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Register a new userSurreal numbers with derivation, Hardy fields and transseries: a survey
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The present article surveys surreal numbers with an informal approach, from their very first definition to their structure of universal real closed analytic and exponential field. Then we proceed to give an overview of the recent achievements on equipping them with a derivation, which is done by proving that surreal numbers can be seen as transseries and by finding the `simplest structure of H-field, the abstract version of a Hardy field. All the latter notions and their context are also addressed, as well as the universality of the resulting structure for surreal numbers.
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Several authors have conjectured that Conways field of surreal numbers, equipped with the exponential function of Kruskal and Gonshor, can be described as a field of transseries and admits a compatible differential structure of Hardy-type. In this paper we give a complete positive solution to both problems. We also show that with this new differential structure, the surreal numbers are Liouville closed, namely the derivation is surjective.
We show that Ecalles transseries and their variants (LE and EL-series) can be interpreted as functions from positive infinite surreal numbers to surreal numbers. The same holds for a much larger class of formal series, here called omega-series. Omega-series are the smallest subfield of the surreal numbers containing the reals, the ordinal omega, and closed under the exp and log functions and all possible infinite sums. They form a proper class, can be composed and differentiated, and are surreal analytic. The surreal numbers themselves can be interpreted as a large field of transseries containing the omega-series, but, unlike omega-series, they lack a composition operator compatible with the derivation introduced by the authors in an earlier paper.
We continue the study of a class of topological $mathcal{L}$-fields endowed with a generic derivation $delta$, focussing on describing definable groups. We show that one can associate to an $mathcal{L}_{delta}$ definable group a type $mathcal{L}$-definable topological group. We use the group configuration tool in o-minimal structures as developed by K. Peterzil.
This paper grew out of the observation that the possibilities of proof by induction and definition by recursion are often confused. The paper reviews the distinctions. The von Neumann construction of the ordinal numbers includes a construction of natural numbers as a special kind of ordinal. In any case, the natural numbers can be understood as composing a free algebra in a certain signature, {0,s}. The paper here culminates in a construction of, for each algebraic signature S, a class ON_S that is to the class of ordinals as S is to {0,s}. In particular, ON_S has a subclass that is a free algebra in the signature S.
We characterize thorn-independence in a variety of structures, focusing on the field of real numbers expanded by predicate defining a dense multiplicative subgroup, G, satisfying the Mann property and whose pth powers are of finite index in G. We also show such structures are super-rosy and eliminate imaginaries up to codes for small sets.
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