Stokes factors and multilogarithms

Faà di Bruno's formula and inversion of power series

It is shown here the compositional inversion of formal power series has a natural formulation in terms of umbral operators. This fact is used to derive a new proof of the Lagrange inversion theorem by operator theoretic methods.

Next article Factorization of Lagrange’s Expansion by Means of Exponential Generating FunctionsR. A. SackR. A. Sackhttps://doi.org/10.1137/0114001PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] P. Appell and , J. Kampé De Fériet, Fonctions hypergéometriques et hypersphériquesPolynomes d' Hermite, Gauthier-Villars, Paris, 1926, Appendix 4 Google Scholar[2] H. Bateman, An extension of Lagrange's expansion, Trans. Amer. Math. Soc., 28 (1926), 346–356 MR1501349 Google Scholar[3] Richard Birkeland, Über die Auflösung algebraischer Gleichungen durch hypergeometrische Funktionen, Math. Z., 26 (1927), 566–578 10.1007/BF01475474 MR1544875 CrossrefGoogle Scholar[4] M. M. Carter and , R. A. Sack, Inverse perturbation expansions, to be published Google Scholar[5] A. Cayley, Note sur une formule pour la réversion des séries, J. Reine Angew. Math., 52 (1856), 276–284, Collected Papers, vol. 4, Cambridge, 1891, pp. 30–37 CrossrefGoogle Scholar[6] A. Cayley, Deuxième note sur une formule pour la réversion des séries, J. Reine Angew. Math., 54 (1857), 156–161, Collected Papers, vol. 4, Cambridge, 1891, pp. 54–59 CrossrefGoogle Scholar[7] A. Cayley, On Wronski's theorem, Quart. J. Pure Appl. Math., 12 (1873), 221–228, Collected Papers, vol. 9, Cambridge, 1896, pp. 96–102 Google Scholar[8] J. S. Frame, Power series expansions for inverse functions, Amer. Math. Monthly, 64 (1957), 236–240 MR0085330 0077.28402 CrossrefGoogle Scholar[9] I. J. Good, Generalizations to several variables of Lagrange's expansion, with applications to stochastic processes, Proc. Cambridge Philos. Soc., 56 (1960), 367–380 MR0123021 0135.18802 CrossrefGoogle Scholar[10] E. Goursat, Cours d' analyse mathématique, Vol. 1, Gauthier-Villars, Paris, 1917, Chap. 9, vol. 2, Chaps. 14, 17; English transl. A Course in Mathe-matical Analysis, E. R. Hedrick and O. Dunkel, translators, New York, 1959 Google Scholar[11] C. G. J. Jacobi, De resolutione aequationum per series infinitas, J. Reine Angew. Math., 6 (1830), 257–286, Ges. Werke, vol. 6, Berlin, 1891, pp. 26–61 CrossrefGoogle Scholar[12] Franz Kamber, Formules exprimant les valeurs des coefficients des séries de puissances inverses, Acta Math., 78 (1946), 193–204 MR0014493 0060.15402 CrossrefISIGoogle Scholar[13] Hr. Mellin, Ein allgemeiner Satz über algebraische Gleichungen, Ann. Acad. Sci. Fenn. Ser. A. I, 7 (1915), Google Scholar[14] R. A. Sack, Interpretation of Lagrange's expansion and its generalization to several variables as integration formulas, J. Soc. Indust. Appl. Math., 13 (1965), 47–59 10.1137/0113004 MR0181723 0143.27402 LinkISIGoogle Scholar[15] R. A. Sack, Generalization of Lagrange's expansion for functions of several implicitly defined variables, J. Soc. Indust. Appl. Math., 13 (1965), 913–926 10.1137/0113059 MR0196020 0143.27403 LinkISIGoogle Scholar[16] R. A. Sack, Rayleigh-Schrödinger perturbation expansions to arbitrary order, to be published Google Scholar[17] P. A. Sturrock, Generalization of the Lagrange expansion with application to physical problems, J. Mathematical Phys., 1 (1960), 405–408 10.1063/1.1703671 MR0116836 0091.43503 CrossrefISIGoogle Scholar[18] J. J. Sylvester, Note on Burman's law for the inversion of the independent variable, Philos. Mag., 8 (1854), 535–540, Collected Papers, vol. 2, Cambridge, 1908, pp. 44–49 CrossrefGoogle Scholar[19] J. J. Sylvester, On the change of systems of independent variables, Quart. J. Math., 1 (1857), 42–56, 126–134, Collected Papers, vol. 2, Cambridge, 1908, pp. 65–85 Google Scholar[20] E. T. Whittaker and , G. N. Watson, A Course of Modern Analysis, Cambridge University Press, Cambridge, 1927, Chaps. 7, 16 Google Scholar Next article FiguresRelatedReferencesCited ByDetails Degenerate RS perturbation theoryThe Journal of Chemical Physics, Vol. 60, No. 3 | 1 Feb 1974 Cross Ref Perturbation Methods and Lagrange’s ExpansionCooperative Phenomena | 1 Jan 1973 Cross Ref Generalization of Lagrange’s Expansion for Functions of Several Implicitly Defined VariablesR. A. SackJournal of the Society for Industrial and Applied Mathematics, Vol. 13, No. 4 | 13 July 2006AbstractPDF (1277 KB)Chapter 3: Approximate MethodsNonlinear Partial Differential Equations in Engineering | 1 Jan 1965 Cross Ref Volume 14, Issue 1| 1966SIAM Journal on Applied Mathematics History Submitted:10 September 1964Published online:13 July 2006 InformationCopyright © 1966 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0114001Article page range:pp. 1-15ISSN (print):0036-1399ISSN (online):1095-712XPublisher:Society for Industrial and Applied Mathematics

An explicit numerical-analytical method is demonstrated for accurate solving the Schrödinger equation in those cases when this equation reducible to a system of n coupled ordinary differential equations with singular points. Fundamental system of solutions is constructed as algebraic combinations of power series, power functions and logarithmic function in the neighbourhood of the regular singular point and as asymptotic expansions of solutions in the neighbourhood of the irregular singular point. The method is based on the calculation of recurrent sequences of constant matrices of coefficients in power series and in inverse power series in asymptotic expansions using derived recurrent relations, that makes possible to calculate solutions at any given point using only algebraic computations and elementary functions. In turn it makes possible to solve accurately the eigenvalue problem and scattering problem and to derive analytical expressions for the wavefunctions. The method is applied to calculations of energies and wavefunctions of the discrete spectrum and wave functions of the continuous spectrum of the hydrogen-like atoms and of acceptors in semiconductors.

Inversion of Power Series

This note shows, using elementary properties of ribbon trees, that the universal formula for the inversion of power series can be obtained by counting strata in the compactified moduli space \overline{\mathcal M}_{0,n} .

Presents algorithms for planning feasible and minimum energy paths. The algorithms rely on series expansions to characterize planning problems for polynomial control systems. The resulting inversion problem is solved through an iterative contraction or a power series inversion. While the design methodology is general, our focus is on a class of polynomial control systems for which we provide explicit convergence guarantees. We demonstrate performance and numerical characteristics using one dimensional and six dimensional systems.

Variations on computing reciprocals of power series

The combinatorial power of the companion matrix

For a ring [Formula: see text] with a derivation [Formula: see text], we determine the Jacobson radical of the pseudo-differential operator ring [Formula: see text] for a large class of rings with a [Formula: see text]-condition. Various types of examples of these rings are provided.

This article draws attention to a comparative study of different members within the inverse Weibull Power Series (IWPS) to analyze the COVID-19 data from South Africa for the period from 27 March to 23 August 2020. A new sibling of the IWPS is introduced, namely the inverse Weibull negative binomial. An EM algorithm is developed for computing the maximum likelihood estimates of the model parameters. The IWPS growth curve model and its special cases are used for prediction of the COVID-19 spread in South Africa. It is found that the IWPS model fits the disease growth of the COVID-19 confirmed cases well with worthy long-term predictions. The IWPS growth curve modeling of South African predicts that the number of confirmed new cases will decrease at the end of November 2020.

We show how the “finite Quot scheme method” applied to Le Potier’s strange duality on del Pezzo surfaces leads to conjectures (valid for all smooth complex projective surfaces) relating two sets of universal power series on Hilbert schemes of points on surfaces: those for top Chern classes of tautological sheaves and those for Euler characteristics of line bundles. We have verified these predictions computationally for low order. We then give an analysis of these conjectures in small ranks. We also give a combinatorial proof of a formula predicted by our conjectures: the top Chern class of the tautological sheaf on $S^{[n]}$ associated to the structure sheaf of a point is equal to $(-1)^n$ times the nth Catalan number.

The action of an affine algebraic group G on an algebraic variety V can be differentiated to a representation of the Lie algebra L(G) of G by derivations on the sheaf of regular functions on V . Conversely, if one has a finite-dimensional Lie algebra L and a homomorphism ρ : L → DerK(K[U]) for an affine algebraic variety U, one may wonder whether it comes from an algebraic group action on U or on a variety V containing U as an open subset. In this paper, we prove two results on this integration problem. First, if L acts faithfully and locally finitely on K[U], then it can be embedded in L(G), for some affine algebraic group G acting on U, in such a way that the representation of L(G) corresponding to that action restricts to ρ on L. In the second theorem, we assume from the start that L = L(G) for some connected affine algebraic group G and show that some technical but necessary conditions on ρ allow us to integrate ρ to an action of G on an algebraic variety V containing U as an open dense subset. In the interesting cases where L is nilpotent or semisimple, there is a natural choice for G, and our technical conditions take a more appealing form.

We use an elementary method to draw analytic conclusions from divergent formal power series solutions of systems of differential equations containing a parameter and give some applications to the theory of turning points. Our main result shows that a divergent formal series transformation of one system into another in which the coefficients satisfy certain estimates is necessarily the asymptotic expansion of an actual transformation. We use it to show the following. Given a two dimensional system ePdy/dx = A(x,e)y with A holomorphic at (x0,0), suppose that x0 is formally not a turning point in the sense that no singularities appear at x0 during the standard formal solution procedure with formal fractional power series in e. Then the formal solution is necessarily a uniform asymptotic representation of a fundamental matrix of the system on a full neighborhood of x0. (This conclusion is known to fail under weaker hypotheses on A). We also obtain similar but less complete results for higher order systems under more specialized hypotheses.

Formal power series and regular operations on fuzzy languages