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Weak convergence of the periodic multiplicative Selmer algorithm

J. Christopher Kops

Abstract

In order to prove weak convergence of the periodic multiplicative Selmer algorithm we ensure that the periodicity matrix is positive and establish a relation between its entries and eigenvalues. Since we can imply that the limit of these relations exist, we arrive at the desired result.

Weak convergence of the periodic multiplicative Selmer algorithm

Abstract

In order to prove weak convergence of the periodic multiplicative Selmer algorithm we ensure that the periodicity matrix is positive and establish a relation between its entries and eigenvalues. Since we can imply that the limit of these relations exist, we arrive at the desired result.

Paper Structure

This paper contains 16 sections, 8 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fibred systems
  3. The concept of cylinders
  4. Selmer's algorithm
  5. Subtractive version
  6. Fibred system and absorbing set
  7. A case of periodicity
  8. Multiplicative algorithms
  9. Selmer's division algorithm
  10. The fibred system
  11. Cylinders and time-1-partition
  12. The matrices
  13. Periodic expansions for Selmer's division algorithm
  14. Weak convergence of the periodic MSA
  15. Example of a periodic MSA
  16. ...and 1 more sections

Key Result

Proposition 1

All cylinders of arbitrary rank $s,\, s\in \mathbb{N}$ are full if all cylinders of rank $1$ are full.

Figures (1)

  • Figure 1: The time-1-partition of the set $B^2$ by the 2-dimensional MSA, where $k\in\mathbb{N}$ indicates the associated cylinder $B(k)$.

Theorems & Definitions (14)

  • Definition 1: fibred system
  • Definition 2: multidimensional continued fraction
  • Definition 3
  • Definition 4: cylinder
  • Proposition 1
  • Proposition 2
  • Theorem 1: absorbing set
  • Definition 5: periodic continued fraction
  • Proposition 3
  • Definition 6: weak convergence
  • ...and 4 more