We describe some of the forms of freeness of group actions on noncommutative C*-algebras that have been used, with emphasis on actions of finite groups. We give some indications of their strengths, weaknesses, applications, and relationships to each other. The properties discussed include the Rokhlin property, K-theoretic freeness, the tracial Rokhlin property, pointwise outerness, saturation, ...

We formally study two privacy-type properties in online auction protocols, bidding-price-secrecy and receipt-freeness. These properties are formalised as observational equivalences in the applied π calculus. We analyse the receipt-free auction protocol by Abe and Suzuki. Bidding-price-secrecy of the protocol is verified using ProVerif, whereas receipt-freeness of the protocol is proved manually.

As in the cases of freeness and monotonic independence, the notion of conditional freeness is meaningful when complex-valued states are replaced by positive conditional expectations. In this framework, the paper presents several positivity results, a version of the central limit theorem and an analogue of the conditionally free R-transform constructed by means of multilinear function series.

As in the cases of freeness and monotonic independence, the notion of conditional freeness is meaningful when complex-valued states are replaced by positive conditional expectations. In this framework, the paper presents several positivity results, a version of the central limit theorem and an analogue of the conditionally free R-transform constructed by means of multilinear function series.

Envy-freeness up to one good (EF1) and envy-freeness any (EFX) are two well-known extensions of for the case indivisible items. It is shown that EF1 can always be guaranteed agents with subadditive valuations. In sharp contrast, it unknown whether or not an EFX allocation exists, even four additive addition, best approximation guarantee (φ − 1) ≃ 0.61 by Amanitidis et al.. order find a middle g...