3401:
2556:
3396:{\displaystyle {\begin{aligned}L&={\begin{pmatrix}g_{1}+h_{2}&g_{2}+h_{1}&g_{3}&h_{3}\\g_{2}+h_{1}&-g_{1}+h_{2}&h_{3}&-g_{3}\\g_{3}&h_{3}&-g_{1}-h_{2}&g_{2}-h_{1}\\h_{3}&-g_{3}&g_{2}-h_{1}&g_{1}+h_{2}\\\end{pmatrix}}\lambda ^{-1}\\&+{\begin{pmatrix}0&0&-l_{2}&-l_{1}\\0&0&l_{1}&-l_{2}\\l_{2}&-l_{1}&-2\lambda &-2l_{3}\\l_{1}&l_{2}&2l_{3}&2\lambda \\\end{pmatrix}},\\P&={\frac {-1}{2}}{\begin{pmatrix}0&-2l_{3}&l_{2}&l_{1}\\2l_{3}&0&-l_{1}&l_{2}\\-l_{2}&l_{1}&2\lambda &2l_{3}+\gamma \\-l_{1}&-l_{2}&-2l_{3}&-2\lambda \\\end{pmatrix}}.\end{aligned}}}
4296:
1564:, the Lax pair refers to matrices that satisfy the zero-curvature equation rather than the Lax equation. Furthermore, the zero-curvature representation makes the link between integrable systems and geometry manifest, culminating in
1989:
1260:
663:
2500:
815:
948:
1816:
3515:
2077:
2561:
1560:
Any PDE which admits a Lax-pair representation also admits a zero-curvature representation. In fact, the zero-curvature representation is more general and for other integrable PDEs, such as the
1480:
505:
868:
764:
2412:
1651:
2308:
186:
1051:
1513:
2224:
2178:
1718:
4185:
1298:
1142:
2548:
117:
2097:
1104:
1387:
2359:
543:
239:
2132:
1683:
1320:
1184:
1542:
1349:
1084:
409:
3848:
2017:
1415:
1164:
888:
377:
323:
303:
283:
263:
3536:. Aside from a factor, observables (without explicit time dependence) in this picture can thus be seen to form Lax pairs together with the Hamiltonian. The
4011:
1825:
4138:
3993:
3969:
2517:
The previous example used an infinite-dimensional
Hilbert space. Examples are also possible with finite-dimensional Hilbert spaces. These include
893:
3606:
1723:
2022:
3702:
1189:
3571:
3861:
3558:
554:
3950:
3841:
3677:
1424:
2423:
771:
4335:
4220:
3611:
4320:
3865:
3525:
3576:
3436:
2240:
3652:
3563:
3772:
A. Sergyeyev, New integrable (3+1)-dimensional systems and contact geometry, Lett. Math. Phys. 108 (2018), no. 2, 359-376,
4016:
4072:
3591:
2415:
4299:
4021:
4006:
3834:
64:
4036:
4281:
4041:
2180:. When a PDE admits a zero-curvature representation but not a Lax equation representation, the connection components
4325:
4235:
4159:
4276:
3552:
4092:
951:
417:
4026:
1569:
834:
4330:
4128:
3929:
4001:
725:
4225:
2370:
1583:
2248:
4256:
4200:
4164:
3540:
is then interpreted as the alternative expression in terms of isospectral evolution of these observables.
3537:
40:
826:
133:
44:
1991:. This differs from the conventional expression by some minus signs, which are ultimately unimportant.
1021:
1489:
4239:
3731:
3619:
3596:
1561:
4205:
4143:
3857:
2183:
2137:
1688:
4230:
4097:
3773:
3755:
3586:
3412:
1265:
1109:
32:
2524:
2505:
where all derivatives act on all objects to the right. This accounts for the infinite number of
78:
4210:
3747:
3698:
3673:
3648:
3416:
2082:
1089:
338:
1354:
4215:
4133:
4102:
4082:
4067:
4062:
4057:
3894:
3804:
3780:
3739:
2518:
2319:
971:
697:
513:
194:
60:
2105:
1656:
1305:
1169:
4077:
4031:
3979:
3974:
3945:
3826:
3533:
2506:
2099:, instead the Lax equation arises as a consistency equation for an overdetermined system.
1565:
1518:
1483:
1325:
1060:
385:
3904:
1544:
By the isospectral property, this curve is preserved under time translation. This is the
3735:
4266:
4118:
3919:
3720:"The Kowalewski top 99 years later: a Lax pair, generalizations and explicit solutions"
3548:
Further examples of systems of equations that can be formulated as a Lax pair include:
2002:
1819:
1549:
1400:
1149:
873:
546:
362:
308:
288:
268:
248:
719:
is generally known better, and to propagate the solution with the following formulas:
4314:
4271:
4195:
3924:
3909:
3899:
3812:
3759:
3581:
120:
3795:
Lax, P. (1968), "Integrals of nonlinear equations of evolution and solitary waves",
4261:
3914:
3884:
3821:
3601:
962:
The above property is the basis for the inverse scattering method. In this method,
681:
3719:
1653:
where the subscripts denote coordinate indices rather than derivatives. Often the
1984:{\displaystyle F_{\mu \nu }==-\partial _{\mu }A_{\nu }+\partial _{\nu }A_{\mu }+}
4190:
4180:
4087:
3889:
3623:
1580:
The zero-curvature equations are described by a pair of matrix-valued functions
1418:
954:
can be written in terms of these traces, the spectrum is preserved by the flow.
355:
28:
4123:
3963:
3959:
3955:
3784:
3420:
334:
242:
17:
3751:
52:
3808:
2226:
are referred to as the Lax pair, and the connection as a Lax connection.
3743:
1568:'s programme to formulate known integrable systems as solutions to the
56:
3670:
Integrable systems : twistors, loop groups, and
Riemann surfaces
3645:
Integrable systems : twistors, loop groups, and
Riemann surfaces
1255:{\displaystyle {\frac {\partial \psi }{\partial t}}(t,x)=P\psi (t,x)}
3718:
Bobenko, A. I.; Reyman, A. G.; Semenov-Tian-Shansky, M. A. (1989).
3777:
2079:
If we instead enforce these, together with time independence of
658:{\displaystyle {\frac {d}{dt}}U(t,s)=P(t)U(t,s),\quad U(s,s)=I,}
3830:
1397:
If the Lax matrix additionally depends on a complex parameter
2495:{\displaystyle P=-4\partial _{x}^{3}+6u\partial _{x}+3u_{x},}
810:{\displaystyle {\frac {\partial \psi }{\partial t}}=P\psi .}
715:, it is possible to solve the same problem at time 0, where
943:{\displaystyle {\frac {d}{dt}}\operatorname {tr} (L^{n})=0}
1818:
It is so called as it corresponds to the vanishing of the
1002:) which is to be determined. It is generally assumed that
1811:{\displaystyle \partial _{t}A_{x}-\partial _{x}A_{t}+=0.}
1720:
and its derivatives. The zero-curvature equation is then
285:
in a prescribed way, so this is a nonlinear equation for
3510:{\displaystyle {\frac {d}{dt}}A(t)={\frac {i}{\hbar }},}
2072:{\displaystyle L\psi =\lambda \psi ,\psi _{t}+A\psi =0.}
3162:
2930:
2579:
67:
makes use of the Lax equations to solve such systems.
3439:
2559:
2527:
2426:
2373:
2322:
2251:
2186:
2140:
2108:
2085:
2025:
2005:
1828:
1726:
1691:
1659:
1586:
1521:
1492:
1427:
1403:
1357:
1328:
1308:
1268:
1192:
1172:
1152:
1112:
1092:
1063:
1024:
896:
876:
837:
774:
728:
557:
516:
420:
388:
365:
311:
291:
271:
251:
197:
136:
81:
3618:
The last is remarkable, as it implies that both the
2521:
and the generalization to include an electric field
4249:
4173:
4152:
4111:
4050:
3992:
3938:
3873:
3697:. Oxford: Oxford University Press. pp. 54â56.
4186:Spectral theory of ordinary differential equations
3509:
3395:
2542:
2494:
2406:
2353:
2302:
2218:
2172:
2126:
2091:
2071:
2011:
1983:
1810:
1712:
1677:
1645:
1536:
1507:
1474:
1409:
1381:
1343:
1314:
1292:
1254:
1178:
1158:
1136:
1098:
1078:
1045:
942:
882:
862:
831:The result can also be shown using the invariants
809:
758:
657:
537:
499:
403:
371:
317:
297:
277:
257:
233:
180:
111:
2134:can be used to define the connection components
1428:
703:In other words, to solve the eigenvalue problem
1685:dependence is through a single scalar function
3797:Communications on Pure and Applied Mathematics
1475:{\displaystyle \det {\big (}wI-L(z){\big )}=0}
75:A Lax pair is a pair of matrices or operators
3842:
1461:
1433:
8:
1031:
1025:
3819:Scattering Theory for Automorphic Functions
500:{\displaystyle L(t)=U(t,s)L(s)U(t,s)^{-1},}
3877:
3849:
3835:
3827:
1053:The method then takes the following form:
863:{\displaystyle \operatorname {tr} (L^{n})}
672:denotes the identity matrix. Note that if
382:The core observation is that the matrices
3568:Integrable systems with contact Lax pairs
3470:
3440:
3438:
3361:
3343:
3328:
3305:
3282:
3270:
3253:
3241:
3221:
3204:
3192:
3180:
3157:
3142:
3103:
3088:
3076:
3062:
3033:
3018:
3004:
2989:
2965:
2950:
2925:
2906:
2888:
2875:
2863:
2850:
2838:
2823:
2809:
2796:
2784:
2771:
2756:
2744:
2730:
2715:
2703:
2690:
2675:
2662:
2648:
2636:
2624:
2611:
2599:
2586:
2574:
2560:
2558:
2529:
2528:
2526:
2483:
2467:
2448:
2443:
2425:
2392:
2387:
2372:
2327:
2321:
2288:
2275:
2256:
2250:
2207:
2194:
2185:
2161:
2148:
2139:
2107:
2084:
2048:
2024:
2004:
1999:For an eigensolution to the Lax operator
1972:
1959:
1943:
1933:
1920:
1910:
1891:
1878:
1865:
1852:
1833:
1827:
1793:
1780:
1764:
1754:
1741:
1731:
1725:
1690:
1658:
1619:
1591:
1585:
1520:
1499:
1495:
1494:
1491:
1460:
1459:
1432:
1431:
1426:
1402:
1356:
1327:
1307:
1267:
1193:
1191:
1171:
1151:
1111:
1091:
1062:
1023:
925:
897:
895:
875:
851:
836:
775:
773:
727:
558:
556:
515:
485:
419:
387:
364:
310:
290:
270:
250:
196:
137:
135:
80:
4139:Group algebra of a locally compact group
2313:can be reformulated as the Lax equation
39:is a pair of time-dependent matrices or
3635:
3475:
958:Link with the inverse scattering method
950:due to the Lax equation, and since the
759:{\displaystyle \lambda (t)=\lambda (0)}
3724:Communications in Mathematical Physics
3607:Lagrange, Euler, and Kovalevskaya tops
2407:{\displaystyle L=-\partial _{x}^{2}+u}
1646:{\displaystyle A_{x}(x,t),A_{t}(x,t),}
1548:. Such curves appear in the theory of
2303:{\displaystyle u_{t}=6uu_{x}-u_{xxx}}
990:)) and depend on an unknown function
119:dependent on time, acting on a fixed
7:
3823:, (1976) Princeton University Press.
3592:Modified Kortewegâde Vries equation
3695:Solitons, instantons, and twistors
2464:
2440:
2384:
1930:
1907:
1875:
1849:
1751:
1728:
1322:in the scattering region, compute
1204:
1196:
1037:
786:
778:
245:. Often, as in the example below,
181:{\displaystyle {\frac {dL}{dt}}=,}
25:
1046:{\displaystyle \|x\|\to \infty .}
4295:
4294:
4221:Topological quantum field theory
1508:{\displaystyle \mathbb {C} ^{2}}
3626:can be understood as solitons.
3572:KadomtsevâPetviashvili equation
3559:non-linear Schrödinger equation
1146:In the scattering region where
1018:in the scattering region where
766: (no change in spectrum),
627:
51:. Lax pairs were introduced by
3501:
3498:
3492:
3480:
3464:
3458:
2534:
2348:
2336:
2213:
2187:
2167:
2141:
2121:
2109:
1978:
1952:
1897:
1845:
1799:
1773:
1707:
1695:
1672:
1660:
1637:
1625:
1609:
1597:
1456:
1450:
1373:
1361:
1338:
1332:
1284:
1272:
1249:
1237:
1225:
1213:
1128:
1116:
1073:
1067:
1034:
931:
918:
857:
844:
753:
747:
738:
732:
643:
631:
621:
609:
603:
597:
588:
576:
532:
520:
482:
469:
463:
457:
451:
439:
430:
424:
398:
392:
333:It can then be shown that the
210:
198:
172:
160:
106:
100:
91:
85:
1:
4017:Uniform boundedness principle
2219:{\displaystyle (A_{x},A_{t})}
2173:{\displaystyle (A_{x},A_{t})}
1713:{\displaystyle \varphi (x,t)}
1556:Zero-curvature representation
411:are all similar by virtue of
43:that satisfy a corresponding
821:Through principal invariants
65:inverse scattering transform
3672:. Oxford: Clarendon Press.
3647:. Oxford: Clarendon Press.
3612:BelinskiâZakharov transform
1293:{\displaystyle \psi (0,x).}
1137:{\displaystyle \psi (0,x).}
4352:
4160:Invariant subspace problem
3817:P. Lax and R.S. Phillips,
3577:Kortewegâde Vries equation
2543:{\displaystyle {\vec {h}}}
2241:Kortewegâde Vries equation
2235:Kortewegâde Vries equation
1995:Lax pair to zero-curvature
1417:(as is the case for, say,
824:
4290:
3880:
3785:10.1007/s11005-017-1013-4
3693:Dunajski, Maciej (2010).
1570:anti-self-dual YangâMills
952:characteristic polynomial
349:. The matrices/operators
112:{\displaystyle L(t),P(t)}
4129:Spectrum of a C*-algebra
3614:, in general relativity.
2416:SturmâLiouville operator
2092:{\displaystyle \lambda }
1822:, which in this case is
1099:{\displaystyle \lambda }
1057:Compute the spectrum of
4336:Exactly solvable models
4226:Noncommutative geometry
3668:Hitchin, N. J. (1999).
3643:Hitchin, N. J. (1999).
3564:DaveyâStewartson system
1576:Zero-curvature equation
1382:{\displaystyle u(t,x).}
1262:with initial condition
545:is the solution of the
337:and more generally the
4321:Differential equations
4282:TomitaâTakesaki theory
4257:Approximation property
4201:Calculus of variations
3809:10.1002/cpa.3160210503
3557:One-dimensional cubic
3511:
3426:without explicit time
3397:
2544:
2496:
2408:
2355:
2354:{\displaystyle L_{t}=}
2304:
2220:
2174:
2128:
2093:
2073:
2013:
1985:
1812:
1714:
1679:
1647:
1538:
1509:
1476:
1411:
1383:
1345:
1316:
1294:
1256:
1180:
1160:
1138:
1100:
1080:
1047:
944:
884:
864:
811:
760:
659:
539:
538:{\displaystyle U(t,s)}
501:
405:
373:
319:
299:
279:
259:
235:
234:{\displaystyle =PL-LP}
182:
113:
4277:BanachâMazur distance
4240:Generalized functions
3553:BenjaminâOno equation
3512:
3430:dependence satisfies
3398:
2545:
2509:of the KdV equation.
2497:
2409:
2356:
2305:
2221:
2175:
2129:
2127:{\displaystyle (L,P)}
2094:
2074:
2014:
1986:
1813:
1715:
1680:
1678:{\displaystyle (x,t)}
1648:
1539:
1510:
1477:
1412:
1384:
1346:
1317:
1315:{\displaystyle \psi }
1295:
1257:
1181:
1179:{\displaystyle \psi }
1161:
1139:
1101:
1081:
1048:
1010:) is known, and that
945:
885:
865:
827:Invariants of tensors
812:
761:
660:
540:
502:
406:
374:
320:
300:
280:
260:
236:
183:
114:
45:differential equation
4022:Kakutani fixed-point
4007:Riesz representation
3620:Schwarzschild metric
3597:Sine-Gordon equation
3437:
2557:
2525:
2424:
2371:
2320:
2249:
2184:
2138:
2106:
2083:
2023:
2003:
1826:
1724:
1689:
1657:
1584:
1562:sine-Gordon equation
1537:{\displaystyle w,z.}
1519:
1490:
1425:
1401:
1355:
1344:{\displaystyle L(t)}
1326:
1306:
1266:
1190:
1170:
1166:is known, propagate
1150:
1110:
1090:
1079:{\displaystyle L(0)}
1061:
1022:
894:
874:
835:
772:
726:
555:
514:
418:
404:{\displaystyle L(t)}
386:
363:
329:Isospectral property
309:
289:
269:
249:
195:
134:
79:
4206:Functional calculus
4165:Mahler's conjecture
4144:Von Neumann algebra
3858:Functional analysis
3736:1989CMaPh.122..321B
3538:Schrödinger picture
2453:
2397:
1572:(ASDYM) equations.
1014:does not depend on
345:are independent of
31:, in the theory of
4231:Riemann hypothesis
3930:Topological vector
3744:10.1007/BF01257419
3587:Marchenko equation
3507:
3413:Heisenberg picture
3407:Heisenberg picture
3393:
3391:
3380:
3119:
2896:
2540:
2492:
2439:
2404:
2383:
2351:
2300:
2216:
2170:
2124:
2089:
2069:
2009:
1981:
1808:
1710:
1675:
1643:
1534:
1505:
1472:
1407:
1379:
1341:
1312:
1290:
1252:
1176:
1156:
1134:
1096:
1076:
1043:
940:
880:
860:
807:
756:
655:
535:
497:
401:
369:
315:
295:
275:
255:
231:
178:
109:
33:integrable systems
4326:Automorphic forms
4308:
4307:
4211:Integral operator
3988:
3987:
3704:978-0-19-857063-9
3478:
3453:
3417:quantum mechanics
3155:
2537:
2012:{\displaystyle L}
1515:with coordinates
1410:{\displaystyle z}
1211:
1186:in time by using
1159:{\displaystyle P}
910:
890:. These satisfy
883:{\displaystyle n}
793:
571:
372:{\displaystyle t}
318:{\displaystyle t}
305:as a function of
298:{\displaystyle L}
278:{\displaystyle L}
258:{\displaystyle P}
155:
123:, and satisfying
16:(Redirected from
4343:
4298:
4297:
4216:Jones polynomial
4134:Operator algebra
3878:
3851:
3844:
3837:
3828:
3811:
3787:
3770:
3764:
3763:
3715:
3709:
3708:
3690:
3684:
3683:
3665:
3659:
3658:
3640:
3544:Further examples
3531:
3523:
3516:
3514:
3513:
3508:
3479:
3471:
3454:
3452:
3441:
3429:
3425:
3402:
3400:
3399:
3394:
3392:
3385:
3384:
3366:
3365:
3348:
3347:
3333:
3332:
3310:
3309:
3287:
3286:
3275:
3274:
3258:
3257:
3246:
3245:
3226:
3225:
3209:
3208:
3197:
3196:
3185:
3184:
3156:
3151:
3143:
3124:
3123:
3108:
3107:
3093:
3092:
3081:
3080:
3067:
3066:
3038:
3037:
3023:
3022:
3009:
3008:
2994:
2993:
2970:
2969:
2955:
2954:
2918:
2914:
2913:
2901:
2900:
2893:
2892:
2880:
2879:
2868:
2867:
2855:
2854:
2843:
2842:
2828:
2827:
2814:
2813:
2801:
2800:
2789:
2788:
2776:
2775:
2761:
2760:
2749:
2748:
2735:
2734:
2720:
2719:
2708:
2707:
2695:
2694:
2680:
2679:
2667:
2666:
2653:
2652:
2641:
2640:
2629:
2628:
2616:
2615:
2604:
2603:
2591:
2590:
2549:
2547:
2546:
2541:
2539:
2538:
2530:
2519:Kovalevskaya top
2513:Kovalevskaya top
2501:
2499:
2498:
2493:
2488:
2487:
2472:
2471:
2452:
2447:
2413:
2411:
2410:
2405:
2396:
2391:
2360:
2358:
2357:
2352:
2332:
2331:
2309:
2307:
2306:
2301:
2299:
2298:
2280:
2279:
2261:
2260:
2225:
2223:
2222:
2217:
2212:
2211:
2199:
2198:
2179:
2177:
2176:
2171:
2166:
2165:
2153:
2152:
2133:
2131:
2130:
2125:
2098:
2096:
2095:
2090:
2078:
2076:
2075:
2070:
2053:
2052:
2018:
2016:
2015:
2010:
1990:
1988:
1987:
1982:
1977:
1976:
1964:
1963:
1948:
1947:
1938:
1937:
1925:
1924:
1915:
1914:
1896:
1895:
1883:
1882:
1870:
1869:
1857:
1856:
1841:
1840:
1820:curvature tensor
1817:
1815:
1814:
1809:
1798:
1797:
1785:
1784:
1769:
1768:
1759:
1758:
1746:
1745:
1736:
1735:
1719:
1717:
1716:
1711:
1684:
1682:
1681:
1676:
1652:
1650:
1649:
1644:
1624:
1623:
1596:
1595:
1543:
1541:
1540:
1535:
1514:
1512:
1511:
1506:
1504:
1503:
1498:
1481:
1479:
1478:
1473:
1465:
1464:
1437:
1436:
1421:), the equation
1416:
1414:
1413:
1408:
1388:
1386:
1385:
1380:
1350:
1348:
1347:
1342:
1321:
1319:
1318:
1313:
1299:
1297:
1296:
1291:
1261:
1259:
1258:
1253:
1212:
1210:
1202:
1194:
1185:
1183:
1182:
1177:
1165:
1163:
1162:
1157:
1143:
1141:
1140:
1135:
1105:
1103:
1102:
1097:
1085:
1083:
1082:
1077:
1052:
1050:
1049:
1044:
972:functional space
949:
947:
946:
941:
930:
929:
911:
909:
898:
889:
887:
886:
881:
869:
867:
866:
861:
856:
855:
816:
814:
813:
808:
794:
792:
784:
776:
765:
763:
762:
757:
664:
662:
661:
656:
572:
570:
559:
544:
542:
541:
536:
506:
504:
503:
498:
493:
492:
410:
408:
407:
402:
378:
376:
375:
370:
324:
322:
321:
316:
304:
302:
301:
296:
284:
282:
281:
276:
264:
262:
261:
256:
240:
238:
237:
232:
187:
185:
184:
179:
156:
154:
146:
138:
118:
116:
115:
110:
61:continuous media
21:
4351:
4350:
4346:
4345:
4344:
4342:
4341:
4340:
4331:Spectral theory
4311:
4310:
4309:
4304:
4286:
4250:Advanced topics
4245:
4169:
4148:
4107:
4073:HilbertâSchmidt
4046:
4037:GelfandâNaimark
3984:
3934:
3869:
3855:
3794:
3791:
3790:
3771:
3767:
3717:
3716:
3712:
3705:
3692:
3691:
3687:
3680:
3667:
3666:
3662:
3655:
3642:
3641:
3637:
3632:
3546:
3534:Planck constant
3529:
3521:
3445:
3435:
3434:
3427:
3423:
3409:
3390:
3389:
3379:
3378:
3367:
3357:
3349:
3339:
3334:
3324:
3318:
3317:
3301:
3296:
3288:
3278:
3276:
3266:
3260:
3259:
3249:
3247:
3237:
3232:
3227:
3217:
3211:
3210:
3200:
3198:
3188:
3186:
3176:
3168:
3158:
3144:
3135:
3129:
3128:
3118:
3117:
3109:
3099:
3094:
3084:
3082:
3072:
3069:
3068:
3058:
3050:
3039:
3029:
3024:
3014:
3011:
3010:
3000:
2995:
2985:
2983:
2978:
2972:
2971:
2961:
2956:
2946:
2941:
2936:
2926:
2916:
2915:
2902:
2895:
2894:
2884:
2871:
2869:
2859:
2846:
2844:
2834:
2829:
2819:
2816:
2815:
2805:
2792:
2790:
2780:
2767:
2762:
2752:
2750:
2740:
2737:
2736:
2726:
2721:
2711:
2709:
2699:
2686:
2681:
2671:
2658:
2655:
2654:
2644:
2642:
2632:
2630:
2620:
2607:
2605:
2595:
2582:
2575:
2567:
2555:
2554:
2523:
2522:
2515:
2507:first integrals
2479:
2463:
2422:
2421:
2369:
2368:
2323:
2318:
2317:
2284:
2271:
2252:
2247:
2246:
2237:
2232:
2203:
2190:
2182:
2181:
2157:
2144:
2136:
2135:
2104:
2103:
2081:
2080:
2044:
2021:
2020:
2001:
2000:
1997:
1968:
1955:
1939:
1929:
1916:
1906:
1887:
1874:
1861:
1848:
1829:
1824:
1823:
1789:
1776:
1760:
1750:
1737:
1727:
1722:
1721:
1687:
1686:
1655:
1654:
1615:
1587:
1582:
1581:
1578:
1558:
1550:Hitchin systems
1517:
1516:
1493:
1488:
1487:
1484:algebraic curve
1423:
1422:
1399:
1398:
1395:
1353:
1352:
1324:
1323:
1304:
1303:
1264:
1263:
1203:
1195:
1188:
1187:
1168:
1167:
1148:
1147:
1108:
1107:
1088:
1087:
1059:
1058:
1020:
1019:
960:
921:
902:
892:
891:
872:
871:
847:
833:
832:
829:
823:
785:
777:
770:
769:
724:
723:
563:
553:
552:
512:
511:
481:
416:
415:
384:
383:
361:
360:
353:are said to be
331:
307:
306:
287:
286:
267:
266:
247:
246:
193:
192:
147:
139:
132:
131:
77:
76:
73:
23:
22:
15:
12:
11:
5:
4349:
4347:
4339:
4338:
4333:
4328:
4323:
4313:
4312:
4306:
4305:
4303:
4302:
4291:
4288:
4287:
4285:
4284:
4279:
4274:
4269:
4267:Choquet theory
4264:
4259:
4253:
4251:
4247:
4246:
4244:
4243:
4233:
4228:
4223:
4218:
4213:
4208:
4203:
4198:
4193:
4188:
4183:
4177:
4175:
4171:
4170:
4168:
4167:
4162:
4156:
4154:
4150:
4149:
4147:
4146:
4141:
4136:
4131:
4126:
4121:
4119:Banach algebra
4115:
4113:
4109:
4108:
4106:
4105:
4100:
4095:
4090:
4085:
4080:
4075:
4070:
4065:
4060:
4054:
4052:
4048:
4047:
4045:
4044:
4042:BanachâAlaoglu
4039:
4034:
4029:
4024:
4019:
4014:
4009:
4004:
3998:
3996:
3990:
3989:
3986:
3985:
3983:
3982:
3977:
3972:
3970:Locally convex
3967:
3953:
3948:
3942:
3940:
3936:
3935:
3933:
3932:
3927:
3922:
3917:
3912:
3907:
3902:
3897:
3892:
3887:
3881:
3875:
3871:
3870:
3856:
3854:
3853:
3846:
3839:
3831:
3825:
3824:
3815:
3803:(5): 467â490,
3789:
3788:
3765:
3730:(2): 321â354.
3710:
3703:
3685:
3678:
3660:
3653:
3634:
3633:
3631:
3628:
3616:
3615:
3609:
3604:
3599:
3594:
3589:
3584:
3579:
3574:
3569:
3566:
3561:
3555:
3545:
3542:
3518:
3517:
3506:
3503:
3500:
3497:
3494:
3491:
3488:
3485:
3482:
3477:
3474:
3469:
3466:
3463:
3460:
3457:
3451:
3448:
3444:
3408:
3405:
3404:
3403:
3388:
3383:
3377:
3374:
3371:
3368:
3364:
3360:
3356:
3353:
3350:
3346:
3342:
3338:
3335:
3331:
3327:
3323:
3320:
3319:
3316:
3313:
3308:
3304:
3300:
3297:
3295:
3292:
3289:
3285:
3281:
3277:
3273:
3269:
3265:
3262:
3261:
3256:
3252:
3248:
3244:
3240:
3236:
3233:
3231:
3228:
3224:
3220:
3216:
3213:
3212:
3207:
3203:
3199:
3195:
3191:
3187:
3183:
3179:
3175:
3172:
3169:
3167:
3164:
3163:
3161:
3154:
3150:
3147:
3141:
3138:
3136:
3134:
3131:
3130:
3127:
3122:
3116:
3113:
3110:
3106:
3102:
3098:
3095:
3091:
3087:
3083:
3079:
3075:
3071:
3070:
3065:
3061:
3057:
3054:
3051:
3049:
3046:
3043:
3040:
3036:
3032:
3028:
3025:
3021:
3017:
3013:
3012:
3007:
3003:
2999:
2996:
2992:
2988:
2984:
2982:
2979:
2977:
2974:
2973:
2968:
2964:
2960:
2957:
2953:
2949:
2945:
2942:
2940:
2937:
2935:
2932:
2931:
2929:
2924:
2921:
2919:
2917:
2912:
2909:
2905:
2899:
2891:
2887:
2883:
2878:
2874:
2870:
2866:
2862:
2858:
2853:
2849:
2845:
2841:
2837:
2833:
2830:
2826:
2822:
2818:
2817:
2812:
2808:
2804:
2799:
2795:
2791:
2787:
2783:
2779:
2774:
2770:
2766:
2763:
2759:
2755:
2751:
2747:
2743:
2739:
2738:
2733:
2729:
2725:
2722:
2718:
2714:
2710:
2706:
2702:
2698:
2693:
2689:
2685:
2682:
2678:
2674:
2670:
2665:
2661:
2657:
2656:
2651:
2647:
2643:
2639:
2635:
2631:
2627:
2623:
2619:
2614:
2610:
2606:
2602:
2598:
2594:
2589:
2585:
2581:
2580:
2578:
2573:
2570:
2568:
2566:
2563:
2562:
2536:
2533:
2514:
2511:
2503:
2502:
2491:
2486:
2482:
2478:
2475:
2470:
2466:
2462:
2459:
2456:
2451:
2446:
2442:
2438:
2435:
2432:
2429:
2419:
2403:
2400:
2395:
2390:
2386:
2382:
2379:
2376:
2362:
2361:
2350:
2347:
2344:
2341:
2338:
2335:
2330:
2326:
2311:
2310:
2297:
2294:
2291:
2287:
2283:
2278:
2274:
2270:
2267:
2264:
2259:
2255:
2236:
2233:
2231:
2228:
2215:
2210:
2206:
2202:
2197:
2193:
2189:
2169:
2164:
2160:
2156:
2151:
2147:
2143:
2123:
2120:
2117:
2114:
2111:
2088:
2068:
2065:
2062:
2059:
2056:
2051:
2047:
2043:
2040:
2037:
2034:
2031:
2028:
2008:
1996:
1993:
1980:
1975:
1971:
1967:
1962:
1958:
1954:
1951:
1946:
1942:
1936:
1932:
1928:
1923:
1919:
1913:
1909:
1905:
1902:
1899:
1894:
1890:
1886:
1881:
1877:
1873:
1868:
1864:
1860:
1855:
1851:
1847:
1844:
1839:
1836:
1832:
1807:
1804:
1801:
1796:
1792:
1788:
1783:
1779:
1775:
1772:
1767:
1763:
1757:
1753:
1749:
1744:
1740:
1734:
1730:
1709:
1706:
1703:
1700:
1697:
1694:
1674:
1671:
1668:
1665:
1662:
1642:
1639:
1636:
1633:
1630:
1627:
1622:
1618:
1614:
1611:
1608:
1605:
1602:
1599:
1594:
1590:
1577:
1574:
1557:
1554:
1546:spectral curve
1533:
1530:
1527:
1524:
1502:
1497:
1471:
1468:
1463:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1435:
1430:
1406:
1394:
1393:Spectral curve
1391:
1390:
1389:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1340:
1337:
1334:
1331:
1311:
1300:
1289:
1286:
1283:
1280:
1277:
1274:
1271:
1251:
1248:
1245:
1242:
1239:
1236:
1233:
1230:
1227:
1224:
1221:
1218:
1215:
1209:
1206:
1201:
1198:
1175:
1155:
1144:
1133:
1130:
1127:
1124:
1121:
1118:
1115:
1095:
1075:
1072:
1069:
1066:
1042:
1039:
1036:
1033:
1030:
1027:
959:
956:
939:
936:
933:
928:
924:
920:
917:
914:
908:
905:
901:
879:
859:
854:
850:
846:
843:
840:
822:
819:
818:
817:
806:
803:
800:
797:
791:
788:
783:
780:
767:
755:
752:
749:
746:
743:
740:
737:
734:
731:
666:
665:
654:
651:
648:
645:
642:
639:
636:
633:
630:
626:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
575:
569:
566:
562:
547:Cauchy problem
534:
531:
528:
525:
522:
519:
508:
507:
496:
491:
488:
484:
480:
477:
474:
471:
468:
465:
462:
459:
456:
453:
450:
447:
444:
441:
438:
435:
432:
429:
426:
423:
400:
397:
394:
391:
368:
330:
327:
314:
294:
274:
254:
230:
227:
224:
221:
218:
215:
212:
209:
206:
203:
200:
189:
188:
177:
174:
171:
168:
165:
162:
159:
153:
150:
145:
142:
125:Lax's equation
108:
105:
102:
99:
96:
93:
90:
87:
84:
72:
69:
24:
18:Spectral curve
14:
13:
10:
9:
6:
4:
3:
2:
4348:
4337:
4334:
4332:
4329:
4327:
4324:
4322:
4319:
4318:
4316:
4301:
4293:
4292:
4289:
4283:
4280:
4278:
4275:
4273:
4272:Weak topology
4270:
4268:
4265:
4263:
4260:
4258:
4255:
4254:
4252:
4248:
4241:
4237:
4234:
4232:
4229:
4227:
4224:
4222:
4219:
4217:
4214:
4212:
4209:
4207:
4204:
4202:
4199:
4197:
4196:Index theorem
4194:
4192:
4189:
4187:
4184:
4182:
4179:
4178:
4176:
4172:
4166:
4163:
4161:
4158:
4157:
4155:
4153:Open problems
4151:
4145:
4142:
4140:
4137:
4135:
4132:
4130:
4127:
4125:
4122:
4120:
4117:
4116:
4114:
4110:
4104:
4101:
4099:
4096:
4094:
4091:
4089:
4086:
4084:
4081:
4079:
4076:
4074:
4071:
4069:
4066:
4064:
4061:
4059:
4056:
4055:
4053:
4049:
4043:
4040:
4038:
4035:
4033:
4030:
4028:
4025:
4023:
4020:
4018:
4015:
4013:
4010:
4008:
4005:
4003:
4000:
3999:
3997:
3995:
3991:
3981:
3978:
3976:
3973:
3971:
3968:
3965:
3961:
3957:
3954:
3952:
3949:
3947:
3944:
3943:
3941:
3937:
3931:
3928:
3926:
3923:
3921:
3918:
3916:
3913:
3911:
3908:
3906:
3903:
3901:
3898:
3896:
3893:
3891:
3888:
3886:
3883:
3882:
3879:
3876:
3872:
3867:
3863:
3859:
3852:
3847:
3845:
3840:
3838:
3833:
3832:
3829:
3822:
3820:
3816:
3814:
3810:
3806:
3802:
3798:
3793:
3792:
3786:
3782:
3779:
3775:
3769:
3766:
3761:
3757:
3753:
3749:
3745:
3741:
3737:
3733:
3729:
3725:
3721:
3714:
3711:
3706:
3700:
3696:
3689:
3686:
3681:
3679:9780198504214
3675:
3671:
3664:
3661:
3656:
3650:
3646:
3639:
3636:
3629:
3627:
3625:
3621:
3613:
3610:
3608:
3605:
3603:
3600:
3598:
3595:
3593:
3590:
3588:
3585:
3583:
3582:KdV hierarchy
3580:
3578:
3575:
3573:
3570:
3567:
3565:
3562:
3560:
3556:
3554:
3551:
3550:
3549:
3543:
3541:
3539:
3535:
3527:
3504:
3495:
3489:
3486:
3483:
3472:
3467:
3461:
3455:
3449:
3446:
3442:
3433:
3432:
3431:
3422:
3418:
3414:
3406:
3386:
3381:
3375:
3372:
3369:
3362:
3358:
3354:
3351:
3344:
3340:
3336:
3329:
3325:
3321:
3314:
3311:
3306:
3302:
3298:
3293:
3290:
3283:
3279:
3271:
3267:
3263:
3254:
3250:
3242:
3238:
3234:
3229:
3222:
3218:
3214:
3205:
3201:
3193:
3189:
3181:
3177:
3173:
3170:
3165:
3159:
3152:
3148:
3145:
3139:
3137:
3132:
3125:
3120:
3114:
3111:
3104:
3100:
3096:
3089:
3085:
3077:
3073:
3063:
3059:
3055:
3052:
3047:
3044:
3041:
3034:
3030:
3026:
3019:
3015:
3005:
3001:
2997:
2990:
2986:
2980:
2975:
2966:
2962:
2958:
2951:
2947:
2943:
2938:
2933:
2927:
2922:
2920:
2910:
2907:
2903:
2897:
2889:
2885:
2881:
2876:
2872:
2864:
2860:
2856:
2851:
2847:
2839:
2835:
2831:
2824:
2820:
2810:
2806:
2802:
2797:
2793:
2785:
2781:
2777:
2772:
2768:
2764:
2757:
2753:
2745:
2741:
2731:
2727:
2723:
2716:
2712:
2704:
2700:
2696:
2691:
2687:
2683:
2676:
2672:
2668:
2663:
2659:
2649:
2645:
2637:
2633:
2625:
2621:
2617:
2612:
2608:
2600:
2596:
2592:
2587:
2583:
2576:
2571:
2569:
2564:
2553:
2552:
2551:
2531:
2520:
2512:
2510:
2508:
2489:
2484:
2480:
2476:
2473:
2468:
2460:
2457:
2454:
2449:
2444:
2436:
2433:
2430:
2427:
2420:
2417:
2401:
2398:
2393:
2388:
2380:
2377:
2374:
2367:
2366:
2365:
2345:
2342:
2339:
2333:
2328:
2324:
2316:
2315:
2314:
2295:
2292:
2289:
2285:
2281:
2276:
2272:
2268:
2265:
2262:
2257:
2253:
2245:
2244:
2243:
2242:
2234:
2229:
2227:
2208:
2204:
2200:
2195:
2191:
2162:
2158:
2154:
2149:
2145:
2118:
2115:
2112:
2102:The Lax pair
2100:
2086:
2066:
2063:
2060:
2057:
2054:
2049:
2045:
2041:
2038:
2035:
2032:
2029:
2026:
2006:
1994:
1992:
1973:
1969:
1965:
1960:
1956:
1949:
1944:
1940:
1934:
1926:
1921:
1917:
1911:
1903:
1900:
1892:
1888:
1884:
1879:
1871:
1866:
1862:
1858:
1853:
1842:
1837:
1834:
1830:
1821:
1805:
1802:
1794:
1790:
1786:
1781:
1777:
1770:
1765:
1761:
1755:
1747:
1742:
1738:
1732:
1704:
1701:
1698:
1692:
1669:
1666:
1663:
1640:
1634:
1631:
1628:
1620:
1616:
1612:
1606:
1603:
1600:
1592:
1588:
1575:
1573:
1571:
1567:
1563:
1555:
1553:
1551:
1547:
1531:
1528:
1525:
1522:
1500:
1485:
1469:
1466:
1453:
1447:
1444:
1441:
1438:
1420:
1404:
1392:
1376:
1370:
1367:
1364:
1358:
1335:
1329:
1309:
1301:
1287:
1281:
1278:
1275:
1269:
1246:
1243:
1240:
1234:
1231:
1228:
1222:
1219:
1216:
1207:
1199:
1173:
1153:
1145:
1131:
1125:
1122:
1119:
1113:
1093:
1070:
1064:
1056:
1055:
1054:
1040:
1028:
1017:
1013:
1009:
1005:
1001:
997:
993:
989:
985:
981:
977:
973:
969:
965:
957:
955:
953:
937:
934:
926:
922:
915:
912:
906:
903:
899:
877:
852:
848:
841:
838:
828:
820:
804:
801:
798:
795:
789:
781:
768:
750:
744:
741:
735:
729:
722:
721:
720:
718:
714:
710:
706:
701:
699:
695:
691:
687:
683:
679:
675:
671:
652:
649:
646:
640:
637:
634:
628:
624:
618:
615:
612:
606:
600:
594:
591:
585:
582:
579:
573:
567:
564:
560:
551:
550:
549:
548:
529:
526:
523:
517:
494:
489:
486:
478:
475:
472:
466:
460:
454:
448:
445:
442:
436:
433:
427:
421:
414:
413:
412:
395:
389:
380:
366:
358:
357:
352:
348:
344:
340:
336:
328:
326:
312:
292:
272:
252:
244:
228:
225:
222:
219:
216:
213:
207:
204:
201:
175:
169:
166:
163:
157:
151:
148:
143:
140:
130:
129:
128:
126:
122:
121:Hilbert space
103:
97:
94:
88:
82:
70:
68:
66:
62:
58:
54:
50:
47:, called the
46:
42:
38:
34:
30:
19:
4262:Balanced set
4236:Distribution
4174:Applications
4027:KreinâMilman
4012:Closed graph
3818:
3800:
3796:
3768:
3727:
3723:
3713:
3694:
3688:
3669:
3663:
3644:
3638:
3617:
3602:Toda lattice
3547:
3532:the reduced
3519:
3410:
2516:
2504:
2363:
2312:
2238:
2101:
1998:
1579:
1559:
1545:
1396:
1015:
1011:
1007:
1003:
999:
995:
991:
987:
983:
979:
975:
967:
963:
961:
830:
716:
712:
708:
704:
702:
693:
689:
685:
682:skew-adjoint
677:
673:
669:
667:
509:
381:
354:
350:
346:
342:
332:
190:
124:
74:
49:Lax equation
48:
36:
26:
4191:Heat kernel
4181:Hardy space
4088:Trace class
4002:HahnâBanach
3964:Topological
3624:Kerr metric
3526:Hamiltonian
1482:defines an
1419:sine-Gordon
356:isospectral
335:eigenvalues
265:depends on
55:to discuss
29:mathematics
4315:Categories
4124:C*-algebra
3939:Properties
3654:0198504217
3630:References
3421:observable
2019:, one has
825:See also:
696:) will be
243:commutator
71:Definition
4098:Unbounded
4093:Transpose
4051:Operators
3980:Separable
3975:Reflexive
3960:Algebraic
3946:Barrelled
3778:1401.2122
3760:121752578
3752:0010-3616
3476:ℏ
3376:λ
3370:−
3352:−
3337:−
3322:−
3315:γ
3294:λ
3264:−
3235:−
3171:−
3146:−
3115:λ
3053:−
3048:λ
3042:−
3027:−
2998:−
2959:−
2944:−
2908:−
2904:λ
2857:−
2832:−
2803:−
2778:−
2765:−
2724:−
2684:−
2535:→
2465:∂
2441:∂
2434:−
2385:∂
2381:−
2282:−
2087:λ
2061:ψ
2046:ψ
2039:ψ
2036:λ
2030:ψ
1974:ν
1961:μ
1945:μ
1935:ν
1931:∂
1922:ν
1912:μ
1908:∂
1904:−
1893:ν
1885:−
1880:ν
1876:∂
1867:μ
1859:−
1854:μ
1850:∂
1838:ν
1835:μ
1752:∂
1748:−
1729:∂
1693:φ
1445:−
1310:ψ
1270:ψ
1235:ψ
1205:∂
1200:ψ
1197:∂
1174:ψ
1114:ψ
1094:λ
1086:, giving
1038:∞
1035:→
1032:‖
1026:‖
1006:(0,
970:act on a
916:
842:
802:ψ
787:∂
782:ψ
779:∂
745:λ
730:λ
487:−
223:−
53:Peter Lax
41:operators
4300:Category
4112:Algebras
3994:Theorems
3951:Complete
3920:Schwartz
3866:glossary
3622:and the
2230:Examples
1302:Knowing
870:for any
711:at time
379:varies.
339:spectrum
57:solitons
37:Lax pair
4103:Unitary
4083:Nuclear
4068:Compact
4063:Bounded
4058:Adjoint
4032:Minâmax
3925:Sobolev
3910:Nuclear
3900:Hilbert
3895:Fréchet
3860: (
3813:archive
3732:Bibcode
3411:In the
1351:and/or
998:,
986:,
698:unitary
692:,
241:is the
4078:Normal
3915:Orlicz
3905:Hölder
3885:Banach
3874:Spaces
3862:topics
3758:
3750:
3701:
3676:
3651:
974:(thus
668:where
510:where
191:where
63:. The
3890:Besov
3774:arXiv
3756:S2CID
3520:with
3419:, an
2364:with
680:) is
4238:(or
3956:Dual
3748:ISSN
3699:ISBN
3674:ISBN
3649:ISBN
3528:and
3524:the
2239:The
1566:Ward
1106:and
966:and
35:, a
3805:doi
3781:doi
3740:doi
3728:122
3415:of
2414:(a
1486:in
1429:det
359:as
341:of
59:in
27:In
4317::
3864:â
3801:21
3799:,
3754:.
3746:.
3738:.
3726:.
3722:.
2550:.
2418:),
2067:0.
1806:0.
1552:.
978:=
913:tr
839:tr
709:λÏ
707:=
705:LÏ
700:.
684:,
325:.
127::
4242:)
3966:)
3962:/
3958:(
3868:)
3850:e
3843:t
3836:v
3807::
3783::
3776::
3762:.
3742::
3734::
3707:.
3682:.
3657:.
3530:ħ
3522:H
3505:,
3502:]
3499:)
3496:t
3493:(
3490:A
3487:,
3484:H
3481:[
3473:i
3468:=
3465:)
3462:t
3459:(
3456:A
3450:t
3447:d
3443:d
3428:t
3424:A
3387:.
3382:)
3373:2
3363:3
3359:l
3355:2
3345:2
3341:l
3330:1
3326:l
3312:+
3307:3
3303:l
3299:2
3291:2
3284:1
3280:l
3272:2
3268:l
3255:2
3251:l
3243:1
3239:l
3230:0
3223:3
3219:l
3215:2
3206:1
3202:l
3194:2
3190:l
3182:3
3178:l
3174:2
3166:0
3160:(
3153:2
3149:1
3140:=
3133:P
3126:,
3121:)
3112:2
3105:3
3101:l
3097:2
3090:2
3086:l
3078:1
3074:l
3064:3
3060:l
3056:2
3045:2
3035:1
3031:l
3020:2
3016:l
3006:2
3002:l
2991:1
2987:l
2981:0
2976:0
2967:1
2963:l
2952:2
2948:l
2939:0
2934:0
2928:(
2923:+
2911:1
2898:)
2890:2
2886:h
2882:+
2877:1
2873:g
2865:1
2861:h
2852:2
2848:g
2840:3
2836:g
2825:3
2821:h
2811:1
2807:h
2798:2
2794:g
2786:2
2782:h
2773:1
2769:g
2758:3
2754:h
2746:3
2742:g
2732:3
2728:g
2717:3
2713:h
2705:2
2701:h
2697:+
2692:1
2688:g
2677:1
2673:h
2669:+
2664:2
2660:g
2650:3
2646:h
2638:3
2634:g
2626:1
2622:h
2618:+
2613:2
2609:g
2601:2
2597:h
2593:+
2588:1
2584:g
2577:(
2572:=
2565:L
2532:h
2490:,
2485:x
2481:u
2477:3
2474:+
2469:x
2461:u
2458:6
2455:+
2450:3
2445:x
2437:4
2431:=
2428:P
2402:u
2399:+
2394:2
2389:x
2378:=
2375:L
2349:]
2346:L
2343:,
2340:P
2337:[
2334:=
2329:t
2325:L
2296:x
2293:x
2290:x
2286:u
2277:x
2273:u
2269:u
2266:6
2263:=
2258:t
2254:u
2214:)
2209:t
2205:A
2201:,
2196:x
2192:A
2188:(
2168:)
2163:t
2159:A
2155:,
2150:x
2146:A
2142:(
2122:)
2119:P
2116:,
2113:L
2110:(
2064:=
2058:A
2055:+
2050:t
2042:,
2033:=
2027:L
2007:L
1979:]
1970:A
1966:,
1957:A
1953:[
1950:+
1941:A
1927:+
1918:A
1901:=
1898:]
1889:A
1872:,
1863:A
1846:[
1843:=
1831:F
1803:=
1800:]
1795:t
1791:A
1787:,
1782:x
1778:A
1774:[
1771:+
1766:t
1762:A
1756:x
1743:x
1739:A
1733:t
1708:)
1705:t
1702:,
1699:x
1696:(
1673:)
1670:t
1667:,
1664:x
1661:(
1641:,
1638:)
1635:t
1632:,
1629:x
1626:(
1621:t
1617:A
1613:,
1610:)
1607:t
1604:,
1601:x
1598:(
1593:x
1589:A
1532:.
1529:z
1526:,
1523:w
1501:2
1496:C
1470:0
1467:=
1462:)
1457:)
1454:z
1451:(
1448:L
1442:I
1439:w
1434:(
1405:z
1377:.
1374:)
1371:x
1368:,
1365:t
1362:(
1359:u
1339:)
1336:t
1333:(
1330:L
1288:.
1285:)
1282:x
1279:,
1276:0
1273:(
1250:)
1247:x
1244:,
1241:t
1238:(
1232:P
1229:=
1226:)
1223:x
1220:,
1217:t
1214:(
1208:t
1154:P
1132:.
1129:)
1126:x
1123:,
1120:0
1117:(
1074:)
1071:0
1068:(
1065:L
1041:.
1029:x
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1012:P
1008:x
1004:u
1000:x
996:t
994:(
992:u
988:x
984:t
982:(
980:Ï
976:Ï
968:P
964:L
938:0
935:=
932:)
927:n
923:L
919:(
907:t
904:d
900:d
878:n
858:)
853:n
849:L
845:(
805:.
799:P
796:=
790:t
754:)
751:0
748:(
742:=
739:)
736:t
733:(
717:L
713:t
694:s
690:t
688:(
686:U
678:t
676:(
674:P
670:I
653:,
650:I
647:=
644:)
641:s
638:,
635:s
632:(
629:U
625:,
622:)
619:s
616:,
613:t
610:(
607:U
604:)
601:t
598:(
595:P
592:=
589:)
586:s
583:,
580:t
577:(
574:U
568:t
565:d
561:d
533:)
530:s
527:,
524:t
521:(
518:U
495:,
490:1
483:)
479:s
476:,
473:t
470:(
467:U
464:)
461:s
458:(
455:L
452:)
449:s
446:,
443:t
440:(
437:U
434:=
431:)
428:t
425:(
422:L
399:)
396:t
393:(
390:L
367:t
351:L
347:t
343:L
313:t
293:L
273:L
253:P
229:P
226:L
220:L
217:P
214:=
211:]
208:L
205:,
202:P
199:[
176:,
173:]
170:L
167:,
164:P
161:[
158:=
152:t
149:d
144:L
141:d
107:)
104:t
101:(
98:P
95:,
92:)
89:t
86:(
83:L
20:)
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