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Stéphane Dugowson

Publications and source records attributed to Stéphane Dugowson.

15 recordsLinked to original sources
arXiv2025

A Brunnian Theorem for Finite Families of Random Variables

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.

math.GN↗
arXiv2020

Interacting open dynamics

This paper presents the basic concepts of a systemic theory of interaction between non-deterministic open dynamics with varying temporalities, which includes three stages: the definition of these dynamics as lax-functors, the notion of interaction -- which uses some notions of requests, synchronizations and social modes (privacy) -- and finally the generation of open global dynamics. Some connectivity structures of an interaction are defined, but the other aspects of dynamical connectivity are left to further work.

math.CT↗
arXiv2018

A Lax Functorial Definition of Open Dynamics

This paper provides a rewording in the language of lax-functors of the definition of open dynamics given in our systemic theory of interactivity exposed in previous papers.

math.DS↗
arXiv2018

Toposes of connectivity spaces. Morita equivalences with topological spaces and partially ordered sets in the finite case

This paper has two parts. First, we recall and detail the definition of the Grothendieck topos of a connectivity space, that is the topos of sheaves on such a space. In the second part, we prove that every finite connectivity space is Morita-equivalent to a finite topological space, and vice versa (we have given this proof in several, but we haven't yet shared this in writing).

math.GN↗
arXiv2016

Espaces connectifs : représentations, feuilletages, ordres, difféologies

This article is a continuation of my former article "On Connectivity Spaces". After some brief historical references relating to the subject, separation spaces and then adjoint notions of connective representation and connective foliation are developed. The connectivity order previously defined only in the finite case is now generalised to all connectivity spaces, and so to connective foliations. Finally, we start the study of some functorial relations between connectivity and diffeological spaces, and we give a characterization of diffeologisable connectivity spaces.

math.GN↗
arXiv2016

Interacting Dynamics

The "theory of open sub-functorial dynamics" is a new theory that defines interacting generalized dynamical systems. The interactions between these dynamics produce new dynamics which, of course, can then enter into other interactions. A major part of this article can already be found in two unpublished texts and it has been partially exposed in conferences. However, we need to give a new, unified and therefore more convenient presentation of this material, and we also need some examples to illustrate it. Moreover, we introduce in this article the new concepts of "normal interaction" and "concrete interaction", and replace the previously used rigid synchronizations by much more general flexible ones.

math.DS↗
arXiv2015

Open sub-categorical dynamics in interaction

The aim of this paper is to define what we call open sub-categorical dynamics, their interactions and the sub-categorical dynamics produced by those interactions, thanks to the stability theorem we prove here and which motivates all this study.

math.DS↗
arXiv2015

Interaction of open graphic dynamics

The aim of this paper is to define what we shall call open graphic dynamics, their interactions and the dynamics produced by those interactions. It prepares the study of "open sub-categorical dynamics" and "open categorical dynamics".

math.DS↗
arXiv2015

Connectivity structure of multiple relations

The prime purpose of this paper is to define the connectivity structure , on a set E, of any multiple relation defined on a family of sets indexed by E, such a relation expressing compatibility between the states of different systems (thus a full compatibility indicates absence of any connection). We then demonstrate a "Brunn's theorem" for those multiple relations, that is the fact that every connectivity structure is the connectivity structure of such a relation.

math.GN↗
arXiv2014

Connectivity structures of quantum entanglement

In this paper, after some recalls about connectivity structures and about the formalisms of quantum mechanics, we associate some families of connectivity structures with any entangled quantum state, and with any "measurement device" on such states. This finally allows us to define a new classification tool for quantum entanglement: the connectivity order. --- Dans ce texte, après des rappels d'une part sur la notion de structure connective et d'autre part sur les formalismes de la mécanique quantique, nous associons certaines familles de structures connectives à tout état quantique intriquant un nombre fini quelconque de particules, ainsi qu'à tout "dispositif de mesure", portant sur de tels états. Cela nous permet finalement de définir un nouvel outil de classification de l'intrication quantique : l'ordre connectif.

quant-ph↗
arXiv2011

Introduction aux dynamiques catégoriques connectives

This text is a continuation to my former article "On Connectivity Spaces". It takes into account that connectivity spaces gives rise to phenomena which are essentially dynamic. In a first stage, the representation of finite connectivity spaces by links (Brunn-Debrunner-Kanenobu's theorem) leads to the notion of connective representation. But examples of connective representations often come from dynamical systems. And this is even more obvious when we study the adjoint notion of connective foliation. To apply those notions to dynamics, we first need to consider dynamical systems in an unified way. This is done with a categorical point of view on temporalities and dynamics. It is then possible to define categorical connective dynamics, and to apply to them the various connective notions, specially the connectivity order of a connectivity space.

math.DS↗
arXiv2010

On Connectivity Spaces

This paper presents some basic facts about the so-called connectivity spaces. In particular, it studies the generation of connectivity structures, the existence of limits and colimits in the main categories of connectivity spaces, the closed monoidal category structure given by the so-called tensor product on integral connectivity spaces; it defines homotopy for connectivity spaces and mention briefly related difficulties; it defines smash product of pointed integral connectivity spaces and shows that this operation results in a closed monoidal category with such spaces as objects. Then, it studies finite connectivity spaces, associating a directed acyclic graph with each such space and then defining a new numerical invariant for links: the connectivity order. Finally, it mentions the not very wellknown Brunn-Debrunner-Kanenobu theorem which asserts that every finite integral connectivity space can be represented by a link.

math.GN↗
arXiv2008

The Connectivity Order of Links

We associate at each link a connectivity space which describes its splittability properties. Then, the notion of order for finite connectivity spaces results in the definition of a new numerical invariant for links, their connectivity order. A section of this short paper presents a theorem which asserts that every finite connectivity structure can be realized by a link : the Brunn-Debrunner-Kanenobu Theorem.

math.GN↗
arXiv2007

Representation of finite connective spaces

After recalling the definition of connectivity spaces and some of their main properties, a way is proposed to represent finite connectivity spaces by directed simple graphs. Then a connectivity structure is associated to each tame link. It is showed that all spaces of a certain class (the iterated Brunnian ones) admit representations by links. Finally, I conjecture that every finite connectivity space is representable by a link. ----- Apres un rappel de la definition des espaces connectifs et de certaines de leurs principales proprietes, nous proposons une maniere de representer les espaces connectifs finis par des graphes simples orientes, puis nous associons a tout entrelacs une structure connective. Nous montrons que tout espace d'une certaine classe (les espaces brunniens iteres) admet une representation par entrelacs, et nous conjecturons finalement que tout espace connectif fini est representable par entrelacs.

math.GN↗