33:
1127:
2373:, respectively, if and only if the induced morphisms on all stalks have the same property. (However it is not true that two sheaves, all of whose stalks are isomorphic, are isomorphic, too, because there may be no map between the sheaves in question.)
1833:, because the latter function is not identically one on any neighborhood of the origin. This example shows that germs contain more information than the power series expansion of a function, because the power series of
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453:
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1505:, we find that the germ at a point determines the function on any connected open set where the function can be everywhere defined. (This does not imply that all the restriction maps of this sheaf are injective!)
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2232:. This idea makes more sense if one adopts the common visualisation of functions mapping from some space above to a space below; with this visualisation, any function that maps
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that is identically one in a neighborhood of the origin and identically zero far away from the origin. On any sufficiently small neighborhood containing the origin,
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As outlined in the introduction, stalks capture the local behaviour of a sheaf. As a sheaf is supposed to be determined by its local restrictions (see
1122:{\displaystyle i^{-1}{\mathcal {F}}(\{x\})=\varinjlim _{U\supseteq \{x\}}{\mathcal {F}}(U)=\varinjlim _{U\ni x}{\mathcal {F}}(U)={\mathcal {F}}_{x}.}
379:
1625:
is a bump function, the germ does not tell us how large its bump is. From what the germ tells us, the bump could be infinitely wide, that is,
1497:, a germ of a function at a point determines the function in a small neighborhood of a point. This is because the germ records the function's
2588:
1884:
is identically one. (This extra information is related to the fact that the stalk of the sheaf of smooth functions at the origin is a non-
2875:
225:. Conceptually speaking, we do this by looking at small neighborhoods of the point. If we look at a sufficiently small neighborhood of
2683:
1581:
is identically one, so at the origin it has the same germ as the constant function with value 1. Suppose that we want to reconstruct
1144:
the direct limit used to define the stalk may not exist. However, it exists for most categories that occur in practice, such as the
2815:
2786:
116:
1521:, germs contain some local information, but are not enough to reconstruct the function on any open neighborhood. For example, let
2357:), it can be expected that the stalks capture a fair amount of the information that the sheaf is encoding. This is indeed true:
2871:
1166:
54:
50:
1782:
On the other hand, germs of smooth functions can distinguish between the constant function with value one and the function
97:
2381:
A sheaf is zero (if we are dealing with sheafs of groups), if and only if all stalks of the sheaf vanish. Therefore, the
69:
2520:
2431:
295:
at that point. Of course, no single neighborhood will be small enough, so we will have to take a limit of some sort.
1524:
1889:
76:
547:
2064:
83:
1404:
649:
181:
consists of points. It is reasonable to attempt to isolate the behavior of a sheaf at a single fixed point
43:
1906:
1138:
826:
2768:
2389:
can be tested on the stalks, which is often easier as one can pass to smaller and smaller neighbourhoods.
1312:, that is, its equivalence class in the direct limit. This is a generalization of the usual concept of a
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expansion, and all analytic functions are by definition locally equal to their power series. Using
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There is another approach to defining a stalk that is useful in some contexts. Choose a point
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1316:, which can be recovered by looking at the stalks of the sheaf of continuous functions on
2878:)Spring 2009. Massachusetts Institute of Technology: MIT OpenCourseWare Creative Commons
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544:) of the direct limit, an element of the stalk is an equivalence class of elements
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2680:(because the sheafification functor is left adjoint to the inclusion functor
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could equal the constant function with value 1. We cannot even reconstruct
448:{\displaystyle {\mathcal {F}}_{x}:=\varinjlim _{U\ni x}{\mathcal {F}}(U).}
2879:
2394:
2319:
153:
2673:{\displaystyle (-)^{+}:\mathbf {Set} ^{{\mathcal {O}}(X)^{op}}\to Sh(X)}
2386:
2339:
2831:
144:
construction capturing the behaviour of a sheaf around a given point.
2330:
space is closed. This feature is the basis of the construction of
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containing the origin, because we cannot tell whether the bump of
271:
on that small neighborhood should be the same as the behavior of
2835:
2749:{\displaystyle Sh(X)\to \mathbf {Set} ^{{\mathcal {O}}(X)^{op}}}
2397:. However, stalks of sheaves and presheaves are tightly linked:
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860:. Notice that the only open sets of the one point space
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if the restrictions of the two sections coincide on some
1206:{\displaystyle {\mathcal {F}}(U)\to {\mathcal {F}}_{x}}
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says that this cannot happen for a
Noetherian ring.)
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2756:) and the fact that left adjoints preserve colimits.
484:, with order relation induced by reverse inclusion
298:The precise definition is as follows: the stalk of
57:. Unsourced material may be challenged and removed.
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2551:{\displaystyle {\mathcal {F}}={\mathcal {P}}^{+}}
2517:agree. This follows from the fact that the sheaf
2462:{\displaystyle {\mathcal {F}}={\mathcal {P}}^{+}}
932:, and there is no data over the empty set. Over
1603:from its germ. Even if we know in advance that
1148:or most categories of algebraic objects such as
1550:{\displaystyle f:\mathbb {R} \to \mathbb {R} }
1401:, (or group, ring, etc) is a sheaf for which
462:is indexed over all the open sets containing
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2318:if the topological space in question is a
583:{\displaystyle f_{U}\in {\mathcal {F}}(U)}
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117:Learn how and when to remove this message
2298:. The same property holds for any point
742:be the inclusion of the one point space
1434:{\displaystyle {\underline {S}}_{x}=S}
7:
2852:"6.27 Skyscraper sheaves and stalks"
1942:{\displaystyle X=\mathrm {Spec} (A)}
853:{\displaystyle i^{-1}{\mathcal {F}}}
55:adding citations to reliable sources
2861:"Introduction to Perverse Sheaves"
1926:
1923:
1920:
1917:
919:
812:{\displaystyle {\mathcal {F}}_{x}}
366:{\displaystyle {\mathcal {F}}_{x}}
25:
1301:{\displaystyle {\mathcal {F}}(U)}
2713:
2710:
2707:
2619:
2616:
2613:
1374:{\displaystyle {\underline {S}}}
31:
2866:. Institute for Advanced Study.
1735:or whether it is so large that
42:needs additional citations for
2732:
2725:
2702:
2699:
2693:
2667:
2661:
2652:
2638:
2631:
2599:
2592:
2575:{\displaystyle {\mathcal {P}}}
2510:{\displaystyle {\mathcal {F}}}
2486:{\displaystyle {\mathcal {P}}}
2418:{\displaystyle {\mathcal {P}}}
2393:Both statements are false for
2242:
2101:On any topological space, the
1970:{\displaystyle {\mathcal {F}}}
1936:
1930:
1877:{\displaystyle 1+e^{-1/x^{2}}}
1826:{\displaystyle 1+e^{-1/x^{2}}}
1539:
1513:In contrast, for the sheaf of
1295:
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1186:
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1177:
1096:
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1045:
1000:
988:
577:
571:
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433:
315:{\displaystyle {\mathcal {F}}}
288:{\displaystyle {\mathcal {F}}}
264:{\displaystyle {\mathcal {F}}}
1:
1669:on a small open neighborhood
1489:For example, in the sheaf of
1485:Sheaves of analytic functions
2850:The Stacks Project authors.
2841:The Stacks Project authors.
2361:A morphism of sheaves is an
1163:There is a natural morphism
531:{\displaystyle U\supseteq V}
247:, the behavior of the sheaf
1509:Sheaves of smooth functions
590:, where two such sections
2912:
2767:Hartshorne, Robin (1977).
2326:, since every point of a T
2278:positioned directly above
1890:Krull intersection theorem
925:{\displaystyle \emptyset }
2779:10.1007/978-1-4757-3849-0
148:Motivation and definition
129:Mathematical construction
2808:10.1017/CBO9780511661761
2796:Tennison, B. R. (1975).
1381:associated to some set,
2349:Properties of the stalk
505:{\displaystyle U\leq V}
152:Sheaves are defined on
2870:Kiran Kedlaya. 18.726
2750:
2674:
2576:
2552:
2511:
2487:
2463:
2419:
2334:, used for example in
2312:
2292:
2272:
2252:
2251:{\displaystyle G\to x}
2228:—hence the name
2222:
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2033:
2012:
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1896:Quasi-coherent sheaves
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1757:is identically one in
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678:Alternative definition
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2675:
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2512:
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2464:
2420:
2343:injective resolutions
2313:
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2273:
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2203:
2183:
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2143:
2123:
2088:
2086:{\displaystyle M_{p}}
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2013:
1993:
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1773:
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1503:analytic continuation
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1456:
1436:
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1376:
1332:
1303:
1270:
1255:: it takes a section
1249:
1228:
1208:
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953:
951:{\displaystyle \{x\}}
927:
907:
905:{\displaystyle \{x\}}
881:
879:{\displaystyle \{x\}}
855:
814:
783:
763:
761:{\displaystyle \{x\}}
737:
717:
697:
668:
639:
637:{\displaystyle f_{V}}
612:
610:{\displaystyle f_{U}}
585:
533:
507:
478:
450:
368:
337:
317:
290:
266:
241:
219:
197:
175:
156:, but the underlying
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2589:
2562:
2521:
2497:
2473:
2432:
2405:
2332:Godement resolutions
2302:
2282:
2262:
2236:
2212:
2192:
2172:
2152:
2132:
2128:and a group or ring
2112:
2070:
2047:
2023:
2002:
1982:
1978:corresponding to an
1957:
1951:quasi-coherent sheaf
1907:
1837:
1786:
1762:
1740:
1718:
1696:
1674:
1652:
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1608:
1586:
1565:
1525:
1465:
1445:
1405:
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1358:
1321:
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1259:
1238:
1217:
1167:
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936:
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827:
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772:
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706:
686:
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621:
594:
548:
516:
490:
467:
380:
346:
326:
302:
275:
251:
230:
208:
186:
164:
132:In mathematics, the
51:improve this article
2876:LEC # 3 - 5 Sheaves
2040:corresponding to a
1156:, which are namely
958:, however, we get:
819:is the same as the
18:Stalk (mathematics)
2872:Algebraic Geometry
2770:Algebraic Geometry
2746:
2670:
2572:
2548:
2507:
2483:
2459:
2415:
2336:algebraic geometry
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2158:
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2118:
2083:
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2029:
2008:
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1967:
1939:
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1491:analytic functions
1471:
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1223:
1203:
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922:
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809:
788:. Then the stalk
778:
758:
732:
712:
692:
663:
634:
607:
580:
542:universal property
540:By definition (or
528:
502:
473:
445:
422:
409:
363:
342:, usually denoted
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312:
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261:
236:
214:
192:
170:
66:"Stalk" sheaf
2401:Given a presheaf
2311:{\displaystyle x}
2291:{\displaystyle x}
2271:{\displaystyle G}
2221:{\displaystyle x}
2201:{\displaystyle G}
2181:{\displaystyle x}
2161:{\displaystyle 0}
2141:{\displaystyle G}
2121:{\displaystyle x}
2056:{\displaystyle p}
2032:{\displaystyle x}
2011:{\displaystyle M}
1991:{\displaystyle A}
1949:, the stalk of a
1771:{\displaystyle U}
1749:{\displaystyle f}
1727:{\displaystyle U}
1713:fits entirely in
1705:{\displaystyle f}
1683:{\displaystyle U}
1661:{\displaystyle f}
1639:{\displaystyle f}
1617:{\displaystyle f}
1595:{\displaystyle f}
1574:{\displaystyle f}
1495:analytic manifold
1474:{\displaystyle X}
1454:{\displaystyle x}
1410:
1394:{\displaystyle S}
1362:
1330:{\displaystyle X}
1268:{\displaystyle s}
1247:{\displaystyle x}
1226:{\displaystyle U}
1213:for any open set
1059:
1057:
1008:
1006:
781:{\displaystyle X}
735:{\displaystyle i}
715:{\displaystyle X}
695:{\displaystyle x}
666:{\displaystyle x}
476:{\displaystyle x}
402:
400:
335:{\displaystyle x}
239:{\displaystyle x}
217:{\displaystyle X}
195:{\displaystyle x}
173:{\displaystyle X}
158:topological space
127:
126:
119:
101:
16:(Redirected from
2903:
2867:
2865:
2855:
2846:
2821:
2792:
2755:
2753:
2752:
2747:
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2744:
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2573:
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2570:
2558:is the image of
2557:
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2516:
2514:
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2508:
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2492:
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2484:
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2481:
2469:, the stalks of
2468:
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2422:
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2255:
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2199:
2187:
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2184:
2179:
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2165:
2164:
2159:
2147:
2145:
2144:
2139:
2127:
2125:
2124:
2119:
2105:associated to a
2103:skyscraper sheaf
2097:Skyscraper sheaf
2092:
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2089:
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2015:
2014:
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1775:
1774:
1769:
1755:
1753:
1752:
1747:
1733:
1731:
1730:
1725:
1711:
1709:
1708:
1703:
1689:
1687:
1686:
1681:
1667:
1665:
1664:
1659:
1645:
1643:
1642:
1637:
1623:
1621:
1620:
1615:
1601:
1599:
1598:
1593:
1580:
1578:
1577:
1572:
1556:
1554:
1553:
1548:
1546:
1538:
1515:smooth functions
1480:
1478:
1477:
1472:
1460:
1458:
1457:
1452:
1440:
1438:
1437:
1432:
1424:
1423:
1418:
1400:
1398:
1397:
1392:
1380:
1378:
1377:
1372:
1370:
1347:Constant sheaves
1336:
1334:
1333:
1328:
1307:
1305:
1304:
1299:
1288:
1287:
1274:
1272:
1271:
1266:
1253:
1251:
1250:
1245:
1232:
1230:
1229:
1224:
1212:
1210:
1209:
1204:
1202:
1201:
1196:
1195:
1176:
1175:
1146:category of sets
1128:
1126:
1125:
1120:
1115:
1114:
1109:
1108:
1089:
1088:
1078:
1067:
1044:
1043:
1033:
1016:
987:
986:
980:
979:
957:
955:
954:
949:
931:
929:
928:
923:
911:
909:
908:
903:
885:
883:
882:
877:
859:
857:
856:
851:
849:
848:
842:
841:
818:
816:
815:
810:
808:
807:
802:
801:
787:
785:
784:
779:
767:
765:
764:
759:
741:
739:
738:
733:
721:
719:
718:
713:
701:
699:
698:
693:
672:
670:
669:
664:
643:
641:
640:
635:
633:
632:
616:
614:
613:
608:
606:
605:
589:
587:
586:
581:
570:
569:
560:
559:
539:
537:
535:
534:
529:
511:
509:
508:
503:
482:
480:
479:
474:
454:
452:
451:
446:
432:
431:
421:
410:
396:
395:
390:
389:
372:
370:
369:
364:
362:
361:
356:
355:
341:
339:
338:
333:
321:
319:
318:
313:
311:
310:
294:
292:
291:
286:
284:
283:
270:
268:
267:
262:
260:
259:
245:
243:
242:
237:
223:
221:
220:
215:
201:
199:
198:
193:
179:
177:
176:
171:
122:
115:
111:
108:
102:
100:
59:
35:
27:
21:
2911:
2910:
2906:
2905:
2904:
2902:
2901:
2900:
2886:
2885:
2863:
2859:Goresky, Mark.
2858:
2849:
2840:
2828:
2818:
2795:
2789:
2766:
2763:
2731:
2705:
2682:
2681:
2637:
2611:
2598:
2587:
2586:
2560:
2559:
2534:
2519:
2518:
2495:
2494:
2471:
2470:
2445:
2430:
2429:
2403:
2402:
2377:In particular:
2351:
2329:
2323:
2300:
2299:
2280:
2279:
2260:
2259:
2234:
2233:
2210:
2209:
2190:
2189:
2170:
2169:
2150:
2149:
2148:has the stalks
2130:
2129:
2110:
2109:
2099:
2073:
2068:
2067:
2045:
2044:
2021:
2020:
2000:
1999:
1980:
1979:
1955:
1954:
1905:
1904:
1898:
1886:Noetherian ring
1862:
1846:
1835:
1834:
1811:
1795:
1784:
1783:
1760:
1759:
1738:
1737:
1716:
1715:
1694:
1693:
1672:
1671:
1650:
1649:
1628:
1627:
1606:
1605:
1584:
1583:
1563:
1562:
1523:
1522:
1519:smooth manifold
1511:
1487:
1463:
1462:
1443:
1442:
1408:
1403:
1402:
1383:
1382:
1356:
1355:
1349:
1344:
1319:
1318:
1277:
1276:
1257:
1256:
1236:
1235:
1215:
1214:
1189:
1165:
1164:
1135:
1102:
968:
963:
962:
934:
933:
914:
913:
888:
887:
862:
861:
830:
825:
824:
795:
790:
789:
770:
769:
744:
743:
724:
723:
704:
703:
684:
683:
680:
655:
654:
644:are considered
624:
619:
618:
597:
592:
591:
551:
546:
545:
514:
513:
488:
487:
485:
465:
464:
383:
378:
377:
349:
344:
343:
324:
323:
300:
299:
273:
272:
249:
248:
228:
227:
206:
205:
184:
183:
162:
161:
150:
130:
123:
112:
106:
103:
60:
58:
48:
36:
23:
22:
15:
12:
11:
5:
2909:
2907:
2899:
2898:
2888:
2887:
2884:
2883:
2868:
2856:
2847:
2838:
2827:
2826:External links
2824:
2823:
2822:
2816:
2793:
2787:
2762:
2759:
2758:
2757:
2741:
2738:
2734:
2730:
2727:
2722:
2715:
2712:
2709:
2704:
2701:
2698:
2695:
2692:
2689:
2669:
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2660:
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2647:
2644:
2640:
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2628:
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2618:
2615:
2610:
2605:
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2597:
2594:
2569:
2545:
2539:
2533:
2528:
2504:
2480:
2456:
2450:
2444:
2439:
2427:sheafification
2412:
2391:
2390:
2375:
2374:
2350:
2347:
2327:
2321:
2307:
2287:
2267:
2247:
2244:
2241:
2217:
2197:
2177:
2157:
2137:
2117:
2098:
2095:
2080:
2076:
2052:
2028:
2007:
1987:
1964:
1938:
1935:
1932:
1928:
1925:
1922:
1919:
1915:
1912:
1897:
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1657:
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1613:
1591:
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1541:
1537:
1533:
1530:
1510:
1507:
1486:
1483:
1470:
1450:
1430:
1427:
1422:
1416:
1413:
1390:
1368:
1365:
1353:constant sheaf
1348:
1345:
1343:
1340:
1326:
1297:
1294:
1291:
1286:
1264:
1243:
1222:
1200:
1194:
1188:
1185:
1182:
1179:
1174:
1150:abelian groups
1134:
1131:
1130:
1129:
1118:
1113:
1107:
1101:
1098:
1095:
1092:
1087:
1082:
1077:
1074:
1071:
1065:
1062:
1056:
1053:
1050:
1047:
1042:
1037:
1032:
1029:
1026:
1023:
1020:
1014:
1011:
1005:
1002:
999:
996:
993:
990:
985:
978:
975:
971:
947:
944:
941:
921:
901:
898:
895:
875:
872:
869:
847:
840:
837:
833:
806:
800:
777:
757:
754:
751:
731:
711:
691:
679:
676:
662:
631:
627:
604:
600:
579:
576:
573:
568:
563:
558:
554:
527:
524:
521:
501:
498:
495:
472:
456:
455:
444:
441:
438:
435:
430:
425:
420:
417:
414:
408:
405:
399:
394:
388:
360:
354:
331:
309:
282:
258:
235:
213:
191:
169:
149:
146:
128:
125:
124:
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2908:
2897:
2894:
2893:
2891:
2881:
2877:
2873:
2869:
2862:
2857:
2853:
2848:
2844:
2843:"6.11 Stalks"
2839:
2837:
2833:
2830:
2829:
2825:
2819:
2817:9780521207843
2813:
2809:
2805:
2801:
2800:
2794:
2790:
2788:9780387902449
2784:
2780:
2776:
2772:
2771:
2765:
2764:
2760:
2739:
2736:
2728:
2696:
2690:
2687:
2664:
2658:
2655:
2645:
2642:
2634:
2608:
2603:
2595:
2585:
2543:
2531:
2454:
2442:
2428:
2400:
2399:
2398:
2396:
2388:
2384:
2380:
2379:
2378:
2372:
2368:
2364:
2360:
2359:
2358:
2356:
2348:
2346:
2344:
2341:
2337:
2333:
2325:
2305:
2285:
2265:
2245:
2239:
2231:
2215:
2195:
2175:
2155:
2135:
2115:
2108:
2104:
2096:
2094:
2078:
2074:
2066:
2050:
2043:
2039:
2026:
2005:
1985:
1977:
1952:
1933:
1913:
1910:
1903:
1902:affine scheme
1895:
1893:
1891:
1887:
1867:
1863:
1858:
1854:
1851:
1847:
1843:
1840:
1816:
1812:
1807:
1803:
1800:
1796:
1792:
1789:
1780:
1778:
1765:
1756:
1743:
1734:
1721:
1712:
1699:
1690:
1677:
1668:
1655:
1646:
1633:
1624:
1611:
1602:
1589:
1568:
1560:
1559:bump function
1531:
1528:
1520:
1516:
1508:
1506:
1504:
1500:
1496:
1492:
1484:
1482:
1468:
1448:
1428:
1425:
1420:
1414:
1411:
1388:
1366:
1363:
1354:
1346:
1341:
1339:
1337:
1324:
1315:
1311:
1292:
1262:
1254:
1241:
1220:
1198:
1180:
1161:
1159:
1155:
1151:
1147:
1143:
1140:
1132:
1116:
1111:
1099:
1093:
1080:
1075:
1072:
1069:
1063:
1060:
1054:
1048:
1035:
1027:
1021:
1018:
1012:
1009:
1003:
994:
976:
973:
969:
961:
960:
959:
942:
896:
870:
838:
835:
831:
822:
821:inverse image
804:
775:
752:
729:
709:
689:
677:
675:
673:
660:
651:
647:
629:
625:
602:
598:
574:
561:
556:
552:
543:
525:
522:
519:
499:
496:
493:
483:
470:
461:
442:
436:
423:
418:
415:
412:
406:
403:
397:
392:
376:
375:
374:
358:
329:
296:
246:
233:
224:
211:
202:
189:
180:
167:
159:
155:
147:
145:
143:
139:
135:
121:
118:
110:
99:
96:
92:
89:
85:
82:
78:
75:
71:
68: –
67:
63:
62:Find sources:
56:
52:
46:
45:
40:This article
38:
34:
29:
28:
19:
2896:Sheaf theory
2799:Sheaf Theory
2798:
2769:
2584:left adjoint
2582:through the
2392:
2376:
2371:monomorphism
2355:gluing axiom
2352:
2345:of sheaves.
2107:closed point
2100:
2065:localization
2063:is just the
2019:
1953:
1899:
1781:
1758:
1736:
1714:
1692:
1670:
1648:
1626:
1604:
1582:
1512:
1499:power series
1488:
1350:
1317:
1309:
1234:
1162:
1141:
1136:
681:
653:
650:neighborhood
463:
460:direct limit
457:
297:
226:
204:
182:
160:
151:
142:mathematical
133:
131:
113:
104:
94:
87:
80:
73:
61:
49:Please help
44:verification
41:
2385:of a given
2367:epimorphism
2363:isomorphism
2042:prime ideal
2018:in a point
1233:containing
2395:presheaves
2340:functorial
2230:skyscraper
1158:cocomplete
1139:categories
722:, and let
646:equivalent
77:newspapers
2761:Reference
2703:→
2653:→
2596:−
2383:exactness
2243:→
1852:−
1801:−
1540:→
1415:_
1367:_
1187:→
1137:For some
1081:
1073:∋
1064:→
1036:
1022:⊇
1013:→
974:−
920:∅
836:−
562:∈
523:⊇
497:≤
458:Here the
424:
416:∋
407:→
154:open sets
107:June 2022
2890:Category
2880:BY-NC-SA
2425:and its
1998:-module
1441:for all
1342:Examples
2387:functor
2338:to get
1888:. The
1308:to its
1133:Remarks
91:scholar
2814:
2785:
1900:On an
1493:on an
823:sheaf
373:, is:
93:
86:
79:
72:
64:
2864:(PDF)
2832:stalk
2369:, or
2324:space
1557:be a
1517:on a
1154:rings
768:into
617:and
512:, if
140:is a
138:sheaf
136:of a
134:stalk
98:JSTOR
84:books
2836:nLab
2812:ISBN
2783:ISBN
2493:and
2258:has
2188:and
2168:off
1351:The
1314:germ
1310:germ
912:and
886:are
70:news
2834:in
2804:doi
2775:doi
2208:on
1461:in
1275:in
1152:or
1061:lim
1010:lim
702:of
652:of
404:lim
322:at
203:of
53:by
2892::
2810:.
2802:.
2781:.
2773:.
2365:,
2093:.
1779:.
1481:.
1338:.
1160:.
674:.
538:).
398::=
2882:.
2874:(
2854:.
2845:.
2820:.
2806::
2791:.
2777::
2740:p
2737:o
2733:)
2729:X
2726:(
2721:O
2714:t
2711:e
2708:S
2700:)
2697:X
2694:(
2691:h
2688:S
2668:)
2665:X
2662:(
2659:h
2656:S
2646:p
2643:o
2639:)
2635:X
2632:(
2627:O
2620:t
2617:e
2614:S
2609::
2604:+
2600:)
2593:(
2568:P
2544:+
2538:P
2532:=
2527:F
2503:F
2479:P
2455:+
2449:P
2443:=
2438:F
2411:P
2328:1
2322:1
2320:T
2306:x
2286:x
2266:G
2246:x
2240:G
2216:x
2196:G
2176:x
2156:0
2136:G
2116:x
2079:p
2075:M
2051:p
2027:x
2006:M
1986:A
1963:F
1937:)
1934:A
1931:(
1927:c
1924:e
1921:p
1918:S
1914:=
1911:X
1868:2
1864:x
1859:/
1855:1
1848:e
1844:+
1841:1
1817:2
1813:x
1808:/
1804:1
1797:e
1793:+
1790:1
1766:U
1744:f
1722:U
1700:f
1678:U
1656:f
1634:f
1612:f
1590:f
1569:f
1544:R
1536:R
1532::
1529:f
1469:X
1449:x
1429:S
1426:=
1421:x
1412:S
1389:S
1364:S
1325:X
1296:)
1293:U
1290:(
1285:F
1263:s
1242:x
1221:U
1199:x
1193:F
1184:)
1181:U
1178:(
1173:F
1142:C
1117:.
1112:x
1106:F
1100:=
1097:)
1094:U
1091:(
1086:F
1076:x
1070:U
1055:=
1052:)
1049:U
1046:(
1041:F
1031:}
1028:x
1025:{
1019:U
1004:=
1001:)
998:}
995:x
992:{
989:(
984:F
977:1
970:i
946:}
943:x
940:{
900:}
897:x
894:{
874:}
871:x
868:{
846:F
839:1
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805:x
799:F
776:X
756:}
753:x
750:{
730:i
710:X
690:x
661:x
630:V
626:f
603:U
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575:U
572:(
567:F
557:U
553:f
526:V
520:U
500:V
494:U
486:(
471:x
443:.
440:)
437:U
434:(
429:F
419:x
413:U
393:x
387:F
359:x
353:F
330:x
308:F
281:F
257:F
234:x
212:X
190:x
168:X
120:)
114:(
109:)
105:(
95:·
88:·
81:·
74:·
47:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
