2074:—is viewed as a map from the base to the total space, and intersects the zero-section (viewed either as a map or as a submanifold) transversely, then the zero set of the section—i.e. the singularities of the vector field—forms a smooth 0-dimensional submanifold of the base, i.e. a set of signed points. The signs agree with the indices of the vector field, and thus the sum of the signs—i.e. the fundamental class of the zero set—is equal to the Euler characteristic of the manifold. More generally, for a
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161:
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22:
1501:
In a three-dimensional space, two curves can be transverse only when they have empty intersection, since their tangent spaces could generate at most a two-dimensional space. Curves transverse to surfaces intersect in points, and surfaces transverse to each other intersect in curves. Curves that are
2062:, whose hypothesis is a special case of the transversality of maps, it can be shown that transverse intersections between submanifolds of a space of complementary dimensions or between submanifolds and maps to a space are themselves smooth submanifolds. For instance, if a smooth
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the manifolds to another transverse intersection. (The intersection points can be counted modulo 2, ignoring the signs, to obtain a coarser invariant.) This descends to a bilinear intersection product on homology classes of any dimension, which is
627:
2078:
over an oriented smooth closed finite-dimensional manifold, the zero set of a section transverse to the zero section will be a submanifold of the base of codimension equal to the rank of the vector bundle, and its homology class will be
1455:
Given any two smooth submanifolds, it is possible to perturb either of them by an arbitrarily small amount such that the resulting submanifold intersects transversally with the fixed submanifold. Such perturbations do not affect the
2090:
An extremely special case of this is the following: if a differentiable function from reals to the reals has nonzero derivative at a zero of the function, then the zero is simple, i.e. it the graph is transverse to the
1460:
class of the manifolds or of their intersections. For example, if manifolds of complementary dimension intersect transversally, the signed sum of the number of their intersection points does not change even if we
637:
The notion of transversality of a pair of submanifolds is easily extended to transversality of a submanifold and a map to the ambient manifold, or to a pair of maps to the ambient manifold, by asking whether the
200:), the condition means that the tangent space to the ambient manifold is the direct sum of the two smaller tangent spaces. If an intersection is transverse, then the intersection will be a submanifold whose
1979:, the transversality condition is frequently used to control the types of solutions found in optimization problems. For example, it is a necessary condition for solution curves to problems of the form:
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Transversality depends on ambient space. The two curves shown are transverse when considered as embedded in the plane, but not if we consider them as embedded in a plane in three-dimensional space
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is equal to the sums of the codimensions of the two manifolds. In the absence of the transversality condition the intersection may fail to be a submanifold, having some sort of
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at their point of intersection, as happens in the case of embedded submanifolds. If the maps are immersions, the intersection of their images will be a manifold of dimension
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of the tangent spaces along the preimage of points of intersection of the images generate the entire tangent space of the ambient manifold. If the maps are
2114:. The zero set of this section consists of holomorphic maps. If the d-bar operator can be shown to be transverse to the zero-section, this
1287:'s tangent space at any point of intersection. Their intersection thus consists of isolated signed points, i.e. a zero-dimensional manifold.
39:
86:
219:, their intersection is oriented. When the intersection is zero-dimensional, the orientation is simply a plus or minus for each point.
58:
2095:-axis at that zero; a zero derivative would mean a horizontal tangent to the curve, which would agree with the tangent space to the
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cannot be transverse. However, non-intersecting manifolds vacuously satisfy the condition, so can be said to intersect transversely.
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In particular, this means that transverse submanifolds of complementary dimension intersect in isolated points (i.e., a
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In many of these problems, the solution satisfies the condition that the solution curve should cross transversally the
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tangent to a surface at a point (for instance, curves lying on a surface) do not intersect the surface transversally.
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transverse. If the manifolds are of complementary dimension (i.e., their dimensions add up to the dimension of the
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144:. It is defined by considering the linearizations of the intersecting spaces at the points of intersection.
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2126:. (Note that for this example, the definition of transversality has to be refined in order to deal with
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it is not a tangency, i.e., their tangent lines inside the tangent plane to the surface are distinct.
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622:{\displaystyle L_{1}\pitchfork L_{2}\iff \forall p\in L_{1}\cap L_{2},T_{p}M=T_{p}L_{1}+T_{p}L_{2}.}
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when that intersection is transverse. In this notation, the definition of transversality reads
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will be a smooth manifold. These considerations play a fundamental role in the theory of
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intersect transversally” or as an alternative notation for the set-theoretic intersection
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The meaning of transversality differs a lot depending on the relative dimensions of
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For an infinite-dimensional example, the d-bar operator is a section of a certain
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Thom, René (1954). "Quelques propriétés globales des variétés differentiables".
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937:. The relationship between transversality and tangency is clearest when
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184:, their separate tangent spaces at that point together generate the
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The simplest non-trivial example of transversality is of arcs in a
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One notation for the transverse intersection of two submanifolds
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where one or both of the endpoints of the curve are not fixed.
1110:'s tangent space at any point. Thus any intersection between
1749:{\displaystyle {\mathfrak {g}}=\oplus {\mathfrak {g}}_{f}}
646:, this is equivalent to transversality of submanifolds.
2139:"Transversal" is a noun; the adjective is "transverse."
1494:. An intersection point between two arcs is transverse
1485:
336:. This notation can be read in two ways: either as “
140:. It formalizes the idea of a generic intersection in
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215:). If both submanifolds and the ambient manifold are
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or some other curve describing terminal conditions.
1478:. Like the cup product, the intersection product is
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at that point. Manifolds that do not intersect are
46:. Unsourced material may be challenged and removed.
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650:Meaning of transversality for different dimensions
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130:; transversality can be seen as the "opposite" of
1505:Here is a more specialised example: suppose that
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164:Non-transverse curves on the surface of a sphere
156:Transverse curves on the surface of a sphere
2035:{\displaystyle \int {F(x,y,y^{\prime })}dx}
521:
517:
126:is a notion that describes how spaces can
2233:Guillemin, Victor; Pollack, Alan (1974).
632:
106:Learn how and when to remove this message
1329:{\displaystyle \ell _{1}+\ell _{2}>m}
1029:{\displaystyle \ell _{1}+\ell _{2}<m}
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159:
151:
1450:
1336:this sum needn't be direct. In fact it
1267:'s tangent spaces must sum directly to
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1954:is known as the "Slodowy slice" after
1440:{\displaystyle \ell _{1}+\ell _{2}-m.}
986:We can consider three separate cases:
2106:bundle over the space of maps from a
1947:{\displaystyle e+{\mathfrak {g}}_{f}}
1890:{\displaystyle e+{\mathfrak {g}}_{f}}
1206:{\displaystyle \ell _{1}+\ell _{2}=m}
976:{\displaystyle \ell _{1}+\ell _{2}=m}
329:{\displaystyle L_{1}\pitchfork L_{2}}
2202:
1686:{\displaystyle {\mathfrak {sl_{2}}}}
1614:{\displaystyle {\mathfrak {sl_{2}}}}
1580:{\displaystyle e\in {\mathfrak {g}}}
1486:Examples of transverse intersections
1036:, it is impossible for the image of
44:adding citations to reliable sources
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847:{\displaystyle \ell _{1},\ell _{2}}
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2143:quote from J.H.C. Whitehead, 1959
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2180:Guillemin and Pollack 1974, p.28.
2171:Guillemin and Pollack 1974, p.30.
55:"Transversality" mathematics
1850:{\displaystyle {\rm {{Ad}(G)e}}}
747:{\displaystyle f_{2}:L_{2}\to M}
701:{\displaystyle f_{1}:L_{1}\to M}
662:Suppose we have transverse maps
20:
1961:
1659:. The representation theory of
1546:{\displaystyle {\mathfrak {g}}}
423:{\displaystyle L_{1}\cap L_{2}}
31:needs additional citations for
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2174:
2165:
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1997:
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814:are manifolds with dimensions
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172:of a given finite-dimensional
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2054:Smoothness of solution spaces
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1977:Pontryagin maximum principle
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1553:is its Lie algebra. By the
787:{\displaystyle L_{1},L_{2}}
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2133:
2066:of an oriented manifold's
1090:'s tangent spaces to span
1917:transversally. The space
2158:
2120:pseudoholomorphic curves
1971:In fields utilizing the
1897:intersects the orbit of
1587:can be included into an
1557:every nilpotent element
1555:Jacobson–Morozov theorem
2112:almost-complex manifold
1652:{\displaystyle (e,h,f)}
903:{\displaystyle M,L_{1}}
178:intersect transversally
2289:Calculus of variations
2153:Transversality theorem
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1973:calculus of variations
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633:Transversality of maps
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136:, and plays a role in
2284:Differential topology
2257:Differential Topology
2235:Differential Topology
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1816:to the adjoint orbit
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1387:{\displaystyle f_{2}}
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1360:{\displaystyle f_{1}}
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1260:{\displaystyle L_{2}}
1235:
1233:{\displaystyle L_{1}}
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1157:{\displaystyle f_{2}}
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1130:{\displaystyle f_{1}}
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1083:{\displaystyle L_{2}}
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1056:{\displaystyle L_{1}}
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978:
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930:{\displaystyle L_{2}}
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477:{\displaystyle L_{2}}
452:
450:{\displaystyle L_{1}}
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383:{\displaystyle L_{2}}
358:
356:{\displaystyle L_{1}}
331:
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269:{\displaystyle L_{2}}
244:
242:{\displaystyle L_{1}}
180:if at every point of
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142:differential topology
2206:Comment. Math. Helv.
2124:Gromov–Witten theory
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40:improve this article
2259:. Springer-Verlag.
2189:Hirsch (1976), p.66
2218:10.1007/BF02566923
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1910:{\displaystyle e}
1809:{\displaystyle e}
1518:{\displaystyle G}
1280:{\displaystyle M}
1103:{\displaystyle M}
867:{\displaystyle m}
807:{\displaystyle M}
289:{\displaystyle M}
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1785:{\displaystyle }
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138:general position
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414:
410:
401:
397:
395:
392:
391:
374:
370:
368:
365:
364:
347:
343:
341:
338:
337:
320:
316:
307:
303:
301:
298:
297:
281:
278:
277:
260:
256:
254:
251:
250:
233:
229:
227:
224:
223:
174:smooth manifold
150:
112:
101:
95:
92:
49:
47:
37:
25:
12:
11:
5:
2307:
2297:
2296:
2291:
2286:
2272:
2271:
2265:
2253:Hirsch, Morris
2249:
2243:
2230:
2198:
2195:
2192:
2191:
2182:
2173:
2163:
2162:
2160:
2157:
2156:
2155:
2148:
2145:
2135:
2132:
2068:tangent bundle
2060:Sard's theorem
2055:
2052:
2044:
2043:
2031:
2028:
2024:
2019:
2015:
2011:
2008:
2005:
2002:
1999:
1996:
1992:
1968:
1965:
1963:
1960:
1941:
1935:
1929:
1926:
1906:
1884:
1878:
1872:
1869:
1844:
1841:
1838:
1835:
1831:
1828:
1805:
1781:
1778:
1775:
1770:
1765:
1743:
1737:
1731:
1728:
1725:
1722:
1717:
1712:
1709:
1704:
1693:tells us that
1678:
1674:
1670:
1648:
1645:
1642:
1639:
1636:
1633:
1630:
1606:
1602:
1598:
1574:
1569:
1566:
1540:
1514:
1496:if and only if
1487:
1484:
1452:
1449:
1448:
1447:
1436:
1433:
1430:
1425:
1421:
1417:
1412:
1408:
1381:
1377:
1354:
1350:
1325:
1322:
1317:
1313:
1309:
1304:
1300:
1288:
1276:
1254:
1250:
1227:
1223:
1202:
1199:
1194:
1190:
1186:
1181:
1177:
1165:
1151:
1147:
1124:
1120:
1099:
1077:
1073:
1050:
1046:
1025:
1022:
1017:
1013:
1009:
1004:
1000:
972:
969:
964:
960:
956:
951:
947:
924:
920:
897:
893:
889:
886:
874:respectively.
863:
841:
837:
833:
828:
824:
803:
781:
777:
773:
768:
764:
743:
740:
735:
731:
727:
722:
718:
697:
694:
689:
685:
681:
676:
672:
651:
648:
634:
631:
630:
629:
618:
613:
609:
603:
599:
595:
590:
586:
580:
576:
572:
569:
564:
560:
556:
551:
547:
543:
538:
534:
530:
527:
524:
520:
514:
510:
506:
501:
497:
471:
467:
444:
440:
417:
413:
409:
404:
400:
377:
373:
350:
346:
323:
319:
315:
310:
306:
285:
263:
259:
236:
232:
206:singular point
149:
146:
124:transversality
114:
113:
28:
26:
19:
9:
6:
4:
3:
2:
2306:
2295:
2292:
2290:
2287:
2285:
2282:
2281:
2279:
2268:
2266:0-387-90148-5
2262:
2258:
2254:
2250:
2246:
2244:0-13-212605-2
2240:
2236:
2231:
2227:
2223:
2219:
2215:
2211:
2208:
2207:
2201:
2200:
2186:
2177:
2168:
2164:
2154:
2151:
2150:
2144:
2140:
2131:
2129:
2128:Banach spaces
2125:
2121:
2117:
2113:
2109:
2105:
2100:
2098:
2094:
2088:
2086:
2082:
2081:Poincaré dual
2077:
2076:vector bundle
2073:
2069:
2065:
2061:
2051:
2049:
2029:
2026:
2013:
2009:
2006:
2003:
2000:
1994:
1990:
1982:
1981:
1980:
1978:
1974:
1959:
1957:
1956:Peter Slodowy
1939:
1927:
1924:
1904:
1882:
1870:
1867:
1860:
1803:
1795:
1794:tangent space
1776:
1773:
1741:
1729:
1723:
1720:
1707:
1643:
1640:
1637:
1634:
1631:
1567:
1564:
1556:
1528:
1512:
1503:
1499:
1497:
1493:
1483:
1481:
1477:
1473:
1469:
1468:Poincaré dual
1464:
1459:
1434:
1431:
1428:
1423:
1419:
1415:
1410:
1406:
1397:
1379:
1375:
1352:
1348:
1340:be direct if
1339:
1323:
1320:
1315:
1311:
1307:
1302:
1298:
1289:
1274:
1252:
1248:
1225:
1221:
1200:
1197:
1192:
1188:
1184:
1179:
1175:
1166:
1149:
1145:
1122:
1118:
1097:
1075:
1071:
1048:
1044:
1023:
1020:
1015:
1011:
1007:
1002:
998:
989:
988:
987:
984:
970:
967:
962:
958:
954:
949:
945:
922:
918:
895:
891:
887:
884:
875:
861:
839:
835:
831:
826:
822:
801:
779:
775:
771:
766:
762:
741:
733:
729:
725:
720:
716:
695:
687:
683:
679:
674:
670:
656:
647:
645:
641:
616:
611:
607:
601:
597:
593:
588:
584:
578:
574:
570:
567:
562:
558:
554:
549:
545:
541:
536:
532:
528:
525:
512:
508:
504:
499:
495:
487:
486:
485:
469:
465:
442:
438:
415:
411:
407:
402:
398:
375:
371:
348:
344:
321:
317:
313:
308:
304:
283:
261:
257:
234:
230:
220:
218:
214:
209:
207:
203:
199:
198:ambient space
195:
191:
187:
186:tangent space
183:
179:
175:
171:
162:
154:
145:
143:
139:
135:
134:
129:
125:
121:
110:
107:
99:
96:December 2009
88:
85:
81:
78:
74:
71:
67:
64:
60:
57: –
56:
52:
51:Find sources:
45:
41:
35:
34:
29:This article
27:
23:
18:
17:
2256:
2234:
2212:(1): 17–86.
2209:
2204:
2185:
2176:
2167:
2142:
2138:
2116:moduli space
2104:Banach space
2101:
2096:
2092:
2089:
2072:vector field
2057:
2045:
1970:
1962:Applications
1859:affine space
1756:. The space
1504:
1500:
1489:
1454:
1337:
985:
876:
661:
640:pushforwards
636:
221:
210:
182:intersection
177:
176:are said to
170:submanifolds
167:
131:
123:
117:
102:
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
2085:Euler class
1857:and so the
1472:cup product
202:codimension
120:mathematics
2278:Categories
2197:References
1476:cohomology
1396:immersions
644:embeddings
213:0-manifold
148:Definition
66:newspapers
2226:120243638
2048:nullcline
2018:′
1991:∫
1983:Minimize
1730:⊕
1568:∈
1429:−
1420:ℓ
1407:ℓ
1312:ℓ
1299:ℓ
1189:ℓ
1176:ℓ
1012:ℓ
999:ℓ
959:ℓ
946:ℓ
836:ℓ
823:ℓ
739:→
693:→
542:∩
529:∈
523:∀
519:⟺
505:⋔
408:∩
314:⋔
194:vacuously
128:intersect
2294:Geometry
2255:(1976).
2147:See also
2110:into an
2070:—i.e. a
1621:-triple
1458:homology
217:oriented
133:tangency
2134:Grammar
2099:-axis.
2083:to the
2064:section
1792:is the
1492:surface
1470:to the
1463:isotope
188:of the
80:scholar
2263:
2241:
2224:
2058:Using
1338:cannot
754:where
82:
75:
68:
61:
53:
2222:S2CID
2159:Notes
1525:is a
1290:When
1167:When
990:When
87:JSTOR
73:books
2261:ISBN
2239:ISBN
2122:and
1529:and
1394:are
1367:and
1321:>
1240:and
1137:and
1063:and
1021:<
910:and
854:and
794:and
708:and
457:and
363:and
249:and
168:Two
59:news
2214:doi
2130:!)
1796:at
1474:on
430:of
296:is
118:In
42:by
2280::
2220:.
2210:28
1958:.
1482:.
983:.
208:.
122:,
2269:.
2247:.
2228:.
2216::
2097:x
2093:x
2030:x
2027:d
2023:)
2014:y
2010:,
2007:y
2004:,
2001:x
1998:(
1995:F
1940:f
1934:g
1928:+
1925:e
1905:e
1883:f
1877:g
1871:+
1868:e
1843:e
1840:)
1837:G
1834:(
1830:d
1827:A
1804:e
1780:]
1777:e
1774:,
1769:g
1764:[
1742:f
1736:g
1727:]
1724:e
1721:,
1716:g
1711:[
1708:=
1703:g
1677:2
1673:l
1669:s
1647:)
1644:f
1641:,
1638:h
1635:,
1632:e
1629:(
1605:2
1601:l
1597:s
1573:g
1565:e
1539:g
1513:G
1435:.
1432:m
1424:2
1416:+
1411:1
1380:2
1376:f
1353:1
1349:f
1324:m
1316:2
1308:+
1303:1
1275:M
1253:2
1249:L
1226:1
1222:L
1201:m
1198:=
1193:2
1185:+
1180:1
1150:2
1146:f
1123:1
1119:f
1098:M
1076:2
1072:L
1049:1
1045:L
1024:m
1016:2
1008:+
1003:1
971:m
968:=
963:2
955:+
950:1
923:2
919:L
896:1
892:L
888:,
885:M
862:m
840:2
832:,
827:1
802:M
780:2
776:L
772:,
767:1
763:L
742:M
734:2
730:L
726::
721:2
717:f
696:M
688:1
684:L
680::
675:1
671:f
617:.
612:2
608:L
602:p
598:T
594:+
589:1
585:L
579:p
575:T
571:=
568:M
563:p
559:T
555:,
550:2
546:L
537:1
533:L
526:p
513:2
509:L
500:1
496:L
470:2
466:L
443:1
439:L
416:2
412:L
403:1
399:L
376:2
372:L
349:1
345:L
322:2
318:L
309:1
305:L
284:M
262:2
258:L
235:1
231:L
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
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