590:. Appel and Haken's approach started by showing that there is a particular set of 1,936 maps, each of which cannot be part of a smallest-sized counterexample to the four color theorem (i.e., if they did appear, one could make a smaller counter-example). Appel and Haken used a special-purpose computer program to confirm that each of these maps had this property. Additionally, any map that could potentially be a counterexample must have a portion that looks like one of these 1,936 maps. Showing this with hundreds of pages of hand analysis, Appel and Haken concluded that no smallest counterexample exists because any must contain, yet do not contain, one of these 1,936 maps. This contradiction means there are no counterexamples at all and that the theorem is therefore true. Initially, their proof was not accepted by mathematicians at all because the
521:
2047:
2059:
38:
2015:
150:": in this approach, all possible cases are considered and shown not to give counterexamples. In some occasions, the number of cases is quite large, in which case a brute-force proof may require as a practical matter the use of a computer algorithm to check all the cases. For example, the validity of the 1976 and 1997 brute-force proofs of the
1389:
From this paper: Definitions: A planar map is a set of pairwise disjoint subsets of the plane, called regions. A simple map is one whose regions are connected open sets. Two regions of a map are adjacent if their respective closures have a common point that is not a corner of the map. A point is a
131:
terminate, has been tested for all integers up to 1.2 × 10 (over a trillion). However, the failure to find a counterexample after extensive search does not constitute a proof that the conjecture is true—because the conjecture might be false but with a very large minimal counterexample.
907:
to systematically excise singular regions as they develop, in a controlled way, but was unable to prove this method "converged" in three dimensions. Perelman completed this portion of the proof. Several teams of mathematicians have verified that
Perelman's proof is correct.
1013:. Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer; it is widely conjectured that the answer is no. It was essentially first mentioned in a 1956 letter written by
548:
if they share a common boundary that is not a corner, where corners are the points shared by three or more regions. For example, in the map of the United States of
America, Utah and Arizona are adjacent, but Utah and New Mexico, which only share a
135:
Nevertheless, mathematicians often regard a conjecture as strongly supported by evidence even though not yet proved. That evidence may be of various kinds, such as verification of consequences of it or strong interconnections with known results.
567:, which has a short elementary proof, states that five colors suffice to color a map and was proven in the late 19th century; however, proving that four colors suffice turned out to be significantly harder. A number of false proofs and false
280:. Few number theorists doubt that the Riemann hypothesis is true. In fact, in anticipation of its eventual proof, some have even proceeded to develop further proofs which are contingent on the truth of this conjecture. These are called
1390:
corner of a map if and only if it belongs to the closures of at least three regions. Theorem: The regions of any simple planar map can be colored with only four colors, in such a way that any two adjacent regions have different colors.
118:
could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the
1087:
been proven that both cannot simultaneously be true (i.e., at least one must be false). It has not been proven which one is false, but it is widely believed that the first conjecture is true and the second one is
201:). In the case of the latter, the first counterexample found for the n=4 case involved numbers in the millions, although it has been subsequently found that the minimal counterexample is actually smaller.
1082:
are a pair of conjectures concerning the distribution of prime numbers, the first of which expands upon the aforementioned twin prime conjecture. Neither one has either been proven or disproven, but it
289:
These "proofs", however, would fall apart if it turned out that the hypothesis was false, so there is considerable interest in verifying the truth or falsity of conjectures of this type.
442:
1025:
in his seminal paper "The complexity of theorem proving procedures" and is considered by many to be the most important open problem in the field. It is one of the seven
1223:
1021:. Gödel asked whether a certain NP-complete problem could be solved in quadratic or linear time. The precise statement of the P=NP problem was introduced in 1971 by
1225:
Shuttling between the particular and the general: reflections on the role of conjecture and hypothesis in the generation of knowledge in science and mathematics
1075:, proposed by Euler in the 18th century but for which counterexamples for a number of exponents (starting with n=4) were found beginning in the mid 20th century
464:
387:
365:
344:
139:
A conjecture is considered proven only when it has been shown that it is logically impossible for it to be false. There are various methods of doing so; see
869:
in 1904, the theorem concerns a space that locally looks like ordinary three-dimensional space but is connected, finite in size, and lacks any boundary (a
1673:
1306:
587:
544:, no more than four colors are required to color the regions of the map—so that no two adjacent regions have the same color. Two regions are called
954:
2084:
252:
be proved using only the axioms of neutral geometry, i.e. without the parallel postulate). The one major exception to this in practice is the
1010:
1772:
1889:
481:
594:
was infeasible for a human to check by hand. However, the proof has since then gained wider acceptance, although doubts still remain.
256:, as the majority of researchers usually do not worry whether a result requires it—unless they are studying this axiom in particular.
488:, and formally published in 1995, after 358 years of effort by mathematicians. The unsolved problem stimulated the development of
624:
have a common refinement, a single triangulation that is a subdivision of both of them. It was originally formulated in 1908, by
1079:
1992:
146:
One method of proof, applicable when there are only a finite number of cases that could lead to counterexamples, is known as "
1604:
1346:
1072:
198:
881:
2094:
1860:
1745:
1502:
1233:
222:
559:
mentioned the problem in his lectures as early as 1840. The conjecture was first proposed on
October 23, 1852 when
1938:
563:, while trying to color the map of counties of England, noticed that only four different colors were needed. The
964:. Along with suitable generalizations, some mathematicians consider it the most important unresolved problem in
1684:
815:
502:
31:
2030:
768:, and should have their zeroes in restricted places. The last two parts were quite consciously modeled on the
93:), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them.
903:
to attempt to solve the problem. Hamilton later introduced a modification of the standard Ricci flow, called
880:
in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An
114:
conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single
2019:
1725:
1275:
973:
226:
17:
268:
when it is used frequently and repeatedly as an assumption in proofs of other results. For example, the
140:
53:, a famous conjecture, says that all non-trivial zeros of the zeta function lie along the critical line.
2037:
1708:
1030:
985:
394:
850:
An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called
556:
41:
The real part (red) and imaginary part (blue) of the
Riemann zeta function along the critical line Re(
1026:
988:
170:
1042:
617:
313:
303:
178:
740:
elements containing that field. The generating function has coefficients derived from the numbers
591:
489:
1338:
1332:
1982:
1497:(Revised color ed.). Princeton, New Jersey: Princeton University Press. pp. 216–222.
497:
244:
In this case, if a proof uses this statement, researchers will often look for a new proof that
111:
82:
911:
The
Poincaré conjecture, before being proven, was one of the most important open questions in
807:
797:
174:
1711:, Bulletin of the European Association for Theoretical Computer Science, vol. 38, pp. 101–107
1552:
1550:
Milnor, John W. (1961). "Two complexes which are homeomorphic but combinatorially distinct".
1124:
1104:
1048:
981:
946:
876:). The Poincaré conjecture claims that if such a space has the additional property that each
769:
550:
2079:
2000:
1975:
1918:
1650:
1581:
1066:
851:
229:
of set theory. It is therefore possible to adopt this statement, or its negation, as a new
210:
1658:
520:
8:
2089:
1620:
1159:
1096:
1006:
1000:
969:
896:
765:
710:
706:
147:
1184:
194:
2051:
1963:
1922:
1884:
1751:
1569:
1475:
1298:
1250:
1054:
942:
930:
924:
891:
presented a proof of the conjecture in three papers made available in 2002 and 2003 on
773:
621:
613:
564:
533:
515:
493:
449:
372:
350:
329:
286:: the conjectures assumed appear in the hypotheses of the theorem, for the time being.
269:
249:
238:
162:
151:
120:
78:
74:
50:
1986:
2046:
1955:
1926:
1906:
1856:
1741:
1600:
1508:
1498:
1467:
1342:
1229:
1164:
1092:
761:
714:
282:
1820:
1803:
1302:
866:
635:
This conjecture is now known to be false. The non-manifold version was disproved by
1947:
1898:
1815:
1786:
1755:
1733:
1654:
1636:
1561:
1459:
1401:
1376:
1290:
1060:
1018:
965:
934:
888:
831:
694:
684:
640:
471:
536:, or the four color map theorem, states that given any separation of a plane into
1996:
1971:
1914:
1646:
1577:
1450:
Swart, E. R. (1980). "The
Philosophical Implications of the Four-Color Problem".
1149:
1144:
877:
870:
835:
629:
560:
321:
253:
214:
155:
537:
480:, where he claimed that he had a proof that was too large to fit in the margin.
248:
require the hypothesis (in the same way that it is desirable that statements in
2063:
1880:
733:
672:
625:
609:
603:
579:
568:
115:
1641:
1624:
1526:
1294:
2073:
1959:
1933:
1910:
1836:
1768:
1512:
1471:
1154:
1100:
977:
961:
575:
524:
A four-coloring of a map of the states of the United States (ignoring lakes).
309:
273:
1790:
1014:
953:
1/2. The name is also used for some closely related analogues, such as the
698:
668:
1936:(1960), "On the rationality of the zeta function of an algebraic variety",
1721:
1022:
843:
718:
571:
have appeared since the first statement of the four color theorem in 1852.
485:
277:
241:
can be taken either as true or false in an axiomatic system for geometry).
218:
154:
by computer was initially doubted, but was eventually confirmed in 2005 by
90:
1737:
1408:, in Mathematical Recreations and Essays, Macmillan, New York, pp 222-232.
276:
that — amongst other things — makes predictions about the distribution of
189:
Conjectures disproven through counterexample are sometimes referred to as
1120:
803:
690:
636:
529:
476:
70:
58:
1967:
1902:
1573:
1479:
1128:
900:
873:
66:
2058:
37:
1853:
Conjectures and refutations : the growth of scientific knowledge
1108:
950:
823:
651:
1951:
1730:
Proceedings of the Third Annual ACM Symposium on Theory of
Computing
1565:
1463:
1785:, Communications of the ACM 52 (2009), no. 9, pp. 78–86.
912:
839:
819:
647:
124:
1276:"Logical probability and the strength of mathematical conjectures"
496:
in the 20th century. It is among the most notable theorems in the
1132:
960:
The
Riemann hypothesis implies results about the distribution of
811:
583:
324:
166:
128:
86:
1985:(1995) , "Formule de Lefschetz et rationalité des fonctions L",
2025:
2014:
1946:(3), American Journal of Mathematics, Vol. 82, No. 3: 631–648,
1033:
to carry a US$ 1,000,000 prize for the first correct solution.
234:
1804:"On the Incompatibility of Two Conjectures Concerning Primes"
1099:' that link different subfields of mathematics (e.g. between
892:
230:
209:
Not every conjecture ends up being proven true or false. The
169:. Many important theorems were once conjectures, such as the
784:, and the analogue of the Riemann hypothesis was proved by
1495:
Four colors suffice : how the map problem was solved
1379:(December 2008). "Formal Proof—The Four-Color Theorem".
574:
The four color theorem was ultimately proven in 1976 by
110:
truth. In mathematics, any number of cases supporting a
1674:"The Riemann Hypothesis – official problem description"
826:
in four-dimensional space. The conjecture states that:
45:) = 1/2. The first non-trivial zeros can be seen at Im(
2035:
452:
397:
375:
353:
332:
1111:). Some of these conjectures have since been proved.
887:
After nearly a century of effort by mathematicians,
553:
that also belongs to
Arizona and Colorado, are not.
884:has been known in higher dimensions for some time.
749:of points over the (essentially unique) field with
1625:"Four-manifolds with positive isotropic curvature"
458:
436:
381:
359:
338:
320:, especially in older texts) states that no three
736:, as well as points over every finite field with
2071:
1375:
955:Riemann hypothesis for curves over finite fields
713:) derived from counting the number of points on
73:that is proffered on a tentative basis without
1726:"The complexity of theorem proving procedures"
1431:Heawood, P. J. (1890). "Map-Colour Theorems".
1395:
1123:pioneered the use of the term "conjecture" in
1599:. New York: New York : Springer-Verlag.
1419:Mechanizing Proof: Computing, Risk, and Trust
1981:
895:. The proof followed on from the program of
781:
506:for "most difficult mathematical problems".
96:
822:, which is the hypersphere that bounds the
27:Proposition in mathematics that is unproven
1709:Gödel, von Neumann, and the P = NP problem
697:were some highly influential proposals by
492:in the 19th century, and the proof of the
204:
1834:
1819:
1640:
1411:
1095:is a far-reaching web of these ideas of '
968:. The Riemann hypothesis, along with the
1801:
1671:
1619:
1597:Geometric Topology in Dimensions 2 and 3
1273:
1221:
941:), is a conjecture that the non-trivial
858:to the 3-sphere, then it is necessarily
519:
297:
213:, which tries to ascertain the relative
123:, which concerns whether or not certain
36:
1879:
1629:Communications in Analysis and Geometry
1430:
1228:. Oxford University Press. p. 93.
938:
785:
500:, and prior to its proof it was in the
14:
2072:
1850:
1549:
1527:"Triangulation and the Hauptvermutung"
1492:
994:
791:
675:in the 1920s and 1950s, respectively.
470:This theorem was first conjectured by
49:) = ±14.135, ±21.022 and ±25.011. The
2085:Concepts in the philosophy of science
1932:
1594:
1449:
1248:
918:
777:
540:regions, producing a figure called a
509:
292:
259:
165:, it is no longer a conjecture but a
1890:Publications Mathématiques de l'IHÉS
1720:
1115:
1036:
1011:unsolved problem in computer science
702:
264:Sometimes, a conjecture is called a
1330:
678:
474:in 1637 in the margin of a copy of
225:from the generally accepted set of
24:
1363:The Guinness Book of World Records
25:
2106:
2007:
1452:The American Mathematical Monthly
1135:refers to a testable conjecture.
597:
437:{\displaystyle a^{n}+b^{n}=c^{n}}
2057:
2045:
2013:
1433:Quarterly Journal of Mathematics
1365:. Guinness Publishing Ltd. 1995.
776:. The rationality was proved by
732:elements has a finite number of
612:(German for main conjecture) of
233:in a consistent manner (much as
199:Euler's sum of powers conjecture
77:. Some conjectures, such as the
1939:American Journal of Mathematics
1844:
1828:
1821:10.1090/S0002-9904-1974-13434-8
1795:
1762:
1714:
1701:
1665:
1613:
1588:
1543:
1519:
1486:
1443:
1424:
1312:from the original on 2017-03-09
616:is the conjecture that any two
106:Formal mathematics is based on
1993:Société Mathématique de France
1873:
1369:
1354:
1324:
1267:
1242:
1215:
1201:
1177:
503:Guinness Book of World Records
32:Conjecture (textual criticism)
13:
1:
1334:Number Theory and Its History
1170:
780:, the functional equation by
221:, was eventually shown to be
141:methods of mathematical proof
30:For text reconstruction, see
1209:Oxford Dictionary of English
1080:Hardy-Littlewood conjectures
7:
1138:
1127:. Conjecture is related to
764:, should satisfy a form of
756:Weil conjectured that such
184:
161:When a conjecture has been
10:
2111:
2031:Unsolved Problems web site
1885:"La conjecture de Weil. I"
1835:Langlands, Robert (1967),
1681:Clay Mathematics Institute
1361:"Science and Technology".
1283:Mathematical Intelligencer
1185:"Definition of CONJECTURE"
1031:Clay Mathematics Institute
998:
986:Clay Mathematics Institute
922:
865:Originally conjectured by
795:
682:
601:
513:
482:The first successful proof
301:
29:
1672:Bombieri, Enrico (2000).
1642:10.4310/CAG.1997.v5.n1.a1
1295:10.1007/s00283-015-9612-3
1027:Millennium Prize Problems
989:Millennium Prize Problems
728:over a finite field with
582:. It was the first major
445:for any integer value of
390:can satisfy the equation
97:Resolution of conjectures
2095:Mathematical terminology
1595:Moise, Edwin E. (1977).
1274:Franklin, James (2016).
984:; it is also one of the
974:Hilbert's eighth problem
484:was released in 1994 by
101:
1983:Grothendieck, Alexander
1791:10.1145/1562164.1562186
1251:"Fermat's Last Theorem"
1189:www.merriam-webster.com
905:Ricci flow with surgery
592:computer-assisted proof
588:proved using a computer
490:algebraic number theory
227:Zermelo–Fraenkel axioms
205:Independent conjectures
1991:, vol. 9, Paris:
1802:Richards, Ian (1974).
1707:Juris Hartmanis 1989,
1493:Wilson, Robin (2014).
1421:(MIT Press, 2004) p103
1406:The Four Color Theorem
1331:Ore, Oystein (1988) ,
848:
525:
498:history of mathematics
460:
438:
383:
361:
340:
171:Geometrization theorem
112:universally quantified
54:
1855:. London: Routledge.
1851:Popper, Karl (2004).
1808:Bull. Amer. Math. Soc
1738:10.1145/800157.805047
1553:Annals of Mathematics
1255:mathworld.wolfram.com
1222:Schwartz, JL (1995).
1125:scientific philosophy
1105:representation theory
1049:twin prime conjecture
1043:Goldbach's conjecture
947:Riemann zeta function
854:: if a 3-manifold is
828:
770:Riemann zeta function
523:
461:
439:
384:
362:
341:
314:Fermat's Last Theorem
304:Fermat's Last Theorem
298:Fermat's Last Theorem
272:is a conjecture from
179:Fermat's Last Theorem
40:
2022:at Wikimedia Commons
1838:Letter to Prof. Weil
1732:. pp. 151–158.
1621:Hamilton, Richard S.
1097:unifying conjectures
1067:Maldacena conjecture
982:23 unsolved problems
935:Bernhard Riemann
929:In mathematics, the
852:homotopy equivalence
711:local zeta-functions
707:generating functions
641:Reidemeister torsion
450:
395:
373:
351:
330:
211:continuum hypothesis
173:(which resolved the
89:, proven in 1995 by
2026:Open Problem Garden
1249:Weisstein, Eric W.
1160:List of conjectures
1007:P versus NP problem
1001:P versus NP problem
995:P versus NP problem
970:Goldbach conjecture
897:Richard S. Hamilton
856:homotopy equivalent
808:Poincaré conjecture
798:Poincaré conjecture
792:Poincaré conjecture
782:Grothendieck (1965)
766:functional equation
715:algebraic varieties
650:version is true in
318:Fermat's conjecture
175:Poincaré conjecture
83:Fermat's conjecture
1995:, pp. 41–55,
1988:SĂ©minaire Bourbaki
1903:10.1007/BF02684373
1774:The status of the
1531:www.maths.ed.ac.uk
1439:. Oxford: 332–338.
1417:Donald MacKenzie,
1381:Notices of the AMS
1337:, Dover, pp.
1055:Collatz conjecture
931:Riemann hypothesis
925:Riemann hypothesis
919:Riemann hypothesis
774:Riemann hypothesis
762:rational functions
622:triangulable space
614:geometric topology
565:five color theorem
534:four color theorem
526:
516:Four color theorem
510:Four color theorem
494:modularity theorem
467:greater than two.
456:
434:
379:
357:
336:
316:(sometimes called
293:Important examples
283:conditional proofs
270:Riemann hypothesis
260:Conditional proofs
250:Euclidean geometry
239:parallel postulate
152:four color theorem
143:for more details.
121:Collatz conjecture
79:Riemann hypothesis
55:
51:Riemann hypothesis
2018:Media related to
1606:978-0-387-90220-3
1387:(11): 1382–1393.
1348:978-0-486-65620-5
1165:Ramanujan machine
1116:In other sciences
1093:Langlands program
1037:Other conjectures
459:{\displaystyle n}
382:{\displaystyle c}
360:{\displaystyle b}
339:{\displaystyle a}
191:false conjectures
16:(Redirected from
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1213:
1212:
1211:(2010 ed.).
1205:
1199:
1198:
1196:
1195:
1181:
1073:Euler conjecture
1061:Manin conjecture
1029:selected by the
1019:John von Neumann
966:pure mathematics
889:Grigori Perelman
882:analogous result
846:to the 3-sphere.
832:simply connected
816:characterization
695:Weil conjectures
685:Weil conjectures
679:Weil conjectures
666:
659:
472:Pierre de Fermat
465:
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195:PĂłlya conjecture
21:
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2099:
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2010:
1952:10.2307/2372974
1897:(43): 273–307,
1881:Deligne, Pierre
1876:
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1145:Bold hypothesis
1141:
1118:
1039:
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997:
927:
921:
800:
794:
748:
734:rational points
687:
681:
667:were proved by
661:
654:
606:
600:
569:counterexamples
561:Francis Guthrie
518:
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156:theorem-proving
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2008:External links
2006:
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1934:Dwork, Bernard
1930:
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999:Main article:
996:
993:
933:, proposed by
923:Main article:
920:
917:
867:Henri Poincaré
796:Main article:
793:
790:
786:Deligne (1974)
758:zeta-functions
744:
699:André Weil
683:Main article:
680:
677:
673:Edwin E. Moise
639:in 1961 using
618:triangulations
610:Hauptvermutung
604:Hauptvermutung
602:Main article:
599:
598:Hauptvermutung
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1692:. Retrieved
1685:the original
1680:
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1534:. Retrieved
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1371:
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1333:
1326:
1314:. Retrieved
1289:(3): 14–19.
1286:
1282:
1269:
1258:. Retrieved
1254:
1244:
1224:
1217:
1208:
1203:
1192:. Retrieved
1188:
1179:
1119:
1084:
1023:Stephen Cook
1004:
959:
928:
910:
904:
886:
864:
860:homeomorphic
859:
855:
849:
844:homeomorphic
829:
801:
778:Dwork (1960)
757:
755:
750:
745:
741:
737:
729:
725:
723:
688:
662:
660:. The cases
655:
645:
634:
607:
573:
555:
545:
541:
527:
501:
486:Andrew Wiles
475:
469:
446:
391:
369:
347:
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208:
190:
188:
160:
145:
138:
134:
107:
105:
91:Andrew Wiles
62:
56:
46:
42:
2080:Conjectures
2052:Mathematics
2020:Conjectures
1874:Works cited
1814:: 419–438.
1635:(1): 1–92.
1131:, which in
1121:Karl Popper
1009:is a major
980:'s list of
899:to use the
804:mathematics
691:mathematics
637:John Milnor
530:mathematics
477:Arithmetica
223:independent
217:of certain
215:cardinality
148:brute force
71:proposition
59:mathematics
18:Conjectures
2090:Statements
2074:Categories
1694:2019-11-12
1659:0892.53018
1536:2019-11-12
1260:2019-11-12
1194:2019-11-12
1171:References
1129:hypothesis
1109:Lie groups
1015:Kurt Gödel
901:Ricci flow
874:3-manifold
814:about the
760:should be
753:elements.
724:A variety
709:(known as
669:Tibor RadĂł
652:dimensions
538:contiguous
266:hypothesis
158:software.
67:conclusion
63:conjecture
1960:0002-9327
1927:123139343
1911:1618-1913
1513:847985591
1472:0002-9890
951:real part
949:all have
824:unit ball
705:) on the
665:= 2 and 3
193:(cf. the
125:sequences
1883:(1974),
1724:(1971).
1623:(1997).
1307:Archived
1303:30291085
1139:See also
913:topology
840:manifold
820:3-sphere
648:manifold
626:Steinitz
546:adjacent
325:integers
322:positive
246:does not
185:Disproof
129:integers
108:provable
2064:Science
2038:Portals
2001:1608788
1976:0140494
1968:2372974
1919:0340258
1782:problem
1778:versus
1756:7573663
1651:1456308
1582:0133127
1574:1970299
1480:2321855
1404:(1960)
1339:203–204
1316:30 June
1133:science
945:of the
937: (
862:to it.
818:of the
812:theorem
701: (
584:theorem
167:theorem
87:theorem
85:(now a
1999:
1974:
1966:
1958:
1925:
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1859:
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1649:
1603:
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1478:
1470:
1345:
1301:
1232:
1088:false.
871:closed
836:closed
830:Every
806:, the
693:, the
630:Tietze
586:to be
557:Möbius
532:, the
368:, and
235:Euclid
163:proven
1964:JSTOR
1923:S2CID
1752:S2CID
1688:(PDF)
1677:(PDF)
1570:JSTOR
1476:JSTOR
1310:(PDF)
1299:S2CID
1279:(PDF)
943:zeros
893:arXiv
810:is a
717:over
620:of a
551:point
231:axiom
102:Proof
75:proof
69:or a
65:is a
1956:ISSN
1907:ISSN
1857:ISBN
1742:ISBN
1601:ISBN
1509:OCLC
1499:ISBN
1468:ISSN
1343:ISBN
1318:2021
1230:ISBN
1103:and
1091:The
1078:The
1071:The
1065:The
1059:The
1053:The
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1005:The
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878:loop
772:and
703:1949
671:and
646:The
628:and
608:The
578:and
197:and
61:, a
1948:doi
1899:doi
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1787:doi
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1460:doi
1291:doi
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1085:has
1017:to
976:in
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689:In
658:≤ 3
542:map
528:In
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1972:MR
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1944:82
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