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A Brunnian Theorem for Finite Families of Random Variables

Stéphane Dugowson

Abstract

In 2014, during a study on the connectivity structures of quantum entanglement, I specifically introduced the notion of ''the connectivity structure of a family of random variables'' -- a structure that expresses the dependency relations between the variables in question -- and I stated the following proposition, which can be described as Brunnian in reference to Hermann Brunn's work on links (1892) : "Every finite connectivity structure is that of a family of random variables". At the time, however, I neglected to write down the proof of this assertion, merely providing an intuitive idea of it. The purpose of this article is to present such a proof.

A Brunnian Theorem for Finite Families of Random Variables

Abstract

In 2014, during a study on the connectivity structures of quantum entanglement, I specifically introduced the notion of ''the connectivity structure of a family of random variables'' -- a structure that expresses the dependency relations between the variables in question -- and I stated the following proposition, which can be described as Brunnian in reference to Hermann Brunn's work on links (1892) : "Every finite connectivity structure is that of a family of random variables". At the time, however, I neglected to write down the proof of this assertion, merely providing an intuitive idea of it. The purpose of this article is to present such a proof.

Paper Structure

This paper contains 31 sections, 13 theorems, 35 equations.

Table of Contents

  1. Notations et rappels
  2. Rappels et notations relatives aux structures connectives.
  3. Dissociations ensemblistes, respect et contestations
  4. Dissociations ensemblistes
  5. Contestations et respect des dissociations ensemblistes
  6. Restrictions des dissociations
  7. Unions jointes.
  8. Dissociations et connexité
  9. Dissociations globales adaptées aux composantes connexes
  10. Validation par les connexes irré-ductibles
  11. Caractérisation des connexes par dissociation
  12. $\Gamma_\mathcal{A}$ est une structure connective contenant $\mathcal{A}$.
  13. $\mathcal{A}\mapsto\Gamma_\mathcal{A}$ est croissant.
  14. $\mathcal{A}\mapsto\Gamma_\mathcal{A}$ laisse les structures connectives intègres invariantes
  15. Somme de structures connectives
  16. ...and 16 more sections

Key Result

Proposition 2.1

Soient $L$ et $J$ deux parties de $I$ telles que $L\subset J$, et soit $\sigma\in\mathcal{D}(J)$. Si $L\dashv \sigma$ alors les traces sur $L$ de $\sigma_{(1)}$ et de $\sigma_{(2)}$ déterminent une dissociation $\sigma_{\vert L}$ de $L$, à savoir $\sigma_{\vert L}=(L=(\sigma_{(1)}\cap L)\sqcup (\sig

Theorems & Definitions (13)

  • Proposition 2.1
  • Proposition 2.3
  • Proposition 2.4
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4: Indépendance des variables associées aux composantes connexes
  • Proposition 4.6
  • ...and 3 more