36:
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210:
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The top half of the second image shows a cover (colored) of a planar shape (dark), where all of the shape's points are contained in anywhere from one to all four of the cover's sets. The bottom illustrates that any attempt to refine said cover such that no point would be contained in more than
202:
The first image shows a refinement (on the bottom) of a colored cover (on the top) of a black circular line. Note how in the refinement, no point on the circle is contained in more than two sets, and also how the sets link to one another to form a "chain".
128:, the Lebesgue covering dimension is just the ordinary Euclidean dimension: zero for points, one for lines, two for planes, and so on. However, not all topological spaces have this kind of "obvious"
743:
The empty set has covering dimension -1: for any open cover of the empty set, each point of the empty set is not contained in any element of the cover, so the order of any open cover is 0.
773:
can be refined so that any point of the disk is contained in no more than three open sets, while two are in general not sufficient. The covering dimension of the disk is thus two.
969:
910:
1597:
810:
1254:
1224:
1144:
1096:
1037:
762:
two open arcs. That is, whatever collection of arcs we begin with, some can be discarded or shrunk, such that the remainder still covers the circle but with simple overlaps.
642:
612:
584:
511:
476:
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309:
1371:
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429:
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223:
sets—ultimately fails at the intersection of set borders. Thus, a planar shape is not "webby": it cannot be covered with "chains", per se. Instead, it proves to be
1451:
1431:
1411:
1391:
1315:
930:
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arcs. The circle has dimension one, by this definition, because any such cover can be further refined to the stage where a given point
175: + 1 covering sets. This is the gist of the formal definition below. The goal of the definition is to provide a number (an
132:, and so a precise definition is needed in such cases. The definition proceeds by examining what happens when the space is covered by
1658:, American Mathematical Society, Collected works series, vol. 4, American Mathematical Society, p. xxiii, footnote 3,
179:) that describes the space, and does not change as the space is continuously deformed; that is, a number that is invariant under
1518:
1798:
17:
1855:. Publications de l'Institut de Mathématique de l'Université de Strasbourg (in French). Vol. III. Paris: Hermann.
2195:
2145:
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2005:
1967:
1901:
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The general idea is illustrated in the diagrams below, which show a cover and refinements of a circle and a square.
79:
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1489:
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1523:
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1997:
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with respect to the covering dimension if every open cover of the space has a refinement consisting of
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1125:
1077:
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open sets, meaning any point in the space is contained in exactly one open set of this refinement.
44:
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682:
414:
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1958:, (1926) Communications to the Amsterdam Academy of Sciences. English translation reprinted in
728:
61:
1825:. North-Holland Mathematical Library. Vol. 19. Amsterdam-Oxford-New York: North-Holland.
1744:. North-Holland Mathematical Library. Vol. 19. Amsterdam-Oxford-New York: North-Holland.
1693:. North-Holland Mathematical Library. Vol. 19. Amsterdam-Oxford-New York: North-Holland.
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2160:
1991:
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Lebesgue's discovery led later to the introduction by E. Čech of the covering dimension
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1436:
1416:
1396:
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1300:
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152:
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in some sense. More rigorously put, its topological dimension must be greater than 1.
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spaces have the same covering dimension. That is, the covering dimension is a
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2247:
1889:
1196:, the covering dimension can be equivalently defined as the minimum value of
863:
180:
1460:, one can strengthen the notion of the multiplicity of a cover: a cover has
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2232:
2125:
1614:
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1500:-dimensional "at large scales", and a space with Assouad–Nagata dimension
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2135:
1972:
1951:
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93:
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261:
144:
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2028:
Encyclopaedia of
Mathematical Sciences, Volume 17, General Topology I
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129:
109:
1919:
Ostrand, Phillip A. (1971). "Covering dimension in general spaces".
376:(if it exists) for which each point of the space belongs to at most
2252:
2215:
2140:
999:
271:
133:
1262: + 1. In particular, this holds for all metric spaces.
2262:
176:
724:
exists, the space is said to have infinite covering dimension.
249:
The first formal definition of covering dimension was given by
1484:
sets in the cover. This idea leads to the definitions of the
1184:
27:
Topologically invariant definition of the dimension of a space
2219:
2070:
147:, in that one can find a collection of open sets such that
1562:"Sur les correspondances entre les points de deux espaces"
1074:+ 1, there exists a family of pairwise disjoint open sets
245:
used closed "bricks" to study covering dimension in 1921.
1789:. Undergraduate Texts in Mathematics (Second ed.).
1439:
1419:
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1303:
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1021:
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will have a refinement consisting of a collection of
685:
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596:
568:
495:
460:
417:
341:
290:
727:
As a special case, a non-empty topological space is
1630:
1268:The Lebesgue covering dimension coincides with the
1785:Edgar, Gerald A. (2008). "Topological Dimension".
1715:
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108:is one of several different ways of defining the
2376:
1962:, Gerald A.Edgar, editor, Addison-Wesley (1993)
155:. The covering dimension is the smallest number
1867:
1872:. Princeton Mathematical Series. Vol. 4.
1492:of a space: a space with asymptotic dimension
2086:
1853:Topologie algébrique et théorie des faisceaux
912:is continuous, then there is an extension of
644:can always be chosen to be finite. Thus, if
1185:Relationships to other notions of dimension
2093:
2079:
284:such that their union is the whole space,
1979:, (1928) B.G Teubner Publishers, Leipzig.
1868:Hurewicz, Witold; Wallman, Henry (1941).
1815:
1734:
1683:
1648:
1613:
1598:"The origins of the concept of dimension"
835:The covering dimension of a normal space
792:
80:Learn how and when to remove this message
1847:
1636:
1556:
237:
208:
192:
43:This article includes a list of general
1918:
1888:
1787:Measure, topology, and fractal geometry
1721:
1550:
1009:Ostrand's theorem on colored dimension.
382:open sets in the cover: in other words
14:
2377:
1433:larger than the covering dimension of
556:is defined to be the minimum value of
260:A modern definition is as follows. An
159:such that for every cover, there is a
119:
2074:
1989:
1784:
1232:(of any size) has an open refinement
2024:The Fundamentals of Dimension Theory
1595:
964:{\displaystyle g:X\rightarrow S^{n}}
905:{\displaystyle f:A\rightarrow S^{n}}
233:
29:
1956:General Spaces and Cartesian Spaces
1589:
1283:is less than or equal to the large
1241:
1211:
1131:
1083:
1024:
629:
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571:
498:
463:
344:
213:Refinement of the cover of a square
197:Refinement of the cover of a circle
24:
2030:, (1993) A. V. Arkhangel'skii and
1993:Dimension Theory of General Spaces
1940:
1655:Collected Works of Witold Hurewicz
1015:is a normal topological space and
686:
562:such that every finite open cover
418:
49:it lacks sufficient corresponding
25:
2396:
2046:
765:Similarly, any open cover of the
620: + 1. The refinement
2159:
2034:(Eds.), Springer-Verlag, Berlin
1519:Carathéodory's extension theorem
805:{\displaystyle \mathbb {E} ^{n}}
253:, based on an earlier result of
139:In general, a topological space
34:
1896:(2nd ed.). Prentice-Hall.
1249:{\displaystyle {\mathfrak {B}}}
1219:{\displaystyle {\mathfrak {A}}}
1139:{\displaystyle {\mathfrak {A}}}
1091:{\displaystyle {\mathfrak {B}}}
1050:} is a locally finite cover of
1032:{\displaystyle {\mathfrak {A}}}
637:{\displaystyle {\mathfrak {B}}}
607:{\displaystyle {\mathfrak {B}}}
579:{\displaystyle {\mathfrak {A}}}
506:{\displaystyle {\mathfrak {B}}}
471:{\displaystyle {\mathfrak {A}}}
352:{\displaystyle {\mathfrak {A}}}
304:{\displaystyle \cup _{\alpha }}
1772:Godement 1973, II.5.12, p. 236
1766:
1727:
1676:
1642:
1508:-dimensional "at every scale".
1477:-ball intersects with at most
1354:
1342:
948:
889:
758:of the circle is contained in
13:
1:
2100:
1945:
1778:
1202:, such that every open cover
819:
718:distinct. If no such minimal
1366:{\displaystyle H^{i}(X,A)=0}
1290:The covering dimension of a
1279:The covering dimension of a
746:Any given open cover of the
7:
2059:Encyclopedia of Mathematics
1816:Engelking, Ryszard (1978).
1735:Engelking, Ryszard (1978).
1684:Engelking, Ryszard (1978).
1524:Geometric set cover problem
1512:
1317:is greater or equal to its
738:
98:Lebesgue covering dimension
10:
2401:
1998:Cambridge University Press
1874:Princeton University Press
1266:Lebesgue covering theorem.
692:{\displaystyle \emptyset }
424:{\displaystyle \emptyset }
2361:
2340:
2276:
2214:
2168:
2157:
2108:
1983:
1921:General Topology and Appl
370:} is the smallest number
1544:
1490:Assouad–Nagata dimension
1190:For a paracompact space
1174:, and together covering
1062:+ 1, then, for each 1 ≤
489:} is another open cover
163:in which every point in
1990:Pears, Alan R. (1975).
1602:Colloquium Mathematicum
1570:Fundamenta Mathematicae
1539:Point-finite collection
1319:cohomological dimension
862:if and only if for any
812:has covering dimension
769:in the two-dimensional
590:has an open refinement
550:of a topological space
264:of a topological space
114:topologically invariant
64:more precise citations.
1615:10.4064/cm-42-1-95-110
1583:10.4064/fm-2-1-256-285
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855:{\displaystyle \leq n}
806:
693:
638:
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472:
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353:
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246:
214:
198:
1733:Proposition 3.2.2 of
1682:Proposition 1.6.9 of
1448:
1428:
1408:
1393:of abelian groups on
1388:
1368:
1312:
1251:
1221:
1141:
1093:
1034:
993:
991:{\displaystyle S^{n}}
966:
927:
907:
857:
830:topological invariant
807:
694:
639:
609:
581:
535:is contained in some
508:
473:
426:
354:
306:
241:
212:
196:
151:lies inside of their
102:topological dimension
18:Topological dimension
2277:Dimensions by number
2054:"Lebesgue dimension"
1960:Classics on Fractals
1486:asymptotic dimension
1437:
1417:
1397:
1377:
1329:
1325:), that is, one has
1301:
1236:
1206:
1126:
1078:
1019:
975:
936:
916:
877:
843:
787:
776:More generally, the
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624:
594:
566:
493:
458:
415:
339:
288:
145:covered by open sets
1793:. pp. 85–114.
1650:Kuperberg, Krystyna
1285:inductive dimension
1003:-dimensional sphere
120:Informal discussion
2206:Degrees of freedom
2109:Dimensional spaces
1443:
1423:
1403:
1383:
1363:
1307:
1274:simplicial complex
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548:covering dimension
524:}, such that each
503:
468:
421:
349:
301:
247:
215:
199:
112:of the space in a
2372:
2371:
2181:Lebesgue covering
2146:Algebraic variety
2022:V. V. Fedorchuk,
1977:Dimensionstheorie
1890:Munkres, James R.
1800:978-0-387-74748-4
1596:Duda, R. (1979).
1534:Metacompact space
1446:{\displaystyle X}
1426:{\displaystyle i}
1406:{\displaystyle X}
1386:{\displaystyle A}
1321:(in the sense of
1310:{\displaystyle X}
925:{\displaystyle f}
454:of an open cover
335:of an open cover
234:Formal definition
231:
230:
106:topological space
90:
89:
82:
16:(Redirected from
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2385:Dimension theory
2169:Other dimensions
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2131:Projective space
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2032:L. S. Pontryagin
2019:
1936:
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1270:affine dimension
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1886:
1865:
1845:
1831:
1813:
1799:
1780:
1777:
1775:
1774:
1765:
1750:
1726:
1714:
1699:
1675:
1664:
1652:, ed. (1995),
1641:
1629:
1588:
1548:
1546:
1543:
1542:
1541:
1536:
1531:
1526:
1521:
1514:
1511:
1510:
1509:
1454:
1442:
1422:
1402:
1382:
1362:
1359:
1356:
1353:
1350:
1347:
1344:
1339:
1335:
1306:
1288:
1277:
1263:
1243:
1213:
1186:
1183:
1182:
1181:
1169:
1152:
1133:
1111:
1098:
1085:
1045:
1026:
1006:
985:
981:
958:
954:
950:
947:
944:
941:
921:
899:
895:
891:
888:
885:
882:
851:
848:
833:
821:
818:
799:
794:
740:
737:
710:
703:
688:
670:
666:
658:
654:
631:
601:
573:
541:
530:
519:
500:
484:
465:
442:
435:
420:
402:
398:
390:
386:
365:
346:
316:
298:
294:
279:
255:Henri Lebesgue
243:Henri Lebesgue
235:
232:
229:
228:
216:
205:
204:
200:
181:homeomorphisms
121:
118:
88:
87:
42:
40:
33:
26:
9:
6:
4:
3:
2:
2397:
2386:
2383:
2382:
2380:
2367:
2366:
2360:
2354:
2351:
2349:
2346:
2345:
2343:
2339:
2333:
2331:
2327:
2325:
2322:
2320:
2317:
2315:
2312:
2310:
2307:
2305:
2302:
2300:
2297:
2295:
2292:
2290:
2287:
2285:
2282:
2281:
2279:
2275:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2248:Demihypercube
2246:
2244:
2241:
2239:
2236:
2234:
2231:
2229:
2226:
2225:
2223:
2221:
2217:
2213:
2207:
2204:
2202:
2199:
2197:
2194:
2192:
2189:
2187:
2184:
2182:
2179:
2177:
2174:
2173:
2171:
2167:
2162:
2152:
2149:
2147:
2144:
2142:
2139:
2137:
2134:
2132:
2129:
2127:
2124:
2122:
2119:
2117:
2114:
2113:
2111:
2107:
2103:
2096:
2091:
2089:
2084:
2082:
2077:
2076:
2073:
2065:
2061:
2060:
2055:
2051:
2050:
2041:
2040:3-540-18178-4
2037:
2033:
2029:
2025:
2021:
2017:
2013:
2009:
2007:0-521-20515-8
2003:
1999:
1995:
1994:
1988:
1987:
1978:
1974:
1971:
1969:
1968:0-201-58701-7
1965:
1961:
1957:
1953:
1950:
1949:
1934:
1930:
1926:
1922:
1917:
1913:
1909:
1905:
1903:0-13-181629-2
1899:
1895:
1891:
1887:
1883:
1879:
1875:
1871:
1866:
1862:
1858:
1854:
1850:
1846:
1842:
1838:
1834:
1832:0-444-85176-3
1828:
1821:
1820:
1814:
1810:
1806:
1802:
1796:
1792:
1788:
1783:
1782:
1769:
1761:
1757:
1753:
1751:0-444-85176-3
1747:
1740:
1739:
1730:
1723:
1718:
1710:
1706:
1702:
1700:0-444-85176-3
1696:
1689:
1688:
1679:
1671:
1667:
1665:9780821800119
1661:
1657:
1656:
1651:
1645:
1638:
1637:Lebesgue 1921
1633:
1625:
1621:
1616:
1611:
1607:
1603:
1599:
1592:
1584:
1580:
1576:
1573:(in French).
1572:
1571:
1563:
1559:
1553:
1549:
1540:
1537:
1535:
1532:
1530:
1527:
1525:
1522:
1520:
1517:
1516:
1491:
1487:
1481:
1470:
1466:
1465:-multiplicity
1459:
1455:
1440:
1420:
1400:
1380:
1360:
1357:
1351:
1348:
1345:
1337:
1333:
1324:
1320:
1304:
1296:
1293:
1289:
1286:
1282:
1278:
1275:
1271:
1267:
1264:
1260:
1230:
1200:
1194:
1189:
1188:
1178:
1167:
1156:
1150:
1115:
1109:
1102:
1072:
1066:
1060:
1054:
1043:
1010:
1007:
1004:
1002:
983:
979:
956:
952:
945:
942:
939:
919:
897:
893:
886:
883:
880:
872:
868:
865:
864:closed subset
849:
846:
838:
834:
831:
827:
824:
823:
817:
815:
797:
783:
780:-dimensional
779:
774:
772:
768:
763:
761:
757:
753:
749:
744:
736:
734:
730:
725:
722:
714:
674:
648:
618:
589:
560:
554:
549:
539:
528:
517:
482:
453:
446:
406:
380:
374:
363:
334:
330:
325:
314:
296:
292:
277:
273:
268:
263:
258:
256:
252:
244:
240:
226:
222:
217:
211:
207:
206:
201:
195:
191:
190:
187:
184:
182:
178:
174:
170:
166:
162:
158:
154:
150:
146:
142:
137:
135:
131:
127:
124:For ordinary
117:
115:
111:
107:
103:
99:
95:
84:
81:
73:
63:
59:
53:
52:
46:
41:
32:
31:
19:
2363:
2329:
2268:Hyperpyramid
2233:Hypersurface
2180:
2126:Affine space
2116:Vector space
2057:
2027:
2023:
1992:
1976:
1959:
1955:
1924:
1920:
1893:
1869:
1852:
1818:
1786:
1768:
1737:
1729:
1722:Ostrand 1971
1717:
1686:
1678:
1669:
1654:
1644:
1632:
1605:
1601:
1591:
1574:
1568:
1552:
1479:
1468:
1461:
1458:metric space
1281:normal space
1272:of a finite
1265:
1258:
1228:
1198:
1192:
1176:
1165:
1154:
1148:
1122:} shrinking
1113:
1107:
1100:
1070:
1064:
1058:
1052:
1041:
1008:
1000:
870:
866:
836:
826:Homeomorphic
813:
777:
775:
764:
759:
755:
745:
742:
726:
720:
712:
672:
646:
616:
587:
558:
552:
547:
537:
526:
515:
480:
450:distinct. A
444:
404:
378:
372:
361:
332:
328:
323:
312:
275:
266:
259:
248:
224:
220:
185:
172:
169:intersection
167:lies in the
164:
156:
148:
140:
138:
123:
101:
97:
91:
76:
67:
48:
2353:Codimension
2332:-dimensions
2253:Hypersphere
2136:Free module
1973:Karl Menger
1952:Karl Menger
1577:: 256–285.
1292:paracompact
1256:with order
1056:of order ≤
748:unit circle
650:is finite,
614:with order
251:Eduard Čech
94:mathematics
62:introducing
2348:Hyperspace
2228:Hyperplane
1946:Historical
1779:References
1608:: 95–110.
1413:and every
820:Properties
452:refinement
262:open cover
161:refinement
70:April 2018
45:references
2238:Hypercube
2216:Polytopes
2196:Minkowski
2191:Hausdorff
2186:Inductive
2151:Spacetime
2102:Dimension
2064:EMS Press
1473:if every
1295:Hausdorff
971:. Here,
949:→
890:→
847:≤
767:unit disk
687:∅
419:∅
297:α
293:∪
272:open sets
134:open sets
130:dimension
110:dimension
2379:Category
2365:Category
2341:See also
2141:Manifold
1894:Topology
1892:(2000).
1851:(1958).
1560:(1921).
1513:See also
739:Examples
733:disjoint
662:∩ ⋅⋅⋅ ∩
394:∩ ⋅⋅⋅ ∩
2263:Simplex
2201:Fractal
2066:, 2001
2016:0394604
1933:0288741
1912:3728284
1882:0006493
1861:0102797
1841:0482697
1809:2356043
1760:0482697
1709:0482697
1624:0567548
1323:sheaves
1146:, i.e.
998:is the
760:at most
706:, ...,
438:, ...,
225:thicker
177:integer
143:can be
58:improve
2220:shapes
2038:
2014:
2004:
1984:Modern
1966:
1931:
1910:
1900:
1880:
1859:
1839:
1829:
1807:
1797:
1758:
1748:
1707:
1697:
1662:
1622:
1297:space
546:. The
327:. The
96:, the
47:, but
2324:Eight
2319:Seven
2299:Three
2176:Krull
1823:(PDF)
1742:(PDF)
1691:(PDF)
1565:(PDF)
1545:Notes
1456:In a
873:, if
771:plane
329:order
153:union
116:way.
104:of a
2309:Five
2304:Four
2284:Zero
2218:and
2036:ISBN
2002:ISBN
1964:ISBN
1898:ISBN
1827:ISBN
1795:ISBN
1746:ISBN
1695:ISBN
1660:ISBN
1488:and
752:open
699:for
431:for
2314:Six
2294:Two
2289:One
1610:doi
1579:doi
1504:is
1496:is
1482:+ 1
1471:+ 1
1226:of
1105:= {
1039:= {
1011:If
932:to
869:of
839:is
586:of
513:= {
478:= {
359:= {
333:ply
331:or
221:two
183:.
100:or
92:In
2381::
2062:,
2056:,
2012:MR
2010:.
2000:.
1996:.
1975:,
1954:,
1929:MR
1923:.
1908:MR
1906:.
1878:MR
1876:.
1857:MR
1837:MR
1835:.
1805:MR
1803:.
1756:MR
1754:.
1705:MR
1703:.
1668:,
1620:MR
1618:.
1606:42
1604:.
1600:.
1567:.
1163:⊆
1068:≤
816:.
716:+2
679:=
676:+2
448:+1
411:=
408:+1
321:=
257:.
136:.
2330:n
2094:e
2087:t
2080:v
2042:.
2018:.
1935:.
1925:1
1914:.
1884:.
1863:.
1843:.
1811:.
1762:.
1724:.
1711:.
1673:.
1639:.
1626:.
1612::
1585:.
1581::
1575:2
1506:n
1502:n
1498:n
1494:n
1480:n
1475:r
1469:n
1463:r
1453:.
1441:X
1421:i
1401:X
1381:A
1361:0
1358:=
1355:)
1352:A
1349:,
1346:X
1343:(
1338:i
1334:H
1305:X
1287:.
1276:.
1259:n
1242:B
1229:X
1212:A
1199:n
1193:X
1180:.
1177:X
1171:α
1166:U
1160:α
1158:,
1155:i
1149:V
1132:A
1119:α
1117:,
1114:i
1108:V
1101:i
1084:B
1071:n
1065:i
1059:n
1053:X
1047:α
1042:U
1025:A
1013:X
1005:.
1001:n
984:n
980:S
957:n
953:S
946:X
943::
940:g
920:f
898:n
894:S
887:A
884::
881:f
871:X
867:A
850:n
837:X
832:.
814:n
798:n
793:E
778:n
756:x
721:n
713:n
708:β
704:1
701:β
673:n
668:β
664:V
659:1
656:β
652:V
647:n
630:B
617:n
600:B
588:X
572:A
559:n
553:X
543:α
538:U
532:β
527:V
521:β
516:V
499:B
486:α
481:U
464:A
445:m
440:α
436:1
433:α
405:m
400:α
396:U
391:1
388:α
384:U
379:m
373:m
367:α
362:U
345:A
324:X
318:α
313:U
281:α
276:U
267:X
173:n
165:X
157:n
149:X
141:X
83:)
77:(
72:)
68:(
54:.
20:)
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