267:
boundaries where terms are to be regarded as approximately leading-order and where not. Instead the terms fade in and out, as the variables change. Deciding whether terms in a model are leading-order (or approximately leading-order), and if not, whether they are small enough to be regarded as negligible, (two different questions), is often a matter of investigation and judgement, and will depend on the context.
312:
457: = 0.001 – this is just its main behaviour in the vicinity of this point. It may be that retaining only the leading-order (or approximately leading-order) terms, and regarding all the other smaller terms as negligible, is insufficient (when using the model for future prediction, for example), and so it may be necessary to also retain the set of next largest terms. These can be called the
69:(regarding the other smaller terms as negligible). This gives the main behaviour – the true behaviour is only small deviations away from this. This main behaviour may be captured sufficiently well by just the strictly leading-order terms, or it may be decided that slightly smaller terms should also be included. In which case, the phrase
275:
Equations with only one leading-order term are possible, but rare. For example, the equation 100 = 1 + 1 + 1 + ... + 1, (where the right hand side comprises one hundred 1's). For any particular combination of values for the variables and parameters, an equation will typically contain at least two
68:
A common and powerful way of simplifying and understanding a wide variety of complicated mathematical models is to investigate which terms are the largest (and therefore most important), for particular sizes of the variables and parameters, and analyse the behaviour produced by just these terms
266:
is that two terms that are within a factor of 10 (one order of magnitude) of each other should be regarded as of about the same order, and two terms that are not within a factor of 100 (two orders of magnitude) of each other should not. However, in between is a grey area, so there are no fixed
510:
Various differential equations may be locally simplified by considering only the leading-order components. Machine learning algorithms can partition simulation or observational data into localized partitions with leading-order equation terms for aerodynamics, ocean dynamics, tumor-induced
280:
terms. In this case, by making the assumption that the lower-order terms, and the parts of the leading-order terms that are the same size as the lower-order terms (perhaps the second or third
390:
and 0.1 terms may be regarded as negligible, and dropped, along with any values in the third significant figure onwards in the two remaining terms. This gives the leading-order balance
308:
of the model for these values of the variables and parameters. The size of the error in making this approximation is normally roughly the size of the largest neglected term.
367:
terms may be regarded as negligible, and dropped, along with any values in the third decimal places onwards in the two remaining terms. This gives the leading-order balance
284:
onwards), are negligible, a new equation may be formed by dropping all these lower-order terms and parts of the leading-order terms. The remaining terms provide the
437:
Note that this description of finding leading-order balances and behaviours gives only an outline description of the process – it is not mathematically rigorous.
1235:
697:
73:
might be used informally to mean this whole group of terms. The behaviour produced by just the group of leading-order terms is called the
1086:
Catani, S.; Seymour, M.H. (1996). "The Dipole
Formalism for the Calculation of QCD Jet Cross Sections at Next-to-Leading Order".
1033:
Campbell, J.; Ellis, R.K. (2002). "Next-to-leading order corrections to W + 2 jet and Z + 2 jet production at hadron colliders".
479:
658:
1289:
576:
1139:
Kidonakis, N.; Vogt, R. (2003). "Next-to-next-to-leading order soft-gluon corrections in top quark hadroproduction".
980:
Kruczenski, M.; Oxman, L.E.; Zaldarriaga, M. (1999). "Large squeezing behaviour of cosmological entropy generation".
805:"Diffusion-Limited Binary Reactions: The Hierarchy of Nonclassical Regimes for Correlated Initial Conditions"
17:
1352:
1347:
1266:
491:
739:
Gorshkov, A. V.; et al. (2008). "Coherent
Quantum Optical Control with Subwavelength Resolution".
725:
1302:
555:
239:, the table shows the sizes of the four terms in this equation, and which terms are leading-order. As
46:
945:
HĂĽseyin, A. (1980). "The leading-order behaviour of the two-photon scattering amplitudes in QCD".
494:
may be considerably simplified by considering only the leading-order components. For example, the
255:
decreases and then becomes more and more negative, which terms are leading-order again changes.
804:
62:
1250:
1208:
1158:
1105:
1052:
999:
954:
903:
884:
Horowitz, G. T.; Tseytlin, A. A. (1994). "Extremal black holes as exact string solutions".
850:
758:
709:
670:
629:
585:
837:Żenczykowski, P. (1988). "Kobayashi–Maskawa matrix from the leading-order solution of the
482:, when the accurate approximate solution in each subdomain is the leading-order solution.
8:
520:
1254:
1212:
1162:
1109:
1056:
1003:
958:
907:
854:
762:
713:
674:
633:
589:
422:. The leading-order behaviour is more complicated when more terms are leading-order. At
1324:
1193:
1174:
1148:
1121:
1095:
1068:
1042:
1015:
989:
927:
893:
782:
748:
614:
499:
304:
to the original equation. Analysing the behaviour given by this new equation gives the
281:
58:
50:
1262:
1328:
1220:
1117:
1072:
1011:
966:
919:
866:
774:
1125:
1019:
931:
544:
461:(NLO) terms or corrections. The next set of terms down after that can be called the
1319:
1314:
1258:
1216:
1178:
1166:
1113:
1060:
1007:
962:
911:
858:
819:
800:
786:
770:
766:
717:
678:
637:
593:
698:"The role of surface tension in the dominant balance in the die swell singularity"
258:
There is no strict cut-off for when two terms should or should not be regarded as
1194:"Vortex motion in the spatially inhomogeneous conservative Ginzburg–Landau model"
351:
Suppose we want to understand the leading-order behaviour of the example above.
915:
1170:
1064:
505:
642:
426:
there is a leading-order balance between the cubic and linear dependencies of
1341:
263:
1301:
Kaiser, Bryan E.; Saenz, Juan A.; Sonnewald, Maike; Livescu, Daniel (2022).
862:
923:
778:
598:
571:
54:
870:
495:
478:
Leading-order simplification techniques are used in conjunction with the
61:. The sizes of the different terms in the equation(s) will change as the
39:
823:
572:"A model of carbon dioxide dissolution and mineral carbonation kinetics"
1153:
1100:
1047:
898:
371: = 0.1. Thus the leading-order behaviour of this equation at
994:
27:
Terms in a mathematical expression with the largest order of magnitude
721:
682:
659:"Onset of Superconductivity in Decreasing Fields for General Domains"
296:, and creating a new equation just involving these terms is known as
311:
615:"A multi-scale model for solute transport in a wavy-walled channel"
42:
753:
65:
change, and hence, which terms are leading-order may also change.
98: + 0.1. (Leading-order terms highlighted in pink.)
506:
Simplification of differential equations by machine learning
1300:
327: + 0.1. The leading order, or main, behaviour at
979:
485:
1303:"Automated identification of dominant physical processes"
490:
For particular fluid flow scenarios, the (very general)
398:. Thus the leading-order behaviour of this equation at
541:
Asymptotic
Analysis and Singular Perturbation Theory
300:. The solutions to this new equation are called the
1307:
243:increases further, the leading-order terms stay as
545:http://www.math.ucdavis.edu/~hunter/notes/asy.pdf
1339:
1191:
883:
656:
523:, an algebraic generalization of "leading order"
473:
511:angiogenesis, and synthetic data applications.
235: + 0.1. For five different values of
1138:
1085:
1032:
836:
498:equations. Also, the thin film equations of
830:
262:the same order, or magnitude. One possible
1185:
973:
799:
565:
563:
270:
1318:
1152:
1132:
1099:
1046:
993:
897:
793:
752:
641:
597:
418:may thus be investigated at any value of
1026:
877:
738:
612:
569:
440:
310:
1233:
1227:
1192:Rubinstein, B.Y.; Pismen, L.M. (1994).
944:
938:
732:
695:
606:
560:
486:Simplifying the Navier–Stokes equations
480:method of matched asymptotic expansions
14:
1340:
1079:
689:
657:Sternberg, P.; Bernoff, A. J. (1998).
650:
1236:"Dynamics of optical vortex solitons"
613:Woollard, H. F.; et al. (2008).
570:Mitchell, M. J.; et al. (2010).
533:
1234:Kivshar, Y.S.; et al. (1998).
696:Salamon, T.R.; et al. (1995).
298:taking an equation to leading-order
24:
622:Journal of Engineering Mathematics
577:Proceedings of the Royal Society A
25:
1364:
86:Sizes of the individual terms in
80:
1294:
1283:
663:Journal of Mathematical Physics
276:leading-order terms, and other
1320:10.1016/j.engappai.2022.105496
1201:Physica D: Nonlinear Phenomena
771:10.1103/PhysRevLett.100.093005
549:
13:
1:
1263:10.1016/S0030-4018(98)00149-7
982:Classical and Quantum Gravity
841:-generation Fritzsch model".
812:Journal of Physical Chemistry
527:
474:Matched asymptotic expansions
465:(NNLO) terms or corrections.
463:next-to-next-to-leading order
1221:10.1016/0167-2789(94)00119-7
1118:10.1016/0370-2693(96)00425-X
967:10.1016/0550-3213(80)90411-3
7:
916:10.1103/PhysRevLett.73.3351
514:
331: = 0.001 is that
10:
1369:
1171:10.1103/PhysRevD.68.114014
1065:10.1103/PhysRevD.65.113007
1012:10.1088/0264-9381/11/9/013
643:10.1007/s10665-008-9239-x
406:increases cubically with
343:increases cubically with
1290:Cornell University notes
468:
359: = 0.001, the
339: = 10 is that
886:Physical Review Letters
863:10.1103/PhysRevD.38.332
741:Physical Review Letters
492:Navier–Stokes equations
453:completely constant at
306:leading-order behaviour
302:leading-order solutions
271:Leading-order behaviour
75:leading-order behaviour
803:; et al. (1994).
599:10.1098/rspa.2009.0349
414:The main behaviour of
386: = 10, the 5
348:
286:leading-order equation
223:Consider the equation
1243:Optics Communications
459:next-to-leading order
441:Next-to-leading order
314:
290:leading-order balance
335:is constant, and at
1353:Asymptotic analysis
1348:Orders of magnitude
1255:1998OptCo.152..198K
1213:1994PhyD...78....1R
1163:2003PhRvD..68k4014K
1110:1996PhLB..378..287C
1057:2002PhRvD..65k3007C
1004:1994CQGra..11.2317K
959:1980NuPhB.163..453A
908:1994PhRvL..73.3351H
855:1988PhRvD..38..332Z
824:10.1021/j100064a020
763:2008PhRvL.100i3005G
714:1995PhFl....7.2328S
675:1998JMP....39.1272B
634:2009JEnMa..64...25W
590:2010RSPSA.466.1265M
584:(2117): 1265–1290.
99:
71:leading-order terms
32:leading-order terms
500:lubrication theory
349:
282:significant figure
85:
59:order of magnitude
1141:Physical Review D
1088:Physics Letters B
1035:Physical Review D
947:Nuclear Physics B
892:(25): 3351–3354.
843:Physical Review D
818:(13): 3389–3397.
708:(10): 2328–2344.
702:Physics of Fluids
221:
220:
57:with the largest
16:(Redirected from
1360:
1333:
1332:
1322:
1298:
1292:
1287:
1281:
1280:
1278:
1277:
1271:
1265:. Archived from
1240:
1231:
1225:
1224:
1198:
1189:
1183:
1182:
1156:
1136:
1130:
1129:
1103:
1083:
1077:
1076:
1050:
1030:
1024:
1023:
997:
988:(9): 2317–2329.
977:
971:
970:
942:
936:
935:
901:
881:
875:
874:
834:
828:
827:
809:
797:
791:
790:
756:
736:
730:
729:
724:. Archived from
722:10.1063/1.868746
693:
687:
686:
683:10.1063/1.532379
669:(3): 1272–1284.
654:
648:
647:
645:
619:
610:
604:
603:
601:
567:
558:
556:NYU course notes
553:
547:
537:
382:Similarly, when
294:dominant balance
100:
84:
21:
1368:
1367:
1363:
1362:
1361:
1359:
1358:
1357:
1338:
1337:
1336:
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1284:
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1232:
1228:
1196:
1190:
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1137:
1133:
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1080:
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1027:
978:
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943:
939:
882:
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798:
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733:
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651:
617:
611:
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568:
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550:
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476:
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443:
273:
83:
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23:
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15:
12:
11:
5:
1366:
1356:
1355:
1350:
1335:
1334:
1293:
1282:
1249:(1): 198–206.
1226:
1184:
1154:hep-ph/0308222
1147:(11): 114014.
1131:
1101:hep-ph/9602277
1094:(1): 287–301.
1078:
1048:hep-ph/0202176
1041:(11): 113007.
1025:
972:
937:
899:hep-th/9408040
876:
849:(1): 332–336.
829:
801:Lindenberg, K.
792:
731:
728:on 2013-07-08.
688:
649:
605:
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94: + 5
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79:
77:of the model.
26:
9:
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1365:
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1351:
1349:
1346:
1345:
1343:
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1312:
1308:
1304:
1297:
1291:
1286:
1272:on 2013-04-21
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1021:
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995:gr-qc/9403024
991:
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394: =
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319: =
318:
313:
309:
307:
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283:
279:
268:
265:
264:rule of thumb
261:
260:approximately
256:
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246:
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227: =
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101:
97:
93:
90: =
89:
81:Basic example
78:
76:
72:
66:
64:
60:
56:
52:
48:
44:
41:
37:
33:
19:
18:Leading-order
1310:
1306:
1296:
1285:
1274:. Retrieved
1267:the original
1246:
1242:
1229:
1204:
1200:
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1140:
1134:
1091:
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747:(9): 93005.
744:
740:
734:
726:the original
705:
701:
691:
666:
662:
652:
628:(1): 25–48.
625:
621:
608:
581:
575:
551:
540:
539:J.K.Hunter,
535:
509:
489:
477:
462:
458:
454:
450:
446:
444:
436:
431:
427:
423:
419:
415:
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407:
403:
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395:
391:
387:
383:
379:is constant.
376:
372:
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328:
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40:mathematical
35:
31:
29:
1207:(1): 1–10.
953:: 453–460.
496:Stokes flow
445:Of course,
278:lower-order
205:0.105000001
134:0.000000001
38:) within a
36:corrections
1342:Categories
1313:: 105496.
1276:2012-10-31
528:References
47:expression
1329:252957864
1073:119355645
754:0706.3879
521:Valuation
315:Graph of
251:, but as
63:variables
1126:15422325
1020:13979794
932:43551044
924:10057359
779:18352706
543:, 2004.
515:See also
451:actually
402:is that
375:is that
53:are the
43:equation
1251:Bibcode
1209:Bibcode
1179:5943465
1159:Bibcode
1106:Bibcode
1053:Bibcode
1000:Bibcode
955:Bibcode
904:Bibcode
871:9959017
851:Bibcode
787:3789664
759:Bibcode
710:Bibcode
671:Bibcode
630:Bibcode
586:Bibcode
449:is not
373:x=0.001
217:1050.1
1327:
1177:
1124:
1071:
1018:
930:
922:
869:
785:
777:
1325:S2CID
1270:(PDF)
1239:(PDF)
1197:(PDF)
1175:S2CID
1149:arXiv
1122:S2CID
1096:arXiv
1069:S2CID
1043:arXiv
1016:S2CID
990:arXiv
928:S2CID
894:arXiv
808:(PDF)
783:S2CID
749:arXiv
618:(PDF)
469:Usage
363:and 5
355:When
292:, or
288:, or
211:2.725
208:0.601
159:0.005
146:1000
140:0.125
137:0.001
110:0.001
55:terms
51:model
920:PMID
867:PMID
775:PMID
400:x=10
247:and
214:18.1
193:0.1
34:(or
30:The
1315:doi
1311:116
1259:doi
1247:152
1217:doi
1167:doi
1114:doi
1092:378
1061:doi
1008:doi
963:doi
951:163
912:doi
859:doi
820:doi
767:doi
745:100
718:doi
679:doi
638:doi
594:doi
582:466
430:on
424:x=2
190:0.1
187:0.1
184:0.1
181:0.1
177:0.1
171:50
165:2.5
162:0.5
122:10
116:0.5
113:0.1
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