Most Probably Intersecting Families of Subsets

Let F be a family of subsets of an n-element set. It is called intersecting if every pair of its members has a non-disjoint intersection. It is well known that an intersecting family satisfies the inequality vertical bar F vertical bar <= 2(n-1). Suppose that vertical bar F vertical bar = 2(n-1) + i. Choose the members of F independently with probability p (delete them with probability 1 - p). The new family is intersecting with a certain probability. We try to maximize this probability by choosing F appropriately. The exact maximum is determined in this paper for some small i. The analogous problem is considered for families consisting of k-element subsets, but the exact solution is obtained only when the size of the family exceeds the maximum size of the intersecting family only by one. A family is said to be inclusion-free if no member is a proper subset of another one. It is well known that the largest inclusion-free family is the one consisting of all [n/2]-element subsets. We determine the most probably inclusion-free family too, when the number of members is (n([n/2])) + 1

Nuclear modification factor of charged particles and light-flavour hadrons in p--Pb collisions measured by ALICE

The hot and dense strongly interacting Quark-Gluon Plasma (sQGP) created in ultra-relativistic heavy-ion collisions can be probed by studying high-$p_{\rm T}$ particle production and parton energy loss. Similar measurements performed in p-Pb collisions may help in determining whether initial or final state nuclear effects play a role in the observed suppression of hadron production at high-$p_{\rm T}$ in Pb--Pb collisions. By examining the nuclear modification factors through the comparison of identified hadron yields in different collision systems one can gain insight into particle production mechanisms and nuclear effects.Comment: 4 pages, 3 figures, to appear in the proceedings of the 51st Rencontres de Moriond (QCD and High Energy Interactions), La Thuile, March 19-26 201

Convergence estimates for the Magnus expansion I. Banach algebras

We review and provide simplified proofs related to the Magnus expansion, and improve convergence estimates. Observations and improvements concerning the Baker--Campbell--Hausdorff expansion are also made. In this Part I, we consider the general Banach algebraic setting. We show that the (cumulative) convergence radius of the Magnus expansion is $2$; and of the Baker--Campbell--Hausdorff series is $\mathrm C_2=2.89847930\ldots$.Comment: Part I of original submission arXiv:1709.01791v1, rewritten and expande