256:
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.
446: = 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
301:
58:(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
264:
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
57:
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
255:
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
499:
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
269:
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
379:
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
297:
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.
356:
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
273:
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
426:
Note that this description of finding leading-order balances and behaviours gives only an outline description of the process – it is not mathematically rigorous.
1224:
686:
468:
62:
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
565:
1075:
Catani, S.; Seymour, M.H. (1996). "The Dipole
Formalism for the Calculation of QCD Jet Cross Sections at Next-to-Leading Order".
1022:
Campbell, J.; Ellis, R.K. (2002). "Next-to-leading order corrections to W + 2 jet and Z + 2 jet production at hadron colliders".
647:
1278:
1341:
1336:
480:
1128:
Kidonakis, N.; Vogt, R. (2003). "Next-to-next-to-leading order soft-gluon corrections in top quark hadroproduction".
969:
Kruczenski, M.; Oxman, L.E.; Zaldarriaga, M. (1999). "Large squeezing behaviour of cosmological entropy generation".
35:
794:"Diffusion-Limited Binary Reactions: The Hierarchy of Nonclassical Regimes for Correlated Initial Conditions"
1255:
51:
728:
Gorshkov, A. V.; et al. (2008). "Coherent
Quantum Optical Control with Subwavelength Resolution".
714:
1291:
544:
228:, the table shows the sizes of the four terms in this equation, and which terms are leading-order. As
1239:
1197:
1147:
1094:
1041:
988:
943:
892:
839:
747:
698:
659:
618:
574:
509:
934:
HĂĽseyin, A. (1980). "The leading-order behaviour of the two-photon scattering amplitudes in QCD".
483:
may be considerably simplified by considering only the leading-order components. For example, the
1313:
1163:
1137:
1110:
1084:
1057:
1031:
1004:
978:
916:
882:
771:
737:
488:
270:
47:
39:
244:
decreases and then becomes more and more negative, which terms are leading-order again changes.
793:
908:
855:
763:
1303:
1247:
1205:
1155:
1102:
1049:
996:
951:
900:
847:
808:
789:
755:
706:
667:
626:
582:
873:
Horowitz, G. T.; Tseytlin, A. A. (1994). "Extremal black holes as exact string solutions".
826:Żenczykowski, P. (1988). "Kobayashi–Maskawa matrix from the leading-order solution of the
471:, when the accurate approximate solution in each subdomain is the leading-order solution.
1243:
1201:
1151:
1098:
1045:
992:
947:
896:
843:
751:
702:
663:
622:
578:
411:. The leading-order behaviour is more complicated when more terms are leading-order. At
1182:
603:
293:
to the original equation. Analysing the behaviour given by this new equation gives the
1251:
1330:
1317:
1209:
1106:
1061:
1000:
955:
252:
1114:
1008:
920:
533:
450:(NLO) terms or corrections. The next set of terms down after that can be called the
1308:
1167:
775:
759:
43:
687:"The role of surface tension in the dominant balance in the die swell singularity"
247:
There is no strict cut-off for when two terms should or should not be regarded as
1183:"Vortex motion in the spatially inhomogeneous conservative Ginzburg–Landau model"
340:
Suppose we want to understand the leading-order behaviour of the example above.
904:
484:
300:
28:
1159:
1053:
631:
415:
there is a leading-order balance between the cubic and linear dependencies of
1290:
Kaiser, Bryan E.; Saenz, Juan A.; Sonnewald, Maike; Livescu, Daniel (2022).
851:
912:
767:
587:
560:
859:
467:
Leading-order simplification techniques are used in conjunction with the
50:. The sizes of the different terms in the equation(s) will change as the
31:
812:
561:"A model of carbon dioxide dissolution and mineral carbonation kinetics"
1142:
1089:
1036:
887:
360: = 0.1. Thus the leading-order behaviour of this equation at
983:
16:
Terms in a mathematical expression with the largest order of magnitude
710:
671:
648:"Onset of Superconductivity in Decreasing Fields for General Domains"
285:, and creating a new equation just involving these terms is known as
604:"A multi-scale model for solute transport in a wavy-walled channel"
742:
299:
54:
change, and hence, which terms are leading-order may also change.
87: + 0.1. (Leading-order terms highlighted in pink.)
495:
Simplification of differential equations by machine learning
316: + 0.1. The leading order, or main, behaviour at
1292:"Automated identification of dominant physical processes"
479:
For particular fluid flow scenarios, the (very general)
387:. Thus the leading-order behaviour of this equation at
530:
Asymptotic
Analysis and Singular Perturbation Theory
289:. The solutions to this new equation are called the
1296:
232:increases further, the leading-order terms stay as
534:http://www.math.ucdavis.edu/~hunter/notes/asy.pdf
512:, an algebraic generalization of "leading order"
500:angiogenesis, and synthetic data applications.
224: + 0.1. For five different values of
8:
487:equations. Also, the thin film equations of
251:the same order, or magnitude. One possible
554:
552:
1307:
1141:
1088:
1035:
982:
886:
741:
630:
586:
407:may thus be investigated at any value of
73:
1181:Rubinstein, B.Y.; Pismen, L.M. (1994).
521:
475:Simplifying the Navier–Stokes equations
469:method of matched asymptotic expansions
646:Sternberg, P.; Bernoff, A. J. (1998).
1225:"Dynamics of optical vortex solitons"
602:Woollard, H. F.; et al. (2008).
559:Mitchell, M. J.; et al. (2010).
7:
1223:Kivshar, Y.S.; et al. (1998).
685:Salamon, T.R.; et al. (1995).
287:taking an equation to leading-order
611:Journal of Engineering Mathematics
566:Proceedings of the Royal Society A
14:
75:Sizes of the individual terms in
652:Journal of Mathematical Physics
265:leading-order terms, and other
1309:10.1016/j.engappai.2022.105496
1190:Physica D: Nonlinear Phenomena
760:10.1103/PhysRevLett.100.093005
1:
1252:10.1016/S0030-4018(98)00149-7
971:Classical and Quantum Gravity
830:-generation Fritzsch model".
801:Journal of Physical Chemistry
463:Matched asymptotic expansions
454:(NNLO) terms or corrections.
452:next-to-next-to-leading order
1210:10.1016/0167-2789(94)00119-7
1107:10.1016/0370-2693(96)00425-X
956:10.1016/0550-3213(80)90411-3
905:10.1103/PhysRevLett.73.3351
320: = 0.001 is that
1358:
1160:10.1103/PhysRevD.68.114014
1054:10.1103/PhysRevD.65.113007
1001:10.1088/0264-9381/11/9/013
632:10.1007/s10665-008-9239-x
395:increases cubically with
332:increases cubically with
1279:Cornell University notes
348: = 0.001, the
328: = 10 is that
875:Physical Review Letters
852:10.1103/PhysRevD.38.332
730:Physical Review Letters
481:Navier–Stokes equations
442:completely constant at
295:leading-order behaviour
291:leading-order solutions
260:Leading-order behaviour
64:leading-order behaviour
792:; et al. (1994).
588:10.1098/rspa.2009.0349
403:The main behaviour of
375: = 10, the 5
337:
275:leading-order equation
212:Consider the equation
1232:Optics Communications
448:next-to-leading order
430:Next-to-leading order
303:
279:leading-order balance
324:is constant, and at
1342:Asymptotic analysis
1337:Orders of magnitude
1244:1998OptCo.152..198K
1202:1994PhyD...78....1R
1152:2003PhRvD..68k4014K
1099:1996PhLB..378..287C
1046:2002PhRvD..65k3007C
993:1994CQGra..11.2317K
948:1980NuPhB.163..453A
897:1994PhRvL..73.3351H
844:1988PhRvD..38..332Z
813:10.1021/j100064a020
752:2008PhRvL.100i3005G
703:1995PhFl....7.2328S
664:1998JMP....39.1272B
623:2009JEnMa..64...25W
579:2010RSPSA.466.1265M
573:(2117): 1265–1290.
88:
60:leading-order terms
21:leading-order terms
489:lubrication theory
338:
271:significant figure
74:
48:order of magnitude
1130:Physical Review D
1077:Physics Letters B
1024:Physical Review D
936:Nuclear Physics B
881:(25): 3351–3354.
832:Physical Review D
807:(13): 3389–3397.
697:(10): 2328–2344.
691:Physics of Fluids
210:
209:
46:with the largest
1349:
1322:
1321:
1311:
1287:
1281:
1276:
1270:
1269:
1267:
1266:
1260:
1254:. Archived from
1229:
1220:
1214:
1213:
1187:
1178:
1172:
1171:
1145:
1125:
1119:
1118:
1092:
1072:
1066:
1065:
1039:
1019:
1013:
1012:
986:
977:(9): 2317–2329.
966:
960:
959:
931:
925:
924:
890:
870:
864:
863:
823:
817:
816:
798:
786:
780:
779:
745:
725:
719:
718:
713:. Archived from
711:10.1063/1.868746
682:
676:
675:
672:10.1063/1.532379
658:(3): 1272–1284.
643:
637:
636:
634:
608:
599:
593:
592:
590:
556:
547:
545:NYU course notes
542:
536:
526:
371:Similarly, when
283:dominant balance
89:
1357:
1356:
1352:
1351:
1350:
1348:
1347:
1346:
1327:
1326:
1325:
1289:
1288:
1284:
1277:
1273:
1264:
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1227:
1222:
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1217:
1185:
1180:
1179:
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1127:
1126:
1122:
1074:
1073:
1069:
1021:
1020:
1016:
968:
967:
963:
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932:
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872:
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825:
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788:
787:
783:
727:
726:
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684:
683:
679:
645:
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558:
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543:
539:
527:
523:
519:
506:
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477:
465:
460:
432:
262:
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17:
12:
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1355:
1353:
1345:
1344:
1339:
1329:
1328:
1324:
1323:
1282:
1271:
1238:(1): 198–206.
1215:
1173:
1143:hep-ph/0308222
1136:(11): 114014.
1120:
1090:hep-ph/9602277
1083:(1): 287–301.
1067:
1037:hep-ph/0202176
1030:(11): 113007.
1014:
961:
926:
888:hep-th/9408040
865:
838:(1): 332–336.
818:
790:Lindenberg, K.
781:
720:
717:on 2013-07-08.
677:
638:
594:
548:
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520:
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312: + 5
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83: + 5
71:
68:
66:of the model.
15:
13:
10:
9:
6:
4:
3:
2:
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1343:
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1319:
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1286:
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1280:
1275:
1272:
1261:on 2013-04-21
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984:gr-qc/9403024
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268:
259:
257:
254:
253:rule of thumb
250:
249:approximately
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216: =
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205:
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199:
196:
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107:
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98:
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95:
91:
90:
86:
82:
79: =
78:
70:Basic example
69:
67:
65:
61:
55:
53:
49:
45:
41:
37:
33:
30:
26:
22:
1299:
1295:
1285:
1274:
1263:. Retrieved
1256:the original
1235:
1231:
1218:
1193:
1189:
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1129:
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736:(9): 93005.
733:
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723:
715:the original
694:
690:
680:
655:
651:
641:
617:(1): 25–48.
614:
610:
597:
570:
564:
540:
529:
528:J.K.Hunter,
524:
498:
478:
466:
451:
447:
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433:
425:
420:
416:
412:
408:
404:
402:
396:
392:
388:
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380:
376:
372:
368:is constant.
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59:
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29:mathematical
24:
20:
18:
1196:(1): 1–10.
942:: 453–460.
485:Stokes flow
434:Of course,
267:lower-order
194:0.105000001
123:0.000000001
27:) within a
25:corrections
1331:Categories
1302:: 105496.
1265:2012-10-31
517:References
36:expression
1318:252957864
1062:119355645
743:0706.3879
510:Valuation
304:Graph of
240:, but as
52:variables
1115:15422325
1009:13979794
921:43551044
913:10057359
768:18352706
532:, 2004.
504:See also
440:actually
391:is that
364:is that
42:are the
32:equation
1240:Bibcode
1198:Bibcode
1168:5943465
1148:Bibcode
1095:Bibcode
1042:Bibcode
989:Bibcode
944:Bibcode
893:Bibcode
860:9959017
840:Bibcode
776:3789664
748:Bibcode
699:Bibcode
660:Bibcode
619:Bibcode
575:Bibcode
438:is not
362:x=0.001
206:1050.1
1316:
1166:
1113:
1060:
1007:
919:
911:
858:
774:
766:
1314:S2CID
1259:(PDF)
1228:(PDF)
1186:(PDF)
1164:S2CID
1138:arXiv
1111:S2CID
1085:arXiv
1058:S2CID
1032:arXiv
1005:S2CID
979:arXiv
917:S2CID
883:arXiv
797:(PDF)
772:S2CID
738:arXiv
607:(PDF)
458:Usage
352:and 5
344:When
281:, or
277:, or
200:2.725
197:0.601
148:0.005
135:1000
129:0.125
126:0.001
99:0.001
44:terms
40:model
909:PMID
856:PMID
764:PMID
389:x=10
236:and
203:18.1
182:0.1
23:(or
19:The
1304:doi
1300:116
1248:doi
1236:152
1206:doi
1156:doi
1103:doi
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1050:doi
997:doi
952:doi
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809:doi
756:doi
734:100
707:doi
668:doi
627:doi
583:doi
571:466
419:on
413:x=2
179:0.1
176:0.1
173:0.1
170:0.1
166:0.1
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154:2.5
151:0.5
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