The Apéry Numbers as a Stieltjes Moment Sequence
The sequence of Apéry numbers is the moment sequence in the sense of Stieltjes. This is the short version of the proof. Appendix added for v.2
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arXiv subjects
G. A. Edgar
Publications and source records attributed to G. A. Edgar.
Fractional Iteration of Series and Transseries
We investigate compositional iteration of fractional order for transseries. For any large positive transseries $T$ of exponentiality 0, there is a family $T^{[s]}$ indexed by real numbers $s$ corresponding to teration of order $s$. It is based on Abel's Equation. We also investigate the question of whether there is a family $T^{[s]}$ all sharing a single support set. A subset of the transseries of exponentiality 0 is divided into three classes ("shallow", "moderate" and "deep") with different properties related to fractional iteration.
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Transseries: Ratios, Grids, and Witnesses
More remarks and questions on transseries. In particular we deal with the system of ratio sets and grids used in the grid-based formulation of transseries. This involves a "witness" concept that keeps track of the ratios required for each computation. There are, at this stage, questions and missing proofs in the development.
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Transseries: Composition, Recursion, and Convergence
Additional remarks and questions for transseries. In particular: properties of composition for transseries; the recursive nature of the construction of R[[[ x ]]]; modes of convergence for transseries. There are, at this stage, questions and missing proofs in the development.
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Transseries for beginners
From the simplest point of view, transseries are a new kind of expansion for real-valued functions. But transseries constitute much more than that--they have a very rich (algebraic, combinatorial, analytic) structure. The set of transseries is a large ordered field, extending the real number field, and endowed with additional operations such as exponential, logarithm, derivative, integral, composition. Over the course of the last 20 years or so, transseries have emerged in several areas of mathematics: asymptotic analysis, model theory, computer algebra, surreal numbers. This paper is an exposition for the non-specialist mathematician. All a mathematician needs to know in order to apply transseries.
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A fractal dimension estimate for a graph-directed IFS of non-similarities
Suppose a graph-directed iterated function system consists of maps f_e with upper estimates of the form d(f_e(x),f_e(y)) <= r_e d(x,y). Then the fractal dimension of the attractor K_v of the IFS is bounded above by the dimension associated to the Mauldin--Williams graph with ratios r_e. Suppose the maps f_e also have lower estimates of the form d(f_e(x),f_e(y)) >= r'_e d(x,y) and that the IFS also satisfies the strong open set condition. Then the fractal dimension of the attractor K_v of the IFS is bounded below by the dimension associated to the Mauldin--Williams graph with ratios r'_e. When r_e = r'_e, then the maps are similarities and this reduces to the dimension computation of Mauldin & Williams for that case.
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