250:
240:
222:
650:
453:
191:
1034:) isn't even defined for n less than 2, not to mention it is the wrong expression for the number of undirected edges in a complete graph (and thus number of edges in a tournament). If something else is meant by "edges" or by the given expression, than this should be pointed out. As it stood, it was plain wrong. -- ManDay
1616:
We show that a tournament of any size has a
Hamilton path. Clearly a tournament with 2 nodes has one. We prove that a trournament of any size must have a Hamilton path because if any tournament with n nodes has such a path then any tournament of size n+1 containing the tournament of size n must have
941:
The diagram in the intro seems to say that the number of edges is "n choose 2", whereas in general it is n*(n-1)/2, the same as K(n). In the case of n=4, these values are the same, but it seems pointless and unhelpful to give the former. I'd be bold and change it if I knew how, though I fear I've
460:
The graph shown on the right is not transitive (at least not according to the definition given in this article). It says a graph with n vertices should have as score sequence {0, 1, ..., n-1}, which means, every outdegree exists exactly once. Nodes 3 and 4 both have the same outdegree (ie. 4: 3-:
904:
I believe the exact relation is that the paradoxical tournaments and the transitive tournaments are disjoint from each other: it is not possible for a tournament to be both paradoxical and transitive. However there exist tournaments that are neither paradoxical nor transitive.
395:
Aren't all tournaments complete graphs? Isn't a complete graph with k vertices always k-connected? Therefore wouldn't it be simpler to write, "Moreover, if the tournament has 4 or more vertices, each pair of vertices can be connected with a
Hamiltonian path (Thomassen 1980)."
1007:
Oops, yes, thanks, and thanks for the reversion. Sloppy pencil & paper work, and brain-softening. Mind you, I think it a bit unkind to call n*(n-1)/2 'random and arbitrary', it is intuitive for the complete graph, and a limiting case of the famous
363:"Every tournament on n vertices has a transitive subtournament on log2n vertices. Reid and Parker showed that this is the best possible result: there exist n-vertex tournaments whose largest transitive subtournament is of size log2n."
1620:
Imagine that a tournament with n nodes has a
Hamilton path. Label the nodes in the path v1, v2, ..., vn. Add another node X to the graph and connect it to all the existing nodes with directed edges. Then there are 3 cases:
153:
1630:
Some of the edges connecting X to the existing graph point from X, while others point to X. Suppose vi is the first node in v1, v2, ... that X points to. Then v1, v2, v(i-1), X, vi, v(i+1), ..., vn is a
Hamiltonian Path.
1133:
Transitive tournaments are the same thing as acyclic tournaments, but they are a special case among all tournaments. This example is not one of these special cases. It is a tournament, but not transitive and not acyclic.
885:
Sorry for the newbye question, but what is the relationship between a paradoxical tournament and a transitive tournament? My first impression is that a paradoxical tournament is the same as a non-transitive tournament.
1609:
More generally, I think this proof should be written without mathematical notation. It's a fantastic inductive proof because it's so simple, and it would be great to make it accessible to the general public.
1592:
993:
There is no requirement of having an even number. (This is not like an actual sports tournament that has to proceed in rounds, and n choose 2 is meaningful for odd numbers as well as even.) —
302:
1327:
424:
It's a directed form of connectivity. I imagine that it means that it remains strongly connected after the removal of any three vertices, but I'd have to check the source to be sure. —
1055:
It's not n over 2, it's n choose 2. And it's perfectly well defined for n less than 2, equalling zero for those n. And it is always the correct number of edges. There is no mistake. —
1410:
1366:
147:
889:
Whatever the exact relation, it would be a good idea to make it explicit in the article, because paradoxical tournaments are explained under the section about transitivity. --
44:
1436:
1605:
In addition, no such i exists if all edges between v0 and v1 ... vn point away from v0. (One cannot say that 0 is the biggest such i, because there is no edge v0-: -->
1490:
1463:
1248:
754:
724:
697:
529:
R. That is, for each x, y in a set E, either x = y, either x R y, either y R x. R can be seen as the domination relation. Would that be worth saying in the article?
1221:
979:
That's really helpful David, thanks. I now see that "n choose 2" is the same as n*(n-1)/2 for all even n - and a tournament must have an even number of players....
1715:
1268:
1195:
1175:
774:
670:
296:
438:
A UCI ICS grad's question fielded by a UCI CS professor on a random obscure
Knowledge Talk page. That's almost a graph theory connectedness anomaly in itself. -
79:
919:
The article would probably benefit from these sentences, and even better, an example of a tournament that is neither paradoxical nor transitive. --
591:"Many of the important properties of tournaments were first investigated by Landau in order to model dominance relations in flocks of chickens."
272:
1710:
1665:
1094:
His initials were already given in the references section. Apparently he was Hyman G. Landau, a mathematician at the
University of Chicago. —
834:
Which example above? I assume the 4-vertices graph drawn on top of the article. If correct, that should be made clear in the article IMHO. --
85:
1041:
373:
392:
It reads: "Moreover, if the tournament is 4‑connected, each pair of vertices can be connected with a
Hamiltonian path (Thomassen 1980)."
729:
In case it interests you, I made an illustration of the proof of the theorem about
Hamiltonian paths in tournaments. Please notice that
1687:
476:
168:
1500:
263:
227:
135:
99:
30:
412:
104:
20:
1118:
1 Condition (3) states: T is asyclic. What is the source of that condition, and which one is correct (graph or condition)?
324:
The intro says the graph is undirected, but the image directly below that statement shows a directed graph. Please fix. --
74:
129:
202:
594:
I would remove "first", as we have other important results from 1934, that is before 1953, date of Landau's article. --
65:
125:
1661:
1647:
24:
1139:
1099:
1060:
998:
970:
910:
854:
800:
618:
560:
429:
366:
This comes with no reference, and is apparently simply wrong. I belive the best known upper bound is ~2log2n.
175:
1273:
1045:
377:
1123:
1691:
1637:
I know that the proof needs to found elsewhere to publish this in WP. Sorry, I don't have time to do that.
480:
109:
1657:
1643:
1624:
All the edges between X and v1, v2, ..., vn point from X. Then X, v1, v2, ..., vn is a
Hamiltonian Path.
1371:
1332:
1119:
1017:
984:
947:
924:
894:
868:
839:
814:
781:
632:
599:
574:
541:
208:
1627:
All the edges between X and v1, v2, ..., vn point to X. Then v1, v2, ..., vn, X is a
Hamiltonian Path.
249:
1683:
1653:
1037:
1031:
472:
400:
369:
141:
1680:
The group of isomorphisms on any tournament graph is solvable. This should be mentioned somewhere.
190:
1135:
1095:
1056:
994:
966:
906:
850:
796:
614:
556:
439:
425:
404:
161:
55:
271:
on Knowledge. If you would like to participate, please visit the project page, where you can join
1084:
408:
255:
70:
239:
221:
1009:
962:
649:
348:
51:
1415:
1013:
980:
943:
920:
890:
864:
835:
810:
777:
628:
595:
570:
537:
1634:(I realize that cases 1 & 3 could be combined, but I like the symmetry of 1 & 2.)
1468:
1441:
1226:
732:
702:
675:
526:
511:
329:
961:
it has that form rather than just being a random and arbitrary quadratic polynomial. See
1602:
First, it's missing the ground case of n=2, which has a Hamiltonian path by inspection.
1200:
452:
792:
610:
552:
1695:
1669:
1253:
1180:
1160:
1143:
1127:
1103:
1088:
1064:
1049:
1021:
1002:
988:
974:
951:
928:
914:
898:
872:
858:
843:
818:
804:
785:
759:
655:
636:
622:
603:
578:
564:
545:
514:
484:
433:
416:
381:
352:
333:
1704:
1080:
525:
Tournaments can also be defined by means of an irreflexive, antisymmetric and total
532:
Other topics that might be added include number of non-isomorphic tournaments over
344:
1114:
The first shown graph is stated as being a tournament graph but is cyclic: 1-: -->
268:
491:
You missed 3 → 8. In fact, the tournament has an extremely simple description:
508:
325:
245:
343:
to edges of a complete undirected graph, thus making it a directed graph. --
1606:
v0.) Nevertheless, a Hamiltonian path still exists, v0, v1, ..., vn.
648:
1587:{\displaystyle v_{1},\ldots ,v_{i},v_{0},v_{i+1},\ldots ,v_{n}}
831:"as the above example shows, there might not be such a player"
339:
Nope, the intro says that a tournament is a graph obtained by
184:
15:
809:
Nope. I decided not to contribute anymore on WP:en, sorry. --
627:
Nope. I decided not to contribute anymore on WP:en, sorry. --
569:
Nope. I decided not to contribute anymore on WP:en, sorry. --
507:. Since < is a linear order, it is clearly transitive.—
160:
1503:
1471:
1444:
1418:
1374:
1335:
1276:
1256:
1229:
1203:
1183:
1163:
957:
They are always the same. And "n choose 2" describes
762:
735:
705:
678:
658:
1079:
Who is Landau? Can we have a first initial or link?
267:, a collaborative effort to improve the coverage of
1586:
1484:
1457:
1430:
1404:
1360:
1321:
1262:
1242:
1215:
1189:
1169:
768:
748:
718:
691:
664:
301:This article has not yet received a rating on the
33:for general discussion of the article's subject.
174:
8:
1399:
1381:
1355:
1342:
1681:
1651:
1035:
216:
1578:
1553:
1540:
1527:
1508:
1502:
1476:
1470:
1449:
1443:
1417:
1373:
1349:
1334:
1322:{\displaystyle v_{1},v_{2},\ldots ,v_{n}}
1313:
1294:
1281:
1275:
1255:
1234:
1228:
1202:
1182:
1162:
761:
740:
734:
710:
704:
683:
677:
657:
451:
1339:
218:
188:
456:A transitive tournament on 8 vertices.
1716:Unknown-priority mathematics articles
1650:) 17:00, 24 June 2017 (UTC) ArthurG
1157:suppose that the statement holds for
7:
1405:{\displaystyle i\in \{0,\ldots ,n\}}
1361:{\displaystyle T\setminus \{v_{0}\}}
261:This article is within the scope of
1640:Diagrams would be nice, of course.
1613:The proof would go something like:
536:vertices, and tournament matrix. --
207:It is of interest to the following
23:for discussing improvements to the
14:
1154:The proof is incorrect. It says:
281:Knowledge:WikiProject Mathematics
849:This has since been clarified. —
359:largest transitive subtournament
284:Template:WikiProject Mathematics
248:
238:
220:
189:
45:Click here to start a new topic.
1599:is a directed path as desired.
1412:be maximal such that for every
645:Theorem about Hamiltonian paths
1438:there is a directed edge from
1177:, and consider any tournament
1022:06:57, 20 September 2014 (UTC)
1003:14:57, 19 September 2014 (UTC)
989:14:15, 19 September 2014 (UTC)
975:16:31, 17 September 2014 (UTC)
952:10:35, 17 September 2014 (UTC)
388:Last Sentence Paths and Cycles
1:
1270:and consider a directed path
929:15:06, 17 February 2013 (UTC)
915:21:57, 14 February 2013 (UTC)
873:14:57, 17 February 2013 (UTC)
859:21:58, 14 February 2013 (UTC)
819:14:56, 17 February 2013 (UTC)
805:21:58, 14 February 2013 (UTC)
637:14:56, 17 February 2013 (UTC)
623:21:58, 14 February 2013 (UTC)
579:14:55, 17 February 2013 (UTC)
565:21:57, 14 February 2013 (UTC)
434:19:49, 25 November 2010 (UTC)
417:19:28, 25 November 2010 (UTC)
382:11:46, 24 November 2010 (UTC)
334:18:36, 22 February 2008 (UTC)
275:and see a list of open tasks.
42:Put new text under old text.
1711:C-Class mathematics articles
899:09:47, 27 January 2013 (UTC)
844:09:45, 27 January 2013 (UTC)
786:09:56, 27 January 2013 (UTC)
604:09:44, 27 January 2013 (UTC)
546:09:42, 27 January 2013 (UTC)
515:12:28, 16 August 2012 (UTC)
485:10:51, 16 August 2012 (UTC)
50:New to Knowledge? Welcome!
1732:
1223:vertices. Choose a vertex
881:Paradoxical vs. Transitive
1696:16:53, 2 March 2020 (UTC)
1670:16:53, 24 June 2017 (UTC)
1144:17:17, 16 July 2016 (UTC)
1128:12:24, 16 July 2016 (UTC)
300:
233:
215:
80:Be welcoming to newcomers
25:Tournament (graph theory)
1104:20:38, 13 May 2016 (UTC)
1089:20:01, 13 May 2016 (UTC)
776:for clarity purposes. --
353:20:21, 5 July 2009 (UTC)
303:project's priority scale
1431:{\displaystyle j\leq i}
1065:16:00, 9 May 2018 (UTC)
1050:09:11, 9 May 2018 (UTC)
264:WikiProject Mathematics
1588:
1486:
1459:
1432:
1406:
1362:
1323:
1264:
1244:
1217:
1191:
1171:
1110:Transitivity - acyclic
942:missed something..:-)
770:
750:
726:
720:
693:
666:
457:
197:This article is rated
75:avoid personal attacks
1589:
1487:
1485:{\displaystyle v_{0}}
1460:
1458:{\displaystyle v_{j}}
1433:
1407:
1363:
1324:
1265:
1245:
1243:{\displaystyle v_{0}}
1218:
1192:
1172:
771:
751:
749:{\displaystyle v_{0}}
721:
719:{\displaystyle v_{3}}
694:
692:{\displaystyle v_{2}}
667:
652:
455:
100:Neutral point of view
1501:
1469:
1442:
1416:
1372:
1333:
1274:
1254:
1227:
1201:
1181:
1161:
1032:binomial coefficient
760:
733:
703:
676:
672:is inserted between
656:
287:mathematics articles
105:No original research
1216:{\displaystyle n+1}
341:assigning direction
1584:
1482:
1455:
1428:
1402:
1358:
1319:
1260:
1240:
1213:
1187:
1167:
766:
746:
727:
716:
689:
662:
458:
256:Mathematics portal
203:content assessment
86:dispute resolution
47:
1698:
1686:comment added by
1672:
1656:comment added by
1617:a Hamilton path.
1263:{\displaystyle T}
1190:{\displaystyle T}
1170:{\displaystyle n}
1052:
1040:comment added by
963:triangular number
827:No winner example
769:{\displaystyle a}
665:{\displaystyle a}
475:comment added by
420:
403:comment added by
372:comment added by
317:
316:
313:
312:
309:
308:
183:
182:
66:Assume good faith
43:
1723:
1593:
1591:
1590:
1585:
1583:
1582:
1564:
1563:
1545:
1544:
1532:
1531:
1513:
1512:
1491:
1489:
1488:
1483:
1481:
1480:
1464:
1462:
1461:
1456:
1454:
1453:
1437:
1435:
1434:
1429:
1411:
1409:
1408:
1403:
1367:
1365:
1364:
1359:
1354:
1353:
1328:
1326:
1325:
1320:
1318:
1317:
1299:
1298:
1286:
1285:
1269:
1267:
1266:
1261:
1249:
1247:
1246:
1241:
1239:
1238:
1222:
1220:
1219:
1214:
1196:
1194:
1193:
1188:
1176:
1174:
1173:
1168:
1150:Paths and cycles
775:
773:
772:
767:
755:
753:
752:
747:
745:
744:
725:
723:
722:
717:
715:
714:
698:
696:
695:
690:
688:
687:
671:
669:
668:
663:
487:
419:
397:
384:
289:
288:
285:
282:
279:
258:
253:
252:
242:
235:
234:
224:
217:
200:
194:
193:
185:
179:
178:
164:
95:Article policies
16:
1731:
1730:
1726:
1725:
1724:
1722:
1721:
1720:
1701:
1700:
1678:
1658:Arthur.Goldberg
1644:Arthur.Goldberg
1574:
1549:
1536:
1523:
1504:
1499:
1498:
1472:
1467:
1466:
1445:
1440:
1439:
1414:
1413:
1370:
1369:
1345:
1331:
1330:
1309:
1290:
1277:
1272:
1271:
1252:
1251:
1230:
1225:
1224:
1199:
1198:
1179:
1178:
1159:
1158:
1152:
1112:
1077:
939:
883:
829:
758:
757:
736:
731:
730:
706:
701:
700:
679:
674:
673:
654:
653:
647:
589:
527:binary relation
523:
470:
450:
398:
390:
367:
361:
322:
286:
283:
280:
277:
276:
254:
247:
201:on Knowledge's
198:
121:
116:
115:
114:
91:
61:
12:
11:
5:
1729:
1727:
1719:
1718:
1713:
1703:
1702:
1677:
1674:
1597:
1596:
1595:
1594:
1581:
1577:
1573:
1570:
1567:
1562:
1559:
1556:
1552:
1548:
1543:
1539:
1535:
1530:
1526:
1522:
1519:
1516:
1511:
1507:
1479:
1475:
1452:
1448:
1427:
1424:
1421:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1357:
1352:
1348:
1344:
1341:
1338:
1316:
1312:
1308:
1305:
1302:
1297:
1293:
1289:
1284:
1280:
1259:
1237:
1233:
1212:
1209:
1206:
1186:
1166:
1151:
1148:
1147:
1146:
1136:David Eppstein
1111:
1108:
1107:
1106:
1096:David Eppstein
1076:
1073:
1072:
1071:
1070:
1069:
1068:
1067:
1057:David Eppstein
1042:88.217.234.135
1028:
1027:
1026:
1025:
1024:
995:David Eppstein
967:David Eppstein
938:
937:Wrong Diagram?
935:
934:
933:
932:
931:
907:David Eppstein
882:
879:
878:
877:
876:
875:
851:David Eppstein
828:
825:
824:
823:
822:
821:
797:David Eppstein
765:
743:
739:
713:
709:
686:
682:
661:
646:
643:
642:
641:
640:
639:
615:David Eppstein
588:
585:
584:
583:
582:
581:
557:David Eppstein
522:
519:
518:
517:
449:
446:
445:
444:
443:
442:
426:David Eppstein
389:
386:
374:132.64.140.208
360:
357:
356:
355:
321:
318:
315:
314:
311:
310:
307:
306:
299:
293:
292:
290:
273:the discussion
260:
259:
243:
231:
230:
225:
213:
212:
206:
195:
181:
180:
118:
117:
113:
112:
107:
102:
93:
92:
90:
89:
82:
77:
68:
62:
60:
59:
48:
39:
38:
35:
34:
28:
13:
10:
9:
6:
4:
3:
2:
1728:
1717:
1714:
1712:
1709:
1708:
1706:
1699:
1697:
1693:
1689:
1685:
1675:
1673:
1671:
1667:
1663:
1659:
1655:
1649:
1645:
1641:
1638:
1635:
1632:
1628:
1625:
1622:
1618:
1614:
1611:
1607:
1603:
1600:
1579:
1575:
1571:
1568:
1565:
1560:
1557:
1554:
1550:
1546:
1541:
1537:
1533:
1528:
1524:
1520:
1517:
1514:
1509:
1505:
1497:
1496:
1495:
1494:
1493:
1477:
1473:
1450:
1446:
1425:
1422:
1419:
1396:
1393:
1390:
1387:
1384:
1378:
1375:
1350:
1346:
1336:
1314:
1310:
1306:
1303:
1300:
1295:
1291:
1287:
1282:
1278:
1257:
1235:
1231:
1210:
1207:
1204:
1184:
1164:
1155:
1149:
1145:
1141:
1137:
1132:
1131:
1130:
1129:
1125:
1121:
1120:Dominikmorgen
1109:
1105:
1101:
1097:
1093:
1092:
1091:
1090:
1086:
1082:
1074:
1066:
1062:
1058:
1054:
1053:
1051:
1047:
1043:
1039:
1033:
1029:
1023:
1019:
1015:
1011:
1006:
1005:
1004:
1000:
996:
992:
991:
990:
986:
982:
978:
977:
976:
972:
968:
964:
960:
956:
955:
954:
953:
949:
945:
936:
930:
926:
922:
918:
917:
916:
912:
908:
903:
902:
901:
900:
896:
892:
887:
880:
874:
870:
866:
862:
861:
860:
856:
852:
848:
847:
846:
845:
841:
837:
832:
826:
820:
816:
812:
808:
807:
806:
802:
798:
794:
790:
789:
788:
787:
783:
779:
763:
741:
737:
711:
707:
684:
680:
659:
651:
644:
638:
634:
630:
626:
625:
624:
620:
616:
612:
608:
607:
606:
605:
601:
597:
592:
586:
580:
576:
572:
568:
567:
566:
562:
558:
554:
550:
549:
548:
547:
543:
539:
535:
530:
528:
520:
516:
513:
510:
506:
502:
498:
494:
490:
489:
488:
486:
482:
478:
474:
465:7 and 4-: -->
454:
447:
441:
437:
436:
435:
431:
427:
423:
422:
421:
418:
414:
410:
406:
402:
393:
387:
385:
383:
379:
375:
371:
364:
358:
354:
350:
346:
342:
338:
337:
336:
335:
331:
327:
319:
304:
298:
295:
294:
291:
274:
270:
266:
265:
257:
251:
246:
244:
241:
237:
236:
232:
229:
226:
223:
219:
214:
210:
204:
196:
192:
187:
186:
177:
173:
170:
167:
163:
159:
155:
152:
149:
146:
143:
140:
137:
134:
131:
127:
124:
123:Find sources:
120:
119:
111:
110:Verifiability
108:
106:
103:
101:
98:
97:
96:
87:
83:
81:
78:
76:
72:
69:
67:
64:
63:
57:
53:
52:Learn to edit
49:
46:
41:
40:
37:
36:
32:
26:
22:
18:
17:
1688:141.20.50.37
1682:— Preceding
1679:
1652:— Preceding
1642:
1639:
1636:
1633:
1629:
1626:
1623:
1619:
1615:
1612:
1608:
1604:
1601:
1598:
1156:
1153:
1113:
1078:
1036:— Preceding
1030:"n over 2" (
1014:John Wheater
981:John Wheater
958:
944:John Wheater
940:
888:
884:
833:
830:
728:
593:
590:
533:
531:
524:
504:
500:
496:
492:
477:141.44.28.72
471:— Preceding
459:
448:Transitivity
394:
391:
365:
362:
340:
323:
320:"undirected"
262:
209:WikiProjects
171:
165:
157:
150:
144:
138:
132:
122:
94:
19:This is the
1676:Solvability
921:MathsPoetry
891:MathsPoetry
865:MathsPoetry
836:MathsPoetry
811:MathsPoetry
791:Please see
778:MathsPoetry
756:is renamed
629:MathsPoetry
609:Please see
596:MathsPoetry
571:MathsPoetry
551:Please see
538:MathsPoetry
399:—Preceding
368:—Preceding
278:Mathematics
269:mathematics
228:Mathematics
148:free images
31:not a forum
1705:Categories
1368:. Now let
521:Definition
468:7, 4-: -->
467:6, 4-: -->
466:5, 4-: -->
464:6, 3-: -->
463:5, 3-: -->
462:4, 3-: -->
1010:handshake
88:if needed
71:Be polite
21:talk page
1684:unsigned
1666:contribs
1654:unsigned
1081:Lena Key
1038:unsigned
473:unsigned
440:Recurve7
413:contribs
405:Recurve7
401:unsigned
370:unsigned
56:get help
29:This is
27:article.
1117:3-: -->
1116:4-: -->
1115:2-: -->
1075:Landau?
793:WP:BOLD
611:WP:BOLD
587:History
553:WP:BOLD
345:Rulatir
199:C-class
154:WP refs
142:scholar
863:OK. --
205:scale.
126:Google
503:<
469:8).
326:AlanH
169:JSTOR
130:books
84:Seek
1692:talk
1662:talk
1648:talk
1140:talk
1124:talk
1100:talk
1085:talk
1061:talk
1046:talk
1018:talk
999:talk
985:talk
971:talk
948:talk
925:talk
911:talk
895:talk
869:talk
855:talk
840:talk
815:talk
801:talk
782:talk
699:and
633:talk
619:talk
600:talk
575:talk
561:talk
542:talk
509:Emil
499:iff
481:talk
430:talk
409:talk
378:talk
349:talk
330:talk
162:FENS
136:news
73:and
1465:to
1329:in
1250:of
1197:on
965:. —
959:why
795:. —
613:. —
555:. —
461:-->
297:???
176:TWL
1707::
1694:)
1668:)
1664:•
1569:…
1518:…
1492:.
1423:≤
1391:…
1379:∈
1340:∖
1304:…
1142:)
1126:)
1102:)
1087:)
1063:)
1048:)
1020:)
1012:.
1001:)
987:)
973:)
950:)
927:)
913:)
897:)
871:)
857:)
842:)
817:)
803:)
784:)
635:)
621:)
602:)
577:)
563:)
544:)
512:J.
495:→
483:)
432:)
415:)
411:•
380:)
351:)
332:)
156:)
54:;
1690:(
1660:(
1646:(
1580:n
1576:v
1572:,
1566:,
1561:1
1558:+
1555:i
1551:v
1547:,
1542:0
1538:v
1534:,
1529:i
1525:v
1521:,
1515:,
1510:1
1506:v
1478:0
1474:v
1451:j
1447:v
1426:i
1420:j
1400:}
1397:n
1394:,
1388:,
1385:0
1382:{
1376:i
1356:}
1351:0
1347:v
1343:{
1337:T
1315:n
1311:v
1307:,
1301:,
1296:2
1292:v
1288:,
1283:1
1279:v
1258:T
1236:0
1232:v
1211:1
1208:+
1205:n
1185:T
1165:n
1138:(
1134:—
1122:(
1098:(
1083:(
1059:(
1044:(
1016:(
997:(
983:(
969:(
946:(
923:(
909:(
905:—
893:(
867:(
853:(
838:(
813:(
799:(
780:(
764:a
742:0
738:v
712:3
708:v
685:2
681:v
660:a
631:(
617:(
598:(
573:(
559:(
540:(
534:n
505:j
501:i
497:j
493:i
479:(
428:(
407:(
376:(
347:(
328:(
305:.
211::
172:·
166:·
158:·
151:·
145:·
139:·
133:·
128:(
58:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
