Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sign sequences of log-atomic numbers

Vincent Bagayoko

Abstract

Log-atomic numbers are surreal numbers whose iterated logarithms are monomials, and consequently have a trivial expansion as transseries. Presenting surreal numbers as sign sequences, we give the sign sequence formula for log-atomic numbers. To that efect, we relate log-atomic numbers to fixed-points of certain surreal functions.

Sign sequences of log-atomic numbers

Abstract

Log-atomic numbers are surreal numbers whose iterated logarithms are monomials, and consequently have a trivial expansion as transseries. Presenting surreal numbers as sign sequences, we give the sign sequence formula for log-atomic numbers. To that efect, we relate log-atomic numbers to fixed-points of certain surreal functions.
Paper Structure (23 sections, 24 theorems, 60 equations)

This paper contains 23 sections, 24 theorems, 60 equations.

Table of Contents

  1. Numbers and sign sequences
  2. Numbers as sign sequences
  3. Monomials
  4. Arithmetic on sign sequences
  5. Sign sequences formulas
  6. Surreal substructures
  7. Surreal substructures
  8. Convexity
  9. Imbrication
  10. Fixed points
  11. Sign sequence formulas for closed structures
  12. Exponentiation
  13. Surreal exponentiation
  14. Log-atomic numbers
  15. Surreal hyperexponentiation
  16. ...and 8 more sections

Key Result

Lemma 1.3

BvdH19 THe laws $\dotplus$ and $\mathbin{\dot{\times}}$ are associative with identities $0$ and $1$ respectively, and $\mathbin{\dot{\times}}$ distribues with $\dotplus$ on the left. Moreover, for $x, y \in \mathbf{No}$ where $y$ is a limit, we have

Theorems & Definitions (47)

  • Definition 1.1
  • Definition 1.2
  • Lemma 1.3
  • Proposition 1.4
  • Lemma 1.5
  • proof
  • Definition 2.1
  • Example 2.2
  • Proposition 2.3
  • Lemma 2.4
  • ...and 37 more