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An exponential formula for polynomial vector fields. II: Lie series, exponential substitution, and rooted trees. (English) Zbl 0969.34006

Summary: Gröbner’s Lie series and the exponential formula provide different explicit formulas for the flow generated by a finite-dimensional polynomial vector field. The author gives (1) a generalization of the Lie series in case of noncommuting variables called exponential substitution, (2) a structural understanding of the three formulas and their mutual relationships in terms of rooted trees, and (3) as a byproduct new results on the enumeration, coding, and statistics of different kinds of rooted trees.
For part I see [ibid. 128, No. 1, 190-216 (1997; Zbl 0888.34006)].

MSC:

34A25 Analytical theory of ordinary differential equations: series, transformations, transforms, operational calculus, etc.
15A69 Multilinear algebra, tensor calculus
34A05 Explicit solutions, first integrals of ordinary differential equations
34C99 Qualitative theory for ordinary differential equations
68W30 Symbolic computation and algebraic computation

Citations:

Zbl 0888.34006

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References:

[1] Bass, H.; Connell, E. H.; Wright, D., The Jacobian conjecture: Reduction of degree and formal expansion of the inverse, Bull. Amer. Math. Soc., 7, 287-330 (1982) · Zbl 0539.13012
[2] Björner, A.; Wachs, M. L., Permutation statistics and linear extensions of posets, J. Combin. Theory Ser. A, 58, 85-114 (1991) · Zbl 0742.05084
[3] Cayley, A., On the theory of analytical forms called trees, Collected Mathematical Papers of Arthur Cayley (1890), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, p. 242-246
[4] Cayley, A., On the theory of analytical forms called trees, second part, Collected Mathematical Papers of Arthur Cayley (1891), Cambridge Univ. Press: Cambridge Univ. Press Cambridge, p. 112-115
[5] de Bruijn, N. G.; Morselt, B. J.M., A note on plane trees, J. Combin. Theory, 2, 27-34 (1967) · Zbl 0147.24005
[6] de Bruijn, N. G.; Knuth, D. E.; Rice, S. O., The average hight of planted plane trees, (Read, R. C., Graph Theory and Computing (1972), Academic Press: Academic Press New York) · Zbl 0247.05106
[7] Dumont, D.; Ramamonjisoa, A., Grammaire de Ramanujan et Arbres de Cayley, Electron. J. Combin., 3 (1996) · Zbl 0852.05018
[8] Derschowitz, N.; Zaks, S., Enumeration of ordered trees, Discrete Math., 31, 9-28 (1980)
[9] Gröbner, W., Die Lie-Reihen und ihre Anwendungen (1960), VEB: VEB Berlin · Zbl 0141.08502
[10] Grossman, R.; Larson, R. G., Hopf-algebraic structure of families of trees, J. Algebra, 126, 184-210 (1989) · Zbl 0717.16029
[11] Grossman, R.; Larson, R. G., Solving nonlinear equations from higher order derivations in linear stages, Adv. Math., 82, 180-202 (1990) · Zbl 0716.17014
[12] Grossman, R.; Larson, R. G., Symbolic computation of derivations using labelled trees, J. Symbolic Comput., 13, 511-523 (1992) · Zbl 0762.68035
[13] Grossman, R., Using trees to compute approximate solutions to ordinary differential equations exactly, Differential Equations and Computer Algebra (1991), Academic Press: Academic Press San Diego · Zbl 0727.34011
[14] Gordon, M.; Kennedy, J. W., The counting and coding of trees of fixed diameter, SIAM J. Appl. Math., 28, 376-398 (1975) · Zbl 0293.05137
[15] Harary, F.; Palmer, E. M., Graphical Enumeration (1973), Academic Press: Academic Press New York · Zbl 0266.05108
[16] Harary, F.; Prins, G., The number of homeomorphically irreducible trees, and other species, Acta Math., 101, 141-162 (1959) · Zbl 0084.19304
[17] Harary, F.; Robinson, R. W.; Schwenk, A. J., Twenty-step algorithm for determining the asymptotic number of trees of various species, J. Austral. Math. Soc., 20, 483-503 (1975) · Zbl 0319.05102
[18] Lie, S., Theorie der Transformationsgruppen (1888), Teubner: Teubner Leipzig · JFM 21.0356.02
[19] Labelle, G., Counting asymmetric enriched trees, J. Symbol. Comput., 14, 211-242 (1992) · Zbl 0753.05044
[20] Leroux, P.; Viennot, X. G., Combinatorial resolution of systems of differential equations. I. Ordinary differential equations, Lecture Notes in Math. (1986), Springer-Verlag: Springer-Verlag New York/Berlin, p. 210-245
[21] Moon, J. W., The distance between nodes in recursive trees, Combinatorics. Combinatorics, London Math. Soc. Lecture Notes, 13 (1974), Cambridge Univ. Press: Cambridge Univ. Press Cambridge · Zbl 0297.05101
[22] Melzak, Z. A., A note on homogeneous dendrits, Canad. Math. Bull., 11, 85-93 (1968) · Zbl 0155.51203
[23] Otter, R., The number of trees, Ann. Math., 49, 583-599 (1948) · Zbl 0032.12601
[24] Polya, G.; Read, R. C., Combinatorial Enumeration of Groups, Graphs, and Chemical Compounds (1987), Springer-Verlag: Springer-Verlag New York
[25] Polya, G., Kombinatorische Anzahlbestimmung für Graphen, und chemische Verbindungen, Acta Math., 68, 145-254 (1937) · JFM 63.0547.04
[26] Read, R. C., Every one a winner or How to avoid isomorphism search when cataloguing combinatorial configurations, Ann. Discrete Math., 2, 107-120 (1978) · Zbl 0392.05001
[27] Read, R. C., A survey of graph generation techniques, Combinatorial Mathematics, VIII. Combinatorial Mathematics, VIII, Lecture Notes in Math., 884 (1981), Springer-Verlag: Springer-Verlag New York/Berlin, p. 77-89
[28] Read, R. C., The coding of various kinds of unlabeled trees, (Read, R. C., Graph Theory and Computing (1972), Academic Press: Academic Press New York) · Zbl 0247.05104
[29] Reutenauer, C., Free Lie Algebras (1993), Clarendon: Clarendon Oxford · Zbl 0798.17001
[30] Rawlings, D., A binary tree decomposition space of permutation statistics, J. Combin. Theory Ser. A, 59, 111-124 (1992) · Zbl 0773.05004
[31] Steinberg, S., Lie series and nonlinear ordinary differential equations, J. Math. Anal. Appl., 101, 39-63 (1984) · Zbl 0598.34009
[32] Steinberg, S., Lie series, Lie transforms, and their applications, Lie Methods in Optics. Lie Methods in Optics, Lecture Notes in Phys., 250 (1986), Springer-Verlag: Springer-Verlag New York/Berlin, p. 45-103
[33] N. J. Sloane, Sloanes on-line encyclopedia of integer sequences, http://www.research.att.com/∼njas/sequences; N. J. Sloane, Sloanes on-line encyclopedia of integer sequences, http://www.research.att.com/∼njas/sequences · Zbl 1274.11001
[34] Stanley, R. P., Enumerative Combinatorics (1986), Wadsworth & BrooksCole: Wadsworth & BrooksCole Monterey · Zbl 0608.05001
[35] Stanley, R. P., Enumerative Combinatorics (1998), Wadsworth & BrooksCole: Wadsworth & BrooksCole Monterey
[36] Wedderburn, J. H.M., The functional equation \(g(x^2)=2 αx +[g(x)]^2\), Ann. Math., 24, 121-140 (1922) · JFM 49.0244.02
[37] Winkel, R., An exponential formula for polynomial vector fields, Adv. Math., 128, 190-216 (1997) · Zbl 0888.34006
[38] Zeilberger, D., Toward a combinatorial proof of the Jacobian conjecture, Lecture Notes in Math. (1986), Springer-Verlag: Springer-Verlag New York/Berlin, p. 370-380
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