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<body class="chap2" onload="jscontent()"><div style="visibility: hidden; overflow: hidden; position: absolute; top: 0px; height: 1px; width: auto; padding: 0px; border: 0px none; margin: 0px; text-align: left; text-indent: 0px; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal;"><div id="MathJax_Hidden"></div></div><div style="display: none;" id="MathJax_Message"></div>
<div class="chlinktop"><span class="chlink1">Goto Chapter: </span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap0_mj.html">Top</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html">1</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html">2</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapA_mj.html">A</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html">B</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapBib_mj.html">Bib</a> <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapInd_mj.html">Ind</a> </div>
<div class="chlinkprevnexttop">&nbsp;<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap0_mj.html">[Top of Book]</a>&nbsp; <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap0_mj.html#contents">[Contents]</a>&nbsp; &nbsp;<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html">[Previous Chapter]</a>&nbsp; &nbsp;<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapA_mj.html">[Next Chapter]</a>&nbsp; </div>
<p id="mathjaxlink" class="pcenter"><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2.html">[MathJax off]</a><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chooser.html?BACK=http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html"> [Style]</a></p>
<p><a id="X82793A7E7A3F2A54" name="X82793A7E7A3F2A54"></a></p>
<div class="ChapSects"><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X82793A7E7A3F2A54">2 <span class="Heading">Other Markup</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X7A480B9A795EF264">2.1 <span class="Heading">Various types of text</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X7AA5BF0279938BE0">2.2 <span class="Heading">Formulae</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X833C410D85CF96A4">2.3 <span class="Heading">Crossreferencing</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X7F10E951789D6EDF">2.4 <span class="Heading">Lists and Tables</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X83A355E68485D6D1">2.5 <span class="Heading">Entities and Special Characters</span></a>
</span>
</div>
</div>
<h3>2 <span class="Heading">Other Markup</span></h3>
<p><a id="X7A480B9A795EF264" name="X7A480B9A795EF264"></a></p>
<h4>2.1 <span class="Heading">Various types of text</span></h4>
<p>[→ <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html#X82A731CA83FB9ADD"><span class="RefLink">B.5</span></a>]</p>
<p>In this section we present examples for all the various types of text that are possible in <strong class="pkg">GAPDoc</strong>:</p>
<ul>
<li><p><em>This</em> is <em>emphasized</em>.</p>
</li>
<li><p><em>Keywords</em> are typeset like <code class="keyw">this</code> and <code class="keyw">that</code>.</p>
</li>
<li><p><em>Arguments</em> of functions have an element. They look like this: <var class="Arg">x</var> and <var class="Arg">y</var>.</p>
</li>
<li><p><em>Code</em> can be written with the Code element: <code class="code">if x = y then Print("Equal"); fi;</code> or <code class="code">while true do Print("Hello"); od;</code>.</p>
</li>
<li><p><em>Filenames</em> have their own element: <code class="file">/usr/local/ca/gap4r2</code> or <code class="file">pkg/xgap/doc</code>.</p>
</li>
<li><p><em>Buttons</em>, <em>menus</em>, <em>menu entries</em>, and such things are also supported: <strong class="button">OK</strong> or <strong class="button">Cancel</strong>.</p>
</li>
<li><p><em>Packages</em> are typeset like this: <strong class="pkg">Small Groups Library</strong></p>
</li>
<li><p><em>Quoted</em> text: "This is a text in quotes."</p>
</li>
</ul>
<p><em>Paragraphs</em> are separated by the empty <code class="code">Par</code> or <code class="code">P</code> element.</p>
<p><em>Alternatives</em> for different output formats: This is other than LaTeX output, namely: <b>HTML</b> output.</p>
<p>There are also three elements to typeset "verbatim-like" text. (→ <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html#X824BD70087820CF0"><span class="RefLink">B.6</span></a>)</p>
<p>The first is a <em>Listing</em>:</p>
<div class="example"><pre>Sieve := function(n)
# Returns the primes less than n
local l,p,i;
l := [1..n]; Unbind(l[1]);
p := 2;
while p^2 &lt;= n do
if IsBound(l[p]) then
i := 2 * p;
while i &lt;= n do Unbind(l[i]); i := i + p; od;
fi;
p := p + 1;
od;
return Compacted(l);
end;
</pre></div>
<p>Here is a <em>Log</em> of a <strong class="pkg">GAP</strong> session using this function:</p>
<div class="example"><pre><span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Sieve(100);</span>
[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Length(last);</span>
25
</pre></div>
<p>Here is a <strong class="pkg">GAP</strong> <em>Example</em> session that is automatically tested:</p>
<div class="example"><pre><span class="GAPprompt">gap&gt;</span> <span class="GAPinput">s := Size(CharacterTable("M"));</span>
808017424794512875886459904961710757005754368000000000
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">s &lt; 10^53; </span>
false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">s &lt; 10^54;</span>
true
</pre></div>
<p><a id="X7AA5BF0279938BE0" name="X7AA5BF0279938BE0"></a></p>
<h4>2.2 <span class="Heading">Formulae</span></h4>
<p>[→ <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html#X8516FAA07A95BBB5"><span class="RefLink">B.7</span></a>]</p>
<p>There are three types of formulae.</p>
<p>The first is the <em>normal math mode</em> of LaTeX: <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;&amp;#x22C5;&lt;/mo&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/munderover&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="position: relative;" tabindex="0" id="MathJax-Element-1-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 10.641em; display: inline-block;" role="math" id="MathJax-Span-1" class="math"><span style="display: inline-block; position: relative; width: 8.173em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.529em, 1008.17em, 3.093em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-2" class="mrow"><span id="MathJax-Span-3" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-4" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-5" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-6" class="mo">⋅</span><span style="padding-left: 0.25em;" id="MathJax-Span-7" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-8" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-9" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-10" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-11" class="munderover"><span style="display: inline-block; position: relative; width: 2.153em; height: 0px;"><span style="position: absolute; clip: rect(3.085em, 1000.86em, 4.396em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; vertical-align: -0.002em;" id="MathJax-Span-12" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1000.45em, 4.144em, -1000em); top: -4.478em; left: 0.914em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-13" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1001.24em, 4.142em, -1000em); top: -3.694em; left: 0.914em;"><span id="MathJax-Span-14" class="texatom"><span id="MathJax-Span-15" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-16" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-17" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-18" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-19" class="msubsup"><span style="display: inline-block; position: relative; width: 1.295em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.48em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-20" class="mi">h</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-21" class="texatom"><span id="MathJax-Span-22" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-23" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-24" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-25" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-26" class="msubsup"><span style="display: inline-block; position: relative; width: 0.901em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-27" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-28" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.458em; border-left: 0px solid; width: 0px; height: 1.783em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>i</mi></msub><mo>⋅</mo><msub><mi>b</mi><mi>j</mi></msub><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><msub><mi>h</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><msub><mi>b</mi><mi>k</mi></msub></math></span></span><script id="MathJax-Element-1" type="math/tex">b_i \cdot b_j = \sum_{k=1}^d h_{ijk} b_k</script></span>. Then there are <em>displayed formulae</em>:</p>
<p class="center"><span style="color: inherit;" class="MathJax_Preview"></span><div style="text-align: center;" class="MathJax_Display"><span role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27F9;&lt;/mo&gt;&lt;mspace width=&quot;1em&quot; /&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/munderover&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;&amp;#x22C5;&lt;/mo&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/munderover&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;munderover&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/mrow&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/munderover&gt;&lt;mrow&gt;&lt;mo&gt;(&lt;/mo&gt;&lt;munder&gt;&lt;mo&gt;&amp;#x2211;&lt;/mo&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/mrow&gt;&lt;/munder&gt;&lt;msub&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;y&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;/msub&gt;&lt;msub&gt;&lt;mi&gt;h&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;mi&gt;j&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/mrow&gt;&lt;/msub&gt;&lt;mo&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;k&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="position: relative;" tabindex="0" id="MathJax-Element-2-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 29.578em; display: inline-block;" role="math" id="MathJax-Span-29" class="math"><span style="display: inline-block; position: relative; width: 22.74em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(2.251em, 1022.74em, 5.723em, -1000em); top: -4.183em; left: 0em;"><span id="MathJax-Span-30" class="mrow"><span style="font-family: STIXGeneral;" id="MathJax-Span-31" class="mo">⟹</span><span style="height: 0em; vertical-align: 0em; width: 1.125em; display: inline-block; overflow: hidden;" id="MathJax-Span-32" class="mspace"></span><span style="padding-left: 0.313em;" id="MathJax-Span-33" class="mrow"><span style="vertical-align: -0.778em;" id="MathJax-Span-34" class="mo"><span style="font-family: STIXSizeFourSym;">(</span></span><span id="MathJax-Span-35" class="munderover"><span style="display: inline-block; position: relative; width: 1.292em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-36" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.96em, 4.242em, -1000em); top: -2.84em; left: 0.129em;"><span id="MathJax-Span-37" class="texatom"><span id="MathJax-Span-38" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-39" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-40" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-41" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.263em, 1000.37em, 4.144em, -1000em); top: -5.19em; left: 0.46em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-42" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-43" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-44" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-45" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-46" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-47" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-48" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.778em;" id="MathJax-Span-49" class="mo"><span style="font-family: STIXSizeFourSym;">)</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-50" class="mo">⋅</span><span style="padding-left: 0.25em;" id="MathJax-Span-51" class="mrow"><span style="vertical-align: -0.778em;" id="MathJax-Span-52" class="mo"><span style="font-family: STIXSizeFourSym;">(</span></span><span id="MathJax-Span-53" class="munderover"><span style="display: inline-block; position: relative; width: 1.292em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-54" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.368em, 1000.96em, 4.381em, -1000em); top: -2.84em; left: 0.128em;"><span id="MathJax-Span-55" class="texatom"><span id="MathJax-Span-56" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-57" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-58" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-59" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.263em, 1000.37em, 4.144em, -1000em); top: -5.19em; left: 0.46em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-60" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-61" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.43em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-62" class="mi">y</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-63" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-64" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-65" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-66" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.778em;" id="MathJax-Span-67" class="mo"><span style="font-family: STIXSizeFourSym;">)</span></span></span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-68" class="mo">=</span><span style="padding-left: 0.313em;" id="MathJax-Span-69" class="munderover"><span style="display: inline-block; position: relative; width: 1.292em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-70" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.363em, 1001.09em, 4.242em, -1000em); top: -2.84em; left: 0.064em;"><span id="MathJax-Span-71" class="texatom"><span id="MathJax-Span-72" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-73" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-74" class="mo">=</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-75" class="mn">1</span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.263em, 1000.37em, 4.144em, -1000em); top: -5.19em; left: 0.46em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-76" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.019em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-77" class="mrow"><span style="vertical-align: -0.778em;" id="MathJax-Span-78" class="mo"><span style="font-family: STIXSizeFourSym;">(</span></span><span id="MathJax-Span-79" class="munderover"><span style="display: inline-block; position: relative; width: 1.292em; height: 0px;"><span style="position: absolute; clip: rect(2.871em, 1001.2em, 4.61em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXSizeOneSym; vertical-align: -0.524em;" id="MathJax-Span-80" class="mo">∑</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; clip: rect(3.384em, 1000.57em, 4.381em, -1000em); top: -2.84em; left: 0.361em;"><span id="MathJax-Span-81" class="texatom"><span id="MathJax-Span-82" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-83" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-84" class="mo">,</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-85" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-86" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-87" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-88" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-89" class="msubsup"><span style="display: inline-block; position: relative; width: 0.716em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.43em, 4.341em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-90" class="mi">y</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.444em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-91" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span id="MathJax-Span-92" class="msubsup"><span style="display: inline-block; position: relative; width: 1.295em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.48em, 4.144em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-93" class="mi">h</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span id="MathJax-Span-94" class="texatom"><span id="MathJax-Span-95" class="mrow"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-96" class="mi">i</span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-97" class="mi">j<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.001em;"></span></span><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-98" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="vertical-align: -0.778em;" id="MathJax-Span-99" class="mo"><span style="font-family: STIXSizeFourSym;">)</span></span></span><span style="padding-left: 0.188em;" id="MathJax-Span-100" class="msubsup"><span style="display: inline-block; position: relative; width: 0.901em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-101" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-102" class="mi">k<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.012em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 4.183em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -1.878em; border-left: 0px solid; width: 0px; height: 4.264em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML MJX_Assistive_MathML_Block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mo stretchy="false">⟹</mo><mspace width="1em"></mspace><mrow><mo>(</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><msub><mi>x</mi><mi>i</mi></msub><msub><mi>b</mi><mi>i</mi></msub><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><msub><mi>y</mi><mi>j</mi></msub><msub><mi>b</mi><mi>j</mi></msub><mo>)</mo></mrow><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mi>d</mi></munderover><mrow><mo>(</mo><munder><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>x</mi><mi>i</mi></msub><msub><mi>y</mi><mi>j</mi></msub><msub><mi>h</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>)</mo></mrow><msub><mi>b</mi><mi>k</mi></msub></math></span></span></div><script id="MathJax-Element-2" type="math/tex; mode=display">
\Longrightarrow \quad \left(\sum_{i=1}^d x_i b_i \right) \cdot
\left(\sum_{j=1}^d y_j b_j \right) =
\sum_{k=1}^d \left( \sum_{i,j} x_i y_j h_{ijk} \right) b_k
</script></p>
<p>If possible, use the <code class="code">Alt</code> element to specify a better readable text version of such a formula as in the following example:</p>
<pre class="normal">
d d d
----- ----- ----- -----
\ \ \ \
==&gt; ( ) x_i b_i )( ) y_j b_j ) = ) ( ) x_i y_j h_ijk ) b_k
/ / / /
----- ----- ----- -----
i = 1 j = 1 k = 1 i,j
</pre>
<p>For small formulae without "difficult" parts use the <code class="code">M</code> element: <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mi&gt;i&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" style="position: relative;" tabindex="0" id="MathJax-Element-3-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.016em; display: inline-block;" role="math" id="MathJax-Span-103" class="math"><span style="display: inline-block; position: relative; width: 0.769em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.625em, 1000.77em, 2.754em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-104" class="mrow"><span id="MathJax-Span-105" class="msubsup"><span style="display: inline-block; position: relative; width: 0.772em; height: 0px;"><span style="position: absolute; clip: rect(3.163em, 1000.47em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-106" class="mi">b</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -3.84em; left: 0.5em;"><span style="font-size: 70.7%; font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-107" class="mi">i</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.268em; border-left: 0px solid; width: 0px; height: 1.218em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>b</mi><mi>i</mi></msub></math></span></span><script id="MathJax-Element-3" type="math/tex">b_i</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" style="position: relative;" tabindex="0" id="MathJax-Element-4-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 1.203em; display: inline-block;" role="math" id="MathJax-Span-108" class="math"><span style="display: inline-block; position: relative; width: 0.913em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.467em, 1000.91em, 2.607em, -1000em); top: -2.452em; left: 0em;"><span id="MathJax-Span-109" class="mrow"><span id="MathJax-Span-110" class="msubsup"><span style="display: inline-block; position: relative; width: 0.926em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-111" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.498em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-112" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.452em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.232em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup></math></span></span><script id="MathJax-Element-4" type="math/tex">x^2</script></span>, <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msup&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;x&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;/math&gt;" style="position: relative;" tabindex="0" id="MathJax-Element-5-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 11.891em; display: inline-block;" role="math" id="MathJax-Span-113" class="math"><span style="display: inline-block; position: relative; width: 9.135em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.659em, 1009.14em, 2.965em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-114" class="mrow"><span id="MathJax-Span-115" class="msubsup"><span style="display: inline-block; position: relative; width: 0.926em; height: 0px;"><span style="position: absolute; clip: rect(3.405em, 1000.45em, 4.146em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-116" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.498em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-117" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-118" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-119" class="mn">2</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-120" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-121" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-122" class="mn">1</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-123" class="mo">=</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-124" class="mo">(</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-125" class="mi">x<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.003em;"></span></span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-126" class="mo">+</span><span style="font-family: STIXGeneral; padding-left: 0.25em;" id="MathJax-Span-127" class="mn">1</span><span id="MathJax-Span-128" class="msubsup"><span style="display: inline-block; position: relative; width: 0.762em; height: 0px;"><span style="position: absolute; clip: rect(3.17em, 1000.28em, 4.312em, -1000em); top: -3.99em; left: 0em;"><span style="font-family: STIXGeneral;" id="MathJax-Span-129" class="mo">)</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span><span style="position: absolute; top: -4.353em; left: 0.333em;"><span style="font-size: 70.7%; font-family: STIXGeneral;" id="MathJax-Span-130" class="mn">2</span><span style="display: inline-block; width: 0px; height: 3.99em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.293em; border-left: 0px solid; width: 0px; height: 1.448em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mn>2</mn></msup></math></span></span><script id="MathJax-Element-5" type="math/tex">x^2 + 2x + 1 = (x + 1)^2</script></span>. Note that here whitespace matters for text (or HTML) output.</p>
<p>Here are two formulae containing less than characters which are special characters for XML: <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mi&gt;b&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mi&gt;c&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;" style="position: relative;" tabindex="0" id="MathJax-Element-6-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 7.578em; display: inline-block;" role="math" id="MathJax-Span-131" class="math"><span style="display: inline-block; position: relative; width: 5.817em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1005.82em, 2.812em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-132" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-133" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-134" class="mo">&lt;</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-135" class="mi">b</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-136" class="mo">&lt;</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-137" class="mi">c</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-138" class="mo">&lt;</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-139" class="mi">d<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.027em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.094em; border-left: 0px solid; width: 0px; height: 1.044em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>&lt;</mo><mi>b</mi><mo>&lt;</mo><mi>c</mi><mo>&lt;</mo><mi>d</mi></math></span></span><script id="MathJax-Element-6" type="math/tex">a < b < c < d</script></span> and <span class="SimpleMath"><span style="color: inherit;" class="MathJax_Preview"></span><span role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;e&lt;/mi&gt;&lt;mo&gt;&amp;lt;&lt;/mo&gt;&lt;mi&gt;f&lt;/mi&gt;&lt;/math&gt;" style="position: relative;" tabindex="0" id="MathJax-Element-7-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 2.828em; display: inline-block;" role="math" id="MathJax-Span-140" class="math"><span style="display: inline-block; position: relative; width: 2.163em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.822em, 1002.16em, 2.995em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-141" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-142" class="mi">e</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-143" class="mo">&lt;</span><span style="font-family: STIXGeneral; font-style: italic; text-rendering: optimizelegibility; padding-left: 0.313em;" id="MathJax-Span-144" class="mi">f<span style="display: inline-block; overflow: hidden; height: 1px; width: 0.146em;"></span></span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.332em; border-left: 0px solid; width: 0px; height: 1.276em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>e</mi><mo>&lt;</mo><mi>f</mi></math></span></span><script id="MathJax-Element-7" type="math/tex">e < f</script></span>.</p>
<p>Using the <code class="code">Mode</code> attribute of a <code class="code">Display</code> element formulae like</p>
<p class="center"><span style="color: inherit;" class="MathJax_Preview"></span><div style="text-align: center;" class="MathJax_Display"><span role="presentation" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot; display=&quot;block&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;&amp;#x27F6;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo lspace=&quot;thickmathspace&quot; rspace=&quot;thickmathspace&quot;&gt;mod&lt;/mo&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mi class=&quot;MJX-variant&quot; mathvariant=&quot;normal&quot;&gt;&amp;#x2032;&lt;/mi&gt;&lt;/math&gt;" style="position: relative;" tabindex="0" id="MathJax-Element-8-Frame" class="MathJax"><nobr aria-hidden="true"><span style="width: 8.578em; display: inline-block;" role="math" id="MathJax-Span-145" class="math"><span style="display: inline-block; position: relative; width: 6.587em; height: 0px; font-size: 130%;"><span style="position: absolute; clip: rect(1.817em, 1006.54em, 2.799em, -1000em); top: -2.644em; left: 0em;"><span id="MathJax-Span-146" class="mrow"><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-147" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.313em;" id="MathJax-Span-148" class="mo">⟶</span><span style="font-family: STIXGeneral; font-style: italic; padding-left: 0.313em;" id="MathJax-Span-149" class="mi">a</span><span style="font-family: STIXGeneral; padding-left: 0.313em; padding-right: 0.313em;" id="MathJax-Span-150" class="mo">mod</span><span style="font-family: STIXGeneral; font-style: italic;" id="MathJax-Span-151" class="mi">m</span><span style="font-family: STIXVariants;" id="MathJax-Span-152" class="mi">′</span></span><span style="display: inline-block; width: 0px; height: 2.644em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.077em; border-left: 0px solid; width: 0px; height: 1.027em;"></span></span></nobr><span role="presentation" class="MJX_Assistive_MathML MJX_Assistive_MathML_Block"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><mi>a</mi><mo stretchy="false">⟶</mo><mi>a</mi><mo lspace="thickmathspace" rspace="thickmathspace">mod</mo><mi>m</mi><mi class="MJX-variant" mathvariant="normal">′</mi></math></span></span></div><script id="MathJax-Element-8" type="math/tex; mode=display">a \longrightarrow a \bmod m\prime</script></p>
<p>can also be displayed nicely in text and HTML output.</p>
<p><a id="X833C410D85CF96A4" name="X833C410D85CF96A4"></a></p>
<h4>2.3 <span class="Heading">Crossreferencing</span></h4>
<p>[→ <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html#X7D19CF4782309661"><span class="RefLink">B.8</span></a>]</p>
<p>In this section we demonstrate various references to parts of this document. Here is a reference to this section: <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X833C410D85CF96A4"><span class="RefLink">2.3</span></a>. Here is a reference to chapter <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink">1</span></a>, to appendix <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapA_mj.html#X7B53252784137533"><span class="RefLink">A</span></a>, and to subsection <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X7E193BD379F58A4C"><span class="RefLink">1.1-1</span></a>.</p>
<p>We distinguish among others references to functions (see <code class="func">f</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X7FA1D0937FA1D093"><span class="RefLink">1.2-1</span></a>)), to methods with tricky name (see <code class="func">\^\{\}\[\]\&lt;\&amp;</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X822B5C487B29E799"><span class="RefLink">1.2-2</span></a>)), to operations (see <code class="func">MyOperation</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X7D33C2597988F481"><span class="RefLink">1.2-3</span></a>)), to methods (see <code class="func">MyOperation</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X783DCD4E826289D4"><span class="RefLink">1.2-4</span></a>) or <code class="func">MyOperation</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X7A5F4A287D06988C"><span class="RefLink">1.2-5</span></a>)), to filters (see <code class="func">IsBla</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X82954B687D2DF3C2"><span class="RefLink">1.2-6</span></a>)), to properties (see <code class="func">IsBlubb</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X80C364DD7C919CCE"><span class="RefLink">1.2-7</span></a>)), to attributes (see <code class="func">NumberBlobbs</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X8052A45E7F9F054C"><span class="RefLink">1.2-8</span></a>)), to variables (<code class="func">AllBlibbs</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X7C00E05A7DDEF003"><span class="RefLink">1.2-9</span></a>)), to families (see <code class="func">BlibbsFamily</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X7CBC935A8142E374"><span class="RefLink">1.2-10</span></a>)), and to info classes (see <code class="func">InfoBlibbs</code> (<a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X84D7D77378AD030A"><span class="RefLink">1.2-11</span></a>)).</p>
<p>There are also references to labels: see <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap2_mj.html#X833C410D85CF96A4"><span class="RefLink">here</span></a>, to other books: see <a href="http://www.gap-system.org/Manuals/pkg/pkg/GAPDoc-1.5.1/doc/chap3_mj.html#X7B76F6F786521F6B"><span class="RefLink">GAPDoc: What is a DTD?</span></a> or <code class="func">IsSubgroup</code> (<a href="http://www.gap-system.org/Manuals/doc/ref/chap40_mj.html#X7839D8927E778334"><span class="RefLink">Reference: IsSubgroup</span></a>) in the <strong class="pkg">GAP</strong> reference manual.</p>
<p>References to sections come in two styles: <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink">1</span></a> or <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink"><span class="Heading">Sectioning Elements</span></span></a>.</p>
<p>Another type of cross referencing is bibliography. Here is a citation: <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapBib_mj.html#biBCR1">[CR81, (5.22)]</a> is an interesting lemma.</p>
<p>There are also URLs:</p>
<p><span class="URL"><a href="http://www.math.rwth-aachen.de/">http://www.math.rwth-aachen.de/</a></span></p>
<p>Email addresses have a special element: <span class="URL"><a href="mailto:Frank.Luebeck@Math.RWTH-Aachen.De">Frank.Luebeck@Math.RWTH-Aachen.De</a></span></p>
<p>and Homepages another one: <span class="URL"><a href="http://www-groups.mcs.st-and.ac.uk/%7Eneunhoef/">http://www-groups.mcs.st-and.ac.uk/~neunhoef/</a></span></p>
<p>And here is a link to the <span class="URL"><a href="http://www.math.rwth-aachen.de/%7EFrank.Luebeck/gap/EDIM/index.html#ARCHS"><strong class="pkg">EDIM</strong> archives</a></span>.</p>
<p>One can generate index entries as follows (look up the words "TeX-UserGroup", "RWTH", "Aachen, Hauptbahnhof", and "<strong class="pkg">GAP</strong>, <strong class="pkg">GAPDoc</strong>").</p>
<p><a id="X7F10E951789D6EDF" name="X7F10E951789D6EDF"></a></p>
<h4>2.4 <span class="Heading">Lists and Tables</span></h4>
<p>[→ <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html#X7BB822947F626E1A"><span class="RefLink">B.9</span></a>]</p>
<p>There are</p>
<ul>
<li><p>lists</p>
</li>
<li><p>enumerations, and</p>
</li>
<li><p>tables</p>
</li>
</ul>
<p>or:</p>
<ol>
<li><p>lists</p>
</li>
<li><p>enumerations, and</p>
</li>
<li><p>tables</p>
</li>
</ol>
<p>or with marks:</p>
<dl>
<dt><strong class="Mark">lists:</strong></dt>
<dd><p>not numbered</p>
</dd>
<dt><strong class="Mark">enumerations:</strong></dt>
<dd><p>numbered</p>
</dd>
<dt><strong class="Mark">tables:</strong></dt>
<dd><p>two-dimensional</p>
</dd>
</dl>
<p>Lists can also be nested:</p>
<ol>
<li><ol>
<li><p>first item of inner enumeration</p>
</li>
<li><p>second item of inner enumeration</p>
</li>
</ol>
</li>
<li>
<ul>
<li><p>first item of inner list</p>
</li>
<li><p>second item of inner list</p>
</li>
</ul>
</li>
</ol>
<p>Here is a <em>table</em>:</p>
<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b>Prices</caption>
<tbody><tr>
<td class="tdright">Object</td>
<td class="tdcenter">Price</td>
<td class="tdleft">available</td>
</tr>
<tr>
<td class="tdright">Shoe</td>
<td class="tdcenter">$1,00</td>
<td class="tdleft">there</td>
</tr>
<tr>
<td class="tdright">Hat</td>
<td class="tdcenter">$2,00</td>
<td class="tdleft">not there</td>
</tr>
</tbody></table><br><p>&nbsp;</p><br>
</div>
<p><a id="X83A355E68485D6D1" name="X83A355E68485D6D1"></a></p>
<h4>2.5 <span class="Heading">Entities and Special Characters</span></h4>
<p>[→ <a href="http://www.gap-system.org/Manuals/pkg/GAPDoc-1.5.1/example/chapB_mj.html#X80B478CD7E584F6F"><span class="RefLink">B.10</span></a>]</p>
<p>Here is a table of special characters, the first two are special for XML and must be typed in by entities in <strong class="pkg">GAPDoc</strong> documents. The other characters are special for LaTeX but in <strong class="pkg">GAPDoc</strong> they can be typed directly.</p>
<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b>Special characters in character data</caption>
<tbody><tr>
<td class="tdcenter"><code class="code">&amp;</code></td>
<td class="tdcenter"><code class="code">&lt;</code></td>
<td class="tdcenter"><code class="code">&gt;</code></td>
<td class="tdcenter"><code class="code">#</code></td>
<td class="tdcenter"><code class="code">$</code></td>
<td class="tdcenter"><code class="code">%</code></td>
<td class="tdcenter"><code class="code">~</code></td>
<td class="tdcenter"><code class="code">\</code></td>
<td class="tdcenter"><code class="code">{</code></td>
<td class="tdcenter"><code class="code">}</code></td>
<td class="tdcenter"><code class="code">_</code></td>
<td class="tdcenter"><code class="code">^</code></td>
<td class="tdcenter"><code class="code">&nbsp;</code></td>
</tr>
</tbody></table><br><p>&nbsp;</p><br>
</div>
<p>And here are the predefined entities in <strong class="pkg">GAPDoc</strong>:</p>
<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b>Predefined Entities in the <strong class="pkg">GAPDoc</strong> system</caption>
<tbody><tr>
<td class="tdleft"><code class="code">&amp;GAP;</code></td>
<td class="tdleft"><strong class="pkg">GAP</strong></td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;GAPDoc;</code></td>
<td class="tdleft"><strong class="pkg">GAPDoc</strong></td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;TeX;</code></td>
<td class="tdleft">TeX</td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;LaTeX;</code></td>
<td class="tdleft">LaTeX</td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;BibTeX;</code></td>
<td class="tdleft">BibTeX</td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;MeatAxe;</code></td>
<td class="tdleft"><strong class="pkg">MeatAxe</strong></td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;XGAP;</code></td>
<td class="tdleft"><strong class="pkg">XGAP</strong></td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;copyright;</code></td>
<td class="tdleft">©</td>
</tr>
</tbody></table><br><p>&nbsp;</p><br>
</div>
<p>And some more for mathematical symbols: ℂ, ℤ, ℕ, ℙ, ℚ, ℍ, ℝ.</p>
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<div class="ChapSects"><a href="chap2_mj.html#X82793A7E7A3F2A54">2 <span class="Heading">Other Markup</span></a>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap2_mj.html#X7A480B9A795EF264">2.1 <span class="Heading">Various types of text</span></a>
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<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap2_mj.html#X7AA5BF0279938BE0">2.2 <span class="Heading">Formulae</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap2_mj.html#X833C410D85CF96A4">2.3 <span class="Heading">Crossreferencing</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap2_mj.html#X7F10E951789D6EDF">2.4 <span class="Heading">Lists and Tables</span></a>
</span>
</div>
<div class="ContSect"><span class="tocline"><span class="nocss">&nbsp;</span><a href="chap2_mj.html#X83A355E68485D6D1">2.5 <span class="Heading">Entities and Special Characters</span></a>
</span>
</div>
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<h3>2 <span class="Heading">Other Markup</span></h3>
<p><a id="X7A480B9A795EF264" name="X7A480B9A795EF264"></a></p>
<h4>2.1 <span class="Heading">Various types of text</span></h4>
<p>[→ <a href="chapB_mj.html#X82A731CA83FB9ADD"><span class="RefLink">B.5</span></a>]</p>
<p>In this section we present examples for all the various types of text that are possible in <strong class="pkg">GAPDoc</strong>:</p>
<ul>
<li><p><em>This</em> is <em>emphasized</em>.</p>
</li>
<li><p><em>Keywords</em> are typeset like <code class="keyw">this</code> and <code class="keyw">that</code>.</p>
</li>
<li><p><em>Arguments</em> of functions have an element. They look like this: <var class="Arg">x</var> and <var class="Arg">y</var>.</p>
</li>
<li><p><em>Code</em> can be written with the Code element: <code class="code">if x = y then Print("Equal"); fi;</code> or <code class="code">while true do Print("Hello"); od;</code>.</p>
</li>
<li><p><em>Filenames</em> have their own element: <code class="file">/usr/local/ca/gap4r2</code> or <code class="file">pkg/xgap/doc</code>.</p>
</li>
<li><p><em>Buttons</em>, <em>menus</em>, <em>menu entries</em>, and such things are also supported: <strong class="button">OK</strong> or <strong class="button">Cancel</strong>.</p>
</li>
<li><p><em>Packages</em> are typeset like this: <strong class="pkg">Small Groups Library</strong></p>
</li>
<li><p><em>Quoted</em> text: "This is a text in quotes."</p>
</li>
</ul>
<p><em>Paragraphs</em> are separated by the empty <code class="code">Par</code> or <code class="code">P</code> element.</p>
<p><em>Alternatives</em> for different output formats: This is other than LaTeX output, namely: <b>HTML</b> output.</p>
<p>There are also three elements to typeset "verbatim-like" text. (→ <a href="chapB_mj.html#X824BD70087820CF0"><span class="RefLink">B.6</span></a>)</p>
<p>The first is a <em>Listing</em>:</p>
<div class="example"><pre>
Sieve := function(n)
# Returns the primes less than n
local l,p,i;
l := [1..n]; Unbind(l[1]);
p := 2;
while p^2 &lt;= n do
if IsBound(l[p]) then
i := 2 * p;
while i &lt;= n do Unbind(l[i]); i := i + p; od;
fi;
p := p + 1;
od;
return Compacted(l);
end;
</pre></div>
<p>Here is a <em>Log</em> of a <strong class="pkg">GAP</strong> session using this function:</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Sieve(100);</span>
[ 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61,
67, 71, 73, 79, 83, 89, 97 ]
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">Length(last);</span>
25
</pre></div>
<p>Here is a <strong class="pkg">GAP</strong> <em>Example</em> session that is automatically tested:</p>
<div class="example"><pre>
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">s := Size(CharacterTable("M"));</span>
808017424794512875886459904961710757005754368000000000
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">s &lt; 10^53; </span>
false
<span class="GAPprompt">gap&gt;</span> <span class="GAPinput">s &lt; 10^54;</span>
true
</pre></div>
<p><a id="X7AA5BF0279938BE0" name="X7AA5BF0279938BE0"></a></p>
<h4>2.2 <span class="Heading">Formulae</span></h4>
<p>[→ <a href="chapB_mj.html#X8516FAA07A95BBB5"><span class="RefLink">B.7</span></a>]</p>
<p>There are three types of formulae.</p>
<p>The first is the <em>normal math mode</em> of LaTeX: <span class="SimpleMath">\(b_i \cdot b_j = \sum_{k=1}^d h_{ijk} b_k\)</span>. Then there are <em>displayed formulae</em>:</p>
<p class="center">\[
\Longrightarrow \quad \left(\sum_{i=1}^d x_i b_i \right) \cdot
\left(\sum_{j=1}^d y_j b_j \right) =
\sum_{k=1}^d \left( \sum_{i,j} x_i y_j h_{ijk} \right) b_k
\]</p>
<p>If possible, use the <code class="code">Alt</code> element to specify a better readable text version of such a formula as in the following example:</p>
<pre class="normal">
d d d
----- ----- ----- -----
\ \ \ \
==> ( ) x_i b_i )( ) y_j b_j ) = ) ( ) x_i y_j h_ijk ) b_k
/ / / /
----- ----- ----- -----
i = 1 j = 1 k = 1 i,j
</pre>
<p>For small formulae without "difficult" parts use the <code class="code">M</code> element: <span class="SimpleMath">\(b_i\)</span>, <span class="SimpleMath">\(x^2\)</span>, <span class="SimpleMath">\(x^2 + 2x + 1 = (x + 1)^2\)</span>. Note that here whitespace matters for text (or HTML) output.</p>
<p>Here are two formulae containing less than characters which are special characters for XML: <span class="SimpleMath">\(a &lt; b &lt; c &lt; d\)</span> and <span class="SimpleMath">\(e &lt; f\)</span>.</p>
<p>Using the <code class="code">Mode</code> attribute of a <code class="code">Display</code> element formulae like</p>
<p class="center">\[a \longrightarrow a \bmod m\prime\]</p>
<p>can also be displayed nicely in text and HTML output.</p>
<p><a id="X833C410D85CF96A4" name="X833C410D85CF96A4"></a></p>
<h4>2.3 <span class="Heading">Crossreferencing</span></h4>
<p>[→ <a href="chapB_mj.html#X7D19CF4782309661"><span class="RefLink">B.8</span></a>]</p>
<p>In this section we demonstrate various references to parts of this document. Here is a reference to this section: <a href="chap2_mj.html#X833C410D85CF96A4"><span class="RefLink">2.3</span></a>. Here is a reference to chapter <a href="chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink">1</span></a>, to appendix <a href="chapA_mj.html#X7B53252784137533"><span class="RefLink">A</span></a>, and to subsection <a href="chap1_mj.html#X7E193BD379F58A4C"><span class="RefLink">1.1-1</span></a>.</p>
<p>We distinguish among others references to functions (see <code class="func">f</code> (<a href="chap1_mj.html#X7FA1D0937FA1D093"><span class="RefLink">1.2-1</span></a>)), to methods with tricky name (see <code class="func">\^\{\}\[\]\&lt;\&amp;</code> (<a href="chap1_mj.html#X822B5C487B29E799"><span class="RefLink">1.2-2</span></a>)), to operations (see <code class="func">MyOperation</code> (<a href="chap1_mj.html#X7D33C2597988F481"><span class="RefLink">1.2-3</span></a>)), to methods (see <code class="func">MyOperation</code> (<a href="chap1_mj.html#X783DCD4E826289D4"><span class="RefLink">1.2-4</span></a>) or <code class="func">MyOperation</code> (<a href="chap1_mj.html#X7A5F4A287D06988C"><span class="RefLink">1.2-5</span></a>)), to filters (see <code class="func">IsBla</code> (<a href="chap1_mj.html#X82954B687D2DF3C2"><span class="RefLink">1.2-6</span></a>)), to properties (see <code class="func">IsBlubb</code> (<a href="chap1_mj.html#X80C364DD7C919CCE"><span class="RefLink">1.2-7</span></a>)), to attributes (see <code class="func">NumberBlobbs</code> (<a href="chap1_mj.html#X8052A45E7F9F054C"><span class="RefLink">1.2-8</span></a>)), to variables (<code class="func">AllBlibbs</code> (<a href="chap1_mj.html#X7C00E05A7DDEF003"><span class="RefLink">1.2-9</span></a>)), to families (see <code class="func">BlibbsFamily</code> (<a href="chap1_mj.html#X7CBC935A8142E374"><span class="RefLink">1.2-10</span></a>)), and to info classes (see <code class="func">InfoBlibbs</code> (<a href="chap1_mj.html#X84D7D77378AD030A"><span class="RefLink">1.2-11</span></a>)).</p>
<p>There are also references to labels: see <a href="chap2_mj.html#X833C410D85CF96A4"><span class="RefLink">here</span></a>, to other books: see <a href="../../pkg/GAPDoc-1.5.1/doc/chap3_mj.html#X7B76F6F786521F6B"><span class="RefLink">GAPDoc: What is a DTD?</span></a> or <code class="func">IsSubgroup</code> (<a href="../../../doc/ref/chap40_mj.html#X7839D8927E778334"><span class="RefLink">Reference: IsSubgroup</span></a>) in the <strong class="pkg">GAP</strong> reference manual.</p>
<p>References to sections come in two styles: <a href="chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink">1</span></a> or <a href="chap1_mj.html#X80E2AD7481DD69D9"><span class="RefLink"><span class="Heading">Sectioning Elements</span></span></a>.</p>
<p>Another type of cross referencing is bibliography. Here is a citation: <a href="chapBib_mj.html#biBCR1">[CR81, (5.22)]</a> is an interesting lemma.</p>
<p>There are also URLs:</p>
<p><span class="URL"><a href="http://www.math.rwth-aachen.de/">http://www.math.rwth-aachen.de/</a></span></p>
<p>Email addresses have a special element: <span class="URL"><a href="mailto:Frank.Luebeck@Math.RWTH-Aachen.De">Frank.Luebeck@Math.RWTH-Aachen.De</a></span></p>
<p>and Homepages another one: <span class="URL"><a href="http://www-groups.mcs.st-and.ac.uk/~neunhoef/">http://www-groups.mcs.st-and.ac.uk/~neunhoef/</a></span></p>
<p>And here is a link to the <span class="URL"><a href="http://www.math.rwth-aachen.de/~Frank.Luebeck/gap/EDIM/index.html#ARCHS"><strong class="pkg">EDIM</strong> archives</a></span>.</p>
<p>One can generate index entries as follows (look up the words "TeX-UserGroup", "RWTH", "Aachen, Hauptbahnhof", and "<strong class="pkg">GAP</strong>, <strong class="pkg">GAPDoc</strong>").</p>
<p><a id="X7F10E951789D6EDF" name="X7F10E951789D6EDF"></a></p>
<h4>2.4 <span class="Heading">Lists and Tables</span></h4>
<p>[→ <a href="chapB_mj.html#X7BB822947F626E1A"><span class="RefLink">B.9</span></a>]</p>
<p>There are</p>
<ul>
<li><p>lists</p>
</li>
<li><p>enumerations, and</p>
</li>
<li><p>tables</p>
</li>
</ul>
<p>or:</p>
<ol>
<li><p>lists</p>
</li>
<li><p>enumerations, and</p>
</li>
<li><p>tables</p>
</li>
</ol>
<p>or with marks:</p>
<dl>
<dt><strong class="Mark">lists:</strong></dt>
<dd><p>not numbered</p>
</dd>
<dt><strong class="Mark">enumerations:</strong></dt>
<dd><p>numbered</p>
</dd>
<dt><strong class="Mark">tables:</strong></dt>
<dd><p>two-dimensional</p>
</dd>
</dl>
<p>Lists can also be nested:</p>
<ol>
<li><ol>
<li><p>first item of inner enumeration</p>
</li>
<li><p>second item of inner enumeration</p>
</li>
</ol>
</li>
<li>
<ul>
<li><p>first item of inner list</p>
</li>
<li><p>second item of inner list</p>
</li>
</ul>
</li>
</ol>
<p>Here is a <em>table</em>:</p>
<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b>Prices</caption>
<tr>
<td class="tdright">Object</td>
<td class="tdcenter">Price</td>
<td class="tdleft">available</td>
</tr>
<tr>
<td class="tdright">Shoe</td>
<td class="tdcenter">$1,00</td>
<td class="tdleft">there</td>
</tr>
<tr>
<td class="tdright">Hat</td>
<td class="tdcenter">$2,00</td>
<td class="tdleft">not there</td>
</tr>
</table><br /><p>&nbsp;</p><br />
</div>
<p><a id="X83A355E68485D6D1" name="X83A355E68485D6D1"></a></p>
<h4>2.5 <span class="Heading">Entities and Special Characters</span></h4>
<p>[→ <a href="chapB_mj.html#X80B478CD7E584F6F"><span class="RefLink">B.10</span></a>]</p>
<p>Here is a table of special characters, the first two are special for XML and must be typed in by entities in <strong class="pkg">GAPDoc</strong> documents. The other characters are special for LaTeX but in <strong class="pkg">GAPDoc</strong> they can be typed directly.</p>
<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b>Special characters in character data</caption>
<tr>
<td class="tdcenter"><code class="code">&amp;</code></td>
<td class="tdcenter"><code class="code">&lt;</code></td>
<td class="tdcenter"><code class="code">&gt;</code></td>
<td class="tdcenter"><code class="code">#</code></td>
<td class="tdcenter"><code class="code">$</code></td>
<td class="tdcenter"><code class="code">%</code></td>
<td class="tdcenter"><code class="code">~</code></td>
<td class="tdcenter"><code class="code">\</code></td>
<td class="tdcenter"><code class="code">{</code></td>
<td class="tdcenter"><code class="code">}</code></td>
<td class="tdcenter"><code class="code">_</code></td>
<td class="tdcenter"><code class="code">^</code></td>
<td class="tdcenter"><code class="code"> </code></td>
</tr>
</table><br /><p>&nbsp;</p><br />
</div>
<p>And here are the predefined entities in <strong class="pkg">GAPDoc</strong>:</p>
<div class="pcenter"><table class="GAPDocTable">
<caption class="GAPDocTable"><b>Table: </b>Predefined Entities in the <strong class="pkg">GAPDoc</strong> system</caption>
<tr>
<td class="tdleft"><code class="code">&amp;GAP;</code></td>
<td class="tdleft"><strong class="pkg">GAP</strong></td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;GAPDoc;</code></td>
<td class="tdleft"><strong class="pkg">GAPDoc</strong></td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;TeX;</code></td>
<td class="tdleft">TeX</td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;LaTeX;</code></td>
<td class="tdleft">LaTeX</td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;BibTeX;</code></td>
<td class="tdleft">BibTeX</td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;MeatAxe;</code></td>
<td class="tdleft"><strong class="pkg">MeatAxe</strong></td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;XGAP;</code></td>
<td class="tdleft"><strong class="pkg">XGAP</strong></td>
</tr>
<tr>
<td class="tdleft"><code class="code">&amp;copyright;</code></td>
<td class="tdleft">©</td>
</tr>
</table><br /><p>&nbsp;</p><br />
</div>
<p>And some more for mathematical symbols: ℂ, ℤ, ℕ, ℙ, ℚ, ℍ, ℝ.</p>
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