1017:, the resulting point set remains a unital in any translation plane whose generating spread contains all of the same lines as the original spread within the quadric surface. In the ovoidal cone case, this forced intersection consists of a single line, and any spread can be mapped onto a spread containing this line, showing that every translation plane of this form admits an embedded unital.
1033:
750:(two Buekenhout, two not), and so another four in the dual Hall plane, and eight in the Hughes plane. However, one of the Buekenhout unitals in the Hall plane is self-dual, and thus gets counted again in the dual Hall plane. Thus, there are 17 distinct embeddable unitals with
979:, either a point-cone over a 3-dimensional ovoid if the line represented by the spread of the Bruck/Bose model meets the unital in one point, or a non-singular quadric otherwise. Because these objects have known intersection patterns with respect to planes of
566:
1702:
1421:
782:
between the point sets which maps blocks to blocks. This concept does not take into account the property of embeddability, so to do so we say that two unitals, embedded in the same ambient plane, are
317:
94:), the classical examples of unitals are given by nondegenerate Hermitian curves. There are also many non-classical examples. The first and the only known unital with non prime power parameters,
1735:
itself) of the variety or lies completely on the variety. One can prove that these lines form a subspace, either a hyperplane of the full space. In the latter case, the point is singular.
415:
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851:. Metz subsequently showed that one of Buekenhout's constructions actually yields non-classical unitals in all finite Desarguesian planes of square order at least 9. These
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50:+ 1 so that every pair of distinct points of the set are contained in exactly one subset. This is equivalent to saying that a unital is a 2-(
102:, was constructed by Bhaskar Bagchi and Sunanda Bagchi. It is still unknown if this unital can be embedded in a projective plane of order
1054:
2357:
2261:
1080:
1338:
2270:
Bagchi, S.; Bagchi, B. (1989), "Designs from pairs of finite fields. A cyclic unital U(6) and other regular steiner 2-designs",
847:, Buekenhout provided two constructions, which together proved the existence of an embedded unital in any finite 2-dimensional
1058:
754:= 3. On the other hand, a nonexhaustive computer search found over 900 mutually nonisomorphic designs which are unitals with
266:
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is a set of points of which the representing vector lines consisting of isotropic points of a non-trivial
Hermitian
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of order 9.), an exhaustive computer search by
Penttila and Royle found 18 unitals (up to equivalence) with
409:
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727:
681:
whose blocks are the fixed point sets of these various involutions μ. Since Γ acts doubly transitively on
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747:
387:
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1114:
941:, the equation of the Hermitian curve satisfied by a classical unital becomes a quadric surface in
561:{\displaystyle {\mathcal {H}}=\{(x_{0},x_{1},x_{2})\colon x_{0}^{q+1}+x_{1}^{q+1}+x_{2}^{q+1}=0\}.}
120:
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1992:
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712:= 3, Grüning proved that a Ree unital can not be embedded in any projective plane of order 9.
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Lüneburg also showed that the Ree unitals can not be embedded in projective planes of order
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1949:
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256:, the set of points of a nondegenerate Hermitian curve form a unital, which is called a
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384:. As all nondegenerate Hermitian curves in the same plane are projectively equivalent,
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on its subspaces that reverses containment. In particular, a correlation interchanges
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with respect to this basis, it is on the
Hermitian variety if and only if :
17:
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1996:
1697:{\displaystyle X={\begin{bmatrix}X_{0}\\X_{1}\\\vdots \\X_{n}\end{bmatrix}}.}
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31:
746:= 3 in these four planes: two in PG(2,9) (both Buekenhout), four in the
221:
The absolute points of a unitary polarity of the projective geometry PG(
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2391:
2141:
2094:
1988:
1094:
2406:
Lüneburg, H. (1966), "Some remarks concerning the Ree group of type (G
2212:
Ebert, G.L. (1992-03-01). "On
Buekenhout-Metz unitals of even order".
2125:
2031:
1925:
858:
The core idea in
Buekenhout's construction is that when one looks at
2256:, Cambridge Tracts in Mathematics #103, Cambridge University Press,
218:
of a polarity if it lies on the image of itself under the polarity.
2316:
Betten, A.; Betten, D.; Tonchev, V.D. (2003), "Unitals and codes",
1895:
1893:
82:, every line of the plane intersects the unital in either 1 or
1026:
1416:{\displaystyle \sum _{i,j=0}^{n}a_{ij}X_{i}X_{j}^{\theta }=0}
2032:"Existence of unitals in finite translation planes of order
421:
393:
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272:
704:
or not) such that the automorphism group Γ is induced by a
903:
in the higher-dimensional Bruck/Bose model, which lies in
1946:
which can be embedded in two different planes of order
1093:
Hermitian varieties are in a sense a generalisation of
2436:) in the affine and projective planes of order nine",
1635:
1567:, the equation can be written in a compact way :
623:(it will be convenient to think of these subgroups as
312:{\displaystyle {\mathcal {H}}={\mathcal {H}}(2,q^{2})}
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A set of n³ + 1 points arranged into subsets of n + 1
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738:of order 9, the dual Hall plane of order 9 and the
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2432:Penttila, T.; Royle, G.F. (1995), "Sets of type (
790:of the plane which maps one unital to the other.
78:points in the projective plane). In this case of
2346:Ergebnisse der Mathematik und ihrer Grenzgebiete
1768:PG(2,9) and the Hughes plane are both self-dual.
2378:Grüning, K. (1986), "Das Kleinste Ree-Unital",
1911:
685:, this will be a 2-design with parameters 2-(
8:
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1061:. Unsourced material may be challenged and
579:was constructed by H. Lüneburg. Let Γ = R(
74:(the subsets of the design become sets of
2423:
2272:Journal of Combinatorial Theory, Series A
2188:
2177:Journal of Combinatorial Theory, Series A
2173:"On Buekenhout-Metz unitals of odd order"
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1262:{\displaystyle e_{0},e_{1},\ldots ,e_{n}}
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1081:Learn how and when to remove this message
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665:, μ. Each such involution fixes exactly
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1469:{\displaystyle a_{ij}=a_{ji}^{\theta }}
855:unitals have been extensively studied.
145:A correlation of order two is called a
319:be a nondegenerate Hermitian curve in
2252:Assmus, E. F. Jr; Key, J. D. (1992),
2171:Baker, R.D; Ebert, G.L (1992-05-01).
1759:≥ 3 to avoid small exceptional cases.
1326:{\displaystyle (X_{0},\ldots ,X_{n})}
798:By examining the classical unital in
7:
2290:Barwick, Susan; Ebert, Gary (2008),
1731:when it is contains only one point (
1715:be a point on the Hermitian variety
1059:adding citations to reliable sources
762:Isomorphic versus equivalent unitals
1608:{\displaystyle X^{t}AX^{\theta }=0}
619:doubly transitively on this set by
575:Another family of unitals based on
119:We review some terminology used in
25:
2214:European Journal of Combinatorics
2018:Betten, Betten & Tonchev 2003
728:four projective planes of order 9
1031:
661:- 1, and thus contains a unique
2438:Designs, Codes and Cryptography
235:nondegenerate Hermitian variety
1755:, p. 28, further require that
1707:Tangent spaces and singularity
1320:
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627:that Γ is acting on.) For any
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401:{\displaystyle {\mathcal {H}}}
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130:of a projective geometry is a
46:arranged into subsets of size
1:
2348:, Band 44, Berlin, New York:
2330:10.1016/s0012-365x(02)00600-3
2030:Buekenhout, F. (1976-07-01).
1926:"A class of unitals of order
1924:Grüning, Klaus (1987-06-01).
1560:{\displaystyle A_{ij}=a_{ij}}
1097:, and occur naturally in the
793:
770:, two unitals are said to be
693:+ 1, 1) called a Ree unital.
408:can be described in terms of
243:nondegenerate Hermitian curve
241:= 2 this variety is called a
2425:10.1016/0021-8693(66)90014-7
2292:Unitals in Projective Planes
2284:10.1016/0097-3165(89)90061-7
2226:10.1016/0195-6698(92)90042-X
2190:10.1016/0097-3165(92)90038-V
1281:has homogeneous coordinates
583:) be the Ree group of type G
163:with companion automorphism
2124:Metz, Rudolf (1979-03-01).
896:{\displaystyle PG(2,q^{2})}
836:{\displaystyle PG(2,q^{2})}
357:{\displaystyle PG(2,q^{2})}
2511:
794:Buekenhout's Constructions
106:, if such a plane exists.
2336:Dembowski, Peter (1968),
2300:10.1007/978-0-387-76366-8
1912:Penttila & Royle 1995
778:between them, that is, a
1900:Barwick & Ebert 2008
1837:Barwick & Ebert 2008
1813:Barwick & Ebert 2008
1801:Barwick & Ebert 2008
1789:Bagchi & Bagchi 1989
1753:Barwick & Ebert 2008
1505:{\displaystyle a_{ij}=0}
2254:Designs and Their Codes
2126:"On a class of unitals"
2059:with a kernel of order
1133:{\displaystyle \theta }
1010:{\displaystyle PG(4,q)}
972:{\displaystyle PG(4,q)}
934:{\displaystyle PG(4,q)}
410:homogeneous coordinates
252:) for some prime power
152:A polarity is called a
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1967:
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1751:Some authors, such as
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1514:If one constructs the
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1160:{\displaystyle \geq 1}
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62:. Some unitals may be
2380:Archiv der Mathematik
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2052:{\displaystyle q^{2}}
1968:
1966:{\displaystyle q^{2}}
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1861:Assmus & Key 1992
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774:if there is a design
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364:for some prime power
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214:A point is called an
2475:Combinatorial design
2318:Discrete Mathematics
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1186:A Hermitian variety
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1099:theory of polarities
1055:improve this section
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2495:Algebraic varieties
2490:Projective geometry
2130:Geometriae Dedicata
2083:Geometriae Dedicata
1977:Journal of Geometry
1914:, pp. 229–245.
1887:, pp. 473–480.
1851:, pp. 256–259.
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1117:with an involutive
1021:Hermitian varieties
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121:projective geometry
88:Desarguesian planes
86:+ 1 points. In the
2485:Incidence geometry
2450:10.1007/bf01388477
2412:Journal of Algebra
2392:10.1007/bf01210788
2142:10.1007/BF00147935
2095:10.1007/BF00145956
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766:Since unitals are
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708:of the plane. For
706:collineation group
607:be the set of all
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204:of the underlying
196:) for all vectors
156:if its associated
2340:Finite geometries
2309:978-0-387-76364-4
2072:{\displaystyle q}
2020:, pp. 23–33.
1939:{\displaystyle q}
1791:, pp. 51–61.
1727:is by definition
1196:sesquilinear form
1091:
1090:
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849:translation plane
677:on the points of
613:Sylow 3-subgroups
377:{\displaystyle q}
158:sesquilinear form
18:Hermitian variety
16:(Redirected from
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1220:
1207:
1206:Representation
1204:
1156:
1153:
1144:be an integer
1129:
1106:
1103:
1089:
1088:
1039:
1037:
1030:
1022:
1019:
1006:
1003:
1000:
997:
994:
991:
988:
968:
965:
962:
959:
956:
953:
950:
930:
927:
924:
921:
918:
915:
912:
892:
887:
883:
879:
876:
873:
870:
867:
832:
827:
823:
819:
816:
813:
810:
807:
795:
792:
786:if there is a
763:
760:
723:
714:
673:. Construct a
669:+ 1 points of
644:
584:
572:
569:
557:
554:
551:
548:
543:
540:
537:
532:
528:
524:
519:
516:
513:
508:
504:
500:
495:
492:
489:
484:
480:
476:
473:
468:
464:
460:
455:
451:
447:
442:
438:
434:
431:
428:
423:
395:
373:
353:
348:
344:
340:
337:
334:
331:
328:
308:
303:
299:
295:
292:
289:
284:
279:
274:
216:absolute point
212:
211:
210:
209:
116:
113:
111:
108:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2507:
2496:
2493:
2491:
2488:
2486:
2483:
2481:
2478:
2476:
2473:
2472:
2470:
2459:
2455:
2451:
2447:
2443:
2439:
2435:
2430:
2426:
2421:
2417:
2413:
2404:
2401:
2397:
2393:
2389:
2385:
2381:
2376:
2373:
2369:
2365:
2361:
2359:3-540-61786-8
2355:
2351:
2347:
2342:
2341:
2334:
2331:
2327:
2323:
2319:
2314:
2311:
2305:
2301:
2297:
2293:
2288:
2285:
2281:
2277:
2273:
2268:
2265:
2263:0-521-41361-3
2259:
2255:
2250:
2249:
2244:
2235:
2231:
2227:
2223:
2219:
2215:
2208:
2205:
2200:
2196:
2191:
2186:
2182:
2178:
2174:
2167:
2164:
2159:
2155:
2151:
2147:
2143:
2139:
2135:
2131:
2127:
2120:
2117:
2112:
2108:
2104:
2100:
2096:
2092:
2088:
2084:
2080:
2066:
2044:
2040:
2026:
2023:
2019:
2014:
2011:
2006:
2002:
1998:
1994:
1990:
1986:
1982:
1978:
1974:
1958:
1954:
1933:
1920:
1917:
1913:
1908:
1905:
1901:
1896:
1894:
1890:
1886:
1881:
1878:
1874:
1869:
1866:
1862:
1857:
1854:
1850:
1849:Lüneburg 1966
1845:
1842:
1838:
1833:
1830:
1826:
1821:
1818:
1814:
1809:
1806:
1802:
1797:
1794:
1790:
1785:
1782:
1775:
1765:
1762:
1758:
1754:
1748:
1745:
1738:
1736:
1734:
1730:
1726:
1722:
1718:
1714:
1706:
1704:
1691:
1686:
1678:
1674:
1666:
1657:
1653:
1643:
1639:
1632:
1627:
1624:
1615:
1602:
1599:
1594:
1590:
1586:
1581:
1577:
1568:
1552:
1549:
1545:
1541:
1536:
1533:
1529:
1520:
1517:
1512:
1499:
1496:
1491:
1488:
1484:
1461:
1456:
1453:
1449:
1445:
1440:
1437:
1433:
1423:
1410:
1407:
1402:
1397:
1393:
1387:
1383:
1377:
1374:
1370:
1364:
1359:
1356:
1353:
1350:
1347:
1343:
1334:
1315:
1311:
1307:
1304:
1301:
1296:
1292:
1280:
1276:
1272:
1254:
1250:
1246:
1243:
1240:
1235:
1231:
1227:
1222:
1218:
1205:
1203:
1201:
1197:
1193:
1189:
1184:
1182:
1178:
1175:-dimensional
1174:
1170:
1154:
1151:
1143:
1127:
1120:
1116:
1112:
1104:
1102:
1100:
1096:
1085:
1082:
1074:
1064:
1060:
1056:
1050:
1049:
1045:
1040:This section
1038:
1034:
1029:
1028:
1025:
1020:
1018:
1001:
998:
995:
989:
986:
963:
960:
957:
951:
948:
925:
922:
919:
913:
910:
885:
881:
877:
874:
868:
865:
856:
854:
850:
846:
825:
821:
817:
814:
808:
805:
791:
789:
785:
781:
777:
773:
769:
768:block designs
761:
759:
757:
753:
749:
745:
741:
737:
734:PG(2,9), the
733:
729:
720:
716:Unitals with
715:
713:
711:
707:
703:
699:
694:
692:
688:
684:
680:
676:
672:
668:
664:
660:
656:
651:
647:
642:
638:
634:
630:
626:
622:
618:
614:
610:
606:
602:
598:
594:
590:
582:
578:
570:
568:
555:
549:
546:
541:
538:
535:
530:
526:
522:
517:
514:
511:
506:
502:
498:
493:
490:
487:
482:
478:
474:
466:
462:
458:
453:
449:
445:
440:
436:
426:
411:
371:
346:
342:
338:
335:
329:
326:
301:
297:
293:
290:
277:
261:
259:
255:
251:
246:
244:
240:
236:
232:
228:
224:
219:
217:
207:
203:
199:
195:
191:
187:
183:
179:
175:
172:
171:
170:
169:
168:
166:
162:
159:
155:
150:
148:
143:
141:
137:
133:
129:
124:
122:
114:
109:
107:
105:
101:
97:
93:
89:
85:
81:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
2441:
2437:
2433:
2415:
2411:
2383:
2379:
2370:– via
2339:
2321:
2317:
2294:, Springer,
2291:
2275:
2271:
2253:
2217:
2213:
2207:
2183:(1): 67–84.
2180:
2176:
2166:
2133:
2129:
2119:
2086:
2082:
2025:
2013:
1983:(1): 61–77.
1980:
1976:
1919:
1907:
1885:Grüning 1986
1880:
1868:
1856:
1844:
1832:
1820:
1808:
1796:
1784:
1764:
1756:
1747:
1732:
1724:
1720:
1716:
1712:
1710:
1616:
1569:
1518:
1513:
1476:and not all
1424:
1335:
1274:
1270:
1209:
1199:
1191:
1187:
1185:
1180:
1177:vector space
1172:
1168:
1141:
1119:automorphism
1110:
1108:
1092:
1077:
1068:
1053:Please help
1041:
1024:
857:
852:
797:
788:collineation
783:
771:
765:
755:
751:
743:
740:Hughes plane
725:
718:
709:
702:Desarguesian
697:
695:
690:
686:
682:
678:
675:block design
670:
666:
658:
649:
645:
636:
632:
628:
624:
608:
604:
600:
596:
592:
588:
580:
574:
412:as follows:
262:
257:
253:
249:
247:
242:
238:
234:
230:
229:), for some
226:
222:
220:
215:
213:
206:vector space
201:
197:
193:
189:
185:
181:
177:
173:
164:
160:
153:
151:
146:
144:
127:
125:
118:
103:
99:
95:
91:
83:
79:
71:
60:block design
55:
51:
47:
39:
38:is a set of
35:
29:
776:isomorphism
621:conjugation
599:− 1) where
571:Ree unitals
140:hyperplanes
128:correlation
2469:Categories
1719:. A line
1179:over
1105:Definition
1071:March 2023
784:equivalent
772:isomorphic
748:Hall plane
736:Hall plane
663:involution
641:stabilizer
587:of order (
577:Ree groups
233:≥ 2, is a
167:satisfies
2400:115302560
2278:: 51–61,
2234:0195-6698
2199:0097-3165
2158:119595725
2150:1572-9168
2111:123037502
2103:1572-9168
2005:117872040
1997:1420-8997
1875:, p. 105.
1863:, p. 209.
1827:, p. 104.
1776:Citations
1667:⋮
1595:θ
1462:θ
1403:θ
1344:∑
1305:…
1244:…
1152:≥
1128:θ
1042:does not
780:bijection
657:of order
603:= 3. Let
475::
237:, and if
132:bijection
115:Classical
76:collinear
70:of order
2458:43638589
1902:, p. 29.
1839:, p. 21.
1815:, p. 18.
1803:, p. 15.
1723:through
1198:on
1095:quadrics
615:of Γ. Γ
248:In PG(2,
147:polarity
64:embedded
58:+ 1, 1)
32:geometry
2368:0233275
2245:Sources
1729:tangent
1277:in the
1140:. Let
1063:removed
1048:sources
843:in the
726:In the
110:Unitals
90:, PG(2,
2456:
2398:
2366:
2356:
2306:
2260:
2232:
2197:
2156:
2148:
2109:
2101:
2003:
1995:
1617:where
1425:where
1171:be an
655:cyclic
625:points
136:points
44:points
36:unital
2454:S2CID
2396:S2CID
2154:S2CID
2107:S2CID
2001:S2CID
1739:Notes
1521:with
1192:PG(V)
1173:(n+1)
1115:field
1113:be a
758:= 3.
730:(the
689:+ 1,
66:in a
54:+ 1,
2410:)",
2354:ISBN
2304:ISBN
2258:ISBN
2230:ISSN
2195:ISSN
2146:ISSN
2099:ISSN
1993:ISSN
1711:Let
1210:Let
1167:and
1109:Let
1046:any
1044:cite
631:and
617:acts
611:+ 1
591:+ 1)
263:Let
184:) =
138:and
42:+ 1
34:, a
2446:doi
2434:m,n
2420:doi
2388:doi
2326:doi
2322:267
2296:doi
2280:doi
2222:doi
2185:doi
2138:doi
2091:doi
1985:doi
1190:in
1057:by
721:= 3
653:is
643:, Γ
635:in
30:In
2471::
2452:,
2440:,
2414:,
2394:,
2384:46
2382:,
2364:MR
2362:,
2352:,
2344:,
2320:,
2302:,
2276:52
2274:,
2228:.
2218:13
2216:.
2193:.
2181:60
2179:.
2175:.
2152:.
2144:.
2132:.
2128:.
2105:.
2097:.
2085:.
2081:.
1999:.
1991:.
1981:29
1979:.
1975:.
1892:^
1202:.
1183:.
1101:.
260:.
245:.
200:,
149:.
142:.
126:A
123:.
104:36
2448::
2442:6
2422::
2416:3
2408:2
2390::
2328::
2298::
2282::
2236:.
2224::
2201:.
2187::
2160:.
2140::
2134:8
2113:.
2093::
2087:5
2079:"
2067:q
2045:2
2041:q
2007:.
1987::
1973:"
1959:2
1955:q
1934:q
1757:n
1733:p
1725:p
1721:L
1717:H
1713:p
1692:.
1687:]
1679:n
1675:X
1658:1
1654:X
1644:0
1640:X
1633:[
1628:=
1625:X
1603:0
1600:=
1591:X
1587:A
1582:t
1578:X
1553:j
1550:i
1546:a
1542:=
1537:j
1534:i
1530:A
1519:A
1500:0
1497:=
1492:j
1489:i
1485:a
1457:i
1454:j
1450:a
1446:=
1441:j
1438:i
1434:a
1411:0
1408:=
1398:j
1394:X
1388:i
1384:X
1378:j
1375:i
1371:a
1365:n
1360:0
1357:=
1354:j
1351:,
1348:i
1321:)
1316:n
1312:X
1308:,
1302:,
1297:0
1293:X
1289:(
1275:p
1271:V
1255:n
1251:e
1247:,
1241:,
1236:1
1232:e
1228:,
1223:0
1219:e
1200:V
1188:H
1181:K
1169:V
1155:1
1142:n
1111:K
1084:)
1078:(
1073:)
1069:(
1065:.
1051:.
1005:)
1002:q
999:,
996:4
993:(
990:G
987:P
967:)
964:q
961:,
958:4
955:(
952:G
949:P
929:)
926:q
923:,
920:4
917:(
914:G
911:P
891:)
886:2
882:q
878:,
875:2
872:(
869:G
866:P
831:)
826:2
822:q
818:,
815:2
812:(
809:G
806:P
756:n
752:n
744:n
719:n
710:q
700:(
698:q
691:q
687:q
683:P
679:P
671:P
667:q
659:q
650:T
648:,
646:S
637:P
633:T
629:S
609:q
605:P
601:q
597:q
595:(
593:q
589:q
585:2
581:q
556:.
553:}
550:0
547:=
542:1
539:+
536:q
531:2
527:x
523:+
518:1
515:+
512:q
507:1
503:x
499:+
494:1
491:+
488:q
483:0
479:x
472:)
467:2
463:x
459:,
454:1
450:x
446:,
441:0
437:x
433:(
430:{
427:=
422:H
394:H
372:q
352:)
347:2
343:q
339:,
336:2
333:(
330:G
327:P
307:)
302:2
298:q
294:,
291:2
288:(
283:H
278:=
273:H
254:q
250:q
239:d
231:d
227:F
225:,
223:d
208:.
202:v
198:u
194:u
192:,
190:v
188:(
186:s
182:v
180:,
178:u
176:(
174:s
165:α
161:s
100:6
98:=
96:n
92:q
84:n
72:n
56:n
52:n
48:n
40:n
20:)
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