Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma


products/Sources/formale Sprachen/GAP/pkg/ugaly/doc/ (Algebra von RWTH Aachen Version 4.15.1^©) Datei vom 7.6.2023 mit Größe 5 kB

Quelle _Chapter_Discreteness.xml Sprache: XML

<?xml version="1.0" encoding="UTF-8"?>


<Chapter Label="Chapter_Discreteness">
<Heading>Discreteness</Heading>

<P/>
This chapter contains functions that are related to the discreteness property (D) presented in Proposition 3.12 of <Cite Key="Tor20"/>.
<Section Label="Section_condition_D">
<Heading>The discreteness condition (D)</Heading>

Said proposition shows that for a given <Math>F\le \mathrm{Aut}(B_{d,k})</Math> the group <Math>\mathrm{U}_{k}(F)</Math> is discrete if and only if the maximal compatible subgroup <Math>C(F)</Math> of <Math>F</Math> satisfies condition (D): <Display>\forall \omega \in \Omega: F_{T_{\omega}}=\{\mathrm{id}\},</Display> where <Math>T_{\omega}</Math> is the <Math>k-1</Math>-neighbourhood of the edge <Math>(b,b_{\omega})</Math> inside <Math>B_{d,k}</Math>. In other words, <Math>F</Math> satisfies (D) if and only if the compatibility set <Math>C_{F}(\mathrm{id},\omega)=\{\mathrm{id}\}</Math>.
We distinguish between <Math>F</Math> satisfying condition (D) and <Math>\mathrm{U}_{k}(F)</Math> being discrete with the methods <Ref Prop="SatisfiesD" Label="for IsLocalAction"/> and <Ref Prop="YieldsDiscreteUniversalGroup" Label="for IsLocalAction"/> below.
</Section>

<Section Label="Chapter_Discreteness_Section_Discreteness">
<Heading>Discreteness</Heading>

<ManSection>
  <Prop Arg="F" Name="SatisfiesD" Label="for IsLocalAction"/>
<Returns> <K>true</K> if <A>F</A> satisfies the discreteness condition (D), and <K>false</K> otherwise.

</Returns>
<Description>
The argument of this attribute is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math> (see <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>).
<P/>
</Description>
</ManSection>

<P/>
<Example><![CDATA[
gap> G:=LocalActionGamma(3,SymmetricGroup(3));
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
gap> SatisfiesD(G);
true
]]></Example>

<ManSection>
  <Prop Arg="F" Name="YieldsDiscreteUniversalGroup" Label="for IsLocalAction"/>
<Returns> <K>true</K> if <Math>\mathrm{U}_{k}(F)</Math> is discrete, and <K>false</K> otherwise.

</Returns>
<Description>
The argument of this attribute is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math> (see <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>).
<P/>
</Description>
</ManSection>

<P/>
<Example><![CDATA[
gap> G:=LocalActionGamma(3,SymmetricGroup(3));
Group([ (1,4,5)(2,3,6), (1,3)(2,4)(5,6) ])
gap> YieldsDiscreteUniversalGroup(G);
true
]]></Example>

<P/>
<Example><![CDATA[
gap> F:=LocalAction(3,2,Group((1,2)));
Group([ (1,2) ])
gap> YieldsDiscreteUniversalGroup(F);
true
gap> SatisfiesD(F);
false
gap> C:=MaximalCompatibleSubgroup(F);
Group(())
gap> SatisfiesD(C);
true
]]></Example>

</Section>

<Section Label="Chapter_Discreteness_Section_Cocycles">
<Heading>Cocycles</Heading>

Subgroups <Math>F\le\mathrm{Aut}(B_{d,k})</Math> that satisfy both (C) and (D) admit an involutive compatibility cocycle, i.e. a map <Math>z:F\times\{1,\ldots,d\}\to F</Math> that satisfies certain properties, see <Cite Key="Tor20" Where="Section 3.2.2"/>. When <Math>F</Math> satisfies just (C), it may still admit an involutive compatibility cocycle. In this case, F admits an extension <Math>\Gamma_{z}(F)\le\mathrm{Aut}(B_{d,k})</Math> that satisfies both (C) and (D). Involutive compatibility cocycles can be searched for using <Ref Attr="InvolutiveCompatibilityCocycle" Label="for IsLocalAction"/> and <Ref Attr="AllInvolutiveCompatibilityCocycles" Label="for IsLocalAction"/> below.
<ManSection>
  <Attr Arg="F" Name="InvolutiveCompatibilityCocycle" Label="for IsLocalAction"/>
<Returns>an involutive compatibility cocycle of <A>F</A>, which is a mapping <A>F</A><Math>\times</Math><C>[1..d]</C><Math>\to</Math><A>F</A> with certain properties, if it exists, and <K>fail</K> otherwise. When <A>k</A> <Math>=1</Math>, the standard cocycle is returned.

</Returns>
<Description>
The argument of this attribute is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math> (see <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>), which is compatible (see <Ref Prop="SatisfiesC" Label="for IsLocalAction"/>).
<P/>
</Description>
</ManSection>

<P/>
<Example><![CDATA[
gap> F:=LocalAction(3,1,AlternatingGroup(3));;
gap> z:=InvolutiveCompatibilityCocycle(F);;
gap> mt:=RandomSource(IsMersenneTwister,1);;
gap> a:=Random(mt,F);; dir:=Random(mt,[1..3]);;
gap> a; Image(z,[a,dir]);
(1,2,3)
(1,2,3)
]]></Example>

<P/>
<Example><![CDATA[
gap> G:=LocalActionGamma(3,AlternatingGroup(3));
Group([ (1,4,5)(2,3,6) ])
gap> InvolutiveCompatibilityCocycle(G) <> fail;
true
gap> InvolutiveCompatibilityCocycle(AutBall(3,2));
fail
]]></Example>

<ManSection>
  <Attr Arg="F" Name="AllInvolutiveCompatibilityCocycles" Label="for IsLocalAction"/>
<Returns>the list of all involutive compatibility cocycles of <Math>F</Math>.

</Returns>
<Description>
The argument of this attribute is a local action <A>F</A> <Math>\le\mathrm{Aut}(B_{d,k})</Math> (see <Ref Filt="IsLocalAction" Label="for IsPermGroup"/>), which is compatible (see <Ref Prop="SatisfiesC" Label="for IsLocalAction"/>).
<P/>
</Description>
</ManSection>

<P/>
<Example><![CDATA[
gap> S3:=LocalAction(3,1,SymmetricGroup(3));;
gap> Size(AllInvolutiveCompatibilityCocycles(S3));
4
gap> Size(AllInvolutiveCompatibilityCocycles(LocalActionGamma(3,SymmetricGroup(3))));
1
]]></Example>

</Section>

</Chapter>

quality98%

¤ Dauer der Verarbeitung: 0.22 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.