
<munderover> under and overbrace
1. accentunder="true | false"
2. accent="true | false"
<mmultiscripts>
1. subscriptshift="number v-unit"
2. superscriptshift="number v-unit"
<none/> is used only as a child element of the mmultiscripts element
<mtable> table or matrix
1. align="(top | bottom | center | baseline | axis)" default is axis
2. rowalign="(top | bottom | center | baseline | axis)+" default is baseline
3. columnalign="(left | center | right)+" default is center
4. columnwidth="(auto | number h-unit | namedspace | fit)" default is auto
5. width="auto | number h-unit" default is auto
6. rowspacing="(number v-unit)+" default is 1.0 ex
7. columnspacing="(number h-unit | namedspace)+" default is 0.8em
8. rowlines="(none | solid | dashed)+" default is none
9. columnlines="(none | solid | dashed)+" default is none
10. equalrows="true | false" default is false
11. equalcolumns="true | false" default is false
12. frame="(none | solid | dashed)+" default is none
13. framespacing="(number h-unit | namedspace) (number v-unit | namedspace)" default is 0.4 em 0.5 ex
14. displaystyle="true | false" default is false
15. side="left | right | leftoverlap | rightoverlap" default is overlap
16. minlabelspacing="number h-unit"
17. groupalign="groupalignment-list-list"
18. alignmentscop="true | false"
<mtr> row of a table
1. rowalign="(top | bottom | center | baseline | axis)+" default is baseline
2. columnalign="(left | center | right)+" default is center
3. groupalign="groupalignment-list-list"default is {left}
<mlabeledtr> labeled row of a table
1. rowalign="(top | bottom | center | baseline | axis)+" default is baseline
2. columnalign="(left | center | right)+" default is center
3. groupalign="groupalignment-list-list"default is {left}
<mtd> cell in a table
1. rowalign="(top | bottom | center | baseline | axis)+" default is baseline
2. columnalign="(left | center | right)+" default is center
3. groupalign="groupalignment-list-list"default is {left}
4. rowspan="number" default is 1
5. columnspan="number" default is 1
<maligngroup> alignment group
<malignmark> alignment marker
1. edge="left | right" default is {left}
<maction> interactive actions
1. selection="number" default is 1
2. actiontype="depends on application"
<mpadded> precise positioning
1. width="[+ | -] number | (%[pseudo-unit] | pseudounit | h-unit | namedspace)"
2. lspace="[+ | -] number | (%[pseudo-unit] | pseudounit | h-unit)"
3. height="[+ | -] number | (%[pseudo-unit] | pseudounit | h-unit)"
4. depth="[+ | -] number | (%[pseudo-unit] | pseudounit | h-unit)"
<mprescripts> used in mmultiscripts to indicate folowing elements in prescript position

Template for an article

The first part of the header is the Document Type Definition (DTD) for web pages that use large amounts of MathML.
The part of the header inside the <style> ... </style> tag is a set of CSS (Cascading Style Sheet) rules for automatic numbering of sections, formats for theorems, propositions and lemmas, and for displayed equations, with and without equation numbers.

<?xml version="1.0" encoding="UTF-8"?>
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0//EN"
   "http://www.w3.org/Math/DTD/mathml2/xhtml-math11-f.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head> 
<meta http-equiv="content-type" content="text/html; charset=UTF-8" />
<title>Your favourite title</title>
<style type="text/css">
body {
    counter-reset: section 0;  
    text-align: justify
}

h2.title {
    text-align:center
}
p.author {
    text-align:center
}
li {
    text-indent: 0em;
}
p.indent {
    text-indent: 2em; 
}

h2.section {
    counter-reset: eqn 0 result 0 subsection 0;  
}
h2.section:before {
    content: counter(section) ". ";
    counter-increment: section 1;  /* Add 1 to section */
}
h3.subsection:before {
    content: counter(section) "." counter(subsection);
    counter-increment: subsection;  /* Add 1 to subsection */
}

div.theorem:before { 
    content: "Theorem " counter(section) "." counter(result) " "; 
    counter-increment: result 1;
    font-style: normal; 
    font-weight: bold; 
}
div.theorem{ font-style:italic }
div.proposition:before { 
    content: "Proposition " counter(section) "." counter(result) " "; 
	counter-increment: result 1;
    font-style: normal; 
    font-weight: bold; 
}
div.proposition{ font-style:italic }
div.lemma:before { 
    content: "Lemma " counter(section) "." counter(result) " "; 
    counter-increment: result;
    font-style: normal; 
    font-weight: bold; 
}
div.lemma{ font-style:italic }

table.dispeq {
   width:100%;
   }
td.eq {
   text-align:center;
     }
td.eqno {
    width: 10px;
    text-align:right;
    }
td.eqno:after { 
     content: "("counter(section)"."counter(eqn)")";
     counter-increment: eqn;
     font-style: normal; 
  }
</style>
</head>

<body>
<h2 class="title"> Template for article </h2>

<p class="author">
Arun Ram <br />
Department of Mathematics <br />
University of Wisconsin, Madison <br />
Madison, WI 53706 USA 
</p>

<p> Last updates: 31 February 1935 </p>


<h2 class="section"> Section title </h2>

<p class="indent"> 
A little introductory text.
</p>
<table class="dispeq">  
<a name="firsteqn" id="firsteqn"></a>
<tr>
<td class="eq"> 
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <mi>G</mi>
  <mo>=</mo>
  <munder>
    <mstyle scriptlevel="-1">
         <mo>&sqcup;</mo>
    </mstyle>
    <mrow>
      <mi>w</mi>
      <mo>&isin;</mo>
      <mover>
        <mi>W</mi>
        <mi>&Tilde;</mi>
      </mover>
    </mrow>
  </munder>
  <mi>I</mi>
  <mi>w</mi>
  <mi>I</mi>
  <mspace width="2em"/>
</math>
</td> 
<td class="eqno"></td>
</tr></table>

<div class="theorem">
<a href="#[BG]">[BG]</a>
A theorem about subscripts and superscripts.
<table class="dispeq">
<a name="fractions" id="fractions"></a>
<tr><td class="eq">
<math xmlns="http://www.w3.org/1998/Math/MathML">
    <msub> <mi>b</mi><mi>n</mi></msub><mo>,</mo>
    <mspace width="2em"/>
    <msup> <mi>x</mi><mn>2</mn></msup><mo>,</mo>
    <mspace width="2em"/>
      <mmultiscripts> 
        <mi> R </mi> 
        <mi> i </mi> 
        <none/> 
        <none/> 
        <mi> j </mi> 
        <mi> k </mi> 
        <none/> 
        <mi> l </mi> 
        <none/> 
      </mmultiscripts> <mo>,</mo>
<mspace width="2em"/>
<mtext>and</mtext>
<mspace width="2em"/>
      <mmultiscripts> 
        <mi> R </mi> 
        <mi> i </mi> 
        <none/> 
        <none/> 
        <mi> j </mi> 
        <mi> k </mi> 
        <none/> 
        <mi> l </mi> 
        <none/> 
  <mprescripts/>
        <mi> ℓ </mi> 
        <none/> 
        <none/> 
        <mi> m </mi> 
        <mi> n </mi> 
        <none/> 
        <mi> p </mi> 
        <none/> 
      </mmultiscripts> 
      <mo>.</mo>
</math>
</td>
<td class="eqno"></td>
</tr></table>
</div>

<p class="indent">A bit of text about hats, tildes, arrows, underbraces
and overbraces.  </p>
<table class="dispeq">
<a name="accents" id="accents"></a>
<tr><td class="eq">
<math xmlns="http://www.w3.org/1998/Math/MathML">
   <mover>
     <mi mathvariant="italic">xxx</mi>
     <mo stretchy="true">&Tilde;</mo>
   </mover>
<mi>&thinsp;</mi>
<mover>
<mo>&rarr;</mo>
<mtext>goes to</mtext>
</mover>
<mi>&thinsp;</mi>
<mover>
<mtext>vector</mtext>
<mo>&rarr;</mo>
</mover>
<mspace width="1em"/>
<mtext>and</mtext>
<mspace width="1em"/>
   <mover>
     <mi mathvariant="italic">xxx</mi>
     <mo stretchy="true">&OverBar;</mo>
   </mover>
<mi>&thinsp;</mi>
<mover>
<mo>&rarr;</mo>
<mrow>
   <mphantom><mtext>goes to</mtext></mphantom>
   <mo>&sim;</mo>
   <mphantom><mtext>goes to</mtext></mphantom>
</mrow>
</mover>
<mi>&thinsp;</mi>
   <mover>
     <mi mathvariant="italic">xxxx</mi>
     <mo>&Hat;</mo>
   </mover>
<mover>
<mo>&larr;</mo>
<mrow><mi>&emsp;</mi><mi>&phiv;</mi><mi>&emsp;</mi></mrow>
</mover>
<mrow>
  <mo stretchy='false'>{</mo>
  <munder>
    <mrow>
      <mover>
        <mrow>
          <mpadded width="0"><mphantom>(</mphantom></mpadded> <!-- \mathstrut -ugly hack a la TeX -->
          <mi>a</mi>
          <mo>,</mo>
          <mo>...</mo>
          <mo>,</mo>
          <mi>a</mi>
        </mrow>
        <mover>
          <mo>&OverBrace;</mo>
          <mrow>
            <mi>k</mi>
            <mtext>&nbsp;</mtext>
            <mi>a</mi>
            <mtext>&rsquo;s</mtext>
          </mrow>
        </mover>
      </mover>
      <mo>,</mo>
      <mover>
        <mrow>
          <mpadded width="0"><mphantom>(</mphantom></mpadded> <!-- \mathstrut -ugly hack a la TeX -->
          <mi>b</mi>
          <mo>,</mo>
          <mo>...</mo>
          <mo>,</mo>
          <mi>b</mi>
        </mrow>
        <mover>
          <mo>&OverBrace;</mo>
          <mrow>
            <mi>&ell;</mi>
            <mtext>&nbsp;</mtext>
            <mi>b</mi>
            <mtext>&rsquo;s</mtext>
          </mrow>
        </mover>
      </mover>
    </mrow>
    <munder>
      <mo>&UnderBrace;</mo>
      <mrow>
        <mi>k</mi>
        <mo>+</mo>
        <mi>&ell;</mi>
        <mtext>&nbsp;elements</mtext>
      </mrow>
    </munder>
  </munder>
  <mo stretchy='false'>}</mo>
</mrow>
</math>
</td>
<td class="eqno"></td>
</tr></table>


<div class="proposition">
A proposition about fractions.
<table class="dispeq">
<a name="fractions" id="fractions"></a>
<tr><td class="eq">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mstyle displaystyle="true">
<mrow>
  <munder>
    <mo>&sum;</mo>
    <mrow>
      <mfrac linethickness="0">
        <mrow>
          <mn>0</mn>
          <mo>&leq;</mo>
          <mi>i</mi>
          <mo>&leq;</mo>
          <mi>m</mi>
        </mrow>
        <mrow>
          <mn>0</mn>
          <mo>&lt;</mo>
          <mi>j</mi>
          <mo>&lt;</mo>
          <mi>n</mi>
        </mrow>
      </mfrac>
    </mrow>
  </munder>
</mrow>
<mrow>
  <mo>&langle;</mo>
<mfrac linethickness="0">
    <mi>n</mi>
    <mrow>
      <mi>k</mi>
      <mo>/</mo>
      <mn>2</mn>
    </mrow>
</mfrac>
  <mo>)</mo>
</mrow>
<mo>=</mo>
  <mrow>
    <mo>[</mo>
    <mfrac>
      <msup>
        <mo>&part;</mo>
        <mn>2</mn>
      </msup>
      <mrow>
        <mo>&part;</mo>
        <msup>
          <mi>x</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mfrac>
    <mo>+</mo>
    <mfrac>
      <msup>
        <mo>&part;</mo>
        <mn>2</mn>
      </msup>
      <mrow>
        <mo>&part;</mo>
        <msup>
          <mi>y</mi>
          <mn>2</mn>
        </msup>
      </mrow>
    </mfrac>
    <mo>&rangle;</mo>
  </mrow>
</mstyle>
<mo>=</mo>
<mrow>
  <mfrac>
    <mrow>
    <mstyle displaystyle="true" scriptlevel="-1">
      <mi>x</mi>
      <mo>+</mo>
      <mfrac>
        <mi>y</mi>
        <mn>2</mn>
      </mfrac>
    </mstyle>
    </mrow>
    <mrow>
      <mi>k</mi>
      <mo>+</mo>
      <mfrac>
         <mn>1</mn>
         <mi>x</mi>
      </mfrac>
    </mrow>
  </mfrac>
</mrow>
</math>
</td>
<td class="eqno"></td>
</tr></table>
</div>

<h3 class="subsection"> Development </h3>

<p class="indent">The following lemma might be useful.</p>

<div class="lemma">
<a name="matrices" id="matrices"></a>
A lemma about matrices and aligned equations
<table class="dispeq">
<tr><td class="eq">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mrow>
  <mo>(</mo>
  <mtable>
    <mtr>
      <mtd columnalign="center">
        <mrow>
          <mo>(</mo>
          <mtable>
            <mtr>
              <mtd columnalign="center">
                <mo>&dtdot;</mo>
              </mtd>
              <mtd columnalign="center">
                <mi>b</mi>
              </mtd>
            </mtr>
            <mtr>
              <mtd columnalign="center">
                <mo>&utdot;</mo>
              </mtd>
              <mtd columnalign="center">
                <mi>d</mi>
              </mtd>
            </mtr>
          </mtable>
          <mo>)</mo>
        </mrow>
      </mtd>
      <mtd columnalign="center">
        <mrow>
          <mo>(</mo>
          <mtable>
            <mtr>
              <mtd columnalign="center">
                <mo>&ctdot;</mo>
              </mtd>
              <mtd columnalign="center">
                <mo>&vellip;</mo>
              </mtd>
            </mtr>
            <mtr>
              <mtd columnalign="center">
                <mi>g</mi>
              </mtd>
              <mtd columnalign="center">
                <mi>h</mi>
              </mtd>
            </mtr>
          </mtable>
          <mo>)</mo>
        </mrow>
      </mtd>
    </mtr>
    <mtr>
      <mtd columnalign="center">
        <mn>0</mn>
      </mtd>
      <mtd columnalign="center">
        <mrow>
          <mo>(</mo>
          <mtable>
            <mtr>
              <mtd columnalign="center">
                <mi>i</mi>
              </mtd>
              <mtd columnalign="center">
                <mi>j</mi>
              </mtd>
            </mtr>
            <mtr>
              <mtd columnalign="center">
                <mi>k</mi>
              </mtd>
              <mtd columnalign="center">
                <mi>l</mi>
              </mtd>
            </mtr>
          </mtable>
          <mo>)</mo>
        </mrow>
      </mtd>
    </mtr>
  </mtable>
  <mo>)</mo>
</mrow>
<mspace width="2em"/>
<mtext>and</mtext>
<mspace width="2em"/>
<mtable columnalign="right center left">
    <mtr>
      <mtd>
        <msub>
          <mi>h</mi>
          <mi>&lambda;</mi>
        </msub>
        <mo>(</mo>
        <mi>f</mi>
        <mo>)</mo>
        <msub>
          <mi>h</mi>
          <mi>&mu;</mi>
        </msub>
        <mo>(</mo>
        <mi>g</mi>
        <mo>)</mo>
      </mtd>
      <mtd><mo>=</mo></mtd>
      <mtd>
        <msub>
          <mi>h</mi>
          <mi>&mu;</mi>
        </msub>
        <mo>(</mo>
        <msup>
          <mi>f</mi>
          <mrow>
            <mo>–</mo>
            <mn>1</mn>
          </mrow>
        </msup>
        <mo>)</mo>
        <msub>
          <mi>h</mi>
          <mrow>
            <mi>&lambda;</mi>
            <mo>+</mo>
            <mi>&mu;</mi>
          </mrow>
        </msub>
        <mo>(</mo>
        <mi>f</mi>
        <mo>)</mo>
        <msub>
          <mi>h</mi>
          <mrow>
            <mi>&lambda;</mi>
            <mo>+</mo>
            <mi>&mu;</mi>
          </mrow>
        </msub>
        <mo>(</mo>
        <mi>g</mi>
        <mo>)</mo>
        <msub>
          <mi>h</mi>
          <mi>&lambda;</mi>
        </msub>
        <mo>(</mo>
        <msup>
          <mi>g</mi>
          <mrow>
            <mo>–</mo>
            <mn>1</mn>
          </mrow>
        </msup>
        <mo>)</mo>
      </mtd>
    </mtr>
    <mtr>
      <mtd></mtd>
      <mtd> <mo>=</mo> </mtd>
      <mtd>
        <msub>
          <mi>h</mi>
          <mi>&mu;</mi>
        </msub>
        <mo>(</mo>
        <msup>
          <mi>f</mi>
          <mrow>
            <mo>–</mo>
            <mn>1</mn>
          </mrow>
        </msup>
        <mo>)</mo>
        <msub>
          <mi>h</mi>
          <mrow>
            <mi>&lambda;</mi>
            <mo>+</mo>
            <mi>&mu;</mi>
          </mrow>
        </msub>
        <mo>(</mo>
        <mi>f</mi><mi>g</mi>
        <mo>)</mo>
        <msub>
          <mi>h</mi>
          <mi>&lambda;</mi>
        </msub>
        <mo>(</mo>
        <msup>
          <mi>g</mi>
          <mrow>
            <mo>–</mo>
            <mn>1</mn>
          </mrow>
        </msup>
        <mo>)</mo>
      </mtd>
    </mtr>
    <mtr>
      <mtd></mtd>
      <mtd>  <mo>=</mo></mtd>
      <mtd>
        <msub>
          <mi>h</mi>
          <mi>&mu;</mi>
        </msub>
        <mo>(</mo>
        <mi>g</mi>
        <mo>)</mo>
        <msub>
          <mi>h</mi>
          <mi>&lambda;</mi>
        </msub>
        <mo>(</mo>
        <mi>f</mi>
        <mo>)</mo>
        <mo>.</mo>
      </mtd>
    </mtr>
  </mtable>
</math>
</td>
<td class="eqno"></td>
</tr></table>
with a link back to equation 
(<a href="#firsteqn">1.1</a>).
</div>

<p>And finally,</p>

<div class="proposition"> <a name="boxes" id="boxes"></a>
A proposition about boxes, backgrounds and colors.
<table class="dispeq">
<tr><td class="eq">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<mtable frame="solid">
<mtr>
 <mtd> 
<msup><mi>x</mi><mn>2</mn></msup>
<mo>+</mo>
<msup><mi>y</mi><mn>2</mn></msup>
<mo>=</mo><mn>1</mn>
 </mtd>
</mtr>
</mtable>
<mspace width="2em"/>
<mtext>and</mtext>
<mspace width="2em"/>
<mstyle background="yellow">
<msup><mi>x</mi><mn>3</mn></msup>
<mo>+</mo>
<msup><mi>y</mi><mn>3</mn></msup>
<mo>=</mo><mn>2</mn>
</mstyle>
<mspace width="2em"/>
<mtext> and </mtext>
<mspace width="2em"/>
      <mi mathcolor="blue">x</mi> 
      <mo>+</mo> 
      <mn mathcolor="#00ff33" color="purple">7</mn> 
      <mo>+</mo> 
      <mn color="yellow">3</mn> 
      <mo>+</mo> 
      <mstyle color="red"> 
        <mi>z</mi> 
        <mo>+</mo> 
        <mi mathcolor="blue">q</mi> 
        <mo>+</mo> 
        <mi>q</mi> 
      </mstyle>
</math>
</td>
<td class="eqno"></td>
</tr></table>
</div>

<p>
<table class="dispeq">
<a name="graphic" id="graphic"></a>
<tr><td class="eq">
<math xmlns="http://www.w3.org/1998/Math/MathML">
  <mi>span</mi>
  <mo>{</mo>
  <msub>
    <mi>v</mi>
    <mn>1</mn>
  </msub>
  <mo>,</mo>
  <mi>&hellip;</mi>
  <mo>,</mo>
  <msub>
    <mi>v</mi>
    <mi>k</mi>
  </msub>
  <mo>}</mo>
  <mover>
    <mo>&map;</mo>
    <mi>&mu;</mi>
  </mover>
  <mi title="convex hull">Conv</mi>
  <mo>(</mo>
  <msup>
    <mi>&lambda;</mi>
    <mrow>
      <mo>(</mo>
      <mn>1</mn>
      <mo>)</mo>
    </mrow>
  </msup>
  <mo>,</mo>
  <mi>&hellip;</mi>
  <mo>,</mo>
  <msup>
    <mi>&lambda;</mi>
    <mrow>
      <mo>(</mo>
      <mi>k</mi>
      <mo>)</mo>
    </mrow>
  </msup>
  <mo>)</mo>
  <mo>=</mo>
  <mi>&emsp;</mi>
<img xmlns="http://www.w3.org/1999/xhtml" align="middle" alt="convex hull" src="convexhull.png" />
</math>
</td> 
<td class="eqno"></td>
</tr></table>
is an illustration of the proposition with a mouseover tooltip on the Conv.
</p>

<h2 class="section"> The main theorem </h2>

<div class="theorem">
<a name="cases" id="cases"></a>
A theorem about cases and toggling content.
<table class="dispeq">
<tr><td class="eq">
<math xmlns="http://www.w3.org/1998/Math/MathML">
<msub> <mi>T</mi><mi>w</mi></msub>
<msub> <mi>T</mi><mrow><msub><mi>s</mi><mi>i</mi></msub></mrow></msub>
<mo>=</mo>
<mrow> <mo>{</mo>
<mtable>
<mtr> <mtd> <msub> <mi>T</mi>
<mrow> 
  <mi>w</mi><msub><mi>s</mi><mi>i</mi></msub>
</mrow>
</msub>
</mtd>
<mtd columnalign="left">
<mtext> if </mtext>
<mspace width="0.5em"/> 
<mi>w</mi> <msub><mi>s</mi><mi>i</mi></msub>
<mo>></mo><mi>w</mi>
</mtd>
</mtr>
<mtr>
<mtd> <mn>0</mn> </mtd>
<mtd columnalign="left"> <mtext>otherwise.</mtext> </mtd>
</mtr>
</mtable>
</mrow>
<mspace width="4em"/>
      <maction actiontype="toggle"> 
        <mrow> 
  <mstyle background="pink">
          <mn>3</mn> 
          <mo>+</mo> 
          <mn>2</mn> 
          <mo>=</mo>
          <mn>32</mn>
  </mstyle>
        </mrow> 
        <mrow> 
          <mtext> fooled you  </mtext> 
        </mrow> 
        <mrow> 
          <mtext>fooled you</mtext> 
            <mo>+</mo> 
            <mn>1</mn> 
        </mrow>
      </maction>
<mspace width="1em"/>
    <mtext>(try it, click on it).</mtext>  
</math>
</td>
<td class="eqno"></td>
</tr></table>
</div>


<h2 class="section"> References </h2>

<p>
<a name="[BG]" id="[BG]"> [BG] </a> 
<a href="http://www.math.brown.edu/faculty/braverman.html">
A. Braverman</a> and 
<a href="http://www.math.harvard.edu/~gaitsgde/"> 
D. Gaitsgory </a>, 
<a href="http://projecteuclid.org/DPubS?service=Repository&amp;version=1.0&amp;verb=Disseminate&amp;handle=euclid.dmj/1091737024&amp;view=body&amp;content-type=pdf_1#" title="View PDF">
<em> Crystals via the affine Grassmanian</em></a>, 
<a href="http://www.dukeupress.edu/dmj/">Duke Math. J.</a> 
<a href="http://projecteuclid.org/DPubS?service=UI&amp;version=1.0&amp;verb=Display&amp;page=toc&amp;handle=euclid.dmj/1091737018">
<strong>107</strong> no. 3,</a> (2001), 561-575;  
<a href="http://arxiv.org/abs/math.AG/9909077v2?front">
arXiv:math/9909077v2</a>, 
<a href="http://www.ams.org/mathscinet-getitem?mr=1828302">
MR1828302 (2002e:20083)</a>
</p>


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