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<p><a id="X7AF8D94A7E56C049" name="X7AF8D94A7E56C049"></a></p>
<div class="ChapSects"><a href="chapA.html#X7AF8D94A7E56C049">A <span class="Heading">Generalities</span></a>
<div class="ContSect"><span class="nocss"> </span><a href="chapA.html#X7A5D608487A8C98F">A.1 <span class="Heading">Bézout sequences</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X86859C84858ECAF1">A.1-1 BezoutSequence</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X86C990AC7F40E8D0">A.1-2 IsBezoutSequence</a></span>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X7C9DCBAF825CF7B2">A.1-3 CeilingOfRational</a></span>
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<div class="ContSect"><span class="nocss"> </span><a href="chapA.html#X7D3D347987953F44">A.2 <span class="Heading">Periodic subadditive functions</span></a>
<span class="ContSS"><br /><span class="nocss"> </span><a href="chapA.html#X8466A4DC82F07579">A.2-1 RepresentsPeriodicSubAdditiveFunction</a></span>
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<h3>A <span class="Heading">Generalities</span></h3>
<p>Here we describe some functions which are not specific for numerical semigroups but are used to do computations with them. As they may have interest by themselves, we decribe them here.</p>
<p><a id="X7A5D608487A8C98F" name="X7A5D608487A8C98F"></a></p>
<h4>A.1 <span class="Heading">Bézout sequences</span></h4>
<p>A sequence of positive rational numbers a_1/b_1 < cdots < a_n/b_n with a_i,b_i positive integers is a <em>Bézout sequence</em> if a_i+1b_i - a_i b_i+1=1 for all iin {1,...,n-1}.</p>
<p>The following function uses an algorithm presented in <a href="chapBib.html#biBRos05">[BR08]</a>.</p>
<p><a id="X86859C84858ECAF1" name="X86859C84858ECAF1"></a></p>
<h5>A.1-1 BezoutSequence</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> BezoutSequence</code>( <var class="Arg">arg</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">arg</var> consits of two rational numbers or a list of two rational numbers. The output is a Bézout sequence with ends the two rational numbers given. (Warning: rational numbers are silently transformed into irreducible fractions.)</p>
<table class="example">
<tr><td><pre>
gap> BezoutSequence(4/5,53/27);
[ 4/5, 1, 3/2, 5/3, 7/4, 9/5, 11/6, 13/7, 15/8, 17/9, 19/10, 21/11, 23/12,
25/13, 27/14, 29/15, 31/16, 33/17, 35/18, 37/19, 39/20, 41/21, 43/22,
45/23, 47/24, 49/25, 51/26, 53/27 ]
</pre></td></tr></table>
<p><a id="X86C990AC7F40E8D0" name="X86C990AC7F40E8D0"></a></p>
<h5>A.1-2 IsBezoutSequence</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> IsBezoutSequence</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">L</var> is a list of rational numbers. <code class="code">IsBezoutSequence</code> returns <code class="keyw">true</code> or <code class="keyw">false</code> according to whether <var class="Arg">L</var> is a Bézout sequence or not.</p>
<table class="example">
<tr><td><pre>
gap> IsBezoutSequence([ 4/5, 1, 3/2, 5/3, 7/4, 9/5, 11/6]);
true
gap> IsBezoutSequence([ 4/5, 1, 3/2, 5/3, 7/4, 9/5, 11/3]);
Take the 6 and the 7 elements of the sequence
false
</pre></td></tr></table>
<p><a id="X7C9DCBAF825CF7B2" name="X7C9DCBAF825CF7B2"></a></p>
<h5>A.1-3 CeilingOfRational</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> CeilingOfRational</code>( <var class="Arg">r</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Returns the smallest integer greater than or equal to the rational <var class="Arg">r</var>.</p>
<table class="example">
<tr><td><pre>
gap> CeilingOfRational(3/5);
1
</pre></td></tr></table>
<p><a id="X7D3D347987953F44" name="X7D3D347987953F44"></a></p>
<h4>A.2 <span class="Heading">Periodic subadditive functions</span></h4>
<p>A periodic function f of period m from the set N of natural numbers into itself may be specified through a list of m natural numbers. The function f is said to be <em>subadditive</em> if f(i+j)<= f(i)+f(j) and f(0)=0.</p>
<p><a id="X8466A4DC82F07579" name="X8466A4DC82F07579"></a></p>
<h5>A.2-1 RepresentsPeriodicSubAdditiveFunction</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">> RepresentsPeriodicSubAdditiveFunction</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p><var class="Arg">L</var> is a list of integers. <code class="code">RepresentsPeriodicSubAdditiveFunction</code> returns <code class="keyw">true</code> or <code class="keyw">false</code> according to whether <var class="Arg">L</var> represents a periodic subAdditive function f periodic of period m or not. To avoid defining f(0) (which we assume to be 0) we define f(m)=0 and so the last element of the list must be 0. This technical need is due to the fact that positions in a list must be positive (not a 0).</p>
<table class="example">
<tr><td><pre>
gap> RepresentsPeriodicSubAdditiveFunction([1,2,3,4,0]);
true
</pre></td></tr></table>
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