We provide a full and rigorous proof of a theorem attributed to Żuk, stating that random groups in the Gromov density model for true0d>13 have property (T) with high probability. The original paper had numerous gaps; in particular, crucial steps involving passing between different models of random groups were not described. We fix the gaps using combinatorial arguments and a recent result concerning perfect matchings in random hypergraphs. We also provide an alternative proof, avoiding combinatorial difficulties and relying solely on the spectral properties of random graphs in the G(n, p) model.

We investigate the relation between unextendible product bases (UPB) and Bell inequalities found recently in [R. Augusiak et al., Phys. Rev. Lett. 107, 070401 (2011)]. We first review the procedure introduced there that associates to any set of mutually orthogonal product vectors in a many-qubit Hilbert space a Bell inequality. We then show that if a set of mutually orthogonal product vectors can be completed to a full basis, then the associated Bell inequality is trivial, in the sense of not being violated by any nonsignalling correlations. This implies that the relevant Bell inequalities that arise from the construction all come from UPBs, which adds additional weight to the significance of UPBs for Bell inequalities. Then, we provide new examples of tight Bell inequalities with no quantum violation constructed from UPBs in this way. Finally, it is proven that the Bell inequalities with no quantum violation introduced recently in [M. Almeida et al., Phys. Rev. Lett. 104, 230404 (2010)] are tight for any odd number of parties.

BackgroundThe paper presents documentation of the traditional use of wild edible mushrooms in Mazovia (33,900 km2), a region of Poland.MethodsA total of 695 semi-structured interviews were carried out among local informants in 38 localities proportionally distributed throughout the study area (one locality approximately every 30 km), asking which mushrooms they collected and how. The species utilized were identified using visual props, morphological identification of voucher specimens, and DNA barcoding.ResultsAltogether, 92 taxa identified to the species or genus level were recorded, among them 76 species used as food, 21 taxa known as toxic, and 11 taxa used for non-culinary purposes. Out of 76 identified edible fungi species, 47% (36 species) were identified using ITS DNA barcode method. Eleven of them were identified exclusively by molecular analysis. The mean number of edible taxa mentioned per interview was 9.5. Two species new to the mycobiota of Poland, Hydnum ellipsosporum and Paxillus cuprinus, were found. Frequent interaction with mushroom collectors enabled the transcription of local folk taxonomy into proper taxonomic classification and the definition of changes in local preferences concerning wild fungi collection.ConclusionsThe list of species utilized is the longest regional list of edible mushrooms ever recorded during ethnomycological field research, putting the inhabitants of the studied region at the top of the mycophilia spectrum.

We give a universal recipe for constructing nonlinear entanglement witnesses able to detect non-classical correlations in arbitrary systems of distinguishable and/or identical particles for an arbitrary number of constituents. The constructed witnesses are expressed in terms of expectation values of observables. As such they are, at least in principle, measurable in experimentsComment: 14 page