A Stronger Bell Argument for (Some Kind of) Parameter Dependence

It is widely accepted that the violation of Bell inequalities excludes local theories of the quantum realm. This paper presents a new derivation of the inequalities from non-trivial non-local theories and formulates a stronger Bell argument excluding also these non-local theories. Taking into account all possible theories, the conclusion of this stronger argument provably is the strongest possible consequence from the violation of Bell inequalities on a qualitative probabilistic level (given usual background assumptions). Among the forbidden theories is a subset of outcome dependent theories showing that outcome dependence is not sufficient for explaining a violation of Bell inequalities. Non-local theories which can violate Bell inequalities (among them quantum theory) are rather characterised by the fact that at least one of the measurement outcomes in some sense (which is made precise) probabilistically depends both on its local as well as on its distant measurement setting ('parameter'). When Bell inequalities are found to be violated, the true choice is not 'outcome dependence or parameter dependence' but between two kinds of parameter dependences, one of them being what is usually called 'parameter dependence'. Against the received view established by Jarrett and Shimony that on a probabilistic level quantum non-locality amounts to outcome dependence, this result confirms and makes precise Maudlin's claim that some kind of parameter dependence is required.Comment: forthcoming in: Studies in the History and Philosophy of Modern Physic

The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

Certain properties of the enhanced power graph associated with a finite group

The enhanced power graph of a finite group $G$, denoted by $\mathcal{P}_E(G)$, is the simple undirected graph whose vertex set is $G$ and two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for some $z \in G$. In this article, we determine all finite groups such that the minimum degree and the vertex connectivity of $\mathcal{P}_E(G)$ are equal. Also, we classify all groups whose (proper) enhanced power graphs are strongly regular. Further, the vertex connectivity of the enhanced power graphs associated to some nilpotent groups is obtained. Finally, we obtain a lower bound and an upper bound for the Wiener index of $\mathcal{P}_E(G)$, where $G$ is a nilpotent group. The finite nilpotent groups attaining these bounds are also characterized.Comment: arXiv admin note: text overlap with arXiv:2207.0464

Homotopy theory for algebras over polynomial monads

We study the existence and left properness of transferred model structures for "monoid-like" objects in monoidal model categories. These include genuine monoids, but also all kinds of operads as for instance symmetric, cyclic, modular, higher operads, properads and PROP's. All these structures can be realised as algebras over polynomial monads. We give a general condition for a polynomial monad which ensures the existence and (relative) left properness of a transferred model structure for its algebras. This condition is of a combinatorial nature and singles out a special class of polynomial monads which we call tame polynomial. Many important monads are shown to be tame polynomial.Comment: Final version. Remark 5.16 extended. Bibliography complete