1382:
2878:
2194:
Proving these statements is an instructive exercise. All directly derive from the ultrametric triangle inequality. Note that, by the second statement, a ball may have several center points that have non-zero distance. The intuition behind such seemingly strange effects is that, due to the strong
1031:
134:
805:
694:
1227:
268:
1784:
1088:
1327:
2060:
2004:
1944:
1271:
932:
1152:
888:
843:
732:
618:
1885:
1637:
1581:
1525:
1371:
1829:
1469:
565:
2164:
1704:
1678:
2115:
1114:
2095:
2495:, ultrametric distances have been applied to measure landscape complexity and to assess the extent to which one landscape function is more important than another.
2188:
2138:
194:
174:
154:
2457:
methods. These algorithms require a constant-rate assumption and produce trees in which the distances from the root to every branch tip are equal. When
2375:(that is, the largest weight of an edge, on a path chosen to minimize this largest weight), then the vertices of the graph, with distance measured by
2665:
Legendre, P. and
Legendre, L. 1998. Numerical Ecology. Second English Edition. Developments in Environmental Modelling 20. Elsevier, Amsterdam.
937:
2882:
2225:
of arbitrary length (finite or infinite), Σ, over some alphabet Σ. Define the distance between two different words to be 2, where
49:
741:
630:
1157:
233:
2675:
Benzi, R.; Biferale, L.; Trovatore, E. (1997). "Ultrametric
Structure of Multiscale Energy Correlations in Turbulent Models".
2860:
2823:
2608:
2434:
exhibits an ultrametric structure, with the solution given by the full replica symmetry breaking procedure first outlined by
2393:
may then be thought of as a way of approximating the final result of a computation (which can be guaranteed to exist by the
2484:
of fluids make use of so-called cascades, and in discrete models of dyadic cascades, which have an ultrametric structure.
1709:
2789:
1036:
1276:
2321:| induces an ultrametric on the space of all complex sequences for which it is finite. (Note that this is not a
17:
2815:
1433:
From the above definition, one can conclude several typical properties of ultrametrics. For example, for all
2009:
1953:
2394:
1893:
1232:
893:
846:
2427:
2337:
1119:
852:
810:
699:
2579:
Centre de Mathématique
Sociale. École Pratique des Hautes Études. Mathématiques et Sciences Humaines
570:
2898:
2419:
1834:
1586:
1530:
1474:
1332:
1793:
1436:
544:
2903:
2379:, form an ultrametric space, and all finite ultrametric spaces may be represented in this way.
2143:
1683:
1657:
2694:
2633:
2586:
2551:
2326:
2170:
of the latter, and the mutual distance of two distinct open balls is (greater or) equal to
2100:
1093:
477:
2080:
8:
2390:
2364:
43:
2698:
2637:
2555:
2807:
2759:
2730:"Mathematical modelling of land use and landscape complexity with ultrametric topology"
2710:
2684:
2442:
2173:
2167:
2123:
1640:
179:
159:
139:
2856:
2829:
2819:
2795:
2785:
2763:
2751:
2604:
2563:
2492:
2446:
2229:
is the first place at which the words differ. The resulting metric is an ultrametric.
217:
2714:
2848:
2818:. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer.
2741:
2702:
2641:
2559:
2348:
2746:
2729:
2582:
2470:
2402:
2233:
2222:
2205:
539:
2706:
2621:
2435:
2423:
2336:
are allowed to be zero, one should use here the rather unusual convention that
2212:
621:
2645:
2892:
2833:
2799:
2784:. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press.
2755:
2477:
2438:
and coworkers. Ultrametricity also appears in the theory of aperiodic solids.
2409:
2398:
1644:
2542:
Osipov, Gutkin (2013), "Clustering of periodic orbits in chaotic systems",
2524:
2431:
39:
476:, which is called the space's associated distance function (also called a
2689:
2291:
278:
224:
31:
2481:
2070:
1381:
1026:{\displaystyle \|x\|=\|(x+y)-y\|\leq \max \left\{\|x+y\|,\|y\|\right\}}
2577:
Leclerc, Bruno (1981), "Description combinatoire des ultramétriques",
2268:) appears the same number of times (which could also be zero) both in
2488:
1947:
1651:
2322:
2074:
2066:
2466:
2305:
2240:
over some alphabet Σ is an ultrametric space with respect to the
2077:. That is, open balls are also closed, and closed balls (replace
487:
satisfies all of the conditions except possibly condition 4 then
2505:
2877:
2469:
data are analyzed, the ultrametricity assumption is called the
2195:
triangle inequality, distances in ultrametrics do not add up.
2454:
2450:
2449:
construction, ultrametric distances are also utilized by the
1639:
holds. That is, every triple of points in the space forms an
129:{\displaystyle d(x,z)\leq \max \left\{d(x,y),d(y,z)\right\}}
800:{\displaystyle \|x+y\|\leq \max \left\{\|x\|,\|y\|\right\}}
689:{\displaystyle \|x+y\|\leq \max \left\{\|x\|,\|y\|\right\}}
2462:
2458:
1222:{\displaystyle \max \left\{\|x+y\|,\|y\|\right\}=\|x+y\|}
263:{\displaystyle d\colon M\times M\rightarrow \mathbb {R} }
1890:
Intersecting balls are contained in each other, i.e. if
2674:
2619:
2176:
2146:
2126:
2103:
2083:
2012:
1956:
1896:
1837:
1796:
1712:
1686:
1660:
1589:
1533:
1477:
1439:
1335:
1279:
1235:
1160:
1122:
1096:
1039:
940:
896:
855:
813:
744:
702:
633:
573:
547:
236:
182:
162:
142:
52:
1385:
In the triangle on the right, the two bottom points
620:), the last property can be made stronger using the
196:. Sometimes the associated metric is also called a
2182:
2158:
2132:
2109:
2089:
2054:
1998:
1938:
1879:
1823:
1779:{\displaystyle B(x;r):=\{y\in M\mid d(x,y)<r\}}
1778:
1698:
1672:
1631:
1575:
1519:
1463:
1365:
1321:
1265:
1221:
1146:
1108:
1082:
1025:
926:
882:
837:
799:
726:
688:
612:
559:
262:
188:
168:
148:
128:
2779:
2511:
2408:makes heavy use of the ultrametric nature of the
1790:Every point inside a ball is its center, i.e. if
1083:{\displaystyle \max \left\{\|x+y\|,\|y\|\right\}}
2890:
1161:
1040:
983:
763:
652:
74:
2620:Rammal, R.; Toulouse, G.; Virasoro, M. (1986).
2065:All balls of strictly positive radius are both
2780:Narici, Lawrence; Beckenstein, Edward (2011).
2806:
2727:
534:is an Abelian group (written additively) and
1773:
1734:
1360:
1354:
1348:
1336:
1316:
1310:
1304:
1292:
1286:
1280:
1260:
1248:
1242:
1236:
1216:
1204:
1193:
1187:
1181:
1169:
1141:
1135:
1129:
1123:
1103:
1097:
1072:
1066:
1060:
1048:
1015:
1009:
1003:
991:
977:
953:
947:
941:
921:
915:
909:
897:
874:
868:
862:
856:
832:
826:
820:
814:
789:
783:
777:
771:
757:
745:
721:
715:
709:
703:
678:
672:
666:
660:
646:
634:
607:
595:
554:
548:
2613:
1322:{\displaystyle \|x\|\leq \|x+y\|\leq \|x\|}
1154:contrary to the initial assumption. Thus,
2847:
2745:
2688:
256:
1380:
1273:. Using the initial inequality, we have
2599:Mezard, M; Parisi, G; and Virasoro, M:
2576:
1471:, at least one of the three equalities
14:
2891:
2541:
2140:and center in a closed ball of radius
2120:The set of all open balls with radius
2055:{\displaystyle B(y;s)\subseteq B(x;r)}
1999:{\displaystyle B(x;r)\subseteq B(y;s)}
2351:, all edge weights are positive, and
1786:, we have the following properties:
207:
1116:, for if that is the case, we have
24:
2841:
2218:form a complete ultrametric space.
25:
2915:
2870:
2525:"Ultrametric Triangle Inequality"
2397:). Similar ideas can be found in
1939:{\displaystyle B(x;r)\cap B(y;s)}
1266:{\displaystyle \|x\|\leq \|x+y\|}
927:{\displaystyle \|x+y\|\leq \|x\|}
2876:
2773:
2622:"Ultrametricity for physicists"
2383:
1147:{\displaystyle \|x\|\leq \|y\|}
883:{\displaystyle \|x\|>\|y\|.}
838:{\displaystyle \|x\|\neq \|y\|}
727:{\displaystyle \|x\|\neq \|y\|}
2721:
2668:
2659:
2593:
2570:
2535:
2517:
2236:with glued ends of the length
2049:
2037:
2028:
2016:
1993:
1981:
1972:
1960:
1933:
1921:
1912:
1900:
1874:
1862:
1853:
1841:
1812:
1800:
1764:
1752:
1728:
1716:
1626:
1614:
1605:
1593:
1570:
1558:
1549:
1537:
1514:
1502:
1493:
1481:
968:
956:
807:, then the equality occurs if
613:{\displaystyle d(x,y)=\|x-y\|}
589:
577:
252:
118:
106:
97:
85:
68:
56:
13:
1:
2728:Papadimitriou, Fivos (2013).
2581:(in French) (73): 5–37, 127,
2512:Narici & Beckenstein 2011
2499:
2426:overlap between spins in the
1880:{\displaystyle B(x;r)=B(y;r)}
1632:{\displaystyle d(x,y)=d(z,x)}
1576:{\displaystyle d(x,z)=d(y,z)}
1520:{\displaystyle d(x,y)=d(y,z)}
1376:
1366:{\displaystyle \|x+y\|=\|x\|}
468:together with an ultrametric
2853:Set Theory and Metric Spaces
2810:; Wolff, Manfred P. (1999).
2747:10.1080/1747423x.2011.637136
2601:SPIN GLASS THEORY AND BEYOND
7:
2734:Journal of Land Use Science
2707:10.1103/PhysRevLett.79.1670
2256:-close if any substring of
2244:-close distance. Two words
2198:
1824:{\displaystyle d(x,y)<r}
1643:, so the whole space is an
10:
2920:
2855:, AMS Chelsea Publishing,
2603:, World Scientific, 1986.
2564:10.1088/0951-7715/26/1/177
2395:Banach fixed-point theorem
2294:decreasing to zero, then |
1464:{\displaystyle x,y,z\in M}
934:. But we can also compute
847:Without loss of generality
560:{\displaystyle \|\cdot \|}
438:strong triangle inequality
2812:Topological Vector Spaces
2782:Topological Vector Spaces
2646:10.1103/RevModPhys.58.765
2626:Reviews of Modern Physics
738:We want to prove that if
519:and an ultrapseudometric
2883:Non-Archimedean geometry
2420:condensed matter physics
2677:Physical Review Letters
2363:) is the weight of the
501:ultrapseudometric space
2184:
2160:
2159:{\displaystyle r>0}
2134:
2111:
2091:
2056:
2000:
1940:
1881:
1825:
1780:
1700:
1699:{\displaystyle x\in M}
1674:
1673:{\displaystyle r>0}
1633:
1577:
1521:
1465:
1430:
1393:violate the condition
1367:
1323:
1267:
1223:
1148:
1110:
1084:
1027:
928:
884:
839:
801:
728:
690:
614:
561:
442:ultrametric inequality
264:
198:non-Archimedean metric
190:
170:
150:
130:
2480:in three dimensional
2260:consecutive letters (
2185:
2161:
2135:
2112:
2110:{\displaystyle \leq }
2092:
2057:
2001:
1941:
1882:
1826:
1781:
1701:
1675:
1634:
1578:
1522:
1466:
1384:
1368:
1324:
1268:
1224:
1149:
1111:
1109:{\displaystyle \|y\|}
1085:
1028:
929:
885:
849:, let us assume that
840:
802:
729:
691:
615:
562:
281:), such that for all
265:
191:
171:
151:
131:
2885:at Wikimedia Commons
2347:is an edge-weighted
2174:
2144:
2124:
2101:
2090:{\displaystyle <}
2081:
2010:
1954:
1894:
1835:
1794:
1710:
1684:
1658:
1587:
1531:
1475:
1437:
1333:
1277:
1233:
1158:
1120:
1094:
1037:
1033:. Now, the value of
938:
894:
853:
811:
742:
700:
631:
571:
545:
515:consisting of a set
464:consisting of a set
234:
180:
160:
140:
50:
27:Type of metric space
2808:Schaefer, Helmut H.
2699:1997PhRvL..79.1670B
2638:1986RvMP...58..765R
2556:2013Nonli..26..177G
2391:contraction mapping
2290:) is a sequence of
46:is strengthened to
44:triangle inequality
2208:is an ultrametric.
2180:
2156:
2130:
2107:
2087:
2052:
1996:
1936:
1877:
1821:
1776:
1696:
1670:
1641:isosceles triangle
1629:
1573:
1517:
1461:
1431:
1363:
1319:
1263:
1219:
1144:
1106:
1080:
1023:
924:
890:This implies that
880:
835:
797:
724:
686:
610:
557:
538:is generated by a
260:
186:
166:
146:
126:
2881:Media related to
2862:978-0-8218-2694-2
2825:978-1-4612-7155-0
2609:978-9971-5-0116-7
2493:landscape ecology
2447:phylogenetic tree
2183:{\displaystyle r}
2133:{\displaystyle r}
696:with equality if
530:In the case when
493:ultrapseudometric
450:ultrametric space
227:-valued function
208:Formal definition
189:{\displaystyle z}
169:{\displaystyle y}
149:{\displaystyle x}
36:ultrametric space
16:(Redirected from
2911:
2880:
2865:
2837:
2803:
2768:
2767:
2749:
2725:
2719:
2718:
2692:
2690:chao-dyn/9705018
2683:(9): 1670–1674.
2672:
2666:
2663:
2657:
2656:
2654:
2652:
2617:
2611:
2597:
2591:
2589:
2574:
2568:
2566:
2539:
2533:
2532:
2521:
2515:
2514:, pp. 1–18.
2509:
2349:undirected graph
2340: = 0.)
2189:
2187:
2186:
2181:
2165:
2163:
2162:
2157:
2139:
2137:
2136:
2131:
2117:) are also open.
2116:
2114:
2113:
2108:
2096:
2094:
2093:
2088:
2061:
2059:
2058:
2053:
2005:
2003:
2002:
1997:
1945:
1943:
1942:
1937:
1886:
1884:
1883:
1878:
1830:
1828:
1827:
1822:
1785:
1783:
1782:
1777:
1705:
1703:
1702:
1697:
1679:
1677:
1676:
1671:
1638:
1636:
1635:
1630:
1582:
1580:
1579:
1574:
1526:
1524:
1523:
1518:
1470:
1468:
1467:
1462:
1372:
1370:
1369:
1364:
1328:
1326:
1325:
1320:
1272:
1270:
1269:
1264:
1228:
1226:
1225:
1220:
1200:
1196:
1153:
1151:
1150:
1145:
1115:
1113:
1112:
1107:
1089:
1087:
1086:
1081:
1079:
1075:
1032:
1030:
1029:
1024:
1022:
1018:
933:
931:
930:
925:
889:
887:
886:
881:
844:
842:
841:
836:
806:
804:
803:
798:
796:
792:
733:
731:
730:
725:
695:
693:
692:
687:
685:
681:
619:
617:
616:
611:
566:
564:
563:
558:
537:
533:
526:
522:
518:
514:
498:
490:
486:
475:
471:
467:
463:
435:
394:
384:
366:
345:
316:
298:
276:
269:
267:
266:
261:
259:
222:
195:
193:
192:
187:
175:
173:
172:
167:
155:
153:
152:
147:
135:
133:
132:
127:
125:
121:
21:
2919:
2918:
2914:
2913:
2912:
2910:
2909:
2908:
2899:Metric geometry
2889:
2888:
2873:
2863:
2844:
2842:Further reading
2826:
2792:
2776:
2771:
2726:
2722:
2673:
2669:
2664:
2660:
2650:
2648:
2618:
2614:
2598:
2594:
2575:
2571:
2550:(26): 177–200,
2540:
2536:
2523:
2522:
2518:
2510:
2506:
2502:
2471:molecular clock
2386:
2334:
2329:— If the
2325:since it lacks
2319:
2313:
2303:
2288:
2206:discrete metric
2201:
2175:
2172:
2171:
2145:
2142:
2141:
2125:
2122:
2121:
2102:
2099:
2098:
2082:
2079:
2078:
2073:in the induced
2011:
2008:
2007:
1955:
1952:
1951:
1895:
1892:
1891:
1836:
1833:
1832:
1795:
1792:
1791:
1711:
1708:
1707:
1685:
1682:
1681:
1659:
1656:
1655:
1588:
1585:
1584:
1532:
1529:
1528:
1476:
1473:
1472:
1438:
1435:
1434:
1379:
1334:
1331:
1330:
1278:
1275:
1274:
1234:
1231:
1230:
1168:
1164:
1159:
1156:
1155:
1121:
1118:
1117:
1095:
1092:
1091:
1047:
1043:
1038:
1035:
1034:
990:
986:
939:
936:
935:
895:
892:
891:
854:
851:
850:
812:
809:
808:
770:
766:
743:
740:
739:
701:
698:
697:
659:
655:
632:
629:
628:
624:sharpening to:
572:
569:
568:
546:
543:
542:
540:length function
535:
531:
524:
520:
516:
504:
496:
488:
484:
473:
469:
465:
453:
410:) ≤ max {
398:
386:
371:
353:
320:
303:
282:
274:
255:
235:
232:
231:
220:
210:
181:
178:
177:
161:
158:
157:
141:
138:
137:
81:
77:
51:
48:
47:
28:
23:
22:
15:
12:
11:
5:
2917:
2907:
2906:
2901:
2887:
2886:
2872:
2871:External links
2869:
2868:
2867:
2861:
2843:
2840:
2839:
2838:
2824:
2804:
2791:978-1584888666
2790:
2775:
2772:
2770:
2769:
2740:(2): 234–254.
2720:
2667:
2658:
2632:(3): 765–788.
2612:
2592:
2569:
2534:
2529:Stack Exchange
2516:
2503:
2501:
2498:
2497:
2496:
2485:
2474:
2439:
2436:Giorgio Parisi
2424:self-averaging
2416:
2406:-adic analysis
2385:
2382:
2381:
2380:
2341:
2332:
2317:
2308:
2299:
2286:
2277:
2230:
2219:
2209:
2200:
2197:
2192:
2191:
2179:
2155:
2152:
2149:
2129:
2118:
2106:
2086:
2063:
2051:
2048:
2045:
2042:
2039:
2036:
2033:
2030:
2027:
2024:
2021:
2018:
2015:
1995:
1992:
1989:
1986:
1983:
1980:
1977:
1974:
1971:
1968:
1965:
1962:
1959:
1935:
1932:
1929:
1926:
1923:
1920:
1917:
1914:
1911:
1908:
1905:
1902:
1899:
1888:
1876:
1873:
1870:
1867:
1864:
1861:
1858:
1855:
1852:
1849:
1846:
1843:
1840:
1820:
1817:
1814:
1811:
1808:
1805:
1802:
1799:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1754:
1751:
1748:
1745:
1742:
1739:
1736:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1695:
1692:
1689:
1669:
1666:
1663:
1628:
1625:
1622:
1619:
1616:
1613:
1610:
1607:
1604:
1601:
1598:
1595:
1592:
1572:
1569:
1566:
1563:
1560:
1557:
1554:
1551:
1548:
1545:
1542:
1539:
1536:
1516:
1513:
1510:
1507:
1504:
1501:
1498:
1495:
1492:
1489:
1486:
1483:
1480:
1460:
1457:
1454:
1451:
1448:
1445:
1442:
1378:
1375:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1341:
1338:
1329:and therefore
1318:
1315:
1312:
1309:
1306:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1282:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1218:
1215:
1212:
1209:
1206:
1203:
1199:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1167:
1163:
1143:
1140:
1137:
1134:
1131:
1128:
1125:
1105:
1102:
1099:
1078:
1074:
1071:
1068:
1065:
1062:
1059:
1056:
1053:
1050:
1046:
1042:
1021:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
989:
985:
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
923:
920:
917:
914:
911:
908:
905:
902:
899:
879:
876:
873:
870:
867:
864:
861:
858:
834:
831:
828:
825:
822:
819:
816:
795:
791:
788:
785:
782:
779:
776:
773:
769:
765:
762:
759:
756:
753:
750:
747:
736:
735:
723:
720:
717:
714:
711:
708:
705:
684:
680:
677:
674:
671:
668:
665:
662:
658:
654:
651:
648:
645:
642:
639:
636:
609:
606:
603:
600:
597:
594:
591:
588:
585:
582:
579:
576:
556:
553:
550:
446:
445:
396:
368:
351:
318:
271:
270:
258:
254:
251:
248:
245:
242:
239:
209:
206:
185:
165:
145:
124:
120:
117:
114:
111:
108:
105:
102:
99:
96:
93:
90:
87:
84:
80:
76:
73:
70:
67:
64:
61:
58:
55:
26:
18:Ultrametricity
9:
6:
4:
3:
2:
2916:
2905:
2904:Metric spaces
2902:
2900:
2897:
2896:
2894:
2884:
2879:
2875:
2874:
2864:
2858:
2854:
2850:
2849:Kaplansky, I.
2846:
2845:
2835:
2831:
2827:
2821:
2817:
2813:
2809:
2805:
2801:
2797:
2793:
2787:
2783:
2778:
2777:
2765:
2761:
2757:
2753:
2748:
2743:
2739:
2735:
2731:
2724:
2716:
2712:
2708:
2704:
2700:
2696:
2691:
2686:
2682:
2678:
2671:
2662:
2647:
2643:
2639:
2635:
2631:
2627:
2623:
2616:
2610:
2606:
2602:
2596:
2588:
2584:
2580:
2573:
2565:
2561:
2557:
2553:
2549:
2545:
2538:
2530:
2526:
2520:
2513:
2508:
2504:
2494:
2490:
2486:
2483:
2479:
2478:intermittency
2475:
2472:
2468:
2464:
2460:
2456:
2452:
2448:
2444:
2440:
2437:
2433:
2429:
2425:
2421:
2417:
2414:
2412:
2407:
2405:
2400:
2399:domain theory
2396:
2392:
2388:
2387:
2378:
2374:
2370:
2366:
2362:
2358:
2354:
2350:
2346:
2342:
2339:
2335:
2328:
2324:
2320:
2311:
2307:
2302:
2297:
2293:
2289:
2282:
2278:
2275:
2271:
2267:
2263:
2259:
2255:
2251:
2247:
2243:
2239:
2235:
2231:
2228:
2224:
2221:Consider the
2220:
2217:
2216:-adic numbers
2215:
2210:
2207:
2203:
2202:
2196:
2177:
2169:
2153:
2150:
2147:
2127:
2119:
2104:
2084:
2076:
2072:
2068:
2064:
2046:
2043:
2040:
2034:
2031:
2025:
2022:
2019:
2013:
1990:
1987:
1984:
1978:
1975:
1969:
1966:
1963:
1957:
1949:
1930:
1927:
1924:
1918:
1915:
1909:
1906:
1903:
1897:
1889:
1871:
1868:
1865:
1859:
1856:
1850:
1847:
1844:
1838:
1818:
1815:
1809:
1806:
1803:
1797:
1789:
1788:
1787:
1770:
1767:
1761:
1758:
1755:
1749:
1746:
1743:
1740:
1737:
1731:
1725:
1722:
1719:
1713:
1693:
1690:
1687:
1667:
1664:
1661:
1653:
1650:Defining the
1648:
1646:
1645:isosceles set
1642:
1623:
1620:
1617:
1611:
1608:
1602:
1599:
1596:
1590:
1567:
1564:
1561:
1555:
1552:
1546:
1543:
1540:
1534:
1511:
1508:
1505:
1499:
1496:
1490:
1487:
1484:
1478:
1458:
1455:
1452:
1449:
1446:
1443:
1440:
1428:
1424:
1420:
1416:
1412:
1408:
1404:
1400:
1396:
1392:
1388:
1383:
1374:
1357:
1351:
1345:
1342:
1339:
1313:
1307:
1301:
1298:
1295:
1289:
1283:
1257:
1254:
1251:
1245:
1239:
1213:
1210:
1207:
1201:
1197:
1190:
1184:
1178:
1175:
1172:
1165:
1138:
1132:
1126:
1100:
1076:
1069:
1063:
1057:
1054:
1051:
1044:
1019:
1012:
1006:
1000:
997:
994:
987:
980:
974:
971:
965:
962:
959:
950:
944:
918:
912:
906:
903:
900:
877:
871:
865:
859:
848:
829:
823:
817:
793:
786:
780:
774:
767:
760:
754:
751:
748:
718:
712:
706:
682:
675:
669:
663:
656:
649:
643:
640:
637:
627:
626:
625:
623:
604:
601:
598:
592:
586:
583:
580:
574:
551:
541:
528:
512:
508:
502:
494:
491:is called an
481:
479:
461:
457:
451:
443:
439:
433:
429:
425:
421:
417:
413:
409:
405:
401:
397:
393:
389:
382:
378:
374:
369:
364:
360:
356:
352:
349:
343:
339:
335:
331:
327:
323:
319:
314:
310:
306:
302:
301:
300:
297:
293:
289:
285:
280:
249:
246:
243:
240:
237:
230:
229:
228:
226:
219:
215:
205:
203:
199:
183:
163:
143:
122:
115:
112:
109:
103:
100:
94:
91:
88:
82:
78:
71:
65:
62:
59:
53:
45:
42:in which the
41:
37:
33:
19:
2852:
2811:
2781:
2774:Bibliography
2737:
2733:
2723:
2680:
2676:
2670:
2661:
2649:. Retrieved
2629:
2625:
2615:
2600:
2595:
2578:
2572:
2547:
2544:Nonlinearity
2543:
2537:
2528:
2519:
2507:
2432:spin glasses
2413:-adic metric
2410:
2403:
2384:Applications
2376:
2372:
2368:
2365:minimax path
2360:
2356:
2352:
2344:
2330:
2315:
2309:
2300:
2295:
2292:real numbers
2284:
2280:
2273:
2269:
2265:
2261:
2257:
2253:
2249:
2245:
2241:
2237:
2234:set of words
2226:
2223:set of words
2213:
2193:
1950:then either
1649:
1432:
1426:
1422:
1418:
1414:
1410:
1406:
1402:
1398:
1394:
1390:
1386:
737:
529:
510:
506:
500:
492:
482:
459:
455:
449:
447:
441:
437:
431:
427:
423:
419:
415:
411:
407:
403:
399:
391:
387:
380:
376:
372:
362:
358:
354:
347:
341:
337:
333:
329:
325:
321:
312:
308:
304:
295:
291:
287:
283:
279:real numbers
272:
213:
211:
202:super-metric
201:
197:
40:metric space
35:
29:
2327:homogeneity
2071:closed sets
1680:centred at
1652:(open) ball
315:) ≥ 0
277:denote the
214:ultrametric
32:mathematics
2893:Categories
2500:References
2482:turbulence
2476:Models of
1654:of radius
1377:Properties
1090:cannot be
503:is a pair
452:is a pair
2834:840278135
2800:144216834
2764:121927387
2756:1747-423X
2489:geography
2304: :=
2168:partition
2105:≤
2032:⊆
1976:⊆
1948:non-empty
1916:∩
1747:∣
1741:∈
1691:∈
1456:∈
1361:‖
1355:‖
1349:‖
1337:‖
1317:‖
1311:‖
1308:≤
1305:‖
1293:‖
1290:≤
1287:‖
1281:‖
1261:‖
1249:‖
1246:≤
1243:‖
1237:‖
1217:‖
1205:‖
1194:‖
1188:‖
1182:‖
1170:‖
1142:‖
1136:‖
1133:≤
1130:‖
1124:‖
1104:‖
1098:‖
1073:‖
1067:‖
1061:‖
1049:‖
1016:‖
1010:‖
1004:‖
992:‖
981:≤
978:‖
972:−
954:‖
948:‖
942:‖
922:‖
916:‖
913:≤
910:‖
898:‖
875:‖
869:‖
863:‖
857:‖
833:‖
827:‖
824:≠
821:‖
815:‖
790:‖
784:‖
778:‖
772:‖
761:≤
758:‖
746:‖
722:‖
716:‖
713:≠
710:‖
704:‖
679:‖
673:‖
667:‖
661:‖
650:≤
647:‖
635:‖
608:‖
602:−
596:‖
567:(so that
555:‖
552:⋅
549:‖
253:→
247:×
241::
72:≤
2851:(1977),
2715:53120932
2443:taxonomy
2428:SK Model
2367:between
2323:seminorm
2199:Examples
2166:forms a
2075:topology
1405:) ≤ max{
348:symmetry
294:∈
136:for all
2695:Bibcode
2651:20 June
2634:Bibcode
2587:0623034
2552:Bibcode
2467:protein
2306:lim sup
273:(where
2859:
2832:
2822:
2798:
2788:
2762:
2754:
2713:
2607:
2585:
2422:, the
1229:, and
478:metric
176:, and
2760:S2CID
2711:S2CID
2685:arXiv
2455:WPGMA
2451:UPGMA
2264:<
2097:with
1831:then
622:Krull
499:. An
385:then
383:) = 0
365:) = 0
223:is a
216:on a
38:is a
34:, an
2857:ISBN
2830:OCLC
2820:ISBN
2796:OCLC
2786:ISBN
2752:ISSN
2653:2011
2605:ISBN
2491:and
2465:and
2453:and
2445:and
2371:and
2272:and
2252:are
2248:and
2232:The
2211:The
2204:The
2151:>
2085:<
2069:and
2067:open
1816:<
1768:<
1665:>
1389:and
866:>
480:).
332:) =
225:real
2816:GTM
2742:doi
2703:doi
2642:doi
2560:doi
2487:In
2463:RNA
2459:DNA
2441:In
2430:of
2418:In
2343:If
2283:= (
2279:If
2006:or
1946:is
1706:as
1583:or
1527:or
1429:)}.
1417:),
1162:max
1041:max
984:max
764:max
653:max
527:.
523:on
495:on
483:If
472:on
448:An
440:or
436:} (
422:),
370:if
218:set
212:An
200:or
75:max
30:In
2895::
2828:.
2814:.
2794:.
2758:.
2750:.
2736:.
2732:.
2709:.
2701:.
2693:.
2681:79
2679:.
2640:.
2630:58
2628:.
2624:.
2583:MR
2558:,
2548:26
2546:,
2527:.
2461:,
2401:.
2389:A
2312:→∞
1732::=
1647:.
1425:,
1413:,
1401:,
1373:.
845:.
509:,
458:,
444:).
430:,
418:,
406:,
390:=
379:,
361:,
350:);
340:,
328:,
311:,
299::
290:,
286:,
204:.
156:,
2866:.
2836:.
2802:.
2766:.
2744::
2738:8
2717:.
2705::
2697::
2687::
2655:.
2644::
2636::
2590:.
2567:.
2562::
2554::
2531:.
2473:.
2415:.
2411:p
2404:p
2377:d
2373:v
2369:u
2361:v
2359:,
2357:u
2355:(
2353:d
2345:G
2338:0
2333:n
2331:r
2318:n
2316:x
2314:|
2310:n
2301:r
2298:|
2296:x
2287:n
2285:r
2281:r
2276:.
2274:y
2270:x
2266:n
2262:p
2258:p
2254:p
2250:y
2246:x
2242:p
2238:n
2227:n
2214:p
2190:.
2178:r
2154:0
2148:r
2128:r
2062:.
2050:)
2047:r
2044:;
2041:x
2038:(
2035:B
2029:)
2026:s
2023:;
2020:y
2017:(
2014:B
1994:)
1991:s
1988:;
1985:y
1982:(
1979:B
1973:)
1970:r
1967:;
1964:x
1961:(
1958:B
1934:)
1931:s
1928:;
1925:y
1922:(
1919:B
1913:)
1910:r
1907:;
1904:x
1901:(
1898:B
1887:.
1875:)
1872:r
1869:;
1866:y
1863:(
1860:B
1857:=
1854:)
1851:r
1848:;
1845:x
1842:(
1839:B
1819:r
1813:)
1810:y
1807:,
1804:x
1801:(
1798:d
1774:}
1771:r
1765:)
1762:y
1759:,
1756:x
1753:(
1750:d
1744:M
1738:y
1735:{
1729:)
1726:r
1723:;
1720:x
1717:(
1714:B
1694:M
1688:x
1668:0
1662:r
1627:)
1624:x
1621:,
1618:z
1615:(
1612:d
1609:=
1606:)
1603:y
1600:,
1597:x
1594:(
1591:d
1571:)
1568:z
1565:,
1562:y
1559:(
1556:d
1553:=
1550:)
1547:z
1544:,
1541:x
1538:(
1535:d
1515:)
1512:z
1509:,
1506:y
1503:(
1500:d
1497:=
1494:)
1491:y
1488:,
1485:x
1482:(
1479:d
1459:M
1453:z
1450:,
1447:y
1444:,
1441:x
1427:z
1423:y
1421:(
1419:d
1415:z
1411:x
1409:(
1407:d
1403:y
1399:x
1397:(
1395:d
1391:y
1387:x
1358:x
1352:=
1346:y
1343:+
1340:x
1314:x
1302:y
1299:+
1296:x
1284:x
1258:y
1255:+
1252:x
1240:x
1214:y
1211:+
1208:x
1202:=
1198:}
1191:y
1185:,
1179:y
1176:+
1173:x
1166:{
1139:y
1127:x
1101:y
1077:}
1070:y
1064:,
1058:y
1055:+
1052:x
1045:{
1020:}
1013:y
1007:,
1001:y
998:+
995:x
988:{
975:y
969:)
966:y
963:+
960:x
957:(
951:=
945:x
919:x
907:y
904:+
901:x
878:.
872:y
860:x
830:y
818:x
794:}
787:y
781:,
775:x
768:{
755:y
752:+
749:x
734:.
719:y
707:x
683:}
676:y
670:,
664:x
657:{
644:y
641:+
638:x
605:y
599:x
593:=
590:)
587:y
584:,
581:x
578:(
575:d
536:d
532:M
525:M
521:d
517:M
513:)
511:d
507:M
505:(
497:M
489:d
485:d
474:M
470:d
466:M
462:)
460:d
456:M
454:(
434:)
432:z
428:y
426:(
424:d
420:y
416:x
414:(
412:d
408:z
404:x
402:(
400:d
395:;
392:y
388:x
381:y
377:x
375:(
373:d
367:;
363:x
359:x
357:(
355:d
346:(
344:)
342:x
338:y
336:(
334:d
330:y
326:x
324:(
322:d
317:;
313:y
309:x
307:(
305:d
296:M
292:z
288:y
284:x
275:ℝ
257:R
250:M
244:M
238:d
221:M
184:z
164:y
144:x
123:}
119:)
116:z
113:,
110:y
107:(
104:d
101:,
98:)
95:y
92:,
89:x
86:(
83:d
79:{
69:)
66:z
63:,
60:x
57:(
54:d
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
