1834:-sigma-finite case, Pertti Mattila proved in 1986 that the conjecture is false, but his proof did not specify which implication of the conjecture fails. Subsequent work by Jones and Muray produced an example of a set with zero Favard length and positive analytic capacity, explicitly disproving one of the directions of the conjecture. As of 2023 it is not known whether the other implication holds but some progress has been made towards a positive answer by Chang and Tolsa.
261:
1546:
427:
1624:
912:
761:
103:
1033:
1810:
667:
488:
1726:
1435:
545:
304:
1403:
819:
1344:
1065:
597:
1306:
1274:
965:) = 0. However, analytic capacity is a purely complex-analytic concept, and much more work needs to be done in order to obtain a more geometric characterization.
1554:
837:
678:
339:
256:{\displaystyle \gamma (K)=\sup\{|f'(\infty )|;\ f\in {\mathcal {H}}^{\infty }(\mathbf {C} \setminus K),\ \|f\|_{\infty }\leq 1,\ f(\infty )=0\}}
984:
1740:
602:
2092:
2073:
433:
1626:
denotes the orthogonal projection in direction θ. By the results described above, Vitushkin's conjecture is true when dim
1541:{\displaystyle \gamma (K)=0\ \iff \ \int _{0}^{\pi }{\mathcal {H}}^{1}(\operatorname {proj} _{\theta }(K))\,d\theta =0}
2025:
1677:
939:
269:
1883:
1202:
1362:
2111:
1878:
496:
1638:
1662:
proved that analytic capacity is countably semiadditive. That is, there exists an absolute constant
1181:
Given the partial correspondence between the 1-dimensional
Hausdorff measure of a compact subset of
789:
74:
1311:
1038:
946:
is removable. This motivated
Painlevé to pose a more general question in 1880: "Which subsets of
943:
550:
1279:
1247:
311:
1873:
1230:
is the union of 4 squares of side length 1/4 and these squares are located in the corners of
8:
1110:
1092:
2014:
1974:
1936:
2088:
2069:
2021:
1992:
1917:
1843:
1815:
David's and Tolsa's theorems together imply that
Vitushkin's conjecture is true when
1118:
39:
2085:
Analytic
Capacity, the Cauchy Transform, and Non-homogeneous Calderón–Zygmund Theory
1984:
1909:
1852:
307:
36:
17:
1619:{\displaystyle \operatorname {proj} _{\theta }(x,y):=x\cos \theta +y\sin \theta }
1206:
2009:
25:
1201:) = 0. However, this conjecture is false. A counterexample was first given by
907:{\displaystyle \gamma (A)=\sup\{\gamma (K):K\subset A,\,K{\text{ compact}}\}.}
756:{\displaystyle \gamma (K)=\sup \left|{\frac {1}{2\pi }}\int _{C}f(z)dz\right|}
422:{\displaystyle f'(\infty ):=\lim _{z\to \infty }z\left(f(z)-f(\infty )\right)}
2105:
1996:
1921:
330:
94:
32:
1898:"Smooth Maps, Null-Sets for Integralgeometric Measure and Analytic Capacity"
2037:
Analytic
Capacity, Rectifiability, Menger Curvature and the Cauchy Integral
1824:
1659:
934:, every function which is bounded and holomorphic on the set Ω \
70:
1962:
1091:. Its existence can be proved by using a normal family argument involving
1988:
1897:
1028:{\displaystyle f\in {\mathcal {H}}^{\infty }(\mathbf {C} \setminus K)}
59:
1913:
1641:
published a proof in 1998 of
Vitushkin's conjecture for the case dim
1098:
1979:
1805:{\displaystyle \gamma (K)\leq C\sum _{i=1}^{\infty }\gamma (K_{i})}
326:
2053:
G. David, Unrectifiable 1-sets have vanishing analytic capacity,
1937:"Positive analytic capacity but zero Buffon needle probability"
1176:
1846: – in Euclidean space, a measure of that set's "size"
662:{\displaystyle f'(\infty )\neq \lim _{z\to \infty }f'(z)}
2043:
J. Garnett, Positive length but zero analytic capacity,
1185:
and its analytic capacity, it might be conjectured that
917:
672:
Equivalently, the analytic capacity may be defined as
1743:
1680:
1557:
1438:
1429:
be a compact set. Vitushkin's conjecture states that
1365:
1314:
1282:
1250:
1041:
987:
840:
792:
681:
605:
553:
499:
436:
342:
272:
106:
1848:
Pages displaying wikidata descriptions as a fallback
483:{\displaystyle f(\infty ):=\lim _{z\to \infty }f(z)}
62:of the space of bounded analytic functions outside
2013:
1804:
1720:
1618:
1540:
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1059:
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906:
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661:
591:
539:
482:
421:
298:
255:
2016:Geometry of sets and measures in Euclidean spaces
1099:Analytic capacity in terms of Hausdorff dimension
2103:
2039:. Lecture Notes in Mathematics. Springer-Verlag.
981:, there exists a unique extremal function, i.e.
856:
697:
627:
453:
364:
122:
73:in the 1940s while studying the removability of
1721:{\displaystyle K=\bigcup _{i=1}^{\infty }K_{i}}
1209:in his 1970 paper. This latter example is the
97:. Then its analytic capacity is defined to be
2087:. Progress in Mathematics. Birkhäuser Basel.
940:Riemann's theorem for removable singularities
1967:Journal of the European Mathematical Society
1048:
1042:
898:
859:
250:
211:
204:
125:
1871:
299:{\displaystyle {\mathcal {H}}^{\infty }(U)}
1960:
1464:
1460:
1177:Positive length but zero analytic capacity
938:has an analytic extension to all of Ω. By
2066:Vitushkin's Conjecture for Removable Sets
1978:
1961:Chang, Alan; Tolsa, Xavier (2020-10-05).
1525:
1416:
930:if, whenever Ω is an open set containing
889:
778:satisfying the same conditions as above:
2063:
2008:
1895:
1223: := × be the unit square. Then,
2104:
1934:
1244:is the union of 4 squares (denoted by
2082:
2034:
918:Removable sets and Painlevé's problem
35:is a number that denotes "how big" a
1398:{\displaystyle H^{1}(K)={\sqrt {2}}}
1155:) > 0. However, the case when dim
831:is an arbitrary set, then we define
1963:"Analytic capacity and projections"
968:
13:
1935:Jones, Peter W.; Murai, Takafumi.
1778:
1703:
1486:
1003:
997:
799:
637:
617:
511:
463:
443:
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374:
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282:
276:
238:
215:
176:
170:
144:
16:In the mathematical discipline of
14:
2123:
2068:. Universitext. Springer-Verlag.
1016:
540:{\displaystyle f'(\infty )=g'(0)}
189:
1012:
774:and the supremum is taken over
185:
1083:). This function is called the
77:of bounded analytic functions.
2020:. Cambridge University Press.
1954:
1944:Pacific Journal of Mathematics
1928:
1889:
1865:
1799:
1786:
1753:
1747:
1583:
1571:
1522:
1519:
1513:
1497:
1461:
1448:
1442:
1382:
1376:
1350:to be the intersection of all
1211:linear four corners Cantor set
1173:) ∈ (0, ∞] is more difficult.
1022:
1008:
871:
865:
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844:
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50:can become. Roughly speaking,
1:
1858:
814:{\displaystyle f(\infty )=0.}
80:
1872:Solomentsev, E. D. (2001) ,
1308:being in the corner of some
1205:, and a much simpler one by
957:is removable if and only if
782:is bounded analytic outside
7:
1879:Encyclopedia of Mathematics
1837:
1339:{\displaystyle Q_{n-1}^{k}}
1060:{\displaystyle \|f\|\leq 1}
592:{\displaystyle g(z)=f(1/z)}
69:It was first introduced by
58:) measures the size of the
10:
2128:
2064:Dudziak, James J. (2010).
1213:, constructed as follows:
1301:{\displaystyle Q_{n}^{j}}
1276:) of side length 4, each
1269:{\displaystyle Q_{n}^{j}}
1896:Mattila, Pertti (1986).
786:, the bound is one, and
953:It is easy to see that
770:is a contour enclosing
2083:Tolsa, Xavier (2014).
2045:Proc. Amer. Math. Soc.
1806:
1782:
1722:
1707:
1620:
1542:
1417:Vitushkin's conjecture
1399:
1340:
1302:
1270:
1061:
1029:
908:
815:
757:
663:
593:
541:
484:
423:
300:
257:
2035:Pajot, Hervé (2002).
1902:Annals of Mathematics
1807:
1762:
1737:is a Borel set, then
1723:
1687:
1674:is a compact set and
1621:
1543:
1400:
1341:
1303:
1271:
1117:denote 1-dimensional
1062:
1030:
909:
816:
758:
664:
594:
542:
485:
424:
301:
258:
1741:
1678:
1666:> 0 such that if
1555:
1436:
1363:
1312:
1280:
1248:
1147:) > 1 guarantees
1039:
985:
838:
790:
679:
603:
551:
497:
434:
340:
270:
104:
2055:Rev. Math. Iberoam.
1658:) < ∞. In 2002,
1482:
1335:
1297:
1265:
1111:Hausdorff dimension
599:. However, usually
306:denotes the set of
2112:Analytic functions
1802:
1718:
1616:
1538:
1468:
1395:
1336:
1315:
1298:
1283:
1266:
1251:
1057:
1025:
904:
811:
753:
659:
641:
589:
537:
480:
467:
419:
378:
296:
253:
2094:978-3-319-00595-9
1989:10.4171/JEMS/1004
1973:(12): 4121–4159.
1844:Capacity of a set
1467:
1459:
1393:
1119:Hausdorff measure
973:For each compact
896:
718:
626:
452:
363:
231:
203:
160:
40:analytic function
22:analytic capacity
2119:
2098:
2079:
2075:978-14419-6708-4
2040:
2031:
2019:
2001:
2000:
1982:
1958:
1952:
1951:
1941:
1932:
1926:
1925:
1893:
1887:
1886:
1869:
1853:Conformal radius
1849:
1811:
1809:
1808:
1803:
1798:
1797:
1781:
1776:
1727:
1725:
1724:
1719:
1717:
1716:
1706:
1701:
1625:
1623:
1622:
1617:
1567:
1566:
1547:
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1544:
1539:
1509:
1508:
1496:
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1489:
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1476:
1465:
1457:
1404:
1402:
1401:
1396:
1394:
1389:
1375:
1374:
1345:
1343:
1342:
1337:
1334:
1329:
1307:
1305:
1304:
1299:
1296:
1291:
1275:
1273:
1272:
1267:
1264:
1259:
1093:Montel's theorem
1085:Ahlfors function
1066:
1064:
1063:
1058:
1034:
1032:
1031:
1026:
1015:
1007:
1006:
1001:
1000:
969:Ahlfors function
950:are removable?"
922:The compact set
913:
911:
910:
905:
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894:
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582:
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140:
132:
18:complex analysis
2127:
2126:
2122:
2121:
2120:
2118:
2117:
2116:
2102:
2101:
2095:
2076:
2050:(1970), 696–699
2028:
2010:Mattila, Pertti
2005:
2004:
1959:
1955:
1939:
1933:
1929:
1914:10.2307/1971273
1894:
1890:
1870:
1866:
1861:
1847:
1840:
1793:
1789:
1777:
1766:
1742:
1739:
1738:
1736:
1712:
1708:
1702:
1691:
1679:
1676:
1675:
1646:
1631:
1562:
1558:
1556:
1553:
1552:
1504:
1500:
1491:
1485:
1484:
1483:
1477:
1472:
1437:
1434:
1433:
1419:
1388:
1370:
1366:
1364:
1361:
1360:
1358:
1330:
1319:
1313:
1310:
1309:
1292:
1287:
1281:
1278:
1277:
1260:
1255:
1249:
1246:
1245:
1242:
1236:
1229:
1222:
1207:John B. Garnett
1203:A. G. Vitushkin
1179:
1160:
1142:
1137:) = 0 while dim
1108:
1101:
1040:
1037:
1036:
1011:
1002:
996:
995:
994:
986:
983:
982:
971:
920:
893:
839:
836:
835:
791:
788:
787:
724:
720:
710:
705:
704:
700:
680:
677:
676:
642:
630:
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604:
601:
600:
578:
552:
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520:
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494:
456:
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432:
431:
386:
382:
367:
343:
341:
338:
337:
281:
275:
274:
273:
271:
268:
267:
214:
210:
184:
175:
169:
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167:
150:
133:
128:
105:
102:
101:
83:
12:
11:
5:
2125:
2115:
2114:
2100:
2099:
2093:
2080:
2074:
2061:
2060:(1998) 269–479
2051:
2041:
2032:
2026:
2003:
2002:
1953:
1927:
1908:(2): 303–309.
1888:
1863:
1862:
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1801:
1796:
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1642:
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1597:
1594:
1591:
1588:
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1579:
1576:
1573:
1570:
1565:
1561:
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1537:
1534:
1531:
1528:
1524:
1521:
1518:
1515:
1512:
1507:
1503:
1499:
1494:
1488:
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1475:
1471:
1463:
1456:
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1450:
1447:
1444:
1441:
1418:
1415:
1392:
1387:
1384:
1381:
1378:
1373:
1369:
1354:
1333:
1328:
1325:
1322:
1318:
1295:
1290:
1286:
1263:
1258:
1254:
1240:
1237:. In general,
1234:
1227:
1220:
1193:) = 0 implies
1178:
1175:
1156:
1138:
1129:) = 0 implies
1104:
1100:
1097:
1056:
1053:
1050:
1047:
1044:
1024:
1021:
1018:
1014:
1010:
1005:
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993:
990:
970:
967:
919:
916:
915:
914:
903:
900:
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888:
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870:
867:
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861:
858:
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849:
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843:
810:
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801:
798:
795:
764:
763:
751:
747:
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741:
738:
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732:
727:
723:
716:
713:
709:
703:
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696:
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687:
684:
658:
655:
652:
648:
645:
639:
636:
633:
629:
625:
622:
619:
616:
612:
609:
588:
585:
581:
577:
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571:
568:
565:
562:
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530:
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389:
385:
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359:
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353:
349:
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329:subset of the
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139:
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131:
127:
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121:
118:
115:
112:
109:
82:
79:
26:compact subset
9:
6:
4:
3:
2:
2124:
2113:
2110:
2109:
2107:
2096:
2090:
2086:
2081:
2077:
2071:
2067:
2062:
2059:
2056:
2052:
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2046:
2042:
2038:
2033:
2029:
2027:0-521-65595-1
2023:
2018:
2017:
2011:
2007:
2006:
1998:
1994:
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1968:
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1931:
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1790:
1783:
1773:
1770:
1767:
1763:
1759:
1756:
1750:
1744:
1735:
1731:
1728:, where each
1713:
1709:
1698:
1695:
1692:
1688:
1684:
1681:
1673:
1669:
1665:
1661:
1657:
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1630:
1613:
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1607:
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1592:
1589:
1586:
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1568:
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1535:
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1529:
1526:
1516:
1510:
1505:
1501:
1492:
1478:
1473:
1469:
1454:
1451:
1445:
1439:
1432:
1431:
1430:
1428:
1424:
1414:
1412:
1408:
1390:
1385:
1379:
1371:
1367:
1357:
1353:
1349:
1331:
1326:
1323:
1320:
1316:
1293:
1288:
1284:
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1256:
1252:
1243:
1233:
1226:
1219:
1214:
1212:
1208:
1204:
1200:
1196:
1192:
1188:
1184:
1174:
1172:
1168:
1164:
1159:
1154:
1150:
1146:
1141:
1136:
1132:
1128:
1124:
1120:
1116:
1112:
1107:
1096:
1094:
1090:
1086:
1082:
1078:
1074:
1070:
1054:
1051:
1045:
1019:
991:
988:
980:
976:
966:
964:
960:
956:
951:
949:
945:
941:
937:
933:
929:
925:
901:
895: compact
890:
886:
883:
880:
877:
874:
868:
862:
853:
847:
841:
834:
833:
832:
830:
826:
821:
808:
805:
793:
785:
781:
777:
773:
769:
749:
745:
742:
736:
730:
725:
721:
714:
711:
707:
701:
694:
688:
682:
675:
674:
673:
670:
653:
646:
643:
631:
623:
610:
607:
583:
579:
575:
569:
566:
560:
554:
531:
524:
521:
517:
504:
501:
474:
468:
457:
449:
437:
430:
415:
402:
399:
393:
387:
383:
379:
368:
360:
347:
344:
336:
335:
334:
332:
331:complex plane
328:
324:
320:
316:
313:
309:
290:
247:
244:
232:
226:
223:
220:
207:
198:
192:
164:
161:
155:
137:
134:
119:
113:
107:
100:
99:
98:
96:
92:
88:
78:
76:
75:singularities
72:
67:
65:
61:
57:
53:
49:
46: \
45:
41:
38:
34:
33:complex plane
30:
27:
23:
19:
2084:
2065:
2057:
2054:
2047:
2044:
2036:
2015:
1970:
1966:
1956:
1950:(1): 99–114.
1947:
1943:
1930:
1905:
1901:
1891:
1877:
1867:
1831:
1829:
1825:sigma-finite
1820:
1816:
1814:
1733:
1729:
1671:
1667:
1663:
1660:Xavier Tolsa
1655:
1651:
1647:
1643:
1637:
1632:
1628:
1550:
1426:
1422:
1420:
1410:
1406:
1355:
1351:
1347:
1238:
1231:
1224:
1217:
1215:
1210:
1198:
1194:
1190:
1186:
1182:
1180:
1170:
1166:
1162:
1157:
1152:
1148:
1144:
1139:
1134:
1130:
1126:
1122:
1114:
1105:
1102:
1088:
1084:
1080:
1076:
1072:
1071:(∞) = 0 and
1068:
978:
974:
972:
962:
958:
954:
952:
947:
935:
931:
927:
923:
921:
828:
824:
822:
783:
779:
775:
771:
767:
765:
671:
492:
322:
318:
314:
265:
90:
86:
84:
71:Lars Ahlfors
68:
63:
55:
51:
47:
43:
28:
21:
15:
1830:In the non
333:. Further,
321:, whenever
1980:1712.00594
1874:"Capacity"
1859:References
1165:) = 1 and
1035:such that
926:is called
493:Note that
81:Definition
1997:1435-9855
1922:0003-486X
1884:EMS Press
1784:γ
1779:∞
1764:∑
1757:≤
1745:γ
1704:∞
1689:⋃
1639:Guy David
1614:θ
1611:
1599:θ
1596:
1569:
1564:θ
1530:θ
1511:
1506:θ
1479:π
1470:∫
1462:⟺
1440:γ
1324:−
1052:≤
1049:‖
1043:‖
1017:∖
1004:∞
992:∈
944:singleton
928:removable
881:⊂
863:γ
842:γ
800:∞
722:∫
715:π
683:γ
638:∞
635:→
624:≠
618:∞
512:∞
464:∞
461:→
444:∞
409:∞
400:−
375:∞
372:→
355:∞
312:functions
310:analytic
283:∞
239:∞
221:≤
216:∞
212:‖
205:‖
190:∖
177:∞
165:∈
145:∞
108:γ
60:unit ball
2106:Category
2012:(1995).
1838:See also
1650:= 1 and
942:, every
647:′
611:′
547:, where
525:′
505:′
348:′
138:′
1413:) = 0.
1346:. Take
1121:. Then
1109:denote
1103:Let dim
308:bounded
95:compact
37:bounded
31:of the
2091:
2072:
2024:
1995:
1920:
1551:where
1466:
1458:
1075:(∞) =
766:where
325:is an
266:Here,
230:
202:
159:
20:, the
1975:arXiv
1940:(PDF)
1635:≠ 1.
1359:then
24:of a
2089:ISBN
2070:ISBN
2022:ISBN
1993:ISSN
1918:ISSN
1560:proj
1502:proj
1421:Let
1405:but
1216:Let
1113:and
327:open
85:Let
1985:doi
1948:133
1910:doi
1906:123
1819:is
1608:sin
1593:cos
1087:of
857:sup
823:If
698:sup
628:lim
454:lim
365:lim
123:sup
93:be
42:on
2108::
2058:14
2048:21
1991:.
1983:.
1971:22
1969:.
1965:.
1946:.
1942:.
1916:.
1904:.
1900:.
1882:,
1876:,
1827:.
1812:.
1670:⊂
1587::=
1425:⊂
1095:.
1073:f′
1067:,
977:⊂
827:⊂
809:0.
669:.
450::=
361::=
317:→
89:⊂
66:.
2097:.
2078:.
2030:.
1999:.
1987::
1977::
1924:.
1912::
1832:H
1823:-
1821:H
1817:K
1800:)
1795:i
1791:K
1787:(
1774:1
1771:=
1768:i
1760:C
1754:)
1751:K
1748:(
1734:i
1730:K
1714:i
1710:K
1699:1
1696:=
1693:i
1685:=
1682:K
1672:C
1668:K
1664:C
1656:K
1654:(
1652:H
1648:K
1644:H
1633:K
1629:H
1605:y
1602:+
1590:x
1584:)
1581:y
1578:,
1575:x
1572:(
1536:0
1533:=
1527:d
1523:)
1520:)
1517:K
1514:(
1498:(
1493:1
1487:H
1474:0
1455:0
1452:=
1449:)
1446:K
1443:(
1427:C
1423:K
1411:K
1409:(
1407:γ
1391:2
1386:=
1383:)
1380:K
1377:(
1372:1
1368:H
1356:n
1352:K
1348:K
1332:k
1327:1
1321:n
1317:Q
1294:j
1289:n
1285:Q
1262:j
1257:n
1253:Q
1241:n
1239:K
1235:0
1232:K
1228:1
1225:K
1221:0
1218:K
1199:K
1197:(
1195:H
1191:K
1189:(
1187:γ
1183:C
1171:K
1169:(
1167:H
1163:K
1161:(
1158:H
1153:K
1151:(
1149:γ
1145:K
1143:(
1140:H
1135:K
1133:(
1131:γ
1127:K
1125:(
1123:H
1115:H
1106:H
1089:K
1081:K
1079:(
1077:γ
1069:f
1055:1
1046:f
1023:)
1020:K
1013:C
1009:(
998:H
989:f
979:C
975:K
963:K
961:(
959:γ
955:K
948:C
936:K
932:K
924:K
902:.
899:}
891:K
887:,
884:A
878:K
875::
872:)
869:K
866:(
860:{
854:=
851:)
848:A
845:(
829:C
825:A
806:=
803:)
797:(
794:f
784:K
780:f
776:f
772:K
768:C
750:|
746:z
743:d
740:)
737:z
734:(
731:f
726:C
712:2
708:1
702:|
695:=
692:)
689:K
686:(
657:)
654:z
651:(
644:f
632:z
621:)
615:(
608:f
587:)
584:z
580:/
576:1
573:(
570:f
567:=
564:)
561:z
558:(
555:g
535:)
532:0
529:(
522:g
518:=
515:)
509:(
502:f
478:)
475:z
472:(
469:f
458:z
447:)
441:(
438:f
416:)
412:)
406:(
403:f
397:)
394:z
391:(
388:f
384:(
380:z
369:z
358:)
352:(
345:f
323:U
319:C
315:U
294:)
291:U
288:(
277:H
251:}
248:0
245:=
242:)
236:(
233:f
227:,
224:1
208:f
199:,
196:)
193:K
186:C
182:(
171:H
162:f
156:;
152:|
148:)
142:(
135:f
130:|
126:{
120:=
117:)
114:K
111:(
91:C
87:K
64:K
56:K
54:(
52:γ
48:K
44:C
29:K
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