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<!DOCTYPE html
  PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
<html>
   <head>
      <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
      <title>Characters, Entities and Fonts</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;
             }
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              padding: 0.5em;
              margin: 0.5em 0 0.5em 0;
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  div.strict-mathml-example h5 {
  margin-top: -0.3em;
  margin-bottom: -0.5em;}
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  var.transmeta {background-color:red}
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padding-left: 1em;
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</style><link rel="stylesheet" type="text/css" href="../../../StyleSheets/TR/W3C-REC.css">
   </head>
   <body>
      
      <h1><a name="chars" id="chars"></a>7 Characters, Entities and Fonts
      </h1>
      <!-- TOP NAVIGATION BAR -->
      <div class="minitoc">
         
         Overview: <a href="Overview-d.html">Mathematical Markup Language (MathML) Version 3.0</a><br>
         Previous: 6 <a href="chapter6-d.html">Interactions with the Host Environment</a><br>
         Next: A <a href="appendixa-d.html">Parsing MathML</a><br><br>7 <a href="chapter7-d.html">Characters, Entities and Fonts</a><br>&nbsp;&nbsp;&nbsp;&nbsp;7.1 <a href="chapter7-d.html#chars.intro">Introduction</a><br>&nbsp;&nbsp;&nbsp;&nbsp;7.2 <a href="chapter7-d.html#chars.unicodechars">Unicode Character Data</a><br>&nbsp;&nbsp;&nbsp;&nbsp;7.3 <a href="chapter7-d.html#chars.entities">Entity Declarations</a><br>&nbsp;&nbsp;&nbsp;&nbsp;7.4 <a href="chapter7-d.html#chars.mglyph">Special Characters Not in Unicode</a><br>&nbsp;&nbsp;&nbsp;&nbsp;7.5 <a href="chapter7-d.html#chars.BMP-SMP">Mathematical Alphanumeric Symbols</a><br>&nbsp;&nbsp;&nbsp;&nbsp;7.6 <a href="chapter7-d.html#chars.nonmark">Non-Marking Characters</a><br>&nbsp;&nbsp;&nbsp;&nbsp;7.7 <a href="chapter7-d.html#chars.anomalous">Anomalous Mathematical Characters</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;7.7.1 <a href="chapter7-d.html#chars.keyboard">Keyboard Characters</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;7.7.1.1 <a href="chapter7-d.html#chars.hyph-minus">Minus</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;7.7.1.2 <a href="chapter7-d.html#chars.prime-quote">Apostrophes, Quotes and Primes</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;7.7.1.3 <a href="chapter7-d.html#id.7.7.1.3">Other Keyboard Substitutions</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;7.7.2 <a href="chapter7-d.html#chars.pseudo-scripts">Pseudo-scripts</a><br>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;7.7.3 <a href="chapter7-d.html#chars.comb-chars">Combining Characters</a><br></div>
      <div class="div1">
         <div class="div2">
            
            <h2><a name="chars.intro" id="chars.intro"></a>7.1 Introduction
            </h2>
            <p>
               Notation and symbols have proved very important for
               mathematics. Mathematics has grown in part because 
               its notation continually changes toward being succinct and 
               suggestive. 
               Many new signs have been developed for use in mathematical notation, 
               and many have been adopted 
               that were  originally introduced elsewhere.The result is
               that mathematics makes use of a very large collection of
               symbols.  It is difficult to write mathematics fluently if these
               characters are not available for use. It is difficult
               to read mathematics if corresponding glyphs are not available
               for presentation on specific display devices.
               
            </p>
            <p>The W3C Math Working Group therefore took on the job
               of specifying part of the mechanism needed to proceed from
               notation to final presentation, and has collaborated with the
               <a href="http://www.unicode.org/">Unicode Technical Committee</a>
               (UTC) and the <a href="http://www.stixfonts.org/">STIX Fonts Project</a>
               in undertaking specification of the rest.
            </p>
            <p>This chapter contains discussion of
               characters for use within MathML, recommendations for their use,
               and warnings concerning the correct form of the
               corresponding code points given in the Universal Multiple-Octet Coded Character
               Set (UCS) <a href="appendixg-d.html#ISO10646">[ISO10646]</a> as codified in Unicode <a href="appendixg-d.html#Unicode">[Unicode]</a>.
               For simplicity we refer to this character set by the short name Unicode.
               Unless otherwise stated, the mappings discussed in this chapter and
               elsewhere in the MathML 3.0 recommendation are based on Unicode 5.2.
               Conformant MathML processors (see <a href="chapter2-d.html#interf.genproc">Section&nbsp;2.3 Conformance</a>) are
               free to use characters defined in Unicode 5.2 or later.
               
            </p>
            <p>While a long process of review and adoption by UTC and ISO/IEC
               of the characters of special interest to mathematics and MathML is
               now complete, more characters may be added in the future. For the latest 
               character tables and font information,
               see the <a href="appendixg-d.html#Entities">[Entities]</a> and the <a href="http://www.unicode.org/">Unicode
                  Home Page</a>, notably 
               <a href="http://www.unicode.org/unicode/alloc/Pipeline.html">Unicode Work
                  in Progress</a> and 
               <a href="http://www.unicode.org/reports/tr25/tr25-8.html">
                  Unicode Technical Report #25 “Unicode Support for Mathematics”</a>.
            </p>
            <p>
               A MathML token element (see <a href="chapter3-d.html#presm.tokel">Section&nbsp;3.2 Token Elements</a>, <a href="chapter4-d.html#contm.cn">Section&nbsp;4.2.1 Numbers <code>&lt;cn&gt;</code></a>,
               <a href="chapter4-d.html#contm.ci">Section&nbsp;4.2.2 Content Identifiers <code>&lt;ci&gt;</code></a>, <a href="chapter4-d.html#contm.csymbol">Section&nbsp;4.2.3 Content Symbols <code>&lt;csymbol&gt;</code></a>) takes as content a
               sequence of MathML characters or <code>mglyph</code> elements.  The latter are
               used to represent characters that do not have a Unicode encoding, as described in <a href="chapter3-d.html#presm.mglyph">Section&nbsp;3.2.1.2  <code>&lt;mglyph/&gt;</code></a>.  The need for <code>mglyph</code> should be rare because Unicode 3.1 
               provided approximately one thousand alphabetic characters for mathematics, and 
               Unicode 3.2 added over 900 more special mathematical symbols.  
               
            </p>
         </div>
         <div class="div2">
            
            <h2><a name="chars.unicodechars" id="chars.unicodechars"></a>7.2 Unicode Character Data
            </h2>
            <p>
               Any character allowed by XML may be used in MathML.  More precisely,
               the legal Unicode characters have the hexadecimal code
               numbers 09 (tab = U+0009), 0A (line feed = U+000A), 0D (carriage
               return = U+000D), 20-D7FF (U+0020..U+D7FF), E000-FFFD
               (U+E000..U+FFFD), and 10000-10FFFF (U+10000..U+10FFFF).  
               The exclusions above code number D7FF are of the blocks used
               in surrogate pairs, and the two characters guaranteed not to be
               Unicode characters at all.  U+FFFE is excluded to allow determination
               of byte order in certain encodings.
               
            </p>
            <p>
               There are essentially three different ways of encoding character data
               in an XML document.
               
               
            </p>
            <ul>
               <li>
                  <p>Using characters directly: For example, the 'é' (character U+00E9 [LATIN SMALL LETTER E WITH ACUTE]) may have been inserted.
                     
                     This option is only useful
                     if the character encoding specified for the XML document includes
                     the character intended.  Note that if the document is, for example, 
                     encoded in
                     Latin-1 (ISO-8859-1) then <em>only</em> the characters in that
                     encoding are available directly; for instance character 
                     U+00E9 (eacute) is, but character 
                     U+03B1 (alpha) is not.   
                     
                  </p>
               </li>
               <li>
                  <p>
                     Using numeric XML character references: For example, 'é' may be
                     represented as &amp;#233; (decimal) or  &amp;#xE9; (hex),
                     or &amp;#0233; (decimal) or  &amp;#x00E9;.
                     Note that the numbers in the character references always refer to the Unicode encoding (and not to
                     the character encoding used in the XML file). By using character
                     references it is always possible to access the entire Unicode range.
                     
                  </p>
               </li>
               <li>
                  <p>
                     Using entity references: The MathML DTD defines internal entities that
                     expand to character data. Thus for example the entity reference
                     &amp;eacute; may be used rather than the character reference
                     &amp;#xE9;. An XML fragment that uses an entity
                     reference which is not defined in a DTD is not well-formed; therefore
                     it will be rejected by an XML parser. For this reason
                     every fragment using entity references must
                     use a DOCTYPE declaration which specifies the MathML DTD, or a DTD
                     that at least declares any entity reference used in the MathML
                     instance.  The need to use a DOCTYPE complicates inclusion of MathML in
                     some documents. However, entity references can be useful for small
                     illustrative examples.
                     
                  </p>
               </li>
            </ul>
         </div>
         <div class="div2">
            
            <h2><a name="chars.entities" id="chars.entities"></a>7.3 Entity Declarations
            </h2>
            <p>Earlier versions of this MathML specification included detailed
               listings of the entity definitions to be used with the MathML DTD.
               These entity definitions are of more general use, and have now been
               separated into an ancillary document, XML Entity Definitions for
               Characters <a href="appendixg-d.html#Entities">[Entities]</a>.  The
               tables there list the entity names and the corresponding 
               Unicode character references.
               That document describes several entity sets; not all of them are used in
               the MathML DTD. The standard MathML DTD references
               the following entity sets:
               
            </p>
            <ul>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isobox.html">isobox</a> Box and Line Drawing
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isocyr1.html">isocyr1</a> Russian Cyrillic
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isocyr2.html">isocyr2</a> Non-Russian Cyrillic
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isodia.html">isodia</a> Diacritical Marks
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isolat1.html">isolat1</a> Added Latin 1
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isolat2.html">isolat2</a> Added Latin 2
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isonum.html">isonum</a> Numeric and Special Graphic
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isopub.html">isopub</a> Publishing
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isoamsa.html">isoamsa</a> Added Math Symbols: Arrow Relations
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isoamsb.html">isoamsb</a> Added Math Symbols: Binary Operators
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isoamsc.html">isoamsc</a> Added Math Symbols: Delimiters
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isoamsn.html">isoamsn</a> Added Math Symbols: Negated Relations
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isoamso.html">isoamso</a> Added Math Symbols: Ordinary
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isoamsr.html">isoamsr</a> Added Math Symbols: Relations
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isogrk3.html">isogrk3</a> Greek Symbols
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isomfrk.html">isomfrk</a> Math Alphabets: Fraktur
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isomopf.html">isomopf</a> Math Alphabets: Open Face
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isomscr.html">isomscr</a> Math Alphabets: Script
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/isotech.html">isotech</a> General Technical
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/mmlextra.html">mmlextra</a> Additional MathML Symbols
                  </p>
               </li>
               <li>
                  <p><a href="http://www.w3.org/TR/xml-entity-names/mmlalias.html">mmlalias</a> MathML Aliases
                  </p>
               </li>
            </ul>
         </div>
         <div class="div2">
            
            <h2><a name="chars.mglyph" id="chars.mglyph"></a>7.4 Special Characters Not in Unicode
            </h2>
            <p>
               For special purposes, one may need a symbol which does not have a
               Unicode representation. 
               In these cases one may use the <a href="chapter3-d.html#presm.mglyph"><code>mglyph</code></a>
               element for direct access to a glyph as an image, or (in some systems)
               from a font that uses a non-Unicode encoding. 
               All MathML token elements accept characters in their content and 
               also accept an <code>mglyph</code> there.
               Beware, however, that use of <code>mglyph</code> to access a font is
               deprecated and the mechanism may not work in all systems. The
               <code>mglyph</code> element should always supply a useful alternative
               representation in its <code>alt</code> attribute.
               
            </p>
         </div>
         <div class="div2">
            
            <h2><a name="chars.BMP-SMP" id="chars.BMP-SMP"></a>7.5 Mathematical Alphanumeric Symbols
            </h2>
            <p>
               In mathematical and scientific writing, single letters often denote 
               variables and constants in a given context.  The increasing complexity 
               of science has led to the use of
               certain common alphabet and font variations to provide enough special
               symbols of this letter-like type.  These denotations are generally 
               <em>not</em> letters that may be used to make up words with
               recognized meanings, but individual carriers of semantics themselves.
               Writing a string of such symbols is usually interpreted in terms of
               some composition law, for instance, multiplication.  Many letter-like
               symbols may be quickly interpreted as
               of a certain mathematical type by specialists in a given area: 
               for instance, bold symbols, whether
               based on Latin or Greek letters, as vectors in physics or engineering,
               or Fraktur symbols as Lie algebras in part of pure mathematics.
               
            </p>
            <p>
               The additional Mathematical Alphanumeric Symbols 
               provided in Unicode 3.1 have
               code points in the range U+1D400 to U+1D7FF in <em>Plane 1</em>, that is, in the first
               plane with Unicode values higher than 2<sup>16</sup>.  This plane of
               characters is also known as the Secondary Multilingual Plane (SMP),
               in contrast to the Basic Multilingual Plane (BMP) which 
               was originally the entire extent of Unicode.  Support for Plane 1 characters in currently
               deployed software is not always reliable, but it should be possible in
               multilingual operating systems, since Plane 2 has many Chinese
               characters that must be displayable in East Asian locales.
            </p>
            <p>
               As discussed in <a href="chapter3-d.html#presm.commatt">Section&nbsp;3.2.2 Mathematics style attributes common to token elements</a>, MathML offers an
               alternative mechanism to specify mathematical alphanumeric characters.
               This alternative mechanism spans the gap between the
               specification of the mathematical alphanumeric symbols as Unicode
               code points, and the deployment of software and fonts that support them.
               Namely, one uses the <code>mathvariant</code> attribute on a token element
               such as <code>mi</code> to indicate that the character data in the token
               element selects a mathematical alphanumeric symbol.
               
            </p>
            <p>
               In principle, any mathvariant value may be used with any character
               data to define a specific symbolic token.  In practice, only certain
               combinations of character data and mathvariant values will be visually
               distinguished by a given renderer.
               In this section we explain the correspondence between certain characters
               in Plane 0 that, when modified by the <code>mathvariant</code> attribute,
               are considered equivalent to mathematical alphanumeric symbol characters.
               
            </p>
            <p>
               The mathematical alphanumeric symbol characters in Plane 1 include
               alphabets for Latin upper-case and lower-case letters, including
               dotless i and j, Greek upper-case and lower-case letters, Greek symbols
               (also known as variants), including upper-case and lower-case digamma,
               and Latin digits.  These alphabets provide Plane 1 Unicode code points
               that differ from corresponding Plane 0 characters only by a variation
               in font that carries mathematical semantics when used in a formula.
               
            </p>
            <p>
               The <code>mathvariant</code> attribute uses exactly this correspondence to
               provide an alternate markup encoding that selects these Plane 1 characters.
               For example, the Mathematical Italic alphabet runs from U+1D434 ("A")
               to U+1D467 ("z").  Thus, a typical example of an identifier for a
               variable, marked up as
               
            </p><pre class="mathml">
&lt;mi&gt;a&lt;/mi&gt;
</pre><p>
               and rendered in a mathematical italic font (as described in
               <a href="chapter3-d.html#presm.mi">Section&nbsp;3.2.3 Identifier <code>&lt;mi&gt;</code></a>) could equivalently be marked up as
               
            </p><pre class="mathml">
&lt;mi&gt;&amp;#x1D44E;&lt;!--MATHEMATICAL ITALIC SMALL A--&gt;&lt;/mi&gt;
</pre><p>
               which invokes the Mathematical Italic lower-case a explicitly.
               
            </p>
            <p>
               An important use of the mathematical alphanumeric symbols in Plane 1
               is for identifiers normally printed in special mathematical fonts,
               such as Fraktur, Greek, Boldface, or Script.  As another example,
               the Mathematical Fraktur alphabet runs from U+1D504 ("A") to U+1D537 ("z").
               Thus, an identifier for a variable that uses Fraktur characters could
               be marked up as
               
            </p><pre class="mathml">
&lt;mi&gt;&amp;#x1D504;&lt;!--BLACK-LETTER CAPITAL A--&gt;&lt;/mi&gt;
</pre><p>
               An alternative, equivalent markup for this example is to use the
               common upper-case A, modified by using the <code>mathvariant</code> attribute:
               
            </p><pre class="mathml">
&lt;mi mathvariant="fraktur"&gt;A&lt;/mi&gt;
</pre><p>
               A MathML processor must treat a mathematical alphanumeric character
               (when it appears) as identical to the corresponding combination of
               the unstyled character and mathvariant attribute value.
               It is important to note that the mathvariant attribute specifies a
               semantic class of characters, each of which has a specific appearance
               that should be protected from document-wide style changes, so the
               intended meaning of the character may be preserved.  The use of a
               mathematical alphanumeric character is also intended to preserve
               this specific appearance, and so these characters are also not to
               be affected by surrounding style changes.
               
            </p>
            <p>
               Not all combinations of character data and mathvariant values have assigned
               Unicode code points.  For example, sans-serif Greek alphabets are
               omitted, while bold sans-serif Greek alphabets are included, and
               bold digits are included, while bold-italic digits are excluded.
               A renderer should visually distinguish those combinations of character data
               and mathvariant attribute values that it can subject to the
               availability of font characters.
               It is intended that renderers distinghish at least those combinations
               that have equivalent Unicode code points, and renderers are free to
               ignore those combinations that have no assigned Unicode code point
               or for which adequate font support is unavailable.
               
            </p>
            <p>
               The exact correspondence between a mathematical alphabetic character
               and an unstyled character is complicated by the fact that certain
               characters that were already present in Unicode in Plane 0 are not in
               the 'expected' sequence in Plane 1.  The table below shows the Plane 0
               mathematical alphanumeric symbols, listing for each character its
               Unicode code point, its Unicode character name, its corresponding
               unstyled alphabetic character, and the code point in Plane 1 where
               one might naturally have sought this character.
               
            </p>
            <table id="chars.letter-like-tables" border="1">
               <thead>
                  <tr>
                     <th>Unicode code point</th>
                     <th>Unicode name</th>
                     <th>BMP code</th>
                     <th>Plane-1 code</th>
                  </tr>
               </thead>
               <tbody>
                  <tr>
                     <td>U+210E</td>
                     <td>PLANCK CONSTANT</td>
                     <td>U+0068</td>
                     <td>U+1D455</td>
                  </tr>
                  <tr>
                     <td>U+212C</td>
                     <td>SCRIPT CAPITAL B</td>
                     <td>U+0042</td>
                     <td>U+1D49D</td>
                  </tr>
                  <tr>
                     <td>U+2130</td>
                     <td>SCRIPT CAPITAL E</td>
                     <td>U+0045</td>
                     <td>U+1D4A0</td>
                  </tr>
                  <tr>
                     <td>U+2131</td>
                     <td>SCRIPT CAPITAL F</td>
                     <td>U+0046</td>
                     <td>U+1D4A1</td>
                  </tr>
                  <tr>
                     <td>U+210B</td>
                     <td>SCRIPT CAPITAL H</td>
                     <td>U+0048</td>
                     <td>U+1D4A3</td>
                  </tr>
                  <tr>
                     <td>U+2110</td>
                     <td>SCRIPT CAPITAL I</td>
                     <td>U+0049</td>
                     <td>U+1D4A4</td>
                  </tr>
                  <tr>
                     <td>U+2112</td>
                     <td>SCRIPT CAPITAL L</td>
                     <td>U+004C</td>
                     <td>U+1D4A7</td>
                  </tr>
                  <tr>
                     <td>U+2133</td>
                     <td>SCRIPT CAPITAL M</td>
                     <td>U+004D</td>
                     <td>U+1D4A8</td>
                  </tr>
                  <tr>
                     <td>U+211B</td>
                     <td>SCRIPT CAPITAL R</td>
                     <td>U+0052</td>
                     <td>U+1D4AD</td>
                  </tr>
                  <tr>
                     <td>U+212F</td>
                     <td>SCRIPT SMALL E</td>
                     <td>U+0065</td>
                     <td>U+1D4BA</td>
                  </tr>
                  <tr>
                     <td>U+210A</td>
                     <td>SCRIPT SMALL G</td>
                     <td>U+0067</td>
                     <td>U+1D4BC</td>
                  </tr>
                  <tr>
                     <td>U+2134</td>
                     <td>SCRIPT SMALL O</td>
                     <td>U+006F</td>
                     <td>U+1D4C4</td>
                  </tr>
                  <tr>
                     <td>U+212D</td>
                     <td>BLACK-LETTER CAPITAL C</td>
                     <td>U+0043</td>
                     <td>U+1D506</td>
                  </tr>
                  <tr>
                     <td>U+210C</td>
                     <td>BLACK-LETTER CAPITAL H</td>
                     <td>U+0048</td>
                     <td>U+1D50B</td>
                  </tr>
                  <tr>
                     <td>U+2111</td>
                     <td>BLACK-LETTER CAPITAL I</td>
                     <td>U+0049</td>
                     <td>U+1D50C</td>
                  </tr>
                  <tr>
                     <td>U+211C</td>
                     <td>BLACK-LETTER CAPITAL R</td>
                     <td>U+0052</td>
                     <td>U+1D515</td>
                  </tr>
                  <tr>
                     <td>U+2128</td>
                     <td>BLACK-LETTER CAPITAL Z</td>
                     <td>U+005A</td>
                     <td>U+1D51D</td>
                  </tr>
                  <tr>
                     <td>U+2102</td>
                     <td>DOUBLE-STRUCK CAPITAL C</td>
                     <td>U+0043</td>
                     <td>U+1D53A</td>
                  </tr>
                  <tr>
                     <td>U+210D</td>
                     <td>DOUBLE-STRUCK CAPITAL H</td>
                     <td>U+0048</td>
                     <td>U+1D53F</td>
                  </tr>
                  <tr>
                     <td>U+2115</td>
                     <td>DOUBLE-STRUCK CAPITAL N</td>
                     <td>U+004E</td>
                     <td>U+1D545</td>
                  </tr>
                  <tr>
                     <td>U+2119</td>
                     <td>DOUBLE-STRUCK CAPITAL P</td>
                     <td>U+0050</td>
                     <td>U+1D547</td>
                  </tr>
                  <tr>
                     <td>U+211A</td>
                     <td>DOUBLE-STRUCK CAPITAL Q</td>
                     <td>U+0051</td>
                     <td>U+1D548</td>
                  </tr>
                  <tr>
                     <td>U+211D</td>
                     <td>DOUBLE-STRUCK CAPITAL R</td>
                     <td>U+0052</td>
                     <td>U+1D549</td>
                  </tr>
                  <tr>
                     <td>U+2124</td>
                     <td>DOUBLE-STRUCK CAPITAL Z</td>
                     <td>U+005A</td>
                     <td>U+1D551</td>
                  </tr>
               </tbody>
            </table>
            <p>
               Mathematical Alphanumeric Symbol characters should not be used for styled prose.
               For example, Mathematical Fraktur A must not be used to just select
               a blackletter font for an uppercase A as it
               would create problems for searching, restyling (e.g. for accessibility), 
               and many other kinds of processing.
               
            </p>
         </div>
         <div class="div2">
            
            <h2><a name="chars.nonmark" id="chars.nonmark"></a>7.6 Non-Marking Characters
            </h2>
            <p>
               Some characters, although important for the quality of print or
               alternative rendering, do not have glyph marks that correspond
               directly to them.  They are called here non-marking characters.
               Their roles are
               discussed in <a href="chapter3-d.html">Chapter&nbsp;3 Presentation Markup</a> and <a href="chapter4-d.html">Chapter&nbsp;4 Content Markup</a>.
            </p>
            <p>In MathML, control of page composition, such as line-breaking, is
               effected by the use of the proper attributes on the <code>mo</code> and <code>mspace</code> elements.
            </p>
            <p>The characters below are not simple spacers.  They are
               especially important new additions to the UCS because they provide
               textual clues which can increase the quality of print rendering,
               permit correct audio rendering, and allow the unique recovery of
               mathematical semantics from text which is visually ambiguous.
               
            </p>
            <table id="chars.table-white">
               <thead>
                  <tr>
                     <th>Unicode code point</th>
                     <th>Unicode name</th>
                     <th>Description</th>
                  </tr>
               </thead>
               <tbody>
                  <tr>
                     <td>U+2061</td>
                     <td>FUNCTION APPLICATION</td>
                     <td>character showing function application in presentation tagging
                        (<a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                           <code>&lt;mo&gt;</code></a>)
                     </td>
                  </tr>
                  <tr>
                     <td>U+2062</td>
                     <td>INVISIBLE TIMES</td>
                     <td>marks multiplication when it is understood without a mark
                        (<a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                           <code>&lt;mo&gt;</code></a>)
                     </td>
                  </tr>
                  <tr>
                     <td>U+2063</td>
                     <td>INVISIBLE SEPARATOR</td>
                     <td>used as a separator, e.g., in indices (<a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                           <code>&lt;mo&gt;</code></a>)
                     </td>
                  </tr>
                  <tr>
                     <td>U+2064</td>
                     <td>INVISIBLE PLUS</td>
                     <td>marks addition, especially in constructs such as 1½
                        (<a href="chapter3-d.html#presm.mo">Section&nbsp;3.2.5 Operator, Fence, Separator or Accent
                           <code>&lt;mo&gt;</code></a>)
                     </td>
                  </tr>
               </tbody>
            </table>
         </div>
         <div class="div2">
            
            <h2><a name="chars.anomalous" id="chars.anomalous"></a>7.7 Anomalous Mathematical Characters
            </h2>
            <p>Some characters which occur fairly often in mathematical texts, and
               have special significance there, are frequently confused with other
               similar characters in the UCS.  In some cases, common keyboard
               characters have become entrenched as alternatives to the more appropriate
               mathematical characters.  In others, characters have legitimate uses in
               both formulas and text, but conflicting rendering and font
               conventions. All these characters are called here anomalous
               characters.
            </p>
            <div class="div3">
               
               <h3><a name="chars.keyboard" id="chars.keyboard"></a>7.7.1 Keyboard Characters
               </h3>
               <p>Typical Latin-1-based keyboards contain several characters that
                  are visually similar to important mathematical characters.
                  Consequently, these characters are frequently substituted,
                  intentionally or unintentionally, for their more correct mathematical
                  counterparts.
               </p>
               <div class="div4">
                  
                  <h4><a name="chars.hyph-minus" id="chars.hyph-minus"></a>7.7.1.1 Minus
                  </h4>
                  <p>The most common ordinary text character which enjoys a special
                     mathematical use is U+002D [HYPHEN-MINUS].  As its Unicode name
                     suggests, it is used as a hyphen in prose contexts, and as a minus
                     or negative sign in formulas.
                     For text use, there is a specific code point U+2010 [HYPHEN] which is
                     intended for prose contexts, and which should render as a hyphen or
                     short dash.
                     For mathematical use, there is another code point U+2212 [MINUS SIGN]
                     which is intended for mathematical formulas, and which should render
                     as a longer minus or negative sign.
                     MathML renderers should treat U+002D [HYPHEN-MINUS] as equivalent to
                     U+2212 [MINUS SIGN] in formula contexts such as <code>mo</code>, and as
                     equivalent to U+2010 [HYPHEN] in text contexts such as <code>mtext</code>.
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="chars.prime-quote" id="chars.prime-quote"></a>7.7.1.2 Apostrophes, Quotes and Primes
                  </h4>
                  <p>On a typical European keyboard there is a key available which is
                     viewed as an apostrophe or a single quotation mark (an upright or
                     right quotation mark).  Thus one key is doing double duty for prose
                     input to enter U+0027 [APOSTROPHE] and U+2019 [RIGHT SINGLE QUOTATION
                     MARK].  In mathematical contexts it is also commonly used for the
                     prime, which should be U+2032 [PRIME].  Unicode recognizes the
                     overloading of this symbol and remarks that it can also signify the
                     units of minutes or feet.  In the unstructured printed text of normal
                     prose the characters are placed next to one another.  The U+0027
                     [APOSTROPHE] and U+2019 [RIGHT SINGLE QUOTATION MARK] are marked with
                     glyphs that are small and raised with respect to the center line of
                     the text.  The fonts used provide small raised glyphs in the
                     appropriate places indexed by the Unicode codes.  The U+2032 [PRIME]
                     of mathematics is similarly treated in fuller Unicode fonts.
                     
                  </p>
                  <p>MathML renderers are encouraged to treat U+0027
                     [APOSTROPHE] as U+2032 [PRIME] when appropriate in
                     formula contexts.
                  </p>
                  <p>A final remark is that a ‘prime’ is often used in
                     transliteration of the Cyrillic character U+044C [CYRILLIC SMALL
                     LETTER SOFT SIGN].  This different use of primes is not part of
                     considerations for mathematical formulas. 
                     
                  </p>
               </div>
               <div class="div4">
                  
                  <h4><a name="id.7.7.1.3" id="id.7.7.1.3"></a>7.7.1.3 Other Keyboard Substitutions
                  </h4>
                  <p>While the minus and prime characters are the most common and
                     important keyboard characters with more precise mathematical
                     counterparts, there are a number of other keyboard character
                     substitutions that are sometime used.  For example some may expect
                     
                  </p><pre class="mathml">&lt;mo&gt;''&lt;/mo&gt;</pre><p> to be treated as 
                     U+2033 [DOUBLE PRIME], and analogous substitutions could perhaps be made for U+2034 [TRIPLE PRIME] and U+2057 [QUADRUPLE PRIME].
                     Similarly, sometimes U+007C [VERTICAL 
                     LINE] is used for U+2223 [DIVIDES].  MathML regards these as
                     application-specific authoring conventions, and recommends that
                     authoring tools generate markup using the more precise mathematical
                     characters for better interoperability. 
                     
                  </p>
               </div>
            </div>
            <div class="div3">
               
               <h3><a name="chars.pseudo-scripts" id="chars.pseudo-scripts"></a>7.7.2 Pseudo-scripts
               </h3>
               <p>There are a number of characters in the UCS that traditionally have
                  been taken to have a natural ‘script’ aspect.  The
                  visual presentation of these characters is similar to a script, that
                  is, raised from the baseline, and smaller than the base font size.  The
                  degree symbol and prime characters are examples.  For use in text, such
                  characters occur in sequence with the identifier they follow, and are
                  typically rendered using the same font.  These characters are called
                  pseudo-scripts here.
               </p>
               <p>In almost all mathematical contexts, pseudo-script characters should
                  be associated
                  with a base expression using explicit script markup in MathML.  For
                  example, the preferred encoding of "x prime" is
                  
                  
               </p><pre class="mathml">&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2032;&lt;!--PRIME--&gt;&lt;/mo&gt;&lt;/msup&gt;</pre><p>
                  
                  and not
                  
                  
               </p><pre class="mathml">&lt;mi&gt;x'&lt;/mi&gt;</pre><p>
                  
                  or any other variants not using an explicit script construct. Note, however, that within
                  text contexts such as <code>mtext</code>, pseudo-scripts may be used in sequence with other character data.
               </p>
               <p>There are two reasons why explicit markup is preferable in
                  mathematical contexts.  First, a problem arises with typesetting, when
                  pseudo-scripts are used with subscripted identifiers.  Traditionally,
                  subscripting of x' would be rendered stacked under the prime. This is
                  easily accomplished with script markup, for example:
                  
                  
               </p><pre class="mathml">&lt;mrow&gt;&lt;msubsup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;mo&gt;&amp;#x2032;&lt;!--PRIME--&gt;&lt;/mo&gt;&lt;/msubsup&gt;&lt;/mrow&gt;</pre><p>
                  
                  By contrast,
                  
                  
               </p><pre class="mathml">&lt;mrow&gt;&lt;msub&gt;&lt;mi&gt;x'&lt;/mi&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/msub&gt;&lt;/mrow&gt;</pre><p>
                  will render with staggered scripts.
               </p>
               <p>Note this means that a renderer of MathML will have to treat
                  pseudo-scripts differently from most other character codes it finds in
                  a superscript position; in most fonts, the glyphs for pseudo-scripts
                  are already shrunk and raised from the baseline.
                  
               </p>
               <p>The second reason that explicit script markup is preferrable to
                  juxtaposition of characters is that it generally better reflects the
                  intended mathematical structure. For example, 
                  
               </p><pre class="mathml">&lt;msup&gt;
  &lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;mrow&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;g&lt;/mi&gt;&lt;/mrow&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;
  &lt;mo&gt;&amp;#x2032;&lt;!--PRIME--&gt;&lt;/mo&gt;
&lt;/msup&gt;</pre><p>
                  accurately reflects that the prime here is operating on an entire
                  expression, and does not suggest that the prime is acting on the final right parenthesis.
                  
               </p>
               <p>However, the data model for all MathML token elements is Unicode text, 
                  so one cannot rule out the possibility of valid MathML markup 
                  containing constructions such as 
                  
               </p><pre class="mathml">&lt;mrow&gt;&lt;mi&gt;x'&lt;/mi&gt;&lt;/mrow&gt;</pre><p>
                  and
                  
               </p><pre class="mathml">&lt;mrow&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;'&lt;/mo&gt;&lt;/mrow&gt;</pre><p>  
                  
                  While the first form may, in some rare situations, legitmately be used
                  to distinguish a multi-character identifer named x' from the
                  derivative of a function x, such forms should generally be avoided.
                  Authoring and validation tools are encouraged to generate the
                  recommended script markup:
                  
               </p><pre class="mathml">&lt;mrow&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;&amp;#x2032;&lt;!--PRIME--&gt;&lt;/mo&gt;&lt;/msup&gt;&lt;/mrow&gt;</pre><p>The U+2032 [PRIME] character is perhaps the most common
                  pseudo-script, but there are many others, as listed below:
                  
                  
               </p>
               <table>
                  <thead>
                     <tr>
                        <th colspan="2">Pseudo-script Characters</th>
                     </tr>
                  </thead>
                  <tbody>
                     <tr>
                        <td>U+0022</td>
                        <td>QUOTATION MARK</td>
                     </tr>
                     <tr>
                        <td>U+0027</td>
                        <td>APOSTROPHE</td>
                     </tr>
                     <tr>
                        <td>U+002A</td>
                        <td>ASTERISK</td>
                     </tr>
                     <tr>
                        <td>U+0060</td>
                        <td>GRAVE ACCENT</td>
                     </tr>
                     <tr>
                        <td>U+00AA</td>
                        <td>FEMININE ORDINAL INDICATOR</td>
                     </tr>
                     <tr>
                        <td>U+00B0</td>
                        <td>DEGREE SIGN</td>
                     </tr>
                     <tr>
                        <td>U+00B2</td>
                        <td>SUPERSCRIPT TWO</td>
                     </tr>
                     <tr>
                        <td>U+00B3</td>
                        <td>SUPERSCRIPT THREE</td>
                     </tr>
                     <tr>
                        <td>U+00B4</td>
                        <td>ACUTE ACCENT</td>
                     </tr>
                     <tr>
                        <td>U+00B9</td>
                        <td>SUPERSCRIPT ONE</td>
                     </tr>
                     <tr>
                        <td>U+00BA</td>
                        <td>MASCULINE ORDINAL INDICATOR</td>
                     </tr>
                     <tr>
                        <td>U+2018</td>
                        <td>LEFT SINGLE QUOTATION MARK</td>
                     </tr>
                     <tr>
                        <td>U+2019</td>
                        <td>RIGHT SINGLE QUOTATION MARK</td>
                     </tr>
                     <tr>
                        <td>U+201A</td>
                        <td>SINGLE LOW-9 QUOTATION MARK</td>
                     </tr>
                     <tr>
                        <td>U+201B</td>
                        <td>SINGLE HIGH-REVERSED-9 QUOTATION MARK</td>
                     </tr>
                     <tr>
                        <td>U+201C</td>
                        <td>LEFT DOUBLE QUOTATION MARK</td>
                     </tr>
                     <tr>
                        <td>U+201D</td>
                        <td>RIGHT DOUBLE QUOTATION MARK</td>
                     </tr>
                     <tr>
                        <td>U+201E</td>
                        <td>DOUBLE LOW-9 QUOTATION MARK</td>
                     </tr>
                     <tr>
                        <td>U+201F</td>
                        <td>DOUBLE HIGH-REVERSED-9 QUOTATION MARK</td>
                     </tr>
                     <tr>
                        <td>U+2032</td>
                        <td>PRIME</td>
                     </tr>
                     <tr>
                        <td>U+2033</td>
                        <td>DOUBLE PRIME</td>
                     </tr>
                     <tr>
                        <td>U+2034</td>
                        <td>TRIPLE PRIME</td>
                     </tr>
                     <tr>
                        <td>U+2035</td>
                        <td>REVERSED PRIME</td>
                     </tr>
                     <tr>
                        <td>U+2036</td>
                        <td>REVERSED DOUBLE PRIME</td>
                     </tr>
                     <tr>
                        <td>U+2037</td>
                        <td>REVERSED TRIPLE PRIME</td>
                     </tr>
                     <tr>
                        <td>U+2057</td>
                        <td>QUADRUPLE PRIME</td>
                     </tr>
                  </tbody>
               </table>
               <p>
                  In addition, the characters in the Unicode Superscript and Subscript block
                  (beginning at U+2070) should be treated as pseudo-scripts when they
                  appear in mathematical formulas.
                  
               </p>
               <p>Note that several of these characters are common on keyboards,
                  including U+002A [ASTERISK], U+00B0 [DEGREE SIGN], U+2033 [DOUBLE PRIME],
                  and U+2035 [REVERSED PRIME] also known as a back prime. 
                  
               </p>
            </div>
            <div class="div3">
               
               <h3><a name="chars.comb-chars" id="chars.comb-chars"></a>7.7.3 Combining Characters
               </h3>
               <p>In the UCS there are many combining characters that are intended 
                  to be used for the many accents of numerous different natural
                  languages.  Some of them may seem to provide markup needed for 
                  mathematical accents. They should not be used in mathematical markup.  
                  Superscript, subscript, underscript, and overscript constructions as
                  just discussed above
                  should be used for this purpose.  Of course, combining
                  characters may be used in multi-character identifiers as they are 
                  needed, or in text contexts.  
                  
               </p>
               <p>
                  There is one more case where combining characters turn up naturally
                  in mathematical markup.  Some relations have associated negations,
                  such as U+226F [NOT GREATER-THAN] for the negation of 
                  U+003E [GREATER-THAN SIGN].  The glyph for U+226F [NOT GREATER-THAN]
                  is usually just that for U+003E [GREATER-THAN SIGN] with a slash through
                  it.  Thus it could also be expressed by U+003E-0338 making use of
                  the combining slash U+0338 [COMBINING LONG SOLIDUS OVERLAY].
                  That is true of 25 other characters in common enough mathematical use
                  to merit their own Unicode code points.  In the other direction there are 
                  31 character entity names listed in <a href="appendixg-d.html#Entities">[Entities]</a> which
                  are to be expressed using U+0338 [COMBINING LONG SOLIDUS OVERLAY].
                  
               </p>
               <p>In a similar way there are mathematical characters which have negations 
                  given by a vertical bar overlay U+20D2 [COMBINING LONG VERTICAL LINE OVERLAY].
                  Some are available in pre-composed forms, and some named character entities 
                  are given explicitly as combinations.  In addition there are examples
                  using U+0333 [COMBINING DOUBLE LOW LINE] and 
                  U+20E5 [COMBINING REVERSE SOLIDUS OVERLAY], and variants specified 
                  by use of the U+FE00 [VARIATION SELECTOR-1].  For fuller listing of
                  these cases see the listings in <a href="appendixg-d.html#Entities">[Entities]</a>.
                  
               </p>
               <p>The general rule is that a base character followed by a string of
                  combining characters should be treated just as though it were the
                  pre-composed character that results from the combination,
                  if such a character exists.
                  
               </p>
            </div>
         </div>
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