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LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <10> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 64. Underfull \hbox (badness 10000) in paragraph at lines 62--66 [] Underfull \hbox (badness 3354) in paragraph at lines 67--70 \OT1/ptm/m/n/10.95 Mathematisches In-sti-tut, Bun-sen-stra^^Ye 3-5, D- [] [1 {/home/laurent/.texmf-var/fonts/map/pdftex/updmap/pdftex.map}] LaTeX Font Info: Font shape `OT1/phv/m/n' will be (Font) scaled to size 8.99994pt on input line 77. Underfull \hbox (badness 10000) in paragraph at lines 83--85 [] LaTeX Font Info: Font shape `OMS/ptm/m/n' in size <10> not available (Font) Font shape `OMS/pzccm/m/n' tried instead on input line 87. Underfull \hbox (badness 10000) in paragraph at lines 87--89 [] Underfull \hbox (badness 10000) in paragraph at lines 91--94 [] Underfull \hbox (badness 10000) in paragraph at lines 109--114 [] [2] (./fr.toc [3 ] [4] [5] [6] [7] [8]) \tf@toc=\write6 \openout6 = `fr.toc'. [9] Chapter 1. LaTeX Font Info: Font shape `OT1/ptm/bx/n' in size <20.74> not available (Font) Font shape `OT1/ptm/b/n' tried instead on input line 122. [10 ] Chapter 2. LaTeX Font Info: Font shape `OT1/phv/m/n' will be (Font) scaled to size 9.85492pt on input line 147. [11 ] [12] LaTeX Font Info: Try loading font information for OML+pcr on input line 218. (/usr/share/texmf-texlive/tex/latex/psnfss/omlpcr.fd File: omlpcr.fd ) LaTeX Font Info: Font shape `OML/pcr/m/n' in size <10.95> not available (Font) Font shape `OML/cmm/m/it' tried instead on input line 218. [13] [14] [15] Chapter 3. Underfull \hbox (badness 6792) in paragraph at lines 364--365 []\OT1/ptm/m/n/10.95 More gen-eral ma-chines are ob-tained if one al-lows the s tate-set $\OML/ptmcm/m/it/10.95 Q$ \OT1/ptm/m/n/10.95 to be a free [] [16 ] LaTeX Font Info: Font shape `OT1/pcr/bx/n' in size <14.4> not available (Font) Font shape `OT1/pcr/b/n' tried instead on input line 399. [17] [18] [19] [20] [21] [22] LaTeX Font Info: Font shape `OT1/pcr/bx/n' in size <12> not available (Font) Font shape `OT1/pcr/b/n' tried instead on input line 850. [23] [24] [25] [26] Underfull \hbox (badness 3449) in paragraph at lines 1142--1143 []\OT1/ptm/m/n/10.95 If a ma-chine [] was cre-ated as a min-i-mized group/monoi d/semigroup ma-chine, then [] Underfull \hbox (badness 10000) in paragraph at lines 1148--1152 []\OT1/ptm/m/n/10.95 If [] was cre-ated us-ing \OT1/pcr/m/n/10.95 AsGroupFRMach ine \OT1/ptm/m/n/10.95 ([][]3.3.4[][]), \OT1/pcr/m/n/10.95 AsMonoidFRMachine \O T1/ptm/m/n/10.95 ([][]3.3.4[][]), [] [27] Chapter 4. [28 ] [29] Underfull \hbox (badness 1983) in paragraph at lines 1322--1324 []\OT1/ptm/m/n/10.95 This func-tion con-structs a new FR el-e-ment. [] is a lis t of FR el-e-ments, and [] is [] Underfull \hbox (badness 1831) in paragraph at lines 1322--1324 \OT1/ptm/m/n/10.95 a per-mu-ta-tion, trans-for-ma-tion or list. In that last ca se, the re-sult-ing el-e-ment \OT1/pcr/m/n/10.95 g \OT1/ptm/m/n/10.95 sat-is-fi es [] [30] Underfull \hbox (badness 10000) in paragraph at lines 1389--1390 []\OT1/ptm/m/n/10.95 In par-tic-u-lar, \OT1/pcr/m/n/10.95 DiagonalElement(0,e) \OT1/ptm/m/n/10.95 is the same as \OT1/pcr/m/n/10.95 VertexElement(1,e)\OT1/ptm /m/n/10.95 ; [] [31] [32] [33] [34] [35] [36] [37] [38] [39] Chapter 5. [40 ] [41] [42] Underfull \hbox (badness 10000) in paragraph at lines 2221--2221 [][]\OT1/ptm/b/n/14.4 Operations and At-tributes for \OT1/pcr/b/n/14.4 MealyMac hine\OT1/ptm/b/n/14.4 s and [] Underfull \hbox (badness 10000) in paragraph at lines 2229--2232 [] [43] [44] [45] [46] [47] [48] Overfull \hbox (55.59563pt too wide) in paragraph at lines 2663--2664 []\OT1/ptm/m/n/10.95 The sec-ond com-mand is a short-hand for \OT1/pcr/m/n/10.9 5 VertexTransformationsFRMachine(UnderlyingFRMachine(e))\OT1/ptm/m/n/10.95 . [] [49] [50] [51] [52] [53] [54] [55] Chapter 6. Underfull \hbox (badness 10000) in paragraph at lines 3147--3147 [][]\OT1/ptm/b/n/14.4 Methods and op-er-a-tions for \OT1/pcr/b/n/14.4 LinearFRM achine\OT1/ptm/b/n/14.4 s and [] [56 ] [57] [58] Overfull \hbox (115.52515pt too wide) in paragraph at lines 3344--3345 []\OT1/ptm/m/n/10.95 In par-tic-u-lar, \OT1/pcr/m/n/10.95 NestedMatrixState(e,[ ],[])=e\OT1/ptm/m/n/10.95 , and \OT1/pcr/m/n/10.95 Activity(e,1)[i][j]=NestedMa trixCoefficient(e,[i],[j])\OT1/ptm/m/n/10.95 , [] [59] [60] Underfull \hbox (badness 1122) in paragraph at lines 3522--3523 []\OT1/ptm/m/n/10.95 Given a lin-ear el-e-ment [], this com-mand at-tempts to f ind a de-com-po-si-tion of the form [] Underfull \hbox (badness 2781) in paragraph at lines 3522--3523 \OML/ptmcm/m/it/10.95 e \OT1/ptmcm/m/n/10.95 = \OML/ptmcm/m/it/10.95 LDU$\OT1/p tm/m/n/10.95 , where $\OML/ptmcm/m/it/10.95 L$ \OT1/ptm/m/n/10.95 is lower tri- an-gu-lar, $\OML/ptmcm/m/it/10.95 D$ \OT1/ptm/m/n/10.95 is di-ag-o-nal, and $\O ML/ptmcm/m/it/10.95 U$ \OT1/ptm/m/n/10.95 is up-per tri-an-gu-lar (see [] [61] Underfull \hbox (badness 1596) in paragraph at lines 3602--3604 []\OT1/ptm/m/n/10.95 If [] is a Mealy ma-chine/element, the re-sult is a vec-to r ma-chine/element. If [] is a [] [62] [63] [64] Chapter 7. Underfull \hbox (badness 1983) in paragraph at lines 3767--3770 []\OT1/ptm/m/n/10.95 The max-i-mal length of words to con-sider in the search i s con-trolled by the vari-able [] [65 ] [66] [67] [68] [69] [70] [71] Underfull \hbox (badness 10000) in paragraph at lines 4233--4235 []\OT1/ptm/m/n/10.95 Furthermore, an op-tion "MealyEle-ment" can be passed to t he func-tion, as [] Underfull \hbox (badness 3612) in paragraph at lines 4233--4235 \OT1/pcr/m/n/10.95 FRGroupByVirtualEndomorphism(f:MealyElement)\OT1/ptm/m/n/10. 95 , to re-quire the re-sult-ing group to be [] [72] [73] [74] [75] [76] [77] [78] [79] Underfull \hbox (badness 4060) in paragraph at lines 4836--4837 []\OT1/ptm/m/n/10.95 This func-tion re-turns \OT1/pcr/m/n/10.95 true \OT1/ptm/m /n/10.95 if all el-e-ments of [] have the \OT1/ptm/m/it/10.95 open set con-di-t ion\OT1/ptm/m/n/10.95 , see [] [80] [81] [82] Underfull \hbox (badness 10000) in paragraph at lines 5094--5095 []\OT1/ptm/m/n/10.95 This is the small-est group/monoid/semigroup \OT1/pcr/m/n/ 10.95 P \OT1/ptm/m/n/10.95 such that [] [] Underfull \hbox (badness 10000) in paragraph at lines 5094--5095 \OT1/ptm/m/n/10.95 is a sub-set of \OT1/pcr/m/n/10.95 FullSCGroup(AlphabetOfFRS emigroup(g),P) \OT1/ptm/m/n/10.95 or [] Overfull \hbox (36.87662pt too wide) in paragraph at lines 5094--5095 \OT1/pcr/m/n/10.95 FullSCMonoid(AlphabetOfFRSemigroup(g),P) \OT1/ptm/m/n/10.95 or \OT1/pcr/m/n/10.95 FullSCSemigroup(AlphabetOfFRSemigroup(g),P)\OT1/ptm/m/n/1 0.95 . [] [83] Underfull \hbox (badness 10000) in paragraph at lines 5201--5202 \OT1/ptm/m/n/10.95 ([][]10.1.2[][]), \OT1/pcr/m/n/10.95 GrigorchukGroup \OT1/pt m/m/n/10.95 ([][]10.1.9[][]), \OT1/pcr/m/n/10.95 GrigorchukOverGroup \OT1/ptm/m /n/10.95 ([][]10.1.10[][]), gen-er-al-ized [] [84] [85] [86] [87] Chapter 8. Underfull \hbox (badness 10000) in paragraph at lines 5441--5444 []\OT1/ptm/m/n/10.95 Each [] is of the form \OT1/pcr/m/n/10.95 "name=[[...],... ,[...]]" \OT1/ptm/m/n/10.95 or of the form [] Underfull \hbox (badness 1629) in paragraph at lines 5441--5444 \OT1/pcr/m/n/10.95 "name=[[...],...,[...]]:out"\OT1/ptm/m/n/10.95 , namely a ma -trix whose en-tries are al-ge-braic ex-pres-sions [] Overfull \hbox (88.59897pt too wide) in paragraph at lines 5445--5448 []\OT1/ptm/m/n/10.95 The op-tion \OT1/pcr/m/n/10.95 IsVectorElement\OT1/ptm/m/n /10.95 , passed e.g. as in \OT1/pcr/m/n/10.95 FRAlgebra(Rationals,"a=[[a,1],[a, 0]]":IsVectorElement)\OT1/ptm/m/n/10.95 , [] [88 ] [89] [90] Chapter 9. [91 ] [92] [93] [94] [95] [96] Chapter 10. Underfull \hbox (badness 10000) in paragraph at lines 5985--6003 [] Underfull \hbox (badness 2401) in paragraph at lines 6005--6008 []\OT1/ptm/m/n/10.95 These are the fini-tary, bounded, polynomial-growth, finit e-state, or un-re-stricted groups [] Underfull \hbox (badness 3930) in paragraph at lines 6005--6008 \OT1/ptm/m/n/10.95 act-ing on the bi-nary tree. They are re-spec-tively short-c uts for \OT1/pcr/m/n/10.95 FullSCGroup([1..2])\OT1/ptm/m/n/10.95 , [] Underfull \hbox (badness 10000) in paragraph at lines 6005--6008 \OT1/pcr/m/n/10.95 FullSCGroup([1..2],l)\OT1/ptm/m/n/10.95 , \OT1/pcr/m/n/10.95 FullSCGroup([1..2],IsFinitaryFRSemigroup)\OT1/ptm/m/n/10.95 , [] Overfull \hbox (62.05121pt too wide) in paragraph at lines 6005--6008 \OT1/pcr/m/n/10.95 FullSCGroup([1..2],IsBoundedFRSemigroup)\OT1/ptm/m/n/10.95 , \OT1/pcr/m/n/10.95 FullSCGroup([1..2],IsPolynomialGrowthFRSemigroup)\OT1/ptm/m /n/10.95 , [] [97 ] Underfull \hbox (badness 10000) in paragraph at lines 6045--6046 []\OT1/ptm/m/n/10.95 As par-tic-u-lar ex-am-ples, \OT1/pcr/m/n/10.95 BinaryKnea dingMachine() \OT1/ptm/m/n/10.95 is the adding ma-chine; [] Underfull \hbox (badness 10000) in paragraph at lines 6045--6046 \OT1/pcr/m/n/10.95 BinaryKneadingGroup() \OT1/ptm/m/n/10.95 is the adding ma-ch ine; and \OT1/pcr/m/n/10.95 BinaryKneadingGroup("1") \OT1/ptm/m/n/10.95 is [] Underfull \hbox (badness 10000) in paragraph at lines 6067--6070 [] Underfull \hbox (badness 10000) in paragraph at lines 6091--6100 [] [98] Underfull \hbox (badness 10000) in paragraph at lines 6136--6145 [] Underfull \hbox (badness 10000) in paragraph at lines 6147--6148 []\OT1/ptm/m/n/10.95 These are re-spec-tively the same as \OT1/pcr/m/n/10.95 Ad dingGroup(2)\OT1/ptm/m/n/10.95 , \OT1/pcr/m/n/10.95 AddingMachine(2) \OT1/ptm/m /n/10.95 and [] Underfull \hbox (badness 1019) in paragraph at lines 6175--6177 []\OT1/ptm/m/n/10.95 The first func-tion con-structs the group gen-er-ated by t he mixer ma-chine. For ex-am-ples [] [99] Underfull \hbox (badness 10000) in paragraph at lines 6233--6234 []\OT1/ptm/m/n/10.95 These groups are \OT1/pcr/m/n/10.95 MixerGroup \OT1/ptm/m/ n/10.95 ([][]10.1.6[][])s. The most fa-mous ex-am-ple is [] Underfull \hbox (badness 7869) in paragraph at lines 6235--6236 []\OT1/ptm/m/n/10.95 These groups are all 4-generated and in-fi-nite. They are de-scribed in [[]][]. [] Underfull \hbox (badness 1484) in paragraph at lines 6235--6236 \OT1/pcr/m/n/10.95 GrigorchukGroups([1]) \OT1/ptm/m/n/10.95 is in-fi-nite di-he -dral. If [] con-tains at least 2 dif-fer-ent dig- [] [100] Underfull \hbox (badness 10000) in paragraph at lines 6260--6266 [] Underfull \hbox (badness 4378) in paragraph at lines 6268--6270 []\OT1/ptm/m/n/10.95 This is Grig-orchuk's first group, in-tro-duced in [[]][]. It is a 4-generated in-fi- [] Underfull \hbox (badness 2894) in paragraph at lines 6268--6270 \OT1/ptm/m/n/10.95 nite tor-sion group, and has in-ter-me-di-ate word growth. I t could have been de-fined as [] Underfull \hbox (badness 10000) in paragraph at lines 6271--6273 []\OT1/ptm/m/n/10.95 The com-mand \OT1/pcr/m/n/10.95 EpimorphismFromFpGroup(Gri gorchukGroup,n) \OT1/ptm/m/n/10.95 will will con- [] Underfull \hbox (badness 5667) in paragraph at lines 6271--6273 \OT1/ptm/m/n/10.95 struct an ap-prox-i-mat-ing pre-sen-ta-tion for the Grig-orc huk group, as proven in [[]][]. [] Underfull \hbox (badness 7308) in paragraph at lines 6271--6273 \OT1/ptm/m/n/10.95 Adding the re-la-tions \OT1/pcr/m/n/10.95 Image(sigma^(n-2), (a*d)^2)\OT1/ptm/m/n/10.95 , \OT1/pcr/m/n/10.95 Image(sigma^(n-1),(a*b)^2) \OT1 /ptm/m/n/10.95 and [] Underfull \hbox (badness 10000) in paragraph at lines 6279--6282 [] Underfull \hbox (badness 10000) in paragraph at lines 6303--6306 [] Underfull \hbox (badness 10000) in paragraph at lines 6308--6309 []\OT1/ptm/m/n/10.95 This is a group with same clo-sure as the Grig-orchuk grou p (see [] Underfull \hbox (badness 3492) in paragraph at lines 6308--6309 \OT1/pcr/m/n/10.95 GrigorchukGroup \OT1/ptm/m/n/10.95 ([][]10.1.9[][])), but no t iso-mor-phic to it. It could have been de-fined as [] [101] Underfull \hbox (badness 10000) in paragraph at lines 6324--6330 [] Underfull \hbox (badness 10000) in paragraph at lines 6364--6365 []\OT1/ptm/m/n/10.95 If \OT1/pcr/m/n/10.95 l=[(1,2),(1,2),()]\OT1/ptm/m/n/10.95 , this is \OT1/pcr/m/n/10.95 AleshinGroup \OT1/ptm/m/n/10.95 ([][]10.1.14[][]) . More gen-er-ally, if [] Underfull \hbox (badness 1320) in paragraph at lines 6364--6365 \OT1/pcr/m/n/10.95 l=[(1,2,(1,2),(),...,()] \OT1/ptm/m/n/10.95 has odd length, this is a free group of rank \OT1/pcr/m/n/10.95 Length(l)\OT1/ptm/m/n/10.95 , s ee [] Underfull \hbox (badness 10000) in paragraph at lines 6375--6381 [] Underfull \hbox (badness 10000) in paragraph at lines 6383--6385 []\OT1/ptm/m/n/10.95 This is the first ex-am-ple of non-abelian free group. It is the group gen-er- [] Underfull \hbox (badness 10000) in paragraph at lines 6383--6385 \OT1/ptm/m/n/10.95 ated by \OT1/pcr/m/n/10.95 AleshinMachine([(1,2),(1,2),()])\ OT1/ptm/m/n/10.95 . It could have been de-fined as [] Underfull \hbox (badness 10000) in paragraph at lines 6383--6385 \OT1/pcr/m/n/10.95 FRGroup("a=\OML/cmm/m/it/10.95 <\OT1/pcr/m/n/10.95 b,c\OML/c mm/m/it/10.95 >\OT1/pcr/m/n/10.95 (1,2)","b=\OML/cmm/m/it/10.95 <\OT1/pcr/m/n/1 0.95 c,b\OML/cmm/m/it/10.95 >\OT1/pcr/m/n/10.95 (1,2)","c=\OML/cmm/m/it/10.95 < \OT1/pcr/m/n/10.95 a,a\OML/cmm/m/it/10.95 >\OT1/pcr/m/n/10.95 ")\OT1/ptm/m/n/10 .95 , but is rather de-fined us-ing [] [102] Underfull \hbox (badness 10000) in paragraph at lines 6391--6397 [] Underfull \hbox (badness 10000) in paragraph at lines 6419--6422 [] Underfull \hbox (badness 10000) in paragraph at lines 6460--6466 [] [103] Underfull \hbox (badness 4647) in paragraph at lines 6468--6470 []\OT1/ptm/m/n/10.95 This is an in-fi-nite, 2-generated, tor-sion 3-group. It c ould have been de-fined as [] Underfull \hbox (badness 1264) in paragraph at lines 6490--6491 \OT1/ptm/m/n/10.95 of non-uniform ex-po-nen-tial growth (see [[]][]), and \OT1/ pcr/m/n/10.95 NeumannGroup(Group((1,2,3))) \OT1/ptm/m/n/10.95 is [] Underfull \hbox (badness 10000) in paragraph at lines 6499--6505 [] Underfull \hbox (badness 10000) in paragraph at lines 6519--6522 [] [104] Underfull \hbox (badness 10000) in paragraph at lines 6574--6577 [] Underfull \hbox (badness 10000) in paragraph at lines 6584--6585 []\OT1/ptm/m/n/10.95 It ad-mits an en-do-mor-phism on its de-rived sub-group. I n-deed [] Underfull \hbox (badness 10000) in paragraph at lines 6584--6585 \OT1/pcr/m/n/10.95 FRElement(1,Comm(a,b))=Comm(c^-1,b/a)\OT1/ptm/m/n/10.95 , \O T1/pcr/m/n/10.95 FRElement(1,Comm(a,c))=Comm(a/b,c)\OT1/ptm/m/n/10.95 , [] Underfull \hbox (badness 10000) in paragraph at lines 6591--6594 [] Underfull \hbox (badness 2894) in paragraph at lines 6596--6598 []\OT1/ptm/m/n/10.95 This group was stud-ied in [[]][]. It is frac-tal, but not con-tract-ing. It could [] Underfull \hbox (badness 10000) in paragraph at lines 6596--6598 \OT1/ptm/m/n/10.95 have been de-fined as \OT1/pcr/m/n/10.95 FRGroup("a=\OML/cmm /m/it/10.95 <\OT1/pcr/m/n/10.95 ,b\OML/cmm/m/it/10.95 >\OT1/pcr/m/n/10.95 (1,2) ","b=\OML/cmm/m/it/10.95 <\OT1/pcr/m/n/10.95 a,c\OML/cmm/m/it/10.95 >\OT1/pcr/m /n/10.95 ","c=\OML/cmm/m/it/10.95 <\OT1/pcr/m/n/10.95 a,a^-1\OML/cmm/m/it/10.95 >\OT1/pcr/m/n/10.95 (1,2)")]]\OML/cmm/m/it/10.95 >\OT1/ptm/m/n/10.95 , [] Underfull \hbox (badness 8075) in paragraph at lines 6596--6598 \OT1/ptm/m/n/10.95 but is rather de-fined us-ing Mealy el-e-ments. It par-tiall y ad-mits branch-ing on its [] Underfull \hbox (badness 10000) in paragraph at lines 6596--6598 \OT1/ptm/m/n/10.95 sub-group \OT1/pcr/m/n/10.95 Subgroup(G,[a^2,(a^2)^b,(a^2)^c ])\OT1/ptm/m/n/10.95 , and, set-ting \OT1/pcr/m/n/10.95 x=Comm(a^2,b)\OT1/ptm/m /n/10.95 , on [] Underfull \hbox (badness 10000) in paragraph at lines 6604--6607 [] [105] [106] [107] Underfull \hbox (badness 10000) in paragraph at lines 6744--6750 [] Underfull \hbox (badness 10000) in paragraph at lines 6762--6768 [] Underfull \hbox (badness 10000) in paragraph at lines 6786--6789 [] Underfull \hbox (badness 10000) in paragraph at lines 6811--6814 [] [108] Underfull \hbox (badness 10000) in paragraph at lines 6827--6830 [] Underfull \hbox (badness 10000) in paragraph at lines 6843--6846 [] [109] [110] [111] [112] [113] [114] Chapter 11. Underfull \hbox (badness 10000) in paragraph at lines 7204--7207 [] [115 ] Underfull \hbox (badness 10000) in paragraph at lines 7220--7223 [] [116] [117] [118] [119] [120] [121] [122] Chapter 12. Underfull \hbox (badness 10000) in paragraph at lines 7782--7785 [] Underfull \hbox (badness 10000) in paragraph at lines 7795--7798 [] Underfull \hbox (badness 10000) in paragraph at lines 7807--7810 [] Underfull \hbox (badness 10000) in paragraph at lines 7819--7822 [] Underfull \hbox (badness 10000) in paragraph at lines 7831--7834 [] Underfull \hbox (badness 10000) in paragraph at lines 7843--7864 [] [123 ] Underfull \hbox (badness 10000) in paragraph at lines 7878--7879 []\OT1/ptm/m/n/10.95 The fourth and fifth com-mand com-press the newly cre-ated pe-ri-odic list, see [] Underfull \hbox (badness 10000) in paragraph at lines 7905--7908 [] [124] [125] LaTeX Font Info: Font shape `OT1/pcr/bx/n' in size <10.95> not available (Font) Font shape `OT1/pcr/b/n' tried instead on input line 8020. Underfull \hbox (badness 2762) in paragraph at lines 8042--8043 \OT1/ptm/m/n/10.95 If the first ar-gu-ment is an in-te-ger \OT1/pcr/m/n/10.95 n \OT1/ptm/m/n/10.95 and not a record, the com-mand is in-ter-preted as [] Underfull \hbox (badness 10000) in paragraph at lines 8044--8045 []\OT1/pcr/m/n/10.95 WordGrowth(...,rec(draw:=true)) \OT1/ptm/m/n/10.95 may be ab-bre-vi-ated as \OT1/pcr/m/n/10.95 Draw(...)\OT1/ptm/m/n/10.95 ; [] Underfull \hbox (badness 10000) in paragraph at lines 8044--8045 \OT1/pcr/m/n/10.95 WordGrowth(...,rec(ball:=n)) \OT1/ptm/m/n/10.95 may be ab-br e-vi-ated as \OT1/pcr/m/n/10.95 Ball(...,n)\OT1/ptm/m/n/10.95 ; [] [126] [127] [128] Underfull \hbox (badness 10000) in paragraph at lines 8281--8282 []\OT1/ptm/m/n/10.95 This func-tion con-structs an en-do-mor-phism of the group ,monoid or semi- [] Underfull \hbox (badness 10000) in paragraph at lines 8281--8282 \OT1/ptm/m/n/10.95 group [] spec-i-fied by send-ing gen-er-a-tor num-ber $\OML/ ptmcm/m/it/10.95 i$ \OT1/ptm/m/n/10.95 to the $\OML/ptmcm/m/it/10.95 i$\OT1/ptm /m/n/10.95 th en-try in [] Underfull \hbox (badness 10000) in paragraph at lines 8281--8282 []\OT1/ptm/m/n/10.95 . It is a short-cut for a call to \OT1/pcr/m/n/10.95 Group HomomorphismByImagesNC \OT1/ptm/m/n/10.95 or [] Underfull \hbox (badness 10000) in paragraph at lines 8296--8297 []\OT1/ptm/m/n/10.95 This func-tion con-structs a ho-mo-mor-phism of the group, monoid or semi- [] Underfull \hbox (badness 10000) in paragraph at lines 8296--8297 \OT1/ptm/m/n/10.95 group [] spec-i-fied by send-ing gen-er-a-tor num-ber $\OML/ ptmcm/m/it/10.95 i$ \OT1/ptm/m/n/10.95 to the $\OML/ptmcm/m/it/10.95 i$\OT1/ptm /m/n/10.95 th en-try in [] Underfull \hbox (badness 10000) in paragraph at lines 8296--8297 []\OT1/ptm/m/n/10.95 . It is a short-cut for a call to \OT1/pcr/m/n/10.95 Group HomomorphismByImagesNC \OT1/ptm/m/n/10.95 or [] [129] Underfull \hbox (badness 10000) in paragraph at lines 8303--8312 [] [130] Underfull \hbox (badness 10000) in paragraph at lines 8392--8395 [] Underfull \hbox (badness 10000) in paragraph at lines 8434--8437 [] [131] Underfull \hbox (badness 10000) in paragraph at lines 8455--8458 [] (./fr.bbl (./fr.brf) \tf@brf=\write7 \openout7 = `fr.brf'. [132] Underfull \hbox (badness 5161) in paragraph at lines 12--12 []\OT1/ptm/m/n/10.95 R. Bacher. De-ter-mi-nants re-lated to bi-no-mial co-ef-f i-cients mod-ulo $\OT1/ptmcm/m/n/10.95 2$ \OT1/ptm/m/n/10.95 and $\OT1/ptmcm/m/ n/10.95 4$\OT1/ptm/m/n/10.95 . [] Underfull \hbox (badness 10000) in paragraph at lines 73--73 []\OT1/ptm/m/n/10.95 L. Bartholdi and I. I. Reznykov. A mealy ma-chine of grow th $\OML/ptmcm/m/it/10.95 n[]$\OT1/ptm/m/n/10.95 . [] [133 ] [134] [135]) (./fr.ind [136] [137 ] [138] [139] [140] [141]) (./fr.aux) ) Here is how much of TeX's memory you used: 4951 strings out of 95086 72076 string characters out of 1183255 132235 words of memory out of 1500000 7267 multiletter control sequences out of 10000+50000 49843 words of font info for 101 fonts, out of 1200000 for 2000 28 hyphenation exceptions out of 8191 25i,7n,29p,673b,625s stack positions out of 5000i,500n,6000p,200000b,5000s {/usr/share/texmf-texlive/fonts/enc/dvip s/base/8r.enc}</usr/share/texmf-texlive/fonts/type1/bluesky/cm/cmex10.pfb></usr /share/texmf-texlive/fonts/type1/bluesky/cm/cmmi10.pfb></usr/share/texmf-texliv e/fonts/type1/bluesky/cm/cmr10.pfb></usr/share/texmf-texlive/fonts/type1/bluesk y/cm/cmsy10.pfb></usr/share/texmf-texlive/fonts/type1/public/cmex/fmex9.pfb></u sr/share/texmf-texlive/fonts/type1/bluesky/ams/msam10.pfb></usr/share/texmf-tex live/fonts/type1/bluesky/ams/msbm10.pfb></usr/share/texmf-texlive/fonts/type1/u rw/courier/ucrb8a.pfb></usr/share/texmf-texlive/fonts/type1/urw/courier/ucrr8a. pfb></usr/share/texmf-texlive/fonts/type1/urw/courier/ucrro8a.pfb></usr/share/t exmf-texlive/fonts/type1/urw/helvetic/uhvr8a.pfb></usr/share/texmf-texlive/font s/type1/urw/symbol/usyr.pfb></usr/share/texmf-texlive/fonts/type1/urw/times/utm b8a.pfb></usr/share/texmf-texlive/fonts/type1/urw/times/utmr8a.pfb></usr/share/ texmf-texlive/fonts/type1/urw/times/utmr8a.pfb></usr/share/texmf-texlive/fonts/ type1/urw/times/utmri8a.pfb> Output written on fr.pdf (141 pages, 796714 bytes). 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