java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 38
<!DOCTYPEhtml PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3div java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 225
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<> ="unc">tableclass""width=100%"<>td class="tdleft>< < =func"><table =func"width"%>tr>tdclass="dleft< class"unc"&<> class=code</ode isa numericalsemigroup returnsthe ofits minimal java.lang.StringIndexOutOfBoundsException: Index 115 out of bounds for length 0
<div class="ChapSects"><a href="chap3_mj.html#X7A9D13C778697F6C">3 <span class="Heading">
Basic operations with numerical semigroups
</span></a>
<iv ="ContSect"> 3 ]
</span3,pan=G"ap&;</>< =""EmbeddingDimensions);/pan>
<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<div class="span class="GAPprompt>/>< =GAPinput>MinimalGeneratingSystemS=MinimalGeneratingSystemOfNumericalSemigroup(;</pan>
<pan class="ContSS"><br /><span class="nocss"> </span><a true
<span>This java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 86
<classjava.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 143
= =func>table= ="%><tr>td =t<span==java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<classjava.lang.StringIndexOutOfBoundsException: Range [51, 16) out of bounds for length 241
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<java.lang.StringIndexOutOfBoundsException: Range [12, 5) out of bounds for length 137
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<span classdivclass="APprompt"gap;<span><span class
<span class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< "ontSS"br/spanclassn>nbspnpvar=Arg><varis java.lang.StringIndexOutOfBoundsException: Range [57, 56) out of bounds for length 503
<span class="ContSS"><br /><span class="nocss"> ><spanclass="APinput"
java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 85
<class"[,<> =Arg>/ar is a <class=Arg"ar functionshichlikesynonyms) <class"Arg><var- element var java.lang.StringIndexOutOfBoundsException: Index 195 out of bounds for length 0
<span class="ContSSdiv =<span class="APprompt><span<class"APinput<span =GAPpromptgt<>span =GAPinput"(,10/
<pan class[ 3952,35443 ,>span =APinput> :7817)<span
<span ="ContSS">br /< class="nocss"> </span><a href="chap3_mj.html<pan =Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 97
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X8022CC477E9BF678">3<panclass=GAPprompt">gap>13>"></a>p
<span class="ContSS"><br /><span class="nocss"> </span><
<=ContSS< >span =nocss"> </span><a href="[1421 ] [15, 18, 21 ],, [1821 ], [21 ]]
<span class=ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7802096584D32795">3.1-21 CocycleOfNumericalSemigroupWRTElement</a></spanjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 span="> >span ="ocss&bsp /span>ahref"hap3_mjh#X847BAD9480D186C0">3122java.lang.StringIndexOutOfBoundsException: Index 129 out of bounds for length 0
<span class="ContSS"><br /><span class="nocss">&java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [38, 5) out of bounds for length 141
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<span class="ContSS"><br /><span class="divclass=[ 0, 11, 7, 18,
<span class="ContSSspan =GAPprompt"gapg<spanspanclass"GAPinput">S :=<>var class"Arg"></var is a numerical semigroup, and <var class="Arg">m</var> is a nonzero elementclass=GAPprompt>>/pan spanclass">AperyListOfNumericalSemigroup(S);</span
<span class="ContSS"><br /><span class="nocss"> </span>/< class="xample">pre>
<panjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class<>a =lt with3 &t
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7CCFC5267FD27DDE">3< class=GAPprompt"gap&<>< class=GAPinput"()<java.lang.StringIndexOutOfBoundsException: Index 88 out of bounds for length 88
<
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AperyList</code>( <var class="Arg">S</var[2,1 ]
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#X7EC438CC7BF539D0">3.1-34 FundamentalGaps</java.lang.StringIndexOutOfBoundsException: Range [0, 133) out of bounds for length 110
<span class 11,0,1]
</div></div>
<div class="ContSect"><span class="tocline"><span class="nocss"> </span
</span>
<div class="ContSSBlock">
<span class="ContSS"><br /><span class
<span h5>3.1-20KunzPolytopeh5>
<span class="ContSS"><br /><span class="java.lang.StringIndexOutOfBoundsException: Range [0, 45) out of bounds for length 0
<span class="ContSS"><br /><span class="nocss"> </span><a href="chap3_mj.html#
<div
</div>
<>3<">
Basic operations with numerical semigroups
h3
<p>This chapter describes some basic functions to java.lang.StringIndexOutOfBoundsException: Range [0, 54) out of bounds for length 12
<div class="span class="GAP">gt/
<div class="func"><java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
>s . themultiplicityof<var=ArgNS</ar is positive to<ar class"Arg"</var> Depending on informationknown < java.lang.StringIndexOutOfBoundsException: Range [0, 240) out of bounds for length 99
<div class="example"><pre>
<span class="GAPprompt">gap></span0,00],[ 0, 34 ] [0,41 ]]
merical semigroup with 2 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">Multiplicity(last 3
<span class="GAPprompt">gap></span> <span class="GAPinput">S := NumericalSemigroup("modular", 7,53);java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<
<spandiv ="classfunc java.lang.StringIndexOutOfBoundsException: Range [45, 43) out of bounds for length 224 8
</pre></div>
<div class=f">table class=func"width"100%"<><tdclass=tdleft>code class="func">#8227; Generators<code>(<var class"Arg"Sp> largest nonnegative belonging to anumerical semigroup <pan class="impleMath">\(S\)</span> is the <em>Frobenius number</em> of <span class="SimpleMath">\(S\)</span>. If <span class="SimpleMath">\(S\)</span>
java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 238
< "nc>table class==func width="%>tr class"dleft"<code"">#; MinimalGeneratorscode> <varclass"Arg">S<var )</td<td class="dright">(&;attributenbsp)<td>/tr</4
< "" class =%< =t"< =>8227 /> <var"></><< "dright"(java.lang.StringIndexOutOfBoundsException: Index 207 out of bounds for length 1
<div class="func"><java.lang.StringIndexOutOfBoundsException: Range [0, 24) out of bounds for length 0
<"Arg"Svar is java.lang.StringIndexOutOfBoundsException: Range [38, 5) out of bounds for length 97
< ""pre
<span class""tableclass"unc"java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 218
M p><a id="X80431F487C71D67Bjava.lang.StringIndexOutOfBoundsException: Range [54, 52) out of bounds for length 62
<span div class=>fd java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 235
[ 11, 12, 13, 32, 53 ]
<span class=GAPprompt"gap&t;/span><span class=re>div class=>< class=unc"idth%<> =>code="unc>#8227 </code>( var class=Arg"S<var </d><td class"tdrightjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<Numerical semigroup with 3 generators>
<span class="GAPprompt">gap></span> <span class="GAPinput">GeneratorsOfNumericalSemigroup
<pan"">apgt;span ="&;/> <pan class="">(s);/>
[ 3,5]
<span class="GAPprompt">gap></span> <
[3
<<div class=< class=G"&;/>
true
<span class="GAPprompt">gap></span> <span class="GAPinput ="X82B1868F7A780B49" <p>The largest nonnegative intege belonging to a numerical semigroup <span class="SimpleMath">\(S\)</span> is the <em>Frobenius number</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<Numerical semigroup with 4
<span class="GAPprompt">gap&<div class=
true
<span classdivclass=example<iv=func>table=func""100"<< class"tdleft><odeclass=f>#; IsOrdinaryNumericalSemigroup/(<varpvarclass=rgNS/ isnumerical semigroup Dectects if the is ,thatis withlessthan deserts.p>
false
<span class="GAPprompt">gap></span> <span class="GAPinput">MinimalGenerators(s);</span>
[ 3, 5, 7 ]
<span class="GAPprompt">gap></spanjava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
[3, 5,7,15 java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 15
<java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<p><a id="X7884AE27790E687F" name="X7884AE27790E687F"></a><java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<h5>3.1-3 EmbeddingDimension</h5>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EmbeddingDimension</java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<div class="func"><table class="func"p<a = class=u"< class" width"100"<tr>td class=t">codeclass=func"&;ConductorOfNumericalSemigroupcode> class="Arg><var )<td<class"tdright>(nbsp;attribute&java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<p>code class="code">NS</code> is a numerical semigroup. It < class"">gapgt</pan <class=GAPinput">(umericalSemigroup3,,);/>
<div class="example">spanclass""gap&;/><panclass""ConductorOfNumericalSemigroup(,57)<span
<span java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 0
<Numerical semigroup with 4 generators>
<span class=GAPprompt">gap></span> <span class="GAPinput">EmbeddingDimension(s)java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 3
<span">gap></span> <span class="GAPinput">EmbeddingDimensionOfNumericalSemigroup(s);</span> 3
</pre></div>
<p><a id="</pre></div>
<h5>3.1-4 SmallElements</h5>
<div class="><a id="X7CCFC5267FD27DDE name="X7CCFC5267FD27DDE"></a></p>
<iv =f"< ="func width=%
<p><code class="code">NS</code> is a <53.-31 Holes</h5>
<div class="p<code ="ode></code isa .Itreturns< class=f>table =func =100%>tr< class=tdleft"< class"unc>#;HolesOfNumericalSemigroup(< class=Arg">S</var </d>td class="tdright">(&p><code class="code">S</code> is a numerical semigroup. Returns the set of gaps <span class="SimpleMath">\(x\)</span> of <code class="code">S</code> such that <span class="SimpleMath">\(F(S gap, where <span class="SimpleMath">\(F(S)\)</span> stands for the Frobenius number of <code class="code">S</code>.</p>
<an class="gap>/><span class=GAPinput"SmallElements(umericalSemigroup,,);/pan
[ 0, 3, 5 ]
<spanclass"GAPprompt">gap>&t;odular numerical satisfying5x 53lt=x>java.lang.StringIndexOutOfBoundsException: Index 65 out of bounds for length 65
</pre></div>
> idXjava.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
<h5>3.1-5 >3.1-32 LatticePatjava.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 0
<fjava.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 215
<< =><>java.lang.StringIndexOutOfBoundsException: Range [47, 46) out of bounds for length 381
<div=func>table class="width=100java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><code class="code">NS</code> is a numerical div class"">gapg<>spanclass"APinput">:NumericalSemigroup(16177172>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <spandiv =""><>
[0 3java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
<span class="gap&;/> <span =GAPinput">FirstElementsOfNumericalSemigroup(10,NumericalSemigroup(3,5,7));</span>
[ 0, 3, 5, </pre<span =GAPprompt>gt<span>< class="Type(S)</pan>
</pre></div>
<>a ="7java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
<h5>3.1-7 java.lang.StringIndexOutOfBoundsException: Range [0, 22) out of bounds for length 1
<div class=<iv class="unc>
<p><var"div class="func<able ="unc width"%>tr<tdclass="tdleft"><code class="func">‣ GenusOfNumericalSemigroup</code>( <var class="Arg">java.lang.StringIndexOutOfBoundsException: Index 154 out of bounds for length 124
<div class="example"><pre>
<span class="< =func">
<span class="div class="example><pre<class=""="00%<< class"tdleft"><code class="func">&#yle='color: green'>8227; GapsOfNumericalSemigroup</code>( <var class="Arg">NS</var> )</td><td class=p ><>of <span class="GAPprompt">gap><"java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 105
[ 0, 5, 7, 10, 12, 14, 15, 17, 19, 20, 21, java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
<span class="GAPprompt">gap></span> <span class="GAPinput">ElementsUpTo(ns,18);</span>
[ 0, 5, 7, 10, Pprompt"gap&;</java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
<class="gap&</><span class"">lementsUpTo(ns,27);&;odular numerical semigroup satisfying 5x mod 53 <= x >
[0, 5, 7,10, 12,14, 1517,1920, 21, 22,24, 25,26,27 ]
</pre></div>
<p><a id="X81A2505E8120F4D7"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<>code>18<span>\</span[ span>\<span>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
< class="xample"><pre>
<span class="GAPprompt">gap></span> <divclass="< class"" =100%"%>tr>java.lang.StringIndexOutOfBoundsException: Range [0, 58) out of bounds for length 0
<span class="p>If <span class="SimpleMath">\(l_1<\cdots <l_g\)</span> are <p>If <span class="SimpleMath">\(l_1<\cdots <l_g\)</span> are theGAPprompt"gap&;/span> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 68
<
<><a id="X84345D5E7CAA9B77 =java.lang.StringIndexOutOfBoundsException: Range [0, 52) out of bounds for length 0
<h5>3.1-10 NextElementOfNumericalSemigroup
<iv =func"><table class="func" width=<pan ="GAPprompt>apg;/pan span"">DesertsOfNumericalSemigroup(s)<spanjava.lang.StringIndexOutOfBoundsException: Index 100 out of bounds for length 100
<p><var class="Arg">S</var<The<m> < ajava.lang.StringIndexOutOfBoundsException: Range [55, 53) out of bounds for length 350
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := NumericalSemigroup(7,8,17);;</span>
<span class="java.lang.StringIndexOutOfBoundsException: Range [93, 22) out of bounds for length 106 14
<span class="GAPprompt">gap></span> <span<p>a id="" name""><a<p 17
<<h5>3.1-29 IsOrdinary/h5>
true
</pre></div>
div class"unc"<tableclass="func" width="100%"><tr><td class="tdleft"><code class="p><var class="Arg">NS</var> is a numerical semigroup. Dectects if the semigroup is ordinary, that is, with less than two deserts.</p[ 40, 41, 42 ]
<div class="func</code>( <var class="Arg">S</var>, <var class=Arg"rvar> )</td><td class="tdright">( operation )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">r</var> isjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div span clas=GAPprompt"gap>/> <span class=GAPinput">:NumericalSemigroup
<span class="GAPprompt">gap>/span<pan class==APprompt"gap&;span span class="GAPinput">IsOrdinary(s);</span>
<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 0 68
<span class="GAPprompt">gap></span> <span class="GAPinput">p<aid"834FE882A79096"name"X83D4AFE882A79096>/>/< 68
</pre></div>
<p><a id="X78F4A7A7797E26D4" name="X78F4A7A7797E26D4<divclass="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="java.lang.StringIndexOutOfBoundsException: Index 89 out of bounds for length 0
<h5>3.1-12 NumberElement_NumericalSemigroup</h5>
<div class="func" width"100%"<tr>td =tdleft"><code class="func">‣ div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAcuteNumericalSemigroup class=func"width=100"<<td class=tdleft">codeclass"func"&8227 </code> var class"rg>NS<var <td>td
<rical semigroup <ar =Arg"r</var>is . returns the < class"GAPprompt"gap&t;<span> <span class="GAPinput">WilfNumber(s);</span>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := NumericalSemigroup(7,8,17);;</span>
<java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0 53
</pre></div>
<p><a id="X867ABF7C7991ED7C" name="X867ABF7C7991ED7C"></a></p>span java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 80
<h5>3.1-13 java.lang.StringIndexOutOfBoundsException: Range [0, 19) out of bounds for length 2
<div class="func"><table</>
<p><var class="Arg">S</var> is a numerical semigroup. <p><var class="Arg">S</var> is a numerical semigroup. It
<div classh5>3.-2EliahouNumber</h5>
<span java.lang.StringIndexOutOfBoundsException: Range [0, 11) out of bounds for length 0
< =GAPprompt>><span> < classclass=GAPinput"iter:Iterator();/span>
<terator>
<span class="GAPprompt">gap></span> <span class="GAPinput">NextIterator(iter);</span> 0
<span class="GAPprompt">gap></span> <span class="GAPinput">NextIterator(iter);</span> 7
<span class="GAPprompt">gap></span> < 8
</pre></div>
<h5>3.1-14 span class="GAPpromptgap&t</pan><span class="APinput">EliahouNumber(s);</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="unc"<table class="func" width="100%"><tr>span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5);;</span>
<div class="func"><table class=java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<p><var class="Arg">S, T</var> are numerical java.lang.StringIndexOutOfBoundsException: Index 53 out of bounds for length 0
<p><a id="X7CB24F5E84793BE1" name="span class="GAPprompt">gap></span> <span classGAPinput"ProfileOfNumericalSemigroup(s);</div ="func"> ="unc"width"100"<java.lang.StringIndexOutOfBoundsException: Range [55, 54) out of bounds for length 301
<5>.115 AperyList</h5>
<div class="func"><table class="func" width= 3, 0, 0]
<div class="func"><table class="func" width="100%"><tr><td class="tdleftjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">n</var> is a positive element of <var class
<div class="example"><pre>
<span class="GAPprompt">gap></span> <spanp> output is div class="func"table ="unc"width"100"<> class=tdleft>codeclass=func>#8227 EliahouSlicesOfNumericalSemigroup/ode(< class=Arg"S<var <td>td =t<=Arg">S<var>,respectively. Let <span class="SimpleMath">(\)/span> and <span class="SimpleMath">\(r\)</span> be the quotient and nonpositive remainder
<span class=GAPprompt">gap></span> <span class="GAPinput">AperyList(S,12);</span>
[ 0, 13, 26, 39, 52, span ="GAPprompt">gap>/> span =GAPinput">(S,12)<span>
[ 0, 13, 26, 39, 52, 53, 54, 43, <pan class=GAPprompt>&t<span> span ="GAPinput"s:(16,17,,72;<span
<span class="GAPprompt">gap></span> <span class="GAPinput">First(S,x-> x mod 12 =1);</span> 13
/>/
<p><a [, ,6, 95 ] , 11312 ,[, ,
<h5>3.1-16 AperyList</h5>
<div class 14, ]
<div class="func"><table class="func" width="100<pre>/div>
<p
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">AperyList(java.lang.StringIndexOutOfBoundsException: Index 82 out of bounds for length 0
[ 0, 11, 7,1814 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">S <iv =func>tablejava.lang.StringIndexOutOfBoundsException: Index 26 out of bounds for length 0
< =GAPprompt"&;/><pan class">AperyListOfNumericalSemigroup(S;/span>
[ 0, 12, 13, 25, 26, 38, 39, 51, 52, 53, 32 ]
</pre>/>< java.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6
<p>
<h5>3.1-17 AperyList</h5>
<div class="func"><table class="func" width="100%"><tr><td <pan =">gap&;<span>< class="GAPinput>:NumericalSemigroup16,,,);/>
<div class="func"><table class="span class="GAPprompt">gap></spa< =GAPinput">Genus(s);</span>
<p><var class="Arg">S</var> is a numerical semigroup and <var class="Arg">m</var> is an integer. Computes the Apéry list of <var class="Arg">S</var> java.lang.StringIndexOutOfBoundsException: Index 153 out of bounds for length 2
<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 105
<span class=GAPprompt">gap></span> <span class="GAPinput">AperyList(s,11);</span>
[ 0, 10, 13, 19, 20, 23, 26, 27, 29, 32, 33, 36, 39, 42, 45, 46, 52, 55 ]
<span class="GAPprompt">gap></span> <span <pan "><>< ">java.lang.StringIndexOutOfBoundsException: Range [68, 67) out of bounds for length 78
[ 0, <>/
<span class="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0 18
<span class="GAPprompt">gap></span> <span class="GAPinput">AperyListOfNumericalSemigroupWRTInteger(s,10);</span>
[ 0, 13, 19, 26, 27, 32, 38, 45, 51, 54
<span class="GAPprompt">gap></span> <span class="<div class="func"><table class="func" width="100%"><tr><td c="tdleft"< class=func"&8227;;FundamentalGaps(< class"rg"S/var> )/td><td class"">(nbspa )/<><table<div>
[ 0, 51, 32, 13, 54, 45, 26, 27, 38, 19 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">AperyList(s,10);</span>
[ 0, 51,,32,1354,4526,27 ,19 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Length(last);</span> 10
</pre
<>< id"" name=X8022CC477E9BF678</>/>
h5>.1- </h5>
<div class="func[ 6 8,,13 ]
<p><var class="Arg">ap</var> is the Apéry list of a numerical semigroup. This function returnsspan class"">gap&;/><class"">S: ("modular,553;/span>
java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput ,,40 41,
[0 ,12 ,14 , 6 ,
<span class="GAPprompt">gap></span> <
[p< "java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
[ 7, 14, 21 ],, [ 9, 12, 15java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ 14, 21 ], [ 15, 18, 21 ],,, [ 18,">= =100"<>class"java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 219
</pre></div>
<p><a id"java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 239
<h5>3.1-19 KunzCoordinates</h5>
<""><java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 251
< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 270
<</var>. The second java.lang.StringIndexOutOfBoundsException: Range [153, 152) out of bounds for length 250
<p>Then the Apéry set of <var class="Arg">m</var> in <var class="Arg">S</var> has the form <span class="SimpleMath">\([0,k_1m+1,...,k_{m-1}m+m-1]\)</span>, and the output is the <span class="SimpleMath">\((m-1)\)</span>-uple <span class="SimpleMath">\([k_1,k_2,...,k_{m-1}]\)</span></p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);</span>
withjava.lang.StringIndexOutOfBoundsException: Range [44, 41) out of bounds for length 45
<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 14
,
<span class="GAPprompt">gap></span> <span class="GAPinput">KunzCoordinatesOfNumericalSemigroup(s);</span>
[ 2, 1 ]
<span class="GAPprompt">gap></span>
[ 1, 1, 0, 1 ]
<span class="GAPprompt">java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
[ 1, 1, 0, 1 ]
</pre>< java.lang.StringIndexOutOfBoundsException: Range [92, 91) out of bounds for length 655
<div class="func"><table class=<<=java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
<p><var class="Arg">m</var> is a java.lang.StringIndexOutOfBoundsException: Range [1, 41) out of bounds for length 25
< of the java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 302
<h5>3.1-22 FrobeniusNumber=>&;/java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 126
<div class="func"><table class="java.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 2
<div class="func"><table class="func" width="100%"><tr><td class="
<Thelargest nonnegativeinteger belongingtonumericalsemigroup span"impleMath">\(S\)</span> is the <em>Frobenius number</em> of<span class="GAPprompt">gap></span> <span class="GAPinput">AperyListOfNumericalSemigroup(S);</span>
<p<div =unc>java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 221
<div class="example"><pre>
<span class="GAPprompt">gap></span> 4
<span class="GAPprompt">gapjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 4
</pre></div>
<p><iv class=func">table ="funcjava.lang.StringIndexOutOfBoundsException: Index 36 out of bounds for length 0
<java.lang.StringIndexOutOfBoundsException: Range [30, 4) out of bounds for length 218
<div class="func"><table class="GAPprompt">gap>< "java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 99
This just <
<div class="example"><pre>
<span class="[ 0, 51, 32, 13, 54, ,iv =>> 5
<span class=< ="gap>/> < class=java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 136 5
</pre></div>
<pa iddivclass">table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ p><var class="Arg">S</var> is a numerical semigroup, and <var class="Arg">m</var> is a nonzero element of <var class The second argument is optional, and if missing it is assumed to be the multiplicity of <var class="Arg">S</var>.</p>
<h5>3.1-24 PseudoFrobenius</h5>
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<spanclass="APprompt>gap>/span> <span class="GAPinput">s:=div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of</>&bsp; a href="chap0_mj.html#contents">[Contents]</a> <a href="chap2_mj.html"&t;Numerical semigroup with 3 generators>
<p>An integer <span class="SimpleMath">\(z
<div class="func"><table java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<div
<Aemgapema <classS>(\<spanis a nonnegative integer not belonging to <span class="SimpleMath">\(S\)</span>. <code class="code">NS</code> is a numerical semigroup. Both return the set of gaps of <code class="code">NS</code>.</p>
<java.lang.StringIndexOutOfBoundsException: Range [0, 4) out of bounds for length 0 5,11)/>
[ 1, 2, 3, 4, 6, 8, 9, 13 ]
<span class="GAPprompt">gap></span> <<p><var class="Arg">S</var> is a numerical s and <varclass==Arg"m/var>isa nonzero element of <var class="Arg">S</var>. The output is the matrix <span class="SimpleMath">\(h(i,j)=(w(i)+w(j)-w((i+j)\bmod m))/m\)</span>, where <span class="SimpleMath">\(w(i)\)</span> is the smallest element in <
[ 1, 2, 4 ]
</pre></div>
<div
<h531- FrobeniusNumber</h5java.lang.StringIndexOutOfBoundsException: Index 31 out of bounds for length 31
<div class="example"><<div class="func"><table width"%>tr>td =tdleft>< class""&8227; FrobeniusNumberOfNumericalSemigroup>(<ar =Arg"NS</td>td class"tdright"(&bsp;ttribute </d<tr</table/div>
<>gap/ class">(NumericalSemigroup(,5,,));/pan> 0
<span class="GAPprompt">gap></span> <span class="GAPinput">Weight(NumericalSemigroup(4,5)); </span> 9
</p><code="ode>NS</code> is a numerical semigroup. It returns the Frobenius number of <code class="code">NS</code>. Of course, the time consumed to return a result may depend on the way the semigroup is given or on the knowledge already produced on the java.lang.StringIndexOutOfBoundsException: Index 267 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsOrdinary< ,,4142
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; IsOrdinaryNumericalSemigroup</code>( <var class="Arg">NS</var> )</td><td class="tdright">( property )</td></tr></table></java.lang.StringIndexOutOfBoundsException: Index 235 out of bounds for length 62
<p, that is, with less than two deserts.</p>
<div java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<><var class="Arg"NSvar>is anumerical semigroup.Dectects ifthe is acute,that is,it is either ordinary or its last desert (the one with the Frobenius number) has less elements than the preceding one (<a href="chapBib_mj.html#biBBr-acute">[Bra04]</a>).</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">IsAcute(s);</span>
java.lang.StringIndexOutOfBoundsException: Range [43, 4) out of bounds for length 4
</pre></div>
java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table class="func" width
<p><code class="code">S</code> is a numerical semigroup and <code class="code">p,q</code> are two coprime elements in
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; Genus</code>( <var<ivclass=""<ableclass"unc "%"<><tdclass=tdleft">="&8227 </code>(var ="Arg>/><td<class"(nbsp;roperty&;)</td>/tr>/table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227;<><varclass="rg">NS/var is anumerical semigroup. Dectects if the semigroup is ordinary, that is, with less than two deserts.</p>
<>code class="code">NS</code> is a numerical semigroup. It returns the number of gaps of <code class="code">NS</code>.</p>
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(16,17,71,72)span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(3,5,7);;</span>
<span class=" 80
<span class="GAPprompt">gap&paid"83" name="X83D4AFE882A79096"></a></p> 80
<span class="GAPprompt">gap>/span><panclass""> := ("odular,553;/>
<Modular numerical semigroup java.lang.StringIndexOutOfBoundsException: Index 35 out of bounds for length 23
<class"APprompt">><span <spanclass"APinput>(S);</span> 26
</pre></div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IsAcuteNumericalSemigroup</code>( <var><ar="><var> is a numerical semigroup. Dectects if the semigroup is acute, that is, it is either ordinary or its last desert (the one with the Frobenius number)
<div class="func"><java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 12
<div class"unc"<tableclassf "100%"<< class=tdleft>code =func>#; FundamentalGapsOfNumericalSemigroup(<class"rg">>S/ )/d<class=""(nbsp;ttributenbsp;<td>/tr<table></div
<p><code class="code">S</code> The <em>fundamental gaps</em> of <span class="SimpleMath">\(S\)</span> are those gaps that are maximal with respect to the partial order induced by division in <span class="SimpleMath">\({\mathbb N}\)</span>. It returns the set of fundamental gaps of <var class="Arg">S</var>.</p>
<div class="func"><table class="func" width="100%"><java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SpecialGapsOfNumericalSemigroup</code>( <var class="Arg">S</var> )<
<p>The <em>special gaps</em> of a numerical semigroup <span class="SimpleMath">\(S\)</span> are those fundamental gaps such div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LatticePathAssociatedToNumericalSemigroupclass="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LatticePathAssociatedToNumericalSemigroup</code>( <var class="Arg">S</var>, <var class="Arg">p</var>, <var class="Arg">q</var> )</td><td class="tdright">( attribute )</td></tr></table></java.lang.StringIndexOutOfBoundsException: Index 298 out of bounds for length 150
<p>Let <span class="SimpleMath">\(S\)</span> be a numerical semigroup, with conductor <span class="SimpleMath">\(c\)</span> and embedding dimension <span class="SimpleMath">\(e\)</span>. Denote by <span class="SimpleMath">\(l\)</span> the cardinality of the set of elements in <java.lang.StringIndexOutOfBoundsException: Range [2, 1) out of bounds for length 75
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EliahouNumber</code>( <var class="Arg">S</var> java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TruncatedWilfNumberOfNumericalSemigroup</code>( <var classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<<code class="code"S/code> is numerical semigroup. Let <span class="SimpleMath">\(c\)</span>, <span class="SimpleMath">\(m\)</span>, <span class="SimpleMath">\(s\)</span> and <span class="SimpleMath">\(l\)</span> be the conductor, multiplicity, number of generators smaller than <span class="SimpleMath">\(c\)</span>, and number of <iv class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FundamentalGapsOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">p< class="ode">S<code The<fundamental gaps</em> of <span class="SimpleMath">\(S\)</span> are those gaps that are maximal with respect to the partial order induced by division in <span class="SimpleMath"java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
unc" width="%>java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 235
<p><code class="code">S</code> is a numerical semigroup. Let <span class="SimpleMath">\(c\)</span> and <span class="SimpleMath">\(java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,7,9);;</span>
<class=&<>java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 100
[ 2, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroupWithGivenElementsAndFrobenius><> < a java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 350
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 100
[ 3, 0, 0 ]
</pre></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; EliahouSlicesOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<p>< class=c>/> is < class"impleMath"\c\</span < class"impleMath"\(\)<span> bejava.lang.StringIndexOutOfBoundsException: Range [148, 147) out of bounds for length 844
<div class="example"><pre>
<span class="GAPprompt=Gjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 84 span"java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 106
[ [ 5, 7 ], [ java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 104
<span class="GAPprompt">gap></span> < 40
[ 0,p< =X =java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62
</pre></div>
<div classjava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="chlinkbot">div >""java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 218
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