135:
3336:
936:
1613:
6630:
3595:
6065:
In any given model, there is usually a finite-dimensional space of complex coupling constants. The complex
Butcher group acts by diffeomorphisms on this space. In particular the renormalization group defines a flow on the space of coupling constants, with the beta function giving the corresponding
1373:
6344:
3008:
676:
2997:
7035:"Algèbre de Hopf des diagrammes de Feynman, renormalisation et factorisation de Wiener-Hopf (d'après A. Connes et D. Kreimer). [Hopf algebra of Feynman diagrams, renormalization and Wiener-Hopf factorization (following A. Connes and D. Kreimer)]"
6768:
4064:
1391:
6367:
2094:
4163:
5958:
3912:
2208:
1186:
4488:
3353:
6073:
with vertices decorated by symbols from a finite index set. Connes and
Kreimer have also defined Hopf algebras in this setting and have shown how they can be used to systematize standard computations in renormalization theory.
4350:
3767:
showed that the homomorphisms defined by the Runge–Kutta method form a dense subgroup of the
Butcher group: in fact he showed that, given a homomorphism φ', there is a Runge–Kutta homomorphism φ agreeing with φ' to order
5068:
1842:
5595:
1047:
1209:
5447:
5226:
612:
6118:
3331:{\displaystyle X_{i}(s)=x_{0}+\sum _{t}{s^{|t|} \over |t|!}\alpha (t)t!\sum _{j=1}^{m}a_{ij}\varphi _{j}(t)\delta _{t}(0),\,\,\,\,x(s)=x_{0}+\sum _{t}{s^{|t|} \over |t|!}\alpha (t)t!\varphi (t)\delta _{t}(0),}
5487:
For the minimal subtraction scheme, this process can be interpreted in terms of
Birkhoff factorization in the complex Butcher group. Φ can be regarded as a map γ of the unit circle into the complexification
4592:
3686:
296:
2775:
2664:
2416:
931:{\displaystyle \delta _{\bullet }^{i}=f^{i},\,\,\,\delta _{}^{i}=\sum _{j_{1},\dots ,j_{n}=1}^{N}(\delta _{t_{1}}^{j_{1}}\cdots \delta _{t_{n}}^{j_{n}})\partial _{j_{1}}\cdots \partial _{j_{n}}f^{i}.}
5141:
5724:
6053:
4917:
3819:
1742:
5285:
2271:
454:
125:
We regard
Butcher’s work on the classification of numerical integration methods as an impressive example that concrete problem-oriented work can lead to far-reaching conceptual results.
2789:
5808:
3750:
6858:
6816:
5645:
4698:
4674:
4626:
4514:
4246:
4222:
6641:
1608:{\displaystyle x^{(4)}=f^{\prime \prime \prime }(f,f,f)+3f^{\prime \prime }(f,f^{\prime }(f))+f^{\prime }(f^{\prime \prime }(f,f))+f^{\prime }(f^{\prime }(f^{\prime }(f))),}
5482:
5364:
5324:
6625:{\displaystyle \displaystyle \int {(|y|^{2})^{-u} \over |y|^{2}+q_{\mu }^{2}}\,d^{D}y=\pi ^{D/2}(q_{\mu }^{2})^{-z-u}{\Gamma (-u+D/2)\Gamma (1+u-D/2) \over \Gamma (D/2)}.}
3930:
489:
2476:
2521:
1951:
4075:
3590:{\displaystyle \varphi _{j}(\bullet )=1.\,\,\,\varphi _{i}()=\sum _{j_{1},\dots ,j_{k}}a_{ij_{1}}\dots a_{ij_{k}}\varphi _{j_{1}}(t_{1})\dots \varphi _{j_{k}}(t_{k})}
5848:
3827:
2117:
1058:
4365:
1885:
4266:
4973:
1751:
6932:
Jackson, K. R.; Kværnø, A.; Nørsett, S.P. (1994), "The use of
Butcher series in the analysis of Newton-like iterations in Runge–Kutta formulas",
5515:
1368:{\displaystyle x^{(4)}=f^{\prime \prime \prime }f^{3}+3f^{\prime \prime }f^{\prime }f^{2}+f^{\prime }f^{\prime \prime }f^{2}+(f^{\prime })^{3}f,}
947:
7546:
7249:
6339:{\displaystyle \displaystyle \Phi ()=\int {\Phi (t_{1})\cdots \Phi (t_{n}) \over |y|^{2}+q_{\mu }^{2}}(|y|^{2})^{-z({c \over 2}-1)}\,d^{D}y,}
5372:
5156:
2279:
Using complex coefficients in the construction of the Hopf algebra of rooted trees one obtains the complex Hopf algebra of rooted trees. Its
7486:"Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group"
512:
7777:
John C. Butcher: "B-Series : Algebraic
Analysis of Numerical Methods", Springer（SSCM, volume 55), ISBN 978-3030709556 (April, 2021).
7426:"Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem"
4528:
3606:
214:
2675:
2564:
2317:
7389:
7805:
44:
5091:
3699:. The corresponding assignment φ is an element of the Butcher group. The homomorphism corresponding to the actual flow has
176:= can be constructed by joining the roots of the trees to a new common root. The number of nodes in a tree is denoted by |
7800:
4249:
5680:
4197:
114:
5994:
4728:
1382:
76:
7795:
4809:
3791:
1709:
5237:
6086:
4964:
2219:
324:
2992:{\displaystyle X_{i}(s)=x_{0}+s\sum _{j=1}^{m}a_{ij}f(X_{j}(s)),\,\,\,x(s)=x_{0}+s\sum _{j=1}^{m}b_{j}f(X_{j}(s))}
4181:
5751:
4798:
3705:
2422:
48:
7810:
7790:
7306:
7301:
7150:
6941:
154:
is removed and the nodes connected to the original node by a single bond are taken as new roots, the tree
68:
192:| to the nodes so that the numbers increase on any path going away from the root. Two heap orderings are
7742:
5973:
72:
7045:
6825:
6763:{\displaystyle \displaystyle \Phi (\bullet )=\pi ^{D/2}(q_{\mu }^{2})^{-zc/2}{\Gamma (1+cz) \over cz}.}
6780:
5621:
7722:
7677:
7628:
7510:
7450:
7353:
7076:
7006:
5605:
4724:
4679:
4655:
4607:
4495:
4227:
4203:
4059:{\displaystyle \varphi \circ f=1+\sum _{t}{s^{|t|} \over |t|!}\alpha (t)t!\varphi (t)\delta _{t}(0),}
1646:
94:
7155:
6946:
5455:
5337:
5297:
56:
2089:{\displaystyle S(t)=-t-\sum _{s\subset t}(-1)^{n(t\backslash s)}S()s,\,\,\,S(\bullet )=-\bullet .}
1649:. It was later discovered that the Hopf algebra was the dual of a Hopf algebra defined earlier by
465:
7759:
7712:
7693:
7667:
7644:
7618:
7595:
7526:
7500:
7466:
7440:
7406:
7369:
7343:
7286:
7226:
7168:
7130:
7112:
7092:
7066:
7022:
6996:
5082:
1938:
1670:
40:
4158:{\displaystyle \varphi _{1}\circ (\varphi _{2}\circ f)=(\varphi _{1}\star \varphi _{2})\circ f.}
2439:
6357:/2 is the regularization parameter. These integrals can be computed explicitly in terms of the
5953:{\displaystyle \displaystyle F_{t}=\lim _{z=0}\gamma _{-}(z)\lambda _{tz}(\gamma _{-}(z)^{-1})}
3907:{\displaystyle {\begin{pmatrix}A&0\\0&A^{\prime }\\\end{pmatrix}},\,\,(b,b^{\prime }).}
2487:
2203:{\displaystyle \varphi _{1}\star \varphi _{2}(t)=(\varphi _{1}\otimes \varphi _{2})\Delta (t).}
1378:
where the four terms correspond to the four rooted trees from left to right in Figure 3 above.
1181:{\displaystyle \displaystyle x(s)=x_{0}+\sum _{t}{s^{|t|} \over |t|!}\alpha (t)\delta _{t}(0).}
7542:
7245:
4483:{\displaystyle (t)=(\theta _{1}\otimes \theta _{2}-\theta _{2}\otimes \theta _{1})\Delta (t).}
201:
150:, in which every other node is connected to the root by a unique path. If the root of a tree
7751:
7737:
7685:
7636:
7587:
7565:
7518:
7458:
7398:
7361:
7278:
7216:
7191:
7160:
7122:
7084:
7034:
7014:
6951:
4636:
of the
Butcher group as an infinite-dimensional Lie group are not the same. The Lie algebra
4180:
is an infinite-dimensional Lie algebra. The existence of this Lie algebra is predicted by a
1704:
113:, is essentially equivalent to the Butcher group, since its dual can be identified with the
7771:
7709:
Factorization in
Quantum Field Theory: An Exercise in Hopf Algebras and Local Singularities
7259:
7767:
7266:
7255:
7237:
7204:
7179:
6070:
4731:
of loops in the character group of the associated Hopf algebra. The models considered by
4720:
1688:
1642:
102:
90:
28:
7164:
7726:
7681:
7632:
7514:
7454:
7357:
7080:
7010:
3759:
th order approximation of the actual flow provided that φ and Φ agree on all trees with
7578:
Hairer, E.; Wanner, G. (1974), "On the
Butcher group and general multi-value methods",
6358:
5613:
4771:
311:
7103:
Bogfjellmo, G.; Schmeding, A. (2015), "The Lie group structure of the Butcher group",
1858:
1385:
of 1855; however in several variables it has to be written more carefully in the form
7784:
7570:
7410:
7387:(1999), "Lessons from quantum field theory: Hopf algebras and spacetime geometries",
7297:
6955:
6059:
106:
7711:, Frontiers in Number Theory, Physics, and Geometry II, Springer, pp. 715–736,
7599:
7530:
7470:
7373:
7290:
7134:
7096:
7026:
134:
7704:
7697:
7655:
7648:
7606:
7556:
Grossman, R.; Larson, R. (1989), "Hopf algebraic structures of families of trees",
7481:
7477:
7421:
7417:
7384:
7380:
7324:
7320:
7172:
6984:
6069:
More general models in quantum field theory require rooted trees to be replaced by
4939:
4747:
has given an account of this renormalization process in terms of Runge–Kutta data.
4716:
1666:
1638:
1627:
495:
197:
143:
98:
86:
60:
5670:
In example, the Feynman rules depend on additional parameter μ, a "unit of mass".
2299:
is an infinite-dimensional complex Lie group which appears as a toy model in the
7733:
7609:(1998), "On the Hopf algebra structure of perturbative quantum field theories",
7141:
Brouder, Christian (2004), "Trees, Renormalization and Differential Equations",
4345:{\displaystyle \theta (ab)=\varepsilon (a)\theta (b)+\theta (a)\varepsilon (b),}
666:) in terms of rooted trees. His formula can be conveniently expressed using the
118:
79:
can be conveniently expressed in terms of rooted trees and their combinatorics.
52:
20:
7689:
7640:
7658:(1999), "Chen's iterated integral represents the operator product expansion",
7402:
7282:
7196:
7126:
7018:
6987:(2005), "The Hopf Algebra of Rooted Trees in Epstein-Glaser Renormalization",
6819:
5063:{\displaystyle \displaystyle R(\sum _{n}a_{n}z^{n})=\sum _{n<0}a_{n}z^{n}.}
2783:
showed that the solution of the corresponding ordinary differential equations
6112:/ μ is a dimensionless constant, Feynman rules can be defined recursively by
3772:; and that if given homomorphims φ and φ' corresponding to Runge–Kutta data (
1837:{\displaystyle \Delta (t)=t\otimes I+I\otimes t+\sum _{s\subset t}s\otimes ,}
7057:
Brouder, Christian (2000), "Runge–Kutta methods and renormalization",
36:
7328:
7269:(2009), "Trees and numerical methods for ordinary differential equations",
7182:(1963), "Coefficients for the study of Runge-Kutta integration processes",
5590:{\displaystyle \displaystyle \gamma (z)=\gamma _{-}(z)^{-1}\gamma _{+}(z),}
138:
Rooted trees with two, three and four nodes, from Cayley's original article
7485:
7462:
7425:
7365:
7088:
6959:(Special issue to honor professor J. C. Butcher on his sixtieth birthday)
4640:
can be identified with the Lie algebra of all derivations in the dual of
1042:{\displaystyle {d^{m}x \over ds^{m}}=\sum _{|t|=m}\alpha (t)\delta _{t},}
7316:(also in Volume 3 of the Collected Works of Cayley, pages 242–246)
1203:
are real-valued functions of a single real variable, the formula yields
7763:
7717:
7672:
7591:
7522:
7505:
7445:
7348:
7230:
7071:
7001:
5442:{\displaystyle \Phi _{S}^{R}=-m(S\otimes \Phi _{S}^{R}\circ P)\Delta .}
5221:{\displaystyle m\circ (S\otimes {\rm {id}})\Delta (x)=\varepsilon (x)1}
7623:
607:{\displaystyle \displaystyle {dx(s) \over ds}=f(x(s)),\,\,x(0)=x_{0},}
5452:
Because of the precise form of Δ, this gives a recursive formula for
1919:
85:
pointed out that the Butcher group is the group of characters of the
7755:
7221:
5736:
The complex Butcher group comes with a natural one-parameter group λ
1657:, i.e. the homomorphisms of the underlying commutative algebra into
89:
of rooted trees that had arisen independently in their own work on
7537:
Gracia-Bondía, José; Várilly, Joseph C.; Figueroa, Héctor (2000),
7117:
5979:
Its infinitesimal generator β is an element of the Lie algebra of
1894:
formed by the rooted trees that arise on erasing all the nodes of
133:
5146:
To define the renormalized Feynman rules, note that the antipode
5655:
corresponds to the renormalized homomorphism. The evaluation at
4587:{\displaystyle \theta _{t}(t^{\prime })=\delta _{tt^{\prime }},}
3681:{\displaystyle \varphi (t)=\sum _{j=1}^{m}b_{j}\varphi _{j}(t).}
291:{\displaystyle \displaystyle \alpha (t)={|t|! \over t!|S_{t}|},}
200:
of rooted trees mapping one of them on the other. The number of
5963:
defines a one-parameter subgroup of the complex Butcher group
5334:
obtained by twisting the homomorphism Φ • S. The homomorphism
3920:
proved that the Butcher group acts naturally on the functions
59:
solutions of the differential equation modeling the flow of a
4359:
at the identity ε. This forms a Lie algebra with Lie bracket
7329:"Hopf Algebras, Renormalization and Noncommutative Geometry"
4574:
4550:
3893:
3861:
3806:
2770:{\displaystyle x_{n}=x_{n-1}+h\sum _{j=1}^{m}b_{j}f(x_{j}).}
2659:{\displaystyle X_{i}=x_{n-1}+h\sum _{j=1}^{m}a_{ij}f(X_{j})}
2041:
2018:
1868:
1822:
1582:
1569:
1556:
1525:
1522:
1509:
1484:
1465:
1462:
1425:
1422:
1419:
1344:
1318:
1315:
1305:
1282:
1272:
1269:
1243:
1240:
1237:
459:
with the tree factorial of an isolated root defined to be 1
2411:{\displaystyle {dx(s) \over ds}=f(x(s)),\,\,\,x(0)=x_{0},}
670:
introduced by Butcher. These are defined inductively by
4700:
turns out to be a (strictly smaller) Lie subalgebra of
2103:
is defined to be the set of algebra homomorphisms φ of
204:
of heap-orderings on a particular tree is denoted by α(
7207:(1972), "An algebraic theory of integration methods",
5136:{\displaystyle \displaystyle P(x)=x-\varepsilon (x)1.}
3836:
658:
gave a method to compute the higher order derivatives
7242:
Numerical methods for ordinary differential equations
6828:
6783:
6645:
6644:
6371:
6370:
6122:
6121:
5997:
5852:
5851:
5754:
5683:
5624:
5519:
5518:
5458:
5375:
5340:
5300:
5240:
5159:
5095:
5094:
4977:
4976:
4812:
4719:
methods to give a simple mathematical formulation of
4682:
4658:
4610:
4531:
4498:
4368:
4269:
4230:
4206:
4078:
3933:
3830:
3794:
3708:
3609:
3356:
3011:
2792:
2678:
2567:
2490:
2442:
2320:
2222:
2120:
1954:
1861:
1754:
1712:
1394:
1212:
1062:
1061:
950:
679:
516:
515:
494:
The ordinary differential equation for the flow of a
468:
327:
218:
217:
6777:
of minimal subtraction, the renormalized quantities
3755:
Butcher showed that the Runge–Kutta method gives an
6884:
1887:is the monomial given by the product the variables
208:) and can be computed using the Butcher's formula:
6852:
6810:
6762:
6624:
6338:
6047:
5952:
5802:
5719:{\displaystyle \partial _{\mu }\gamma _{\mu -}=0,}
5718:
5639:
5589:
5476:
5441:
5358:
5318:
5279:
5220:
5135:
5062:
4954:satisfies the Rota–Baxter identity if and only if
4911:
4692:
4668:
4620:
4586:
4508:
4482:
4344:
4240:
4216:
4157:
4058:
3906:
3813:
3744:
3680:
3589:
3330:
2991:
2769:
2658:
2515:
2470:
2410:
2265:
2202:
2088:
1879:
1836:
1736:
1607:
1367:
1180:
1041:
930:
606:
483:
448:
290:
6048:{\displaystyle \beta =\partial _{t}F_{t}|_{t=0}.}
1847:where the sum is over all proper rooted subtrees
1381:In a single variable this formula is the same as
318:and the tree factorial is defined recursively by
51:. It arose from an algebraic formalism involving
5867:
4912:{\displaystyle R(fg)+R(f)R(g)=R(fR(g))+R(R(f)g)}
3814:{\displaystyle \varphi \star \varphi ^{\prime }}
1737:{\displaystyle \Delta :H\rightarrow H\otimes H}
123:
7302:"On the theory of analytic forms called trees"
5280:{\displaystyle S=-m\circ (S\otimes P)\Delta ,}
4176:showed that associated with the Butcher group
2311:The non-linear ordinary differential equation
1650:
6905:
6903:
5671:
5608:on the interior of the closed unit disk and γ
4712:
4629:
4173:
2266:{\displaystyle \varphi ^{-1}(t)=\varphi (St)}
2213:The inverse in the Butcher group is given by
1634:
1622:Definition using Hopf algebra of rooted trees
449:{\displaystyle !=||\cdot t_{1}!\cdots t_{n}!}
82:
8:
7244:(2nd ed.), John Wiley & Sons Ltd.,
5634:
5628:
3917:
2283:-valued characters form a group, called the
32:
27:, named after the New Zealand mathematician
5663:or the renormalized homomorphism gives the
5509:). As such it has a Birkhoff factorization
4188:: the commutativity and natural grading on
4185:
2307:Butcher series and Runge–Kutta method
188:is an allocation of the numbers 1 through |
1653:in a different context. The characters of
648:is the starting point of the flow at time
7716:
7671:
7622:
7569:
7504:
7444:
7347:
7220:
7195:
7154:
7116:
7070:
7000:
6945:
6844:
6839:
6827:
6793:
6788:
6782:
6721:
6711:
6701:
6691:
6686:
6669:
6665:
6643:
6604:
6582:
6550:
6529:
6514:
6504:
6499:
6482:
6478:
6462:
6457:
6448:
6443:
6430:
6425:
6416:
6405:
6395:
6390:
6381:
6375:
6369:
6323:
6318:
6297:
6287:
6277:
6272:
6263:
6251:
6246:
6233:
6228:
6219:
6208:
6186:
6173:
6155:
6136:
6120:
6030:
6025:
6018:
6008:
5996:
5937:
5921:
5905:
5886:
5870:
5857:
5850:
5790:
5782:
5781:
5759:
5753:
5698:
5688:
5682:
5623:
5568:
5555:
5539:
5517:
5468:
5463:
5457:
5418:
5413:
5385:
5380:
5374:
5350:
5345:
5339:
5310:
5305:
5299:
5239:
5176:
5175:
5158:
5093:
5050:
5040:
5024:
5008:
4998:
4988:
4975:
4811:
4684:
4683:
4681:
4660:
4659:
4657:
4644:(i.e. the space of all linear maps from
4612:
4611:
4609:
4573:
4565:
4549:
4536:
4530:
4500:
4499:
4497:
4456:
4443:
4430:
4417:
4389:
4376:
4367:
4268:
4232:
4231:
4229:
4224:. Connes and Kreimer explicitly identify
4208:
4207:
4205:
4137:
4124:
4099:
4083:
4077:
4038:
3993:
3985:
3977:
3969:
3968:
3962:
3956:
3932:
3892:
3878:
3877:
3860:
3831:
3829:
3805:
3793:
3724:
3707:
3660:
3650:
3640:
3629:
3608:
3578:
3563:
3558:
3542:
3527:
3522:
3510:
3502:
3487:
3479:
3467:
3448:
3443:
3424:
3405:
3389:
3384:
3383:
3382:
3361:
3355:
3310:
3265:
3257:
3249:
3241:
3240:
3234:
3228:
3215:
3195:
3194:
3193:
3192:
3174:
3155:
3142:
3132:
3121:
3088:
3080:
3072:
3064:
3063:
3057:
3051:
3038:
3016:
3010:
2971:
2955:
2945:
2934:
2918:
2898:
2897:
2896:
2875:
2856:
2846:
2835:
2819:
2797:
2791:
2755:
2739:
2729:
2718:
2696:
2683:
2677:
2647:
2628:
2618:
2607:
2585:
2572:
2566:
2504:
2489:
2456:
2441:
2399:
2379:
2378:
2377:
2321:
2319:
2227:
2221:
2176:
2163:
2138:
2125:
2119:
2061:
2060:
2059:
2008:
1983:
1953:
1944:can be defined recursively by the formula
1902:. The number of such trees is denoted by
1860:
1798:
1753:
1711:
1581:
1568:
1555:
1521:
1508:
1483:
1461:
1418:
1399:
1393:
1353:
1343:
1327:
1314:
1304:
1291:
1281:
1268:
1252:
1236:
1217:
1211:
1159:
1132:
1124:
1116:
1108:
1107:
1101:
1095:
1082:
1060:
1030:
1001:
993:
992:
976:
958:
951:
949:
919:
907:
902:
887:
882:
867:
862:
855:
850:
835:
830:
823:
818:
805:
792:
773:
768:
755:
745:
726:
718:
713:
712:
711:
702:
689:
684:
678:
594:
574:
573:
517:
514:
467:
437:
421:
409:
400:
381:
369:
354:
335:
326:
276:
270:
261:
245:
237:
234:
216:
121:of the Butcher group. As they commented:
77:derivatives of a composition of functions
7105:Foundations of Computational Mathematics
5803:{\displaystyle \lambda _{w}(t)=w^{|t|}t}
5612:is holomorphic on its complement in the
4676:is obtained from the graded dual. Hence
172:, ... Reversing this process a new tree
6968:
6920:
6909:
6895:
6872:
6082:
4744:
4732:
3764:
2780:
1674:
7336:Communications in Mathematical Physics
4762:given by an algebra homomorphism Φ of
3745:{\displaystyle \Phi (t)={1 \over t!}.}
655:
146:with a distinguished node, called the
109:. This Hopf algebra, often called the
64:
5838:have the same negative part and, for
4727:. Renormalization was interpreted as
4715:provided a general context for using
4604:The infinite-dimensional Lie algebra
1618:where the tree structure is crucial.
7:
6880:
6878:
6876:
3347:and φ are determined recursively by
2425:. This iterative scheme requires an
2300:
7738:"On the structure of Hopf algebras"
7539:Elements of noncommutative geometry
4685:
4661:
4613:
4501:
4233:
4209:
2421:can be solved approximately by the
7165:10.1023/B:BITN.0000046809.66837.cc
6785:
6773:Taking the renormalization scheme
6724:
6646:
6595:
6561:
6532:
6198:
6176:
6123:
6085:has given a "toy model" involving
6005:
5742:of automorphisms, dual to that on
5685:
5631:
5460:
5433:
5410:
5377:
5342:
5302:
5271:
5188:
5180:
5177:
5073:In addition there is a projection
4962:does. An important example is the
4465:
3709:
3691:The power series above are called
2276:and the identity by the counit ε.
2185:
1755:
1713:
1052:giving the power series expansion
899:
879:
14:
6853:{\displaystyle \log q_{\mu }^{2}}
4516:is generated by the derivations θ
43:to study solutions of non-linear
7033:Boutet de Monvel, Louis (2003),
6811:{\displaystyle \Phi _{S}^{R}(t)}
5640:{\displaystyle \cup \{\infty \}}
7390:Letters in Mathematical Physics
6885:Bogfjellmo & Schmeding 2015
4693:{\displaystyle {\mathfrak {g}}}
4669:{\displaystyle {\mathfrak {g}}}
4621:{\displaystyle {\mathfrak {g}}}
4509:{\displaystyle {\mathfrak {g}}}
4241:{\displaystyle {\mathfrak {g}}}
4217:{\displaystyle {\mathfrak {g}}}
3002:has the power series expansion
1633:of rooted trees was defined by
45:ordinary differential equations
6805:
6799:
6742:
6727:
6698:
6679:
6655:
6649:
6612:
6598:
6590:
6564:
6558:
6535:
6511:
6492:
6426:
6417:
6402:
6391:
6382:
6378:
6313:
6294:
6284:
6273:
6264:
6260:
6229:
6220:
6214:
6201:
6192:
6179:
6164:
6161:
6129:
6126:
6026:
5946:
5934:
5927:
5914:
5898:
5892:
5791:
5783:
5771:
5765:
5580:
5574:
5552:
5545:
5529:
5523:
5430:
5400:
5268:
5256:
5212:
5206:
5197:
5191:
5185:
5166:
5126:
5120:
5105:
5099:
5014:
4981:
4906:
4900:
4894:
4888:
4879:
4876:
4870:
4861:
4852:
4846:
4840:
4834:
4825:
4816:
4754:has two pieces of input data:
4750:In this simplified setting, a
4555:
4542:
4474:
4468:
4462:
4410:
4404:
4398:
4395:
4369:
4336:
4330:
4324:
4318:
4309:
4303:
4297:
4291:
4282:
4273:
4143:
4117:
4111:
4092:
4050:
4044:
4031:
4025:
4013:
4007:
3994:
3986:
3978:
3970:
3898:
3879:
3718:
3712:
3672:
3666:
3619:
3613:
3584:
3571:
3548:
3535:
3433:
3430:
3398:
3395:
3373:
3367:
3322:
3316:
3303:
3297:
3285:
3279:
3266:
3258:
3250:
3242:
3205:
3199:
3186:
3180:
3167:
3161:
3108:
3102:
3089:
3081:
3073:
3065:
3028:
3022:
2986:
2983:
2977:
2964:
2908:
2902:
2890:
2887:
2881:
2868:
2809:
2803:
2761:
2748:
2653:
2640:
2510:
2497:
2465:
2449:
2389:
2383:
2371:
2368:
2362:
2356:
2336:
2330:
2260:
2251:
2242:
2236:
2194:
2188:
2182:
2156:
2150:
2144:
2071:
2065:
2050:
2047:
2035:
2032:
2024:
2012:
2005:
1995:
1964:
1958:
1874:
1862:
1828:
1816:
1764:
1758:
1722:
1599:
1596:
1593:
1587:
1574:
1561:
1545:
1542:
1530:
1514:
1498:
1495:
1489:
1470:
1448:
1430:
1406:
1400:
1350:
1336:
1224:
1218:
1171:
1165:
1152:
1146:
1133:
1125:
1117:
1109:
1072:
1066:
1023:
1017:
1002:
994:
875:
811:
751:
719:
584:
578:
567:
564:
558:
552:
532:
526:
410:
406:
374:
370:
360:
328:
277:
262:
246:
238:
228:
222:
130:Differentials and rooted trees
35:, is an infinite-dimensional
16:Infinite dimensional Lie group
1:
6934:Applied Numerical Mathematics
5667:values for each rooted tree.
5477:{\displaystyle \Phi _{S}^{R}}
5359:{\displaystyle \Phi _{S}^{R}}
5319:{\displaystyle \Phi _{S}^{R}}
4260:, i.e. linear maps such that
4196:* can be identified with the
4192:implies that the graded dual
7571:10.1016/0021-8693(89)90328-1
6956:10.1016/0168-9274(94)00031-X
5294:are given by a homomorphism
4355:the formal tangent space of
4198:universal enveloping algebra
3788:), the product homomorphism
2542:by first finding a solution
2290:. The complex Butcher group
1682:Hopf algebra of rooted trees
1651:Grossman & Larson (1989)
484:{\displaystyle \bullet !=1.}
158:breaks up into rooted trees
115:universal enveloping algebra
5672:Connes & Kreimer (2001)
4785:given by a linear operator
4778:with poles of finite order;
4713:Connes & Kreimer (1998)
4630:Connes & Kreimer (1998)
4174:Connes & Kreimer (1998)
2533:The scheme defines vectors
2303:of quantum field theories.
1699:runs through rooted trees.
1661:, form a group, called the
1635:Connes & Kreimer (1998)
83:Connes & Kreimer (1999)
75:, who first noted that the
7827:
7690:10.4310/ATMP.1999.v3.n3.a7
7641:10.4310/ATMP.1998.v2.n2.a4
6101:is a positive integer and
6087:dimensional regularization
5974:renormalization group flow
5292:renormalized Feynman rules
4965:minimal subtraction scheme
3918:Hairer & Wanner (1974)
2471:{\displaystyle A=(a_{ij})}
633:is a smooth function from
71:on change of variables in
67:, prompted by the work of
33:Hairer & Wanner (1974)
7736:; Moore, John C. (1965),
7283:10.1007/s11075-009-9285-0
7197:10.1017/S1446788700027932
7143:BIT Numerical Mathematics
7127:10.1007/s10208-015-9285-5
7019:10.1007/s00023-005-0210-3
5665:dimensionally regularized
5366:is uniquely specified by
4186:Milnor & Moore (1965)
3763:nodes or less. Moreover,
2516:{\displaystyle b=(b_{i})}
1922:is the homomorphism ε of
1898:and connected links from
3821:corresponds to the data
1669:structure discovered in
1665:. It corresponds to the
668:elementary differentials
49:Runge–Kutta method
7660:Adv. Theor. Math. Phys.
7611:Adv. Theor. Math. Phys.
7403:10.1023/A:1007523409317
2285:complex Butcher group G
7307:Philosophical Magazine
7184:J. Austral. Math. Soc.
6989:Annales Henri Poincaré
6983:Bergbauer, Christoph;
6854:
6812:
6764:
6626:
6340:
6049:
5954:
5804:
5720:
5641:
5591:
5478:
5443:
5360:
5320:
5281:
5222:
5137:
5064:
4913:
4783:renormalization scheme
4729:Birkhoff factorization
4694:
4670:
4622:
4588:
4510:
4484:
4346:
4242:
4218:
4159:
4060:
3908:
3815:
3746:
3682:
3645:
3591:
3332:
3137:
2993:
2950:
2851:
2771:
2734:
2660:
2623:
2517:
2472:
2412:
2301:§ Renormalization
2267:
2204:
2090:
1930:sending each variable
1881:
1838:
1738:
1609:
1383:Faà di Bruno's formula
1369:
1182:
1043:
932:
810:
608:
485:
450:
292:
139:
127:
111:Connes–Kreimer algebra
7806:Renormalization group
7743:Annals of Mathematics
7463:10.1007/s002200050779
7366:10.1007/s002200050499
7089:10.1007/s100529900235
6855:
6813:
6765:
6627:
6341:
6050:
5955:
5805:
5733:is independent of μ.
5721:
5642:
5592:
5479:
5444:
5361:
5321:
5282:
5223:
5138:
5065:
4914:
4743:, the Butcher group.
4695:
4671:
4623:
4597:for each rooted tree
4589:
4511:
4485:
4347:
4243:
4219:
4160:
4061:
3909:
3816:
3747:
3683:
3625:
3592:
3333:
3117:
2994:
2930:
2831:
2772:
2714:
2661:
2603:
2518:
2473:
2413:
2268:
2205:
2111:with group structure
2091:
1882:
1839:
1739:
1687:is defined to be the
1610:
1370:
1183:
1044:
933:
764:
609:
486:
451:
293:
137:
73:differential calculus
7801:Quantum field theory
7734:Milnor, John Willard
7271:Numerical Algorithms
6826:
6781:
6642:
6368:
6119:
5995:
5849:
5752:
5681:
5622:
5516:
5456:
5373:
5338:
5298:
5238:
5157:
5092:
4974:
4931:lies in the algebra
4810:
4799:Rota–Baxter identity
4752:renormalizable model
4739:and character group
4725:quantum field theory
4680:
4656:
4632:and the Lie algebra
4608:
4529:
4496:
4366:
4267:
4228:
4204:
4076:
3931:
3828:
3792:
3706:
3607:
3354:
3009:
2790:
2676:
2565:
2488:
2440:
2318:
2220:
2118:
1952:
1859:
1752:
1710:
1647:quantum field theory
1641:'s previous work on
1392:
1210:
1059:
948:
677:
513:
466:
325:
215:
95:quantum field theory
39:first introduced in
7727:2003hep.th....6020K
7682:1999hep.th....1099K
7633:1997q.alg.....7029K
7515:2001CMaPh.216..215C
7493:Commun. Math. Phys.
7455:2000CMaPh.210..249C
7433:Commun. Math. Phys.
7358:1998CMaPh.199..203C
7081:2000EPJC...12..521B
7011:2005AnHP....6..343B
6849:
6798:
6696:
6509:
6453:
6256:
5651:(∞) = 1. The loop γ
5473:
5423:
5390:
5355:
5315:
1637:in connection with
1191:As an example when
941:With this notation
874:
842:
760:
694:
202:equivalence classes
142:A rooted tree is a
57:formal power series
7796:Numerical analysis
7592:10.1007/BF02268387
7558:Journal of Algebra
7523:10.1007/PL00005547
7046:Séminaire Bourbaki
6860:when evaluated at
6850:
6835:
6808:
6784:
6760:
6759:
6682:
6622:
6621:
6495:
6439:
6361:using the formula
6336:
6335:
6242:
6045:
5988:and is defined by
5950:
5949:
5881:
5800:
5716:
5637:
5587:
5586:
5474:
5459:
5439:
5409:
5376:
5356:
5341:
5316:
5301:
5277:
5218:
5133:
5132:
5083:augmentation ideal
5060:
5059:
5035:
4993:
4909:
4690:
4666:
4618:
4584:
4506:
4480:
4342:
4238:
4214:
4155:
4056:
3961:
3924:. Indeed, setting
3904:
3868:
3811:
3742:
3678:
3587:
3474:
3328:
3233:
3056:
2989:
2767:
2656:
2513:
2468:
2423:Runge–Kutta method
2408:
2263:
2200:
2086:
1994:
1877:
1834:
1809:
1734:
1671:numerical analysis
1605:
1365:
1178:
1177:
1100:
1039:
1013:
928:
846:
814:
714:
680:
625:) takes values in
604:
603:
498:on an open subset
481:
446:
288:
287:
140:
41:numerical analysis
7746:, Second Series,
7548:978-0-8176-4124-5
7251:978-0-470-72335-7
6754:
6616:
6455:
6305:
6258:
6058:It is called the
5866:
5020:
4984:
4923:and the image of
4766:into the algebra
4735:had Hopf algebra
4200:of a Lie algebra
4069:they proved that
4002:
3952:
3737:
3439:
3274:
3224:
3097:
3047:
2669:and then setting
2348:
1979:
1794:
1691:in the variables
1141:
1091:
988:
983:
544:
282:
196:, if there is an
184:of a rooted tree
7818:
7774:
7729:
7720:
7700:
7675:
7651:
7626:
7602:
7574:
7573:
7551:
7533:
7508:
7490:
7473:
7448:
7430:
7413:
7376:
7351:
7333:
7315:
7293:
7277:(2–3): 153–170,
7262:
7238:Butcher, John C.
7233:
7224:
7200:
7199:
7175:
7158:
7137:
7120:
7099:
7074:
7053:
7039:
7029:
7004:
6971:
6966:
6960:
6958:
6949:
6929:
6923:
6918:
6912:
6907:
6898:
6893:
6887:
6882:
6859:
6857:
6856:
6851:
6848:
6843:
6817:
6815:
6814:
6809:
6797:
6792:
6769:
6767:
6766:
6761:
6755:
6753:
6745:
6722:
6720:
6719:
6715:
6695:
6690:
6678:
6677:
6673:
6631:
6629:
6628:
6623:
6617:
6615:
6608:
6593:
6586:
6554:
6530:
6528:
6527:
6508:
6503:
6491:
6490:
6486:
6467:
6466:
6456:
6454:
6452:
6447:
6435:
6434:
6429:
6420:
6414:
6413:
6412:
6400:
6399:
6394:
6385:
6376:
6345:
6343:
6342:
6337:
6328:
6327:
6317:
6316:
6306:
6298:
6282:
6281:
6276:
6267:
6259:
6257:
6255:
6250:
6238:
6237:
6232:
6223:
6217:
6213:
6212:
6191:
6190:
6174:
6160:
6159:
6141:
6140:
6093:and the algebra
6071:Feynman diagrams
6054:
6052:
6051:
6046:
6041:
6040:
6029:
6023:
6022:
6013:
6012:
5959:
5957:
5956:
5951:
5945:
5944:
5926:
5925:
5913:
5912:
5891:
5890:
5880:
5862:
5861:
5809:
5807:
5806:
5801:
5796:
5795:
5794:
5786:
5764:
5763:
5725:
5723:
5722:
5717:
5706:
5705:
5693:
5692:
5646:
5644:
5643:
5638:
5596:
5594:
5593:
5588:
5573:
5572:
5563:
5562:
5544:
5543:
5483:
5481:
5480:
5475:
5472:
5467:
5448:
5446:
5445:
5440:
5422:
5417:
5389:
5384:
5365:
5363:
5362:
5357:
5354:
5349:
5325:
5323:
5322:
5317:
5314:
5309:
5286:
5284:
5283:
5278:
5227:
5225:
5224:
5219:
5184:
5183:
5142:
5140:
5139:
5134:
5069:
5067:
5066:
5061:
5055:
5054:
5045:
5044:
5034:
5013:
5012:
5003:
5002:
4992:
4918:
4916:
4915:
4910:
4699:
4697:
4696:
4691:
4689:
4688:
4675:
4673:
4672:
4667:
4665:
4664:
4627:
4625:
4624:
4619:
4617:
4616:
4593:
4591:
4590:
4585:
4580:
4579:
4578:
4577:
4554:
4553:
4541:
4540:
4515:
4513:
4512:
4507:
4505:
4504:
4489:
4487:
4486:
4481:
4461:
4460:
4448:
4447:
4435:
4434:
4422:
4421:
4394:
4393:
4381:
4380:
4351:
4349:
4348:
4343:
4248:with a space of
4247:
4245:
4244:
4239:
4237:
4236:
4223:
4221:
4220:
4215:
4213:
4212:
4164:
4162:
4161:
4156:
4142:
4141:
4129:
4128:
4104:
4103:
4088:
4087:
4065:
4063:
4062:
4057:
4043:
4042:
4003:
4001:
3997:
3989:
3983:
3982:
3981:
3973:
3963:
3960:
3913:
3911:
3910:
3905:
3897:
3896:
3873:
3872:
3865:
3864:
3820:
3818:
3817:
3812:
3810:
3809:
3751:
3749:
3748:
3743:
3738:
3736:
3725:
3687:
3685:
3684:
3679:
3665:
3664:
3655:
3654:
3644:
3639:
3596:
3594:
3593:
3588:
3583:
3582:
3570:
3569:
3568:
3567:
3547:
3546:
3534:
3533:
3532:
3531:
3517:
3516:
3515:
3514:
3494:
3493:
3492:
3491:
3473:
3472:
3471:
3453:
3452:
3429:
3428:
3410:
3409:
3394:
3393:
3366:
3365:
3337:
3335:
3334:
3329:
3315:
3314:
3275:
3273:
3269:
3261:
3255:
3254:
3253:
3245:
3235:
3232:
3220:
3219:
3179:
3178:
3160:
3159:
3150:
3149:
3136:
3131:
3098:
3096:
3092:
3084:
3078:
3077:
3076:
3068:
3058:
3055:
3043:
3042:
3021:
3020:
2998:
2996:
2995:
2990:
2976:
2975:
2960:
2959:
2949:
2944:
2923:
2922:
2880:
2879:
2864:
2863:
2850:
2845:
2824:
2823:
2802:
2801:
2776:
2774:
2773:
2768:
2760:
2759:
2744:
2743:
2733:
2728:
2707:
2706:
2688:
2687:
2665:
2663:
2662:
2657:
2652:
2651:
2636:
2635:
2622:
2617:
2596:
2595:
2577:
2576:
2522:
2520:
2519:
2514:
2509:
2508:
2477:
2475:
2474:
2469:
2464:
2463:
2417:
2415:
2414:
2409:
2404:
2403:
2349:
2347:
2339:
2322:
2272:
2270:
2269:
2264:
2235:
2234:
2209:
2207:
2206:
2201:
2181:
2180:
2168:
2167:
2143:
2142:
2130:
2129:
2095:
2093:
2092:
2087:
2028:
2027:
1993:
1886:
1884:
1883:
1880:{\displaystyle }
1878:
1843:
1841:
1840:
1835:
1808:
1743:
1741:
1740:
1735:
1705:comultiplication
1614:
1612:
1611:
1606:
1586:
1585:
1573:
1572:
1560:
1559:
1529:
1528:
1513:
1512:
1488:
1487:
1469:
1468:
1429:
1428:
1410:
1409:
1374:
1372:
1371:
1366:
1358:
1357:
1348:
1347:
1332:
1331:
1322:
1321:
1309:
1308:
1296:
1295:
1286:
1285:
1276:
1275:
1257:
1256:
1247:
1246:
1228:
1227:
1187:
1185:
1184:
1179:
1164:
1163:
1142:
1140:
1136:
1128:
1122:
1121:
1120:
1112:
1102:
1099:
1087:
1086:
1048:
1046:
1045:
1040:
1035:
1034:
1012:
1005:
997:
984:
982:
981:
980:
967:
963:
962:
952:
937:
935:
934:
929:
924:
923:
914:
913:
912:
911:
894:
893:
892:
891:
873:
872:
871:
861:
860:
859:
841:
840:
839:
829:
828:
827:
809:
804:
797:
796:
778:
777:
759:
754:
750:
749:
731:
730:
707:
706:
693:
688:
613:
611:
610:
605:
599:
598:
545:
543:
535:
518:
490:
488:
487:
482:
455:
453:
452:
447:
442:
441:
426:
425:
413:
405:
404:
386:
385:
373:
359:
358:
340:
339:
297:
295:
294:
289:
283:
281:
280:
275:
274:
265:
253:
249:
241:
235:
7826:
7825:
7821:
7820:
7819:
7817:
7816:
7815:
7781:
7780:
7756:10.2307/1970615
7732:
7703:
7654:
7605:
7577:
7555:
7549:
7536:
7488:
7476:
7428:
7416:
7379:
7331:
7319:
7296:
7265:
7252:
7236:
7222:10.2307/2004720
7215:(117): 79–106,
7203:
7178:
7156:10.1.1.180.7535
7140:
7102:
7059:Eur. Phys. J. C
7056:
7037:
7032:
6982:
6979:
6974:
6967:
6963:
6931:
6930:
6926:
6919:
6915:
6908:
6901:
6894:
6890:
6883:
6874:
6870:
6824:
6823:
6779:
6778:
6746:
6723:
6697:
6661:
6640:
6639:
6594:
6531:
6510:
6474:
6458:
6424:
6415:
6401:
6389:
6377:
6366:
6365:
6319:
6283:
6271:
6227:
6218:
6204:
6182:
6175:
6151:
6132:
6117:
6116:
6107:
6080:
6024:
6014:
6004:
5993:
5992:
5987:
5971:
5933:
5917:
5901:
5882:
5853:
5847:
5846:
5837:
5833:
5827:
5777:
5755:
5750:
5749:
5741:
5732:
5694:
5684:
5679:
5678:
5662:
5654:
5650:
5620:
5619:
5611:
5603:
5564:
5551:
5535:
5514:
5513:
5496:
5454:
5453:
5371:
5370:
5336:
5335:
5296:
5295:
5236:
5235:
5155:
5154:
5090:
5089:
5085:ker ε given by
5046:
5036:
5004:
4994:
4972:
4971:
4937:
4808:
4807:
4721:renormalization
4710:
4708:Renormalization
4678:
4677:
4654:
4653:
4606:
4605:
4569:
4561:
4545:
4532:
4527:
4526:
4521:
4494:
4493:
4452:
4439:
4426:
4413:
4385:
4372:
4364:
4363:
4265:
4264:
4226:
4225:
4202:
4201:
4171:
4133:
4120:
4095:
4079:
4074:
4073:
4034:
3984:
3964:
3929:
3928:
3888:
3867:
3866:
3856:
3854:
3848:
3847:
3842:
3832:
3826:
3825:
3801:
3790:
3789:
3729:
3704:
3703:
3656:
3646:
3605:
3604:
3574:
3559:
3554:
3538:
3523:
3518:
3506:
3498:
3483:
3475:
3463:
3444:
3420:
3401:
3385:
3357:
3352:
3351:
3346:
3306:
3256:
3236:
3211:
3170:
3151:
3138:
3079:
3059:
3034:
3012:
3007:
3006:
2967:
2951:
2914:
2871:
2852:
2815:
2793:
2788:
2787:
2751:
2735:
2692:
2679:
2674:
2673:
2643:
2624:
2581:
2568:
2563:
2562:
2557:
2548:
2541:
2500:
2486:
2485:
2452:
2438:
2437:
2395:
2340:
2323:
2316:
2315:
2309:
2298:
2288:
2223:
2218:
2217:
2172:
2159:
2134:
2121:
2116:
2115:
2004:
1950:
1949:
1893:
1857:
1856:
1750:
1749:
1708:
1707:
1689:polynomial ring
1643:renormalization
1624:
1577:
1564:
1551:
1517:
1504:
1479:
1457:
1414:
1395:
1390:
1389:
1349:
1339:
1323:
1310:
1300:
1287:
1277:
1264:
1248:
1232:
1213:
1208:
1207:
1155:
1123:
1103:
1078:
1057:
1056:
1026:
972:
968:
954:
953:
946:
945:
915:
903:
898:
883:
878:
863:
851:
831:
819:
788:
769:
741:
722:
698:
675:
674:
647:
590:
536:
519:
511:
510:
506:can be written
464:
463:
433:
417:
396:
377:
350:
331:
323:
322:
309:
266:
254:
236:
213:
212:
171:
164:
132:
91:renormalization
29:John C. Butcher
17:
12:
11:
5:
7824:
7822:
7814:
7813:
7808:
7803:
7798:
7793:
7783:
7782:
7779:
7778:
7775:
7750:(2): 211–264,
7730:
7718:hep-th/0306020
7701:
7673:hep-th/9901099
7666:(3): 627–670,
7652:
7617:(2): 303–334,
7603:
7575:
7553:
7547:
7541:, Birkhäuser,
7534:
7506:hep-th/0003188
7499:(1): 215–241,
7474:
7446:hep-th/9912092
7439:(1): 249–273,
7414:
7377:
7349:hep-th/9808042
7342:(1): 203–242,
7317:
7298:Cayley, Arthur
7294:
7263:
7250:
7234:
7201:
7190:(2): 185–201,
7176:
7149:(3): 425–438,
7138:
7111:(1): 127–159,
7100:
7072:hep-th/9904014
7065:(3): 521–534,
7054:
7030:
7002:hep-th/0403207
6995:(2): 343–367,
6978:
6975:
6973:
6972:
6961:
6947:10.1.1.42.8612
6940:(3): 341–356,
6924:
6913:
6899:
6888:
6871:
6869:
6866:
6847:
6842:
6838:
6834:
6831:
6807:
6804:
6801:
6796:
6791:
6787:
6771:
6770:
6758:
6752:
6749:
6744:
6741:
6738:
6735:
6732:
6729:
6726:
6718:
6714:
6710:
6707:
6704:
6700:
6694:
6689:
6685:
6681:
6676:
6672:
6668:
6664:
6660:
6657:
6654:
6651:
6648:
6635:In particular
6633:
6632:
6620:
6614:
6611:
6607:
6603:
6600:
6597:
6592:
6589:
6585:
6581:
6578:
6575:
6572:
6569:
6566:
6563:
6560:
6557:
6553:
6549:
6546:
6543:
6540:
6537:
6534:
6526:
6523:
6520:
6517:
6513:
6507:
6502:
6498:
6494:
6489:
6485:
6481:
6477:
6473:
6470:
6465:
6461:
6451:
6446:
6442:
6438:
6433:
6428:
6423:
6419:
6411:
6408:
6404:
6398:
6393:
6388:
6384:
6380:
6374:
6359:Gamma function
6347:
6346:
6334:
6331:
6326:
6322:
6315:
6312:
6309:
6304:
6301:
6296:
6293:
6290:
6286:
6280:
6275:
6270:
6266:
6262:
6254:
6249:
6245:
6241:
6236:
6231:
6226:
6222:
6216:
6211:
6207:
6203:
6200:
6197:
6194:
6189:
6185:
6181:
6178:
6172:
6169:
6166:
6163:
6158:
6154:
6150:
6147:
6144:
6139:
6135:
6131:
6128:
6125:
6105:
6083:Kreimer (2007)
6079:
6076:
6066:vector field.
6062:of the model.
6056:
6055:
6044:
6039:
6036:
6033:
6028:
6021:
6017:
6011:
6007:
6003:
6000:
5983:
5967:
5961:
5960:
5948:
5943:
5940:
5936:
5932:
5929:
5924:
5920:
5916:
5911:
5908:
5904:
5900:
5897:
5894:
5889:
5885:
5879:
5876:
5873:
5869:
5865:
5860:
5856:
5835:
5829:
5825:
5811:
5810:
5799:
5793:
5789:
5785:
5780:
5776:
5773:
5770:
5767:
5762:
5758:
5737:
5730:
5727:
5726:
5715:
5712:
5709:
5704:
5701:
5697:
5691:
5687:
5660:
5652:
5648:
5636:
5633:
5630:
5627:
5614:Riemann sphere
5609:
5601:
5598:
5597:
5585:
5582:
5579:
5576:
5571:
5567:
5561:
5558:
5554:
5550:
5547:
5542:
5538:
5534:
5531:
5528:
5525:
5522:
5492:
5471:
5466:
5462:
5450:
5449:
5438:
5435:
5432:
5429:
5426:
5421:
5416:
5412:
5408:
5405:
5402:
5399:
5396:
5393:
5388:
5383:
5379:
5353:
5348:
5344:
5313:
5308:
5304:
5288:
5287:
5276:
5273:
5270:
5267:
5264:
5261:
5258:
5255:
5252:
5249:
5246:
5243:
5229:
5228:
5217:
5214:
5211:
5208:
5205:
5202:
5199:
5196:
5193:
5190:
5187:
5182:
5179:
5174:
5171:
5168:
5165:
5162:
5144:
5143:
5131:
5128:
5125:
5122:
5119:
5116:
5113:
5110:
5107:
5104:
5101:
5098:
5071:
5070:
5058:
5053:
5049:
5043:
5039:
5033:
5030:
5027:
5023:
5019:
5016:
5011:
5007:
5001:
4997:
4991:
4987:
4983:
4980:
4948:
4947:
4935:
4921:
4920:
4919:
4908:
4905:
4902:
4899:
4896:
4893:
4890:
4887:
4884:
4881:
4878:
4875:
4872:
4869:
4866:
4863:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4836:
4833:
4830:
4827:
4824:
4821:
4818:
4815:
4802:
4801:
4797:satisfies the
4779:
4772:Laurent series
4745:Brouder (2000)
4733:Kreimer (1999)
4717:Hopf algebraic
4709:
4706:
4687:
4663:
4615:
4595:
4594:
4583:
4576:
4572:
4568:
4564:
4560:
4557:
4552:
4548:
4544:
4539:
4535:
4517:
4503:
4491:
4490:
4479:
4476:
4473:
4470:
4467:
4464:
4459:
4455:
4451:
4446:
4442:
4438:
4433:
4429:
4425:
4420:
4416:
4412:
4409:
4406:
4403:
4400:
4397:
4392:
4388:
4384:
4379:
4375:
4371:
4353:
4352:
4341:
4338:
4335:
4332:
4329:
4326:
4323:
4320:
4317:
4314:
4311:
4308:
4305:
4302:
4299:
4296:
4293:
4290:
4287:
4284:
4281:
4278:
4275:
4272:
4235:
4211:
4170:
4167:
4166:
4165:
4154:
4151:
4148:
4145:
4140:
4136:
4132:
4127:
4123:
4119:
4116:
4113:
4110:
4107:
4102:
4098:
4094:
4091:
4086:
4082:
4067:
4066:
4055:
4052:
4049:
4046:
4041:
4037:
4033:
4030:
4027:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4000:
3996:
3992:
3988:
3980:
3976:
3972:
3967:
3959:
3955:
3951:
3948:
3945:
3942:
3939:
3936:
3915:
3914:
3903:
3900:
3895:
3891:
3887:
3884:
3881:
3876:
3871:
3863:
3859:
3855:
3853:
3850:
3849:
3846:
3843:
3841:
3838:
3837:
3835:
3808:
3804:
3800:
3797:
3765:Butcher (1972)
3753:
3752:
3741:
3735:
3732:
3728:
3723:
3720:
3717:
3714:
3711:
3697:Butcher series
3689:
3688:
3677:
3674:
3671:
3668:
3663:
3659:
3653:
3649:
3643:
3638:
3635:
3632:
3628:
3624:
3621:
3618:
3615:
3612:
3598:
3597:
3586:
3581:
3577:
3573:
3566:
3562:
3557:
3553:
3550:
3545:
3541:
3537:
3530:
3526:
3521:
3513:
3509:
3505:
3501:
3497:
3490:
3486:
3482:
3478:
3470:
3466:
3462:
3459:
3456:
3451:
3447:
3442:
3438:
3435:
3432:
3427:
3423:
3419:
3416:
3413:
3408:
3404:
3400:
3397:
3392:
3388:
3381:
3378:
3375:
3372:
3369:
3364:
3360:
3342:
3339:
3338:
3327:
3324:
3321:
3318:
3313:
3309:
3305:
3302:
3299:
3296:
3293:
3290:
3287:
3284:
3281:
3278:
3272:
3268:
3264:
3260:
3252:
3248:
3244:
3239:
3231:
3227:
3223:
3218:
3214:
3210:
3207:
3204:
3201:
3198:
3191:
3188:
3185:
3182:
3177:
3173:
3169:
3166:
3163:
3158:
3154:
3148:
3145:
3141:
3135:
3130:
3127:
3124:
3120:
3116:
3113:
3110:
3107:
3104:
3101:
3095:
3091:
3087:
3083:
3075:
3071:
3067:
3062:
3054:
3050:
3046:
3041:
3037:
3033:
3030:
3027:
3024:
3019:
3015:
3000:
2999:
2988:
2985:
2982:
2979:
2974:
2970:
2966:
2963:
2958:
2954:
2948:
2943:
2940:
2937:
2933:
2929:
2926:
2921:
2917:
2913:
2910:
2907:
2904:
2901:
2895:
2892:
2889:
2886:
2883:
2878:
2874:
2870:
2867:
2862:
2859:
2855:
2849:
2844:
2841:
2838:
2834:
2830:
2827:
2822:
2818:
2814:
2811:
2808:
2805:
2800:
2796:
2781:Butcher (1963)
2778:
2777:
2766:
2763:
2758:
2754:
2750:
2747:
2742:
2738:
2732:
2727:
2724:
2721:
2717:
2713:
2710:
2705:
2702:
2699:
2695:
2691:
2686:
2682:
2667:
2666:
2655:
2650:
2646:
2642:
2639:
2634:
2631:
2627:
2621:
2616:
2613:
2610:
2606:
2602:
2599:
2594:
2591:
2588:
2584:
2580:
2575:
2571:
2553:
2546:
2537:
2524:
2523:
2512:
2507:
2503:
2499:
2496:
2493:
2479:
2478:
2467:
2462:
2459:
2455:
2451:
2448:
2445:
2419:
2418:
2407:
2402:
2398:
2394:
2391:
2388:
2385:
2382:
2376:
2373:
2370:
2367:
2364:
2361:
2358:
2355:
2352:
2346:
2343:
2338:
2335:
2332:
2329:
2326:
2308:
2305:
2294:
2286:
2274:
2273:
2262:
2259:
2256:
2253:
2250:
2247:
2244:
2241:
2238:
2233:
2230:
2226:
2211:
2210:
2199:
2196:
2193:
2190:
2187:
2184:
2179:
2175:
2171:
2166:
2162:
2158:
2155:
2152:
2149:
2146:
2141:
2137:
2133:
2128:
2124:
2097:
2096:
2085:
2082:
2079:
2076:
2073:
2070:
2067:
2064:
2058:
2055:
2052:
2049:
2046:
2043:
2040:
2037:
2034:
2031:
2026:
2023:
2020:
2017:
2014:
2011:
2007:
2003:
2000:
1997:
1992:
1989:
1986:
1982:
1978:
1975:
1972:
1969:
1966:
1963:
1960:
1957:
1946:
1945:
1935:
1891:
1876:
1873:
1870:
1867:
1864:
1845:
1844:
1833:
1830:
1827:
1824:
1821:
1818:
1815:
1812:
1807:
1804:
1801:
1797:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1757:
1746:
1745:
1733:
1730:
1727:
1724:
1721:
1718:
1715:
1675:Butcher (1972)
1623:
1620:
1616:
1615:
1604:
1601:
1598:
1595:
1592:
1589:
1584:
1580:
1576:
1571:
1567:
1563:
1558:
1554:
1550:
1547:
1544:
1541:
1538:
1535:
1532:
1527:
1524:
1520:
1516:
1511:
1507:
1503:
1500:
1497:
1494:
1491:
1486:
1482:
1478:
1475:
1472:
1467:
1464:
1460:
1456:
1453:
1450:
1447:
1444:
1441:
1438:
1435:
1432:
1427:
1424:
1421:
1417:
1413:
1408:
1405:
1402:
1398:
1376:
1375:
1364:
1361:
1356:
1352:
1346:
1342:
1338:
1335:
1330:
1326:
1320:
1317:
1313:
1307:
1303:
1299:
1294:
1290:
1284:
1280:
1274:
1271:
1267:
1263:
1260:
1255:
1251:
1245:
1242:
1239:
1235:
1231:
1226:
1223:
1220:
1216:
1189:
1188:
1176:
1173:
1170:
1167:
1162:
1158:
1154:
1151:
1148:
1145:
1139:
1135:
1131:
1127:
1119:
1115:
1111:
1106:
1098:
1094:
1090:
1085:
1081:
1077:
1074:
1071:
1068:
1065:
1050:
1049:
1038:
1033:
1029:
1025:
1022:
1019:
1016:
1011:
1008:
1004:
1000:
996:
991:
987:
979:
975:
971:
966:
961:
957:
939:
938:
927:
922:
918:
910:
906:
901:
897:
890:
886:
881:
877:
870:
866:
858:
854:
849:
845:
838:
834:
826:
822:
817:
813:
808:
803:
800:
795:
791:
787:
784:
781:
776:
772:
767:
763:
758:
753:
748:
744:
740:
737:
734:
729:
725:
721:
717:
710:
705:
701:
697:
692:
687:
683:
645:
615:
614:
602:
597:
593:
589:
586:
583:
580:
577:
572:
569:
566:
563:
560:
557:
554:
551:
548:
542:
539:
534:
531:
528:
525:
522:
492:
491:
480:
477:
474:
471:
457:
456:
445:
440:
436:
432:
429:
424:
420:
416:
412:
408:
403:
399:
395:
392:
389:
384:
380:
376:
372:
368:
365:
362:
357:
353:
349:
346:
343:
338:
334:
330:
312:symmetry group
305:
299:
298:
286:
279:
273:
269:
264:
260:
257:
252:
248:
244:
240:
233:
230:
227:
224:
221:
169:
162:
131:
128:
107:index theorems
55:that provides
15:
13:
10:
9:
6:
4:
3:
2:
7823:
7812:
7811:Hopf algebras
7809:
7807:
7804:
7802:
7799:
7797:
7794:
7792:
7791:Combinatorics
7789:
7788:
7786:
7776:
7773:
7769:
7765:
7761:
7757:
7753:
7749:
7745:
7744:
7739:
7735:
7731:
7728:
7724:
7719:
7714:
7710:
7706:
7705:Kreimer, Dirk
7702:
7699:
7695:
7691:
7687:
7683:
7679:
7674:
7669:
7665:
7661:
7657:
7656:Kreimer, Dirk
7653:
7650:
7646:
7642:
7638:
7634:
7630:
7625:
7624:q-alg/9707029
7620:
7616:
7612:
7608:
7607:Kreimer, Dirk
7604:
7601:
7597:
7593:
7589:
7585:
7581:
7576:
7572:
7567:
7563:
7559:
7554:
7552:, Chapter 14.
7550:
7544:
7540:
7535:
7532:
7528:
7524:
7520:
7516:
7512:
7507:
7502:
7498:
7494:
7487:
7483:
7482:Kreimer, Dirk
7479:
7478:Connes, Alain
7475:
7472:
7468:
7464:
7460:
7456:
7452:
7447:
7442:
7438:
7434:
7427:
7423:
7422:Kreimer, Dirk
7419:
7418:Connes, Alain
7415:
7412:
7408:
7404:
7400:
7396:
7392:
7391:
7386:
7385:Kreimer, Dirk
7382:
7381:Connes, Alain
7378:
7375:
7371:
7367:
7363:
7359:
7355:
7350:
7345:
7341:
7337:
7330:
7326:
7325:Kreimer, Dirk
7322:
7321:Connes, Alain
7318:
7313:
7309:
7308:
7303:
7299:
7295:
7292:
7288:
7284:
7280:
7276:
7272:
7268:
7264:
7261:
7257:
7253:
7247:
7243:
7239:
7235:
7232:
7228:
7223:
7218:
7214:
7210:
7209:Math. Comput.
7206:
7202:
7198:
7193:
7189:
7185:
7181:
7177:
7174:
7170:
7166:
7162:
7157:
7152:
7148:
7144:
7139:
7136:
7132:
7128:
7124:
7119:
7114:
7110:
7106:
7101:
7098:
7094:
7090:
7086:
7082:
7078:
7073:
7068:
7064:
7060:
7055:
7051:
7047:
7043:
7036:
7031:
7028:
7024:
7020:
7016:
7012:
7008:
7003:
6998:
6994:
6990:
6986:
6985:Kreimer, Dirk
6981:
6980:
6976:
6970:
6965:
6962:
6957:
6953:
6948:
6943:
6939:
6935:
6928:
6925:
6922:
6917:
6914:
6911:
6906:
6904:
6900:
6897:
6892:
6889:
6886:
6881:
6879:
6877:
6873:
6867:
6865:
6863:
6845:
6840:
6836:
6832:
6829:
6821:
6802:
6794:
6789:
6776:
6756:
6750:
6747:
6739:
6736:
6733:
6730:
6716:
6712:
6708:
6705:
6702:
6692:
6687:
6683:
6674:
6670:
6666:
6662:
6658:
6652:
6638:
6637:
6636:
6618:
6609:
6605:
6601:
6587:
6583:
6579:
6576:
6573:
6570:
6567:
6555:
6551:
6547:
6544:
6541:
6538:
6524:
6521:
6518:
6515:
6505:
6500:
6496:
6487:
6483:
6479:
6475:
6471:
6468:
6463:
6459:
6449:
6444:
6440:
6436:
6431:
6421:
6409:
6406:
6396:
6386:
6372:
6364:
6363:
6362:
6360:
6356:
6352:
6332:
6329:
6324:
6320:
6310:
6307:
6302:
6299:
6291:
6288:
6278:
6268:
6252:
6247:
6243:
6239:
6234:
6224:
6209:
6205:
6195:
6187:
6183:
6170:
6167:
6156:
6152:
6148:
6145:
6142:
6137:
6133:
6115:
6114:
6113:
6111:
6104:
6100:
6096:
6092:
6088:
6084:
6077:
6075:
6072:
6067:
6063:
6061:
6060:beta function
6042:
6037:
6034:
6031:
6019:
6015:
6009:
6001:
5998:
5991:
5990:
5989:
5986:
5982:
5977:
5975:
5970:
5966:
5941:
5938:
5930:
5922:
5918:
5909:
5906:
5902:
5895:
5887:
5883:
5877:
5874:
5871:
5863:
5858:
5854:
5845:
5844:
5843:
5841:
5832:
5822:
5820:
5816:
5797:
5787:
5778:
5774:
5768:
5760:
5756:
5748:
5747:
5746:
5745:
5740:
5734:
5713:
5710:
5707:
5702:
5699:
5695:
5689:
5677:
5676:
5675:
5673:
5668:
5666:
5658:
5625:
5618:
5615:
5607:
5583:
5577:
5569:
5565:
5559:
5556:
5548:
5540:
5536:
5532:
5526:
5520:
5512:
5511:
5510:
5508:
5504:
5500:
5495:
5491:
5485:
5469:
5464:
5436:
5427:
5424:
5419:
5414:
5406:
5403:
5397:
5394:
5391:
5386:
5381:
5369:
5368:
5367:
5351:
5346:
5333:
5329:
5311:
5306:
5293:
5274:
5265:
5262:
5259:
5253:
5250:
5247:
5244:
5241:
5234:
5233:
5232:
5215:
5209:
5203:
5200:
5194:
5172:
5169:
5163:
5160:
5153:
5152:
5151:
5149:
5129:
5123:
5117:
5114:
5111:
5108:
5102:
5096:
5088:
5087:
5086:
5084:
5080:
5076:
5056:
5051:
5047:
5041:
5037:
5031:
5028:
5025:
5021:
5017:
5009:
5005:
4999:
4995:
4989:
4985:
4978:
4970:
4969:
4968:
4967:
4966:
4961:
4957:
4953:
4945:
4941:
4934:
4930:
4926:
4922:
4903:
4897:
4891:
4885:
4882:
4873:
4867:
4864:
4858:
4855:
4849:
4843:
4837:
4831:
4828:
4822:
4819:
4813:
4806:
4805:
4804:
4803:
4800:
4796:
4792:
4788:
4784:
4780:
4777:
4773:
4769:
4765:
4761:
4760:Feynman rules
4757:
4756:
4755:
4753:
4748:
4746:
4742:
4738:
4734:
4730:
4726:
4722:
4718:
4714:
4707:
4705:
4703:
4651:
4647:
4643:
4639:
4635:
4631:
4602:
4600:
4581:
4570:
4566:
4562:
4558:
4546:
4537:
4533:
4525:
4524:
4523:
4520:
4477:
4471:
4457:
4453:
4449:
4444:
4440:
4436:
4431:
4427:
4423:
4418:
4414:
4407:
4401:
4390:
4386:
4382:
4377:
4373:
4362:
4361:
4360:
4358:
4339:
4333:
4327:
4321:
4315:
4312:
4306:
4300:
4294:
4288:
4285:
4279:
4276:
4270:
4263:
4262:
4261:
4259:
4255:
4251:
4199:
4195:
4191:
4187:
4183:
4179:
4175:
4168:
4152:
4149:
4146:
4138:
4134:
4130:
4125:
4121:
4114:
4108:
4105:
4100:
4096:
4089:
4084:
4080:
4072:
4071:
4070:
4053:
4047:
4039:
4035:
4028:
4022:
4019:
4016:
4010:
4004:
3998:
3990:
3974:
3965:
3957:
3953:
3949:
3946:
3943:
3940:
3937:
3934:
3927:
3926:
3925:
3923:
3919:
3901:
3889:
3885:
3882:
3874:
3869:
3857:
3851:
3844:
3839:
3833:
3824:
3823:
3822:
3802:
3798:
3795:
3787:
3783:
3779:
3775:
3771:
3766:
3762:
3758:
3739:
3733:
3730:
3726:
3721:
3715:
3702:
3701:
3700:
3698:
3694:
3675:
3669:
3661:
3657:
3651:
3647:
3641:
3636:
3633:
3630:
3626:
3622:
3616:
3610:
3603:
3602:
3601:
3579:
3575:
3564:
3560:
3555:
3551:
3543:
3539:
3528:
3524:
3519:
3511:
3507:
3503:
3499:
3495:
3488:
3484:
3480:
3476:
3468:
3464:
3460:
3457:
3454:
3449:
3445:
3440:
3436:
3425:
3421:
3417:
3414:
3411:
3406:
3402:
3390:
3386:
3379:
3376:
3370:
3362:
3358:
3350:
3349:
3348:
3345:
3325:
3319:
3311:
3307:
3300:
3294:
3291:
3288:
3282:
3276:
3270:
3262:
3246:
3237:
3229:
3225:
3221:
3216:
3212:
3208:
3202:
3196:
3189:
3183:
3175:
3171:
3164:
3156:
3152:
3146:
3143:
3139:
3133:
3128:
3125:
3122:
3118:
3114:
3111:
3105:
3099:
3093:
3085:
3069:
3060:
3052:
3048:
3044:
3039:
3035:
3031:
3025:
3017:
3013:
3005:
3004:
3003:
2980:
2972:
2968:
2961:
2956:
2952:
2946:
2941:
2938:
2935:
2931:
2927:
2924:
2919:
2915:
2911:
2905:
2899:
2893:
2884:
2876:
2872:
2865:
2860:
2857:
2853:
2847:
2842:
2839:
2836:
2832:
2828:
2825:
2820:
2816:
2812:
2806:
2798:
2794:
2786:
2785:
2784:
2782:
2764:
2756:
2752:
2745:
2740:
2736:
2730:
2725:
2722:
2719:
2715:
2711:
2708:
2703:
2700:
2697:
2693:
2689:
2684:
2680:
2672:
2671:
2670:
2648:
2644:
2637:
2632:
2629:
2625:
2619:
2614:
2611:
2608:
2604:
2600:
2597:
2592:
2589:
2586:
2582:
2578:
2573:
2569:
2561:
2560:
2559:
2556:
2552:
2545:
2540:
2536:
2531:
2529:
2505:
2501:
2494:
2491:
2484:
2483:
2482:
2481:and a vector
2460:
2457:
2453:
2446:
2443:
2436:
2435:
2434:
2432:
2428:
2424:
2405:
2400:
2396:
2392:
2386:
2380:
2374:
2365:
2359:
2353:
2350:
2344:
2341:
2333:
2327:
2324:
2314:
2313:
2312:
2306:
2304:
2302:
2297:
2293:
2289:
2282:
2277:
2257:
2254:
2248:
2245:
2239:
2231:
2228:
2224:
2216:
2215:
2214:
2197:
2191:
2177:
2173:
2169:
2164:
2160:
2153:
2147:
2139:
2135:
2131:
2126:
2122:
2114:
2113:
2112:
2110:
2106:
2102:
2101:Butcher group
2083:
2080:
2077:
2074:
2068:
2062:
2056:
2053:
2044:
2038:
2029:
2021:
2015:
2009:
2001:
1998:
1990:
1987:
1984:
1980:
1976:
1973:
1970:
1967:
1961:
1955:
1948:
1947:
1943:
1940:
1936:
1933:
1929:
1925:
1921:
1917:
1916:
1915:
1913:
1909:
1905:
1901:
1897:
1890:
1871:
1865:
1854:
1850:
1831:
1825:
1819:
1813:
1810:
1805:
1802:
1799:
1795:
1791:
1788:
1785:
1782:
1779:
1776:
1773:
1770:
1767:
1761:
1748:
1747:
1744:is defined by
1731:
1728:
1725:
1719:
1716:
1706:
1702:
1701:
1700:
1698:
1694:
1690:
1686:
1683:
1678:
1676:
1672:
1668:
1664:
1663:Butcher group
1660:
1656:
1652:
1648:
1644:
1640:
1636:
1632:
1629:
1621:
1619:
1602:
1590:
1578:
1565:
1552:
1548:
1539:
1536:
1533:
1518:
1505:
1501:
1492:
1480:
1476:
1473:
1458:
1454:
1451:
1445:
1442:
1439:
1436:
1433:
1415:
1411:
1403:
1396:
1388:
1387:
1386:
1384:
1379:
1362:
1359:
1354:
1340:
1333:
1328:
1324:
1311:
1301:
1297:
1292:
1288:
1278:
1265:
1261:
1258:
1253:
1249:
1233:
1229:
1221:
1214:
1206:
1205:
1204:
1202:
1198:
1195:= 1, so that
1194:
1174:
1168:
1160:
1156:
1149:
1143:
1137:
1129:
1113:
1104:
1096:
1092:
1088:
1083:
1079:
1075:
1069:
1063:
1055:
1054:
1053:
1036:
1031:
1027:
1020:
1014:
1009:
1006:
998:
989:
985:
977:
973:
969:
964:
959:
955:
944:
943:
942:
925:
920:
916:
908:
904:
895:
888:
884:
868:
864:
856:
852:
847:
843:
836:
832:
824:
820:
815:
806:
801:
798:
793:
789:
785:
782:
779:
774:
770:
765:
761:
756:
746:
742:
738:
735:
732:
727:
723:
715:
708:
703:
699:
695:
690:
685:
681:
673:
672:
671:
669:
665:
661:
657:
656:Cayley (1857)
653:
651:
644:
640:
636:
632:
628:
624:
620:
600:
595:
591:
587:
581:
575:
570:
561:
555:
549:
546:
540:
537:
529:
523:
520:
509:
508:
507:
505:
501:
497:
478:
475:
472:
469:
462:
461:
460:
443:
438:
434:
430:
427:
422:
418:
414:
401:
397:
393:
390:
387:
382:
378:
366:
363:
355:
351:
347:
344:
341:
336:
332:
321:
320:
319:
317:
313:
308:
304:
284:
271:
267:
258:
255:
250:
242:
231:
225:
219:
211:
210:
209:
207:
203:
199:
195:
191:
187:
183:
182:heap-ordering
179:
175:
168:
161:
157:
153:
149:
145:
136:
129:
126:
122:
120:
116:
112:
108:
104:
100:
96:
92:
88:
84:
80:
78:
74:
70:
66:
65:Cayley (1857)
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
25:Butcher group
22:
7747:
7741:
7708:
7663:
7659:
7614:
7610:
7583:
7579:
7561:
7557:
7538:
7496:
7492:
7436:
7432:
7394:
7388:
7339:
7335:
7311:
7305:
7274:
7270:
7267:Butcher, J.C
7241:
7212:
7208:
7205:Butcher, J.C
7187:
7183:
7180:Butcher, J.C
7146:
7142:
7108:
7104:
7062:
7058:
7049:
7041:
6992:
6988:
6969:Kreimer 2007
6964:
6937:
6933:
6927:
6921:Brouder 2000
6916:
6910:Butcher 2008
6896:Brouder 2004
6891:
6861:
6774:
6772:
6634:
6354:
6350:
6348:
6109:
6102:
6098:
6094:
6090:
6081:
6068:
6064:
6057:
5984:
5980:
5978:
5968:
5964:
5962:
5839:
5830:
5823:
5818:
5814:
5812:
5743:
5738:
5735:
5728:
5674:showed that
5669:
5664:
5656:
5616:
5599:
5506:
5502:
5498:
5493:
5489:
5486:
5451:
5331:
5327:
5291:
5289:
5230:
5147:
5145:
5078:
5074:
5072:
4963:
4959:
4955:
4951:
4949:
4943:
4940:power series
4932:
4928:
4924:
4794:
4790:
4786:
4782:
4775:
4767:
4763:
4759:
4751:
4749:
4740:
4736:
4711:
4701:
4649:
4645:
4641:
4637:
4633:
4603:
4598:
4596:
4518:
4492:
4356:
4354:
4257:
4253:
4193:
4189:
4177:
4172:
4068:
3921:
3916:
3785:
3781:
3777:
3773:
3769:
3760:
3756:
3754:
3696:
3692:
3690:
3599:
3343:
3340:
3001:
2779:
2668:
2554:
2550:
2543:
2538:
2534:
2532:
2530:components.
2527:
2525:
2480:
2430:
2426:
2420:
2310:
2295:
2291:
2284:
2280:
2278:
2275:
2212:
2108:
2104:
2100:
2098:
1941:
1931:
1927:
1923:
1911:
1907:
1903:
1899:
1895:
1888:
1852:
1848:
1846:
1696:
1692:
1684:
1681:
1679:
1667:formal group
1662:
1658:
1654:
1630:
1628:Hopf algebra
1625:
1617:
1380:
1377:
1200:
1196:
1192:
1190:
1051:
940:
667:
663:
659:
654:
649:
642:
638:
634:
630:
626:
622:
618:
616:
503:
499:
496:vector field
493:
458:
315:
310:denotes the
306:
302:
300:
205:
198:automorphism
193:
189:
185:
181:
177:
173:
166:
159:
155:
151:
147:
141:
124:
110:
101:' work with
87:Hopf algebra
81:
61:vector field
53:rooted trees
24:
18:
7564:: 184–210,
6820:polynomials
5972:called the
5824:The loops γ
5606:holomorphic
5505:instead of
5501:(maps into
4652:), whereas
4522:defined by
4250:derivations
4169:Lie algebra
119:Lie algebra
21:mathematics
7785:Categories
7042:Astérisque
6977:References
5150:satisfies
4950:Note that
4793:such that
194:equivalent
7580:Computing
7411:117848361
7397:: 85–96,
7314:: 172–176
7151:CiteSeerX
7118:1410.4761
7052:: 149–165
6942:CiteSeerX
6841:μ
6833:
6786:Φ
6725:Γ
6703:−
6688:μ
6663:π
6653:∙
6647:Φ
6596:Γ
6577:−
6562:Γ
6539:−
6533:Γ
6522:−
6516:−
6501:μ
6476:π
6445:μ
6407:−
6373:∫
6308:−
6289:−
6248:μ
6199:Φ
6196:⋯
6177:Φ
6171:∫
6146:…
6124:Φ
6006:∂
5999:β
5939:−
5923:−
5919:γ
5903:λ
5888:−
5884:γ
5757:λ
5729:so that γ
5703:−
5700:μ
5696:γ
5690:μ
5686:∂
5659:= 0 of γ
5632:∞
5626:∪
5566:γ
5557:−
5541:−
5537:γ
5521:γ
5461:Φ
5434:Δ
5425:∘
5411:Φ
5407:⊗
5395:−
5378:Φ
5343:Φ
5303:Φ
5272:Δ
5263:⊗
5254:∘
5248:−
5204:ε
5189:Δ
5173:⊗
5164:∘
5118:ε
5115:−
5081:onto the
5022:∑
4986:∑
4758:a set of
4575:′
4563:δ
4551:′
4534:θ
4466:Δ
4454:θ
4450:⊗
4441:θ
4437:−
4428:θ
4424:⊗
4415:θ
4387:θ
4374:θ
4328:ε
4316:θ
4301:θ
4289:ε
4271:θ
4147:∘
4135:φ
4131:⋆
4122:φ
4106:∘
4097:φ
4090:∘
4081:φ
4036:δ
4023:φ
4005:α
3954:∑
3938:∘
3935:φ
3894:′
3862:′
3807:′
3803:φ
3799:⋆
3796:φ
3710:Φ
3658:φ
3627:∑
3611:φ
3556:φ
3552:…
3520:φ
3496:…
3458:…
3441:∑
3415:⋯
3387:φ
3371:∙
3359:φ
3308:δ
3295:φ
3277:α
3226:∑
3172:δ
3153:φ
3119:∑
3100:α
3049:∑
2932:∑
2833:∑
2716:∑
2701:−
2605:∑
2590:−
2249:φ
2229:−
2225:φ
2186:Δ
2174:φ
2170:⊗
2161:φ
2136:φ
2132:⋆
2123:φ
2081:∙
2078:−
2069:∙
2042:∖
2019:∖
1999:−
1988:⊂
1981:∑
1977:−
1971:−
1869:∖
1823:∖
1814:⊗
1803:⊂
1796:∑
1786:⊗
1774:⊗
1756:Δ
1729:⊗
1723:→
1714:Δ
1583:′
1570:′
1557:′
1526:′
1523:′
1510:′
1485:′
1466:′
1463:′
1426:′
1423:′
1420:′
1345:′
1319:′
1316:′
1306:′
1283:′
1273:′
1270:′
1244:′
1241:′
1238:′
1157:δ
1144:α
1093:∑
1028:δ
1015:α
990:∑
900:∂
896:⋯
880:∂
848:δ
844:⋯
816:δ
783:…
766:∑
736:…
716:δ
686:∙
682:δ
470:∙
431:⋯
415:⋅
391:…
345:…
220:α
105:on local
103:Moscovici
69:Sylvester
63:. It was
37:Lie group
7707:(2007),
7600:21392760
7586:: 1–15,
7531:10349737
7484:(2001),
7471:17448874
7424:(2000),
7374:10371164
7327:(1998),
7300:(1857),
7291:41661943
7240:(2008),
7135:27789611
7097:16539907
7027:16100842
5600:where γ
5231:so that
3693:B-series
2549:, ... ,
1939:antipode
1934:to zero.
1695:, where
7772:0174052
7764:1970615
7723:Bibcode
7698:1174142
7678:Bibcode
7649:7018827
7629:Bibcode
7511:Bibcode
7451:Bibcode
7354:Bibcode
7260:2401398
7231:2004720
7173:7977686
7077:Bibcode
7007:Bibcode
6078:Example
5817:≠ 0 in
4182:theorem
3780:) and (
3341:where φ
2433:matrix
1639:Kreimer
117:of the
47:by the
23:, the
7770:
7762:
7696:
7647:
7598:
7545:
7529:
7469:
7409:
7372:
7289:
7258:
7248:
7229:
7171:
7153:
7133:
7095:
7025:
6944:
6353:= 1 –
6349:where
5976:(RG).
5842:real,
5647:with γ
1920:counit
617:where
301:where
99:Connes
7760:JSTOR
7713:arXiv
7694:S2CID
7668:arXiv
7645:S2CID
7619:arXiv
7596:S2CID
7527:S2CID
7501:arXiv
7489:(PDF)
7467:S2CID
7441:arXiv
7429:(PDF)
7407:S2CID
7370:S2CID
7344:arXiv
7332:(PDF)
7287:S2CID
7227:JSTOR
7169:S2CID
7131:S2CID
7113:arXiv
7093:S2CID
7067:arXiv
7038:(PDF)
7023:S2CID
6997:arXiv
6868:Notes
6864:= 0.
6097:. If
5828:and λ
5330:into
4628:from
4256:into
4252:θ of
2526:with
2107:into
1926:into
652:= 0.
180:|. A
144:graph
7543:ISBN
7312:XIII
7246:ISBN
6818:are
6089:for
5813:for
5290:The
5029:<
4702:L(G)
4638:L(G)
4634:L(G)
3600:and
2099:The
1937:Its
1918:Its
1703:Its
1680:The
1626:The
1199:and
641:and
637:to
148:root
97:and
7752:doi
7686:doi
7637:doi
7588:doi
7566:doi
7519:doi
7497:216
7459:doi
7437:210
7399:doi
7362:doi
7340:199
7279:doi
7217:doi
7192:doi
7161:doi
7123:doi
7085:doi
7050:290
7015:doi
6952:doi
6830:log
6822:in
5868:lim
5834:· γ
5604:is
5497:of
5326:of
5077:of
4958:–
4942:in
4938:of
4789:on
4774:in
4770:of
4723:in
4648:to
4184:of
3786:b'
3782:A'
3695:or
2558:of
1914:).
1851:of
1673:by
1645:in
502:of
314:of
93:in
31:by
19:In
7787::
7768:MR
7766:,
7758:,
7748:81
7740:,
7721:,
7692:,
7684:,
7676:,
7662:,
7643:,
7635:,
7627:,
7613:,
7594:,
7584:13
7582:,
7562:26
7560:,
7525:,
7517:,
7509:,
7495:,
7491:,
7480:;
7465:,
7457:,
7449:,
7435:,
7431:,
7420:;
7405:,
7395:48
7393:,
7383:;
7368:,
7360:,
7352:,
7338:,
7334:,
7323:;
7310:,
7304:,
7285:,
7275:53
7273:,
7256:MR
7254:,
7225:,
7213:26
7211:,
7186:,
7167:,
7159:,
7147:44
7145:,
7129:,
7121:,
7109:17
7107:,
7091:,
7083:,
7075:,
7063:12
7061:,
7048:,
7044:,
7040:,
7021:,
7013:,
7005:,
6991:,
6950:,
6938:15
6936:,
6902:^
6875:^
6108:=
5821:.
5731:μ–
5484:.
5130:1.
4956:id
4929:id
4927:–
4781:a
4704:.
4601:.
3784:,
3776:,
3380:1.
2429:x
1855:;
1677:.
629:,
479:1.
165:,
7754::
7725::
7715::
7688::
7680::
7670::
7664:3
7639::
7631::
7621::
7615:2
7590::
7568::
7521::
7513::
7503::
7461::
7453::
7443::
7401::
7364::
7356::
7346::
7281::
7219::
7194::
7188:3
7163::
7125::
7115::
7087::
7079::
7069::
7017::
7009::
6999::
6993:6
6954::
6862:z
6846:2
6837:q
6806:)
6803:t
6800:(
6795:R
6790:S
6775:R
6757:.
6751:z
6748:c
6743:)
6740:z
6737:c
6734:+
6731:1
6728:(
6717:2
6713:/
6709:c
6706:z
6699:)
6693:2
6684:q
6680:(
6675:2
6671:/
6667:D
6659:=
6656:)
6650:(
6619:.
6613:)
6610:2
6606:/
6602:D
6599:(
6591:)
6588:2
6584:/
6580:D
6574:u
6571:+
6568:1
6565:(
6559:)
6556:2
6552:/
6548:D
6545:+
6542:u
6536:(
6525:u
6519:z
6512:)
6506:2
6497:q
6493:(
6488:2
6484:/
6480:D
6472:=
6469:y
6464:D
6460:d
6450:2
6441:q
6437:+
6432:2
6427:|
6422:y
6418:|
6410:u
6403:)
6397:2
6392:|
6387:y
6383:|
6379:(
6355:D
6351:z
6333:,
6330:y
6325:D
6321:d
6314:)
6311:1
6303:2
6300:c
6295:(
6292:z
6285:)
6279:2
6274:|
6269:y
6265:|
6261:(
6253:2
6244:q
6240:+
6235:2
6230:|
6225:y
6221:|
6215:)
6210:n
6206:t
6202:(
6193:)
6188:1
6184:t
6180:(
6168:=
6165:)
6162:]
6157:n
6153:t
6149:,
6143:,
6138:1
6134:t
6130:[
6127:(
6110:q
6106:μ
6103:q
6099:c
6095:V
6091:H
6043:.
6038:0
6035:=
6032:t
6027:|
6020:t
6016:F
6010:t
6002:=
5985:C
5981:G
5969:C
5965:G
5947:)
5942:1
5935:)
5931:z
5928:(
5915:(
5910:z
5907:t
5899:)
5896:z
5893:(
5878:0
5875:=
5872:z
5864:=
5859:t
5855:F
5840:t
5836:μ
5831:w
5826:μ
5819:C
5815:w
5798:t
5792:|
5788:t
5784:|
5779:w
5775:=
5772:)
5769:t
5766:(
5761:w
5744:H
5739:w
5714:,
5711:0
5708:=
5661:+
5657:z
5653:+
5649:–
5635:}
5629:{
5617:C
5610:–
5602:+
5584:,
5581:)
5578:z
5575:(
5570:+
5560:1
5553:)
5549:z
5546:(
5533:=
5530:)
5527:z
5524:(
5507:R
5503:C
5499:G
5494:C
5490:G
5470:R
5465:S
5437:.
5431:)
5428:P
5420:R
5415:S
5404:S
5401:(
5398:m
5392:=
5387:R
5382:S
5352:R
5347:S
5332:V
5328:H
5312:R
5307:S
5275:,
5269:)
5266:P
5260:S
5257:(
5251:m
5245:=
5242:S
5216:1
5213:)
5210:x
5207:(
5201:=
5198:)
5195:x
5192:(
5186:)
5181:d
5178:i
5170:S
5167:(
5161:m
5148:S
5127:)
5124:x
5121:(
5112:x
5109:=
5106:)
5103:x
5100:(
5097:P
5079:H
5075:P
5057:.
5052:n
5048:z
5042:n
5038:a
5032:0
5026:n
5018:=
5015:)
5010:n
5006:z
5000:n
4996:a
4990:n
4982:(
4979:R
4960:R
4952:R
4946:.
4944:z
4936:+
4933:V
4925:R
4907:)
4904:g
4901:)
4898:f
4895:(
4892:R
4889:(
4886:R
4883:+
4880:)
4877:)
4874:g
4871:(
4868:R
4865:f
4862:(
4859:R
4856:=
4853:)
4850:g
4847:(
4844:R
4841:)
4838:f
4835:(
4832:R
4829:+
4826:)
4823:g
4820:f
4817:(
4814:R
4795:R
4791:V
4787:R
4776:z
4768:V
4764:H
4741:G
4737:H
4686:g
4662:g
4650:R
4646:H
4642:H
4614:g
4599:t
4582:,
4571:t
4567:t
4559:=
4556:)
4547:t
4543:(
4538:t
4519:t
4502:g
4478:.
4475:)
4472:t
4469:(
4463:)
4458:1
4445:2
4432:2
4419:1
4411:(
4408:=
4405:)
4402:t
4399:(
4396:]
4391:2
4383:,
4378:1
4370:[
4357:G
4340:,
4337:)
4334:b
4331:(
4325:)
4322:a
4319:(
4313:+
4310:)
4307:b
4304:(
4298:)
4295:a
4292:(
4286:=
4283:)
4280:b
4277:a
4274:(
4258:R
4254:H
4234:g
4210:g
4194:H
4190:H
4178:G
4153:.
4150:f
4144:)
4139:2
4126:1
4118:(
4115:=
4112:)
4109:f
4101:2
4093:(
4085:1
4054:,
4051:)
4048:0
4045:(
4040:t
4032:)
4029:t
4026:(
4020:!
4017:t
4014:)
4011:t
4008:(
3999:!
3995:|
3991:t
3987:|
3979:|
3975:t
3971:|
3966:s
3958:t
3950:+
3947:1
3944:=
3941:f
3922:f
3902:.
3899:)
3890:b
3886:,
3883:b
3880:(
3875:,
3870:)
3858:A
3852:0
3845:0
3840:A
3834:(
3778:b
3774:A
3770:n
3761:n
3757:n
3740:.
3734:!
3731:t
3727:1
3722:=
3719:)
3716:t
3713:(
3676:.
3673:)
3670:t
3667:(
3662:j
3652:j
3648:b
3642:m
3637:1
3634:=
3631:j
3623:=
3620:)
3617:t
3614:(
3585:)
3580:k
3576:t
3572:(
3565:k
3561:j
3549:)
3544:1
3540:t
3536:(
3529:1
3525:j
3512:k
3508:j
3504:i
3500:a
3489:1
3485:j
3481:i
3477:a
3469:k
3465:j
3461:,
3455:,
3450:1
3446:j
3437:=
3434:)
3431:]
3426:k
3422:t
3418:,
3412:,
3407:1
3403:t
3399:[
3396:(
3391:i
3377:=
3374:)
3368:(
3363:j
3344:j
3326:,
3323:)
3320:0
3317:(
3312:t
3304:)
3301:t
3298:(
3292:!
3289:t
3286:)
3283:t
3280:(
3271:!
3267:|
3263:t
3259:|
3251:|
3247:t
3243:|
3238:s
3230:t
3222:+
3217:0
3213:x
3209:=
3206:)
3203:s
3200:(
3197:x
3190:,
3187:)
3184:0
3181:(
3176:t
3168:)
3165:t
3162:(
3157:j
3147:j
3144:i
3140:a
3134:m
3129:1
3126:=
3123:j
3115:!
3112:t
3109:)
3106:t
3103:(
3094:!
3090:|
3086:t
3082:|
3074:|
3070:t
3066:|
3061:s
3053:t
3045:+
3040:0
3036:x
3032:=
3029:)
3026:s
3023:(
3018:i
3014:X
2987:)
2984:)
2981:s
2978:(
2973:j
2969:X
2965:(
2962:f
2957:j
2953:b
2947:m
2942:1
2939:=
2936:j
2928:s
2925:+
2920:0
2916:x
2912:=
2909:)
2906:s
2903:(
2900:x
2894:,
2891:)
2888:)
2885:s
2882:(
2877:j
2873:X
2869:(
2866:f
2861:j
2858:i
2854:a
2848:m
2843:1
2840:=
2837:j
2829:s
2826:+
2821:0
2817:x
2813:=
2810:)
2807:s
2804:(
2799:i
2795:X
2765:.
2762:)
2757:j
2753:x
2749:(
2746:f
2741:j
2737:b
2731:m
2726:1
2723:=
2720:j
2712:h
2709:+
2704:1
2698:n
2694:x
2690:=
2685:n
2681:x
2654:)
2649:j
2645:X
2641:(
2638:f
2633:j
2630:i
2626:a
2620:m
2615:1
2612:=
2609:j
2601:h
2598:+
2593:1
2587:n
2583:x
2579:=
2574:i
2570:X
2555:m
2551:X
2547:1
2544:X
2539:n
2535:x
2528:m
2511:)
2506:i
2502:b
2498:(
2495:=
2492:b
2466:)
2461:j
2458:i
2454:a
2450:(
2447:=
2444:A
2431:m
2427:m
2406:,
2401:0
2397:x
2393:=
2390:)
2387:0
2384:(
2381:x
2375:,
2372:)
2369:)
2366:s
2363:(
2360:x
2357:(
2354:f
2351:=
2345:s
2342:d
2337:)
2334:s
2331:(
2328:x
2325:d
2296:C
2292:G
2287:C
2281:C
2261:)
2258:t
2255:S
2252:(
2246:=
2243:)
2240:t
2237:(
2232:1
2198:.
2195:)
2192:t
2189:(
2183:)
2178:2
2165:1
2157:(
2154:=
2151:)
2148:t
2145:(
2140:2
2127:1
2109:R
2105:H
2084:.
2075:=
2072:)
2066:(
2063:S
2057:,
2054:s
2051:)
2048:]
2045:s
2039:t
2036:[
2033:(
2030:S
2025:)
2022:s
2016:t
2013:(
2010:n
2006:)
2002:1
1996:(
1991:t
1985:s
1974:t
1968:=
1965:)
1962:t
1959:(
1956:S
1942:S
1932:t
1928:R
1924:H
1912:s
1910:\
1908:t
1906:(
1904:n
1900:t
1896:s
1892:i
1889:t
1875:]
1872:s
1866:t
1863:[
1853:t
1849:s
1832:,
1829:]
1826:s
1820:t
1817:[
1811:s
1806:t
1800:s
1792:+
1789:t
1783:I
1780:+
1777:I
1771:t
1768:=
1765:)
1762:t
1759:(
1732:H
1726:H
1720:H
1717::
1697:t
1693:t
1685:H
1659:R
1655:H
1631:H
1603:,
1600:)
1597:)
1594:)
1591:f
1588:(
1579:f
1575:(
1566:f
1562:(
1553:f
1549:+
1546:)
1543:)
1540:f
1537:,
1534:f
1531:(
1519:f
1515:(
1506:f
1502:+
1499:)
1496:)
1493:f
1490:(
1481:f
1477:,
1474:f
1471:(
1459:f
1455:3
1452:+
1449:)
1446:f
1443:,
1440:f
1437:,
1434:f
1431:(
1416:f
1412:=
1407:)
1404:4
1401:(
1397:x
1363:,
1360:f
1355:3
1351:)
1341:f
1337:(
1334:+
1329:2
1325:f
1312:f
1302:f
1298:+
1293:2
1289:f
1279:f
1266:f
1262:3
1259:+
1254:3
1250:f
1234:f
1230:=
1225:)
1222:4
1219:(
1215:x
1201:f
1197:x
1193:N
1175:.
1172:)
1169:0
1166:(
1161:t
1153:)
1150:t
1147:(
1138:!
1134:|
1130:t
1126:|
1118:|
1114:t
1110:|
1105:s
1097:t
1089:+
1084:0
1080:x
1076:=
1073:)
1070:s
1067:(
1064:x
1037:,
1032:t
1024:)
1021:t
1018:(
1010:m
1007:=
1003:|
999:t
995:|
986:=
978:m
974:s
970:d
965:x
960:m
956:d
926:.
921:i
917:f
909:n
905:j
889:1
885:j
876:)
869:n
865:j
857:n
853:t
837:1
833:j
825:1
821:t
812:(
807:N
802:1
799:=
794:n
790:j
786:,
780:,
775:1
771:j
762:=
757:i
752:]
747:n
743:t
739:,
733:,
728:1
724:t
720:[
709:,
704:i
700:f
696:=
691:i
664:s
662:(
660:x
650:s
646:0
643:x
639:R
635:U
631:f
627:U
623:s
621:(
619:x
601:,
596:0
592:x
588:=
585:)
582:0
579:(
576:x
571:,
568:)
565:)
562:s
559:(
556:x
553:(
550:f
547:=
541:s
538:d
533:)
530:s
527:(
524:x
521:d
504:R
500:U
476:=
473:!
444:!
439:n
435:t
428:!
423:1
419:t
411:|
407:]
402:n
398:t
394:,
388:,
383:1
379:t
375:[
371:|
367:=
364:!
361:]
356:n
352:t
348:,
342:,
337:1
333:t
329:[
316:t
307:t
303:S
285:,
278:|
272:t
268:S
263:|
259:!
256:t
251:!
247:|
243:t
239:|
232:=
229:)
226:t
223:(
206:t
190:t
186:t
178:t
174:t
170:2
167:t
163:1
160:t
156:t
152:t
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