20:
846:
512:
87:) and dubbed by G. Ewald, D. G. Larman and C. A. Rogers in 1970. Macbeath regions have been used to solve certain complex problems in the study of the boundaries of convex bodies. Recently they have been used in the study of convex approximations and other aspects of
304:
684:
1017:
1761:
2144:
628:
319:
1448:
2022:
1529:
1594:
1217:
1075:
1641:
901:
675:
125:
1342:
2200:
Ewald, G.; Larman, D. G.; Rogers, C. A. (June 1970). "The directions of the line segments and of the r-dimensional balls on the boundary of a convex body in
Euclidean space".
1871:
1816:
77:
1964:
1297:
2081:
1703:
1670:
1474:
1261:
1166:
1375:
2044:
556:
1911:
943:
1139:
841:{\displaystyle B_{H}\left(x,{\frac {1}{2}}\ln(1+\lambda )\right)\subset M^{\lambda }(x)\subset B_{H}\left(x,{\frac {1}{2}}\ln {\frac {1+\lambda }{1-\lambda }}\right)}
1891:
1941:
1109:
948:
1708:
2086:
507:{\displaystyle M_{K}^{\lambda }(x)=x+\lambda ((K-x)\cap (x-K))=\{(1-\lambda )x+\lambda k'|k'\in K,\exists k\in K{\text{ and }}k'-x=x-k\}}
565:
2305:
Arya, Sunil; da
Fonseca, Guilherme D.; Mount, David M. (December 2017). "On the Combinatorial Complexity of Approximating Polytopes".
1392:
1969:
2352:
Vernicos, Constantin; Walsh, Cormac (2021). "Flag-approximability of convex bodies and volume growth of
Hilbert geometries".
2407:
2307:
1479:
23:
The
Macbeath region around a point x in a convex body K and the scaled Macbeath region around a point x in a convex body K
2455:
1553:
1171:
1022:
299:{\displaystyle {M_{K}}(x)=K\cap (2x-K)=x+((K-x)\cap (x-K))=\{k'\in K|\exists k\in K{\text{ and }}k'-x=x-k\}}
865:
648:
1599:
1302:
1821:
1766:
53:
1946:
1270:
80:
2053:
1675:
1649:
1453:
1233:
1144:
2450:
2445:
1354:
1077:. Essentially if two Macbeath regions intersect, you can scale one of them up to contain the other.
2029:
541:
88:
1896:
916:
1118:
2403:"Shallow Packings, Semialgebraic Set Systems, Macbeath Regions, and Polynomial Partitioning"
1876:
1919:
1087:
8:
2424:
2379:
2361:
2334:
2316:
2182:
2402:
2428:
2383:
2416:
2371:
2338:
2326:
2274:
2234:
2211:
2174:
559:
108:
48:
36:
2420:
2279:
2262:
634:
2330:
2215:
2439:
2237:(June 8, 2001). "The techhnique of M-regions and cap-coverings: a survey".
2165:
Macbeath, A. M. (September 1952). "A Theorem on Non-Homogeneous
Lattices".
1012:{\displaystyle M^{\frac {1}{2}}(x)\cap M^{\frac {1}{2}}(y)\neq \emptyset }
2267:
16th
Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)
2202:
28:
19:
2375:
2186:
907:
104:
40:
1756:{\displaystyle O\left({\frac {1}{\epsilon ^{\frac {d-1}{2}}}}\right)}
2401:
Dutta, Kunal; Ghosh, Arijit; Jartoux, Bruno; Mustafa, Nabil (2019).
2178:
2366:
2321:
2139:{\displaystyle C_{i}^{\frac {1}{\beta ^{2}}}\subset C\subset C_{i}}
623:{\displaystyle O(\log ^{\frac {d+1}{2}}({\frac {1}{\epsilon }}))}
115:
and a scaler λ the λ-scaled the
Macbeath region around a point
1223:, the center of gravity of K in the bounding hyper-plane of
1443:{\displaystyle M^{\lambda }(x)\cap K\subset C^{1+\lambda }}
2017:{\displaystyle R_{i}\subset C_{i}\subset R_{i}^{\lambda }}
1115:, with the half-space disjoint from the ball, and the cap
633:
Macbeath regions can be used to approximate balls in the
2400:
2263:"Economical Delone Sets for Approximating Convex Bodies"
2089:
2056:
2032:
1972:
1949:
1922:
1899:
1879:
1824:
1769:
1711:
1678:
1652:
1602:
1556:
1524:{\displaystyle M^{\lambda }(x)\subset C^{1+\lambda }}
1482:
1456:
1395:
1357:
1305:
1273:
1236:
1174:
1147:
1121:
1090:
1025:
951:
919:
868:
687:
651:
568:
544:
322:
128:
56:
1141:
of our convex set has a width less than or equal to
1111:
containing both a ball of radius r and a half-space
2354:
2304:
1705:in canonical form, there exists some collection of
2138:
2075:
2038:
2016:
1958:
1935:
1905:
1885:
1865:
1810:
1755:
1697:
1664:
1635:
1588:
1523:
1468:
1442:
1369:
1336:
1291:
1255:
1211:
1160:
1133:
1103:
1069:
1011:
937:
895:
840:
669:
622:
550:
506:
298:
71:
2199:
1344:, where x is the centroid of the base of the cap.
2437:
2260:
2351:
630:combinatorial complexity of the lower bound.
501:
401:
293:
228:
2261:Abdelkader, Ahmed; Mount, David M. (2018).
1763:centrally symmetric disjoint convex bodies
517:This can be seen to be the intersection of
1589:{\displaystyle C\cap M'(x)\neq \emptyset }
2365:
2320:
2278:
59:
2164:
1212:{\displaystyle K\cap H\subset M^{3d(x)}}
1070:{\displaystyle M^{1}(y)\subset M^{5}(x)}
84:
18:
538:Macbeath regions can be used to create
2438:
2233:
562:, of convex shapes within a factor of
2408:Discrete & Computational Geometry
2308:Discrete & Computational Geometry
2300:
2298:
2296:
2294:
2292:
2290:
16:Brief description on Macbeath Regions
2256:
2254:
2252:
2229:
2227:
2225:
558:approximations, with respect to the
1263:in canonical form, then any cap of
896:{\displaystyle M_{K}^{\lambda }(x)}
670:{\displaystyle 0\leq \lambda <1}
35:is an explicitly defined region in
13:
2394:
2287:
1636:{\displaystyle M'(x)\subset C^{2}}
1583:
1337:{\displaystyle C\subset M^{3d}(x)}
1006:
458:
250:
14:
2467:
2249:
2222:
1866:{\displaystyle C_{1},....,C_{k}}
1811:{\displaystyle R_{1},....,R_{k}}
903:is centrally symmetric around x.
72:{\displaystyle \mathbb {R} ^{d}}
1959:{\displaystyle \beta \epsilon }
1292:{\displaystyle {\frac {1}{6d}}}
532:
2345:
2193:
2158:
2076:{\displaystyle R_{i}\subset C}
1698:{\displaystyle K\subset R^{d}}
1665:{\displaystyle \epsilon >0}
1617:
1611:
1577:
1571:
1499:
1493:
1469:{\displaystyle \lambda \leq 1}
1412:
1406:
1331:
1325:
1256:{\displaystyle K\subset R^{d}}
1204:
1198:
1161:{\displaystyle {\frac {r}{2}}}
1064:
1058:
1042:
1036:
1000:
994:
973:
967:
890:
884:
764:
758:
737:
725:
617:
614:
601:
572:
437:
416:
404:
395:
392:
380:
374:
362:
359:
344:
338:
309:The scaled Macbeath region at
246:
222:
219:
207:
201:
189:
186:
174:
159:
147:
141:
1:
2151:
1370:{\displaystyle \lambda >0}
856:
94:
79:. The idea was introduced by
1873:such that for some constant
7:
10:
2472:
2421:10.1007/s00454-019-00075-0
2280:10.4230/LIPIcs.SWAT.2018.4
2331:10.1007/s00454-016-9856-5
2216:10.1112/S0025579300002655
2167:The Annals of Mathematics
2039:{\displaystyle \epsilon }
2026:If C is any cap of width
551:{\displaystyle \epsilon }
1913:depending on d we have:
1906:{\displaystyle \lambda }
938:{\displaystyle x,y\in K}
637:, e.g. given any convex
1134:{\displaystyle K\cap H}
521:with the reflection of
2456:Computational geometry
2140:
2077:
2040:
2018:
1960:
1937:
1907:
1887:
1886:{\displaystyle \beta }
1867:
1812:
1757:
1699:
1666:
1637:
1590:
1525:
1470:
1444:
1371:
1338:
1293:
1257:
1213:
1162:
1135:
1105:
1071:
1013:
939:
897:
842:
671:
624:
552:
508:
300:
89:computational geometry
81:Alexander Macbeath
73:
24:
2239:Rendiconti di Palermo
2141:
2078:
2041:
2019:
1961:
1938:
1936:{\displaystyle C_{i}}
1908:
1888:
1868:
1813:
1758:
1700:
1667:
1638:
1591:
1526:
1471:
1450:. In particular when
1445:
1377:, then for any point
1372:
1339:
1294:
1258:
1214:
1163:
1136:
1106:
1104:{\displaystyle R^{d}}
1072:
1014:
940:
906:Macbeath regions are
898:
843:
672:
625:
553:
509:
301:
74:
22:
2087:
2054:
2046:there must exist an
2030:
1970:
1947:
1920:
1897:
1877:
1822:
1767:
1709:
1676:
1650:
1600:
1554:
1534:Given a convex body
1480:
1454:
1393:
1355:
1303:
1271:
1234:
1230:Given a convex body
1172:
1145:
1119:
1088:
1023:
949:
917:
866:
685:
649:
566:
542:
320:
126:
54:
2376:10.24033/asens.2482
2116:
2013:
1267:with width at most
883:
337:
2136:
2090:
2073:
2036:
2014:
1999:
1956:
1933:
1903:
1883:
1863:
1808:
1753:
1695:
1662:
1633:
1586:
1521:
1466:
1440:
1367:
1351:and some constant
1334:
1289:
1253:
1209:
1158:
1131:
1101:
1067:
1009:
935:
893:
869:
838:
667:
620:
560:Hausdorff distance
548:
504:
323:
296:
69:
25:
2114:
1747:
1744:
1287:
1156:
991:
964:
831:
799:
717:
612:
595:
473:
265:
2463:
2432:
2388:
2387:
2369:
2360:(5): 1297–1314.
2349:
2343:
2342:
2324:
2302:
2285:
2284:
2282:
2258:
2247:
2246:
2231:
2220:
2219:
2197:
2191:
2190:
2162:
2145:
2143:
2142:
2137:
2135:
2134:
2115:
2113:
2112:
2100:
2098:
2082:
2080:
2079:
2074:
2066:
2065:
2045:
2043:
2042:
2037:
2023:
2021:
2020:
2015:
2012:
2007:
1995:
1994:
1982:
1981:
1965:
1963:
1962:
1957:
1942:
1940:
1939:
1934:
1932:
1931:
1912:
1910:
1909:
1904:
1892:
1890:
1889:
1884:
1872:
1870:
1869:
1864:
1862:
1861:
1834:
1833:
1817:
1815:
1814:
1809:
1807:
1806:
1779:
1778:
1762:
1760:
1759:
1754:
1752:
1748:
1746:
1745:
1740:
1729:
1720:
1704:
1702:
1701:
1696:
1694:
1693:
1671:
1669:
1668:
1663:
1642:
1640:
1639:
1634:
1632:
1631:
1610:
1595:
1593:
1592:
1587:
1570:
1530:
1528:
1527:
1522:
1520:
1519:
1492:
1491:
1475:
1473:
1472:
1467:
1449:
1447:
1446:
1441:
1439:
1438:
1405:
1404:
1376:
1374:
1373:
1368:
1343:
1341:
1340:
1335:
1324:
1323:
1298:
1296:
1295:
1290:
1288:
1286:
1275:
1262:
1260:
1259:
1254:
1252:
1251:
1218:
1216:
1215:
1210:
1208:
1207:
1167:
1165:
1164:
1159:
1157:
1149:
1140:
1138:
1137:
1132:
1110:
1108:
1107:
1102:
1100:
1099:
1076:
1074:
1073:
1068:
1057:
1056:
1035:
1034:
1018:
1016:
1015:
1010:
993:
992:
984:
966:
965:
957:
944:
942:
941:
936:
902:
900:
899:
894:
882:
877:
847:
845:
844:
839:
837:
833:
832:
830:
819:
808:
800:
792:
779:
778:
757:
756:
744:
740:
718:
710:
697:
696:
676:
674:
673:
668:
641:, containing an
629:
627:
626:
621:
613:
605:
597:
596:
591:
580:
557:
555:
554:
549:
513:
511:
510:
505:
482:
474:
471:
448:
440:
435:
336:
331:
305:
303:
302:
297:
274:
266:
263:
249:
238:
140:
139:
138:
111:. Given a point
78:
76:
75:
70:
68:
67:
62:
2471:
2470:
2466:
2465:
2464:
2462:
2461:
2460:
2451:Convex analysis
2446:Metric geometry
2436:
2435:
2397:
2395:Further reading
2392:
2391:
2350:
2346:
2303:
2288:
2259:
2250:
2232:
2223:
2198:
2194:
2179:10.2307/1969800
2163:
2159:
2154:
2130:
2126:
2108:
2104:
2099:
2094:
2088:
2085:
2084:
2061:
2057:
2055:
2052:
2051:
2031:
2028:
2027:
2008:
2003:
1990:
1986:
1977:
1973:
1971:
1968:
1967:
1948:
1945:
1944:
1927:
1923:
1921:
1918:
1917:
1898:
1895:
1894:
1878:
1875:
1874:
1857:
1853:
1829:
1825:
1823:
1820:
1819:
1802:
1798:
1774:
1770:
1768:
1765:
1764:
1730:
1728:
1724:
1719:
1715:
1710:
1707:
1706:
1689:
1685:
1677:
1674:
1673:
1651:
1648:
1647:
1627:
1623:
1603:
1601:
1598:
1597:
1563:
1555:
1552:
1551:
1509:
1505:
1487:
1483:
1481:
1478:
1477:
1455:
1452:
1451:
1428:
1424:
1400:
1396:
1394:
1391:
1390:
1356:
1353:
1352:
1347:Given a convex
1316:
1312:
1304:
1301:
1300:
1279:
1274:
1272:
1269:
1268:
1247:
1243:
1235:
1232:
1231:
1191:
1187:
1173:
1170:
1169:
1148:
1146:
1143:
1142:
1120:
1117:
1116:
1095:
1091:
1089:
1086:
1085:
1080:If some convex
1052:
1048:
1030:
1026:
1024:
1021:
1020:
983:
979:
956:
952:
950:
947:
946:
918:
915:
914:
878:
873:
867:
864:
863:
859:
820:
809:
807:
791:
784:
780:
774:
770:
752:
748:
709:
702:
698:
692:
688:
686:
683:
682:
650:
647:
646:
604:
581:
579:
575:
567:
564:
563:
543:
540:
539:
535:
475:
472: and
470:
441:
436:
428:
332:
327:
321:
318:
317:
313:is defined as:
267:
264: and
262:
245:
231:
134:
130:
129:
127:
124:
123:
109:Euclidean space
97:
63:
58:
57:
55:
52:
51:
49:Euclidean space
37:convex analysis
33:Macbeath region
17:
12:
11:
5:
2469:
2459:
2458:
2453:
2448:
2434:
2433:
2415:(4): 756–777.
2396:
2393:
2390:
2389:
2344:
2315:(4): 849–870.
2286:
2248:
2221:
2192:
2173:(2): 269–293.
2156:
2155:
2153:
2150:
2149:
2148:
2147:
2146:
2133:
2129:
2125:
2122:
2119:
2111:
2107:
2103:
2097:
2093:
2072:
2069:
2064:
2060:
2035:
2024:
2011:
2006:
2002:
1998:
1993:
1989:
1985:
1980:
1976:
1955:
1952:
1930:
1926:
1902:
1882:
1860:
1856:
1852:
1849:
1846:
1843:
1840:
1837:
1832:
1828:
1805:
1801:
1797:
1794:
1791:
1788:
1785:
1782:
1777:
1773:
1751:
1743:
1739:
1736:
1733:
1727:
1723:
1718:
1714:
1692:
1688:
1684:
1681:
1661:
1658:
1655:
1646:Given a small
1644:
1630:
1626:
1622:
1619:
1616:
1613:
1609:
1606:
1585:
1582:
1579:
1576:
1573:
1569:
1566:
1562:
1559:
1532:
1518:
1515:
1512:
1508:
1504:
1501:
1498:
1495:
1490:
1486:
1465:
1462:
1459:
1437:
1434:
1431:
1427:
1423:
1420:
1417:
1414:
1411:
1408:
1403:
1399:
1366:
1363:
1360:
1345:
1333:
1330:
1327:
1322:
1319:
1315:
1311:
1308:
1285:
1282:
1278:
1250:
1246:
1242:
1239:
1228:
1206:
1203:
1200:
1197:
1194:
1190:
1186:
1183:
1180:
1177:
1155:
1152:
1130:
1127:
1124:
1098:
1094:
1078:
1066:
1063:
1060:
1055:
1051:
1047:
1044:
1041:
1038:
1033:
1029:
1008:
1005:
1002:
999:
996:
990:
987:
982:
978:
975:
972:
969:
963:
960:
955:
934:
931:
928:
925:
922:
911:
904:
892:
889:
886:
881:
876:
872:
858:
855:
854:
853:
852:Dikin’s Method
849:
848:
836:
829:
826:
823:
818:
815:
812:
806:
803:
798:
795:
790:
787:
783:
777:
773:
769:
766:
763:
760:
755:
751:
747:
743:
739:
736:
733:
730:
727:
724:
721:
716:
713:
708:
705:
701:
695:
691:
679:
678:
666:
663:
660:
657:
654:
635:Hilbert metric
631:
619:
616:
611:
608:
603:
600:
594:
590:
587:
584:
578:
574:
571:
547:
534:
531:
515:
514:
503:
500:
497:
494:
491:
488:
485:
481:
478:
469:
466:
463:
460:
457:
454:
451:
447:
444:
439:
434:
431:
427:
424:
421:
418:
415:
412:
409:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
376:
373:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
335:
330:
326:
307:
306:
295:
292:
289:
286:
283:
280:
277:
273:
270:
261:
258:
255:
252:
248:
244:
241:
237:
234:
230:
227:
224:
221:
218:
215:
212:
209:
206:
203:
200:
197:
194:
191:
188:
185:
182:
179:
176:
173:
170:
167:
164:
161:
158:
155:
152:
149:
146:
143:
137:
133:
96:
93:
66:
61:
15:
9:
6:
4:
3:
2:
2468:
2457:
2454:
2452:
2449:
2447:
2444:
2443:
2441:
2430:
2426:
2422:
2418:
2414:
2410:
2409:
2404:
2399:
2398:
2385:
2381:
2377:
2373:
2368:
2363:
2359:
2355:
2348:
2340:
2336:
2332:
2328:
2323:
2318:
2314:
2310:
2309:
2301:
2299:
2297:
2295:
2293:
2291:
2281:
2276:
2272:
2268:
2264:
2257:
2255:
2253:
2244:
2240:
2236:
2230:
2228:
2226:
2217:
2213:
2209:
2205:
2204:
2196:
2188:
2184:
2180:
2176:
2172:
2168:
2161:
2157:
2131:
2127:
2123:
2120:
2117:
2109:
2105:
2101:
2095:
2091:
2070:
2067:
2062:
2058:
2049:
2033:
2025:
2009:
2004:
2000:
1996:
1991:
1987:
1983:
1978:
1974:
1953:
1950:
1928:
1924:
1915:
1914:
1900:
1880:
1858:
1854:
1850:
1847:
1844:
1841:
1838:
1835:
1830:
1826:
1803:
1799:
1795:
1792:
1789:
1786:
1783:
1780:
1775:
1771:
1749:
1741:
1737:
1734:
1731:
1725:
1721:
1716:
1712:
1690:
1686:
1682:
1679:
1672:and a convex
1659:
1656:
1653:
1645:
1628:
1624:
1620:
1614:
1607:
1604:
1580:
1574:
1567:
1564:
1560:
1557:
1549:
1546:, if x is in
1545:
1541:
1537:
1533:
1516:
1513:
1510:
1506:
1502:
1496:
1488:
1484:
1463:
1460:
1457:
1435:
1432:
1429:
1425:
1421:
1418:
1415:
1409:
1401:
1397:
1388:
1384:
1380:
1364:
1361:
1358:
1350:
1346:
1328:
1320:
1317:
1313:
1309:
1306:
1283:
1280:
1276:
1266:
1248:
1244:
1240:
1237:
1229:
1226:
1222:
1201:
1195:
1192:
1188:
1184:
1181:
1178:
1175:
1153:
1150:
1128:
1125:
1122:
1114:
1096:
1092:
1083:
1079:
1061:
1053:
1049:
1045:
1039:
1031:
1027:
1003:
997:
988:
985:
980:
976:
970:
961:
958:
953:
932:
929:
926:
923:
920:
912:
909:
905:
887:
879:
874:
870:
861:
860:
851:
850:
834:
827:
824:
821:
816:
813:
810:
804:
801:
796:
793:
788:
785:
781:
775:
771:
767:
761:
753:
749:
745:
741:
734:
731:
728:
722:
719:
714:
711:
706:
703:
699:
693:
689:
681:
680:
664:
661:
658:
655:
652:
644:
640:
636:
632:
609:
606:
598:
592:
588:
585:
582:
576:
569:
561:
545:
537:
536:
530:
529:scaled by λ.
528:
524:
520:
498:
495:
492:
489:
486:
483:
479:
476:
467:
464:
461:
455:
452:
449:
445:
442:
432:
429:
425:
422:
419:
413:
410:
407:
398:
389:
386:
383:
377:
371:
368:
365:
356:
353:
350:
347:
341:
333:
328:
324:
316:
315:
314:
312:
290:
287:
284:
281:
278:
275:
271:
268:
259:
256:
253:
242:
239:
235:
232:
225:
216:
213:
210:
204:
198:
195:
192:
183:
180:
177:
171:
168:
165:
162:
156:
153:
150:
144:
135:
131:
122:
121:
120:
118:
114:
110:
106:
103:be a bounded
102:
92:
90:
86:
82:
64:
50:
47:-dimensional
46:
42:
41:convex subset
39:on a bounded
38:
34:
30:
21:
2412:
2406:
2357:
2353:
2347:
2312:
2306:
2273:: 4:1–4:12.
2270:
2266:
2242:
2238:
2235:Bárány, Imre
2207:
2201:
2195:
2170:
2166:
2160:
2047:
1547:
1543:
1539:
1538:, and a cap
1535:
1386:
1382:
1378:
1348:
1264:
1224:
1220:
1112:
1081:
642:
638:
533:Example uses
526:
522:
518:
516:
310:
308:
116:
112:
100:
98:
44:
32:
26:
2210:(1): 1–20.
2203:Mathematika
908:convex sets
29:mathematics
2440:Categories
2367:1809.09471
2322:1604.01175
2152:References
1943:has width
857:Properties
95:Definition
2429:127559205
2124:⊂
2118:⊂
2106:β
2068:⊂
2034:ϵ
2010:λ
1997:⊂
1984:⊂
1954:ϵ
1951:β
1901:λ
1881:β
1818:and caps
1735:−
1726:ϵ
1683:⊂
1654:ϵ
1621:⊂
1584:∅
1581:≠
1561:∩
1517:λ
1503:⊂
1489:λ
1476:, we get
1461:≤
1458:λ
1436:λ
1422:⊂
1416:∩
1402:λ
1381:in a cap
1359:λ
1310:⊂
1241:⊂
1185:⊂
1179:∩
1168:, we get
1126:∩
1046:⊂
1007:∅
1004:≠
977:∩
930:∈
880:λ
828:λ
825:−
817:λ
805:
768:⊂
754:λ
746:⊂
735:λ
723:
659:λ
656:≤
610:ϵ
599:
546:ϵ
496:−
484:−
465:∈
459:∃
450:∈
426:λ
414:λ
411:−
387:−
378:∩
369:−
357:λ
334:λ
288:−
276:−
257:∈
251:∃
240:∈
214:−
205:∩
196:−
169:−
157:∩
107:set in a
2384:53689683
2245:: 21–38.
2050:so that
1608:′
1568:′
1389:we know
480:′
446:′
433:′
272:′
236:′
2339:1841737
2187:1969800
1596:we get
525:around
83: (
2427:
2382:
2337:
2185:
1966:, and
645:and a
105:convex
2425:S2CID
2380:S2CID
2362:arXiv
2335:S2CID
2317:arXiv
2183:JSTOR
1916:Each
1299:then
1019:then
677:then:
2083:and
1893:and
1657:>
1550:and
1362:>
1219:for
945:and
862:The
662:<
119:is:
99:Let
85:1952
31:, a
2417:doi
2372:doi
2327:doi
2275:doi
2271:101
2212:doi
2175:doi
1542:of
1385:of
1084:in
913:If
577:log
43:of
27:In
2442::
2423:.
2413:61
2411:.
2405:.
2378:.
2370:.
2358:54
2356:.
2333:.
2325:.
2313:58
2311:.
2289:^
2269:.
2265:.
2251:^
2243:65
2241:.
2224:^
2208:17
2206:.
2181:.
2171:56
2169:.
802:ln
720:ln
91:.
2431:.
2419::
2386:.
2374::
2364::
2341:.
2329::
2319::
2283:.
2277::
2218:.
2214::
2189:.
2177::
2132:i
2128:C
2121:C
2110:2
2102:1
2096:i
2092:C
2071:C
2063:i
2059:R
2048:i
2005:i
2001:R
1992:i
1988:C
1979:i
1975:R
1929:i
1925:C
1859:k
1855:C
1851:,
1848:.
1845:.
1842:.
1839:.
1836:,
1831:1
1827:C
1804:k
1800:R
1796:,
1793:.
1790:.
1787:.
1784:.
1781:,
1776:1
1772:R
1750:)
1742:2
1738:1
1732:d
1722:1
1717:(
1713:O
1691:d
1687:R
1680:K
1660:0
1643:.
1629:2
1625:C
1618:)
1615:x
1612:(
1605:M
1578:)
1575:x
1572:(
1565:M
1558:C
1548:K
1544:K
1540:C
1536:K
1531:.
1514:+
1511:1
1507:C
1500:)
1497:x
1494:(
1485:M
1464:1
1433:+
1430:1
1426:C
1419:K
1413:)
1410:x
1407:(
1398:M
1387:K
1383:C
1379:x
1365:0
1349:K
1332:)
1329:x
1326:(
1321:d
1318:3
1314:M
1307:C
1284:d
1281:6
1277:1
1265:K
1249:d
1245:R
1238:K
1227:.
1225:H
1221:x
1205:)
1202:x
1199:(
1196:d
1193:3
1189:M
1182:H
1176:K
1154:2
1151:r
1129:H
1123:K
1113:H
1097:d
1093:R
1082:K
1065:)
1062:x
1059:(
1054:5
1050:M
1043:)
1040:y
1037:(
1032:1
1028:M
1001:)
998:y
995:(
989:2
986:1
981:M
974:)
971:x
968:(
962:2
959:1
954:M
933:K
927:y
924:,
921:x
910:.
891:)
888:x
885:(
875:K
871:M
835:)
822:1
814:+
811:1
797:2
794:1
789:,
786:x
782:(
776:H
772:B
765:)
762:x
759:(
750:M
742:)
738:)
732:+
729:1
726:(
715:2
712:1
707:,
704:x
700:(
694:H
690:B
665:1
653:0
643:x
639:K
618:)
615:)
607:1
602:(
593:2
589:1
586:+
583:d
573:(
570:O
527:x
523:K
519:K
502:}
499:k
493:x
490:=
487:x
477:k
468:K
462:k
456:,
453:K
443:k
438:|
430:k
423:+
420:x
417:)
408:1
405:(
402:{
399:=
396:)
393:)
390:K
384:x
381:(
375:)
372:x
366:K
363:(
360:(
354:+
351:x
348:=
345:)
342:x
339:(
329:K
325:M
311:x
294:}
291:k
285:x
282:=
279:x
269:k
260:K
254:k
247:|
243:K
233:k
229:{
226:=
223:)
220:)
217:K
211:x
208:(
202:)
199:x
193:K
190:(
187:(
184:+
181:x
178:=
175:)
172:K
166:x
163:2
160:(
154:K
151:=
148:)
145:x
142:(
136:K
132:M
117:x
113:x
101:K
65:d
60:R
45:d
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