Finitely generated dyadic convex sets

Dyadic rationals are rationals whose denominator is a power of $2$. We define dyadic $n$-dimensional convex sets as the intersections with $n$-dimensional dyadic space of an $n$-dimensional real convex set. Such a dyadic convex set is said to be a dyadic $n$-dimensional polytope if the real convex set is a polytope whose vertices lie in the dyadic space. Dyadic convex sets are described as subreducts (subalgebras of reducts) of certain faithful affine spaces over the ring of dyadic numbers, or equivalently as commutative, entropic and idempotent groupoids under the binary operation of arithmetic mean. The paper contains two main results. First, it is proved that, while all dyadic polytopes are finitely generated, only dyadic simplices are generated by their vertices. This answers a question formulated in an earlier paper. Then, a characterization of finitely generated subgroupoids of dyadic convex sets is provided, and it is shown how to use the characterization to determine the minimal number of generators of certain convex subsets of the dyadic plane

Tourism of Polish cannabis consumers

The aim of this study is to characterize the tourism activity of Polish cannabis consumers in terms of (i) the level of their participation in tourism, (ii) parameters describing this participation, (iii) the effect of legal access to cannabis on choosing tourism destinations. The study is based on an anonymous online survey in which 886 voluntary respondents answered a series of questions about their tourist travels, their attitude to cannabis consumption, and their demographic, socio-economic and geographic metrics. Results of the survey were analyzed using several statistical indicators of variability, structure, correlation, and structure similarity. For the respondents declaring cannabis consumption, the level of their participation in tourism is close to the national level. Other parameters describing the domestic and foreign tourism of these respondents differ quite significantly from those reported for the general public of Poland. This indicates that the possibility of cannabis consumption significantly affects the nature and directions of travels undertaken by tourists interested in cannabis. Furthermore, there is a strong connection between the respondents’ personal preferences and the nature of their tourism, especially the destinations of their foreign trips. The conclusions from this study mostly apply to current and recent cannabis consumers because the vast majority of respondents (90%) rank among such kinds of cannabis users

The lattice of quasivarietes of modules over a Dedekind ring

In 1995 D. V. Belkin described the lattice of quasivarieties of modules over principal ideal domains [1]. The following paper provides a description of the lattice of subquasivarieties of the variety of modules over a given Dedekind ring. It also shows which subvarieties of these modules are deductive (a variety is deductive if every subquasivariety is a variety)

1-Ammonio-1-phosphono­pentane-1-phospho­nic acid

The title compound, C5H15NO6P2, was obtained by the reaction of penta­nenitrile with PCl3 followed by the dropwise addition of water. The asymmetric unit contains one mol­ecule, which exists as a zwitterion with a positive charge on the –NH3 group and a negative charge on one of the phospho­nic O atoms. The crystal structure displays N—H⋯O and O—H⋯O hydrogen bonding, which creates a three-dimensional network

Oxonium ammonio­(cyclo­prop­yl)methyl­enebis(hydrogenphospho­nate) monohydrate

The title compound, H3O+·C4H10NO6P2 −·H2O, was obtained from the reaction of cyclo­propane­carbonitrile with PCl3, followed by dropwise addition of water. The asymmetric unit comprises an oxonium cation, a zwitterionic monoanion containing a positively charged ammonium group and two negatively charged phospho­nic acid residues and a water mol­ecule of crystallization. The hydroxonium cation and water mol­ecule are hydrogen bonded to the anion and further N—H⋯O and O—H⋯O bonds create a three-dimensional network