Transseries as germs of surreal functions

Berarducci, Alessandro; Mantova, Vincenzo

doi:10.1090/tran/7428

Cited by 16 publications

(26 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let K be a truncation closed, cross sectional logarithmic subfield of a transserial Hahn field and let be a transserial embedding into . Then is a truncation closed logarithmic subfield of , so K is a field of transseries in the sense of Berarducci and Mantova [11, page 3561]. Schmeling refers to any logarithmic subfield of a transserial Hahn field as a field of transseries , though most natural examples (increasing unions of transserial Hahn fields, grid-based transseries, etc.)…”

Section: Transserial Embeddingsmentioning

confidence: 99%

“…As a byproduct of this result, we show that every transserial Hahn field admits a truncation closed logarithmic field embedding into , but that this embedding can not in general be taken to be initial. In light of the growing body of work relating transseries and surreal numbers [3–8, 10, 11, 13, 14], this result on embeddings of transserial Hahn fields is of independent interest.…”

Section: Introductionmentioning

confidence: 99%

“…The proof of this uses the precursory result that trigonometric-exponential initial subfields of and trigonometric ordered initial subfields of , more generally, admit canonical sine and cosine functions. This is shown to apply to the members of a distinguished family of initial exponential subfields of isolated by van den Dries and Ehrlich ([46], Corollary 5.5), to the image of the canonical map of the ordered exponential field of transseries into [4], which is shown to be initial, and to the ordered exponential fields and , considered by Berarducci and Mantova ([11]; also see [34–36]), which are likewise shown to be initial.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Surreal Ordered Exponential Fields

Ehrlich¹,

Kaplan²

2021

J. symb. log.

View full text Add to dashboard Cite

In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field (ordered -vector space) to be isomorphic to an initial subfield ( -subspace) of , i.e. a subfield ( -subspace) of that is an initial subtree of . In this sequel, analogous results are established for ordered exponential fields, making use of a slight generalization of Schmeling’s conception of a transseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of . These include all models of , where is the reals expanded by a convergent Weierstrass system W. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of , which includes itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field of logarithmic-exponential transseries into is shown to be initial, as are the ordered exponential fields and .

show abstract

Section: Transserial Embeddingsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Surreal Ordered Exponential Fields

Ehrlich¹,

Kaplan²

2021

J. symb. log.

View full text Add to dashboard Cite

show abstract

“…The field R ω of omega-series is the smallest subfield of the surreal numbers containing R(ω) and closed under log, exp and sums of arbitrary summable sequences. In [11] it is shown that this field has a unique natural derivation and composition operator and contains, as differential subfields, the various variants of transseries fields (LE and EL-series). Each f ∈ R ω (hence in particular any transseries) determines a functionf : No >R → No on positive infinite surreal numbers.…”

Section: Open Problems and Questionsmentioning

confidence: 99%

“…Here we try to reconcile the algebraic and the analytic approach to surreal derivation and integration through a notion of composition [11]. The special session on surreal numbers at the joint AMS-MAA meeting in Seattle (6-9 Jan. 2016) was a timely occasion to discuss these developments and some of the results of this contribution were presented during that meeting.…”

Section: Initial Embeddings In the Surreal Numbers Antongiulio Fornasmentioning

confidence: 99%

Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations

2017

Self Cite

View full text Add to dashboard Cite

New striking analogies between H. Hahn's fields of generalised series with real coefficients, G. H. Hardy's field of germs of real valued functions, and J. H. Conway's field No of surreal numbers, have been lately discovered and exploited. The aim of the workshop was to bring quickly together experts and young researchers, to articulate and investigate current key questions and conjectures regarding these fields, and to explore emerging applications of this recent discovery.

show abstract

Supremum, infimum and hyperlimits in the non-Archimedean ring of Colombeau generalized numbers

et al. 2021

View full text Add to dashboard Cite

It is well-known that the notion of limit in the sharp topology of sequences of Colombeau generalized numbers $$\widetilde{\mathbb {R}}$$ R ~ does not generalize classical results. E.g. the sequence $$\frac{1}{n}\not \rightarrow 0$$ 1 n ↛ 0 and a sequence $$(x_{n})_{n\in \mathbb {N}}$$ ( x n ) n ∈ N converges if and only if $$x_{n+1}-x_{n}\rightarrow 0$$ x n + 1 - x n → 0 . This has several deep consequences, e.g. in the study of series, analytic generalized functions, or sigma-additivity and classical limit theorems in integration of generalized functions. The lacking of these results is also connected to the fact that $$\widetilde{\mathbb {R}}$$ R ~ is necessarily not a complete ordered set, e.g. the set of all the infinitesimals has neither supremum nor infimum. We present a solution of these problems with the introduction of the notions of hypernatural number, hypersequence, close supremum and infimum. In this way, we can generalize all the classical theorems for the hyperlimit of a hypersequence. The paper explores ideas that can be applied to other non-Archimedean settings.

show abstract

Transseries as germs of surreal functions

Cited by 16 publications

References 23 publications

Surreal Ordered Exponential Fields

Surreal Ordered Exponential Fields

Mini-Workshop: Surreal Numbers, Surreal Analysis, Hahn Fields and Derivations

Supremum, infimum and hyperlimits in the non-Archimedean ring of Colombeau generalized numbers

Contact Info

Product

Resources

About