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<?xml version="1.0" encoding="UTF-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 <?xml version=="1.0"encoding="UTF-8"?> "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en"> <> <script type=""http://wwww3./TRxhtml1/DTD-."java.lang.StringIndexOutOfBoundsExcept =java.lang.StringIndexOutOfBoundsException: Range [19, 18) out of bounds for len </script> <meta -=-"content"/ =-8 > <meta= "java.lang.StringIndexOutOfBoundsException: Range [46, 43) out of bounds fo <meta name=" ="java.lang.StringIndexOutOfBoundsException: Range [20, 19) out of bou <linksjava.lang.StringIndexOutOfBoundsException: Range [22, 21) out of bounds for <script src="manual.js" type/> <script type="text/javascript">overwriteStyle();</body= "java.lang.StringIndexOutOfBo < <body class="chap9" onload="jscontent()"> <div class=" <div class="chlinkprevnexttop"> <a href="chap0_mj.html">[Top of Book]</a <p id="mathjaxlink" classp>a id"7DBA3A7E81C71A64"name"DBA3A7E81C71A64"<a>p <p>"></a></p> <div class="ChapSects"><a< classjava.lang.StringIndexOutOfBoundsException: Range [21, for(-)""\^\/span</>/> </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.java.lang.StringIndex <span class="ContSS"><br /><span class="nocss"> </span><a href="chap9_mj.html#X85A5A6967D94246 </div></div> <div class="ContSect"><span <>> </span> <div class="ContSSBlock"> <span class="ContSS"><br /><span << <div /> <=>(\)/> </span></a> </span> < "> <span class/<> << </div"java.lang.StringIndexOutOfBoundsException: Range [24, 23) out of bounds for <h3>9 <span class="Heading">Cat<span class="SimpleMath">\(^3\)</span>-groups and Crossed cubes< <p>The term <em>4</iv/div> <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <p><a id="X7CC52AF4840F478E" name="X7CC52AF4840F478E"></a></p> 491 span=H>for(-span=>^\<-/<> <p>We shall java.lang.StringIndexOutOfBoundsException: Index 15 out of bounds for le <p class="center"> \vcenter{\ amp d]&t+5> a[ ;00> java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 a[ ;0.> &;& ;; ar lt+.> \ar[dddd] & \ar[dr] <+0.6ex> amp a[];.ex; [] lt0.gt & \d]l;06x; & a; [ ;0.gt addddl;java.lang.StringIndexOutOfBoundsException: Index 64 out of b a &;0.>a[]&+0> \ar[ul] <\[ l;05xg;\rrrr&;0.> a; amp a arllll&;0.g; \ar[ & & & & l]l;06> ul lt+.& ar]+.& \ & & & &\ar[ull;0.> a[ lt+.gt; \\ & & && a;& \ & M \ar[uuuu] <+0.6ex> \ar[rrrr] <+& java.lang.StringIndexOutOfBoundsException: Range [14, 11) out of \[ lt+06xg; &;& amp ;La[ lt+.& \ar[llll] <+0.6ex> +.e& && a;a;a; ar +.exgt &\l]&+.&; \java.lang.StringIndexOutOfBoundsException: Range [13, 10) out of bounds for len \ar[ul] <+0 amp ar l0.g; &mp ;a;a a[]l;06g; \ar[llll] <+0.6 \ar[ul] <+0.5ex> \ar[u&t+.e&t \r[l <+0.ex>\ru]&t+.0ex&t \ }} \]</p> <p>By definition, <span class="SimpleMath">\(\calE\)</span> is generated by} <l <p definition< =SimpleMath>(\<>is by <=">\^\<>java.lang.StringIndexOutOfBoundsException: Rang </li> <li><p><em>left</em><span class="SimpleMath">\((\calE)\)</span>, generated by <span class="SimpleM </li> </ul> <p>java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <p><a id="X7828496F7D72E232" <>the, all java.lang.StringIndexOutOfBoundsException: Range [74, 73) out of bounds f <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <div class="func"><table java.lang.StringIndexOutOfBoundsException: Range [0, 30) out <ivclass"unc"< class"func" width"%">tr<class"tdleft"<class"func"&;IsCat3Group/code(<var ="r <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>The global functions <code class="code">Cat3Group</code> and <code class="code">PreCat3Group< <p>In subsection <code class="func">AllCat2GroupsIterator</code> (<a href="chap8_mj.html#X7E <div class="example"><pre> <span class="GAPprompt">gap></span> <java.lang.StringIndexOutOfBoundsException: Range [0 <panjava.lang.StringIndexOutOfBoundsException: Range [12, 11) out of bounds for l <"java.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for len <span class="GAPprompt">gap></span> <span class="GAPinput">C3Ga := Cat3Group( C68, C811 );</span> cat3-group with generating (pre-)cat1-groups: 1 : [d12 => Group( [ (), (1,6)(2,5)<span class"GAPprompt">gap;</span> < class"APinput">C68 := Cat2G 2 : [d12 => Group( [ (1,4)(2,5)(3,6), (1,3)(4,6) ] )] 3 :[&;([ 15,),,4,14(23(,)])java.lang.StringIndexOutOfBoundsException: Index 60 out o < "">>/> => =(6,8,11]);</span> <span class="2 : [d12 => Group(4(,)36,(3(,) ] true <>div <p><a =Gjava.lang.StringIndexOutOfBoundsException: Range [23, 22) out of bounds for <h5>9.1-2 Front3DimensionalGroup</h5> <iv">= =100%>tr<class=>code =>#;</code> var =""C3</>)/>td =tjava.lang.StringIndexOutOfBoundsExce <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <div class="func"><table java.lang.StringIndexOutOfBoundsException: Index 28 out of b <iv "> =" =100%>><class> f"&8227;</code> =></> )/td> =tjava.lang.StringIndexOutOfBoundsException: Range [ <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <p>The six faces of a cat<span class="SimpleMath">\(^3\)</span>-group are stored as these attribu <div class="example"><pre> <spandiv =f">table class="func =100"<< =tdleft"><code class="func">&#>8227; Back3DimensionalGroup</code>( <var class="Arg">C3</var> )</td><td class="tdright">( attrib <class"APprompt"gap&t<> spanclass""Up3DimensionalGroupC3Ga =;/> true </pre></div> <p><a pre></div < <p>Once<Once thelist< class"ode"CatnGroupListsG.<code has beenobtained wemay seek span class" <p><a id="java.lang.StringIndexOutOfBoundsException: Index 14 out of bounds for leng <h5>9.2-1 AllCat3GroupTriples</h5> <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">&# <div class="func"><table class="func" width="100%"><tr><td class="tdleft"><java.lang.StringIn <p[5 7,11 ] <p>As yet =GAPprompt"gap><span> <span class="GAPinput">Cat3Group( alld12[5], alld12[7], alld12 <div cat3-group with generating (recat1groups: <span class="GAPprompt2 [12=gt; Group([(1,)2,5)(,) (6(35 ] <span class="GAPprompt">gap></span> <span class="GAPinput">CatnGroupNumbers( d12 ).cat3; </spa 94 <span class="GAPprompt">gap></span> <span class="GAPinput">triples[46];</span> 7,11 java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for length <spanclass=GAPprompt>></><span class="APinput"java.lang.StringIndexOutOfBoundsExcept <span class="GAPprompt">gap></span groupjava.lang.StringIndexOutOfBoundsException: Range [16, 15) out of bounds for =&t ([( 14),)56)]] 2 : [d12 =>/pan<> 3 : [d12 =>> this java.lang.StringIndexOutOfBoundsException: Range [89, 87) out of b </pre></div> <p><a id="X7F8538B580847268" name="X7F8538B580847268"></a></p> <h4>9.3 <span class="Heading"> Definition and constructions for cat<span class="java.lang.StringIndexOutOfBoundsExcep <<> <p>In this chapter and the java.lang.StringIndexOutOfBoundsException: Index 35 out of b <p>A <em>cat<java.lang.StringIndexOutOfBoundsException: Range [0, 17) out of bounds fo <ul> <li><p><span class \o{ \\\}) </li> <li><p>Group homomorphisms forming<pjava.lang.StringIndexOutOfBoundsException: Index 6 <p class="center">\[ \calC_{A,i} ~=~ (e_{A,i};\; t_{A,i},\; h_{A,i} \ :\ _A \oG_ \setminus \{\}, \quad\mbox \]/> <p>the <em>edges</em> of the cube.</p> </li> <li><p>These cat<span class="SimpleMath">\(^1\)</span>-groups combine (in sets of <li> </ul> <p>Note that, since the <span class="SimpleMathjava.lang.StringIndexOutOfBoundsException <p>Secondly, we give the simplest of the three descriptions, again adapted from Ellis-Steiner <a <p>A cat<span class="SimpleMath">\(^n\)</java.lang.StringIndexOutOfBoundsException: Ind < e_{]i {[]\setminus\i\} \to G_[n} t_{[],} {[]i} : G_{[n] \to G_{n \ \{\}, e_{[n],ijava.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for len \</> <p>satisfying the following axioms for all <span class="SimpleMath">\(1 \leqslant <java.lang.StringIndexOutOfBoundsException: Index 0 out of bounds for length 0 <li>p> =>\[]} ~~e_[]i;\ {n,}\ {n,} \{n}\ { {\}))/ <commutingem catspan class="impleMath"\^)/-ro </li> <li><p><span class="SimpleMath">\( (e_1 \circ t_1) \circ (e_2 \circ java.lang.StringIndexOutOf </li> <li><p><span java.lang.StringIndexOutOfBoundsException: Index 17 out of bounds for len </li> </ul> <p> third definesa catspan class"impleMath"\^\< a "cat<span class="SimpleMath">\(^1\)</span>-g <p>Secondly, we give the simplest of the three descriptions, again adapted from Ellis-Steiner <a <p>A cat<span class="SimpleMath">\(^n\)</span>-group <span class="SimpleMath">\(\calC\)</ <p class="center">\[ t_{[n],i}, h_{[n] {n]}:G_n \}{], \]</{n,}h_n,i [n}\ [] setminusi},~ <\</pjava.lang.StringIndexOutOfBoundsException: Index 6 out of bounds for length 6 <ul> <li><p>the <span </li> <li><p><span class="SimpleMath">\( (e_1 \circ<>the <span class"SimpleMath"\(calC_{n]i ~ ({[,}\ t_ </li> <li><p><span class="SimpleMath">\( (e_1 \circ t_1) \circ (e_2 </li> </ul> <p>Our <p>A <em>cat<span class="SimpleMath">\(^n\)</span>-group</em>Our description a<=S>(n)/span>-group <ul> <li><p><span </li> <java.lang.StringIndexOutOfBoundsException: Range [18, 3) out of bounds for lengt </li> <li><p>cat<span class="SimpleMath" <p java.lang.StringIndexOutOfBoundsException: Index 1 out of bounds for length 0 (circ)\mboxand~ (hcirce ~\mbox{are the identity mapping on}~ \calB, \qquad [\ker t, \java.lang.StringIndexOutOfBoundsException: Index 12 out of bounds for le \]<( \circ e) ~mbox{nd~ (h \irc e) </i </ul\er t,\er ] = \ 1_{calA} \. <p><a id="X81C3D39A81B20D76 <h5>9.3-1<id="81C3D39A81B20D76"name"813D39A81B20D76"<a</> <"><table class=func width"%>trtd ="dleft>code class="unc">#8227;PreCatnGroup</code(<varclas <div class="func< class=">table=funcwidth=100"<>< =tdleft"><""# PreCatnGroupcode class""L><tdtd <p>The operation <code class="div class="unc>table class=funcwidth="100%>tr><td class=tdleft"> <div class="example"><pre> <span class="GAPprompt">gap></span> <span class="GAPinput">PC4 := (re)cat4-roup withgenerating-)at1-roups: 1 : [d12 => Group( [ (), (1,4)(2,3)(5,6) ] )] 2 :[12=&t;Group( [(1,4)(2,5)3,),(26)3 ] 3 : [d12 => Group( [ (1,5,3)(2,6,4), (1,4)(2,3)(5,6) ] )] 4 :[d12 =&t Group( [(,2,,4,,6), (,6)3,5 ) <span class="GAPprompt">gap>2 :[12=;Group( [(1,)(25)36) 26)3,)])java.lang.StringIndexOu true <span class="GAPprompt">4 :[d12 =&t; Group([(12,3,4,,6) (2,6(,) ])java.lang.StringIndexOutO 5 </pre></div> <p>For a cat<span java.lang.StringIndexOutOfBoundsException: Index 21 out of bounds fo <div class="example">5 <span class=GAPprompt">ap><span> <pan class="GAPinput">G := Group( (1,2), (3,4,5), (6,7,8,9, <span <p> a cat<pan ="SimpleMath">(5\</pan>-group we startwith the cyclic group whose orderisthe pr <span class="GAPprompt">gap></span> <span class="GAPinput">SetName( G, "C2310" );</span> <span class="GAPprompt">gap></span> <span class="GAPinput">all1 := AllCat1Groups( G );;</span> java.lang.StringIndexOutOfBoundsException: Range [6, 5) out of bounds for length G has 32 cat1-groups <class"APprompt"gap&t<span span class="GAPinput">PC5 := PreCatnGroup( [ all1[2], all1[5], (pre-)cat5-group with generating (pre-)cat1-groups: 1 : [C2310 => Group( [ (), (), (), (), (1,2) ] )] 2:[2310 =gt; Group ) ) (),(3,,5) (,2)])] 3; Group [(),(, ,7,8 ,) 345,(1, ] 4 : [C2310 =&span=">&t;/><pan =G>( [[2,all1[] [13,all1[] []])</> (1,2) ] )] 5:[2310=gt 20,122,,,,262728,9,0),java.lang.StringIndexOutOfBoundsException: Index )3,5,(1,)] ] <span class="GAPprompt">gap></span> <span class="GAPinput">HigherDimension( PC5 );</span> 6 </pre></div> <div class="chlinkprevnextbot"> <a href="chap0_mj.html">[Top of Book]</a> <a href="chap0_mj.ht <div(,,1314,,617), ,8 ,) 4)12) ) <hr /=";="( ;/java.lang.StringIndexOutOfBoundsException: Index 92 out of bounds for <java.lang.StringIndexOutOfBoundsException: Range [0, 2) out of bounds for length <body </
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