The N-variable Hopf algebra introduced by Brouder, Fabretti, and Krattenaler (BFK) in the context of non-commutative Lagrange inversion can be identified with the inverse of the incidence algebra of N-colored interval partitions. The (BFK) antipode and its reflection determine the (generally distinct) left and right inverses of power series with non-commuting coefficients and N non-commuting va...

Given a non-abelian finite group G, let π(G) denote the set of prime divisors of the order of G and denote by Z(G) the center of G. The prime graph of G is the graph with vertex set π(G) where two distinct primes p and q are joined by an edge if and only if G contains an element of order pq and the non-commuting graph of G is the graph with the vertex set G−Z(G) where two non-central elements x...

We study the existence of jointly measurable POVM approximations to two non-commuting sharp spin observables. We compare two different ways to specify optimal approximations.

We describe a variational framework for non-commuting flows, extending the theories of Lagrangian multiforms and pluri-Lagrangian systems, which have gained prominence in recent years as description integrable systems sense multidimensional consistency. In context manifold independent variables, often called multi-time, is Lie group whose bracket structure corresponds to commutation relations b...