one of unsolved problems in quantum measurement theory is to characterize coexistence of quantum effects. in this paper, applying positive operator matrix theory, we give a mathematical characterization of the witness set of coexistence of quantum effects and obtain a series of properties of coexistence. we also devote to characterizing bijective morphisms on quantum effects leaving the witness...

We prove that if an nth degree rational Bézier curve has a singular point, then it belongs to the two (n− 1)th degree rational Bézier curves defined in the (n− 1)th step of the de Casteljau algorithm. Moreover, both curves are tangent at the singular point. A procedure to construct Bézier curves with singularities of any order is given.  2001 Elsevier Science B.V. All rights reserved.

Considering operators defined using Structural Operational Semantics (SOS), commutativity axioms are intuitive properties that hold for many of them. Proving this intuition is usually a laborious task, requiring several pages of boring and standard proof. To save this effort, we propose a syntactic SOS format which guarantees commutativity for a set of composition operators.