597:
308:
301:
1592:
1810:
78:
592:{\displaystyle f(x,y)=\sum _{i=0}^{1}\sum _{j=0}^{1}a_{ij}x^{i}y^{j}=a_{00}+a_{10}x+a_{01}y+a_{11}xy={\begin{pmatrix}1&x\end{pmatrix}}{\begin{pmatrix}a_{00}&a_{01}\\a_{10}&a_{11}\end{pmatrix}}{\begin{pmatrix}1\\y\end{pmatrix}}}
1359:
1249:
The value of the polynomial at an arbitrary point can be found by repeated linear interpolation along each coordinate axis. Equivalently, it is a weighted mean of the vertex values, where the weights are the
952:
1617:
772:
1202:
points with minimal and maximal coordinate values. The value of the function on these points completely determines the function, since the value on the edges of the boundary can be found by
2124:
1244:
1113:
1002:
1925:
1879:
1354:
2042:
1287:
1200:
839:
651:
2015:
1612:
1169:
1133:
1077:
1046:
1022:
879:
859:
812:
792:
624:
296:{\displaystyle f(x)=\sum _{i_{1}=0}^{1}\sum _{i_{2}=0}^{1}\cdots \sum _{i_{n}=0}^{1}a_{i_{1}i_{2}\cdots i_{n}}x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{n}^{i_{n}}}
895:
1587:{\displaystyle f(x)=\sum _{v}f(v)\prod _{i|v_{i}=b_{i}}{\frac {x_{i}-a_{i}}{b_{i}-a_{i}}}\prod _{i|v_{i}=a_{i}}{\frac {b_{i}-x_{i}}{b_{i}-a_{i}}}}
656:
1971:
a linear function of the coordinates (its degree can be higher than 1), but it is a linear function of the fitted data values.
2215:
47:
is a constant times a product of distinct variables. For example f(x,y,z) = 3xy + 2.5 y - 7z is a multilinear polynomial of
1255:
1805:{\displaystyle f(x)={\frac {1}{V}}\sum _{v}f(v)\prod _{i|v_{i}=b_{i}}(x_{i}-a_{i})\prod _{i|v_{i}=a_{i}}(b_{i}-x_{i})}
1289:
smaller hyperrectangles, and the weight of each vertex is the (fractional) volume of the hyperrectangle opposite it.
2053:
2072:
1960:
2100:
2065:
1213:
1082:
971:
1882:
55:
of a multilinear polynomial is the maximum number of distinct variables occurring in any monomial.
1888:
2088:
2057:
1979:
1956:
1845:
1295:
2084:
2049:
1987:
1952:
1831:
at the center are proportional to the balance of the vertex values along each coordinate axis.
881:
is linear in the "shaped like a line" sense, but not in the "directly proportional" sense of a
52:
48:
1835:
1207:
32:
21:
2233:
2094:
2020:
1948:
1265:
1203:
1178:
1053:
817:
629:
43:. It is a polynomial in which no variable occurs to a power of 2 or higher; that is, each
8:
2144:
1251:
1839:
2000:
1824:
1597:
1154:
1118:
1062:
1031:
1007:
889:
864:
844:
797:
777:
609:
72:
2211:
2189:
1172:
1049:
1025:
2179:
2128:
2061:
1991:
966:
68:
2132:
1820:
882:
64:
36:
2184:
2167:
1827:
over the domain boundary, and the mean over the interior. The components of the
2068:
expressed as a multilinear polynomial (up to a choice of domain and codomain).
1259:
1144:
955:
2135:, multilinear functions that are strictly linear (not affine) in each variable
2227:
2193:
959:
2045:
1834:
The vertex values and the coefficients of the polynomial are related by a
2138:
1975:
1936:
1206:, and the value on the rest of the boundary and the interior is fixed by
71:) applied to the vectors , , etc. The general form can be written as a
40:
24:
51:
2 (because of the monomial 3xy) whereas f(x,y,z) = x² +4y is not. The
1148:
1983:
1828:
44:
1135:
is harmonic on every "slice" of the domain along coordinate axes.
1115:
still holds when one or more variables are fixed. In other words,
1292:
Algebraically, the multilinear interpolant on the hyperrectangle
1990:
multilinear polynomials in the elements of the matrix (and also
1959:
to an arbitrary number of variables. This is a specific form of
28:
2166:
Laneve, Cosimo; Lascu, Tudor A.; Sordoni, Vania (2010-10-01).
947:{\displaystyle {\frac {\partial ^{2}f}{\partial x_{k}^{2}}}=0}
2091:, using multivariate polynomials with two or three variables
767:{\displaystyle f(x_{1},x_{2},...,x_{k},...,x_{n})=ax_{k}+b}
1947:
interpolation on a rectangular grid, a generalization of
1079:
to a subset of its coordinates is also multilinear, so
2071:
Multilinear polynomials are important in the study of
568:
504:
480:
2103:
2023:
2003:
1891:
1848:
1620:
1600:
1362:
1298:
1268:
1216:
1181:
1157:
1121:
1085:
1065:
1034:
1010:
974:
898:
867:
847:
820:
800:
780:
659:
632:
612:
311:
81:
1262:. Geometrically, the point divides the domain into
626:is linear (affine) when varying only one variable,
2168:"The Interval Analysis of Multilinear Expressions"
2165:
2118:
2036:
2009:
1967:linear interpolation. The resulting polynomial is
1919:
1873:
1804:
1606:
1586:
1348:
1281:
1238:
1194:
1163:
1127:
1107:
1071:
1040:
1016:
996:
946:
873:
853:
833:
806:
786:
766:
645:
618:
591:
295:
2225:
2172:Electronic Notes in Theoretical Computer Science
1823:of the value at the vertices, which is also the
63:Multilinear polynomials can be understood as a
2054:Fourier analysis of (pseudo-)Boolean functions
2208:Polynomial Identities and Asymptotic Methods.
1908:
1892:
1862:
1849:
1138:
2183:
2106:
1885:if the domain is the symmetric hypercube
1594:where the sum is taken over the vertices
1254:. These weights also constitute a set of
1171:will have maxima and minima only on the
2226:
1175:of the domain, i.e. the finite set of
1816:is the volume of the hyperrectangle.
1059:More generally, every restriction of
2161:
2159:
1842:if the domain is the unit hypercube
1256:generalized barycentric coordinates
13:
1252:Lagrange interpolation polynomials
1218:
1145:rectangular in the coordinate axes
1087:
976:
917:
903:
14:
2245:
2156:
2141:, a multivariate linear function
2119:{\displaystyle \mathbb {F} _{2}}
2097:, a multilinear polynomial over
1935:Multilinear polynomials are the
1883:Walsh-Hadamard-Fourier transform
1997:The multilinear polynomials in
1930:
1819:The value at the center is the
2206:A. Giambruno, Mikhail Zaicev.
2200:
1799:
1773:
1744:
1732:
1706:
1677:
1665:
1659:
1630:
1624:
1501:
1409:
1397:
1391:
1372:
1366:
1326:
1299:
1239:{\displaystyle \nabla ^{2}f=0}
1108:{\displaystyle \nabla ^{2}f=0}
997:{\displaystyle \nabla ^{2}f=0}
739:
663:
327:
315:
305:For example, in two variables:
91:
85:
1:
2150:
601:
58:
1920:{\displaystyle \{-1,1\}^{n}}
1052:only on the boundary of the
7:
2185:10.1016/j.entcs.2010.09.017
2078:
2073:polynomial identity testing
1874:{\displaystyle \{0,1\}^{n}}
35:) in each of its variables
10:
2250:
1963:, not to be confused with
1961:multivariate interpolation
1349:{\displaystyle _{i=1}^{n}}
861:is generally not zero, so
1994:in the rows or columns).
606:A multilinear polynomial
2089:trilinear interpolation
1957:trilinear interpolation
1139:On a rectangular domain
2120:
2038:
2011:
1953:bilinear interpolation
1921:
1875:
1806:
1608:
1588:
1350:
1283:
1240:
1196:
1165:
1129:
1109:
1073:
1042:
1018:
998:
948:
875:
855:
835:
808:
788:
768:
647:
620:
593:
374:
353:
297:
183:
152:
124:
39:, but not necessarily
18:multilinear polynomial
2121:
2039:
2037:{\displaystyle 2^{n}}
2012:
1922:
1876:
1836:linear transformation
1807:
1609:
1589:
1351:
1284:
1282:{\displaystyle 2^{n}}
1241:
1197:
1195:{\displaystyle 2^{n}}
1166:
1130:
1110:
1074:
1043:
1019:
999:
949:
876:
856:
836:
834:{\displaystyle x_{k}}
809:
789:
769:
648:
646:{\displaystyle x_{k}}
621:
594:
354:
333:
298:
156:
125:
97:
2210:AMS Bookstore, 2005
2101:
2095:Zhegalkin polynomial
2048:, which is also the
2021:
2001:
1949:linear interpolation
1889:
1846:
1618:
1598:
1360:
1296:
1266:
1214:
1204:linear interpolation
1179:
1155:
1119:
1083:
1063:
1032:
1008:
972:
954:In other words, its
896:
888:All repeated second
865:
845:
818:
798:
778:
657:
630:
610:
309:
79:
2145:Harmonic polynomial
1345:
1143:When the domain is
965:In particular, the
934:
890:partial derivatives
292:
267:
245:
2116:
2034:
2007:
1917:
1871:
1802:
1772:
1705:
1655:
1604:
1584:
1529:
1437:
1387:
1346:
1325:
1279:
1236:
1208:Laplace's equation
1192:
1161:
1125:
1105:
1069:
1038:
1014:
994:
944:
920:
871:
851:
831:
804:
784:
764:
643:
616:
589:
583:
557:
493:
293:
271:
246:
224:
73:tensor contraction
2216:978-0-8218-3829-7
2017:variables form a
2010:{\displaystyle n}
1992:multilinear forms
1838:(specifically, a
1735:
1668:
1646:
1644:
1607:{\displaystyle v}
1582:
1492:
1490:
1400:
1378:
1164:{\displaystyle f}
1128:{\displaystyle f}
1072:{\displaystyle f}
1050:maxima and minima
1041:{\displaystyle f}
1026:harmonic function
1017:{\displaystyle f}
936:
874:{\displaystyle f}
854:{\displaystyle b}
814:do not depend on
807:{\displaystyle b}
787:{\displaystyle a}
619:{\displaystyle f}
67:(specifically, a
2241:
2219:
2204:
2198:
2197:
2187:
2163:
2129:Multilinear form
2125:
2123:
2122:
2117:
2115:
2114:
2109:
2062:Boolean function
2043:
2041:
2040:
2035:
2033:
2032:
2016:
2014:
2013:
2008:
1986:of a matrix are
1926:
1924:
1923:
1918:
1916:
1915:
1880:
1878:
1877:
1872:
1870:
1869:
1840:Möbius transform
1811:
1809:
1808:
1803:
1798:
1797:
1785:
1784:
1771:
1770:
1769:
1757:
1756:
1747:
1731:
1730:
1718:
1717:
1704:
1703:
1702:
1690:
1689:
1680:
1654:
1645:
1637:
1613:
1611:
1610:
1605:
1593:
1591:
1590:
1585:
1583:
1581:
1580:
1579:
1567:
1566:
1556:
1555:
1554:
1542:
1541:
1531:
1528:
1527:
1526:
1514:
1513:
1504:
1491:
1489:
1488:
1487:
1475:
1474:
1464:
1463:
1462:
1450:
1449:
1439:
1436:
1435:
1434:
1422:
1421:
1412:
1386:
1355:
1353:
1352:
1347:
1344:
1339:
1324:
1323:
1311:
1310:
1288:
1286:
1285:
1280:
1278:
1277:
1245:
1243:
1242:
1237:
1226:
1225:
1201:
1199:
1198:
1193:
1191:
1190:
1170:
1168:
1167:
1162:
1134:
1132:
1131:
1126:
1114:
1112:
1111:
1106:
1095:
1094:
1078:
1076:
1075:
1070:
1047:
1045:
1044:
1039:
1023:
1021:
1020:
1015:
1003:
1001:
1000:
995:
984:
983:
953:
951:
950:
945:
937:
935:
933:
928:
915:
911:
910:
900:
880:
878:
877:
872:
860:
858:
857:
852:
840:
838:
837:
832:
830:
829:
813:
811:
810:
805:
793:
791:
790:
785:
773:
771:
770:
765:
757:
756:
738:
737:
713:
712:
688:
687:
675:
674:
652:
650:
649:
644:
642:
641:
625:
623:
622:
617:
598:
596:
595:
590:
588:
587:
562:
561:
554:
553:
542:
541:
528:
527:
516:
515:
498:
497:
465:
464:
449:
448:
433:
432:
420:
419:
407:
406:
397:
396:
387:
386:
373:
368:
352:
347:
302:
300:
299:
294:
291:
290:
289:
279:
266:
265:
264:
254:
244:
243:
242:
232:
223:
222:
221:
220:
208:
207:
198:
197:
182:
177:
170:
169:
151:
146:
139:
138:
123:
118:
111:
110:
69:multilinear form
2249:
2248:
2244:
2243:
2242:
2240:
2239:
2238:
2224:
2223:
2222:
2205:
2201:
2164:
2157:
2153:
2133:multilinear map
2110:
2105:
2104:
2102:
2099:
2098:
2081:
2028:
2024:
2022:
2019:
2018:
2002:
1999:
1998:
1933:
1911:
1907:
1890:
1887:
1886:
1865:
1861:
1847:
1844:
1843:
1821:arithmetic mean
1793:
1789:
1780:
1776:
1765:
1761:
1752:
1748:
1743:
1739:
1726:
1722:
1713:
1709:
1698:
1694:
1685:
1681:
1676:
1672:
1650:
1636:
1619:
1616:
1615:
1614:. Equivalently,
1599:
1596:
1595:
1575:
1571:
1562:
1558:
1557:
1550:
1546:
1537:
1533:
1532:
1530:
1522:
1518:
1509:
1505:
1500:
1496:
1483:
1479:
1470:
1466:
1465:
1458:
1454:
1445:
1441:
1440:
1438:
1430:
1426:
1417:
1413:
1408:
1404:
1382:
1361:
1358:
1357:
1340:
1329:
1319:
1315:
1306:
1302:
1297:
1294:
1293:
1273:
1269:
1267:
1264:
1263:
1221:
1217:
1215:
1212:
1211:
1186:
1182:
1180:
1177:
1176:
1156:
1153:
1152:
1141:
1120:
1117:
1116:
1090:
1086:
1084:
1081:
1080:
1064:
1061:
1060:
1033:
1030:
1029:
1028:. This implies
1009:
1006:
1005:
979:
975:
973:
970:
969:
958:is a symmetric
929:
924:
916:
906:
902:
901:
899:
897:
894:
893:
883:multilinear map
866:
863:
862:
846:
843:
842:
825:
821:
819:
816:
815:
799:
796:
795:
779:
776:
775:
752:
748:
733:
729:
708:
704:
683:
679:
670:
666:
658:
655:
654:
637:
633:
631:
628:
627:
611:
608:
607:
604:
582:
581:
575:
574:
564:
563:
556:
555:
549:
545:
543:
537:
533:
530:
529:
523:
519:
517:
511:
507:
500:
499:
492:
491:
486:
476:
475:
460:
456:
444:
440:
428:
424:
415:
411:
402:
398:
392:
388:
379:
375:
369:
358:
348:
337:
310:
307:
306:
285:
281:
280:
275:
260:
256:
255:
250:
238:
234:
233:
228:
216:
212:
203:
199:
193:
189:
188:
184:
178:
165:
161:
160:
147:
134:
130:
129:
119:
106:
102:
101:
80:
77:
76:
65:multilinear map
61:
12:
11:
5:
2247:
2237:
2236:
2221:
2220:
2218:. Section 1.3.
2199:
2154:
2152:
2149:
2148:
2147:
2142:
2136:
2126:
2113:
2108:
2092:
2080:
2077:
2031:
2027:
2006:
1932:
1929:
1914:
1910:
1906:
1903:
1900:
1897:
1894:
1868:
1864:
1860:
1857:
1854:
1851:
1801:
1796:
1792:
1788:
1783:
1779:
1775:
1768:
1764:
1760:
1755:
1751:
1746:
1742:
1738:
1734:
1729:
1725:
1721:
1716:
1712:
1708:
1701:
1697:
1693:
1688:
1684:
1679:
1675:
1671:
1667:
1664:
1661:
1658:
1653:
1649:
1643:
1640:
1635:
1632:
1629:
1626:
1623:
1603:
1578:
1574:
1570:
1565:
1561:
1553:
1549:
1545:
1540:
1536:
1525:
1521:
1517:
1512:
1508:
1503:
1499:
1495:
1486:
1482:
1478:
1473:
1469:
1461:
1457:
1453:
1448:
1444:
1433:
1429:
1425:
1420:
1416:
1411:
1407:
1403:
1399:
1396:
1393:
1390:
1385:
1381:
1377:
1374:
1371:
1368:
1365:
1343:
1338:
1335:
1332:
1328:
1322:
1318:
1314:
1309:
1305:
1301:
1276:
1272:
1260:hyperrectangle
1235:
1232:
1229:
1224:
1220:
1189:
1185:
1160:
1140:
1137:
1124:
1104:
1101:
1098:
1093:
1089:
1068:
1037:
1013:
993:
990:
987:
982:
978:
956:Hessian matrix
943:
940:
932:
927:
923:
919:
914:
909:
905:
870:
850:
828:
824:
803:
783:
763:
760:
755:
751:
747:
744:
741:
736:
732:
728:
725:
722:
719:
716:
711:
707:
703:
700:
697:
694:
691:
686:
682:
678:
673:
669:
665:
662:
640:
636:
615:
603:
600:
586:
580:
577:
576:
573:
570:
569:
567:
560:
552:
548:
544:
540:
536:
532:
531:
526:
522:
518:
514:
510:
506:
505:
503:
496:
490:
487:
485:
482:
481:
479:
474:
471:
468:
463:
459:
455:
452:
447:
443:
439:
436:
431:
427:
423:
418:
414:
410:
405:
401:
395:
391:
385:
382:
378:
372:
367:
364:
361:
357:
351:
346:
343:
340:
336:
332:
329:
326:
323:
320:
317:
314:
288:
284:
278:
274:
270:
263:
259:
253:
249:
241:
237:
231:
227:
219:
215:
211:
206:
202:
196:
192:
187:
181:
176:
173:
168:
164:
159:
155:
150:
145:
142:
137:
133:
128:
122:
117:
114:
109:
105:
100:
96:
93:
90:
87:
84:
60:
57:
41:simultaneously
16:In algebra, a
9:
6:
4:
3:
2:
2246:
2235:
2232:
2231:
2229:
2217:
2213:
2209:
2203:
2195:
2191:
2186:
2181:
2177:
2173:
2169:
2162:
2160:
2155:
2146:
2143:
2140:
2137:
2134:
2130:
2127:
2111:
2096:
2093:
2090:
2086:
2083:
2082:
2076:
2074:
2069:
2067:
2063:
2059:
2055:
2051:
2047:
2044:-dimensional
2029:
2025:
2004:
1995:
1993:
1989:
1985:
1981:
1977:
1972:
1970:
1966:
1962:
1958:
1954:
1950:
1946:
1942:
1938:
1928:
1912:
1904:
1901:
1898:
1895:
1884:
1866:
1858:
1855:
1852:
1841:
1837:
1832:
1830:
1826:
1822:
1817:
1815:
1794:
1790:
1786:
1781:
1777:
1766:
1762:
1758:
1753:
1749:
1740:
1736:
1727:
1723:
1719:
1714:
1710:
1699:
1695:
1691:
1686:
1682:
1673:
1669:
1662:
1656:
1651:
1647:
1641:
1638:
1633:
1627:
1621:
1601:
1576:
1572:
1568:
1563:
1559:
1551:
1547:
1543:
1538:
1534:
1523:
1519:
1515:
1510:
1506:
1497:
1493:
1484:
1480:
1476:
1471:
1467:
1459:
1455:
1451:
1446:
1442:
1431:
1427:
1423:
1418:
1414:
1405:
1401:
1394:
1388:
1383:
1379:
1375:
1369:
1363:
1341:
1336:
1333:
1330:
1320:
1316:
1312:
1307:
1303:
1290:
1274:
1270:
1261:
1257:
1253:
1247:
1233:
1230:
1227:
1222:
1209:
1205:
1187:
1183:
1174:
1158:
1150:
1146:
1136:
1122:
1102:
1099:
1096:
1091:
1066:
1057:
1055:
1051:
1035:
1027:
1011:
991:
988:
985:
980:
968:
963:
961:
960:hollow matrix
957:
941:
938:
930:
925:
921:
912:
907:
891:
886:
884:
868:
848:
826:
822:
801:
781:
761:
758:
753:
749:
745:
742:
734:
730:
726:
723:
720:
717:
714:
709:
705:
701:
698:
695:
692:
689:
684:
680:
676:
671:
667:
660:
638:
634:
613:
599:
584:
578:
571:
565:
558:
550:
546:
538:
534:
524:
520:
512:
508:
501:
494:
488:
483:
477:
472:
469:
466:
461:
457:
453:
450:
445:
441:
437:
434:
429:
425:
421:
416:
412:
408:
403:
399:
393:
389:
383:
380:
376:
370:
365:
362:
359:
355:
349:
344:
341:
338:
334:
330:
324:
321:
318:
312:
303:
286:
282:
276:
272:
268:
261:
257:
251:
247:
239:
235:
229:
225:
217:
213:
209:
204:
200:
194:
190:
185:
179:
174:
171:
166:
162:
157:
153:
148:
143:
140:
135:
131:
126:
120:
115:
112:
107:
103:
98:
94:
88:
82:
74:
70:
66:
56:
54:
50:
46:
42:
38:
34:
30:
26:
23:
19:
2207:
2202:
2178:(2): 43–53.
2175:
2171:
2070:
2052:used in the
2046:vector space
1996:
1973:
1968:
1964:
1944:
1940:
1937:interpolants
1934:
1931:Applications
1833:
1818:
1813:
1291:
1248:
1142:
1058:
964:
887:
841:. Note that
605:
304:
62:
22:multivariate
17:
15:
2234:Polynomials
2139:Linear form
1988:homogeneous
1976:determinant
1941:multilinear
2151:References
1982:and other
602:Properties
59:Definition
37:separately
25:polynomial
2194:1571-0661
2056:. Every (
1984:immanants
1980:permanent
1965:piecewise
1896:−
1787:−
1737:∏
1720:−
1670:∏
1648:∑
1569:−
1544:−
1494:∏
1477:−
1452:−
1402:∏
1380:∑
1219:∇
1149:hypercube
1088:∇
977:∇
967:Laplacian
918:∂
904:∂
892:are zero:
356:∑
335:∑
269:⋯
210:⋯
158:∑
154:⋯
127:∑
99:∑
31:(meaning
2228:Category
2085:Bilinear
2079:See also
2066:uniquely
1945:n-linear
1881:, and a
1829:gradient
1258:for the
1173:vertices
1147:(e.g. a
45:monomial
27:that is
2064:can be
2058:pseudo-
2214:
2192:
1812:where
1054:domain
774:where
53:degree
49:degree
33:affine
29:linear
2050:basis
1024:is a
1004:, so
20:is a
2212:ISBN
2190:ISSN
2131:and
2087:and
1974:The
1955:and
1825:mean
1048:has
794:and
2180:doi
2176:267
1969:not
1943:or
1939:of
1927:).
1356:is:
1246:.
1151:),
962:.
2230::
2188:.
2174:.
2170:.
2158:^
2075:.
1978:,
1951:,
1210:,
1056:.
885:.
551:11
539:10
525:01
513:00
462:11
446:01
430:10
417:00
2196:.
2182::
2112:2
2107:F
2060:)
2030:n
2026:2
2005:n
1913:n
1909:}
1905:1
1902:,
1899:1
1893:{
1867:n
1863:}
1859:1
1856:,
1853:0
1850:{
1814:V
1800:)
1795:i
1791:x
1782:i
1778:b
1774:(
1767:i
1763:a
1759:=
1754:i
1750:v
1745:|
1741:i
1733:)
1728:i
1724:a
1715:i
1711:x
1707:(
1700:i
1696:b
1692:=
1687:i
1683:v
1678:|
1674:i
1666:)
1663:v
1660:(
1657:f
1652:v
1642:V
1639:1
1634:=
1631:)
1628:x
1625:(
1622:f
1602:v
1577:i
1573:a
1564:i
1560:b
1552:i
1548:x
1539:i
1535:b
1524:i
1520:a
1516:=
1511:i
1507:v
1502:|
1498:i
1485:i
1481:a
1472:i
1468:b
1460:i
1456:a
1447:i
1443:x
1432:i
1428:b
1424:=
1419:i
1415:v
1410:|
1406:i
1398:)
1395:v
1392:(
1389:f
1384:v
1376:=
1373:)
1370:x
1367:(
1364:f
1342:n
1337:1
1334:=
1331:i
1327:]
1321:i
1317:b
1313:,
1308:i
1304:a
1300:[
1275:n
1271:2
1234:0
1231:=
1228:f
1223:2
1188:n
1184:2
1159:f
1123:f
1103:0
1100:=
1097:f
1092:2
1067:f
1036:f
1012:f
992:0
989:=
986:f
981:2
942:0
939:=
931:2
926:k
922:x
913:f
908:2
869:f
849:b
827:k
823:x
802:b
782:a
762:b
759:+
754:k
750:x
746:a
743:=
740:)
735:n
731:x
727:,
724:.
721:.
718:.
715:,
710:k
706:x
702:,
699:.
696:.
693:.
690:,
685:2
681:x
677:,
672:1
668:x
664:(
661:f
653::
639:k
635:x
614:f
585:)
579:y
572:1
566:(
559:)
547:a
535:a
521:a
509:a
502:(
495:)
489:x
484:1
478:(
473:=
470:y
467:x
458:a
454:+
451:y
442:a
438:+
435:x
426:a
422:+
413:a
409:=
404:j
400:y
394:i
390:x
384:j
381:i
377:a
371:1
366:0
363:=
360:j
350:1
345:0
342:=
339:i
331:=
328:)
325:y
322:,
319:x
316:(
313:f
287:n
283:i
277:n
273:x
262:2
258:i
252:2
248:x
240:1
236:i
230:1
226:x
218:n
214:i
205:2
201:i
195:1
191:i
186:a
180:1
175:0
172:=
167:n
163:i
149:1
144:0
141:=
136:2
132:i
121:1
116:0
113:=
108:1
104:i
95:=
92:)
89:x
86:(
83:f
75::
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