3151:
565:
3794:
2862:
3440:
3805:
This algebraic approach to size functions leads to the definition of new similarity measures between shapes, by translating the problem of comparing size functions into the problem of comparing formal series. The most studied among these metrics between size function is the
3560:
3146:{\displaystyle \mu (p){\stackrel {\rm {def}}{=}}\min _{\alpha >0,\beta >0}\ell _{({M},\varphi )}(x+\alpha ,y-\beta )-\ell _{({M},\varphi )}(x+\alpha ,y+\beta )-\ell _{({M},\varphi )}(x-\alpha ,y-\beta )+\ell _{({M},\varphi )}(x-\alpha ,y+\beta )}
2625:
2753:
188:
1449:
3259:
2779:) of such formal series encode the information about discontinuities of the corresponding size functions, while their multiplicities contain the information about the values taken by the size function.
2771:
An algebraic representation of size functions in terms of collections of points and lines in the real plane with multiplicities, i.e. as particular formal series, was furnished in . The points (called
1566:
in order to get the wanted invariance. Moreover, size functions show properties of relative resistance to noise, depending on the fact that they distribute the information all over the half-plane
3789:{\displaystyle \ell _{({M},\varphi )}({\bar {x}},{\bar {y}})=\sum _{p=(x,y) \atop x\leq {\bar {x}},y>{\bar {y}}}\mu {\big (}p{\big )}+\sum _{r:x=k \atop k\leq {\bar {x}}}\mu {\big (}r{\big )}}
613:
526:
237:
922:
869:
816:
720:
389:
1990:
667:
2438:
2102:
1872:
1789:
1684:
336:
3549:
2491:
2312:
1004:
282:
2022:
2500:
1904:
1493:
555:
445:
2630:
1239:
82:
1591:
1140:
1304:
3918:
2372:
2210:
2142:
1213:
945:
743:
465:
3473:
3184:
2825:
2851:
2048:
1930:
1818:
46:
3248:
2273:
2237:
1190:
1113:
1036:
3435:{\displaystyle \mu (r){\stackrel {\rm {def}}{=}}\min _{\alpha >0,k+\alpha <y}\ell _{({M},\varphi )}(k+\alpha ,y)-\ell _{({M},\varphi )}(k-\alpha ,y)>0.}
3497:
3208:
2352:
2332:
2186:
2162:
2122:
1732:
1708:
1619:
1533:
1299:
1279:
1259:
1163:
1086:
1056:
965:
763:
633:
485:
416:
3852:
1535:-th persistent homology group, while the relation between the persistent homology group and the size homotopy group is analogous to the one existing between
3802:
This representation contains the same amount of information about the shape under study as the original size function does, but is much more concise.
3962:
49:
4194:
572:
490:
196:
3935:, Proc. SPIE, Intelligent Robots and Computer Vision X: Algorithms and Techniques, Boston, MA, 1607:122–133, 1991.
874:
821:
768:
672:
341:
4174:
1935:
638:
2384:
2053:
1823:
1740:
1635:
287:
3510:
2443:
1687:
2620:{\displaystyle \ell _{(N,\psi )}({\bar {x}},{\bar {y}})>\ell _{(M,\varphi )}({\tilde {x}},{\tilde {y}})}
4149:
2760:
2379:
2278:
1711:
970:
242:
1998:
1880:
1469:
531:
421:
2748:{\displaystyle d((M,\varphi ),(N,\psi ))\geq \min\{{\tilde {x}}-{\bar {x}},{\bar {y}}-{\tilde {y}}\}}
183:{\displaystyle \ell _{(M,\varphi )}:\Delta ^{+}=\{(x,y)\in \mathbb {R} ^{2}:x<y\}\to \mathbb {N} }
4159:
1551:
1512:
1504:
1459:
392:
57:
1444:{\displaystyle d((P_{1},\ldots ,P_{k}),(Q_{1}\ldots ,Q_{k}))=\max _{1\leq i\leq k}\|P_{i}-Q_{i}\|}
1218:
22:
are shape descriptors, in a geometrical/topological sense. They are functions from the half-plane
1569:
1563:
1559:
1463:
1455:
1193:
1118:
925:
723:
4189:
4169:
4068:
3976:
3900:
3896:
3807:
2357:
2195:
2127:
1622:
1558:. The main point is that size functions are invariant for every transformation preserving the
1198:
930:
728:
450:
396:
53:
3449:
3160:
2792:
1546:
Size functions have been initially introduced as a mathematical tool for shape comparison in
2830:
2027:
1909:
1797:
1281:-tuples of points in a submanifold of a Euclidean space is considered. Here the topology on
25:
3221:
2246:
2215:
1168:
1091:
1009:
2189:
1625:
1547:
1496:
1142:
487:. The concept of size function can be easily extended to the case of a measuring function
3482:
3193:
2337:
2317:
2171:
2147:
2107:
1717:
1693:
1604:
1540:
1536:
1518:
1284:
1264:
1244:
1148:
1071:
1041:
950:
748:
618:
470:
401:
4183:
3903:, Patrizio Frosini, Daniela Giorgi, Claudia Landi, Laura Papaleo, Michela Spagnuolo,
1562:. Hence, they can be adapted to many different applications, by simply changing the
4154:
4115:
1508:
1500:
4043:
The use of size functions for comparison of shapes through differential invariants
4144:
1555:
558:
73:
3873:, International Journal of Imaging Systems and Technology, 16(5):154–161, 2006.
557:
is endowed with the usual partial order . A survey about size functions (and
2763:
and is one of the main motivation to introduce the concept of size function.
4164:
1511:
are strictly related to the concept of persistent homology group studied in
191:
4114:, Proc. SPIE Vol. 3168, pp. 52–60, Vision Geometry VI, Robert A. Melter,
4056:
Retrieval of trademark images by means of size functions
Graphical Models
3905:
Describing shapes by geometrical-topological properties of real functions
61:
3922:, Bulletin of the Australian Mathematical Society, 42(3):407–416, 1990.
3919:
A distance for similarity classes of submanifolds of a
Euclidean space
2378:
A strong link between the concept of size function and the concept of
1515:. It is worth to point out that the size function is the rank of the
3989:
Alessandro Verri, Claudio Uras, Patrizio
Frosini and Massimo Ferri,
564:
4017:
Metric-topological approach to shape representation and recognition
391:
that contain at least one point at which the measuring function (a
4045:, Journal of Mathematical Imaging and Vision, 21(2):107–118, 2004.
2759:
The previous result gives an easy way to get lower bounds for the
563:
4088:, Mathematical Methods in the Applied Sciences, 19:555–569, 1996.
3882:
Silvia
Biasotti, Andrea Cerri, Patrizio Frosini, Claudia Landi,
3886:, Journal of Mathematical Imaging and Vision 32:161–179, 2008.
3849:
Size homotopy groups for computation of natural size distances
818:, is depicted in red. (D) Two connected components of the set
4101:, Archives of Inequalities and Applications, 2(1):1–12, 2004.
3835:, Pattern Recognition And Image Analysis, 9(4):596–603, 1999.
1068:
Size functions were introduced in for the particular case of
4099:
Lower bounds for natural pseudodistances via size functions
338:
is equal to the number of connected components of the set
3957:
722:
is depicted in green. (C) The set of points at which the
4041:
Françoise Dibos, Patrizio
Frosini and Denis Pasquignon,
3907:
ACM Computing
Surveys, vol. 40 (2008), n. 4, 12:1–12:87.
4131:, Appl. Algebra Engrg. Comm. Comput., 12:327–349, 2001.
3977:
Describing and recognising shape through size functions
4086:
Connections between size functions and critical points
3948:, Acta Applicandae Mathematicae, 67(3):225–235, 2001.
3869:
Michele d'Amico, Patrizio
Frosini and Claudia Landi,
3833:
3563:
3513:
3485:
3452:
3262:
3224:
3196:
3163:
2865:
2833:
2795:
2633:
2503:
2446:
2387:
2360:
2340:
2320:
2281:
2249:
2218:
2198:
2174:
2150:
2130:
2110:
2056:
2030:
2001:
1938:
1912:
1883:
1826:
1800:
1743:
1720:
1696:
1638:
1607:
1572:
1521:
1472:
1307:
1287:
1267:
1247:
1221:
1201:
1171:
1151:
1121:
1094:
1074:
1044:
1012:
973:
953:
933:
877:
824:
771:
751:
731:
675:
641:
621:
575:
534:
493:
473:
453:
424:
404:
344:
290:
245:
199:
85:
28:
16:
Shape descriptions in a geometrical/topological sense
4067:
Silvia
Biasotti, Daniela Giorgi, Michela Spagnuolo,
4006:, Acta Applicandae Mathematicae, 49(1):85–104, 1997.
3884:
Multidimensional size functions for shape comparison
1145:
embedded in a Euclidean space. Here the topology on
4032:, Internat. J. Comput. Vision, 23(2):169–183, 1997.
3944:Francesca Cagliari, Massimo Ferri and Paola Pozzi,
3788:
3543:
3491:
3467:
3434:
3242:
3202:
3178:
3145:
2845:
2819:
2747:
2619:
2485:
2432:
2366:
2346:
2326:
2306:
2267:
2231:
2204:
2180:
2156:
2136:
2116:
2096:
2042:
2016:
1984:
1924:
1898:
1866:
1812:
1783:
1726:
1702:
1678:
1613:
1585:
1527:
1487:
1443:
1293:
1273:
1253:
1233:
1207:
1184:
1157:
1134:
1107:
1080:
1050:
1030:
998:
959:
939:
916:
863:
810:
757:
737:
714:
661:
627:
607:
549:
520:
479:
459:
439:
410:
383:
330:
276:
231:
182:
40:
4004:Size functions and morphological transformations
3980:ICSI Technical Report TR-92-057, Berkeley, 1992.
3871:Using matching distance in Size Theory: a survey
3300:
2903:
2679:
2239:-function, the following useful property holds:
2008:
1890:
1791:is locally right-constant in both its variables.
1454:An extension of the concept of size function to
1391:
1088:equal to the topological space of all piecewise
4112:New pseudodistances for the size function space
3991:On the use of size functions for shape analysis
1261:equal to the topological space of all ordered
608:{\displaystyle (M,\varphi :M\to \mathbb {R} )}
521:{\displaystyle \varphi :M\to \mathbb {R} ^{k}}
232:{\displaystyle (M,\varphi :M\to \mathbb {R} )}
4054:Andrea Cerri, Massimo Ferri, Daniela Giorgi,
3781:
3771:
3712:
3702:
2104:equals the number of connected components of
569:An example of size function. (A) A size pair
8:
3853:Bulletin of the Belgian Mathematical Society
2742:
2682:
1438:
1412:
917:{\displaystyle \{p\in M:\varphi (p)\leq a\}}
911:
878:
864:{\displaystyle \{p\in M:\varphi (p)\leq b\}}
858:
825:
811:{\displaystyle \{p\in M:\varphi (p)\leq a\}}
805:
772:
715:{\displaystyle \{p\in M:\varphi (p)\leq b\}}
709:
676:
384:{\displaystyle \{p\in M:\varphi (p)\leq y\}}
378:
345:
169:
124:
3946:Size functions from a categorical viewpoint
1985:{\displaystyle \ell _{(M,\varphi )}(x,y)=0}
239:is defined in the following way. For every
3993:, Biological Cybernetics, 70:99–107, 1993.
3959:Topological Persistence and Simplification
3865:
3863:
3861:
662:{\displaystyle \varphi :M\to \mathbb {R} }
4019:, Image Vision Comput., 14:189–207, 1996.
3843:
3841:
3827:
3825:
3823:
3780:
3779:
3770:
3769:
3751:
3750:
3724:
3711:
3710:
3701:
3700:
3682:
3681:
3661:
3660:
3628:
3607:
3606:
3592:
3591:
3572:
3568:
3562:
3530:
3529:
3515:
3514:
3512:
3484:
3451:
3389:
3385:
3341:
3337:
3303:
3284:
3283:
3278:
3276:
3275:
3261:
3223:
3195:
3162:
3100:
3096:
3046:
3042:
2992:
2988:
2938:
2934:
2906:
2887:
2886:
2881:
2879:
2878:
2864:
2832:
2794:
2731:
2730:
2716:
2715:
2701:
2700:
2686:
2685:
2632:
2603:
2602:
2588:
2587:
2566:
2545:
2544:
2530:
2529:
2508:
2502:
2445:
2433:{\displaystyle d((M,\varphi ),(N,\psi ))}
2386:
2359:
2339:
2319:
2286:
2280:
2248:
2223:
2217:
2197:
2173:
2149:
2129:
2109:
2097:{\displaystyle \ell _{(M,\varphi )}(x,y)}
2061:
2055:
2029:
2000:
1943:
1937:
1911:
1882:
1867:{\displaystyle \ell _{(M,\varphi )}(x,y)}
1831:
1825:
1799:
1784:{\displaystyle \ell _{(M,\varphi )}(x,y)}
1748:
1742:
1719:
1695:
1679:{\displaystyle \ell _{(M,\varphi )}(x,y)}
1643:
1637:
1606:
1577:
1571:
1520:
1479:
1475:
1474:
1471:
1432:
1419:
1394:
1375:
1359:
1340:
1321:
1306:
1286:
1266:
1246:
1220:
1200:
1176:
1170:
1150:
1126:
1120:
1099:
1093:
1073:
1043:
1011:
978:
972:
952:
932:
876:
823:
770:
750:
730:
674:
655:
654:
640:
620:
598:
597:
574:
541:
537:
536:
533:
512:
508:
507:
492:
472:
452:
431:
427:
426:
423:
403:
343:
331:{\displaystyle \ell _{(M,\varphi )}(x,y)}
295:
289:
268:
244:
222:
221:
198:
176:
175:
151:
147:
146:
115:
90:
84:
48:to the natural numbers, counting certain
27:
3847:Patrizio Frosini and Michele Mulazzani,
3544:{\displaystyle {\bar {x}}<{\bar {y}}}
2486:{\displaystyle (M,\varphi ),\ (N,\psi )}
924:, that is, at least one point where the
4075:Pattern Recognition 41:2855–2873, 2008.
4030:Computing size functions from edge maps
3819:
947:takes a value smaller than or equal to
745:takes a value smaller than or equal to
467:takes a value smaller than or equal to
4097:Pietro Donatini and Patrizio Frosini,
4073:Size functions for comparing 3D models
1503:) was introduced in . The concepts of
967:. (E) The value of the size function
7:
4127:Patrizio Frosini and Claudia Landi,
4110:Claudia Landi and Patrizio Frosini,
4002:Patrizio Frosini and Claudia Landi,
3831:Patrizio Frosini and Claudia Landi,
2307:{\displaystyle \ell _{(M,\varphi )}}
999:{\displaystyle \ell _{(M,\varphi )}}
669:is the height function. (B) The set
277:{\displaystyle (x,y)\in \Delta ^{+}}
4028:Alessandro Verri and Claudio Uras,
4015:Alessandro Verri and Claudio Uras,
3974:Claudio Uras and Alessandro Verri,
3963:Discrete and Computational Geometry
2017:{\displaystyle y\geq \max \varphi }
1554:, and have constituted the seed of
3933:Measuring shapes by size functions
3725:
3629:
3291:
3288:
3285:
2894:
2891:
2888:
1899:{\displaystyle x<\min \varphi }
1574:
1127:
265:
112:
14:
4118:, Longin J. Latecki (eds.), 1997.
1628:. The following statements hold:
1458:was made in where the concept of
4129:Size functions and formal series
2354:or both are critical values for
1488:{\displaystyle \mathbb {R} ^{k}}
550:{\displaystyle \mathbb {R} ^{k}}
440:{\displaystyle \mathbb {R} ^{k}}
3218:and are defined as those lines
2767:Representation by formal series
3756:
3687:
3666:
3649:
3637:
3618:
3612:
3597:
3588:
3583:
3569:
3535:
3520:
3462:
3456:
3423:
3405:
3400:
3386:
3375:
3357:
3352:
3338:
3272:
3266:
3173:
3167:
3140:
3116:
3111:
3097:
3086:
3062:
3057:
3043:
3032:
3008:
3003:
2989:
2978:
2954:
2949:
2935:
2875:
2869:
2814:
2802:
2736:
2721:
2706:
2691:
2673:
2670:
2658:
2652:
2640:
2637:
2614:
2608:
2593:
2584:
2579:
2567:
2556:
2550:
2535:
2526:
2521:
2509:
2480:
2468:
2459:
2447:
2427:
2424:
2412:
2406:
2394:
2391:
2299:
2287:
2262:
2250:
2124:on which the minimum value of
2091:
2079:
2074:
2062:
1973:
1961:
1956:
1944:
1861:
1849:
1844:
1832:
1778:
1766:
1761:
1749:
1673:
1661:
1656:
1644:
1384:
1381:
1352:
1346:
1314:
1311:
1241:to its length. In the case of
1025:
1013:
991:
979:
902:
896:
871:contain at least one point in
849:
843:
796:
790:
700:
694:
651:
602:
594:
576:
503:
369:
363:
325:
313:
308:
296:
258:
246:
226:
218:
200:
172:
139:
127:
103:
91:
1:
2275:is a discontinuity point for
1495:are allowed. An extension to
2789:are defined as those points
2314:it is necessary that either
2144:is smaller than or equal to
1234:{\displaystyle \gamma \in M}
1586:{\displaystyle \Delta ^{+}}
1135:{\displaystyle C^{\infty }}
4211:
4175:Topological data analysis
3157:is positive. The number
1301:is induced by the metric
2367:{\displaystyle \varphi }
2205:{\displaystyle \varphi }
2137:{\displaystyle \varphi }
1208:{\displaystyle \varphi }
1064:History and applications
940:{\displaystyle \varphi }
738:{\displaystyle \varphi }
460:{\displaystyle \varphi }
3468:{\displaystyle \mu (r)}
3179:{\displaystyle \mu (p)}
2820:{\displaystyle p=(x,y)}
2440:between the size pairs
2168:If we also assume that
1712:non-increasing function
1688:non-decreasing function
4150:Natural pseudodistance
3965:, 28(4):511–533, 2002.
3790:
3545:
3505:Representation Theorem
3493:
3469:
3436:
3244:
3204:
3180:
3147:
2853:, such that the number
2847:
2846:{\displaystyle x<y}
2821:
2761:natural pseudodistance
2749:
2621:
2487:
2434:
2380:natural pseudodistance
2368:
2348:
2328:
2308:
2269:
2233:
2206:
2182:
2158:
2138:
2118:
2098:
2044:
2043:{\displaystyle x<y}
2018:
1986:
1926:
1925:{\displaystyle y>x}
1900:
1868:
1814:
1813:{\displaystyle x<y}
1785:
1728:
1704:
1680:
1615:
1587:
1529:
1489:
1445:
1295:
1275:
1255:
1235:
1209:
1186:
1159:
1136:
1109:
1082:
1060:
1052:
1032:
1000:
961:
941:
918:
865:
812:
759:
739:
716:
663:
635:is the blue curve and
629:
609:
551:
522:
481:
461:
441:
412:
385:
332:
278:
233:
184:
42:
41:{\displaystyle x<y}
3791:
3546:
3494:
3470:
3437:
3245:
3243:{\displaystyle r:x=k}
3205:
3181:
3148:
2848:
2822:
2750:
2622:
2488:
2435:
2369:
2349:
2329:
2309:
2270:
2268:{\displaystyle (x,y)}
2234:
2232:{\displaystyle C^{1}}
2207:
2183:
2159:
2139:
2119:
2099:
2045:
2019:
1987:
1927:
1901:
1869:
1815:
1786:
1729:
1705:
1681:
1616:
1588:
1530:
1490:
1462:was introduced. Here
1446:
1296:
1276:
1256:
1236:
1210:
1187:
1185:{\displaystyle C^{0}}
1160:
1137:
1110:
1108:{\displaystyle C^{1}}
1083:
1053:
1033:
1031:{\displaystyle (a,b)}
1001:
962:
942:
919:
866:
813:
760:
740:
717:
664:
630:
610:
567:
552:
523:
482:
462:
442:
413:
386:
333:
279:
234:
185:
43:
3561:
3511:
3483:
3450:
3260:
3222:
3194:
3161:
2863:
2831:
2793:
2775:) and lines (called
2631:
2501:
2444:
2385:
2358:
2338:
2318:
2279:
2247:
2216:
2196:
2172:
2148:
2128:
2108:
2054:
2028:
1999:
1936:
1910:
1881:
1824:
1798:
1741:
1737:every size function
1718:
1694:
1636:
1632:every size function
1605:
1570:
1519:
1470:
1305:
1285:
1265:
1245:
1219:
1199:
1169:
1149:
1119:
1092:
1072:
1042:
1010:
971:
951:
931:
875:
822:
769:
749:
729:
673:
639:
619:
573:
532:
491:
471:
451:
422:
402:
342:
288:
243:
197:
190:associated with the
83:
50:connected components
26:
4160:Size homotopy group
1552:pattern recognition
1513:persistent homology
1505:size homotopy group
1464:measuring functions
1460:size homotopy group
561:) can be found in.
393:continuous function
58:pattern recognition
56:. They are used in
4195:Algebraic topology
4084:Patrizio Frosini,
3931:Patrizio Frosini,
3916:Patrizio Frosini,
3786:
3765:
3696:
3541:
3489:
3465:
3432:
3332:
3240:
3200:
3186:is said to be the
3176:
3143:
2929:
2843:
2817:
2745:
2617:
2483:
2430:
2364:
2344:
2324:
2304:
2265:
2229:
2202:
2178:
2154:
2134:
2114:
2094:
2040:
2014:
1982:
1922:
1896:
1864:
1810:
1781:
1724:
1700:
1676:
1611:
1583:
1564:measuring function
1560:measuring function
1525:
1485:
1456:algebraic topology
1441:
1411:
1291:
1271:
1251:
1231:
1205:
1194:measuring function
1182:
1165:is induced by the
1155:
1132:
1115:closed paths in a
1105:
1078:
1061:
1048:
1028:
996:
957:
937:
926:measuring function
914:
861:
808:
755:
735:
724:measuring function
712:
659:
625:
605:
547:
518:
477:
457:
437:
408:
381:
328:
274:
229:
180:
38:
4170:Matching distance
4069:Bianca Falcidieno
4058:68:451–471, 2006.
3901:Bianca Falcidieno
3897:Leila De Floriani
3895:Silvia Biasotti,
3855:, 6:455–464 1999.
3808:matching distance
3763:
3759:
3720:
3694:
3690:
3669:
3624:
3615:
3600:
3538:
3523:
3492:{\displaystyle r}
3475:is sad to be the
3299:
3296:
3203:{\displaystyle p}
2902:
2899:
2739:
2724:
2709:
2694:
2611:
2596:
2553:
2538:
2467:
2347:{\displaystyle y}
2327:{\displaystyle x}
2181:{\displaystyle M}
2157:{\displaystyle x}
2117:{\displaystyle M}
1727:{\displaystyle y}
1703:{\displaystyle x}
1623:locally connected
1614:{\displaystyle M}
1528:{\displaystyle 0}
1466:taking values in
1390:
1294:{\displaystyle M}
1274:{\displaystyle k}
1254:{\displaystyle M}
1192:-norm, while the
1158:{\displaystyle M}
1081:{\displaystyle M}
1051:{\displaystyle 2}
960:{\displaystyle a}
758:{\displaystyle a}
628:{\displaystyle M}
480:{\displaystyle x}
411:{\displaystyle M}
397:topological space
68:Formal definition
54:topological space
4202:
4132:
4125:
4119:
4108:
4102:
4095:
4089:
4082:
4076:
4065:
4059:
4052:
4046:
4039:
4033:
4026:
4020:
4013:
4007:
4000:
3994:
3987:
3981:
3972:
3966:
3955:
3949:
3942:
3936:
3929:
3923:
3914:
3908:
3893:
3887:
3880:
3874:
3867:
3856:
3845:
3836:
3829:
3795:
3793:
3792:
3787:
3785:
3784:
3775:
3774:
3764:
3762:
3761:
3760:
3752:
3742:
3716:
3715:
3706:
3705:
3695:
3693:
3692:
3691:
3683:
3671:
3670:
3662:
3652:
3617:
3616:
3608:
3602:
3601:
3593:
3587:
3586:
3576:
3550:
3548:
3547:
3542:
3540:
3539:
3531:
3525:
3524:
3516:
3498:
3496:
3495:
3490:
3474:
3472:
3471:
3466:
3441:
3439:
3438:
3433:
3404:
3403:
3393:
3356:
3355:
3345:
3331:
3298:
3297:
3295:
3294:
3282:
3277:
3249:
3247:
3246:
3241:
3209:
3207:
3206:
3201:
3185:
3183:
3182:
3177:
3152:
3150:
3149:
3144:
3115:
3114:
3104:
3061:
3060:
3050:
3007:
3006:
2996:
2953:
2952:
2942:
2928:
2901:
2900:
2898:
2897:
2885:
2880:
2852:
2850:
2849:
2844:
2826:
2824:
2823:
2818:
2754:
2752:
2751:
2746:
2741:
2740:
2732:
2726:
2725:
2717:
2711:
2710:
2702:
2696:
2695:
2687:
2626:
2624:
2623:
2618:
2613:
2612:
2604:
2598:
2597:
2589:
2583:
2582:
2555:
2554:
2546:
2540:
2539:
2531:
2525:
2524:
2492:
2490:
2489:
2484:
2465:
2439:
2437:
2436:
2431:
2373:
2371:
2370:
2365:
2353:
2351:
2350:
2345:
2333:
2331:
2330:
2325:
2313:
2311:
2310:
2305:
2303:
2302:
2274:
2272:
2271:
2266:
2238:
2236:
2235:
2230:
2228:
2227:
2211:
2209:
2208:
2203:
2187:
2185:
2184:
2179:
2163:
2161:
2160:
2155:
2143:
2141:
2140:
2135:
2123:
2121:
2120:
2115:
2103:
2101:
2100:
2095:
2078:
2077:
2049:
2047:
2046:
2041:
2023:
2021:
2020:
2015:
1991:
1989:
1988:
1983:
1960:
1959:
1931:
1929:
1928:
1923:
1905:
1903:
1902:
1897:
1873:
1871:
1870:
1865:
1848:
1847:
1819:
1817:
1816:
1811:
1790:
1788:
1787:
1782:
1765:
1764:
1733:
1731:
1730:
1725:
1714:in the variable
1709:
1707:
1706:
1701:
1690:in the variable
1685:
1683:
1682:
1677:
1660:
1659:
1620:
1618:
1617:
1612:
1592:
1590:
1589:
1584:
1582:
1581:
1534:
1532:
1531:
1526:
1494:
1492:
1491:
1486:
1484:
1483:
1478:
1450:
1448:
1447:
1442:
1437:
1436:
1424:
1423:
1410:
1380:
1379:
1364:
1363:
1345:
1344:
1326:
1325:
1300:
1298:
1297:
1292:
1280:
1278:
1277:
1272:
1260:
1258:
1257:
1252:
1240:
1238:
1237:
1232:
1215:takes each path
1214:
1212:
1211:
1206:
1191:
1189:
1188:
1183:
1181:
1180:
1164:
1162:
1161:
1156:
1141:
1139:
1138:
1133:
1131:
1130:
1114:
1112:
1111:
1106:
1104:
1103:
1087:
1085:
1084:
1079:
1057:
1055:
1054:
1049:
1037:
1035:
1034:
1029:
1005:
1003:
1002:
997:
995:
994:
966:
964:
963:
958:
946:
944:
943:
938:
923:
921:
920:
915:
870:
868:
867:
862:
817:
815:
814:
809:
764:
762:
761:
756:
744:
742:
741:
736:
721:
719:
718:
713:
668:
666:
665:
660:
658:
634:
632:
631:
626:
614:
612:
611:
606:
601:
556:
554:
553:
548:
546:
545:
540:
527:
525:
524:
519:
517:
516:
511:
486:
484:
483:
478:
466:
464:
463:
458:
446:
444:
443:
438:
436:
435:
430:
417:
415:
414:
409:
390:
388:
387:
382:
337:
335:
334:
329:
312:
311:
283:
281:
280:
275:
273:
272:
238:
236:
235:
230:
225:
189:
187:
186:
181:
179:
156:
155:
150:
120:
119:
107:
106:
47:
45:
44:
39:
4210:
4209:
4205:
4204:
4203:
4201:
4200:
4199:
4180:
4179:
4141:
4136:
4135:
4126:
4122:
4109:
4105:
4096:
4092:
4083:
4079:
4066:
4062:
4053:
4049:
4040:
4036:
4027:
4023:
4014:
4010:
4001:
3997:
3988:
3984:
3973:
3969:
3956:
3952:
3943:
3939:
3930:
3926:
3915:
3911:
3894:
3890:
3881:
3877:
3868:
3859:
3846:
3839:
3830:
3821:
3816:
3743:
3726:
3653:
3630:
3564:
3559:
3558:
3509:
3508:
3481:
3480:
3448:
3447:
3381:
3333:
3258:
3257:
3220:
3219:
3192:
3191:
3159:
3158:
3092:
3038:
2984:
2930:
2861:
2860:
2829:
2828:
2791:
2790:
2769:
2629:
2628:
2562:
2504:
2499:
2498:
2442:
2441:
2383:
2382:
2356:
2355:
2336:
2335:
2316:
2315:
2282:
2277:
2276:
2245:
2244:
2219:
2214:
2213:
2194:
2193:
2190:closed manifold
2170:
2169:
2146:
2145:
2126:
2125:
2106:
2105:
2057:
2052:
2051:
2026:
2025:
1997:
1996:
1939:
1934:
1933:
1908:
1907:
1879:
1878:
1827:
1822:
1821:
1796:
1795:
1744:
1739:
1738:
1716:
1715:
1692:
1691:
1639:
1634:
1633:
1626:Hausdorff space
1603:
1602:
1599:
1597:Main properties
1573:
1568:
1567:
1548:computer vision
1541:homotopy groups
1537:homology groups
1517:
1516:
1497:homology theory
1473:
1468:
1467:
1428:
1415:
1371:
1355:
1336:
1317:
1303:
1302:
1283:
1282:
1263:
1262:
1243:
1242:
1217:
1216:
1197:
1196:
1172:
1167:
1166:
1147:
1146:
1143:closed manifold
1122:
1117:
1116:
1095:
1090:
1089:
1070:
1069:
1066:
1040:
1039:
1008:
1007:
974:
969:
968:
949:
948:
929:
928:
873:
872:
820:
819:
767:
766:
747:
746:
727:
726:
671:
670:
637:
636:
617:
616:
571:
570:
535:
530:
529:
506:
489:
488:
469:
468:
449:
448:
425:
420:
419:
400:
399:
340:
339:
291:
286:
285:
264:
241:
240:
195:
194:
145:
111:
86:
81:
80:
70:
24:
23:
17:
12:
11:
5:
4208:
4206:
4198:
4197:
4192:
4182:
4181:
4178:
4177:
4172:
4167:
4162:
4157:
4152:
4147:
4140:
4137:
4134:
4133:
4120:
4103:
4090:
4077:
4060:
4047:
4034:
4021:
4008:
3995:
3982:
3967:
3950:
3937:
3924:
3909:
3888:
3875:
3857:
3837:
3818:
3817:
3815:
3812:
3800:
3799:
3798:
3797:
3783:
3778:
3773:
3768:
3758:
3755:
3749:
3746:
3741:
3738:
3735:
3732:
3729:
3723:
3719:
3714:
3709:
3704:
3699:
3689:
3686:
3680:
3677:
3674:
3668:
3665:
3659:
3656:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3627:
3623:
3620:
3614:
3611:
3605:
3599:
3596:
3590:
3585:
3582:
3579:
3575:
3571:
3567:
3553:
3552:
3537:
3534:
3528:
3522:
3519:
3501:
3500:
3488:
3464:
3461:
3458:
3455:
3444:
3443:
3442:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3407:
3402:
3399:
3396:
3392:
3388:
3384:
3380:
3377:
3374:
3371:
3368:
3365:
3362:
3359:
3354:
3351:
3348:
3344:
3340:
3336:
3330:
3327:
3324:
3321:
3318:
3315:
3312:
3309:
3306:
3302:
3293:
3290:
3287:
3281:
3274:
3271:
3268:
3265:
3252:
3251:
3239:
3236:
3233:
3230:
3227:
3212:
3211:
3199:
3175:
3172:
3169:
3166:
3155:
3154:
3153:
3142:
3139:
3136:
3133:
3130:
3127:
3124:
3121:
3118:
3113:
3110:
3107:
3103:
3099:
3095:
3091:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3067:
3064:
3059:
3056:
3053:
3049:
3045:
3041:
3037:
3034:
3031:
3028:
3025:
3022:
3019:
3016:
3013:
3010:
3005:
3002:
2999:
2995:
2991:
2987:
2983:
2980:
2977:
2974:
2971:
2968:
2965:
2962:
2959:
2956:
2951:
2948:
2945:
2941:
2937:
2933:
2927:
2924:
2921:
2918:
2915:
2912:
2909:
2905:
2896:
2893:
2890:
2884:
2877:
2874:
2871:
2868:
2855:
2854:
2842:
2839:
2836:
2816:
2813:
2810:
2807:
2804:
2801:
2798:
2768:
2765:
2757:
2756:
2744:
2738:
2735:
2729:
2723:
2720:
2714:
2708:
2705:
2699:
2693:
2690:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2648:
2645:
2642:
2639:
2636:
2616:
2610:
2607:
2601:
2595:
2592:
2586:
2581:
2578:
2575:
2572:
2569:
2565:
2561:
2558:
2552:
2549:
2543:
2537:
2534:
2528:
2523:
2520:
2517:
2514:
2511:
2507:
2482:
2479:
2476:
2473:
2470:
2464:
2461:
2458:
2455:
2452:
2449:
2429:
2426:
2423:
2420:
2417:
2414:
2411:
2408:
2405:
2402:
2399:
2396:
2393:
2390:
2376:
2375:
2363:
2343:
2323:
2301:
2298:
2295:
2292:
2289:
2285:
2264:
2261:
2258:
2255:
2252:
2243:in order that
2226:
2222:
2201:
2177:
2166:
2165:
2153:
2133:
2113:
2093:
2090:
2087:
2084:
2081:
2076:
2073:
2070:
2067:
2064:
2060:
2039:
2036:
2033:
2013:
2010:
2007:
2004:
1993:
1981:
1978:
1975:
1972:
1969:
1966:
1963:
1958:
1955:
1952:
1949:
1946:
1942:
1921:
1918:
1915:
1895:
1892:
1889:
1886:
1875:
1863:
1860:
1857:
1854:
1851:
1846:
1843:
1840:
1837:
1834:
1830:
1809:
1806:
1803:
1792:
1780:
1777:
1774:
1771:
1768:
1763:
1760:
1757:
1754:
1751:
1747:
1735:
1723:
1699:
1675:
1672:
1669:
1666:
1663:
1658:
1655:
1652:
1649:
1646:
1642:
1610:
1598:
1595:
1580:
1576:
1524:
1482:
1477:
1440:
1435:
1431:
1427:
1422:
1418:
1414:
1409:
1406:
1403:
1400:
1397:
1393:
1389:
1386:
1383:
1378:
1374:
1370:
1367:
1362:
1358:
1354:
1351:
1348:
1343:
1339:
1335:
1332:
1329:
1324:
1320:
1316:
1313:
1310:
1290:
1270:
1250:
1230:
1227:
1224:
1204:
1179:
1175:
1154:
1129:
1125:
1102:
1098:
1077:
1065:
1062:
1047:
1027:
1024:
1021:
1018:
1015:
993:
990:
987:
984:
981:
977:
956:
936:
913:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
860:
857:
854:
851:
848:
845:
842:
839:
836:
833:
830:
827:
807:
804:
801:
798:
795:
792:
789:
786:
783:
780:
777:
774:
754:
734:
711:
708:
705:
702:
699:
696:
693:
690:
687:
684:
681:
678:
657:
653:
650:
647:
644:
624:
604:
600:
596:
593:
590:
587:
584:
581:
578:
544:
539:
515:
510:
505:
502:
499:
496:
476:
456:
434:
429:
407:
380:
377:
374:
371:
368:
365:
362:
359:
356:
353:
350:
347:
327:
324:
321:
318:
315:
310:
307:
304:
301:
298:
294:
271:
267:
263:
260:
257:
254:
251:
248:
228:
224:
220:
217:
214:
211:
208:
205:
202:
178:
174:
171:
168:
165:
162:
159:
154:
149:
144:
141:
138:
135:
132:
129:
126:
123:
118:
114:
110:
105:
102:
99:
96:
93:
89:
69:
66:
37:
34:
31:
20:Size functions
15:
13:
10:
9:
6:
4:
3:
2:
4207:
4196:
4193:
4191:
4188:
4187:
4185:
4176:
4173:
4171:
4168:
4166:
4163:
4161:
4158:
4156:
4153:
4151:
4148:
4146:
4143:
4142:
4138:
4130:
4124:
4121:
4117:
4113:
4107:
4104:
4100:
4094:
4091:
4087:
4081:
4078:
4074:
4070:
4064:
4061:
4057:
4051:
4048:
4044:
4038:
4035:
4031:
4025:
4022:
4018:
4012:
4009:
4005:
3999:
3996:
3992:
3986:
3983:
3979:
3978:
3971:
3968:
3964:
3960:
3954:
3951:
3947:
3941:
3938:
3934:
3928:
3925:
3921:
3920:
3913:
3910:
3906:
3902:
3898:
3892:
3889:
3885:
3879:
3876:
3872:
3866:
3864:
3862:
3858:
3854:
3850:
3844:
3842:
3838:
3834:
3828:
3826:
3824:
3820:
3813:
3811:
3809:
3803:
3776:
3766:
3753:
3747:
3744:
3739:
3736:
3733:
3730:
3727:
3721:
3717:
3707:
3697:
3684:
3678:
3675:
3672:
3663:
3657:
3654:
3646:
3643:
3640:
3634:
3631:
3625:
3621:
3609:
3603:
3594:
3580:
3577:
3573:
3565:
3557:
3556:
3555:
3554:
3532:
3526:
3517:
3507:: For every
3506:
3503:
3502:
3486:
3478:
3459:
3453:
3445:
3429:
3426:
3420:
3417:
3414:
3411:
3408:
3397:
3394:
3390:
3382:
3378:
3372:
3369:
3366:
3363:
3360:
3349:
3346:
3342:
3334:
3328:
3325:
3322:
3319:
3316:
3313:
3310:
3307:
3304:
3279:
3269:
3263:
3256:
3255:
3254:
3253:
3237:
3234:
3231:
3228:
3225:
3217:
3214:
3213:
3197:
3189:
3170:
3164:
3156:
3137:
3134:
3131:
3128:
3125:
3122:
3119:
3108:
3105:
3101:
3093:
3089:
3083:
3080:
3077:
3074:
3071:
3068:
3065:
3054:
3051:
3047:
3039:
3035:
3029:
3026:
3023:
3020:
3017:
3014:
3011:
3000:
2997:
2993:
2985:
2981:
2975:
2972:
2969:
2966:
2963:
2960:
2957:
2946:
2943:
2939:
2931:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2882:
2872:
2866:
2859:
2858:
2857:
2856:
2840:
2837:
2834:
2811:
2808:
2805:
2799:
2796:
2788:
2785:
2784:
2783:
2780:
2778:
2774:
2766:
2764:
2762:
2733:
2727:
2718:
2712:
2703:
2697:
2688:
2676:
2667:
2664:
2661:
2655:
2649:
2646:
2643:
2634:
2605:
2599:
2590:
2576:
2573:
2570:
2563:
2559:
2547:
2541:
2532:
2518:
2515:
2512:
2505:
2496:
2495:
2494:
2477:
2474:
2471:
2462:
2456:
2453:
2450:
2421:
2418:
2415:
2409:
2403:
2400:
2397:
2388:
2381:
2361:
2341:
2321:
2296:
2293:
2290:
2283:
2259:
2256:
2253:
2242:
2241:
2240:
2224:
2220:
2199:
2191:
2175:
2151:
2131:
2111:
2088:
2085:
2082:
2071:
2068:
2065:
2058:
2037:
2034:
2031:
2011:
2005:
2002:
1994:
1979:
1976:
1970:
1967:
1964:
1953:
1950:
1947:
1940:
1919:
1916:
1913:
1893:
1887:
1884:
1876:
1858:
1855:
1852:
1841:
1838:
1835:
1828:
1807:
1804:
1801:
1793:
1775:
1772:
1769:
1758:
1755:
1752:
1745:
1736:
1721:
1713:
1697:
1689:
1670:
1667:
1664:
1653:
1650:
1647:
1640:
1631:
1630:
1629:
1627:
1624:
1621:is a compact
1608:
1596:
1594:
1578:
1565:
1561:
1557:
1553:
1549:
1544:
1542:
1538:
1522:
1514:
1510:
1506:
1502:
1498:
1480:
1465:
1461:
1457:
1452:
1433:
1429:
1425:
1420:
1416:
1407:
1404:
1401:
1398:
1395:
1387:
1376:
1372:
1368:
1365:
1360:
1356:
1349:
1341:
1337:
1333:
1330:
1327:
1322:
1318:
1308:
1288:
1268:
1248:
1228:
1225:
1222:
1202:
1195:
1177:
1173:
1152:
1144:
1123:
1100:
1096:
1075:
1063:
1059:
1045:
1022:
1019:
1016:
1006:in the point
988:
985:
982:
975:
954:
934:
927:
908:
905:
899:
893:
890:
887:
884:
881:
855:
852:
846:
840:
837:
834:
831:
828:
802:
799:
793:
787:
784:
781:
778:
775:
752:
732:
725:
706:
703:
697:
691:
688:
685:
682:
679:
648:
645:
642:
622:
591:
588:
585:
582:
579:
566:
562:
560:
542:
513:
500:
497:
494:
474:
454:
432:
405:
398:
394:
375:
372:
366:
360:
357:
354:
351:
348:
322:
319:
316:
305:
302:
299:
292:
269:
261:
255:
252:
249:
215:
212:
209:
206:
203:
193:
166:
163:
160:
157:
152:
142:
136:
133:
130:
121:
116:
108:
100:
97:
94:
87:
79:
78:size function
75:
67:
65:
63:
59:
55:
51:
35:
32:
29:
21:
4155:Size functor
4128:
4123:
4116:Angela Y. Wu
4111:
4106:
4098:
4093:
4085:
4080:
4072:
4063:
4055:
4050:
4042:
4037:
4029:
4024:
4016:
4011:
4003:
3998:
3990:
3985:
3975:
3970:
3958:
3953:
3945:
3940:
3932:
3927:
3917:
3912:
3904:
3891:
3883:
3878:
3870:
3848:
3832:
3804:
3801:
3504:
3477:multiplicity
3476:
3446:The number
3215:
3188:multiplicity
3187:
2787:cornerpoints
2786:
2781:
2776:
2773:cornerpoints
2772:
2770:
2758:
2377:
2188:is a smooth
2167:
1601:Assume that
1600:
1545:
1509:size functor
1501:size functor
1453:
1067:
1038:is equal to
568:
77:
71:
19:
18:
4145:Size theory
3216:cornerlines
2777:cornerlines
1556:size theory
765:, that is,
559:size theory
74:size theory
4184:Categories
3814:References
3551:, it holds
2782:Formally:
2024:and every
1995:for every
1906:and every
1877:for every
1874:is finite.
1794:for every
4165:Size pair
3767:μ
3757:¯
3748:≤
3722:∑
3698:μ
3688:¯
3667:¯
3658:≤
3626:∑
3613:¯
3598:¯
3581:φ
3566:ℓ
3536:¯
3521:¯
3454:μ
3415:α
3412:−
3398:φ
3383:ℓ
3379:−
3367:α
3350:φ
3335:ℓ
3323:α
3305:α
3264:μ
3250:such that
3165:μ
3138:β
3126:α
3123:−
3109:φ
3094:ℓ
3084:β
3081:−
3072:α
3069:−
3055:φ
3040:ℓ
3036:−
3030:β
3018:α
3001:φ
2986:ℓ
2982:−
2976:β
2973:−
2964:α
2947:φ
2932:ℓ
2920:β
2908:α
2867:μ
2737:~
2728:−
2722:¯
2707:¯
2698:−
2692:~
2677:≥
2668:ψ
2650:φ
2609:~
2594:~
2577:φ
2564:ℓ
2551:¯
2536:¯
2519:ψ
2506:ℓ
2478:ψ
2457:φ
2422:ψ
2404:φ
2362:φ
2297:φ
2284:ℓ
2200:φ
2132:φ
2072:φ
2059:ℓ
2012:φ
2006:≥
1954:φ
1941:ℓ
1894:φ
1842:φ
1829:ℓ
1759:φ
1746:ℓ
1654:φ
1641:ℓ
1575:Δ
1439:‖
1426:−
1413:‖
1405:≤
1399:≤
1366:…
1331:…
1226:∈
1223:γ
1203:φ
1128:∞
989:φ
976:ℓ
935:φ
906:≤
894:φ
885:∈
853:≤
841:φ
832:∈
800:≤
788:φ
779:∈
733:φ
704:≤
692:φ
683:∈
652:→
643:φ
595:→
586:φ
504:→
495:φ
455:φ
373:≤
361:φ
352:∈
306:φ
293:ℓ
266:Δ
262:∈
219:→
210:φ
192:size pair
173:→
143:∈
113:Δ
101:φ
88:ℓ
4190:Topology
4139:See also
2827:, with
2493:exists.
615:, where
528:, where
62:topology
395:from a
2466:
1710:and a
76:, the
2627:then
2212:is a
1686:is a
1499:(the
52:of a
3679:>
3527:<
3479:of
3427:>
3326:<
3308:>
2923:>
2911:>
2838:<
2560:>
2192:and
2035:<
1917:>
1888:<
1805:<
1550:and
1539:and
1507:and
164:<
60:and
33:<
3301:min
3190:of
2904:min
2680:min
2497:if
2334:or
2009:max
1891:min
1392:max
418:to
72:In
4186::
4071:,
3961:,
3899:,
3860:^
3851:,
3840:^
3822:^
3810:.
3430:0.
2050:,
1932:,
1820:,
1593:.
1543:.
1451:.
447:)
284:,
64:.
3796:.
3782:)
3777:r
3772:(
3754:x
3745:k
3740:k
3737:=
3734:x
3731::
3728:r
3718:+
3713:)
3708:p
3703:(
3685:y
3676:y
3673:,
3664:x
3655:x
3650:)
3647:y
3644:,
3641:x
3638:(
3635:=
3632:p
3622:=
3619:)
3610:y
3604:,
3595:x
3589:(
3584:)
3578:,
3574:M
3570:(
3533:y
3518:x
3499:.
3487:r
3463:)
3460:r
3457:(
3424:)
3421:y
3418:,
3409:k
3406:(
3401:)
3395:,
3391:M
3387:(
3376:)
3373:y
3370:,
3364:+
3361:k
3358:(
3353:)
3347:,
3343:M
3339:(
3329:y
3320:+
3317:k
3314:,
3311:0
3292:f
3289:e
3286:d
3280:=
3273:)
3270:r
3267:(
3238:k
3235:=
3232:x
3229::
3226:r
3210:.
3198:p
3174:)
3171:p
3168:(
3141:)
3135:+
3132:y
3129:,
3120:x
3117:(
3112:)
3106:,
3102:M
3098:(
3090:+
3087:)
3078:y
3075:,
3066:x
3063:(
3058:)
3052:,
3048:M
3044:(
3033:)
3027:+
3024:y
3021:,
3015:+
3012:x
3009:(
3004:)
2998:,
2994:M
2990:(
2979:)
2970:y
2967:,
2961:+
2958:x
2955:(
2950:)
2944:,
2940:M
2936:(
2926:0
2917:,
2914:0
2895:f
2892:e
2889:d
2883:=
2876:)
2873:p
2870:(
2841:y
2835:x
2815:)
2812:y
2809:,
2806:x
2803:(
2800:=
2797:p
2755:.
2743:}
2734:y
2719:y
2713:,
2704:x
2689:x
2683:{
2674:)
2671:)
2665:,
2662:N
2659:(
2656:,
2653:)
2647:,
2644:M
2641:(
2638:(
2635:d
2615:)
2606:y
2600:,
2591:x
2585:(
2580:)
2574:,
2571:M
2568:(
2557:)
2548:y
2542:,
2533:x
2527:(
2522:)
2516:,
2513:N
2510:(
2481:)
2475:,
2472:N
2469:(
2463:,
2460:)
2454:,
2451:M
2448:(
2428:)
2425:)
2419:,
2416:N
2413:(
2410:,
2407:)
2401:,
2398:M
2395:(
2392:(
2389:d
2374:.
2342:y
2322:x
2300:)
2294:,
2291:M
2288:(
2263:)
2260:y
2257:,
2254:x
2251:(
2225:1
2221:C
2176:M
2164:.
2152:x
2112:M
2092:)
2089:y
2086:,
2083:x
2080:(
2075:)
2069:,
2066:M
2063:(
2038:y
2032:x
2003:y
1992:.
1980:0
1977:=
1974:)
1971:y
1968:,
1965:x
1962:(
1957:)
1951:,
1948:M
1945:(
1920:x
1914:y
1885:x
1862:)
1859:y
1856:,
1853:x
1850:(
1845:)
1839:,
1836:M
1833:(
1808:y
1802:x
1779:)
1776:y
1773:,
1770:x
1767:(
1762:)
1756:,
1753:M
1750:(
1734:.
1722:y
1698:x
1674:)
1671:y
1668:,
1665:x
1662:(
1657:)
1651:,
1648:M
1645:(
1609:M
1579:+
1523:0
1481:k
1476:R
1434:i
1430:Q
1421:i
1417:P
1408:k
1402:i
1396:1
1388:=
1385:)
1382:)
1377:k
1373:Q
1369:,
1361:1
1357:Q
1353:(
1350:,
1347:)
1342:k
1338:P
1334:,
1328:,
1323:1
1319:P
1315:(
1312:(
1309:d
1289:M
1269:k
1249:M
1229:M
1178:0
1174:C
1153:M
1124:C
1101:1
1097:C
1076:M
1058:.
1046:2
1026:)
1023:b
1020:,
1017:a
1014:(
992:)
986:,
983:M
980:(
955:a
912:}
909:a
903:)
900:p
897:(
891::
888:M
882:p
879:{
859:}
856:b
850:)
847:p
844:(
838::
835:M
829:p
826:{
806:}
803:a
797:)
794:p
791:(
785::
782:M
776:p
773:{
753:a
710:}
707:b
701:)
698:p
695:(
689::
686:M
680:p
677:{
656:R
649:M
646::
623:M
603:)
599:R
592:M
589::
583:,
580:M
577:(
543:k
538:R
514:k
509:R
501:M
498::
475:x
433:k
428:R
406:M
379:}
376:y
370:)
367:p
364:(
358::
355:M
349:p
346:{
326:)
323:y
320:,
317:x
314:(
309:)
303:,
300:M
297:(
270:+
259:)
256:y
253:,
250:x
247:(
227:)
223:R
216:M
213::
207:,
204:M
201:(
177:N
170:}
167:y
161:x
158::
153:2
148:R
140:)
137:y
134:,
131:x
128:(
125:{
122:=
117:+
109::
104:)
98:,
95:M
92:(
36:y
30:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
