2979:
2111:
2974:{\displaystyle {\begin{array}{c|cccccc}&{\color {Blue}C_{id}}&{\color {Blue}C_{1}}&{\color {Blue}C_{g}}&{\color {Blue}C_{g^{2}}}&{\color {Gray}\dots }&{\color {Blue}C_{g^{p-2}}}\\\hline {\color {Blue}\chi _{1}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Blue}e^{\frac {2\pi i}{p-1}}}&{\color {Blue}e^{\frac {4\pi i}{p-1}}}&{\color {Gray}\dots }&{\color {Blue}e^{\frac {2\pi (p-2)i}{p-1}}}\\{\color {Blue}\chi _{2}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Blue}e^{\frac {4\pi i}{p-1}}}&{\color {Blue}e^{\frac {8\pi i}{p-1}}}&{\color {Gray}\dots }&{\color {Blue}e^{\frac {4\pi (p-2)i}{p-1}}}\\{\color {Blue}\chi _{3}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Blue}e^{\frac {6\pi i}{p-1}}}&{\color {Blue}e^{\frac {12\pi i}{p-1}}}&{\color {Gray}\dots }&{\color {Blue}e^{\frac {6\pi (p-2)i}{p-1}}}\\{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }&{\color {Gray}\dots }\\{\color {Blue}\chi _{p-1}}&{\color {Gray}1}&{\color {Gray}1}&{\color {Gray}1}&{\color {Gray}1}&{\color {Gray}\dots }&{\color {Gray}1}\\{\color {Blue}\chi _{p}}&{\color {Gray}p-1}&{\color {Gray}-1}&{\color {Gray}0}&{\color {Gray}0}&{\color {Gray}\dots }&{\color {Gray}0}\end{array}}}
3584:
3059:
1718:
3579:{\displaystyle {\begin{aligned}{\text{1.}}&&(x,y)&\mapsto (x+a,y+b),\\{\text{2.}}&&(x,y)&\mapsto (ax,by),&\qquad {\text{where }}ab\neq 0,\\{\text{3.}}&&(x,y)&\mapsto (ax,y+b),&\qquad {\text{where }}a\neq 0,\\{\text{4.}}&&(x,y)&\mapsto (ax+y,ay),&\qquad {\text{where }}a\neq 0,\\{\text{5.}}&&(x,y)&\mapsto (x+y,y+a)\\{\text{6.}}&&(x,y)&\mapsto (a(x\cos t+y\sin t),a(-x\sin t+y\cos t)),&\qquad {\text{where }}a\neq 0.\end{aligned}}}
1376:
1713:{\displaystyle {\begin{aligned}C_{id}&=\left\{{\begin{pmatrix}1&0\\0&1\end{pmatrix}}\right\}\,,\\C_{1}&=\left\{{\begin{pmatrix}1&b\\0&1\end{pmatrix}}{\Bigg |}b\in \mathbf {F} _{p}^{*}\right\}\,,\\{\Bigg \{}C_{a}&=\left\{{\begin{pmatrix}a&b\\0&1\end{pmatrix}}{\Bigg |}b\in \mathbf {F} _{p}\right\}{\Bigg |}a\in \mathbf {F} _{p}\setminus \{0,1\}{\Bigg \}}\,.\end{aligned}}}
1348:
974:
1151:
1100:
1950:
444:
1821:
3064:
675:
282:
3983:
The subgroup of the special affine group consisting of those transformations whose linear part has determinant 1 is the group of orientation- and volume-preserving maps. Algebraically, this group is a semidirect product
870:
1343:{\displaystyle {\begin{pmatrix}c&d\\0&1\end{pmatrix}}{\begin{pmatrix}a&b\\0&1\end{pmatrix}}{\begin{pmatrix}c&d\\0&1\end{pmatrix}}^{-1}={\begin{pmatrix}a&(1-a)d+bc\\0&1\end{pmatrix}}\,,}
182:
4446:
1381:
3855:
4252:
3027:
362:(which is uniquely defined), however, no particular subgroup is a natural choice, since no point is special – this corresponds to the multiple choices of transverse subgroup, or splitting of the
3771:
602:
2073:
985:
2116:
1843:
3053:
are real numbers (the given conditions insure that transformations are invertible, but not for making the classes distinct; for example, the identity belongs to all the classes).
65:
Over any field, the affine group may be viewed as a matrix group in a natural way. If the associated field of scalars is the real or complex field, then the affine group is a
4212:
4188:
4154:
4088:
4290:
4020:
4335:
3970:
3935:
371:
4359:
4040:
2092:
is the dimension of the last irreducible representation. Finally using the orthogonality of irreducible representations, we can complete the character table of
1757:
627:
969:{\displaystyle A=\left({\begin{array}{cc}1&0\\0&0\end{array}}\right),\qquad B=\left({\begin{array}{cc}0&1\\0&0\end{array}}\right)\,,}
212:
3660:
The proof may be done by first remarking that if an affine transformation has no fixed point, then the matrix of the associated linear map has an
3803:
131:
4392:
4670:
4634:
4609:
4584:
763:, namely the stabilizer of this affine plane; the above matrix formulation is the (transpose of) the realization of this, with the
4223:
3643:
belong to cases 1, 3, and 5. The transformations that do not preserve the orientation of the plane belong to cases 2 (with
4545:
2992:
4361:
with the translations. Geometrically, it is the subgroup of the affine group generated by the orthogonal reflections.
3732:
636:
62:), the affine group consists of those functions from the space to itself such that the image of every line is a line.
4662:
3940:
1095:{\displaystyle e^{aA+bB}=\left({\begin{array}{cc}e^{a}&{\tfrac {b}{a}}(e^{a}-1)\\0&1\end{array}}\right)\,.}
523:
316:
3640:
2009:
1945:{\displaystyle \rho _{k}{\begin{pmatrix}a&b\\0&1\end{pmatrix}}=\exp \left({\frac {2ikj\pi }{p-1}}\right)}
4690:
3030:
4516:(the addition and origin), but not necessarily scalar multiplication, and these groups differ if working over
4477:
4126:
3590:
4695:
3633:
51:
4464:
4452:
4193:
4169:
4135:
4069:
3877:
3727:
3693:
363:
118:
4271:
3987:
4700:
4305:
4059:
453:
with a vector space, the subgroup that stabilizes the origin (of the vector space) is the original
47:
3895:
3665:
3597:
482:
323:
is isomorphic to the general linear group of the same dimension (so the stabilizer of a point in
200:
102:
4666:
4630:
4605:
4580:
4376:
4370:
843:
807:
791:
4554:
4338:
439:{\displaystyle 1\to V\to V\rtimes \operatorname {GL} (V)\to \operatorname {GL} (V)\to 1\,.}
4472:
4468:
3797:
3609:
3601:
55:
4344:
4025:
3880:. In terms of the semi-direct product, the special affine group consists of all pairs
3041:, an affine coordinate system exists on which it has one of the following forms, where
4508:. Note that this containment is in general proper, since by "automorphisms" one means
4302:
is a subgroup of the affine group. Algebraically, this group is a semidirect product
4684:
4655:
4572:
4380:
4043:
3626:
3619:
3605:
1816:{\displaystyle \rho _{k}:\operatorname {Aff} (\mathbf {F} _{p})\to \mathbb {C} ^{*}}
17:
4650:
4558:
4055:
3615:
Case 3 corresponds to a scaling in one direction and a translation in another one.
3034:
670:{\displaystyle \left({\begin{array}{c|c}M&v\\\hline 0&1\end{array}}\right)}
619:
87:
43:
1019:
931:
885:
354:
All these subgroups are conjugate, where conjugation is given by translation from
199:
is the natural one (linear transformations are automorphisms), so this defines a
3865:
3038:
59:
31:
842:
matrices representing the affine group in one dimension. It is a two-parameter
277:{\displaystyle \operatorname {Aff} (n,K)=K^{n}\rtimes \operatorname {GL} (n,K)}
4119:
3661:
4062:, the affine group can be easily specified. For example, Günter Ewald wrote:
3800:
by a vector representation", and, as above, one has the short exact sequence
828:
Each of these two classes of matrices is closed under matrix multiplication.
846:
351:: recall that if one fixes a point, an affine space becomes a vector space.
66:
3864:
The subset of all invertible affine transformations that preserve a fixed
4294:
4163:
3976:
is a linear transformation of whose determinant has absolute value 1 and
177:{\displaystyle \operatorname {Aff} (V)=V\rtimes \operatorname {GL} (V)}
825:
identity matrix with the bottom row replaced by a row of all ones.
4579:. Vol. 1. Berlin Heidelberg: Springer-Verlag. Section 2.7.6.
4441:{\displaystyle \mathbf {R} ^{1,3}\rtimes \operatorname {O} (1,3)}
4268:
is a
Euclidean space (over the field of real numbers), the group
3600:
that may differ in two different directions. When working with a
339:); formally, it is the general linear group of the vector space
1751:
one-dimensional representations, decided by the homomorphism
4277:
4512:
automorphisms, i.e., they preserve the group structure on
4543:
Poole, David G. (November 1995). "The
Stochastic Group".
806:
matrix in which the entries in each column sum to 1. The
469:
Representing the affine group as a semidirect product of
3872:. (The transformations themselves are sometimes called
849:, so with merely two generators (Lie algebra elements),
3850:{\displaystyle 1\to V\to V\rtimes _{\rho }G\to G\to 1.}
3033:. More precisely, given an affine transformation of an
777:) blocks corresponding to the direct sum decomposition
4467:– certain discrete subgroups of the affine group on a
4247:{\displaystyle {\mathfrak {A}}\subset {\mathfrak {P}}}
1862:
1591:
1483:
1413:
1281:
1233:
1196:
1160:
1035:
27:
Group of all affine transformations of an affine space
4395:
4347:
4308:
4274:
4226:
4196:
4172:
4138:
4072:
4028:
3990:
3943:
3898:
3806:
3735:
3696:, one can produce an affine group, sometimes denoted
3062:
2995:
2114:
2012:
1846:
1760:
1379:
1154:
988:
873:
630:
526:
449:
In the case that the affine group was constructed by
374:
215:
134:
3022:{\displaystyle \operatorname {Aff} (2,\mathbb {R} )}
813:
for passing from the above kind to this kind is the
3796:: one can say that the affine group obtained is "a
4654:
4440:
4353:
4329:
4284:
4246:
4206:
4182:
4148:
4082:
4034:
4014:
3964:
3929:
3849:
3765:
3578:
3021:
2973:
2067:
1944:
1815:
1712:
1342:
1094:
968:
669:
596:
438:
276:
176:
3766:{\displaystyle \rho :G\to \operatorname {GL} (V)}
1740:irreducible representations. By above paragraph (
1697:
1654:
1624:
1559:
1516:
3904:
97:acting by translations, and the affine group of
4042:with the translations. It is generated by the
3722:More generally and abstractly, given any group
58:(where the associated field of scalars is the
54:from the space into itself. In the case of a
3876:.) This group is the affine analogue of the
597:{\displaystyle (v,M)\cdot (w,N)=(v+Mw,MN)\,.}
8:
3666:Jordan normal form theorem for real matrices
3636:when the coordinate axes are perpendicular.
1692:
1680:
2068:{\displaystyle p(p-1)=p-1+\chi _{p}^{2}\,,}
831:The simplest paradigm may well be the case
1741:
311:Given the affine group of an affine space
93:obtained by "forgetting" the origin, with
4402:
4397:
4394:
4346:
4307:
4276:
4275:
4273:
4238:
4237:
4228:
4227:
4225:
4198:
4197:
4195:
4174:
4173:
4171:
4140:
4139:
4137:
4074:
4073:
4071:
4027:
3989:
3942:
3916:
3899:
3897:
3826:
3805:
3734:
3558:
3438:
3377:
3356:
3293:
3272:
3212:
3188:
3131:
3067:
3063:
3061:
3012:
3011:
2994:
2960:
2951:
2942:
2933:
2921:
2906:
2896:
2890:
2879:
2870:
2861:
2852:
2843:
2834:
2818:
2812:
2801:
2792:
2783:
2774:
2765:
2756:
2747:
2699:
2693:
2684:
2653:
2647:
2616:
2610:
2601:
2592:
2582:
2576:
2528:
2522:
2513:
2482:
2476:
2445:
2439:
2430:
2421:
2411:
2405:
2357:
2351:
2342:
2311:
2305:
2274:
2268:
2259:
2250:
2240:
2234:
2214:
2209:
2203:
2194:
2182:
2177:
2171:
2161:
2155:
2145:
2139:
2126:
2120:
2115:
2113:
2061:
2055:
2050:
2011:
1906:
1857:
1851:
1845:
1807:
1803:
1802:
1789:
1784:
1765:
1759:
1702:
1696:
1695:
1671:
1666:
1653:
1652:
1641:
1636:
1623:
1622:
1586:
1568:
1558:
1557:
1549:
1538:
1533:
1528:
1515:
1514:
1478:
1460:
1448:
1408:
1388:
1380:
1378:
1336:
1276:
1264:
1228:
1191:
1155:
1153:
1088:
1053:
1034:
1026:
1018:
993:
987:
962:
930:
884:
872:
731:is naturally isomorphic to a subgroup of
635:
629:
590:
525:
483:by construction of the semidirect product
432:
373:
244:
214:
133:
3029:can take a simple form on a well-chosen
4535:
4489:
3639:The affine transformations without any
1677:
3937:, that is, the affine transformations
3781:, one gets an associated affine group
303:is matrix multiplication of a vector.
78:Construction from general linear group
2961:
2952:
2943:
2934:
2922:
2907:
2891:
2880:
2871:
2862:
2853:
2844:
2835:
2813:
2802:
2793:
2784:
2775:
2766:
2757:
2748:
2694:
2685:
2648:
2611:
2602:
2593:
2577:
2523:
2514:
2477:
2440:
2431:
2422:
2406:
2352:
2343:
2306:
2269:
2260:
2251:
2235:
2204:
2195:
2172:
2156:
2140:
2121:
7:
4604:. Belmont: Wadsworth. Section 4.12.
4629:. Belmont: Wadsworth. p. 241.
4239:
4229:
4199:
4175:
4141:
4090:of all projective collineations of
4075:
101:can be described concretely as the
4451:This example is very important in
4417:
2985:Planar affine group over the reals
206:In terms of matrices, one writes:
25:
4096:is a group which we may call the
3980:is any fixed translation vector.
3672:Other affine groups and subgroups
3664:equal to one, and then using the
1992:. Then compare with the order of
517:, and multiplication is given by
287:where here the natural action of
82:Concretely, given a vector space
4398:
1785:
1667:
1637:
1529:
838:, that is, the upper triangular
73:Relation to general linear group
4207:{\displaystyle {\mathfrak {P}}}
4183:{\displaystyle {\mathfrak {P}}}
4149:{\displaystyle {\mathfrak {A}}}
4083:{\displaystyle {\mathfrak {P}}}
4022:of the special linear group of
3557:
3355:
3271:
3187:
919:
607:This can be represented as the
4559:10.1080/00029890.1995.12004664
4435:
4423:
4318:
4312:
4285:{\displaystyle {\mathcal {E}}}
4190:consisting of all elements of
4015:{\displaystyle SL(V)\ltimes V}
4003:
3997:
3947:
3917:
3913:
3907:
3900:
3841:
3835:
3816:
3810:
3760:
3754:
3745:
3549:
3546:
3513:
3504:
3474:
3468:
3465:
3458:
3446:
3431:
3407:
3404:
3397:
3385:
3347:
3323:
3320:
3313:
3301:
3263:
3242:
3239:
3232:
3220:
3179:
3161:
3158:
3151:
3139:
3121:
3097:
3094:
3087:
3075:
3016:
3002:
2720:
2708:
2549:
2537:
2378:
2366:
2028:
2016:
1798:
1795:
1780:
1301:
1289:
1065:
1046:
587:
563:
557:
545:
539:
527:
426:
423:
417:
408:
405:
399:
384:
378:
271:
259:
234:
222:
171:
165:
147:
141:
1:
4546:American Mathematical Monthly
4330:{\displaystyle O(V)\ltimes V}
4292:of distance-preserving maps (
4260:Isometries of Euclidean space
3965:{\displaystyle x\mapsto Mx+v}
3604:these directions need not be
747:embedded as the affine plane
4058:and the projective group of
1976:is a generator of the group
1742:§ Matrix representation
4379:is the affine group of the
3930:{\displaystyle |\det(M)|=1}
3612:need not be perpendicular.
716:row of zeros, and 1 is the
4717:
4663:Cambridge University Press
4368:
3629:combined with a dilation.
3622:combined with a dilation.
1370:conjugacy classes, namely
4627:Geometry: An Introduction
4602:Geometry: An Introduction
3868:up to sign is called the
509:is a linear transform in
485:, the elements are pairs
4653:(1985). "Section VI.1".
3625:Case 5 corresponds to a
3618:Case 4 corresponds to a
3031:affine coordinate system
4054:Presuming knowledge of
720:identity block matrix.
86:, it has an underlying
4625:Ewald, Günter (1971).
4600:Ewald, Günter (1971).
4442:
4355:
4331:
4286:
4264:When the affine space
4248:
4208:
4184:
4150:
4127:hyperplane at infinity
4084:
4036:
4016:
3966:
3931:
3851:
3767:
3632:Case 6 corresponds to
3596:Case 2 corresponds to
3589:Case 1 corresponds to
3580:
3023:
2975:
2069:
1946:
1817:
1714:
1344:
1096:
970:
794:representation is any
709:column vector, 0 is a
671:
598:
440:
278:
178:
52:affine transformations
4443:
4356:
4332:
4287:
4249:
4209:
4185:
4151:
4106:. If we proceed from
4085:
4037:
4017:
3967:
3932:
3852:
3768:
3581:
3024:
2976:
2070:
1947:
1818:
1715:
1345:
1097:
971:
672:
599:
465:Matrix representation
441:
307:Stabilizer of a point
279:
179:
4465:Affine Coxeter group
4393:
4345:
4306:
4272:
4224:
4194:
4170:
4136:
4112:to the affine space
4070:
4026:
3988:
3941:
3896:
3878:special linear group
3870:special affine group
3860:Special affine group
3804:
3733:
3694:general linear group
3060:
2993:
2112:
2010:
1844:
1758:
1377:
1152:
986:
871:
628:
524:
372:
364:short exact sequence
213:
132:
119:general linear group
40:general affine group
18:Special affine group
4657:Groups and Geometry
4060:projective geometry
4050:Projective subgroup
3681:Given any subgroup
2060:
1543:
1106:Character table of
4438:
4351:
4327:
4282:
4244:
4204:
4180:
4146:
4080:
4032:
4012:
3962:
3927:
3847:
3777:on a vector space
3763:
3576:
3574:
3019:
2971:
2969:
2965:
2956:
2947:
2938:
2929:
2917:
2902:
2884:
2875:
2866:
2857:
2848:
2839:
2830:
2806:
2797:
2788:
2779:
2770:
2761:
2752:
2741:
2689:
2680:
2643:
2606:
2597:
2588:
2570:
2518:
2509:
2472:
2435:
2426:
2417:
2399:
2347:
2338:
2301:
2264:
2255:
2246:
2228:
2199:
2190:
2167:
2151:
2135:
2065:
2046:
1942:
1887:
1813:
1723:Then we know that
1710:
1708:
1616:
1527:
1508:
1438:
1340:
1330:
1258:
1221:
1185:
1092:
1082:
1044:
966:
956:
910:
667:
661:
594:
436:
274:
201:semidirect product
174:
103:semidirect product
50:of all invertible
4354:{\displaystyle V}
4035:{\displaystyle V}
3704:, analogously as
3561:
3441:
3380:
3359:
3296:
3275:
3215:
3191:
3134:
3070:
2738:
2677:
2640:
2567:
2506:
2469:
2396:
2335:
2298:
1936:
1043:
331:is isomorphic to
16:(Redirected from
4708:
4676:
4660:
4641:
4640:
4622:
4616:
4615:
4597:
4591:
4590:
4569:
4563:
4562:
4540:
4523:
4521:
4515:
4507:
4494:
4471:that preserve a
4447:
4445:
4444:
4439:
4413:
4412:
4401:
4385:
4360:
4358:
4357:
4352:
4339:orthogonal group
4336:
4334:
4333:
4328:
4301:
4291:
4289:
4288:
4283:
4281:
4280:
4267:
4253:
4251:
4250:
4245:
4243:
4242:
4233:
4232:
4217:
4213:
4211:
4210:
4205:
4203:
4202:
4189:
4187:
4186:
4181:
4179:
4178:
4161:
4155:
4153:
4152:
4147:
4145:
4144:
4129:, we obtain the
4124:
4117:
4111:
4105:
4098:projective group
4095:
4089:
4087:
4086:
4081:
4079:
4078:
4041:
4039:
4038:
4033:
4021:
4019:
4018:
4013:
3979:
3975:
3971:
3969:
3968:
3963:
3936:
3934:
3933:
3928:
3920:
3903:
3891:
3856:
3854:
3853:
3848:
3831:
3830:
3795:
3780:
3776:
3772:
3770:
3769:
3764:
3725:
3718:
3703:
3691:
3656:
3649:
3585:
3583:
3582:
3577:
3575:
3562:
3559:
3444:
3442:
3439:
3383:
3381:
3378:
3360:
3357:
3299:
3297:
3294:
3276:
3273:
3218:
3216:
3213:
3192:
3189:
3137:
3135:
3132:
3073:
3071:
3068:
3052:
3048:
3044:
3028:
3026:
3025:
3020:
3015:
2989:The elements of
2980:
2978:
2977:
2972:
2970:
2966:
2957:
2948:
2939:
2930:
2918:
2903:
2901:
2900:
2885:
2876:
2867:
2858:
2849:
2840:
2831:
2829:
2828:
2807:
2798:
2789:
2780:
2771:
2762:
2753:
2742:
2740:
2739:
2737:
2726:
2700:
2690:
2681:
2679:
2678:
2676:
2665:
2654:
2644:
2642:
2641:
2639:
2628:
2617:
2607:
2598:
2589:
2587:
2586:
2571:
2569:
2568:
2566:
2555:
2529:
2519:
2510:
2508:
2507:
2505:
2494:
2483:
2473:
2471:
2470:
2468:
2457:
2446:
2436:
2427:
2418:
2416:
2415:
2400:
2398:
2397:
2395:
2384:
2358:
2348:
2339:
2337:
2336:
2334:
2323:
2312:
2302:
2300:
2299:
2297:
2286:
2275:
2265:
2256:
2247:
2245:
2244:
2229:
2227:
2226:
2225:
2224:
2200:
2191:
2189:
2188:
2187:
2186:
2168:
2166:
2165:
2152:
2150:
2149:
2136:
2134:
2133:
2118:
2104:
2091:
2074:
2072:
2071:
2066:
2059:
2054:
2002:
1991:
1990:
1989:
1975:
1971:
1961:
1951:
1949:
1948:
1943:
1941:
1937:
1935:
1924:
1907:
1892:
1891:
1856:
1855:
1836:
1822:
1820:
1819:
1814:
1812:
1811:
1806:
1794:
1793:
1788:
1770:
1769:
1750:
1744:), there exist
1739:
1735:
1719:
1717:
1716:
1711:
1709:
1701:
1700:
1676:
1675:
1670:
1658:
1657:
1651:
1647:
1646:
1645:
1640:
1628:
1627:
1621:
1620:
1573:
1572:
1563:
1562:
1548:
1544:
1542:
1537:
1532:
1520:
1519:
1513:
1512:
1465:
1464:
1447:
1443:
1442:
1396:
1395:
1369:
1365:
1349:
1347:
1346:
1341:
1335:
1334:
1272:
1271:
1263:
1262:
1226:
1225:
1190:
1189:
1144:
1133:
1118:
1101:
1099:
1098:
1093:
1087:
1083:
1058:
1057:
1045:
1036:
1031:
1030:
1010:
1009:
975:
973:
972:
967:
961:
957:
915:
911:
863:
856:
852:
841:
837:
824:
812:
805:
786:
776:
772:
762:
746:
742:
730:
719:
715:
708:
701:
697:
693:
683:
676:
674:
673:
668:
666:
662:
618:
603:
601:
600:
595:
516:
508:
504:
500:
496:
480:
472:
460:
445:
443:
442:
437:
361:
357:
350:
338:
330:
322:
314:
302:
298:
283:
281:
280:
275:
249:
248:
198:
194:
183:
181:
180:
175:
124:
116:
108:
100:
96:
92:
85:
21:
4716:
4715:
4711:
4710:
4709:
4707:
4706:
4705:
4691:Affine geometry
4681:
4680:
4679:
4673:
4649:
4645:
4644:
4637:
4624:
4623:
4619:
4612:
4599:
4598:
4594:
4587:
4571:
4570:
4566:
4542:
4541:
4537:
4532:
4527:
4526:
4517:
4513:
4497:
4495:
4491:
4486:
4469:Euclidean space
4461:
4396:
4391:
4390:
4383:
4373:
4367:
4343:
4342:
4304:
4303:
4299:
4270:
4269:
4265:
4262:
4222:
4221:
4215:
4192:
4191:
4168:
4167:
4157:
4134:
4133:
4122:
4118:by declaring a
4113:
4107:
4101:
4091:
4068:
4067:
4052:
4024:
4023:
3986:
3985:
3977:
3973:
3939:
3938:
3894:
3893:
3881:
3862:
3822:
3802:
3801:
3798:group extension
3791:
3782:
3778:
3774:
3731:
3730:
3723:
3705:
3697:
3682:
3679:
3674:
3651:
3644:
3610:coordinate axes
3602:Euclidean plane
3573:
3572:
3555:
3461:
3443:
3435:
3434:
3400:
3382:
3374:
3373:
3353:
3316:
3298:
3290:
3289:
3269:
3235:
3217:
3209:
3208:
3185:
3154:
3136:
3128:
3127:
3090:
3072:
3058:
3057:
3050:
3046:
3042:
2991:
2990:
2987:
2968:
2967:
2958:
2949:
2940:
2931:
2919:
2904:
2892:
2887:
2886:
2877:
2868:
2859:
2850:
2841:
2832:
2814:
2809:
2808:
2799:
2790:
2781:
2772:
2763:
2754:
2744:
2743:
2727:
2701:
2695:
2691:
2682:
2666:
2655:
2649:
2645:
2629:
2618:
2612:
2608:
2599:
2590:
2578:
2573:
2572:
2556:
2530:
2524:
2520:
2511:
2495:
2484:
2478:
2474:
2458:
2447:
2441:
2437:
2428:
2419:
2407:
2402:
2401:
2385:
2359:
2353:
2349:
2340:
2324:
2313:
2307:
2303:
2287:
2276:
2270:
2266:
2257:
2248:
2236:
2231:
2230:
2210:
2205:
2201:
2192:
2178:
2173:
2169:
2157:
2153:
2141:
2137:
2122:
2110:
2109:
2102:
2093:
2084:
2079:
2008:
2007:
2001:
1993:
1988:
1983:
1982:
1981:
1977:
1973:
1963:
1956:
1925:
1908:
1902:
1886:
1885:
1880:
1874:
1873:
1868:
1858:
1847:
1842:
1841:
1827:
1801:
1783:
1761:
1756:
1755:
1745:
1737:
1733:
1724:
1707:
1706:
1665:
1635:
1615:
1614:
1609:
1603:
1602:
1597:
1587:
1585:
1581:
1574:
1564:
1554:
1553:
1507:
1506:
1501:
1495:
1494:
1489:
1479:
1477:
1473:
1466:
1456:
1453:
1452:
1437:
1436:
1431:
1425:
1424:
1419:
1409:
1404:
1397:
1384:
1375:
1374:
1367:
1363:
1354:
1329:
1328:
1323:
1317:
1316:
1287:
1277:
1257:
1256:
1251:
1245:
1244:
1239:
1229:
1227:
1220:
1219:
1214:
1208:
1207:
1202:
1192:
1184:
1183:
1178:
1172:
1171:
1166:
1156:
1150:
1149:
1135:
1131:
1122:
1120:
1116:
1107:
1081:
1080:
1075:
1069:
1068:
1049:
1032:
1022:
1014:
989:
984:
983:
955:
954:
949:
943:
942:
937:
926:
909:
908:
903:
897:
896:
891:
880:
869:
868:
858:
854:
850:
839:
832:
814:
810:
795:
778:
774:
764:
748:
744:
732:
724:
717:
710:
703:
699:
695:
685:
681:
660:
659:
654:
648:
647:
642:
631:
626:
625:
608:
522:
521:
510:
506:
502:
501:is a vector in
498:
486:
474:
470:
467:
454:
370:
369:
359:
355:
340:
332:
324:
320:
312:
309:
300:
288:
240:
211:
210:
196:
188:
130:
129:
122:
110:
106:
98:
94:
90:
83:
80:
75:
56:Euclidean space
28:
23:
22:
15:
12:
11:
5:
4714:
4712:
4704:
4703:
4698:
4693:
4683:
4682:
4678:
4677:
4671:
4646:
4643:
4642:
4635:
4617:
4610:
4592:
4585:
4564:
4553:(9): 798–801.
4534:
4533:
4531:
4528:
4525:
4524:
4488:
4487:
4485:
4482:
4481:
4480:
4475:
4460:
4457:
4449:
4448:
4437:
4434:
4431:
4428:
4425:
4422:
4419:
4416:
4411:
4408:
4405:
4400:
4377:Poincaré group
4371:Poincaré group
4369:Main article:
4366:
4365:Poincaré group
4363:
4350:
4326:
4323:
4320:
4317:
4314:
4311:
4279:
4261:
4258:
4257:
4256:
4255:
4254:
4241:
4236:
4231:
4201:
4177:
4143:
4077:
4051:
4048:
4044:shear mappings
4031:
4011:
4008:
4005:
4002:
3999:
3996:
3993:
3961:
3958:
3955:
3952:
3949:
3946:
3926:
3923:
3919:
3915:
3912:
3909:
3906:
3902:
3874:equiaffinities
3861:
3858:
3846:
3843:
3840:
3837:
3834:
3829:
3825:
3821:
3818:
3815:
3812:
3809:
3787:
3762:
3759:
3756:
3753:
3750:
3747:
3744:
3741:
3738:
3728:representation
3678:
3675:
3673:
3670:
3587:
3586:
3571:
3568:
3565:
3556:
3554:
3551:
3548:
3545:
3542:
3539:
3536:
3533:
3530:
3527:
3524:
3521:
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3494:
3491:
3488:
3485:
3482:
3479:
3476:
3473:
3470:
3467:
3464:
3462:
3460:
3457:
3454:
3451:
3448:
3445:
3437:
3436:
3433:
3430:
3427:
3424:
3421:
3418:
3415:
3412:
3409:
3406:
3403:
3401:
3399:
3396:
3393:
3390:
3387:
3384:
3376:
3375:
3372:
3369:
3366:
3363:
3354:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3317:
3315:
3312:
3309:
3306:
3303:
3300:
3292:
3291:
3288:
3285:
3282:
3279:
3270:
3268:
3265:
3262:
3259:
3256:
3253:
3250:
3247:
3244:
3241:
3238:
3236:
3234:
3231:
3228:
3225:
3222:
3219:
3211:
3210:
3207:
3204:
3201:
3198:
3195:
3186:
3184:
3181:
3178:
3175:
3172:
3169:
3166:
3163:
3160:
3157:
3155:
3153:
3150:
3147:
3144:
3141:
3138:
3130:
3129:
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3091:
3089:
3086:
3083:
3080:
3077:
3074:
3066:
3065:
3018:
3014:
3010:
3007:
3004:
3001:
2998:
2986:
2983:
2982:
2981:
2964:
2959:
2955:
2950:
2946:
2941:
2937:
2932:
2928:
2925:
2920:
2916:
2913:
2910:
2905:
2899:
2895:
2889:
2888:
2883:
2878:
2874:
2869:
2865:
2860:
2856:
2851:
2847:
2842:
2838:
2833:
2827:
2824:
2821:
2817:
2811:
2810:
2805:
2800:
2796:
2791:
2787:
2782:
2778:
2773:
2769:
2764:
2760:
2755:
2751:
2746:
2745:
2736:
2733:
2730:
2725:
2722:
2719:
2716:
2713:
2710:
2707:
2704:
2698:
2692:
2688:
2683:
2675:
2672:
2669:
2664:
2661:
2658:
2652:
2646:
2638:
2635:
2632:
2627:
2624:
2621:
2615:
2609:
2605:
2600:
2596:
2591:
2585:
2581:
2575:
2574:
2565:
2562:
2559:
2554:
2551:
2548:
2545:
2542:
2539:
2536:
2533:
2527:
2521:
2517:
2512:
2504:
2501:
2498:
2493:
2490:
2487:
2481:
2475:
2467:
2464:
2461:
2456:
2453:
2450:
2444:
2438:
2434:
2429:
2425:
2420:
2414:
2410:
2404:
2403:
2394:
2391:
2388:
2383:
2380:
2377:
2374:
2371:
2368:
2365:
2362:
2356:
2350:
2346:
2341:
2333:
2330:
2327:
2322:
2319:
2316:
2310:
2304:
2296:
2293:
2290:
2285:
2282:
2279:
2273:
2267:
2263:
2258:
2254:
2249:
2243:
2239:
2233:
2232:
2223:
2220:
2217:
2213:
2208:
2202:
2198:
2193:
2185:
2181:
2176:
2170:
2164:
2160:
2154:
2148:
2144:
2138:
2132:
2129:
2125:
2119:
2117:
2098:
2082:
2076:
2075:
2064:
2058:
2053:
2049:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
1997:
1984:
1953:
1952:
1940:
1934:
1931:
1928:
1923:
1920:
1917:
1914:
1911:
1905:
1901:
1898:
1895:
1890:
1884:
1881:
1879:
1876:
1875:
1872:
1869:
1867:
1864:
1863:
1861:
1854:
1850:
1824:
1823:
1810:
1805:
1800:
1797:
1792:
1787:
1782:
1779:
1776:
1773:
1768:
1764:
1729:
1721:
1720:
1705:
1699:
1694:
1691:
1688:
1685:
1682:
1679:
1674:
1669:
1664:
1661:
1656:
1650:
1644:
1639:
1634:
1631:
1626:
1619:
1613:
1610:
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1605:
1604:
1601:
1598:
1596:
1593:
1592:
1590:
1584:
1580:
1577:
1575:
1571:
1567:
1561:
1556:
1555:
1552:
1547:
1541:
1536:
1531:
1526:
1523:
1518:
1511:
1505:
1502:
1500:
1497:
1496:
1493:
1490:
1488:
1485:
1484:
1482:
1476:
1472:
1469:
1467:
1463:
1459:
1455:
1454:
1451:
1446:
1441:
1435:
1432:
1430:
1427:
1426:
1423:
1420:
1418:
1415:
1414:
1412:
1407:
1403:
1400:
1398:
1394:
1391:
1387:
1383:
1382:
1359:
1351:
1350:
1339:
1333:
1327:
1324:
1322:
1319:
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1286:
1283:
1282:
1280:
1275:
1270:
1267:
1261:
1255:
1252:
1250:
1247:
1246:
1243:
1240:
1238:
1235:
1234:
1232:
1224:
1218:
1215:
1213:
1210:
1209:
1206:
1203:
1201:
1198:
1197:
1195:
1188:
1182:
1179:
1177:
1174:
1173:
1170:
1167:
1165:
1162:
1161:
1159:
1127:
1119:
1112:
1104:
1103:
1102:
1091:
1086:
1079:
1076:
1074:
1071:
1070:
1067:
1064:
1061:
1056:
1052:
1048:
1042:
1039:
1033:
1029:
1025:
1021:
1020:
1017:
1013:
1008:
1005:
1002:
999:
996:
992:
977:
976:
965:
960:
953:
950:
948:
945:
944:
941:
938:
936:
933:
932:
929:
925:
922:
918:
914:
907:
904:
902:
899:
898:
895:
892:
890:
887:
886:
883:
879:
876:
678:
677:
665:
658:
655:
653:
650:
649:
646:
643:
641:
638:
637:
634:
605:
604:
593:
589:
586:
583:
580:
577:
574:
571:
568:
565:
562:
559:
556:
553:
550:
547:
544:
541:
538:
535:
532:
529:
466:
463:
447:
446:
435:
431:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
398:
395:
392:
389:
386:
383:
380:
377:
308:
305:
285:
284:
273:
270:
267:
264:
261:
258:
255:
252:
247:
243:
239:
236:
233:
230:
227:
224:
221:
218:
187:The action of
185:
184:
173:
170:
167:
164:
161:
158:
155:
152:
149:
146:
143:
140:
137:
79:
76:
74:
71:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4713:
4702:
4699:
4697:
4694:
4692:
4689:
4688:
4686:
4674:
4672:0-521-31694-4
4668:
4664:
4659:
4658:
4652:
4651:Lyndon, Roger
4648:
4647:
4638:
4636:9780534000349
4632:
4628:
4621:
4618:
4613:
4611:9780534000349
4607:
4603:
4596:
4593:
4588:
4586:9780534000349
4582:
4578:
4574:
4568:
4565:
4560:
4556:
4552:
4548:
4547:
4539:
4536:
4529:
4520:
4511:
4505:
4501:
4493:
4490:
4483:
4479:
4476:
4474:
4470:
4466:
4463:
4462:
4458:
4456:
4454:
4432:
4429:
4426:
4420:
4414:
4409:
4406:
4403:
4389:
4388:
4387:
4382:
4381:Lorentz group
4378:
4372:
4364:
4362:
4348:
4340:
4324:
4321:
4315:
4309:
4297:
4296:
4259:
4234:
4220:
4219:
4165:
4160:
4132:
4128:
4121:
4116:
4110:
4104:
4099:
4094:
4065:
4064:
4063:
4061:
4057:
4049:
4047:
4045:
4029:
4009:
4006:
4000:
3994:
3991:
3981:
3959:
3956:
3953:
3950:
3944:
3924:
3921:
3910:
3889:
3885:
3879:
3875:
3871:
3867:
3859:
3857:
3844:
3838:
3832:
3827:
3823:
3819:
3813:
3807:
3799:
3794:
3790:
3785:
3757:
3751:
3748:
3742:
3739:
3736:
3729:
3720:
3717:
3713:
3709:
3701:
3695:
3689:
3685:
3676:
3671:
3669:
3667:
3663:
3658:
3654:
3650:) or 3 (with
3647:
3642:
3637:
3635:
3630:
3628:
3627:shear mapping
3623:
3621:
3620:shear mapping
3616:
3613:
3611:
3607:
3606:perpendicular
3603:
3599:
3594:
3592:
3569:
3566:
3563:
3552:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3510:
3507:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3477:
3471:
3463:
3455:
3452:
3449:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3402:
3394:
3391:
3388:
3370:
3367:
3364:
3361:
3350:
3344:
3341:
3338:
3335:
3332:
3329:
3326:
3318:
3310:
3307:
3304:
3286:
3283:
3280:
3277:
3266:
3260:
3257:
3254:
3251:
3248:
3245:
3237:
3229:
3226:
3223:
3205:
3202:
3199:
3196:
3193:
3182:
3176:
3173:
3170:
3167:
3164:
3156:
3148:
3145:
3142:
3124:
3118:
3115:
3112:
3109:
3106:
3103:
3100:
3092:
3084:
3081:
3078:
3056:
3055:
3054:
3040:
3036:
3032:
3008:
3005:
2999:
2996:
2984:
2962:
2953:
2944:
2935:
2926:
2923:
2914:
2911:
2908:
2897:
2893:
2881:
2872:
2863:
2854:
2845:
2836:
2825:
2822:
2819:
2815:
2803:
2794:
2785:
2776:
2767:
2758:
2749:
2734:
2731:
2728:
2723:
2717:
2714:
2711:
2705:
2702:
2696:
2686:
2673:
2670:
2667:
2662:
2659:
2656:
2650:
2636:
2633:
2630:
2625:
2622:
2619:
2613:
2603:
2594:
2583:
2579:
2563:
2560:
2557:
2552:
2546:
2543:
2540:
2534:
2531:
2525:
2515:
2502:
2499:
2496:
2491:
2488:
2485:
2479:
2465:
2462:
2459:
2454:
2451:
2448:
2442:
2432:
2423:
2412:
2408:
2392:
2389:
2386:
2381:
2375:
2372:
2369:
2363:
2360:
2354:
2344:
2331:
2328:
2325:
2320:
2317:
2314:
2308:
2294:
2291:
2288:
2283:
2280:
2277:
2271:
2261:
2252:
2241:
2237:
2221:
2218:
2215:
2211:
2206:
2196:
2183:
2179:
2174:
2162:
2158:
2146:
2142:
2130:
2127:
2123:
2108:
2107:
2106:
2101:
2097:
2089:
2085:
2062:
2056:
2051:
2047:
2043:
2040:
2037:
2034:
2031:
2025:
2022:
2019:
2013:
2006:
2005:
2004:
2000:
1996:
1987:
1980:
1970:
1966:
1959:
1938:
1932:
1929:
1926:
1921:
1918:
1915:
1912:
1909:
1903:
1899:
1896:
1893:
1888:
1882:
1877:
1870:
1865:
1859:
1852:
1848:
1840:
1839:
1838:
1834:
1830:
1808:
1790:
1777:
1774:
1771:
1766:
1762:
1754:
1753:
1752:
1748:
1743:
1732:
1728:
1703:
1689:
1686:
1683:
1672:
1662:
1659:
1648:
1642:
1632:
1629:
1617:
1611:
1606:
1599:
1594:
1588:
1582:
1578:
1576:
1569:
1565:
1550:
1545:
1539:
1534:
1524:
1521:
1509:
1503:
1498:
1491:
1486:
1480:
1474:
1470:
1468:
1461:
1457:
1449:
1444:
1439:
1433:
1428:
1421:
1416:
1410:
1405:
1401:
1399:
1392:
1389:
1385:
1373:
1372:
1371:
1362:
1358:
1337:
1331:
1325:
1320:
1313:
1310:
1307:
1304:
1298:
1295:
1292:
1284:
1278:
1273:
1268:
1265:
1259:
1253:
1248:
1241:
1236:
1230:
1222:
1216:
1211:
1204:
1199:
1193:
1186:
1180:
1175:
1168:
1163:
1157:
1148:
1147:
1146:
1142:
1138:
1130:
1126:
1115:
1111:
1105:
1089:
1084:
1077:
1072:
1062:
1059:
1054:
1050:
1040:
1037:
1027:
1023:
1015:
1011:
1006:
1003:
1000:
997:
994:
990:
982:
981:
980:
963:
958:
951:
946:
939:
934:
927:
923:
920:
916:
912:
905:
900:
893:
888:
881:
877:
874:
867:
866:
865:
862:
848:
845:
835:
829:
826:
822:
818:
809:
803:
799:
793:
788:
785:
781:
771:
767:
760:
756:
752:
740:
736:
728:
721:
714:
706:
692:
688:
663:
656:
651:
644:
639:
632:
624:
623:
622:
621:
616:
612:
591:
584:
581:
578:
575:
572:
569:
566:
560:
554:
551:
548:
542:
536:
533:
530:
520:
519:
518:
514:
494:
490:
484:
478:
464:
462:
458:
452:
433:
429:
420:
414:
411:
402:
396:
393:
390:
387:
381:
375:
368:
367:
366:
365:
352:
348:
344:
336:
328:
318:
306:
304:
296:
292:
268:
265:
262:
256:
253:
250:
245:
241:
237:
231:
228:
225:
219:
216:
209:
208:
207:
204:
202:
192:
168:
162:
159:
156:
153:
150:
144:
138:
135:
128:
127:
126:
120:
114:
104:
89:
77:
72:
70:
68:
63:
61:
57:
53:
49:
45:
41:
37:
33:
19:
4696:Group theory
4656:
4626:
4620:
4601:
4595:
4576:
4567:
4550:
4544:
4538:
4518:
4509:
4503:
4499:
4492:
4450:
4374:
4293:
4263:
4158:
4131:affine group
4130:
4114:
4108:
4102:
4097:
4092:
4056:projectivity
4053:
3982:
3887:
3883:
3873:
3869:
3863:
3792:
3788:
3783:
3721:
3715:
3711:
3707:
3699:
3687:
3683:
3680:
3677:General case
3659:
3652:
3645:
3638:
3634:similarities
3631:
3624:
3617:
3614:
3608:, since the
3595:
3591:translations
3588:
3035:affine plane
2988:
2099:
2095:
2087:
2080:
2077:
1998:
1994:
1985:
1978:
1968:
1964:
1957:
1954:
1832:
1828:
1825:
1746:
1730:
1726:
1722:
1360:
1356:
1352:
1140:
1136:
1128:
1124:
1121:
1113:
1109:
978:
860:
857:, such that
833:
830:
827:
820:
816:
801:
797:
789:
783:
779:
769:
765:
758:
754:
750:
738:
734:
726:
722:
712:
704:
694:matrix over
690:
686:
679:
620:block matrix
614:
610:
606:
512:
492:
488:
476:
468:
456:
450:
448:
353:
346:
342:
334:
326:
310:
294:
290:
286:
205:
190:
186:
112:
88:affine space
81:
64:
60:real numbers
44:affine space
39:
36:affine group
35:
29:
4502:) < Aut(
4214:that leave
3866:volume form
3641:fixed point
3560:where
3358:where
3274:where
3190:where
844:non-Abelian
319:of a point
32:mathematics
4701:Lie groups
4685:Categories
4573:Berger, M.
4530:References
4453:relativity
4295:isometries
4120:hyperplane
3710:) :=
3662:eigenvalue
2003:, we have
1134:has order
808:similarity
723:Formally,
317:stabilizer
4478:Holomorph
4421:
4415:⋊
4322:⋉
4235:⊂
4007:⋉
3948:↦
3842:→
3836:→
3828:ρ
3824:⋊
3817:→
3811:→
3752:
3746:→
3737:ρ
3567:≠
3541:
3526:
3517:−
3499:
3484:
3466:↦
3405:↦
3365:≠
3321:↦
3281:≠
3240:↦
3200:≠
3159:↦
3095:↦
3037:over the
3000:
2954:…
2924:−
2912:−
2894:χ
2873:…
2823:−
2816:χ
2804:…
2795:…
2786:…
2777:…
2768:…
2759:…
2750:…
2732:−
2715:−
2706:π
2687:…
2671:−
2660:π
2634:−
2623:π
2580:χ
2561:−
2544:−
2535:π
2516:…
2500:−
2489:π
2463:−
2452:π
2409:χ
2390:−
2373:−
2364:π
2345:…
2329:−
2318:π
2292:−
2281:π
2238:χ
2219:−
2197:…
2048:χ
2038:−
2023:−
1930:−
1922:π
1900:
1849:ρ
1831:= 1, 2,…
1809:∗
1799:→
1778:
1763:ρ
1678:∖
1663:∈
1633:∈
1540:∗
1525:∈
1296:−
1266:−
1060:−
864:, where
847:Lie group
543:⋅
427:→
415:
409:→
397:
391:⋊
385:→
379:→
257:
251:⋊
220:
163:
157:⋊
139:
67:Lie group
4577:Geometry
4575:(1987).
4459:See also
4164:subgroup
4125:to be a
4066:The set
3686:< GL(
3598:scalings
1837:, where
1353:we know
1145:. Since
979:so that
819:+ 1) × (
800:+ 1) × (
613:+ 1) × (
497:, where
451:starting
4473:lattice
4337:of the
4218:fixed.
4162:as the
3692:of the
792:similar
753:, 1) |
743:, with
481:, then
325:Aff(2,
46:is the
42:of any
4669:
4633:
4608:
4583:
4496:Since
4384:O(1,3)
3972:where
3726:and a
3655:< 0
3648:< 0
3049:, and
2078:hence
684:is an
680:where
333:GL(2,
315:, the
117:, the
34:, the
4510:group
4484:Notes
4298:) of
3892:with
3039:reals
840:2 × 2
775:1 × 1
718:1 × 1
48:group
4667:ISBN
4631:ISBN
4606:ISBN
4581:ISBN
4375:The
3706:Aff(
3698:Aff(
2094:Aff(
1960:= −1
1955:and
1826:for
1736:has
1725:Aff(
1366:has
1355:Aff(
1143:− 1)
1123:Aff(
1108:Aff(
853:and
823:+ 1)
804:+ 1)
773:and
725:Aff(
711:1 ×
617:+ 1)
505:and
4555:doi
4551:102
4498:GL(
4341:of
4166:of
4156:of
4100:of
3905:det
3773:of
3657:).
3538:cos
3523:sin
3496:sin
3481:cos
2997:Aff
2090:− 1
1897:exp
1835:− 1
1775:Aff
1749:− 1
836:= 1
733:GL(
707:× 1
702:an
511:GL(
475:GL(
473:by
455:GL(
358:to
299:on
289:GL(
217:Aff
195:on
189:GL(
136:Aff
121:of
111:GL(
109:by
105:of
38:or
30:In
4687::
4665:.
4661:.
4549:.
4455:.
4386::
4046:.
3886:,
3845:1.
3749:GL
3719:.
3714:⋊
3668:.
3646:ab
3593:.
3570:0.
3440:6.
3379:5.
3295:4.
3214:3.
3133:2.
3069:1.
3045:,
2657:12
2105::
2086:=
1972:,
1967:=
1962:,
859:=
790:A
787:.
782:⊕
768:×
757:∈
749:{(
737:⊕
698:,
689:×
491:,
461:.
412:GL
394:GL
345:,
293:,
254:GL
203:.
160:GL
125::
69:.
4675:.
4639:.
4614:.
4589:.
4561:.
4557::
4522:.
4519:R
4514:V
4506:)
4504:V
4500:V
4436:)
4433:3
4430:,
4427:1
4424:(
4418:O
4410:3
4407:,
4404:1
4399:R
4349:V
4325:V
4319:)
4316:V
4313:(
4310:O
4300:A
4278:E
4266:A
4240:P
4230:A
4216:ω
4200:P
4176:P
4159:A
4142:A
4123:ω
4115:A
4109:P
4103:P
4093:P
4076:P
4030:V
4010:V
4004:)
4001:V
3998:(
3995:L
3992:S
3978:v
3974:M
3960:v
3957:+
3954:x
3951:M
3945:x
3925:1
3922:=
3918:|
3914:)
3911:M
3908:(
3901:|
3890:)
3888:v
3884:M
3882:(
3839:G
3833:G
3820:V
3814:V
3808:1
3793:G
3789:ρ
3786:⋊
3784:V
3779:V
3775:G
3761:)
3758:V
3755:(
3743:G
3740::
3724:G
3716:G
3712:V
3708:G
3702:)
3700:G
3690:)
3688:V
3684:G
3653:a
3564:a
3553:,
3550:)
3547:)
3544:t
3535:y
3532:+
3529:t
3520:x
3514:(
3511:a
3508:,
3505:)
3502:t
3493:y
3490:+
3487:t
3478:x
3475:(
3472:a
3469:(
3459:)
3456:y
3453:,
3450:x
3447:(
3432:)
3429:a
3426:+
3423:y
3420:,
3417:y
3414:+
3411:x
3408:(
3398:)
3395:y
3392:,
3389:x
3386:(
3371:,
3368:0
3362:a
3351:,
3348:)
3345:y
3342:a
3339:,
3336:y
3333:+
3330:x
3327:a
3324:(
3314:)
3311:y
3308:,
3305:x
3302:(
3287:,
3284:0
3278:a
3267:,
3264:)
3261:b
3258:+
3255:y
3252:,
3249:x
3246:a
3243:(
3233:)
3230:y
3227:,
3224:x
3221:(
3206:,
3203:0
3197:b
3194:a
3183:,
3180:)
3177:y
3174:b
3171:,
3168:x
3165:a
3162:(
3152:)
3149:y
3146:,
3143:x
3140:(
3125:,
3122:)
3119:b
3116:+
3113:y
3110:,
3107:a
3104:+
3101:x
3098:(
3088:)
3085:y
3082:,
3079:x
3076:(
3051:t
3047:b
3043:a
3017:)
3013:R
3009:,
3006:2
3003:(
2963:0
2945:0
2936:0
2927:1
2915:1
2909:p
2898:p
2882:1
2864:1
2855:1
2846:1
2837:1
2826:1
2820:p
2735:1
2729:p
2724:i
2721:)
2718:2
2712:p
2709:(
2703:6
2697:e
2674:1
2668:p
2663:i
2651:e
2637:1
2631:p
2626:i
2620:6
2614:e
2604:1
2595:1
2584:3
2564:1
2558:p
2553:i
2550:)
2547:2
2541:p
2538:(
2532:4
2526:e
2503:1
2497:p
2492:i
2486:8
2480:e
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2032:=
2029:)
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2017:(
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1894:=
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1704:.
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1618:)
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1607:0
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1589:(
1583:{
1579:=
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1560:{
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1475:{
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1406:{
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1274:=
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1223:)
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1047:(
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1038:b
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1024:e
1016:(
1012:=
1007:B
1004:b
1001:+
998:A
995:a
991:e
964:,
959:)
952:0
947:0
940:1
935:0
928:(
924:=
921:B
917:,
913:)
906:0
901:0
894:0
889:1
882:(
878:=
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861:B
855:B
851:A
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817:n
815:(
811:P
802:n
798:n
796:(
784:K
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770:n
766:n
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745:V
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729:)
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611:n
609:(
592:.
588:)
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570:+
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561:=
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555:N
552:,
549:w
546:(
540:)
537:M
534:,
531:v
528:(
515:)
513:V
507:M
503:V
499:v
495:)
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489:v
487:(
479:)
477:V
471:V
459:)
457:V
434:.
430:1
424:)
421:V
418:(
406:)
403:V
400:(
388:V
382:V
376:1
360:q
356:p
349:)
347:p
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341:(
337:)
335:R
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272:)
269:K
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246:n
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238:=
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232:K
229:,
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223:(
197:V
193:)
191:V
172:)
169:V
166:(
154:V
151:=
148:)
145:V
142:(
123:V
115:)
113:V
107:V
99:A
95:V
91:A
84:V
20:)
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