Quelle databases.xml Sprache: XML

<Chapter Label="ch:Databases">
<Heading>
 Databases of Residue-Class-Wise Affine Groups and -Mappings
</Heading>

The &RCWA; package contains a number of databases of rcwa groups and rcwa
mappings. They can be loaded into a &GAP; session by the functions described
in this chapter.



<Section Label="sec:Examples">
<Heading>The collection of examples</Heading>

<ManSection>
 <Func Name="LoadRCWAExamples" Arg = ""/>
 <Returns>
 the name of the variable to which the record containing the
 collection of examples of rcwa groups and -mappings loaded from the file
 <F>pkg/rcwa/examples/examples.g</F> got bound.
 </Returns>
 <Description>
 The components of the examples record are records which contain the
 individual groups and mappings.
 A detailed description of some of the examples can be found in
 Chapter <Ref Label="ch:Examples"/>.
<Example>
<![CDATA[
gap> LoadRCWAExamples();
"RCWAExamples"
gap> Set(RecNames(RCWAExamples));
[ "AbelianGroupOverPolynomialRing", "Basics", "CT3Z", "CTPZ",
 "CheckingForSolvability", "ClassSwitches",
 "ClassTranspositionProducts", "ClassTranspositionsAsCommutators",
 "CollatzFactorizationOld", "CollatzMapping", "CollatzlikePerms",
 "CoprimeMultDiv", "F2_PSL2Z", "Farkas", "FiniteQuotients",
 "FiniteVsDenseCycles", "GF2xFiniteCycles", "GrigorchukQuotients",
 "Hexagon", "HicksMullenYucasZavislak", "HigmanThompson",
 "LongCyclesOfPrimeLength", "MatthewsLeigh",
 "MaybeInfinitelyPresentedGroup", "ModuliOfPowers",
 "OddNumberOfGens_FiniteOrder", "Semilocals",
 "SlowlyContractingMappings", "Syl3_S9", "SymmetrizingCollatzTree",
 "TameGroupByCommsOfWildPerms", "Venturini", "ZxZ" ]
gap> AssignGlobals(RCWAExamples.CollatzMapping);
The following global variables have been assigned:
[ "T", "T5", "T5m", "T5p", "Tm", "Tp" ]
]]>
</Example>
 </Description>
</ManSection>

</Section>



<Section Label="sec:DatabasesOfRcwaGroups">
<Heading>Databases of rcwa groups</Heading>

<ManSection>
 <Func Name="LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions"
 Arg = "" Label = "small database"/>
 <Returns>
 the name of the variable to which the record containing the
 database of all groups generated by 3 class transpositions which
 interchange residue classes with moduli <M>\leq 6</M> got bound.
 </Returns>
 <Description>
 The database record has at least the following components (the index
 <C>i</C> is always an integer in the range <C>[1..52394]</C>, and the
 term <Q>indices</Q> always refers to list indices in that range):
 <List>

 <C>cts</C>
 <Item>
 The list of all 69 class transpositions which interchange residue
 classes with moduli <M>\leq 6</M>.
 </Item>

 <C>grps</C>
 <Item>
 The list of the 52394 groups -- 21948 finite and 30446 infinite ones.
 </Item>

 <C>sizes</C>
 <Item>
 The list of group orders --
 it is <C>Size(grps[i]) = sizes[i]</C>.
 </Item>

 <C>mods</C>
 <Item>
 The list of moduli of the groups --
 it is <C>Mod(grps[i]) = mods[i]</C>.
 </Item>

 <C>equalityclasses</C>
 <Item>
 A list of lists of indices <C>i</C> of groups which are known
 to be equal, i.e. if <C>i</C> and <C>j</C> lie in the same list,
 then <C>grps[i] = grps[j]</C>.
 </Item>

 <C>samegroups</C>
 <Item>
 A list of lists, where <C>samegroups[i]</C> is a list of indices
 of groups which are known to be equal to <C>grps[i]</C>.
 </Item>

 <C>conjugacyclasses</C>
 <Item>
 A list of lists of indices of groups which are known to be conjugate
 in RCWA(&ZZ;).
 </Item>

 <C>subgroups</C>
 <Item>
 A list of lists, where <C>subgroups[i]</C> is a list of indices
 of groups which are known to be proper subgroups of <C>grps[i]</C>.
 </Item>

 <C>supergroups</C>
 <Item>
 A list of lists, where <C>supergroups[i]</C> is a list of indices
 of groups which are known to be proper supergroups of <C>grps[i]</C>.
 </Item>

 <C>chains</C>
 <Item>
 A list of lists, where each list contains the indices of the groups
 in a descending chain of subgroups.
 </Item>

 <C>respectedpartitions</C>
 <Item>
 The list of shortest respected partitions.
 If <C>grps[i]</C> is finite, then <C>respectedpartitions[i]</C>
 is a list of pairs (residue, modulus) for the residue classes
 in the shortest respected partition <C>grps[i]</C>. If <C>grps[i]</C>
 is infinite, then <C>respectedpartitions[i] = fail</C>.
 </Item>

 <C>partitionlengths</C>
 <Item>
 The list of lengths of shortest respected partitions.
 If the group <C>grps[i]</C> is finite, then <C>partitionlengths[i]</C>
 is the length of the shortest respected partition of <C>grps[i]</C>.
 If <C>grps[i]</C> is infinite, then <C>partitionlengths[i] = 0</C>.
 </Item>

 <C>degrees</C>
 <Item>
 The list of permutation degrees, i.e. numbers of moved points,
 in the action of the finite groups on their shortest respected
 partitions. If there is no respected partition, i.e. if
 <C>grps[i]</C> is infinite, then <C>degrees[i] = 0</C>.
 </Item>

 <C>orbitlengths</C>
 <Item>
 The list of lists of orbit lengths in the action of the finite groups
 on their shortest respected partitions.
 If <C>grps[i]</C> is infinite, then <C>orbitlengths[i] = fail</C>.
 </Item>

 <C>permgroupgens</C>
 <Item>
 The list of lists of generators of the isomorphic permutation groups
 induced by the finite groups on their shortest respected partitions.
 If <C>grps[i]</C> is infinite, then <C>permgroupgens[i] = fail</C>.
 </Item>

 <C>stabilize_digitsum_base2_mod2</C>
 <Item>
 The list of indices of groups which stabilize the digit sum
 in base 2 modulo 2.
 </Item>
 <C>stabilize_digitsum_base2_mod3</C>
 <Item>
 The list of indices of groups which stabilize the digit sum
 in base 2 modulo 3.
 </Item>

 <C>stabilize_digitsum_base3_mod2</C>
 <Item>
 The list of indices of groups which stabilize the digit sum
 in base 3 modulo 2.
 </Item>

 <C>freeproductcandidates</C>
 <Item>
 A list of indices of groups which may be isomorphic to the free
 product of 3 copies of the cyclic group of order 2.
 </Item>

 <C>freeproductlikes</C>
 <Item>
 A list of indices of groups which are not isomorphic to the free
 product of 3 copies of the cyclic group of order 2, but
 where the shortest relation indicating this is relatively long.
 </Item>

 <C>abc_torsion</C>
 <Item>
 A list of pairs (index, order of product of generators) for all
 infinite groups for which the product of the generators has
 finite order.
 </Item>

 <C>cyclist</C>
 <Item>
 A list described in the comments in
 <F>rcwa/data/3ctsgroups6/spheresizecycles.g</F>.
 </Item>

 <C>finiteorbits</C>
 <Item>
 A record described in the comments in
 <F>rcwa/data/3ctsgroups6/finite-orbits.g</F>.
 </Item>

 <C>intransitivemodulo</C>
 <Item>
 For every modulus <C>m</C> from 1 to 60, <C>intransitivemodulo[m]</C>
 is the list of indices of groups none of whose orbits on &ZZ;
 has nontrivial intersection with all residue classes
 modulo <C>m</C>.
 </Item>

 <C>trsstatus</C>
 <Item>
 A list of strings which describe what is known about whether the
 groups <C>grps[i]</C> act transitively on the nonnegative integers
 in their support, or how the computation has failed.
 </Item>

 <C>orbitgrowthtype</C>
 <Item>
 A list of integers and lists of integers which encode what has been
 observed heuristically on the growth of the orbits of the groups
 <C>grps[i]</C> on &ZZ;.
 </Item>

 </List>
 Note that the database contains an entry for every unordered
 3-tuple of distinct class transpositions in <C>cts</C>, which means
 that it contains multiple copies of equal groups -- cf. the components
 <C>equalityclasses</C> and <C>samegroups</C> described above. 

 To mention an example, the group <C>grps[44132]</C> might be called
 the <Q>Collatz group</Q> -- its action on the set of positive integers
 which are not multiples of 6 is transitive if and only if the Collatz
 conjecture holds.
<Example>
<![CDATA[
gap> LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions();
"3CTsGroups6"
gap> AssignGlobals(3CTsGroups6); # for convenience
The following global variables have been assigned:
[ "3CTsGroupsWithGivenOrbit", "Id3CTsGroup",
 "ProbablyFixesDigitSumsModulo", "ProbablyStabilizesDigitSumsModulo",
 "TriangleTypes", "abc_torsion", "chains", "conjugacyclasses", "cts",
 "cyclist", "degrees", "epifromfpgroupto_ct23z",
 "epifromfpgrouptocollatzgroup_c", "epifromfpgrouptocollatzgroup_t",
 "equalityclasses", "finiteorbits", "freeproductcandidates",
 "freeproductlikes", "groups", "grps", "intransitivemodulo",
 "minwordlengthcoprimemultdiv", "minwordlengthnonbalanced", "mods",
 "orbitgrowthtype", "orbitlengths", "partitionlengths", "permgroupgens",
 "redundant_generator", "refinementseqlngs", "respectedpartitions",
 "samegroups", "shortresidueclassorbitlengths", "sizes", "sizespos",
 "sizesset", "spheresizebound_12", "spheresizebound_24",
 "spheresizebound_4", "spheresizebound_6",
 "stabilize_digitsum_base2_mod2", "stabilize_digitsum_base2_mod3",
 "stabilize_digitsum_base3_mod2", "subgroups", "supergroups",
 "trsstatus", "trsstatuspos", "trsstatusset" ]
gap> grps[44132]; # the "3n+1 group"
<(2(3),4(6)),(1(3),2(6)),(1(2),4(6))>
gap> trsstatus[44132]; # deciding this would solve the 3n+1 problem
"exceeded memory bound"
gap> Length(Set(sizes));
1066
gap> Maximum(Filtered(sizes,IsInt)); # order of largest finite group stored
7165033589793852697531456980706732548435609645091822296777976465116824959\
2135499174617837911754921014138184155204934961004073853323458315539461543\
4480515260818409913846161473536000000000000000000000000000000000000000000\
000000
gap> PrintFactorsInt(last);
2^200*3^103*5^48*7^28*11^16*13^13*17^8*19^6*23^6*29
gap> Positions(sizes,last);
[ 33814, 36548 ]
gap> grps{last};
[ <(1(5),4(5)),(0(3),1(6)),(3(4),0(6))>,
 <(0(5),3(5)),(2(3),4(6)),(0(4),5(6))> ]
gap> samegroups[1];
[ 1, 2, 68 ]
gap> grps[1] = grps[68];
true
gap> Maximum(mods);
77760
gap> Positions(mods,last);
[ 26311, 26313, 26452, 26453, 26455, 26456, 26457, 26459, 26461, 26462,
 27781, 27784, 27785, 27786, 27788, 27789, 27790, 27791, 27829, 27832,
 30523, 30524, 30525, 30526, 30529, 30530, 30532, 30534, 32924, 32927,
 32931, 32933 ]
gap> Set(sizes{last});
[ 45509262704640000 ]
gap> Collected(mods);
[ [ 0, 30446 ], [ 3, 1 ], [ 4, 37 ], [ 5, 120 ], [ 6, 1450 ], [ 8, 18 ],
 [ 10, 45 ], [ 12, 3143 ], [ 15, 165 ], [ 18, 484 ], [ 20, 528 ],
 [ 24, 1339 ], [ 30, 2751 ], [ 36, 2064 ], [ 40, 26 ], [ 48, 515 ],
 [ 60, 2322 ], [ 72, 2054 ], [ 80, 44 ], [ 90, 108 ], [ 96, 108 ],
 [ 108, 114 ], [ 120, 782 ], [ 144, 310 ], [ 160, 26 ], [ 180, 206 ],
 [ 192, 6 ], [ 216, 72 ], [ 240, 304 ], [ 270, 228 ], [ 288, 14 ],
 [ 360, 84 ], [ 432, 36 ], [ 480, 218 ], [ 540, 18 ], [ 720, 120 ],
 [ 810, 112 ], [ 864, 8 ], [ 960, 94 ], [ 1080, 488 ], [ 1620, 44 ],
 [ 1920, 38 ], [ 2160, 506 ], [ 3240, 34 ], [ 3840, 12 ],
 [ 4320, 218 ], [ 4860, 16 ], [ 6480, 282 ], [ 7680, 10 ],
 [ 8640, 16 ], [ 12960, 120 ], [ 14580, 2 ], [ 25920, 34 ],
 [ 30720, 2 ], [ 38880, 12 ], [ 51840, 8 ], [ 77760, 32 ] ]
gap> Collected(trsstatus);
[ [ "> 1 orbit (mod m)", 593 ],
 [ "Mod(U DecreasingOn) exceeded ", 23 ],
 [ "U DecreasingOn stable and exceeded memory bound", 11 ],
 [ "U DecreasingOn stable for steps", 5753 ],
 [ "exceeded memory bound", 497 ], [ "finite", 21948 ],
 [ "intransitive, but finitely many orbits", 8 ],
 [ "seemingly only finite orbits (long)", 1227 ],
 [ "seemingly only finite orbits (medium)", 2501 ],
 [ "seemingly only finite orbits (short)", 4816 ],
 [ "seemingly only finite orbits (very long)", 230 ],
 [ "seemingly only finite orbits (very long, very unclear)", 76 ],
 [ "seemingly only finite orbits (very short)", 208 ],
 [ "there are infinite orbits which have exponential sphere size growth"
 , 2934 ],
 [ "there are infinite orbits which have linear sphere size growth",
 10881 ],
 [ "there are infinite orbits which have unclear sphere size growth",
 86 ], [ "transitive", 562 ],
 [ "transitive up to one finite orbit", 40 ] ]
]]>
</Example>
 </Description>
</ManSection>

<ManSection>
 <Func Name="LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions"
 Arg = "max_m" Label = "both databases"/>
 <Returns>
 the name of the variable to which the record containing the
 database of all groups generated by 3 class transpositions which
 interchange residue classes with moduli less than or equal to
 <A>max_m</A> got bound, where <A>max_m</A> is either 6 or 9.
 </Returns>
 <Description>
 If <A>max_m</A> is 6, this is equivalent to the call of the function
 without argument described above. If <A>max_m</A> is 9, the function
 returns a record with at least the following components
 (in the sequel, the indices <C>i > j > k</C> are always integers in
 the range <C>[1..264]</C>):
 <List>

 <C>cts</C>
 <Item>
 The list of all 264 class transpositions which interchange residue
 classes with moduli <M>\leq 9</M>.
 </Item>

 <C>mods</C>
 <Item>
 The list of moduli of the groups, i.e.
 <C>Mod(Group(cts{[i,j,k]})) = mods[i][j][k]</C>.
 </Item>

 <C>partlengths</C>
 <Item>
 The list of lengths of shortest respected partitions of the groups
 in the database, i.e.
 <C>Length(RespectedPartition(Group(cts{[i,j,k]})))</C> <C>=</C>
 <C>partlengths[i][j][k]</C>.
 </Item>

 <C>sizes</C>
 <Item>
 The list of orders of the groups, i.e.
 <C>Size(Group(cts{[i,j,k]}))</C> <C>=</C> <C>sizes[i][j][k]</C>.
 </Item>

 <C>All3CTs9Indices</C>
 <Item>
 A selector function which takes as argument a function <A>func</A>
 of three arguments <A>i</A>, <A>j</A> and <A>k</A>. It returns a
 list of all triples of indices <C>[<A>i</A>,<A>j</A>,<A>k</A>]</C>
 where <M>264 \geq i > j > k \geq 1</M> for which <A>func</A>
 returns <C>true</C>.
 </Item>

 <C>All3CTs9Groups</C>
 <Item>
 A selector function which takes as argument a function <A>func</A>
 of three arguments <A>i</A>, <A>j</A> and <A>k</A>. It returns a
 list of all groups <C>Group(cts{[<A>i</A>,<A>j</A>,<A>k</A>]})</C>
 from the database for which
 <C><A>func</A>(<A>i</A>,<A>j</A>,<A>k</A>)</C> returns <C>true</C>.
 </Item>

 </List>
<Example>
<![CDATA[
gap> LoadDatabaseOfGroupsGeneratedBy3ClassTranspositions(9);
"3CTsGroups9"
gap> AssignGlobals(3CTsGroups9);
The following global variables have been assigned:
[ "All3CTs9Groups", "All3CTs9Indices", "cts", "mods", "partlengths",
 "sizes" ]
gap> PrintFactorsInt(Maximum(Filtered(Flat(sizes),n->n<>infinity)));
2^1283*3^673*5^305*7^193*11^98*13^84*17^50*19^41*23^25*29^13*31^4
]]>
</Example>
 </Description>
</ManSection>

<ManSection>
 <Func Name="LoadDatabaseOfGroupsGeneratedBy4ClassTranspositions"
 Arg = ""/>
 <Returns>
 the name of the variable to which the record containing the
 database of all groups generated by 4 class transpositions which
 interchange residue classes with moduli <M>\leq 6</M> for which
 all subgroups generated by 3 out of the 4 generators are finite
 got bound.
 </Returns>
 <Description>
 The record has at least the following components (the index <C>i</C>
 is always an integer in the range <C>[1..140947]</C>, and the term
 <Q>indices</Q> always refers to list indices in that range):
 <List>

 <C>cts</C>
 <Item>
 The list of all 69 class transpositions which interchange residue
 classes with moduli <M>\leq 6</M>.
 </Item>

 <C>grps4_3finite</C>
 <Item>
 The list of all 140947 groups in the database.
 </Item>

 <C>grps4_3finitepos</C>
 <Item>
 The list obtained from <C>grps4_3finite</C> by replacing every group
 by the list of positions of its generators in the list <C>cts</C>.
 </Item>

 <C>sizes4</C>
 <Item>
 The list of group orders --
 it is <C>Size(grps4_3finite[i]) = sizes4[i]</C>.
 </Item>

 <C>mods4</C>
 <Item>
 The list of moduli of the groups --
 it is <C>Mod(grps4_3finite[i]) = mods4[i]</C>.
 </Item>

 <C>conjugacyclasses4cts</C>
 <Item>
 A list of lists of indices of groups which are known to be conjugate
 in RCWA(&ZZ;).
 </Item>

 <C>grps4_3finite_reps</C>
 <Item>
 Tentative conjugacy class representatives from the list
 <C>grps4_3finite</C> -- <E>tentative</E> in the sense that likely
 some of the groups in the list are still conjugate.
 </Item>

 </List>
 Note that the database contains an entry for every suitable unordered
 4-tuple of distinct class transpositions in <C>cts</C>, which means
 that it contains multiple copies of equal groups.
<Example>
<![CDATA[
gap> LoadDatabaseOfGroupsGeneratedBy4ClassTranspositions();
"4CTsGroups6"
gap> AssignGlobals(4CTsGroups6);
The following global variables have been assigned:
[ "conjugacyclasses4cts", "cts", "grps4_3finite", "grps4_3finite_reps",
 "grps4_3finitepos", "mods4", "sizes4", "sizes4pos", "sizes4set" ]
gap> Length(grps4_3finite);
140947
gap> Length(sizes4);
140947
gap> Size(grps4_3finite[1]);
518400
gap> sizes4[1];
518400
gap> Maximum(Filtered(sizes4,IsInt));
<integer 420...000 (3852 digits)>
gap> Modulus(grps4_3finite[1]);
12
gap> mods4[1];
12
gap> Length(Set(sizes4));
7339
gap> Length(Set(mods4));
91
gap> conjugacyclasses4cts{[1..4]};
[ [ 1, 23, 563, 867 ], [ 2, 859 ], [ 3, 622 ], [ 4, 16, 868, 873 ] ]
gap> grps4_3finite[1] = grps4_3finite[23];
true
gap> grps4_3finite[4] = grps4_3finite[16];
false
]]>
</Example>
 </Description>
</ManSection>

</Section>



<Section Label="sec:DatabasesOfRcwaMappings">
<Heading>Databases of rcwa mappings</Heading>

<ManSection>
 <Func Name="LoadDatabaseOfProductsOf2ClassTranspositions"
 Arg = ""/>
 <Returns>
 the name of the variable to which the record containing
 the database of products of 2 class transpositions got bound.
 </Returns>
 <Description>
 There are 69 class transpositions which interchange residue
 classes with moduli <M>\leq 6</M>, thus there is a total of
 <M>(69 \cdot 68)/2 = 2346</M> unordered pairs of distinct
 such class transpositions. Looking at intersection-
 and subset relations between the 4 involved residue classes,
 we can distinguish 17 different <Q>intersection types</Q>
 (or 18, together with the trivial case of equal class transpositions).
 The intersection type does not fully determine the cycle
 structure of the product. -- In total, we can distinguish
 88 different cycle types of products of 2 class transpositions
 which interchange residue classes with moduli <M>\leq 6</M>. 

 The components of the database record are a list <C>CTPairs</C>
 of all 2346 pairs of distinct class transpositions which interchange
 residue classes with moduli <M>\leq 6</M>, functions
 <C>CTPairsIntersectionTypes</C>, <C>CTPairIntersectionType</C> and
 <C>CTPairProductType</C>, as well as data lists <C>OrdersMatrix</C>,
 <C>CTPairsProductClassification</C>, <C>CTPairsProductType</C>,
 <C>CTProds12</C> and <C>CTProds32</C>.
 -- For the description of these components, see the file
 <F>pkg/rcwa/data/ctproducts/ctprodclass.g</F>.
<Example>
<![CDATA[
gap> LoadDatabaseOfProductsOf2ClassTranspositions();
"CTProducts"
gap> Set(RecNames(CTProducts));
[ "CTPairIntersectionType", "CTPairProductType", "CTPairs",
 "CTPairsIntersectionTypes", "CTPairsProductClassification",
 "CTPairsProductType", "CTProds12", "CTProds32", "OrdersMatrix" ]
gap> Length(CTProducts.CTPairs);
2346
gap> Collected(List(CTProducts.CTPairsProductType,l->l[2])); # order stats
[ [ 2, 165 ], [ 3, 255 ], [ 4, 173 ], [ 6, 693 ], [ 10, 2 ],
 [ 12, 345 ], [ 15, 4 ], [ 20, 10 ], [ 30, 120 ], [ 60, 44 ],
 [ infinity, 535 ] ]
]]>
</Example>
 </Description>
</ManSection>

<ManSection>
 <Func Name="LoadDatabaseOfNonbalancedProductsOfClassTranspositions"
 Arg = ""/>
 <Returns>
 the name of the variable to which the record containing the database
 of non-balanced products of class transpositions got bound.
 </Returns>
 <Description>
 This database contains a list of the 24 pairs of class
 transpositions which interchange residue classes with moduli
 <M>\leq 6</M> and whose product is not balanced, as well as a list
 of all 36 essentially distinct triples of such class transpositions
 whose product has coprime multiplier and divisor.
<Example>
<![CDATA[
gap> LoadDatabaseOfNonbalancedProductsOfClassTranspositions();
"CTProductsNB"
gap> Set(RecNames(CTProductsNB));
[ "PairsOfCTsWhoseProductIsNotBalanced",
 "TriplesOfCTsWhoseProductHasCoprimeMultiplierAndDivisor" ]
gap> CTProductsNB.PairsOfCTsWhoseProductIsNotBalanced;
[ [ ( 1(2), 2(4) ), ( 2(4), 3(6) ) ], [ ( 1(2), 2(4) ), ( 2(4), 5(6) ) ],
 [ ( 1(2), 2(4) ), ( 2(4), 1(6) ) ], [ ( 1(2), 0(4) ), ( 0(4), 1(6) ) ],
 [ ( 1(2), 0(4) ), ( 0(4), 3(6) ) ], [ ( 1(2), 0(4) ), ( 0(4), 5(6) ) ],
 [ ( 0(2), 1(4) ), ( 1(4), 2(6) ) ], [ ( 0(2), 1(4) ), ( 1(4), 4(6) ) ],
 [ ( 0(2), 1(4) ), ( 1(4), 0(6) ) ], [ ( 0(2), 3(4) ), ( 3(4), 4(6) ) ],
 [ ( 0(2), 3(4) ), ( 3(4), 2(6) ) ], [ ( 0(2), 3(4) ), ( 3(4), 0(6) ) ],
 [ ( 1(2), 2(6) ), ( 3(4), 2(6) ) ], [ ( 1(2), 2(6) ), ( 1(4), 2(6) ) ],
 [ ( 1(2), 4(6) ), ( 3(4), 4(6) ) ], [ ( 1(2), 4(6) ), ( 1(4), 4(6) ) ],
 [ ( 1(2), 0(6) ), ( 1(4), 0(6) ) ], [ ( 1(2), 0(6) ), ( 3(4), 0(6) ) ],
 [ ( 0(2), 1(6) ), ( 2(4), 1(6) ) ], [ ( 0(2), 1(6) ), ( 0(4), 1(6) ) ],
 [ ( 0(2), 3(6) ), ( 2(4), 3(6) ) ], [ ( 0(2), 3(6) ), ( 0(4), 3(6) ) ],
 [ ( 0(2), 5(6) ), ( 2(4), 5(6) ) ], [ ( 0(2), 5(6) ), ( 0(4), 5(6) ) ]
]
]]>
</Example>
 </Description>
</ManSection>

</Section>



</Chapter>

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