超实数子结构

Recursive definitions on surreal numbers

Let No be Conway's class of surreal numbers. I will make explicit the notion of a function f on No recursively defined over some family of functions. Under some "tameness" and uniformity condition, f must satisfy some interesting properties; in particular, the supremum of the class of element greater or equal to a fixed d in No is actually an element of No. For similar reasons, the concatenation function x:y cannot be defined recursively in a uniform way over polynomial functions.

The surreal numbers as a universal H-field

We show that the natural embedding of the differential field of transseries into Conway's field of surreal numbers with the Berarducci-Mantova derivation is an elementary embedding. We also prove that any Hardy field embeds into the field of surreals with the Berarducci-Mantova derivation.

指数域与康威的 omega-映射

Exponential fields and Conway's omega-map

本文受康威超现实数启发，提出 omega-域概念，证明具有实系数的有界 Hahn 级数 omega-域可配备指数函数成为实指数域理论模型，并推广到更一般的系数域情形。

surreal 数的自同构群分解

Decomposing the automorphism group of the surreal numbers

本文研究surreal数域的自同构群，通过分解成五个因素来揭示其结构，利用Conway正常形式和Hahn场理论，提供自同构的构造方法，并扩展到强线性自同构和指数结构保持自同构。

Surreal fields stable under exponential, logarithmic, derivative and anti-derivative functions

The class of surreal numbers, denoted by $\textbf{No}$, initially proposed by Conway, is a universal ordered field in the sense that any ordered field can be embedded in it. They include in particular the real numbers and the ordinal numbers. They have strong relations with other fields such as field of transseries. Following Gonshor, surreal numbers can be seen as signs sequences of ordinal length, with some exponential and logarithmic functions that extend the usual functions over the reals. $\textbf{No}$ can actually be seen as an elegant (generalized) power series field with real coefficie…

The exponential-logarithmic equivalence classes of surreal numbers

In his monograph, H. Gonshor showed that Conway's real closed field of surreal numbers carries an exponential and logarithmic map. Subsequently, L. van den Dries and P. Ehrlich showed that it is a model of the elementary theory of the field of real numbers with the exponential function. In this paper, we give a complete description of the exponential equivalence classes in the spirit of the classical Archimedean and multiplicative equivalence classes. This description is made in terms of a recursive formula as well as a sign sequence formula for the family of representatives of minimal length …

Surreal ordered exponential fields

In [26], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field $\mathbf{No}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of $\mathbf{No}$, i.e. a subfield ($K$-subspace) of $\mathbf{No}$ that is an initial subtree of $\mathbf{No}$. In this sequel to [15], piggybacking on the just-said results, analogous results are established for ordered exponential fields, making use of a slight generalizati…

Homogeneous Universal H-fields

We consider derivations $\partial$ on Conway's field $\mathbf{No}$ of surreal numbers such that the ordered differential field $(\mathbf{No},\partial)$ has constant field $\mathbb{R}$ and is a model of the model companion of the theory of $H$-fields with small derivation. We show that this determines $(\mathbf{No},\partial)$ uniquely up to isomorphism, and that this structure is absolutely homogeneous universal for models of this theory with constant field $\mathbb{R}$.

Analysis on Surreal Numbers

The class $\mathbf{No}$ of surreal numbers, which John Conway discovered while studying combinatorial games, possesses a rich numerical structure and shares many arithmetic and algebraic properties with the real numbers. Some work has also been done to develop analysis on $\mathbf{No}$. In this paper, we extend this work with a treatment of functions, limits, derivatives, power series, and integrals. We propose surreal definitions of the arctangent and logarithm functions using truncations of Maclaurin series. Using a new representation of surreals, we present a formula for the limit of a sequ…

超实数上的积分

Integration on the Surreals

本文证明实际应用中的多数函数，尤其是复苏函数的一大子类，可构造性扩展到超实数域并定义积分，同时指出更一般函数的扩展受基础数学限制。

Number Systems with Simplicity Hierarchies II

In [15], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field No of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of No, i.e. a subfield of No that is an initial subtree of No. In this sequel to [15], analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of No that are themselves initial. It is further show…