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Logarithms of iteration matrices, and proof of a conjecture by Shadrin and Zvonkine
Abstract
A proof for a conjecture by Shadrin and Zvonkine, relating the entries of a
matrix arising in the study of Hurwitz numbers to a certain sequence of
rational numbers, is given. The main tools used are iteration matrices of
formal power series and their (matrix) logarithms.
... This formula was first proved (with difficulty) by Aschenbrenner [3], in the process of showing a conjecture, and inspired the result of Lemma 4.1. ...
In this paper, we explore the effectiveness of purely operational methods in the study of umbral calculus. To accomplish this goal, we systematically reconstruct the theory \emph{operationally}, offering new proofs and results throughout. Our approach is applied to the study of invertible power series, where we notably offer a concise two-line proof of Lagrange's inversion theorem and derive new formulas for both regular and fractional compositional iterates. In addition, we will develop several new insights into umbral operators, including their fractional exponents and novel expressions, in connection with the previously discussed applications. Finally, we discuss both established and new concepts and examples, such as pseudoinverses of delta operators and a natural generalization of Laguerre polynomials.
... AHP has many techniques that can be applied, but for discussion in this paper using Multi-criteria Decision Making (MCDM) (Marhavilas et al., 2022). MCDM has a very different technique and this difference is one of the advantages of other techniques, the advantage is the level of decision accuracy produced gives very optimal results, when compared to other methods with the concept of mathematical algebra matrices (Aschenbrenner, 2012). The mathematic algebra matrices method will apply iterative eigenvector values up to many times with the aim of obtaining eigenvector values without differences as a result of optimal eigenvectors. ...
The progress of smartphone technology is now very rapid, supported by many renewable features, even many users are competing to get the latest products without regard to the costs that have been incurred. The problem that arises is that it is increasingly difficult to select technology-based products with many criteria. The purpose of writing this paper is to provide the best solution for selecting technology-based products with multi-criteria to suit user needs by taking into account the costs incurred effectively and the use of contradictory multi-criteria applications. The presence of technology products always has many criteria that make it more difficult for users to choose products as the right choice according to their needs, thus the right method is needed as a solution to obtain technology-based products such as smartphones. The Analytic Hierarchy Process (AHP) method is used for the evaluation and selection process. This AHP method will collaborate with the ELECTRE and PROMETHEE methods as a comparison solution for smartphone product selection. The resulting comparison will be an applied model for smartphone selection that produces the best decision-making support according to user needs. The results of the collaborative implementation process of the ELECTRE and PROMETHEE methods provide a decision on the rating system. The collaborative application of the AHP method to the ELECTRE and PROMETHEE methods provides optimal decision support for the selection process, so that this can be used as a comparison material in making decisions regarding the selection of smartphones as technology-based products.
... of rational numbers which recently arose both in a conjecture made by Shadrin and Zvonkine [34] (and proved in [2]) in connection with a generating series for Hurwitz numbers, and also in another context (ongoing joint work of the first-named author with van den Dries and van der Hoeven on asymptotic differential algebra [3] ...
We show that the iterative logarithm of each non-linear entire function is
differentially transcendental over the ring of entire functions, and we give a
sufficient criterion for such an iterative logarithm to be differentially
transcendental over the ring of convergent power series.
A new class of lifetime distributions, called tetration distribution, is presented based on the continuous iteration of the exponential‐minus‐one function. In particular, this distribution encompasses and extends the Weibull, Pareto, and Gompertz distributions. In addition, a characterization of the tail of the distribution is proposed through two indices, one of them encompassing the Pareto and Weibull tail indices. Finally, an inference procedure based on rank regression is implemented for the tetration distribution and applied to multiple datasets from the reliability field.
We all recognize 0, 1, 1, 2, 3, 5, 8, 13,... but what about 1, 2, 4, 6, 3, 9, 12, 8, 10, 5, 15,...? If you come across a number sequence and want to know if it has been studied before, there is only one place to look, the On-Line Encyclopedia of Integer Sequences (or OEIS). Now in its 49th year, the OEIS contains over 220,000 sequences and 20,000 new entries are added each year. This article will briefly describe the OEIS and its history. It will also discuss some sequences generated by recurrences that are less familiar than Fibonacci's, due to Greg Back and Mihai Caragiu, Reed Kelly, Jonathan Ayres, Dion Gijswijt, and Jan Ritsema van Eck.
This is a historical note on the habilitation thesis of the famous logician Gott- lob Frege (1848-1925): "Rechnungsmethoden, die sich auf eine Erweiterung des Grossenbegriffes grunden," Jena 1874. In this paper of Frege one can find astonishingly rich results on iteration theory. Frege investigates the translation equation in explicit form and he uses differential equations and the infinitesimal generator to determine solutions of the translation equation.
This article is an extended version of preprint math.AG/9902104. We find an explicit formula for the number of topologically different ramified coverings of a sphere by a genus g surface with only one complicated branching point in terms of Hodge integrals over the moduli space of genus g curves with marked points. Comment: 30 pages (AMSTeX). Minor typos are corrected
The theory of formal power series and derivation is developed from the point of view of the power matrix. A Loewner equation for formal power series is introduced. We then show that the matrix exponential is surjective onto the group of power matrices, and the coefficients are entire functions of finitely many coefficients of the infinitesimal generator. Furthermore coefficients of the solution to the Loewner equation with constant infinitesimal generator can be obtained by exponentiating an infinitesimal power matrix. We also use the formal Loewner equations to investigate the relation between holomorphicity of an infinitesimal generator to holomorphicity of the exponentiated matrix. Comment: 25 pages
In [5] I.P. Goulden, D.M. Jackson, and R. Vakil formulated a conjecture relating certain Hurwitz numbers (enumerating ramified coverings of the sphere) to the intersection theory on a conjectural Picard variety. We are going to use their formula to study the intersection theory on this variety (if it is ever to be constructed) by methods close to those of M. Kazarian and S. Lando in [7]. In particular, we prove a Witten-Kontsevich-type theorem relating the intersection theory and integrable hierarchies. We also extend the results of [7] to include the Hodge integrals over the moduli spaces, involving one lambda-class.
We say that a family of functions is coherent if they all satisfy one and the same algebraic differential equation. After proving some general theorems on coherent families, we show that many special families (like the Jacobi polynomials) that arise in applications are indeed coherent, while other families (like the Bernoulli polynomials) are totally incoherent--that is, they have no infinite coherent subfamilies. For certain polynomials, the sequence of their compositional iterates is coherent, while for “most” polynomials, it is totally incoherent.
We believe there is room for a subject named as in the title of this paper. Motivating examples are Hardy fields and fields of transseries. Assuming no previous knowledge of these notions, we introduce both, state some of their basic properties, and explain connections to o-minimal structures. We describe a common algebraic framework for these examples: the category of H-fields. This unified setting leads to a better understanding of Hardy fields and transseries from an algebraic and model-theoretic perspective.
Let Ω be the set of the analytic functions F(z), regular in some neighbourhood of the origin with the expansion
This paper is concerned with the ring A of all complex formal power series and the group G of substitution-invertible formal series. The two main questions of interest will be these. How can one tell whether two members of A represent the same “function” up to “change of variables” by a member of G? Which members of G can be embedded in a one-parameter subgroup of G? Sections 1–4 give a self-contained account of the solution to these problems. This account is largely expository, most of the partial results having appeared elsewhere. Some of the proofs of the known results are new and simpler; some are just paraphrases. The sources of this material are given in a remark at the end of Section 4. Section 5 gives a new proof of the characterization of those one-parameter subgroups of G which consist wholly of convergent series.
Preface Symbols and conventions 1. Introduction 2. Iteration 3. Linear equations and branching processes 4. Regularity of solutions of linear equations 5. Analytic and integrable solutions of linear equations 6. Theory of nonlinear equations 7. Equations of higher orders and systems of linear equations 8. Equations of infinite order and systems of nonlinear equations 9. On conjugacy 10. More on the Schroder and Abel equations 11. Characterization of functions 12. Iterative roots and invariant curves 13. Linear iterative functional inequalities References Author index Subject index.
We shall be concerned with the behaviour of the fractional iterates of analytic functions which have a fixpoint ζ with multiplier 11. The general form of such a function is
An analytic function f(z) is said to have a fixpoint ξ ≠ ∞ of multiplier 1 if f(ξ) = ξ, f'(ξ) = 1. The function then has an expansion
It is shown that there does not exist a transcendental entire function with the property that all its iterates satisfy the same algebraic differential equation. This answers a question of L. A. Rubel.
H-fields are fields with an ordering and a derivation subject to some compatibilities. (Hardy fields extending R and fields of transseries over R are H-fields.) We prove basic facts about the location of zeros of differential polynomials in Liouville closed H-fields, and study various constructions in the category of H-fields: closure under powers, constant field extension, completion, and building H-fields with prescribed constant field and H-couple. We indicate difficulties in obtaining a good model theory of H-fields, including an undecidability result. We finish with open questions that motivate our work.
The polynomials that arise as coefficients when a power series is raised to the power x include many important special cases, which have surprising properties that are not widely known. This paper explains how to recognize and use such properties, and it closes with a general result about approximating such polynomials asymptotically.
Asymptotic differential algebra Analyzable Functions and Applications
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D. Gronau, Gottlob Frege, a pioneer in iteration theory, in: L. Reich, J. Smítal, and G. Targonski, Iteration Theory (ECIT 94 ), pp. 105–119, Grazer Math. Ber., vol. 334, Karl-Franzens-Univ. Graz, Graz, 1997.
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ApplicationàApplicationà l'itération de e z et de e z − 1
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