28:
884:
876:
A non-holonomic solution to this relation would consist in the data of two functions, a differentiable function f(x), and a continuous function g(x), with g(x) nowhere vanishing. A holonomic solution gives rise to a non-holonomic solution by taking g(x) = f'(x). The space of non-holonomic solutions
791:
into a holonomic one in the class of non-holonomic solutions. Thus in the presence of h-principle, a differential topological problem reduces to an algebraic topological problem. More explicitly this means that apart from the topological obstruction there is no other obstruction to the existence of a
1152:
to holonomic ones but also can be arbitrarily well approximated by the holonomic ones (by going back and forth, like parallel parking in a limited space) – note that this approximates both the position and the angle of the car arbitrarily closely. This implies that, theoretically, it is possible to
799:
Many underdetermined partial differential equations satisfy the h-principle. However, the falsity of an h-principle is also an interesting statement, intuitively this means the objects being studied have non-trivial geometry that cannot be reduced to topology. As an example, embedded
969:
in complex numbers). Note that here there are no immersions of order 0, as those would need to turn back on themselves. Extending this to circles immersed in the plane – the immersion condition is precisely the condition that the derivative does not vanish – the
779:
In order to check whether a solution to our original equation exists, one can first check if there is a non-holonomic solution. Usually this is quite easy, and if there is no non-holonomic solution, then our original equation did not have any solutions.
632:
868:
A holonomic solution to this relation is a function whose derivative is nowhere vanishing, i.e. a strictly monotone differentiable function, either increasing or decreasing. The space of such functions consists of two disjoint
763:
529:
1346:-embedding (respectively, immersion). This is also a dense h-principle, and can be proven by an essentially similar "wrinkling" – or rather, circling – technique to the car in the plane, though it is much more involved.
997:
which requires the partial derivatives in each direction to not vanish and to be linearly independent, and the resulting analog of the Gauss map is a map to the
Stiefel manifold, or more generally between frame bundles.
1132:
264:
1547:
169:
401:
934:
1317:
859:
1700:
967:
1148:
A non-holonomic solution in this case, roughly speaking, corresponds to a motion of the car by sliding in the plane. In this case the non-holonomic solutions are not only
1255:
339:
299:
1488:
1452:
1064:
1887:
1851:
1785:
1758:
1727:
1671:
1644:
1610:
1583:
1419:
1344:
1282:
1209:
1178:
1508:
1044:
1024:
540:
1898:
David Spring, Convex integration theory - solutions to the h-principle in geometry and topology, Monographs in
Mathematics 92, Birkhauser-Verlag, 1998
643:
409:
1006:
As another simple example, consider a car moving in the plane. The position of a car in the plane is determined by three parameters: two coordinates
2028:
993:
are much further-reaching generalizations, and much more involved, but similar in principle – immersion requires the derivative to have rank
899:
This trivial example has nontrivial generalizations: extending this to immersions of a circle into itself classifies them by order (or
1968:
1946:
1072:
2033:
177:
1956:
71:
971:
888:
79:
1153:
parallel park in any space longer than the length of your car. It also implies that, in a contact 3 manifold, any curve is
52:
1729:
because a small oscillating sphere would provide a large lower bound for the principal curvatures, and therefore for the
56:
27:
1513:
989:, and Hirsch's generalization of this to immersions of manifolds being classified as homotopy classes of maps of
907:
and applying the above analysis to the resulting monotone map – the linear map corresponds to multiplying angle:
109:
904:
804:
in a symplectic manifold do not satisfy an h-principle, to prove this one needs to find invariants coming from
344:
1223:
910:
805:
63:
PDEs or PDRs, such as the immersion problem, isometric immersion problem, fluid dynamics, and other areas.
1219:
1142:
801:
60:
1287:
74:
and
Anthony V. Phillips. It was based on earlier results that reduced partial differential relations to
1676:
1046:
for the location (a good choice is the location of the midpoint between the back wheels) and an angle
1817:
S. Smale, The classification of immersions of spheres in
Euclidean spaces. Ann. of Math(2) 69 (1959)
939:
1616:
796:
is much easier to handle and can be addressed with the obstruction theory for topological bundles.
824:
1995:
1233:
1552:
Any open manifold admits a (non-complete) Riemannian metric of positive (or negative) curvature.
1370:
Here we list a few counter-intuitive results which can be proved by applying the h-principle:
312:
272:
1964:
1942:
1788:
1457:
883:
1424:
1049:
2005:
986:
67:
1865:
1829:
1763:
1736:
1705:
1649:
1622:
1588:
1561:
1397:
1322:
1260:
1187:
1156:
1730:
1555:
1181:
32:
1066:
which describes the orientation of the car. The motion of the car satisfies the equation
880:
Thus the inclusion of holonomic into non-holonomic solutions satisfies the h-principle.
877:
again consists of two disjoint convex sets, according as g(x) is positive or negative.
627:{\displaystyle y_{j}={\partial ^{k}f \over \partial u_{j_{1}}\ldots \partial u_{j_{k}}}.}
982:
and showing that this satisfies an h-principle; here again order 0 is more complicated.
873:: the increasing ones and the decreasing ones, and has the homotopy type of two points.
821:
Perhaps the simplest partial differential relation is for the derivative to not vanish:
1493:
1029:
1009:
975:
900:
892:
891:
shows that immersions of the circle in the plane satisfy an h-principle, expressed by
772:, and a solution of the system which is also solution of our original PDE is called a
2022:
1184:
curve. This last property is stronger than the general h-principle; it is called the
2010:
1979:
990:
1932:
1924:
78:, particularly for immersions. The first evidence of h-principle appeared in the
758:{\displaystyle \Psi _{}^{}(u_{1},u_{2},\dots ,u_{m},y_{1},y_{2},\dots ,y_{N})=0}
524:{\displaystyle \Psi _{}^{}(u_{1},u_{2},\dots ,u_{m},y_{1},y_{2},\dots ,y_{N})=0}
40:
870:
1227:
1149:
979:
17:
1760:
this has to be equal to 1 everywhere, the Gauss curvature of the standard
985:
Smale's classification of immersions of spheres as the homotopy groups of
1889:
Isometric
Imbeddings I, II. Nederl. Acad. Wetensch. Proc. Ser A 58 (1955)
1138:
788:
75:
1808:
M. W. Hirsch, Immersions of manifold. Trans. Amer. Math. Soc. 93 (1959)
1137:
since a non-skidding car must move in the direction of its wheels. In
1355:
Removal of
Singularities technique developed by Gromov and Eliashberg
31:
The homotopy principle generalizes such results as Smale's proof of
2000:
1733:
of the immersed sphere, but on the other hand if the immersion is
882:
26:
1394:|. Then there is a continuous one-parameter family of functions
1319:
or larger can be arbitrarily well approximated by an isometric
865:
differential relation, as this is a function in one variable.
1127:{\displaystyle {\dot {x}}\sin \alpha ={\dot {y}}\cos \alpha .}
1909:
Mathematical
Omnibus: Thirty Lectures on Classic Mathematics
259:{\displaystyle \Psi (u_{1},u_{2},\dots ,u_{m},J_{f}^{k})=0}
102:
which satisfies a partial differential equation of degree
86:
embedding theorem and the Smale–Hirsch immersion theorem.
792:
holonomic solution. The topological problem of finding a
403:
Then our original equation can be thought as a system of
1361:
Convex integration based on the work of Nash and Kuiper.
1358:
Sheaf technique based on the work of Smale and Hirsch.
1868:
1832:
1766:
1739:
1708:
1679:
1652:
1625:
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1516:
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1427:
1400:
1325:
1290:
1263:
1236:
1190:
1159:
1075:
1052:
1032:
1012:
942:
913:
827:
646:
543:
412:
347:
315:
275:
180:
112:
1978:
De Lellis, Camillo; Székelyhidi, László Jr. (2012).
534:and some number of equations of the following type
1881:
1845:
1779:
1752:
1721:
1694:
1665:
1638:
1604:
1577:
1541:
1502:
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1413:
1338:
1311:
1276:
1249:
1203:
1172:
1126:
1058:
1038:
1018:
961:
928:
853:
757:
626:
523:
395:
333:
293:
258:
163:
1931:Eliashberg, Y.; Mishachev, N.; Ariki, S. (2002).
82:. This was followed by the Nash–Kuiper isometric
1984:-principle and the equations of fluid dynamics"
1853:Isometric Imbedding. Ann. of Math(2) 60 (1954)
1558:without creasing or tearing can be done using
1218:While this example is simple, compare to the
8:
1542:{\displaystyle \operatorname {grad} (f_{t})}
1141:terms, not all paths in the task space are
164:{\displaystyle (u_{1},u_{2},\dots ,u_{m})}
2009:
1999:
1873:
1867:
1837:
1831:
1771:
1765:
1744:
1738:
1713:
1707:
1686:
1682:
1681:
1678:
1657:
1651:
1630:
1624:
1617:Nash-Kuiper C isometric embedding theorem
1596:
1590:
1569:
1563:
1530:
1515:
1495:
1465:
1459:
1432:
1426:
1405:
1399:
1330:
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1262:
1241:
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1195:
1189:
1164:
1158:
1101:
1100:
1077:
1076:
1074:
1051:
1031:
1011:
978:by considering the homotopy class of the
953:
941:
912:
826:
740:
721:
708:
695:
676:
663:
653:
651:
645:
610:
605:
587:
582:
564:
557:
548:
542:
506:
487:
474:
461:
442:
429:
419:
417:
411:
396:{\displaystyle y_{1},y_{2},\dots ,y_{N}.}
384:
365:
352:
346:
325:
320:
314:
285:
280:
274:
241:
236:
223:
204:
191:
179:
152:
133:
120:
111:
1619:, in particular implies that there is a
1801:
929:{\displaystyle \theta \mapsto n\theta }
301:stands for all partial derivatives of
787:if any non-holonomic solution can be
59:(PDRs). The h-principle is good for
7:
309:. Let us exchange every variable in
1673:into an arbitrarily small ball of
1374:Cone eversion. Consider functions
1312:{\displaystyle \mathbf {R} ^{m+1}}
1242:
648:
598:
575:
561:
414:
181:
94:Assume we want to find a function
25:
1941:. American Mathematical Society.
1646:isometric immersion of the round
51:) is a very general way to solve
1695:{\displaystyle \mathbb {R} ^{3}}
1293:
2011:10.1090/S0273-0979-2012-01376-9
2029:Partial differential equations
1961:Partial differential relations
1536:
1523:
962:{\displaystyle z\mapsto z^{n}}
946:
917:
842:
836:
746:
656:
512:
422:
341:for new independent variables
247:
184:
158:
113:
57:partial differential relations
53:partial differential equations
1:
1350:Ways to prove the h-principle
903:), by lifting the map to the
1257:) embedding or immersion of
854:{\displaystyle f'(x)\neq 0.}
1702:. This immersion cannot be
1250:{\displaystyle C^{\infty }}
55:(PDEs), and more generally
2050:
1907:D. Fuchs, S. Tabachnikov,
66:The theory was started by
1927:, translation Kiki Hudson
1925:Embeddings and immersions
1549:is not zero at any point.
972:Whitney–Graustein theorem
889:Whitney–Graustein theorem
806:pseudo-holomorphic curves
785:satisfies the h-principle
334:{\displaystyle J_{f}^{k}}
294:{\displaystyle J_{f}^{k}}
80:Whitney–Graustein theorem
1483:{\displaystyle f_{1}=-f}
905:universal covering space
171:. One can rewrite it as
2034:Mathematical principles
1447:{\displaystyle f_{0}=f}
1059:{\displaystyle \alpha }
1883:
1847:
1781:
1754:
1723:
1696:
1667:
1640:
1606:
1579:
1543:
1504:
1484:
1448:
1415:
1340:
1313:
1278:
1251:
1226:, which says that any
1220:Nash embedding theorem
1205:
1174:
1128:
1060:
1040:
1020:
963:
930:
896:
855:
794:non-holonomic solution
770:non-holonomic solution
759:
628:
525:
397:
335:
295:
260:
165:
36:
1988:Bull. Amer. Math. Soc
1884:
1882:{\displaystyle C^{1}}
1848:
1846:{\displaystyle C^{1}}
1782:
1780:{\displaystyle S^{2}}
1755:
1753:{\displaystyle C^{2}}
1724:
1722:{\displaystyle C^{2}}
1697:
1668:
1666:{\displaystyle S^{2}}
1641:
1639:{\displaystyle C^{1}}
1607:
1605:{\displaystyle S^{2}}
1580:
1578:{\displaystyle C^{1}}
1544:
1505:
1485:
1449:
1416:
1414:{\displaystyle f_{t}}
1341:
1339:{\displaystyle C^{1}}
1314:
1279:
1277:{\displaystyle M^{m}}
1252:
1206:
1204:{\displaystyle C^{0}}
1175:
1173:{\displaystyle C^{0}}
1129:
1061:
1041:
1021:
964:
931:
886:
861:Properly, this is an
856:
760:
629:
526:
398:
336:
296:
261:
166:
30:
1934:Introduction to the
1866:
1830:
1764:
1737:
1706:
1677:
1650:
1623:
1589:
1562:
1514:
1494:
1458:
1425:
1398:
1323:
1288:
1261:
1234:
1188:
1157:
1073:
1050:
1030:
1010:
974:classified these by
940:
911:
825:
644:
541:
410:
345:
313:
273:
178:
110:
1224:Nash–Kuiper theorem
1222:, specifically the
655:
421:
330:
290:
246:
1879:
1843:
1777:
1750:
1719:
1692:
1663:
1636:
1602:
1575:
1539:
1500:
1480:
1444:
1411:
1336:
1309:
1274:
1247:
1201:
1170:
1124:
1056:
1036:
1016:
1002:A car in the plane
959:
926:
897:
851:
817:Monotone functions
774:holonomic solution
755:
647:
624:
521:
413:
393:
331:
316:
291:
276:
256:
232:
161:
106:, in co-ordinates
45:homotopy principle
37:
1923:Masahisa Adachi,
1789:Theorema Egregium
1503:{\displaystyle t}
1213:dense h-principle
1109:
1085:
1039:{\displaystyle y}
1019:{\displaystyle x}
987:Stiefel manifolds
619:
305:up to order
16:(Redirected from
2041:
2015:
2013:
2003:
1974:
1952:
1911:
1905:
1899:
1896:
1890:
1888:
1886:
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1778:
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1111:
1110:
1102:
1087:
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1017:
968:
966:
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958:
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933:
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927:
860:
858:
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852:
835:
764:
762:
761:
756:
745:
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713:
712:
700:
699:
681:
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420:
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209:
208:
196:
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170:
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167:
162:
157:
156:
138:
137:
125:
124:
68:Yakov Eliashberg
21:
2049:
2048:
2044:
2043:
2042:
2040:
2039:
2038:
2019:
2018:
1977:
1971:
1955:
1949:
1930:
1920:
1918:Further reading
1915:
1914:
1906:
1902:
1897:
1893:
1869:
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1861:
1857:
1833:
1828:
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1816:
1812:
1807:
1803:
1798:
1767:
1762:
1761:
1740:
1735:
1734:
1731:Gauss curvature
1709:
1704:
1703:
1680:
1675:
1674:
1653:
1648:
1647:
1626:
1621:
1620:
1592:
1587:
1586:
1565:
1560:
1559:
1556:Sphere eversion
1526:
1512:
1511:
1492:
1491:
1461:
1456:
1455:
1428:
1423:
1422:
1401:
1396:
1395:
1390:) = |
1382:without origin
1368:
1352:
1326:
1321:
1320:
1291:
1286:
1285:
1264:
1259:
1258:
1237:
1232:
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1191:
1186:
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1160:
1155:
1154:
1071:
1070:
1048:
1047:
1028:
1027:
1008:
1007:
1004:
949:
938:
937:
909:
908:
828:
823:
822:
819:
814:
812:Simple examples
736:
717:
704:
691:
672:
659:
642:
641:
606:
601:
583:
578:
574:
560:
559:
544:
539:
538:
502:
483:
470:
457:
438:
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408:
407:
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361:
348:
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311:
310:
271:
270:
219:
200:
187:
176:
175:
148:
129:
116:
108:
107:
92:
61:underdetermined
33:sphere eversion
23:
22:
15:
12:
11:
5:
2047:
2045:
2037:
2036:
2031:
2021:
2020:
2017:
2016:
1975:
1969:
1953:
1947:
1928:
1919:
1916:
1913:
1912:
1900:
1891:
1876:
1872:
1862:N. Kuiper, On
1855:
1840:
1836:
1819:
1810:
1800:
1799:
1797:
1794:
1793:
1792:
1774:
1770:
1747:
1743:
1716:
1712:
1689:
1684:
1660:
1656:
1633:
1629:
1613:
1599:
1595:
1585:immersions of
1572:
1568:
1553:
1550:
1538:
1533:
1529:
1525:
1522:
1519:
1499:
1479:
1476:
1473:
1468:
1464:
1443:
1440:
1435:
1431:
1408:
1404:
1367:
1366:Some paradoxes
1364:
1363:
1362:
1359:
1356:
1351:
1348:
1333:
1329:
1306:
1303:
1300:
1295:
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1267:
1244:
1240:
1198:
1194:
1167:
1163:
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1134:
1123:
1120:
1117:
1114:
1108:
1105:
1099:
1096:
1093:
1090:
1084:
1081:
1055:
1035:
1015:
1003:
1000:
976:turning number
956:
952:
948:
945:
925:
922:
919:
916:
901:winding number
893:turning number
850:
847:
844:
841:
838:
834:
831:
818:
815:
813:
810:
766:
765:
754:
751:
748:
743:
739:
735:
732:
729:
724:
720:
716:
711:
707:
703:
698:
694:
690:
687:
684:
679:
675:
671:
666:
662:
658:
650:
637:A solution of
635:
634:
623:
613:
609:
604:
600:
597:
590:
586:
581:
577:
572:
567:
563:
556:
551:
547:
532:
531:
520:
517:
514:
509:
505:
501:
498:
495:
490:
486:
482:
477:
473:
469:
464:
460:
456:
453:
450:
445:
441:
437:
432:
428:
424:
416:
392:
387:
383:
379:
376:
373:
368:
364:
360:
355:
351:
328:
323:
319:
288:
283:
279:
267:
266:
255:
252:
249:
244:
239:
235:
231:
226:
222:
218:
215:
212:
207:
203:
199:
194:
190:
186:
183:
160:
155:
151:
147:
144:
141:
136:
132:
128:
123:
119:
115:
91:
88:
72:Mikhail Gromov
24:
14:
13:
10:
9:
6:
4:
3:
2:
2046:
2035:
2032:
2030:
2027:
2026:
2024:
2012:
2007:
2002:
1997:
1993:
1989:
1985:
1983:
1976:
1972:
1970:3-540-12177-3
1966:
1962:
1958:
1954:
1950:
1948:9780821832271
1944:
1940:
1939:
1935:
1929:
1926:
1922:
1921:
1917:
1910:
1904:
1901:
1895:
1892:
1874:
1870:
1859:
1856:
1838:
1834:
1823:
1820:
1814:
1811:
1805:
1802:
1795:
1790:
1772:
1768:
1745:
1741:
1732:
1714:
1710:
1687:
1658:
1654:
1631:
1627:
1618:
1614:
1597:
1593:
1570:
1566:
1557:
1554:
1551:
1531:
1527:
1520:
1517:
1497:
1477:
1474:
1471:
1466:
1462:
1441:
1438:
1433:
1429:
1406:
1402:
1393:
1389:
1385:
1381:
1377:
1373:
1372:
1371:
1365:
1360:
1357:
1354:
1353:
1349:
1347:
1331:
1327:
1304:
1301:
1298:
1269:
1265:
1238:
1229:
1225:
1221:
1216:
1214:
1196:
1192:
1183:
1165:
1161:
1151:
1146:
1144:
1140:
1121:
1118:
1115:
1112:
1106:
1103:
1097:
1094:
1091:
1088:
1082:
1079:
1069:
1068:
1067:
1053:
1033:
1013:
1001:
999:
996:
992:
991:frame bundles
988:
983:
981:
977:
973:
954:
950:
943:
923:
920:
914:
906:
902:
894:
890:
885:
881:
878:
874:
872:
866:
864:
848:
845:
839:
832:
829:
816:
811:
809:
807:
803:
797:
795:
790:
786:
781:
777:
775:
771:
752:
749:
741:
737:
733:
730:
727:
722:
718:
714:
709:
705:
701:
696:
692:
688:
685:
682:
677:
673:
669:
664:
660:
640:
639:
638:
621:
611:
607:
602:
595:
588:
584:
579:
570:
565:
554:
549:
545:
537:
536:
535:
518:
515:
507:
503:
499:
496:
493:
488:
484:
480:
475:
471:
467:
462:
458:
454:
451:
448:
443:
439:
435:
430:
426:
406:
405:
404:
390:
385:
381:
377:
374:
371:
366:
362:
358:
353:
349:
326:
321:
317:
308:
304:
286:
281:
277:
253:
250:
242:
237:
233:
229:
224:
220:
216:
213:
210:
205:
201:
197:
192:
188:
174:
173:
172:
153:
149:
145:
142:
139:
134:
130:
126:
121:
117:
105:
101:
97:
89:
87:
85:
81:
77:
73:
69:
64:
62:
58:
54:
50:
46:
42:
34:
29:
19:
1991:
1987:
1981:
1963:. Springer.
1960:
1937:
1933:
1908:
1903:
1894:
1858:
1822:
1813:
1804:
1787:, by Gauss'
1490:and for any
1391:
1387:
1383:
1379:
1375:
1369:
1217:
1212:
1180:-close to a
1147:
1136:
1005:
994:
984:
898:
879:
875:
867:
862:
820:
798:
793:
784:
782:
778:
773:
769:
768:is called a
767:
636:
533:
306:
302:
268:
103:
99:
95:
93:
83:
65:
48:
44:
38:
1994:: 347–375.
1826:John Nash,
871:convex sets
802:Lagrangians
49:h-principle
41:mathematics
18:H-principle
2023:Categories
1957:Gromov, M.
1938:-principle
1796:References
1421:such that
1182:Legendrian
90:Rough idea
2001:1111.2700
1521:
1475:−
1243:∞
1150:homotopic
1143:holonomic
1119:α
1116:
1107:˙
1095:α
1092:
1083:˙
1054:α
980:Gauss map
947:↦
924:θ
918:↦
915:θ
846:≠
731:…
686:…
649:Ψ
599:∂
596:…
576:∂
562:∂
497:…
452:…
415:Ψ
375:…
214:…
182:Ψ
143:…
1959:(1986).
1230:smooth (
1139:robotics
863:ordinary
833:′
789:deformed
76:homotopy
1967:
1945:
783:A PDE
303:ƒ
269:where
96:ƒ
43:, the
1996:arXiv
1980:"The
1228:short
1965:ISBN
1943:ISBN
1615:The
1518:grad
1026:and
887:The
47:(or
2006:doi
1378:on
1284:in
1113:cos
1089:sin
98:on
39:In
2025::
2004:.
1992:49
1990:.
1986:.
1510:,
1454:,
1215:.
1145:.
995:k,
849:0.
808:.
776:.
70:,
2014:.
2008::
1998::
1982:h
1973:.
1951:.
1936:h
1875:1
1871:C
1839:1
1835:C
1791:.
1773:2
1769:S
1746:2
1742:C
1715:2
1711:C
1688:3
1683:R
1659:2
1655:S
1632:1
1628:C
1612:.
1598:2
1594:S
1571:1
1567:C
1537:)
1532:t
1528:f
1524:(
1498:t
1478:f
1472:=
1467:1
1463:f
1442:f
1439:=
1434:0
1430:f
1407:t
1403:f
1392:x
1388:x
1386:(
1384:f
1380:R
1376:f
1332:1
1328:C
1305:1
1302:+
1299:m
1294:R
1270:m
1266:M
1239:C
1211:-
1197:0
1193:C
1166:0
1162:C
1122:.
1104:y
1098:=
1080:x
1034:y
1014:x
955:n
951:z
944:z
936:(
921:n
895:.
843:)
840:x
837:(
830:f
753:0
750:=
747:)
742:N
738:y
734:,
728:,
723:2
719:y
715:,
710:1
706:y
702:,
697:m
693:u
689:,
683:,
678:2
674:u
670:,
665:1
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657:(
622:.
612:k
608:j
603:u
589:1
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580:u
571:f
566:k
555:=
550:j
546:y
519:0
516:=
513:)
508:N
504:y
500:,
494:,
489:2
485:y
481:,
476:1
472:y
468:,
463:m
459:u
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444:2
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436:,
431:1
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423:(
391:.
386:N
382:y
378:,
372:,
367:2
363:y
359:,
354:1
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327:k
322:f
318:J
307:k
287:k
282:f
278:J
254:0
251:=
248:)
243:k
238:f
234:J
230:,
225:m
221:u
217:,
211:,
206:2
202:u
198:,
193:1
189:u
185:(
159:)
154:m
150:u
146:,
140:,
135:2
131:u
127:,
122:1
118:u
114:(
104:k
100:R
84:C
35:.
20:)
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