2229:
2737:
610:
3233:
441:
4084:. 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs). Vol. 99. Dagstuhl, Germany: Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik. pp. 31:1–31:15.
1168:
2594:
3613:
3387:
164:
1430:
1942:
2441:
522:
3477:
695:
3626:
Discrete Morse theory finds its application in molecular shape analysis, skeletonization of digital images/volumes, graph reconstruction from noisy data, denoising noisy point clouds and analysing
2485:
645:
476:
2070:
2058:
954:
827:
2360:
2903:
1556:
1036:
860:
733:
1490:
1066:
890:
763:
266:
3272:
3115:
3501:
3419:
3296:
3046:
2947:
2849:
2825:
2622:
1994:
1817:
1789:
1765:
1741:
1709:
1660:
1588:
1363:
1307:
1251:
1225:
1195:
1002:
362:
228:
113:
1682:
1613:
529:
2998:
3439:
338:
3521:
2785:
978:
910:
783:
294:
184:
3072:
3122:
2315:
2295:
2255:
1636:
1339:
370:
314:
204:
2275:
1106:
1086:
3018:
2967:
2923:
2760:
1510:
1453:
1283:
85:
2630:
1119:
3537:
3304:
2494:
121:
4261:
3823:
4099:
3989:
1368:
3903:
3783:
3748:
1837:
481:
4348:
4318:
4291:
4183:
3653:
40:
2365:
4398:
3444:
662:
3663:
2224:{\displaystyle \nu (\rho )={\frac {\prod _{m=1}^{M-1}-\kappa (K_{m},Q_{m+1})}{\prod _{m=1}^{M}\kappa (K_{m},Q_{m})}}.}
618:
449:
1999:
915:
788:
2446:
3729:
3658:
269:
56:
52:
2854:
1515:
1108:
value. The cells which have no pairs, i.e., whose function values are strictly higher than their boundary cells
960:
It can be shown that the cardinalities in the two conditions cannot both be one simultaneously for a fixed cell
1011:
835:
708:
1458:
1041:
865:
738:
233:
4229:
3648:
2324:
3244:
4388:
3528:
3080:
1716:
44:
4393:
1116:
cells. Thus, a discrete Morse function partitions the CW complex into three distinct cell collections:
3862:
3643:
3239:
605:{\displaystyle \sum _{\tau \in {\mathcal {X}}}\kappa (\sigma ,\tau )\kappa (\tau ,\tau ^{\prime })=0}
3482:
3400:
3277:
3027:
2928:
2830:
2806:
2603:
1950:
1798:
1770:
1746:
1722:
1690:
1641:
1569:
1344:
1288:
1232:
1206:
1176:
983:
343:
209:
94:
32:
1665:
1596:
28:
4197:
4140:
4105:
4060:
3995:
3854:
3792:
1563:
1559:
615:
which is a consequence of the above definition of the boundary operator and the requirement that
2976:
3824:"Applications of Forman's discrete Morse theory to topology visualization and mesh compression"
3424:
323:
4344:
4314:
4287:
4189:
4179:
4095:
4052:
4044:
3985:
3846:
3524:
3228:{\displaystyle m_{N}-m_{N-1}+\dots \pm m_{0}\geq \beta _{N}-\beta _{N-1}+\dots \pm \beta _{0}}
1262:
436:{\displaystyle \partial (\sigma )=\sum _{\tau \in {\mathcal {X}}}\kappa (\sigma ,\tau )\tau .}
317:
3684:
3506:
963:
895:
768:
279:
169:
4336:
4238:
4171:
4132:
4085:
4078:
Dey, Tamal K.; Wang, Jiayuan; Wang, Yusu (2018). Speckmann, Bettina; Tóth, Csaba D. (eds.).
4034:
4026:
3977:
3912:
3838:
3802:
3757:
3699:
3051:
36:
4358:
4301:
4274:
3713:
2300:
2280:
2237:
1618:
1312:
299:
189:
4354:
4297:
4270:
3709:
2260:
1091:
1071:
2795:
Many of the familiar results from continuous Morse theory apply in the discrete setting.
4253:
3969:
2765:
4163:
3970:"Molecular shape analysis based upon the morse-smale complex and the connolly function"
3003:
2952:
2908:
2745:
2732:{\displaystyle \Delta (A)=\kappa (A,A')+\sum _{A{\stackrel {\rho }{\to }}A'}m(\rho )A'}
1495:
1438:
1433:
1268:
1163:{\displaystyle {\mathcal {X}}={\mathcal {A}}\sqcup {\mathcal {K}}\sqcup {\mathcal {Q}}}
70:
3725:
4382:
4366:
4144:
1795:
between adjacent critical cells which can be used to obtain the boundary operator on
273:
20:
4201:
4175:
4079:
3608:{\displaystyle H_{*}({\mathcal {X}},\partial )\simeq H_{*}({\mathcal {A}},\Delta ),}
4109:
4064:
3999:
3858:
3382:{\displaystyle \chi ({\mathcal {X}})=m_{0}-m_{1}+\dots \pm m_{\dim {\mathcal {X}}}}
3021:
24:
2589:{\displaystyle m(\rho )=\kappa (A,Q_{1})\cdot \nu (\rho )\cdot \kappa (K_{M},A')}
2234:
The division here makes sense because the incidence between paired cells must be
159:{\displaystyle \kappa \colon {\mathcal {X}}\times {\mathcal {X}}\to \mathbb {Z} }
4328:
4159:
4090:
4015:"Skeletonization and Partitioning of Digital Images Using Discrete Morse Theory"
3631:
1712:
1687:
The fundamental result of discrete Morse theory establishes that the CW complex
1638:. This technical requirement is guaranteed, for instance, when one assumes that
656:
4221:
4136:
4030:
4014:
3899:"Morse Theory for Filtrations and Efficient computation of Persistent Homology"
3744:"Morse Theory for Filtrations and Efficient computation of Persistent Homology"
3704:
4340:
3917:
3898:
3807:
3778:
3762:
3743:
3627:
88:
4193:
4048:
4168:
Proceedings of
Eurographics Workshop on Graphics and Cultural Heritage (GCH)
1258:
48:
4243:
4056:
3850:
4013:
Delgado-Friedrichs, Olaf; Robins, Vanessa; Sheppard, Adrian (March 2015).
3981:
3842:
3974:
Proceedings of the nineteenth annual symposium on
Computational geometry
4123:
Mukherjee, Soham (2021-09-01). "Denoising with discrete Morse theory".
3879:
2257:. Note that by construction, the values of the discrete Morse function
1591:
4039:
3779:"Optimal Topological Simplification of Discrete Functions on Surfaces"
1819:. Some details of this construction are provided in the next section.
1425:{\displaystyle p^{k}\colon {\mathcal {K}}^{k}\to {\mathcal {Q}}^{k-1}}
4286:. Algorithms and Computation in Mathematics. Vol. 21. Springer.
31:. The theory has various practical applications in diverse fields of
4333:
Algebraic
Combinatorics: Lectures at a Summer School In Nordfjordeid
4164:"Lithic Feature Identification in 3D based on Discrete Morse Theory"
1937:{\displaystyle \rho =(Q_{1},K_{1},Q_{2},K_{2},\ldots ,Q_{M},K_{M})}
3797:
478:. In more axiomatic definitions one can find the requirement that
517:{\displaystyle \forall \sigma ,\tau ^{\prime }\in {\mathcal {X}}}
4370:
4170:, Delft, Netherlands: Eurographics Association, pp. 55–58,
2742:
where the sum is taken over all gradient path connections from
4019:
IEEE Transactions on
Pattern Analysis and Machine Intelligence
3503:
be the associated Morse complex with Morse boundary operator
3588:
3556:
3488:
3456:
3406:
3372:
3316:
3283:
3256:
3033:
2934:
2879:
2836:
2812:
2609:
2436:{\displaystyle \kappa (A,Q_{1})\neq 0\neq \kappa (K_{M},A')}
2347:
1804:
1776:
1752:
1728:
1696:
1647:
1575:
1543:
1471:
1405:
1388:
1350:
1294:
1238:
1212:
1182:
1155:
1145:
1135:
1125:
1053:
1023:
989:
877:
847:
750:
720:
674:
588:
546:
509:
499:
402:
349:
215:
143:
133:
100:
1743:
consisting of only the critical cells. The paired cells in
1455:. It is an additional technical requirement that for each
3472:{\displaystyle \mu \colon {\mathcal {X}}\to \mathbb {R} }
690:{\displaystyle \mu \colon {\mathcal {X}}\to \mathbb {R} }
3831:
1492:, the degree of the attaching map from the boundary of
1227:
denotes cells which are paired with boundary cells, and
1112:
strictly lower than their co-boundary cells are called
3777:
Bauer, Ulrich; Lange, Carsten; Wardetzky, Max (2012).
1253:
denotes cells which are paired with co-boundary cells.
3540:
3509:
3485:
3447:
3427:
3403:
3307:
3280:
3247:
3125:
3083:
3054:
3030:
3006:
2979:
2955:
2931:
2911:
2857:
2833:
2809:
2768:
2748:
2633:
2606:
2497:
2449:
2368:
2327:
2303:
2283:
2263:
2240:
2073:
2002:
1953:
1840:
1801:
1773:
1749:
1725:
1693:
1668:
1644:
1621:
1599:
1572:
1518:
1498:
1461:
1441:
1371:
1347:
1315:
1291:
1271:
1235:
1209:
1179:
1122:
1094:
1074:
1044:
1014:
986:
966:
918:
898:
868:
838:
791:
771:
741:
711:
665:
621:
532:
484:
452:
446:
It is a defining property of boundary operators that
373:
346:
326:
302:
282:
236:
212:
192:
172:
124:
97:
73:
2064:of this gradient path is defined to be the integer
3685:"(Discrete) Morse theory for Configuration spaces"
3607:
3515:
3495:
3471:
3433:
3413:
3381:
3290:
3266:
3227:
3109:
3066:
3040:
3012:
2992:
2961:
2941:
2917:
2897:
2843:
2819:
2779:
2754:
2731:
2616:
2588:
2479:
2435:
2354:
2309:
2289:
2269:
2249:
2223:
2052:
1988:
1936:
1811:
1783:
1759:
1735:
1703:
1676:
1654:
1630:
1607:
1582:
1550:
1504:
1484:
1447:
1424:
1357:
1333:
1301:
1277:
1245:
1219:
1189:
1162:
1100:
1080:
1060:
1030:
996:
972:
948:
904:
884:
854:
821:
777:
757:
727:
689:
639:
604:
516:
470:
435:
356:
332:
308:
288:
260:
222:
198:
178:
158:
107:
79:
2827:be a Morse complex associated to the CW complex
2491:of this connection is defined to be the integer
1038:can be paired with at most one exceptional cell
640:{\displaystyle \partial \circ \partial \equiv 0}
471:{\displaystyle \partial \circ \partial \equiv 0}
3421:be a regular CW complex with boundary operator
701:if it satisfies the following two properties:
4081:Graph Reconstruction by Discrete Morse Theory
3897:Mischaikow, Konstantin; Nanda, Vidit (2013).
3742:Mischaikow, Konstantin; Nanda, Vidit (2013).
2053:{\displaystyle \kappa (K_{m},~Q_{m+1})\neq 0}
949:{\displaystyle \mu (\sigma )\geq \mu (\tau )}
822:{\displaystyle \mu (\sigma )\leq \mu (\tau )}
8:
3968:Cazals, F.; Chazal, F.; Lewiner, T. (2003).
3822:Lewiner, T.; Lopes, H.; Tavares, G. (2004).
2480:{\displaystyle A{\stackrel {\rho }{\to }}A'}
4158:Bullenkamp, Jan Philipp; Linsel, Florian;
2898:{\displaystyle m_{q}=|{\mathcal {A}}_{q}|}
1551:{\displaystyle p^{k}(K)\in {\mathcal {Q}}}
1088:value, or a co-boundary cell with smaller
4254:"A user's guide to discrete Morse theory"
4242:
4089:
4038:
3916:
3806:
3796:
3761:
3703:
3683:Mori, Francesca; Salvetti, Mario (2011),
3587:
3586:
3577:
3555:
3554:
3545:
3539:
3508:
3487:
3486:
3484:
3465:
3464:
3455:
3454:
3446:
3426:
3405:
3404:
3402:
3393:Discrete Morse homology and homotopy type
3371:
3370:
3363:
3344:
3331:
3315:
3314:
3306:
3282:
3281:
3279:
3255:
3254:
3246:
3219:
3194:
3181:
3168:
3143:
3130:
3124:
3101:
3088:
3082:
3053:
3032:
3031:
3029:
3005:
2984:
2978:
2954:
2933:
2932:
2930:
2910:
2890:
2884:
2878:
2877:
2871:
2862:
2856:
2835:
2834:
2832:
2811:
2810:
2808:
2767:
2747:
2691:
2686:
2684:
2683:
2679:
2632:
2608:
2607:
2605:
2566:
2529:
2496:
2461:
2456:
2454:
2453:
2448:
2413:
2385:
2367:
2346:
2345:
2326:
2302:
2282:
2262:
2239:
2206:
2193:
2177:
2166:
2145:
2132:
2107:
2096:
2089:
2072:
2029:
2013:
2001:
1977:
1958:
1952:
1925:
1912:
1893:
1880:
1867:
1854:
1839:
1803:
1802:
1800:
1775:
1774:
1772:
1751:
1750:
1748:
1727:
1726:
1724:
1695:
1694:
1692:
1670:
1669:
1667:
1646:
1645:
1643:
1620:
1601:
1600:
1598:
1574:
1573:
1571:
1542:
1541:
1523:
1517:
1497:
1476:
1470:
1469:
1460:
1440:
1410:
1404:
1403:
1393:
1387:
1386:
1376:
1370:
1349:
1348:
1346:
1314:
1293:
1292:
1290:
1270:
1237:
1236:
1234:
1211:
1210:
1208:
1181:
1180:
1178:
1154:
1153:
1144:
1143:
1134:
1133:
1124:
1123:
1121:
1093:
1073:
1052:
1051:
1043:
1031:{\displaystyle \sigma \in {\mathcal {X}}}
1022:
1021:
1013:
988:
987:
985:
965:
917:
897:
876:
875:
867:
855:{\displaystyle \sigma \in {\mathcal {X}}}
846:
845:
837:
790:
770:
749:
748:
740:
728:{\displaystyle \sigma \in {\mathcal {X}}}
719:
718:
710:
683:
682:
673:
672:
664:
620:
587:
545:
544:
537:
531:
508:
507:
498:
483:
451:
401:
400:
393:
372:
348:
347:
345:
325:
301:
281:
235:
214:
213:
211:
191:
171:
152:
151:
142:
141:
132:
131:
123:
99:
98:
96:
72:
2443:. This relationship may be expressed as
3675:
3618:and similarly for the homotopy groups.
1485:{\displaystyle K\in {\mathcal {K}}^{k}}
1061:{\displaystyle \tau \in {\mathcal {X}}}
885:{\displaystyle \tau \in {\mathcal {X}}}
758:{\displaystyle \tau \in {\mathcal {X}}}
340:of the free abelian group generated by
261:{\displaystyle \kappa (\sigma ,~\tau )}
4262:Séminaire Lotharingien de Combinatoire
3955:
3943:
3931:
2355:{\displaystyle A,A'\in {\mathcal {A}}}
166:in the following way: given two cells
3904:Discrete & Computational Geometry
3784:Discrete & Computational Geometry
3749:Discrete & Computational Geometry
3267:{\displaystyle \chi ({\mathcal {X}})}
1068:: either a boundary cell with larger
7:
3110:{\displaystyle m_{N}\geq \beta _{N}}
1008:CW complex. In this case, each cell
3596:
3564:
3510:
3428:
3074:, the following inequalities hold
2634:
628:
622:
485:
459:
453:
374:
327:
14:
4222:"Morse theory for cell complexes"
4284:Combinatorial algebraic topology
912:in their boundary which satisfy
3976:. ACM Press. pp. 351–360.
63:Notation regarding CW complexes
4311:Simplicial complexes of graphs
3664:Discrete differential geometry
3599:
3583:
3567:
3551:
3496:{\displaystyle {\mathcal {A}}}
3461:
3441:and a discrete Morse function
3414:{\displaystyle {\mathcal {X}}}
3321:
3311:
3291:{\displaystyle {\mathcal {X}}}
3261:
3251:
3041:{\displaystyle {\mathcal {X}}}
2942:{\displaystyle {\mathcal {A}}}
2891:
2872:
2844:{\displaystyle {\mathcal {X}}}
2820:{\displaystyle {\mathcal {A}}}
2718:
2712:
2687:
2669:
2652:
2643:
2637:
2617:{\displaystyle {\mathcal {A}}}
2583:
2559:
2550:
2544:
2535:
2516:
2507:
2501:
2457:
2430:
2406:
2391:
2372:
2212:
2186:
2157:
2125:
2083:
2077:
2041:
2006:
1989:{\displaystyle Q_{m}=p(K_{m})}
1983:
1970:
1931:
1847:
1831:is a sequence of paired cells
1812:{\displaystyle {\mathcal {A}}}
1784:{\displaystyle {\mathcal {Q}}}
1760:{\displaystyle {\mathcal {K}}}
1736:{\displaystyle {\mathcal {A}}}
1704:{\displaystyle {\mathcal {X}}}
1655:{\displaystyle {\mathcal {X}}}
1615:, the only allowed values are
1583:{\displaystyle {\mathcal {X}}}
1535:
1529:
1399:
1358:{\displaystyle {\mathcal {Q}}}
1328:
1316:
1302:{\displaystyle {\mathcal {K}}}
1246:{\displaystyle {\mathcal {Q}}}
1220:{\displaystyle {\mathcal {K}}}
1190:{\displaystyle {\mathcal {A}}}
997:{\displaystyle {\mathcal {X}}}
943:
937:
928:
922:
816:
810:
801:
795:
679:
593:
574:
568:
556:
424:
412:
383:
377:
357:{\displaystyle {\mathcal {X}}}
255:
240:
223:{\displaystyle {\mathcal {X}}}
148:
108:{\displaystyle {\mathcal {X}}}
1:
3692:Mathematical Research Letters
1662:is a regular CW complex over
115:its set of cells. Define the
1677:{\displaystyle \mathbb {Z} }
1608:{\displaystyle \mathbb {Z} }
1257:By construction, there is a
4176:10.2312/VAST/VAST10/131-138
4091:10.4230/LIPIcs.SoCG.2018.31
4415:
4335:. Universitext. Springer.
4331:; Welker, Volkmar (2007).
4137:10.1007/s00371-021-02255-7
4031:10.1109/TPAMI.2014.2346172
3705:10.4310/MRL.2011.v18.n1.a4
2993:{\displaystyle \beta _{q}}
1365:, which can be denoted by
4341:10.1007/978-3-540-68376-6
3946:, Corollaries 3.5 and 3.6
3918:10.1007/s00454-013-9529-6
3808:10.1007/s00454-011-9350-z
3763:10.1007/s00454-013-9529-6
3659:Topological combinatorics
3434:{\displaystyle \partial }
1590:. For instance, over the
1201:cells which are unpaired,
333:{\displaystyle \partial }
57:topological data analysis
651:Discrete Morse functions
4367:"Discrete Morse theory"
4309:Jonsson, Jakob (2007).
4282:Kozlov, Dmitry (2007).
4230:Advances in Mathematics
3649:Stratified Morse theory
3516:{\displaystyle \Delta }
2598:Morse boundary operator
973:{\displaystyle \sigma }
905:{\displaystyle \sigma }
778:{\displaystyle \sigma }
699:discrete Morse function
289:{\displaystyle \sigma }
179:{\displaystyle \sigma }
4399:Computational topology
4252:Forman, Robin (2002).
4244:10.1006/aima.1997.1650
4220:Forman, Robin (1998).
3880:"the Topology ToolKit"
3609:
3517:
3497:
3473:
3435:
3415:
3383:
3292:
3268:
3229:
3111:
3068:
3067:{\displaystyle N>0}
3042:
3014:
2994:
2963:
2943:
2919:
2899:
2845:
2821:
2799:The Morse inequalities
2781:
2756:
2733:
2618:
2600:on the critical cells
2590:
2481:
2437:
2356:
2311:
2291:
2271:
2251:
2225:
2182:
2118:
2054:
1990:
1938:
1813:
1785:
1761:
1737:
1705:
1678:
1656:
1632:
1609:
1584:
1552:
1506:
1486:
1449:
1426:
1359:
1341:-dimensional cells in
1335:
1303:
1285:-dimensional cells in
1279:
1247:
1221:
1191:
1164:
1102:
1082:
1062:
1032:
998:
974:
950:
906:
886:
862:, the number of cells
856:
823:
779:
759:
735:, the number of cells
729:
691:
641:
606:
518:
472:
437:
358:
334:
310:
290:
262:
224:
200:
180:
160:
109:
81:
3982:10.1145/777792.777845
3610:
3518:
3498:
3474:
3436:
3416:
3384:
3293:
3269:
3230:
3112:
3069:
3043:
3015:
2995:
2964:
2944:
2920:
2900:
2846:
2822:
2782:
2757:
2734:
2619:
2591:
2482:
2438:
2357:
2312:
2310:{\displaystyle \rho }
2292:
2290:{\displaystyle \rho }
2277:must decrease across
2272:
2252:
2250:{\displaystyle \pm 1}
2226:
2162:
2092:
2055:
1991:
1939:
1814:
1786:
1762:
1738:
1706:
1679:
1657:
1633:
1631:{\displaystyle \pm 1}
1610:
1585:
1553:
1507:
1487:
1450:
1427:
1360:
1336:
1334:{\displaystyle (k-1)}
1304:
1280:
1248:
1222:
1192:
1165:
1103:
1083:
1063:
1033:
999:
975:
951:
907:
887:
857:
824:
780:
760:
730:
692:
642:
607:
519:
473:
438:
359:
335:
311:
309:{\displaystyle \tau }
291:
276:from the boundary of
263:
225:
201:
199:{\displaystyle \tau }
181:
161:
110:
82:
17:Discrete Morse theory
4269:: Art. B48c, 35 pp.
3843:10.1109/TVCG.2004.18
3644:Digital Morse theory
3538:
3523:. Then, there is an
3507:
3483:
3445:
3425:
3401:
3305:
3278:
3245:
3240:Euler characteristic
3123:
3081:
3052:
3028:
3004:
2977:
2953:
2929:
2909:
2855:
2831:
2807:
2766:
2746:
2631:
2604:
2495:
2447:
2366:
2325:
2301:
2281:
2270:{\displaystyle \mu }
2261:
2238:
2071:
2000:
1951:
1838:
1799:
1771:
1747:
1723:
1691:
1666:
1642:
1619:
1597:
1570:
1516:
1496:
1459:
1439:
1369:
1345:
1313:
1289:
1269:
1233:
1207:
1177:
1120:
1101:{\displaystyle \mu }
1092:
1081:{\displaystyle \mu }
1072:
1042:
1012:
984:
964:
916:
896:
866:
836:
789:
769:
739:
709:
663:
619:
530:
482:
450:
371:
344:
324:
320:is the endomorphism
300:
280:
234:
210:
190:
170:
122:
95:
71:
41:configuration spaces
4125:The Visual Computer
3730:Persistent Homology
2321:two critical cells
1512:to its paired cell
765:in the boundary of
33:applied mathematics
3605:
3513:
3493:
3469:
3431:
3411:
3379:
3288:
3264:
3225:
3107:
3064:
3038:
3010:
2990:
2959:
2939:
2915:
2895:
2841:
2817:
2780:{\displaystyle A'}
2777:
2752:
2729:
2708:
2614:
2586:
2477:
2433:
2352:
2307:
2287:
2267:
2247:
2221:
2050:
1986:
1934:
1809:
1781:
1757:
1733:
1701:
1674:
1652:
1628:
1605:
1580:
1562:in the underlying
1548:
1502:
1482:
1445:
1422:
1355:
1331:
1299:
1275:
1243:
1217:
1187:
1160:
1098:
1078:
1058:
1028:
994:
970:
946:
902:
882:
852:
819:
775:
755:
725:
687:
637:
602:
552:
514:
468:
433:
408:
354:
330:
306:
286:
258:
220:
196:
176:
156:
117:incidence function
105:
77:
4101:978-3-95977-066-8
3991:978-1-58113-663-0
3013:{\displaystyle q}
2962:{\displaystyle q}
2918:{\displaystyle q}
2755:{\displaystyle A}
2696:
2675:
2466:
2216:
2024:
1823:The Morse complex
1719:to a new complex
1505:{\displaystyle K}
1448:{\displaystyle k}
1278:{\displaystyle k}
659:-valued function
533:
389:
318:boundary operator
251:
80:{\displaystyle X}
4406:
4374:
4362:
4324:
4305:
4278:
4258:
4248:
4246:
4226:
4211:
4210:
4209:
4208:
4155:
4149:
4148:
4120:
4114:
4113:
4093:
4075:
4069:
4068:
4042:
4010:
4004:
4003:
3965:
3959:
3953:
3947:
3941:
3935:
3929:
3923:
3922:
3920:
3894:
3888:
3887:
3876:
3870:
3869:
3867:
3861:. Archived from
3828:
3819:
3813:
3812:
3810:
3800:
3774:
3768:
3767:
3765:
3739:
3733:
3723:
3717:
3716:
3707:
3689:
3680:
3614:
3612:
3611:
3606:
3592:
3591:
3582:
3581:
3560:
3559:
3550:
3549:
3522:
3520:
3519:
3514:
3502:
3500:
3499:
3494:
3492:
3491:
3478:
3476:
3475:
3470:
3468:
3460:
3459:
3440:
3438:
3437:
3432:
3420:
3418:
3417:
3412:
3410:
3409:
3388:
3386:
3385:
3380:
3378:
3377:
3376:
3375:
3349:
3348:
3336:
3335:
3320:
3319:
3297:
3295:
3294:
3289:
3287:
3286:
3273:
3271:
3270:
3265:
3260:
3259:
3234:
3232:
3231:
3226:
3224:
3223:
3205:
3204:
3186:
3185:
3173:
3172:
3154:
3153:
3135:
3134:
3116:
3114:
3113:
3108:
3106:
3105:
3093:
3092:
3073:
3071:
3070:
3065:
3048:. Then, for any
3047:
3045:
3044:
3039:
3037:
3036:
3019:
3017:
3016:
3011:
2999:
2997:
2996:
2991:
2989:
2988:
2968:
2966:
2965:
2960:
2948:
2946:
2945:
2940:
2938:
2937:
2924:
2922:
2921:
2916:
2904:
2902:
2901:
2896:
2894:
2889:
2888:
2883:
2882:
2875:
2867:
2866:
2850:
2848:
2847:
2842:
2840:
2839:
2826:
2824:
2823:
2818:
2816:
2815:
2786:
2784:
2783:
2778:
2776:
2761:
2759:
2758:
2753:
2738:
2736:
2735:
2730:
2728:
2707:
2706:
2698:
2697:
2695:
2690:
2685:
2668:
2623:
2621:
2620:
2615:
2613:
2612:
2595:
2593:
2592:
2587:
2582:
2571:
2570:
2534:
2533:
2486:
2484:
2483:
2478:
2476:
2468:
2467:
2465:
2460:
2455:
2442:
2440:
2439:
2434:
2429:
2418:
2417:
2390:
2389:
2361:
2359:
2358:
2353:
2351:
2350:
2341:
2316:
2314:
2313:
2308:
2296:
2294:
2293:
2288:
2276:
2274:
2273:
2268:
2256:
2254:
2253:
2248:
2230:
2228:
2227:
2222:
2217:
2215:
2211:
2210:
2198:
2197:
2181:
2176:
2160:
2156:
2155:
2137:
2136:
2117:
2106:
2090:
2059:
2057:
2056:
2051:
2040:
2039:
2022:
2018:
2017:
1995:
1993:
1992:
1987:
1982:
1981:
1963:
1962:
1943:
1941:
1940:
1935:
1930:
1929:
1917:
1916:
1898:
1897:
1885:
1884:
1872:
1871:
1859:
1858:
1818:
1816:
1815:
1810:
1808:
1807:
1790:
1788:
1787:
1782:
1780:
1779:
1766:
1764:
1763:
1758:
1756:
1755:
1742:
1740:
1739:
1734:
1732:
1731:
1715:on the level of
1710:
1708:
1707:
1702:
1700:
1699:
1683:
1681:
1680:
1675:
1673:
1661:
1659:
1658:
1653:
1651:
1650:
1637:
1635:
1634:
1629:
1614:
1612:
1611:
1606:
1604:
1589:
1587:
1586:
1581:
1579:
1578:
1557:
1555:
1554:
1549:
1547:
1546:
1528:
1527:
1511:
1509:
1508:
1503:
1491:
1489:
1488:
1483:
1481:
1480:
1475:
1474:
1454:
1452:
1451:
1446:
1431:
1429:
1428:
1423:
1421:
1420:
1409:
1408:
1398:
1397:
1392:
1391:
1381:
1380:
1364:
1362:
1361:
1356:
1354:
1353:
1340:
1338:
1337:
1332:
1308:
1306:
1305:
1300:
1298:
1297:
1284:
1282:
1281:
1276:
1252:
1250:
1249:
1244:
1242:
1241:
1226:
1224:
1223:
1218:
1216:
1215:
1196:
1194:
1193:
1188:
1186:
1185:
1169:
1167:
1166:
1161:
1159:
1158:
1149:
1148:
1139:
1138:
1129:
1128:
1107:
1105:
1104:
1099:
1087:
1085:
1084:
1079:
1067:
1065:
1064:
1059:
1057:
1056:
1037:
1035:
1034:
1029:
1027:
1026:
1003:
1001:
1000:
995:
993:
992:
980:, provided that
979:
977:
976:
971:
955:
953:
952:
947:
911:
909:
908:
903:
891:
889:
888:
883:
881:
880:
861:
859:
858:
853:
851:
850:
828:
826:
825:
820:
784:
782:
781:
776:
764:
762:
761:
756:
754:
753:
734:
732:
731:
726:
724:
723:
696:
694:
693:
688:
686:
678:
677:
646:
644:
643:
638:
611:
609:
608:
603:
592:
591:
551:
550:
549:
523:
521:
520:
515:
513:
512:
503:
502:
477:
475:
474:
469:
442:
440:
439:
434:
407:
406:
405:
363:
361:
360:
355:
353:
352:
339:
337:
336:
331:
315:
313:
312:
307:
295:
293:
292:
287:
267:
265:
264:
259:
249:
229:
227:
226:
221:
219:
218:
205:
203:
202:
197:
185:
183:
182:
177:
165:
163:
162:
157:
155:
147:
146:
137:
136:
114:
112:
111:
106:
104:
103:
86:
84:
83:
78:
53:mesh compression
37:computer science
4414:
4413:
4409:
4408:
4407:
4405:
4404:
4403:
4379:
4378:
4377:
4365:
4351:
4327:
4321:
4308:
4294:
4281:
4256:
4251:
4224:
4219:
4215:
4214:
4206:
4204:
4186:
4157:
4156:
4152:
4122:
4121:
4117:
4102:
4077:
4076:
4072:
4012:
4011:
4007:
3992:
3967:
3966:
3962:
3954:
3950:
3942:
3938:
3930:
3926:
3896:
3895:
3891:
3878:
3877:
3873:
3865:
3826:
3821:
3820:
3816:
3776:
3775:
3771:
3741:
3740:
3736:
3724:
3720:
3687:
3682:
3681:
3677:
3672:
3640:
3624:
3573:
3541:
3536:
3535:
3505:
3504:
3481:
3480:
3443:
3442:
3423:
3422:
3399:
3398:
3395:
3359:
3340:
3327:
3303:
3302:
3276:
3275:
3243:
3242:
3215:
3190:
3177:
3164:
3139:
3126:
3121:
3120:
3097:
3084:
3079:
3078:
3050:
3049:
3026:
3025:
3002:
3001:
2980:
2975:
2974:
2951:
2950:
2927:
2926:
2907:
2906:
2876:
2858:
2853:
2852:
2829:
2828:
2805:
2804:
2801:
2793:
2769:
2764:
2763:
2744:
2743:
2721:
2699:
2661:
2629:
2628:
2602:
2601:
2596:. Finally, the
2575:
2562:
2525:
2493:
2492:
2469:
2445:
2444:
2422:
2409:
2381:
2364:
2363:
2334:
2323:
2322:
2299:
2298:
2279:
2278:
2259:
2258:
2236:
2235:
2202:
2189:
2161:
2141:
2128:
2091:
2069:
2068:
2025:
2009:
1998:
1997:
1973:
1954:
1949:
1948:
1921:
1908:
1889:
1876:
1863:
1850:
1836:
1835:
1825:
1797:
1796:
1769:
1768:
1745:
1744:
1721:
1720:
1689:
1688:
1664:
1663:
1640:
1639:
1617:
1616:
1595:
1594:
1568:
1567:
1519:
1514:
1513:
1494:
1493:
1468:
1457:
1456:
1437:
1436:
1402:
1385:
1372:
1367:
1366:
1343:
1342:
1311:
1310:
1287:
1286:
1267:
1266:
1231:
1230:
1205:
1204:
1175:
1174:
1118:
1117:
1090:
1089:
1070:
1069:
1040:
1039:
1010:
1009:
982:
981:
962:
961:
956:is at most one.
914:
913:
894:
893:
864:
863:
834:
833:
829:is at most one.
787:
786:
767:
766:
737:
736:
707:
706:
661:
660:
653:
617:
616:
583:
528:
527:
494:
480:
479:
448:
447:
369:
368:
342:
341:
322:
321:
298:
297:
278:
277:
232:
231:
208:
207:
188:
187:
168:
167:
120:
119:
93:
92:
69:
68:
65:
12:
11:
5:
4412:
4410:
4402:
4401:
4396:
4391:
4381:
4380:
4376:
4375:
4363:
4350:978-3540683759
4349:
4325:
4320:978-3540758587
4319:
4306:
4293:978-3540719618
4292:
4279:
4249:
4216:
4213:
4212:
4184:
4150:
4131:(9): 2883–94.
4115:
4100:
4070:
4025:(3): 654–666.
4005:
3990:
3960:
3948:
3936:
3924:
3911:(2): 330–353.
3889:
3871:
3868:on 2012-04-26.
3837:(5): 499–508.
3814:
3791:(2): 347–377.
3769:
3756:(2): 330–353.
3734:
3718:
3674:
3673:
3671:
3668:
3667:
3666:
3661:
3656:
3654:Shape analysis
3651:
3646:
3639:
3636:
3623:
3620:
3616:
3615:
3604:
3601:
3598:
3595:
3590:
3585:
3580:
3576:
3572:
3569:
3566:
3563:
3558:
3553:
3548:
3544:
3512:
3490:
3467:
3463:
3458:
3453:
3450:
3430:
3408:
3394:
3391:
3390:
3389:
3374:
3369:
3366:
3362:
3358:
3355:
3352:
3347:
3343:
3339:
3334:
3330:
3326:
3323:
3318:
3313:
3310:
3285:
3263:
3258:
3253:
3250:
3238:Moreover, the
3236:
3235:
3222:
3218:
3214:
3211:
3208:
3203:
3200:
3197:
3193:
3189:
3184:
3180:
3176:
3171:
3167:
3163:
3160:
3157:
3152:
3149:
3146:
3142:
3138:
3133:
3129:
3118:
3104:
3100:
3096:
3091:
3087:
3063:
3060:
3057:
3035:
3009:
2987:
2983:
2958:
2949:is called the
2936:
2914:
2893:
2887:
2881:
2874:
2870:
2865:
2861:
2838:
2814:
2800:
2797:
2792:
2789:
2775:
2772:
2751:
2740:
2739:
2727:
2724:
2720:
2717:
2714:
2711:
2705:
2702:
2694:
2689:
2682:
2678:
2674:
2671:
2667:
2664:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2624:is defined by
2611:
2585:
2581:
2578:
2574:
2569:
2565:
2561:
2558:
2555:
2552:
2549:
2546:
2543:
2540:
2537:
2532:
2528:
2524:
2521:
2518:
2515:
2512:
2509:
2506:
2503:
2500:
2475:
2472:
2464:
2459:
2452:
2432:
2428:
2425:
2421:
2416:
2412:
2408:
2405:
2402:
2399:
2396:
2393:
2388:
2384:
2380:
2377:
2374:
2371:
2349:
2344:
2340:
2337:
2333:
2330:
2306:
2286:
2266:
2246:
2243:
2232:
2231:
2220:
2214:
2209:
2205:
2201:
2196:
2192:
2188:
2185:
2180:
2175:
2172:
2169:
2165:
2159:
2154:
2151:
2148:
2144:
2140:
2135:
2131:
2127:
2124:
2121:
2116:
2113:
2110:
2105:
2102:
2099:
2095:
2088:
2085:
2082:
2079:
2076:
2049:
2046:
2043:
2038:
2035:
2032:
2028:
2021:
2016:
2012:
2008:
2005:
1985:
1980:
1976:
1972:
1969:
1966:
1961:
1957:
1945:
1944:
1933:
1928:
1924:
1920:
1915:
1911:
1907:
1904:
1901:
1896:
1892:
1888:
1883:
1879:
1875:
1870:
1866:
1862:
1857:
1853:
1849:
1846:
1843:
1824:
1821:
1806:
1793:gradient paths
1778:
1754:
1730:
1698:
1672:
1649:
1627:
1624:
1603:
1577:
1545:
1540:
1537:
1534:
1531:
1526:
1522:
1501:
1479:
1473:
1467:
1464:
1444:
1434:natural number
1419:
1416:
1413:
1407:
1401:
1396:
1390:
1384:
1379:
1375:
1352:
1330:
1327:
1324:
1321:
1318:
1296:
1274:
1255:
1254:
1240:
1228:
1214:
1202:
1184:
1157:
1152:
1147:
1142:
1137:
1132:
1127:
1097:
1077:
1055:
1050:
1047:
1025:
1020:
1017:
991:
969:
958:
957:
945:
942:
939:
936:
933:
930:
927:
924:
921:
901:
879:
874:
871:
849:
844:
841:
830:
818:
815:
812:
809:
806:
803:
800:
797:
794:
785:which satisfy
774:
752:
747:
744:
722:
717:
714:
685:
681:
676:
671:
668:
652:
649:
636:
633:
630:
627:
624:
613:
612:
601:
598:
595:
590:
586:
582:
579:
576:
573:
570:
567:
564:
561:
558:
555:
548:
543:
540:
536:
511:
506:
501:
497:
493:
490:
487:
467:
464:
461:
458:
455:
444:
443:
432:
429:
426:
423:
420:
417:
414:
411:
404:
399:
396:
392:
388:
385:
382:
379:
376:
351:
329:
305:
285:
257:
254:
248:
245:
242:
239:
217:
195:
175:
154:
150:
145:
140:
135:
130:
127:
102:
91:and denote by
76:
64:
61:
23:adaptation of
13:
10:
9:
6:
4:
3:
2:
4411:
4400:
4397:
4395:
4392:
4390:
4389:Combinatorics
4387:
4386:
4384:
4372:
4368:
4364:
4360:
4356:
4352:
4346:
4342:
4338:
4334:
4330:
4326:
4322:
4316:
4312:
4307:
4303:
4299:
4295:
4289:
4285:
4280:
4276:
4272:
4268:
4264:
4263:
4255:
4250:
4245:
4240:
4237:(1): 90–145.
4236:
4232:
4231:
4223:
4218:
4217:
4203:
4199:
4195:
4191:
4187:
4185:9783038681786
4181:
4177:
4173:
4169:
4165:
4161:
4154:
4151:
4146:
4142:
4138:
4134:
4130:
4126:
4119:
4116:
4111:
4107:
4103:
4097:
4092:
4087:
4083:
4082:
4074:
4071:
4066:
4062:
4058:
4054:
4050:
4046:
4041:
4036:
4032:
4028:
4024:
4020:
4016:
4009:
4006:
4001:
3997:
3993:
3987:
3983:
3979:
3975:
3971:
3964:
3961:
3958:, Theorem 7.3
3957:
3952:
3949:
3945:
3940:
3937:
3933:
3928:
3925:
3919:
3914:
3910:
3906:
3905:
3900:
3893:
3890:
3885:
3881:
3875:
3872:
3864:
3860:
3856:
3852:
3848:
3844:
3840:
3836:
3832:
3825:
3818:
3815:
3809:
3804:
3799:
3794:
3790:
3786:
3785:
3780:
3773:
3770:
3764:
3759:
3755:
3751:
3750:
3745:
3738:
3735:
3731:
3727:
3722:
3719:
3715:
3711:
3706:
3701:
3697:
3693:
3686:
3679:
3676:
3669:
3665:
3662:
3660:
3657:
3655:
3652:
3650:
3647:
3645:
3642:
3641:
3637:
3635:
3633:
3629:
3621:
3619:
3602:
3593:
3578:
3574:
3570:
3561:
3546:
3542:
3534:
3533:
3532:
3530:
3526:
3451:
3448:
3392:
3367:
3364:
3360:
3356:
3353:
3350:
3345:
3341:
3337:
3332:
3328:
3324:
3308:
3301:
3300:
3299:
3248:
3241:
3220:
3216:
3212:
3209:
3206:
3201:
3198:
3195:
3191:
3187:
3182:
3178:
3174:
3169:
3165:
3161:
3158:
3155:
3150:
3147:
3144:
3140:
3136:
3131:
3127:
3119:
3102:
3098:
3094:
3089:
3085:
3077:
3076:
3075:
3061:
3058:
3055:
3023:
3007:
2985:
2981:
2972:
2956:
2912:
2885:
2868:
2863:
2859:
2851:. The number
2798:
2796:
2791:Basic results
2790:
2788:
2773:
2770:
2749:
2725:
2722:
2715:
2709:
2703:
2700:
2692:
2680:
2676:
2672:
2665:
2662:
2658:
2655:
2649:
2646:
2640:
2627:
2626:
2625:
2599:
2579:
2576:
2572:
2567:
2563:
2556:
2553:
2547:
2541:
2538:
2530:
2526:
2522:
2519:
2513:
2510:
2504:
2498:
2490:
2473:
2470:
2462:
2450:
2426:
2423:
2419:
2414:
2410:
2403:
2400:
2397:
2394:
2386:
2382:
2378:
2375:
2369:
2342:
2338:
2335:
2331:
2328:
2320:
2304:
2284:
2264:
2244:
2241:
2218:
2207:
2203:
2199:
2194:
2190:
2183:
2178:
2173:
2170:
2167:
2163:
2152:
2149:
2146:
2142:
2138:
2133:
2129:
2122:
2119:
2114:
2111:
2108:
2103:
2100:
2097:
2093:
2086:
2080:
2074:
2067:
2066:
2065:
2063:
2047:
2044:
2036:
2033:
2030:
2026:
2019:
2014:
2010:
2003:
1978:
1974:
1967:
1964:
1959:
1955:
1926:
1922:
1918:
1913:
1909:
1905:
1902:
1899:
1894:
1890:
1886:
1881:
1877:
1873:
1868:
1864:
1860:
1855:
1851:
1844:
1841:
1834:
1833:
1832:
1830:
1829:gradient path
1822:
1820:
1794:
1718:
1714:
1685:
1625:
1622:
1593:
1565:
1561:
1538:
1532:
1524:
1520:
1499:
1477:
1465:
1462:
1442:
1435:
1417:
1414:
1411:
1394:
1382:
1377:
1373:
1325:
1322:
1319:
1272:
1264:
1260:
1229:
1203:
1200:
1173:
1172:
1171:
1150:
1140:
1130:
1115:
1111:
1095:
1075:
1048:
1045:
1018:
1015:
1007:
967:
940:
934:
931:
925:
919:
899:
872:
869:
842:
839:
832:For any cell
831:
813:
807:
804:
798:
792:
772:
745:
742:
715:
712:
705:For any cell
704:
703:
702:
700:
669:
666:
658:
650:
648:
634:
631:
625:
599:
596:
584:
580:
577:
571:
565:
562:
559:
553:
541:
538:
534:
526:
525:
524:
504:
495:
491:
488:
465:
462:
456:
430:
427:
421:
418:
415:
409:
397:
394:
390:
386:
380:
367:
366:
365:
319:
303:
283:
275:
274:attaching map
271:
252:
246:
243:
237:
193:
173:
138:
128:
125:
118:
90:
74:
62:
60:
58:
54:
50:
47:computation,
46:
42:
38:
34:
30:
27:developed by
26:
22:
21:combinatorial
18:
4394:Morse theory
4332:
4329:Orlik, Peter
4313:. Springer.
4310:
4283:
4266:
4260:
4234:
4228:
4205:, retrieved
4167:
4160:Mara, Hubert
4153:
4128:
4124:
4118:
4080:
4073:
4022:
4018:
4008:
3973:
3963:
3951:
3939:
3927:
3908:
3902:
3892:
3883:
3874:
3863:the original
3834:
3830:
3817:
3788:
3782:
3772:
3753:
3747:
3737:
3721:
3698:(1): 39–57,
3695:
3691:
3678:
3628:lithic tools
3625:
3622:Applications
3617:
3396:
3237:
3022:Betti number
2971:Morse number
2970:
2802:
2794:
2741:
2597:
2489:multiplicity
2488:
2318:
2233:
2061:
1946:
1828:
1826:
1792:
1686:
1256:
1198:
1197:denotes the
1113:
1109:
1005:
959:
698:
654:
614:
445:
116:
66:
29:Robin Forman
25:Morse theory
16:
15:
3956:Forman 1998
3944:Forman 1998
3934:, Lemma 2.5
3932:Forman 1998
3632:archaeology
3525:isomorphism
3000:denote the
2317:is said to
2297:. The path
1947:satisfying
892:containing
364:defined by
4383:Categories
4207:2022-10-05
4040:1885/12873
3670:References
3298:satisfies
2925:-cells in
1713:isomorphic
89:CW complex
39:, such as
4194:2312-6124
4145:237426675
4049:1939-3539
3884:GitHub.io
3798:1001.1269
3732:software.
3597:Δ
3579:∗
3571:≃
3565:∂
3547:∗
3511:Δ
3462:→
3452::
3449:μ
3429:∂
3368:
3357:±
3354:⋯
3338:−
3309:χ
3249:χ
3217:β
3213:±
3210:⋯
3199:−
3192:β
3188:−
3179:β
3175:≥
3162:±
3159:⋯
3148:−
3137:−
3099:β
3095:≥
2982:β
2716:ρ
2693:ρ
2688:→
2677:∑
2650:κ
2635:Δ
2557:κ
2554:⋅
2548:ρ
2542:ν
2539:⋅
2514:κ
2505:ρ
2463:ρ
2458:→
2404:κ
2401:≠
2395:≠
2370:κ
2343:∈
2305:ρ
2285:ρ
2265:μ
2242:±
2184:κ
2164:∏
2123:κ
2120:−
2112:−
2094:∏
2081:ρ
2075:ν
2045:≠
2004:κ
1903:…
1842:ρ
1791:describe
1623:±
1539:∈
1466:∈
1432:for each
1415:−
1400:→
1383::
1323:−
1259:bijection
1170:, where:
1151:⊔
1141:⊔
1096:μ
1076:μ
1049:∈
1046:τ
1019:∈
1016:σ
968:σ
941:τ
935:μ
932:≥
926:σ
920:μ
900:σ
873:∈
870:τ
843:∈
840:σ
814:τ
808:μ
805:≤
799:σ
793:μ
773:σ
746:∈
743:τ
716:∈
713:σ
680:→
670::
667:μ
632:≡
629:∂
626:∘
623:∂
589:′
585:τ
578:τ
572:κ
566:τ
560:σ
554:κ
542:∈
539:τ
535:∑
505:∈
500:′
496:τ
489:σ
486:∀
463:≡
460:∂
457:∘
454:∂
428:τ
422:τ
416:σ
410:κ
398:∈
395:τ
391:∑
381:σ
375:∂
328:∂
304:τ
284:σ
253:τ
244:σ
238:κ
194:τ
174:σ
149:→
139:×
129::
126:κ
49:denoising
4202:17294591
4162:(2022),
4057:26353267
3851:15794132
3638:See also
3529:homology
2774:′
2726:′
2704:′
2666:′
2580:′
2474:′
2427:′
2339:′
1717:homology
1592:integers
1309:and the
1265:between
1199:critical
1114:critical
45:homology
4359:2322081
4302:2361455
4275:1939695
4110:3994099
4065:7406197
4000:1570976
3859:2185198
3726:Perseus
3714:2770581
3531:groups
2319:connect
1006:regular
272:of the
268:be the
4357:
4347:
4317:
4300:
4290:
4273:
4200:
4192:
4182:
4143:
4108:
4098:
4063:
4055:
4047:
3998:
3988:
3857:
3849:
3728:: the
3712:
3479:. Let
2973:. Let
2487:. The
2060:. The
2023:
316:. The
270:degree
250:
230:, let
55:, and
4257:(PDF)
4225:(PDF)
4198:S2CID
4141:S2CID
4106:S2CID
4061:S2CID
3996:S2CID
3866:(PDF)
3855:S2CID
3827:(PDF)
3793:arXiv
3688:(PDF)
3117:, and
2062:index
1558:is a
1004:is a
697:is a
87:be a
19:is a
4371:nLab
4345:ISBN
4315:ISBN
4288:ISBN
4190:ISSN
4180:ISBN
4096:ISBN
4053:PMID
4045:ISSN
3986:ISBN
3847:PMID
3397:Let
3059:>
3020:-th
2969:-th
2803:Let
1996:and
1767:and
1564:ring
1560:unit
1263:sets
657:real
186:and
67:Let
35:and
4337:doi
4239:doi
4235:134
4172:doi
4133:doi
4086:doi
4035:hdl
4027:doi
3978:doi
3913:doi
3839:doi
3803:doi
3758:doi
3700:doi
3630:in
3527:of
3365:dim
3274:of
3024:of
2905:of
2762:to
2362:if
1711:is
1566:of
1261:of
1110:and
296:to
206:in
4385::
4369:.
4355:MR
4353:.
4343:.
4298:MR
4296:.
4271:MR
4267:48
4265:.
4259:.
4233:.
4227:.
4196:,
4188:,
4178:,
4166:,
4139:.
4129:37
4127:.
4104:.
4094:.
4059:.
4051:.
4043:.
4033:.
4023:37
4021:.
4017:.
3994:.
3984:.
3972:.
3909:50
3907:.
3901:.
3882:.
3853:.
3845:.
3835:10
3833:.
3829:.
3801:.
3789:47
3787:.
3781:.
3754:50
3752:.
3746:.
3710:MR
3708:,
3696:18
3694:,
3690:,
3634:.
2787:.
1827:A
1684:.
655:A
647:.
59:.
51:,
43:,
4373:.
4361:.
4339::
4323:.
4304:.
4277:.
4247:.
4241::
4174::
4147:.
4135::
4112:.
4088::
4067:.
4037::
4029::
4002:.
3980::
3921:.
3915::
3886:.
3841::
3811:.
3805::
3795::
3766:.
3760::
3702::
3603:,
3600:)
3594:,
3589:A
3584:(
3575:H
3568:)
3562:,
3557:X
3552:(
3543:H
3489:A
3466:R
3457:X
3407:X
3373:X
3361:m
3351:+
3346:1
3342:m
3333:0
3329:m
3325:=
3322:)
3317:X
3312:(
3284:X
3262:)
3257:X
3252:(
3221:0
3207:+
3202:1
3196:N
3183:N
3170:0
3166:m
3156:+
3151:1
3145:N
3141:m
3132:N
3128:m
3103:N
3090:N
3086:m
3062:0
3056:N
3034:X
3008:q
2986:q
2957:q
2935:A
2913:q
2892:|
2886:q
2880:A
2873:|
2869:=
2864:q
2860:m
2837:X
2813:A
2771:A
2750:A
2723:A
2719:)
2713:(
2710:m
2701:A
2681:A
2673:+
2670:)
2663:A
2659:,
2656:A
2653:(
2647:=
2644:)
2641:A
2638:(
2610:A
2584:)
2577:A
2573:,
2568:M
2564:K
2560:(
2551:)
2545:(
2536:)
2531:1
2527:Q
2523:,
2520:A
2517:(
2511:=
2508:)
2502:(
2499:m
2471:A
2451:A
2431:)
2424:A
2420:,
2415:M
2411:K
2407:(
2398:0
2392:)
2387:1
2383:Q
2379:,
2376:A
2373:(
2348:A
2336:A
2332:,
2329:A
2245:1
2219:.
2213:)
2208:m
2204:Q
2200:,
2195:m
2191:K
2187:(
2179:M
2174:1
2171:=
2168:m
2158:)
2153:1
2150:+
2147:m
2143:Q
2139:,
2134:m
2130:K
2126:(
2115:1
2109:M
2104:1
2101:=
2098:m
2087:=
2084:)
2078:(
2048:0
2042:)
2037:1
2034:+
2031:m
2027:Q
2020:,
2015:m
2011:K
2007:(
1984:)
1979:m
1975:K
1971:(
1968:p
1965:=
1960:m
1956:Q
1932:)
1927:M
1923:K
1919:,
1914:M
1910:Q
1906:,
1900:,
1895:2
1891:K
1887:,
1882:2
1878:Q
1874:,
1869:1
1865:K
1861:,
1856:1
1852:Q
1848:(
1845:=
1805:A
1777:Q
1753:K
1729:A
1697:X
1671:Z
1648:X
1626:1
1602:Z
1576:X
1544:Q
1536:)
1533:K
1530:(
1525:k
1521:p
1500:K
1478:k
1472:K
1463:K
1443:k
1418:1
1412:k
1406:Q
1395:k
1389:K
1378:k
1374:p
1351:Q
1329:)
1326:1
1320:k
1317:(
1295:K
1273:k
1239:Q
1213:K
1183:A
1156:Q
1146:K
1136:A
1131:=
1126:X
1054:X
1024:X
990:X
944:)
938:(
929:)
923:(
878:X
848:X
817:)
811:(
802:)
796:(
751:X
721:X
684:R
675:X
635:0
600:0
597:=
594:)
581:,
575:(
569:)
563:,
557:(
547:X
510:X
492:,
466:0
431:.
425:)
419:,
413:(
403:X
387:=
384:)
378:(
350:X
256:)
247:,
241:(
216:X
153:Z
144:X
134:X
101:X
75:X
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