1054:
1279:
855:
210:
830:
19:
This article is about functions that are invariant under all permutations of their variables. For the generalization of symmetric polynomials to infinitely many variables (in algebraic combinatorics), see
1777:
1654:
1063:
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118:
364:
314:
1335:
1806:
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264:
237:
663:
617:
504:
391:
1914:
1894:
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1850:
1694:
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1454:
1434:
850:
683:
637:
586:
562:
542:
481:
453:
430:
57:
701:
2714:
2417:
2032:
123:
596:. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions
2601:
2280:
2060:
1049:{\displaystyle f(x_{1},x_{2},\ldots ,x_{n})=f(x_{2},x_{1},\ldots ,x_{n})=f(x_{3},x_{1},\ldots ,x_{n},x_{n-1}),\quad {\text{ etc.}}}
2482:
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1942:
1459:
1342:
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1954:
2709:
2358:
2596:
2023:
1965:
21:
2407:
2227:
2007:
1873:
2079:
588:
over all permutations of the arguments. Similarly, an anti-symmetric function can be constructed by summing over
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2535:
2242:
70:
319:
269:
2633:
2320:
2237:
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1960:
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1979:
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1937:
507:
405:
64:
36:
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2222:
398:
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2187:
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2128:
1274:{\displaystyle (x-x_{1})(x-x_{2})(x-x_{3})=(x-x_{2})(x-x_{1})(x-x_{3})=(x-x_{3})(x-x_{1})(x-x_{2})}
394:
2658:
2515:
2368:
2182:
2118:
1973: – process that converts any function in n variables to a symmetric function in n variables
2653:
2561:
2422:
2397:
2212:
2123:
2103:
2028:
1782:
412:
that act as functions of several vectors can be symmetric, and in fact the space of symmetric
1536:
2668:
2566:
2343:
2310:
2295:
2177:
2046:
589:
409:
25:
242:
215:
2638:
2586:
2530:
2510:
2412:
2300:
2167:
2138:
2014:
1925:
593:
642:
599:
486:
408:, which change sign under an interchange of variables. Aside from polynomial functions,
373:
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2643:
2623:
2540:
2373:
2363:
2353:
2275:
2247:
2232:
2217:
2133:
1970:
1899:
1879:
1855:
1835:
1679:
1659:
1439:
1419:
835:
668:
622:
571:
547:
527:
519:
466:
438:
415:
42:
2698:
2615:
2520:
2432:
2305:
2018:
1999:
639:
can be recovered if both its symmetrization and antisymmetrization are known is when
565:
2683:
2487:
2472:
2437:
2285:
2270:
433:
2571:
2545:
2467:
2156:
2095:
1995:
1921:
1917:
1823:
1057:
695:
32:
2452:
1829:
456:
2427:
2378:
685:
is equal to half the sum of its symmetrization and its antisymmetrization.
2457:
2442:
665:
and the abelian group admits a division by 2 (inverse of doubling); then
2151:
2113:
2477:
2069:
205:{\displaystyle f\left(x_{1},x_{2}\right)=f\left(x_{2},x_{1}\right)}
2027:, §5.1 Symmetric functions, pp 222–5, Cambridge University Press,
16:
Function that is invariant under all permutations of its variables
825:{\displaystyle f(x_{1},x_{2},x_{3})=(x-x_{1})(x-x_{2})(x-x_{3}).}
2042:
568:, a symmetric function can be constructed by summing values of
24:. For symmetric functions on elements of a vector space, see
2038:
120:
of two arguments is a symmetric function if and only if
1951: – Functions such that f(–x) equals f(x) or –f(x)
1533:
which yields exactly the same results as the original
393:
The most commonly encountered symmetric functions are
1902:
1882:
1858:
1838:
1785:
1702:
1682:
1662:
1579:
1539:
1462:
1442:
1422:
1345:
1287:
1066:
858:
838:
704:
671:
645:
625:
602:
574:
550:
530:
489:
469:
441:
418:
376:
322:
272:
245:
218:
126:
73:
45:
1984:
Pages displaying wikidata descriptions as a fallback
1975:
Pages displaying wikidata descriptions as a fallback
1896:-sample statistic, yielding a symmetric function in
1056:
In general, the function remains the same for every
63:
if its value is the same no matter the order of its
2614:
2554:
2503:
2496:
2388:
2319:
2256:
2200:
2147:
2094:
2087:
1908:
1888:
1864:
1844:
1800:
1771:
1688:
1668:
1648:
1563:
1525:
1448:
1428:
1408:
1329:
1273:
1048:
844:
824:
677:
657:
631:
611:
580:
556:
536:
498:
475:
447:
424:
385:
358:
308:
258:
231:
204:
112:
51:
1779:This function is not the same as the original if
1060:of its variables. This means that, in this case,
506:Symmetric functions should not be confused with
2054:
8:
1772:{\displaystyle f(y,x)=ay^{2}+bx^{2}-r^{2}.}
1649:{\displaystyle f(x,y)=ax^{2}+by^{2}-r^{2}.}
510:, which have a different sort of symmetry.
2500:
2091:
2061:
2047:
2039:
1982: – determinant of Vandermonde matrix
1901:
1881:
1857:
1837:
1784:
1760:
1747:
1731:
1701:
1681:
1661:
1637:
1624:
1608:
1578:
1538:
1526:{\displaystyle f(y,x)=y^{2}+x^{2}-r^{2},}
1514:
1501:
1488:
1461:
1441:
1421:
1409:{\displaystyle f(x,y)=x^{2}+y^{2}-r^{2}.}
1397:
1384:
1371:
1344:
1318:
1305:
1292:
1286:
1262:
1240:
1218:
1193:
1171:
1149:
1124:
1102:
1080:
1065:
1041:
1022:
1009:
990:
977:
955:
936:
923:
901:
882:
869:
857:
837:
832:By definition, a symmetric function with
810:
788:
766:
741:
728:
715:
703:
670:
644:
624:
601:
573:
549:
529:
488:
468:
440:
417:
375:
345:
332:
321:
295:
282:
271:
250:
244:
223:
217:
191:
178:
152:
139:
125:
113:{\displaystyle f\left(x_{1},x_{2}\right)}
99:
86:
72:
44:
2418:Covariance and contravariance of vectors
359:{\displaystyle \left(x_{2},x_{1}\right)}
309:{\displaystyle \left(x_{1},x_{2}\right)}
1696:are interchanged, the function becomes
1456:are interchanged the function becomes
7:
2004:Symmetric Function and Allied Tables
1281:and so on, for all permutations of
2281:Tensors in curvilinear coordinates
1330:{\displaystyle x_{1},x_{2},x_{3}.}
14:
1852:-sample statistic (a function in
852:variables has the property that
2715:Properties of binary operations
1943:Elementary symmetric polynomial
1872:variables) that is obtained by
1040:
1808:which makes it non-symmetric.
1718:
1706:
1595:
1583:
1555:
1543:
1478:
1466:
1361:
1349:
1268:
1249:
1246:
1227:
1224:
1205:
1199:
1180:
1177:
1158:
1155:
1136:
1130:
1111:
1108:
1089:
1086:
1067:
1034:
970:
961:
916:
907:
862:
816:
797:
794:
775:
772:
753:
747:
708:
1:
2334:Exterior covariant derivative
2266:Tensor (intrinsic definition)
1957: – Concept in statistics
1955:Exchangeable random variables
1945: – Mathematical function
592:and subtracting the sum over
2359:Raising and lowering indices
619:The only general case where
564:variables with values in an
2597:Gluon field strength tensor
2024:Combinatorics: The Rota Way
1966:Ring of symmetric functions
22:ring of symmetric functions
2731:
2408:Cartan formalism (physics)
2228:Penrose graphical notation
2008:Cambridge University Press
2002:& D. E. Barton (1966)
1821:
1573:Consider now the function
517:
67:. For example, a function
18:
2080:Glossary of tensor theory
2076:
397:, which are given by the
2664:Gregorio Ricci-Curbastro
2536:Riemann curvature tensor
2243:Van der Waerden notation
1801:{\displaystyle a\neq b,}
2634:Elwin Bruno Christoffel
2567:Angular momentum tensor
2238:Tetrad (index notation)
2208:Abstract index notation
1961:Quasisymmetric function
1920:. Examples include the
1916:variables, is called a
1564:{\displaystyle f(x,y).}
461:homogeneous polynomials
406:alternating polynomials
2448:Levi-Civita connection
1980:Vandermonde polynomial
1949:Even and odd functions
1938:Alternating polynomial
1910:
1890:
1866:
1846:
1802:
1773:
1690:
1670:
1650:
1565:
1527:
1450:
1430:
1410:
1339:Consider the function
1331:
1275:
1050:
846:
826:
679:
659:
633:
613:
582:
558:
538:
508:even and odd functions
500:
477:
449:
426:
387:
360:
310:
260:
233:
206:
114:
53:
2674:Jan Arnoldus Schouten
2629:Augustin-Louis Cauchy
2109:Differential geometry
1911:
1891:
1867:
1847:
1803:
1774:
1691:
1671:
1651:
1566:
1528:
1451:
1431:
1411:
1332:
1276:
1051:
847:
827:
680:
660:
634:
614:
583:
559:
539:
501:
478:
450:
427:
399:symmetric polynomials
388:
361:
311:
261:
259:{\displaystyle x_{2}}
234:
232:{\displaystyle x_{1}}
207:
115:
54:
2649:Carl Friedrich Gauss
2582:stress–energy tensor
2577:Cauchy stress tensor
2329:Covariant derivative
2291:Antisymmetric tensor
2223:Multi-index notation
1900:
1880:
1876:symmetrization of a
1856:
1836:
1783:
1700:
1680:
1660:
1577:
1537:
1460:
1440:
1420:
1343:
1285:
1064:
856:
836:
702:
669:
643:
623:
600:
572:
548:
528:
487:
467:
439:
416:
404:A related notion is
395:polynomial functions
374:
320:
270:
243:
216:
124:
71:
43:
2710:Symmetric functions
2526:Nonmetricity tensor
2381:(2nd-order tensors)
2349:Hodge star operator
2339:Exterior derivative
2188:Transport phenomena
2173:Continuum mechanics
2129:Multilinear algebra
2013:Joseph P. S. Kung,
658:{\displaystyle n=2}
524:Given any function
2659:Tullio Levi-Civita
2602:Metric tensor (GR)
2516:Levi-Civita symbol
2369:Tensor contraction
2183:General relativity
2119:Euclidean geometry
1906:
1886:
1862:
1842:
1798:
1769:
1686:
1666:
1646:
1561:
1523:
1446:
1426:
1406:
1327:
1271:
1046:
842:
822:
675:
655:
629:
612:{\displaystyle f.}
609:
578:
554:
534:
499:{\displaystyle V.}
496:
473:
445:
422:
386:{\displaystyle f.}
383:
356:
306:
256:
229:
202:
110:
49:
2692:
2691:
2654:Hermann Grassmann
2610:
2609:
2562:Moment of inertia
2423:Differential form
2398:Affine connection
2213:Einstein notation
2196:
2195:
2124:Exterior calculus
2104:Coordinate system
2033:978-0-521-73794-4
1909:{\displaystyle n}
1889:{\displaystyle k}
1865:{\displaystyle n}
1845:{\displaystyle n}
1689:{\displaystyle y}
1669:{\displaystyle x}
1449:{\displaystyle y}
1429:{\displaystyle x}
1044:
845:{\displaystyle n}
678:{\displaystyle f}
632:{\displaystyle f}
590:even permutations
581:{\displaystyle f}
557:{\displaystyle n}
537:{\displaystyle f}
476:{\displaystyle k}
448:{\displaystyle V}
425:{\displaystyle k}
52:{\displaystyle n}
2722:
2669:Bernhard Riemann
2501:
2344:Exterior product
2311:Two-point tensor
2296:Symmetric tensor
2178:Electromagnetism
2092:
2063:
2056:
2049:
2040:
2019:Catherine H. Yan
1985:
1976:
1915:
1913:
1912:
1907:
1895:
1893:
1892:
1887:
1871:
1869:
1868:
1863:
1851:
1849:
1848:
1843:
1807:
1805:
1804:
1799:
1778:
1776:
1775:
1770:
1765:
1764:
1752:
1751:
1736:
1735:
1695:
1693:
1692:
1687:
1675:
1673:
1672:
1667:
1655:
1653:
1652:
1647:
1642:
1641:
1629:
1628:
1613:
1612:
1570:
1568:
1567:
1562:
1532:
1530:
1529:
1524:
1519:
1518:
1506:
1505:
1493:
1492:
1455:
1453:
1452:
1447:
1435:
1433:
1432:
1427:
1415:
1413:
1412:
1407:
1402:
1401:
1389:
1388:
1376:
1375:
1336:
1334:
1333:
1328:
1323:
1322:
1310:
1309:
1297:
1296:
1280:
1278:
1277:
1272:
1267:
1266:
1245:
1244:
1223:
1222:
1198:
1197:
1176:
1175:
1154:
1153:
1129:
1128:
1107:
1106:
1085:
1084:
1055:
1053:
1052:
1047:
1045:
1042:
1033:
1032:
1014:
1013:
995:
994:
982:
981:
960:
959:
941:
940:
928:
927:
906:
905:
887:
886:
874:
873:
851:
849:
848:
843:
831:
829:
828:
823:
815:
814:
793:
792:
771:
770:
746:
745:
733:
732:
720:
719:
684:
682:
681:
676:
664:
662:
661:
656:
638:
636:
635:
630:
618:
616:
615:
610:
594:odd permutations
587:
585:
584:
579:
563:
561:
560:
555:
543:
541:
540:
535:
505:
503:
502:
497:
482:
480:
479:
474:
459:to the space of
454:
452:
451:
446:
431:
429:
428:
423:
392:
390:
389:
384:
365:
363:
362:
357:
355:
351:
350:
349:
337:
336:
315:
313:
312:
307:
305:
301:
300:
299:
287:
286:
265:
263:
262:
257:
255:
254:
238:
236:
235:
230:
228:
227:
211:
209:
208:
203:
201:
197:
196:
195:
183:
182:
162:
158:
157:
156:
144:
143:
119:
117:
116:
111:
109:
105:
104:
103:
91:
90:
58:
56:
55:
50:
26:symmetric tensor
2730:
2729:
2725:
2724:
2723:
2721:
2720:
2719:
2695:
2694:
2693:
2688:
2639:Albert Einstein
2606:
2587:Einstein tensor
2550:
2531:Ricci curvature
2511:Kronecker delta
2497:Notable tensors
2492:
2413:Connection form
2390:
2384:
2315:
2301:Tensor operator
2258:
2252:
2192:
2168:Computer vision
2161:
2143:
2139:Tensor calculus
2083:
2072:
2067:
2015:Gian-Carlo Rota
1992:
1983:
1974:
1934:
1926:sample variance
1898:
1897:
1878:
1877:
1854:
1853:
1834:
1833:
1826:
1820:
1815:
1781:
1780:
1756:
1743:
1727:
1698:
1697:
1678:
1677:
1658:
1657:
1633:
1620:
1604:
1575:
1574:
1535:
1534:
1510:
1497:
1484:
1458:
1457:
1438:
1437:
1418:
1417:
1393:
1380:
1367:
1341:
1340:
1314:
1301:
1288:
1283:
1282:
1258:
1236:
1214:
1189:
1167:
1145:
1120:
1098:
1076:
1062:
1061:
1018:
1005:
986:
973:
951:
932:
919:
897:
878:
865:
854:
853:
834:
833:
806:
784:
762:
737:
724:
711:
700:
699:
691:
667:
666:
641:
640:
621:
620:
598:
597:
570:
569:
546:
545:
526:
525:
522:
516:
485:
484:
465:
464:
437:
436:
414:
413:
372:
371:
341:
328:
327:
323:
318:
317:
291:
278:
277:
273:
268:
267:
246:
241:
240:
219:
214:
213:
187:
174:
173:
169:
148:
135:
134:
130:
122:
121:
95:
82:
81:
77:
69:
68:
41:
40:
29:
17:
12:
11:
5:
2728:
2726:
2718:
2717:
2712:
2707:
2697:
2696:
2690:
2689:
2687:
2686:
2681:
2679:Woldemar Voigt
2676:
2671:
2666:
2661:
2656:
2651:
2646:
2644:Leonhard Euler
2641:
2636:
2631:
2626:
2620:
2618:
2616:Mathematicians
2612:
2611:
2608:
2607:
2605:
2604:
2599:
2594:
2589:
2584:
2579:
2574:
2569:
2564:
2558:
2556:
2552:
2551:
2549:
2548:
2543:
2541:Torsion tensor
2538:
2533:
2528:
2523:
2518:
2513:
2507:
2505:
2498:
2494:
2493:
2491:
2490:
2485:
2480:
2475:
2470:
2465:
2460:
2455:
2450:
2445:
2440:
2435:
2430:
2425:
2420:
2415:
2410:
2405:
2400:
2394:
2392:
2386:
2385:
2383:
2382:
2376:
2374:Tensor product
2371:
2366:
2364:Symmetrization
2361:
2356:
2354:Lie derivative
2351:
2346:
2341:
2336:
2331:
2325:
2323:
2317:
2316:
2314:
2313:
2308:
2303:
2298:
2293:
2288:
2283:
2278:
2276:Tensor density
2273:
2268:
2262:
2260:
2254:
2253:
2251:
2250:
2248:Voigt notation
2245:
2240:
2235:
2233:Ricci calculus
2230:
2225:
2220:
2218:Index notation
2215:
2210:
2204:
2202:
2198:
2197:
2194:
2193:
2191:
2190:
2185:
2180:
2175:
2170:
2164:
2162:
2160:
2159:
2154:
2148:
2145:
2144:
2142:
2141:
2136:
2134:Tensor algebra
2131:
2126:
2121:
2116:
2114:Dyadic algebra
2111:
2106:
2100:
2098:
2089:
2085:
2084:
2077:
2074:
2073:
2068:
2066:
2065:
2058:
2051:
2043:
2037:
2036:
2011:
1991:
1988:
1987:
1986:
1977:
1971:Symmetrization
1968:
1963:
1958:
1952:
1946:
1940:
1933:
1930:
1905:
1885:
1861:
1841:
1822:Main article:
1819:
1816:
1814:
1811:
1810:
1809:
1797:
1794:
1791:
1788:
1768:
1763:
1759:
1755:
1750:
1746:
1742:
1739:
1734:
1730:
1726:
1723:
1720:
1717:
1714:
1711:
1708:
1705:
1685:
1665:
1645:
1640:
1636:
1632:
1627:
1623:
1619:
1616:
1611:
1607:
1603:
1600:
1597:
1594:
1591:
1588:
1585:
1582:
1571:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1522:
1517:
1513:
1509:
1504:
1500:
1496:
1491:
1487:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1445:
1425:
1405:
1400:
1396:
1392:
1387:
1383:
1379:
1374:
1370:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1337:
1326:
1321:
1317:
1313:
1308:
1304:
1300:
1295:
1291:
1270:
1265:
1261:
1257:
1254:
1251:
1248:
1243:
1239:
1235:
1232:
1229:
1226:
1221:
1217:
1213:
1210:
1207:
1204:
1201:
1196:
1192:
1188:
1185:
1182:
1179:
1174:
1170:
1166:
1163:
1160:
1157:
1152:
1148:
1144:
1141:
1138:
1135:
1132:
1127:
1123:
1119:
1116:
1113:
1110:
1105:
1101:
1097:
1094:
1091:
1088:
1083:
1079:
1075:
1072:
1069:
1039:
1036:
1031:
1028:
1025:
1021:
1017:
1012:
1008:
1004:
1001:
998:
993:
989:
985:
980:
976:
972:
969:
966:
963:
958:
954:
950:
947:
944:
939:
935:
931:
926:
922:
918:
915:
912:
909:
904:
900:
896:
893:
890:
885:
881:
877:
872:
868:
864:
861:
841:
821:
818:
813:
809:
805:
802:
799:
796:
791:
787:
783:
780:
777:
774:
769:
765:
761:
758:
755:
752:
749:
744:
740:
736:
731:
727:
723:
718:
714:
710:
707:
690:
687:
674:
654:
651:
648:
628:
608:
605:
577:
553:
533:
520:Symmetrization
518:Main article:
515:
514:Symmetrization
512:
495:
492:
472:
444:
432:-tensors on a
421:
382:
379:
354:
348:
344:
340:
335:
331:
326:
304:
298:
294:
290:
285:
281:
276:
253:
249:
226:
222:
200:
194:
190:
186:
181:
177:
172:
168:
165:
161:
155:
151:
147:
142:
138:
133:
129:
108:
102:
98:
94:
89:
85:
80:
76:
48:
15:
13:
10:
9:
6:
4:
3:
2:
2727:
2716:
2713:
2711:
2708:
2706:
2705:Combinatorics
2703:
2702:
2700:
2685:
2682:
2680:
2677:
2675:
2672:
2670:
2667:
2665:
2662:
2660:
2657:
2655:
2652:
2650:
2647:
2645:
2642:
2640:
2637:
2635:
2632:
2630:
2627:
2625:
2622:
2621:
2619:
2617:
2613:
2603:
2600:
2598:
2595:
2593:
2590:
2588:
2585:
2583:
2580:
2578:
2575:
2573:
2570:
2568:
2565:
2563:
2560:
2559:
2557:
2553:
2547:
2544:
2542:
2539:
2537:
2534:
2532:
2529:
2527:
2524:
2522:
2521:Metric tensor
2519:
2517:
2514:
2512:
2509:
2508:
2506:
2502:
2499:
2495:
2489:
2486:
2484:
2481:
2479:
2476:
2474:
2471:
2469:
2466:
2464:
2461:
2459:
2456:
2454:
2451:
2449:
2446:
2444:
2441:
2439:
2436:
2434:
2433:Exterior form
2431:
2429:
2426:
2424:
2421:
2419:
2416:
2414:
2411:
2409:
2406:
2404:
2401:
2399:
2396:
2395:
2393:
2387:
2380:
2377:
2375:
2372:
2370:
2367:
2365:
2362:
2360:
2357:
2355:
2352:
2350:
2347:
2345:
2342:
2340:
2337:
2335:
2332:
2330:
2327:
2326:
2324:
2322:
2318:
2312:
2309:
2307:
2306:Tensor bundle
2304:
2302:
2299:
2297:
2294:
2292:
2289:
2287:
2284:
2282:
2279:
2277:
2274:
2272:
2269:
2267:
2264:
2263:
2261:
2255:
2249:
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2229:
2226:
2224:
2221:
2219:
2216:
2214:
2211:
2209:
2206:
2205:
2203:
2199:
2189:
2186:
2184:
2181:
2179:
2176:
2174:
2171:
2169:
2166:
2165:
2163:
2158:
2155:
2153:
2150:
2149:
2146:
2140:
2137:
2135:
2132:
2130:
2127:
2125:
2122:
2120:
2117:
2115:
2112:
2110:
2107:
2105:
2102:
2101:
2099:
2097:
2093:
2090:
2086:
2082:
2081:
2075:
2071:
2064:
2059:
2057:
2052:
2050:
2045:
2044:
2041:
2034:
2030:
2026:
2025:
2020:
2016:
2012:
2009:
2005:
2001:
2000:M. G. Kendall
1997:
1994:
1993:
1989:
1981:
1978:
1972:
1969:
1967:
1964:
1962:
1959:
1956:
1953:
1950:
1947:
1944:
1941:
1939:
1936:
1935:
1931:
1929:
1927:
1923:
1919:
1903:
1883:
1875:
1874:bootstrapping
1859:
1839:
1831:
1825:
1817:
1812:
1795:
1792:
1789:
1786:
1766:
1761:
1757:
1753:
1748:
1744:
1740:
1737:
1732:
1728:
1724:
1721:
1715:
1712:
1709:
1703:
1683:
1663:
1643:
1638:
1634:
1630:
1625:
1621:
1617:
1614:
1609:
1605:
1601:
1598:
1592:
1589:
1586:
1580:
1572:
1558:
1552:
1549:
1546:
1540:
1520:
1515:
1511:
1507:
1502:
1498:
1494:
1489:
1485:
1481:
1475:
1472:
1469:
1463:
1443:
1423:
1403:
1398:
1394:
1390:
1385:
1381:
1377:
1372:
1368:
1364:
1358:
1355:
1352:
1346:
1338:
1324:
1319:
1315:
1311:
1306:
1302:
1298:
1293:
1289:
1263:
1259:
1255:
1252:
1241:
1237:
1233:
1230:
1219:
1215:
1211:
1208:
1202:
1194:
1190:
1186:
1183:
1172:
1168:
1164:
1161:
1150:
1146:
1142:
1139:
1133:
1125:
1121:
1117:
1114:
1103:
1099:
1095:
1092:
1081:
1077:
1073:
1070:
1059:
1037:
1029:
1026:
1023:
1019:
1015:
1010:
1006:
1002:
999:
996:
991:
987:
983:
978:
974:
967:
964:
956:
952:
948:
945:
942:
937:
933:
929:
924:
920:
913:
910:
902:
898:
894:
891:
888:
883:
879:
875:
870:
866:
859:
839:
819:
811:
807:
803:
800:
789:
785:
781:
778:
767:
763:
759:
756:
750:
742:
738:
734:
729:
725:
721:
716:
712:
705:
697:
694:Consider the
693:
692:
688:
686:
672:
652:
649:
646:
626:
606:
603:
595:
591:
575:
567:
566:abelian group
551:
531:
521:
513:
511:
509:
493:
490:
470:
462:
458:
442:
435:
419:
411:
407:
402:
400:
396:
380:
377:
369:
352:
346:
342:
338:
333:
329:
324:
302:
296:
292:
288:
283:
279:
274:
251:
247:
224:
220:
198:
192:
188:
184:
179:
175:
170:
166:
163:
159:
153:
149:
145:
140:
136:
131:
127:
106:
100:
96:
92:
87:
83:
78:
74:
66:
62:
59:variables is
46:
38:
34:
27:
23:
2684:Hermann Weyl
2488:Vector space
2473:Pseudotensor
2438:Fiber bundle
2391:abstractions
2286:Mixed tensor
2271:Tensor field
2078:
2022:
2003:
1827:
1818:U-statistics
1813:Applications
523:
434:vector space
403:
60:
30:
2624:Élie Cartan
2572:Spin tensor
2546:Weyl tensor
2504:Mathematics
2468:Multivector
2259:definitions
2157:Engineering
2096:Mathematics
1996:F. N. David
1922:sample mean
1918:U-statistic
1824:U-statistic
1058:permutation
366:are in the
33:mathematics
2699:Categories
2453:Linear map
2321:Operations
1990:References
1830:statistics
1043: etc.
463:of degree
457:isomorphic
266:such that
2592:EM tensor
2428:Dimension
2379:Transpose
1790:≠
1754:−
1631:−
1508:−
1391:−
1256:−
1234:−
1212:−
1187:−
1165:−
1143:−
1118:−
1096:−
1074:−
1027:−
1000:…
946:…
892:…
804:−
782:−
760:−
698:function
65:arguments
61:symmetric
2458:Manifold
2443:Geodesic
2201:Notation
2017:, &
1932:See also
689:Examples
212:for all
37:function
2555:Physics
2389:Related
2152:Physics
2070:Tensors
2021:(2009)
410:tensors
2483:Vector
2478:Spinor
2463:Matrix
2257:Tensor
2031:
368:domain
2403:Basis
2088:Scope
1832:, an
2029:ISBN
1924:and
1676:and
1436:and
696:real
316:and
239:and
35:, a
1828:In
1656:If
1416:If
544:in
483:on
455:is
370:of
39:of
31:In
2701::
2006:,
1998:,
1928:.
401:.
2062:e
2055:t
2048:v
2035:.
2010:.
1904:n
1884:k
1860:n
1840:n
1796:,
1793:b
1787:a
1767:.
1762:2
1758:r
1749:2
1745:x
1741:b
1738:+
1733:2
1729:y
1725:a
1722:=
1719:)
1716:x
1713:,
1710:y
1707:(
1704:f
1684:y
1664:x
1644:.
1639:2
1635:r
1626:2
1622:y
1618:b
1615:+
1610:2
1606:x
1602:a
1599:=
1596:)
1593:y
1590:,
1587:x
1584:(
1581:f
1559:.
1556:)
1553:y
1550:,
1547:x
1544:(
1541:f
1521:,
1516:2
1512:r
1503:2
1499:x
1495:+
1490:2
1486:y
1482:=
1479:)
1476:x
1473:,
1470:y
1467:(
1464:f
1444:y
1424:x
1404:.
1399:2
1395:r
1386:2
1382:y
1378:+
1373:2
1369:x
1365:=
1362:)
1359:y
1356:,
1353:x
1350:(
1347:f
1325:.
1320:3
1316:x
1312:,
1307:2
1303:x
1299:,
1294:1
1290:x
1269:)
1264:2
1260:x
1253:x
1250:(
1247:)
1242:1
1238:x
1231:x
1228:(
1225:)
1220:3
1216:x
1209:x
1206:(
1203:=
1200:)
1195:3
1191:x
1184:x
1181:(
1178:)
1173:1
1169:x
1162:x
1159:(
1156:)
1151:2
1147:x
1140:x
1137:(
1134:=
1131:)
1126:3
1122:x
1115:x
1112:(
1109:)
1104:2
1100:x
1093:x
1090:(
1087:)
1082:1
1078:x
1071:x
1068:(
1038:,
1035:)
1030:1
1024:n
1020:x
1016:,
1011:n
1007:x
1003:,
997:,
992:1
988:x
984:,
979:3
975:x
971:(
968:f
965:=
962:)
957:n
953:x
949:,
943:,
938:1
934:x
930:,
925:2
921:x
917:(
914:f
911:=
908:)
903:n
899:x
895:,
889:,
884:2
880:x
876:,
871:1
867:x
863:(
860:f
840:n
820:.
817:)
812:3
808:x
801:x
798:(
795:)
790:2
786:x
779:x
776:(
773:)
768:1
764:x
757:x
754:(
751:=
748:)
743:3
739:x
735:,
730:2
726:x
722:,
717:1
713:x
709:(
706:f
673:f
653:2
650:=
647:n
627:f
607:.
604:f
576:f
552:n
532:f
494:.
491:V
471:k
443:V
420:k
381:.
378:f
353:)
347:1
343:x
339:,
334:2
330:x
325:(
303:)
297:2
293:x
289:,
284:1
280:x
275:(
252:2
248:x
225:1
221:x
199:)
193:1
189:x
185:,
180:2
176:x
171:(
167:f
164:=
160:)
154:2
150:x
146:,
141:1
137:x
132:(
128:f
107:)
101:2
97:x
93:,
88:1
84:x
79:(
75:f
47:n
28:.
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