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<?xml version="1.0" encoding="utf-8"?><!DOCTYPE html
SYSTEM "mathml.dtd">
<html xmlns="http://www.w3.org/1999/xhtml">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
<title>Mathematical Markup Language (MathML) Version 3.0 -- XHTML + MathML Version</title><style type="text/css">
.egmeta {
color:#5555AA;font-style:italic;font-family:serif;font-weight:bold;
}
table.syntax {
font-size: 75%;
background-color: #DDDDDD;
border: thin solid;
}
table.syntax td {
border: solid thin;
}
table.syntax th {
text-align: left;
}
table.attributes td { padding-left:0.5em; padding-right:0.5em; }
table.attributes td.attname { white-space:nowrap; vertical-align:top;}
table.attributes td.attdesc { background-color:#F0F0FF; padding-left:2em; padding-right:2em}
th.uname {font-size: 50%; text-align:left;}
code { font-family: monospace; }
div.constraint,
div.issue,
div.note,
div.notice { margin-left: 2em; }
li p { margin-top: 0.3em;
margin-bottom: 0.3em; }
div.exampleInner pre { margin-left: 1em;
margin-top: 0em; margin-bottom: 0em}
div.exampleOuter {border: 4px double gray;
margin: 0em; padding: 0em}
div.exampleInner { background-color: #d5dee3;
border-top-width: 4px;
border-top-style: double;
border-top-color: #d3d3d3;
border-bottom-width: 4px;
border-bottom-style: double;
border-bottom-color: #d3d3d3;
padding: 4px; margin: 0em }
div.exampleWrapper { margin: 4px }
div.exampleHeader { font-weight: bold;
margin: 4px}
a.mainindex {font-weight: bold;}
li.sitem {list-style-type: none;}
.error { color: red }
div.mathml-example {border:solid thin black;
padding: 0.5em;
margin: 0.5em 0 0.5em 0;
}
div.strict-mathml-example {border:solid thin black;
padding: 0.5em;
margin: 0.5em 0 0.5em 0;
}
div.strict-mathml-example h5 {
margin-top: -0.3em;
margin-bottom: -0.5em;}
var.meta {background-color:green}
var.transmeta {background-color:red}
pre.mathml {padding: 0.5em;
background-color: #FFFFDD;}
pre.mathml-fragment {padding: 0.5em;
background-color: #FFFFDD;}
pre.strict-mathml {padding: 0.5em;
background-color: #FFFFDD;}
.minitoc { border-style: solid;
border-color: #0050B2;
border-width: 1px ;
padding: 0.3em;}
.attention { border-style: solid;
border-width: 1px ;
color: #5D0091;
background: #F9F5DE;
border-color: red;
margin-left: 1em;
margin-right: 1em;
margin-top: 0.25em;
margin-bottom: 0.25em; }
.attribute-Name { background: #F9F5C0; }
.method-Name { background: #C0C0F9; }
.IDL-definition { border-style: solid;
border-width: 1px ;
color: #001000;
background: #E0FFE0;
border-color: #206020;
margin-left: 1em;
margin-right: 1em;
margin-top: 0.25em;
margin-bottom: 0.25em; }
.baseline {vertical-align: baseline}
#eqnoc1 {width: 10%}
#eqnoc2 {width: 80%; text-align: center; }
#eqnoc3 {width: 10%; text-align: right; }
div.div1 {margin-bottom: 1em;}
.h3style {
text-align: left;
font-family: sans-serif;
font-weight: normal;
color: #0050B2;
font-size: 125%;
}
h4 { text-align: left;
font-family: sans-serif;
font-weight: normal;
color: #0050B2; }
h5 { text-align: left;
font-family: sans-serif;
font-weight: bold;
color: #0050B2; }
th {background: #E0FFE0;}
p, blockquote, h4 { font-family: sans-serif; }
dt, dd, dl, ul, li { font-family: sans-serif; }
pre, code { font-family: monospace }
a.termref {
text-decoration: none;
color: black;
}
.mathml-render {
font-family: serif;
font-size: 130%;
border: solid 4px green;
padding-left: 1em;
padding-right: 1em;
}
</style><link rel="stylesheet" type="text/css"
href="../../../StyleSheets/TR/W3C-REC.css" />
</head>
<body>
<div>
<p>The presentation of this document has been augmented to
use XHTML rather than HTML, and to include all MathML
examples as inlined MathML markup that should be rendered by
a MathML enabled browser. The MathML examples that have
been added, and which should be displayed in your browser,
are marked <span class="mathml-render">like this</span>.
</p>
<hr />
</div>
<div class="head">
<p><a href="http://www.w3.org/"><img src="../../../Icons/w3c_home" alt="W3C" height="48" width="72" /></a></p>
<h1><a id="title"></a>Mathematical Markup Language (MathML) Version 3.0
</h1>
<h2><a id="w3c-doctype"></a>W3C Recommendation 21 October 2010
</h2>
<dl>
<dt>This version:</dt>
<dd>
<a href="index.html">http://www.w3.org/TR/2010/REC-MathML3-20101021/</a>
</dd>
<dt>Latest MathML 3 version:</dt>
<dd><a href="http://www.w3.org/TR/MathML3/">http://www.w3.org/TR/MathML3/</a></dd>
<dt>Latest MathML Recommendation:</dt>
<dd><a href="http://www.w3.org/TR/MathML/">http://www.w3.org/TR/MathML/</a></dd>
<dt>Previous version:</dt>
<dd><a href="http://www.w3.org/TR/2010/PR-MathML3-20100810/">http://www.w3.org/TR/2010/PR-MathML3-20100810/</a></dd>
<dt>Editors:</dt>
<dd>David Carlisle, NAG</dd>
<dd>Patrick Ion, Mathematical Reviews, American Mathematical Society</dd>
<dd>Robert Miner, Design Science, Inc.</dd>
<dt>Principal Authors:</dt>
<dd>Ron Ausbrooks, Stephen Buswell, David Carlisle, Giorgi Chavchanidze, Stéphane Dalmas, Stan Devitt, Angel Diaz, Sam Dooley, Roger Hunter, Patrick Ion, Michael Kohlhase, Azzeddine Lazrek, Paul Libbrecht, Bruce Miller, Robert Miner, Chris Rowley, Murray Sargent, Bruce Smith, Neil Soiffer, Robert Sutor, Stephen Watt</dd>
</dl>
<p>Please refer to the <a href="http://www.w3.org/Math/Documents/mathml3-errata.html"><strong>errata</strong></a>
for this document, which may include some normative corrections.
</p>
<p>In addition to the <a href="Overview.html">HTML</a> version, this document is also available in these non-normative formats: <a href="Overview-d.html">diff marked HTML version</a>, <a href="Overview.xml.html">XHTML+MathML version</a>, and <a href="http://www.w3.org/TR/2010/REC-MathML3-20101021/mathml.pdf">PDF version</a>.
</p>
<p>See also <a href="http://www.w3.org/2005/11/Translations/Query?titleMatch=MathML3"><strong>translations</strong></a>.
</p>
<p class="copyright"><a href="http://www.w3.org/Consortium/Legal/ipr-notice#Copyright">Copyright</a> © 1998-2010 <a href="http://www.w3.org/"><acronym title="World Wide Web Consortium">W3C</acronym></a><sup>®</sup> (<a href="http://www.csail.mit.edu/"><acronym title="Massachusetts Institute of Technology">MIT</acronym></a>, <a href="http://www.ercim.eu/"><acronym title="European Research Consortium for Informatics and Mathematics">ERCIM</acronym></a>, <a href="http://www.keio.ac.jp/">Keio</a>), All Rights Reserved. W3C <a href="http://www.w3.org/Consortium/Legal/ipr-notice#Legal_Disclaimer">liability</a>, <a href="http://www.w3.org/Consortium/Legal/ipr-notice#W3C_Trademarks">trademark</a> and <a href="http://www.w3.org/Consortium/Legal/copyright-documents">document use</a> rules apply.
</p>
</div>
<hr class="html-compat" />
<div>
<h2><a id="abstract"></a>Abstract
</h2>
<p>
This specification defines the Mathematical Markup Language, or
<a id="td-mathml"></a>MathML. MathML is an
XML application for describing mathematical notation and capturing
both its structure and content. The goal of MathML is to enable
mathematics to be served, received, and processed on the World Wide
Web, just as <a id="td-html"></a>HTML has
enabled this functionality for text.
</p>
<p>
This specification of the markup language MathML is intended
primarily for a readership consisting of those who will be
developing or implementing renderers or editors using it, or
software that will communicate using MathML as a protocol for input
or output. It is <em>not</em> a User's Guide but rather a
reference document.
</p>
<p>
MathML can be used to encode both mathematical notation and
mathematical content. About thirty-eight of the MathML tags describe
abstract notational structures, while another about one hundred and
seventy provide a way of unambiguously specifying the intended
meaning of an expression. Additional chapters discuss how the MathML
content and presentation elements interact, and how MathML renderers
might be implemented and should interact with browsers. Finally,
this document addresses the issue of special characters used for
mathematics, their handling in MathML, their presence in Unicode,
and their relation to fonts.
</p>
<p>
While MathML is human-readable, authors typically will
use equation editors, conversion
programs, and other specialized software tools to generate
MathML. Several versions of such MathML tools exist,
both freely available software and commercial
products, and more are under development.
</p>
</div>
<div>
<h2><a id="status"></a>Status of this Document
</h2>
<p><em> This section describes the status of this document at the time
of its publication. Other documents may supersede this document. A
list of current W3C publications and the latest revision of this
technical report can be found in the <a href="http://www.w3.org/TR/">W3C technical reports index</a> at
http://www.w3.org/TR/.</em>
</p>
<p>
This document was produced by the <a href="http://www.w3.org/Math/">W3C Math Working Group</a> as
a <a href="http://www.w3.org/Consortium/Process/tr#RecsW3C">Recommendation</a>
and is part of the
W3C <a href="http://www.w3.org/Math/Activity">Math Activity</a>.
The goals of the W3C Math Working Group are discussed in the <a href="http://www.w3.org/Math/Documents/Charter2006.html"> W3C Math WG
Charter</a> (revised July 2006). The authors of this document are
the W3C Math Working Group members. A list of <a href="appendixe.xml.html">participants in the W3C Math Working Group</a>
is available.
</p>
<p>This document has been reviewed by W3C Members, by software
developers, and by other W3C groups and interested parties, and is
endorsed by the Director as a W3C Recommendation. It is a stable
document and may be used as reference material or cited from another
document. W3C's role in making the Recommendation is to draw attention
to the specification and to promote its widespread deployment. This
enhances the functionality and interoperability of the Web.
</p>
<p>
The Working Group maintains a comprehensive
<a href="http://www.w3.org/Math/testsuite/">Test Suite</a>. This
is publicly available and developers are encouraged to submit their results for display.
The <a href="http://www.w3.org/Math/testsuite/results/tests.html">Test Results</a>
are public. They show at least two interoperable implementations for each
essential test. Further details may be found in the
<a href="http://www.w3.org/Math/Documents/mml3-implementation-report.html">Implementation Report</a>.
</p>
<p>Closely associated with the MathML 3.0 specification is
<a href="http://www.w3.org/TR/mathml-for-css/">A MathML for CSS Profile</a>, also approved to be published as a Recommendation. This
describes a profile of MathML 3.0 that can be well formatted with Cascading Style Sheets.
</p>
<p>
The MathML 2.0 (Second Edition) specification has been a W3C
Recommendation since 2001. After its
recommendation, a W3C Math Interest Group collected
reports of experience
with the deployment of MathML and identified issues with MathML that
might be ameliorated. The rechartering of a Math Working Group
did not signal any change in the overall design of MathML.
The major additions in MathML 3 are support for
bidirectional layout, better linebreaking and explicit positioning,
elementary math notations, and a new strict content MathML vocabulary
with well-defined semantics. The MathML 3 Specification has also been
restructured.
</p>
<p> This document was produced by a group operating under
the <a href="http://www.w3.org/Consortium/Patent-Policy-20040205/">5 February 2004 W3C Patent Policy</a>. W3C maintains a
<a href="http://www.w3.org/2004/01/pp-impl/35549/status" rel="disclosure">public list of any patent disclosures</a> made in connection with
the deliverables of the group; that page also includes instructions
for disclosing a patent. An individual who has actual knowledge of
a patent which the individual believes contains
<a href="http://www.w3.org/Consortium/Patent-Policy-20040205/#def-essential">Essential Claim(s)</a>
must disclose the information in accordance with
<a href="http://www.w3.org/Consortium/Patent-Policy-20040205/#sec-Disclosure">section 6 of the W3C Patent Policy</a>.
</p>
<p>Public discussion of MathML and issues of support through the W3C
for mathematics on the Web takes place on <a href="mailto:www-math@w3.org"> the public mailing list of the Math Working
Group</a> (<a href="http://lists.w3.org/Archives/Public/www-math/">list archives</a>).
To subscribe send an email to <a href="mailto:www-math-request@w3.org">www-math-request@w3.org</a>
with the word <code>subscribe</code> in the subject line.
</p>
<p>
The basic chapter structure of this document is based on the earlier
MathML 2.0 Recommendation <a href="appendixh.xml.html#MathML2">[MathML2]</a>. That MathML 2.0
itself was a revision of the earlier W3C Recommendation MathML 1.01
<a href="appendixh.xml.html#MathML1">[MathML1]</a>; MathML 3.0 is a revision of the W3C
Recommendation MathML 2.0. It differs from it in that all previous
chapters have been updated, some new
elements and attributes added and some
deprecated.
Much has been moved to separate documents containing explanatory
material, material on characters and entities and on the MathML DOM.
The discussion of character entities has led to the document
<em>XML Entity Definitions for Characters</em> <a href="appendixg.xml.html#Entities">[Entities]</a>,
which is now a W3C Recommendation. The concern with
use of CSS with MathML has led to the document <em>A MathML for CSS Profile</em>
<a href="appendixh.xml.html#MathMLforCSS">[MathMLforCSS]</a>, which is a W3C Recommendation
accompanying this one.
</p>
<p>
The biggest differences from MathML 2.0 (Second Edition) are in Chapters 4 and 5, although there have been
smaller improvements throughout the specification. A more detailed
description of changes from the previous Recommendation follows.
</p>
<ul>
<li>
<p>
Much of the non-normative explication
that formerly was found in Chapters 1 and 2, and many examples from
elsewhere in the previous MathML specifications, were removed
from the MathML3 specification and planned to be incorporated into a MathML Primer
to be prepared as a separate document. It is expected this will help
the use of this formal MathML3 specification as a reference document
in implementations, and offer the new user better help in
understanding MathML's deployment. The remaining content of Chapters 1
and 2 has been edited to reflect the changes elsewhere in the
document, and in the rapidly evolving Web environment. Some of the
text in them went back to early days of the Web and XML, and its
explanations are now commonplace.
</p>
</li>
<li>
<p>
Chapter 3, on presentation-oriented markup, adds new material on linebreaking,
and on markup for elementary math notations used in many
countries (<code>mstack</code>, <code>mlongdiv</code> and other
associated elements). Other changes include revisions to
the <code>mglyph</code>, <code>mpadded</code> and <code>maction</code>
elements and significant unification and cleanup of attribute
values.
Earlier work, as recorded in the W3C Note <a href="http://www.w3.org/TR/2006/NOTE-arabic-math-20060131/">Arabic
mathematical notation</a>, has allowed clarification of the
relationship with bidirectional text and examples
with RTL text have been added.
</p>
</li>
<li>
<p>
Chapter 4, on content-oriented markup, contains major changes and additions.
The meaning of the actual content remains as before in principle,
but a lot of work has been done on expressing it better. A few new elements have been added.
</p>
</li>
<li>
<p>
Chapter 5 has been refined as its purpose has been further
clarified to deal with the mixing of markup languages. This chapter deals,
in particular, with interrelations of parts of the
MathML specification, especially with presentation and content
markup.
</p>
</li>
<li>
<p>
Chapter 6 is a new addition which deals with the issues of interaction
of MathML with a host environment. This chapter deals with interrelations
of the MathML specification with XML and HTML, but in the context of deployment
on the Web. In particular there is a discussion of the interaction of CSS
with MathML.
</p>
</li>
<li>
<p>
Chapter 7 replaces the previous Chapter 6, and has been rewritten
and reorganized to reflect the new
situation in regard to Unicode, and the changed
W3C context with regard to named character entities. The
new W3C specification <em>XML Entity Definitions for
Characters</em>, which incorporates those used for mathematics has
become a a W3C Recommendation, <a href="appendixg.xml.html#Entities">[Entities]</a>.
</p>
</li>
<li>
<p>
The Appendices, of which there are eight shown, have been
reworked.
Appendix A now contains the new RelaxNG schema
for MathML3 as well as discussion of MathML3 DTD issues.
Appendix B addresses media types associated with MathML and
implicitly constitutes a request for the registration of three new ones,
as is now standard for work from the W3C.
Appendix C contains a new simplified and reconsidered Operator Dictionary.
Appendices D, E, F, G and H contain similar non-normative material to
that in the previous specification, now appropriately updated.
</p>
</li>
<li>
<p>
A fuller discussion of the document's evolution can be found in
<a href="appendixf.xml.html">Appendix F Changes</a>.
</p>
</li>
</ul>
</div>
<div class="toc">
<h2><a id="contents"></a>Table of Contents
</h2>
<p class="toc">1 <a href="chapter1.xml.html">Introduction</a><br class="html-compat" /> 1.1 <a href="chapter1.xml.html#intro.notation">Mathematics and its Notation</a><br class="html-compat" /> 1.2 <a href="chapter1.xml.html#intro.origin">Origins and Goals</a><br class="html-compat" /> 1.2.1 <a href="chapter1.xml.html#intro.goals">Design Goals of MathML</a><br class="html-compat" /> 1.3 <a href="chapter1.xml.html#intro.overview">Overview</a><br class="html-compat" /> 1.4 <a href="chapter1.xml.html#intro.example">A First Example</a><br class="html-compat" />2 <a href="chapter2.xml.html">MathML Fundamentals</a><br class="html-compat" /> 2.1 <a href="chapter2.xml.html#fund.syntax">MathML Syntax and Grammar</a><br class="html-compat" /> 2.1.1 <a href="chapter2.xml.html#fund.xmlgeneral">General Considerations</a><br class="html-compat" /> 2.1.2 <a href="chapter2.xml.html#interf.namespace">MathML and Namespaces</a><br class="html-compat" /> 2.1.3 <a href="chapter2.xml.html#fund.xmlsyntax">Children versus Arguments</a><br class="html-compat" /> 2.1.4 <a href="chapter2.xml.html#fund.renderingmodel">MathML and Rendering</a><br class="html-compat" /> 2.1.5 <a href="chapter2.xml.html#fund.attval">MathML Attribute Values</a><br class="html-compat" /> 2.1.6 <a href="chapter2.xml.html#fund.globatt">Attributes Shared by all MathML Elements</a><br class="html-compat" /> 2.1.7 <a href="chapter2.xml.html#fund.collapse">Collapsing Whitespace in Input</a><br class="html-compat" /> 2.2 <a href="chapter2.xml.html#interf.toplevel">The Top-Level
<code>math</code> Element</a><br class="html-compat" /> 2.2.1 <a href="chapter2.xml.html#interf.toplevel.atts">Attributes</a><br class="html-compat" /> 2.2.2 <a href="chapter2.xml.html#id.2.2.2">Deprecated Attributes</a><br class="html-compat" /> 2.3 <a href="chapter2.xml.html#interf.genproc">Conformance</a><br class="html-compat" /> 2.3.1 <a href="chapter2.xml.html#fund.mathmlconf">MathML Conformance</a><br class="html-compat" /> 2.3.2 <a href="chapter2.xml.html#interf.error">Handling of Errors</a><br class="html-compat" /> 2.3.3 <a href="chapter2.xml.html#interf.unspecified">Attributes for unspecified data</a><br class="html-compat" />3 <a href="chapter3.xml.html">Presentation Markup</a><br class="html-compat" /> 3.1 <a href="chapter3.xml.html#presm.intro">Introduction</a><br class="html-compat" /> 3.1.1 <a href="chapter3.xml.html#id.3.1.1">What Presentation Elements Represent</a><br class="html-compat" /> 3.1.2 <a href="chapter3.xml.html#id.3.1.2">Terminology Used In This Chapter</a><br class="html-compat" /> 3.1.3 <a href="chapter3.xml.html#presm.reqarg">Required Arguments</a><br class="html-compat" /> 3.1.4 <a href="chapter3.xml.html#id.3.1.4">Elements with Special Behaviors</a><br class="html-compat" /> 3.1.5 <a href="chapter3.xml.html#presm.bidi">Directionality</a><br class="html-compat" /> 3.1.6 <a href="chapter3.xml.html#presm.scriptlevel">Displaystyle and Scriptlevel</a><br class="html-compat" /> 3.1.7 <a href="chapter3.xml.html#presm.linebreaking">Linebreaking of Expressions</a><br class="html-compat" /> 3.1.8 <a href="chapter3.xml.html#presm.warnfinetuning">Warning about fine-tuning of presentation</a><br class="html-compat" /> 3.1.9 <a href="chapter3.xml.html#presm.summary">Summary of Presentation Elements</a><br class="html-compat" /> 3.1.10 <a href="chapter3.xml.html#presm.presatt">Mathematics style attributes common to presentation elements</a><br class="html-compat" /> 3.2 <a href="chapter3.xml.html#presm.tokel">Token Elements</a><br class="html-compat" /> 3.2.1 <a href="chapter3.xml.html#presm.tokenchars">MathML characters in token elements</a><br class="html-compat" /> 3.2.2 <a href="chapter3.xml.html#presm.commatt">Mathematics style attributes common to token elements</a><br class="html-compat" /> 3.2.3 <a href="chapter3.xml.html#presm.mi">Identifier <code><mi></code></a><br class="html-compat" /> 3.2.4 <a href="chapter3.xml.html#presm.mn">Number <code><mn></code></a><br class="html-compat" /> 3.2.5 <a href="chapter3.xml.html#presm.mo">Operator, Fence, Separator or Accent
<code><mo></code></a><br class="html-compat" /> 3.2.6 <a href="chapter3.xml.html#presm.mtext">Text <code><mtext></code></a><br class="html-compat" /> 3.2.7 <a href="chapter3.xml.html#presm.mspace">Space <code><mspace/></code></a><br class="html-compat" /> 3.2.8 <a href="chapter3.xml.html#presm.ms">String Literal <code><ms></code></a><br class="html-compat" /> 3.3 <a href="chapter3.xml.html#presm.genlayout">General Layout Schemata</a><br class="html-compat" /> 3.3.1 <a href="chapter3.xml.html#presm.mrow">Horizontally Group Sub-Expressions
<code><mrow></code></a><br class="html-compat" /> 3.3.2 <a href="chapter3.xml.html#presm.mfrac">Fractions <code><mfrac></code></a><br class="html-compat" /> 3.3.3 <a href="chapter3.xml.html#presm.mroot">Radicals <code><msqrt></code>, <code><mroot></code></a><br class="html-compat" /> 3.3.4 <a href="chapter3.xml.html#presm.mstyle">Style Change <code><mstyle></code></a><br class="html-compat" /> 3.3.5 <a href="chapter3.xml.html#presm.merror">Error Message <code><merror></code></a><br class="html-compat" /> 3.3.6 <a href="chapter3.xml.html#presm.mpadded">Adjust Space Around Content
<code><mpadded></code></a><br class="html-compat" /> 3.3.7 <a href="chapter3.xml.html#presm.mphantom">Making Sub-Expressions Invisible <code><mphantom></code></a><br class="html-compat" /> 3.3.8 <a href="chapter3.xml.html#presm.mfenced">Expression Inside Pair of Fences
<code><mfenced></code></a><br class="html-compat" /> 3.3.9 <a href="chapter3.xml.html#presm.menclose">Enclose Expression Inside Notation
<code><menclose></code></a><br class="html-compat" /> 3.4 <a href="chapter3.xml.html#presm.scrlim">Script and Limit Schemata</a><br class="html-compat" /> 3.4.1 <a href="chapter3.xml.html#presm.msub">Subscript <code><msub></code></a><br class="html-compat" /> 3.4.2 <a href="chapter3.xml.html#presm.msup">Superscript <code><msup></code></a><br class="html-compat" /> 3.4.3 <a href="chapter3.xml.html#presm.msubsup">Subscript-superscript Pair <code><msubsup></code></a><br class="html-compat" /> 3.4.4 <a href="chapter3.xml.html#presm.munder">Underscript <code><munder></code></a><br class="html-compat" /> 3.4.5 <a href="chapter3.xml.html#presm.mover">Overscript <code><mover></code></a><br class="html-compat" /> 3.4.6 <a href="chapter3.xml.html#presm.munderover">Underscript-overscript Pair
<code><munderover></code></a><br class="html-compat" /> 3.4.7 <a href="chapter3.xml.html#presm.mmultiscripts">Prescripts and Tensor Indices
<code><mmultiscripts></code></a><br class="html-compat" /> 3.5 <a href="chapter3.xml.html#presm.tabmat">Tabular Math</a><br class="html-compat" /> 3.5.1 <a href="chapter3.xml.html#presm.mtable">Table or Matrix
<code><mtable></code></a><br class="html-compat" /> 3.5.2 <a href="chapter3.xml.html#presm.mtr">Row in Table or Matrix <code><mtr></code></a><br class="html-compat" /> 3.5.3 <a href="chapter3.xml.html#presm.mlabeledtr">Labeled Row in Table or Matrix
<code><mlabeledtr></code></a><br class="html-compat" /> 3.5.4 <a href="chapter3.xml.html#presm.mtd">Entry in Table or Matrix <code><mtd></code></a><br class="html-compat" /> 3.5.5 <a href="chapter3.xml.html#presm.malign">Alignment Markers
<code><maligngroup/></code>, <code><malignmark/></code></a><br class="html-compat" /> 3.6 <a href="chapter3.xml.html#presm.elementary">Elementary Math</a><br class="html-compat" /> 3.6.1 <a href="chapter3.xml.html#presm.mstack">Stacks of Characters <code><mstack></code></a><br class="html-compat" /> 3.6.2 <a href="chapter3.xml.html#presm.mlongdiv">Long Division <code><mlongdiv></code></a><br class="html-compat" /> 3.6.3 <a href="chapter3.xml.html#presm.msgroup">Group Rows with Similiar Positions <code><msgroup></code></a><br class="html-compat" /> 3.6.4 <a href="chapter3.xml.html#presm.msrow">Rows in Elementary Math <code><msrow></code></a><br class="html-compat" /> 3.6.5 <a href="chapter3.xml.html#presm.mscarries">Carries, Borrows, and Crossouts <code><mscarries></code></a><br class="html-compat" /> 3.6.6 <a href="chapter3.xml.html#presm.mscarry">A Single Carry <code><mscarry></code></a><br class="html-compat" /> 3.6.7 <a href="chapter3.xml.html#presm.msline">Horizontal Line <code><msline/></code></a><br class="html-compat" /> 3.6.8 <a href="chapter3.xml.html#presm.elemmath.examples">Elementary Math Examples</a><br class="html-compat" /> 3.7 <a href="chapter3.xml.html#presm.enliven">Enlivening Expressions</a><br class="html-compat" /> 3.7.1 <a href="chapter3.xml.html#presm.maction">Bind Action to Sub-Expression
<code><maction></code></a><br class="html-compat" /> 3.8 <a href="chapter3.xml.html#presm.semantics">Semantics and Presentation</a><br class="html-compat" />4 <a href="chapter4.xml.html">Content Markup</a><br class="html-compat" /> 4.1 <a href="chapter4.xml.html#contm.intro">Introduction</a><br class="html-compat" /> 4.1.1 <a href="chapter4.xml.html#id.4.1.1">The Intent of Content Markup</a><br class="html-compat" /> 4.1.2 <a href="chapter4.xml.html#contm.rendering">The Structure and Scope of Content MathML Expressions</a><br class="html-compat" /> 4.1.3 <a href="chapter4.xml.html#contm.strict">Strict Content MathML</a><br class="html-compat" /> 4.1.4 <a href="chapter4.xml.html#contm.cds">Content Dictionaries</a><br class="html-compat" /> 4.1.5 <a href="chapter4.xml.html#id.4.1.5">Content MathML Concepts</a><br class="html-compat" /> 4.2 <a href="chapter4.xml.html#contm.core">Content MathML Elements Encoding Expression Structure</a><br class="html-compat" /> 4.2.1 <a href="chapter4.xml.html#contm.cn">Numbers <code><cn></code></a><br class="html-compat" /> 4.2.2 <a href="chapter4.xml.html#contm.ci">Content Identifiers <code><ci></code></a><br class="html-compat" /> 4.2.3 <a href="chapter4.xml.html#contm.csymbol">Content Symbols <code><csymbol></code></a><br class="html-compat" /> 4.2.4 <a href="chapter4.xml.html#contm.cs">String Literals <code><cs></code></a><br class="html-compat" /> 4.2.5 <a href="chapter4.xml.html#contm.apply">Function Application <code><apply></code></a><br class="html-compat" /> 4.2.6 <a href="chapter4.xml.html#contm.binding">Bindings and Bound Variables <code><bind></code>
and <code><bvar></code></a><br class="html-compat" /> 4.2.7 <a href="chapter4.xml.html#contm.sharing">Structure Sharing <code><share></code></a><br class="html-compat" /> 4.2.8 <a href="chapter4.xml.html#contm.semantics">Attribution via <code>semantics</code></a><br class="html-compat" /> 4.2.9 <a href="chapter4.xml.html#contm.cerror">Error Markup <code><cerror></code></a><br class="html-compat" /> 4.2.10 <a href="chapter4.xml.html#contm.cbytes">Encoded Bytes <code><cbytes></code></a><br class="html-compat" /> 4.3 <a href="chapter4.xml.html#contm.structure.extended">Content MathML for Specific Structures</a><br class="html-compat" /> 4.3.1 <a href="chapter4.xml.html#contm.container">Container Markup</a><br class="html-compat" /> 4.3.2 <a href="chapter4.xml.html#contm.bind.apply">Bindings with <code><apply></code></a><br class="html-compat" /> 4.3.3 <a href="chapter4.xml.html#contm.qualifiers">Qualifiers</a><br class="html-compat" /> 4.3.4 <a href="chapter4.xml.html#contm.opclasses">Operator Classes</a><br class="html-compat" /> 4.3.5 <a href="chapter4.xml.html#id.4.3.5">Non-strict Attributes</a><br class="html-compat" /> 4.4 <a href="chapter4.xml.html#contm.opel">Content MathML for Specific Operators and Constants</a><br class="html-compat" /> 4.4.1 <a href="chapter4.xml.html#contm.basicfun">Functions and Inverses</a><br class="html-compat" /> 4.4.2 <a href="chapter4.xml.html#id.4.4.2">Arithmetic, Algebra and Logic</a><br class="html-compat" /> 4.4.3 <a href="chapter4.xml.html#id.4.4.3">Relations</a><br class="html-compat" /> 4.4.4 <a href="chapter4.xml.html#id.4.4.4">Calculus and Vector Calculus</a><br class="html-compat" /> 4.4.5 <a href="chapter4.xml.html#contm.sets">Theory of Sets</a><br class="html-compat" /> 4.4.6 <a href="chapter4.xml.html#id.4.4.6">Sequences and Series</a><br class="html-compat" /> 4.4.7 <a href="chapter4.xml.html#contm.elemclass">Elementary classical functions</a><br class="html-compat" /> 4.4.8 <a href="chapter4.xml.html#id.4.4.8">Statistics</a><br class="html-compat" /> 4.4.9 <a href="chapter4.xml.html#id.4.4.9">Linear Algebra</a><br class="html-compat" /> 4.4.10 <a href="chapter4.xml.html#contm.constantsandsymbols">Constant and Symbol Elements</a><br class="html-compat" /> 4.5 <a href="chapter4.xml.html#contm.deprecated">Deprecated Content Elements</a><br class="html-compat" /> 4.5.1 <a href="chapter4.xml.html#contm.declare">Declare <code><declare></code></a><br class="html-compat" /> 4.6 <a href="chapter4.xml.html#contm.p2s">The Strict Content MathML Transformation</a><br class="html-compat" />5 <a href="chapter5.xml.html">Mixing Markup Languages for Mathematical Expressions</a><br class="html-compat" /> 5.1 <a href="chapter5.xml.html#mixing.semantic.annotations">Annotation Framework</a><br class="html-compat" /> 5.1.1 <a href="chapter5.xml.html#mixing.annotation.elements">Annotation elements</a><br class="html-compat" /> 5.1.2 <a href="chapter5.xml.html#mixing.annotation.keys">Annotation keys</a><br class="html-compat" /> 5.1.3 <a href="chapter5.xml.html#mixing.alternate.representations">Alternate representations</a><br class="html-compat" /> 5.1.4 <a href="chapter5.xml.html#mixing.content.equiv">Content equivalents</a><br class="html-compat" /> 5.1.5 <a href="chapter5.xml.html#mixing.annotation.references">Annotation references</a><br class="html-compat" /> 5.2 <a href="chapter5.xml.html#mixing.semantic.elements">Elements for Semantic Annotations</a><br class="html-compat" /> 5.2.1 <a href="chapter5.xml.html#mixing.elements.semantics">The <code>semantics</code> element</a><br class="html-compat" /> 5.2.2 <a href="chapter5.xml.html#mixing.elements.annotation">The <code>annotation</code> element</a><br class="html-compat" /> 5.2.3 <a href="chapter5.xml.html#mixing.elements.annotation.xml">The <code>annotation-xml</code> element</a><br class="html-compat" /> 5.3 <a href="chapter5.xml.html#mixing.markup">Combining Presentation and Content Markup</a><br class="html-compat" /> 5.3.1 <a href="chapter5.xml.html#mixing.pmincm">Presentation Markup in Content Markup</a><br class="html-compat" /> 5.3.2 <a href="chapter5.xml.html#mixing.cminpm">Content Markup in Presentation Markup</a><br class="html-compat" /> 5.4 <a href="chapter5.xml.html#mixing.parallel">Parallel Markup</a><br class="html-compat" /> 5.4.1 <a href="chapter5.xml.html#mixing.top.level">Top-level Parallel Markup</a><br class="html-compat" /> 5.4.2 <a href="chapter5.xml.html#mixing.cr">Parallel Markup via Cross-References</a><br class="html-compat" />6 <a href="chapter6.xml.html">Interactions with the Host Environment</a><br class="html-compat" /> 6.1 <a href="chapter6.xml.html#world.introduction">Introduction</a><br class="html-compat" /> 6.2 <a href="chapter6.xml.html#world.invoking.processors">Invoking MathML Processors</a><br class="html-compat" /> 6.2.1 <a href="chapter6.xml.html#world.recognizing.mathml">Recognizing MathML in XML</a><br class="html-compat" /> 6.2.2 <a href="chapter6.xml.html#world.resource.types">Resource Types for MathML Documents</a><br class="html-compat" /> 6.2.3 <a href="chapter6.xml.html#encoding-names">Names of MathML Encodings</a><br class="html-compat" /> 6.3 <a href="chapter6.xml.html#world-int-transfers">Transferring MathML</a><br class="html-compat" /> 6.3.1 <a href="chapter6.xml.html#world-int-transf-flavors">Basic Transfer Flavor Names and Contents</a><br class="html-compat" /> 6.3.2 <a href="chapter6.xml.html#world-int-transf-recommend">Recommended Behaviors when Transferring</a><br class="html-compat" /> 6.3.3 <a href="chapter6.xml.html#world-int-transf-discuss">Discussion</a><br class="html-compat" /> 6.3.4 <a href="chapter6.xml.html#world-int-transf-exa">Examples</a><br class="html-compat" /> 6.4 <a href="chapter6.xml.html#world-int-combine-other">Combining MathML and Other Formats</a><br class="html-compat" /> 6.4.1 <a href="chapter6.xml.html#interf.xhtml">Mixing MathML and XHTML</a><br class="html-compat" /> 6.4.2 <a href="chapter6.xml.html#interf.html">Mixing MathML and HTML, and other non-XML contexts</a><br class="html-compat" /> 6.4.3 <a href="chapter6.xml.html#interf.link">Linking</a><br class="html-compat" /> 6.4.4 <a href="chapter6.xml.html#interf.graphics">MathML and Graphical Markup</a><br class="html-compat" /> 6.5 <a href="chapter6.xml.html#world-int-style">Using CSS with MathML</a><br class="html-compat" /> 6.5.1 <a href="chapter6.xml.html#id.6.5.1">Order of processing attributes versus style sheets</a><br class="html-compat" /> 6.5.2 <a href="chapter6.xml.html#id.6.5.2">Layout engines that lack native MathML support</a><br class="html-compat" />7 <a href="chapter7.xml.html">Characters, Entities and Fonts</a><br class="html-compat" /> 7.1 <a href="chapter7.xml.html#chars.intro">Introduction</a><br class="html-compat" /> 7.2 <a href="chapter7.xml.html#chars.unicodechars">Unicode Character Data</a><br class="html-compat" /> 7.3 <a href="chapter7.xml.html#chars.entities">Entity Declarations</a><br class="html-compat" /> 7.4 <a href="chapter7.xml.html#chars.mglyph">Special Characters Not in Unicode</a><br class="html-compat" /> 7.5 <a href="chapter7.xml.html#chars.BMP-SMP">Mathematical Alphanumeric Symbols</a><br class="html-compat" /> 7.6 <a href="chapter7.xml.html#chars.nonmark">Non-Marking Characters</a><br class="html-compat" /> 7.7 <a href="chapter7.xml.html#chars.anomalous">Anomalous Mathematical Characters</a><br class="html-compat" /> 7.7.1 <a href="chapter7.xml.html#chars.keyboard">Keyboard Characters</a><br class="html-compat" /> 7.7.2 <a href="chapter7.xml.html#chars.pseudo-scripts">Pseudo-scripts</a><br class="html-compat" /> 7.7.3 <a href="chapter7.xml.html#chars.comb-chars">Combining Characters</a><br class="html-compat" /></p>
<h3><a id="appendices"></a>Appendices
</h3>
<p class="toc">A <a href="appendixa.xml.html">Parsing MathML</a><br class="html-compat" /> A.1 <a href="appendixa.xml.html#parsing.wellformed">Use of MathML as Well-Formed
XML</a><br class="html-compat" /> A.2 <a href="appendixa.xml.html#parsing.usingrnc">Using the RelaxNG Schema for MathML3</a><br class="html-compat" /> A.2.1 <a href="appendixa.xml.html#parsing.rnc.full">Full MathML</a><br class="html-compat" /> A.2.2 <a href="appendixa.xml.html#parsing.rnc.common">Elements Common to Presentation and Content MathML</a><br class="html-compat" /> A.2.3 <a href="appendixa.xml.html#parsing.rnc.pres">The Grammar for Presentation MathML</a><br class="html-compat" /> A.2.4 <a href="appendixa.xml.html#parsing.rnc.strict">The Grammar for Strict Content MathML3</a><br class="html-compat" /> A.2.5 <a href="appendixa.xml.html#parsing.rnc.content">The Grammar for Content MathML</a><br class="html-compat" /> A.2.6 <a href="appendixa.xml.html#parsing.rnc.module">MathML as a module in a RelaxNG Schema</a><br class="html-compat" /> A.3 <a href="appendixa.xml.html#parsing.usingdtd">Using the MathML DTD</a><br class="html-compat" /> A.3.1 <a href="appendixa.xml.html#interf.validation">Document Validation Issues</a><br class="html-compat" /> A.3.2 <a href="appendixa.xml.html#id.A.3.2">Attribute values in the MathML DTD</a><br class="html-compat" /> A.3.3 <a href="appendixa.xml.html#parsing.doctype">DOCTYPE declaration for MathML</a><br class="html-compat" /> A.4 <a href="appendixa.xml.html#parsing.usingxsd">Using the MathML XML Schema</a><br class="html-compat" /> A.4.1 <a href="appendixa.xml.html#parsing.schemalocation">Associating the MathML schema with MathML fragments</a><br class="html-compat" />B <a href="appendixb.xml.html">Media Types Registrations</a><br class="html-compat" /> B.1 <a href="appendixb.xml.html#media-type-choosing">Selection of Media Types for
MathML Instances</a><br class="html-compat" /> B.2 <a href="appendixb.xml.html#media-types-mathml">Media type for Generic MathML</a><br class="html-compat" /> B.3 <a href="appendixb.xml.html#media-types-mathml-p">Media type for Presentation MathML</a><br class="html-compat" /> B.4 <a href="appendixb.xml.html#media-types-mathml-c">Media type for Content MathML</a><br class="html-compat" />C <a href="appendixc.xml.html">Operator Dictionary (Non-Normative)</a><br class="html-compat" /> C.1 <a href="appendixc.xml.html#oper-dict.index">Indexing of the operator dictionary</a><br class="html-compat" /> C.2 <a href="appendixc.xml.html#oper-dict.format">Format of operator dictionary entries</a><br class="html-compat" /> C.3 <a href="appendixc.xml.html#oper-dict.space">Notes on lspace and
rspace attributes</a><br class="html-compat" /> C.4 <a href="appendixc.xml.html#oper-dict.entries-table">Operator dictionary entries</a><br class="html-compat" />D <a href="appendixd.xml.html">Glossary (Non-Normative)</a><br class="html-compat" />E <a href="appendixe.xml.html">Working Group Membership and Acknowledgments (Non-Normative)</a><br class="html-compat" /> E.1 <a href="appendixe.xml.html#contrib">The Math Working Group Membership</a><br class="html-compat" /> E.2 <a href="appendixe.xml.html#contrib.ack">Acknowledgments</a><br class="html-compat" />F <a href="appendixf.xml.html">Changes (Non-Normative)</a><br class="html-compat" /> F.1 <a href="appendixf.xml.html#changes.mathml2.0e-3.0">Changes between MathML 2.0 Second Edition and MathML 3.0</a><br class="html-compat" />G <a href="appendixg.xml.html">Normative References</a><br class="html-compat" />H <a href="appendixh.xml.html">References (Non-Normative)</a><br class="html-compat" />I <a href="appendixi.xml.html">Index (Non-Normative)</a><br class="html-compat" /> I.1 <a href="appendixi.xml.html#index.elem">MathML Elements</a><br class="html-compat" /> I.2 <a href="appendixi.xml.html#index.att">MathML Attributes</a><br class="html-compat" /></p>
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