1993:
329:
1908:
477:
161:
680:
849:
953:
1048:
1484:
1182:
1296:
1643:
Note that all of the parameter restrictions have the same basic source: The exponent of non-negative quantity must be non-negative in order for the function to be log-concave.
729:
709:
215:
1803:
1506:
1761:
382:
66:
763:
524:
Every concave function that is nonnegative on its domain is log-concave. However, the reverse does not necessarily hold. An example is the
1305:
2001:
1690:(CDF) of all log-concave distributions is also log-concave. However, some non-log-concave distributions also have log-concave CDF's:
2209:
2180:
2151:
2124:
1976:
1547:
867:
593:
1922:
1791:
1728:
1687:
1533:
2089:
1932:
1927:
1376:
2236:
1794:
since the derivative of the logarithm of the survival function is the negative hazard rate, and by concavity is monotone i.e.
1586:
1096:
2119:. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley \& Sons, Ltd. pp. ix+238 pp.
1769:
1502:
1651:
1765:
1565:
1756:, which always has a shape parameter ≥ 1) will be log-concave. This property is heavily used in general-purpose
1753:
974:
2241:
2204:. Mathematics in Science and Engineering. Vol. 187. Boston, MA: Academic Press, Inc. pp. xiv+467 pp.
1773:
1540:
1521:
962:
1745:
is log-concave. This follows from the fact that the convolution of two log-concave functions is log-concave.
1695:
1672:
1622:
1608:
1514:
514:
2112:
1558:
1242:
1205:
737:
A twice differentiable, nonnegative function with a convex domain is log-concave if and only if for all
731:
2077:
1709:
1636:
1593:
1572:
53:
1702:
1665:
1658:
1529:
714:
694:
2056:
1749:
1716:
1615:
360:
2205:
2176:
2147:
2146:. Probability and Mathematical Statistics. Boston, MA: Academic Press, Inc. pp. xiv+278.
2120:
1972:
1780:
1629:
525:
364:
2048:
2010:
1735:
1579:
689:
206:
2219:
2190:
2161:
2134:
2215:
2186:
2157:
2130:
1960:
1937:
1742:
20:
1757:
1679:
324:{\displaystyle \log f(\theta x+(1-\theta )y)\geq \theta \log f(x)+(1-\theta )\log f(y)}
2230:
2060:
24:
2033:
2081:
1787:
1314:
1903:{\displaystyle {\frac {d}{dx}}\log \left(1-F(x)\right)=-{\frac {f(x)}{1-F(x)}}}
1061:
Products: The product of log-concave functions is also log-concave. Indeed, if
2052:
1551:
517:. This follows from the fact that the logarithm is monotone implying that the
372:
57:
518:
195:
2014:
472:{\displaystyle f(\theta x+(1-\theta )y)\leq f(x)^{\theta }f(y)^{1-\theta }}
156:{\displaystyle f(\theta x+(1-\theta )y)\geq f(x)^{\theta }f(y)^{1-\theta }}
1964:
1748:
The product of two log-concave functions is log-concave. This means that
2082:"Logarithmic concave measures with application to stochastic programming"
1501:
Log-concave distributions are necessary for a number of algorithms, e.g.
363:
of convex sets (which requires the more flexible definition), and the
1723:
The following are among the properties of log-concave distributions:
844:{\displaystyle f(x)\nabla ^{2}f(x)\preceq \nabla f(x)\nabla f(x)^{T}}
1752:
densities formed by multiplying two probability densities (e.g. the
1646:
The following distributions are non-log-concave for all parameters:
1992:
Grechuk, Bogdan; Molyboha, Anton; Zabarankin, Michael (May 2009).
1910:
which is decreasing as it is the derivative of a concave function.
2202:
Convex functions, partial orderings, and statistical applications
948:{\displaystyle f(x)\nabla ^{2}f(x)-\nabla f(x)\nabla f(x)^{T}}
675:{\displaystyle f''(x)=e^{-{\frac {x^{2}}{2}}}(x^{2}-1)\nleq 0}
2171:
Pfanzagl, Johann; with the assistance of R. Hamböker (1994).
965:. For functions of one variable, this condition simplifies to
578:
is not concave since the second derivative is positive for |
194:
is strictly positive, this is equivalent to saying that the
1994:"Maximum Entropy Principle with General Deviation Measures"
2117:
Information and exponential families in statistical theory
2200:
Pečarić, Josip E.; Proschan, Frank; Tong, Y. L. (1992).
1479:{\displaystyle (f*g)(x)=\int f(x-y)g(y)dy=\int h(x,y)dy}
1177:{\displaystyle \log \,f(x)+\log \,g(x)=\log(f(x)g(x))}
1806:
1379:
1245:
1099:
977:
870:
766:
717:
697:
596:
385:
218:
69:
1734:
If a multivariate density is log-concave, so is the
1505:. Every distribution with log-concave density is a
1902:
1478:
1290:
1176:
1042:
947:
843:
723:
703:
674:
471:
323:
155:
2142:Dharmadhikari, Sudhakar; Joag-Dev, Kumar (1988).
1971:. Cambridge University Press. pp. 104–108.
1776:derived from the product of other distributions.
1786:If a density is log-concave, it has a monotone
2034:"Log-Concave Probability and Its Applications"
359:Examples of log-concave functions are the 0-1
8:
1625:if the number of degrees of freedom is ≥ 2,
1043:{\displaystyle f(x)f''(x)\leq (f'(x))^{2}}
2072:
2070:
1862:
1807:
1805:
1378:
1244:
1122:
1103:
1098:
1034:
976:
939:
887:
869:
835:
783:
765:
716:
696:
651:
631:
625:
621:
595:
457:
438:
384:
217:
141:
122:
68:
2144:Unimodality, convexity, and applications
1507:maximum entropy probability distribution
1949:
1779:If a density is log-concave, so is its
1741:The sum of two independent log-concave
1727:If a density is log-concave, so is its
376:if it satisfies the reverse inequality
2032:Bagnoli, Mark; Bergstrom, Ted (2005).
1965:"Log-concave and log-convex functions"
1632:if both shape parameters are ≥ 1, and
60:, and if it satisfies the inequality
7:
2026:
2024:
1955:
1953:
1712:when the shape parameter < 1, and
1087:are concave by definition. Therefore
1056:Operations preserving log-concavity
2002:Mathematics of Operations Research
1291:{\displaystyle g(x)=\int f(x,y)dy}
923:
908:
884:
819:
804:
780:
14:
1534:multivariate normal distributions
1524:are log-concave. Some examples:
1923:logarithmically concave sequence
1768:, which are thereby able to use
1729:cumulative distribution function
1719:when the shape parameter < 1.
1688:cumulative distribution function
1073:are log-concave functions, then
2090:Acta Scientiarum Mathematicarum
1963:; Vandenberghe, Lieven (2004).
1933:logarithmically convex function
1928:logarithmically concave measure
1317:preserves log-concavity, since
513:A log-concave function is also
1894:
1888:
1874:
1868:
1848:
1842:
1639:if the shape parameter is ≥ 1.
1618:if the shape parameter is ≥ 1,
1587:hyperbolic secant distribution
1520:. As it happens, many common
1467:
1455:
1437:
1431:
1425:
1413:
1401:
1395:
1392:
1380:
1367:are log-concave, and therefore
1279:
1267:
1255:
1249:
1171:
1168:
1162:
1156:
1150:
1144:
1132:
1126:
1113:
1107:
1031:
1027:
1021:
1010:
1004:
998:
987:
981:
936:
929:
920:
914:
902:
896:
880:
874:
832:
825:
816:
810:
798:
792:
776:
770:
718:
698:
663:
644:
611:
605:
454:
447:
435:
428:
419:
413:
401:
389:
318:
312:
300:
288:
282:
276:
258:
252:
240:
228:
138:
131:
119:
112:
103:
97:
85:
73:
1:
2173:Parametric Statistical Theory
1738:over any subset of variables.
16:Type of mathematical function
1611:, if all parameters are ≥ 1,
724:{\displaystyle \Rightarrow }
704:{\displaystyle \Rightarrow }
521:of this function are convex.
1770:adaptive rejection sampling
1503:adaptive rejection sampling
1306:Prékopa–Leindler inequality
1190:is concave, and hence also
546:which is log-concave since
2258:
1566:extreme value distribution
2053:10.1007/s00199-004-0514-4
1774:conditional distributions
1754:normal-gamma distribution
1522:probability distributions
1497:Log-concave distributions
566:is a concave function of
370:Similarly, a function is
1652:Student's t-distribution
1541:exponential distribution
1772:over a wide variety of
1696:log-normal distribution
1673:log-normal distribution
1623:chi-square distribution
688:From above two points,
46:logarithmically concave
2113:Barndorff-Nielsen, Ole
2015:10.1287/moor.1090.0377
1904:
1609:Dirichlet distribution
1515:Deviation risk measure
1480:
1292:
1178:
1044:
963:negative semi-definite
949:
845:
725:
705:
676:
473:
325:
157:
2237:Mathematical analysis
2175:. Walter de Gruyter.
1905:
1559:logistic distribution
1481:
1293:
1179:
1045:
950:
846:
726:
706:
677:
474:
326:
158:
1804:
1792:regular distribution
1710:Weibull distribution
1637:Weibull distribution
1594:Wishart distribution
1573:Laplace distribution
1548:uniform distribution
1509:with specified mean
1377:
1304:is log-concave (see
1243:
1233:is log-concave, then
1097:
975:
868:
764:
715:
695:
594:
383:
216:
67:
1969:Convex Optimization
1703:Pareto distribution
1666:Pareto distribution
1659:Cauchy distribution
1530:normal distribution
1228: →
584:| > 1:
361:indicator functions
1900:
1717:gamma distribution
1616:gamma distribution
1476:
1355:is log-concave if
1313:This implies that
1288:
1174:
1040:
945:
841:
752:) > 0
721:
701:
672:
469:
321:
153:
52:for short) if its
1898:
1820:
1781:survival function
1760:programs such as
1630:beta distribution
640:
526:Gaussian function
502: < 1
498:0 <
365:Gaussian function
354: < 1
350:0 <
198:of the function,
186: < 1
182:0 <
2249:
2223:
2194:
2165:
2138:
2099:
2098:
2086:
2074:
2065:
2064:
2038:
2028:
2019:
2018:
1998:
1989:
1983:
1982:
1957:
1909:
1907:
1906:
1901:
1899:
1897:
1877:
1863:
1855:
1851:
1821:
1819:
1808:
1790:(MHR), and is a
1743:random variables
1736:marginal density
1580:chi distribution
1485:
1483:
1482:
1477:
1366:
1360:
1354:
1331:
1297:
1295:
1294:
1289:
1232:
1222:
1199:
1183:
1181:
1180:
1175:
1086:
1079:
1072:
1066:
1049:
1047:
1046:
1041:
1039:
1038:
1020:
997:
954:
952:
951:
946:
944:
943:
892:
891:
850:
848:
847:
842:
840:
839:
788:
787:
753:
742:
730:
728:
727:
722:
710:
708:
707:
702:
681:
679:
678:
673:
656:
655:
643:
642:
641:
636:
635:
626:
604:
583:
577:
571:
565:
557:
545:
537:
503:
495:
478:
476:
475:
470:
468:
467:
443:
442:
355:
347:
330:
328:
327:
322:
204:
193:
187:
179:
162:
160:
159:
154:
152:
151:
127:
126:
43:
2257:
2256:
2252:
2251:
2250:
2248:
2247:
2246:
2242:Convex analysis
2227:
2226:
2212:
2199:
2183:
2170:
2154:
2141:
2127:
2111:
2108:
2103:
2102:
2097:(3–4): 301–316.
2084:
2078:Prékopa, András
2076:
2075:
2068:
2041:Economic Theory
2036:
2031:
2029:
2022:
1996:
1991:
1990:
1986:
1979:
1959:
1958:
1951:
1946:
1938:convex function
1919:
1878:
1864:
1832:
1828:
1812:
1802:
1801:
1499:
1492:is log-concave.
1375:
1374:
1362:
1356:
1333:
1318:
1241:
1240:
1224:
1209:
1200:is log-concave.
1191:
1095:
1094:
1081:
1074:
1068:
1062:
1058:
1030:
1013:
990:
973:
972:
935:
883:
866:
865:
831:
779:
762:
761:
744:
738:
713:
712:
693:
692:
647:
627:
617:
597:
592:
591:
579:
573:
567:
559:
547:
539:
528:
519:superlevel sets
510:
497:
483:
453:
434:
381:
380:
349:
335:
214:
213:
199:
189:
181:
167:
137:
118:
65:
64:
42:
28:
21:convex analysis
17:
12:
11:
5:
2255:
2253:
2245:
2244:
2239:
2229:
2228:
2225:
2224:
2210:
2196:
2195:
2181:
2167:
2166:
2152:
2139:
2125:
2107:
2104:
2101:
2100:
2066:
2047:(2): 445–469.
2020:
2009:(2): 445–467.
1984:
1977:
1948:
1947:
1945:
1942:
1941:
1940:
1935:
1930:
1925:
1918:
1915:
1914:
1913:
1912:
1911:
1896:
1893:
1890:
1887:
1884:
1881:
1876:
1873:
1870:
1867:
1861:
1858:
1854:
1850:
1847:
1844:
1841:
1838:
1835:
1831:
1827:
1824:
1818:
1815:
1811:
1796:
1795:
1784:
1777:
1758:Gibbs sampling
1746:
1739:
1732:
1721:
1720:
1713:
1706:
1699:
1686:Note that the
1684:
1683:
1680:F-distribution
1676:
1669:
1662:
1655:
1641:
1640:
1633:
1626:
1619:
1612:
1605:
1590:
1583:
1576:
1569:
1562:
1555:
1544:
1537:
1498:
1495:
1494:
1493:
1489:
1488:
1487:
1486:
1475:
1472:
1469:
1466:
1463:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1439:
1436:
1433:
1430:
1427:
1424:
1421:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1397:
1394:
1391:
1388:
1385:
1382:
1369:
1368:
1310:
1309:
1301:
1300:
1299:
1298:
1287:
1284:
1281:
1278:
1275:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1235:
1234:
1202:
1201:
1187:
1186:
1185:
1184:
1173:
1170:
1167:
1164:
1161:
1158:
1155:
1152:
1149:
1146:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1121:
1118:
1115:
1112:
1109:
1106:
1102:
1089:
1088:
1057:
1054:
1053:
1052:
1051:
1050:
1037:
1033:
1029:
1026:
1023:
1019:
1016:
1012:
1009:
1006:
1003:
1000:
996:
993:
989:
986:
983:
980:
967:
966:
959:
958:
957:
956:
942:
938:
934:
931:
928:
925:
922:
919:
916:
913:
910:
907:
904:
901:
898:
895:
890:
886:
882:
879:
876:
873:
860:
859:
855:
854:
853:
852:
838:
834:
830:
827:
824:
821:
818:
815:
812:
809:
806:
803:
800:
797:
794:
791:
786:
782:
778:
775:
772:
769:
756:
755:
735:
732:quasiconcavity
720:
711:log-concavity
700:
685:
684:
683:
682:
671:
668:
665:
662:
659:
654:
650:
646:
639:
634:
630:
624:
620:
616:
613:
610:
607:
603:
600:
586:
585:
522:
509:
506:
480:
479:
466:
463:
460:
456:
452:
449:
446:
441:
437:
433:
430:
427:
424:
421:
418:
415:
412:
409:
406:
403:
400:
397:
394:
391:
388:
332:
331:
320:
317:
314:
311:
308:
305:
302:
299:
296:
293:
290:
287:
284:
281:
278:
275:
272:
269:
266:
263:
260:
257:
254:
251:
248:
245:
242:
239:
236:
233:
230:
227:
224:
221:
164:
163:
150:
147:
144:
140:
136:
133:
130:
125:
121:
117:
114:
111:
108:
105:
102:
99:
96:
93:
90:
87:
84:
81:
78:
75:
72:
40:
15:
13:
10:
9:
6:
4:
3:
2:
2254:
2243:
2240:
2238:
2235:
2234:
2232:
2221:
2217:
2213:
2211:0-12-549250-2
2207:
2203:
2198:
2197:
2192:
2188:
2184:
2182:3-11-013863-8
2178:
2174:
2169:
2168:
2163:
2159:
2155:
2153:0-12-214690-5
2149:
2145:
2140:
2136:
2132:
2128:
2126:0-471-99545-2
2122:
2118:
2114:
2110:
2109:
2105:
2096:
2092:
2091:
2083:
2079:
2073:
2071:
2067:
2062:
2058:
2054:
2050:
2046:
2042:
2035:
2027:
2025:
2021:
2016:
2012:
2008:
2004:
2003:
1995:
1988:
1985:
1980:
1978:0-521-83378-7
1974:
1970:
1966:
1962:
1961:Boyd, Stephen
1956:
1954:
1950:
1943:
1939:
1936:
1934:
1931:
1929:
1926:
1924:
1921:
1920:
1916:
1891:
1885:
1882:
1879:
1871:
1865:
1859:
1856:
1852:
1845:
1839:
1836:
1833:
1829:
1825:
1822:
1816:
1813:
1809:
1800:
1799:
1798:
1797:
1793:
1789:
1785:
1782:
1778:
1775:
1771:
1767:
1763:
1759:
1755:
1751:
1747:
1744:
1740:
1737:
1733:
1730:
1726:
1725:
1724:
1718:
1714:
1711:
1707:
1704:
1700:
1697:
1693:
1692:
1691:
1689:
1681:
1677:
1674:
1670:
1667:
1663:
1660:
1656:
1653:
1649:
1648:
1647:
1644:
1638:
1634:
1631:
1627:
1624:
1620:
1617:
1613:
1610:
1606:
1603:
1599:
1595:
1591:
1588:
1584:
1581:
1577:
1574:
1570:
1567:
1563:
1560:
1556:
1553:
1549:
1545:
1542:
1538:
1535:
1531:
1527:
1526:
1525:
1523:
1519:
1516:
1512:
1508:
1504:
1496:
1491:
1490:
1473:
1470:
1464:
1461:
1458:
1452:
1449:
1446:
1443:
1440:
1434:
1428:
1422:
1419:
1416:
1410:
1407:
1404:
1398:
1389:
1386:
1383:
1373:
1372:
1371:
1370:
1365:
1359:
1352:
1348:
1344:
1340:
1336:
1332: =
1329:
1325:
1321:
1316:
1312:
1311:
1307:
1303:
1302:
1285:
1282:
1276:
1273:
1270:
1264:
1261:
1258:
1252:
1246:
1239:
1238:
1237:
1236:
1231:
1227:
1223: :
1220:
1216:
1212:
1207:
1204:
1203:
1198:
1194:
1189:
1188:
1165:
1159:
1153:
1147:
1141:
1138:
1135:
1129:
1123:
1119:
1116:
1110:
1104:
1100:
1093:
1092:
1091:
1090:
1085:
1078:
1071:
1065:
1060:
1059:
1055:
1035:
1024:
1017:
1014:
1007:
1001:
994:
991:
984:
978:
971:
970:
969:
968:
964:
961:
960:
940:
932:
926:
917:
911:
905:
899:
893:
888:
877:
871:
864:
863:
862:
861:
857:
856:
836:
828:
822:
813:
807:
801:
795:
789:
784:
773:
767:
760:
759:
758:
757:
751:
747:
741:
736:
733:
691:
687:
686:
669:
666:
660:
657:
652:
648:
637:
632:
628:
622:
618:
614:
608:
601:
598:
590:
589:
588:
587:
582:
576:
570:
563:
558: =
555:
551:
543:
538: =
535:
531:
527:
523:
520:
516:
515:quasi-concave
512:
511:
507:
505:
501:
494:
490:
486:
464:
461:
458:
450:
444:
439:
431:
425:
422:
416:
410:
407:
404:
398:
395:
392:
386:
379:
378:
377:
375:
374:
368:
366:
362:
357:
353:
346:
342:
338:
315:
309:
306:
303:
297:
294:
291:
285:
279:
273:
270:
267:
264:
261:
255:
249:
246:
243:
237:
234:
231:
225:
222:
219:
212:
211:
210:
208:
203:
197:
192:
185:
178:
174:
170:
148:
145:
142:
134:
128:
123:
115:
109:
106:
100:
94:
91:
88:
82:
79:
76:
70:
63:
62:
61:
59:
55:
51:
47:
39:
35:
31:
26:
22:
2201:
2172:
2143:
2116:
2094:
2088:
2044:
2040:
2006:
2000:
1987:
1968:
1722:
1685:
1645:
1642:
1601:
1597:
1517:
1510:
1500:
1363:
1357:
1350:
1346:
1342:
1338:
1334:
1327:
1323:
1319:
1229:
1225:
1218:
1214:
1210:
1196:
1192:
1083:
1076:
1069:
1063:
749:
745:
739:
580:
574:
568:
561:
553:
549:
541:
533:
529:
499:
492:
488:
484:
481:
371:
369:
358:
351:
344:
340:
336:
333:
201:
190:
183:
176:
172:
168:
165:
49:
45:
37:
33:
29:
25:non-negative
18:
1788:hazard rate
1315:convolution
743:satisfying
540:exp(−
209:; that is,
50:log-concave
2231:Categories
2106:References
1552:convex set
508:Properties
373:log-convex
58:convex set
1883:−
1860:−
1837:−
1826:
1550:over any
1450:∫
1420:−
1408:∫
1387:∗
1262:∫
1206:Marginals
1142:
1082:log
1075:log
1008:≤
924:∇
909:∇
906:−
885:∇
820:∇
805:∇
802:⪯
781:∇
719:⇒
699:⇒
690:concavity
667:≰
658:−
623:−
465:θ
462:−
440:θ
423:≤
411:θ
408:−
393:θ
307:
298:θ
295:−
271:
265:θ
262:≥
250:θ
247:−
232:θ
223:
196:logarithm
149:θ
146:−
124:θ
107:≥
95:θ
92:−
77:θ
27:function
2115:(1978).
2080:(1971).
1917:See also
1018:′
995:″
602:″
482:for all
334:for all
166:for all
32: :
2220:1162312
2191:1291393
2162:0954608
2135:0489333
2061:1046688
1345:)
560:−
207:concave
2218:
2208:
2189:
2179:
2160:
2150:
2133:
2123:
2059:
1975:
1731:(CDF).
1195:
572:. But
500:θ
491:∈ dom
352:θ
343:∈ dom
200:log ∘
184:θ
175:∈ dom
54:domain
2085:(PDF)
2057:S2CID
2037:(PDF)
1997:(PDF)
1944:Notes
1750:joint
1675:, and
1596:, if
1208:: if
205:, is
188:. If
56:is a
2206:ISBN
2177:ISBN
2148:ISBN
2121:ISBN
2030:See
1973:ISBN
1766:JAGS
1764:and
1762:BUGS
1715:the
1708:the
1701:the
1694:the
1678:the
1671:the
1664:the
1657:the
1650:the
1635:the
1628:the
1621:the
1614:the
1607:the
1604:+ 1,
1592:the
1585:the
1578:the
1571:the
1564:the
1557:the
1546:the
1539:the
1532:and
1528:the
1513:and
1361:and
1080:and
1067:and
858:i.e.
548:log
496:and
348:and
180:and
48:(or
23:, a
2049:doi
2011:doi
1823:log
1139:log
1120:log
1101:log
544:/2)
304:log
268:log
220:log
44:is
19:In
2233::
2216:MR
2214:.
2187:MR
2185:.
2158:MR
2156:.
2131:MR
2129:.
2095:32
2093:.
2087:.
2069:^
2055:.
2045:26
2043:.
2039:.
2023:^
2007:34
2005:.
1999:.
1967:.
1952:^
1600:≥
1308:).
955:is
564:/2
504:.
367:.
356:.
36:→
2222:.
2193:.
2164:.
2137:.
2063:.
2051::
2017:.
2013::
1981:.
1895:)
1892:x
1889:(
1886:F
1880:1
1875:)
1872:x
1869:(
1866:f
1857:=
1853:)
1849:)
1846:x
1843:(
1840:F
1834:1
1830:(
1817:x
1814:d
1810:d
1783:.
1705:,
1698:,
1682:.
1668:,
1661:,
1654:,
1602:p
1598:n
1589:,
1582:,
1575:,
1568:,
1561:,
1554:,
1543:,
1536:,
1518:D
1511:μ
1474:y
1471:d
1468:)
1465:y
1462:,
1459:x
1456:(
1453:h
1447:=
1444:y
1441:d
1438:)
1435:y
1432:(
1429:g
1426:)
1423:y
1417:x
1414:(
1411:f
1405:=
1402:)
1399:x
1396:(
1393:)
1390:g
1384:f
1381:(
1364:g
1358:f
1353:)
1351:y
1349:(
1347:g
1343:y
1341:-
1339:x
1337:(
1335:f
1330:)
1328:y
1326:,
1324:x
1322:(
1320:h
1286:y
1283:d
1280:)
1277:y
1274:,
1271:x
1268:(
1265:f
1259:=
1256:)
1253:x
1250:(
1247:g
1230:R
1226:R
1221:)
1219:y
1217:,
1215:x
1213:(
1211:f
1197:g
1193:f
1172:)
1169:)
1166:x
1163:(
1160:g
1157:)
1154:x
1151:(
1148:f
1145:(
1136:=
1133:)
1130:x
1127:(
1124:g
1117:+
1114:)
1111:x
1108:(
1105:f
1084:g
1077:f
1070:g
1064:f
1036:2
1032:)
1028:)
1025:x
1022:(
1015:f
1011:(
1005:)
1002:x
999:(
992:f
988:)
985:x
982:(
979:f
941:T
937:)
933:x
930:(
927:f
921:)
918:x
915:(
912:f
903:)
900:x
897:(
894:f
889:2
881:)
878:x
875:(
872:f
851:,
837:T
833:)
829:x
826:(
823:f
817:)
814:x
811:(
808:f
799:)
796:x
793:(
790:f
785:2
777:)
774:x
771:(
768:f
754:,
750:x
748:(
746:f
740:x
734:.
670:0
664:)
661:1
653:2
649:x
645:(
638:2
633:2
629:x
619:e
615:=
612:)
609:x
606:(
599:f
581:x
575:f
569:x
562:x
556:)
554:x
552:(
550:f
542:x
536:)
534:x
532:(
530:f
493:f
489:y
487:,
485:x
459:1
455:)
451:y
448:(
445:f
436:)
432:x
429:(
426:f
420:)
417:y
414:)
405:1
402:(
399:+
396:x
390:(
387:f
345:f
341:y
339:,
337:x
319:)
316:y
313:(
310:f
301:)
292:1
289:(
286:+
283:)
280:x
277:(
274:f
259:)
256:y
253:)
244:1
241:(
238:+
235:x
229:(
226:f
202:f
191:f
177:f
173:y
171:,
169:x
143:1
139:)
135:y
132:(
129:f
120:)
116:x
113:(
110:f
104:)
101:y
98:)
89:1
86:(
83:+
80:x
74:(
71:f
41:+
38:R
34:R
30:f
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.