1582:
625:
829:
1381:
834:
The advantage of this formulation (apart from its simplicity) is that it can be computed without performing two simulations, one in each specific ensemble. Indeed, it is possible to define an extra kind of "potential switching" Metropolis trial move (taken every fixed number of steps), such that the
317:
Suppose that two super states of interest, A and B, are given. We assume that they have a common configuration space, i.e., they share all of their micro states, but the energies associated to these (and hence the probabilities) differ because of a change in some parameter (such as the strength of a
1896:
439:
667:
1705:
1577:{\displaystyle \Delta F\approx \left\langle U_{\text{B}}-U_{\text{A}}\right\rangle _{\text{A}}-{\frac {\beta }{2}}\left(\left\langle (U_{\text{B}}-U_{\text{A}})^{2}\right\rangle _{\text{A}}-\left(\left\langle (U_{\text{B}}-U_{\text{A}})\right\rangle _{\text{A}}\right)^{2}\right)}
991:(or FEP), involves sampling from state A only. It requires that all the high probability configurations of super state B are contained in high probability configurations of super state A, which is a much more stringent requirement than the overlap condition stated above.
161:
2040:
1197:
1093:
1316:
950:
The densities of the two super states (in their common configuration space) should have a large overlap. Otherwise, a chain of super states between A and B may be needed, such that the overlap of each two consecutive super states is
620:{\displaystyle e^{-\beta (\Delta F-C)}={\frac {\left\langle f\left(\beta (U_{\text{B}}-U_{\text{A}}-C)\right)\right\rangle _{\text{A}}}{\left\langle f\left(\beta (U_{\text{A}}-U_{\text{B}}+C)\right)\right\rangle _{\text{B}}}}}
1808:
824:{\displaystyle e^{-\beta \Delta F}={\frac {\left\langle M\left(\beta (U_{\text{B}}-U_{\text{A}})\right)\right\rangle _{\text{A}}}{\left\langle M\left(\beta (U_{\text{A}}-U_{\text{B}})\right)\right\rangle _{\text{B}}}}}
1801:
644:
are the potential energies of the same configurations, calculated using potential function A (when the system is in superstate A) and potential function B (when the system is in the superstate B) respectively.
1609:
974:) is a generalization of the Bennett acceptance ratio that calculates the (relative) free energies of several multi states. It essentially reduces to the BAR method when only two super states are involved.
1952:
49:
1960:
312:
1373:
905:
2049:(or TI) result. It can be approximated by dividing the range between states A and B into many values of λ at which the expectation value is estimated, and performing numerical integration.
957:
The cost of simulating both ensembles should be approximately equal - and then, in fact, the system is sampled roughly equally in both super states. Otherwise, the optimal expression for
1104:
268:
340:) on moving between the two super states be calculated from sampling in both ensembles? The kinetic energy part in the free energy is equal between states so can be ignored. Also the
1002:
423:
1229:
940:
286:
ensemble), the resulting states along the simulated trajectory are likewise distributed. Averaging along the trajectory (in either formulation) is denoted by angle brackets
1234:
27:) is an algorithm for estimating the difference in free energy between two systems (usually the systems will be simulated on the computer). It was suggested by
1891:{\displaystyle {\frac {\partial F(\lambda )}{\partial \lambda }}=\left\langle {\frac {\partial U(\lambda )}{\partial \lambda }}\right\rangle _{\lambda }}
954:
The sample size should be large. In particular, as successive states are correlated, the simulation time should be much larger than the correlation time.
847:
is the most efficient, in the sense of yielding the smallest standard error for a given simulation time. He shows that the optimal choice is to take
942:. This value, of course, is not known (it is exactly what one is trying to compute), but it can be approximately chosen in a self-consistent manner.
28:
1721:
1700:{\displaystyle \langle U_{\text{B}}-U_{\text{A}}\rangle _{\text{B}}\leq \Delta F\leq \langle U_{\text{B}}-U_{\text{A}}\rangle _{\text{A}}}
2101:
1599:, gives an inequality in the linear level; combined with the analogous result for the B ensemble one gets the following version of the
1710:
Note that the inequality agrees with the negative sign of the coefficient of the (positive) variance term in the second order result.
1901:
156:{\displaystyle p({\text{State}}_{x}\rightarrow {\text{State}}_{y})=\min \left(e^{-\beta \,\Delta U},1\right)=M(\beta \,\Delta U)}
1898:
This can either be directly verified from definitions or seen from the limit of the above Gibbs-Bogoliubov inequalities when
2148:
2035:{\displaystyle \Delta F=\int _{0}^{1}\left\langle {\frac {\partial U(\lambda )}{\partial \lambda }}\right\rangle \,d\lambda }
1202:
This exact result can be obtained from the general BAR method, using (for example) the
Metropolis function, in the limit
289:
1587:
Note that the first term is the expected value of the energy difference, while the second is essentially its variance.
1375:
and Taylor expanding the second exact perturbation theory expression to the second order, one gets the approximation
1231:. Indeed, in that case, the denominator of the general case expression above tends to 1, while the numerator tends to
961:
is modified, and the sampling should devote equal times (rather than equal number of time steps) to the two ensembles.
1329:
908:
853:
1192:{\displaystyle \Delta F=-kT\cdot \log \left\langle e^{\beta (U_{\text{A}}-U_{\text{B}})}\right\rangle _{\text{A}}}
2046:
2127:
1088:{\displaystyle e^{-\beta \Delta F}=\left\langle e^{-\beta (U_{\text{B}}-U_{\text{A}})}\right\rangle _{\text{A}}}
988:
213:
363:
2143:
1595:
Using the convexity of the log function appearing in the exact perturbation analysis result, together with
1205:
1596:
271:
40:
1600:
319:
43:
walk it is possible to sample the landscape of states that the system moves between, using the equation
1311:{\displaystyle e^{\beta C}\left\langle e^{-\beta (U_{\text{B}}-U_{\text{A}})}\right\rangle _{\text{A}}}
916:
657:
the
Metropolis function defined above (which satisfies the detailed balance condition), and setting
2121:
2096:
Charles H. Bennett (1976) Efficient estimation of free energy differences from Monte Carlo data.
2077:
2058:
279:
207:
341:
426:
2115:
2137:
270:
is the
Metropolis function. The resulting states are then sampled according to the
318:
certain interaction). The basic question to be addressed is, then, how can the
1318:. A direct derivation from the definitions is more straightforward, though.
16:
Algorithm for estimating the difference in free energy between two systems
835:
single sampling from the "mixed" ensemble suffices for the computation.
2066:
2062:
1796:{\displaystyle U_{\text{A}}=U(\lambda =0),U_{\text{B}}=U(\lambda =1),}
1718:
writing the potential energy as depending on a continuous parameter,
199:
39:
Take a system in a certain super (i.e. Gibbs) state. By performing a
2065:. Python-based code for MBAR and BAR is available for download at
2128:
Weighted
Histogram Analysis Method (MBAR being the unbinned case)
1947:{\displaystyle {\text{A}}=\lambda ^{+},{\text{B}}=\lambda ^{-}}
278:. Alternatively, if the system is dynamically simulated in the
946:
Some assumptions needed for the efficiency are the following:
2057:
The
Bennett acceptance ratio method is implemented in modern
1963:
1904:
1811:
1724:
1612:
1384:
1332:
1237:
1208:
1107:
1005:
919:
856:
670:
442:
366:
292:
216:
52:
911:(satisfying indeed the detailed balance condition).
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1946:
1890:
1795:
1699:
1576:
1367:
1310:
1223:
1191:
1087:
934:
899:
823:
619:
417:
307:{\displaystyle \left\langle \cdots \right\rangle }
306:
262:
155:
843:Bennett explores which specific expression for Δ
232:
92:
1368:{\displaystyle U_{\text{B}}-U_{\text{A}}\ll kT}
900:{\displaystyle f(x)\equiv {\frac {1}{1+e^{x}}}}
190:) is the difference in potential energy, β = 1/
8:
1688:
1661:
1640:
1613:
2025:
1992:
1982:
1977:
1962:
1938:
1926:
1917:
1905:
1903:
1882:
1849:
1812:
1810:
1763:
1729:
1723:
1691:
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1611:
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1183:
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1106:
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1004:
970:The multistate Bennett acceptance ratio (
918:
888:
872:
855:
813:
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719:
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669:
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263:{\displaystyle M(x)\equiv \min(e^{-x},1)}
242:
215:
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111:
104:
80:
75:
65:
60:
51:
429:condition), and for every energy offset
2089:
418:{\displaystyle f(x)/f(-x)\equiv e^{-x}}
356:Bennett shows that for every function
1322:The second order (approximate) result
1224:{\displaystyle C\rightarrow -\infty }
7:
1714:The thermodynamic integration method
2122:Multistate Bennett Acceptance Ratio
966:Multistate Bennett acceptance ratio
2012:
1995:
1964:
1869:
1852:
1832:
1815:
1652:
1385:
1218:
1108:
1017:
926:
682:
457:
274:of the super state at temperature
144:
112:
14:
995:The exact (infinite order) result
935:{\displaystyle C\approx \Delta F}
433:, one has the exact relationship
2098:Journal of Computational Physics
2007:
2001:
1864:
1858:
1827:
1821:
1787:
1775:
1753:
1741:
1545:
1519:
1485:
1458:
1293:
1267:
1212:
1174:
1148:
1070:
1044:
983:The perturbation theory method
866:
860:
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774:
738:
712:
596:
564:
528:
496:
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454:
396:
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370:
257:
235:
226:
220:
150:
137:
86:
71:
56:
1:
1591:The first order inequalities
1601:Gibbs-Bogoliubov inequality
907:, which is essentially the
2165:
1805:one has the exact result
425:(which is essentially the
2047:thermodynamic integration
1954:. we can therefore write
987:This method, also called
978:Relation to other methods
360:satisfying the condition
2116:Bennett Acceptance Ratio
2100:22 : 245–268
989:Free energy perturbation
909:Fermi–Dirac distribution
21:Bennett acceptance ratio
839:The most efficient case
2036:
1948:
1892:
1797:
1701:
1578:
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1312:
1225:
1193:
1089:
936:
901:
825:
621:
419:
308:
272:Boltzmann distribution
264:
198:is the temperature in
157:
41:Metropolis Monte Carlo
2149:Statistical mechanics
2130:from AlchemistryWiki.
2124:from AlchemistryWiki.
2118:from AlchemistryWiki.
2037:
1949:
1893:
1798:
1702:
1579:
1370:
1313:
1226:
1194:
1090:
937:
902:
826:
622:
420:
320:Helmholtz free energy
309:
265:
158:
1961:
1902:
1809:
1722:
1610:
1382:
1330:
1235:
1206:
1105:
1003:
917:
854:
668:
440:
364:
290:
214:
180:) −
50:
1987:
1597:Jensen's inequality
344:corresponds to the
333: −
2078:Parallel tempering
2059:molecular dynamics
2032:
1973:
1944:
1888:
1793:
1697:
1574:
1365:
1308:
1221:
1189:
1085:
932:
897:
821:
617:
415:
304:
280:canonical ensemble
260:
208:Boltzmann constant
153:
29:Charles H. Bennett
2061:systems, such as
2019:
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1908:
1876:
1839:
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1529:
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1407:
1353:
1340:
1305:
1290:
1277:
1186:
1171:
1158:
1082:
1067:
1054:
895:
819:
816:
797:
784:
754:
735:
722:
653:Substituting for
615:
612:
587:
574:
544:
519:
506:
342:Gibbs free energy
282:(also called the
78:
63:
2156:
2103:
2094:
2041:
2039:
2038:
2033:
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2020:
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2010:
1993:
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1504:
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1418:
1409:
1408:
1405:
1374:
1372:
1371:
1366:
1355:
1354:
1351:
1342:
1341:
1338:
1317:
1315:
1314:
1309:
1307:
1306:
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1297:
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1292:
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1288:
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1275:
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1222:
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1094:
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1084:
1083:
1080:
1078:
1074:
1073:
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1068:
1065:
1056:
1055:
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1023:
941:
939:
938:
933:
906:
904:
903:
898:
896:
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893:
892:
873:
830:
828:
827:
822:
820:
818:
817:
814:
812:
808:
807:
803:
799:
798:
795:
786:
785:
782:
756:
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750:
746:
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741:
737:
736:
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724:
723:
720:
694:
689:
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623:
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546:
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542:
540:
536:
535:
531:
521:
520:
517:
508:
507:
504:
478:
473:
472:
427:detailed balance
424:
422:
421:
416:
414:
413:
383:
352:The general case
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305:
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269:
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154:
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119:
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85:
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79:
76:
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69:
64:
61:
2164:
2163:
2159:
2158:
2157:
2155:
2154:
2153:
2134:
2133:
2112:
2107:
2106:
2095:
2091:
2086:
2074:
2055:
2011:
1994:
1988:
1959:
1958:
1934:
1913:
1900:
1899:
1868:
1851:
1845:
1844:
1831:
1814:
1807:
1806:
1759:
1725:
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1716:
1687:
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1535:
1522:
1518:
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766:
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728:
715:
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661:to zero, gives
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567:
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288:
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238:
212:
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189:
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100:
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59:
48:
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37:
17:
12:
11:
5:
2162:
2160:
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2144:Thermodynamics
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2125:
2119:
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2110:External links
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2053:Implementation
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1326:Assuming that
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649:The basic case
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68:
58:
55:
36:
33:
15:
13:
10:
9:
6:
4:
3:
2:
2161:
2150:
2147:
2145:
2142:
2141:
2139:
2129:
2126:
2123:
2120:
2117:
2114:
2113:
2109:
2102:
2099:
2093:
2090:
2083:
2079:
2076:
2075:
2071:
2069:
2067:
2064:
2060:
2052:
2050:
2048:
2045:which is the
2029:
2026:
2021:
2015:
2004:
1998:
1989:
1983:
1978:
1974:
1970:
1967:
1957:
1956:
1955:
1939:
1935:
1931:
1923:
1918:
1914:
1910:
1883:
1878:
1872:
1861:
1855:
1846:
1841:
1835:
1824:
1818:
1803:
1790:
1784:
1781:
1778:
1772:
1769:
1760:
1756:
1750:
1747:
1744:
1738:
1735:
1726:
1713:
1711:
1678:
1674:
1665:
1658:
1655:
1649:
1630:
1626:
1617:
1606:
1605:
1604:
1602:
1598:
1590:
1588:
1570:
1564:
1559:
1549:
1536:
1532:
1523:
1515:
1510:
1505:
1495:
1489:
1475:
1471:
1462:
1454:
1448:
1442:
1439:
1434:
1424:
1414:
1410:
1401:
1396:
1391:
1388:
1378:
1377:
1376:
1362:
1359:
1356:
1347:
1343:
1334:
1321:
1319:
1298:
1284:
1280:
1271:
1264:
1261:
1257:
1253:
1246:
1243:
1239:
1215:
1209:
1179:
1165:
1161:
1152:
1145:
1141:
1137:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1101:
1100:
1099:
1075:
1061:
1057:
1048:
1041:
1038:
1034:
1030:
1025:
1020:
1014:
1011:
1007:
999:
998:
994:
992:
990:
982:
977:
975:
973:
965:
960:
956:
953:
949:
948:
947:
929:
923:
920:
913:
910:
889:
885:
881:
878:
874:
869:
863:
857:
850:
849:
848:
846:
838:
836:
809:
804:
791:
787:
778:
771:
767:
763:
759:
747:
742:
729:
725:
716:
709:
705:
701:
697:
690:
685:
679:
676:
672:
664:
663:
662:
660:
656:
648:
646:
640:
633:
605:
600:
593:
590:
581:
577:
568:
561:
557:
553:
549:
537:
532:
525:
522:
513:
509:
500:
493:
489:
485:
481:
474:
466:
463:
460:
451:
448:
444:
436:
435:
434:
432:
428:
410:
407:
403:
399:
393:
390:
384:
380:
373:
367:
359:
351:
349:
347:
343:
336:
329:
326: =
325:
321:
315:
300:
297:
294:
285:
281:
277:
273:
254:
251:
246:
243:
239:
229:
223:
217:
209:
205:
201:
197:
193:
188:
183:
178:
173:
169:
147:
140:
134:
131:
127:
123:
120:
115:
108:
105:
101:
96:
89:
81:
66:
53:
46:
45:
44:
42:
35:Preliminaries
34:
32:
30:
26:
22:
2097:
2092:
2056:
2044:
1804:
1717:
1709:
1594:
1586:
1325:
1201:
1097:
986:
971:
969:
958:
945:
844:
842:
833:
658:
654:
652:
638:
631:
629:
430:
357:
355:
345:
334:
327:
323:
316:
283:
275:
203:
195:
191:
186:
181:
176:
171:
167:
165:
38:
24:
20:
18:
2138:Categories
2084:References
348:ensemble.
2030:λ
2016:λ
2013:∂
2005:λ
1996:∂
1975:∫
1965:Δ
1940:−
1936:λ
1915:λ
1884:λ
1873:λ
1870:∂
1862:λ
1853:∂
1836:λ
1833:∂
1825:λ
1816:∂
1779:λ
1745:λ
1689:⟩
1675:−
1662:⟨
1659:≤
1653:Δ
1650:≤
1641:⟩
1627:−
1614:⟨
1533:−
1506:−
1472:−
1440:β
1435:−
1411:−
1392:≈
1386:Δ
1357:≪
1344:−
1281:−
1265:β
1262:−
1244:β
1219:∞
1216:−
1213:→
1162:−
1146:β
1133:
1127:⋅
1118:−
1109:Δ
1058:−
1042:β
1039:−
1018:Δ
1015:β
1012:−
951:adequate.
927:Δ
924:≈
870:≡
788:−
772:β
726:−
710:β
683:Δ
680:β
677:−
578:−
562:β
523:−
510:−
494:β
464:−
458:Δ
452:β
449:−
408:−
400:≡
391:−
298:⋯
244:−
230:≡
145:Δ
141:β
113:Δ
109:β
106:−
72:→
31:in 1976.
2072:See also
2022:⟩
1990:⟨
1879:⟩
1847:⟨
1550:⟩
1516:⟨
1496:⟩
1455:⟨
1425:⟩
1397:⟨
1299:⟩
1254:⟨
1180:⟩
1138:⟨
1076:⟩
1031:⟨
810:⟩
760:⟨
748:⟩
698:⟨
606:⟩
550:⟨
538:⟩
482:⟨
301:⟩
295:⟨
202:, while
23:method (
2063:Gromacs
210:), and
206:is the
200:kelvins
170:=
166:where Δ
630:where
184:(State
174:(State
77:State
62:State
1098:or
972:MBAR
637:and
19:The
1130:log
346:NpT
284:NVT
233:min
93:min
25:BAR
2140::
2068:.
1603::
314:.
192:kT
2027:d
2008:)
2002:(
1999:U
1984:1
1979:0
1971:=
1968:F
1932:=
1928:B
1924:,
1919:+
1911:=
1907:A
1865:)
1859:(
1856:U
1842:=
1828:)
1822:(
1819:F
1791:,
1788:)
1785:1
1782:=
1776:(
1773:U
1770:=
1765:B
1761:U
1757:,
1754:)
1751:0
1748:=
1742:(
1739:U
1736:=
1731:A
1727:U
1693:A
1683:A
1679:U
1670:B
1666:U
1656:F
1645:B
1635:A
1631:U
1622:B
1618:U
1571:)
1565:2
1560:)
1555:A
1546:)
1541:A
1537:U
1528:B
1524:U
1520:(
1511:(
1501:A
1490:2
1486:)
1480:A
1476:U
1467:B
1463:U
1459:(
1449:(
1443:2
1430:A
1419:A
1415:U
1406:B
1402:U
1389:F
1363:T
1360:k
1352:A
1348:U
1339:B
1335:U
1304:A
1294:)
1289:A
1285:U
1276:B
1272:U
1268:(
1258:e
1247:C
1240:e
1210:C
1185:A
1175:)
1170:B
1166:U
1157:A
1153:U
1149:(
1142:e
1124:T
1121:k
1115:=
1112:F
1081:A
1071:)
1066:A
1062:U
1053:B
1049:U
1045:(
1035:e
1026:=
1021:F
1008:e
959:C
930:F
921:C
890:x
886:e
882:+
879:1
875:1
867:)
864:x
861:(
858:f
845:F
815:B
805:)
801:)
796:B
792:U
783:A
779:U
775:(
768:(
764:M
753:A
743:)
739:)
734:A
730:U
721:B
717:U
713:(
706:(
702:M
691:=
686:F
673:e
659:C
655:f
642:B
639:U
635:A
632:U
611:B
601:)
597:)
594:C
591:+
586:B
582:U
573:A
569:U
565:(
558:(
554:f
543:A
533:)
529:)
526:C
518:A
514:U
505:B
501:U
497:(
490:(
486:f
475:=
470:)
467:C
461:F
455:(
445:e
431:C
411:x
404:e
397:)
394:x
388:(
385:f
381:/
377:)
374:x
371:(
368:f
358:f
338:A
335:F
331:B
328:F
324:F
276:T
258:)
255:1
252:,
247:x
240:e
236:(
227:)
224:x
221:(
218:M
204:k
196:T
194:(
187:x
182:U
177:y
172:U
168:U
151:)
148:U
138:(
135:M
132:=
128:)
124:1
121:,
116:U
102:e
97:(
90:=
87:)
82:y
67:x
57:(
54:p
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