<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 describe them here.</p>
<p>A sequence of positive rational numbers <span class="SimpleMath">a_1/b_1 < ⋯ < a_n/b_n</span> with <span class="SimpleMath">a_i,b_i</span> positive integers is a <em>Bézout sequence</em> if <span class="SimpleMath">a_i+1b_i - a_i b_i+1=1</span> for all <span class="SimpleMath">i∈ {1,...,n-1}</span>.</p>
<p>The following function uses an algorithm presented in <a href="chapBib.html#biBRos05">[BR09]</a>.</p>
<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> consists 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>
<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>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBezoutSequence([ 4/5, 1, 3/2, 5/3, 7/4, 9/5, 11/6]);</span>
true
<span class="GAPprompt">gap></span> <span class="GAPinput">IsBezoutSequence([ 4/5, 1, 3/2, 5/3, 7/4, 9/5, 11/3]);</span>
Take the 6 and the 7 elements of the sequence
false
</pre></div>
<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>
<p>A periodic function <span class="SimpleMath">f</span> of period <span class="SimpleMath">m</span> from the set <span class="SimpleMath">N</span> of natural numbers into itself may be specified through a list of <span class="SimpleMath">m</span> natural numbers. The function <span class="SimpleMath">f</span> is said to be <em>subadditive</em> if <span class="SimpleMath">f(i+j)≤ f(i)+f(j)</span> and <span class="SimpleMath">f(0)=0</span>.</p>
<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 <span class="SimpleMath">f</span> periodic of period <span class="SimpleMath">m</span> or not. To avoid defining <span class="SimpleMath">f(0)</span> (which we assume to be 0) we define <span class="SimpleMath">f(m)=0</span> 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>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsListOfIntegersNS</code>( <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Detects whether <var class="Arg">L</var> is a nonempty list of integers.</p>
Die Informationen auf dieser Webseite wurden
nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit,
noch Qualität der bereit gestellten Informationen zugesichert.
Bemerkung:
Die farbliche Syntaxdarstellung ist noch experimentell.
