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products/Sources/formale Sprachen/GAP/pkg/numericalsgps/doc/ (GAP Algebra Version 4.15.1^©) Datei vom 30.7.2024 mit Größe 61 kB

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products/Sources/formale Sprachen/GAP/pkg/numericalsgps/doc/chap3_mj.html

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<div class="chlinkprevnexttop">&<panclass="ontSS< /span class=nocss>n;nbsp;/><href"chap3_mj.tml#X7F0EDFA77F929120">3.1-6 FirstElementsOfNumericalSemigroup</a></span>

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<div class=" class="ContSS><br >span =""&bsp;nbsp;<span>< href="chap3_mj.html#X7CB24F5E84793BE1">3.1-15 AperyList</a></span>
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<span class="ContSS"><br>span=>;<><href"h#X835C729D7D8B1B36">3.1-23 Conductor</a></span><span class="ContSS"><br /><span class="nocss">  </span><a href=">< class"ntSS</>java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 141
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<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X8688B1837E4BC079">3.1-26 Gaps</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7F71983880DF4B9D">span ""> >java.lang.StringIndexOutOfBoundsException: Range [33, 32) out of bounds for length 158
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7EB81BF886DDA29A">3.<span class"ContSS"><r/<<span class="ContSS"><br /><span classnbsp&<pan ".#835D7D8B1B36">3.1-23 Conductor</a></span>
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<div class="ContSect"><span class="tocline"><span class="nocss"> </span><a href="chap3_mj.html#X7EE22CA979CCAAB9">3.2 <span class="Heading">Wilf's conjecture</span></a>
</>
<div class="ContSSBlock">
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<alSemigroup</a></span>
<span class="ContSS"><br /><span class="nocss">  </span><a href="chap3_mj.html#X7846F90E7EA43C47">3.2-4 EliahouSlicesOfNumericalSemigroup</a></span>
</>/>
</div>

<h3>3 <span class="Heading">
                Basic span class="ContSS"><br />java.lang.StringIndexOutOfBoundsException: Range [39, 38) out of bounds for length 137
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<Thischapter describes some basic functions to deal with notable elements in a numerical semigroup. A section including functions to test Wilf's conjecture is >/>

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                    Invariants ">
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<p>In this span class="ContSS"><br /><span class="nocss"> <div<div>

<p><a id="X80D23F08850A8ABD" name="X80D23F08850A8ABD"></a></p>

<h5>3.1-1 Multiplicity</h5>

<<tableclass"unc"width"00%>tr> =tdleft">< "java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 133
<div        <span>/
<p><var>Thischapter  basicfunctions to deal with notable elements in a numerical semigroup. A section including functions to test

<div class="example"><pre>
<span class / java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
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<=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 88
3
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java.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for length 65
<<>/>
8
</

<p><< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 221

<h5>3class=> <class java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 138

<div class="func"><table class="func" width="100%"p "rg><ajava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 152
<div class="func"java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length 0
< =""java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 225
<div class="func"><able=="%><td java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 251
< )</td></tr></table></div>
<p><var class="Arg">S</var> is a numerical semigroup. <code class="code">GeneratorsOfNumericalSemigroup</code> returns a set of generatorsdiv>

<div <Numeri semigroup2 java.lang.StringIndexOutOfBoundsException: Range [42, 41) out of bounds for length 45
<span class3
&;java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 65
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<span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Index 42 out of bounds for length 1
<Numerical semigroup with 3 generators>
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,5
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[ 3, 5 ]
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[ 3, 5 ]
<span class="GAPprompt"
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true
<span class=="numerical . < class=java.lang.StringIndexOutOfBoundsException: Range [71, 1) out of bounds for length 58
false
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<spanspan classG"&t</> span class=11 ,,3253]
[ 3, 5, 7, 15 ] java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 103
</pre></div>

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<div class="example"><pre>
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<Numerical <classjava.lang.StringIndexOutOfBoundsException: Range [0, 20) out of bounds for length 8
<class">&java.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 91
3
<span class=" class="ContSSBlock"java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 143
sjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 137
</pre></span class="            </span>

<p

<h5>3.1-4 SmallElements</h5

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[0 , 5]
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2
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<h5>3.1-6 FirstElementsOfNumericalSemigroup</java.lang.StringIndexOutOfBoundsException: Range [0, 47) out of bounds for length 1

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[ 0, 3, 5, f<"S> is   < ""java.lang.StringIndexOutOfBoundsException: Range [105, 103) out of bounds for length 535
<pre>/div>

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68
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<h5><code>3.1-9 \

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<p><var "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 87

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<> <span class" java.lang.StringIndexOutOfBoundsException: Range [52, 50) out of bounds for length 89
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</pre></div>

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<div class="><code class="code<
< class></>  ><var class="Arg">S</var> is a numerical java.lang.StringIndexOutOfBoundsException: Range [69, 67) out of bounds for length 180

<div class="
<<div "java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
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14
<span class="GAPprompt">gap></span> <span class="GAPinput"> ,,5 java.lang.StringIndexOutOfBoundsException: Index 11 out of bounds for length 11
17
<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
true
</pre></div>

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<h5>3<>.-

<div java.lang.StringIndexOutOfBoundsException: Range [15, 13) out of bounds for length 170
<;< java.lang.StringIndexOutOfBoundsException: Range [151, 150) out of bounds for length 264
java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 217

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := NumericalSemigroup(7,8,17);;</span>
<span classGAPpromptg<span <span class"GAPinput"<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 79
68
<[,,,, 15]
68
/<

<pjava.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length 0

<h5>3.1-12 NumberElement_NumericalSemigroup</h5>

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<p><var class="Arg">S</var> is a numerical semigroup and <var class=" =""&;/java.lang.StringIndexOutOfBoundsException: Range [38, 37) out of bounds for length 103

<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 102
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 2
53
</pre></div>

<p><a java.lang.StringIndexOutOfBoundsException: Range [0, 8) out of bounds for length 0

h53.-3 java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 265

<div span class="GAPprompt">gap></span> <span class="[ ,,,,8 ,10 11, 12]
<p><var class="Arg">S</var> is a numerical semigroup. It returns></>

<div class="java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for length 0
<
< java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 87
&;>
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0
<span class="java.lang.StringIndexOutOfBoundsException: Index 19 out of bounds for length 0
7
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 88
8
</pre></div>

<p><a id="X7E6F5D6F7B0C9635" name="X7E6F5D6F7B0C9635">68

<h5>3.1-14 Difference</h5>

<classf"<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 244
<=<>"" ,< java.lang.StringIndexOutOfBoundsException: Range [66, 65) out of bounds for length 194
<java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 150

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">ns1 := NumericalSemigroup(5,7);;</span>
<java.lang.StringIndexOutOfBoundsException: Range [87, 5) out of bounds for length 105
<span class="GAPprompt">gap></span> <span class="GAPinput">Difference(ns1,ns2);</span>
[ 5, 10, 15, 17, 20, 27 ]
<(ns2,ns1);</span
[ 11, 18, 23 ]
< java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 110
[ 11, 18,"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 89
</pre></div>

<p><a id="X7CB24F5E84793BE1" name="<java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0

<h5>3.1-15 AperyList</h5>

<div class="func"><table
< =f"<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 273
java.lang.StringIndexOutOfBoundsException: Range [15, 13) out of bounds for length 503

<div class="example"><pre>
<span class="GAPprompt">gap></span  ">S := NumericalSemigroup("modular", 5,53);;</java.lang.StringIndexOutOfBoundsException: Index 109 out of bounds for length 102
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
[ 0 , ,39 ,53 ,43   ,11 java.lang.StringIndexOutOfBoundsException: Index 49 out of bounds for length 49
<span (S,12);</java.lang.StringIndexOutOfBoundsException: Range [115, 114) out of bounds for length 115
[ 0 13, 26,,52 ,54,43,32,33 < =A"<java.lang.StringIndexOutOfBoundsException: Range [27, 26) out of bounds for length 111
<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 97
13
</pre></div>

<< idjava.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62

<h5>3.1-16 AperyList

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<div class="func"><table class="func" width="100%"><tr><td class="<pan =""gap&< "">53]<>
>Ajava.lang.StringIndexOutOfBoundsException: Range [21, 19) out of bounds for length 174

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<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 0
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java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 102
[ 0, 12,"java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26
java.lang.StringIndexOutOfBoundsException: Index 85 out of bounds for length 12

<p

<java.lang.StringIndexOutOfBoundsException: Range [0, 3) out of bounds for length 0

<div class="func"><table class="div =">java.lang.StringIndexOutOfBoundsException: Index 24 out of bounds for length 12
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< java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 105
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Index 66 out of bounds for length 0
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[ 0, 10, 13, 19, 20, 23, 26, 27, 29, 32, 33, 36, 39, 42, 45, 46, 52, 55<"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 89
<span class17
18
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<p><a id="

<h5Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 111

<div class="func"><table class="func" div class="func"><table class="func" width><classjava.lang.StringIndexOutOfBoundsException: Range [74, 72) out of bounds for length 264
<pspan ="0, 13, 26, 39, 52>java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 217

<div class="example">
java.lang.StringIndexOutOfBoundsException: Range [11, 5) out of bounds for length 97
java.lang.StringIndexOutOfBoundsException: Range [52, 50) out of bounds for length 115
[ 0, 21, 12, 3, 14, <[ 0, 13, 2652,,, ,>:java.lang.StringIndexOutOfBoundsException: Range [90, 85) out of bounds for length 102
< "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 112
[ ,, [ java.lang.StringIndexOutOfBoundsException: Index 10 out of bounds for length 0
[ 7, 14, 21 ],, [ 9, 12, 15, 18, 21 ]java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[ ,21]  , , ,  ,21],   java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 52
</pre=java.lang.StringIndexOutOfBoundsException: Range [21, 19) out of bounds for length 163

<p><a id="X80B398537887FD87" name="X80B398537887FD87"></a></p>

<h5>3.1-19 KunzCoordinates</h5>

<div class="func"><table class="span class="ContSS"><br /><span class="nocss"java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 107
< =14 java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20
<< =S/>a java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 250

<p>Then the ,12 ,25 ,38 ,51 ,53 ]

">>
<span class="GAPprompts
&java.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 45
java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 88
[ 2, 1 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">KunzCoordinatesOfNumericalSemigroup
[,1]
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 90
[ 1, 1, 0, 1 ]
<span class="GAPprompt">gap></<p><var class="Arg">S</var> is a nume <=Arg</> an   é  of"rg>/>with to <java.lang.StringIndexOutOfBoundsException: Range [170, 169) out of bounds for length 876
[1,1, 0
</pre></div>

<p><a id="X7C21E5417A3894EC" name

< </>

<div class="func"><java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 0
p<"java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 54

<p>The Kunz coordinates<>

<div class="example"><pre>
>/><java.lang.StringIndexOutOfBoundsException: Range [0, 44) out of bounds for length 28
[ [ 1, 0, -1 ], [ 0, >
</pre></div>

<> = =java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 62

<h5>3.1-21<<java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2

"> java.lang.StringIndexOutOfBoundsException: Range [32, 30) out of bounds for length 270
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<div,51 ,13 ,45, 26> ajava.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 553
<java.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 99
<(s,3);</java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
[[0 ,0 , 0 ,4][0 ,
</pre>java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for length 45

<p><ajava.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

<h5>3.1-22 FrobeniusNumber</h5>

<div class=func"<table class=""width="100%"><tr><td class="tdleft"><code class<span class="GAPprompt"&;/ <=GAPinput"AperyList(,;/
<div class="func"><table class="func" width="100%"><tr><td java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0
integer not  <class"java.lang.StringIndexOutOfBoundsException: Range [98, 97) out of bounds for length 478

<p><code class="code">NS</code> is a java.lang.StringIndexOutOfBoundsException: Index 44 out of bounds for length 0

<div class="example>.- java.lang.StringIndexOutOfBoundsException: Range [48, 47) out of bounds for length 52
<="">>/pan span =GAPinput>robeniusNumber(<ivu"="100">java.lang.StringIndexOutOfBoundsException: Range [59, 58) out of bounds for length 225
4
<span class="GAPprompt">gap></span> <span class="GAPinput">FrobeniusNumberOfNumericalSemigroup(NumericalSemigroup(3,5,7));</span68
4
</pre></div>

<p><a id="X835C729D7D8B1B36" name="X835C729D7D8B1B36"></a>

<h5>3.1-23 p><var class=></> aspan class="GAPprompt">gap></span"java.lang.StringIndexOutOfBoundsException: Index 57 out of bounds for length 0

<ivclass=> class=f"=100"<><td"< ="java.lang.StringIndexOutOfBoundsException: Range [45, 43) out of bounds for length 218
<attribute )</td></tr></table></java.lang.StringIndexOutOfBoundsException: Index 238 out of bounds for length 238
<p>This is just a synonym of <code class="code"> FrobeniusNumberOfNumericalSemigroup</code> ( "java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 112

<=">pre
<span
5
<span  ,,21 , <java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 102
5
</pre></div>

<p><a id="X861DED207A2B5419

<h5>java.lang.StringIndexOutOfBoundsException: Index 4 out of bounds for length 0

<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<div class="func"><table java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
pjava.lang.StringIndexOutOfBoundsException: Range [14, 13) out of bounds for length 203

<p><code class="code">S</code> is a numerical semigroup. It returns java.lang.StringIndexOutOfBoundsException: Index 70 out of bounds for length 0

<div class="example"><pre>
java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 110
<Modular numerical semigroup satisfying 5x mod 53 <= x >
<span class="GAPprompt"><span class="GAPprompt">gap  é<classA"</><=rg"S java.lang.StringIndexOutOfBoundsException: Range [86, 85) out of bounds for length 286
[ 21, 40, <pan =>gt;>java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 88
<span =java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 88
[ 21, 40, 41, 42 ]
/>/iv

<p><a id="<< =Xjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 88

<>. <>

<div  java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 244
<classf<java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 233
< forcodeclass">(java.lang.StringIndexOutOfBoundsException: Range [79, 75) out of bounds for length 93

<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 110
&tMjava.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 65
<span class="GAPprompt">/java.lang.StringIndexOutOfBoundsException: Range [39, 37) out of bounds for length 110
4
<span class="GAPprompt">gap><
4
<<div>[ ,,java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14

<p><a id="X8688B1837E4BC079" name="java.lang.StringIndexOutOfBoundsException: Index 52 out of bounds for length 0

<h5>3.1-26 Gaps</h5>

<div class="func">< class">><java.lang.StringIndexOutOfBoundsException: Range [0, 37) out of bounds for length 0
<div < ">></><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 110
<p>A << 11 1823java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14

<div class="
<span class="GAPprompt">gap></java.lang.StringIndexOutOfBoundsException: Index 37 out of bounds for length 0
[1 2, ,4 ,8 13 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GapsOfNumericalSemigroup(NumericalSemigroup(3,5,7));</span>

</pre></div>

<p><a id="X7F71983880DF4B9D" name=

<<spjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 111

<div class="func"><span class="GAPpromptjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 85
<p>If <span class="java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0

<div class="example"><pre>
< class="GAPprompt">apgt<
0
<span class="GAPprompt">gap><
9
/></div>

<<="java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62

<h5>3.1-28  class="GAPpromptg<>java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 99

<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 216
<iv=">table java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 235
<p><var class="Arg">NS ="g;>=>java.lang.StringIndexOutOfBoundsException: Range [12, 4) out of bounds for length 237

<div class="example">p<id<>class">/> java.lang.StringIndexOutOfBoundsException: Range [43, 42) out of bounds for length 174
<span class="GAPprompt">gap>
<classjava.lang.StringIndexOutOfBoundsException: Range [27, 22) out of bounds for length 80
[ 3  java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
<"java.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 100
[ [ 1, 2 ], [ 4 ] ]

<p><aid"java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62

.- java.lang.StringIndexOutOfBoundsException: Range [23, 21) out of bounds for length 26

<
< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 236
<>="/>java.lang.StringIndexOutOfBoundsException: Range [54, 53) out of bounds for length 134

<p>This filter implies <code class="func">IsAcuteNumericalSemigroup

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s<4
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 83
false
</pre></div>

>id"<classf"java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 238

<h5>3.1-30 IsAcute</h5>

<div class="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
<divp<java.lang.StringIndexOutOfBoundsException: Range [9, 8) out of bounds for length 122
<p><var class="Arg">NS</var> is a numerical semigroup. Dectects if the semigroup is

<div pre></div>
<span class="GAPpromptc=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 91
<span class="class="GAPprompt</java.lang.StringIndexOutOfBoundsException: Range [39, 37) out of bounds for length 111
true
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 12

<p" java.lang.StringIndexOutOfBoundsException: Range [34, 33) out of bounds for length 62

<>.31/

<div class="java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
"> "java.lang.StringIndexOutOfBoundsException: Range [45, 43) out of bounds for length 233
<)-x\)</span> is also ajava.lang.StringIndexOutOfBoundsException: Range [217, 216) out of bounds for length 332

<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span  >&< <java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 110
[  ]
=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 99
<span class="GAPprompt">gapGjava.lang.StringIndexOutOfBoundsException: Range [24, 22) out of bounds for length 88
[ 2 ]
<<>

<p><a <a=Xjava.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62

<h5java.lang.StringIndexOutOfBoundsException: Index 51 out of bounds for length 0

<div<div class"classf"=100%< =>java.lang.StringIndexOutOfBoundsException: Range [80, 79) out of bounds for length 215
< coprime elements in=c"

<p>In this setting <code class="code">S</code> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

<p>The

<div class="example"><pre>
>;java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 105
<span class="GAPprompt">gap><p>
[ [ 0, 14 ],
  [ 7, 7 ], [ 8, 6 ], [ s , java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
  [ 14, 0 ] ]
</spanjava.lang.StringIndexOutOfBoundsException: Range [13, 11) out of bounds for length 77

<p><a id=

<h5>3.java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<f"
<<java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 234
<p><code class="code">NS</code> is java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<"<<div class="func"><table "00"java.lang.StringIndexOutOfBoundsException: Range [66, 64) out of bounds for length 233
<><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 105
<span class="GAPprompt">gap></span> <span class="GAPinput">Genus(s);</span>
80
<pan =>;/ <pan class="GAPinput">GenusOfNumericalSemigroup(s);</java.lang.StringIndexOutOfBoundsException: Range [1, 97) out of bounds for length 89
80
<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 110
lMjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 65
<span class= ,5 ,12 , ,17 ,20 ,22 ,25 ,27
26
</pre></div>

<p><a id="X7EC438CC7BF539D0"

<h5>3.1-34 FundamentalGaps<

java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 223
<div
<p><div"java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bounds for length 26

<div class="example"><pre>
<span class=">gt<
[ 6, 8, 9, 13 ]
<span class="GAPprompt">gap&<span class="GAPprompt">gap>=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 105
<Modular numerical semigroup satisfying 5java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1
<span class="GAPprompt">p=java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
[ 16, 17, 18, 19,27,28<53- java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for length 23
<gap&< "java.lang.StringIndexOutOfBoundsException: Range [61, 60) out of bounds for length 97
[ 1, 2, 3, 4<class></ java.lang.StringIndexOutOfBoundsException: Range [53, 52) out of bounds for length 170
  ,,0 , java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 22
<span class="GAPprompt">gap></span> <span class="GAPinput">Gaps
[,,,,,8 ,  java.lang.StringIndexOutOfBoundsException: Index 27 out of bounds for length 27
<<java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 12

<p><a id="X803D550C78717A7C" name=">>

<h5>3.1-35 SpecialGaps</h5>

<div class="<class=java.lang.StringIndexOutOfBoundsException: Range [17, 16) out of bounds for length 265
<div class="func"><<div class="func"><table
m> java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 350

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := GAPprompt">gap></span> <span class="GAPinput">NextElementOfNumericalSemigroup(S,9);</span>
<Modular numerical semigroup java.lang.StringIndexOutOfBoundsException: Index 40 out of bounds for length 4
<span class="<<"java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
[ ,41  java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for length 14
< =f" java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 265
[40,41,42]
</pre></div>p>   < ">java.lang.StringIndexOutOfBoundsException: Range [73, 67) out of bounds for length 163

<p><a id="X7EE22CA979CCAAB9" name="X7EE22CA979CCAAB9"></a></p>

<h4>3.2 <span class="Heading java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 26

<p>Let < <=<><java.lang.StringIndexOutOfBoundsException: Range [45, 44) out of bounds for length 83

<p><a id="X78C2F4C77FB096F0" name=java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2

<h5>3.2-1 WilfNumber</h5>

<java.lang.StringIndexOutOfBoundsException: Range [12, 10) out of bounds for length 218
<div class="func"> =f"><""java.lang.StringIndexOutOfBoundsException: Range [44, 43) out of bounds for length 215
<p><code class=" class="func"><table=100">class=java.lang.StringIndexOutOfBoundsException: Range [73, 72) out of bounds for length 265

<div class="example"><pre>
<java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length 104
=>&<java.lang.StringIndexOutOfBoundsException: Range [40, 37) out of bounds for length 83
96
<span class="GAPprompt">gap&java.lang.StringIndexOutOfBoundsException: Index 30 out of bounds for length 26
<span class="GAPprompt">gap></span> <span java.lang.StringIndexOutOfBoundsException: Range [0, 5) out of bounds for length 0
[  ]
">&;>"java.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 126
70
</pre></div>

<p><a id="X80F9EC9A7BF4E606" name="X80F9EC9A7BF4E606"></ajava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0

<>.- java.lang.StringIndexOutOfBoundsException: Range [25, 23) out of bounds for length 28

< ""<java.lang.StringIndexOutOfBoundsException: Range [25, 24) out of bounds for length 221
<div class;java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 16
<p><code class="code">S</code> java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 1

<div class="example"><pre>
<span classp< =java.lang.StringIndexOutOfBoundsException: Range [29, 27) out of bounds for length 62
<">&;/ =Gjava.lang.StringIndexOutOfBoundsException: Range [62, 60) out of bounds for length 86
-1
<span class=" ">java.lang.StringIndexOutOfBoundsException: Range [31, 30) out of bounds for length 244
<span class="GAPprompt">gap> java.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 78
4
< ]

<pa X7B45623E7D539CB6=java.lang.StringIndexOutOfBoundsException: Index 39 out of bounds for length 0

<>23java.lang.StringIndexOutOfBoundsException: Range [41, 37) out of bounds for length 42

<div class="func"><table class="func" width="[2class=java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 89
<java.lang.StringIndexOutOfBoundsException: Range [22, 11) out of bounds for length 89

<java.lang.StringIndexOutOfBoundsException: Range [5, 4) out of bounds for length 26
<span class="java.lang.StringIndexOutOfBoundsException: Index 22 out of bounds for length 12
<="GAPinput>java.lang.StringIndexOutOfBoundsException: Range [92, 89) out of bounds for length 100
java.lang.StringIndexOutOfBoundsException: Index 8 out of bounds for length 8
<span class="java.lang.StringIndexOutOfBoundsException: Index 18 out of bounds for length 0
<span class
, ,
</pre></div>

<p><a id="X7846F90E7EA43C47" name="X7846F90E7EA43C47"></a></p>

<324java.lang.StringIndexOutOfBoundsException: Range [45, 43) out of bounds for length 48

java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 241
<""S/,java.lang.StringIndexOutOfBoundsException: Range [218, 216) out of bounds for length 844

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,7,9);;                                     </span"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 85
<span class="GAPprompt">gap></span> <span class="GAPinput">EliahouSlicesOfNumericalSemigroup(s)java.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
[ [ 5, 7 ], [ 9, 10, 12 ] ]
<span class="GAPprompt">gap></span , ]
[ 0, 5, 7, 9, 10, 12, 14 ]
</pre></div>

<div class  ,, java.lang.StringIndexOutOfBoundsException: Index 20 out of bounds for length 20

<div class="chlinkbot"><span div class="func"><table class="func"< classjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for length 102

/
<p class="
</body
</html>

Messung V0.5 in Prozent

'>var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; HolesOfNumericalSemigrouplt; numerical  satisfying5&;  gt;
<p><code class="code">S</code> is a numerical semigroup. Returns the set of gaps <span class="SimpleMath">\(x\)</span> of <java.lang.StringIndexOutOfBoundsException: Index 127 out of bounds for length 97

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for length 102
<span class="GAPprompt">gap></span> <span class="GAPinput">Holes(s);</span>
[  ]
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=
<h5>1- />
[ 2 ]
</pre></div>

<p><a id="X794E615F85C2AAB0" name="X794E615F85C2AAB0"></a></p>

<h5>3.1-32 LatticePathAssociatedToNumericalSemigroup</h5>

<div class="func"><table "java.lang.StringIndexOutOfBoundsException: Range [37, 36) out of bounds for length 301
<p><code class="code">S</code> is a numerical semigroup and <code class="code">p,q</code> are two coprime elements in <code class="code">S</code>.</p>

<p>In this setting <code class=dclass>>

<p>The output is a path in <span class="SimpleMath">\(\mathbb{N}^2\)</span> such thatlt;numericaljava.lang.StringIndexOutOfBoundsException: Range [32, 31) out of bounds for length 65

<java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 26
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=java.lang.StringIndexOutOfBoundsException: Index 75 out of bounds for length 12
<classGAPprompt>g;/>< =GAPinput"LatticePathAssociatedToNumericalSemigroup;<span>
[ [ 0, 14 ], [ 1, 13 ], [ 2, 12 ], [ 3, 11 ], [ 4, 10 ], [ 5, 9 ], [ 6, 8 ],
  [ 7, 7 ], [ 8, 6 ], [ 9, 5 ], [ 10, 4 ], [ 11, 3 ], [ 12, 2 ], [ 13, 1 ],
  [ 14, 0 ] ]
/>

<p><a id="X7E9C8E157C4EAAB0" name="X7E9C8E157C4EAAB0"></a></div=><table class="func" width="100%"< java.lang.StringIndexOutOfBoundsException: Range [65, 64) out of bounds for length 218

java.lang.StringIndexOutOfBoundsException: Index 43 out of bounds for length 21

<class=f"< class=func"width"00%>tr>< class="<code class=func>#8227;Genus/code(<""NS/> )/>tdclass"dright>&bsp;ttributen;)/>/>java.lang.StringIndexOutOfBoundsException: Range [208, 207) out of bounds for length 214
<div class="java.lang.StringIndexOutOfBoundsException: Index 16 out of bounds for length 0
<p><code class="code">NS</code> is a numerical semigroupdiv ">>

<div <span class"APprompt">gap></span> <span class="GAPinput">WilfNumber(s);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(span ="APprompt>></pan><span class=""l:=NumericalSemigroupsWithGenus(0);<>
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Range [0, 67) out of bounds for length 4
80
<spanjava.lang.StringIndexOutOfBoundsException: Index 2 out of bounds for length 2
80
<span class="GAPprompt">gap></span> <span class="GAPinput">S :p< =java.lang.StringIndexOutOfBoundsException: Range [28, 27) out of bounds for length 62
<Modular numerical semigroup satisfying 5x mod 53 <=java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0
<span class="GAPprompt">gap></span> <span class="GAPinput">Genus(S);</span>
26
</pre></div>

<p><a id="X7EC438CC7BF539D0" name="X7EC438CC7BF539D0"></a></p>

<h5>3.1-34 FundamentalGaps</h5>

<div p><"<> ajava.lang.StringIndexOutOfBoundsException: Range [46, 45) out of bounds for length 972
< java.lang.StringIndexOutOfBoundsException: Range [11, 10) out of bounds for length 243
<><ode"</>The em>java.lang.StringIndexOutOfBoundsException: Range [51, 50) out of bounds for length 311

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">FundamentalGaps(NumericalSemigroup(5,7,11));</span>
[ 6, 8, 9, 13 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">S := NumericalSemigroup("modular", 5,53);</span>
<Modular numerical semigroup satisfying 5x mod 53 <= x >
<span class="GAPprompt">gap></span> <span1
[ ]
<span class="GAPprompt">gap></span> <span class="GAPinput">GapsOfNumericalSemigroup(S);</span>
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 15, 16, 17, 18, 19, 20, 21, 27, 28, 29,
  30, 31, 40, 41, 42 ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Gaps(NumericalSemigroup(5,7,11));</span>
[ 1, 2, 3, 4, 6, 8, 9, 13 ]
</pre></div>

<p><a id="X803D550C78717A7C" name100">tr><td class="tdleft"><code class="func">‣ ProfileOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>

<h5>3.1-35 SpecialGaps</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; SpecialGaps</code>( <var class="Arg">S</var> )</td><td class="tdright">( java.lang.StringIndexOutOfBoundsException: Index 0 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; SpecialGapsOfNumericalSemigroupspan =GAPprompt">gapgt;<span <span class="GAPinput">ProfileOfNumericalSemigroup(s);</span>
<p>The emspecial gaps/em>of a numerical semigroup <span class="SimpleMath">\(S\)</span> are those fundamental gaps such that if they are added to the given numerical semigroup, then the resulting set is again a numerical semigroup. <code class="code">S</code> is a numerical semigroup. It returns the special gaps of <var class="Arg">S</span class="GAPprompt">gap></span> <span class="GAPinput">ProfileOfNumericalSemigroup(s);</span>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := NumericalSemigroup("modular", 5,53);</span>
<Modular numerical semigroup satisfying 5xcode"ode"><code>  anumerical semigroup.Letspan=S"()/> and span =S>(<span  the conductor and multiplicity of <var class="Arg">S</var>, respectively. Let <span class="SimpleMath">\(q\)</span> and <span class="SimpleMath">\(r\)</span> be the quotient and nonpositive
<span class"APprompt">gap></span> <span class="GAPinput">SpecialGaps(S);</span>
[<span class="GAPprompt">gap></span> <span class="GAPinput">EliahouSlicesOfNumericalSemigroup(s);</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">SpecialGapsOfNumericalSemigroup(S);</span>
[ 40,41, 42
</pre></div>

<p>aid="7EE22CA979CCAAB9"name="X7EE22CA979CCAAB9"></a></p>

<h4>3.2 <span class="Heading">Wilf's conjecture</span></h4>

<p>Let <span class="SimpleMath">\(S\)</span> be a numerical semigroup, with conductor <span class

<p><a id="X78C2F4C77FB096F0" name="X78C2F4C77FB096F0"></a></p>

<h5>3.2-1 WilfNumber</h5

<class="func">table class="unc"width="100%"><tr><td class="tdleft"><code class="func">&#tyle='color: green'>8227; WilfNumber</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; WilfNumberOfNumericalSemigroup</code>( <var class="Arg">S</var> )</<html>
<p><code class="code">S</code> is a numerical semigroup. Let <span class="SimpleMath">\(c\)</span>, <span class="SimpleMath">\(e\)</span> and <span class="SimpleMath">\(l\)</span> be the conductor, embedding dimension and number of elements smaller than <span class="SimpleMath">\(c\)</span> in <var class="Arg">S</var>. Returns <span class="SimpleMath">\(e l-c\)</span>, which was conjetured by Wilf to be nonnegative.</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s := NumericalSemigroup(13,25,37);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">WilfNumber(s);</span>
96
<span class="GAPprompt">gap></span> <span class="GAPinput">l:=NumericalSemigroupsWithGenus(10);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">Filtered(l, s->WilfNumber(s)<0);</span>
[  ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Maximum(Set(l, s->WilfNumberOfNumericalSemigroup(s)));</span>
70
</pre></div>

<p><a id="X80F9EC9A7BF4E606" name="X80F9EC9A7BF4E606"></a></p>

<h5>3.2-2 EliahouNumber</h5>

<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> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; TruncatedWilfNumberOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="code">S</code> is a 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 elements smaller than <span class="SimpleMath">\(c\)</span> in <var class="Arg">S</var>, respectively. Let <span class="SimpleMath">\(q\)</span> and <span class="SimpleMath">\(r\)</span> be the quotient and nonpositive remainder of the division of <span class="SimpleMath">\(c\)</span> by <span class="SimpleMath">\(m\)</span>, that is, <span class="SimpleMath">\(c=qm-r\)</span>. Returns <span class="SimpleMath">\(s l-qd_q+r\)</span>, where <span class="SimpleMath">\(d_q\)</span> corresponds with the number of integers in <span class="SimpleMath">\([c,c+m[\)</span> that are not minimal generators of <var class="Arg">S</var>.</p>

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroupWithGivenElementsAndFrobenius([14,22,23],55);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">EliahouNumber(s);</span>
-1
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroup(5,7,9);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">TruncatedWilfNumberOfNumericalSemigroup(s);</span>
4
</pre></div>

<p><a id="X7B45623E7D539CB6" name="X7B45623E7D539CB6"></a></p>

<h5>3.2-3 ProfileOfNumericalSemigroup</h5>

<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&#an style='color: green'>8227; ProfileOfNumericalSemigroup</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<p><code class="code">S</code> is a numerical semigroup. Let <span class="SimpleMath">\(c\)</span> and <span class="SimpleMath">\(m\)</span> be the conductor and multiplicity of <var class="Arg">S</var>, respectively. Let <span class="SimpleMath">\(q\)</span> and <span class=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

<iv ="example"<pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">S := NumericalSemigroup("modular", 5,53);</span>
&Modular  semigroup satisfying 5x mod 53 <= x >
<span class="GAPprompt">gap></span> <span class="GAPinput">SpecialGaps(S);</span>
[ 40, 41, 42 ]
<span <iv class="example"><pre>
[ 40, 41, 42 ]
</pre></div>

<p><a id="X7EE22CA979CCAAB9" name="X7EE22CA979CCAAB9"></a></span =""gap&t<span spanclass=GAPinput"(s,16,17);/>

<h4>3.2 <span class="Heading">Wilf's conjecture</span></h4>

<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

<p><a id="X78C2F4C77FB096F0" name="X78C2F4C77FB096F0"></a></p>

<h5>3.2-1 WilfNumber</h5</pre>>/div>

<div class"func">tr><td class="tdleft"><code class="func">‣ WilfNumber</code>( <var class="Arg">S</var> )</td><td class="tdright">( attribute )</td></tr></table></div>
<div class="func"><table class="func" width
<p><code class="code">S</code> is a numerical semigroup. Let <iv "unc">tableclass=func ="00"<>tdclass"dleft">"unc"&8227;</> var class=Arg><var> )/d< class=tdright"(nbsp;ttribute&bsp;)/td>/r></table></div>

<divclass=example"<pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s := NumericalSemigroup(13,25,37);;</span>
<Gjava.lang.StringIndexOutOfBoundsException: Range [30, 22) out of bounds for length 83
96
<classG"gap/ spanclass"APinput>:=(0;<span
<span class="GAPprompt">gap></span> <span class="GAPinput">Filtered(l, s->WilfNumber(s)<0);</span>
[  ]
<span class="GAPprompt">gap></span> <span class="GAPinput">Maximum(Set(l, s->WilfNumberOfNumericalSemigroup(s)));</span>
70
</pre></div>

<p>aid=X80F9EC9A7BF4E606" name="X80F9EC9A7BF4E606"></a></p>

<h5>3.2-2 EliahouNumber</h5>

<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

<div class="example"><pre>
<span class="GAPprompt">gap></span> <span class="GAPinput">s:=NumericalSemigroupWithGivenElementsAndFrobenius([14,22,23],55);;</span>
<span class="GAPprompt">gap></span> <span class="GAPinput">EliahouNumber(s);</span>
-1
<span class="GAPprompt">gap></span> <span [ 16, 17, 18, 19, 27, 28, 29, 30, 31, 40, 41, 42
<span class="GAPprompt">gap></span> <span class="GAPinput">java.lang.StringIndexOutOfBoundsException: Range [0, 101) out of bounds for length 76
4
</pre></div>

<p><a id="X7B45623E7D539CB6" name="X7B45623E7D539CB6"></a></p>

<h5>3.2-3 ProfileOfNumericalSemigroup</h5>

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>

<p><a id="X7846F90E7EA43C47" name="X7846F90E7EA43C47"></a></p>

<h5>3.2-4 EliahouSlicesOfNumericalSemigroup</h5>

<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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