arXiv subjects
Publications and source records attributed to Tomasz Sobieszek.
We present a somewhat different way of looking on Shannon entropy. This leads to an axiomatisation of Shannon entropy that is essentially equivalent to that of Fadeev. In particular we give a new proof of Fadeev theorem.
We investigate the following generalisation of the entropy of quantum measurement. Let H be an infinite-dimensional separable Hilbert space with a 'density' operator ρ, tr ρ=1. Let I(P)\in R be defined for any partition P = (P_1,...,P_m), P_1+ ... +P_m=1_H, P_i \in proj H$ and let I(P_i Qj, i \leq m, j \leq n) = I(P) + I(Q) for Q =(Q_1,..., Q_n), \sum Q_j = 1_H and P_iQ_j = Q_j P_i, tr ρ P_iQ_j = tr ρ P_i tr ρ Q_j (P, Q are physically independent). Assuming some continuity properties we give a general form of generalised information I.
We provide, under minimal continuity assumptions, a description of \textsl{additive partition entropies}. They are real functions $I$ on the set of finite partitions that are additive on stochastically independent partitions in a given probability space.
Given a convergent sequence of nodes we present a one-dimensional-holomorphic-function version of the Newton interpolation method of polynomials. It also generalises the Taylor and the Laurent formula. In other words, we present an effective identity theorem for complex domains.
In a previous paper: A. Paszkiewicz, T. Sobieszek, Additive Entropies of Partitions, we have given a description of additive partition entropies that is real functions $I$ on the set of finite partitions that are additive on stochastically independent partitions in a given probability space. We now present an analogical result, this time without assuming continuity. As a by-product of our efforts we solve a 2-cocycle functional equation for certain subsets of convex cones.